DK342X Half Title pg 5/18/05 9:51 PM Page 1
The
Hilbert-Huang
Transform
in
Engineering
© 2005 by Taylor & Francis Group, LLC
DK342X Title pg 5/18/05 9:50 PM Page 1
The
Hilbert-Huang
Transform
in
Engineering
Edited by
Norden Huang
Nii O. Attoh-Okine
Boca Raton London New York Singapore
A CRC title, part of the Taylor & Francis imprint, a member of the
Taylor & Francis Group, the academic division of T&F Informa plc.
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Preface
Data analysis serves two purposes: to determine the parameters needed to construct
a model and to confirm that the model constructed actually represents the phenomenon. Unfortunately, the data — whether from physical measurements or numerical
modeling — most likely will have one or more of the following problems:
•
•
•
The total data are too limited.
The data are nonstationary.
The data represent nonlinear processes.
These problems dictate how the data can be analyzed and interpreted.
This book describes the formulation and application of the Hilbert-Huang transform (HHT) in various areas of engineering, including structural, seismic, and ocean
engineering.
The primary objective of this book is to present the theory of the Hilbert-Huang
Transform (HHT) and its application to engineering. The presentation of the book
is such that it can be used as both a reference and a teaching text. The authors of
the individual chapters provide a strong theoretical background and some new
developments before addressing their specific application. This approach demonstrates the versatility of the HHT.
The book comprises 13 chapters, covering more than 300 pages. These chapters
were written by 30 invited experts from 6 different countries.
The book begins with an introduction and some recent developments in HHT.
Chapter 2 uses HHT to interpret nonlinear wave systems and provides a comprehensive analysis on the assessment of rogue waves. Chapter 3 discusses HHT
applications in oceanography and ocean-atmosphere remote sensing data and presents some examples of these applications. Chapter 4 presents a comparison of the
energy flux computation for shooting waves of HHT and wavelet analysis techniques.
In Chapter 5, HHT is applied to nearshore waves, and the results are compared to
field data. In Chapter 6, the author uses HHT to characterize the underwater electromagnetic environment and to identify transient manmade electromagnetic disturbances, where the HHT was able to act as a filter effectively discriminating different
dipole components. Chapter 7 presents a comparative analysis of HHT and wavelet
transforms applied to turbulent open channel flow data.
In Chapter 8, nonlinear soil amplification is quantified by using HHT. Chapter
9 extends the application of HHT to nonstationary random processes. Chapter 10
presents a comparative analysis of HHT wavelet and Fourier transforms in some
structural health-monitoring applications. In Chapter 11, HHT is applied to molecular dynamics simulations. Chapter 12 presents a straightforward application of HHT
to decomposition of wave jumps. Chapter 13, the concluding chapter, presents
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perspectives on the theory and practices of HHT. It attempts to review HHT applications in biomedical engineering, chemistry and chemical engineering, financial
engineering, meteorological and atmospheric studies, ocean engineering, seismic
studies, structural analysis, health monitoring, and system identification. It also
indicates some directions for future research.
One important feature of the book is the inclusion of a variety of modern topics.
The examples presented are real-life engineering problems, as well as problems that
can be useful for benchmarking new techniques.
The studies reported in this book clearly indicate an increasing interest in HHT
and analysis for diversified applications. These studies are expected to stimulate the
interest of other researchers around the world who are facing new challenges in new
theoretical studies and innovative applications.
Norden Huang
NASA
Nii O. Attoh-Okine
University of Delaware
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Acknowledgments
The editors are grateful to the contributing authors. We also wish to express our
thanks to Tao Woolfe, B. J. Clark, and Michael Masiello of Taylor & Francis for
providing useful feedback and guiding the editors throughout the complex editorial
phase.
Norden E. Huang would like to thank Professors Theodore T. Y. Wu of the
California Institute of Technology, and Owen M. Phillips of the Johns Hopkins
University for their guidance and encouragement throughout the years, without
which the Hilbert-Huang Transform would not be what it is today.
Nii O. Attoh-Okine, the co-editor of the book, wishes to express his gratitude
to his parents, Madam Charkor Quaynor and Richard Attoh-Okine, for their support
and encouragement through the years.
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Contributors
Nii O. Attoh-Okine
Civil Engineering Department
University of Delaware
Newark, Delaware
Brad Battista
Information Systems Laboratories, Inc.
San Diego, California
Rodney R. Buntzen
Information Systems Laboratories, Inc.
San Diego, California
Marcus Dätig
Civil Engineering Department
Bergische University Wuppertal
Wuppertal, Germany
Brian Dzwonkowski
Graduate College of Marine Studies
University of Delaware
Newark, Delaware
Ping Gu
Department of Civil Engineering
University of Illinois at UrbanaChampaign
Urbana, Illinois
Norden E. Huang
Goddard Institute for Data Analysis
NASA Goddard Space Flight Center
Greenbelt, Maryland
Paul A. Hwang
Oceanography Division
Naval Research Laboratory
Stennis Space Center, Mississippi
Lide Jiang
Graduate College of Marine
Studies
University of Delaware
Newark, Delaware
Colin M. Edge
GlaxoSmithKline Pharmaceuticals
Harlow, United Kingdom
Young-Heon Jo
Graduate College of Marine Studies
University of Delaware
Newark, Delaware
Jonathan W. Essex
School of Chemistry
University of Southampton
Southampton, United Kingdom
James M. Kaihatu
Oceanography Division
Naval Research Laboratory
Stennis Space Center, Mississippi
Michael Gabbay
Information Systems Laboratories, Inc.
San Diego, California
Michael L. Larsen
Information Systems Laboratories, Inc.
San Diego, California
Robert J. Gledhill
School of Chemistry
University of Southampton
Southampton, United Kingdom
Stephen C. Phillips
School of Chemistry
University of Southampton
Southampton, United Kingdom
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Panayotis Prinos
Department of Civil Engineering
Aristotle University of Thessaloniki
Thessaloniki, Greece
David W. Wang
Oceanography Division
Naval Research Laboratory
Stennis Space Center, Mississippi
Ser-Tong Quek
Department of Civil Engineering
National University of Singapore
Singapore
Quan Wang
Department of Civil Engineering
National University of Singapore
Singapore
Jeffrey Ridgway
Information Systems Laboratories, Inc.
San Diego, California
Wei Wang
Physical Oceanography Laboratory
Ocean University of China
Shandong, P. R. China
Torsten Schlurmann
Civil Engineering Department
Bergische University Wuppertal
Wuppertal, Germany
Martin T. Swain
School of Chemistry
University of Southampton
Southampton, United Kingdom
Puat-Siong Tua
Department of Civil Engineering
National University of Singapore
Singapore
Albena Dimitrova Veltcheva
Port and Airport Research Institute
Yokosuka, Japan
Cye H. Waldman
Information Systems Laboratories, Inc.
San Diego, California
Yi-Kwei Wen
Department of Civil Engineering
University of Illinois at UrbanaChampaign
Urbana, Illinois
Adrian P. Wiley
School of Chemistry
University of Southampton
Southampton, United Kingdom
Xiao-Hai Yan
Graduate College of Marine
Studies
University of Delaware
Newark, Delaware
Athanasios Zeris
Department of Civil Engineering
Aristotle University of Thessaloniki
Thessaloniki, Greece
Ray Ruichong Zhang
Division of Engineering
Colorado School of Mines
Golden, Colorado
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Contents
Chapter 1
Introduction to Hilbert-Huang Transform and Some
Recent Developments...........................................................................1
Norden E. Huang
Chapter 2
Carrier and Riding Wave Structure of Rogue Waves........................25
Torsten Schlurmann and Marcus Dätig
Chapter 3
Applications of Hilbert-Huang Transform to Ocean-Atmosphere
Remote Sensing Research..................................................................59
Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and
Lide Jiang
Chapter 4
A Comparison of the Energy Flux Computation of Shoaling
Waves Using Hilbert and Wavelet Spectral Analysis Techniques ....83
Paul A. Hwang, David W. Wang, and James M. Kaihatu
Chapter 5
An Application of HHT Method to Nearshore Sea Waves...............97
Albena Dimitrova Veltcheva
Chapter 6
Transient Signal Detection Using the Empirical Mode
Decomposition .................................................................................121
Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman,
Michael Gabbay, Rodney R. Buntzen, and Brad Battista
Chapter 7
Coherent Structures Analysis in Turbulent Open Channel Flow
Using Hilbert-Huang and Wavelets Transforms..............................141
Athanasios Zeris and Panayotis Prinos
Chapter 8
An HHT-Based Approach to Quantify Nonlinear Soil
Amplification and Damping............................................................. 159
Ray Ruichong Zhang
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Chapter 9
Simulation of Nonstationary Random Processes Using
Instantaneous Frequency and Amplitude from Hilbert-Huang
Transform .........................................................................................191
Ping Gu and Y. Kwei Wen
Chapter 10 Comparison of Hilbert-Huang, Wavelet, and Fourier
Transforms for Selected Applications .............................................213
Ser-Tong Quek, Puat-Siong Tua, and Quan Wang
Chapter 11 The Analysis of Molecular Dynamics Simulations
by the Hilbert-Huang Transform .....................................................245
Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips,
Martin T. Swain, Colin M. Edge, and Jonathan W. Essex
Chapter 12 Decomposition of Wave Groups with EMD Method......................267
Wei Wang
Chapter 13 Perspectives on the Theory and Practices of the Hilbert-Huang
Transform .........................................................................................281
Nii O. Attoh-Okine
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1
Introduction to
Hilbert-Huang
Transform and Some
Recent Developments
Norden E. Huang
CONTENTS
1.1
1.2
1.3
Introduction ......................................................................................................1
The Hilbert-Huang Transform .........................................................................9
The Recent Developments .............................................................................16
1.3.1 The Normalized Hilbert Transform ...................................................16
1.3.2 The Confidence Limit ........................................................................19
1.3.3 The Statistical Significance of IMFs .................................................21
1.4 Conclusion......................................................................................................22
References................................................................................................................22
1.1 INTRODUCTION
Hilbert-Huang transform (HHT) is the designated name for the result of empirical
mode decomposition (EMD) and the Hilbert spectral analysis (HSA) methods, which
were both introduced recently by Huang et al. (1996, 1998, 1999, and 2003),
specifically for analyzing data from nonlinear and nonstationary processes. Data
analysis is an indispensable step in understanding the physical processes, but traditionally the data analysis methods were dominated by Fourier-based analysis. The
problems of such an approach were discussed in detail by Huang et al. (1998). As
data analysis is important for both theoretical and experimental studies (for data is
the only real link between theory and reality), we desperately need new methods in
order to gain a deeper insight into the underlying processes that actually generate
the data. The method we really need should not be limited to linear and stationary
processes, and it should yield physically meaningful results.
1
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The Hilbert-Huang Transform in Engineering
The development of the HHT is motivated precisely by such needs: first, because
the natural physical processes are mostly nonlinear and nonstationary, there are very
limited options in data analysis methods that can correctly handle data from such
processes. The available methods are either for linear but nonstationary processes
(such as the wavelet analysis, Wagner-Ville, and various short-time Fourier spectrograms as summarized by Priestley [1988], Cohen [1995], Daubechies [1992], and
Flandrin [1999]) or for nonlinear but stationary and statistically deterministic processes (such as the various phase plane representations and time-delayed imbedded
methods as summarized by Tong [1990], Diks [1997], and Kantz and Schreiber
[1997]). To examine data from real-world nonlinear, nonstationary, and stochastic
processes, we urgently need new approaches.
Second, the nonlinear processes need special treatment. Other than periodicity,
we want to learn the detailed dynamics in the processes from the data. One of the
typical characteristics of nonlinear processes, proposed by Huang et al. (1998), is
the intra-wave frequency modulation, which indicates that the instantaneous frequency changes within one oscillation cycle. Let us examine a very simple nonlinear
system given by the nondissipative Duffing equation as
∂2 x
+ x + ε x 3 = γ cos ωt ,
∂t 2
(1.1)
where ε is a parameter not necessarily small and γ is the amplitude of a periodic
forcing function with a frequency ω.
In Equation 1.1, if the parameter ε were zero, the system is linear, and the
solution would be easy. For a small, however, the system is nonlinear, but it could
be solved easily with perturbation methods. If ε is not small compared to unity, then
the system is highly nonlinear. No known general analytic method is available for
this condition; we have to resort to numerical solutions, where all kinds of complications such as bifurcations and chaos can result.
Even with these complications, let us examine the qualitative nature of the
solution for Equation 1.1 by rewriting it in a slightly different way, as
∂2 x
+ x (1 + ε x 2 ) = γ cos ωt ,
∂t 2
(1.2)
where the symbols are defined as in Equation 1.1. Then the quantity within the
parentheses can be regarded as a variable spring constant or a variable pendulum
length. With this view, we can see that the frequency should be changing from
location to location and from time to time, even within one oscillation cycle.
As Huang et al. (1998) pointed out, this intra-frequency frequency variation is
the hallmark of nonlinear systems. In the past, there has been no clear way to depict
this intra-wave frequency variation by using Fourier-based analysis methods, except
to resort to the harmonics. Even by the classical Hamiltonian approach, in which
the frequency is defined as the rate of change of the Hamiltonian with respect to
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Introduction to Hilbert-Huang Transform and Some Recent Developments
3
the action, we still cannot gain any more insight, for the definition of the action is
an integration of generalized momentum along the generalized coordinates; therefore, there is no instantaneous value. Thus, the best we could do for any nonlinear
distorted waveform in the earlier approaches was to refer to harmonic distortions.
Harmonic distortion is, in fact, a rather poor alternative, for it is the result obtained
by imposing a linear structure on a nonlinear system. Consequently, the results may
make perfect mathematical sense, but at the same time they have absolutely no
physical meaning. The physically meaningful way to describe the system should be
in terms of the instantaneous frequency.
The easiest way to compute the instantaneous frequency is by the Hilbert transform, through which we can find the complex conjugate, y(t), of any real valued
function x(t) of Lp class,
1
y(t ) = P
π
∞
x (τ)
∫ t − τ dτ ,
(1.3)
−∞
in which the P indicates the principal value of the singular integral. With the Hilbert
transform, we have
z (t ) = x (t ) + j y(t ) = a (t ) e j θ(t ) ,
(1.4)
where
(
a(t ) = x 2 + y 2
)
1/ 2
; θ(t ) = tan −1
y
.
x
(1.5)
Here a is the instantaneous amplitude and θ is the phase function; thus the
instantaneous frequency is simply
ω= −
dθ
.
dt
(1.6)
As the instantaneous frequency is defined through a derivative, it is very local
and can be used to describe the detailed variation of frequency, including the
intra-wave frequency variation. As simple as this principle is, the implementation is
not at all trivial. To represent the function in terms of a meaningful amplitude and
phase, however, requires that the function satisfy certain conditions. Let us take the
length-of-day (LOD) data, given in Figure 1.1, as an example to explain this requirement. The daily data covers from 1962 to 2002, for roughly 40 years. After performing the Hilbert transform, the polar representation of the data is given in Figure
1.2, which is a random collection of intertwined looping curves. The corresponding
phase function is given in Figure 1.3, where random finite jumps intersperse within
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4
The Hilbert-Huang Transform in Engineering
FIGURE 1.1 (See color insert following page 20). Data of Length-of-Day measure the
deviation from the mean 24-hour-day.
FIGURE 1.2 (See color insert following page 20). Analytic function in complex phase
plane formed by the real data and it Hilbert Transform. It shows not apparent order. After the
EMD, the annual cycle is extract and plotted also.
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Introduction to Hilbert-Huang Transform and Some Recent Developments
5
FIGURE 1.3 The phase function of the analytic function based on the Length-of-Day data.
the data span. If we follow through the definition of the instantaneous frequency as
given in Equation 1.6 literally, we would have a totally nonsensical result, as given
in Figure 1.4, where the instantaneous frequency is equally likely to be positive or
negative. Unfortunately, this is exactly the procedure recommended by Hahn (1996).
To show how this should not be the case, let us consider the three curves given
by the following three expressions
x1 = sin ωt ;
x 2 = 0.5 + sin ωt ;
(1.7)
x3 = 1.5 + sin ωt ;
shown in Figure 1.5. All three curves are perfect sine functions, but with the mean
displaced: for x1, its mean is exactly zero; for x2, its mean is moved up by half of
its amplitude; for x3, its mean is moved up by 1.5 times its amplitude. As a result,
the curve represented by x3 is totally above the zero reference axis. If we perform
the Hilbert transform to all three functions given in Equation 1.7, we would get
three different circles in the phase plane, with the centers of two of the circles
displaced by the amount of the added constants, as shown in Figure 1.6. Consequently, the phase functions from the three circles will be different, as shown in
Figure 1.7: for x1, the phase function is a straight line; for x2, the phase function is
a wavy line, but the general trend still agrees with the straight line; for x3, the phase
function is also wavy, but the variation is always within ± π.
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The Hilbert-Huang Transform in Engineering
FIGURE 1.4 Instantaneous frequency obtained form derivative of the phase function without
decomposition first. The values are equally like to be positive as negative.
FIGURE 1.5 (See color insert following page 20). Model data to illustrate the fallacy of
the instantaneous frequency without decomposition.
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Introduction to Hilbert-Huang Transform and Some Recent Developments
7
FIGURE 1.6 (See color insert following page 20). The analytic function in complex
phase plane of the data given in Figure 1.5.
Phase Angles for Test Data
20
sin x
0.5 + sin x
1.5 + sin x
Phase Angle: radians
15
10
5
0
−5
0
100
200
300
400
500
Time: (100) second
600
700
800
FIGURE 1.7 (See color insert following page 20). Phase function of the model function
given in Figure 1.5.
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The Hilbert-Huang Transform in Engineering
FIGURE 1.8 (See color insert following page 20). Instantaneous frequency computed
from the cosine model functions consist of the identical cosine function with different displacements.
Based on these phase functions, the instantaneous frequency values are very
different for the three expressions, as shown in Figure 1.8: for x1, the instantaneous
frequency is a constant value, which is exactly what we expected; for x2, the
instantaneous frequency is a variable curve with all positive values; for x3, the
instantaneous frequency is a highly variable curve fluctuating from positive to
negative values. Even if we are prepared to accept negative frequency, the result of
three different values for the same sine wave with only a displaced mean is very
unsettling: some of the results are, of course, nonsensical. The only meaningful
result is from the sine curve with a zero mean.
What went wrong was the fact that two of the curves do not have a zero mean,
or the envelopes of the curves are not symmetric with respect to the zero axis. Thus,
before performing the Hilbert transform, we have to preprocess the data. In the past,
any preprocessing usually consisted of band-pass filtering. For some of the data
from linear and stationary processes, this band-pass filtering method will give the
correct results. For data from nonlinear and nonstationary processes, however, the
band-pass filter will alter the characteristics of the filtered curve. The problem with
the filtering approach is that all the frequency domain filters are Fourier-based, which
means they are established under linear and stationary assumptions. When the data
are from nonlinear and nonstationary processes, such Fourier-based analysis will
surely generate spurious harmonics, which are mathematically necessary but physically meaningless, as discussed by Huang et al. (1998, 1999).
Considering these points leads us to this conclusion: The correct preprocessing
for data from nonlinear and nonstationary processes will have to be adaptive and
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Introduction to Hilbert-Huang Transform and Some Recent Developments
9
FIGURE 1.9 Test data to illustrate the procedures of Empirical Mode Decomposition also
known as sifting.
implemented in the time domain. The only method presently known to achieve this
is based on the Hilbert-Huang transform proposed by Huang et al. (1996, 1998,
1999, and 2003). This method is the subject of the next section.
1.2 THE HILBERT-HUANG TRANSFORM
The Hilbert-Huang transform is the result of the empirical mode decomposition and
the Hilbert spectral analysis. As the EMD method is more fundamental, and it is a
necessary step to reduce any given data into a collection of intrinsic mode functions
(IMF) to which the Hilbert analysis can be applied (see, for example, Huang et al.,
1996, 1998, and 1999), we will discuss it first. An IMF represents a simple oscillatory
mode as a counterpart to the simple harmonic function, but it is much more general:
by definition, an IMF is any function with the same number of extrema and zero
crossings, with its envelopes, as defined by all the local maxima and minima, being
symmetric with respect to zero. Obtaining the EMD consists of the following steps:
For any data as given in Figure 1.9, we first identify all the local extrema and
then connect all the local maxima by a cubic spline line as the upper envelope. We
repeat the procedure for the local minima to produce the lower envelope. The upper
and lower envelopes should cover all the data between them. Their mean is designated as m1, as shown in Figure 1.10, and the difference between the data and m1
is the first proto-IMF (PIMF) component, h1:
x (t ) − m1 = h1 .
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10
The Hilbert-Huang Transform in Engineering
FIGURE 1.10 (See color insert following page 20). The cubic spline upper and the lower
envelopes and their mean, m1.
This result is shown in Figure 1.11. The procedure of extracting an IMF is called
sifting. By construction, this PIMF, h1, should satisfy the definition of an IMF, but
the change of its reference frame from rectangular coordinate to a curvilinear one
can cause anomalies, as shown in Figure 1.11, where multi-extrema between successive zero-crossings still existed. To eliminate such anomalies, the sifting process
has to be repeated as many times as necessary to eliminate all the riding waves. In
the subsequent sifting process steps, h1 is treated as the data. Then
h1 − m11 = h11 ,
(1.9)
where m11 is the mean of the upper and lower envelopes of h1. This process can be
repeated up to k times; then, h1k is given by
h1( k −1) − m1k = h1k .
(1.10)
Each time the procedure is repeated, the mean moves closer to zero, as shown
in Figures 1.12a, b, and c. Theoretically, this step can go on for many iterations, but
each time, as the effects of the iterations make the mean approach zero, they also
make amplitude variations of the individual waves more even. Yet the variation of
the amplitude should represent the physical meaning of the processes. Thus this
iteration procedure, though serving the useful purpose of making the mean to be
zero, also drains the physical meaning out of the resulting components if carried
too far. Theoretically, if one insists on achieving a strictly zero mean, one would
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Introduction to Hilbert-Huang Transform and Some Recent Developments
11
Data and h1
10
h1
data
8
6
Amplitude
4
2
0
−2
−4
−6
−8
−10
200
250
300
350
400
Time: second
450
500
550
600
FIGURE 1.11 (See color insert following page 20). Comparison between data and h1 , as
given by Equation (1.9). Note most, but not all, riding waves are eliminated in h1.
Envelopes and the Mean: h1
10
Data: h1
Envelope
Envelope
Mean: m2
8
6
Amplitude
4
2
0
−2
−4
−6
−8
−10
200
250
300
350
400
Time: second
450
500
550
600
FIGURE 1.12 (A) (See color insert following page 20). Repeat the sifting using h1 as
data.
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The Hilbert-Huang Transform in Engineering
Envelopes and the Mean: h2
10
Data: h2
Envelope
Envelope
Mean: m3
8
6
Amplitude
4
2
0
−2
−4
−6
−8
−10
200
250
300
350
400
Time: second
450
500
550
600
FIGURE 1.12 (B) (See color insert following page 20). Repeat the sifting using h2 as data.
FIGURE 1.12 (C) (See color insert following page 20). After 12 iterations, the first
Intrinsic Mode Function is found.
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Introduction to Hilbert-Huang Transform and Some Recent Developments
13
probably have to make the resulting components purely frequency-modulated functions,
with the amplitude becoming constant. Then the resulting component would not
retain any physically meaningful information. Thus to attain the delicate balance of
achieving a reasonably small mean and also retaining enough physical meaning in
the resulting component, we have proposed two stoppage criteria. The “stoppage
criterion” actually determines the number of sifting steps to produce an IMF; it is
thus of critical importance in a successful implementation of the EMD method.
The first stoppage criterion is similar to the Cauchy convergence test, where we
first define a sum of the difference, SD, as
T
∑ h (t) − h (t)
k −1
SD = t = 0
2
k
;
T
∑ h (t)
(1.11)
2
k −1
t =0
then the sifting will stop when SD is smaller than a preassigned value. This definition
is a slight modification from the original one proposed by Huang et al. (1998), where
the SD was defined simply as
T
SD =
∑
t =0
hk −1 (t ) − hk (t )
hk2−1 (t )
2
.
(1.12)
The shortcoming of this old definition as given in Equation 1.12 is that the value
of SD can be dominated by local small values of hk–1, while the definition given in
Equation 1.11 sums up all the contributions over the whole duration of the data.
Even with this modification, there is still a problem with this seemingly mathematically sound approach: in this definition, the important criterion that the number of
extrema has to equal the number of zero-crossings has not been checked. To overcome this practical difficulty, Huang et al. (1999, 2003) proposed an alternative in
a second stoppage criterion.
The second stoppage criterion is based on a number called the S-number, which
is defined as the number of consecutive siftings when the numbers of zero-crossings
and extrema are equal or at most differing by one; it requires that number shall
remain unchanged. Through exhaustive testing, Huang et al. (2003) used this S-number method of defining a stoppage criterion to establish a confidence limit for the
EMD, to be discussed later.
When the resulting function satisfies either of the criteria given above, this
component is designated as the first IMF, c1, as shown in Figure 1.12c. We can then
separate c1 from the rest of the data by
X (t ) − c1 = r1 .
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14
The Hilbert-Huang Transform in Engineering
Data and residue: r1
10
Data
Residue: r1
8
6
Amplitude
4
2
0
−2
−4
−6
−8
−10
200
250
300
350
400
Time: second
450
500
550
600
FIGURE 1.13 (See color insert following page 20). Comparison between data and the
residue, r1, after the first IMF, c1, is removed. Notice the residue behaves like a moving mean
to the data that bisect all the waves.
This resulting residue is shown in Figure 1.13. Since the residue, r1, still contains
information with longer periods, it is treated as the new data and subjected to the
same process as described above. This procedure can be repeated to all the subsequent rj’s, and the result is
r1 − c2 = r2 ,
.
...
(1.14)
rn−1 − cn = rn
By summing up Equation 1.13 and Equation 1.14, we finally obtain
n
X (t ) =
∑ c +r .
j
n
(1.15)
j =1
Thus, we achieve a decomposition of the data into n IMF modes, and a residue,
rn, which can be either a constant, a monotonic mean trend, or a curve having only
one extremum. Recent studies by Flandrin et al. (2004) and Wu and Huang (2004)
established that the EMD is a dyadic filter, and it is equivalent to an adaptive wavelet.
Since it is adaptive, we avoid the shortcomings of using an a priori–defined wavelet
basis, and we also avoid the spurious harmonics that would have resulted. The
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Introduction to Hilbert-Huang Transform and Some Recent Developments
15
components of the EMD are usually physically meaningful, for the characteristic
scales are defined by the physical data. The sifting process is, in fact, a Reynolds-type
decomposition: separating variations from the mean, except that the mean is a local
instantaneous mean, so that the different modes are almost orthogonal to each other,
except for the nonlinearity in the data.
Having obtained the intrinsic mode function components, we can apply the
Hilbert transform to each IMF component and compute the instantaneous frequency
as the derivative of the phase function. After performing the Hilbert transform to
each IMF component, we can express the original data as the real part, RP, in the
following form:
n
X(t) = RP
∑ a (t) e
j
i ∫ ω j (t) dt
.
(1.16)
j=1
Equation 1.16 gives both amplitude and frequency of each component as a
function of time. The same data, if expanded in a Fourier representation, would have
a constant amplitude and frequency for each component. The contrast between EMD
and Fourier decomposition is clear: the IMF represents a generalized Fourier expansion with a time-varying function for amplitude and frequency. This frequency–time
distribution of the amplitude is designated as the Hilbert amplitude spectrum, H(,
t), or simply the Hilbert spectrum.
With the Hilbert spectrum defined, we can also define the marginal spectrum,
h(), as
T
h( ω ) =
∫ H(ω ,t)dt.
(1.17)
0
The marginal spectrum offers a measure of total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the
entire data span in a probabilistic sense.
The combination of the EMD and the HSA is known as the Hilbert-Huang
transform for short. Empirically, all tests indicate that HHT is a superior tool for
time–frequency analysis of nonlinear and nonstationary data. It has an adaptive basis,
and the frequency is defined through the Hilbert transform. Consequently, there is
no need for the spurious harmonics to represent nonlinear waveform deformations
as in any of the a priori basis methods, and there is no uncertainty principle limitation
on time or frequency resolution resulting from the convolution pairs possessing a
priori bases. Table 1.1 compares Fourier, wavelet, and HHT analyses.
From this table, we can see that the HHT approach is indeed a powerful method
for the analysis of data from nonlinear and nonstationary processes: it has an adaptive
basis; the frequency is derived by differentiation rather than convolution — therefore,
it is not limited by the uncertainty principle; it is applicable to nonlinear and
nonstationary data; and it presents the results in time–frequency–energy space for
feature extraction. This basic development of the HHT method has been followed
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The Hilbert-Huang Transform in Engineering
TABLE 1.1
Comparisons between Fourier, Wavelet, and Hilbert-Huang Transform
in Data Analysis
Fourier
Wavelet
Hilbert
Presentation
Nonlinear
Nonstationary
Feature Extraction
A priori
Convolution: global,
uncertainty
Energy–frequency
No
No
No
Adaptive
Differentiation: local,
certainty
Energy–time–frequency
Yes
Yes
Yes
Theoretical Base
Theory complete
A priori
Convolution: regional,
uncertainty
Energy–time–frequency
No
Yes
Discrete: no
Continuous: yes
Theory complete
Basis
Frequency
Empirical
by recent developments that have either added insight to the results or enhanced
their statistical significance. Some of the recent developments are summarized in
the following section.
1.3 THE RECENT DEVELOPMENTS
We will discuss in some detail recent developments in the areas of the normalized
Hilbert transform, the confidence limit, and the statistical significance of IMFs.
1.3.1 THE NORMALIZED HILBERT TRANSFORM
It is well known that, although the Hilbert transform exists for any function of Lp
class, the phase function of the transformed function will not always yield physically
meaningful instantaneous frequencies, as discussed above. In addition to the requirement of being an IMF, which is only a necessary condition, additional limitations
have been summarized succinctly in two theorems.
First, the Bedrosian theorem (1963) states that the Hilbert transform for the
product of two functions, f(t) and h(t), can be written as
H [ f (t ) h(t )] = f (t ) H [ h(t )] ,
(1.18)
only if the Fourier spectra for f(t) and h(t) are totally disjoint in frequency space,
and if the frequency content of the spectrum for h(t) is higher than that of f(t). This
limitation is critical, for we need to have
H [ a(t ) cos θ(t ) ] = a(t ) H [cos θ(t )] ;
(1.19)
otherwise, we cannot use Equation 1.5 to define the phase function. According to
the Bedrosian theorem, Equation 1.19 is true only if the amplitude is varying so
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Introduction to Hilbert-Huang Transform and Some Recent Developments
17
slowly that the frequency spectra of the envelope and the carrier waves are disjoint.
This has made the application of the Hilbert transform even to IMFs problematic.
To satisfy this requirement, Huang and Long (2003) have proposed the normalization
of the IMFs in the following steps: starting from an IMF, we first find all the maxima
of the IMFs, defining the envelope by spline through all the maxima and designating
the envelope as E(t). Now, we normalize the IMF by dividing the IMF by E(t). Thus,
we have the normalized function with amplitude always equal to unity.
Even with this normalization, we have not resolved all the limitations on the
Hilbert transform. The new restriction is given by the Nuttall theorem (1966). This
theorem states that the Hilbert transform of cosine is not necessarily the sine with
the same phase function for a cosine with an arbitrary phase function. Nuttall gave
an error bound, ∆E, defined as the difference between y(t), the Hilbert transform of
the data, and Q(t), the quadrature (with phase shift of exactly 90°) of the function:
T
∆E =
0
∫ y(t) − Q(t) dt = ∫ S (ω) dω ,
2
q
t =0
(1.20)
−∞
where Sq is Fourier spectrum of the quadrature function. The proof of this theorem
is rigorous, but the result is hardly useful, for it gives a constant error bound over
the whole data range. For a nonstationary time series, such a constant bound will
not reveal the location of the error on the time axis.
With the normalized IMF, Huang and Long (2003) have proposed a variable
error bound based on a simple argument, which goes as follows: let us compute the
difference between the squared amplitude of the normalized IMF and unity. If the
Hilbert transform is exactly the quadrature, then the squared amplitude of the
normalized IMF should be unity; therefore, the difference between it and unity
should be zero. If the squared amplitude is not exactly unity, then the Hilbert
transform cannot be exactly the quadrature. Consequently, the error can be measured
simply by the difference between the squared normalized IMF and unity, which is
a function of time. Huang and Long (2003) and Huang et al. (2005) have conducted
detailed comparisons and found the result quite satisfactory.
Even with the error indicator, we can only know that the Hilbert transform is
not exactly the quadrature; we still do not have the correct answer. This prompts
the suggestion of a drastic alternative, eschewing the Hilbert transform totally. To
this end, Huang et al. (2005) suggest that the phase function can be found by
computing the arc-cosine of the normalized function. A checking of the results so
obtained has also proved to be satisfactory. The only problem is that the imperfect
normalization will give some values greater than unity. Under that condition, the
arc-cosine will break down.
An example of the normalized and regular Hilbert transforms is given in Figure
1.14, from the data given in Figure 1.12c. There are three different instantaneous
frequency values: the instantaneous frequency from regular Hilbert transform, the
normalized Hilbert transform, and the generalized zero-crossing, which can serve
as the standard in the mean. It is easy to see that the normalized instantaneous
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18
The Hilbert-Huang Transform in Engineering
Comparison of Instantaneous frequency from Different Methods
0.2
0.15
Data
IF-H
IF-NH
IF-Z
Amplitude
0.1
0.05
0
–0.05
0
50
100
150
200
Time : second
250
300
350
400
FIGURE 1.14 (See color insert following page 20). Comparison of the instantaneous frequency values derived from different methods. Note that the Instantaneous from IMF is still
not correct when the amplitude fluctuates too much. The normalized Hilbert Transform,
however, gives a much better instantaneous frequency when compared with the values derived
from the generalized-zero-crossing method.
frequency is very close to the zero-crossing values, while the regular Hilbert transform result gives large undulations that will never result in the mean as given by
the zero-crossing method. The high undulation results from the large changes of
amplitude and some nonlinear distortions of the waveform, both of which will cause
the envelope to fluctuate as shown in Figure 1.15. In the normalization scheme, the
smooth spline helps to eliminate many of the undulations in the resulting instantaneous frequency. One can also see that the problem of the regular Hilbert transform
occurs always at the location where either the amplitudes change drastically or the
amplitude is very low, as predicted by the Nuttall theorem. The normalized Hilbert
transform alleviates the problems substantially.
Finally, the error index is given in Figure 1.16; here we can see that the error
is also small in general, unless the waveform is locally distorted. Even over the large
error location, the index values are smaller than 10%, except for the end region,
where the end effect of the Hilbert transform causes additional problems. Thus the
normalized Hilbert transform has helped to overcome many of the difficulties of the
regular Hilbert transform, and it should be used all the time.
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Introduction to Hilbert-Huang Transform and Some Recent Developments
19
Comparison of Amplitude from Different Methods
10
Data
A-H
A-NH
A-Z
9
8
Amplitude : cm
7
6
5
4
3
2
1
0
0
50
100
150
200
Time : second
250
300
350
400
FIGURE 1.15 (See color insert following page 20). The instantaneous amplitude (or the
envelope) of the test data. Note the improvement in adopting the spline envelope over the
simple analytic function.
1.3.2 THE CONFIDENCE LIMIT
The confidence limit for the Fourier spectral analysis is routinely computed. The
computation, however, is based on the practice of cutting the data into N sections
and computing spectra from each section. The confidence limit is defined as the
statistical spread of the N different spectra. This practice is based on the ergodic
theory, where the temporal average is treated as the ensemble average. The ergodic
condition is satisfied only if the processes are stationary; otherwise, averaging them
will not make sense. Huang et al. (2003) have proposed a different approach, using
the fact that there are infinitely many ways to decompose one given function into
different components. Even using EMD, we can still obtain many different sets of
IMFs by changing the stoppage criteria. For example, Huang et al. (2003) explored
the stoppage criterion by changing the S-number. Using the length-of-day (LOD)
data, they varied the S-number from 1 to 20 and found the mean and the standard
deviation for the Hilbert spectrum given in Figure 1.15. The confidence limit so
derived does not depend on the ergodic theory. By using the same data length, there
is also no downgrading of the spectral resolution in frequency space through subdividing of the data into sections.
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The Hilbert-Huang Transform in Engineering
Error Index for Normalized Hilbert Transform
0.1
0.08
0.06
0.04
Amplitude
0.02
0
–0.02
–0.04
–0.06
–0.08
–0.1
0
50
100
150
200
Time : second
250
300
350
400
FIGURE 1.16 (See color insert following page 20). The Error Index of the normalized
Hilbert transform; it has large value whenever the wave form deviated from a smooth sinusoidal form. But their values are, in general, small except near the ends.
Additionally, Huang et al. (2003) invoked the intermittence criterion and forced
the number of IMFs to be the same for different S-numbers. As a result, they were
able to find the mean for specific IMFs. Figure 1.17 shows the IMF representing
variations of the annual cycle of the length of day. The peak and valley of the
envelope represent the El Niño events. Of particular interest are the periods of high
standard deviations, from 1965 to 1970 and from 1990 to 1995. These periods turn
out to be the anomaly periods of the El Niño phenomena, when the sea surface
temperature readings in the equatorial region were consistently high based on observations, indicating a prolonged heating of the ocean, rather than the changes from
warm to cool during the El Niño to La Niña changes.
Finally, from the confidence limit study, an unexpected result was the determination of the optimal S-number. Huang et al. (2003) computed the difference between
the individual cases and the overall mean and found that there is always a range
where the differences reach a local minimum. Based on their limited experience
from different data sets, they concluded that an S-number in the range of 4 to 8
performed well. Logic also dictates that the S-number should not be too high (which
would drain all the physical meaning out of the IMF) nor too low (which would
leave some riding waves remaining in the resulting IMFs).
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Introduction to Hilbert-Huang Transform and Some Recent Developments
21
Error Index for Normalized Hilbert Transform
0.1
MeanAnnual Envelope
M+std
M–std
std
0.08
0.06
0.04
Amplitude
0.02
0
–0.02
–0.04
–0.06
–0.08
–0.1
1965
1970
1975
1980
1985
Time : second
1990
1995
2000
FIGURE 1.17 (See color insert following page 20). The mean envelope of the annual
cycle IMF component from LOD data. The peaks of the envelope are all aligned with El Nio
events, when the additional angular momentum imparted to the atmosphere from the over
heated Equatorial ocean water. The large scatter of the envelope periods in 1065-70 and 199095 represent periods of El Nio anomalies.
1.3.3 THE STATISTICAL SIGNIFICANCE OF IMFS
The EMD is a method of separating data into different components by their scales.
There is always the question: on what is the statistical significance of the IMFs
based? In data that contains noise, how can we separate the noise from information
with confidence? This question was addressed by both Flandrin et al. (2004) and
Wu and Huang (2004) through the study of signals consisting of noise only.
Flandrin et al. (2004) studied the fractal Gaussian noises and found that the
EMD is a dyadic filter. They also found that when one plotted the mean period and
root-mean-square (RMS) values of the IMFs derived from the fractal Gaussian noise
on log–log scale, the results formed a straight line. The slope of the straight line for
white noise is –1; however, the values change regularly with the different Hurst
indices. Based on these results, Flandrin et al. (2004) suggested that the EMD results
could be used to discriminate what kind of noise one was encountering.
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22
The Hilbert-Huang Transform in Engineering
Instead of fractal Gaussian noise, Wu and Huang (2004) studied the Gaussian
white noise only. They also found the relationship between the mean period and
RMS values of the IMFs. Additionally, they have also studied the statistical properties of the scattering of the data and found the bounds of the data distribution
analytically. From the scattering, they deduced a 95% bound for the white noise.
Therefore, they concluded that when a data set is analyzed with EMD, if the mean
period-RMS values exist within the noise bounds, the components most likely
represent noise. On the other hand, if the mean period-RMS values exceed the noise
bounds, then those IMFs must represent statistically significant information.
1.4 CONCLUSION
HHT is a relatively new method in data analysis. Its power is in the totally adaptive
approach that it takes, which results in the adaptive basis, the IMFs, from which the
instantaneous frequency can be defined. This offers a totally new and valuable view
of nonstationary and nonlinear data analysis methods. With the recent developments
on the normalized Hilbert transform, the confidence limit, and the statistical significance test for the IMFs, the HHT has become a more robust tool for data analysis,
and it is now ready for a wide variety of applications. The development of HHT,
however, is not over yet. We still need a more rigorous mathematical foundation for
the general adaptive methods for data analysis, and the end effects must be improved
as well.
REFERENCES
Bedrosian, E. (1963). On the quadrature approximation to the Hilbert transform of modulated
signals. Proc. IEEE, 51, 868–869.
Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall, Englewood Cliffs, NJ.
Diks, C. (1997). Nonlinear Time Series Analysis. World Scientific Press, Singapore.
Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia.
Flandrin, P. (1999). Time-Frequency/Time-Scale Analysis. Academic Press, San Diego, CA.
Flandrin, P., Rilling, G., and Gonçalves, P. (2004). Empirical mode decomposition as a
filterbank. IEEE Signal Proc. Lett. 11 (2): 112–114.
Hahn, S. L. (1996). Hilbert Transforms in Signal Processing. Artech House, Boston.
Huang, N. E., and Long, S. R. (2003). A generalized zero-crossing for local frequency
determination. U.S. Patent pending.
Huang N. E., Long, S. R., and Shen, Z. (1996). Frequency downshift in nonlinear water wave
evolution. Advances in Appl. Mech. 32, 59–117.
Huang, N. E., Shen, Z., Long, S. R. (1999). A new view of nonlinear water waves — the
Hilbert spectrum. Ann. Rev. Fluid Mech. 31, 417–457.
Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C., and
Liu, H. H. (1998). The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis. Proc. Roy. Soc. London, A454, 903–995.
Huang, N. E., Wu, Z., Long, S. R., Arnold, K. C., Blank, K., Liu, T. W. (2005). On instantaneous frequency. Proc. Roy. Soc. London (submitted).
© 2005 by Taylor & Francis Group, LLC
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Introduction to Hilbert-Huang Transform and Some Recent Developments
23
Huang, N. E., Wu, M. L., Long, S. R., Shen, S. S. P., Qu, W. D., Gloersen, P., and Fan, K.
L. (2003). A confidence limit for the empirical mode decomposition and the Hilbert
spectral analysis. Proc. Roy. Soc. London, A459, 2317–2345.
Kantz, H., and Schreiber, T. (1997). Nonlinear Time Series Analysis. Cambridge University
Press, Cambridge.
Nuttall, A. H. (1966). On the quadrature approximation to the Hilbert transform of modulated
signals. Proc. IEEE, 54, 1458–1459.
Priestley, M. B. (1988). Nonlinear and nonstationary time series analysis. Academic Press,
London.
Tong, H. (1990). Nonlinear Time Series Analysis. Oxford University Press, Oxford.
Wu, Z., and Huang, N. E. (2004). A study of the characteristics of white noise using the
empirical mode decomposition method. Proc. Roy. Soc. London, A460, 1597–1611.
© 2005 by Taylor & Francis Group, LLC
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3
Applications of
Hilbert-Huang Transform
to Ocean-Atmosphere
Remote Sensing
Research
Xiao-Hai Yan, Young-Heon Jo,
Brian Dzwonkowski, and Lide Jiang
CONTENTS
3.1
3.2
Introduction ....................................................................................................60
Analyses of TOPEX/Poseidon Sea Level Anomaly Interannual
Variation Using HHT and EOF .....................................................................62
3.3 Application of HHT to Ocean Color Remote Sensing
of the Delaware Bay ......................................................................................64
3.4 Mediterranean Outflow and Meddies (O & M)from Satellite
Multisensor Remote Sensing .........................................................................71
3.5 Conclusion......................................................................................................78
Acknowledgments....................................................................................................79
References................................................................................................................79
ABSTRACT
The Hilbert-Huang transform (HHT) is a newly developed method for analyzing nonlinear and nonstationary processes. Its application in oceanography and oceanatmosphere remote sensing research is still in its infancy. In this chapter, we briefly introduce
the application of this method in oceanatmosphere remote sensing data analyses and
present a few examples of such applications.
59
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The Hilbert-Huang Transform in Engineering
3.1 INTRODUCTION
Spectral analysis is a very useful tool to analyze a time series signal. However, this
method does not fully describe a data set that changes with time. The spectrum gives
us the frequencies that exist over the entire duration of the data set. On the other
hand, time–frequency analysis allows us to determine the frequencies at a particular
time. Hence, the fundamental idea of time–frequency analysis is to understand and
describe phenomena where the frequency content of a signal is changing in time.
Scientists traditionally use short Fourier transform by sliding the window along
the time axis to get a time–frequency distribution. Since it relies on the traditional
Fourier spectral analysis, one has to assume the data to be piecewise stationary.
Currently, the most famous time–frequency analysis method is wavelet transform.
The most common method used is Morlet wavelet, defined as Gaussian enveloped
sine and cosine wave groups with 5.5 waves (1). The problem with Morlet wavelet
is the leakage generated by the limited length of the basic wavelet function, which
makes the quantitative definition of the energy–frequency–time distribution difficult.
Once the basic wavelet is selected, one has to apply it to analysis of all the data (2).
Recently Huang et al. (3) introduced a new and potentially more robust method
for time–frequency analysis. This method, the empirical mode decomposition–Hilbert-Huang transform (EMD-HHT), is applicable to both nonstationary and nonlinear signals. In real ocean and ocean atmosphere coupling, most processes are nonlinear and nonstationary. One example is that at the onset of El Nino: nonlinear
Kelvin waves carry warm water from the western Pacific to the east (4). This process
is exhibited as a nonlinear pattern in altimeter data. For this reason, we use the
EMD-HHT technique in our El Nino study and in many of our other studies.
Since the EMD-HHT is relatively new to the ocean remote sensing community,
a brief summary of the technique based on Huang et al. (3) is given in this section.
Basically, the EMD-HHT method requires two steps in analyzing the data. The first
step is to decompose time series data into a number of intrinsic mode functions
(IMFs). These functions must satisfy the following two conditions: (a) within the
entire data set, the total number of extrema (as a function of time) and the total
number of zero-crossings must either be equal or differ at most by one, and (b) at
any point, the mean value of the envelope defined by the local minima (as a function
of time) and the envelope defined by the local maxima (as a function of time) must
be zero. The second step is to apply the Hilbert transform to the decomposed IMFs
and construct the energy–frequency–time distribution, designated as the Hilbert
spectrum. The presentation of the final results of the time–frequency analysis is
similar to the wavelet transform method, which is a spectrogram (time–frequency–energy plot). For clarity, a spectral analysis of the corresponding signal is
shown next to the spectrogram when we present our results in the next sections.
The decomposition of the time series data (i.e. H(t)) into IMFs uses separately
defined envelopes of local maxima and minima. Once the extrema are identified, all
the local maxima are connected by a cubic spline to form the upper envelope. The
procedure is repeated for the local minima to produce the lower envelope. Their
mean is designated as m1(t), and the difference between the time series data and
m1(t) is the first component, h1(t). One can repeat this procedure k times, until hk(t)
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Applications of Hilbert-Huang Transform
61
is an IMF. Then, hk(t) = c1(t) is the first IMF component of the data. c1(t) should
contain the finest scale or the shortest period component of the signal. Then, c1(t)
is separated from the data, and the process is repeated until either the component
cn(t) or the residue rn(t) becomes so small that it is less than a predetermined value,
or when the residue rn(t) becomes a monotonic function from which no IMF can be
extracted.
After all IMFs have been determined, one can check the original data with the
sum of the IMF components
n
( ) ∑ c (t ) + r (t ) .
H t =
1
n
(3.1)
i =1
Thus, a decomposition of the data into n empirical modes and a residue rn(t) is
achieved. The rn(t) can be either the mean trend or a constant.
After IMFs of the data have been generated, the next step is to apply the Hilbert
transform to each IMF time series. For an arbitrary time series X(t), one can define
its Hilbert transform, Y(t), as:
()
1
Y t =
π
()
∞
X τ
∫ t − τ dτ.
(3.2)
−∞
With this definition, X(t) and Y(t) form a complex conjugate pair, so one has an
analytic signal Z(t) as
()
()
()
Z t = X t + iY t ,
(3.3)
where its amplitude function a(t) is
()
()
()
1
a t =  X 2 t + Y 2 t  2 ,
(3.4)
and its phase function (t) is
()
()
Y t 
θ t = arctan 
.
 X t 
()
(3.5)
The instantaneous frequency is defined as
ω=
© 2005 by Taylor & Francis Group, LLC
( ).
dθ t
dt
(3.6)
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62
The Hilbert-Huang Transform in Engineering
Once the instantaneous frequency (as function of time) of each IMF has been
generated, the final result of the time–frequency analysis is similar to the wavelet
transform method, time–frequency–energy plot, or spectrogram. The energy density
similar to Fourier transform can be generated by summing the spectrogram for the
whole time series along a constant frequency for each frequency.
From the definition of the phase function and instantaneous frequency, we can
clearly see that for a simple function such as a = sin(t), the Hilbert spectrum is
simply cos(t), and the phase function is a monotonic straight line. Hence, the
spectrogram is simply a horizontal line for a whole time along fixed frequency, and
its spectrum is simply a single peak at that particular frequency.
In the next sections, we describe a few examples of applications of the HHT in
ocean-atmosphere remote sensing data processing and research. These examples
include an analysis of TOPEX/Poseidon sea level anomaly interannual variation
using HHT and empirical orthogonal function (EOF), application of HHT to ocean
color remote sensing of the Delaware Bay, and Mediterranean outflow and Meddies
determined from satellite multisensor remote sensing.
3.2 ANALYSES OF TOPEX/POSEIDON SEA LEVEL ANOMALY
INTERANNUAL VARIATION USING HHT AND EOF
The measurement of sea surface height provides a unique opportunity to make
significant contributions to many of the science objectives of oceanatmosphere
remote sensing scientists working in the area of climate variability, oceanatmosphere
interactions, and upper ocean dynamics. Satellite altimeter data can be employed to
gain vital new understanding of the nature of the ocean’s circulation and ocean
atmosphere coupling, both in terms of mean state and variability and interrelationships between this circulation, its variability, and global climate change.
We used TOPEX/Poseidon (T/P) monthly sea level anomaly (SLA) data from
January 1993 to August 2002, obtained from the University of Texas’ Center for
Space Research (UT/CSR). The deviation of the sea surface was removed from a
mean surface and was preceded by instrument corrections (ionosphere, wet and dry
troposphere, and electromagnetic bias) and geophysical corrections (tides and
inverted barometer). The availability of accurate altimeter data, such as that from
T/P, with the root-mean-square (RMS) error as small as 2 cm (5), allows one to
examine the annual and interannual sea surface variability due to El Niño Southern
Oscillation (ENSO) and global climate change. The original spatial range was 65°S
to 65°N, 180°E to 180°W; however, we limited the latitude range to 60°S to 60°N.
The resolution of the data set was 1° by 1°.
To isolate the signal of the interannual variation of SLA, the annual fluctuation
was removed by subtracting the overall monthly mean for a given month from the
individual monthly data for that respective month. According to calculation, the
interannual energy accounted for about 60% of the total energy. Furthermore, to
analyze the resulting time series, EOF method was applied. The first three EOFs,
EOF-1, EOF-2, and EOF-3, accounted for 35.4%, 12.6%, and 9.8% of the total
interannual energy, respectively.
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Applications of Hilbert-Huang Transform
Interannual EOF-1 (35.4%)
63
IMF-1 of EOF-1
0.02
60N
0.01
40N
0
20N
−0.01
EQ
−0.02
93
94
95
96
97 98 99 00
IMF-2 of EOF-1
01
02
94
95
96
97 98 99 00
IMF-3 of EOF-1
01
02
94
95
96
97 98 99 00
Residual of EOF-1
01
02
94
95
96
01
02
20S
0.1
40S
0.05
60S
60E
120E
0
180
120W
60W
0
0.005 0.01 0.015 0.02
−0.05
−0.02 −0.015 −0.01 −0.005
−0.1
93
0.2
0.1
0.3
0.25
0
0.2
−0.1
0.15
−0.2
93
0.1
0.08
0.05
0.06
0
0.04
−0.05
0.02
−0.1
0
−0.15
−0.02
93
93
94
95
96
97
98
99
00
01
02
97
98
99
00
FIGURE 3.1 The spatial (upper) and temporal (lower) interannual EOF-1 (left) and EMD
of the temporal mode (right) of the T/P-SLA.
Due to its important role in interannual variation, EOF1 was further analyzed
using the HHT. Empirical mode decomposition (EMD) was applied, and EOF1 was
decomposed into three IMFs plus a residue (Figure 3.1). The temporal mode of
EOF1 shows a peak in late ‘97, which correspond to the strong 1997–1998 El Niño.
To examine the effect of ENSO events on EOF1, the Southern Oscillation Index
(SOI) was also used as a reference by decomposing it using EMD. The IMF4 of the
SOI appeared to be a smoothed curve of the SOI, and Salisbury and Wimbush (6)
used this to predict future ENSO events.
Here a comparison was made between the EOF-1 and the SOI by calculating
the correlation coefficient between the IMF-3 of the EOF-1 and IMF-4 of the SOI.
The correlation coefficient was found to be as high as 0.85 (Figure 3.2). Considering
this, the EOF-1 can be considered to behave under the impact of the ENSO, and we
can further infer that the ENSO contributes to about one third of the interannual
SLA variation.
Moreover, we can reexamine the spatial mode of EOF1 (Figure 3.1) to see the
global impact of the ENSO events on human activities. Besides the coastal areas of
America, Australia, and Indonesia, which are most strongly and directly affected,
those of east Africa and Japan also are undergoing remarkable ENSO variation. The
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64
The Hilbert-Huang Transform in Engineering
Correlation = 0.85
0.1
0.05
0
−0.05
−0.1
−0.15
93
94
95
96
97
98
99
00
01
02
03
FIGURE 3.2 Correlation between negative IMF-4 of SOI (red) and IMF-3 of EOF-1 (blue)
is 0.85. It suggests that the primary interannual EOF is impacted by ENSO.
impact even spread as far as Antarctica. On the other hand, coastal areas of the
Atlantic Ocean are less affected by the ENSO events.
In addition, the HHT spectrum of the EOF1 was investigated to extract the
frequency information (Figure 3.3). The frequency distribution of the EOF1 is
between 0 and 4 cycle/yr. The 2.5 to 4 cycle/yr range corresponds to the spectrum
of the IMF1, which behaves in a relatively random pattern. The 0.5 to 2 cycle/yr
range corresponds to the spectrum of IMF2, which has more energy within the
frequency range 0.5 to 0.8 cycle/yr. The 0.2 to 0.4 cycle/yr range has much higher
energy, which is shown as a dark red curve at the bottom part of the spectrum. This
curve corresponds to the frequency feature of IMF3, i.e., a period of 2.5 to 5 yr,
which can be considered as the typical frequency of ENSO events. The lowest
frequency part of the 0 to 0.1 cycle/yr range corresponds to the frequency of the
residue, which is the longterm trend. It has a period of tens of years or longer and,
therefore, requires much longer time series to analyze.
3.3 APPLICATION OF HHT TO OCEAN COLOR REMOTE
SENSING OF THE DELAWARE BAY
The use of satellite-based ocean color data has become an integral part of oceanographic studies, aiding in the exploration of a number of important topics. The ability
to collect ocean color data at high temporal and spatial resolutions, which only
satellite sensors can provide, has given the oceanographic community a rich data
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Applications of Hilbert-Huang Transform
65
HHT Spectrum of EOF-1
Hilbert Spectrum (cycles per year)
4
3.5
3
2.5
2
1.5
1
0.5
93
94
0
0.005
95
96
0.01
97
0.015
98
Year
99
00
0.02
0.025
0.03
01
02
0.035
0.04
FIGURE 3.3 The HHT spectrum of EOF-1. The dark red curve indicates high energy at
frequencies between 0.2 and 0.4 cycle/yr, which corresponds to the Hilbert spectrum of IMF-3
of interannual EOF-1. It indicates that ENSO events have a typical period of 2.5 to 5 yr.
source. However, use of this data source is dependent on the premise that there are
discernable relationships between reflectance exiting the water column and the
constituents in it. A primary constituent of study has been chlorophyll-a on account
of its connection to phytoplankton and thus to primary production and biomass (7).
As coastal management strategies begin to focus on large-scale ecosystem based
programs, the relationship between chlorophyll-a and primary production and biomass has the potential to provide a cost-effective alternative to traditional ship-based
or point source sampling. Monitoring and assessing the health of ecosystem-size
regions will be much more feasible with satellite-based parameters due to the
frequent repeat periods and large spatial areas covered by satellite sensors. Thus,
the quantitative assessment of water constituents from satellite-based ocean color
data holds great promise for implementing dynamic management strategies. However, to interpret ocean color data appropriately, a full understanding of the periodicity of chlorophyll concentrations in a given area is essential. The purpose of this
study is to examine the seasonal and interannual chlorophyll-a cycles from satellite
data of the coastal region at the mouth of the Delaware Bay.
Although quantifying chlorophyll concentration from satellite ocean color data
has been successful in the open ocean, coastal areas (case II water) can still present
problems. Water constituents associated with coastal regions, such as excessive
chlorophyll-a concentrations, suspended sediments, and colored dissolved organic
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66
The Hilbert-Huang Transform in Engineering
material, create an optically complex area in which traditional empirical models
typically perform poorly (8). However, several studies have used an innovative neural
network (NN) methodology successfully to develop ocean color algorithms in Case
II water that outperform empirical methods (9, 10, 11, 12). NN based algorithms
are advantageous on account of their ability to model complex, nonlinear geophysical
transfer functions like those required to relate chlorophyll-a concentrations to the
reflectance measured by a satellite sensor (13). The NN requires training data
consisting of input data paired with the desired output, from which it “learns” the
geophysical relationship. The ability of a NN to generalize (i.e. accurately predict
the geophysical parameter) is assessed through a validation set of pair data that the
NN has never encountered. Given the success of NNs in coastal water algorithm
development, the chlorophyll-a data used in this study was produced from a NN
trained and validated using the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS)
remotely sensed reflectance data paired with in situ chlorophyll data in the Delaware
Bay and its adjacent coastal water. These paired data points were collected during
a number of different days and seasons. The neural network showed significant
improvement over the current SeaWiFS processing algorithm (Ocean Color 4 [OC4])
and was used to reprocess the satellite data used in this study (14).
Over the course of the SeaWiFS project, daily high-resolution picture transmission (HRPT) images of the Delaware Bay and the adjacent coastal zone were
subjectively viewed to identify all the cloud-free images of the Delaware Bay and
the adjacent coastal zone from September 1997 to July 2003. This period covers
almost six years of data, from which 468 images were collected, with an average
of 6.6 images per month. The daily images were obtained at the L1A processing
level and converted to a remapped L2 remote sensing reflectance (Rrs) product. The
remapping used a standard Mercator projection with a fixed grid of 890 × 890 pixels.
Each image ranges from 34°N to 42°N latitude and from 68°W to 78°W longitude,
which covers approximately 9.58 × 105 km2. The Rrs product was then used as input
for the optimal NN model mentioned earlier to produce daily chlorophyll-a maps
of the Delaware Bay and the adjacent coastal zone. Since NNs are extremely good
at interpolation but poor at extrapolation, only a limited subset (the 121 × 121 pixel
Delaware Bay and adjacent coastal zone) of the entire SeaWiFS image was processed
using the NN. This ensured that the model would only be applied to the region
where it was specifically trained with in situ data. The SeaWiFS imagery was binned
by month and geometrically averaged to produce an image of the mean monthly
values of chlorophyll-a concentrations for each month. This organizational process
created spatially coherent patterns of chlorophyll-a values at evenly spaced time
scales that help reduce the effects of sensor and algorithm errors and minimize the
influence of high frequency variability on seasonal distributions of chlorophyll-a
concentrations (15). Furthermore, due to the fact that chlorophyll pigments tend to
be lognormally distributed, the geometric means were used in the various statistical
procedures applied to chlorophyll-a values in this study (15). A station at the mouth
of the Delaware Bay (38.916°N, –75.100°W) was used as the geographic point from
which a monthly times series of chlorophyll-a concentrations was produced. This
monthly time series is shown in Figure 3.4.
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Applications of Hilbert-Huang Transform
67
Chlorophyll-a (mg/m3)
25
20
15
10
5
0
1998
1999
2000
2001
2002
2003
1998
1999
2000
2001
Years
2002
2003
2
NAO Index
1
0
−1
−2
−3
FIGURE 3.4 The time series of the monthly means of the chlorophyll-a concentrations (top
panel) and the North Atlantic Oscillation index (bottom panel) from September 1997 to July
2003.
Hilbert-Huang transform (HHT) analysis provides an innovative way to study
the time–frequency characteristics of time series, due to its ability to handle nonlinear
and nonstationary data and to account for changes in frequency over time. This novel
technique has been applied to the time series of chlorophyll-a concentrations at the
mouth of the Delaware Bay, in order to better understand the forcing mechanisms
associated with seasonal and interannual variations. The initial chlorophyll-a time
series was decomposed into IMFs, to which the Hilbert transform can be applied.
The resulting Hilbert spectrum and its associated marginal spectrum are shown in
Figure 3.5.
The marginal spectrum is conceptually similar to a spectral density graph produced by Fourier analysis; however, it is produced by summing the instantaneous
energy associated with a constant frequency over the entire time period. Thus, the
marginal spectrum provides a measurement of the frequencies that contain the largest
amounts of energy. As expected, the dominant energy peak occurs at about 0.09,
which is the approximate annual forcing frequency and corresponds to period of
11.1 months. This result is supported by the fact that several annual physical forcing
factors, such as sea surface temperature, solar insolation, and stream flow, have been
shown to strongly influence chlorophyll-a concentrations in the Delaware Bay and
its adjacent coastal zone (8, 16). The fact that the chlorophyll-a periodicity, typically
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The Hilbert-Huang Transform in Engineering
0.4
Frequency (cycles/month)
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
1998
1999
2000
2001
Years
2002
2003
0
0
10
20
Energy
FIGURE 3.5 The Hilbert spectrum (left panel) and marginal spectrum (right panel) for the
IMFs of the chlorophyll-a times series.
presumed to be annual, is not precisely 12 months is not unexpected, since the
biological processes are not simply controlled by physical processes. Examining the
Hilbert spectrum along the 0.09 frequency shows that there is generally high energy
along this signal, with the exception of two time periods. From January 1999 to
October 1999, the annual signal completely degenerates, and in the summer of 2001
there is a weakening of this signal. Without the ability of the HHT to ascertain
instantaneous frequency, these breakdowns in the annual periodicity would be difficult to determine.
In addition to the strong annual signal, there is a significant energy peak in the
lower frequencies. The marginal spectrum indicates an energy peak at the 0.02
frequency, which corresponds to a 4.2-yr period. However, looking at the Hilbert
spectrum shows that the energy at the 0.02 frequency is not constant; instead, it
appears to be roughly oscillating around the 0.02 frequency. This energy begins to
increase in frequency from mid-1999 to late 2001, when the signal briefly weakens
and starts to decrease in frequency. This interannual signal is primarily captured in
IMF 4 of the empirical mode decomposition (Figure 3.6). The relatively long period
of the mode suggests that a slowly varying basin-scale process could be a forcing
candidate.
One such basin-scale process is the North Atlantic Oscillation (NAO). NAO is
a measure of atmospheric conditions focused around the North Atlantic. The NAO
can have a positive or negative phase, as determined by the NAO index, which is
the surface sea-level pressure difference between the Subtropical (Azores) High and
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69
0.5
2
0
0
−0.5
1998
1999
2000
2001
Years
2002
2003
Interannual variation of Chlorophyll-a
Interannual variation of NAO
Applications of Hilbert-Huang Transform
−2
FIGURE 3.6 Interannual variation (IMF 4) for both the NAO index (blue line) and the
chlorophyll-a (green line) time series.
the Subpolar Low. Both positive and negative phases affect basin-wide variations in
the North Atlantic jet stream and storm tracks, as well as in large-scale modulations
of the typical patterns of zonal and meridional heat and moisture transport. These
alterations will consequently affect temperature and precipitation patterns and can
extend from eastern North America to western and central Europe (17). Furthermore,
recent studies have shown links between the NAO and hydrographic properties along
the northeast Atlantic shelf slope and the Gulf of Maine, and zooplankton and
chlorophyll-a concentrations within the Gulf of Maine (18). Thus, the interannual
frequency of the NAO was examined.
Monthly mean data of the NAO index from September 1997 to July 2003 were
obtained from the NOAA Climate Prediction Center. The NAO index data are shown
in Figure 3.4. This time series was also decomposed into its component IMFs in
order to isolate the NAO interannual mode. After decomposing the NAO index time
series, the fourth IMF contained an interannual signal that exhibited features very
similar to the fourth chlorophyll-a IMF (Figure 3.6).
To quantify this relationship, cross-correlation analysis was performed on the
IMF 4 of both the chlorophyll-a concentrations and the NAO index. The results are
shown in Figure 3.7. This study focused only on positive lag time (chlorophyll-a
time series following the NAO time series), because it would not be sensible to
assume that negative lags (chlorophyll-a time series leading the NAO time series)
were meaningful. The correlation analysis reveals that the NAO and chlorophyll-a
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70
The Hilbert-Huang Transform in Engineering
Maximum correlation is r = −0.67 at tiag = 10
(95% significance level: rsig = 0.25)
1
Correlation coefficient (r)
95% significance level
0.8
0.6
Correlation coefficient
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
10
40
50
Time lag
Chlorophyll-a follows NAO Index for lag > 0
20
30
60
70
FIGURE 3.7 Correlation coefficient (blue line) and the related 95 % confidence interval (red
dashed line). The maximum correlation coefficient with a positive lag (i.e. chlorophyll-a
interannual variation following NAO index interannual variation) is –0.67, corresponding to
a lag of 10 months.
concentrations have their highest correlation (correlation coefficient of –0.67) when
chlorophyll-a concentration lags (follows) the NAO by 10 months. The result is
significant at the 95% confidence level, as shown in Figure 3.7. Although a high
correlation coefficient does not guarantee a relationship between the two times series,
the fact that previous studies (mentioned earlier) have shown that the NAO can affect
the interannual time scales of biological and physical processes in oceanic regions
as distant as the Gulf of Maine suggests that this correlation is important.
This study examined the seasonal and interannual variability in satellite derived
chlorophyll-a data from the mouth of the Delaware Bay for a 6-yr period. By
applying the HHT to the chlorophyll-a time series, a more complete understanding
of its time–frequency characteristics was obtained. The resulting Hilbert spectrum
displayed a strong annual signal, with an unexpected weakening at two brief intervals. In addition, the energy at interannual frequencies was compared to the NAO.
The fourth IMF of the NAO index and the fourth IMF of the chlorophyll-a concentrations displayed strong similarities and had a cross-correlation coefficient of –0.67
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Applications of Hilbert-Huang Transform
71
that was significant at the 95% confidence level. This relatively high correlation
coefficient occurred at a lag of 10 months, which suggests that the possible interannual link between chlorophyll-a concentrations at the mouth of the Delaware Bay
and the NAO index may result from chlorophyll-a concentrations responding to
variations in the NAO index. Finally, the observations of both the weakening of the
seasonal signal and the suggested teleconnection of the interannual variation between
chlorophyll-a concentrations and the NAO are topics worthy of future investigation
that would have been difficult to identify with traditional time series analysis techniques.
3.4 MEDITERRANEAN OUTFLOW AND MEDDIES (O & M)
FROM SATELLITE MULTISENSOR REMOTE SENSING
One of the most interesting and prominent features of the North Atlantic Ocean is
the salt tongues originating from an exchange flow between the Mediterranean Sea
and the Atlantic through the Strait of Gibraltar. The Mediterranean outflow through
the strait is heavier than Atlantic water due to its higher salt content. Evaporation
in the Mediterranean Sea raises water salinity to approximately 38.4 ppt, compared
to 36.4 ppt in the eastern North Atlantic. After leaving the strait under the incoming
lighter North Atlantic water, the Mediterranean outflow sinks and turns to the right,
due to the Coriolis force, following the continental slope of Spain and Portugal.
Eventually, the Mediterranean water leaves the coast and spreads out into the middle
North Atlantic, forming a tongue of salty water and generating clockwise eddies in
the process. These Mediterranean eddies, or Meddies, are rapidly rotating double
convex lenses that contain a warm, highly saline core of Mediterranean water. They
are typically 100 km in diameter, extend over 800 m vertically, and are located at
a depth of 1000 m.
Generally speaking, most of the remotely sensed oceanographic observations
are confined to either the sea surface or the top part of the upper mixed layer. This
limitation is not always true. Yan et al. (19, 20, 21) and Yan and Okubo (22)
developed methods to infer the upper ocean mixed layer depths from multisensor
satellite data. Stammer et al. (23) applied Geosat Exact Repeat Mission altimeter
data to investigate the possible surface signal of the Meddies. In this section, we
report a new method to study the O&M by satellite integration analyses of altimeter, scatterometer, SST, and XBT data with help of HHT. We computed the sea
level difference, namely ∆η′ = η′Total − ηUL
′ = η′Total − ( η′T + η′S + ηW
′ ). η′Total is the deviation of the sea surface topography of the whole vertical water column, which can
be measured by TOPEX/Poseidon (T/P) altimetry. ηUL
′ is the sea surface variation
in the upper layer (400 m) due to thermal expansion (η′T ), calculated from XBT.
Salt expansion ( η′S ) and wind stress (ηW
′ ) are computed from satellite scatterometer
data. The HHT was applied to help interpret ∆η′.
To estimate the sea level variation due to thermal expansion in the upper layer,
sea surface height anomaly was calculated using monthly mean XBT (η′T ) data from
the Joint Environmental Data Analysis (JEDA) Center by the integration of temperature down to the 400 m depth from January 1993 to December 1999, i.e.,
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The Hilbert-Huang Transform in Engineering
0
η′T = α
∫ ∆Tdz.
−400
∆T is the temperature difference between two different depths (dz), and α is the
thermal expansion coefficient, which is calculated as function of salinity (S) and
water pressure (P) based on empirical measurements (24);
α=−
1 ∆ρ
( s , p) .
ρ0 ∆T
The spatial resolution of XBT data is 73° × 61° longitude/latitude, which is interpolated to 1° × 1° longitude/latitude data for this study. It is possible that errors
occurred due to interpolation of the time rates of change of the integrated upper
ocean heat storage anomaly (5 W/m2, on average); this is discussed by White and
Tai (25). We also checked interpolation error using 1° × 1° longitude/latitude optimum interpolation sea surface temperature (OISST). We compared interpolated sea
surface temperature (SST) from XBT data and OISST by calculating correlation
coefficients. Correlation coefficients between OISST and SST from XBT were over
95% with RMS 0.2°C, except for two small areas, and correlation coefficient 80%
with RMS 1.2°C. We conclude that interpolation error of XBTs is less than 1°C at
the sea surface.
Because of scarcity of temporal and spatial salinity data, we estimated the effect
of η′S in the upper layer indirectly. Two different correlation and RMS calculations
were made: the first correlation is between η′T and OISST, and the second correlation
is between η′Total and OISST. Good temporal and spatial agreement between SST
and η′Total or η′T suggests that a robust regression between fields may have some
physical significance with respect to thermal expansion, but a low correlation with
high RMS may have some other variability in the mixed layer or below the mixed
layer due to the salinity. It appears that the difference is due to a change in ocean
salinity, which is reflected in the T/P sea level measurements but not in the η′T
measurements. The first correlation coefficients were all over 90% with smaller RMS
than the second ones, but the second ones were also over 60% to 80%, with larger
RMS than the first ones. This result is caused by a warm and salty core below the
mixed layer. Using Levitus 94 (26), the vertical structure of the salinity was examined
to see the salinity distribution. The whole upper layer in our study domain has a
uniform salinity distribution above 1000 m. We conclude that the anomaly of η′S is
spatially uniform with only small variability.
Wind effect on the η′Total signal was considered using scatterometer data from
European Remote Sensing Satellite-1/2 (ERS-1/2) from January 1993 to December
1999. First we calculated the correlations between wind stress curl and the η′Total for
the temporal variation. Negative correlation would be expected; rising sea level is
associated with negative wind stress curl. Correlation coefficients showed a relation
of about –30% in our study area, with southern areas having larger RMS than
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Applications of Hilbert-Huang Transform
73
northern areas. Second, we considered the magnitude of sea level variation due to
wind stress based on a linear barotropic vorticity equation, i.e.,
d η − g′
≈
∇x τ,
dt
f0ρg
where η is the sea level height, ρ is the water density, g′ is the reduced gravity, f0
is the Coriolis parameter, and τ is the wind stress. We also estimated ηW
′ by using
NASA’s scatterometer: NSCAT (August 1996 to June 1997) and Quikscat (June
1999 to present). The mean sea level variation due to wind stress was around 1 cm.
The result of ∆η′ = η′T P − ηUL
′ is apparently some variation of the vertical water
column, which is a response according to the different incoming or outgoing water
mass under the upper layer. Sea level variation due to salinity (temperature) change
using mass conservation with 800 m thick is around 60 cm (32 cm), with 1-ppt
differences from background fluid (2°C), derived from the salt expansion coefficient
7.5 × 10–4/°C and thermal expansion coefficient 2.0 × 10–4/°C, respectively. The
bottom pressure (deep current) variation is negligible at this region (26, 27). However, since the Meddy was a weakly stratified result of extensive salt fingering (28),
there were regions of high stratification above and below the lens, where the background isopycnal surface becomes increasingly broader as it moves above the Meddy
toward the sea surface. Because of this isopycnal compensation, the O&M are not
revealed in the η′Total signal. This is why we cannot detect a Meddy with the altimeter
observation alone.
We computed the absolute differences of the sea surface height anomaly of ηUL
′
and η′Total to examine the trajectories of the O&M from January 1993 to December
1999. Figure 3.8 is an example of such trajectories from July 1998 to June 1999.
One can see a strong signature (|∆η′|) toward west and south from July to November.
Generally, southward Meddies are formed near 36°N by separation of the frictional
boundary layer at sharp corners (29) and in the Canary Basin (30, 31, 32). The
southward travelling Meddies can be explained by the strongest low frequency zonal
motions driven by baroclinic instability (33) and the influence of the neighboring
mesoscale features (cyclonic vortices or Azores Current meanders) in the regions
(34). The northward O&M over 36° to 40°N are also shown from August to October.
Furthermore, a strong signature over 40°N in February is considered to be the
influence of the returning Gulf Stream. From March to June the weak O&M signatures are found because of the weak salinity deviation in 1000 m depth examined
from climatological data.
To investigate the reasons for the change in the direction of propagation of
Meddies, the stream function (ψ) was computed by using the T/P altimeter. This
representation of the stream function permits observations of the interactions
between the sea surface gradient and the Meddies’ propagation. The computation
of the sea surface height anomaly η′ in terms of the usual dynamic variables is
straightforward, if the flow is assumed to be quasigeostrophic: The stream function
(ψ) is defined as
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74
The Hilbert-Huang Transform in Engineering
Jly 1998
Jan 1999
45
45
40
40
35
35
30
30W
25W
20W 15W
Aug 1998
10W
30
30W
05W
45
45
40
40
35
35
30
30W
25W
20W 15W
Sep 1998
10W
30
30W
05W
45
45
40
40
35
35
30
30W
25W
20W 15W
Oct 1998
10W
30
30W
05W
45
45
40
40
35
35
30
30W
25W
20W 15W
Nov 1998
10W
30
30W
05W
45
45
40
40
35
35
30
30W
25W
20W 15W
Dec 1998
10W
30
30W
05W
45
45
40
40
35
35
30
30W
25W
0
20W
15W
1
10W
30
30W
05W
2
3
(
4
25W
20W 15W
Feb 1999
10W
05W
25W
20W 15W
Mar 1999
10W
05W
25W
20W 15W
Apr 1999
10W
05W
25W
20W 15W
May 1999
10W
05W
25W
20W 15W
Jun 1999
10W
05W
25W
20W
15W
10W
05W
6
7
8
5
)
FIGURE 3.8 Calculation of the ∆η′ ∆η′ = η′Total − ηUL
from July 1998 to June 1999.
′
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Applications of Hilbert-Huang Transform
75
Jan 1994
Apr 1994
45
45
40
40
35
35
30
30W 25W 20W 15W 10W 05W
30
30W 25W 20W 15W 10W 05W
0W
Jly 1994
Oct 1994
45
45
40
40
35
35
30
30W 25W 20W 15W 10W 05W
−1.5
−1.0
0W
30
30W 25W 20W 15W 10W 05W
0W
−0.5
0.0
0.5
1.0
0W
1.5
FIGURE 3.9 Comparisons of Meddy trajectories with stream functions (Equation 3.7) for
January, April, July, and October 1994.
ψ= g
f0
η′Total ,
(3.7)
where ψ is the surface quasigeostrophic stream function (35). This is illustrated in
Figure 3.9. Due to the seasonal migration of the tropical water toward the north and
polar water toward the south, we can see that the seasonal signals divide into a
northern part and a southern part at 37°N. We compared the Meddies measured by
the floats and the stream functions in all the months in the experimental periods and
found that the Meddies preferred to travel toward the low streamlines and that they
stayed within the northmost and southmost streamlines in April and October, respectively.
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The Hilbert-Huang Transform in Engineering
3rd EMD mode using HHT at A(30W, 30N) with other places
5
B(20W, 30N)
0
−5
93
5
94
95
96
97
98
99
00
95
96
97
98
99
00
95
96
97
98
99
00
95
96
97
98
99
00
95
96
97
98
99
00
95
96
97
98
99
00
95
96
97
98
99
00
C(10W, 30N)
0
−5
93
5
94
D(10W, 37N)
0
−5
93
5
94
E(10W, 42N)
0
−5
93
5
94
F(20W, 42N)
0
−5
93
5
94
G(30W, 42N)
0
−5
93
5
94
H(30W, 37N)
0
−5
93
94
FIGURE 3.10 A comparison of the third EMD modes using HHT are shown at given
locations. The solid curve is for location A (refer to Figure 3.11 for the location) in all panels
for comparison. The dotted curve is the individual EMD modes for those locations.
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Applications of Hilbert-Huang Transform
77
Annual mean of |η′| with two experiments in 1994
5
45
AMUSE
4.5
SEMAPHORE
G
F
4
E
3.5
40
3
2.5
D
H
2
35
1.5
1
0.5
30 A
30W
25W
B
20W
15W
C
10W
05W
0
FIGURE 3.11 Comparisons of annual mean |∆η′| signal from two float experiments in 1994.
The gray scale and contours show the annual mean of the |∆η′| signal estimated from our
method. All Meddies were discovered during the AMUSE experiment (stars) and SEMAPHORE experiment (circles) in 1994.
To find out the dominant signal of the sea surface interaction with the Meddies,
the HHT was applied to their EMD modes. We chose eight places to investigate the
dominant signal in our study area. Figure 3.10 showed that Place Group 1 (B, C,
and D, shown in Figure 3.11) and Group 2 (H and G, shown in Figure 3.11) had a
similar signal. However, Group 3 (E and F, shown in Figure 3.11) had slightly
different signals from Groups 1 and 2. Consequently, there were seasonal fluctuations
between location A and Group 3. To compute the dominant signal of the power for
locations A and E, HHT was also employed; this is shown in Figure 3.12. In both,
the dominant frequency was around f = 0.082 (1 year). However, location A had a
lower frequency, f = 0.03 (33.3 months), which seems to indicate that wind stress
with 33.3-month period produces sea surface forcing. The surface variations produce
the baroclinic instability on the Meddies, which is related to southward translation.
The southward translation of the Meddies due to the baroclinic effects were consistent with that discussed by Müler and Siedler (36) and by Käse and Zenk (37). If
the current is vertically sheared in a stratified fluid, baroclinic instability can occur.
Figure 3.12 also demonstrates that the comparison between field observations from
two float experiments in 1994 and our computation was excellent.
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The Hilbert-Huang Transform in Engineering
Frequency-Spectrum-Energy of the streamfuction at A
Energy
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
10
20
30
40
50
60
70
80
0
Frequency-Spectrum-Energy of the streamfuction at E
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
10
20
30
40
50
60
70
40
Energy
0.4
0
20
80
0
20
40
FIGURE 3.12 Time–frequency–energy spectrum of the stream functions at locations A and
E. Refer to Figure 3.11 for location A and E. One can see the high energy at the 33-month
period at location A, but not at location E.
3.5 CONCLUSION
Three examples of applications of the HHT method in ocean-atmosphere remote
sensing research were illustrated in this chapter. These examples show that the HHT
method is indeed a potentially very useful and powerful tool for ocean engineering
and science studies.
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Applications of Hilbert-Huang Transform
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ACKNOWLEDGMENTS
This research was supported partially by the National Aeronautics and Space Administration (NASA) through Grant NAG5-12745 and NGT5-40024, by the Office of
Naval Research (ONR) through Grant N00014-03-1-0337, and by the National
Oceanic and Atmospheric Administration (NOAA) through Grants NA17EC2449
and NA96RG0029.
REFERENCES
1. Chan, Y. T. (1995). Wavelet Basics. Boston, MA: Academic.
2. Huang, N., Long, S., Shen, Z. (1996). The mechanism for frequency downshift in
nonlinear wave evolution. Adv. Appl. Mech. 32:59–111.
3. Huang, N., Shen, Z., Long, S., Wu, M., Shin, H., Zheng, Q., Yen, N.-C, Tung, C.,
Liu H. (1998). The empirical mode decomposition and Hilbert spectrum for nonlinear
and nonstationary time seriesanalysis. Proc. Royal Soc. Mathematical, Phys. Eng.
Science 454:903–995.
4. Yan, X.-H., Zheng, Q., Ho, C.-R., Tai, C.-K., Cheney, R. (1994). Development of the
pattern recognition and the spatial integration filtering methods for analyzing satellite
altimeter data. Remote Sens. Environ. 48:147–158.
5. Cheney, R., Miller, L., Agreen, R., Doyle, N., Lillibridge, J. (1994). TOPEX/POSEIDON: 2 cm solution. J. Geophys. Res. 99:24555–24563.
6. Salisbury, J. I., Wimbush, M. (2002). Using modern time series analysis techniques
to predict ENSO events from the SOI times series. Nonlinear Proc. Geophys.
9:341–345.
7. Kiddon, J. A., Paul, J. F., Buffum, H. W., Strobel, C. S., Hale, S. S., Cobb, D., Brown,
B. S. (2003). Ecological condition of U.S. mid-Atlantic estuaries. 1997–1998. Marine
Pollution Bulletin 46(10):1224–1244.
8. Keiner, L. E., Yan, X-H. (1998). A neural network model for estimating sea surface
chlorophyll and sediments from thematic mapper imagery. Remote Sens. Environ.
66(2):153–165.
9. Zhang, Y., Pulliainen, J., Koponen, S., Hallikainen, M. (2002). Application of an
empirical neural network to surface water quality estimation in the Gulf of Finland
using combined optical data and microwave data. Remote Sens. of Environ.
81:327–336.
10. Baruah, P. J., Oki., K., Nishimura, H. (2000). A neural network model for estimating
surface chlorophyll and sediment content at Lake Kasumi Gaura of Japan. In: Conference Processing of GIS Development. Taipei, Taiwan, December.
11. Buckton, D., O’Mongain, E., Danaher, S. (1999). The use of neural networks for the
estimation of oceanic constituents based on the MERIS instrument. Int. J. Remote
Sensing 20(9):1841–1851.
12. Keiner, L. E., Yan, X.-H. (1997). Empirical orthogonal function analysis of sea surface
temperature patterns in Delaware Bay. IEEE Trans. Geoscience Remote Sensing
25(5):1299–1306.
13. Zhang, T., Fell, F., Liu Z.-S., Preusker, R., Fischer, J., He, M.-X. (2003). Evalating
the performance of artificial neural network techniques for pigment retrieval from
ocean color in Case I Water. J. Geophys. Res. 108(C9):3286–3298.
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14. Dzwonkowski, B. (2003). Development and application of a neural network based
ocean color algorithm in coastal waters. Masters thesis, University of Delaware,
Newark, DE.
15. Barnard, A. H., Stegmann, P. M., Yoder, J. A. (1997). Seasonal surface ocean variability in the South Atlantic Bight derived from CZCS and AVHRR imagery. Continental Shelf Res. 17:1181–1206.
16. Pennock, J. R. (1985). Chlorophyll distributions in the Delaware estuary: Regulation
by light-limitation. Estuarine, Coastal and Shelf Science 21:711–725.
17. Hurrell, J. W. (1995). Decadel trends in the North Atlantic Oscillation regional
temperatures and precipitation. www.cgd.ucar.edu/cas/papers/science1995/sci.htm.
18. Thomas, A. C., Townsend, D. W., Weatherbee, R. (2003). Satellite-measured phytoplankton variability in the Gulf of Marine. Continential Shelf Res. 23:971–989.
19. Yan, X.-H., Schubel, J. R., Pritchard, D. W. (1990). Oceanic upper mixed layer depth
determination by the use of satellite data. Remote Sens. Environ. 32 (1):55–74.
20. Yan, X.-H., Okubo, A., Shubel, J. R., Pritchard, D. W. (1991). An analytical mixed
layer remote sensing model. Deep Sea Res. 38:267–287.
21. Yan, X.-H., Niller, P. P, Stewart, R. H. (1991). Construction and accuracy analysis
of images of the daily-mean mixed layer depth. Int. J. Remote Sensing
12(12):2573–2584.
22. Yan, X.-H., Okubo, A. (1992). Three-dimensional analytical model for the mixed
layer depth. J. Geophy. Res., 97(C12):20201–20226.
23. Stammer D.,. Hinrichsen, H. H, Käse, R. H. (1991). Can Meddies be detected by
satellite altimetry? J. Geophys. Res. 96:7005–7014.
24. Wilson, W., Bradley, D. (1996). Technical Report NOLTR. 66–103.
25. White, W. B., Tai, C.-K. (1995). Inferring interannual changes in global upper ocean
heat storage from TOPEX altimetry J. Geophys. Res. 15:24943–24954.
26. Levitus, S., Boyer, T. P. (1994). World Ocean Atlas, Vol. 4, Temperature, NOAA Atlas
NESDIS 4, U.S. Govt. Print. Off., Washington, D.C., 117.
27. Iorga, M. C., Lozier, M. S. (1999). Signature of the Mediterranean outflow from a
North Atlantic climatology, 1. Salinity and density fields. J. Geophys. Res.
104:25985–26009.
28. Hebert, D. L. (1988). A Mediterranean salt lens. Ph.D. dissertation, Dalhousie University, Halifax, Nova Scotia.
29. D’Asaro, E. A. (1993). Generation of submesoscale vortices: A new mechanism. J.
Geophys. Res. 93:6685–6693.
30. Armi, L., Stommel, H. (1983). Four views of a portion of the North Atlantic subtropical gyre. J. Phys. Oceanogr. 13:828–857.
31. Armi, L., Zenk, W. (1984). Large lenses of highly saline Mediterranean salt lens. J.
Phys. Oceanogr. 14:1560–1576.
32. Richardson, P. L., Tychensky, A. (1998). Meddy trajectories in the Canary Basin
measured during the SEMAPHORE experiment, 1993–1995. J. Geophys. Res.
103:25029–25045.
33. Spall, M. A., Richardson, P. L., Price, J. (1993). Advection and eddy mixing in the
Mediterranean salt tongue. J. Mar. Res. 51:797–818.
34. Tychensky, A., Carton, X. (1998). Hydrological and dynamical characterization of
Meddies in the Azores region: A paradigm for baroclinic vortex dynamics. J. Geophys.
Res. 103:25061–25079.
35. Douglas, B. C., Cheney, R. E, Agreen, R. W. (1983). Eddy energy of the northwest
Atlantic and Gulf of Mexico determined from GEOS 3 altimetry. J. Geophys. Res.
88:9595–9603.
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36. Müler, T. J., Siedler, G. (1992). Multi-year current time series in the eastern North
Atlantic Ocean, J. Mar. Res. 50:63–98.
37. Käse, R. H., Zenk W. (1996). Structure of the Mediterranean water and Meddy
characteristics in the northeastern Atlantic. In: Ocean Circulation and Climate:
Observation and Modeling the Global Ocean, W. Krauss, Ed. Berlin: Gebrüder
Bornträger, Berlin, pp. 365–395.
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4
A Comparison of the
Energy Flux Computation
of Shoaling Waves
Using Hilbert and
Wavelet Spectral
Analysis Techniques
Paul A. Hwang, David W. Wang, and
James M. Kaihatu
CONTENTS
4.1
4.2
4.3
Introduction ....................................................................................................84
The Huang-Hilbert Spectral Analysis............................................................84
Shoaling Waves and Energy Flux Computation............................................85
4.3.1 Field Measurement.............................................................................85
4.3.2 Numerical Simulations.......................................................................88
4.4 Discussions.....................................................................................................91
4.4.1 Resolution and Nonlinearity ..............................................................91
4.4.2 Edge Effect.........................................................................................93
4.5 Summary ........................................................................................................94
Acknowledgments....................................................................................................94
References................................................................................................................94
83
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The Hilbert-Huang Transform in Engineering
ABSTRACT
The HHT analysis interprets wave nonlinearity in terms of frequency modulation
instead of harmonic generation. The resulting spectrum energy concentrates in the
neighborhood of the fundamental frequency in comparison with the FFT or wavelet
spectrum. The energy flux computation from the HHT spectrum is considerably larger
than with the FFT or wavelet spectrum.
4.1 INTRODUCTION
Fourier transform decomposes a nonlinear waveform into a fundamental frequency
component and its harmonics. As a result, the energy of a nonlinear wave is spread
into higher frequencies. This may produce difficulties in the interpretation of wave
propagation and quantification of wave dynamic properties such as the energy flux
of the wave field. Recently, Norden Huang and his colleagues developed a new
analysis technique, the Hilbert-Huang transformation (HHT). Through analytical
examples, they demonstrated the excellent frequency and temporal resolution of
HHT for analyzing nonstationary and nonlinear signals [1, 2].
With the HHT analysis, the physical interpretation of nonlinearity is frequency
modulation, which is fundamentally different from harmonic generation. The HHT
spectrum therefore retains its energy density near the fundamental frequency of the
wave motion in comparison with the FFT or wavelet spectrum [3]. In this article,
we briefly describe the HHT technique and investigate its use in the calculation of
the energy flux of ocean waves. The resulting HHT spectrum is compared with the
counterpart obtained by the wavelet method. The wavelet technique is based on
Fourier spectral analysis but uses adjustable frequency-dependent window functions
— the mother wavelets — to provide temporal/spatial resolution for nonstationary
signals [4–7]. As expected, the Fourier-based analysis interprets wave nonlinearity
in terms of harmonic generation; thus the spectral energy leaks to higher frequency
components. The HHT interprets wave nonlinearity as frequency modulation, and
the spectral energy remains near the base frequency. The computed energy flux from
the HHT method is much higher than that from the wavelet method.
4.2 THE HUANG-HILBERT SPECTRAL ANALYSIS
Hilbert transformation was introduced to water wave analysis in the 1980s. Applications include the study of wave modulation leading to wave breaking [8], local
properties of sea waves such as the group and pulse structure, fluctuation of wave
energy and energy flux [9], and the measurement and quantification of breaking
events of wind-generated surface waves [10]. A key function of the Hilbert analysis
is the derivation of local wave number in a spatial series or instantaneous frequency
in a time series. To use the Hilbert transformation, proper preprocessing of the signals
is critical. Large errors in the computed local frequency or wave number can occur
when small waves are riding on longer waves or when sharp changes of the oscillation frequencies occur in the wave signal. Quantitative discussions on the riding
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A Comparison of the Energy Flux Computation of Shoaling Waves
85
wave problem [1] and on the rapid change of the oscillation frequencies [11] have
been published and will not be repeated here.
The key ingredient in the HHT is empirical mode decomposition (EMD),
designed to reposition the riding waves at the mean water level. This is achieved
through a sifting process that repeatedly subtracts the local mean from the original
signals. The procedure decomposes the signal into many modes with different
frequency characteristics, and thus also alleviates the problem of sharp frequency
change in the original signal. Huang and his coworkers have provided extensive
discussions on the EMD [1, 2]. From experience, even for very complicated random
signals, a time series can usually be decomposed into a relatively small number of
modes, M < log2 N. Each mode is free of riding waves and is suitable for Hilbert
transformation to yield accurate local frequency of the mode. The spectrum of the
original signal can be obtained from the sum of the Hilbert spectra of all modes.
Extensive tests have been carried out, and the HHT technique proves to deliver very
high frequency and temporal/spatial resolutions in analyzing nonstationary signals.
It is also shown to be able to handle the task of analyzing nonlinear signals produced
by exact solution with modulating oscillation frequencies, whereas Fourier-based
techniques interpret the signals as superposition of harmonics [1–3].
4.3 SHOALING WAVES AND ENERGY FLUX COMPUTATION
The Hilbert view of nonlinear waves is significantly different from the conventional
Fourier view. In particular, intra-wave modulation is a key signature of nonlinearity
based on the Hilbert spectral analysis, while harmonic generation is typical of the
Fourier spectrum. The difference in the interpretations of nonlinearity will lead to
quantitative differences in the energy flux computation.
4.3.1 FIELD MEASUREMENT
An example of intra-wave modulation highlighted by the HHT analysis is illustrated
in the analysis of shoaling swell shown in Figure 4.1. The three-dimensional (3D)
surface wave topography was acquired by an airborne topographic mapper (ATM,
an airborne scanning laser system) near Duck, N.C., three days after an extratropical
storm passed through the area. The wind condition at the time of data acquisition
was low, resulting in a relatively simple two-dimensional (2D) swell system propagating on mild slope bathymetry. More discussions of the environmental conditions,
measurement techniques, and the bathymetry of the region are reported by Hwang
et al. [12]. Figure 4.1 illustrates the spatial evolution of the airborne measured
shoaling swell (Plot a), the Hilbert spectrum (Plot b), and the wavelet spectrum (Plot
c) in the near coast region. For reference, the water depth based on the bathymetry
database is shown in Plot d. The spectral energy is distributed narrowly about the
dominant frequency. The distinctive intra-wave modulation in the neighborhood of
the peak frequency of the wave spectrum can be clearly identified in the HHT
spectrum (Figure 4.1b). In contrast, the wavelet spectrum distributes the energy into
harmonics, as a result of interpreting nonlinearity as harmonic generation in the
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The Hilbert-Huang Transform in Engineering
FIGURE 4.1 (a) A 3D surface wave topography of shoaling swell in the coastal region. (b)
The HHT spectrum of the waves shown in (a). (c) The wavelet spectrum of the waves shown
in (a). (d) The water depth from bathymetry database.
Fourier-based spectral processing (Figure 4.1c), as discussed earlier. It is noticeable
that the second harmonic of the wavelet spectrum displays a spatial modulation
characteristic similar to the intra-wave modulation of the HHT spectrum at the peak
frequency.
The energy flux, F, of the wave field can be computed from the wave spectrum,
S, by
( ) ∫ S (ω; x )C (ω; x ) d ω ,
F x =
g
(4.1)
where x is the spatial coordinate in the propagation direction, ω is angular frequency,
and Cg is the wave group velocity. The computed results based on HHT and wavelet
analyses are shown in Figure 4.2. The magnitude derived from the Hilbert spectrum
is in general larger than that obtained by the wavelet spectrum. As explained in the
last section, this is partly caused by the much higher spectral level in the lower
frequency of the Hilbert spectrum compared to the wavelet spectrum. Because phase
velocity and group velocity of gravity waves increase monotonically with wavelength, the increased low frequency spectral density translates to a higher level of
the energy flux in the wave field.
The source function, Q, of the wave system can be derived from the spatial rate
of change of the energy flux:
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A Comparison of the Energy Flux Computation of Shoaling Waves
87
(a)
F
3
2
1
d(F)/dxWavelet
d(F)/dxHHT
0
200
400
600
800
1000
1200
1400
1600
<EF(HHT)>=−6.373e−004, <EF(wavelet)>=−3.083e−004, offset=30
1800
2000
2200
(b)
0.2
0
−0.2
0
0.02
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
(c)
0
−0.02
0
200
400
600
800
1000 1200
x (m)
1400
1600
1800
2000
2200
FIGURE 4.2 (a) Energy flux computed from the shoaling swell shown in Figure 4.1. (b) The
source function computed from the HHT spectrum. (b) The source function computed from
the wavelet spectrum.
∂F
=Q .
∂x
(4.2)
Figure 4.2b and c shows the derived source function based on the Hilbert and wavelet
spectra, respectively. The magnitude of the Hilbert source term is considerably larger
than the wavelet source term (notice the difference in scale of the two ordinates).
The oscillatory nature of the source function highlights the complex nature of the
wave conditions in the field. The group structure can be introduced by many processes, such as wave nonlinearity, interaction among different wave components,
and bathymetry-induced wave scattering. The group structure causes the source
function to fluctuate between positive and negative values, as can be visualized from
Equation 4.1 and Equation 4.2. Further discussion of the group structure will be
presented later.
In appearance, the source functions obtained by the HHT and wavelet methods,
as shown in Figure 4.2b and c, are considerably different. In reality, the apparent
difference reflects the much higher temporal/spatial resolution of the HHT method.
Figure 4.3 shows the results of the running average of HHT computation, with the
wavelet result superimposed. With an averaging bin width of 60 elements, the
running average of the HHT source function is almost identical to the wavelet source
function. The spatial average (excluding the leading and trailing 90 m of the spatial
coverage shown) of the source term is –2.49 × 10–4 for the wavelet processing, –5.64
× 10–4 for the HHT processing, and –4.78 × 10–4 for the running average of the HHT
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The Hilbert-Huang Transform in Engineering
<EF(HHT)>=−6.373e−004, <EF(HHT)> =−5.522e−004, <EF(wavelet)>=−3.083e−004, offset=30
60
0.1
HHT
<HHT>
0.08
60
wavelet
0.06
0.04
d(F)/dx
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
0
200
400
600
800
1000 1200
x (m)
1400
1600
1800
2000
2200
FIGURE 4.3 A comparison of the wavelet source function, the HHT source function, and
the ensemble average of HHT source function with a bin width of 60 spatial elements.
processing. From this limited investigation, we conclude that the energy flux computation may be off by a factor of two using FFT based processing techniques.
As mentioned earlier, harmonic generation has created major difficulties for the
analysis of wave dynamics due to the dispersive nature of water waves. For example,
the phase and group velocities of each free wave component are frequency dependent, yet the harmonics (of nonlinear waves) are not dispersive. Therefore, the energy
flux (the product of group velocity and spectral density) of the harmonics associated
with the nonlinear waves cannot be distinguished from the energy flux of the free
waves at the same frequency, and FFT based processing will always underestimate
the energy flux of the wave field. The underestimation will reflect on the magnitude
of parameters such as the dissipation rate or growth rate of a wave field.
4.3.2 NUMERICAL SIMULATIONS
To investigate further the influence of processing methods and wave nonlinearity on
the quantitative results of energy flux computation, numerical simulations were
carried out using the refraction–diffraction (REFDEF) model [13]. The case simulated is a train of monochromatic waves (10 sec wave period and 0.25 m initial wave
amplitude) propagating from offshore along a plane sloping beach. In the first
scenario, the nonlinear terms are turned off, and the waveform remains sinusoidal
along the full computational domain (Figure 4.4Aa). In the second scenario, the
nonlinear terms are turned on, and wave asymmetry becomes obvious in the nearshore region (Figure 4.4Ba). The spatial evolution of the wave spectra for the two
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A Comparison of the Energy Flux Computation of Shoaling Waves
89
(a)
(b)
FIGURE 4.4 Numerical simulations of shoaling waves. Left panels: linear simulation. Right
panels: nonlinear simulation. (a) Waveform and water depth, (b) HHT spectrum, (c) wavelet
spectrum, and (d) characteristic wave numbers kp and k2.
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The Hilbert-Huang Transform in Engineering
linear
nonnolear
(a)
(a)
EC
E Cg
0.4
g
0.4
0.2
0.2
HHT
wavelet
500
1000
d(EF)/dx
(b)
0
500
1000
(c)
<d(EF)/dx>
0
−1
0
500
1000
x (m)
1500
500
1000
1500
(b)
0
−2
0
−3
x 10
1
1500
x (m)
60
−2
0
−3
x 10
1
0
0
−3
x 10
2
1500
<d(EF)/dx>60
d(EF)/dx
0
0
−3
x 10
2
500
1000
1500
x (m)
(c)
0
−1
0
500
1000
1500
x (m)
FIGURE 4.5 Same as Figure 4.4 but for the energy flux computation. (a) Energy flux, (b)
gradient of energy flux, and (c) same as (b) but phase averaged. A: linear simulation. B:
nonlinear simulation.
scenarios by the HHT processing method is shown in Figure 4.4b. The spectral
density is concentrated in a very narrow frequency band, as expected from a monochromatic wave train. The lack of frequency modulation in the absence of nonlinearity is clear from a comparison of Figure 4.4Ab with Figure 4.4Bb. The result
based on the wavelet method is shown in Figure 4.4c. The spectral energy is more
spread out because of the finite window size, and the effect of nonlinearity is
harmonic generation.
The characteristic wave number is typically defined as the peak wave number,
kp, or weighted wave number computed from the spectral moment, kn = ( ∫knS(k)dk/
∫S(k)dk)1/n. With the HHT analysis, kp and kn are almost identical for both linear and
nonlinear scenarios because the spectral energy is confined in a narrow frequency
range. With the wavelet analysis, kp and kn are also very similar in the linear scenario,
but they differ considerably in the nonlinear scenario as a result of harmonic generation. The results of phase-average kp and k2 for the two scenarios are shown in
Figure 4.4d. Intra-wave frequency modulation as reflected by the oscillation of the
characteristic wave number is clearly seen in the HHT analysis of the nonlinear
simulation; it is also somewhat noticeable but much weaker in the wavelet analysis.
For the linear case, the computed energy flux and the gradient of energy flux
derived from the two processing methods are very similar (Figure 4.5a,b,c). The
HHT results show small intra-wave modulation due to finite amplitude of the waveform. For the nonlinear case, the intra-wave modulation is greatly amplified by the
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nonlinearity of the wave system. The differentiation process in calculating the gradient of energy flux sometimes produces large spikes in the computation. The results
become very sensitive to small modifications in the data processing procedure. Figure
4.5 shows an example of the processed results.
4.4 DISCUSSIONS
4.4.1 RESOLUTION AND NONLINEARITY
Fourier-based spectral analysis methods have been widely used for studying random
waves. One major weakness of the Fourier-based spectral analysis methods is the
assumption of linear superposition of wave components. As a result, the energy of
a nonlinear wave is spread into many harmonics, which are phase-coupled via the
nonlinear dynamics inherent in ocean waves. In addition to the nonlinearity issue,
strictly speaking Fourier spectral analysis should be used for periodic and stationary
processes only. Wave propagation in the ocean is certainly neither stationary nor
periodic. Using the HHT analysis, the physical interpretation of nonlinearity is
frequency modulation, which is fundamentally different from the commonly
accepted concept associating nonlinearity with harmonic generation. Huang et al.
[1, 2] argued that harmonic generation results from the perturbation method used in
solving the nonlinear equation governing the physical processes, thus the harmonics
are produced by the mathematical tools used for the solution rather than being a
true physical phenomenon. Through analytical examples, they demonstrated the
excellent frequency and temporal resolutions of HHT for analyzing nonstationary
and nonlinear signals. Here we give a few computational examples to illustrate the
points discussed earlier.
Figure 4.6 presents a comparison of HHT and wavelet analysis of three different
cases [3]. Case 1 is an example of an ideal time (t) or space (x) series of sinusoidal
oscillations of constant amplitude; the frequency (f) or wave number (k) of the first
half of the signal is twice that of the second half (Figure 4.6a). The spectra computed
by the HHT and wavelet techniques are displayed in Figure 6b and c, respectively.
The HHT spectrum yields very precise frequency resolution as well as high temporal
resolution in identifying the sudden change of signal frequency at about the half
point of the time series. In comparison, the wavelet spectrum has only a mediocre
temporal resolution of the frequency change. There is also a serious leakage problem,
and the spectral energy of the simple oscillations spreads over a broad frequency
range (the contour interval is 3 dB in the spectral plots).
Case 2 is a single cycle sinusoidal oscillation occurring at the middle of the
otherwise quiescent signal stream (Figure 4.6d). The precise temporal resolution of
the HHT method is clearly demonstrated by the sharp rise and fall of the HHT
spectrum coincident with the transient signal, as illustrated in Figure 4.6e. In comparison, the wavelet spectrum is much more smeared, both in the frequency and
temporal resolutions (Figure 4.6f).
Case 3 is a sinusoidal function, y, with its oscillating frequencies subject to
periodic modulation (Figure 4.6g)
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The Hilbert-Huang Transform in Engineering
1000
0
−1
500
1500
0.3
(d)
y
0
−1
500
1
(a)
y
y
1
1000
1500
(b) 0.2
0
−1
500
550
3dB contours
600
(h)
k|f
k|f
k|f
0.2
0.1
0.1
0.1
1000
0
500
1500
0.3
1000
1500
(c) 0.2
0
500
550
(f) 0.3
600
(i)
k|f
k|f
0.2
k|f
0.2
0.1
0.1
0
500
(g)
(e) 0.3
0.2
0
500
1
0.1
1000
x|t
1500
0
500
1000
x|t
1500
0
500
550
x|t
600
FIGURE 4.6 Examples comparing HHT (middle row) and wavelet (bottom row) analysis of
nonlinear and nonstationary signals. (a,b,c) Time series of the harmonic motion with the
oscillation period doubled in the second half, and its HHT and wavelet spectra. (d,e,f) Transient
sinusoidal function of one cycle. (g,h,i) Oscillatory motion with modulated frequencies.
()
(
)
y t = a cos ωt + ε sin ωt ,
(4.3)
where a is the amplitude, ω is the angular frequency, and ε is a small perturbation
parameter. This is the exact solution for the nonlinear differential equation [1]
(
)
(
)
0.5
2
d2y
+ ω + εω cos ωt y − 1 − x 2 εω 2 sin ωt = 0.
2
dt
(4.4)
If the perturbation method is used to solve Equation 4.2, the solution to the first
order of ε is
1

y1 t = cos ωt − ε sin 2ωt = cos ωt − ε  1 − cos 2ωt  .
2

()
(
)
(4.5)
The HHT spectrum (Figure 4.6h) correctly reveals the nature of oscillatory frequencies of the exact solution (Equation 4.3). In contrast, the wavelet spectrum (Figure
4.6i) shows a dominant component at the base frequency and periodic oscillations
of the second harmonic component.
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A Comparison of the Energy Flux Computation of Shoaling Waves
f(t)
1
f(t)
f(t)
50
100
150
t
200
150
t
200
150
t
200
150
t
200
250
(a)
300
(b)
0
−1
0
1
50
100
50
100
250
300
exp(−0.01t)cosθ (expanded)
Hilbert transform
exp(−0.01t)sinθ (expanded)
0
−1
0
1
f(t)
cosθ (expanded)
Hilbert transform
sinθ (expanded)
0
−1
0
1
93
250
(c)
300
(d)
0
−1
0
50
100
250
300
FIGURE 4.7 An example illustrating the mirror-imaging approach to alleviate edge problems.
These three examples illustrate the excellent temporal (spatial) and frequency
(wave number) resolution of the HHT method for processing nonlinear and nonstationary signals. The result from Case 3 is especially interesting, as it illustrates the
unique ability of the HHT technique to reveal more accurately the nature of nonlinear
processes with nonstationary oscillation frequencies. Solutions obtained through the
perturbation method and analysis results obtained through Fourier-based techniques
represent those processes as superpositions of harmonic motions.
4.4.2 EDGE EFFECT
One problem frequently encountered in data processing involves the treatment of
the two edges of the data segment. This problem is especially serious when the data
segment is short, as in many transient processes. Figure 4.7a shows an example of
one sinusoidal cycle (cos t) between 100 and 300 time marks, shown by the solid
curve. The Hilbert transformation of the data segment (shown by x) deviates from
its exact solution, sin t, at the two edges. Padding zeros to expand the data segment
does not alleviate the problem (Figure 4.7a), but repeating the signal sometimes
helps, as shown in Figure 4.7b. However, if the expanded data sequence contains
discontinuities (Figure 4.7c), the edge problem persists. A technique to avoid discontinuities at the two edges when expanding the data segment is to mirror image
the data, as illustrated in Figure 4.7d. Our experience indicates that the edge problem
can be significantly alleviated with about 30% expansion of the original data segment
with the mirror-imaging method. More elaborate mirror-imaging techniques may
include applying windowing (e.g., exponential decay) to the imaged segments (private communication, N. E. Huang, 2004).
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The Hilbert-Huang Transform in Engineering
4.5 SUMMARY
Analyzing nonlinear and nonstationary signals remains a very challenging task.
Many methods developed to deal with nonstationarity are based on the concept of
Fourier decomposition; therefore, all the shortcomings associated with Fourier transformation are also inherent in those methods. The recent introduction of empirical
mode decomposition [1, 2] represents a fundamentally different approach for processing nonlinear and nonstationary signals. The associated spectral analysis (HHT)
provides excellent spatial (temporal) and wave number (frequency) resolution for
handling nonstationarity and nonlinearity [1–3]. The HHT spectrum also results in
a considerably different interpretation of nonlinearity (frequency modulation, as
compared to the traditional view of harmonic generation). Applying the technique
to the problems of ocean waves, we found that the spectral function derived from
HHT is markedly different from those obtained by the Fourier-based techniques.
The difference in the resulting spectral functions is attributed to the interpretation
of nonlinearity. The Fourier techniques decompose a nonlinear signal into sinusoidal
harmonics; therefore, some of the spectral energy at the base frequency is distributed
to the higher frequency components. The HHT interprets nonlinearity in terms of
frequency modulation, and the spectral energy remains in the neighborhood of the
base frequency. This results in a considerably higher spectral energy at lower frequencies and a sharper dropoff at higher frequencies in the HHT spectrum in
comparison with the Fourier-based spectra [3]. The energy flux computed by using
the HHT spectrum is much higher than that obtained from the FFT or wavelet
methods.
ACKNOWLEDGMENTS
This work is supported by the Office of Naval Research (Naval Research Laboratory
Program Elements N61153 and N62435). This is NRL contribution BC/7330.
REFERENCES
1. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yuen, N.
C., Tung, C. C., and Liu, H. H. (1998). The empirical mode decomposition and the
Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc.
Lond. 454A: 903–995.
2. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear water
waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457.
3. Hwang, P. A., Huang, N. E., and Wang, D. W. (2003). A note on analyzing nonlinear
and nonstationary ocean wave data. Appl. Ocean Res. 25: 187–193.
4. Shen, Z., and Mei, L. (1993). Equilibrium spectra of water waves forced by intermittent wind turbulence. J. Phys. Oceanogr. 23(9): 2019–2026.
5. Shen, Z., Wang, W., and Mei, L. (1994). Finestructure of wind waves analyzed with
wavelet transform. J. Phys. Oceanogr. 24(5): 1085–1094.
6. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27:
577–588.
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95
7. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records.
Ocean Eng. 28: 957–987.
8. Melville, W. K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128:
489–506.
9. Bitner-Gregersen, E. M., and Gran S. (1983). Local properties of sea waves derived
from a wave record. Appl. Ocean Res. 5: 210–214.
10. Hwang, P. A., Xu, D., and Wu, J. (1989). Breaking of wind-generated waves: measurements and characteristics. J. Fluid Mech. 202: 177–200.
11. Guillaume, D. W. (2002). A comparison of peak frequency-time plots produced with
Hilbert and wavelet transforms. Rev. Scientific Inst. 73(1): 98–101.
12. Hwang, P. A., Walsh, E. J., Krabill, W. B., Swift, R. N., Manizade, S. S., Scott, J. F.,
and Earle, M. D. (1998). Airborne remote sensing applications to coastal wave
research. J. Geophys. Res. 103(C9): 18791–18800.
13. Kaihatu, J. M., and Kirby, J. T. (1995). Nonlinear transformation of waves in finite
water depth. Phys. Fluids 7: 1903–1914.
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5
An Application of HHT
Method to Nearshore
Sea Waves
Albena Dimitrova Veltcheva
CONTENTS
5.1
5.2
5.3
5.4
5.5
Introduction ....................................................................................................98
Field Data.......................................................................................................99
Conventional Methods for Wave Data Analysis..........................................100
Hilbert-Huang Transform Method ...............................................................106
Application of the HHT Method .................................................................108
5.5.1 Offshore Waves During Different Sea Stages .................................112
5.5.2 Cross-Shore Transformation of Sea Waves .....................................114
5.6 Conclusions ..................................................................................................117
References..............................................................................................................118
ABSTRACT
Real sea waves are nonlinear by nature, and this nonlinearity increases considerably
in the shoreward direction. The presence of wave groups in the sea wave records shows
that the wave process is also not stationary, as it is usually considered in conventional
wave data analysis. In this work, the Hilbert-Huang transform (HHT) method for
nonlinear and nonstationary time series analysis is applied to wave data from the
nearshore area. The field data were collected at Hazaki Oceanographical Research
Station (HORS), Port and Airport Research Institute, Japan, and covered different sea
states. The importance of a proper choice of an analyzing technique for the correct
understanding of the examined phenomenon is discussed. The ability of EMD, a key
part of the HHT method, to extract into IMFs the different oscillation modes embedded
in the sea surface elevation records is demonstrated. The variation of IMFs along the
beach profile is examined in an attempt to investigate cross-shore transformation of
sea waves. The dominant oscillations for each data record are determined, and their
variations during different sea stages are investigated.
97
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The Hilbert-Huang Transform in Engineering
5.1 INTRODUCTION
Real sea waves are nonlinear and nonstationary by nature, and their profile is
asymmetrical with respect to the zero level, with sharp crests and flat troughs,
especially well pronounced in shallow water. The existence of small ripples riding
longer waves, as well as an appearance of many small waves due to wave breaking,
complicates the wave profile considerably. Real sea waves also appear in groups,
which are time localized transient events. The wave data records always register a
mixture of many wave systems with different periods and energy, generated by
different sources. Through analysis of sea surface elevation record, we seek to
determine characteristic time–frequency scales of the energy. The widely used methods of wave analysis are burdened with the traditional concept of sea waves: as a
sine wave with definition domain from minus infinity to plus infinity. It is also
assumed, according to linear wave theory, that within the framework of one wave
the period is constant and the wave height is equal to twice the wave amplitude. In
addition to the positive maxima and negative minima of the wave profile, positive
minima and negative maxima are also often observed in the profile of real sea waves.
The neglect of these actual features of sea waves leads to incompleteness and
misinterpretation of the recorded sea state, and thus the analyzing method can distort
the investigated phenomenon.
The dominant approach in the existing methods of wave analysis is decomposition of the time series into component basis functions. Global sinusoidal components of fixed amplitude are the basis for data expansion in the commonly used
Fourier spectral analysis methods. The parameters of water waves, determined by
Fourier spectra, are integral characteristics of the sea state. Despite its being widespread and useful, the Fourier spectral technique has an important drawback: the
Fourier spectrum contains no information about the timing of the analyzed process.
Consequently, the local time variations of determined wave characteristics are impossible to track.
To understand the sea wave process correctly, we need an adequate method for
analysis. In the last decade the attention of scientists started to turn from globally
estimated wave parameters to local properties of a wave system. The attention has
been addressed to the time distribution of the wave energy, complimentary to the
frequency distribution information provided by usual Fourier spectral analysis. Liu
(1) suggested that the wave frequency spectrum approach is outdated, presenting
the advantages of the wavelet spectrum in the investigation of wave growth. The
differences between the Fourier and the wavelet spectrum are widely discussed by
Massel (2) on the basis of the application of the wavelet transform for the processing
of sea wave data. Huang et al. (3) also point out the necessity to break down the
earlier paradigm of wave analysis, emphasizing that Fourier analysis is not a good
method for studying nonlinear and nonstationary water waves.
Several Fourier-based methods for frequency–time distribution of the energy are
reviewed by Huang et al. (4): time window-width Fourier transform, evolutionary
spectrum, principal component analysis, and wavelet analysis. These methods suffer
from the demerits of Fourier spectral analysis — global definition of harmonic
components and the use of linear superposition for decomposition of the data
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An Application of HHT Method to Nearshore Sea Waves
99
(assuming the stationarity and linearity of data). There is also a resolution problem
inherent in all Fourier-based analysis due to the Heisenberg Uncertainty Principle.
Wavelet analysis provides some improvement in the resolution by using adjustable
windows, but the problems of the nonadaptive nature of these windows are still not
solved completely. It has to be mentioned in addition that in these methods for
frequency–time distribution of the energy, the basis for expansion of the data must
be chosen in advance. An exception is the principal component analysis, in which
the basis is derived empirically from the data.
In this paper, we discuss the importance of the analyzing technique for the correct
understanding of the examined phenomenon and pay special attention to the nonlinear and nonstationary behavior of real sea waves. We propose a new method for
nonlinear and nonstationary time series analysis, the Hilbert-Huang transform (HHT)
method, introduced by Huang et al. (4), as an alternative one for investigation of
real sea waves. The empirical mode decomposition (EMD), a key part of the HHT
method, extracts the energy associated with various intrinsic time scales into a set
of intrinsic mode functions (IMFs). By virtue of their derivation, the IMFs have
well-behaved Hilbert transform results, and the instantaneous frequencies can be
calculated. The frequency–time distribution of energy is designated as a Hilbert
spectrum. Any event in the data series can be localized in time as well as in frequency.
We apply the HHT to the wave data from the nearshore area and demonstrate
the ability of EMD to extract into IMFs the different oscillation modes embedded
in the sea surface elevation records. The variation of IMFs along the beach profile
is examined in an attempt to investigate cross-shore transformation of sea waves.
The dominant oscillations for each data record are determined, and their variations
during different sea stages are investigated.
5.2 FIELD DATA
The wave data used in this work were collected at the Hazaki Oceanographical
Research Station (HORS), Port and Airport Research Institute, Japan, during the
period from 25 February to 1 March 1989. The HORS is located on the Pacific coast
of Japan. The measurements were performed by six wave gauges mounted on a 427
m long pier and by three deepwater bottom wave gauges in an extension line of the
pier, as shown schematically in Figure 5.1. The offshore distance and water depth
of the wave gauges are listed in Table 5.1. The sea surface elevation was recorded
every 6 h, and the records are 2 h long at a sampling data rate of 0.5 sec.
Different sea stages were observed during the measurement period. The variations of offshore significant wave height (Hs) and mean wave period (Tmean) during
the observation are presented in Figure 5.2. The sea conditions are classified into
four different sea stages: calm, wave growth, wave decay, and post-storm stage, after
Katoh et al. (5). The initial three terms of measurements were in a comparatively
calm sea stage, when sea waves had a period of less than 6 sec and significant wave
height was less than 2 m. Under the direct influence of a passing atmospheric
depression with a strong north wind, growth of the sea waves was observed in the
next three terms. The waves became greater than 3 m in significant wave height, but
their periods remained short. In the wave decay stage, when the wind speed dropped,
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The Hilbert-Huang Transform in Engineering
FIGURE 5.1 Location of wave gauges during the observations (HORS, PARI, Japan).
TABLE 5.1
Location of Wave Gauges
Number of Wave Gauge
Offshore distance (m)
Water depth (m)
1
2
3
4
5
6
7
8
3000
24.00
2000
14.00
1000
9.00
380
5.65
320
5.05
215
2.33
145
1.81
80
0.30
the wave periods became longer than 9 sec. At the end of the observation, noted
here as the post-storm stage, the wind changed its direction and the registered waves
were small in amplitude, but with long periods. The wave gauges 7, 8, and 9 were
located inside the surf zone during the calm sea stage; in other sea stages, the surf
zone became wider and also included wave gauges 4, 5, and 6.
5.3 CONVENTIONAL METHODS FOR WAVE DATA ANALYSIS
An example of sea surface elevation, recorded in the surf zone at 5.05 m water
depth, is shown in Figure 5.3. The wave profile is irregular and considerably asymmetrical. There are small waves, resulting from wave breaking as well as ripples,
riding the longer waves. The question is how to analyze properly such considerably
nonlinear data time series, in view of the fact that the right choice of method is very
important for the correct understanding of the investigated phenomenon.
The conventional methods for wave data analysis propose estimation of wave
characteristics by means of Fourier spectra or by analysis of time series of individual
waves, determined by some criteria. The stochastic properties of sea waves, derived
by different discretization methods, are presented and discussed in detail by Ochi (6).
The identification of discrete waves from the record of sea surface elevation is
nontrivial work, especially for the shallow water wave data. The three criteria most
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An Application of HHT Method to Nearshore Sea Waves
101
FIGURE 5.2 Offshore characteristics of sea waves during the observation period.
Sea surface elevatiom (m)
2
1
0
-1
-2
80
90
100
110
Time (s)
FIGURE 5.3 Example of a wave record from surf zone
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130
140
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The Hilbert-Huang Transform in Engineering
TABLE 5.2
Discretization Criteria
Criterion
Definition of Individual Waves
Zero-crossing criterion
A wave is determined by two consecutive (up–up or
down–down) crossings of mean water level.
A wave is determined by two consecutive maxima of surface
elevation.
A wave is determined as corresponding to a 2 advance of
the phase angle in the complex plane.
Crest to crest criterion
Orbital criterion
Mean Wave
Frequency
ω02
ω24
ω01
often used for determination of individual waves from the sea surface elevation
records are briefly described in Table 5.2. The estimation of mean frequency depends
on the discretization criterion and is determined as ω ij = 2π j −i m j mi , where mk is
a kth spectral moment
∞
mk =
∫ f S( f )df ,
k
0
and S( f ) is a spectral density function.
The application of the zero-crossing (ZC) method for determination of individual
waves often meets difficulties in the treatment of small amplitude waves. They were
considered as “non real” or false waves by Gimenez et al. (7), Kitano et al. (8), and
Veltcheva and Nakamura (9), and were eliminated by some of the criteria proposed
by Mizuguchi (10), Hamm and Peronnard (11), and Gimenez et al. (12). The main
goal of removing the small waves, especially from the surf zone data, is to ensure
the constancy of wave period from deep to shallow water. It must be stressed that
the procedure of removing the small waves affects the estimation of wave characteristics, assuming a priori the linearity of sea waves.
The orbital criterion, introduced by Gimenez et al. (12), uses the complex
presentation of wave process. The analytical function
ξ(t ) = X (t ) + jXˆ (t ) ,
(5.1)
corresponds to the vertical displacement of sea surface elevation X(t). Here X̂ (t) is
the Hilbert transform of the sea surface elevation,
X t′
( ) π1 P ∫ t −( t′)dt′ ,
X̂ t =
(5.2)
where P indicates the Cauchy principal values. The Hilbert transform is a convolution
of the sea surface elevation X(t) with 1/t and thus emphasizes the local properties of X(t).
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An Application of HHT Method to Nearshore Sea Waves
103
The wave envelope, phase, and frequency are defined as
()
() ,
2
A(t ) = X t + Xˆ t
ϕ(t ) = arctg
Xˆ (t )
,
X (t )
2
(5.3)
(5.4)
and
ω (t ) =
dϕ
,
dt
(5.5)
respectively. The envelope A(t), defined by Equation 5.3, provides a direct measure
of the wave amplitude and is widely used in the statistical analysis of the sea waves
(13, 14, 15, 16, 17), while the phase information is usually not considered. Being
an important physical quantity, the phase φ(t) associated with the sea wave process
can also bear fundamental information about the dynamics of sea waves. But unfortunately, the phase is totally disregarded when conventional methods for wave
analysis are employed to study the wave system.
The phase φ(t) is a wrapped phase φ(t) ∈ [– π, π] or an unwrapped phase (t) ∈
[–∞, ∞], depending on its interval of definition. An example of the unwrapped and
wrapped phase of sea surface elevation X(t) is shown in Figure 5.4. The unwrapped
phase in Figure 5.4a increases smoothly with time, in contrast to the wrapped phase,
which varies widely in the range [– π, π].
The frequency ω(t), defined by Equation 5.5 as the rate of phase change, is
called the instantaneous frequency, and its value changes with time. This definition
of frequency was adopted before by Huang et al. (18) and used by Cherneva and
Veltcheva (19) for investigation of the local properties of sea waves and their group
structure. Later, Huang et al. (4) emphasized the considerable controversy in this
definition of instantaneous frequency, even with utilization of the Hilbert transform.
The instantaneous frequency is, as mentioned by Cohen (20), “one of the most
intuitive concepts, since we are surrounded by light of changing color, by sounds
of varying pitch, and by many other phenomena whose periodicity changes. The
exact mathematical description and understanding of the concept of changing frequency is far from obvious.…” Blindly applying the definition Equation 5.5 of
instantaneous frequency to any analytic function may lead to one of the five paradoxes listed by Cohen (20). One of these paradoxes concerns the appearance of
negative frequency: “Although the spectrum of the analytic signal is zero for negative
frequencies, the instantaneous frequency may be negative.” The enlarged portion of
the unwrapped phase shown in the upper left-hand corner of Figure 5.4a reveals the
jumps up and down in the time variation of the unwrapped phase. The unwrapped
phase function ϕ(t) is not always increasing function of time, and consequently the
instantaneous frequency ω(t) will have negative values, which have no physical
meaning.
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104
The Hilbert-Huang Transform in Engineering
1000
Unwrapped phase [rad]
100
Unwrapped phase [rad]
800
600
90
80
70
60
50
80
100
120
Time [s]
140
400
200
0
0
200
400
600
Time [s]
800
1000
1200
0
200
400
600
800
1000
1200
Wrapped phase [rad]
4
2
0
-2
-4
Time [s]
FIGURE 5.4 The phase φ(t) of the analytical process. (a) Unwrapped phase; (b) wrapped
phase.
The negative advances of the phase ϕ(t) produce loops or phase reversals in a
polar diagram of the wave process (Figure 5.5), where the analytical function ξ(t)
is presented as a radius-vector rotating in the complex plane. The magnitude of the
vector is equal to the wave envelope A(t), estimated by Huang et al. (3), and the
angle of rotation is equal to the unwrapped phase function ϕ(t). The arrows show
the direction of rotation of the radius-vector. The rotation of the radius-vector with
positive instantaneous frequency is called a proper rotation by Yalcinkaya and Lai
(21) in their investigation of the phase dynamics of continuous chaotic flow.
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An Application of HHT Method to Nearshore Sea Waves
105
Sea surface elevation (m)
1.5
0
-1.5
0
1.5
-1 .5
Hilbert transform of sea surface elevation (m)
FIGURE 5.5 Presentation of sea surface elevation in complex plane.
For a clearer illustration, Figure 5.5 shows the polar diagram of the part of the
wave record (thick, solid line) of Figure 5.3 between 85 sec and 112 sec. The
trajectory of ξ(t) in the complex plane generally may have multiple centers of
rotation. The majority of the individual zero-crossing waves in Figure 5.3 correspond
to a completely closed trajectory of rotation around the origin of the coordinate
system in Figure 5.5. The small ZC waves, around 90 sec and between 105 sec and
110 sec in Figure 5.3, are represented by loops in Figure 5.5. The centers of rotation
of these loops differ from the origin of the coordinate system. The small riding
waves in the wave profile also produce loops in the polar diagram, like the riding
wave between 91 sec and 95 sec in Figure 5.3. These phase reversals, as seen in
Equation 5.5, produce negative frequencies, which have no physical meaning.
According Gimenez et al.’s (12) orbital criterion, any discrete waves that do not
correspond to a 2π advance of the phase in the complex plane are considered as
false waves and are simply removed. In this way, the wave process is again assumed
to be a linear one, since only the completely closed orbit of water particles with 2π
advance of phase angle is accepted as an individual wave.
The small zero-crossing waves and ripples are indeed distinctive features of real
sea waves from the surf zone, and thus it is incorrect to eliminate them by the orbital
criterion for the purpose of achieving the positive frequency. Band-pass filtering,
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106
The Hilbert-Huang Transform in Engineering
proposed by Melville (22), also does not resolve the problem with negative frequencies. Huang et al. (4) especially stressed the fact that the filtering operation itself
contaminated the data with spurious harmonics. These unsuccessful attempts to
resolve the problem of negative frequencies are probably attributable to an inappropriately chosen tool for investigation of nonlinear and nonstationary sea waves.
5.4 HILBERT-HUANG TRANSFORM METHOD
Generally, the method of analysis affects the investigated phenomenon. The choice
of an appropriate analyzing technique is very important for a correct understanding
of the examined phenomenon, since the functions used in the analysis can be
misinterpreted as characteristics of the investigated phenomenon (23). To reduce
this risk, the analyzing function has to be chosen in accordance with the intrinsic
structure of the field to be analyzed. The HHT method, introduced by Huang et al.
(4), completely covers this requirement, since the expansion of the data is based on
the local characteristics of the particular data. This method offers a unique and
different approach for data processing. The HHT method was motivated “from the
simple assumption that any data consists of different intrinsic mode oscillations” (3).
The main idea of EMD, a key part of the HHT method, is to identify the time
scale that will reveal the physical characteristics of the studied process, recorded in
time series, and to extract these time scales into IMFs. A time lapse between
successive extrema in the time series is defined as a time scale, and this is the essence
of EMD. The EMD is a data sifting process to eliminate locally riding waves as
well as to eliminate local asymmetry of the time series profile. Huang et al. (3, 4)
present a procedure of sifting and several applications of the HHT method. On the
basis of the results of EMD of the data, Veltcheva (24, 25) investigated the group
structure of sea waves in the coastal zone.
The time series X(t) is first decomposed by EMD into a finite number n IMFs
(denoted Cj), which extract the energy associated with various intrinsic time scales,
and residual Rn. The superposition of the IMF and residue reconstruct the data:
n
( ) ∑ C (t ) + R (t ) .
X t =
j
n
(5.6)
j =1
By definition, the IMF has to satisfy two conditions. First, the number of extrema
must be equal or differ at most by one from the number of zero-crossings; this is
similar to the traditional narrow band requirements for the stationary Gaussian
process. Practically, this condition corresponds to elimination of riding waves or
small waves in the time series with multiple centers of rotation. The derived IMF
corresponds to a proper rotation in the complex plane, as the center of rotation is
the origin of the coordinate system. Second, the IMF has to have symmetric envelopes,
defined by the local maxima and minima respectively. These upper and lower envelopes
are determined by using cubic splines, as suggested by Huang et al. (4). This second
condition ensures that the instantaneous frequency will not have fluctuations arising
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An Application of HHT Method to Nearshore Sea Waves
107
from an asymmetric wave profile. The important condition for the IMF is that only
one maximum or minimum exists between successive zeros. Each IMF is determined
by a sifting procedure, which is repeated several times in order to ensure that all
the requirements of IMFs are satisfied.
The set of IMFs obtained in this way is unique and specific to the particular
time series, since it is based on and derived from the local characteristics of these
data. IMFs could be considered as a more general case of the simple harmonic
functions, but IMFs are claimed to have a physical meaning in additional to a
mathematical one due to their specific derivation.
In the second step, the Hilbert transform is applied to these IMFs:
()
1
Ĉ j t = P
π
∞
( )
C j t′
∫ t − t′ dt′
(5.7)
−∞
where P indicates the Cauchy principal value. The amplitude aj, the phase ϕj, and
the instantaneous frequency ωj are calculated by
a j (t ) = C 2j t + Cˆ 2j t ,
()
()
(5.8)
 Cˆ (t ) 
ϕ j (t ) = arctg 
 ,
 C (t ) 
(5.9)
and
ω j (t ) =
( ).
dϕ j t
dt
(5.10)
By using Equation 5.8, Equation 5.9, and Equation 5.10, we can express the original
data X(t) as a real part (Re) of the complex expansion
()
X t = Re
n
∑ a (t ) e ∫ ( ) ,
i ω j t dt
j
(5.11)
j =1
which is considered a generalized form of the Fourier expansion. Here, both amplitude aj(t) and instantaneous frequency ωj(t) are functions of time t. Whereas the
Fourier expansion is made in a global sense, the expansion by HHT is made in a
local sense.
It must be mentioned that, by definition, IMFs always have positive frequencies,
because the oscillations in IMFs are symmetric with respect to the local mean. Thus,
the HHT method solves the problem of negative frequencies, and there is no need
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108
The Hilbert-Huang Transform in Engineering
for any additional elimination of small waves, which are indeed observed peculiarities of real sea waves. All the distinguishing features of sea surface elevation are
taken into account, and the subjectivity in estimation of wave characteristics probably
will be reduced. This is a direct result of the use of a more appropriate tool for
examining nonlinear and nonstationary real sea waves.
The frequency–time distribution of the amplitude or squared amplitude is designated as a Hilbert amplitude spectrum or Hilbert energy spectrum. In the present
work, we use the Hilbert energy spectrum, determined as the frequency–time distribution of the squared amplitude. For simplicity, the Hilbert energy spectrum is
denoted as the Hilbert spectrum H(ω,t). The frequency–time distribution of the
energy allows us to determine which frequency exists at a particular time, whereas
the Fourier frequency spectrum provides information as to which frequency exists
generally in given data series.
The time resolution of the Hilbert spectrum can be as precise as the sampling
data rate. Since the Hilbert spectrum gives the best fit of a local sine or cosine
function to the data, the frequency is uniformly defined by a local derivative of the
phase function. The lowest extractable frequency is 1/T, where T is the duration of
the record, and the highest frequency is 1/(l∆t), where l = 5 is the minimum number
of points necessary to define frequency accurately, and ∆t is the sampling rate.
Based on the Hilbert spectrum, the marginal Hilbert spectrum h(ω) can be
defined as
T
h(ω ) =
∫ H (ω, t)dt .
(5.12)
0
The marginal Hilbert spectrum represents the cumulated squared amplitude over
the entire data span and offers a measure of total energy contribution from each
frequency.
5.5 APPLICATION OF THE HHT METHOD
For each wave record, a set of IMFs is obtained by EMD. Figure 5.6 presents detailed
results of decomposition of sea surface elevation data, recorded at an offshore point
with 24 m water depth at wave decay stage, with significant wave height of 3.18 m
and mean period 9.96 sec. Figure 5.6a shows the data time series. The EMD method
yields eight IMFs and residue, shown in Figure 5.6b through 5.6j. Since time scales
are identified as intervals between the successive alternations of local maxima and
minima, IMFs represent oscillation modes embedded in the data. The different IMFs
correspond to the different physical time scales that characterize the various dynamical oscillations in the time series. A nearly constant (as to time scale) wave component dominates in each IMF, representing the carrier wave constituent at the
specific time scale.
The sifting procedure of EMD generates modes Cj(t), j = 1, …, n with degree of
time variations in decreasing order. By construction, each mode Cj(t) generates a proper
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An Application of HHT Method to Nearshore Sea Waves
3
109
0.2
(f )
C5
X(t)
(a)
0
−3
0
C1
C6
0
0
C8
C3
−1.5
0
0
1000 2000 3000 4000 5000 6000 7000
0.1
(d)
0
1000 2000 3000 4000 5000 6000 7000
0
−0.1
1000 2000 3000 4000 5000 6000 7000
1.5
0
(h)
C7
C2
−3
(g)
0.1
(c)
0
1000 2000 3000 4000 5000 6000 7000
0
−0.1
1000 2000 3000 4000 5000 6000 7000
3
0
0.1
(b)
0
−3
−0.2
1000 2000 3000 4000 5000 6000 7000
3
0
0
−0.1
1000 2000 3000 4000 5000 6000 7000
(i)
0
1000 2000 3000 4000 5000 6000 7000
0.2
0.5
(j)
0
−0.5
R
C4
(e)
0
1000 2000 3000 4000 5000 6000 7000
Time (s)
0
−0.2
0
1000 2000 3000 4000 5000 6000 7000
Time (s)
FIGURE 5.6 EMD of wave record. (a) Data of sea surface elevation; (b) through (i) eight
IMFs (C1 through C8); (j) residue R.
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110
The Hilbert-Huang Transform in Engineering
rotation in the complex plane of its own analytic function ψj(t) = aj(t) exp [iφj(t)]. The
average rotation frequencies ω j (t ) = d φ j (t ) dt obey the order ω1 ≥ ω 2 … ≥ ω n because
the sifting procedure picks the components with the fastest variations embedded in the
original data first and those with the slowest variations last. The mean periods of the
IMFs, shown in Figure 5.6, are correspondingly 3.4 sec, 8.4 sec, 16.6 sec, 35.3 sec,
89.2 sec, 235 sec, 572 sec, and 1540 sec.
These oscillations not only have different time scales, but they also have a
different range of energy. The limits of the vertical axes in Figure 5.6b through 5.6i
are different for different IMFs. The most energetic in the decomposition is the
second IMF, C2. The shortest oscillations in decomposition, presented by the first
IMF, C1, have smaller amplitudes than those of C2. Physically, these high frequency
oscillations, extracted in C1, may represent short period waves, such as wind waves
in the growing stage, as well as the small ripples that ride longer waves. The third
IMF, C3, takes second place in the energy hierarchy. The first three constituents in
the decomposition, C1, C2, and C3, probably represent the contribution of the wind
waves oscillations, while the lower frequency oscillations are characterized by the
IMFs with higher index: C4, C5, and so forth. The complete presentation of the sea
wave state recorded in this data can be considered as a composition of dominant
wave oscillations, extracted in the second IMF and several (but finite number)
components with smaller amplitudes.
The contribution of different Intrinsic Mode Functions to the energy and frequency contents of wave data is investigated. The spectrum of wave record is
compared with spectra of its Intrinsic Mode Functions in Figure 5.7 as estimated
Fourier spectrum is shown in Figure 5.7a, while a marginal Hilbert spectrum h(ω)
is presented in Figure 5.7b. Under a sampling rate of these data, the highest frequency
that can be extracted by Empirical Mode Decomposition is 0.4 Hz. The highest
frequency of Fourier spectrum is 1 Hz, but for simplicity and easy comparison with
marginal spectrum, only until 0.35 Hz is shown in Figure 5.7a. The Fourier spectrum
is estimated as an average of the raw spectra, calculated in the overlapped segments,
by which the wave record is divided. The spectral estimations are also smoothed
with a Hamming window.
The Intrinsic Mode Functions represent different as period and energy oscillations, extracted by EMD from the wave data. The spectrum of the first IMF C1 (stars
line) covers very well the tail of wave spectrum (thick solid line) in the both ñ
Fourier and marginal spectrum. The peak of second IMF C2 (open circle line)
coincides with the major peak of the spectrum of sea surface elevation, both Fourier
and marginal spectrum.
The marginal Hilbert spectra are projections of real frequency-time distribution
of the energy, provided by Hilbert spectrum. There is basic difference in the meaning
of frequency in Fourier and Hilbert spectrum. The presence of energy at a given
frequency in Fourier spectrum means that a trigonometric component with this
frequency and amplitude exists through the whole time span of the data. In the
marginal Hilbert spectrum the existence of energy at a given frequency means that
in the whole time span of the data, there is a higher probability for local appearance
of such a wave.
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An Application of HHT Method to Nearshore Sea Waves
101
(a)
Fourier spectrum
100
10–1
111
Wave data
C1
C2
C3
C4
C5
C6
C7
C8
10–2
10–3
10–4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frequency (Hz)
101
(b)
Marginal spectrum
100
10–1
Wave data
C1
C2
C3
C4
C5
C6
C7
C8
10–2
10–3
10–4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frequency (Hz)
FIGURE 5.7 Spectra of wave data and its IMFs. (a) Fourier spectra; (b) marginal Hilbert
spectra.
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The Hilbert-Huang Transform in Engineering
5.5.1 OFFSHORE WAVES DURING DIFFERENT SEA STAGES
The peculiarities of the decomposition of offshore waves during different sea stages
were examined. The energy and time characteristics of individual IMFs were estimated and compared. The zero-th moment m0Cj of the marginal Hilbert spectrum
hj(ω) is proposed as a measure of the integrally determined energy of the jth IMF:
C
m0 j =
∫ h (ω)dω .
j
(5.13)
The mean value of the instantaneous frequency ω j , estimated by Equation 5.10, is
used as a measure of the frequency of the jth IMF.
Four cases, representative of each of the four different sea stages, were chosen.
The characteristics of their decomposition are shown in Table 5.3. The energy of
the IMFs with index higher than 4 is significantly lower than the energy of the first
four IMFs. This tendency is confirmed by the analysis of all offshore wave records.
From an energetic point of view, the first four IMFs contribute mainly to the energy
contents of the wave data, measured at 24 m depth. For a complete description of
the analyzed data, a complete set of all IMFs is necessary.
In the wave growing stage, C2 has the highest energy, followed by C1, in
agreement with the prevailing short period waves in the wave growing stage. The
swell type of sea waves starts to dominate in the wave decay stage; consequently,
C3 has energy comparable with or higher than the energy of C1. A similar tendency
in the energy distribution of the IMFs is also observed in the post-storm stage, when
the majority of the registered sea waves have a longer period. The energy hierarchy
of the IMFs in decomposition of a wave record reflects clearly the specific peculiarities of the particular wave data.
The different IMFs have different significance in view of the purpose of the
specific study. If the signal of interest is captured in a single IMF, the investigation
and eventual prediction of the examined phenomenon is facilitated, since the dynamical system is simpler. Salisbury and Wimbush (26) demonstrated this advantage of
the EMD, analyzing only one IMF, which had the same mean frequency as a
phenomenon of interest, instead of considering the original data series.
An understanding of the characteristics of large sea waves is essential for many
coastal engineering problems; therefore, special attention is paid here to the most
energetic components in the decomposition. In this connection, the IMF most dominant in energy among the whole set of IMFs is determined and denoted Cm. In
Figure 5.8, the energy of m0Cm is investigated as a function of the energy of the
original wave data m0; the data from all observation terms and all water depths are
used. The zero-th moments Equation 5.13 of the marginal Hilbert spectrum are again
used as a measure of energy. The dominant IMF, Cm, contained about 55% of the
energy of the wave data.
The specific peculiarities of the wave process are well captured by EMD and
reproduced by IMFs. Easy separation of the different time and energy associated
with the oscillations in the data is well achieved by EMD.
© 2005 by Taylor & Francis Group, LLC
Growing Stage
(Hs = 3.25 m, Tmean = 7.4 sec)
Decay Stage
(Hs = 2.86 m, Tmean = 10sec)
Post-Storm Stage
(Hs = 2.21 m, Tmean = 8.81 sec)
IMF
fp (Hz)
m0 (m2)
fp (Hz)
m0 (m2)
fp (Hz)
m0 (m2)
fp (Hz)
m0 (m2)
C1
C2
C3
C4
C5
C6
C7
C8
0.1920
0.1627
0.0987
0.0373
0.0187
0.0107
0.0053
0.0027
0.0797
0.0912
0.0309
0.0027
0.0007
0.0005
0.0002
0.0003
0.1120
0.1093
0.0907
0.0373
0.0160
0.0080
0.0053
0.0027
0.3257
0.5288
0.0865
0.0133
0.0044
0.0018
0.0013
0.0005
0.1053
0.0787
0.0693
0.0240
0.0107
0.0053
0.0027
0.0027
0.0693
0.5382
0.1580
0.0130
0.0030
0.0015
0.0007
0.0007
0.1947
0.0747
0.0800
0.0400
0.0187
0.0053
0.0027
0.0027
0.0473
0.2456
0.1232
0.0138
0.0050
0.0021
0.0004
0.0004
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Calm Stage
(Hs = 1.53 m, Tmean = 5.2 sec)
An Application of HHT Method to Nearshore Sea Waves
© 2005 by Taylor & Francis Group, LLC
TABLE 5.3
Characteristics of Offshore Sea Waves and Their IMFs
during Calm, Wave Growing, Decay, and Post-Storm Conditions
113
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114
The Hilbert-Huang Transform in Engineering
1.4
C
2 −1
m0 m(m s )
1.2
1
0.8
2 −1
m0(m s )
Cm
↑
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
m0 (m2s-1)
FIGURE 5.8 Comparison of energy of the dominant IMF (m0Cm) and wave data (m0).
5.5.2 CROSS- SHORE TRANSFORMATION OF SEA WAVES
After the wave data, measured at different points of the nearshore, were disintegrated
into IMFs, the cross-shore variations of the energy and frequency of the IMFs were
traced in order to examine the transformation of sea waves. The energy of the IMFs,
defined by the zero-th moment of the marginal Hilbert spectrum, is presented in
Figure 5.9a, while the frequency of the IMFs, defined as a spectral peak frequency,
is shown in Figure 5.9b. The horizontal axes in Figure 5.9 are an index of the first
eight decomposition components Cj (j = 1, …, 8). The data from 24 m and 14 m
water depth were collected in the region before breaking, while the other observation
points, at 5.65 m water depth and less, were located inside the surf zone. The wave
process in the surf zone is commonly considered to be a complicated one and
associated with a broadband Fourier spectrum. Here, by virtue of the EMD method,
these significantly nonlinear and complicated wave data series are practically decomposed into a small number of IMFs with a properly defined phase ϕ(t) respectively
frequency ω(t).
The energy of the first IMF, C1, characterizing the high frequency oscillations
in the data, is significantly smaller than that of the second component for the case
of offshore waves. However, in the shoreward region of the surf zone (2.33 m, 1.81
m, and 0.30 m water depths), the energy of C1 becomes comparable to the energy
of C2, C3, and C4. The second IMF, C2, has the highest energy in the decomposition
of the offshore waves. After breaking, the energy of the second IMF decreases
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An Application of HHT Method to Nearshore Sea Waves
115
0.8
24.0m
14.0m
9.00m
5.65m
5.05m
2.33m
1.81m
0.80m
0.7
Energy ( m 2s-1)
0.6
0.5
0.4
(a)
0.3
0.2
0.1
0
0
1
2
3
4
5
Ci, i =1,8
6
7
8
9
Peak frequency(Hz)
0.3
(b)
24.0m
14.0m
9.00m
5.65m
5.05m
2.33m
1.81m
0.80m
0.2
0.1
0
0
1
2
3
4
5
Ci, i =1,8
6
7
8
9
FIGURE 5.9 Decomposition of wave data measured in the cross-shore. (a) Energy of IMFs;
(b) frequency of IMFs.
considerably. At the same time, the oscillations extracted in C2 become shorter. The
energy and frequency of the lower frequency IMFs, C4, C5, C6, and C7, remain
approximately constant in the process of cross-shore transformation.
These peculiarities in the cross-shore variations of IMFs are reflected in the
frequency–time distribution of the energy. A section of the Hilbert spectrum of sea
waves in three different zones — offshore with water depth 24 m, in the seaward
surf zone at 5.05 m depth, and in the shoreward surf zone at 1.81 m — is shown in
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116
The Hilbert-Huang Transform in Engineering
(a) 24 m water depth
0.4
16
0.35
12
0.3
Fr 0.25
eq
0.2
ue
nc 0.15
y(
0.1
H
z)
0.05
10
8
1000
6
0.3
Frequency (Hz)
H (m2)
14
15
10
5
0
0.4
0.35
4
600
s)
400
m
Ti
e(
0
0
2
1
1.5
1000
800
1000
800
1000
0.25
0.2
0.15
0.05
0.5
600
0
s)
(
me
Ti
0
(c) 1.81 m water depth
1.5
200
800
0.1
1
800
400
600
0.3
0.4
1000
400
0.35
Frequency (Hz)
2.5
4
0
200
0.4
3
3
0
Time (s)
(b) 5.05 m water depth
H (m2)
0.15
0.05
2
0 0
0.3
Fr
eq
ue 0.2
nc
y(
H 0.1
z)
0.2
0.1
800
200
0.25
0
200
400
600
Time (s)
0 0
0.4
0.35
H (m2)
1
1
0.5
0
0.4
0.3
Fr
eq
ue 0.2
nc
y(
Hz 0.1
)
1000
0.5
800
600
0 0
Tim
0.25
0.2
0.15
0.1
0.05
s)
e(
400
200
Frequency (Hz)
0.3
1.5
0
0
0
200
400
600
Time (s)
FIGURE 5.10 A section of the Hilbert spectrum of sea waves, as a 3D plot in the left panels
and as a 9 × 9 Gaussian filtered color coded map in the right panels. (a) Offshore zone; (b)
seaward surf zone; (c) shoreward surf zone.
Figure 5.10. The Hilbert spectrum is presented as a three-dimensional plot in the
left side panels. The color scale maps of the same sections of H(ω,t) in a smoothed
form are shown in the right side panels. The Hilbert spectrum is smoothed with 9
× 9 weighted Gaussian filter. The smoothing operation degrades both the frequency
and time resolution of H(ω,t). The color bars next to the diagrams denote the energy
scale.
The frequency–time distribution of the wave energy changes considerably due
to wave transformation in a cross-shore direction. The Hilbert spectrum in Figure
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An Application of HHT Method to Nearshore Sea Waves
117
5.10a shows an energy concentration around 0.092 Hz, corresponding with the
second component, C2, which has the highest energy in the decomposition in Figure
5.6. The magnitude of energy decreases, as revealed by changes in the energy scale
of the color bars. A wide variation in the local frequency in the surf zone is evident
in Figure 10b and c, in contrast with the offshore region. This broadening of the
wave frequency range is partially due to wave breaking. The strong nonlinearity of
the sea waves in shallow water, shown by the intra-wave frequency modulation or
the variation of the local frequency within one wave, also contributes to a large
variation in the local frequency in the Hilbert spectrum.
On the basis of the results of EMD, real sea waves could be considered as a
system of oscillations with different periods and amplitudes. It has to be stressed
that the amplitudes and periods of these oscillations are not constant but instead are
functions of time according to Equation 5.11. The wave system, as a whole, changes
its content in cross-shore transformation. The process of wave breaking mainly
affects the IMFs with the highest energy in the decomposition of offshore waves.
Inside the surf zone, the frequency of each IMF tends to increase. These peculiarities
in the spatial distribution of different IMFs agree with general principles of sea wave
transformation in the cross-shore direction. The analysis of IMFs and the Hilbert
spectrum provides information on the cross-shore transformation of sea waves.
The EMD method, with its great ability to extract specific oscillation data
embedded in the data, can facilitate the investigation of sea waves. Attention can be
concentrated on a finite number of IMFs, each characterizing a dynamical system
that is much simpler than the original one. The application of the HHT method to
nearshore waves is a fertile area of research, as future studies will shed more light
on the complicated dynamics of the coastal zone.
5.6 CONCLUSIONS
We have discussed the importance of the analyzing technique for the correct understanding of the examined phenomenon and paid special attention to the nonlinear
and nonstationary behavior of real sea waves. A review of widely used conventional
methods for wave analysis shows that all of them assume a priori linearity and
stationarity of sea waves; these assumptions consequently affect the estimation of
wave characteristics.
The Hilbert-Huang transform method for the analysis of nonlinear and nonstationary time series is proposed as an alternative for the investigation of the nonlinear
and nonstationary nature of real sea waves. In this method, the oscillations embedded
in the data are extracted by EMD into a set of IMFs without placing any subjective
preliminary limitations on the nature of the investigated phenomenon. Easy separation of the different, as time and as energy associated with the oscillation in the data
is well achieved by EMD.
The dominant oscillations for each data record were determined and their variations during different sea stages were investigated. The energy hierarchies of the
IMFs in decomposition of wave records observed during different sea stages reflect
the specific peculiarities of the particular wave data. The process of wave breaking
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118
The Hilbert-Huang Transform in Engineering
mainly affects the IMFs with the highest energy in the decomposition of offshore
waves. Inside the surf zone, the frequency of each IMF tends to increase.
The EMD method, with its great ability to extract oscillation modes embedded
in the data, can facilitate the investigation of sea waves. Attention can be concentrated
on a finite number of IMFs, each characterizing a dynamical system that is much
simpler than the original one. The application of the HHT method to nearshore
waves is a fertile area of research, as future studies will shed more light on the
complicated dynamics of the coastal zone.
REFERENCES
1. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27:
577–588.
2. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records.
Ocean Eng. 28: 957–987.
3. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear water
waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457.
4. Huang, N. E., Shen, Z., Long, S..R., Wu, M. C., Shin, H. S., Zheng, Q., Yuen, Y.,
Tung, C. C., and Liu, H. H. (1998). The EMD and Hilbert spectrum for nonlinear
and non-stationary time series analysis. Proc. R. Soc. London. Vol. 454: 903–995.
5. Katoh, K., Nakamura, S., and Ikeda, N. (1991). Estimation of infragravity waves in
consideration of wave groups: An examination on basis of field observation at HORF.
Rep. Port Harbour Res. Inst. Vol. 30, 1: 137–163.
6. Ochi, M. K. (1998). Ocean Waves: The Stochastic Approach. Cambridge Ocean
Technology Series 6, 319 p. New York: Cambridge University Press.
7. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1998). Influence of
false waves on wave records statistics. Proc. ICCE 1998, pp. 920–933.
8. Kitano, T., Mase, H., and Nakano, S. (1998): Statistical properties of random wave
periods in shallow water: Analysis by utilizing false waves. Proc. Coast. Eng, JSCE,
Vol. 45, 221–225.
9. Veltcheva, A., and Nakamura, S. (2000). Wave groups and low frequency waves in
the coastal zone. Rep. Port Harbour Res. Inst. Vol. 39, 4: 75–94.
10. Mizuguchi, M. (1982). Individual wave analysis of irregular wave deformation in the
nearshore zone. Proc. ICCE 82, pp. 485–504.
11. Hamm, L., and Peronnard, C. (1997). Wave parameters in the nearshore: A clarification. Coastal Eng. 32: 119–135.
12. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1994). Analysis of
false waves in numerical sea simulations. Ocean Eng. Vol. 21, 8: 751–764.
13. Longuet-Higgins, M. S. (1958). On the intervals between successive zeros of a
random function. Proc. R. Soc. Ser. A, 246: 99–118.
14. Tayfun, M. A. (1983). Frequency analysis of wave heights based on wave envelope.
J. Geophys. Res. Vol. 88, C12: 7573–7587.
15. Bitner-Gregersen, E. M., and Gran, S. (1983). Local properties of sea waves derived
from a wave record. Appl. Ocean Res. Vol. 5, 4: 210–214.
16. Hudspeth, R. T., and Medina, J. R. (1988). Wave group analysis by Hilbert transform.
Coastal Eng. Vol. 1: 885–898.
17. Tayfun, M. A., and Lo, J. M. (1989). Envelope, phase, and narrow-band models of
sea waves. J. Waterway Port Coastal Ocean Eng. ASCE, Vol. 115, 5: 594–613.
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18. Huang, N. E., Long, S. R., Tung, C. C., Donelan, M. A., Yuan, Y., and Lai, R. J.
(1992). The local properties of ocean surface waves by the phase-time method.
Geophysical Res. Lett. Vol. 19, 7: 685–688.
19. Cherneva, Z., and Veltcheva, A. (1993). Wave group analysis based on phase properties. Proc. 1st Int. Conf. Mediterranean Coastal Environ. (MEDCOAST ’93), Turkey, pp. 1213–1220.
20. Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall Signal Processing Series,
Alan V. Oppenheim, series editor, 299 p.
21. Yalcinkaya, T., and Lai, Ying-Cheng (1997). Phase characterization of chaos. Phys.
Review Lett. Vol. 79, 20: 3885–3888.
22. Melville, K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128: 489–506.
23. Farge, M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev.
Fluid Mech. 24: 395–457.
24. Veltcheva, A. D. (2001). Wave groupiness in the nearshore by Hilbert spectrum. Proc.
4th Int. Symp. Ocean Wave Meas. Anal. (WAVES 2001), San Francisco, pp. 367–376.
25. Veltcheva, A. D. (2002). Wave and group transformation by a Hilbert spectrum.
Coastal Eng. J. Vol. 44, 4: 283–300.
26. Salisbury, J. I., and Wimbush, M. (2002). Using modern time series analysis techniques to predict ENSO events from the SOI time series. Nonlinear Processes Geophysics, Vol. 9: 341–345.
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6
Transient Signal
Detection Using the
Empirical Mode
Decomposition
Michael L. Larsen, Jeffrey Ridgway,
Cye H. Waldman, Michael Gabbay,
Rodney R. Buntzen, and Brad Battista
CONTENTS
6.1 Introduction ..................................................................................................122
6.2 Empirical Mode Decomposition..................................................................125
6.3 EMD-Based Signal Processing....................................................................126
6.4 Experimental Validation Using Towed-Source Signals...............................135
6.5 Conclusion....................................................................................................137
Acknowledgment ...................................................................................................138
References..............................................................................................................138
ABSTRACT
In this paper, we report on efforts to develop signal processing methods appropriate
for the detection of manmade electromagnetic signals in the nonlinear and nonstationary
underwater electromagnetic noise environment of the littoral. Using recent advances
in time series analysis methods [Huang et al., 1998], we present new techniques for
signal detection and compare their effectiveness with conventional signal processing
methods, using experimental data from recent field experiments. These techniques are
based on an empirical mode decomposition which is used to isolate signals to be
detected from noise without a priori assumptions. The decomposition generates a
physically motivated basis for the data.
121
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The Hilbert-Huang Transform in Engineering
6.1 INTRODUCTION
In this paper, we discuss application of nonlinear signal processing techniques to
underwater electromagnetic data. Electromagnetic measurements provide a potentially useful complement to acoustic detection methods in littoral zones where high
noise levels and reverberation in the littoral limit acoustic detection ranges. In
shallow waters, for example, the ambient noise levels from wind-generated breaking
waves are typically 10 to 100 times greater than those in deep water. Electromagnetic
measurements are affected by noise as well. One source of electromagnetic fields
in seawater is electrical currents induced by hydrodynamic flow in the Earth’s
magnetic field. The ocean is a constantly moving conducting fluid [26], and ocean
flows caused by gravity waves (wind waves and far-field swell), internal waves,
turbulence, tides, and currents all result in electromagnetic field fluctuations.
While the standard way of viewing the complexity of wave phenomena is to
consider the ocean to be a random, Gaussian process, the importance of nonlinear
interactions in water wave dynamics has been conclusively demonstrated [28]. Furthermore, turbulence is also often present at the bottom boundary layer due to
nonlinear interactions between flows arising from tidal and wind-driven currents,
wind waves, swell, and internal waves [4]. In the random sea assumption (used
primarily in the deep ocean), the evolution of the sea surface is obtained as a
superposition of waves of different frequencies and directions with random phases.
The wave amplitudes are determined according to a measured power spectrum, the
sea surface is modeled as a linear system, and energy transfer between wave modes
is not accounted for. In littoral regimes, however, nonlinear wave interactions are
significant and cannot be neglected. Higher-order statistics show that wave behavior
in shallow water zones displays a distinctly non-Gaussian regime, where wave–wave
interaction plays an important role and energy transfer between wave components
occurs [18, 12].
A typical power spectrum of naturally occurring ocean-bottom magnetic and
electric field data is shown in Figure 6.1. The most obvious peaks in both electric
and magnetic spectra in Figure 6.1 are visible at 62 MHz, the dominant oceanic
swell frequency. Swells have characteristic frequencies in the 50 to 100 MHz range,
and surface waves driven by local winds are found at frequencies above the swell
frequency, up to about 0.5 Hz. The effect of ocean swell on the magnetic field is
significantly more pronounced than its effect on the electric field. An additional peak
is also seen in the spectrum of the magnetic field data at twice the swell frequency
(125 MHz). Frequency doubling effects due to the nonlinear interference of the swell
with wave fields reflecting off local land masses have been previously observed [27,
24]. Spectral peaks above these frequencies are likely due to local turbulence, with
the exception of the Schumann resonance discussed below.
Another major component of underwater electromagnetic noise is actually due
to sources of non-oceanic origin — geomagnetic, atmospheric, and ionospheric
electromagnetic radiation. The Schumann resonance at 8 Hz is caused by worldwide
lightning, which resonates in the earth-ionosphere cavity. Geomagnetic noise at
lower frequencies originates in the ionosphere, is nonstationary, and increases in
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Transient Signal Detection Using the Empirical Mode Decomposition
Electric field spectrum, with and
without ionospheric mitigation
Magnetic field spectrum, with and
without ionospheric mitigation
3
3
10
10
ionosphere unmitigated
102
101
0
Power (nT2/Hz)
Power (uV/m2/Hz)
ionosphere unmitigated
ocean swell
102
ocean swell
101
10
instr. artifact
−1
10
10−2
123
ionosphere mitigated
10−3
Schmn res.
−4
10
−5
10
10−6
10−3
double freq.
10−1
10−2
10−3
10−1
Frequency (Hz)
0
10
101
instrument artifacts.
ionosphere
mitigated
−4
10
10−5
(a) E-field
10−2
100
10−6
10−3
(b) B-field
10−2
10−1
0
10
101
Frequency (Hz)
FIGURE 6.1 Power spectra for ocean-bottom electromagnetic fields before and after noise
mitigation.
power at lower frequencies, imparting the strong slope to the spectrum seen in Figure
6.1. These effects were observed in our experiments and in nearshore tests done by
other researchers [16]. Geomagnetic noise is typically coherent over large distances
and can be mitigated by using a remote reference station [17]. A transfer function
can be calculated to account for the difference in the conductivity of the underlying
geology and used to cancel out the effect of magnetospheric sources observed in
the local sensor data. Figure 6.1 shows the spectrum before and after such noise
reduction was performed.
Characterizing the nonstationary and nonlinear character of ocean-bottom electromagnetic noise is important for detection of manmade electromagnetic (EM)
anomalies. Ships and submarines generate electric fields in the surrounding water
due to corrosion currents caused by the dissimilar metals used in their construction
and also by onboard cathodic protection systems designed to mitigate such currents.
The bronze propeller acts as one node of the vessel’s galvanic cell, while the
sacrificial zincs on the hull act as nodes of opposite polarity. A current flows from
the propeller through the drive shaft, bearings, and hull into the water, returning to
the propeller. This current flow in the surrounding water can be well modeled as an
electric dipole. The horizontal motion of the ship or submarine imparts characteristic
ultra-low frequency signals to the field components, which change with the speed
and range of the submarine. This signature is described as the “quasi-static” horizontal
electric dipole (HED) signature. Impressed on the quasi-static dipole field is an AC
modulation, which is caused by variations in the electrical resistance in the shaft as it
rotates. A vessel passing a stationary electric field sensor generates a transient signal
in the nonstationary background ambient electric field. Figure 6.2 shows an example
of simulated horizontal electric field (E-field) components (Ex and Ey).
For a stationary sensor, the measured dipole signal is a function of the vessel
speed and its lateral distance from the sensor at the point of closest approach. The
spectra of moving dipole signatures are generally confined to ultra-low frequencies
(0.5 to 5 MHz) and thus fall into the underwater electromagnetic frequency regime
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The Hilbert-Huang Transform in Engineering
Electric Field: Track8, Rec’r ‘k’
1.5
x
y
Electric Field, E (µV/m)
1
ELFE Detail
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (mins)
30
35
40
45
FIGURE 6.2 Simulated time series of the two horizontal components of the electric field
from a moving vessel. The dipole appears as a slow transient. The inset is an enlarged view
of the cyclic modulation of the dipole.
dominated by ionospheric noise. When available, ionospheric noise removal can
greatly aid detection of the moving dipole signal, as any reduction in spurious signal
power translates directly into extended ranges [5, 1]. While remote reference mitigation techniques can remove much of this noise, specialized detection and extraction
techniques are still necessary. A number of techniques for processing underwater
electromagnetic data exist in the literature. A generalized likelihood ratio test was
developed for HED detection using a 3-axis E-field sensor [6], but under simplifying
Gaussian white noise assumptions. The use of higher-order statistics in detection
algorithms has been investigated [3,20], and the wavelet transform has also proved
useful for detection purposes [21,22].
In this paper, we investigate the use of signal processing techniques for underwater electromagnetic data which use the empirical mode decomposition (EMD)
reported by Huang et al. [13]. The EMD was developed specifically to deal with the
nonstationary and nonlinear characteristics of ocean wave data. In Section 6.2, we
review the properties of the EMD. In Section 6.3, we describe a detection algorithm
based on the conventional matched filter that employs the EMD for electromagnetic
anomaly detection. Experimental results are presented in Section 6.4.
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Transient Signal Detection Using the Empirical Mode Decomposition
125
6.2 EMPIRICAL MODE DECOMPOSITION
The EMD facilitates calculation of physically meaningful instantaneous frequencies
[13] by using the Hilbert transform. The Hilbert transform can be used to define the
instantaneous amplitude and phase of an arbitrary time series x(t). The Hilbert
transform of x(t) yields a time series y(t) given by
y(t ) =
1
PV
π
∞
x (τ)
∫ t − τ d τ,
−∞
(6.1)
where PV denotes the Cauchy principal value. Hence, the Hilbert transform is the
convolution of x(t) with 1/t, which consequently stresses the local nature of the
signal. From the complex analytic signal formed via z(t) = x(t) + iy(t), an instantaneous amplitude and phase can be calculated. The instantaneous frequency corresponds to the time derivative of the phase of z(t).
Direct application of the Hilbert transform to data may yield negative frequencies. Only positive instantaneous frequencies, however, correspond to physically
meaningful oscillatory behavior. In order to avoid nonphysical instantaneous frequencies, the empirical mode decomposition is used to separate the data into
well-behaved intrinsic mode functions (IMFs) from which the Hilbert transform will
yield positive instantaneous frequencies. An IMF is a function satisfying two conditions: i) the number of extrema and the number of zero-crossings of the function
must either be equal or differ at most by one, and ii) at any instant in time, the mean
value of the envelopes defined by the function’s local maxima and minima must be
zero. Each IMF has a characteristic frequency, although IMFs can overlap in frequency content. IMFs are found by using a recursive sifting procedure that generates
the highest-frequency IMF first. This IMF is subtracted from the time series, and
the process is iteratively applied to the difference until only a non-oscillatory residual, r, which represents the trend in the data, remains. Each IMF xn(t) has a variable
instantaneous amplitude, an(t), and frequency, ωn(t), calculated via the Hilbert transform. The time–frequency distribution H(ωn,t) of the amplitude is known as the Hilbert
spectrum. The original time series can be written as a sum of a finite number of IMFs:
 N

x (t ) = Re 
an (t )eiω n (t ) dt + r  .
 n=1

∑
(6.2)
Completeness of the EMD is assured in principle and depends only on the
numerical accuracy of the sifting process. Orthogonality of the IMFs, while not
guaranteed theoretically, is satisfied in practice. The Hilbert spectrum can be integrated over time to yield the Hilbert marginal spectrum,
HMS (ω ) =
∫ H (ω, t)dt,
T
0
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126
The Hilbert-Huang Transform in Engineering
which represents the accumulated energy at each frequency over the entire data span
and is related to the fraction of time that a given frequency can be observed in the
system. The marginal spectrum is analogous to the Fourier power spectral density.
The EMD leads to compact representation of nonlinear or nonstationary signals
without higher-order harmonics or a proliferation of global modes needed to account
for rapid transients. It is often observed that individual IMFs often correspond to
identifiable physical processes in the data [13,14].
Recent work is yielding greater understanding of the behavior of the EMD and
its potential for signal processing applications [8,7]. It appears to operate on fractional Gaussian noise as an almost dyadic filterbank, resembling a wavelet decomposition [10,9]. Furthermore, numerical experiments indicate that its impulse
response is similar to a cubic spline wavelet [25].
6.3 EMD-BASED SIGNAL PROCESSING
Electromagnetic noise in a littoral, oceanic environment is generally nonstationary
and can also be nonlinear. The spectrum of such noise is colored, insofar as it rises
with lower frequencies according to a power law. Hence, traditional detection methods based upon the assumption of the noise being white, stationary, and Gaussian
may not be optimal. The empirical mode decomposition appears to distinguish
signals originating from different physical processes by grouping them into a limited
number of intrinsic mode functions [24]. This behavior suggests the use of empirical
mode decomposition for increasing the effectiveness of detection of horizontal
electric dipole in a littoral oceanic EM noise background. Such anomalies generated
by moving sources are transient, with time scales lasting between 10 and 30 min,
depending upon the ratio of the closest point of approach (CPA) to the velocity.
The traditional method for detecting a deterministic signal in a noise background
is the matched filter, which uses an estimate of the expected signal to filter the data.
The matched filter is the optimal filter for detection of a deterministic signal in white
Gaussian noise [15]. It has been used for HED detection by Blampain [2]. Other
methods in the literature for detecting transient HEDs include a generalized likelihood ratio test [6] and a method based on the eigen-decomposition of the sample
covariance matrix [23]. All three of these methods assume that the noise is spatially
and temporally both white and Gaussian, an assumption that is not representative
of the littoral oceanic environment. A method of transient magnetic signal detection
in geomagnetic noise [22], based upon the application of wavelet packets, yielded
mixed results. As a starting point for our analysis, we have used the matched filter
method as our standard of comparison, with a priori knowledge of the target (i.e.,
bearing, CPA, velocity, and time of CPA).
Electromagnetic signal anomalies generated by moving sea vessels are transient
in nature. Results from simulated and actual data show that the transient quasi-static
HED signature is often captured in one or two IMFs. This behavior is demonstrated
with a simulated example without noise of a HED passing by at two different ranges,
with a simulated double-frequency cyclic component whose propagation is modeled
with the algorithm found in Orr [19]. The EMD of Ex generates five IMFs (see
Figure 6.3).
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Transient Signal Detection Using the Empirical Mode Decomposition
127
c0
Time Series and IMF Components
0.5
0
−0.5
c1
0.02
0
−0.02
c2
0.02
0
c3
0.5
0
−0.5
c4
−0.02
0.2
0
−0.2
c5
0.2
0
−0.2
0
100
200
300
400
500
Time
FIGURE 6.3 Simulated, combined dipole and cyclic signals at different ranges and their
derived IMF components. The lower five graphs display the individual IMFs, showing the
ability of the EMD procedure to separate signals with different characteristics.
Similar results are obtained for noisy signals. A typical track time series from
the experiment, containing both the HED and cyclic signal generated by a towed
source, is shown in the top graph of Figure 6.4. The cyclic signal (in the second)
and the HED (in the seventh panel), highlighted in red, are easily identified in this
close-pass track, which still contains noise.
We investigated the performance of the matched filter for detecting a moving
HED in the ocean using synthetic marine signals, which simulate a single dipole
moving in a straight path past a single receiver. The expected signal is a function
of the dipole amplitude and target range, bearing, and speed. In this study we elected
to simplify the situation by assuming a priori knowledge of speed and range, in
order to focus on the comparison between the matched filter and the empirical mode
(EMD)-assisted matched filter (EMMF) to be described later. In practice, one would
sweep through the set of realizations of speed, range, and HED dipole strength, and
determine the parameters that maximize the matched filter output. The two components of the electric field were processed together in quadrature as a complex signal,
with Ex as the real component and Ey as the imaginary component. A correlation
filter was used with the magnitude of the output being the detection criterion,
normalized to a maximum unity output. Through simulation it was determined that
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The Hilbert-Huang Transform in Engineering
r
x11
x11
x7–10
x6
µ
x5
x4
x3
x1–2
x (t)
128
2
0
−2
0.2
0
−0.2
0.1
0
−0.1
0.2
0
−0.2
0.5
0
−0.5
0.5
0
−0.5
2
0
−2
0.5
0
−0.5
0.1
0
−0.1
0.1
0
−0.1
0
500
1000
1500
2000
2500
3000
Time (sec.)
FIGURE 6.4 EMD analysis of underwater electromagnetic data.
s(t)
Known target
signature
x(t)
Target
signal
EMD
N
i(t)
Constrained
LSQ
x̂(t)
Matched
Filter
Detection γ(t)
statistic
i=1
FIGURE 6.5 Block diagram of EMD-assisted matched filter, where the matched filter process
is preceded by a least-squares fit of the IMFs (generated from the time series data).
the expected signal magnitude depends on the CPA range and target speed but not
the target bearing. All bearings will have the same magnitude response, with differing
phase responses. Thus the matched filter output depends only upon using the correct
range and speed of the vessel.
Building on the matched filter, an improved algorithm, the empirical mode
matched filter, was developed to exploit the EMD’s ability to capture the signal in
just a few of the IMFs. This method applies the matched filter to a signal generated
using the IMFs as basis functions, rather than to the target signal x(t) itself. This
improved detector performance. A block diagram of the processing steps of the
algorithm is shown in Figure 6.5. Empirical mode decomposition was first applied
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Transient Signal Detection Using the Empirical Mode Decomposition
129
to the time series to generate the IMFs αi(t), i = 1, …, N, where N is the number
of IMFs. In the next step, the linear combination of the IMFs containing the target
signal s(t) which best approximates the known target signature in the least-square
sense was determined:
N
min || s(t ) −
ci ≥ 0
∑ c α || .
i
i
i =1
The least-squares optimization was constrained so that the weights ci are positive.
A matched filter based on the target signature s(t) was then applied to the signal
x̂ (t), which is composed of the weighted linear combination of the IMFs,
N
xˆ (t ) =
∑ cα ,
i
i
i =1
to generate a detection statistic λ(t).
The underlying motivation for this technique is the tendency of the EMD to
generate a physically motivated basis for the data. Individual IMFs can be associated
with the dipole signal. Often the signal is captured in a single IMF. In such a case,
it would be advantageous to apply a matched filter to this IMF, rather than to the
whole time series, which may contain corrupting background signals. In general, it
is not known a priori which IMF the dipole will be in; it may even be spread across
several IMFs. Thus we employ the least-squares criterion above. The positive weight
constraint stems from the fact that the IMFs contain no sign ambiguity — their sum
is the complete time series. This constraint also reduces false alarms. A sample time
series of this process from the experiment is shown in Figure 6.6. The process of
weighting the IMFs results in an improved correlation with the target signal as
compared to the full time series.
We examined the performance of the EMMF using a statistical Monte Carlo
analysis with real electromagnetic background data from the experiment and simulated target signals (which are shown above to match the actual towed source signals
very well). This procedure validated that the EMMF robustly improves the probability of detection of the HED dipole signal in realistic oceanic/geomagnetic noise.
A number of signal-conditioning and pre-processing steps were used as well. The
simulation analysis was confirmed with detection results for real signals generated
by the DC electric towed source.
The first type of noise background tested for comparison between the linear
matched filter (LMF) and the EMMF was white Gaussian noise generated by a
random signal generator. The noise was band-limited by using a high-pass filter with
a frequency cutoff of 0.5 MHz (2000-sec period), because this low-frequency cutoff
does not change the moving dipole signals we wish to detect. A dipole signal was
created with a source strength equal to that used in the experiment and numerically
embedded in the white Gaussian noise background, which usually resulted in about
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The Hilbert-Huang Transform in Engineering
1.5
raw signal
LSQ wt signal
model
1
0.5
µV/m
0
0.5
1
1.5
2
2.5
0
100
200
300
400
500
600
700
time (sec)
FIGURE 6.6 Comparison of matched filter and EMMF techniques.
3000 overlapping samples in the simulation. The window size was set to τ = 2.7
(τ = V/RCPA), which from experience yields an effective template for the matched
filter. The desired Ex and Ey signals are shown in Figure 6.7.
A τ parameter of 2.7 yields signals that are 9 min long for this geometry. Longer
τ values include the tails of the signals, which contain little information, and shorter
τ values may cut out valuable information about a signal’s shape that makes it
different from the background. For this τ and constant source speed V, the signals
lengthen in time proportional to their CPA offset RCPA.
Receiver operating curves (ROC) curves were generated for the white Gaussian
noise and signal by calculating distributions of the matched filter output (normalized
to 0–1) for inputs of noise only and the noise plus signal. Once the distributions
were created, ROC curves were extracted, utilizing a sliding threshold, to determine
the probability of detection (PD) at different probabilities of false alarm (PFAs). The
result, shown in Figure 6.8, confirms the expected theoretical behavior: that the
standard LMF technique yields the best detection statistics for this type of noise.
The LMF procedure outperforms the EMMF (also labeled as the “LSQ” in the
figures that follow because the method involves a least-squares fit of the IMFs to
the desired matched filter). As the source strength and signal-to-noise ratio decrease,
both the LMF and EMMF ROC performances deteriorate, but the LMF maintains
a slight advantage, as predicted by theory. Signal-to-noise ratio (SNR) is defined
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Transient Signal Detection Using the Empirical Mode Decomposition
131
E-field X/Y components, Tau = 2.7
1.5
1
E-field (uV/m)
0.5
0
−0.5
−1
−1.5
−2
0
100
200
300
400
Time (seconds)
500
600
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
PD
PD
FIGURE 6.7 Ex (solid) and Ey (dotted) signals for a north–south track, where τ = 2.7, which
yields a 9-min signal length at this CPA and speed.
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0.1
LSQ
LMF
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
0
LSQ
LMF
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
PFA
1
PFA
FIGURE 6.8 Comparative ROC curves for HED moving dipole detection in Gaussian white
noise, for LMF method (lower curve) and EMMF/LSQ method (upper curve). The original
source strength has been reduced to correspond to SNR values of –14 and –16.4. The LMF
method outperforms the EMMF in Gaussian white noise, in accordance with signal-processing
theory.
here as the peak excursion of the target signal versus the standard deviation of the
background noise [11], normalized to a logarithmic dB number:
SNR = 20 log10 ( P / σ ) ,
where P is the peak excursion of the target signal (of a specified component), and
σ is the standard deviation of the background noise. This broadband definition is
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132
The Hilbert-Huang Transform in Engineering
Source strength effect on ROC curves, LMF vs. EMMF
1
SNR 10
0.9
SNR 7
0.8
0.7
PD (01)
SNR 5
0.6
0.5
0.4
0.3
0.2
EMMF/LSQ
0.1
LMF
0
0
0.2
0.4
0.6
0.8
1
PFA (01)
FIGURE 6.9 ROC curves for LMF process (light) and EMMF/LSQ process (dark), for
several different SNRs. When the SNR is strong, the two processes achieve nearly identical
detection results, but when the SNR is weaker, the EMMF process has a definite advantage.
calculated independently of frequency. The SNR for the ROC curves in Figure 6.8
is negative (printed on curves) and means that the signal amplitude is lower than
the standard deviation of the noise. A SNR of –14, for example, corresponds to the
peak signal value being one-fifth the noise standard deviation.
The advantage of the LMF over the EMMF method is lost when an actual noise
background is used. The background used in the following analysis consists of 52
h of experimentally measured underwater electromagnetic data, which is non-Gaussian with a colored spectrum and a nonstationary variance. In actual oceanic and
geomagnetic noise, the EMMF method is superior. This is demonstrated in Figure
6.9, where we have displayed three representative curves together on one graph,
utilizing SNRs between –5 and +14, much higher than the SNRs used in the white
Gaussian noise process. A major characteristic of the real-world noise is its nonstationarity and high power at low frequencies, which can lead to anomalies that mimic
the transient HED.
The ROC performance curves in Figure 6.9 have been enhanced by two signal-conditioning procedures: high-pass filtering of the data to produce first-order
stationarity (see above), and removal of a linear trend within each window being
examined. The window length is determined by the τ value being sought. (For the
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Transient Signal Detection Using the Empirical Mode Decomposition
133
PD versus source strength (dB)
1
0.9
0.8
0.7
Black: LSQ
Red: LMF
Blue: LSQ, 2x range
Green: LMF, 2x range
PD (01)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
Source Strength (dB Am)
40
50
FIGURE 6.10 Relative gain for HED detection using EMMF at set PFA (= 0.1) for two
different ranges. PD is plotted as a function of source strength in dB. The EMMF method is
in blue. For a given PD, the necessary source strength is lowered by 2 to 3 dB, for moderate PD.
simulations performed, we assume knowledge of such parameters; for detection in
a field situation, one would sweep through a predetermined set of moving dipole
parameters and choose the set that yields the highest matched-filter output.) Other
signal-conditioning techniques were also explored and found to be less effective.
For example, when we remove a mean value from the window rather than de-trending, the overall ROC curve degrades, but the EMMF displays a relative advantage
over the linear matched filter.
Figure 6.10 shows the effects of source strength and SNR on ROC performance
curves. The source strength is represented in dB relative to 1 A-m. For low-tomedium source strengths (up to 35 dB), the EMMF method yields superior PDs at
a given strength. Conversely, for a given PD, the necessary source strength to achieve
that PD is reduced by 2 to 3 dB in the PD range of 0.4 to 0.8. This detection gain
vanishes at high source strengths (high SNRs), where the two methods yield identical
detection statistics. It must be emphasized that the analysis here has been done on
“unreduced” data, which has not been differenced versus a remote reference and is
a high-noise environment. Removing coherent geomagnetic noise by using a remote
reference sensor yields a greater range.
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The Hilbert-Huang Transform in Engineering
PD versus CPA; unreduced data
EMMF
LMF
1
PD (01)
0.8
0.6
0.4
0.2
Range
FIGURE 6.11 Probability of detection versus CPA, for LMF (light) and EMMF (dark)
methods. As the range is further increased and SNR decreases, the EMMF method excels,
extending the detection range by 10 to 15% for a set PD, or increasing the PD by 10 to 25%
for a given range.
We repeated this procedure for a greater range for unreduced data, which yields
a nearly identical curve, but shifted to the right about 15 dB. Thus the EMMF method
maintains its increase in effectiveness across range. The detection versus range
behavior is summarized in Figure 6.11, where we plot PD versus CPA range for a
set PFA = 0.1. Up to a certain range, the two methods again yield an identical high
probability of detection, but as the range is further increased (and SNR decreases),
the EMMF method (dark curve) is more effective, extending the detection range by
10 to 15% for a set PD, or increasing the PD by 0.1 to 0.2 for a given range. We
see that the EMMF method flattens the falloff curve for low SNR regimes, thereby
extending detection ranges. The inclusion of the EMD into the matched filter process
adds detection gain in high-amplitude, colored, and nonstationary noise such as
observed in the underwater electromagnetic environment. This effectiveness
enhancement is seen generally in low to medium SNRs, whether caused by weaker
source strength or a farther CPA.
We also investigated the effectiveness of the EMMF method in a noise regime
where the common-mode geomagnetic noise has been partially mitigated by using
remote-reference techniques. This yields improved noise reduction so that an algorithm
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Transient Signal Detection Using the Empirical Mode Decomposition
ROC curves, REDUCED data
Unreduced and reduced background noise
3
0.9
sigma = 0.23
2
0.8
1.5
0.7
1
0.6
PD (01)
E-field amplitude (uV/m)
1
Unreduced
reduced
2.5
0.5
0
0.5
0.4
0.3
−0.5
0.2
−1
−1.5
135
sigma = 0.090
0
5
10
15
20
0.1
25
30
35
40
0
EMMF/LSQ
LMF
0
Time (hours)
0.2
0.4
0.6
0.8
1
PFA (01)
FIGURE 6.12 Effect of remote-reference noise reduction on matched filter methods. (A)
Unreduced (top) and reduced (bottom) time series for background, X component, over 40 h.
The geomagnetic noise removal not only lowers the variance, but also makes the resulting
series more stationary. (B) ROC curves generated from reduced data, for EMMF (dark) and
LMF (light) methods. The LMF method is more effective at low ranges/high SNR, whereas
the EMMF method increasingly dominates at greater ranges.
comparison can be made in a background where oceanic and other processes dominate, rather than ionospheric sources. Figure 6.12A shows a plot of the unreduced
and reduced time series. Visually, the reduced time series is more uniform across
its entire length and has both lower variance and fewer outliers than the original
time series, which is simply high-pass filtered. Histogram analysis at various positions in the time series also indicates that it has more a stationary variance, and
spectral analysis shows it to have a flatter power spectrum at low frequencies and
thus to be closer to white Gaussian noise.
We repeated the Monte Carlo procedure for generating ROC curves as a function
of range, with the results shown in Figure 6.12B. Detection ranges are noticeably
extended. Also, the LMF method is actually slightly superior to the EMMF, at ranges
where the SNR is high. However, as the range increases and the SNR decreases, the
EMMF method becomes more effective than the linear matched filter, providing a
flatter rolloff of PD versus range.
Figure 6.13 shows PD performance versus range for the EMMF and LMF
methods for reduced data. Here we have set the PFA to a constant (= 0.1), and
calculated the PD as a function of range for both methods. The linear method has
a slight advantage at lower ranges and higher SNR, whereas the HHT-enhanced
method is more effective at long ranges and a small SNR. Overall, the EMMF
method provides superior performance with flatter rolloff versus range.
6.4 EXPERIMENTAL VALIDATION USING
TOWED-SOURCE SIGNALS
We utilized a database of 17 tracks of a towed electromagnetic source collected over
6 sensors with various CPAs to test the relative effectiveness of the EMMF method
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The Hilbert-Huang Transform in Engineering
PD versus CPA, REDUCED data
1
0.9
0.8
0.7
PD (01)
0.6
0.5
0.4
0.3
0.2
0.1
0
Range
FIGURE 6.13 Summary graph of behavior of PD versus range for EMMF (dark) and LMF
(light), after coherent geomagnetic noise removal. The LMF method is slightly improved at
lower ranges, but overall the EMMF method provides a superior curve with flatter rolloff
versus range.
versus the LMF method of detection for the moving HED signal on unreduced data.
A total of 36 track/sensor combinations existed in this CPA range. For each individual
track, assuming knowledge of the track bearing and CPA, we performed the algorithms discussed above and measured the correlation signal. Such signals generally
produced a peak very near the CPA. To determine whether a detection occurred, we
applied the matched filter to data measured at a remote sensor which contains only
background noise, and calculated a threshold for a given false alarm level (a PFA
level of 5% was used).
The results are displayed in Figure 6.14A for the LMF method and Figure 6.14B
for the EMMF method. As expected from the ROC analysis, the EMMF method is
superior. EMMF correlations are uniformly higher with a much flatter falloff versus
CPA than for the LMF method. Additionally, the threshold (actually a mean value
of the thresholds over all 36 cases) for the EMMF method raises only slightly, which
results in an overall increase in detectability using the EMMF. As displayed in Figure
6.15, in the 36 cases tested, the EMD-based method results in 24 detections, versus
only 14 for the linear matched filter (points very near to the detection threshold were
counted as detections), using a PFA of 5%.
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Transient Signal Detection Using the Empirical Mode Decomposition
LMF-based correlation vs. threshold
EMMF-based correlation vs. threshold
1
(a)
0.9
0.9
0.8
0.8
Correlation (0–1)
Correlation (0–1)
1
137
0.7 threshold for PFA = 0.05
0.6
0.5
0.4
(b)
0.7 threshold for PFA = 0.05
0.6
0.5
0.4
0.3 A): 14/36 detections
0.2
0.3 A): 24/36 detections
0.2
0.1
0.1
FIGURE 6.14 Comparison of (A) the LMF method with (B) the EMMF method, utilizing
signals created by the towed electric source. The EMMF has a flatter falloff with range and
more detections (24/36 versus only 14 for the LMF).
Detections vs. range: LMF vs EMMF for data with coherent noise reduction
Number of detections, chances
Possible chances
Detections EMD
Detections LMF
Range
FIGURE 6.15 Comparison of number of detections of the LMF method (light) with the
EMMF method (medium), utilizing signals created by the towed electric source.
6.5 CONCLUSION
In summary, we have exploited the empirical mode decomposition to isolate physical
signals into a discrete number of intrinsic mode functions in order to enhance the
detection of transient HED signals in a background of geomagnetic and hydrodynamic noise. We have numerically compared this method to the standard matched
filter by deriving ROC curves via a Monte Carlo simulation for experimental data.
In white Gaussian noise, the LMF method yields superior ROC curves, in accordance
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The Hilbert-Huang Transform in Engineering
with signal-processing theory. In a realistic, non-Gaussian, high-noise environment,
the EMMF method outperforms its linear counterpart by enhancing detection ranges
and minimum detectable source strengths. When it is possible to mitigate much of
the geomagnetic noise by using remote reference methods, then the traditional LMF
provides slightly superior detection at high SNRs, but the EMMF method yields a
flatter PD versus range curve and significantly extends detection ranges for a given
PD. The results for unreduced data were confirmed by applying the matched filter
methods to experimental data, where the EMMF method increases detection ranges
and flattens the detection curve with range. Further investigation of the optimal use
of such methods is warranted.
ACKNOWLEDGMENT
This work was funded by the Office of Naval Research and others. The authors are
grateful for the experimental assistance of the Swedish Defense Ministry and the
FOI, and for the assistance of Dave Rees of SPAWARSYSCEN San Diego.
REFERENCES
1. J. Berthier, F. Robach, B. Flament, and R. Blanpain. Geomagnetic noise reduction
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gradients. In Proc. 1998 Oceans Conf., July 2001.
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Doctoral thesis of engineering, INPG, Grenoble, 1979.
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4. C.S. Cox, X. Zhang, S.C. Webb, and D.C. Jacobs. Electro-magnetic fluctuations
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In MARELEC, 3rd Int. Conf. Marine Electromagnetics, July 2001.
5. B. Deniel. Undersea magnetic noise reduction. In Proc. 1997 MTS/IEEE Conf., vol.
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6. R. Donati. Detection of the electric field due to corrosion phenomena. In Proc. 2nd
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7. P. Flandrin, P. Gonclaves, and G. Rilling. Detrending and denoising with empirical
mode decompositions. In Proc. IEEE-EURASIP Workshop Nonlinear Signal Image
Process. NSIP-04, to appear in 2004, available at http://www.inrialpes.fr/is2/people/pgoncalv/pub/emd-eusipco04.pdf.
8. P. Flandrin and P. Gonclaves. Sur la decomposition modale empirique. In Proc.
GRETSI-03, 2003.
9. P. Flandrin, G. Rilling, and P. Gonclaves. Empirical mode decomposition as a filter
bank. IEEE Sig. Proc. Lett., 11(2):112–114, 2004.
10. P. Flandrin. Empirical mode decompositions as data-driven wavelet-like expansions
for stochastic processes. In Proc. Conf. Wavelets Statistics, 2003.
11. B.C. Fowler, H.W. Smith, and F.W. Bostick Jr. Magnetic anomaly detection utilizing
component differencing techniques. Technical Report 154, Electrical Geophysics
Research Lab., Univerisity of Austin Texas, 1973.
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139
12. K. Hasselmann. On the nonlinear energy transfer in a gravity wave spectrum. Jour.
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13. N.E. Huang, Z. Shen, S.R. Long, M.L. Wu, H.H. Shih, Q. Zheng, N. Yen, C.C. Tung,
and H.H. Liu. The empirical mode decomposition and Hilbert spectrum for nonlinear
and non-stationary time series analysis. Proc. R. Soc. London A, 454: 903–995, 1998.
14. N.E. Huang, Z. Shen, and S.R. Long. A new view of nonlinear water waves: the
Hilbert spectrum. Annu. Rev. Fluid Mech., 31:417–57, 1999.
15. S.M. Kay. Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory.
Prentice Hall, Englewood Cliffs, N.J., 1998.
16. M.J. Morey, T. Bentson, and C. Osborn. TET E-field data analysis, summer 1991
experiment. Tech. Document 2482, NCCOSC RDT & E Divsion, 1993.
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19. A.M. Orr. Computational techniques for evaluating extremely low frequency electromagnetic fields produced by a horizontal electrical dipole in seawater. Dept. of
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7
Coherent Structures
Analysis in Turbulent
Open Channel Flow
Using Hilbert-Huang and
Wavelets Transforms
Athanasios Zeris and Panayotis Prinos
CONTENTS
7.1
7.2
Introduction ..................................................................................................142
Results ..........................................................................................................143
7.2.1 Academic Signals.............................................................................143
7.2.2 Turbulent Signals .............................................................................143
7.2.3 Coherent Structures.........................................................................143
7.3 Conclusions ..................................................................................................150
References..............................................................................................................156
ABSTRACT
In this study the Hilbert-Huang transform with empirical mode decomposition is
applied for analyzing and quantifing turbulent velocity data from open channel flow.
This novel technique, proposed by Huang et al. [1,2], is applied using experimental
velocity signals in turbulent, fully developed open channel flow, with and without the
imposition of bed suction. Signals were obtained with laser Doppler and hot film
techniques [3,4]. Velocity data for the above flow were analyzed in previous studies,
with traditional methods, for various suction rates for the description of the flow field
and for the identification of the intense events related with coherent structures that
contribute to the organized motion. Here, an attempt is made to exploit the novel
decomposition method in the domain of joint frequency–time analysis of the turbulence
data.
141
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The Hilbert-Huang Transform in Engineering
7.1 INTRODUCTION
Details for the instrumentation and experimental results from measurements of the
application of suction from the bed in open channel have been presented in previous
articles [3,4,5]. We mention for completeness of the present study that measurements
were conducted with LDA (tracker) and hot film methods, the suction rate is based
on discharge ratio (Qs/Qtot, Qs = suction discharge, Qtot = total discharge), the length
of suction region L = 0.035 m, and the Re number based on upstream velocity Um
and flow depth was Re = Umh/ = 59,000.
The empirical mode decomposition (EMD) method proposed by Huang [1] and
the Hilbert transform of the decomposed signal are summarized as follows:
All the extrema of the signal x(t) are identified, and an interpolation for the
maxima and the minima is attempted with cubic splines curves, following recommendations and proposals from previous work [1,2]. The local mean m(t) is computed, and the difference between the local mean m(t) and the initial signal is
extracted. Until a stopping criterion [2] is achieved, an iteration process takes place.
In this study the criterion that we adopt is that the number of extrema equals (or
differs by one from) the number of zero-crossings. When this criterion has been
achieved, the remainder is considered as an intrinsic mode function (IMF) and
denoted c1; this IMF contains the shortest periods of the signal. The residual
()
()
x t − c1 = r t ,
is considered as a new signal and subjected to the iterative process repeatedly, until the
last residue is smaller than a predetermined small value or is a monotonic function.
Preliminarily, the problem of extrema interpolation is checked from the viewpoint of (a) it is time consuming especially when the signal belongs to the multifrequency ones as turbulent signals that needs a large number of iterations for a
correct decomposition and (b) because of end effects propagation of the splines into
the interior of the signal. We used different approaches as linear, linear with empirical
modifications, and iterative process with preselected iteration numbers. With the
criterion of the most perfect composition of the original signal we chose, for the
present article, to extract IMFs with the originally proposed method [2] cubic spline
with the addition of sine waveforms at the beginning and at the end of the analyzed
data.
With the aid of the Hilbert transform, Huang et al. [1,2] proposed the construction
of the analytic signal for every IMF component. From them we get the time distributions of amplitude and phase
()
()
()
h t = c t + H c t 
( ( ))  ,
H c t
φ t = tan −1 
 c t
()
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( ) 
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143
where h(t) and φ(t) are the instantaneous amplitude and phase respectively. From
this quantity, values for the instantaneous frequency are obtained as
()
f t =
()
dφ τ
dt
With the extraction of IMFs as a time series, the tools for the HHT amplitude–frequency–time representation of a signal — the instantaneous frequency and
the amplitude — are available, and, depending on the signal’s nature, their quality
is better than or equal to the results of other well documented methods, such as
short FFT, proper orthogonal decomposition, Wigner-Ville, and the wavelet decomposition method.
7.2 RESULTS
7.2.1 ACADEMIC SIGNALS
Some academic signals are used for the estimation of the quality of decomposition
applied in the present study. The traditional frequency shift sine signal (Figure 7.1a)
is examined through frequency time variation of the first IMF with the point of
frequency shift well localized. Figure 7.1c also gives the Hilbert spectrum and
Figure 7.1d presents the FFD distribution.
In Figure 7.2a the sawtooth signal is presented along with the distribution of
the instantaneous frequency (Figure 7.1c) because this kind of variation is a candidate
(and as the edge detection problem) as part of an average representative pattern
indicating the presence of organised structures.
7.2.2 TURBULENT SIGNALS
Figure 7.3 shows a typical velocity signal, and Figure 7.4 illustrates its analysis in
eight IMF components. Figure 7.5 shows the distribution of these components in
the frequency domain. Figure 7.5b presents the filtering approach of the Ci components divided by the root-mean-square (RMS) value of every component.
Figure 7.6a and b gives the variation of frequency with time. For clarification
only, Figure 7.6a shows the frequency–time distribution with a median filter of high
order for a better visual observation of the different components. In addition, Figure
7.6c gives a representative energy–frequency–time distribution. Figure 7.7 presents
the time variations in instantaneous energy IE(t) for a short time period for a velocity
signal at y+ = 100 from hot film measurements.
7.2.3
COHERENT STRUCTURES
In turbulent wall flows, a great portion of the momentum transport is connected with
the presence of organized structures in the form of burst structures in the region
near the wall. According to well-documented models, streaks of low velocity start
to oscillate and to lift up from the wall. The ejections from the decomposition of
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The Hilbert-Huang Transform in Engineering
(a)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
1
2
3
4
5
3
4
5
t (sec)
(b)
40
35
30
f (Hz)
25
20
15
10
5
0
0
1
2
t (sec)
(c)
(d)
2
E (t)
EH (f )
1.5
1
0.5
0
1
10
f (Hz)
100
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
f (Hz)
FIGURE 7.1 (a) Frequency shift signal; (b) instantaneous frequency variation; (c) Hilbert
spectrum; (d) FFT distribution.
these streaks, appropriately grouped, are characterized as burst structures. In the
search for quantitative information, many techniques have been introduced during
the past decades toward the identification of patterns and levels in the signal velocity
connected with these organized structures. These techniques include the variable
© 2005 by Taylor & Francis Group, LLC
FIGURE 7.2 Sawtooth signal with (a) instantaneous frequency (b) variation with time.
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f (t)
1.5
16
1
14
2
0
t (sec)
4
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Coherent Structures Analysis in Turbulent Open Channel Flow
18
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4
-1.5
10
0
6
-1
12
0.5
8
6
4
2
-0.5 0
(b)
(a)
2
0
145
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The Hilbert-Huang Transform in Engineering
0.4
0.2
0
u(m/s)
0
signal
5
10
15
20
25
30
35 t(sec) 40
FIGURE 7.3 Typical velocity signal.
interval time average (VITA) technique [6,7], quadrant splitting [8] of Re stresses,
the temporal pattern average (TPAV) [10], linear stochastic estimation [9], and the
proper orthogonal decomposition (POD) and wavelet decomposition–based techniques [11].
Traditional techniques, such as VITA (focusing on local variance that exceeds
part of the whole signal variance) and quadrant splitting for velocity and Re stress,
with empirical threshold criteria k = 1.0, predetermined time scale T* = 10, and
H = 1 for quadrant splitting, respectively, have been applied in previous studies [3,4]
to extract intense events in the flow field inside the porous suction region (x/L =
0.5) and beyond the suction region (x/L = 1.2) from hot film and LDA results. Figure
7.8 gives an example of results from the VITA technique. Here the effect of suction
is evident at the exit of the suction region from the frequency appearance of the
intense events.
With the tool of signal decomposition, it is possible to overcome the shortcomings resulting from the single scale limitations of the VITA method. The VITA
method does not permit the identification of events separated by a time smaller than
that imposed by the VITA treatment. Also, a main disadvantage for VITA is the fact
that it requires the selection of two rather subjective criteria (time scale and threshold).
Wavelets decomposition allows intense events to be identified based on the value
of the wavelet coefficients in the representation of the translation — scale plane
using a nonsubjective threshold criterion.
Analysis in the scale time plane of turbulent statistical measures gives scale
dependent behavior for quantities such skewness and flatness factors. Strong
non-Gaussian behavior of the smaller scales is an indication of the intermittency.
For intermittency estimation at each scale, Farge [11] proposed the local intermittency index Im,n , the ratio of local energy to the mean energy at the respective scale.
As this index takes an extreme value at a time instant, a strong percentage of energy
and intermittency is found in the corresponding time and scale.
With the above in mind, it is possible to use EMD to determine the ratio of local
energy (amplitude squared) to mean energy for every IMF in the energy–frequency–time distribution. Figure 7.9 shows the above ratios with the 4th order Daub4
wavelet and for the HHT method.
The next step in the identification of coherent events is to apply the appropriate
thresholds. Universal proposed relations from wavelets such as Donoho and
Johnstone [12] is considered that cannot contribute to these turbulent data because
of the Gaussian consideration of the incoherent part.
In this study, we applied different values for thresholding the ratio of instantaneous energy for every IMF mode. We propose the selection of a threshold value in
the region where the number of the detected events are not independent from the
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0.1
0.05
0
–0.05
–0.1
IMF1
0.1
0.05
0
–0.05
–0.1
IMF2
147
IMF3
0.1
0.05
0
–0.05
–0.1
0.1
0.05
0
–0.05
–0.1
IMF4
0.1
0.05
0
–0.05
–0.1
IMF5
0.1
0.05
0
–0.05
–0.1
IMF6
0.1
0.05
0
–0.05
–0.1
IMF7
0.1
0.05
0
–0.05
–0.1
IMF8
0.1
0.05
0
–0.05
–0.1
IMF9
0.1
0.05
0
–0.05
–0.1
IMF10
m/s
0.1
0.05
0
0
Residual
5
10
FIGURE 7.4 (a) IMFs; (b) residual.
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15
20
25
30
35
t(sec)
40
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The Hilbert-Huang Transform in Engineering
a)
E*103(m2/s)
1.0E+00
IMF1
IMF3
IMF5
IMF7
IMF9
ALL
1.1E-00
IMF2
IMF4
IMF6
IMF8
IMF10
1.2E-00
1.3E-00
1.4E-00
1.5E-00
1
b)
1.E+02
10
(E/rms)*103(m2/s)
f(H) 100
IMF1
IMF3
IMF5
IMF7
IMF9
ALL
1.E+01
1.E+00
IMF2
IMF4
IMF6
IMF8
IMF10
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
1
10
f(H) 100
FIGURE 7.5 Frequency distribution of Ci.
threshold. Instead of using arbitrary predefined thresholds, we used relative thresholds (local value as percentage of the maximum, predefining only the existence of
structure). Figure 7.10b presents instantaneous frequencies (that localize the corresponding amplitude squared) for a range of IMFs and for values 2 < IMF < 5, and
Figure 7.10c shows Hilbert-Huang transforms of local intermittency thresholds based
on these relative criteria.
Having defined the points in time and IMF mode where strong events are
identified, and having checked the decomposition process for the quality of reconstruction of the original time series and the degree of orthogonality, it is possible to
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149
60
50
f(Hz)
40
30
20
10
200
400
600
800
1000
1200
1400
1600
1800
2000
N
FIGURE 7.6 ( a,b ) Frequency–time distributions of velocity signal; (c ) energy–frequency–time distribution.
select the corresponding points at every IMF that allow the coherent signal to be
distinguished from the noncoherent signal in the reconstruction formula. As an
example, Figure 7.11 presents the reconstruction of the organized and disorganized
part of the signal applying a hard thresholding method for the Hilbert-Huang transform.
With well-known strategies [14,15], it is possible to average the coherent pattern
[6,7,13,14] and correct the phase jitter (through an iterative process with cross-correlation and a time shift between the ensemble average pattern and each individual
pattern), in order to extract the averaged patterns for every turbulent quantity as
streamwise, vertical velocities, and Re stresses.
Figure 7.12 illustrates the shortcomings of the corresponding wavelets reconstruction in the case of the Daub4 wavelet, where the signature of the wavelet itself
is observed in the reconstruction formula (N = 1024, m = 2, scale reconstruction).
Figure 7.13 presents an IMF analysis (sum from coarser to finer scales Ri) of the
intense events for two regions of the flow field where suction is applied in open
channel flow. A stronger effect of suction is observed near the wall and at the exit
of the suction region.
Because of the indication of organized activity of the non-Gaussian behaviour of
the turbulent data, we propose the analysis of the skewness and kirtosis coefficients,
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The Hilbert-Huang Transform in Engineering
(a)
1
u (t)
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
2
4
6
8
10
(b)
0.5
IE (t)
0.15
0.1
0.05
0
0
FIGURE 7.7 (a) Velocity signal; (b) instantaneous energy distribution.
from the reconstruction formula (starting the composition of the initial signal from
residual and the larger scales). Figure 7.14 a presents the skewness coefficient Sk
for a short signal of N = 8192. The value of the whole signal is –0.23. Figure 7.14b
shows a scale analysis of the same signal with Daub4 wavelet.
Figure 7.15 shows, analysis with distributions for every IMF, of the turbulent
quantities with the same length of signal for comparative reasons. Figure 7.15a
presents the energy content of streamwise component Eiu and the vertical component
Eiv, as percentage of the whole energy resulting from hot film measurements that
lead to conclusions for the principal time scales. Figure 7.15b shows the ratio Eiu/Eiv.
Figure 7.15c gives the percentage of energy for every IMF, from LDA measurements
with and without suction application near the wall at the exit of the suction region.
7.3 CONCLUSIONS
Having established an approach using empirical mode decomposition to study open
channel flow, we must mention the need for a systematic application in different
flow situations. Nonstationary flows, vortex shedding over obstacle, and wake flows
may comprise an ideal domain for application of this new technique. Further documentation of the sifting process is necessary, including the stopping criteria and
extrema interpolation, especially for difficult environments such as turbulent flow
where the appearance of all the frequency bands makes the overall process time
consuming. Applications that demand real-time processing, such as active control,
must be investigated under this new method with concepts such as the modulated
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151
Cq = 0
Cq = 0.056
0.9
0.8
0.7
f
0.6
0.5
0.4
0.3
0.2
0.1
0
0.001
0.01
0.1
1
y/h
FIGURE 7.8 Frequency of events with VITA analysis (y+ = 8, x/L = 1.2).
intra-wave frequency. Nonstationary flows with short time variations would be the
field for comparison of HHT with other older techniques, such as those of entropy
(MEM)–based methods. However, the final general impression remains the ability
of this novel technique to exploit the Hilbert transform and the analytic signal for
multifrequency signals to achieve better localization in the frequency time domain
and avoid the shortcomings of wavelets coefficients, such as the strong smear effect.
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The Hilbert-Huang Transform in Engineering
(a)
700
600
HHTLIM
500
400
300
200
100
0
(b)
100
90
80
70
LIM
60
50
40
30
20
10
0
FIGURE 7.9 Local intermittency measure with IMF and Daub4.
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Coherent Structures Analysis in Turbulent Open Channel Flow
153
(a)
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
t (sec)
f (Hz)
(b)
35
30
25
20
15
10
5
0
IMF2
IMF3
IMF4
IMF5
0
2
4
6
8
10
12
14
16
t (sec)
(c)
HHTLIM IMF2
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
10
12
14
16
10
12
14
16
10
12
14
16
HHTLIM IMF3
t (sec)
10
8
6
4
2
0
0
2
4
6
8
HHTLIM IMF4
t (sec)
10
8
6
4
2
0
0
2
4
6
8
HHTLIM IMF5
t (sec)
10
8
6
4
2
0
0
2
4
6
8
t (sec)
FIGURE 7.10 (a) Velocity signal; (b) instantaneous frequency variations; (c) HHTLIM with
thresholds.
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The Hilbert-Huang Transform in Engineering
(a)
0.2
0.1
0
−0.1
−0.2
(b)
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
FIGURE 7.11 (a) Disorganized part; (b) organized part.
0.4
u (m/s)
0.35
0.3
0.25
0.2
0
0.5
1
1.5
t (sec)
FIGURE 7.12 Reconstruction with Daub4 (N = 1024, m = 2).
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2.5
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Coherent Structures Analysis in Turbulent Open Channel Flow
1.6
155
Y+ = 8, x/L = 1.2
1.4
Y+ = 100, x/L = 0.5
Ns/N0 events
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
IMF
4
5
FIGURE 7.13 Ratio of intense events for suction and no-suction cases.
1
Sk
1
a)
Sk
b)
0.5
0.5
Ri
0
1
2
3
4
5
6
7
8
m
0
1
9 10 11
-0.5
-0.5
-1
-1
2
3
4
5
6
7
8
9 10 11 12
FIGURE 7.14 (a) Sk distributions with residuals; (b) Sk distribution with scales Daub4.
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The Hilbert-Huang Transform in Engineering
30
0.2
Ei/Eiall
u
v
0.16
20
0.12
15
0.08
10
0.04
IMF
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.1 Ei/Eall
no suction
0.08
suction
Eiu/Eiv
25
5
IMF
0
1 2 3 4 5 6
7 8 9 10 11 12 13
0.06
0.04
0.02
IMF
0
1 2 3 4 5 6 7 8 9 10 11 12 13
FIGURE 7.15 (a) Percentage contribution of Eiu and Eiv; (b) ratio Eiu/Eiv; (c) effect of
suction near the wall (y+ = 10).
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REFERENCES
1. Huang N., Shen Z., Long S. (1999). A new view of nonlinear water waves: The
Hilbert Spectrum. Annu. Rev. Fluid Mech. 31: 417–457.
2. Huang N. E., Shen Z., Long S., Wu M., Shih H., Zheng Q. et al. (1998). The Empirical
Mode Decomposition and the Hilbert Spectrum for non linear and non stationary
time series analysis. Proc. R. Soc. London A 454, 903–995.
3. Zeris A. (2001). Turbulent flow in open channel with suction from the bed. Ph.D.
thesis, Aristotle University, Thessaloniki, Greece, 2001.
4. Zeris A., Prinos P. (2001). Measurements of bed suction effects on an open channel
flow with Laser–Doppler/Hot-Film Anemometry. EFHT Congr., pp. 1099–1103.
5. Zeris A., Prinos P. Open channel flow with suction from the bed. J. Hydraulic Eng.,
submitted 2004.
6. Blackwelder R., Haritonidis J. Scaling of the bursting frequency in turbulent boundary
layers. J. Fluid Mech. Vol. 132, 87–103, 1983.
7. Bogard, D. G., Tiederman W. G. (1986). Burst detection with single point velocimetry
measurements. J. Fluid Mech. Vol. 162, 389–413.
8. Lu, S. S., Wilmarth, W. W. (1973). Measurements of structure of Reynolds stress in
a turbulent boundary layer. J. Fluid Mech. Vol. 60, 481–511.
9. Adrian R. J. On the role of conditional averages in turbulence theory. In Turbulence
in Liquids, Zakin and Patterson, Eds., Princeton Science Press, 1977.
10. Wallace, J. M., Eckelmann, H., Brodkey, R. S. (1972). The wall region in turbulent
shear flow. J. Fluid Mech. Vol. 54, 39–48.
11. Farge M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev.
Fluid Mech. Vol. 24, 395–497.
12. Donoho D., Johnstone I. M. (1994). Ideal spatial adaptation by wavelet shrinkage.
Biometrika, Vol. 81, 425–455.
13. Camussi R., Gui G. (1997). Orthonormal wavelet decomposition of turbulent flows:
intermittency and coherent structure. J. Fluid Mech. Vol. 348, 177–199.
14. Raupach, M. (1981). Conditional statistics of Reynolds stress in rough wall and
smooth wall turbulent flow. J. Fluid Mech. Vol. 108, 363–382.
15. Subramanian C. S., Rajagopalan S., Antonia R., Chambers A. (1982). Comparison
of conditional sampling and averaging techniques in a turbulent boundary layer. J.
Fluid Mech. Vol. 23, 335–362.
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8
An HHT-Based Approach
to Quantify Nonlinear
Soil Amplification and
Damping
Ray Ruichong Zhang
CONTENTS
8.1
8.2
8.3
8.4
8.5
Introduction ..................................................................................................160
Symptoms of Soil Nonlinearity ...................................................................161
Fourier-Based Approach for Characterizing Nonlinearity ..........................162
HHT-Based Approach for Characterizing Nonlinearity ..............................166
Applications to 2001 Nisqually Earthquake Data.......................................172
8.5.1 Detection of Nonlinear Soil Sites....................................................173
8.5.2 HHT-Based Factor of Site Amplification ........................................176
8.5.3 Influences of Window Length of Data ............................................178
8.5.4 Comparison of HHT- and Fourier-Based Factors for Site
Amplification....................................................................................180
8.5.5 HHT-Based Factor for Site Damping ..............................................184
8.6 Concluding Remarks and Discussion ..........................................................185
Acknowledgments..................................................................................................187
References..............................................................................................................187
ABSTRACT
This study proposes to use a method of nonlinear, nonstationary data processing and
analysis, i.e., the Hilbert-Huang transform (HHT), to quantify influences of soil nonlinearity in earthquake recordings. The paper first summarizes symptoms of soil nonlinearity shown in earthquake ground motion recordings. It also reviews the Fourier-based approach to characterizing the nonlinearity in the recordings and
demonstrates the deficiencies. It then offers the justifications of the HHT in addressing
the nonlinearity issues. With the use of the 2001 Nisqually earthquake recordings and
results of the Fourier-based approach as a reference, this study shows that the
159
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The Hilbert-Huang Transform in Engineering
HHT-based approach is effective in characterizing soil nonlinearity and quantifying
the influences of nonlinearity in seismic site amplification. Primary results are as
follows:
The first HHT-based component and the Hilbert amplitude spectra can identify
abnormal high-frequency spikes in the recording at sites where strong soil
nonlinearity occurs; this can help to detect the nonlinear sites at a glance.
The HHT-based factor for site amplification is defined as the ratio of marginal
Hilbert amplitude spectra, similar to the Fourier-based one that is the ratio of
Fourier amplitude spectra. The HHT-based factor is effective in quantifying soil
nonlinearity in terms of frequency downshift in the low-frequency range and
amplitude downshift in the intermediate-frequency range.
Hilbert and marginal damping spectra are identified in ways similar to Hilbert
and marginal amplitude spectra. Consequently, the HHT-based factor for site
damping is found as the difference of marginal Hilbert damping spectra, which
can be extracted from the HHT-based factor for site amplification and used as
an alternative index to measure the influences of soil nonlinearity in seismic
ground responses.
8.1 INTRODUCTION
Site amplification is the phenomenon in which the amplitude of seismic waves
increases significantly when they pass through soil layers near the earth’s surface.
It can be illustrated by considering the seismic energy flux along a tube of seismic
rays, which is proportional to the impedance (density × wave speed) and squared
shaking velocity. Since the energy should be constant in the absence of damping,
any reduction in the impedance is compensated by an increase in the shaking velocity,
thus yielding site amplification for seismic waves in soil layers. Site amplification
is a key factor in mapping seismic hazard in urban areas (e.g., [1]) and designing
geotechnical and structural engineering systems on soils (e.g., [2]).
In general, site amplification is not linearly proportional to the intensity of input
seismic motion at bedrock because of soil nonlinearity under large-amplitude earthquakes. The extent of soil nonlinearity can be characterized by the change of two
dynamic features of soil layer, i.e., soil resonant frequency and damping, in the
frequency-dependent site amplification. Consensus has been building that the
site-amplification factors in the current codes overemphasize the extent of soil
nonlinearity and thus potentially underestimate the level of site amplification. It has
been demonstrated [3] that the recording-based amplification factors are larger than
those in codes for a certain range of base acceleration intensity. In addition, some
features of site-amplification factors used in codes and guidance for structural design
contradict recent findings from the 1994 Northridge ground motion data set [4].
The aforementioned problem might exist partly because seismologists and engineers lack sufficient understanding of the underlying causes in nonlinear soil. For
example, the influence of soil heterogeneity does not scale linearly, even when the
soil is perfectly linear [5]. In other words, a linear elastic medium with random
heterogeneity can change ground motion in a way similar to that caused by medium
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161
nonlinearity. Consequently, it is possible to interpret the motion influenced by random heterogeneous media as soil nonlinearity (e.g., in the form of damping [6]),
which can distort the quantification of site amplification. The problem may also exist
partly because there is lack of an effective approach to properly characterize nonlinear features of ground motion in recordings and then to quantify them.
The objective of this study is to propose the use of a method for nonlinear,
nonstationary data processing [7], referred to as the Hilbert-Huang transform (HHT)
method, to characterize the soil nonlinearity from earthquake recordings. In particular, this paper first summarizes symptoms of soil nonlinearity shown in earthquake
ground motion recordings from previous studies and literature. It will also review
Fourier-based approaches to characterizing nonlinearity from earthquake motion and
demonstrate their deficiencies. It then proposes to use the HHT method for characterizing soil nonlinearity in the motion. For illustration, the study analyzes a hypothetical recording to show the HHT-based characterization of nonlinearity. Finally,
it examines the mainshock and aftershock of the 2001 Nisqually earthquake recordings to demonstrate the validity and effectiveness of the proposed approach for
characterizing soil nonlinearity and the influences.
8.2 SYMPTOMS OF SOIL NONLINEARITY
In general, the stress–strain relationship of a soil becomes nonlinear and hysteretic
for a large-amplitude input excitation. Such nonlinearity and hysteresis correspond
to a reduction of soil strength, increased soil damping, and deformed waveform of
response in comparison with those of the linear case (e.g., [8–12]).
With a reduction of soil strength such as shear modulus (G) for nonlinear soil,
the shear-wave velocity (v = (G/ρ)½, where ρ is the soil density) and thus the
fundamental resonant frequency (f = v/4h) of the soil layer with thickness h decrease.
Therefore, seismic motion recordings over a nonlinear soil layer could show strong
wave response at a lower resonant frequency than for the same layer with linear
excitation. Accordingly, increased site amplification at the downshifted soil resonant
frequency can be regarded as a signature of soil nonlinearity observable in ground
motion records (e.g., [13, 14]).
On the other hand, increased damping for nonlinear soil will decrease ground
motion, thus moderating the site amplification. Since soil damping is typically
frequency dependent, so is the change of damping for nonlinear soil. The increased
damping of nonlinear soil is likely lower at higher frequencies [15]. The increased
site amplification at the downshifted soil resonant frequency and the frequency-dependent increased damping imply that the change in site amplification
due to soil nonlinearity should be strongly dependent upon frequency.
Soil nonlinearity can also sometimes be inferred from abnormal high-frequency
spikes and recorded waveforms that appear only in recordings that are close to the
locations where strong nonlinearity occurred (e.g., [16]). The cusped waveforms and
high-frequency spikes observed in the 2001 Nisqually earthquake recordings, for
example, are symptomatic of a nonlinear response at some soil sites (e.g., [17]).
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The Hilbert-Huang Transform in Engineering
8.3 FOURIER-BASED APPROACH FOR CHARACTERIZING
NONLINEARITY
In practice, the Fourier series expansion is frequently used for representing and
analyzing recorded digital data of earthquake acceleration X(t), i.e.,
N
X (t ) = ℜ
∑
N
Aje
iΩ j t
=ℜ
j =1
∑[ A sin(Ω t) + iA cos(Ω t)] ,
j
j
j
j
(8.1)
j =1
where ℜ denotes the real part of the value to be calculated, i = (–1)∫ is an imaginary
unit, amplitudes Aj are a function of time-independent frequency Ωj that is defined
over the window in which the data is analyzed, and the Fourier amplitude spectrum
is defined as
N
F (Ω) =
∑A .
(8.2)
j
j =1
To apply this Fourier spectral analysis to estimate the influences of soil nonlinearity
in the seismic wave responses at soil site or simply site amplification, two sets of
recordings are typically needed [18], one at a soil site and the other at a referenced
site such as bedrock or outcrop. For a frequency, the Fourier-based factor of site
amplification (FF) for an earthquake event (either mainshock or aftershock) can then
be found by
( )
FFs Ω =
Fs2,h1 + Fs2,h 2
Fr2,h1 + Fr2,h 2
,
(8.3)
where subscripts s and r denote respectively the soil and referenced sites, and
subscripts h1 and h2 denote the two horizontal components. Note that Equation 8.3
is one of many representatives for site-amplification factor that can be the ratio of
characteristics of seismic waves or spectral responses at a site versus referenced site.
Since the wave paths and earth structures except the soil layer are almost the
same for the soil and referenced sites, the factor for site amplification in Equation
8.3 eliminates approximately the influences of source from the earthquake event and
thus provides essentially the dynamic characteristics of the soil. In addition, the
recordings at the referenced site are generally believed to be the results of linear
wave responses and the recordings at the soil site subject to the large-amplitude
mainshock to be the results of nonlinear wave responses. Accordingly, comparing
the factors from the mainshock and the aftershock could help us explore and quantify
the influences of soil nonlinearity in site amplification.
While the Fourier-based approach given here and similar methods are widely
used, they have the following deficiencies in characterizing the nonstationarity of
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An HHT-Based Approach to Quantify Nonlinear Soil Amplification
163
the earthquake motion that is caused by source, different types of propagating waves,
and soil nonlinearity if the earthquake magnitude is large enough.
A Fourier-based approach defines harmonic components globally and thus yields
average characteristics over the entire duration of the data. However, some characteristics of data, such as the downshift of soil resonant frequency at a nonlinear site,
may occur only over a short portion of a record. This is particularly true when the
intensity of the seismic input to a soil layer is not strong, such that the soil becomes
nonlinear over only a portion of the entire duration of motion and in only a certain
frequency band. As a result, the averaging characteristic in Fourier spectral analysis
makes it insensitive for identifying time-dependent frequency content. While a
windowed (or short-time) Fourier-based approach can be used to improve the above
analysis to a certain extent, it also reduces frequency resolution as the length of the
window shortens. Thus, one is faced with a tradeoff. The shorter the window, the
better the temporal localization of the Fourier amplitude spectrum, but the poorer
the frequency resolution, which directly influences the measurement of downshift
of soil resonance that typically arises in a low to intermediate frequency band.
More important, a Fourier-based approach explains data in terms of a linear
superposition of harmonic functions. Therefore, it is an appropriate, effective method
for characterizing linear phenomena such as waves with time-independent frequency,
rather than nonlinear phenomena with time-dependent frequency. An example of
time-independent and time-dependent frequency waves is a hypothetical wave record
y(t) = y1(t) + y2(t), where decaying waves y1(t) = cos[2πt + εsin(2πt)]e–0.2t have
time-dependent frequency of 1 + εcos(2πt) Hz, with ε denoting a constant factor,
and noise y2(t) = 0.05sin(30t) has time-independent frequency of 15 Hz. Note that
the waves shown in Figure 8.1 with ε = 0.5 are physically related to one type of
1.5
1
Amplitude
0.5
0
−0.5
−1
−1.5
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
FIGURE 8.1 A hypothetical wave recording, consisting of nonlinear waves and noise with
frequencies 1 + 0.5cos(2πt) Hz and 15 Hz, respectively.
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The Hilbert-Huang Transform in Engineering
FIGURE 8.2 Fourier and marginal Hilbert amplitude spectra of the recording in Figure 8.1.
water waves that result from a nonlinear dynamic process and are also representative
of seismic responses at a nonlinear soil site (to be elaborated).
The time-dependent frequency waves can be expanded into and thus interpreted
by a series of time-independent frequency waves, as done by the Fourier spectral
analysis in which y(t), or y1(t) in particular, can be interpreted as to contain Fourier
components at all frequencies (see Equation 8.1, Figure 8.2, and Figure 8.3).
Alternatively, the expansion of y1(t), i.e.,
y1 (t ) ≈ [−0.5ε + cos(2πt ) + 0.5ε cos(4 πt )]e −0.2t , for
ε << 1
(8.4)
suggests that the Fourier transform of y1(t) consists primarily of two harmonic
functions centered respectively at 1 Hz and 2 Hz for ε << 1, and the widths of these
harmonic functions are proportional to the exponential parameter 0.2, which is
related to the damping factor. Note that Figure 8.1 and Figure 8.2 use ε = 0.5, which
is not a small number in comparison to unity, and thus they have the third observable
harmonic function at 3 Hz in Figure 8.2. Therefore, one can equally well describe
y1(t) by saying that it consists of just two frequency components, each component
having a time-varying amplitude that is proportional to e–0.2t. Indeed, if one were to
examine the local behavior of y1(t) in the neighborhood of a given time instant, say
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165
FIGURE 8.3 Fourier components (fj, j = 1, 2, 3, 4, 5) of the recording in Figure 8.1 at selected
frequencies (i.e., 10 Hz, 5 Hz, 2 Hz, 1 Hz, and 0.5 Hz).
t0, this is precisely what one would observe. The Fourier-based analysis or interpretation given here can also be seen in Priestley [19] and Zhang et al. [20], among
others.
Because the true frequency content of the waves y1(t) is bounded between 1 –
ε and 1 + ε, much less than 2 Hz, analysis of the above example suggests that Fourier
spectral analysis typically needs higher-frequency harmonics (at least 2 Hz for the
example) to simulate the nonlinear waveform of the data. Stated differently, Fourier
spectral analysis distorts the nonlinear data. Consequently, the Fourier-based
approach in Equation 8.3 twists the influences of soil nonlinearity in site amplification. The above assertions are confirmed in Huang et al. [21] and Worden and
Tomlinson [22], among others, with the aid of solutions to classic nonlinear systems
such as the Duffing equation in general, and in Zhang et al. [23] with the nonlinear
site amplification in particular.
In theory, Fourier spectral analysis in general and Fourier-based approaches for
site amplification in particular can be further used for evaluating damping factor.
For example, the resonant amplification method or half-power method uses the
amplitude change or width of the peaks at a certain frequency in the Fourier amplitude spectrum to find the damping factor of dynamic systems such as a soil layer
(e.g., [24]). However, the distorted Fourier amplitude spectrum for nonlinear data
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The Hilbert-Huang Transform in Engineering
will mislead the subsequent use for damping evaluation with nonlinear soil. For
example, the damping factor evaluated at the first and second peaks in the Fourier
amplitude spectrum in Figure 8.2 suggests that the damping is associated with
frequency at 1 Hz and 2 Hz. In fact, the damping of the hypothetical record is
dependent only on the true frequency content of the waves y1(t) bounded between
1 – ε and 1 + ε, or 0.5 Hz and 1.5 Hz with ε = 0.5 in Figure 8.1. Accordingly, a
Fourier-based approach would misrepresent the influences of damping factor as it
relates to soil nonlinearity.
8.4 HHT-BASED APPROACH FOR CHARACTERIZING
NONLINEARITY
A method for nonlinear, nonstationary data processing [7] can be used as an alternative to the Fourier-based approach for characterizing soil nonlinearity. This
method, referred to as the Hilbert-Huang transform, consists of empirical mode
decomposition (EMD) and Hilbert spectral analysis (HSA). Any complicated time
domain record can be decomposed via EMD into a finite, often small, number of
intrinsic mode functions (IMFs) that admit a well-behaved Hilbert transform. The
IMF is defined by the following conditions: (1) over the entire time series, the number
of extrema and the number of zero-crossings must be equal or differ at most by one,
and (2) the mean value of the envelope defined by the local maxima and the envelope
defined by the local minima is zero at any point. An IMF represents a simple
oscillatory mode similar to a component in the Fourier-based harmonic function,
but more general.
The EMD explores temporal variation in the characteristic time scale of the data
and thus is adaptive to nonlinear, nonstationary data processes. The HSA defines an
instantaneous or time-dependent frequency of the data via Hilbert transformation of
each IMF component. These two unique features endow the HHT with a possibly
enhanced interpretive value, making it a useful alternative to Fourier components
and amplitude spectra.
The HHT representation of data X(t) is
n
n
X (t ) = ℜ
∑ a (t)e
j
iθ j ( t )
=ℜ
j =1
∑[C (t) + iY (t)] ,
j
j
(8.5)
j =1
where Cj(t) and Yj(t) are respectively the jth IMF component of X(t) and its Hilbert
transform,
Yj (t ) =
C j (t ′)
1
P
dt ′ ,
t − t′
π
∫
where P denotes the Cauchy principal value, and the time-dependent amplitudes
aj(t) and phases θj(t) are the polar-coordinate expression of Cartesian-coordinate
expression of Cj(t) and Yj(t), from which the instantaneous frequency is defined as
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An HHT-Based Approach to Quantify Nonlinear Soil Amplification
ω j (t ) =
d θ j (t )
.
dt
167
(8.6)
It should be emphasized here that the instantaneous frequency has physical
meaning only through its definition on each IMF component as in Equation 8.5 and
Equation 8.6; by contrast, the instantaneous frequency defined through the Hilbert
transformation of the original data is generally less directly related to frequency
content [7].
Equation 8.5 and Equation 8.6 indicate that the amplitudes aj(t) are associated
with ωj(t) at time t, or, in general, functions of ω and t. Similar to the Fourier
amplitude spectrum, the Hilbert amplitude spectrum is defined as
n
H (ω , t ) =
∑ a (t) ,
j
(8.7)
j =1
and its square gives the temporal evolution of the energy distribution. The marginal
Hilbert amplitude spectrum, h(ω), defined as
T
h(ω ) =
∫ H (ω, t)dt ,
(8.8)
0
provides a measure of the total amplitude or energy contribution from each frequency
value, in which T denotes the time duration of the data.
In comparison with Equation 8.2, the Hilbert amplitude spectrum H(ω,t) provides
an extra dimension by including time t in motion frequency and is thus more general
than the Fourier amplitude spectrum F(Ω). While the marginal amplitude spectrum
h(ω) provides information similar to the Fourier amplitude spectrum, its frequency
term is different. The Fourier-based frequency (Ω) is constant over the sinusoidal
harmonics persisting through the data window, as seen in Equation 8.1, while the
HHT-based frequency ω varies with time based on Equation 8.6. As the Fourier
transformation window length reduces to zero, the Fourier-based frequency (Ω)
approaches the HHT-based frequency (ω). The Fourier-based frequency is locally
averaged and not truly instantaneous, for it depends on window length, which is
controlled by the uncertainty principle and the sampling rate of data.
Recordings that are stationary and linear can typically be decomposed or represented by a series of time-independent frequency waves through the Fourier-based
approach in Equation 8.1. If the jth IMF component, i.e., Cj(t) in Equation 8.5,
corresponds to a Fourier component with a sine function at a time-independent
frequency, the Hilbert transform of the sine function, i.e., Yj(t) in Equation 8.5, can
be found to equal the cosine function at the same frequency in opposite sign. Because
the sign can be changed with adding a constant phase, the above analysis essentially
leads to the consistence between Fourier- and HHT-based approaches in general,
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The Hilbert-Huang Transform in Engineering
and Fourier and marginal Hilbert amplitude spectra in particular, in characterizing
linear phenomena with time-independent frequency.
For recordings that are nonstationary and nonlinear, such as large-magnitude
earthquakes, many studies have showed [21] that the marginal Hilbert amplitude
spectra can truthfully represent the nonlinear data in comparison with Fourier amplitude spectra. As a result, an HHT-based approach for characterizing nonlinear site
amplification can be proposed that is similar to a Fourier-based approach. The
HHT-based factor of site amplification (FH) is defined
( )
FH s ω =
hs2,h1 + hs2,h 2
hr2,h1 + hr2,h 2
.
(8.9)
While this HHT-based factor could provide an alternative insight in characterizing and quantifying the influences of nonlinear soil in site amplification, the role
of damping in nonlinear site responses is not discerned from the general features of
soil nonlinearity, which should be implicitly involved in the factor.
To single out the damping factor from the HHT-based factor in Equation 8.9,
which is associated with amplitude aj(t) in Equation 8.7 and Equation 8.8, the
physical meanings of the jth IMF component that forms the amplitude aj(t) in
Equation 8.5 is first examined below. Since all the IMF components are extracted
from acceleration records that are the result of seismic waves generated by a seismic
source and propagating in the earth, they should reflect the wave characteristics
inherent to the rupture process and the earth medium properties. Indeed, with the
aid of a finite-fault inversion method, Zhang et al. [25] examined signatures of the
seismic source of the 1994 Northridge earthquake in the ground acceleration recordings. That study looks over only the second to fifth IMF components because they
are much larger in amplitude than the remaining higher-order, low-frequency IMF
components. The first IMF component was not investigated in that study because it
contains information that is not simply or easily related to the seismic source (e.g.,
wave scattering in the heterogeneous media). That study shows that the second IMF
component is predominantly wave motion generated near the hypocenter, with
high-frequency content that might be related to a large stress drop associated with
the initiation of the earthquake. As one progresses from the second to the fifth IMF
component, there is a general migration of the source region away from the hypocenter with associated longer-period signals as the rupture propagates. In addition,
that study shows that some IMF components (e.g., the fifth IMF) can exhibit motion
features reflecting the influences of nonlinear site condition.
Because of the significance of IMF components that directly relate amplitudes
aj(t), Equation 8.5 can be rewritten as
n
n
X (t ) = ℜ
∑
j =1
© 2005 by Taylor & Francis Group, LLC
iθ ( t )
a j (t )e j
=ℜ
∑ Λ (t)e
j
j =1
− ϕ j ( t )+ iθ j ( t )
(8.10)
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An HHT-Based Approach to Quantify Nonlinear Soil Amplification
169
where time-dependent amplitudes Λ j(t) can be interpreted as the source-related
intensity, ϕj(t) are the exponential factors characterizing the time-dependent decay
of the waves in the jth IMF component due to damping, and
a j (t ) = Λ j (t )e
− ϕ j (t )
.
(8.11)
Similar to the description of the instantaneous frequency in Equation 8.6, the
instantaneous damping factor can be defined as
η j (t ) =
d ϕ j (t )
.
dt
(8.12)
With the aid of Equation 8.11, the Hilbert damping spectrum can be found as
n
D(ω , t ) =
∑
η j (t ) =
j =1
n
 a j (t )
(t ) 
Λ
j
j =1
j
j
∑ − a (t) + Λ (t)  .
(8.13)
The marginal Hilbert damping spectrum is
T
d (ω ) =
∫
(t ) 
 a j (t ) Λ
+ j  dt = d a (ω ) + d Λ (ω ) .
−
 a j (t ) Λ j (t ) 
j =1 0 
n
D(ω , t )dt =
0
T
∑∫
(8.14)
Equation 8.14 indicates that the marginal Hilbert damping spectrum consists of
two terms: one is from the time-dependent amplitudes aj(t) that are related to
marginal and Hilbert amplitude spectra, and the other is from source-related intensity,
i.e., time-dependent amplitudes Λ j(t).
It is of interest to note that the definition of instantaneous damping factor in
Equation 8.12 and subsequent spectra in Equation 8.13 and Equation 8.14 are
different from those in Salvino [26] and Loh et al. [27]. For recordings of
impulse-induced or ambient linear vibration responses, some IMF components can
be extracted from the data that are related to certain vibration modes [28–30].
Consequently, Λj(t) are constant and ηj(t) are proportional to the damping ratio and
damped frequency. The modal damping ratio can then be found. This is essentially
the same as those in Salvino [26] and Loh et al. [27], if the latter can judicially
relate the IMF components to the vibration/wave modes.
For recordings to an earthquake, Λj are functions of time and unknown, which
are dependent upon the seismic source. Their influences in the site amplification,
however, can be removed if two recordings at soil and referenced sites are used.
Similar to the HHT-based factor of site amplification, the difference of marginal
Hilbert damping spectra at soil and referenced sites, or HHT-based factor of site
damping, could approximately eliminate the influences of the source that is associated
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170
The Hilbert-Huang Transform in Engineering
c1
0.5
0
c2
−0.5
0.5
0
−0.5
c3
0.02
0
−0.02
c4
0.02
0
c5
−0.02
×10
−3
10
5
0
0
1
2
3
4
5
Time (sec)
6
7
8
9
10
FIGURE 8.4 The five IMF components of the recording in Figure 8.1.
with Λj and thus provide essentially the characterization of the damping in the soil
site. The HHT-based factor of site damping can be found as
d ∆ (ω ) = [d s ,h1 (ω ) − dr ,h1 (ω )]2 + [d s ,h 2 (ω ) − dr ,h 2 (ω )]2
(8.15)
≈ [d
a
s ,h1
(ω ) − d
a
r ,h1
(ω )] + [d
2
a
s ,h 2
(ω ) − d
a
r ,h 2
(ω )]
2
where use has been made in the last approximation of the fact that the source-related
damping terms at the soil and referenced sites are approximately equal, i.e., dsΛ (ω)
drΛ (ω).
Finally, comparing the HHT-based factors of site damping from the mainshock
and the aftershock could help quantify the influences of nonlinear soil damping in
site responses.
To illustrate the HHT-based characterization of nonlinearity, the hypothetical
record in Figure 8.1 is analyzed again. Figure 8.4 shows the five IMF components
decomposed from the data by EMD. The first and second components (c1 and c2)
capture the noise and primary waveform, while the other three (c3 to c5), with
negligible amplitudes, represent the numerical error in the EMD process. Comparing
Figure 8.3 and Figure 8.4 suggests that some IMF components can be not only more
physically meaningful than the Fourier components, they can also be in principle
used to explore the damping factor with the use of, e.g., the consecutive peak values
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171
Hilbert Spectrum (Hypothetical Water Wave)
Frequency (Hz)
20
18
0.9
16
0.8
14
0.7
12
0.6
10
0.5
8
0.4
6
0.3
4
0.2
2
0.1
0
0
1
2
3
4
5
6
Time (sec)
7
8
9
10
FIGURE 8.5 Hilbert amplitude spectra of the recording in Figure 8.1.
and corresponding time elapse in the second IMF component, if that component is
related to a free-vibration response or forced response at a certain mode.
The Hilbert amplitude spectrum in Figure 8.5 shows a clear picture of temporal–frequency energy distribution of the data, i.e., primary waves with frequency
dependence modulated around 1 Hz and bounded by 0.5 Hz and 1.5 Hz, noise at
15 Hz, and the decaying energy of the primary waves with the color changing from
the red/yellow at the beginning to the dark blue at the end of the record. In contrast,
the Fourier amplitude spectrum in Figure 8.2 not only loses the information pertaining to temporal characteristics of the motion, but, more important, it also distorts
the information of the record by introducing higher-order harmonics, notably at 2
Hz and 3 Hz. For comparison, the marginal amplitude spectrum of the recording is
also plotted in Figure 8.2, showing truthfully the energy distribution of the motion
in frequency.
Figure 8.6 shows the marginal Hilbert damping spectrum, calculated without
using any smooth function, which was not so done in the calculation of the above
marginal Hilbert amplitude spectrum with the use of Hilbert-Huang Transformation
Toolbox [31]. While oscillation of the curve in Figure 8.6 is originally due to the
numerical calculation of a· j(t) and a· j(t)/aj(t) that subsequently influences the computation of the spectra in Equation 8.13 and Equation 8.14, the estimated mean damping
factor at frequency 0.5 to 1 Hz is around the true value of 0.2. Compared with no
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The Hilbert-Huang Transform in Engineering
Marginal Damping Spectrum
Wave Data–Marginal Damping Spectrum
100
10−1
10−2
100
10
1
Frequency (Hz)
FIGURE 8.6 Marginal Hilbert damping spectrum of the recording in Figure 8.1.
damping in the record at frequency larger than 1.5 Hz, the very small mean damping
factor in Figure 8.6 (about 0.04, lower at a factor of five than the damping factor of
waves) due to the aforementioned unavoidable, cumulated numerical error is still
acceptable. It is believed that the oscillated curve in Figure 8.6 can be improved by
replacing it with the instantaneous mean curve. The mean curve can be found by
many methods, one of which is the use of the summation of all the IMF components,
excluding the first IMF component, that are extracted from the data of the oscillated
curve with one sifting process. The large damping factors at around 1.5 Hz are due
to the numerical error caused by the transition of two damping factors from 0.2 to
0.04, although such abrupt damping change is unlikely in practice. The large damping factor below 0.5 Hz is caused by the high-order, low-frequency IMF components
(primarily from the third to fifth IMFs). The error in overestimating damping factor
at very low frequency can be theoretically minimized if high-order, low-frequency
IMF components with very small amplitudes are judicially not used in the damping
calculation.
8.5 APPLICATIONS TO 2001 NISQUALLY EARTHQUAKE DATA
In this section, the HHT-based approach is used to analyze the recordings of the
M6.8 mainshock and the ML3.4 aftershock of the 2001 Nisqually earthquake at SDS
and LAP. The SDS is a station on artificial fill with nearby liquefaction, and the
average shear-wave velocity in the top 30 m for the soft soil is estimated as Vs30 =
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173
Mainshock Time History Sds1
Acceleration (g)
0.3
0.2
0.1
0
−0.1
−0.2
0
5
10
15
20
25
30
35
40
30
35
40
1st Component Sds1
Acceleration (g)
0.1
0.05
0
−0.05
−0.1
0
5
10
15
20
Time (sec)
25
FIGURE 8.7 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisqually
mainshock at SDS (soft soil).
148 m/s. The LAP is located over a stiff soil with Vs30 = 367 m/s. To examine the
soil nonlinearity from recordings at the two stations, recordings at SEW are used
as referenced ones, because Vs30 = 433 m/s at SEW is within the range of Vs30
values for typical rock sites in the western United States. Previous Fourier-based
studies (e.g., [17]) suggest that SDS experienced strong soil nonlinearity during the
mainshock while the LAP did not, with recordings at SEW as a reference.
8.5.1 DETECTION OF NONLINEAR SOIL SITES
Figure 8.7a shows the NS-acceleration recording of the 2001 Nisqually earthquake
at SDS. The one-sided cusped waves and high-frequency spikes in the S-coda waves,
notably between 20 sec and 25 sec, are similar to the waveform with sharper crests
and rounded-off troughs in Figure 8.1, and have been concluded [17] to be symptomatic of nonlinear response at the soft soil site.
Indeed, the high-frequency spikes are believed to appear only in recordings that
are close to the locations where strong nonlinearity occurred (e.g., [16]). Therefore,
singling out the spikes will help detect strongly nonlinear sites. Frankel et al. [17]
used Fourier-based high-frequency band-pass filtering (10 to 20 Hz) to identify the
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The Hilbert-Huang Transform in Engineering
Mainshock Time History Sds2
Acceleration (g)
0.4
0.2
0
−0.2
−0.4
0
5
10
15
20
25
30
35
40
30
35
40
1st Component Sds1
0.06
Acceleration (g)
0.04
0.02
0
−0.02
−0.04
−0.06
0
5
10
15
20
Time (sec)
25
FIGURE 8.8 (a, top) EW-acceleration recording and (b, bottom) its first IMF of the Nisqually
mainshock at SDS (soft soil).
spikes. It should be noted that there exist other tools to identify the spikes. For
example, Hou et al. [32] used a wavelet-based approach to characterize the spikes
from nonlinear vibration recordings in the vicinity of a damaged-structure location
subject to a severe earthquake. The disadvantages of these approaches, however,
reside with the subjective selection of frequency band in a Fourier-based approach
and subjective selection of another wavelet in a wavelet-based approach, among
others.
This study presents the effectiveness of the HHT-based approach in identifying
the spikes. Figure 8.7b depicts the first IMF component of the NS-component of
motion, clearly showing the two largest spikes between 20 sec and 25 sec. While
the spikes in the recording of Figure 8.7a can be visualized without using any tools,
the first IMF component in Figure 8.7b is shown simply for validation of the
HHT-based approach in identifying the spikes. Indeed, the EW-component of the
same recording in Figure 8.8a does not clearly show the spikes between 20 sec and
25 sec. The corresponding first IMF component in Figure 8.8b is, however, able to
reveal them explicitly, suggesting that the first IMF component is effective at detecting the high-frequency spikes. This study also analyzed the horizontal recordings
of the ML3.4 aftershock at the same location in Figure 8.9 and Figure 8.10, in which
one does not observe the spikes in the S-coda waves in the corresponding first IMF
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An HHT-Based Approach to Quantify Nonlinear Soil Amplification
Acceleration (g)
1
×10−3
175
Aftershock Time History Sds1
0.5
0
−0.5
−1
Acceleration (g)
2
0
5
10
×10−4
15
20
25
30
35
40
30
35
40
1st Component Sds1
1
0
−1
−2
0
5
10
15
20
Time (sec)
25
FIGURE 8.9 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisqually
aftershock at SDS (soft soil).
components. These observations partially confirm that the nonlinearity-related spikes
can be detected via the first IMF component.
It should be pointed out here that the high-frequency spikes before the S-coda
waves in the first IMF component (such as those before 20 sec in Figure 8.7b, Figure
8.8b, Figure 8.9b, and Figure 8.10b), which also show up in the Fourier-based
high-frequency band-pass (10 to 20 Hz) approach, may not be simply, directly related
to the soil nonlinearity.
The aforementioned nonlinearity-related spikes can also be identified via Hilbert
amplitude spectra. Specifically, the Hilbert amplitude spectra of the mainshock in
Figure 8.11 and Figure 8.12 show two beams of high-frequency energy (5 Hz and
above) between 20 sec and 25 sec, when the dominant energy of the motion is
primarily below 5 Hz. In contrast, those of the aftershock in Figure 8.13 and Figure
8.14 do not show the distinguishing high-frequency energy beams in the same time
period. This suggests that the first IMF component and Hilbert amplitude spectra
can be used effectively to single out nonlinearity-related high-frequency spikes.
Accordingly, the results will help us, at the first glance, to detect strong nonlinear
soil sites.
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The Hilbert-Huang Transform in Engineering
Acceleration (g)
1
×10−3
Aftershock Time History Sds2
0.5
0
−0.5
−1
8
0
5
10
×10−4
15
20
25
30
35
40
30
35
40
1st Component Sds2
Acceleration (g)
6
4
2
0
−2
−4
0
5
10
15
20
Time (sec)
25
FIGURE 8.10 (a, top) EW-acceleration recording and (b, bottom) its first IMF of the
Nisqually aftershock at SDS (soft soil).
8.5.2 HHT-BASED FACTOR OF SITE AMPLIFICATION
Figure 8.15a shows the HHT-based factors of site amplification of the mainshock
and aftershock at SDS. In calculating the factors in Equation 8.9, the correction for
1/R geometrical spreading in the recordings at SDS and SEW is not carried out since
the hypocentral distances for the sites under investigation are similar. In addition,
the marginal Hilbert amplitude spectra are not smoothed in the calculation, for the
nonsmoothed spectra more clearly show the characteristics of the HHT-based
approach. Examining Figure 8.15a shows the following:
•
The profile of the HHT-based factor in the frequency band up to 2.5 Hz
(referred to as the low-frequency range) is generally downshifted in frequency from the aftershock to the mainshock, with an average shift of
approximately 0.7 Hz. This downshift is exemplified by the peaks of the
aftershock at 0.95 Hz and 2.1 Hz downshifted to those of the mainshock
at 0.25 Hz and 1.4 Hz, respectively. The degree of soil nonlinearity under
the mainshock motion can be quantified by the frequency downshift.
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Nisqually Mainshock – 2-D Hilbert Spectrum - SDSns
0.18
101
0.16
0.14
Frequency (Hz)
0.12
0.1
100
0.08
0.06
0.04
0.02
10−1
0
5
10
15
20
Time (s)
25
30
35
40
FIGURE 8.11 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisqually
earthquake mainshock at SDS (soft soil).
•
•
The profile of the HHT-based factor in the frequency band 2.5 to 7 Hz
(intermediate-frequency range) is generally reduced in amplitude from
the aftershock to the mainshock, with an average difference of approximately a factor of 0.43. Note that the measure is carried out in the way
that the averaged factor of 7 for the aftershock is reduced to 3 for the
mainshock in frequency range 3 to 4 Hz.
There is no evidence to support a difference in HHT-based factor from
about 7 Hz up (high-frequency range) between the mainshock and aftershock.
To facilitate understanding of these observations, this study analyzes the HHT-based
factor of the mainshock and aftershock at LAP in Figure 8.16a, which shows the
following:
•
In the low-frequency range (below 2.5 Hz), Figure 8.16a shows a downshift profile in both frequency and amplitude from the aftershock to
mainshock that is similar to Figure 8.15a in the low to intermediate
frequency range, but the former shows a smaller shift (about 0.1 Hz in
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The Hilbert-Huang Transform in Engineering
Nisqually Mainshock – 2-D Hilbert Spectrum - SDSnew
0.11
0.1
101
0.09
Frequency (Hz)
0.08
0.07
0.06
100
0.05
0.04
0.03
0.02
0.01
10
−1
0
5
10
15
20
Time (s)
25
30
35
40
FIGURE 8.12 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisqually
earthquake mainshock at SDS (soft soil).
•
frequency and a factor of 0.2) than the latter (about 0.7 Hz and a factor
of 0.43).
In the intermediate to high frequency range, there is almost no difference
in the two factors between the mainshock and the aftershock.
Comparison of the HHT-based factors at SDS and LAP suggests that SDS had
severe soil nonlinearity during the mainshock and LAP had slight soil nonlinearity.
Site LAP can also be regarded as having no soil nonlinearity under the mainshock
if the aforementioned small downshift in both frequency and amplitude in the
low-frequency range is the result of variation of data collection and sampling.
8.5.3 INFLUENCES OF WINDOW LENGTH OF DATA
The above results were obtained on the basis of 40-sec record lengths. It can be
seen from Figure 8.7 through Figure 8.14 that the characteristics of the motion
during the 40 sec change from one time interval to another. The noticeable features
in different sections of the 40 sec (see Figure 8.7a and Figure 8.8a) are the dominant
P-wave signals with high frequencies and low amplitude from 0 to 10 sec, followed
by the dominant S-wave signals with intermediate frequencies and high amplitude
from 10 to 20 sec, and finally the surface and coda waves with low frequencies and
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−4
×10
Nisqually Aftershock – 2-D Hilbert Spectrum - SDSns
6
101
5
Frequency (Hz)
4
3
100
2
1
10−1
0
5
10
15
20
Time (s)
25
30
35
40
FIGURE 8.13 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisqually
earthquake aftershock at SDS (soft soil).
intermediate amplitude from 20 to 40 sec. Accordingly, the HHT-based factor calculated over 40 sec is a characteristic averaged in frequency and amplitude, i.e.,
mixing low to high frequencies and amplitudes. It is also noticed that the soil
nonlinearity is likely most prevalent in the strong S-wave motion. To characterize
the soil nonlinearity precisely, one should select an appropriate window of motion
for analysis in which the soil nonlinearity shows up most strongly. However, the
selection of an appropriate window is subjective.
Nevertheless, this study next examines the influence of window length on the
HHT-based factor by selecting two windows, 0 to 10 sec and 10 to 20 sec. In the
first window, 0 to 10 sec, where the P-wave signals are dominant with relatively
small amplitudes and high frequencies, the soil is likely linear under both mainshock
and aftershock motions, which should lead to the same factor in a broad spectrum
of frequency for the mainshock and aftershock. This is verified in Figure 8.17a,
except for a large difference in amplitude between 3 Hz and 6 Hz. The difference
might be caused by unknown factors such as a lack of recorded frequency content
in the referenced site or indeed by a change of physical properties in the soil. Further
research is needed along this line.
In the second window, 10 to 20 sec, where the S-wave signals are dominant with
large amplitudes and intermediate frequencies, the soil is likely strongly nonlinear under
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The Hilbert-Huang Transform in Engineering
−4
×10
Nisqually Aftershock – 2-D Hilbert Spectrum - SDSew
5.5
101
5
4.5
Frequency (Hz)
4
3.5
3
10
0
2.5
2
1.5
1
0.5
10−1
0
5
10
15
20
Time (s)
25
30
35
40
FIGURE 8.14 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisqually
earthquake aftershock at SDS (soft soil).
the mainshock. The downshift in both amplitude and frequency from the HHT-based
factors between mainshock and aftershock is clearly seen in Figure 8.17b.
8.5.4 COMPARISON OF HHT- AND FOURIER- BASED FACTORS
FOR SITE AMPLIFICATION
To further illustrate the characteristics of the HHT approach, this study compares
the HHT-based factors of site amplification at SDS shown in Figure 8.15a with the
Fourier-based ones shown in Figure 8.15b (i.e., Figure 8.7 in Frankel et al. [17]).
In the low-frequency range, Figure 8.15b shows a frequency-downshift profile
from the aftershock to mainshock that is similar to Figure 8.15a, but the former
shows a smaller shift (about 0.2 Hz) than the latter (about 0.7 Hz). Because of the
averaging characteristic in Fourier spectral analysis, as indicated in Section 8.3, the
frequency downshift measured from the HHT-based factors in Figure 8.15a may
give a more truthful indication of the soil nonlinearity than that measured from
Fourier-based factors in Figure 8.15b. In addition, the factor in the low-frequency
range in Figure 8.15a is generally somewhat larger than the factors in Figure 8.15b.
In the intermediate-frequency range, Figure 8.15b shows an amplitude-reduction
profile from the aftershock to mainshock that is similar to Figure 8.15a, but the
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Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW)
1
Spectral Ratio
10
0
10
Aftershock
Mainshock
−1
10
10−1
100
Frequency (Hz)
10
1
10
1
Nisqually Earthquake - Fourier Spectral Ratio (SDS/SEW)
1
Spectral Ratio
10
100
Aftershock
Mainshock
10−1
10−1
100
Frequency (Hz)
FIGURE 8.15 (A) HHT-based factor for site amplification at SDS (soft soil) for mainshock
and aftershock of the 2001 Nisqually earthquake. (B) Fourier-based factor for site amplification at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake.
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The Hilbert-Huang Transform in Engineering
Nisqually Earthquake - Marginal Spectral Ratio (LAP/SEW)
1
Spectral Ratio
10
0
10
Aftershock
Mainshock
10−1
10−1
100
Frequency (Hz)
101
Nisqually Earthquake - Fourier Spectral Ratio (LAP/SEW)
Spectral Ratio
101
100
Aftershock
Mainshock
10−1
10−1
100
Frequency (Hz)
101
FIGURE 8.16 (A) HHT-based factor for site amplification at LAP (stiff soil) for mainshock
and aftershock of the 2001 Nisqually earthquake. (B) Fourier-ased factor for site amplification
at LAP (stiff soil) for mainshock and aftershock of the 2001 Nisqually earthquake.
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Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW) ==> 0–10s
2
Spectral Ratio
10
1
10
10
0
Aftershock
Mainshock
−1
0
10
10
Frequency (Hz)
101
Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW) ==> 10–20s
2
Spectral Ratio
10
1
10
10
0
Aftershock
Mainshock
−1
10
0
10
Frequency (Hz)
101
FIGURE 8.17 (A) HHT-based factor for site amplification with the use of recordings in
window 0 to 10 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually
earthquake. (B) HHT-based factor for site amplification with the use of recordings in window
10 to 20 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake.
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The Hilbert-Huang Transform in Engineering
Nisqually Earthquake - Difference of Damping Factor (SDS-SEW)
Aftershock
Mainshock
Difference of Damping Factor
10−1
10−2
0
2
4
6
8
10
12
Frequency (Hz)
14
16
18
20
FIGURE 8.18 Difference of marginal Hilbert damping spectra at SDS and SEW for mainshock and aftershock of the 2001 Nisqually earthquake.
former shows a smaller reduction (about a factor of 0.2) than the latter (about a
factor of 0.43).
In the high-frequency range, Figure 8.15b shows a significant increase in the
Fourier-based factor from the aftershock to mainshock, while Figure 8.15a does not.
Following the discussion in Section 8.3 and numerical illustration in Figure 8.1 and
Figure 8.2, the high-frequency content in the Fourier amplitude spectra may be
influenced by higher-order harmonics used to represent a nonlinear waveform, which
consequently increases the Fourier-based factor of the mainshock in the high-frequency range.
This study also compares HHT- and Fourier-based factors at LAP in Figure 8.16a
and b. Almost no fundamental difference in the factors is observed in terms of overall
profile, amplitude value, frequency downshift, and amplitude reduction between the
mainshock and aftershock, indicating that the two approaches are essentially consistent
with each other in estimating linear or approximately linear site amplification.
8.5.5 HHT-BASED FACTOR FOR SITE DAMPING
Figure 8.18 shows the HHT-based factors for site damping at SDS calculated by
Equation 8.15, an alternative perspective of soil nonlinearity from recordings. It
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185
reveals that the site damping during the mainshock is much larger than that in the
aftershock at frequency 0.5 to 5 Hz, suggesting that strong soil nonlinearity occurred
during the mainshock in this frequency band. The increased damping will decrease
the amplified magnitude of seismic wave responses through the nonlinear soil and
thus reduce the site amplification factor. This can be confirmed from Figure 8.15a,
which shows that the HHT-based factor for site amplification is observably reduced
for the mainshock from the aftershock in the similar frequency band of 0.5 to 7 Hz.
Figure 8.18 also shows that the second largest increased site damping for the
mainshock is in the frequency band 8 to 16 Hz. However, the HHT-based factors
for site amplification in Figure 8.15a do not appear to change between the mainshock
and aftershock. This can be explained as follows: On the one hand, the increased
site damping for the mainshock over the frequencies 8 to 16 Hz will reduce the
seismic wave responses in the same frequency band. Note that the soil nonlinearity
typically occurs under large-amplitude seismic waves incident to the soil layer, and
that the amplitude of the motion changes with time. This suggests that the soil
nonlinearity is likely most prevalent in the strong S-wave motion or following surface
or S-coda waves, but not in the P-wave motion during the mainshock. Figure 8.7a
and Figure 8.8a reveal that except for the abnormal high-frequency spikes between
20 sec and 25 sec, the mainshock accelerations after the S-wave arrival at about 10
sec contain less high-frequency motion than does the aftershock in general, and in
the high-frequency band of about 8 to 16 Hz in particular. This fact is consistent
with the increased damping of the nonlinear soil during the mainshock in the
high-frequency band of 8 to 16 Hz, which damped out the high-frequency wave
responses in the recordings.
On the other hand, the soil nonlinearity indeed introduces large-amplitude
high-frequency spikes from 20 to 25 sec in the mainshock recordings in Figure 8.7a
and Figure 8.8a.
Together with the fact that the HHT-based factors in Figure 8.15a show the
averaged characteristics of soil linearity and nonlinearity, these observations suggest
that the overall high-frequency (around 8 to 16 Hz) content of the mainshock may
be equivalent to that of the aftershock. This yields the same factors in Figure 8.15a
in the frequency range 8 to 16 Hz.
For further clarification, the HHT-based factor for site damping at LAP is
examined. Figure 8.19 shows that the profiles of soil damping are essentially no
different between the mainshock and aftershock, suggesting that site LAP is linear
during the mainshock. This can be further verified from the HHT-based site-amplification factors shown in Figure 8.16a.
8.6 CONCLUDING REMARKS AND DISCUSSION
This study investigates the role of the HHT method in effectively detecting soil
nonlinearity and precisely quantifying soil nonlinearity in terms of site amplification
and damping. In particular, it reveals that the first IMF component can help identify
the high-frequency spikes related to soil nonlinearity, which can also be further
confirmed by Hilbert amplitude spectra.
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The Hilbert-Huang Transform in Engineering
Nisqually Earthquake - Difference of Damping Factor (LAP-SEW)
Aftershock
Mainshock
Difference of Damping Factor
10−1
10−2
0
2
4
6
8
10
12
Frequency (Hz)
14
16
18
20
FIGURE 8.19 Difference of marginal Hilbert damping spectra at LAP and SEW for mainshock and aftershock of the 2001 Nisqually earthquake.
The HHT-based factor of site amplification is defined as the ratio of marginal
Hilbert amplitude spectra, similar to the Fourier-based factor that is the ratio of
Fourier amplitude spectra. The HHT-based factor has the following relationships
with Fourier-based one:
•
•
•
•
The HHT-based factor is essentially equivalent to the Fourier-based factor
in quantifying linear site amplification.
The HHT-based factor is more effective than the Fourier-based factor in
quantifying soil nonlinearity in terms of frequency downshift in the
low-frequency range and amplitude reduction in the intermediate-frequency range.
In the low to intermediate frequency range, the HHT-based factor is
generally larger than the Fourier-based factor, which is also dependent
upon the window length in which the recordings are used.
For nonlinear site amplification, the Fourier-based factor is increased with
respect to the linear one at high frequencies; this phenomenon is not seen
in the HHT-based factor. This difference may be due to the introduction
of high frequencies in Fourier spectral analysis in the characterization of
a nonlinear waveform.
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The HHT-based factor for site damping can be extracted from the HHT-based
factor for site amplification and used as an alternative index to measure the influences
of soil nonlinearity in seismic wave responses at sites.
It should be pointed out that the results from this study rely mainly on the use
of advanced signal processing techniques to explore and then quantify the signature
of soil nonlinearity from recordings. They must, therefore, be validated by
model-based simulation. Recently, significant advances have been made in borehole
data collection and simulation techniques. These include data from 17 borehole
arrays in Southern California [33] and from the Port Island vertical array for the
1995 Hyogoken Nanbu earthquake, geophysical data including the S-wave velocity
profile in the top layer(s) at key strong motion station sites (Rosrine 2002 at
http://geoinfo.usc.edu/rosrine, USGS open file reports), and simulated broadband
ground motion for scenario earthquakes including nonlinear soil effects [34]. The
above information should allow the further validation of the observations and results
from this study.
ACKNOWLEDGMENTS
The author would like to express sincere gratitude to Norden E. Huang at NASA;
Stephen Hartzell, Authur Frankel, and Erdal Safak at USGS; Lance VanDemark
from Colorado School of Mines; and Yuxian Hu from China Seismological Bureau
and Jianwen Liang from Tianjin University of China for providing data, Fourier
analysis and calculations, and more important, constructive suggestions. This work
was supported by the National Science Foundation with Grant Nos. 0085272 and
0414363, and by the US-PRC Researcher Exchange Program administered by Multidisciplinary Center for Earthquake Engineering Research. The opinions, findings,
and conclusions expressed herein are those of the author and do not necessarily
reflect the views of the sponsors.
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decomposition-Hilbert spectrum method to identify near-fault ground-motion characteristics and structural responses, Bull. Seism. Soc. Am. 91, 1339–1357.
28. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002a). “System identification of linear
structures based on Hilbert-Huang spectral analysis. Part I: Normal modes,” Earthquake Eng. Structural Dynamics, Vol. 32, pp. 1443–1467.
29. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002b). “System identification of linear
structures based on Hilbert-Huang spectral analysis. Part II: Complex modes,” Earthquake Eng. Structural Dynamics, Vol. 32, pp. 1533–1554.
30. Xu, Y.L., S.W. Chen, and R. Zhang (2003) “Modal identification of Di Wang building
under typhoon York using HHT method,” The Structural Design of Tall and Special
Buildings, Vol. 12, No. 1, pp. 21–47.
31. Hilbert-Huang Transformation Toolbox (2000). Professional Edition V1.0, Princeton
Satellite Systems, Inc., Princeton, New Jersey.
32. Hou, Z., M. Noori and R. St. Amand (2000) “Wavelet-based approach for structural
damage detection,” ASCE Journal of Engineering Mechanics, 126(7), pp. 677–683.
33. Archuleta, R.J. and J.H. Steidl (1998). “ESG studies in the United States: results
from borehole arrays,” The Effects of Surface Geology on Seismic Motion, (Irikura,
Kudo, Okada and Sasatani, eds.)., Balkema, Rotterdam, pp. 3–14.
34. Hartzell, S., A. Leeds, A. Frankel, R.A. Williams, J. Odum, W. Stephenson and W.
Silva (2002). “Simulation of broadband ground motion including nonlinear soil
effects for a magnitude 6.5 earthquake on the Seattle fault, Seattle, Washington,” Bull.
Seism. Soc. Am. Vol. 92, No. 2, pp. 831–853.
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9
Simulation of
Nonstationary Random
Processes Using
Instantaneous Frequency
and Amplitude from
Hilbert-Huang Transform
Ping Gu and Y. Kwei Wen
CONTENTS
9.1 Introduction ..................................................................................................191
9.2 Hilbert-Huang Transform (HHT).................................................................194
9.3 IMF Recombination Method .......................................................................197
9.4 Improved Wen-Yeh Method.........................................................................200
9.5 Conclusions ..................................................................................................203
Acknowledgments..................................................................................................207
References..............................................................................................................207
Appendix: Relation between Hilbert Marginal Spectrum and
Fourier Energy Spectrum ......................................................................................209
9.1 INTRODUCTION
With the ever-increasing power of the computer and of the capacity of computational
mechanics, simulation has become an indispensable tool in engineering. It is more
so in earthquake engineering since earthquake ground motions are highly random and
structural behaviors are nonlinear and inelastic, making random vibration solutions
191
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The Hilbert-Huang Transform in Engineering
difficult to obtain. Generally speaking, structural responses are functions of the entire
ground excitation time history. Current codes use a scalar intensity measure such as
spectral acceleration or displacement for the earthquake demand on structures. These
measures do not reflect satisfactorily the effects of near source or effects due to
higher modes and other important structural response behaviors caused by ground
motions. For performance evaluation of complex structural systems, simulation of
earthquake ground motions and time history response analysis have played a more
and more important role.
Many models for simulation based on earthquake records have been proposed.
For example, a stationary white noise model was first proposed [1], followed by
filtered white noise models to reflect the site characteristics [2, 3, 4]. A high-pass
filter was later added to remove the undesirable zero frequency content of the power
spectral density function [5]. To account for the intensity variation with time, a
uniformly (amplitude) modulated random process was first used [6, 7]. Nonstationary
models with frequency content variation with time — in particular, the long-duration
acceleration pulses observed in many near-fault earthquake records — have also
been developed.
Saragoni and Hart [8] proposed a modulated filtered Gaussian white noise model
with different intensity and frequency content in three consecutive time intervals.
The arbitrary selection of time intervals and the resultant abrupt change of frequency
content in this model, however, are difficult to justify physically. The oscillatory
process described by an evolutionary power spectral density suggested by Priestley
[9], allowing both amplitude and frequency content variation with time, has been
widely used in earthquake ground motion simulation. For example, Lin and Yong
[10] formulated an evolutionary Kanai-Tajimi [2, 3] model as a filtered pulse train,
where the filter represents the ground medium and the pulse train represents intermittent ruptures at the earthquake source.
Deodatis and Shinozuka [11] simulated the earthquake ground motions using
an autoregressive (AR) model based on selected parametric forms of the evolutionary
power spectral density for given earthquake records. Li and Kareem [12, 13] further
extended the method using the fast Fourier transform (FFT) and digital filtering
technique to make it more computationally efficient. Der Kiureghian and Crempien
[14] proposed an evolutionary earthquake model composed of individually modulated band-limited white noise processes and estimated the parameters of evolutionary power spectral density by matching the moments of each component process.
Papadimitriou [15] produced a parsimonious nonstationary model by applying a
second-order filter with slowly varying parameters to time-modulated white noise.
Conte and Peng [16] proposed to use Thomson’s spectrum to estimate the evolutionary power spectra and proposed a model for simulation based on this. By
extending the energy principle of wavelet transform developed by Iyama and Kuwamura [17], Spanos and Failla [18] and Spanos et al. [19] proposed methods using
the wavelet spectrum to estimate the evolutionary power spectral density.
The main challenge in evolutionary process-based methods lies in parameter
estimation. In addition to estimating the power spectral density function, one also
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193
needs to know the modulation functions of frequency and time, which are not known
a priori and are difficult to estimate. Motivated by an amplitude- and frequency-modulated process proposed by Grigoriu et al. [20], Yeh and Wen [21] proposed to model
ground motion as I(t)ς(φ(t)), where I(t) is a deterministic intensity envelope that
controls the amplitude of the ground motion, ς(φ(t)) is a stationary filtered white
noise in time scale φ and depicts the spectral density form of the process, and φ(t)
is a smooth, strictly increasing function that depicts the frequency modulation. The
stationary white noise was filtered to have a Clough-Penzien spectrum [5] with zero
mean and unit variance. By changing the time scale of the stationary filtered white
noise, the spectral density of the stationary process becomes time varying:
Sςς (t , ω ) =
 ω 
1
.
SCP 
φ′(t )
 φ′(t ) 
(9.1)
This method does not require the “slowly time-varying” assumption implied in
the evolutionary spectrum method, and it gives a time-dependent power spectral
density as a simple modification of the spectral density of the stationary process.
However, the frequency modulation function is estimated by a zero-crossing rate
procedure and is independent of the amplitude variation. It is therefore somewhat
restrictive in nature and only suited for a certain special class of random processes.
It has been applied to earthquake ground motion simulation with some success but
lacks general theoretical basis for wider applications.
Recently, Huang et al. [22] introduced a new method of spectral analysis of
nonstationary time series based on empirical modal decomposition (EMD), Hilbert
transform, and the concept of instantaneous frequency, commonly referred to as the
method of Hilbert-Huang transform (HHT). Unlike the Fourier transform, the Hilbert
transform emphasizes local behavior and does not decompose the signal into sine
and cosine waves of fixed frequencies; thus it is suitable for nonstationary processes.
HHT decomposes the signal into orthogonal functions called intrinsic mode functions (IMFs) directly based on data. The Hilbert transform of IMFs yields the
instantaneous amplitude and instantaneous frequency. HHT can be used to overcome
the difficulties of the Wen-Yeh simulation method since EMD provides a reasonable
and physically meaningful decomposition of the nonstationary signal. The Hilbert
transform yields instantaneous amplitude and frequency of each IMF, which can be
used directly in the Wen-Yeh method. It bypasses the difficult tasks of estimating
and constructing analytical functions for frequency and amplitude modulation. Also,
since these two modulation functions are obtained from Hilbert transform of IMFs,
the dependence between the two functions is preserved.
In this study, an IMF recombination method for simulation of nonstationary
processes is first proposed, followed by an improved Wen-Yeh method. After a brief
introduction of HHT, we describe the IMF recombination method, followed by the
improved Wen-Yeh Method. We then compare the two. These methods are demonstrated by numerical examples of simulation of earthquake ground motions.
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The Hilbert-Huang Transform in Engineering
9.2 HILBERT-HUANG TRANSFORM (HHT)
What follows is a brief description of the background of the HHT. Details can be
found in Huang et al. [22]. First, note that a time series can be decomposed into a
series of IMFs. An IMF is defined as a time series that satisfies two requirements:
1. Its number of extrema and number of zero-crossings are either equal or
differ by one.
2. The mean value of the envelopes defined by the local maxima and minima
is zero.
The extraction of IMFs out of a time series requires a repeated “sifting” procedure called empirical mode decomposition. EMD is described briefly as follows:
Connect all the local maxima by a cubic spline as the upper envelope and connect
all the local minima by a cubic spline as the lower envelope. Then calculate the
mean of the envelopes and subtract the mean from the time series. The result is the
first component. Ideally it should have satisfied the requirements for an IMF, but in
practice a gentle hump on a slope can be amplified to become a local extreme. So
the procedure is repeated until an IMF is obtained. This is the first IMF. Subtract
the first IMF from the original time series, then continue the procedure on the residue
to obtain the second IMF. Repeat the process to obtain the third, fourth, fifth, and
subsequent IMFs, until the residue is too small to be of any consequence or becomes
a monotonic function from which no more IMF can be extracted.
After the EMD, the time series X(t) can be expressed in terms of IMFs as follows:
n
X (t ) =
∑ C (t) + r (t)
j
(9.2)
n
j =1
in which Cj(t) is the jth IMF, and rn(t) is the residue that can be the mean trend or
a constant. Taking Hilbert transform of Cj(t),
D j (t ) =
C j (τ)
1
P
dτ
π
t−τ
∫
(9.3)
in which P indicates Cauchy principal value. One can then form an analytical
function
iθ ( t )
Zj(t) = Cj(t) + iDj(t) = aj(t) e j .
(9.4)
The instantaneous frequency of the IMF is given by
ω j (t ) =
© 2005 by Taylor & Francis Group, LLC
d θ j (t )
.
dt
(9.5)
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195
One can then express X(t) as follows:
n
∑ a (t)e
X (t ) = Re{
j
iθ j ( t )
} + rn (t )
(9.6)
j =1
in which Re denotes the real part. Therefore, aj(t), j = 1, …, n, therefore control the
amplitude variation; each has an instantaneous frequency at a given time. Collectively, the functions aj(t) and j(t) define the Hilbert amplitude spectrum. Similarly,
aj2(t) and j(t) define the energy distribution with respect to time and frequency, called
the Hilbert energy spectrum or simply Hilbert spectrum, H(,t). The marginal Hilbert
energy spectrum is obtained by integration over the duration:
T
h(ω ) =
∫ H (ω, t)dt .
(9.7)
0
Because of the local nature of the Hilbert transform and the concept of instantaneous frequency, the Hilbert marginal spectrum describes an energy contribution
corresponding to a given frequency at a certain time but not the entire duration of
the process, as would be the case for a Fourier spectrum. This is a fundamental
difference between these two approaches.
As an example, the method is applied to the well-known N-S component of the
El Centro earthquake record of 1940 and the Newhall ground motion of the 1994
Northridge earthquake in SAC-2 ground motion records [23], representing far-source
and near-source records respectively. The time histories of the ground accelerations
(in cm/sec2) are shown at the top of Figure 9.1 (At the bottom of the figure, four
simulated samples of the ground accelerations are shown based on the IMF recombination method described later in this paper.) The IMFs of the Newhall record are
shown in Figure 9.2. The Hilbert spectra are calculated and shown in Figure 9.3.
As can be seen, one distinct advantage of the Hilbert spectrum is that the energy
distribution over time and frequency is always sharp. For example, a darker shade
of color in the plot indicates higher energy, which can be clearly seen in the Newhall
record from 4 to 10 seconds and below 20 rad/sec (3 cps), a frequently observed
feature of near-source records; it can also be seen over a longer time period in the
El Centro record. Figure 9.4 compares the marginal Hilbert energy spectrum with
the Fourier energy spectrum obtained from the same record. There are some obvious
differences due to the two different methods of spectral description. The scale of
the Hilbert marginal spectra has been adjusted by a factor of π for better comparison.
Details of comparison of these two spectra and the conversion factor are shown in
the Appendix. Note that the Hilbert transform can be carried out by first performing
the Fourier transform on the original signal, changing the phase of each positive
frequency component by /2 and the phase of each negative frequency component
by –/2, and finally performing an inverse Fourier transform to obtain the Hilbert
transform of the sample [24]. One can use FFT to make the procedure more efficient.
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The Hilbert-Huang Transform in Engineering
FIGURE 9.1 Recorded (top) and simulated El Centro and Newhall ground motions using
IMF recombination method.
In spite of its numerical difficulties — such as in end swing caused by spline
fitting, convergence in the sifting procedure; ambiguity of the local symmetry
requirement of IMFs; and dealing with intermittent signals, negative frequency, and
end distortion in Hilbert transform — HHT represents a new approach to the
challenging problem of time–frequency analysis of nonstationary signals, especially
for slowly varying signals, that has many advantages over existing methods. Instead
of decomposing the signal into predetermined analytical functions, it obtains the
orthogonal functions (IMFs) directly from the data; thus it is more adaptive, and the
resulting IMFs usually carry physical meaning. For example, it has been shown that
IMFs of earthquake recordings have physical meanings related to seismic source
mechanism [25]. Unlike the Fourier transform, the Hilbert transform emphasizes
local behavior, as the Hilbert transform is the convolution of the signal with a window
function of 1/t. Because of these characteristics, HHT represents an attractive alternative to other methods currently available, such as short-time Fourier transform,
wavelet transform, instantaneous spectrum, double-frequency spectrum, Karhunen-Loeve decomposition, and so forth.
The emphasis of HHT study so far has been on data analysis and interpretation
of a deterministic time series of a real physical phenomenon [22, 26–33]. Much less
has been done to further development of the method as a simulation tool for non-
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Simulation of Nonstationary Random Processes
197
FIGURE 9.2 IMFs of the 1994 Northridge Newhall record (SAC LA14).
stationary random processes. Such extension is essential for performance evaluation
of engineering systems, since the demands need to be modeled as random processes.
This paper proposes to use the new approach for simulation of nonstationary random
processes and to apply the simulation methods to earthquake engineering.
9.3 IMF RECOMBINATION METHOD
The Hilbert spectral representation of a signal,

 n
iθ ( t ) 
x (t ) = Re 
a j (t )e j  + rn (t )

 j =1
∑
suggests that the underlying random process can be represented by introducing
random elements as follows:
 n

i [ θ ( t )+ φ j ] 
X (t ) = Re 
a j (t )e j
 + rn (t ) ,
 j =1

∑
© 2005 by Taylor & Francis Group, LLC
(9.8)
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The Hilbert-Huang Transform in Engineering
Hilbert Energy Spectrum for EI Centro Record
140
Frequency (rad/s)
120
100
80
60
40
20
0
5
10
15
Time (s)
20
25
Hilbert Energy Spectrum for Newhall Record
140
Frequency (rad/s)
120
100
80
60
40
20
0
5
10
15
Time (s)
20
25
30
FIGURE 9.3 Hilbert spectra of 1940 El Centro and 1994 Northridge Newhall ground motion.
in which φj is an independent random phase angle uniformly distributed between 0
and 2π. X(t) is therefore a random process. The process has the following mean,
covariance, and variance functions:
 n

iθ ( t )
iφ 
µ X (t ) Re 
a j (t )e j E (e j )  + rn (t ) = rn (t )
 j =1

∑
© 2005 by Taylor & Francis Group, LLC
(9.9)
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Simulation of Nonstationary Random Processes
106
199
Hilbert Marginal Spectrum and Fourier Energy Spectrum for Newhall Record
105
104
103
10
2
101
100
10−1
10−2
10
−3
Fourier Energy Spectrum
Hilbert Marginal Spectrum (adjusted)
10−2
10−1
100
101
102
103
rad/s
10−1
Hilbert Marginal Spectrum and Fourier Energy Spectrum for EI Centro Record
10−2
10−3
10−4
10−5
10−6
10−7
Fourier Energy Spectrum
Hilbert Marginal Spectrum (adjusted)
10
−1
100
101
102
rad/s
FIGURE 9.4 Comparison of Hilbert marginal spectrum (solid line) with Fourier energy
spectrum (dashed line).
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The Hilbert-Huang Transform in Engineering
n
∑
1
K XX (t1 , t2 ) =
a j (t1 )a j (t2 ) cos[θ j (t1) − θ j (t2 )]
2 j =1
{
()
()
σ 2x (t) = E  X t − uX t 
2
} ∑ ()
(9.10)
n
=
1
a 2j t
2 j =1
(9.11)
X(t), as given in Equation 9.8, has a Hilbert energy spectrum characterized by aj2(t)
with instantaneous frequency d(j(t))/dt, for j = 1, …, n. Due to the Central Limit
Theorem, X(t) approaches a Gaussian process for large n. This Hilbert spectral
representation model suggests a simple method for simulation of a nonstationary
process as shown in Equation 9.8. One can generate the random phase angles on
computer and use Equation 9.8 to simulate the process. The name reflects the fact
that it is essentially a method of recombination of the IMFs. The Hilbert spectrum
of each simulated sample can be shown to be the same as that of the recorded ground
motion.
The physical basis for this method is that the frequency and amplitude of each
IMF represent those of a sub-wave caused by a given physical mechanism of the
rupture surface of the earthquake, as shown by Zhang [25]. For comparison with
the traditional Fourier-based source inversion solution obtained by Hartzell et al.
[34] for the Northridge earthquake based on 70 records, Zhang studied the same 70
earthquake records using HHT. The spatial distributions of the slip amplitude over
the finite fault, each of which corresponds respectively to the second to fifth IMFs
of those records, were plotted. It was found that the large slip amplitude regions are
located at almost the same regions as those in Hartzell’s study, indicating that wave
motions in each IMF are generated by a fraction of the source. In addition, the
rupture with large-slip region scatters out to the northwest from the hypocenter, with
the IMF changing from the second to the fifth, indicating that the seismic waves are
generated sequentially from the dominant short-period signals to main long-period
signals as the rupture propagates. Therefore, the IMFs are closely related to the
source mechanism in general and the source heterogeneity and rupture process in
particular. For future earthquakes of similar characteristics, the rupture surface spatial
features are random and unpredictable. The IMF recombination method represents
an efficient way to capture these variabilities.
Simulations of the N-S component of the El Centro and the Northridge Newhall
records (SAC LA14) were carried out, and sample time histories are shown in Figure
9.1. The simulated samples represent a reasonably statistical image of the intensity
and frequency variation with time of the records. The method has also been extended
to vector processes.
9.4 IMPROVED WEN-YEH METHOD
The IMF recombination method has certain shortcomings compared to the Wen-Yeh
method. They are as follows:
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Simulation of Nonstationary Random Processes
•
•
•
201
In the context of the Wen-Yeh method, the underlying stationary process
for each IMF has only one frequency component.
There is only a small number of random elements. For example, Equation
9.8 has only n random variables, n being the number of IMFs, which is
usually not very large (usually <15 for earthquake accelerograms).
No smoothing of aj(t) and j(t) is done to separate noise from the underlying
true physical characteristics.
Taking advantage of both methods, an improved Wen-Yeh method is proposed.
In this method, the recorded ground motion is first decomposed into IMFs. Then
the smoothing of aj(t) and j(t) of each IMF is performed to separate noise from the
underlying true physical characteristics, from which the underlying stationary random process of the IMF is obtained and simulated by using the well-known spectral
representation method [35]. The frequency and amplitude of the simulated sample
of the stationary random process are then modulated in time. The frequency modulation is done by changing the time scale as in the Wen-Yeh method. The frequency
modulation function and amplitude modulation function are obtained directly from
the Hilbert transform. Finally, the sample function of the process is simulated by
combining the simulated IMFs. The procedure is described in detail as follows:
1. Decompose the record into IMFs Cj(t), j = 1, …, n, by EMD, and determine
the aj(t) and j(t) for each IMF by Hilbert transform.
2. Smooth aj(t) and j(t) and denote the smoothed functions respectively as
ajs(t) and js(t).
3. Obtain the reduced process Cjr(t) from Cj(t) by removing the amplitude
modulation ajs(t) from Cj(t) and changing the time scale by ωjs–1 (t). Changing time scale is done by making the signal a function of θ instead of t,
θ being the integration of js(t). Then resample the signal using an evenly
spaced.
4. Simulate the reduced process Cjr(t) as stationary process and obtain samples of the process Sjr(t).
5. Restore the time scale by using the function ωjs(t) in Sjr(t), and restore the
amplitude modulation by using ajs(t). The result is the simulated jth IMF
Sj(t). Restoring time scale is done by first resampling the underlying
stationary process of θ and then expressing the signal as a function of
time t.
6. Add all Sj(t), j = 1, …, n.
The procedure is entirely numerical and based on sample data. No analytical
forms are assumed for the frequency and amplitude modulation or for the spectrum
of the underlying stationary random process. It is therefore conceptually simple and
easy to implement.
Figure 9.5 shows eight simulated samples using this method along with the
record for the Newhall Northridge earthquake. Figure 9.6 shows the response spectra
of the samples. As can be seen, the long-duration pulses of the near-fault ground
motion are reproduced well. Both the amplitude and frequency changes with time
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The Hilbert-Huang Transform in Engineering
Simulated samples of Newhall LA14
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
1000
0
−1000
Original record of Newhall LA 14
0
5
10
15
20
25
30
FIGURE 9.5 Simulation of Newhall Northridge 1994 by improved Wen-Yeh method.
are reflected well in the simulated samples. The response spectra generally fluctuate
around that of the record. Unlike samples generated by methods based on analytical
models of the spectra, the response spectra of the simulated samples retain the jagged
look of those of the records. The artificial smoothness of the response spectra of the
simulated ground motions obtained by existing power spectrum-based methods has
been criticized by seismologists and practicing engineers.
To further illustrate the procedure, the steps leading to the first sample in Figure
9.5, shown below by figures, are as follows:
1. Decompose the record into IMFs (Figure 9.2).
2. Smooth aj(t) and j(t). Figure 9.7 shows a2(t) and a2s(t), and Figure 9.8
shows 2(t) and 2s(t) as an example. Here, 24-point rectangular window
smoothing with zero-padding is used.
3. Obtain the reduced process Cjr(t). The reduced process C2r(t) is shown in
Figure 9.9 as an example. As can be seen, both the amplitude and frequency are much more uniform than the original time history of the second
IMF. Figure 9.10 illustrates the effect of changing time scale: it removes
the frequency modulation from the signal and makes it stationary.
4. Simulate the reduced process Cjr(t) as a stationary process. A sample of
S2r(t) is shown in Figure 9.11.
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Simulation of Nonstationary Random Processes
203
Response Spectra of Newhall LA 14 Simulations using
Improved Wen-Yeh Method (5% damping)
3.5
Simulation 1
Simulation 2
Simulation 3
Simulation 4
Simulation 5
Simulation 6
Simulation 7
Simulation 8
Original record
3
Pseudo Acceleration (g)
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
Period (sec)
6
7
8
9
10
FIGURE 9.6 Response spectra of Newhall LA14 simulations by improved Wen-Yeh method.
5. Restore the time scale and the amplitude modulation to obtain the simulated jth IMF Sj(t). The restored sample in Step 4 of the second IMF is
shown in Figure 9.12 and compared with the original second IMF.
6. Add all Sj(t), j =1 to n. The result is Sample 1 in Figure 9.5.
Further investigation is needed, including refinement of the smoothing process,
modeling of the possible dependence among the IMFs. This method is being used
in a study of the effects of near-fault ground motions on buildings and structures
and in simulation of suites of uniform hazard ground motions for performance
evaluation.
9.5 CONCLUSIONS
Two new methods of simulation of nonstationary random processes are presented
based on the Hilbert-Huang transform of sample observations, namely the IMF recombination method and the improved Wen-Yeh method. Both take advantage of EMD and
the instantaneous frequency and amplitude of the Hilbert transform, thus overcoming
difficulties in the estimation of the frequency modulation and interdependence of
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The Hilbert-Huang Transform in Engineering
Instantaneous amplitude function of the 2nd IMF
500
Amplitude function
Smoothed amplitude function
450
400
350
300
250
200
150
100
50
0
0
5
10
15
20
25
30
FIGURE 9.7 Original (dashed line) and smoothed (solid line) instantaneous amplitude function of the second IMF.
Instantaneous frequency function of the 2nd IMF
120
Frequency function
Smoothed frequency function
100
80
60
40
20
0
0
5
10
15
20
25
30
FIGURE 9.8 Original (dashed line) and smoothed (solid line) instantaneous frequency function of the second IMF.
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Simulation of Nonstationary Random Processes
205
Reduced 2nd IMF
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
50
100
150
200
250
300
350
FIGURE 9.9 Reduced (stationary) second IMF.
1.5
1
0.5
0
−0.5
−1
−1.5
3
4
5
6
7
8
9
10
1.5
1
0.5
0
−0.5
−1
−1.5
40
50
60
70
80
90
100
110
120
130
FIGURE 9.10 Effect of change in time scale to remove frequency modulation: original (top);
frequency modulation removed (bottom).
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The Hilbert-Huang Transform in Engineering
Simulated sample of reduced 2nd IMF
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
50
100
150
200
250
300
350
25
30
FIGURE 9.11 Simulated sample of reduced (stationary) second IMF.
Simulated (solid) and original (dotted) 2nd IMF
500
400
300
200
100
0
−100
−200
−300
−400
−500
0
5
10
15
20
FIGURE 9.12 Comparison of original (dotted line) and simulated (solid line) second IMF.
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Simulation of Nonstationary Random Processes
207
frequency and amplitude modulation functions faced by most currently available
methods. These new methods extract and preserve the true physical features of the
process. They have advantages over most currently available methods since assumptions about any functional forms of the spectra or about piecewise stationarity of
the process are unnecessary; difficulties associated with estimation of these analytical
functions from data are therefore bypassed. These methods are largely numerical
and therefore suited for computer-aided model-based simulations. Numerical examples on earthquake ground motion simulation were carried out. The sample ground
motions and records are shown to have the same time-varying intensity and spectral
characteristics as in the samples from the underlying nonstationary process. The
response spectra of the simulated ground motions retain the jagged features of the
spectra of the real records. The proposed HHT method has been shown to have great
potential in engineering applications when dealing with nonstationary processes.
ACKNOWLEDGMENTS
This research is supported by the University of Illinois Research Board and the NSF
Earthquake Engineering Research Center Program under Award Number
EEC-9701785. Information exchange with C. H. Loh of National Taiwan University
is greatly appreciated.
REFERENCES
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ASCE, Vol. 86, EM2: 1–16.
2. Kanai, K. (1957). Semi-empirical formula for the seismic characteristics of the
ground. Bull. Earthquake Res. Inst., Univ. Tokyo, Vol. 35, 309–325.
3. Tajimi, H. (1960). A statistical method of determining the maximum responses of a
building structure during an earthquake. Proc. Second World Conf. Earthquake Eng.,
Tokyo and Kyoto, Japan, Vol. II.
4. Housner, G. W. and Jenning, P. C. (1964). Generation of Artificial Earthquakes, J.
Eng. Mech. Div., ASCE, Vol. 90, EM1: 113–150.
5. Clough, R. W. and Penzien, J. (1975). Dynamics of Structures. McGraw-Hill Book
Co., New York.
6. Shinozuka, M. and Sato, Y. (1967). Simulation of nonstationary random processes.
J. Eng. Mech. Div., ASCE, Vol. 93, EM1: 11–40.
7. Amin, M. and Ang, A. H.-S. (1968). Nonstationary stochastic model for earthquake
motions. J. Eng. Mech. Div., ASCE, Vol. 94, EM2: 559–583.
8. Saragoni, G. R. and Hart, G. C. (1974). Simulation of artificial earthquakes. Earthquake Eng. Structural Dynamics, Vol. 2, 249–267.
9. Priestley, M. B. (1967). Power spectral analysis of non-stationary random processes.
J. Sound Vibration, Vol. 6, 1: 86–97.
10. Lin Y. K. and Yong, Y. (1987). Evolutionary Kanai-Tajimi earthquake models. J. Eng.
Mech. Div., ASCE, Vol. 113, 8: 1119–1137.
11. Deodatis, G. and Shinozuka, M. (1988). Auto-regressive model for nonstationary
stochastic processes. J. Eng. Mech., ASCE, Vol. 114, 4: 1995–2012.
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The Hilbert-Huang Transform in Engineering
12. Li, Y. and Kareem, A. (1991). Simulation of multivariate nonstationary random
processes by FFT. J. Eng. Mech., ASCE, Vol. 117, 5: 1037–1058.
13. Li, Y. and Kareem, A. (1997). Simulation of multivariate nonstationary random
processes: Hybrid DFT and digital filtering approach. J. Eng. Mech., ASCE, Vol. 123,
12: 1302–1310.
14. Der Kiureghian, A. and Crempien, J. (1989). An evolutionary model for earthquake
ground motion. Struct. Safety, 6: 235–246.
15. Papadimitriou, C. (1990). Stochastic characterization of strong ground motion and
application to structural response, Rep. No. EERL 90-03, California Institute of
Technology, Pasadena, Calif., U.S.A.
16. Conte, J. P. and Peng, B. F. (1997). Fully nonstationary analytical earthquake
ground-motion model. J. Eng. Mech., Vol. 123, 1: 15–24.
17. Iyama, J. and Kuwamura, H. (1999). Application of wavelets to analysis and simulation of earthquake motions. Earthquake Eng. Structural Dynamics, 28: 255–272.
18. Spanos, P. D. and Failla, G. (2002). Multi-scale modelling via wavelets of evolutionary spectra in mechanics applications. Proc. Int. Symp. Multiscaling in Mechanics,
Messini, Greece.
19. Spanos, P. D., Tratskas, P. and Failla, G. (2002). Wavelets applications in structural
dynamics. In: Structural Dynamics, EURODYN2002, Grundmann & Schueller (eds.).
Swets & Zeitinger, Lisse, ISBN 90 5809 510 X.
20. Grigoriu, M., Ruiz, S. E., and Rosenblueth, E. (1988). Nonstationary models of
seismic ground acceleration. Earthquake Spectra, 4: 551–568.
21. Yeh, C. H. and Wen, Y. K. (1990). Modelling of nonstationary ground motion and
analysis of inelastic structural response. Structural Safety, 8: 281–298.
22. Huang, N. E., et al. (1998). The empirical mode decomposition and the Hilbert
spectrum for nonlinear, nonstationary time series analysis. Proc. R. Soc. Lond. A,
454: 903–995.
23. Somerville, P., Smith, N., Punyamurthula, S., and Sun, J. (1997). Development of
ground motion time histories for Phase 2 of the FEMA/SAC steel project. Report
No. SAC/BD-97/04, SAC Joint Venture, Sacramento, California, U.S.A.
24. Bendat, J. S. and Piersol, A. G. (2000). Random Data, Analysis and Measurement
Procedures, 3rd ed. John Wiley & Sons, Inc., New York.
25. Zhang R. (2001). The role of Hilbert-Huang transform in earthquake engineering.
Proc. World Multiconference Systemics, Cybernetics Informatics, Vol. XVII, Orlando,
Florida, U.S.A.
26. Huang, et al. (2001). A new spectral representation of earthquake data: Hilbert spectral
analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seismological Soc. Am., Vol. 91, 5: 1310–1338.
27. Huang, N. E., Shen, Z. and Long, S. R. (1999). New view of nonlinear water waves:
The Hilbert spectrum. Annu. Rev. Fluid Mech., Vol. 31, 417–457.
28. Loh, C. H., Wu, T.C. and Huang, N. E. (2001). Application of the empirical mode
decomposition-Hilbert spectrum method to identify near-fault ground-motion characteristics and structural responses. Bull. Seismological Soc. Am., Vol. 91, 5:
1339–1357.
29. Yang, J. N. and Lei, Y. (2000). System identification of linear structures using Hilbert
transform and empirical mode decomposition. Proc. Int. Modal Anal. Conf., Vol. 1,
213–219.
30. Zhang, R. and Ma, S. (2000). HHT analysis of earthquake motion recordings and its
implications to simulation of ground motion. In: Monto Carlo Simulation, G. I. Schueller
and P.D. Spanos (eds.), 483–490. A. A. Balkema Publishers, Lisse, the Netherlands.
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Simulation of Nonstationary Random Processes
209
31. Zhang R. R. (2001). The signature of structural damage: An HHT view. In: Structural
Safety and Reliability, Corotis et al. (eds.), Swets & Zeitlinger, ISBN 90 5809 197 X.
32. Salvino, L. W. (2000). Empirical mode analysis of structural response and damping.
Proc. SPIE Int. Soc. Optical Eng., Vol. 4062, 503–509.
33. Gravier, B. M. et al. (2001). An assessment of the application of the Hilbert spectrum
to the fatigue analysis of marine risers. Proc. Int. Offshore Polar Eng. Conf., Vol. 2,
268–275.
34. Hartzell, S., Liu, P., and Mendoza, C. (1996). The 1994 Northridge, California
earthquake: Investigation of rupture velocity, risetime, and high-frequency radiation.
J. Geophys. Res., Vol. 101, 20091–20108.
35. Shinozuka, M. and Deodatis, G. (1991). Simulation of stochastic processes by spectral
representation. App. Mech. Reviews, Vol. 44, 4: 191–203
APPENDIX: RELATION BETWEEN HILBERT MARGINAL
SPECTRUM AND FOURIER ENERGY SPECTRUM
Consider a time series y(t) which allows Fourier transform, defined as follows:
y(ω ) =
∞
∫ y(t)e dt
− iωt
(9A.1)
−∞
Its complex conjugate is then
y∗ (ω ) =
∞
∫ y(t)e dt
iωt
(9A.2)
−∞
Now if we define the Fourier energy spectrum by
E f (ω ) = y(ω )
2
(9A.3)
then the area under the Fourier energy spectrum for all frequency is given by
∞
E=
∞ ∞ ∞
∫ y(ω) y (ω)dω = ∫ ∫ ∫ y(t ) y(t )e
∗
1
−∞
2
−∞ −∞ −∞
∞ ∞
=
∫ ∫ y(t ) y(t )2πδ(t − t )dt dt
1
−∞ −∞
© 2005 by Taylor & Francis Group, LLC
2
1
2
1
2
− iωt1 + iωt2
d ωdt1dt2
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210
The Hilbert-Huang Transform in Engineering
∞
= 2π
T
∫ y (t)dt = 2π ∫ y (t)dt
2
2
−∞
(9A.4)
0
if y(t) is defined for 0 < t < T, and zero otherwise. Therefore, the area under the
Fourier energy spectrum defined by Equation 9A.3 is equal to 2π times the total
energy of the time series, defined by the integral of the square of the time series.
The area under the Hilbert energy spectrum can be illustrated by a simple case
of a time series that is a summation of n sine functions defined for a given time
period 0 to T,
n
y(t ) =
∑ A sin(ω t) for 0 < t < T
j
j
(9A.5)
j =1
where T = mπ/2ωj for all j, m an integer.
When we apply EMD to y(t), we in effect recover the n sine functions as n
IMFs. From Equation 9.1, Equation 9.2, and Equation 9.3 (in the text), one can see
that if Cj(t) = Ajsinωjt, the Hilbert transform of each of the sine functions produces
Dj(t) = –Ajcosωjt, and a2j (t) = A2j . The Hilbert energy spectrum is therefore equal to
A2j at ωj for all time. For a finite period of time T, the discrete marginal spectrum is
equal to A2j T at ωj. The total area under the Hilbert marginal spectrum is therefore
n
∑ A T.
2
j
j =1
The total energy over the time period is given by
∫
n
∑
T
1
y (t )dt = T
Aj 2
2 j =1
0
2
(9A.6)
Hence, the total area under Hilbert marginal spectrum is
n
∑
∫ y (t)dt
T
A2j T = 2
2
(9A.7)
0
j =1
More generally, for an arbitrary time series,
n
y(t ) =
∑ a (t) ⋅ cos[θ (t)]
j
j =1
© 2005 by Taylor & Francis Group, LLC
j
(9A.8)
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Simulation of Nonstationary Random Processes
211
if we neglect the residual term from Equation 9.5. Its total area under the Hilbert
marginal spectrum is
n
T
∑ ∫ a (t)dt.
2
j
j =1 0
Its total energy over time is
∫
T
n
n
T
i =1
j =1
0
∑ ∑ ∫ a (t)a (t) cos[θ (t)]cos[θ (t)]dt
y 2 (t )dt =
i
0
j
j
i
(9A.9)
Due to the orthogonality of IMFs (see Huang et al. [22]), the cross-terms (i ≠ j) in
Equation 9A.9 can be neglected. Equation 9A.9 reduces to
∫
n
T
y 2 (t )dt =
0
∑ ∫ a (t) cos [θ (t)]dt
T
2
j
j =1
2
j
(9A.10)
0
When the IMFs have relatively smooth and sinusoidal shapes, for integer number
of quarter-waves, we have
∫
T
a 2j (t ) cos2 [θ j (t )]dt ≈
0
1
2
∫ a (t)dt
T
2
j
(9A.11)
0
For high-frequency IMFs, the number of quarter-waves is usually large, so the
portion that does not contain a complete quarter-wave is small and can be neglected.
For low-frequency IMFs, this portion cannot be neglected, but low-frequency IMFs
usually have small amplitudes and contribute insignificantly to the total energy.
Therefore one arrives at
∫
T
n
y 2 (t )dt ≈
0
∑ ∫ a (t)dt
1
2 j =1
T
2
j
(9A.12)
0
The total area under Hilbert marginal spectrum is
n
∑ ∫ a (t)dt ≈ 2 ∫ y (t)dt
T
T
2
j
j =1
0
which is the same as in Equation 9A.7.
© 2005 by Taylor & Francis Group, LLC
2
0
(9A.13)
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212
The Hilbert-Huang Transform in Engineering
Comparing Equation 9A.13 with Equation 9A.4, we conclude that the total area
under the Hilbert marginal spectrum needs to be multiplied by a factor, to be
comparable with the area under the Fourier energy spectrum defined by Equation
9A.3.
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10
Comparison of
Hilbert-Huang, Wavelet,
and Fourier Transforms
for Selected Applications
Ser-Tong Quek, Puat-Siong Tua, and Quan Wang
CONTENTS
10.1 Introduction ..................................................................................................214
10.2 Hilbert-Huang Transform.............................................................................215
10.3 Applications of HHT, WT, and FT to Experimental Data..........................216
10.3.1 Locating Crack in Aluminum Beam Using HHT ...........................216
10.3.2 Detection of Edges in Aluminum Plate...........................................219
10.3.3 Determination of Modal Frequencies of Aluminum Beam ............222
10.4 Concluding Remarks....................................................................................236
Acknowledgments..................................................................................................239
References..............................................................................................................243
ABSTRACT
This paper illustrates the suitability of three different signal-processing techniques,
namely, the Hilbert-Huang transform (HHT), the wavelet transform (WT), and the fast
Fourier transform (FFT), under different practical situations in relation to structural
health monitoring. Three experimentally obtained signals are analyzed. First, in the
time-of-flight analysis of flexural wave propagation in an aluminum beam, HHT is
found to be a more direct method compared to WT, as no knowledge of the actuation
frequency is required. However in the case of WT, if prior knowledge of the actuation
frequency is utilized combined with a refined scale search, WT produces more accurate
results. In another experiment involving the time-of-flight analysis of acoustic Lamb
wave propagation in an aluminum plate, piezoelectric actuators and sensors (PZTs) are
used to excite and receive direct and reflected waves at a specific frequency. HHT and
213
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214
The Hilbert-Huang Transform in Engineering
WT give similar results in terms of precision and accuracy. Last, a modified Fourier
transform (FT) is investigated based on the concept of HHT and applied to the signals
collected from the free vibration of an aluminum beam under three different support
conditions. The proposed method is able to reveal the presence of higher vibration
modes than can be revealed by conventional FT of the original signal. The results
produced are close to that of the more common WT. It is believed that supporters of
FT will be more receptive to this modified HHT method.
Keywords Hilbert-Huang transform, wavelet transform, fast Fourier transform,
structural health monitoring, flexural wave, Lamb wave, free vibration
10.1 INTRODUCTION
Signal processing has gradually become an indispensable tool in the field of engineering for extraction of important information from raw signals. In civil engineering, many practical and robust nondestructive evaluation (NDE) techniques developed for assessment of structural performance involve two key components, namely:
(1) data acquisition and (2) signal processing and interpretation. Although the use
of vibration measurements is a simpler and less costly method with respect to
instrumentation system in comparison with infrared thermography, ground-penetrating radar, acoustic emission monitoring, and eddy current detection, a key factor for
identifying damage precisely lies in having an appropriate data analysis method.
The most well known conventional signal processing technique is the Fourier
transform (FT). Fourier spectral analysis provides a general method for examining
the global energy–frequency distributions of a given signal. FT has high resolution
in the frequency domain but loses the resolution in the time domain. Despite this,
FT has been employed in health monitoring. For example, Crema et al. [1] used a
fast Fourier transform (FFT) analyzer to perform modal analysis to detect damage
in composite material structures. A broadband impulse input was applied, and the
damage that was induced by loads well above the working load caused a small
decrease in the eigen-frequencies for several modes. However, it can be problematic
to locate damage by using FT-based modal analysis techniques.
A few methods are available for processing signals of nonlinear and nonstationary systems, of which the short-time Fourier transform (STFT) method and wavelet
transform (WT) method are widely adopted. Cook and Berthelot [2] used the
time-dependent energy spectrum from STFT to distinguish backscattered signals
from a single crack against those from a series of closely spaced cracks on steel
surface. Hurlebaus et al. [3] used the time–frequency spectrum obtained by performing STFT on the Lamb waves generated by laser to obtain the group velocity–frequency domain. The notches are located from the autocorrelation plot of a series of
group velocity spectra at different assumed propagation distances.
Wavelet analysis is another adjustable window signal-processing technique to
handle nonstationary signals. Although WT appears similar to STFT, the basic wave
components are not limited to sinusoidal functions. Rather, classes of functions have
been proposed, from the simplest Haar wavelet to the more complicated higher order
Debauchies wavelet functions, to address various problems of time–frequency
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
215
resolution. WT has been widely used in the field of damage detection, such as in
modal-based detection (Staszewski [4]). An early work using WT to obtain a dispersive relation of group velocity of flexural waves in beam was presented by
Kishimoto et al. [5]. Lee and Liew [6] studied the sensitivity of WT in space domain
for identifying the damage location of a beam for small damage not easily detected
via conventional eigenvalue analysis. Quek et al. [7], on the other hand, used wavelet
coefficients over the time domain to locate cracks in beams via the time-of-flight
analysis of the wave propagation along the beam. However, WT has its shortcomings.
Experience shows that it can produce many spurious harmonics under different scales
that makes the analysis difficult or sometimes meaningless.
The most recently introduced signal-processing technique for analyzing nonlinear and nonstationary time series data is the Hilbert-Huang transform (HHT) [8].
Empirical mode decomposition (EMD) of the data is first performed to segregate
the signals into narrow band components, and the Hilbert transform (HT) is then
applied on each mode. Its ability to analyze nonlinear and nonstationary time series
data [8–10] has attracted many applications. For example, Zhang et al. [11] adopted
HHT to assess structural damage from vibration recordings. Shim et al. [12]
exploited HHT’s ability to exhibit nonlinear and nonstationary behavior to locate
damages on bridges by using frequency sweep and controlled excitations. Quek et
al. [13] also employed HHT to analyze structural dynamic responses to detect and
locate anomalies in beams and plates. In addition, HHT has been used to analyze
experimental data collected by oscilloscope to investigate the dominance of the
lowest antisymmetric (A0) and symmetric (S0) modes of Lamb wave vary across the
frequency range adopted for excitation and good agreement with, which agrees well
with theoretical response prediction in terms of their relative dominance [14].
Individual signal-processing techniques have their own strength and weaknesses.
Suitability of a particular technique may, therefore, be problem dependent, especially
with respect to structural health monitoring problems. This paper compares the three
signal-processing techniques, namely, HHT, WT, and FFT, for signals obtained
experimentally. The first example concerns the time-of-flight analysis of flexural
wave propagation in an aluminum beam, where WT and HHT are compared. Next,
we compare the analysis from the two techniques on a two-dimensional problem
involving the propagation of acoustic Lamb waves in an aluminum plate. Last, we
apply a modified FT based on the EMD concept of HHT to the free vibration signals
of an aluminum beam under three different support conditions, we compare these
results with results from using conventional direct FFT, HHT, and WT to obtain the
modal frequencies.
10.2 HILBERT-HUANG TRANSFORM
Hilbert transform was first developed to process nonstationary narrow-band signals [15].
To apply it to signals in general, Huang et al. [8] proposed using the empirical mode
decomposition technique to decompose any given signal into a set of narrow-band
signals. Each component (which may be nonstationary) of the set is termed an intrinsic
mode function (IMF) of the original signal, on the IMF where HT can be carried out.
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216
The Hilbert-Huang Transform in Engineering
The distinct features of a narrow-band signal, and hence an IMF, are that (a)
over its entire length, the number of extrema and the number of zero-crossings either
are equal or differ at most by one; and (b) at any point, the mean value of the
envelope of the signal defined by the local maxima and the envelope defined by the
local minima is zero. This forms the basis on which the EMD technique was
developed. For the EMD to be applicable, it is assumed that (a) the signal has at
least one maximum and one minimum; (b) the characteristic time scale is defined
by the time lapse between the extrema; and (c) if the data are totally devoid of
extrema but contain only inflection points, then the signal can be differentiated one
or more times to reveal the extrema. Hilbert spectral analysis can then be performed
on each IMF. MATLAB 6.1 is adopted for the implementation of the HHT in this
study.
10.3 APPLICATIONS OF HHT, WT, AND FT
TO EXPERIMENTAL DATA
An accurate signal-processing technique to obtain the flight times is crucial to locate
damages in structures. Both WT and HHT are potentially applicable. The advantages
of HHT for damage detection in beams using flight times from narrow-band nonstationary signals have been discussed by Quek et al. [13]. Waves in plates are even
more complex due to the dispersive nature of Lamb waves; hence, a technique that
provides good representation of localized events in both the frequency and energy
is vital for determining the location of the damage precisely. Another practical
problem of interest is the determination of higher frequencies from free vibration
data. In this section we examine a proposal to combine FT with HHT, using an
aluminum beam under three different boundary conditions as illustration.
10.3.1 LOCATING CRACK IN ALUMINUM BEAM USING HHT
Quek et al. [7] presented a wavelet analysis on experimental data to locate a crack
in an aluminum beam based on simple wave propagation considerations. The beam
considered has geometrical and material properties given in Table 10.1. Three sets
TABLE 10.1
Geometrical and Material Properties
of Aluminum Beam/Plate
Dimensions (mm)
Young’s modulus, E (GPa)
Shear modulus, G (GPa)
Mass density, ρ (kg/m3)
© 2005 by Taylor & Francis Group, LLC
Beam
Plate
650 × 32 × 6
73.1
—
2790
600 × 600 × 2
102
26
2700
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
217
FIGURE 10.1 Schematic view of wave propagation for impact at 200 mm position.
of data with different boundary conditions are analyzed, namely, fixed-fixed, simply
supported, and cantilever conditions. The basic concept is shown in Figure 10.1, in
a schematic view of wave propagation due to an impact. The piezoelectric sensor
captures the wave signal, which contains timings of the direct, damage-reflected,
and boundary-reflected waves. By estimating these timings with a signal-processing
technique, the damage location can be deduced. In view of the dispersive nature of
the wave, only a wave at a particular frequency, with a corresponding wave propagation velocity, is used so that accurate results can be obtained.
HHT is applied in this study on the same three sets of signals. One of the signals
obtained (and shown in Figure 10.2a) was decomposed via EMD into its IMF
components, shown in Figure 10.2b. It can be observed that the first IMF component
has higher amplitude than the other IMFs. This IMF component is then subjected
to HT, and the corresponding frequency–time and energy–time distributions plotted
in Figure 10.2c and d, respectively. From the energy–time distribution, the timings
of the energy peaks in the signal, which are due to the propagating wave passing
the sensor, are obtained. The frequencies corresponding to the energy peaks are also
noted so as to establish that they do not deviate significantly from each other. It
should be pointed out that not all IMFs are used, as the first IMF is distinct from
the others and contains only the frequency of interest, whereas other IMFs do not.
The results for the estimated damage position are summarized in Table 10.2 and
compared with those obtained with Gabor wavelet analysis with coarse (discrete
WT, scale = 12) and fine (“continuous” WT or CWT, scale = 12.2) resolution. The
wavelet results have been discussed in detail by Quek et al. [7], including the 3D
wavelet plot, and will not be repeated here. It can be observed that the HHT method
is able to identify the damage position with reasonable accuracy; the results are
better than the discrete wavelet results but worse than the continuous wavelet results.
However, using CWT for this application requires determining an appropriate scale
to process the final results. From past experience, this requires a fair amount of skill,
as the selection can be rather subjective and may lead to varied end results (see, for
© 2005 by Taylor & Francis Group, LLC
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(a)
Frequency (Hz)
10
0
−10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
−3
c(2)
0
×10
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.5
1
1.5
2
2.5
3
3.5
4
4.5 5
−3
×10
5
0
−5
0
1
2
3
4
5
5
−3
×10
0
−5
(d)
1
Residue
Amplitude
1
0
−5
(c)
Time, t(s)
5
4
3
2
1
0
0 0.5
5
Time, t(s)
5000
4000
3000
2000
1000
0
(b)
c(3)
10
8
6
4
2
0
−2
−4
−6
−8
−10
c(1)
The Hilbert-Huang Transform in Engineering
c(4)
Input
218
0
1
2
3
4
5
×10
Time, t(s)
0
−1
−2
−3
Time, t(s)
FIGURE 10.2 (a) Dynamic response of a beam; (b) IMFs of response signal after EMD; (c)
frequency; (d) energy spectra of first IMF component.
TABLE 10.2
Comparison of Results Based on HHT Method and Wavelet Analysis
Wavelet Analysis
HHT Analysis
Scale 12
Scale 12.2
End Condition
Estimated
Damage
Position
Error
(%)
Estimated
Damage
Position
Error
(%)
Estimated
Damage
Position
*Error
(%)
Fixed ended
Simply Supported
Cantilever
437.5
471.7
403.8
–2.78
4.83
–10.26
492.0
493.0
542.0
9.33
9.56
20.00
450
457
457
0.00
2.00
2.00
Note: Actual damage position is 450 mm from left of the beam.
© 2005 by Taylor & Francis Group, LLC
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
D
219
C
600
500
Sensor
400
Actuator
50 mm
300
75 mm
Line crack
200
100
A
y
0
0
X
100
200
300
400
500
600 B
FIGURE 10.3 Schematic view of PZT positions on plate.
example, Table 3 of Quek et al. [7]). This is due to the shortcomings of WT, which
produces false harmonics depending on the scale adopted, although WT also has
the ability to achieve high resolution in both frequency and time domain. The
advantage of HHT over WT in this case is that the HHT procedure is more direct
and can be performed to obtain the desired accuracy for the results without prior
knowledge of the excitation frequency (which in this case of impact load is broadband). It should be noted that there are other applications where CWT may be more
appropriate.
10.3.2 DETECTION OF EDGES IN ALUMINUM PLATE
In this example, we adopted the time-of-flight analysis of acoustic Lamb wave
propagation in a 600 mm × 600 mm aluminum plate to detect the plate’s edges, In
the experiment, a through crack, 1 mm wide and 30 mm long, inclined at 30º to one
side of the plate (side AB in Figure 10.3), was cut with a milling machine. The
location of the center of the crack is (270, 290) mm. Piezoelectric sensors and
actuators (PZTs) were used to excite and receive direct and reflected waves at a
specific frequency. A Lamb wave with a narrow-band (600 kHz) actuation pulse
(Figure 10.4) was activated based on the following equation:
(
)
X (t ) = cos 2πω t e − a (t −t0 )
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2
(10.1)
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220
The Hilbert-Huang Transform in Engineering
20
15
Amplitude (V)
10
5
0
−5
−10
−15
−20
1.5
2
2.5
Time (s)
3
3.5
4
×10−5
FIGURE 10.4 Actuation pulse using Equation 10.1 at frequency 600 kHz.
where ω is the actuation frequency (Hz), and a and t0 are constants. This was done
with a function generator (Yokogawa FG300 15 MHz Synthesized FG) to feed a
PZT actuator (8.0 mm × 8.0 mm × 0.5 mm). The responses were captured by a PZT
sensor placed at a distance away and channeled to an oscilloscope (Yokogawa DL716
16-Channel) for monitoring and analysis. Figure 10.3 shows the positions of the
actuator and sensor. By estimating the flight time due to the propagation of either
a S0 or an A0 Lamb wave, the distance traveled by the wave from the actuator to
the sensor can be computed as
li = ∆ ti cg,
(10.2)
where ti is the time of flight referenced from the actuation time, and cg is the group
velocity of either the S0 or A0 mode, determined either theoretically or experimentally.
The response signal captured by the PZT sensor (Figure 10.5a) was processed
by both HHT and Gabor WT. The resulting energy spectra are given in Figure 10.5d
and Figure 10.6. In the HHT analysis, the peaks in the energy–time spectrum for
the IMF component containing the highest energy (in this case the first IMF) give
the wave arrival times of interest. Based on these energy peaks, the distances traveled
by the wave from the actuator to the sensor can be computed; these are summarized
in Table 10.3. It can also be observed from the frequency spectrum (see Figure
10.5b) that the frequency corresponding to the energy peaks is very close to the
actuation frequency, which indicates the accurate representation of localization
events by HHT. On the other hand, Gabor WT analysis on the same signal response
was performed based on the prior knowledge that the actuation frequency is 600
kHz, which corresponds to a scale of 20.2. Similar energy peaks were obtained,
which accounted for the wave pulses on the different paths received by the sensor.
The results obtained by WT analysis are also given in Table 10.3. These indicate
© 2005 by Taylor & Francis Group, LLC
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0
1
2
−4
×10
c(1)
0.05
0
−0.05
0.02
0
−0.02
c(5)
Time, t(s)
1
0
−1
c(2)
(b)
(a)
c(3)
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
c(4)
Amplitude
Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
5
(c)
c(7)
10
5
c(8)
0
1
2
×10
Time, t(s)
1
A0/S0
(incident)
0.8
(d)
A0
(crack)
0.6
−4
c(9)
0
c(10)
−5
Amplitude
c(6)
×10
c(11)
Frequency, (Hz)
15
A0
(boundary, CD)
0.4
0
Res
0.2
0
1
2
Time, t(s)
×10
−4
221
0
1
2
0
1
2
0
0.02
0
−0.02
0
0.01
0
−0.01
0
0.01
0
−0.01
0 ×10−3
5
0
−5
0
0.01
0
−0.01
0
0.2
0
−0.2
0
0.02
0
−0.02
0
0.1
0
−0.1
0
0.1
0
−0.1
0
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
Time, t(s)
2
−4
×10
FIGURE 10.5 (a) Response signal captured by sensor on aluminum plate; (b) IMFs of
response signal after EMD; (c) frequency; (d) energy spectra of first IMF component.
1.2E–03
Amplitude
1.0E–03
A0/S0
(incident)
8.0E–04
A0
(crack)
6.0E–04
A0
(boundary, CD)
4.0E–04
2.0E–04
0.0E+00
0.0E+00
5.0E–05
1.0E–04
Time (s)
1.5E–04
FIGURE 10.6 Gabor wavelet analysis of response signal given in Figure 10.5a.
© 2005 by Taylor & Francis Group, LLC
2.0E–04
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222
The Hilbert-Huang Transform in Engineering
TABLE 10.3
Results of Wave Propagation in Aluminum Plate
Signal
Analysis
Technique
HHT
WT
Theoretical
Velocity
(m/s)
Distance Traveled
by Incident Wave
(mm)a
Distance Traveled
by Reflected Wave
from Line Crack
(mm)b
Distance Traveled
by Reflected Wave
from Boundary
(CD; mm)c
A0
A0
Error
(%)
A0
Error
(%)
A0
Error
(%)
3134
52.9
52.7
5.80
5.40
196.1
196.4
–1.95
–1.80
425.0
424.0
–1.62
–1.85
a
Actual distance traveled by incident wave is 50 mm.
Actual distance traveled by crack-reflected wave is 200 mm.
c Actual distance traveled by boundary (CD)-reflected wave is 432 mm.
b
that HHT and WT give similar results in terms of precision and accuracy. HHT does
not require prior knowledge of the activation frequency.
10.3.3 DETERMINATION OF MODAL FREQUENCIES OF ALUMINUM BEAM
In this part of the study, we consider an aluminum beam under three support
conditions, shown in Figure 10.7, with material properties similar to the plate in
Section 10.3.2 (see Table 10.1). The theoretical modal frequencies of the beam were
first obtained (see Table 10.4) by using the following equation [16]:
fn =
(β nl )2
2πl 2
EI
,
ρA
n = 1, 2, 3, ...
(10.3)
where l, E, I, ρ, and A are the length, Young’s modulus, moment of inertia, density,
and cross-sectional area of the beam, respectively; n corresponds to the vibration
modes; and βn is solved from the following equations, respectively, for cantilever,
simply supported, and fixed-fixed conditions.
cos β n l cosh β n l = −1
(10.4a)
sinβ n l = 0
(10.4b)
cos β n l cosh β n l = 1
(10.4c)
For the experiment, the free vibration responses of the beam under the three different
support conditions (Figure 10.7) were monitored and collected by an oscilloscope
(Yokogawa DL716 16-Channel; see Figure 10.8). FFT was first performed directly on
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
Piezoelectric
x Sensor
223
Aluminum
beam
6 mm
(a)
32 mm
y
0 mm
150 mm
Piezoelectric
x Sensor
(b)
920 mm
Aluminum
beam
6 mm
32 mm
y
0 mm
800 mm
150 mm
Piezoelectric
x Sensor
Aluminum
beam
6 mm
(c)
32 mm
y
0 mm
150 mm
800 mm
FIGURE 10.7 Schematic setup of aluminum beam under different support conditions: (a)
cantilever; (b) simply supported; (c) fixed-fixed for modal frequency analysis.
the response signal. For example, Figure 10.9a and b shows the linear and logarithmic
plots of the Fourier spectrum, respectively, for the cantilevered beam. In the linear
plot of the Fourier spectrum (Figure 10.9a), the first two modal frequencies of the
beam can be easily identified (5.99 Hz and 34.93 Hz respectively). However, the
third mode may be easily overlooked due to the low energy of the displayed peak.
On the other hand, the logarithmic scale plot of the spectrum (Figure 10.9b) exhibits
the first three modal frequencies clearly. This is also true for the fixed-fixed support
condition (see Figure 10.10). However, for the simply supported condition, it is not
easy to distinguish the modes from the logarithmic scale plot of the Fourier spectrum.
It can be observed in Figure 10.11b that besides the three fundamental modal
frequencies, there are also a number of energy peaks at other frequency ranges
having amplitudes of order close to the modal frequencies. Hence for beams with
closely spaced frequencies, the Fourier spectrum may not be appropriate.
EMD was performed on the responses given in Figure 10.8. Consider the results
for the cantilevered beam: three decomposed IMF components and one residual
component are obtained, as depicted in Figure 10.12a. The individual IMFs are
subjected to FT, and the resulting spectra are shown in Figure 10.12b through d.
The residual component is not subjected to FT as it represents the mean or residual
trend of the captured response signal. Figure 10.12b through d clearly identifies the
© 2005 by Taylor & Francis Group, LLC
FFT
Mode
Cantilever
1st
2nd
3rd
1st
2nd
3rd
1st
2nd
3rd
4th
5.86
36.73
102.86
22.77
87.03
195.84
49.33
135.97
266.56
440.64
Simply supported
Fixed-fixed
EMD + FT
HHT
WT
Freq
(Hz)
Error
(%)
Freq
(Hz)
Error
(%)
Freq
(Hz)
Error
(%)
Freq
(Hz)
Error
(%)
5.99
34.93
96.80
22.89
81.80
165.80
48.00
138.00
268.00
446.00
2.20
–4.91
–5.89
5.17
–6.01
–15.34
–2.69
1.49
0.54
1.22
5.76
35.23
98.00
22.95
82.85
167.50
48.00
138.00
—
460.00
-1.72
–4.09
–4.73
5.44
–4.81
–14.47
–2.69
1.49
—
4.39
5.77
34.63
96.62
22.80
77.88
159.18
46.96
135.34
263.19
—
1.52
5.72
6.06
4.75
–10.52
–18.72
4.80
0.47
1.26
—
5.64
35.86
101.56
23.20
81.25
162.50
49.85
135.42
270.83
451.39
3.73
2.39
1.26
6.59
–6.65
–17.02
1.06
–0.41
1.60
2.44
The Hilbert-Huang Transform in Engineering
Support Conditions
Theoretical
Frequency (Hz)
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224
© 2005 by Taylor & Francis Group, LLC
TABLE 10.4
Analysis of Modal Frequencies of Aluminum Beam under Different Support Conditions Obtained
by FFT, EMD Followed by FT, HHT, and WT
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
25
225
(a)
20
15
Amplitude
10
5
0
−5
−10
−15
−20
−25
0
0.2
0.4
0.6
0.8
1
Time, t(s)
15
(b)
10
Amplitude
5
0
−5
−10
−15
0
0.2
0.4
0.6
0.8
1
Time, t(s)
25
(c)
20
15
Amplitude
10
5
0
−5
−10
−15
−20
0
0.1
0.2
0.3
0.4
0.5
Time, t(s)
FIGURE 10.8 Free vibration response of aluminum under (a) cantilever; (b) simply supported; (c) fixed-fixed conditions.
© 2005 by Taylor & Francis Group, LLC
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226
The Hilbert-Huang Transform in Engineering
2.5
×104
1st Mode
= 5.99 Hz
(a)
Amplitude
2
1.5
1
0.5
0
2nd Mode
= 34.93 Hz
0
50
rd
3 Mode
= 96.8 Hz
100
150
Frequency (Hz)
200
105 1st Mode
= 5.99 Hz
10
4
(b)
2nd Mode
= 34.93 Hz
103
Amplitude
250
rd
3 Mode
= 96.8 Hz
102
101
100
10
−1
0
50
100
150
Frequency (Hz)
200
250
FIGURE 10.9 FFT analysis of response of cantilever beam given in Figure 10.8a, plotted on
(a) linear scale and (b) log-scale.
first three modal frequencies of the beam (5.76 Hz, 35.23 Hz, and 96.8 Hz respectively). Even for the third mode, the appearance of the model frequencies is very
distinctive because the EMD process has effectively decomposed the response signal
into its respective components, which are narrow band. Hence, the energy peak of
the third mode is not obscured by the first and second modes, which contain relatively
higher energies. Similar analyses are done for the same beam under the two other
support conditions, and the results (see Figure 10.13 and Figure 10.14) obtained are
comparable to those obtained by the conventional direct FT. However, for the case
of the fixed-fixed condition, the third modal frequency could not be identified by
the FT method (Figure 10.14). Instead, the first and second modal frequencies are
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
1200
227
(a)
st
1 Mode
= 22.89 Hz
1000
Amplitude
800
600
400
200
0
nd
2 Mode
= 81.8 Hz
0
50
100
150
Frequency (Hz)
Amplitude
10
10
0
1 Mode
= 22.89 Hz
200
nd
2 Mode
= 81.8 Hz
250
(b)
rd
st
2
rd
3 Mode
= 165.8 Hz
3 Mode
= 165.8 Hz
−2
10
−4
10
0
50
100
150
Frequency (Hz)
200
250
FIGURE 10.10 FFT analysis of response of fixed-fixed beam given in Figure 10.8c, plotted
on (a) linear scale and (b) log-scale.
each contained in two IMF components (fourth and fifth IMFs and second and third
IMFs respectively; see Figure 10.14). On closer observation, the third and fourth
modal frequencies can be spotted in the Fourier spectrum of the second IMF component, but they are obscured by the stronger presence of the second modal frequency. This setback of the proposed method can be explained by the fact that IMFs
may contain mix modes. This may occur when the frequency of interest in an IMF
is only contained along a particular segment of the signal and not throughout the
entire time domain. This is the so-called intermittency problem discussed by Huang
et al [17].
The original HHT was also performed on the decomposed IMF components of
the response signals for the beam under the different support conditions. However,
as mentioned earlier, the frequency of interest in an IMF may be contained only
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228
The Hilbert-Huang Transform in Engineering
1400
nd
2 Mode
= 138 Hz
st
1 Mode
= 48 Hz
1200
(a)
Amplitude
1000
800
600
400
3rd Mode
= 268 Hz
200
0
10
4
10
3
10
2
10
1
10
0
0
100
st
200
300
Frequency (Hz)
400
500
nd
1 Mode
= 48 Hz
(b)
2 Mode
= 138 Hz
rd
3 Mode
= 268 Hz
Amplitude
th
4 Mode
= 446 Hz
−1
10
−2
10
10−3
0
100
200
300
Frequency (Hz)
400
500
FIGURE 10.11 FFT analysis of response of simply supported beam given in Figure 10.8b,
plotted on (a) linear scale and (b) log-scale.
along a particular segment of the signal; hence, a weighted average using the
amplitude is performed on segments of the frequency spectrum from HT for each
IMF to obtain the modal frequencies of the beam. To illustrate, spectra for the beam
under the three support conditions will be presented in detail individually.
First, consider the HT of the IMFs for the cantilever beam (Figure 10.15). From
the Hilbert spectra of the three IMF components, it can be observed that the EMD
has effectively decomposed the original response signal into its different components, which are well contained within a narrow band, as the fluctuation of the
frequency along the time axis is rather small. For this case, weighted averaging was
performed over the entire time domain for each of the IMF components. The modal
frequencies obtained are 101.56 Hz, 35.86 Hz, and 5.64 Hz from the first, second,
and third IMFs, respectively, which corresponds well with the theoretical values.
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
229
c(1)
5
0
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 (a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c(2)
10
0
−10
c(3)
20
0
−20
Residue
10
5
0
−5
Time, t(s)
180
rd
(c)
300
120
100
80
60
250
200
150
40
100
20
50
0
nd
2 Mode
= 35.23 Hz
350
Amplitude
Amplitude
140
400
(b)
3 Mode
= 96.8 Hz
160
0
50
100
150
200
250
0
0
50
100
150
200
250
Frequency (Hz)
Frequency (Hz)
4
3.5
×10
st
Amplitude
(d)
1 Mode
= 5.76 Hz
3
2.5
2
1.5
1
0.5
0
0
50
100
150
200
250
Frequency (Hz)
FIGURE 10.12 (a) IMF components of response of cantilever beam given in Figure 10.8a;
FT analysis of (b) first; (c) second; (d) third IMF component.
Next, for the simply supported beam, it can be observed in the HT spectra shown
in Figure 10.16 that the frequency is stable only over a certain time range for each
IMF. For example, for the first IMF, the frequency has a small fluctuation between
© 2005 by Taylor & Francis Group, LLC
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230
The Hilbert-Huang Transform in Engineering
c(1)
5
0
−5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 (a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c(2)
5
0
−5
c(3)
10
0
−10
Residue
1
0
−1
−2
Time, t(s)
45
rd
3.5
30
25
20
15
2
1.5
1
0.5
50
100
150
200
250
900
rd
700
0
50
100
150
200
250
(d)
1 Mode
= 22.95 Hz
800
0
Frequency (Hz)
Frequency (Hz)
Amplitude
3
5
0
(c)
2.5
10
0
nd
2 Mode
= 82.85 Hz
4
Amplitude
Amplitude
35
4.5
(b)
3 Mode
= 167.5 Hz
40
600
500
400
300
200
100
0
0
50
100
150
200
250
Frequency (Hz)
FIGURE 10.13 (a) IMF components of response of simply supported beam given in Figure
10.8b; FT analysis of (b) first; (c) second; (d) third IMF component.
0.1 sec and 0.3 sec; hence, a weighted average on the frequency is done over this
time range. The frequency obtained is 159.18 Hz, which is closer to the third
vibration mode of the beam. Similar analysis is done for the second IMF (between
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
231
c(1)
10
0
c(2)
−10
0
5
c(3)
c(4)
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5 (a)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
−10
0
5
0
−5
0
2
c(5)
0.1
0
−5
0
10
0
−2
0
0.5
Residue
0.05
0
−0.5
0
Time, t(s)
0.7
90
(b)
Amplitude
Amplitude
70
0.5
0.4
0.3
0.2
60
rd
3 Mode
(268 Hz)
50
40
th
30
4 Mode
(460 Hz)
10
0
100
200
300
400
500
0
600
0
100
600
nd
300
400
500
st
600
400
300
200
(e)
1 Mode
= 48 Hz
300
Amplitude
Amplitude
350
(d)
2 Mode
= 138 Hz
500
200
Frequency (Hz)
Frequency (Hz)
250
200
150
100
100
0
(c)
20
0.1
0
nd
2 Mode
= 138 Hz
80
0.6
50
0
100
200
300
400
Frequency (Hz)
500
600
0
0
100
200
300
400
500
600
Frequency (Hz)
FIGURE 10.14 (a) IMF components of response of fixed-fixed beam given in Figure 10.8c;
and FT analysis of (b) first; (c) second; (d) third; (e) fourth IMF; (f) fifth IMF component.
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The Hilbert-Huang Transform in Engineering
45
st
(f )
1 Mode
= 48 Hz
40
35
Amplitude
30
25
20
15
10
5
0
0
100
200
300
400
Frequency (Hz)
500
600
FIGURE 10.14 (continued)
0.12 sec and 0.28 sec), and the result (77.88 Hz) gives the second vibration mode.
For the third IMF, a rather stable fluctuation of the frequency is observed over the
entire time domain; hence, a weighted average is taken over the entire data range. This
obviously gave the result corresponding to the first fundamental frequency (22.80 Hz).
The Hilbert spectra for the IMF components of the fixed-fixed condition are
shown in Figure 10.17. Considering the first IMF, a large fluctuation of the frequency
over the entire time domain is observed. This can be explained by the fact that the
filtered component effectively contains the noise in the original response signal;
hence, meaningful modal parameters of the beam can be extracted from it. The
second IMF reveals a stable frequency region with high amplitude over the time
domain between 0.05 sec and 0.12 sec. Taking the weighted average over this time
range gives the second modal frequency (135.34 Hz). For the third IMF, two stable
frequency fluctuation regions with high amplitude can be identified, namely, from
0 to 0.16 sec and from 0.16 to 0.46 sec. By performing a weighted average on the
two regions, two fundamental frequencies are obtained (138 Hz and 263.19 Hz). A
similar stable region is observed for the fourth IMF, from 0.04 to 0.15 sec; this
corresponds to an averaged frequency of 46.96 Hz, which gives the first modal
frequency of the beam. In the fifth IMF, the two stable regions can be pinpointed,
from 0 to 0.15 sec and from 0.24 to 0.48 sec. These two regions give a weighted
average frequency of 42 Hz and 43.3 Hz respectively.
Last, wavelet analysis using the Morlet function was also performed on the three
signals of the beam under the three support conditions. WT was first done on the
original response signals for the beam over a large scale range. By observing the
dominant energy contents in this overall spectrum, different scales were zoomed in
to obtain the dominant frequency contents. For example, WT was first performed
for the vibration response of the cantilevered beam given in Figure 10.8a over a
wide scale range of 1 to 256. This spectrum plot (see Figure 10.18a) shows three
regions with a consistent existence of a corresponding frequency throughout the
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Frequency, (Hz)
Frequency, (Hz)
Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
500
450
400
350
300
250
200
150
100
50
0
100
90
80
70
60
50
40
30
20
10
0
233
2.0
1.0 (a)
0
0
0.2
0.4
0.6
Time, t(s)
0.8
1
8.0
4.0 (b)
0
0
0.2
0.4
0.6
Time, t(s)
0.8
1
60
20
Frequency, (Hz)
50
40
10
30
(c)
20
10
0
0
0
0.2
0.4
0.6
Time, t(s)
0.8
1
FIGURE 10.15 HT spectrum for (a) first, (b) second, and (c) third IMF component for
response of cantilever beam as given in Figure 10.12a.
entire time domain, namely, in scales 105 to 183, 14 to 27, and 4 to 12. By focusing
toward these regions, a more refined dominant frequency can be deduced, namely
at scales of 144, 22.66, and 8, which corresponds to frequencies of 5.64 Hz, 35.86
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Frequency, (Hz)
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The Hilbert-Huang Transform in Engineering
500
450
400
350
300
250
200
150
100
50
0
4.0
2.0 (a)
0
0
0.2
0.4
0.6
0.8
1
Time, t(s)
300
3.0
Frequency, (Hz)
250
200
1.5 (b)
150
100
50
0
0
0
0.2
0.4
0.6
0.8
1
Time, t(s)
350
8
Frequency, (Hz)
300
250
200
4
(c)
150
100
50
0
0
0
0.2
0.4
0.6
0.8
1
Time, t(s)
FIGURE 10.16 HT spectrum for (a) first, (b) second, and (c) third IMF component for
response of simply supported beam as given in Figure 10.13a.
Hz, and 101.56 Hz respectively. These results give the three fundamental frequencies
of the beam. The wavelet analyses for the other two support conditions are presented
in Figure 10.19 and Figure 10.20.
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Frequency, (Hz)
Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
235
10
5
(a)
0
0
0.1
0.2
0.3
0.4
0.5
Time, t(s)
2500
5
Frequency, (Hz)
2000
1500
2.5 (b)
1000
500
0
0
0
0.1
0.3
0.2
0.4
0.5
Frequency, (Hz)
Time, t(s)
2000
1800
1600
1400
1200
1000
800
600
400
200
0
10
5
(c)
0
0
0.1
0.2
0.3
0.4
0.5
Time, t(s)
FIGURE 10.17 HT spectrum for (a) first, (b) second, (c) third, (d) fourth, and (e) fifth IMF
component for response of fixed-fixed beam as given in Figure 10.14a.
The results obtained by the different signal analysis methods are compared in
Table 10.4. It can be observed that all the methods produce results of approximately
the same order of error. The methods are equally feasible in determining the modal
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The Hilbert-Huang Transform in Engineering
800
6
Frequency, (Hz)
700
600
500
3
400
(d)
300
200
100
0
0
0
0.1
0.2
0.3
0.4
0.5
Time, t(s)
300
2
Frequency, (Hz)
250
200
1
150
(e)
100
50
0
0
0
0.1
0.2
0.3
0.4
0.5
Time, t(s)
FIGURE 10.17 (continued)
frequencies of the beam. However, these methods do contain imperfections. For
example, the direct FT method may not be very effective for cases where the modal
frequencies have energy of same order as that of other frequency ranges. Similarly,
for the modified FT with EMD process, the identification of certain modal frequencies may not be possible when the IMF components contain mixed modes. For HHT,
the definition of the stable frequency region over the time domain, although subjective, is least problematic. For WT, identification of the precise scale from a range
of scales for the dominant frequency can be ambiguous.
10.4 CONCLUDING REMARKS
A comparison of three signal-processing techniques is presented. In the first study of
flexural wave propagation in an aluminum beam, HHT is shown to be a more direct
method compared to WT when no knowledge of the actuation frequency is available.
In the second experiment, involving acoustic Lamb wave propagation in an aluminum
plate, HHT and WT give similar results for the analysis of the propagating Lamb waves
actuated and received by the PZTs. Good WT results hinge on the fact that the actuation
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Scale
Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
237
248
235
222
209
196
183
170
157
144
131
118
105
92
79
66
53
40
27
14
1
(a)
Scale
Scale of colors from MIN to MAX
Time
144
Scale of colors from MIN to MAX
Amplitude
(b)
300
200
100
0
−100
−200
−300
Coefficients Line - Ca, b for scale a = 144 (frequency = 5.642)
100
200
300
400
500
Time
600
700
800
900
1000
FIGURE 10.18 Morlet WT analysis for response of cantilever beam given in Figure 10.8a
at scales (a) 1–256, (b) 144, (c) 22.66, and (d) 8 with sampling rate 1000 samples/sec.
frequency is narrowband and known a priori. For the last case, the combination of the
EMD process from HHT with FT on the decomposed IMF components has proven to
be capable of revealing modal frequencies of the aluminum beam having relatively
lower energy contents. These modal frequencies may be overlooked by the traditional
direct FFT, unless a logarithmic plot is done for the energy spectrum. However, the
problem of mixed modes within individual IMFs needs to be addressed. The original
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Scale
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The Hilbert-Huang Transform in Engineering
22.66
Scale of colors from MIN to MAX
(c)
Coefficients Line - Ca, b for scale a = 22.66 (frequency = 35.856)
Amplitude
50
0
Scale
−50
100
200
300
400
500
Time
600
700
800
900
1000
8
Scale of colors from MIN to MAX
(d)
Coefficients Line - Ca, b for scale a = 8 (frequency = 101.563)
Amplitude
15
10
5
0
−5
−10
100
200
300
400
500
Time
600
700
800
900
1000
FIGURE 10.18 (continued)
HHT with knowledge of the problem does provide good results when it is used with
an averaging concept. These examples show that HHT is a suitable tool for processing
nonstationary wave propagation signals encountered in damage detection studies. It
provides a good representation of localized events in both the frequency and energy of
any transient signal collected. Indeed, it is a simple signal-processing tool to use and
provides reasonably good results.
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
239
Scale
248
235
222
209
196
183
170
157
144
131
118
105
92
79
66
53
40
27
14
1
(a)
Scale
Scale of colors from MIN to MAX
Time
35
Scale of colors from MIN to MAX
(b)
Coefficients Line - Ca, b for scale a = 35 (frequency = 23.214)
Amplitude
40
20
0
−20
−40
100
200
300
400
500
Time
600
700
800
900
1000
FIGURE 10.19 Morlet WT analysis for response of simply supported beam given in Figure
10.8b at scales (a) 1–256, (b) 33.5, (c) 10, and (d) 5 with sampling rate 1000 samples/sec.
ACKNOWLEDGMENTS
The authors wish to thank the National University of Singapore for providing the
support to perform this study.
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The Hilbert-Huang Transform in Engineering
10
4
3
2
1
0
−1
−2
−3
−4
Scale
Amplitude
Scale of colors from MIN to MAX
(c)
Coefficients Line - Ca, b for scale a = 10 (frequency = 81.250)
100
200
300
400
500
Time
600
700
800
900
1000
5
Scale of colors from MIN to MAX
Amplitude
(d)
8
6
4
2
0
−2
−4
−6
−8
Coefficients Line - Ca, b for scale a = 5 (frequency = 162.500)
100
200
FIGURE 10.19 (continued)
© 2005 by Taylor & Francis Group, LLC
300
400
500
Time
600
700
800
900
1000
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
241
1958
1855
1752
1649
1546
1443
1340
1237
1134
1031
928
825
722
619
516
413
310
207
104
1
(a)
Scale
Scale of colors from MIN to MAX
Time
163
Scale of colors from MIN to MAX
Amplitude
(b)
80
60
40
20
0
−20
−40
−60
−80
Coefficients Line - Ca, b for scale a = 163 (frequency = 49.847)
500
1000
1500
2000
2500
Time
3000
3500
4000
4500
5000
FIGURE 10.20 Morlet WT analysis for response of fixed-fixed beam given in Figure 10.8c
at scales (a) 1–2042, (b) 163, (c) 60, (d) 28, and (e) 18 with sampling rate at 10000 samples/sec.
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The Hilbert-Huang Transform in Engineering
60
Scale of colors from MIN to MAX
80
60
40
20
0
−20
−40
−60
−80
Scale
Amplitude
(c)
Coefficients Line - Ca, b for scale a = 60 (frequency = 135.417)
500
1000
1500
2000
2500
Time
3000
3500
4000
4500
5000
30
Scale of colors from MIN to MAX
(d)
Coefficients Line - Ca, b for scale a = 30 (frequency = 270.833)
15
Amplitude
10
5
0
−5
−10
−15
500
1000
FIGURE 10.20 (continued)
© 2005 by Taylor & Francis Group, LLC
1500
2000
2500
Time
3000
3500
4000
4500
5000
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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms
243
163
Scale of colors from MIN to MAX
Amplitude
(e)
80
60
40
20
0
−20
−40
−60
−80
Coefficients Line - Ca, b for scale a = 163 (frequency = 49.847)
500
1000
1500
2000
2500
Time
3000
3500
4000
4500
5000
FIGURE 10.20 (continued)
REFERENCES
1. Crema L B, Peroni I and Castellani A (1985). Modal Tests on Composite Material
Structures Application in Damage Detection. Proc. Int. Modal Anal. Conf. Exhibit
Vol. 2, pp. 708–713.
2. Cook D A and Berthelot Y H (2001). Detection of Small Surface-Breaking Fatigue
Cracks in Steel Using Scattering of Rayleigh Waves. NDT & E Int. 34:483–492.
3. Hurlebaus S, Niethammer, Jacobs L J and Valle C (2001). Automated Methodology
to Locate Notches with Lamb Waves. Acous. Soc. Am. ARLO 2(4):97–102.
4. Staszewski W J (1998) Structural and Mechanical Damage Detection Using Wavelets.
Shock Vibr. Digest 30(6):457–472.
5. Kishimoto K, Inoue H, Hamada M and Shibuya T (1995). Time Frequency Analysis
of Dispersive Waves by Means of Wavelet Transform. ASME J. Appl. Mech. 62:
841–846.
6. Lee Y Y and Liew K M (2001). Detection of Damage Locations in a Beam Using
the Wavelet Analysis. Int. J. Struct. Stab. Dyn. 1(3):455–465.
7. Quek S T, Wang Q, Zhang L and Ong K H (2001). Practical Issues in Detection of
Damage in Beams Using Wavelets. Smart Mater. Struct. 10(5):1009–1017.
8. Huang H, Norden E, Zheng S, Long S R, Wu M C, Shih H H, Zheng Q, Yen N C,
Tung C C and Liu H H (1998). The Empirical Mode Decomposition and Hilbert
Spectrum for NonLinear and Nonstationary Time Series Analysis. Proc. R. Soc. Lond.
A 454:903–995.
9. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear Water
Waves: The Hilbert Spectrum. Annu. Rev. Fluid Mech. 31:417–457.
© 2005 by Taylor & Francis Group, LLC
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10. Huang H, Norden E, Chern C C, Huang K, Salvino L W, Long S R and Fan K L
(2001). A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysis
of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seism. Soc. Am.
91(5):1310–1338.
11. Zhang R R, King R, Olson L and Xu Y (2001). A HHT View of Structural Damage
from Vibration Recordings Proc. ICOSSAR ’01 (San Diego) – CD.
12. Shim S H, Jang S A, Lee J J and Yun C B (2002). Damage Detection Method for
Bridge Structures Using Hilbert-Huang Transform Technique. 2nd Int. Conf. Struct.
Stab. Dynam., Dec. 16–18, 2002, Singapore.
13. Quek S T, Tua P S and Wang Q (2003). Detecting Anomaly in Beams and Plate
Based on Hilbert-Huang Transform of Real Signals. Smart Mater. Struct. 12(3):
447–460.
14. Tua P S, Jin J, Quek S T and Wang Q (2002). Analysis of Lamb Modes Dominance
in Plates via Huang Hilbert Transform for Health Monitoring. Proc. 15 KKCNN Symp.
Civil Eng., Ed. S T Quek and D W S Ho, Dec. 19–20, 2002, Singapore.
15. Hahn S L (1996). Hilbert Transform in Signal Processing. London: Artech House
Boston.
16. Graff K F (1975). Wave Motion in Elastic Solids. Oxford: Clarendon Press.
17. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear Water
Waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31:417–457.
© 2005 by Taylor & Francis Group, LLC
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11
The Analysis of
Molecular Dynamics
Simulations by the
Hilbert-Huang Transform
Adrian P. Wiley, Robert J. Gledhill,
Stephen C. Phillips, Martin T. Swain,
Colin M. Edge, and Jonathan W. Essex
CONTENTS
11.1 Introduction ..................................................................................................246
11.2 The Hilbert-Huang Transform .....................................................................249
11.3 Molecular Dynamics ....................................................................................251
11.4 Reversible Digitally Filtered Molecular Dynamics.....................................254
11.5 Limitations ...................................................................................................262
11.6 Conclusions ..................................................................................................262
References..............................................................................................................263
ABSTRACT
Proteins are an integral component of all living systems, and obtaining a detailed
understanding of their structure and dynamics is of considerable importance. Molecular
dynamics computer simulations are an important tool in this process. However, the
length of simulation required to probe the full range of motions available to proteins
245
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The Hilbert-Huang Transform in Engineering
is prohibitively expensive. As a solution to this problem, we have developed a technique,
Reversible Digitally Filtered Molecular Dynamics (RDFMD), that can enhance or
suppress specific frequencies of motion. We have demonstrated that RDFMD is able
to amplify the low frequency vibrations in the system that are responsible for largescale changes in structure.
In this paper we describe the use of empirical mode decomposition (EMD) and the
Hilbert-Huang transform (HHT) to probe the effect of RDFMD on the internal vibrations of molecular systems, including a simple protein, in a molecular dynamics (MD)
simulation. EMD allows the decomposition of the MD trajectory into intrinsic mode
functions (IMFs) that are shown to be of physical relevance to the input signal. In
particular, spontaneous changes in molecular shape are shown to be accompanied by
increased energy in low frequency IMFs. Moreover, we show that the HHT enables a
superior analysis of the MD trajectories than Fourier-based techniques and provides
essential information that may be used to determine the optimum parameters for the
RDFMD method.
11.1 INTRODUCTION
Proteins are polymer chains made up from 20 possible monomer units (amino acids),
each sharing a common backbone structure with differing functional side chains (1).
YPGDV, a simple protein used in later discussion, consists of five amino acids:
tyrosine, proline, glycine, aspartic acid, and valine (Figure 11.1).
It is convenient to consider the amino acid chains (the primary structure) in
terms of relatively rigid and stable secondary structure units. These are classified as
either α-helices, where a section of coiled amino acids forms a rod-like helix, or
β-sheets, where amino acid strands form layered structures. These units of secondary
structure are connected by flexible loop regions, and secondary structure units can
themselves associate to form compact globular units called domains that move in a
correlated fashion.
Understanding the functions and mechanisms of biological molecules requires
detailed knowledge of their structure and internal motions. The arrangement, or
conformation, of a protein is of considerable importance to its biological function
(2, 3). For example, Creutzfeldt-Jakob Disease (CJD) is a fatal neurodegenerative
FIGURE 11.1 Pentapentide YPGDV (Tyr-Pro-Gly-Asp-Val) with backbone dihedral angles
φi and ψi labeled.
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The Analysis of Molecular Dynamics Simulations
247
disorder that is generally believed to be caused by conformational change in the
prion protein (4). The prion (proteinaceous infectious particle) theory proposes that
the prion protein (a normal constituent of mammalian cells) undergoes a conformational change to an active form that induces the same change in further prion proteins.
Protein aggregation then occurs, which is associated with cell death.
Large-scale conformational motion occurs predominantly by rotation about the
single bonds in the protein, the so-called dihedral angles, since a small change in
these angles can have a significant effect on the structure of the protein. The
important dihedral angles in the main-chain of YPGDV are marked in Figure 11.1.
Anharmonic, low frequency motions in dihedral angles are responsible for conformational changes (5, 6).
There are several experimental methods for determining three-dimensional structures of molecules, of which X-ray crystallography (7) is undoubtedly the most
common. The molecular arrangements seen in X-ray crystallography are, however,
“snapshots” of a protein that is not in its natural environment. Often, proteins can
be crystallized in a variety of arrangements that differ significantly (8, 9), and it is
believed that crystal-packing forces can have a significant influence on the conformer
observed (10). Analytical techniques, such as essential dynamics (11) or normal
mode analysis (12), can be applied to experimentally obtained structures in order to
give limited information about the flexibility of the protein (13).
Another prominent experimental technique for studying protein structures is
nuclear magnetic resonance (NMR) (14). Here data is gathered about the protein
structure when it is in solution, so that the crystal-packing artifacts associated with
X-ray crystallography are avoided. Unfortunately the method is time consuming
owing to the need for significant isotopic replacements to allow for the spectra to
be assigned.
Circular dichroism (CD) is a form of light absorption spectroscopy that is very
sensitive to the secondary structure of proteins (15). The relative amounts of disordered, helical, and sheet secondary structure can be determined, although problems
can be encountered if side chains also contribute to the spectrum.
Experimental methods are generally unable to follow atomic coordinates with
sufficient time resolution to describe the nature of many protein motions. For example, the T4 lysozyme (T4L) enzyme consists of two main domains linked by a hinge
region, and genetic mutants have been crystallized with a range of hinge angles (8).
Unfortunately, the natural form of the enzyme can only be crystallized in a closed
state in which there is no space for known substrates to bind. Although NMR data
suggest that the closed state is abundant in solution (16), there is significant evidence
that an opening event must occur. The structure of this open conformer is thought
to be similar to that crystallized from a mutated form of the protein (17). Experimental methods are unable to observe this opening/closing event. To increase our
understanding of the mechanisms of conformational change, a technique is therefore
required that can follow atomistic detail on the time scales over which the relevant
motions occur.
Molecular dynamics (MD) is a computational modeling tool that follows the
trajectories of atoms through time. Improved software and the decreasing cost of
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The Hilbert-Huang Transform in Engineering
computer power now allow routine simulations of large protein systems of around
10 ns. Over this time scale, conformational events such as the opening of the two
large flaps in HIV-1 protease can be observed (18).
A classical description of atoms is generally used in MD; the energy of the
system is assumed to be independent of the location of electrons (BornOppenheimer
approximation), so atoms are not polarizable and are described by a point location
only. MD effectively solves Newton’s Second Law,F = ma, for each atom, using a
finite difference integrator to overcome the many body problem. One such integrator,
the Verlet algorithm (19), is shown in Equation 11.1. The time step, δt, must be
sufficiently small to keep the system energy stable, yet as large as possible for
efficient time evolution of the system (a 2 fs time step is commonly employed).
Forces are obtained from the gradient of a potential energy function. This function
has separate terms for bond stretching, angle deformations, dihedral motion, and
non-bonded interactions. The parameters used for the potential energy function are
optimized with data from vibrational spectra, experimental geometry information,
and ab initio quantum mechanical calculations (20).
(
)
() (
)
()
r t + δt = 2r t − r t − δt + δt 2a t .
(11.1)
Simulations can be run with constant energy, although various thermostats (21,
22, 23, 24) and barostats (22, 23) can be applied to reproduce more biologically
relevant constant temperature and constant pressure ensembles.
Frequency analysis of molecular dynamics trajectories yields spectra that can
be compared to infra-red experimental data. The majority of intramolecular motions
are wavelike in nature and exist either as localized oscillations in single bonds or
angles, or as collective motions (or modes) characterized by coupled vibrations. For
example, the amide I mode in proteins that originates from the carbonyl stretching
vibration also includes a contribution from the backbone carbon–nitrogen bond (25).
Postsimulation digital filtering has been applied to molecular dynamics simulations to remove high frequency motions that are irrelevant to conformational changes.
One simple approach Fourier transforms the atomic trajectories, applies a frequencydomain filter, and inverse Fourier transforms the result (26). Alternatively the filtering
can be done in the time-domain by using non-recursive digital filters (27). These
methods have proved useful for the visualization of low frequency motions that are
important for conformational change.
Although MD has been used to study protein conformational change, the time
scales over which these motions occur are generally longer than those accessible by
conventional simulation routes. Methods to overcome this problem are therefore
required. Reversible Digitally Filtered Molecular Dynamics (RDFMD) is a method
that applies digital filters to the velocities in an evolving simulation to promote low
frequency motions. RDFMD has been successfully applied to a range of systems
(28, 29). The analysis of an RDFMD simulation cannot be achieved with sufficient
time resolution by using traditional Fourier-based methods, especially given the
discovery of nonstationary frequency targets (30). The Hilbert-Huang transform
provides the necessary analysis tools, as presented here.
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11.2 THE HILBERT-HUANG TRANSFORM
The Hilbert-Huang transform (HHT) is a combination of empirical mode decomposition (EMD) and the Hilbert transform (HT). The method has been applied across
a broad range of fields, including biology (31, 32), geophysics (33), and solar physics
(34). Numerous applications to simple systems and comparisons to Fourier-based
techniques have also been performed (30, 31).
The Hilbert transform takes a time-domain signal and transforms it into another
time-domain signal, unlike the Fourier transform, which moves from the time to the
frequency domain. The HT of a real-valued function, x(t), over the range –∞ < t < +∞
is another realvalued function, h(t), which is the convolution of x(t) with 1/πt
(Equation 11.2). In this equation, P signifies the Cauchy principal value. The rapid
decay of 1/(t–u) biases the transform heavily to points close to t, and thus the HT
acts on time-localized data.
∞
h(t ) = Ρ
∫ π (t − u) du
x (u)
(11.2)
−∞
In practice, the Hilbert transform can be computed by taking the Fourier transform of the data, setting the negative frequency components to zero, doubling the
positive frequency components, and back-transforming (35). This method relies on
the infinite replication of the data set required by the Fourier transform, yielding
unreliable regions at the start and end of the transformed data due to the discontinuities introduced. These regions are excluded from our analysis.
It can be shown that the signal magnitude is unchanged by HT, but the phase is
adjusted by π/2 (35). The original signal, x(t), and its Hilbert transform, h(t), may
be considered part of a complex signal, z(t):
() ()
()
z t = x t + ih t
(11.3)
This can also be written in terms of amplitude, A, and phase, φ:
()
()
()
()
z t =A t e ()
iφ t
(11.4)
where
A t = x 2 t + h2 t
()
(11.5)
h t 
φ t = tan −1 
.
x t 
()
()
(11.6)
and
()
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The Hilbert-Huang Transform in Engineering
The rate of change of the phase angle with time is the frequency of motion
occurring. The instantaneous frequency, f(t), is thus defined as:
dφ t
( ) 2π1 dt( ) .
f t =
(11.7)
The minimum frequency that can be reliably sampled by this method is 1/T (31),
where T is the time length of the data set. For an instantaneous frequency to be
physically meaningful, the signal must carry no riding waves and must be locally
symmetrical about its mean point, as defined by the envelope of the local maxima
and minima. Real-world phenomena that are described by signals matching these
criteria are not common.
EMD is a recently developed signal-processing method. It decomposes a signal,
which may be nonstationary, into a set of intrinsic mode functions (IMFs) suitable
for the Hilbert transform (31). An IMF is defined as a wave in which the number
of extrema and the number of zero-crossings differ by a maximum of one and where,
at any point, the mean of the envelope defined by the local maxima and minima is
zero.
To generate an IMF, the local mean is repeatedly determined and subtracted
from the data set until the number of extrema and zerocrossings in the residual data
differ by at most one (this process is termed “sifting”). The local mean is found by
taking the mean of a curve through all of the maxima and a curve through all of the
minima. These curves are defined, in this work, by cubic spline functions. The
algorithm proceeds by subtracting each generated IMF from the remaining signal
until either the recovered IMF or the residual data is small, or the residual has
become a trend component with only a single maximum and minimum. The final
residual of the data is similar in concept to the DC component of a Fourier transform,
except that it contains the overall trend of the data.
Once a signal has been decomposed into a set of IMFs, each IMF is separately
Hilbert transformed, giving the amplitude and instantaneous frequency at discrete
points in time. If the IMF is a harmonic oscillator of variable amplitude and frequency, then its signal energy at time t can be written in terms of the signal amplitude,
A, and frequency, v:
( ) ( ) A (t ) .
E t ∝v t
2
2
(11.8)
In the HHT spectra presented in this work, the energy in each IMF is calculated
and presented using Equation 11.8.
This paper presents the application of the HHT method to the analysis of MD
and RDFMD simulation trajectories, with a view to obtaining a better understanding
of system dynamics, validating the assumptions underlying the RDFMD methods,
and deriving some of the parameters associated with the RDFMD approach.
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FIGURE 11.2 The bond length (C=O) and angle (CA-C-N) used in the frequency analysis.
11.3 MOLECULAR DYNAMICS
The YPGDV protein is a common target for conformational studies by molecular
dynamics methods (36) and NMR (37), and it has been used as an RDFMD test
case (29). Results included here have been derived from previously published constant energy (30) and RDFMD (29) simulations.
The versatility of computer simulation allows the selection of specific internal
coordinates for analysis. For example, it is trivial to extract the progression of an
angle or bond during a constant energy YPGDV simulation and to locate the frequencies of motion with a windowed Fourier transform. Figure 11.2 illustrates a
bond-stretching and angle-bending motion in a section of YPGDV, and Figure 11.3
shows the Fourier transform of the simulation trajectory for these degrees of freedom.
These harmonic motions are stationary throughout the simulation, and the analysis
extracts the expected results: the amide I vibration between 1600 cm–1 and 1800 cm–1,
the angular motion at around 500 cm–1, and collective backbone motions between
800 cm–1 and 1200 cm–1. The angular motion has a large component at the bondstretching frequency of 1600 to 1800 cm–1, showing that the angles are coupled to
the bond stretches. However, the bond motion has no low frequency component.
The time integral of the HHT spectrum, the marginal spectrum (data not shown),
is broadly similar to the Fourier spectrum. However, the HHT analysis can be
continued to greater depth. Figure 11.4 shows the HHT for the angle-bending
CA-C-N vibration between glycine and aspartate, and Figure 11.5 presents the
spectrum for C=O bondstretching vibration. The Hilbert spectra show energy flowing
into and out from the high frequency vibrations at the same times, demonstrating
the coupled nature of these vibrations. Figure 11.6 plots the highest frequency IMFs
of the glycine–aspartic acid backbone angle (CACN) and of the glycine carbonyl
length (C=O); these are clearly in phase. Subtracting the first IMF from the original
signal and taking the Fourier transform of the result shows that this IMF is responsible for all the vibrational motion above approximately 1250 cm–1.
The signals of most relevance to conformational change are the low frequency
dihedral motions. These are generally nonstationary, and Fourier analysis provides
a poor description of conformational events (30). By looking at the energy in low
frequency regions as a function of time, Hilbert techniques allow recognition of
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The Hilbert-Huang Transform in Engineering
FIGURE 11.3 Fourier transform of backbone degrees of freedom using a Hanning window
and a 50-point running average.
events that occur simultaneously across several dihedrals. Figure 11.7 plots the eight
main-chain dihedral angles of YPGDV that are most important for conformational
change (see Figure 11.1) as a function of time in a region where a spontaneous
conformational transition occurred. The energy in the 0 to 10 cm–1 region obtained
by integrating the HHT spectrum is also shown, and the three highest energy peaks
occur from 15 to 22 ps, corresponding to a rearrangement of dihedrals ψ2, φ3, and
ψ3. Several dihedrals are often seen moving at once, as this reduces the need for
significant solvent reorganization.
It is necessary to show that the EMD method produces physically relevant IMFs.
Results for the φ3 data of the spontaneous transition are shown in Figure 11.8. The
low frequency IMFs follow physical motions from the signal, and there is an obvious
separation between the high frequency (IMFs 1 to 5) and low frequency motions
(IMFs 6 to 10). The application of EMD as a signal-smoothing technique is also clear.
It is a requirement of the HHT method that frequencies of motion be separable.
Without this, artifacts may enter the data (31). This is inherent in the method because,
in each sifting iteration, the extracted IMF tries to follow the highest frequency
motion in the remaining signal. Meaningful separation is not always guaranteed for
dihedral data due to the flexible nature of the simulation model, and so EMD results
must always be interpreted with caution. However, we have shown the physical
relevance of IMFs and, as Figure 11.8 shows, we find that a single IMF describes
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253
2000
Frequency/wavenumbers
1800
1600
1400
0
10
20
30
Time/ps
FIGURE 11.4 HHT of bond angle trajectory CA-C-N.
conformational events. Sometimes the frequency separation of IMFs is not complete,
and the physical relevance of a single IMF is not clear. A blind analysis of a single
IMF is therefore not sensible, but by averaging results over several signals or by
integrating over long time periods, a more robust analysis may be obtained.
In this section the ability of the HHT to yield extra information over and above
that available by a conventional Fourier analysis has been demonstrated. The coupled
nature of vibrational motion in molecular systems has been clearly shown, as has
the ability of EMD to smooth signals of high frequency data that are not required
and can indeed obscure overall trends. The analysis of conformational change events
has revealed that these motions are associated with low frequency vibrations, vindicating the assumptions underpinning the RDFMD method. The physical relevance
of the IMFs obtained from these signals has been shown, although the separability
of scales required for HHT does not always apply. Consequently, the EMD results
must always be interpreted with caution.
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The Hilbert-Huang Transform in Engineering
2000
Frequency/wavenumbers
1800
1600
1400
0
10
20
30
Time/ps
FIGURE 11.5 HHT of bond length trajectory C=O.
11.4 REVERSIBLE DIGITALLY FILTERED MOLECULAR DYNAMICS
The basis of the RDFMD method is the application of digital filters to the atomic
velocities in an ongoing molecular dynamics simulation to promote low frequency
motion and hence increase the rate of conformational change (29). The digital filters
used in RDFMD are linear phase, non-recursive filters designed in MATLAB (38).
A time series of input velocity data is required, and a mathematical function is
applied to the data to produce modified output velocities. For example, to apply a
filter to a single atom in the simulation, the atom’s velocity is stored for every time
step until a buffer of 2m + 1 velocities, v, is obtained. The filter is a list of 2m + 1
real coefficients, ci, which is applied to the velocity buffer to produce a single output
velocity, v ′, as shown in Equation 11.9. The application of a filter to a signal (the
velocity buffer in this case) may affect the frequencies in the signal in different ways
— it may enhance the amplitudes of some frequency components and reduce the
amplitudes of others. The filter coefficients can be chosen to control the effect of
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Signal Amplitude
1st IMF of Gly/Asp CA-C-N
1st IMF of Gly C = 0
0
0.1
0.2
0.3
Time/ps
FIGURE 11.6 The first IMFs of the coupled angle, Gly / Asp CA-C-N, and bond motion,
Gly C=0.
Angle / degrees
300
200
Energy in 0 -10 cm-1 region
100
0
0
10
0
10
20
30
20
30
Time / ps
FIGURE 11.7 Top: The eight dihedral angle trajectories around a spontaneous conformational transition. Bottom: The energy in the 0 to 10 cm–1 region extracted using HHT.
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FIGURE 11.8 YPGDV ϕ3 data during a trajectory containing a spontaneous conformational
transition. Top: Signal (solid line) and residual (dashed line). Upper middle: Sum of high
frequency IMFs (1 to 5). Lower middle: Low frequency IMFs (6 to 10). Bottom: Sum of low
frequency IMFs (6 to 10).
the filter on the input signal. For our purposes, the desired frequency response is an
amplification of low frequencies (typically those lower than 100 cm–1) with no
change in the amplitude of higher frequencies. The coefficients of the digital filter
are chosen such that the actual frequency response of the filter matches the desired
response as closely as possible (see Figure 11.9). The longer the list of coefficients,
the closer the actual frequency response is to the desired frequency response. However, short filters are desirable as they require fewer data points and hence less
simulation. We have found filters of 1001 coefficients (i.e., m = 500) to be sufficient.
m
v′t =
∑c v .
i t −i
(11.9)
i =− m
The RDFMD method generally applies a digital filter to a set of atoms in an
MD simulation that are of particular interest to the simulator (e.g., just the solute
and not the solvent, or just a region of a protein). A normal MD simulation is
performed to fill a buffer of length 2m + 1 (corresponding to the number of coefficients in the filter) for each atom of interest, and the filter is applied to each of the
atoms to enhance or suppress motion in the chosen frequency ranges. A digital filter
affects the phase of each frequency component in the signal as well as the frequency,
but, by using a property of linear phase filters, the phase can also be controlled. The
output of a linear phase filter is in phase with the midpoint of the filter input buffer.
It is for this reason that the system state (coordinates, accelerations, and forces) is
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1.5
Actual frequency response
Desired frequency response
Frequency Response
1
0.5
0
−0.5
0
25
50
Frequency/wavenumbers
75
100
FIGURE 11.9 The desired and actual frequency responses for a 1001 coefficient filter.
saved as the midpoint and then restored after the application of the filter — ensuring
that the new velocities are in phase with the atomic positions.
After applying a filter, we allow the system to evolve for a number of steps d.
Another filter may then be applied in order to build up gradually the kinetic energy
in the low frequency modes. We have been using a value of d = 20 time steps (29),
and as a safeguard against unrealistic, high energy events, the filter series is terminated early if the kinetic energy of the system exceeds that equivalent to a temperature of 2000 K. Since new velocities are applied at the midpoint of a buffer, if d
is less than m, molecular dynamics must be run backward in time to fill the buffer,
as shown in Figure 11.10. This is possible with the velocity Verlet MD integrator (39).
This flexible method allows many filters to be applied, with subsequent filter
applications positioned anywhere in time relative to the first. It is, however, possible
that the system is not approaching a conformational transition, and amplifying low
frequency motions will have little effect. It is for this reason that the number of filter
applications in a series is limited to ten, after which the simulation proceeds as
normal from the end of the final filter buffer. Once filter applications have been
terminated (either by reaching the maximum number of filters allowed or by heating
to the temperature cut-off), 20 ps of MD are performed, giving a new starting
structure from which the filtering process can be repeated.
Prior to the RDFMD simulation, a desired frequency target must be found. A
short MD simulation of YPGDV has been used for this purpose. Since conformational motions are complex with non-stationary frequencies, Fourier methods yield
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FIGURE 11.10 RDFMD sequence showing the delay parameter, d, and the filter length, 2m + 1.
little information. Figure 11.11 presents a spontaneous conformational change and
shows the energy present in defined frequency ranges. These data are derived by
integrating the associated HHT spectrum over frequency. It is clear that the spontaneous conformational change is associated not only with vibrations in the 0 to
10 cm–1 region but also with vibrations up to 50 cm–1. Furthermore, there are other
locations in the trajectory where energy is present in the region 25 to 50 cm–1. These
data suggest that although the conformational change itself is associated with very
low (< 10 cm–1) frequency motion, to enhance conformational change, the digital
filter should be designed to amplify vibrations up to 50 cm–1.
For this paper, a set of RDFMD simulations of YPGDV in water solvent has
been performed. The digital filter was applied to all atoms of YPGDV and not to
the solvent. A starting conformation was chosen, and ten sets of random initial
velocities were generated. From each set, seven filter application series were run,
using digital filters to amplify the 0 to 25 cm–1 range by two, three, four, five, six,
seven, and eight times. The filters used 1001 coefficients, and each filter series
consisted of up to ten filter applications using a delay, d, of 20 steps and a temperature
cut-off of 2000 K.
Once a filter has been applied to a buffer, the next buffer should contain an
amplification of any low frequency motions present in the system. The HHT method
can help demonstrate this, as Figure 11.12 shows. The top of the figure shows a
representative example of a dihedral angle trajectory progressing through two filter
buffers. Only the data corresponding to the middle section of the buffers are shown;
the filter is applied at step 500. The bottom graph of the figure is the sum of the
EMD trend data and the IMFs with components in the low frequency (0 to 50 cm–1)
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150
Signal
0–10
0–25
0–50
100
Dihedral Angle/degrees
Signal Energy/Arbitrary Units
200
50
0
10
20
30
0
Time/ps
Dihedral angle/degrees
FIGURE 11.11 YPGDV ψ2 data during a trajectory showing a spontaneous conformational
transition. The signal and the energy in low frequency bands (cm–1), derived from the HHT
spectrum, are shown.
1st buffer
2nd buffer
170
160
Displacement/degrees
150
170
160
150
450
500
Timesteps through buffer
550
FIGURE 11.12 YPGDV ψ2 data. Top: Signal before and after the application of a digital
filter. Bottom: Sum of the residual and the low frequency IMFs showing the trend in the data.
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Dihedral angle change (RMSD)
30
Filter 1
Filter 2
Filter 3
Filter 4
Filter 5
20
10
0
Energy
FIGURE 11.13 Change in YPGDV backbone dihedral angles as low frequency kinetic energy
is successively amplified.
region. The dihedral motion is clearly amplified either side of the central point in
the direction of the low frequency trend.
The amount of additional motion in the backbone dihedral angles caused by the
application of a filter to the system should correlate with the amount of energy in
the low frequency region at the point of filter application. To test this hypothesis,
we define a measure of the additional dihedral motion going from one buffer to the
next as the average of the root-mean-square deviations (RMSDs) of the eight dihedral
angles in the central region of the buffer (the midpoint ± 100 steps). The amount of
energy in the low frequency region is calculated by taking the instantaneous frequencies and amplitudes of the dihedral IMFs at the midpoint of the buffer and
calculating the energy at each frequency, as described previously. To estimate the
amount of energy in the filter amplification region, these energies are multiplied by
the filter’s frequency response, scaled from unity at 0 cm–1 to zero at 100 cm–1.
Finally, the energies are summed to provide our measure of the kinetic energy
available for amplification.
Figure 11.13 plots the dihedral angle RMSDs against the energy amplified for
five successive applications of a ×5 amplifying filter. The energy and the dihedral
motion are clearly correlated, though the data is quite spread. We also see a steady
increase in the energy as the digital filter is applied to successive filter buffers. Fewer
points are displayed for the later buffers due to simulations terminating as they
exceed the temperature cut-off. A trend line has been fitted to the entire data set by
a least-squares procedure, plotted to the extremes of the data set only.
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FIGURE 11.14 Change in YPGDV backbone dihedral angles as low frequency kinetic
energy is successively amplified using different digital filters.
This analysis can be extended to investigate the effect of different amplification
filters. High amplification filters would intuitively yield greater induced conformational change, although the temperature cutoff will be reached more quickly. In
Figure 11.14, the analysis reported in Figure 11.13 has been extended to include
different amounts of amplification by the digital filters. Only the trend lines obtained
by least-squares fitting for all the filters are reported. As expected, as the filter’s
amplification factor increases, so does the induced conformational change. It was
found that the data for amplification in excess of ×6 were poorly correlated, presumably as a result of rapid simulation termination due to the temperature cut-off
being exceeded, and consequently these data are not presented.
The controlled increase of energy in low frequency motions is likely to be of
greater importance to larger protein systems, for which lower temperature cutoffs
may be required. The energy in the low frequency region reached a maximum at an
amplification factor of ×3. However, factors of ×4 to ×6 all reach significant levels
of induced conformational change, and higher factors do this in fewer buffers and
thus in less computer time. It is believed that the tradeoff between slowly building
up energy in low frequency motions and efficiently inducing a conformational
response will have to be determined for each system to which RDFMD is applied.
In this section, HHT has been applied to examine the hypothesis underpinning
the RDFMD technique, namely that the amplification of low frequency vibrations
in a molecular dynamics simulation of a protein increases the amplitude of dihedral
angle vibrations and hence conformational change. Supporting this hypothesis, a
correlation has been observed between the energy available for amplification by the
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The Hilbert-Huang Transform in Engineering
digital filter and the resulting RMSD in dihedral motion. As expected, using larger
amplifying filters increases the energy in the critical degrees of freedom more rapidly,
although there will inevitably be a compromise between the need for efficiency by
amplifying as rapidly as possible and the requirement that the conformational transitions be physically sensible without significantly disrupting the protein structure,
for which gentle amplification will be required. Finally, more detailed analysis of
spontaneous conformational transitions has confirmed that, although such transitions
are indeed associated with very low frequency motions (< 10 cm–1), they are also
associated with higher frequencies (25 to 50 cm–1). Analysis of the remainder of the
simulations trajectory has identified other occasions where significant energy resides
in this frequency region, but without associated conformational change. We suspect
that these conditions are associated with possible conformational change events.
Amplification of frequencies of up to 50 cm–1 may therefore be more efficient in
terms of driving conformational change.
11.5 LIMITATIONS
The limitations of the HHT method as applied to molecular dynamics simulations
are essentially threefold. First, analysis of short (1001 time step) RDFMD filter
buffers results in a low frequency limit in the Hilbert transform of 16.7 cm–1. Given
that this limit lies in the frequency range over which amplification is taking place,
it restricts our ability to analyze the simulation behavior over the content of the filter
buffer. Second, conformational transitions often leave large residuals, giving dihedral
angle plots similar to step functions. EMD produces physically unrealistic IMFs in
an attempt to recreate the unusual signal given by the data less the residual, as shown
in Figure 11.15. Although the data could be windowed to reduce the size of the
affected regions, low frequency resolution would be lost. This limitation is due to
the nature of the data and suggests that the HHT must be used with care, with
constant checks of the physical relevance of the IMFs. Third, the HHT requirement
of separable scales is not necessarily met by signals from molecular dynamics
simulations. Great care must therefore be taken when performing analyses by, for
example, careful examination of the IMFs to confirm their physical relevance,
repeated analysis over many different simulations, averaging over a number of
simulation signals, and integration of the HHT spectra over frequency and time.
11.6 CONCLUSIONS
In this paper the application of the HHT method to the analysis of molecular
dynamics computer simulations has been reported. The non-stationary data associated with these simulations are unsuitable for analysis with Fourier methods. However, it has been shown that the HHT method does yield useful insight into the
dynamics. In particular, the assumptions underpinning the RDFMD method of
enhancing conformational change, namely that low frequency vibrations are associated with conformational change and that non-recursive digital filters may be used
to amplify these vibrations, have been confirmed. HHT analysis has also suggested
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FIGURE 11.15 YPGDV ψ2 data. Top: Signal and residual (dotted line). Middle: Signal after
the residual is removed (equivalent to the sum of the IMFs). Bottom: IMFs generated by EMD.
that frequencies greater than those earlier identified with spontaneous conformational
change may in fact be more appropriate targets for RDFMD. However, the presence
of large residuals and an incomplete separation of scales indicate that HHT cannot
be applied to these simulations as a “black box” and that care must be taken.
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932–935.
18. Scott, W. R. P., Schiffer, C. A. (2000). Curling of flap tips in HIV1 protease as a
mechanism for substrate entry and tolerance of drug resistance. Structure 8:
1259–1265.
19. Verlet, L. (1967). Computer experiments on classical fluids. I. Thermodynamical
properties of Lennard-Jones molecules. Phys. Rev. 159:98–103.
20. Weiner, P.W., Kollman, P.A. (1981). AMBER: assisted model building with energy
refinement. A general program for modelling molecules and their interactions. J.
Comput. Chem., 4:287–303.
21. Woodcock, L. V. (1971). Isothermal molecular dynamics calculations for liquid salts.
Chem. Phys. Lett. 10:257–261.
22. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., Di Nola, A., Haak, J. R.
(1984). Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81:
3684–3690.
23. Anderson, H. C. (1980). Molecular dynamics simulations at constant pressure and/or
temperature. J. Chem. Phys. 72:2384–2393.
24. Hoover, W. G. (1985). Canonical dynamics: equilibrium phase-space distributions.
Phys. Rev. A31:1695–1697.
25. Creighton, T. E. (1993). Conformational properties of polypeptide chains. In: Proteins
structures and molecular properties. 2nd ed. New York: W. H. Freeman and Company,
pp. 192–193.
26. Sessions, R. B., Dauber-Osguthorpe, P., Osguthorpe, D. J. (1989). Filtering moleculardynamics trajectories to reveal low-frequency collective motions — phospholipaseA2. J. Mol. Biol. 210(3):617–633.
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The Analysis of Molecular Dynamics Simulations
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27. Levitt, M. (1991). Real-time interactive frequency filtering of moleculardynamics
trajectories. J. Mol. Biol. 220(1):1–4.
28. Phillips, S. C., Essex, J. W., Edge, C. M. (2000). Digitally filtered molecular dynamics: the frequency specific control of molecular dynamics simulations. J. Chem. Phys.
112(6):2586–2597.
29. Phillips, S. C., Swain, M. T., Wiley, A. P., Essex, J. W., Edge, C. M. (2003). Reversible
digitally filtered molecule dynamics. J. Phys. Chem. B. 107(9):2098–2110.
30. Phillips, S. C., Gledhill, R. J., Essex, J. W., Edge, C. M. (2003). Application of the
Hilbert-Huang transform to the analysis of molecular dynamics simulations. J. Phys.
Chem. A. 107(24):4869–4876.
31. Huang, N. E., Chen, Z., Long, S. R., Wu, M. L. C., Shih, H. H., Zheng, Q. N., Yen,
N. C., Tung, C. C., Liu, H. H. (1971). The empirical mode decomposition and the
Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc.
London Ser. A Math. Phys. Eng. Sci. 454:903–995.
32. Echeverria, J. C., Crowe, J. A., Woolfson, M. S., HayesGill, B. R. (2001). Application
of empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng.
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33. Chen, K. Y., Yeh, H. C., Su, S. Y., Liu, C. H., Huang, N. E. (2001). Anatomy of
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34. Komm, R. W., Hill, F., Howe, R. (2001). Empirical mode decomposition and Hilbert
analysis applied to rotation residuals of the solar convection zone. Astrophys. J.
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35. Bendat, J. S. (1985). The Hilbert transform and application to correlation measurements. Denmark: Brüel & Kjær.
36. Wu, X. W., Wang, S. M. (2000). Folding studies of a linear pentamer peptide adopting
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37. Dyson, H. J., Rance, M., Houghten, R. A., Wright, P. E., Lerner, R. A. (1988). Folding
of immunogenic peptide-fragments of proteins in water solution. 2. The nascent helix.
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38. MATLAB 5.3.0. Natick, MA: The MathWorks Inc. (1999).
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physical clusters of molecules — application to small water clusters. J. Chem. Phys.
76(1):637–649.
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12
Decomposition of Wave
Groups with EMD
Method
Wei Wang
CONTENTS
12.1 Introduction ..................................................................................................267
12.2 Decomposition of Intermittent Small-Scale Fluctuations
from the Large Wave ...................................................................................269
12.3 Method for Decomposing Wave Groups .....................................................272
12.4 Validations ....................................................................................................274
12.5 Conclusions ..................................................................................................280
References..............................................................................................................280
ABSTRACT
By adding a proper real temporary time series that can induce a series of additional
extrema, intermittent small fluctuations can be accurately decomposed from the large
waves they ride on. We can thus effectively solve the mode mixing problem that arises
when the amplitudes of the components of a time series are very different. Another
kind of mode mixing arises when the frequencies of the wave components are close
to each other, such as in wave groups. Here a temporary complex time series is
introduced to shift down the frequencies of the components. This will greatly enlarge
the ratio of the frequencies in a wave group, and the wave groups can then easily be
decomposed with the empirical mode decomposition method.
Keywords wave groups, EMD method
12.1 INTRODUCTION
Being among the most energetic events in the ocean wave field, wave groups play
an important role in ocean wave research. The significant characteristic of a wave
267
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The Hilbert-Huang Transform in Engineering
group is that the group velocity Cg is equal to the energy propagation speed of the
wave field; this is widely used in ocean wave models such as WAM. The formation
of wave groups is assumed analytically to be the result of the superposition of several
cosine waves with almost the same frequency. In practice, this kind of analysis often
cannot be achieved since no method up to now has been available to decompose a
wave group efficiently into several simple components. This problem has been the
main obstacle in wave group studies.
Feasible methods that might be used to decompose wave groups are Fourier
transform, wavelet transform (Chan 1995), and empirical mode decomposition
(EMD) analysis (Huang et al. 1998, 1999). For a continuous uniform wave group,
in which the wave group arises from the superposition of continuous constant-amplitude cosine waves, Fourier transform can distinguish the frequencies of all the
components accurately. For an intermittent wave group, in which the wave group
arises from the superposition of intermittent cosine waves, Fourier transform can
yield only a collection of frequencies that will not represent the actual components.
The reason is that Fourier transform is a high-resolution frequency detector with no
resolution in the time domain; as a result, the intermittency of wave groups will
show as additional frequencies in the spectrum, and the actual frequencies of the
wave components will be altered to new locations.
Compared with Fourier transform, wavelet transform is more suitable for dealing
with intermittent signals. It has higher resolution in the time domain and lower
resolution in the frequency domain. Theoretically, intermittent wave groups can be
decomposed by wavelet transform when the frequencies of the wave components
are widely different, but this method would fail when the frequencies are only slightly
different. In practice, the lower resolution of wavelet transform in the frequency
domain will also induce a rich distribution of harmonics around the real components
and will ultimately result in difficulties in the distinction of the actual wave group
components.
In consideration of the resolution in the time–frequency domain, the most acceptable method is EMD. Compared to the Fourier transform and the wavelet transform,
the EMD method has higher resolution in both the time and frequency domains. As
an example, Huang et al. (1998) decomposed a wave group that came from the linear
sum of two cosine waves into eight components with up to 3000 sifting processes
and an extremely stringent criterion. Although the first two components contained
most of the energy and were similar to the original components, the results are not
acceptable for the following reasons:
1. The extra components would not have any physical meaning.
2. Using too many sifting cycles is not suitable for intermittent wave groups
because “too many sifting cycles could reduce all components to a constant-amplitude signal with frequency modulation only and the components would lose all their physical significance” (Huang et al. 1999). A
signal can be decomposed by the EMD method only when the ratio of
the frequencies is larger than approximately 1.5 and when the number of
sifting cycles is no more than 20. This is a limitation of the EMD method
and reflects a kind of mode mixing.
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Decomposition of Wave Groups with EMD Method
2
C1
2
1
0
−2
0
0
500
1000
1500
2000
2500
3000
0
500
1000
1500
2000
2500
3000
2
C2
−1
−2
269
0
500
1000
1500
2000
2500
3000
0
−2
FIGURE 12.1 The original time series and the first two IMF components decomposed with
the EMD method. The solid line in left panel is the original signal, the dashed line in the left
panel is the envelope, and the lines in the right panel are the IMF components.
Another kind of mode mixing is introduced by the intermittency of the
small-scale fluctuations riding on large waves. Huang et al. (1999) gave a method
for solving this kind of mode mixing, but defining which signal in the intrinsic mode
function (IMF) components is the one we want may be difficult, as shown in Figure
12.1.
In the first part of the present paper, we improve the limitation on the ratio of
the frequencies of the wave components to 1.2 by introducing a high frequency
temporary wave. However, decomposition of a wave group remains impossible with
this method, since the ratio of the frequencies for a typical wave group is always
smaller than 1.1.
In the second part of this paper, instead of improving the frequency resolution
of the EMD method, we introduce a temporary complex signal to shift the frequencies of the components down. This will greatly increase the ratio of the frequencies
in the wave group, allowing it to be decomposed easily with the EMD method.
12.2 DECOMPOSITION OF INTERMITTENT SMALL-SCALE
FLUCTUATIONS FROM THE LARGE WAVE
In general, a fluctuation with high frequency and small amplitude can only change
the large-scale wave profile slightly, but not alter the stationarity of the large scale
wave (not introduce additional extreme value). In such a case, the EMD method will
take no account of the existence of the small-scale fluctuations. Moreover, if we
carry out the Hilbert transform, the Hilbert spectrum will treat this small-scale wave
as a nonlinear characteristic of the large-scale wave itself, which seriously distorts
the original physical meaning. Only when the small wave is superimposed on the
wave crest or trough of the large-scale wave can it affect the stationarity of the
large-scale wave, and in this case the EMD method can decompose it.
Apparently, if the envelope of this kind of signal is very close to the dominant
wave profile, the intermittent signal with high frequency and small amplitude can
be effectively separated. Based on this idea and the characteristics of the EMD
method, we introduced a temporary signal with high frequency and large amplitude
(called the temp-signal hereafter). As a result, the IMF components with high
frequency derived through the EMD method will contain the temp-signal along with
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The Hilbert-Huang Transform in Engineering
the high frequency fluctuations in the original signal, and it can easily be separated
by subtracting the temp-signal from the IMFs. But an optionally selected temp-signal
will not match the original signal. Hence, the selection of the temp-signal is very
important.
For the case of a large wave with single frequency, containing an intermittent
signal with high frequency and small amplitude, we can write:
a cos ω1t
f (t ) =  1
a1 cos ω1t + a2 cos ω 2t
t ∈(0, t1 ) ∪ (t1 + 2π / ω 2 , T1 )
t ∈[t1 , t1 + 2π / ω 2 ]
(12.1)
where ω2 ω1, a1 > a2.
The temp-signal is defined as
X tmp (t ) = a3 cos ω 3t
t ∈(t1 , T1 )
(12.2)
The data series with the temp-signal is
a1 cos ω1t + a3 cos ω 3t

X (t ) = 
a1 cos ω1t + a2 cos ω 2t + a3 cos ω 3t
t ∈(0, t1 ) ∪ (t1 + 2π / ω 2 , T1 )
(12.3)
t ∈[t1 , t1 + 2π / ω 2 ]
The temp-signal should satisfy the following requirements:
•
•
•
The temp-signal can introduce a series of additional extrema, so its frequency should not be near that of the dominant wave; that is, ω3 ω1.
a3 should be large enough to introduce a series of extrema to conceal the
discontinuity caused by the intermittency.
The envelope of X(t) calculated by spline function S(t), must satisfy ε a1, where ε = S(t) – a1 cos ω1t.
It is easily found that in order to restrict ε to smaller than 5%, the suitable choice
of the temp-signal satisfies 4T2/5 < T3 T1, in which T1, T2, and T3, are the periods
of the signal of the dominant wave, the fluctuations, and the temp-signal, respectively.
In addition, the amplitude of the temp-signal should be large enough to form a series
of extrema in the original signal.
For the signal of Figure 12.1, the mathematical description is:
  2π 
t  , as t < 1264 or t > 1296
 cos 
  640 
X (t ) = 
cos  2π t  − 0.02 sin  2π t  , as 1264 ≤ t ≤ 1296
 32 
  640 

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271
1.5
X(t) + Xtmp (t)
1
0.5
0
−0.5
−1
−1.5
0
500
1000
1500
t
2000
2500
3000
FIGURE 12.2 The shape of the original signal with the superposition of the temp-signal.
0.03
(a)
0.02
0.01
0.03
0.01
0
0
−0.01
−0.01
−0.02
−0.02
−0.03
1200
(b)
0.02
1250
1300
1350
1400
−0.03
1200
1250
1300
1350
1400
FIGURE 12.3 Comparison of the original small-scale fluctuation and the one decomposed
with the EMD method. The solid line represents the decomposed signal, and the dashed line
represents the original signal. In the left panel, the temp-signal is defined by atmp = 0.2 and
Ttmp = 60. In the right panel, the temp-signal is defined by atmp = 50 and Ttmp = 120.
The signal includes a small-scale fluctuation with frequency 50 times larger and
amplitude 20 times smaller than those of the dominant wave. Based on the discussion
above, the temp-signal
 2π 
X tmp (t ) = atmp cos 
t
 Ttmp 
should be selected in the range 26 < Ttmp < 640. Here we chose atmp = 0.2 and Ttmp
= 60. The composed signal is shown in Figure 12.2.
After processing with the EMD method and Deng et al.’s (2001) method of
dealing with the boundary condition, the intermittent small-scale fluctuation is
decomposed successfully, as shown in Figure 12.3a. As a comparison, different
choices of atmp and Ttmp were used, and the result was shown in Figure 12.3b.
The above example represents a case where the small-scale intermittent fluctuation is located on the crest of the dominant wave. For a case where the small-scale
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The Hilbert-Huang Transform in Engineering
1
0.03
(a)
A
0.5
0.01
0
0
−0.01
−0.5
−1
(b)
0.02
−0.02
0
500
1000
1500
2000
2500
3000
−0.03
1950
2000
2050
2100
2150
FIGURE 12.4 Same as Figure 12.3, but the intermittent small-scale fluctuation is changed
to the position marked by A in the left panel. The temp-signal is selected as atmp = 50 and
Ttmp = 120.
2
1
(a)
1.5
1
(b)
0.5
0.5
0
0
−0.5
−1
0
500
1000
1500
2000
2500
3000
−0.5
1100
1200
1300
1400
1500
1600
FIGURE 12.5 Decomposition of signals with singularities. The temp-signal is selected as
atmp = 50 and Ttmp = 120.
intermittent fluctuation is located at an arbitrary location other than crest or trough,
the stationarity of the dominant wave will not be changed. As a result, decomposing
by the EMD method will give no result. Even under such a condition, the method
we propose can work. Figure 12.4 shows the original signal and the decomposed
results obtained with our method; the original signal is defined by Equation 12.4,
but the position of the intermittent small-scale fluctuation is changed. Signals in
which singularities exist, such as the one shown in Figure 12.5, can also be decomposed with this method.
In addition to offering the advantages already mentioned, this method can be
used to decompose wave groups when the ratio of the frequencies of the components
is not less than 1.2 and the distortion of each component is quite small even after a
large number of sifting cycles.
12.3 METHOD FOR DECOMPOSING WAVE GROUPS
As mentioned above, the key point in wave group decomposition is the ratio of the
frequencies of wave group components. Let us consider the wave group as
y(t ) = cos(ω1t ) + cos(ω 2t + φ),
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Decomposition of Wave Groups with EMD Method
273
in which ω2 > ω1. The ratio of the two frequencies is α = ω2/ω1. Typically α is
smaller than 1.1, but it can be greatly enlarged when we are subtracting a temporary
frequency, ω0, in the numerator and denominator simultaneously. For a real signal,
such a downshifting is hard to achieve. Theoretically, Equation 12.5 can be rewritten
in a complex form as
y(t ) = Re Y (t )  = Re  exp(iω1t ) + exp(i(ω 2t + φ))  ,
(12.6)
where Y(t) is the complex form of y(t), which can easily be constructed by using
the Hilbert transform. For an arbitrary time series, y(t), we can always have its
Hilbert transform, ỹ (t), as
y (t ) =
∞
y( τ )
1
P
d τ,
π
−∞ t − τ
∫
(12.7)
where P indicates the Cauchy principal value. This transform exists for all functions
of class Lp (Huang et al. 1998). Therefore, the complex function Y(t) can be formed as
Y (t ) = y(t ) + iy (t ).
(12.8)
The downshifting can easily be achieved by multiplying Y(t) by a function, exp
(–iω0t), to yield a new complex signal:
Z (t ) = Zr (t ) + iZi (t )
= Y (t ) ⋅ exp(−iω 0t )
=  exp(iω1t ) + exp(i(ω 2t + φ))  ⋅ exp(−iω 0t )
(12.9)
= exp i(ω1 − ω 0 )t  + exp i((ω 2 − ω 0 )t + φ)  ,
The ratio of the frequencies for signal Z(t) is α = (ω2 – ω0)/(ω1 – ω0).
For a well-selected ω0, α can be much larger than 1.5, the limitation in the EMD
method. Zr(t) and Zi(t) can easily be decomposed, either with the EMD method or
the method described in Section 12.2, and represented by their IMF components as:
n
Zr (t ) =
∑ C (t) + r ,
rk
nr
k =1
n
Zi (t ) =
∑ C (t) + r .
ik
k =1
© 2005 by Taylor & Francis Group, LLC
ni
(12.10)
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The Hilbert-Huang Transform in Engineering
The combination of the corresponding IMF components of Zr(t) and Zi(t) will
reconstruct Z (t) as a sum of complex components Ck(t):
Z (t ) = Zr (t ) + iZi (t )
n
=
∑ (C (t) + iC (t)) + (r + ir )
r
i
k =1
k
nr
ni
(12.11)
n
=
∑ C (t) + R .
k
n
k =1
According to Equation 12.9, Y(t) can be written as
Y (t ) = Z (t ) ⋅ exp(iω 0t )
n
=
∑
Ck (t ) ⋅ exp(iω 0t ) + Rn ⋅ exp(iω 0t ).
(12.12)
k =1
Finally, the original signal y(t) may be represented as:
y(t ) = Re Y (t ) 
 n

= Re 
Ck (t ) ⋅ exp(iω 0t ) + Rn ⋅ exp(iω 0t ) 
 k =1

∑
(12.13)
n
=
∑ c (t) + r ,
k
n
k =1
where ck(t) are IMF components of the original signal, and rn is the trend or mean
value.
12.4 VALIDATIONS
We offer two examples to verify the performance of the present method. The first
is a continuous uniform wave group with a phase shift in one of the components.
For ω1 = 2π/30, ω2 = 2π/32, and φ = π/6, the wave profile represented by Equation
12.5 is given in Figure 12.6. In this case α = 1.067. With ω0 = 2π/28, the wave
profile of Z(t), governed by Equation 12.9, is shown in Figure 12.7. Here, a and b
represent the real and imaginary components of Z(t), respectively.
Under such conditions, the ratio of the frequencies of the components is approximately 2. Thus, Zr(t) and Zi(t) can easily be decomposed into a series of IMF
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Decomposition of Wave Groups with EMD Method
275
2
1.5
1
y (t)
0.5
0
−0.5
−1
−1.5
−2
0
200
400
600
800
1000
t
FIGURE 12.6 Wave profile represented by Equation 12.9 with ω1 = 2π/30, ω2 = 2π/32, and
φ = π/6.
Zr (t)
2
0
−2
Zi (t)
2
0
−2
0
200
400
600
800
1000
t
FIGURE 12.7 Wave profile represented by Equation 12.10 with ω0 = 2π/28. (a) The real
component of Z(t); (b) the imaginary component of Z(t).
components. Figure 12.8 shows the wave profiles of all the IMF components of the
real and imaginary parts.
Results calculated from Equation 12.11, Equation 12.12, and Equation 12.13
are shown in Figure 12.9 and Figure 12.10. The solid lines in Figure 12.9 and Figure
12.10 correspond to the first and second modes in Equation 12.5, respectively. For
comparison, the actual modes are also plotted as dotted lines in the same figures.
The agreement is excellent.
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The Hilbert-Huang Transform in Engineering
2
2
1
1
C1i
C1r
276
0
−1
−2
0
−1
0
200
400
600
800
−2
1000
0
200
400
2
2
1
1
0
−1
−2
600
800
1000
600
800
1000
t
C2i
C2r
t
0
−1
0
200
400
600
800
−2
1000
0
200
400
t
t
FIGURE 12.8 The IMF components of Zr(t) and Zi(t). (a,b) IMF components of Zr(t); (c,d)
IMF components of Zi(t).
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
t
FIGURE 12.9 Comparison of the first IMF component calculated from Equation 12.11, Equation 12.11, and Equation 12.13 (solid line) and the first mode of Equation 12.5 (dotted line).
Figure 12.11 illustrates the differences between the IMF components and the
actual modes. It is clear that the differences are negligible.
The second example is an intermittent wave group. The intermittent wave group
is formed as
0, as t < t1 or t > t2
y(t ) = 
cos ω1t + cos ω 2t + φ , as t1 ≤ t ≤ t2
( )
© 2005 by Taylor & Francis Group, LLC
(
)
(12.14)
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Decomposition of Wave Groups with EMD Method
277
1.5
1
0.5
0
−0.5
−1
−1.5
0
200
400
600
800
1000
t
FIGURE 12.10 Comparison of the second IMF component calculated from Equation 12.11,
Equation 12.11, and Equation 12.13 (solid line) and the second mode of Equation 12.5 (dotted
line).
1
0.5
0
−0.5
−1
0
200
400
0
200
400
600
800
1000
600
800
1000
1
0.5
0
−0.5
−1
t
FIGURE 12.11 Differences between IMF components and actual modes. The upper panel
shows the differences between the first IMF component and the first mode in Equation 12.5.
The lower panel shows the differences between the second IMF component and the second
mode in Equation 12.5.
For ω1 = 2π/30, ω2 = 2π/32, φ = π/6, t1 = 1740, and t2 = 3150, the wave profile
represented by Equation 12.14 is shown in Figure 12.12. With ω0 = 2π/28, the wave
profile of Z(t), governed by Equation 12.10, is shown in Figure 12.13, where a and
b represent the real and imaginary components of Z(t), respectively.
In the first example (shown in Figure 12.9, Figure 12.10, and Figure 12.11),
only negligible errors exist at the ends because we have confined end effects with
a method of adding characteristic waves at the ends. In the second example, however,
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The Hilbert-Huang Transform in Engineering
2
1.5
1
y (t)
0.5
0
−0.5
−1
−1.5
−2
0
500
1000
1500
2000
2500
t
3000
3500
4000
4500
5000
FIGURE 12.12 Wave profile represented by Equation 12.14 with ω1 = 2π/30, ω2 = 2π/32,
φ = π/6, t1 = 1740, and t2 = 3150.
2
(a)
Zr (t)
1
0
−1
−2
2
(b)
Zi (t)
1
0
−1
−2
0
500
1000
1500
2000
2500
t
3000
3500
4000
4500
5000
FIGURE 12.13 Wave profile represented by Equation 12.10 with ω0 = 2π/28. (a) The real
component of Z(t); (b) the imaginary component of Z(t).
end effects are hardly confined because the ends of the wave groups are not at the
ends of the data set. If the ends of the wave groups are left unattended, wide swings
caused by spline fitting will exist at the ends and will propagate to the interior and
ulterior as the number of sifting iterations increases. Figure 12.14 and Figure 12.15
show the final results.
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Decomposition of Wave Groups with EMD Method
2
2
(a)
0
0
−1
−1
−2
−2
2
2
(b)
1
(d)
1
Ci2
Cr2
(c)
1
Ci1
Cr1
1
0
−1
−2
279
0
−1
0
1000
2000
3000
4000
−2
5000
0
1000
t
2000
3000
4000
5000
t
FIGURE 12.14 The IMF components of Zr(t) and Zi(t): (a,b) IMF components of Zr(t); (c,d)
IMF components of Zi(t).
2
(a)
1
0
−1
−2
2
(b)
1
0
−1
−2
0
500
1000
1500
2000
2500
t
3000
3500
4000
4500
5000
FIGURE 12.15 Comparison of the IMF components calculated from Equation 12.11, Equation 12.12, and Equation 12.13 (solid line) and the two actual modes of Equation 12.14 (dotted
line).
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The Hilbert-Huang Transform in Engineering
12.5 CONCLUSIONS
The present paper introduces two methods for decomposing a wave group into a
series of components of relatively single scale. The first method was used not only
in solving the problem of decomposition of wave groups but also as an improved
method of EMD for a variety of uses. The second method can decompose wave
groups of small α, but the sensitivity to the selection of the temporary frequency is
still an open question.
REFERENCES
Chan, Y.T. (1995). Wavelet basics. Boston: Kluwer.
Deng, Y.J., W. Wang, C.C. Qian, Z. Wang and D.J. Dai. (2001). Boundary-processing technique in EMD method and Hilbert transform. Chinese Science Bulletin, 46(11),
954–960.
Huang, N.E., Z. Shen, S.R. Long, et al. (1998). The empirical mode decomposition and the
Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc.
Lond. A, 454, 899–995.
Huang, N.E., Z. Shen and S.R. Long. (1999). A new view of nonlinear water waves: the
Hilbert spectrum. Annu. Rev. Fluid Mech., 31, 417–457.
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13
Perspectives on the
Theory and Practices
of the Hilbert-Huang
Transform
Nii O. Attoh-Okine
CONTENTS
13.1 Introduction ..................................................................................................282
13.2 Basic Ideas ...................................................................................................282
13.2.1 The Hilbert-Huang Transform .........................................................282
13.2.2 Starting Point....................................................................................288
13.3 Current Applications ....................................................................................289
13.3.1 Biomedical Applications ..................................................................289
13.3.2 Chemistry and Chemical Engineering.............................................289
13.3.3 Financial Applications......................................................................290
13.3.4 Meteorological and Atmospheric Applications ...............................290
13.3.5 Ocean Engineering...........................................................................291
13.3.6 Seismic Studies ................................................................................292
13.3.7 Structural Applications.....................................................................295
13.3.8 Health Monitoring............................................................................296
13.3.9 System Identification........................................................................296
13.4 Some Limitations .........................................................................................297
13.5 Potential Future Research ............................................................................298
References..............................................................................................................299
Addendum: Perspectives on the Theory and Practices
of the Hilbert-Huang Transform............................................................................302
13A.1 Analytical .........................................................................................302
13A.2 Hybrid Method.................................................................................302
13A.3 Bidimensional EMD ........................................................................303
13A.4 Some Observations...........................................................................304
References..............................................................................................................304
281
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The Hilbert-Huang Transform in Engineering
ABSTRACT
The theory and practice of the Hilbert-Huang transform (HHT) provide another way
of analyzing nonlinear and nonstationary time series. The central idea of HHT is the
empirical mode decomposition (EMD), which is then decomposed into a set of intrinsic
mode functions (IMFs) and analyzed by using the Hilbert spectrum. This chapter is
primarily a summary of current applications of HHT and suggests areas for future
research based on my personal views.
13.1 INTRODUCTION
The Hilbert-Huang transform (HHT) provides a new method of analyzing nonstationary and nonlinear time series data. It allows the exploration of intermittent and
amplitude-varying motions. The classical signal analysis theory and practice have
been dominated by the Fourier transform, which represents the signal system in the
frequency domain. The Fourier works well for strictly periodic or stationary random
functions of time. To address the issues of nonstationarity and nonperiodic functions,
various methods have been introduced in the literature. These include wavelet analysis, the Wigner-Ville distribution, and the evolutionary spectrum. Unfortunately,
these methods failed in one way or another [Huang et al., 1998].
HHT is emerging as a powerful signal-processing tool. During the past five
years, a great deal has been learned about the computation, interpretation, and
implementation of the HHT technique. This work has been scattered in a wide variety
of fields, including signal processing, structural health monitoring, earthquake engineering, nondestructive testing, transportation engineering, ocean engineering, and
financial and biomedical applications. No one has drawn the new work together in
a comprehensive manner.
This chapter is primarily a summary of my own current views. My failure to
mention a particular contribution should not be taken as an indication of disinterest
or disagreement. After this general introduction, basic ideas of HHT are presented
in a manner that should be accessible to readers with no previous familiarity with
HHT. The paper then discusses various application areas, outlines some perceived
“difficulties,” and finally highlights a future research direction of HHT.
13.2 BASIC IDEAS
13.2.1 THE HILBERT- HUANG TRANSFORM
Huang et al. [1998] developed the Hilbert-Huang transform to decompose any
time-dependent data series into individual characteristic oscillations by using the
empirical mode decomposition (EMD). The decomposition is developed from the
assumption that any data or signal consists of different intrinsic mode functions
(IMFs), each representing embedded characteristics of the data/signal. The decomposition can be viewed as an expansion of the data in terms of the IMFs. The HHT
method is a two-step data analyzing method that consists of EMD and Hilbert
spectral analysis (HSA). The first step is the EMD, by which a completed time
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history of the signal is decomposed into a finite and often small number of IMFs
such that they may be Hilbert transformed. The EMD is based on local characteristics
in the time scale of the data, making the approach direct, posterior, and adaptive.
The IMFs admit well-behaved Hilbert transforms.
An IMF is defined by the following two criteria:
1. The number of extrema and zero-crossings must either be equal or differ
by no more than one.
2. At any instant in time, the mean value of the envelope defined by the local
maxima and the envelope of the local minima is zero.
With the above definition, any function can be decomposed as follows:
•
•
Identify all the local extrema, then connect all the local maxima by cubic
spline as upper envelope.
Repeat the procedure for the minima to produce the lower envelope. The
upper and lower envelope should cover all the data. If the mean is designated as m1 and the difference between the data and m1 is the first component h1, then
x (t ) − m1 = h1
(13.1)
where h1 is an IMF.
The mean m1 is given by the sum of local extrema connected by cubic spline:
m1 =
L +U
2
where U is the local maximum and L is the local minimum.
The IMF can have both amplitude and frequency modulations. In many cases,
there are overshoots and undershoots after the first round of processing, which is
termed sifting. The sifting process serves two purposes [Huang et al., 2001]: eliminating riding signals/profile and making the signals or profile more symmetric. The
sifting process has to be repeated many times. In the second sifting process, h1 is
treated as the data and as the first component. h1 is almost an IMF, except some
error might be introduced by the spline curve-fitting process. To treat h1as new set
of data, a new mean is computed. Then,
h1 − m11 = h11 .
(13.2)
After repeating the sifting process up to k times, h1k becomes an IMF. That is:
h1( k −1) − m1k = h1k .
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Let h1k = c1, the first IMF from the data. c1 should contain the finest scale or the
shortest period component of the data/profile. Now c1 can be separated from the rest
of the data by
x (t ) − c1 = r1 .
(13.4)
Since r1 is the residue, it contains information of longer period component; it
is now treated as the new data and subjected to the same sifting process. The
procedure is repeated for all subsequent rjs, and the result is
r1 − c2 = r2; ..., rn−1 − cn = rn;
(13.5)
c2 is now the second IMF of the data.
The sifting process can be stopped by predetermined criteria: either when the
component of cn or the residue rn becomes so small that it has a predetermined value
of substantial consequence, or when the residue rn becomes a monotonic function
from which no IMF can be extracted.
Summing Equation 13.4 and Equation 13.5 yields the following equation:
n
x (t ) =
∑ c +r ,
j
n
(13.6)
j =1
where cj is the jth IMF and n is the number of sifted IMFs. rn can be interpreted as
a trend in the signal/profile. The cj has zero mean. Due to the iterative procedure,
none of the sifted IMFs derived is closed analytical form [Schlurmann, 2002]. The
IMFs can be linear or nonlinear based on the characteristics of the data. The IMFs
are almost orthogonal and form a complete basis. Their sum equals the original data
[Salisbury and Wimbush, 2002]. The EMD then picks out the highest-frequency
oscillation that remains in the signal.
Flandrin et al. [2003] established how EMD can be used as a filterbank. They
represented the signal as follows:
K
x (t ) = mk (t ) +
∑ d (t)
k
(13.7)
k =1
where mk(t) stands for a residual “trend,” and the modes {dk(t), k = 1, …, K} and
dk(t) stand for details.
The orthogonality of the IMF components can be checked as follows, if Equation
13.6 is expressed as
n +1
x (t ) =
∑ c (t).
j
j =1
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That is, one has included the last residual or data trend as an additional element:
n +1
x (t ) =
2
n +1
n +1
j =1
k =1
∑ c (t) + 2∑ ∑ c (t)c (t).
2
j
j
j =1
k
(13.9)
If the decomposition is orthogonal, then the cross terms given in the second part
on the right hand side should be zero when they are integrated over time. The overall
index of orthogonality OI can be defined as follows:

c (t )c (t )
∑
∑
.
∑


IO =

t =o 
T
n +1
n +1
j =1
k =1
2
j
k

x (t )
(13.10)
To ensure that the IMF components retain enough physical sense of both amplitude and frequency modulations, we can limit the size of the standard deviation
(SD), computed from two consecutive sifting results as
T
SD =
∑ | h (ht) −(ht) (t) |
2
k −1
k
2
k −1
t =0
A value of SD between 0.2 and 0.3 was used.
In the next step, the Hilbert transform is applied to each of the IMFs, subsequently providing the Hilbert amplitude spectra a significant instantaneous frequency. The Hilbert transform of each IMF is represented by
∑ a (t) exp(i ∫ ω(t)dt)
n
x (t ) = Re
j
(13.11)
j =1
Equation 13.7 gives both the amplitude and frequency of each component as a
function of time. The Fourier representation would be as follows:
∞
x (t ) = Re
∑ a exp
j
iω j t
(13.12)
j =1
with both aj and j constants.
The Fourier transform represents the global rather than any local properties of
the data/signal. The major difference between the conventional Hilbert transform
and HHT is the definition of instantaneous frequency. The instantaneous frequency
has more physical meaning through its definition of each IMF component, while
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the classical Hilbert transform of the original data might possess unrealistic features
[Huang et al., 1998]. This implies that the IMF represents a generalized Fourier
expansion. The variable amplitude and instantaneous frequency enable the expansion
to accommodate nonstationary data. Huang et al. [2001] demonstrated that the
Fourier- and wavelet-based components and spectra may not have a clear physical
meaning as in HHT. For example, the wavelet-based interpretation of a pavement
profile is meaningful relative to selected mother wavelets [Attoh-Okine, 1999].
Furthermore, Hilbert analysis [Long et al., 1995] is based on almost noncausal
singular information. Therefore, at any given time, the data or signal has only one
amplitude and one frequency, both of which can be determined locally. These
represent the best information at that particular time. Physically, the definition of
instantaneous frequency has true meaning for monocomponent signals, where there
is one frequency, or at least a narrow range of frequencies, varying as a function of
time. Since most data do not show these necessary characteristics, sometimes the
Hilbert transform makes little physical sense in practical applications. In the present
method, the EMD is used to decompose the signal into a series of monocomponent
signals. Furthermore, to extract significant information, the time–frequency–amplitude joint distribution [a(t), (t), t] can be developed. This joint distribution in 3D
space can be replaced by [H(,t)], where x(t) = H(,t). This final representation is
referred to as the Hilbert spectrum.
With the Hilbert spectrum defined, we can define a marginal spectrum:
h(ω ) =
∫ H (ω, t)dt .
t
(13.13)
0
The marginal spectrum offers a measure of the total amplitude (or energy) contribution from each of the frequency values [Huang et al., 1999a]. The degree of
stationarity is defined as follows:
D(ω ) =
1
T
H (ω, t )
∫ (1 − h(ω) / T )dt .
T
(13.14)
0
The closer the above equation is to zero, the more stationary the system is and the
instantaneous energy:
IE (t ) =
n
H (ω, t ) =
∫ H (ω, t)dω
∑
j =1
2
(13.15)
ω
_
H j (ω, t ) ≈
n
∑ A (t)
j
(13.16)
j =1
≅ by definition. In Equation 13.16, H j is the jth component of the total Hilbert
spectrum. The equation
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Perspectives on the Theory and Practices of the Hilbert-Huang Transform
n
h(ω ) =
∑
(ω ) ≈
_
hj
j =1
n
T
j =1
0
∑ ∫ a (t)dt
j
287
(13.17)
provides a measure of the total amplitude contribution for each frequency value
[Zhang et al., 2003].
By using the Hilbert spectrum, first and second order time-moments and the
frequency-dependent order can be derived. The first and second order of time-dependent moments are as follows:
∞
∫ ωH (ω, t)dω
∫ H (ω, t)dω
mω ,1 (t ) =
−∞
∞
(13.18)
−∞
∞
∫ ω H (ω, t)dω − m (t)
m (t ) =
∫ H (ω, t)dω
ω ,2
2
−∞
∞
ω ,1
(13.19)
−∞
The first and second frequency-dependent distributions are as follows:
∞
∫ tH (ω, t)dt
∫ H (ω, t)dt
mt,1 (ω ) =
−∞
∞
(13.20)
−∞
∞
∫ t H (ω, t)dt − m (ω)
m (ω ) =
∫ H (ω, t)dt
t ,2
2
−∞
∞
t ,1
(13.21)
−∞
The marginal in the time and frequency domains can be calculated as follows:
∞
mω ,0 (t )
∫ H (ω, t)dω
mt,0 (ω ) =
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∞
(13.22)
∞
∫ H (ω, t)dt
∞
(13.23)
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The Hilbert-Huang Transform in Engineering
The information content of the marginal mt,0() is much reduced [Huang et al., 2001].
The time marginal corresponds to the instantaneous power |x(t)|2 of the signal, and
the frequency marginal corresponds to the spectral energy density |X()|2.
Huang et al. [2003a] developed a confidence limit for HHT. The ensemble mean
and standard deviation of IMF sample sets obtained with different stopping criteria
are calculated and form a simple random set. The confidence limit for EMD/HSA
is then defined as a range of standard deviations from the assembled mean. The
authors attempted to establish a confidence limit for the EMD/HSA approach as a
statistical measure of the validity of the results. The method of establishing a
confidence limit depends on factors like stopping criteria, the maximum number of
siftings, intermittence, sifting methods, and the nomenclature needed for the confidence limit. The method does not invoke the ergodic assumption. Qu and Jarzynski
[2001] did a comparative study of different time–frequency representations of Lamb
waves. The authors noted that the time–frequency accuracy of the Hilbert spectrum
is dependent on the accuracy of the EMD. That is, if the decomposition into IMFs
does not capture the signal’s behavior, the resulting Hilbert spectrum will not give
precise time–frequency results.
13.2.2 STARTING POINT
The starting point of HHT is the Hilbert transform. The Hilbert transforms were
originally developed to integrate equations [Duffy 2004]. Instead of expressing a
function of time with its Fourier transform, which depended only on frequency, the
Hilbert transform yields another temporal function:
1
fˆ =
π
∞
f (τ)
∫ t − τ dτ
−∞
(13.24)
Gabor (1946) developed the concept of analytic signal:
Y (t ) = f (t ) + i ∧ f (t )
(13.25)
Y (t ) = a(t )eiθ(t )
(13.26)
or
which is a local time-varying wave with amplitude a(t) and phase (t). The Hilbert
transform can be considered to be a filter that simply shifts the phase of its input
by –/2 radians. Unfortunately, most signals are not band limited. One of Huang’s
notable contributions was to devise a method — the sifting method — that transforms
a wide class of signals into set of band-limited functions. The instantaneous frequency may possess negative values. Lai (1998) developed a proper method to
implement positive frequencies. The Hilbert spectrum is similar to the Fourier
spectrum in a physical sense, but the Fourier approach is more global. In the Hilbert
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spectrum, at any given time, the signal has only one amplitude and one frequency,
both determined locally [Long et al., 1995].
13.3 CURRENT APPLICATIONS
13.3.1 BIOMEDICAL APPLICATIONS
Huang et al. [1999b] analyzed the pulmonary arterial pressure on conscious and
unrestrained rats. The authors analyzed the signal by using HHT and compared the
Hilbert spectrum to the Fourier spectrum. The Hilbert spectrum clearly depicts the
fluctuations of the frequency with time, whereas the Fourier spectrum gives only
the distribution of energy over frequencies without allowing the frequency of the
oscillations to be variable in the whole time window. Huang et al. [1999c] also
studied the signals obtained from pulmonary hypertension. Their analysis attempted
to address the linearity and nonlinearity of the dependencies of blood pressure on
other variables. Huang et al. [1999c] developed a generalized function connecting
the IMFs and an analytic equation:
M k (t ) = IMFk + IMFk −1 + … + IMFn
M k (t ) = A + B(t − t0 )e
− t −t 0
T1
− t −t 0
+ C (1 − e T2 ); t0 ≤ t t1
(13.27)
(13.28)
where t0 is the instant in time when oxygen concentration drops suddenly, t1 is the
instant when oxygen concentration increases suddenly, A is the mean value of Mk(t)
before time t0, and B, C, T1, and T2 are constants.
13.3.2 CHEMISTRY AND CHEMICAL ENGINEERING
Phillips et al. [2003] performed a comparative analysis of HHT and wavelet methods.
They investigated a conformational change in Brownian dynamics (BD) and molecular dynamics (MD) simulations. Most of the MD trajectories were generated by
using a reversible digitally filtered MD method. The authors used HHT to analyze
the trajectories. The HHT analysis of BD trajectories was able to isolate characteristic
frequencies of motion. The authors compared their results with the Morlet wavelet,
and the results yielded similar information. The authors also tried to use Akima
spline, but this did not enhance the algorithm.
Wiley et al. [2004] used HHT to investigate the effect of reversible digitally
filtered molecular dynamics (RDFMD), which can enhance or suppress specific
frequencies of motion. HHT was applied to conformational motion, which occurs
predominantly by rotation about single bonds in a protein, called dihedral angles.
A small change in these angles can have significant effects on the structure of the
protein. Unfortunately, there were limitations of the HHT method as applied to the
MD simulations. These were:
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The Hilbert-Huang Transform in Engineering
•
•
•
The analysis of short RDFMD filters buffers results in a low frequency
limit in the HHT.
EMD produces physically unreliable IMFs. In an attempt to recreate the
usual signal given by the data, less the residual.
The HHT requirement of separable scales is not necessarily met by signals
from MD simulations.
Montesinos et al. [2002] applied HHT to signals obtained from BWR neuron
stability. The authors used data from a nuclear power plant that were recorded during
startup of the plant. They show a sharp peak produced by sudden activity insertion
due to the change of the recirculation pump velocity from low to high speed. The
authors attempted to recognize the IMF that is related to the stability of the BWR.
They further used an autoregressive model to analyze the fourth IMF and showed
that the IMFs are mutually orthogonal.
13.3.3 FINANCIAL APPLICATIONS
Huang et al. [2003b] applied HHT to nonstationary financial time series and used
a weekly mortgage rate data. The latter authors illustrated the differences among
Fourier, wavelet, and Hilbert spectral analysis. The authors relate the slope to the
global level of the data. By using the HHT, the authors address the question of
volatility, introducing a new measure of volatility, designated as “variability,” based
on EMD applied to produce IMF components. The variability was defined as the
ratio of the absolute value of the component(s) to the signal at any time:
V (t; T ) =
Sh (t )
S (t )
(13.29)
where T corresponds to the period at the Hilbert spectrum peak of the high-passed
signal up to h terms, and
h
Sh (t ) =
∑ c (t).
j
(13.30)
j =1
The unit of this variability parameter is the fraction of the market value. This is
a simple and direct measure of the market volatility.
13.3.4 METEOROLOGICAL AND ATMOSPHERIC APPLICATIONS
Salisbury and Wimbush [2002], using Southern Oscillation Index (SOI) data, applied
the HHT technique to determine whether the SOI data are sufficiently noise free
that useful predictions can be made and whether future El Niño southern oscillation
(ENSO) events can be predicted from SOI data. The authors were able to identify
an IMF that roughly corresponded to a 3.6-yr mean interval between ENSO warm
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events. The authors further developed a dynamical model using selected IMFs and
SOI data that can make a predictive estimate.
Pan et al. [2002] used HHT to analyze satellite scatterometer wind data over the
northwestern Pacific and compared the results to vector empirical orthogonal function (VEOF) results. Duffy [2004] used HHT to analyze three different data files:
•
•
•
Sea level heights
Incoming solar radiation
Barographic observations
The HHT persistently detected periodic features such as tides, snowmelt, and
heavy precipitation events in terms of sea level heights. Using the radiation data,
Duffy [2004] observed anomalies during ENSO and its effect on the aerosol concentration in the troposphere. Some of the IMFs of the observation data could be
paired with meteorological events, such as the passage of a cold front, a warm front,
and a trough. Generally, the HHT is capable of discerning mesoscale and synoptic
events.
With a five-day time series of temperature, Lundquist [2003] demonstrated how
the HHT allows simultaneous identification of the stationary diurnal temperature
cycle as well as intermittent and nonstationary cooling events such as frontal passages and density currents. One of the IMFs obtained by the authors appears to be
associated with inertia gravity waves.
13.3.5 OCEAN ENGINEERING
Schlurmann [2002] introduced the application of HHT to characterize nonlinear
water waves from two different perspectives, using laboratory experiments. The
author examined monochromatic and transient (freak) waves and compared the
results with wavelet analysis. The HHT was able to identify superharmonic frequencies, which were not evident in the wavelet spectrum, in both cases. Schlurmann
and Datig [2005] applied HHT to rogue waves (freak waves). The authors determined
statistical characteristics of the IMFs and then deduced that the natural data sets do
not behave like white noise. The statistical analysis demonstrated that the number
of extrema, the mean periods, and the corresponding standard deviation of the IMFs
are strongly related to each other. The authors developed an equation relating the
average period and mode number. Various coefficients were defined for different
ranges of modes.
Veltcheva [2002] applied HHT to wave data from nearshore sea. The IMF was
able to group the waves into calm and wave-decay stages at the offshore. Veltcheva
[2004] also compared the HHT and Fourier transforms for nearshore sea waves. The
author noted that the process of wave breaking mainly affects the IMF with the
highest energy in the decomposition of offshore waves. Wang [2004] presented an
improved EMD for the decomposition of wave groups. Hwang et al. [2004] compared
the energy flux computation of shoaling waves using HHT, FFT, and wavelet techniques. They found that the associated HHT spectral analysis provides superior
spatial (temporal) and wavenumber (frequency) resolution for handling nonstationarity
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and nonlinearity. They also found that the energy flux computed using the HHT
spectrum is higher than that from wavelet and FFT techniques.
Larsen et al. [2004] used HHT to characterize the underwater electromagnetic
environment and identify transient manmade electromagnetic disturbances. The
authors collected field data based on in-water magnetic data. The HHT was able to
act as a filter, effectively discriminating different dipole components. Therefore, the
method helps in detecting moving electric dipoles in the ocean. The authors further
developed an empirical mode–matched filter. The main idea is to apply a matched
filter to a signal generated by using IMFs as basis function, rather than to the target
signal. In the empirical mode–matched filter, EMD is initially applied to the signal
(time series) to generate the IMFs, i(t), i = 1, …, N, where N is the number of IMFs.
The second step involves determining the linear combination of the IMFs containing
the target signal s(t) that is best approximated to the known target signature in the
least-square sense:
N
min s(t ) −
ci ≥ 0
∑ c α .
i
i
(13.31)
i =1
A matched filter based on the target signature s(t) is applied to the modeled
signal, which is composed of weighted linear combination of the IMFs. The authors
attempted to use receiver operator characteristics (ROC) curves, HHT, and the
matched filter to address the probability of detection. Zeris and Prinos [2004]
compared the results of using HHT and wavelet transform for turbulent open channel
flow data. Laser Doppler and hot films techniques were used to obtain signals in
turbulent open channel flow, with and without the imposed bed suction. The authors
were fairly successful in the application. Unfortunately, it is not clear how the authors
did the comparative analysis.
13.3.6 SEISMIC STUDIES
Huang et al. [2001] used HHT to develop a spectral representation of earthquake
data. The HHT offers a better representation of the earthquake signals in the low
frequency range. The authors were able to address the shortcomings of various
methods typically used in earthquake spectral analysis. For example, the use of
Fourier-based methods and Duhanle integral assumes stationarity in the data; in
addition, the selection of an absolute maximum in the responses makes the HHT
spectrum less dependent on the record length. Therefore, the design spectrum,
defined as the envelope of a collection of response spectra, will inherit the shortcomings of the response spectrum: the underrepresentation of the most critical low
frequency range and its dangerous energy content. In the HHT approach, the frequency is instantaneous and is not determined by convolution. Loh et al. [2001]
used EMD and HHT to analyze the free field ground motion and to estimate the
global structural characteristics of building and bridge infrastructures. The process
was used to identify damage from the response data.
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Chen et al. [2002a] used HHT to determine the dispersion curves of seismic
surface waves and compared their results to Fourier-based time–frequency analysis.
The surface-wave dispersion measurement is essential for the study of the crust and
upper mantle structures, earthquake source mechanisms, and inelastic properties of
the earth. The authors applied EMD to the seismogram and obtained the IMFs from
which the Hilbert spectrum is calculated. Then, for each frequency, the time of the
maximum amplitude corresponds to the group arrival at which the group velocity
can be obtained. The result of group velocity and phase velocity determined by using
HHT is more accurate than by Fourier-based methods.
Shen et al. [2003] applied HHT to ground motion and compared the HHT result
with the Fourier spectrum. The HHT is capable of identifying the occurrence and
properties of destruction near fault ground motions. The authors used the method
as a demonstration tool in the estimation of seismic loading in bridge design.
Magrin-Chagnolleau and Baraniuk [undated] applied HHT to decompose a seismic
trace into different oscillatory modes and were able to extract instantaneous frequencies of these modes. Chun-xiang and Qi-feng [2003] made a comparative study of
wavelet and HHT using earthquake record. The IMF directly decomposed from the
original data, whereas wavelet components are decomposed according to mother
wavelets, which means the results can be influenced by the selected mother wavelets.
The Hilbert spectrum clearly presents the energy distribution with time and frequency.
Zhang et al. [2003] examined the applicability of HHT to dynamic and earthquake loading motion recordings. The IMF was capable of capturing low and high
frequency components. The authors commented how EMD can act like a filter by
grouping the IMFs. The authors defined EMD-based high frequency motions as the
summation of the first few IMF components, whereas the EMD-based low frequency
motion is the summation of the remaining components. Unfortunately, the authors
failed to give the exact number of IMFs needed to be summed. This choice, highlighted by the authors, is very subjective.
Wen and Gu [2004; Gu and Wen, 2004] extended the HHT and applied their
methods to earthquake ground motions. The authors express x(t) as follows:
n
x (t ) = Re
∑ a (t) exp
j
iθ j ( t )
+ rn (t )
(13.32)
j =1
and introduced j, an independent random phase angle uniformly distributed between
0 and 2:
n
x (t ) = Re
∑ a (t) exp
j
i [ θ j ( t )+ φ j ]
+ rn (t ) .
(13.33)
j =1
This makes x(t) a random process. The process then has the following mean, covariance, and variance functions:
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The Hilbert-Huang Transform in Engineering
 n

iθ
iφ 
µ x (t ) = Re 
a j (t )e j (t ) E (e j )  + rn (t ) = rn (t ) ,
 j =1

∑
(13.34)
n
K XX (t1, t2 ) =
∑
1
a j (t1 )a j (t2 ) cos[θ j (t1 ) − θ j (t2 ) ,
2 j =1
(13.35)
n
∑
1
σ x (t ) = E [ x (t ) − µx (t )] =
a 2j (t ) .
2 j =1



2
2


(13.36)
The authors extended the approach to a vector process and developed the
cross-correlation covariance as follows:
n
x k (t ) = Re
∑ a (t) exp
i [ θ jk ( t )+ φ jk ]
jk
+ rnk (t ) .
(13.37)
j =1
With the modified approach, the spectra of the simulated samples compare well
with those of the record.
Zhang [2005] used a modified HHT to estimate the damping factor of nonlinear
soil and its role in seismic wave responses at soil sites, using data from earthquake
recordings. Zhang presented the following equation:
n
x (t ) = Re
∑
n
iθ ( t )
a j (t ) exp j
= x (t ) = Re
j =1
∑ Λ (t) exp
[ − ϕ j ( t )+ iθ j ( t )]
j
(13.38)
j =1
where j(t) can be interpreted as the source-related intensity, j(t) is exponential factor
characterizing the time-dependent decay of waves in the jth IMF component due to
damping, and
n
a j (t ) = Re
∑ Λ (t) exp
j
− ϕ j (t )
.
(13.39)
j =1
The authors defined the instantaneous damping factor as:
η j (t ) =
© 2005 by Taylor & Francis Group, LLC
d ϕ j (t )
.
dt
(13.40)
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Perspectives on the Theory and Practices of the Hilbert-Huang Transform
295
The Hilbert damping spectrum is defined as follows:
D(ω, t ) =
n
 .a j (t )
.Λ j (t ) 
j =1
j
j
∑ − a (t) + Λ (t)  dt
(13.41)
whereas the marginal Hilbert damping spectrum is:
d (ω ) =
∫
T
D(ω, t )dt =
0
n
T
 .a j (t )
.Λ j (t ) 
j =1
0
j
j
∑ ∫ − a (t) + Λ (t)  dt = d (ω) + d (ω) .
a
Λ
(13.42)
Equation 13.4 consists of two components, the time-dependent amplitude, which
is related to marginal and Hilbert amplitude spectra, and the source-related intensity
time-dependent amplitude. The new approach can be used to evaluate site damping
and to address the site amplification factor for the influences of soil nonlinearity in
seismic analysis. The authors suggested that their approach must be validated by
model-based simulation.
13.3.7 STRUCTURAL APPLICATIONS
Quek et al. [2003] illustrate the feasibility of the HHT as a signal process tool for
locating an anomaly in the form of a crack, delamination, or stiffness loss in beams
and plates based on physically acquired propagating wave signals. The authors
considered four different cases:
1. Experimental data of a damaged aluminum beam to estimate the damage
and boundary conditions.
2. Delamination in a sandwiched aluminium plate.
3. Healthy and damaged states of reinforced concrete (RC) slab based on
experimental response time history.
4. Boundary and damage location of an aluminum square.
It appears that the authors were successful in all cases. Quek et al. [2004] did
a comparative analysis of HHT, wavelet transform, and FFT in damage detection.
Although the HHT was found to be a more direct method compared to the wavelet
transform, the authors provide similar results in terms of precision and accuracy.
The authors advocated the combination of EMD with Fourier transform on the
decomposed IMF components. They ascertained that this combination is capable of
revealing modal frequencies with relatively low energy content.
Using HHT, Li et al. [2003] analyzed the results of a pseudodynamic test of two
rectangular reinforced concrete bridge columns. The authors conducted extensive
laboratory investigation to model a near-fault ground acceleration of the Chi-Chi
Earthquake. The HHT was used to analyze the signal output from various columns
tested. The aim of these analyses was to understand the structural behaviors of the
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The Hilbert-Huang Transform in Engineering
columns under near-fault ground acceleration by observing the characteristics of the
Hilbert spectrum. The authors compared their results with Fourier transform results.
The authors drew various conclusions, notably that HHT can be used to obtain the
instantaneous natural frequencies of the bridge columns and to understand the
relationship between frequency changes and bridge conditions. Furthermore, the
HHT method outperforms the FFT method. Finally, the authors noted during the
experimental investigation that the more wide spread the distribution of the Hilbert
spectrum, the more severe the damage of the column.
13.3.8 HEALTH MONITORING
Pines and Salvino [2002] applied HHT in structural health monitoring. The authors
analyzed a civil building model with and without the presence of damage. The
authors also used the same model to simulate a seismic loading. The authors investigated two case studies: one looked at ground floor damage and the other at top
floor damage. To correctly identify damage structure, the authors used the concept
of phase dereverberation and applied EMD to dereverberation time series. To address
characteristic frequency change as a result of damage, the authors suggest that the
reverberated/dereverberated time domain response and the phase angles of both
dereverberated/reverberated signal can be used as indicators of damage.
Yang et al. [2004] used HHT for damage detection, applying EMD to extract
damage spikes due to sudden changes in structural stiffness. This enabled them to
identify damage time and locations. They also used the method to identify natural
frequencies and damping ratios of the structure before and after damage. The authors
noted that the capability of detecting damage spikes in a signal is a statistical variable
depending on the severity of damage and level of noise pollution. This implies that
when the severity of damage is low or the level of noise pollution is high, the current
method cannot detect the damage.
HHT was also used to detect cracks at rivet holes in thin plates by using Lamb
waves [Osegueda et al., 2003]. The authors conducted experimental studies on a
plate with a hole and with a notch. Area scanning was used to obtain time series
information on the wave propagation characteristics. The location of the notch was
detected accurately, and the extent of flaw was accurately predicted for all the
notches, except for the 3.2 mm long notch. The extent of flaw was overpredicted
for the shortest notch.
Yu et al. [2003] used HHT for fault diagnosis of roller bearings. The accelerative
vibration signal of roller bearings was analyzed. A wavelet was used to decompose
the signal to decrease the influence of lower frequency noise, after EMD was applied
to the signals.
13.3.9 SYSTEM IDENTIFICATION
Chen and Xu [2002] explored the possibility of using HHT to identify the modal
damping ratios of a structure with closely spaced modal frequencies and compared
their results to FFT. The damping ratios determined by using the combined HHT
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Perspectives on the Theory and Practices of the Hilbert-Huang Transform
297
and random decrement technique (RDT) are much better than those resulting from
the fast Fourier approach. In their studies, to identify instantaneous frequency and
damping ratio, the measured response time history of the MDOF system is decomposed into IMFs. The authors checked the validity of model by applying it to a real
civil engineering structure. The results show that the FFT-based method fails to
identify modal damping ratios when two modal frequencies are too close.
Xu et al. [2003] compared the modal frequencies and damping ratios in various
time increments and different winds for one of the tallest composite buildings in the
world. The results were then compared to FFT. Once again, the HHT method
outperformed the FFT. The FFT-based results significantly overestimate the first
lateral and longitudinal modal damping ratios of the building.
Yang et al. [2002] developed a damage identification–based HHT. The authors
applied the technique to the ASCE structural health monitoring benchmark structure.
The approach was quite successful in identifying damage sections. Yang et al.
[2003a] used HHT to identify multi-degree-of-freedom linear systems by using
measured free vibration time histories. The authors proposed band filter and EMD
cases with polluted signals and high modal frequencies. Failure to introduce the
band-pass filtering approach will result in a large number of siftings. The Fourier
spectrum of the acceleration signal helps to identify the range of natural frequencies.
The acceleration signal is processed through band-pass filters. The time history
obtained from each band-pass proceeds through EMD. A linear least-square fit
procedure is proposed to identify natural frequencies, damping ratios, mode shapes,
mass matrix, damping matrix, and stiffness matrix. Yang et al. [2003b] extended
their work [Yang et al., 2003a] to identify general linear structures with complex
modes by using free vibration response data polluted by noise. Simulated results
show that the approach is capable of identifying the complex mode shapes as well
as the mass, stiffness, and damping matrices. This can help with the complete
dynamic characteristics of general linear structure.
13.4 SOME LIMITATIONS
Quek et al. [2003] noted the following:
•
•
The approximate envelopes obtained using the local maxima and minima
do not always encompass all the data.
The use of discrete signals instead of continuous introduces numerical
uncertainty in the true extrema.
The spline at the beginning and the end of the envelope can result in large
variation, which is very critical in a low frequency signal.
Deng et al. [2001] discussed the use of a cubic spline in calculating the lower
and upper envelopes. The authors mentioned the data extension method as key in
addressing this issue. They used a two-step extension method based on a neural
network approach. The original signal was extended forward and backward, and a
neural network was used to learn the envelopes.
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The Hilbert-Huang Transform in Engineering
Chen and Feng [undated] proposed a technique to improve the HHT procedure.
The authors noted that the EMD has its limitation in distinguishing different
components in narrow-band signals. The narrow band may contain either (a) components that have adjacent frequencies or (b) components where the frequencies
may not be adjacent, but one of the components has a dominant energy intensity
much higher than the other components. The improved technique is based on beating-phenomenon waves.
Olhede and Walden [2004] compared HHT to a maximal-overlap discrete wavelet transform (MODWT) and wavelet packet transforms (MODWPT). The authors
replaced EMD projections by a wavelet-based one and developed a Hilbert spectrum
via wavelet packets. The authors claimed the MODWPT and MODWT to be superior
to the HHT using a particular class of Fejer-Korovkin wavelet filters. It appears that
their assertion may be true only for nonlinear data. Attoh-Okine [2004] did a
comparative analysis of HHT, MODWPT, and MODWT on pavement profile data
and deduced that the HHT outperforms the MODWPT and MODWT in the case of
nonlinear and nonstationary data.
Datig and Schlurmann [2004] did the most comprehensive studies on the performance and limitations of HHT with particular applications to irregular waves.
The authors did extensive investigation into the spline interpolation. The authors
discussed using additional points, both forward and backward, to determine better
envelopes. They also did a parametric study on the proposed improvement and
showed significant improvement in the overall EMD computations. The authors
noted that HHT is capable of differentiating between time-variant components from
any given data. Their study also showed that HHT was able to distinguish between
riding and carrier waves.
13.5 POTENTIAL FUTURE RESEARCH
The HHT is becoming one of the most useful signal-processing techniques. The idea
of cubic spline needs a critical look, although it has been addressed by a few authors.
Another area that needs further research is proper interpretation of all of the IMFs.
This depends on a variety of factors, especially the proper understanding of the
system. Researchers have to address what criteria should be used to eliminate IMFs
within a particular signal. Is the statistical analysis and distribution of the IMFs the
answer? What are proper approaches for using the HHT technique as a filter method?
Comparative analyses of HHT, FFT, and wavelet techniques have been well
addressed, but more research is needed to investigate the MODWPT and MODWT,
and which of these approaches is more applicable than the other. The use of computational intelligence has not been addressed in the overall HHT applications. For
example, how can neural networks be used effectively in constructing the upper and
lower envelopes? The use of HHT in image analysis and two-dimensional problems
also needs to be explored.
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In this Volume.
Zhang, R. R., Demark, L. V., Liang, J., and Hu, Y. (2004). On Estimating Site Damping with
Soil Non-Linearity from Earthquake Recordings. Int. J. Non-Linear Mech. 39,
1501–1517.
Zhang, R. R., Ma, S., Safak, E., and Hartzell, S. (2003). Hilbert-Huang Transform Analysis
of Dynamic and Earthquake Motion Recordings. J. Eng. Mech. 129, 8: 861–875.
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The Hilbert-Huang Transform in Engineering
ADDENDUM: PERSPECTIVES ON THE THEORY AND PRACTICES
OF THE HILBERT-HUANG TRANSFORM
Recently there have been innovative applications and major improvement to EMDHHT. These improvements have spread to general analytical approaches, hybrid
applications, and bidimensional HHT.
13A.1 ANALYTICAL
Coughlin and Tung [2004] used EMD to extract the solar cycle signal from stratospheric data. The authors highlighted some difficulties in using EMD — especially
the influence of the end point in the sifting process. They addressed a method of
reducing the influence of the end effects on the internal solution. Coughlin and Tung
[2004] extended both the beginning and end of the spline by introducing a wave
equation of the form:

 2πt
Wave Extension = A sin 
+ phase + local mean 

 p
(13A.1)
The typical amplitude, A, and period p, are determined by the nearest local extrema.
()
()
( )
( )
Abeginning = max 1 − min 1
Aend = max N − min N
()
()
( )
( )
(13A.2)
pbeginning = 2 time max 1 − time min 1
pend = 2 time max N − time min N
where max(1) and min(1) are the first two local extrema in the time series and
max(N) and min(N) are the last two local extrema. This approach reduces the large
swings in the spline calculation. This approach is quite different from the one
proposed by Dätig and Schlurmman [2004].
13A.2 HYBRID METHOD
Iyengar and Kanth [2004] used HHT to decompose India monsoon data. The authors
argued that the first IMF is nonlinear and the remaining IMFs are linear. This is a
little questionable, since it is not clear how the authors determined these properties.
The authors used a neural network to forecast the nonlinear part of the data. Lee et
al. [2003] explored the vibration mechanism of a bridge at both global and local
behavior. They proposed a combination of EMD, HHT, and a nonlinear parametric
model as an identification methodology. The approach was successful in identifying
abnormal signals indicating the structural condition of the bridge decks based on
the time–frequency spectrum.
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Perspectives on the Theory and Practices of the Hilbert-Huang Transform
303
Loutridis [2004] used EMD to decompose vibration signals from gear systems.
The author developed an empirical law that relates the energy of the intrinsic modes
to the crack magnitude of the gear. Using this technique, the degree of crack can be
predicted based on energy obtained from the IMFs.
13A.3 BIDIMENSIONAL EMD
Nunes and coworkers developed a method for analyzing two-dimensional (2D) time
series which was used for textural analysis [2003a]. The bidimensional sifting
process they presented in an earlier paper [Nunes et al., 2003] is as follows:
•
•
•
•
•
Identify the extrema of the image I by morphological reconstruction based
on geodesic operators.
Generate the 2D “envelope” by connecting the maxima points with a
radial basis function (RBF).
Determine the local mean m_{i}, by averaging the two envelopes.
Subtract I – m_{i} = h_{i}
Repeat the process.
The authors used a radial basis function in the form:
()
N
( ) ∑ λ Φ( x − x )
s x = pm x +
i
i
(13A.3)
i =1
radial basis function (RBF);
pm is mth degree polynomial in d variables;
λ are RBF coefficients;
xi are the RBF centers.
The stopping criteria for Nunes et al. [2003] approach is based on similar to 1D
case used, using the standard deviation.
Linderhed [2002] developed a 2D EMD and presented a coding scheme for
image compression purposes. In the coding scheme, only the maximum and minimum values of the IMFs are used. Linderhed [2004] also developed the concept of
empiquency, short for empirical mode frequency as a frequency measure.
Empiquency is defined as one half the reciprocal distance between two consecutive
extrema points. Linderhed [2004] developed a sifting process for 2D time series.
The symmetry criterion on the IMF envelope is relaxed, and the stop criterion is
based on the condition that the IMF envelope is close to zero. Hence, there is no
need to check for symmetry. The new sifting process is as follows:
1. Find the positions and amplitudes of all local maxima and minima in the
input signal in_{lk}(m,n), where m and n denote spatial dimensions of the
image.
2. Create upper and lower envelopes by spline interpolation, denoted by
e_{max}(m,n) and e_{min}(m,n).
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The Hilbert-Huang Transform in Engineering
3. For each position, calculate the mean of the upper and lower envelopes:
(
(
)
(
emax m, n + emin m, n
)
emlk m, n =
)
2
(13A.4)
4. Subtract the envelope mean signal from the input signal:
(
)
(
)
(
hlk m, n = inlk m, n − emlk m, n
)
(13A.5)
The process is terminated if the envelope mean signal is close to zero:
(
)
emlk m, n ≺ ε
(13A.6)
for all (m,n). If it is not, the process repeated until:
(
)
(
hl( k +1) m, n = hlk m, n
)
(13A.7)
When the termination criterion is achieved, k = K, and the IMF is defined
as:
(
)
(
)
)
(
cl m, n = hlK m, n
(13A.8)
5. The residue is defined as r_{l}(m,n):
(
)
(
rl m, n = inl1 m, n = cl m, n
)
(13A.9)
6. The next IMF is found as follows:
(
) (
in(l +1)1 m, n = rl m, n
)
(13A.10)
7. Just as the 1D series the signal can be represented as:
(
)
(
L
) ∑ c ( m, n )
x m, n = rL m, n +
j
(13A.11)
i =1
Liu and Peng [2005] proposed a boundary-processing technique for bidimensional EMD. The authors proposed two techniques, but unfortunately the two algorithms developed by these authors are similar to algorithms proposed by Nunes et
al. [2003] and Linderhead [2004].
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Perspectives on the Theory and Practices of the Hilbert-Huang Transform
305
13A.4 SOME OBSERVATIONS
The use of bidimensional EMD-HHT is beginning to emerge. Few results indicate
that the 2D HHT outperforms traditional wavelet and Fourier analysis. As with the
1D HHT, there are difficulties regarding the type of spline and the effect of the end
points on the overall sifting process. The use of different RBFs, such as thin plate
splines, cubic spline, and multiquadrics, will be worth exploring in the bidimensional
HHT.
REFERENCES
Coughlin, K. T. and Tung, K. K. (2004). 11-Year Solar Cycle in the Stratosphere Extracted
by the Empirical Mode Decomposition Method. Adv. Space Res. 34, 323–329.
Dätig, M. and Schlurmann, T. (2004). Performance and Limitations of the Hilbert-Huang
Transformation with an Application to Irregular Water Waves. Ocean Eng. 31, 14/15:
1783–1834.
Iyengar, R. N. and Kanth R. S. T. G (2004). Intrinsic Mode Functions and a Strategy for
Forecasting Indian Monsoon Rainfall. Meteorol. Atmospheric Phys. (online publication).
Lee, Z. K., Wu, T. H. and Loh, C. H. (2003). System Identification on Seismic Behavior of
an Isolated Bridge. Earthquake Eng. Structural Dynamics 32, 1797–1812.
Linderhed, A. (2003). 2-D Empirical Mode Decomposition: In Spirit of Image Compression.
SPIE Proc. 4738, 1–8.
Linderhed, A. (2003). Image Compression Based on Empirical Mode Decomposition. Proc.
SSBA Symp. Image Anal. 110–113.
Linderhed, A. (2004). Adaptive Image Compression with Wavelet Packets and Empirical
Mode Decomposition. Dissertation No. 909 submitted to Linkoping Studies in Science
and Technology.
Liu, Z. and Peng, S. (2005). Boundary Processing of Bidimensional EMD Using Texture
Synthesis. IEEE Signal Process. Lett. 12, 1: 33–36.
Loutridis, S. J. (2004). Damage Detection in Gear Systems Using Empirical Mode Decomposition. Eng. Structures 26, 1833–1841.
Nunes, J. C., Bouaoune, Y., Delechelle, E., Niang, O. and Bunel P. (2003). Image Analysis
by Bidimensional Empirical Mode Decomposition. Image Vision Comput. 21,
1019–1026.
Nunes, J. C., Niang, O., Bouaoune, Y., Delechelle, E., and Bunel P. (2003a). Bidimensional
Empirical Mode Decomposition Modified for Texture Analysis. SCIA 2003, Lecture
Notes on Computer Science 2749, 171–177.
© 2005 by Taylor & Francis Group, LLC