Seismic Signal Processing by Wavelet Based Multifractal Analysis
SOUNDARARAJAN EZEKIEL
ROBERT A.HOVIS
Ohio Northern University
ADA, Ohio, USA
and
Kramer James M
Abdullah A. Alsheri
Mine Safety & Health Administration
Pittsburgh, PA 15236
ABSTRACT
Fractal geometry and wavelets are a new and promising
approach to analyze and characterize non-stationary
signals such as seismic signals, ECG, stock prices, etc.
Short-term Fourier transform analysis imposes the
assumption that signals are stationary over small temporal
segments. Such an assumption is inappropriate for seismic
signals because conditions inside mines undergo constant
change. Because they do not impose this assumption,
wavelet-based techniques are proving useful for
processing seismic data. In this paper, we propose a
wavelet-based method to analyze seismic signals, which
is superior to many conventional methods used in the
industry today. The fractal dimension and other measures
are calculated by using wavelets, in which time is
irrelevant. Another measure, Multifractal Spectrum is
computed with the help of a scaling exponent. Using this
strategy, our method distinguishes between a seismic
signal and a noisy signal or truly chaotic signal by fractal
dimension. Further our result shows that the larger fractal
dimension indicates a rock burst in mines. The approach
introduced here should be useful in the analysis of other
non-stationary biological signals.
Keywords: Fractal dimension, Wavelet, Multifractal,
Seismic signals, Scaling exponent
1. INTRODUCTION
One of the most important tasks in seismic signal
processing is to be able to both detect and identify the
seismic signals. Detection consists of recognizing that a
seismic event has occurred and locating the source of the
seismic signals. Once a seismic event has been detected,
the next task is to determine if an underground nuclear
explosion created it or if it was created by other seismic
events including natural earthquakes, rock bursts in
mines, and chemical explosions conducted for mining,
quarry blasting, and construction. Rock bursts are
experienced in underground mining at various localities in
the world, causing death and injury to underground
miners and damaging mine structures. To date, many
studies have been conducted to understand the cause of
rock bursts and outbursts to predict their occurrence. The
approaches made to detect rock bursts include the
traditional time-frequency analysis such as the windowed
Fourier Transform (WFT), or any standard statistical
methods like the microgravity method, rheological
method, rebound method, drilling-yield method,
microseismic method and so on. Although all of these
methods have been used, none is completely reliable and
few are useful in the rapidly advancing mining
environment. The U.S. Bureau of mines recognized
microsesismic technology as a potential tool for rock
burst prediction as early as 1939. However, to date only a
few successful predictions have been achieved. In this
paper, we introduce a different and efficient detection
method, which is based on fractal dimension by using
wavelets. In section 2 we recall some fundamental
definitions of fractal measure, multifractal analysis, and
wavelets. Section 3 introduces the Multidimensional Nonlinear Spectral Estimation (MNSE) method. Experimental
results and discussion are given in section 4. Section 5
concludes the paper.
2. WAVELET BASED MULTIFRACTAL
ANALYSIS
Multifractal theory was described for the first time by B.
B. Mandelbrot [2] in the context of fully developed
turbulence. Since then, mathematicians have increasingly
studied multifractals. Multifractal analysis has recently
drawn much attention as a tool for studying singular
measures and functions in both theory and applications
[3]. Multifractal analysis has been extended as an
application of Choquet capacities in[4]. In the multifractal
scheme, the pointwise structure of a singular measure is
analyzed through the so called ''multifractal spectrum'',
which gives either geometrical or probabilistic
information about the distribution of points having the
same degree of singularity. Several definitions of a
multifractal spectrum exist. In this section, we take an
approach to multifractal analysis pioneered by Levy
Vehel et al[5][6] However, instead of computing the
local singularity exponent by using Choquet capacities
[6], we develop an efficient wavelet-based calculation of
local fractal dimension[9]. This method is explained in
Section 3. Using this method, we capture the dynamics of
seismic signals.
3. MULTIDIMENSIONAL NON-LINEAR
SPECTRAL ESTIMATION (MNSE)
The two signals shown in Figures 1 and 2 look quite
similar. They have approximately the same statistical
properties, such as mean, standard deviation, and
variance. But they are quite different.
Wavelets
Wavelets [1] are presently used in many disciplines in
science and engineering. In the last few years, the
wavelet transform has become a cutting edge technology
in the field of image and signal processing. Jean Morlet
and Alex Grossmann introduced the concept of wavelets.
It was mainly developed by Y. Meyer [7]. Stephen Mallet
developed the first algorithm in 1988[8]. After that, many
scientists like Ingrid Daubchies and Ronald Coifmen
contributed to this field. A wavelet is a waveform of
effectively limited duration that has an average value that
is zero. So wavelet analysis is done by breaking up of a
signal into shifted and scaled versions of the original
(mother) wavelet. From this, we can define a continuous
wavelet transform as the sum over all time of the signal
multiplied by a scaled and shifted version of the wavelet
function . i.e.
Figure 1
where scaling means stretching(or compressing) and
position means shifting the wavelet.
Fractal Measure
The basic approach of Vehel et al. is as follows. A
measure  on Rn(with 0<( Rn)<) can give rise to a
hierarchy of fractal sets. For 0 we define sets
where
is the local dimension of  at x, thus E() is the set of
points at which  has local dimension . One problem
that arises is how big the sets E() are for various ’s.
There are two approaches for this: a) we may consider
(E()) as  varies and b) find the Hausdorff dimension
of E(). The function
is termed the multifractal spectrum of .
Figure 2
The one on the top is random and the one on the bottom is
derived from Xn+1=CXn(1-Xn), where C is any constant
number (say 3.95). Many signals look random. Examples
are seismic signals, heart rate, blood pressure in the
arteries, and stock prices. It has always been assumed that
random processes can describe these fluctuations.
However, if these fluctuations are not random, we might
then be able to understand these mechanisms and to
control them. This will lead to an increase in our
understanding of physiological systems. To analyze such
kinds of signals we develop a new wavelet based
approach to estimate a multifractal spectrum and its
important
parameters.
We
call
this
method
Multidimensional Non-linear Spectral Estimation
(MNSE).
Methodology
Our method is wavelet based, because wavelets are useful
in many frameworks for approximation and they are also
a better tool to analyze seismic signals. First, we start with
the original seismic signal which has some abnormal
values (spikes) and then truncate (or remove) the
abnormal values. Let n be the length of the truncated
signal. Apply a continuous wavelet transformation for
multilevel say l (32 levels) to get wavelet coefficients Cji
where i varies from 1 to n and j varies from 1 to l.
Secondly, we construct the slope signal S as follows: For
each i, set:
Figure 4 (a)
Fit Linear regression Y=a+bX and set Si=b.
Next, cluster the slope signal S into N segments, where
each segment consists of about 1000 elements. For each
segment, compute E() for different  (say 15 bins),
and the following measures: fractal dimension (FD)
[9][10], average fractal dimension, mean alpha, second
moment about zero for alpha, third moment about zero
for alpha, standard deviation for alpha, maximum alpha,
and maximum fractal dimension etc,. Finally, draw the
multifractal spectrum by plotting fractal dimension of
each set E().
4. RESULTS AND DISCUSSIONS
Twenty-four signals were included in the study (n=24).
We used the seismic data collected over a 24-hour period
for the analysis. We divided the signal into segments of
about 1000 points. For each segment, we computed fractal
dimension and its important parameters [11], such as
Hurst exponent, correlation dimension, etc. Figures 4(a)Figure 4(d) shows two sample seismic signals, fractal
dimension spectrum, and Hurst exponent spectrum.
Figure 4 (b)
Figure 4 (c)
Figure 4 (d)
For each signal, we then computed the average fractal
dimension, Hurst exponent and correlation dimension.
The average values are plotted against number of days.
Refer to Figure 4(e) and Figure (f). We noticed that
smaller the Hurst exponent (correlation dimension) - that
is, the larger the fractal dimension, corresponds to the
signal with some kind of local burst. This method can also
provide a significant amount of additional information
that is not covered by the standard measures
Figure 4 (f)
5. CONCLUSION
A framework using wavelets and multifractal spectrum
was presented. This framework is suitable to analyze nonstationary signals especially seismic signals. Although in
principle the techniques of non-linear dynamics have been
shown to be powerful tools for characterization of various
non-stationary signals, no major breakthrough has yet
been achieved by their application to seismic signal
analysis. However, further experimental analysis needs to
be carried on to fine tune the various parameters in this
method.
6. REFERENCES
Figure 4 (e)
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