Fourier Transform: Applications in seismology
• Estimation of spectra
– windowing
– resampling
•
•
•
•
Seismograms – frequency content
Eigenmodes of the Earth
„Seismo-weather“ with FFTs
Derivative using FFTs – pseudospectral method for wave
propagation
• Convolutional operators: finite-difference operators using
FFTs
Scope: Understand how to calculate the spectrum from time series
and interpret both phase and amplitude part. Learn other
applications of the FFT in seismology
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Spectra: Applications
Computational Geophysics and Data Analysis
Fourier: Space and Time
Space
x
space variable
L
spatial wavelength
k=2p/l spatial wavenumber
F(k)
wavenumber spectrum
t
T
f
w=2pf
Time
Time variable
period
frequency
angular frequency
Fourier integrals
With the complex representation of sinusoidal functions eikx (or (eiwt)
the Fourier transformation can be written as:
1
f ( x) 
2p
1
F (k ) 
2p

ikx
F
(
k
)
e
dx




f ( x)eikxdx

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Spectra: Applications
Computational Geophysics and Data Analysis
The Fourier Transform
discrete vs. continuous
Whatever we do on the
computer with data will be
based on the discrete Fourier
transform
discrete
1
Fk 
N
N 1

j 0
continuous
1
f ( x) 
2p
1
F (k ) 
2p

ikx
F
(
k
)
e
dx




f ( x)eikxdx

f j e  2pikj / N , k  0,1,..., N  1
N 1
f k   F j e 2pikj / N , k  0,1,..., N  1
j 0
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Spectra: Applications
Computational Geophysics and Data Analysis
Phase and amplitude spectrum
The spectrum consists of two real-valued functions of angular
frequency, the amplitude spectrum mod (F(w)) and the phase
spectrum f(w)
i (w )
F (w)  F (w) e
In many cases the amplitude spectrum is the most important
part to be considered. However there are cases where the
phase spectrum plays an important role (-> resonance,
seismometer)
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Spectra: Applications
Computational Geophysics and Data Analysis
Spectral synthesis
The red trace is the sum of all blue traces!
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Spectra: Applications
Computational Geophysics and Data Analysis
The spectrum
Phase spectrum
Fourier space
Amplitude spectrum
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Spectra: Applications
Computational Geophysics and Data Analysis
The Fast Fourier Transform (FFT)
Most processing tools
(e.g. octave, Matlab,
Mathematica,
Fortran, etc) have
intrinsic functions
for FFTs
>> help fft
FFT Discrete Fourier transform.
FFT(X) is the discrete Fourier transform (DFT) of vector X. For
matrices, the FFT operation is applied to each column. For N-D
arrays, the FFT operation operates on the first non-singleton
dimension.
FFT(X,N) is the N-point FFT, padded with zeros if X has less
than N points and truncated if it has more.
FFT(X,[],DIM) or FFT(X,N,DIM) applies the FFT operation across the
dimension DIM.
Matlab FFT
For length N input vector x, the DFT is a length N vector X,
with elements
N
X(k) =
sum x(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.
n=1
The inverse DFT (computed by IFFT) is given by
N
x(n) = (1/N) sum X(k)*exp( j*2*pi*(k-1)*(n-1)/N), 1 <= n <= N.
k=1
See also IFFT, FFT2, IFFT2, FFTSHIFT.
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Spectra: Applications
Computational Geophysics and Data Analysis
Good practice – for estimating spectra
1. Filter the analogue record to avoid aliasing
2. Digitise such that the Nyquist lies above the
highest frequency in the original data
3. Window to appropriate length
4. Detrend (e.g., by removing a best-fitting line)
5. Taper to smooth ends to avoid Gibbs
6. Pad with zeroes (to smooth spectrum or to use
2n points for FFT)
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Spectra: Applications
Computational Geophysics and Data Analysis
Resampling (Decimating)
• Often it is useful to down-sample a time series
(e.g., from 100Hz to 1Hz, when looking at
surface waves).
• In this case the time series has to be
preprocessed to avoid aliasing effects
• All frequencies above twice the new sampling
interval have to be filtered out
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Spectra: Applications
Computational Geophysics and Data Analysis
Spectral leakage, windowing, tapering
Care must be taken when extracting time
windows when estimating spectra:
– as the FFT assumes periodicity, both ends
must have the same value
– this can be achieved by „tapering“
– It is useful to remove drifts as to avoid any
discontinuities in the time series -> Gibbs
phenonemon
-> practicals
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Spectra: Applications
Computational Geophysics and Data Analysis
Spectral leakage
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Spectra: Applications
Computational Geophysics and Data Analysis
Fourier Spectra: Main Cases
random signals
Random signals may contain all frequencies. A spectrum with
constant contribution of all frequencies is called a white spectrum
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Spectra: Applications
Computational Geophysics and Data Analysis
Fourier Spectra: Main Cases
Gaussian signals
The spectrum of a Gaussian function will itself be a Gaussian
function. How does the spectrum change, if I make the Gaussian
narrower and narrower?
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Spectra: Applications
Computational Geophysics and Data Analysis
Fourier Spectra: Main Cases
Transient waveform
A transient wave form is a wave form limited in time (or space) in
comparison with a harmonic wave form that is infinite
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Spectra: Applications
Computational Geophysics and Data Analysis
Puls-width and Frequency Bandwidth
spectrum
Narrowing physical signal
Widening frequency band
time (space)
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Spectra: Applications
Computational Geophysics and Data Analysis
Frequencies in seismograms
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Spectra: Applications
Computational Geophysics and Data Analysis
Amplitude spectrum
Eigenfrequencies
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Spectra: Applications
Computational Geophysics and Data Analysis
Sound of an instrument
a‘ - 440Hz
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Spectra: Applications
Computational Geophysics and Data Analysis
Instrument Earth
26.-29.12.2004 (FFB )
0S2 –
Earth‘s gravest tone
T=3233.5s =53.9min
Theoretical eigenfrequencies
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Spectra: Applications
Computational Geophysics and Data Analysis
The 2009 earthquake „swarm“
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Spectra: Applications
Computational Geophysics and Data Analysis
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Spectra: Applications
Computational Geophysics and Data Analysis
Rotations …
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Spectra: Applications
Computational Geophysics and Data Analysis
First observations of eigenmodes with ring laser!
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Spectra: Applications
Computational Geophysics and Data Analysis
16 hr time window (Samoa)
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Spectra: Applications
Computational Geophysics and Data Analysis
36 hr time window (Chile quake)
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Spectra: Applications
Computational Geophysics and Data Analysis
Time – frequency analysis
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Spectra: Applications
Computational Geophysics and Data Analysis
Spectral analysis: windowed spectra
24 hour ground motion, do you see any signal?
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Spectra: Applications
Computational Geophysics and Data Analysis
Seismo-“weather“
Running spectrum of the same data
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Spectra: Applications
Computational Geophysics and Data Analysis
Going beyond ray tomography …
Vz [m/s]
Shear wave speed variations
Gorbatov & Kennett (2003)
Western Samoa, 1993/5/16, D=34.97°
t [s]
Vz [m/s]
Western Samoa, 1993/5/16, D=34.97°
t [s]
Spectra: Applications
Computational Geophysics and Data Analysis
?
Time-frequency representation of seismograms
comparing data with synthetics
Kristeková et al., 2006 : ... the most complete and informative characterization of
a signal can be obtained by its decomposition in the time-frequency plane ... .
û(w , t) : G(u) 
1
2π



1
û 0 (w , t) : G(u 0 ) 
2π
u( ) g(τ  t) e iw dτ



u 0 ( ) g(τ  t) e iw dτ
[ t-w representation of synthetics, u(t) ]
[ t-w representation of data, u0(t) ]
… an elegant way of separating phase (i.e., travel time) and amplitude
(envelope) information!
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Spectra: Applications
Computational Geophysics and Data Analysis
Time-frequency misfits
... individually weighted …
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Spectra: Applications
Computational Geophysics and Data Analysis
Some properties of FT
• FT is linear
signals can be treated as the sum
of several signals, the transform
will be the sum of their transforms
-> stacking is possible!
• FT of a real signals
has symmetry properties
F (w )  F * (w )
the negative frequencies can be
obtained from symmetry
properties
• Shifting corresponds to
changing the phase (shift
theorem)
f (t  a)  e iwa F (w )
F (w  a)  e iwa f (t )
• Derivative
dn
f (t )  (iw ) n F (w )
dt
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Spectra: Applications
Computational Geophysics and Data Analysis
Summary
• Care has to be taken when estimating spectra
for finite-length signals (detrending, downsampling, etc. -> practicals)
• The Fourier transform is extremely useful in
understanding the behaviour of numerical
operators (like finite-difference operators)
• The Fourier transform allows the calculation of
exact (to machine precision) derivatives and
interpolations (-> practicals)
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Spectra: Applications
Computational Geophysics and Data Analysis