spectra: applicationscomputational geophysics and data analysis 1 fourier transform: applications in...

Computational Geophysics and Data Analysis1

Spectra: Applications

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



Fourier: Space and Time

Spacex space variableL spatial wavelengthk=2p/ spatial wavenumberF(k) wavenumber spectrum

Spacex space variableL spatial wavelengthk=2p/ spatial wavenumberF(k) wavenumber spectrum

Timet Time variableT periodf frequency=2f angular frequency

Timet Time variableT periodf frequency=2f angular frequency

Fourier integralsFourier integrals

With the complex representation of sinusoidal functions eikx

(or (eit) the Fourier transformation can be written as:

With the complex representation of sinusoidal functions eikx

(or (eit) the Fourier transformation can be written as:

dxexfkF

dxekFxf

ikx

ikx

)(2

1)(

)(2

1)(



The Fourier Transformdiscrete vs. continuous

dxexfkF

dxekFxf

ikx

ikx

)(2

1)(

)(2

1)(

1,...,1,0,

1,...,1,0,1

/21

0

/21

0

NkeFf

NkefN

F

NikjN

jjk

NikjN

jjk

discrete

continuousWhatever we do on the computer with data will be based on the discrete Fourier transform

Whatever we do on the computer with data will be based on the discrete Fourier transform



Phase and amplitude spectrum

)()()( ieFF

The spectrum consists of two real-valued functions of angular frequency, the amplitude spectrum mod (F()) and the phase spectrum

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)



Spectral synthesis

The red trace is the sum of all blue traces!



The spectrum

Amplitude spectrum Phase spectrumFo

uri

er

space

Physi

cal sp

ace



The Fast Fourier Transform (FFT)

Most processing tools (e.g. octave, Matlab, Mathematica, Fortran, etc) have intrinsic functions for FFTs

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. 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.

>> 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. 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.

Matlab FFT



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)



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



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



Spectral leakage



Fourier Spectra: Main Casesrandom signals

Random signals may contain all frequencies. A spectrum with constant contribution of all frequencies is called a white

spectrum

Random signals may contain all frequencies. A spectrum with constant contribution of all frequencies is called a white

spectrum



Fourier Spectra: Main CasesGaussian 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?

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?



Fourier Spectra: Main CasesTransient waveform

A transient wave form is a wave form limited in time (or space) in comparison with a harmonic wave form that is

infinite

A transient wave form is a wave form limited in time (or space) in comparison with a harmonic wave form that is

infinite



Puls-width and Frequency Bandwidth

time (space) spectrumN

arr

ow

ing

ph

ysi

cal si

gn

al

Wid

en

ing

fre

qu

en

cy b

an

d



Frequencies in seismograms



Amplitude spectrumEigenfrequencies



Sound of an instrument

a‘ - 440Hz



Instrument Earth

26.-29.12.2004 (FFB )

0S2 – Earth‘s gravest tone T=3233.5s =53.9min

Theoretical eigenfrequencies



The 2009 earthquake „swarm“



Rotations …



First observations of eigenmodes with ring laser!



16 hr time window (Samoa)



36 hr time window (Chile quake)



Time – frequency analysis



Spectral analysis: windowed spectra

24 hour ground motion, do you see any signal?



Seismo-“weather“

Running spectrum of the same data

Computational Geophysics and Data AnalysisSpectra: Applications

Shear w

ave sp

eed

varia

tions

Gorb

ato

v &

Kennett

(20

03

)

?

Western Samoa, 1993/5/16, D=34.97°

Western Samoa, 1993/5/16, D=34.97°

Vz [m

/s]

Vz [m

/s]

t [s]

t [s]

Going beyond ray tomography …



Kristeková et al., 2006 : ... the most complete and informative characterization of

a signal can be obtained by its decomposition in the time-frequency plane ... .

dτet)g(τ)u(2π

1G(u):t),û( i

dτet)g(τ)(u2π

1)G(u:t),(û i

000

[ t-w representation of synthetics, u(t) ]

[ t-w representation of data, u0(t) ]

Time-frequency representation of seismogramscomparing data with synthetics

… an elegant way of separating phase (i.e., travel time) and amplitude (envelope) information!



Time-frequency misfits... individually weighted …



Some properties of FT

• FT is linearsignals 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

the negative frequencies can be obtained from symmetry properties

• Shifting corresponds to changing the phase (shift theorem)

• Derivative)(*)( FF

)()(

)()(

tfeaF

Featfai

ai

)()()( Fitfdt

d nn



Summary

• Care has to be taken when estimating spectra for finite-length signals (detrending, down-sampling, 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)

spectra: applicationscomputational geophysics and data analysis 1 fourier transform: applications in...

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