Fourier Transform: Applications in seismology

1. 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 Computational Geophysics and Data Analysis

2. Fourier: Space and Time Time t Time variable T period f frequency w=2pf angular frequency Space x space variable L spatial wavelength k=2p/l spatial wavenumber F(k) wavenumber spectrum Fourier integrals With the complex representation of sinusoidal functions eikx (or (eiwt) the Fourier transformation can be written as: Computational Geophysics and Data Analysis

3. The Fourier Transformdiscrete vs. continuous Whatever we do on the computer with data will be based on the discrete Fourier transform continuous discrete Computational Geophysics and Data Analysis

4. 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) 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) Computational Geophysics and Data Analysis

5. Spectral synthesis The red trace is the sum of all blue traces! Computational Geophysics and Data Analysis

6. The spectrum Amplitude spectrum Phase spectrum Fourier space Physical space Computational Geophysics and Data Analysis

7. The Fast Fourier Transform (FFT) >> 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. Most processing tools (e.g. octave, Matlab, Mathematica, Fortran, etc) have intrinsic functions for FFTs Matlab FFT Computational Geophysics and Data Analysis

8. Good practice – for estimating spectra • Filter the analogue record to avoid aliasing • Digitise such that the Nyquist lies above the highest frequency in the original data • Window to appropriate length • Detrend (e.g., by removing a best-fitting line) • Taper to smooth ends to avoid Gibbs • Pad with zeroes (to smooth spectrum or to use 2n points for FFT) Computational Geophysics and Data Analysis

9. 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 Computational Geophysics and Data Analysis

10. 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 Computational Geophysics and Data Analysis

11. Spectral leakage Computational Geophysics and Data Analysis

12. Fourier Spectra: Main Casesrandom signals Random signals may contain all frequencies. A spectrum with constant contribution of all frequencies is called a white spectrum Computational Geophysics and Data Analysis

13. 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? Computational Geophysics and Data Analysis

14. 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 Computational Geophysics and Data Analysis

15. Puls-width and Frequency Bandwidth spectrum time (space) Widening frequency band Narrowing physical signal Computational Geophysics and Data Analysis

16. Frequencies in seismograms Computational Geophysics and Data Analysis

17. Amplitude spectrumEigenfrequencies Computational Geophysics and Data Analysis

18. Sound of an instrument a‘ - 440Hz Computational Geophysics and Data Analysis

19. Instrument Earth 26.-29.12.2004 (FFB ) 0S2 – Earth‘s gravest tone T=3233.5s =53.9min Theoretical eigenfrequencies Computational Geophysics and Data Analysis

20. The 2009 earthquake „swarm“ Computational Geophysics and Data Analysis

21. Computational Geophysics and Data Analysis

22. Rotations … Computational Geophysics and Data Analysis

23. First observations of eigenmodes with ring laser! Computational Geophysics and Data Analysis

24. 16 hr time window (Samoa) Computational Geophysics and Data Analysis

25. 36 hr time window (Chile quake) Computational Geophysics and Data Analysis

26. Time – frequency analysis Computational Geophysics and Data Analysis

27. Spectral analysis: windowed spectra 24 hour ground motion, do you see any signal? Computational Geophysics and Data Analysis

28. Seismo-“weather“ Running spectrum of the same data Computational Geophysics and Data Analysis

29. Going beyond ray tomography … Western Samoa, 1993/5/16, D=34.97° Vz [m/s] Shear wave speed variations Gorbatov & Kennett (2003) t [s] Western Samoa, 1993/5/16, D=34.97° ? Vz [m/s] t [s] Computational Geophysics and Data Analysis

30. Time-frequency representation of seismogramscomparing 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 ... . [ 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! Computational Geophysics and Data Analysis

31. Time-frequency misfits... individually weighted … Computational Geophysics and Data Analysis

32. 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 the negative frequencies can be obtained from symmetry properties Shifting corresponds to changing the phase (shift theorem) Derivative Computational Geophysics and Data Analysis

33. 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) Computational Geophysics and Data Analysis