Spectra: ApplicationsComputational Geophysics and Data Analysis 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

Spectra: ApplicationsComputational Geophysics and Data Analysis 2 Fourier: Space and Time Space x space variable L spatial wavelength k=2p/ spatial wavenumber F(k) wavenumber spectrum Space x space variable L spatial wavelength k=2p/ spatial wavenumber F(k) wavenumber spectrum Time tTime variable Tperiod ffrequency  =2  fangular frequency Time tTime variable Tperiod ffrequency  =2  fangular frequency Fourier integrals With the complex representation of sinusoidal functions e ikx (or (e i  t ) the Fourier transformation can be written as:

Spectra: ApplicationsComputational Geophysics and Data Analysis 3 The Fourier Transform discrete vs. continuous discrete continuous Whatever we do on the computer with data will be based on the discrete Fourier transform

Spectra: ApplicationsComputational 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(  )) 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)

Spectra: ApplicationsComputational Geophysics and Data Analysis 5 Spectral synthesis The red trace is the sum of all blue traces!

Spectra: ApplicationsComputational Geophysics and Data Analysis 6 The spectrum Amplitude spectrum Phase spectrum Fourier space Physical space

Spectra: ApplicationsComputational Geophysics and Data Analysis 7 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. 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

Spectra: ApplicationsComputational Geophysics and Data Analysis 8 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 2 n points for FFT)

Spectra: ApplicationsComputational 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

Spectra: ApplicationsComputational 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

Spectra: ApplicationsComputational Geophysics and Data Analysis 11 Spectral leakage

Spectra: ApplicationsComputational Geophysics and Data Analysis 12 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

Spectra: ApplicationsComputational Geophysics and Data Analysis 13 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?

Spectra: ApplicationsComputational Geophysics and Data Analysis 14 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

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

Spectra: ApplicationsComputational Geophysics and Data Analysis 16 Frequencies in seismograms

Spectra: ApplicationsComputational Geophysics and Data Analysis 17 Amplitude spectrum Eigenfrequencies

Spectra: ApplicationsComputational Geophysics and Data Analysis 18 Sound of an instrument a‘ - 440Hz

Spectra: ApplicationsComputational Geophysics and Data Analysis 19 Instrument Earth (FFB ) 0 S 2 – Earth‘s gravest tone T=3233.5s =53.9min Theoretical eigenfrequencies

Spectra: ApplicationsComputational Geophysics and Data Analysis 20 The 2009 earthquake „swarm“

Spectra: ApplicationsComputational Geophysics and Data Analysis 21

Spectra: ApplicationsComputational Geophysics and Data Analysis 22 Rotations …

Spectra: ApplicationsComputational Geophysics and Data Analysis 23 First observations of eigenmodes with ring laser!

Spectra: ApplicationsComputational Geophysics and Data Analysis hr time window (Samoa)

Spectra: ApplicationsComputational Geophysics and Data Analysis hr time window (Chile quake)

Spectra: ApplicationsComputational Geophysics and Data Analysis 26 Time – frequency analysis

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

Spectra: ApplicationsComputational Geophysics and Data Analysis 28 Seismo-“weather“ Running spectrum of the same data

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

Spectra: ApplicationsComputational Geophysics and Data Analysis 30 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, u 0 (t) ] Time-frequency representation of seismograms comparing data with synthetics … an elegant way of separating phase (i.e., travel time) and amplitude (envelope) information!

Spectra: ApplicationsComputational Geophysics and Data Analysis 31 Time-frequency misfits... individually weighted …

Spectra: ApplicationsComputational 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

Spectra: ApplicationsComputational 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)