Filtering Geophysical Data Be careful Filtering basic concepts
Filtering Geophysical Data: Be careful! Ø Ø Filtering: basic concepts Seismogram examples, high-low-bandpass filters The crux with causality Windowing seismic signals Ø Various window functions Ø Multitaper approach Ø Wavelets (principle) Scope: Understand the effects of filtering on time series (seismograms). Get to know frequently used windowing functions. Filtering Computational Geophysics and Data Analysis 1
Why filtering 1. 2. 3. 4. Get rid of unwanted frequencies Highlight signals of certain frequencies Identify harmonic signals in the data Correcting for phase or amplitude characteristics of instruments 5. Prepare for down-sampling 6. Avoid aliasing effects Filtering Computational Geophysics and Data Analysis 2
Amplitude A seismogram Spectral amplitude Time (s) Frequency (Hz) Filtering Computational Geophysics and Data Analysis 3
Digital Filtering Often a recorded signal contains a lot of information that we are not interested in (noise). To get rid of this noise we can apply a filter in the frequency domain. The most important filters are: Filtering • High pass: cuts out low frequencies • Low pass: cuts out high frequencies • Band pass: cuts out both high and low frequencies and leaves a band of frequencies • Band reject: cuts out certain frequency band leaves all other frequencies Computational Geophysics and Data Analysis 4
Cutoff frequency Filtering Computational Geophysics and Data Analysis 5
Cut-off and slopes in spectra Filtering Computational Geophysics and Data Analysis 6
Digital Filtering Computational Geophysics and Data Analysis 7
Low-pass filtering Filtering Computational Geophysics and Data Analysis 8
Lowpass filtering Filtering Computational Geophysics and Data Analysis 9
High-pass filter Filtering Computational Geophysics and Data Analysis 10
Band-pass filter Filtering Computational Geophysics and Data Analysis 11
The simplemost filter gets rid of all frequencies above a certain cut-off frequency (low-pass), „boxcar“ Filtering Computational Geophysics and Data Analysis 12
The simplemost filter … and its brother … (high-pass) Filtering Computational Geophysics and Data Analysis 13
… let‘s look at the consequencse … but what does H(w) look like in the time domain … remember the convolution theorem? Filtering Computational Geophysics and Data Analysis 14
… surprise … Filtering Computational Geophysics and Data Analysis 15
Zero phase and causal filters Zero phase filters can be realised by Ø Convolve first with a chosen filter Ø Time reverse the original filter and convolve again Ø First operation multiplies by F(w), the 2 nd operation is a multiplication by F*(w) Ø The net multiplication is thus | F(w)|2 Ø These are also called two-pass filters Filtering Computational Geophysics and Data Analysis 16
The Butterworth Filter (Low-pass, 0 -phase) Filtering Computational Geophysics and Data Analysis 17
… effect on a spike … Filtering Computational Geophysics and Data Analysis 18
… on a seismogram … … varying the order … Filtering Computational Geophysics and Data Analysis 19
… on a seismogram … … varying the cut-off frequency… Filtering Computational Geophysics and Data Analysis 20
The Butterworth Filter (High-Pass) Filtering Computational Geophysics and Data Analysis 21
… effect on a spike … Filtering Computational Geophysics and Data Analysis 22
… on a seismogram … … varying the order … Filtering Computational Geophysics and Data Analysis 23
… on a seismogram … … varying the cut-off frequency… Filtering Computational Geophysics and Data Analysis 24
The Butterworth Filter (Band-Pass) Filtering Computational Geophysics and Data Analysis 25
… effect on a spike … Filtering Computational Geophysics and Data Analysis 26
… on a seismogram … … varying the order … Filtering Computational Geophysics and Data Analysis 27
… on a seismogram … … varying the cut-off frequency… Filtering Computational Geophysics and Data Analysis 28
Zero phase and causal filters When the phase of a filter is set to zero (and simply the amplitude spectrum is inverted) we obtain a zero-phase filter. It means a peak will not be shifted. Such a filter is acausal. Why? Filtering Computational Geophysics and Data Analysis 29
Butterworth Low-pass (20 Hz) on spike Filtering Computational Geophysics and Data Analysis 30
(causal) Butterworth Low-pass (20 Hz) on spike Filtering Computational Geophysics and Data Analysis 31
Butterworth Low-pass (20 Hz) on data Filtering Computational Geophysics and Data Analysis 32
Other windowing functions Ø So far we only used the Butterworth filtering window Ø In general if we want to extract time windows from (permanent) recordings we have other options in the time domain. Ø The key issues are Ø Do you want to preserve the main maxima at the expense of side maxima? Ø Do you want to have as little side lobes as posible? Filtering Computational Geophysics and Data Analysis 33
Example Filtering Computational Geophysics and Data Analysis 34
Possible windows Plain box car (arrow stands for Fourier transform): Bartlett Filtering Computational Geophysics and Data Analysis 35
Possible windows Hanning The spectral representations of the boxcar, Bartlett (and Parzen) functions are: Filtering Computational Geophysics and Data Analysis 36
Examples Filtering Computational Geophysics and Data Analysis 37
Examples Filtering Computational Geophysics and Data Analysis 38
The Gabor transform: t-f misfits phase information: amplitude information: • can be measured reliably • ± linearly related to Earth structure • physically interpretable • hard to measure (earthquake magnitude often unknown) • non-linearly related to structure [ t-w representation of synthetics, u(t) ] [ t-w representation of data, u 0(t) ] Filtering Computational Geophysics and Data Analysis 39
The Gabor time window The Gaussian time windows is given by Filtering Computational Geophysics and Data Analysis 40
Example Filtering Computational Geophysics and Data Analysis 41
Multitaper Goal: „obtaining a spectrum with little or no bias and small uncertainties“. problem comes down to finding the right tapering to reduce the bias (i. e, spectral leakage). In principle we seek: This section follows Prieto eet al. , GJI, 2007. Ideas go back to a paper by Thomson (1982). Filtering Computational Geophysics and Data Analysis 42
Multi-taper Principle • Data sequence x is multiplied by a set of orthgonal sequences (tapers) • We get several single periodograms (spectra) that are then averaged • The averaging is not even, various weights apply • Tapers are constructed to optimize resistance to spectral leakage • Weighting designed to generate smooth estimate with less variance than with single tapers Filtering Computational Geophysics and Data Analysis 43
Spectrum estimates We start with To maintain total power. Filtering Computational Geophysics and Data Analysis 44
Condition for optimal tapers N is the number of points, W is the resolution bandwith (frequency increment) One seeks to maximize l the fraction of energy in the interval (–W, W). From this equation one finds a‘s by an eigenvalue problem -> Slepian function Filtering Computational Geophysics and Data Analysis 45
Slepian functions The tapers (Slepian functions) in time and frequency domains Filtering Computational Geophysics and Data Analysis 46
Final assembly Slepian sequences (tapers) Final averaging of spectra Filtering Computational Geophysics and Data Analysis 47
Example Filtering Computational Geophysics and Data Analysis 48
Classical Periodogram Filtering Computational Geophysics and Data Analysis 49
… and its power … Filtering Computational Geophysics and Data Analysis 50
… multitaper spectrum … Filtering Computational Geophysics and Data Analysis 51
Wavelets – the principle Motivation: Ø Time-frequency analysis Ø Multi-scale approach Ø „when do we hear what frequency? “ Filtering Computational Geophysics and Data Analysis 52
Continuous vs. local basis functions Filtering Computational Geophysics and Data Analysis 53
Filtering Computational Geophysics and Data Analysis 54
Some maths A wavelet can be defined as With the transform pair: Filtering Computational Geophysics and Data Analysis 55
Filtering Computational Geophysics and Data Analysis 56
Resulting wavelet representation Filtering Computational Geophysics and Data Analysis 57
Shifting and scaling Filtering Computational Geophysics and Data Analysis 58
Filtering Computational Geophysics and Data Analysis 59
Application to seismograms http: //users. math. uni-potsdam. de/~hols/DFG 1114/projectseis. html Filtering Computational Geophysics and Data Analysis 60
Graphical comparison Filtering Computational Geophysics and Data Analysis 61
Summary Ø Filtering is not necessarily straight forward, even the fundamental operations (LP, HP, BP, etc) require some thinking before application to data. Ø The form of the filter decides upon the changes to the waveforms of the time series you are filtering Ø For seismological applications filtering might drastically influence observables such as travel times or amplitudes Ø „Windowing“ the signals in the right way is fundamental to obtain the desired filtered sequence Filtering Computational Geophysics and Data Analysis 62
Filtering Computational Geophysics and Data Analysis 63
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