1. Filtering Geophysical Data: Be careful!
Filtering: basic concepts
Seismogram examples, high-low-bandpass filters
The crux with causality
Windowing seismic signals
Various window functions
Multitaper approach
Wavelets (principle)
Scope: Understand the effects of filtering on time series (seismograms). Get to know frequently used windowing functions.
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Why filtering
Get rid of unwanted frequencies
Highlight signals of certain frequencies
Identify harmonic signals in the data
Correcting for phase or amplitude characteristics of instruments
Prepare for down-sampling
Avoid aliasing effects
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A seismogram
Amplitude
Time (s)
Spectral amplitude
Frequency (Hz)
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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:
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 and leaves all other frequencies
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Cutoff frequency
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6. Cut-off and slopes in spectra
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Digital Filtering
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Low-pass filtering
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Lowpass filtering
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High-pass filter
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Band-pass filter
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The simplemost filter
The simplemost filter gets rid of all frequencies above a certain cut-off frequency (low-pass), „box-car“
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The simplemost filter
… and its brother … (high-pass)
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14. … let‘s look at the consequencse
… but what does H(w) look like in the time domain … remember the convolution theorem?
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… surprise …
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16. Zero phase and causal filters
Zero phase filters can be realised by
Convolve first with a chosen filter
Time reverse the original filter and convolve again
First operation multiplies by F(w), the 2nd operation is a multiplication by F*(w)
The net multiplication is thus | F(w)|2
These are also called two-pass filters
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17. The Butterworth Filter (Low-pass, 0-phase)
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… effect on a spike …
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19. … on a seismogram … … varying the order …
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20. … on a seismogram … … varying the cut-off frequency…
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21. The Butterworth Filter (High-Pass)
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… effect on a spike …
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23. … on a seismogram … … varying the order …
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24. … on a seismogram … … varying the cut-off frequency…
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25. The Butterworth Filter (Band-Pass)
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… effect on a spike …
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27. … on a seismogram … … varying the order …
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28. … on a seismogram … … varying the cut-off frequency…
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29. 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?
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30. Butterworth Low-pass (20 Hz) on spike
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31. (causal) Butterworth Low-pass (20 Hz) on spike
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32. Butterworth Low-pass (20 Hz) on data
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33. Other windowing functions
So far we only used the Butterworth filtering window
In general if we want to extract time windows from (permanent) recordings we have other options in the time domain.
The key issues are
Do you want to preserve the main maxima at the expense of side maxima?
Do you want to have as little side lobes as posible?
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Example
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Possible windows
Plain box car (arrow stands for Fourier transform):
Bartlett
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Possible windows
Hanning
The spectral representations of the boxcar, Bartlett (and Parzen) functions are:
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Examples
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Examples
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39. The Gabor transform: t-f misfits
phase information:
can be measured reliably
± linearly related to Earth structure
physically interpretable
amplitude information:
hard to measure (earthquake
magnitude often unknown)
non-linearly related to structure
[ t-w representation of synthetics, u(t) ]
[ t-w representation of data, u0(t) ]
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The Gabor time window
The Gaussian time windows is given by
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Example
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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, Ideas go back to a paper by Thomson (1982).
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43. 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
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Spectrum estimates
We start with
with
To maintain total power.
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45. 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
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Slepian functions
The tapers (Slepian functions) in time and frequency domains
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Final assembly
Slepian sequences (tapers)
Final averaging of spectra
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Example
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49. Classical Periodogram
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… and its power …
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51. … multitaper spectrum …
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52. Wavelets – the principle
Motivation:
Time-frequency analysis
Multi-scale approach
„when do we hear what frequency?“
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53. Continuous vs. local basis functions
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Some maths
A wavelet can be defined as
With the transform pair:
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57. Resulting wavelet representation
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Shifting and scaling
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60. Application to seismograms
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Graphical comparison
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Summary
Filtering is not necessarily straight forward, even the fundamental operations (LP, HP, BP, etc) require some thinking before application to data.
The form of the filter decides upon the changes to the waveforms of the time series you are filtering
For seismological applications filtering might drastically influence observables such as travel times or amplitudes
„Windowing“ the signals in the right way is fundamental to obtain the desired filtered sequence
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