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FilteringComputational Geophysics and Data Analysis 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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FilteringComputational Geophysics and Data Analysis 2 Why filtering 1.Get rid of unwanted frequencies 2.Highlight signals of certain frequencies 3.Identify harmonic signals in the data 4.Correcting for phase or amplitude characteristics of instruments 5.Prepare for down-sampling 6.Avoid aliasing effects

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FilteringComputational Geophysics and Data Analysis 3 A seismogram Time (s) Frequency (Hz) Amplitude Spectral amplitude

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FilteringComputational Geophysics and Data Analysis 4 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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FilteringComputational Geophysics and Data Analysis 5 Cutoff frequency

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FilteringComputational Geophysics and Data Analysis 6 Cut-off and slopes in spectra

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FilteringComputational Geophysics and Data Analysis 7 Digital Filtering

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FilteringComputational Geophysics and Data Analysis 8 Low-pass filtering

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FilteringComputational Geophysics and Data Analysis 9 Lowpass filtering

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FilteringComputational Geophysics and Data Analysis 10 High-pass filter

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FilteringComputational Geophysics and Data Analysis 11 Band-pass filter

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FilteringComputational Geophysics and Data Analysis 12 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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FilteringComputational Geophysics and Data Analysis 13 The simplemost filter … and its brother … (high-pass)

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FilteringComputational Geophysics and Data Analysis 14 … let‘s look at the consequencse … but what does H( ) look like in the time domain … remember the convolution theorem?

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FilteringComputational Geophysics and Data Analysis 15 … surprise …

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FilteringComputational Geophysics and Data Analysis 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( ), the 2nd operation is a multiplication by F*( ) The net multiplication is thus | F(w)| 2 These are also called two-pass filters

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FilteringComputational Geophysics and Data Analysis 17 The Butterworth Filter (Low-pass, 0-phase)

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FilteringComputational Geophysics and Data Analysis 18 … effect on a spike …

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FilteringComputational Geophysics and Data Analysis 19 … on a seismogram … … varying the order …

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FilteringComputational Geophysics and Data Analysis 20 … on a seismogram … … varying the cut-off frequency…

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FilteringComputational Geophysics and Data Analysis 21 The Butterworth Filter (High-Pass)

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FilteringComputational Geophysics and Data Analysis 22 … effect on a spike …

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FilteringComputational Geophysics and Data Analysis 23 … on a seismogram … … varying the order …

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FilteringComputational Geophysics and Data Analysis 24 … on a seismogram … … varying the cut-off frequency…

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FilteringComputational Geophysics and Data Analysis 25 The Butterworth Filter (Band-Pass)

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FilteringComputational Geophysics and Data Analysis 26 … effect on a spike …

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FilteringComputational Geophysics and Data Analysis 27 … on a seismogram … … varying the order …

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FilteringComputational Geophysics and Data Analysis 28 … on a seismogram … … varying the cut-off frequency…

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FilteringComputational Geophysics and Data Analysis 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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FilteringComputational Geophysics and Data Analysis 30 Butterworth Low-pass (20 Hz) on spike

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FilteringComputational Geophysics and Data Analysis 31 (causal) Butterworth Low-pass (20 Hz) on spike

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FilteringComputational Geophysics and Data Analysis 32 Butterworth Low-pass (20 Hz) on data

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FilteringComputational Geophysics and Data Analysis 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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FilteringComputational Geophysics and Data Analysis 34 Example

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FilteringComputational Geophysics and Data Analysis 35 Possible windows Plain box car (arrow stands for Fourier transform): Bartlett

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FilteringComputational Geophysics and Data Analysis 36 Possible windows Hanning The spectral representations of the boxcar, Bartlett (and Parzen) functions are:

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FilteringComputational Geophysics and Data Analysis 37 Examples

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FilteringComputational Geophysics and Data Analysis 38 Examples

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FilteringComputational Geophysics and Data Analysis 39 The Gabor transform: t-f misfits [ t- representation of synthetics, u(t) ] [ t- representation of data, u 0 (t) ] 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

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FilteringComputational Geophysics and Data Analysis 40 The Gabor time window The Gaussian time windows is given by

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FilteringComputational Geophysics and Data Analysis 41 Example

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FilteringComputational Geophysics and Data Analysis 42 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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FilteringComputational Geophysics and Data Analysis 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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FilteringComputational Geophysics and Data Analysis 44 Spectrum estimates We start with with To maintain total power.

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FilteringComputational Geophysics and Data Analysis 45 Condition for optimal tapers N is the number of points, W is the resolution bandwith (frequency increment) One seeks to maximize 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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FilteringComputational Geophysics and Data Analysis 46 Slepian functions The tapers (Slepian functions) in time and frequency domains

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FilteringComputational Geophysics and Data Analysis 47 Final assembly Final averaging of spectra Slepian sequences (tapers)

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FilteringComputational Geophysics and Data Analysis 48 Example

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FilteringComputational Geophysics and Data Analysis 49 Classical Periodogram

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FilteringComputational Geophysics and Data Analysis 50 … and its power …

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FilteringComputational Geophysics and Data Analysis 51 … multitaper spectrum …

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FilteringComputational Geophysics and Data Analysis 52 Wavelets – the principle Motivation: Time-frequency analysis Multi-scale approach „when do we hear what frequency?“

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FilteringComputational Geophysics and Data Analysis 53 Continuous vs. local basis functions

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FilteringComputational Geophysics and Data Analysis 54

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FilteringComputational Geophysics and Data Analysis 55 Some maths A wavelet can be defined as With the transform pair:

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FilteringComputational Geophysics and Data Analysis 56

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FilteringComputational Geophysics and Data Analysis 57 Resulting wavelet representation

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FilteringComputational Geophysics and Data Analysis 58 Shifting and scaling

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FilteringComputational Geophysics and Data Analysis 59

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FilteringComputational Geophysics and Data Analysis 60 Application to seismograms

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FilteringComputational Geophysics and Data Analysis 61 Graphical comparison

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FilteringComputational Geophysics and Data Analysis 62 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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FilteringComputational Geophysics and Data Analysis 63

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