Calculation of inverse filters

Problems in Exploration Seismology and their Solutions

Series	Geophysical References Series
Title	Problems in Exploration Seismology and their Solutions
Author	Lloyd P. Geldart and Robert E. Sheriff
Chapter	9
Pages	295 - 366
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 9.18

Assuming that the signature of an air-gun array is a unit impulse and that the recorded wavelet after transmission through the earth is $[-12,\;-4,\;+3,\;+1]$, find the inverse filter that will remove the earth filtering. How many terms should the filter include?

Solution

The inverse filter $i_{t}$ (see problem 9.7) is a filter that will restore the source impulse, i.e.,

${\begin{aligned}g_{t}*i_{t}=\delta _{t},\end{aligned}}$

or in the frequency domain where $i_{t}\leftrightarrow I(z)$,

${\begin{aligned}G(z)I(z)=1.\end{aligned}}$

Thus,

${\begin{aligned}I(z)=1/G(z)=1/[-12-4z+3z^{2}+z^{3}]\\=-{\frac {1}{12}}\left[1+{\frac {z}{3}}-{\frac {z^{2}}{4}}-{\frac {z^{3}}{12}}\right]^{-1}=-{\frac {1}{12}}(1+A)^{-1},\end{aligned}}$

(9.18a)

where $A={\frac {z}{3}}-{\frac {z^{2}}{4}}-{\frac {z^{3}}{12}}$.

Since $z$ has the magnitude $|z|=1$, the magnitude of $A<1$ for all values of $z$, and we can expand equation (9.18a) using the binomial theorem [see equation (4.1b)] and Sheriff and Geldart, 1995, equation (15.43):

${\begin{aligned}I(z)=-{\frac {1}{12}}(1-A+A^{2}-A^{3}+A^{4}-\ldots ).\end{aligned}}$

(9.18b)

We first find $(1-A+A^{2}-A^{3}+A^{4})$ neglecting powers higher than $z^{4}$:

${\begin{aligned}-A&=-0.3333z+0.2500z^{2}+0.0833z^{3},\\A^{2}&=+0.1111z^{2}-0.1667z^{3}+0.0069z^{4},\\-A^{3}&=-0.0370z^{3}+0.0833z^{4},\\A^{4}&=+0.0123z^{4}\\\mathrm {Sum} &=-0.3333z+0.3611z^{2}-0.1204z^{3}+0.1025z^{4}\\I(z)&=-{\frac {1}{12}}(1-A+A^{2}-A^{3}+A^{4})\\&=-{\frac {1}{12}}(1-0.3333z+0.3611z^{2}-0,1204z^{3}+0.1025z^{4}).\end{aligned}}$

We can verify the accuracy of $I(z)$ by multiplying $G(z)$ by $I(z)$. We have

${\begin{aligned}I(z):1-0.3333z+0.3611z^{2}-0,1204z^{3}+0.1025z^{4},\\G(z):1+0.3333z-0.2500z^{2}-0.0833z^{3}\\1-0.3333z+0.3611z^{2}-0.1204z^{3}+0.1025z^{4}\\+0.3333z-0.1111z^{2}+0.1204z^{3}-0.0401z^{4}+0.0342z^{5}\\-0.2500z^{2}+0.0833z^{3}-0.0903z^{4}+0.0301z^{5}-0.0256z^{6}\\{-0.0833z^{3}+0.0278z^{4}-0.0301z^{5}+0.0100z^{6}-0.0085z^{7}}\\{1+0+0+0-0.0001z^{4}+0.0342z^{5}-0.0156z^{6}-0.0085z^{7}}.\end{aligned}}$

We see that the inverse filter is exact as far as the term $z^{3}$ and terms for higher powers are small. The overall effect is to create a small tail whose energy is 0.00149 or 0.1%.

To determine how the accuracy depends on the number of terms used in $I(z)$, we observe the effect on the product $I(z)G(z)$ as we successively drop high powers in $I(z)$. Dropping the $z^{4}$ term in $I(z)$ yields the product $1-0.1026z^{4}+0.0100z^{6}$ and the energy of the tail is now 0.01063 or 1.1%. If we want accuracy of at least 1%, we must therefore retain the $z^{4}$ term.

If we also delete the $z^{3}$ term in $I(z)$ [but not in $G(z)$] the product becomes $1+0.1204z^{3}-0.0625z^{4}-0.0100z^{5}$ and the energy of the tail is 0.01850 or 1.8%. If we go one step further and drop the $z^{2}$ term in $I(z)$, we get $1-0.3611z^{2}+0.0278z^{4}$ and the energy of the tail is 0.13107 or 13.1%.

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Table of Contents (book)

Also in this chapter

• Fourier series
• Space-domain convolution and vibroseis acquisition
• Fourier transforms of the unit impulse and boxcar
• Extension of the sampling theorem
• Alias filters
• The convolutional model
• Water reverberation filter
• Calculating crosscorrelation and autocorrelation
• Digital calculations
• Semblance
• Convolution and correlation calculations
• Properties of minimum-phase wavelets
• Phase of composite wavelets
• Tuning and waveshape
• Making a wavelet minimum-phase
• Zero-phase filtering of a minimum-phase wavelet
• Deconvolution methods
• Calculation of inverse filters
• Inverse filter to remove ghosting and recursive filtering
• Ghosting as a notch filter
• Autocorrelation
• Wiener (least-squares) inverse filters
• Interpreting stacking velocity
• Effect of local high-velocity body
• Apparent-velocity filtering
• Complex-trace analysis
• Kirchhoff migration
• Using an upward-traveling coordinate system
• Finite-difference migration
• Effect of migration on fault interpretation
• Derivative and integral operators
• Effects of normal-moveout (NMO) removal
• Weighted least-squares

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