The convolutional model – book

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 9.6a

The earth acts as a filter to seismic energy to change the waveshape so that what we record is a distorted version of the waveshape we send into the earth. Assume that the signal generated by a seismic source is so short that it can be represented adequately by an impulse; what aspects of passage through the earth will distort the impulse and what will be the effects?

Solution

The signal from a seismic source will be distorted: (a) by the region near the source where the stresses are so high as to produce nonlinear effects $s_{i}$, (b) by the near-surface region where absorption is especially large $n_{i}$, (c) by absorption in passage through the main body of the earth $p_{i}$, and (d) perhaps by other factors. The first of these will change an impulsive signal to a waveform $s_{i}$, which we can think of as a series of impulses with amplitudes that give the shape $s_{i}$. Passage through the near surface will replace each of these impulses by $n_{i}$ multiplied by the scaled value $s_{i}$, that is, $s_{i}*n_{i}$. Absorption will change this series to $s_{i}*n_{i}*p_{i}$ (where the absorption $p_{i}$ is frequency dependent). Reflection from interfaces in the earth can also be thought of as additional convolutions changing the waveform to $s_{i}*n_{i}*p_{i}*r_{i}$, etc. Thus the effect on the waveform can be thought of as a sequence of convolutions. In ordinary reflection seismic work the objective is to determine the reflectivity $r_{i}$ so we combine all the other waveshape-changing factors into an equivalent wavelet, generally called the embedded wavelet $w_{i}$:

${\begin{aligned}s_{i}*n_{i}*p_{i}*r_{i}=w_{i}*r_{i},\end{aligned}}$

where $w_{i}=s_{i}*n_{i}*p_{i}$. This concept is called the convolutional model.

Problem 9.6b

What is the effect if the source is not impulsive?

Solution

A nonimpulsive source $v_{i}$, such as the vibroseis, has the effect of replacing $s_{i}$ with $v_{i}$, but otherwise the reasoning in part (a) is unchanged.

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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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