# The convolutional model – book

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 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 ${\displaystyle s_{i}}$, (b) by the near-surface region where absorption is especially large ${\displaystyle n_{i}}$, (c) by absorption in passage through the main body of the earth ${\displaystyle p_{i}}$, and (d) perhaps by other factors. The first of these will change an impulsive signal to a waveform ${\displaystyle s_{i}}$, which we can think of as a series of impulses with amplitudes that give the shape ${\displaystyle s_{i}}$. Passage through the near surface will replace each of these impulses by ${\displaystyle n_{i}}$ multiplied by the scaled value ${\displaystyle s_{i}}$, that is, ${\displaystyle s_{i}*n_{i}}$. Absorption will change this series to ${\displaystyle s_{i}*n_{i}*p_{i}}$ (where the absorption ${\displaystyle p_{i}}$ is frequency dependent). Reflection from interfaces in the earth can also be thought of as additional convolutions changing the waveform to ${\displaystyle s_{i}*n_{i}*p_{i}*r_{i}}$, 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 ${\displaystyle r_{i}}$ so we combine all the other waveshape-changing factors into an equivalent wavelet, generally called the embedded wavelet ${\displaystyle w_{i}}$:

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

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

## Problem 9.6b

What is the effect if the source is not impulsive?

### Solution

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