The convolutional model – book

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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 , (b) by the near-surface region where absorption is especially large , (c) by absorption in passage through the main body of the earth , and (d) perhaps by other factors. The first of these will change an impulsive signal to a waveform , which we can think of as a series of impulses with amplitudes that give the shape . Passage through the near surface will replace each of these impulses by multiplied by the scaled value , that is, . Absorption will change this series to (where the absorption is frequency dependent). Reflection from interfaces in the earth can also be thought of as additional convolutions changing the waveform to , 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 so we combine all the other waveshape-changing factors into an equivalent wavelet, generally called the embedded wavelet :

where . This concept is called the convolutional model.

Problem 9.6b

What is the effect if the source is not impulsive?

Solution

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

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