# Convolutional model

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 9 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

In any active remote-sensing problem, a source signal s is sent into an unknown subsurface region, and the resulting reflected signal or transmitted signal, as the case may be, is recorded as a trace x. Exploration seismology uses reflection records. For a sharp energy source, such as an explosion, a large amount of high-amplitude energy is injected into the earth over a short time span. Use of dynamite generally is frowned on, so alternative sources have been developed. In cases for which minimal damage to the ground is desirable, a low-amplitude source is needed. By spreading a low-amplitude input signal over a comparatively long time span, one can achieve a total desired input energy level.

For land work with a dynamite source, recording the source wavelet can present difficulties, so this step usually is avoided. For vibroseis work, the source is a long low-amplitude swept-frequency signal, which is causal but not minimum phase (Tyapkin and Robinson, 2003[1]). The vibroseis source is recorded as a separate entity along with all the resulting traces.

In marine seismic work, various types of devices can be used as an energy source. One of the most popular is the air gun, which was designed to minimize damage to marine life. Typically, an array of air guns is used to produce the source wavelet. Such low-amplitude source signals generally are not minimum phase. Detailed knowledge of the signatures of air-gun arrays can be obtained by directly measuring their signatures. Such an approach can be expensive if it is implemented at every shot point. As a result, computer models often are used to estimate the signatures.

The basic building block of such a model is a description of the oscillating bubble produced by an air gun. The model must be able to cope with the interactions of the bubbles produced by the array, including the problem of heat transfer between the bubble and the water. Such models are required to predict accurately the shape of the pressure wave generated by the air-gun array. It is well known that the surface reflection coefficient (for an upgoing pressure wave) of the sea-to-air interface approximates the value of -1. This nearly perfect reflection leads to a strong ghost at both the source and the receiver. At different frequencies, ghosts will interfere destructively or constructively with the primary reflection arrival. The source wavelet produced by the air-gun array, along with the ghosting effects and any other near-surface effects, result in a waveform called the signature wavelet. Because it is produced by a physical phenomenon, a signature wavelet necessarily must be causal (i.e., one-sided). However, because the bubble pulses of the air guns have large oscillations well after the time of initiation (i.e., time zero), the source signature that results from an air-gun array generally is not minimum phase.

The source wavelet does its required job of injecting input energy into the medium. However, its presence on the recorded traces can only cloud the desired subsurface picture. In the final analysis, the effects of the source wavelet usually must be eliminated to obtain a clearer picture of the subsurface. In the mathematical treatment given here, we assumed that the signature wavelet is causal but not minimum phase. In cases in which the source wavelet is known either by direct measurements or by estimation techniques, it can be eliminated by the process called signature deconvolution.

Let x be the trace, let s be the source signal or signature, let m be the multiple response, and let ${\displaystyle \varepsilon }$ be the reflectivity. As we saw in Chapter 8, in the case of small reflection coefficients, the recorded trace (without absorption but with multiples) can be approximated as the linear time-invariant convolutional model of equation 2 of Chapter 8: ${\displaystyle x=s*m*\varepsilon }$.

The symbol z can be used for different things. Now let z denote the signature-free trace ${\displaystyle m*\varepsilon }$. Then the trace can be written as

 {\displaystyle {\begin{aligned}x=s*z.\end{aligned}}} (31)

The seismic source signature s is a causal wavelet. Signature deconvolution is the process of removing the signature wavelet from the seismic trace by means of the inverse of the signature. When the signature is not minimum phase, the inverse of the signature is a noncausal (two-sided) filter. The input to the signature deconvolution filter is the field trace; the desired output is the signature-free trace.

## References

1. Tyapkin, Y. K., and E. A. Robinson, 2003, Optimum pilot sweep: Geophysical Prospecting, 51, 15-22.

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