Introduction - Chapter 8

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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 8 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store
No rock so hard but that a little wave may beat admission in a thousand years.

— Alfred, Lord Tennyson


We assume that a seismic trace has been corrected for amplitude decay resulting from spherical spreading over the seismic time scale of interest (say, for example, from 0 to 6 s). However, in reality, other effects also must be considered. One such effect is inelastic absorption — the loss of seismic energy to frictionally generated heat. (We will treat inelastic absorption in Chapter 14.) We must consider the effect of the seismic energy’s source. In addition, effects result from the instrumentation; source and instrument effects are manmade at or near the surface of the ground. We lump these surface effects together in the form of a source wavelet, which we denote by s.

However, the most important effect is that of the deep earth itself. The deep earth is represented by the sequence ${\displaystyle \varepsilon }$ of reflection coefficients. This sequence is called the reflectivity (Robinson, 1957[1]; Ulrych, 1999[2]). In an ideal seismic experiment, an impulsive source (i.e., a sharp spike of energy imposed at time 0) produces the impulse response of the earth, which we denote by h. In the elastic range, the earth is a linear system, so we can write the synthetic trace (also called the synthetic seismogram) X as the convolution

 {\displaystyle {\begin{aligned}x=s*h.\end{aligned}}} (1)

Equation 1 is called the dynamic convolutional model (Robinson, 1999[3]). The model is called dynamic because the reflection impulse response h is a highly nonlinear function of the reflection coefficients. To deconvolve the dynamic convolutional model, dynamic deconvolution must be used (Robinson, 1975[4], 1999).

The basic problem that we confront in this chapter is the discovery of adequate and useful approximations of the impulse response h in equation 1. We arrive at two approximations, which we call 1 and 2. This chapter will give the details. Approximation number 1, which holds in the case of small reflection coefficients, says that h can be approximated by ${\displaystyle m\;*\;\varepsilon }$, where m is a wavelet representing the multiple reflections and ${\displaystyle \varepsilon }$ is the reflectivity. Under this approximation, equation 1 reduces to the synthetic seismogram with multiples, as given by

 {\displaystyle {\begin{aligned}x=s*m*\varepsilon .\end{aligned}}} (2)

Approximation number 2, which holds in the case of small and white reflection coefficients, says that h can be approximated by the reflectivity ${\displaystyle \varepsilon }$. Under this approximation, equation 1 reduces to the synthetic seismogram without multiples, as given by

 {\displaystyle {\begin{aligned}x=s*\varepsilon .\end{aligned}}} (3)

By white, we mean that the reflectivity is made up of independent random variables all with the same distribution. In other words, by white, we mean white noise. Equation 3 represents the classic synthetic seismogram that has been in constant use for the past 50 years.

References

1. Robinson, E. A., 1957, Predictive decomposition of seismic traces: Geophysics, 22, 767–778.
2. Ulrych, T. J., 1999, The whiteness hypothesis: Reflectivity, inversion, chaos, and Enders: Geophysics, 64, no. 5, 1512–1523.
3. Robinson, E. A., 1999, Seismic inversion and deconvolution: Handbook of geophysical exploration, 4B: Elsevier.
4. Robinson, E. A., 1975, Dynamic predictive deconvolution: Geophysical Prospecting, 23, 779–797.