View source for Introduction - Chapter 8

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{{publication
 | name    = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | image   = [[File:DigitalImaging.png|frameless|180px]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | part    = 
 | chapter = 8
 | frompg  = 193
 | topg    = 216
 | author  = [[Special:MyLanguage/Enders A. Robinson|Enders A. Robinson]] and [[Special:MyLanguage/Sven Treitel|Sven Treitel]]
 | doi     = 10.1190/1.9781560801610
 | pdf     = 
 | store   = https://seg.org/shop/products/detail/898
 | isbn    = 9781560801481
}}
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<pre>No rock so hard but that a little wave may beat admission in a thousand years.

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— Alfred, Lord Tennyson
</pre>

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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''.

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However, the most important effect is that of the deep earth itself. The deep earth is represented by the sequence <math>\varepsilon </math> of reflection coefficients. This sequence is called the ''reflectivity'' (Robinson, 1957<ref name=ch08r9>Robinson, E. A., 1957, Predictive decomposition of seismic traces: Geophysics, '''22''', 767–778.</ref>; Ulrych, 1999<ref name=ch08r17>Ulrych, T. J., 1999, The whiteness hypothesis: Reflectivity, inversion, chaos, and Enders: Geophysics, '''64''', no. 5, 1512–1523.</ref>). 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

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{{NumBlk|:|<math>\begin{align}
x=s*h.
\end{align}</math>|{{EquationRef|1}}}}

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Equation {{EquationNote|1}} is called the ''dynamic convolutional model'' (Robinson, 1999<ref name=ch08r12>Robinson, E. A., 1999, Seismic inversion and deconvolution: Handbook of geophysical exploration, 4B: Elsevier.</ref>). 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<ref name=ch08r10>Robinson, E. A., 1975, Dynamic predictive deconvolution: Geophysical Prospecting, '''23''', 779–797.</ref>, 1999).

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The basic problem that we confront in this chapter is the discovery of adequate and useful approximations of the impulse response ''h'' in equation {{EquationNote|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 <math>m\;*\;\varepsilon </math>, where ''m'' is a wavelet representing the multiple reflections and <math>\varepsilon </math> is the reflectivity. Under this approximation, equation {{EquationNote|1}} reduces to the ''[[Special:MyLanguage/synthetic seismogram with multiples|synthetic seismogram with multiples]]'', as given by

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{{NumBlk|:|<math>\begin{align}
x=s*m*\varepsilon .
\end{align}</math>|{{EquationRef|2}}}}

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Approximation number 2, which holds in the case of ''small and white'' reflection coefficients, says that ''h'' can be approximated by the reflectivity <math>\varepsilon </math>. Under this approximation, equation {{EquationNote|1}} reduces to the ''[[Special:MyLanguage/synthetic seismogram without multiples|synthetic seismogram without multiples]]'', as given by

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{{NumBlk|:|<math>\begin{align}
x=s*\varepsilon .
\end{align}</math>|{{EquationRef|3}}}}

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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 {{EquationNote|3}} represents the classic synthetic seismogram that has been in constant use for the past 50 years.
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==References== <!--T:12-->

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{{reflist}}
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[[Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing|Table of Contents (book)]]
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== Also in this chapter == <!--T:15-->

<!--T:16-->
* [[Special:MyLanguage/Polarity|Polarity]]
* [[Special:MyLanguage/Reflection coefficients and transmission coefficients|Reflection coefficients and transmission coefficients]]
* [[Special:MyLanguage/Ghost reflection|Ghost reflection]]
* [[Special:MyLanguage/Layer-cake model|Layer-cake model]]
* [[Special:MyLanguage/Synthetic seismogram without multiples|Synthetic seismogram without multiples]]
* [[Special:MyLanguage/Water reverberations|Water reverberations]]
* [[Special:MyLanguage/Synthetic seismogram with multiples|Synthetic seismogram with multiples]]
* [[Special:MyLanguage/Examples|Examples]]
* [[Special:MyLanguage/Small and white reflection coefficients|Small and white reflection coefficients]]
* [[Appendix H: Exercises|Appendix H: Exercises]]

== External links == <!--T:17-->

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