View source for Model-driven predictive deconvolution

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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 = 11
 | frompg  = 289
 | topg    = 328
 | 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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Model-driven predictive deconvolution (Robinson, 1998<ref name=ch11r12>Robinson, E. A., 1998, Model-driven predictive deconvolution: Geophysics, '''63''', 713-722.</ref>) is a method that uses a model both for theory and for application to empirical data. Model-driven deconvolution employs a strategy for effective use of both spike deconvolution and gap deconvolution in conjunction with each other. It is assumed that the given trace ''x ''already has undergone significant processing operations such as [[Special:MyLanguage/surface-consistent deconvolution|surface-consistent deconvolution]] and/or signature deconvolution to remove everything except the minimum-delay reverberation train ''b''. However, it also is assumed that these previous processing steps have not been done perfectly, so an unwanted nonminimum-delay orphan signal ''u'' also remains on the given trace. Thus, the given trace ''x'' consists of the white reflectivity <math>\varepsilon</math>, colored by the unwanted minimum-delay reverberation train ''b'' and the unwanted nonminimum-delay orphan ''u''. The seismic wavelet ''w'' is the convolution of the reverberation train ''b'' and the unwanted orphan ''u''; that is, the wavelet is the nonminimum-delay signal

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{{NumBlk|:|<math>\begin{align}
w= b*u.
\end{align}</math>|{{EquationRef|10}}}}

<!--T:3-->
The convolutional model says that the trace ''x'' is the convolution of the wavelet ''w'' and the white reflectivity <math>\varepsilon</math>; that is,

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{{NumBlk|:|<math>\begin{align}
x= w*\varepsilon.
\end{align}</math>|{{EquationRef|11}}}}

<!--T:5-->
Let <math>w_{{\rm M}}</math> be the minimum-delay counterpart of the wavelet ''w''. The minimum-delay wavelet <math>w_{{\rm M}}</math> is equal to the convolution of the minimum-delay reverberation ''b'' and the minimum-delay counterpart <math>u_{{\rm M}}</math> of the orphan ''u''; that is,

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<translate><!--T:6-->
{{NumBlk|:|<math>\begin{align}
w_{{\rm M}}
= b*u_{{\rm M}}
\end{align}</math>|{{EquationRef|12}}}}

<!--T:7-->
For a [[Special:MyLanguage/prediction distance|prediction distance]] <math>\alpha</math> that is equal to the reverberation cycle time ''T'', the head <math>h_{{\rm M}}</math> of the minimum-delay wavelet <math>w_{{\rm M}}</math> is equal to the minimum-delay counterpart <math>u_{{\rm M}}</math> of the orphan ''u''. Because

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{{NumBlk|:|<math>\begin{align}
w= p*w_{{\rm M}}
\end{align}</math>|{{EquationRef|13}}}}

<!--T:9-->
or

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{{NumBlk|:|<math>\begin{align}
b*u= p*b*u_{{\rm M}}
,
\end{align}</math>|{{EquationRef|14}}}}

<!--T:11-->
it follows that

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{{NumBlk|:|<math>\begin{align}
u= p*u_{{\rm M}}
.
\end{align}</math>|{{EquationRef|15}}}}

<!--T:13-->
Two cases can occur. In the first case, the phase-shift filter ''p'' is a spike, so the unwanted orphan ''u'' is in fact minimum delay. In the second case, the phase-shift filter ''p'' is not a spike but is a stable dispersive all-pass filter. In the first case, it follows that the wavelet ''w'' is a minimum-delay wavelet, so predictive deconvolution works in the conventional way. In the second case, the wavelet ''w'' is a nonminimum-delay wavelet; that is,

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<translate><!--T:14-->
{{NumBlk|:|<math>\begin{align}
w= p*b*u_{{\rm M}}
.
\end{align}</math>|{{EquationRef|16}}}}

<!--T:15-->
This section deals with the second case. Because wavelet ''w'' is a nonminimum-delay wavelet, it cannot be removed from the trace by conventional deconvolution. This case requires model-driven predictive deconvolution. For our purposes here, it is assumed that the preprocessing has been done so well that the unwanted orphan ''u'' is a signal of short length. More specifically, it is assumed that the length of the orphan is less than or equal to the reverberation cycle time ''T''.

<!--T:16-->
Model-driven deconvolution comprises three steps. In step 1, the spike-deconvolution filter ''f'' is computed from the trace by the method of least squares. As is well known, the spike-deconvolution filter ''f'' is necessarily minimum delay. Next, the inverse <math>f^{-1}</math> of the spike-deconvolution filter ''f ''is computed. This computation can be made by the least-squares method, by polynomial division, or by some other method. The inverse <math>f^{-1}</math> is the minimum-delay counterpart <math>w_{{\rm M}}</math>. The spike-deconvolved trace ''y'' then is obtained by convolving the spike-deconvolution filter ''f'' and the trace ''x''.

<!--T:17-->
In step 2, the spike-deconvolution filter ''f'' and the minimum-delay counterpart <math>w_{{\rm M}}</math> are used to determine the cycle time ''T''. The optimum gap-deconvolution filter <math>g_0</math> is defined as the gap-deconvolution filter that has [[Special:MyLanguage/prediction distance|prediction distance]] <math>\alpha</math> equal to the cycle time ''T''. Convolution of the head <math>h_{{\rm M}}</math> (for <math>\alpha = T</math>) and the spike-deconvolved trace ''y'' yields the optimum gap-deconvolved trace ''z''; that is,

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{{NumBlk|:|<math>\begin{align}
z= h_{{\rm M}}
*y= h_{{\rm M}}*p*\varepsilon .
\end{align}</math>|{{EquationRef|17}}}}

<!--T:19-->
Recall that for [[Special:MyLanguage/prediction distance|prediction distance]] <math>\alpha</math> equal to the cycle time ''T'', the head <math>h_{{\rm M}}</math> is equal to <math>u_{{\rm M}}</math>. Hence,

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<translate><!--T:20-->
{{NumBlk|:|<math>\begin{align}
z= u_{{\rm M}}
*p*\varepsilon = u*\varepsilon .
\end{align}</math>|{{EquationRef|18}}}}

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That is, the optimum gap-deconvolved trace ''z'' is equal to the reflectivity smoothed by the nonminimum-delay orphan ''u''.

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In step 3, the phase-correcting filter <math>p^{-1}</math> is estimated. This filter is applied to both the spike-deconvolved trace ''y'' and the optimum gap-deconvolved trace ''z''. The resulting phase-corrected traces then can be subjected to wavelet processing to yield the final results.

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Let us summarize. A spike-deconvolution filter is the special case of a gap-deconvolution filter with the gap equal to one time unit. It often is stated that gap deconvolution with gap <math>\alpha</math> shortens an input wavelet of arbitrary length to an output wavelet of length <math>\alpha</math> (or less). Because an arbitrary value of <math>\alpha</math> can be chosen, it follows that resolution or wavelet contraction can be controlled by use of gap deconvolution. In general, this characterization of gap deconvolution is true for arbitrary <math>\alpha</math> if and only if the wavelet is a minimum-delay wavelet (i.e., minimum phase).

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The method of model-driven deconvolution can be used in the case of a nonminimum-delay wavelet. The wavelet is the convolution of a minimum-delay reverberation and a short nonminimum-delay orphan. The model specifies that the given trace is the convolution of the white reflectivity and this nonminimum-delay wavelet. The given trace yields the spike-deconvolution filter and its inverse. These two signals then are used to compute the gap-deconvolution filters and their inverses for various prediction distances. The inverses are examined, and a stable one is picked as the most likely minimum-delay reverberation. The corresponding gap-deconvolution filter is the optimum one for removal of this minimum-delay reverberation from the given trace. As a by-product, the minimum-delay counterpart of the orphan can be obtained.

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The optimum gap-deconvolved trace is examined for zones that contain little activity, and the leading edge of the wavelet following such a zone is chosen. Next, the phase of the minimum-delay counterpart of the orphan is rotated until it fits the extracted leading edge. From the amount of phase rotation, the required phase-correcting filter can be estimated. Alternatively, downhole information, if available, can be used to estimate the phase-correcting filter. Application of the phase-correcting filter to the spike-deconvolved trace gives the required approximation to the reflectivity. As a final step, wavelet processing can be applied to yield a final interpreter trace made up of zero-phase wavelets.
__NOTOC__

==References== <!--T:26-->

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| align="center" | [[Special:MyLanguage/Prediction distance|Prediction distance]]
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| align="center" | [[:Category:Deconvolution - Chapter 10|Deconvolution]]
| align="center" | [[:Category:Attributes|Attributes]]
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[[Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing|Table of Contents (book)]]
|}


== Also in this chapter == <!--T:29-->

 
<!--T:30-->
* [[Special:MyLanguage/Prediction-error filters|Prediction-error filters]]
* [[Special:MyLanguage/Water reverberations - book|Water reverberations]]
* [[Special:MyLanguage/Gap deconvolution of a mixed-delay wavelet|Gap deconvolution of a mixed-delay wavelet]]
* [[Special:MyLanguage/Prewhitening - book|Prewhitening]]
* [[Special:MyLanguage/Prediction distance|Prediction distance]]
* [[Special:MyLanguage/Convolutional model in the frequency domain|Convolutional model in the frequency domain]]
* [[Special:MyLanguage/Time-variant spectral whitening - book|Time-variant spectral whitening]]
* [[Special:MyLanguage/Model-based deconvolution|Model-based deconvolution]]
* [[Special:MyLanguage/Surface-consistent deconvolution|Surface-consistent deconvolution]]
* [[Special:MyLanguage/Interactive earth-digital processing|Interactive earth-digital processing]]
* [[Appendix K: Exercises|Appendix K: Exercises]]


== External links == <!--T:31-->

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