# Model-driven predictive deconvolution

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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 11 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Model-driven predictive deconvolution (Robinson, 1998[1]) 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 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 ${\displaystyle \varepsilon }$, 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

 {\displaystyle {\begin{aligned}w=b*u.\end{aligned}}} (10)

The convolutional model says that the trace x is the convolution of the wavelet w and the white reflectivity ${\displaystyle \varepsilon }$; that is,

 {\displaystyle {\begin{aligned}x=w*\varepsilon .\end{aligned}}} (11)

Let ${\displaystyle w_{\rm {M}}}$ be the minimum-delay counterpart of the wavelet w. The minimum-delay wavelet ${\displaystyle w_{\rm {M}}}$ is equal to the convolution of the minimum-delay reverberation b and the minimum-delay counterpart ${\displaystyle u_{\rm {M}}}$ of the orphan u; that is,

 {\displaystyle {\begin{aligned}w_{\rm {M}}=b*u_{\rm {M}}\end{aligned}}} (12)

For a prediction distance ${\displaystyle \alpha }$ that is equal to the reverberation cycle time T, the head ${\displaystyle h_{\rm {M}}}$ of the minimum-delay wavelet ${\displaystyle w_{\rm {M}}}$ is equal to the minimum-delay counterpart ${\displaystyle u_{\rm {M}}}$ of the orphan u. Because

 {\displaystyle {\begin{aligned}w=p*w_{\rm {M}}\end{aligned}}} (13)

or

 {\displaystyle {\begin{aligned}b*u=p*b*u_{\rm {M}},\end{aligned}}} (14)

it follows that

 {\displaystyle {\begin{aligned}u=p*u_{\rm {M}}.\end{aligned}}} (15)

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,

 {\displaystyle {\begin{aligned}w=p*b*u_{\rm {M}}.\end{aligned}}} (16)

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.

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 ${\displaystyle f^{-1}}$ 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 ${\displaystyle f^{-1}}$ is the minimum-delay counterpart ${\displaystyle w_{\rm {M}}}$. The spike-deconvolved trace y then is obtained by convolving the spike-deconvolution filter f and the trace x.

In step 2, the spike-deconvolution filter f and the minimum-delay counterpart ${\displaystyle w_{\rm {M}}}$ are used to determine the cycle time T. The optimum gap-deconvolution filter ${\displaystyle g_{0}}$ is defined as the gap-deconvolution filter that has prediction distance ${\displaystyle \alpha }$ equal to the cycle time T. Convolution of the head ${\displaystyle h_{\rm {M}}}$ (for ${\displaystyle \alpha =T}$) and the spike-deconvolved trace y yields the optimum gap-deconvolved trace z; that is,

 {\displaystyle {\begin{aligned}z=h_{\rm {M}}*y=h_{\rm {M}}*p*\varepsilon .\end{aligned}}} (17)

Recall that for prediction distance ${\displaystyle \alpha }$ equal to the cycle time T, the head ${\displaystyle h_{\rm {M}}}$ is equal to ${\displaystyle u_{\rm {M}}}$. Hence,

 {\displaystyle {\begin{aligned}z=u_{\rm {M}}*p*\varepsilon =u*\varepsilon .\end{aligned}}} (18)

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

In step 3, the phase-correcting filter ${\displaystyle p^{-1}}$ 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.

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 ${\displaystyle \alpha }$ shortens an input wavelet of arbitrary length to an output wavelet of length ${\displaystyle \alpha }$ (or less). Because an arbitrary value of ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha }$ if and only if the wavelet is a minimum-delay wavelet (i.e., minimum phase).

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.

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.

## References

1. Robinson, E. A., 1998, Model-driven predictive deconvolution: Geophysics, 63, 713-722.