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2-D deconvolution

In the case of deconvolution, the relation between 2-D and 1-D (plus stacking), if it exists, is not straightforward to illustrate. This is because deconvolution in the (xs,t) plane should be implemented by recursive filtering. However, we know that 2-D deconvolution in the (xs,t) plane compresses the information in both the shot and the time dimensions. This is a better way of compressing information than 1-D deconvolution followed by a stack in the shot dimension.

In practice, however, 2-D deconvolution can be computed as a polynomial division in the Fourier domain. But in this case, the wavefields that are divided, ${\bf U}(x,z,\omega,k_{x_s})$ and ${\bf D}(x,z,\omega,k_{x_s})$, are the two-dimensional Fourier transforms of the receiver and the source wavefields, respectively.

The zero lag coefficient of the 2-D deconvolution is computed as:  
 \begin{displaymath}
{\bf r}(x,z)= \sum_{k_{x_s}}\sum_{\omega}\frac{{\bf U}(x,z,\...
 ...a,k_{x_s}) {\bf D}^{*}(x,z,\omega,k_{x_s})+\varepsilon^2(x,z)}.\end{displaymath} (8)
Note that whereas we sum over the shot positions (xs) in equation (3), we now sum over the shot position spatial frequency (kxs) in equation (8).

Also notice that the regularization parameter $\varepsilon(x,z)$ is spatially variable but constant in the $(\omega,k_{x_s})$ plane. It is defined as
\begin{displaymath}
\varepsilon^2(x,z)=\lambda < {\bf D}(x,z,\omega,k_{x_s}) {\bf D}^{*}(x,z,\omega,k_{x_s}) \gt\end{displaymath} (9)
where <> is the mean in the $(\omega,k_{x_s})$ plane.

Analogous to Rickett and Sava (2000), we extend the 2-D deconvolution imaging condition to compute a range of offsets. This is done by shifting the source and the receiver wavefields in the x dimension. Then, the nonzero-offset reflectivity can be computed as follows:  
 \begin{displaymath}
{\bf r}(x,z,h)= \sum_{k_{x_s}}\sum_{\omega}\frac{{\bf U}(x-h...
 ...{x_s}) {\bf D}^{*}(x+h,z,\omega,k_{x_s})+\varepsilon^2(x+h,z)}.\end{displaymath} (10)
where h is the subsurface offset.

In the following section we show the advantages of 2-D deconvolution imaging condition over the crosscorrelation and the 1-D deconvolution imaging conditions.

 
mig_tune2
mig_tune2
Figure 4
Comparison of 3 different imaging conditions.(a) Image obtained with the crosscorrelation imaging condition. (b) Image obtained with the 1-D deconvolution imaging condition in the time dimension. (c) Image obtained with the 2-D deconvolution imaging condition in the (xs,t) dimensions.
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next up previous print clean
Next: Results with synthetic data Up: 2-D imaging conditions in (xs,t) Previous: 2-D crosscorrelation
Stanford Exploration Project
7/8/2003