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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 (*x*_{s},*t*) plane should be implemented by recursive filtering. However, we know that 2-D deconvolution in the (*x*_{s},*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, and , 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:

| |
(8) |

Note that whereas we sum over the shot positions (*x*_{s}) in equation (3), we now sum over the shot position spatial frequency (*k*_{xs}) in equation (8).
Also notice that the regularization parameter is spatially variable but constant in the plane. It is defined as

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(9) |

where <> is the mean in the 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:

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(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
**

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 (*x*_{s},*t*) dimensions.

** Next:** Results with synthetic data
** Up:** 2-D imaging conditions in (x_{s},t)
** Previous:** 2-D crosscorrelation
Stanford Exploration Project

7/8/2003