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1-D Deconvolution

The division in equation (1) is better approximated by implementing a 1-D deconvolution imaging condition in the time dimension. It adds more complexity and potential instability in the computation of the image, but better approximates the definition of the reflection coefficient (ratio between incoming and reflected wave amplitude).

In practice, however, the 1-D deconvolution can be computed in the Fourier domain as a polynomial division. The zero lag coefficient is computed as the sum over frequencies:  
 \begin{displaymath}
{\bf r}(x,z)= \sum_{x_s}\sum_{\omega}\frac{{\bf U}(x,z,\omeg...
 ...omega,x_s) {\bf D}^{*}(x,z,\omega,x_s)+\varepsilon^2(x,z,x_s)}.\end{displaymath} (3)
In equation (3), ${\bf U}(x,z,\omega)$ and ${\bf D}(x,z,\omega)$ are the one-dimensional Fourier transforms of the receiver and the source wavefields respectively. The contribution of each shot (located at xs) is added to form the final image.

Notice that the regularization parameter $\varepsilon(x,z,x_s)$ can vary spatially. Jacobs (1982) discuss the difficulties of choosing a spatially variable $\varepsilon$. In practice, defining $\varepsilon$ as a function of a dimensionless parameter $\lambda$ makes its selection easier Claerbout (1991). We define this dependence as
\begin{displaymath}
\varepsilon^2(x,z,x_s)=\lambda < {\bf D}(x,z,\omega,x_s) {\bf D}^{*}(x,z,\omega,x_s) \gt, \end{displaymath} (4)
where <> indicates the mean in the frequency dimension. $\lambda$ can be set constant for the whole image because it is independent of the data scale.

A 1-D imaging conditions implicitly make the assumption that each shot contributes to the image with the same weight [equations (2) and (3)]. This assumption is far from reality, since even when the subsurface geometry is not complex, reflectors are illuminated in a different way according to the source position.


next up previous print clean
Next: 2-D imaging conditions in (xs,t) Up: 1-D imaging conditions Previous: 1-D Crosscorrelation
Stanford Exploration Project
7/8/2003