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Regularization in the prestack image space

The previous subsection solve equation 3 in a poststack image domain (zero subsurface-offset). But a prestack regularization is necessary to reduce the noise in the inversion result. If subsurface offset is included in the computation of the Hessian, a generalization to the prestack image domain of equation 4 is possible.

Three different regularization schemes for wave-equation inversion have been discussed in the literature. First, an identity operator which is customary in many scientific applications (damping). Second, a geophysical regularization which penalizes the roughness of the image in the offset ray parameter dimension (which is equivalent the reflection angle dimension) Kuehl and Sacchi (2001); Prucha et al. (2000). Third, a differential semblance operator to penalize the energy in the image not focused at zero subsurface-offset Shen et al. (2003). In this paper I use the third regularization scheme, the regularization in the reflection angle domain.

A generalization to the prestack image domain of equation 3 needs regularization to obtain a stable solution. The first option for regularization is a customary damping that can be stated as follows:
 (5)
where is a point in the image, and is the half subsurface-offset. The subsurface-offset Hessian is where and are the Green functions from shot position and receiver position to a model space point .

The third regularization option for the prestack generalization of equation 3, is penalizing the energy in the image not focused at zero subsurface-offset. This is obtained using the fitting goals,
 (6)
where is the differential semblance operator Shen et al. (2003). The only difference between equations 5 and 6 is in the regularization operator.

In the next section I compare the numerical solution of the inversion problems stated in equations 5 and 6 to the imaging of Sigsbee model.

Next: Numerical results: Sigsbee model Up: Linear least-squares inversion Previous: Regularization in the postack
Stanford Exploration Project
5/6/2007