Inversion in the RAD

We want to fill the model null space with information that is as reasonable as possible. We do not want to allow artificial amplitude variations that could lead to disastrous AVA analysis. To put information in the model's null space that is based on the known information, we can use regularization in the inversion process.

Regularization is a familiar process that can be represented by these fitting goals:

$$\begin{eqnarray} {\bf d \approx Lm} \ 0 \approx \epsilon {\bf A m}\end{eqnarray}$$ (1) (2)

where d is the data, m is the model, L is a linear operator, A is the regularization operator, and $\epsilon$ determines the strength of the regularization. To reduce the number of iterations needed, we can change this regularized problem into a preconditioned one with this substitution: ${\bf m = Sp}$, where ${\bf S = A^{-1}}$ so our fitting goals become:

$$\begin{eqnarray} {\bf d \approx LSp} \ 0 \approx \epsilon {\bf p} .\end{eqnarray}$$ (3) (4)

In this paper, L is the modeling operator that is the adjoint to the wave-equation method of creating RAD CIGs explained in Prucha et al. (1999b). The preconditioning operator S was chosen based on two known facts. First, assuming that we have the correct velocity function when we carry out the migration, the resultant model will have horizontal events along the reflection angle axis. Second, the AVA response is expected to be smoothly varying for a particular point in the subsurface. Therefore, we chose an operator that would smooth horizontally along the reflection angle axis. In this paper, we use a steering filter Clapp et al. (1997) that will smooth in a narrow path along the reflection angle axis.

This process has already been introduced by Prucha et al. (2000) but we are now interested in the amplitudes that result from it. To do so, we experimented with a simple synthetic model.