Example

We illustrate this diffraction focusing technique with a synthetic model simulating the diffracting points at the top of a rough salt body. The model is depicted in Figure : the reflectivity at the top, and the reference slowness in the middle. The model consists of several diffractions, and the reference slowness is smoothly spatially varying.

Figure  shows at the bottom the image perturbation $\Delta \r$ caused by the ideal slowness perturbation $\Delta {\bf S}$ shown in the top panel in Figure . $\Delta \r$ is created by subtracting the reference image from the perfectly focused one $\Delta \r=\r-\r_0$.

We take the image perturbation shown in the bottom panel of Figure  and compute the corresponding slowness perturbation using Equation (). Figure  shows in the middle the result we obtain by applying the adjoint of the operator ${\bf L}$ to the image perturbation in Figure , and at the bottom the result of applying the least-squares inverse of ${\bf L}$ to the same $\Delta \r$.

Despite the inherent vertical smearing caused by the limited angular coverage, the slowness perturbations are nicely focused at their correct locations. Obviously, the result obtained with the least-squares inverse is much better focused than the one obtained by the simple adjoint operator, although we have only used the zero-offset and not the entire prestack data. The simple backprojection (top panel in Figure ) creates ``fat rays,'' also discussed by () and ().