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 1: 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 1 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 2. $\Delta \r$ 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 1 and compute the corresponding slowness perturbation using Equation (1). Figure 2 shows in the middle the result we obtain by applying the adjoint of the operator ${\bf L}$ to the image perturbation in Figure 1, 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 2) creates ``fat rays,'' also discussed by Woodward (1992) and Sava (2000).