Methodology

In its original formulation Biondi and Sava (1999), wave-equation migration velocity analysis relates a perturbation of the slowness model ($\Delta {\bf S}$) to its corresponding perturbation of the seismic image ($\Delta \r$). Mathematically, this relation can be expressed as the linear fitting goal  

$${\bf L}\Delta {\bf S}\approx \Delta \r.$$ (1)

${\bf L}$ is the WEMVA operator and is constructed as a linearization of downward continuation operators involving the Born approximation Sava and Fomel (2002). We obtain the slowness perturbation $\Delta {\bf S}$ from Equation (1) by applying either the adjoint or the least-squares inverse of ${\bf L}$ to the image perturbation $\Delta \r$.

The critical quantity in Equation (1) is the perturbation of the seismic image $\Delta \r$. For the purpose of this equation, this is the known quantity and various techniques can be used to derive it.

Suppose we can isolate all the diffractions from a given dataset. The migration velocity is correct if all diffractions are focused, both as a function of space and as a function of offset. Any inaccuracy of the velocity model leaves undiffracted energy in the image.

This simple observation gives us a mechanism to define image perturbations usable for WEMVA. First, we migrate the data using the reference slowness model and obtain a reference image $\r_0$. Second, we correct all unfocused diffractions using a residual technique (residual migration for example), and obtain an improved image $\r$. Finally we take the difference between $\r$ and $\r_0$ as the image perturbation $\Delta \r$, which can be inverted for slowness perturbation $\Delta {\bf S}$using Equation (1).

This method is in principle usable for diffraction data analyzed in a prestack volume. However, as we will demonstrate with the example in the next section, a substantial part of information usable for MVA is present in the zero offset section. Our example concerns WEMVA purely at zero offset with the goal of isolating focusing effects from moveout-based effects.