VTI migration velocity analysis using RTM

Migration Velocity Analysis Gradients

In this section, we derive the MVA gradients of objective function 15 by two different approaches: the adjoint method from the perturbation theory, and the Lagrangian augmented function.

WEMVA is a non-linear inversion process that aims to find the velocity model that minimizes the residual field $\Delta \bf I$ in the image space. Without losing any generality, we define our objective function by DSO (Shen and Symes, 2008) in the subsurface offset domain:

$$J = \frac{1}{2} \sum_{\bf h} \langle {\bf h} I_{\bf h}, {\bf h} I_{\bf h} \rangle.$$ (15)

Although we don't use this DSO objective function in the example, the derivation follows the same logic, and readers can easily substitute their desired image-space objective function into the derivation.