VTI migration velocity analysis using RTM

Extension to update anisotropic parameters

The extension from isotropic model updates to anisotropic updates is straightforward. Built on the derivations in the last section, we can easily get the gradients for anisotropic parameters $\epsilon$ and $\delta$ as follows:

$$\nabla_{\epsilon} J = \left < \bf\lambda,~ -\frac{\partial {\bf L}}{\partial \epsilon} ... ...\left < \bf\mu,~ -\frac{\partial {\bf L}^*}{\partial \epsilon} {\bf q} \right >$$

$$= \left ( -\frac{\partial {\bf L}}{\partial \epsilon} {\bf p} \righ... ...+ {\bf q}^* \left ( -\frac{\partial {\bf L}}{\partial \epsilon} \right )\bf\mu,$$ (29)

$$\nabla_{\delta} J = \left < \bf\lambda,~ -\frac{\partial {\bf L}}{\partial \delta} {\... ...+ \left < \bf\mu,~ -\frac{\partial {\bf L}^*}{\partial \delta} {\bf q} \right >$$

$$= \left ( -\frac{\partial {\bf L}}{\partial \delta} {\bf p} \right ... ...a + {\bf q}^* \left ( -\frac{\partial {\bf L}}{\partial \delta} \right )\bf\mu,$$ (30)

where

$$\frac{\partial {\bf L}} {\partial \epsilon} = \left \vert \begin{... ...0 & 0 & 0 \\ 2\partial_x & 2\partial_y & 0 & 0 & 0 \\ \end{array} \right \vert,$$ (31)

$$\frac{\partial {\bf L}} {\partial \delta} = \left \vert \begin{ar... ... & 0 & \frac{\partial_z}{\sqrt{1+2\delta}} & 0 & 0 \\ \end{array} \right \vert.$$ (32)

It is well-known that $\delta$ is the parameter most poorly constrained by surface seismic. Therefore, in our study, we assume that $\delta$ is obtained from well logs or seismic-well ties, and we invert only for velocity and $\epsilon$ .

VTI migration velocity analysis using RTM