Wave-equation migration velocity analysis by residual moveout fitting

Derivatives of objective function (${J}$ ) with respect to moveout parameters ( ${\boldsymbol \mu}_{\vec x}$ )

The derivatives of 4 with respect to the vector of moveout parameters is easily evaluated using the following expression:

$${\frac {\partial {J}} {{\boldsymbol \mu}_{\vec x}} }' = {\frac {\... ...M_{\gamma }\left[\bar{{\boldsymbol \mu}}_{\vec x}\right]{\bf\bar I_{\gamma }} .$$ (A-8)

The linear operator ${\frac {\partial \mathcal M_{\gamma }} {\partial {\boldsymbol \mu}_{\vec x}}}$ can be represented as a $N_{\vec x}N_{\gamma }\times {N_{{\mu}}}_{{\overline{x}}}$ matrix. The elements of this matrix are given by:

$$\frac {\partial \mathcal M_{\gamma }} {\partial {\boldsymbol \mu}... ...rm I} \underbrace{ \frac{\partial {\zeta}}{\partial {\mu_{\vec x}}} }_{\rm II}.$$ (A-9)

The first term (I) is given by the depth-derivative of the image $\partial I_{\gamma }\left(\vec x,\gamma ;\bar {s}\right) / \partial z$ after moveout. This term can be numerically evaluated by applying to the moved-out image a finite-difference approximation of the first-derivative operator. The second term (II) is different from zero only when the spatial coordinate $\vec x$ of the image element $I_{\gamma }\left(\vec x,\gamma \right)$ is the same as the coordinate corresponding to the moveout parameter ${\mu_{\vec x}}$ . When they do, and for the choice of moveout parameters expressed in equation 7, we have $\partial {\zeta}/\partial {\mu_{\vec x}}= \tan^2 \gamma$ .

The preceding expression simplifies when the gradient is evaluated for ${\boldsymbol \mu}_{\vec x}=0$ . This simplifying condition is actually always fulfilled unless the optimization algorithm includes inner iterations for fitting the moveout parameters using a linearized approach. Under this condition, equation 8 becomes

$$\left. \frac {\partial {J}} {{\boldsymbol \mu}_{\vec x}} \right\v... ... \mu}_{\vec x}=0} ' {\bf S}_{\gamma }' {\bf S}_{\gamma } {\bf\bar I_{\gamma }},$$ (A-10)

and equation 9 becomes

$$\frac {\partial \mathcal M_{\gamma }} {\partial {\boldsymbol \mu}... ... ;\bar {s}\right) \frac{\partial {\zeta}}{\partial {\boldsymbol \mu}_{\vec x}}.$$ (A-11)

Wave-equation migration velocity analysis by residual moveout fitting