Wave-equation tomography by beam focusing

Derivatives with respect to slowness

The evaluation of the derivatives of the moveout parameters with respect to slowness follows a slightly different procedure from the one above because the moveout parameters are solutions of the optimization problems 5 and 9. We take advantage of the fact that we need to evaluate the derivatives only at the solution points, where the objective functions are maximized and thus their derivatives with respect to the moveout parameters are zero. We can therefore write:

$$\left. \frac {\partial {J_{\rm FL}}\left({\boldsymbol \mu}_{\over... ...vert _{{\boldsymbol \mu}_{\overline{x}}=\bar{{\boldsymbol \mu}}_{\overline{x}}} = \stackrel{.}{{J_{\rm FL}}}\left(\bar{{\boldsymbol \mu}}_{\overline{x}}\right) =0=$$

$$\left\langle \left. \frac {\partial\mathcal M_{\overline{x}}\left... ...erline{x}}} , {{\bf B}_{\overline{x}} C\left({\tau};{s}\right) } \right\rangle,$$

and

$${ \left. \frac {\partial {J_{\rm FG}}\left({\boldsymbol \mu}\righ... ...ol \mu}}} =\stackrel{.}{{J_{\rm FG}}}\left(\bar{{\boldsymbol \mu}}\right) =0= }$$

$$\left\langle \left. \frac {\partial { \mathcal M\left\{ {\theta}\... ...ight) ,{\bf B}_{\overline{x}} C\left({\tau};{s}\right) \right] } \right\rangle.$$

Using the rule for differentiating implicit functions, and taking advantage that the fitting problems are all independent from each other (i.e. the cross derivatives with respect to the moveout parameters are all zero), we can formally write:

$$\left. \frac {\partial {\boldsymbol \mu}_{\overline{x}}} {\partia... ...bol \mu}_{\overline{x}}\right) } {\partial {\boldsymbol \mu}_{\overline{x}}} },$$ (A-20)

and

$$\left. \frac {\partial {\boldsymbol \mu}} {\partial {s}} \right\v... ...}{{J_{\rm FG}}}\left({\boldsymbol \mu}\right) } {\partial {\boldsymbol \mu}} }.$$ (A-21)

Appendix A presents the analytical development of these expressions to compute the derivatives of the moveout parameters with respect to slowness. As for the derivatives of the main objective function with respect to moveout parameters, the final results for the special case of $\bar{{\boldsymbol \mu}}_{\overline{x}}=0$ and $\bar{{\boldsymbol \mu}}=0$ have a fairly simple analytical expression. The derivative of the local moveout parameters are (A-1):

$$\left. \frac {\partial {\boldsymbol \mu}_{\overline{x}}} {\partia... ...line{x}}} , {{\bf B}_{\overline{x}} C\left({\tau};{s}\right) } \right\rangle },$$ (A-22)

and the derivative of the global moveout parameters are (A-2):

$$\left. \frac {\partial {\boldsymbol \mu}} {\partial {s}} \right\v... ...{\overline{x}} {\bf B}_{\overline{x}} C\left({\tau};{s}\right) \right\rangle },$$ (A-23)

in which $\stackrel{..}{C}\left({\tau}\right) = C\left({\tau}\right)\left[ {\widetilde{{P}} }\left({t}\right) , {\stackrel{..}{{P}}}_{D}\left({t}\right) \right]$ . In both equations 22 and 23 the operator ${\bf {P}_{D} }$ represents a convolution with the recorded data, whereas the operator ${\partial{\widetilde{{P}} }}/{\partial {s}}$ is the basic wave-equation tomography operator that that links perturbations in the slowness model to perturbations in the modeled data.

Combining the derivatives in equation 22 with the derivatives in equations 16-17 we can compute the gradient of the local objective function 3 with respect to slowness as:

$$\nabla{{J_{\rm Local}}}= - \underbrace { {\frac{\partial{\widetil... ...{{\bf B}_{\overline{x}} C\left({\tau};{s}\right) } \right\rangle }. }_{\rm III}$$ (A-24)

I will now examine the effects of each of the terms in equation 24 starting from the rightmost one. The third term (III) produces a scalar for each local curvature parameter ${\mu _C}$ . This scalar multiplies the traces in each beam, after they have been differentiated in time and scaled by ${\partial {\theta}}/{\partial {\boldsymbol \mu}_{\overline{x}}}$ , as described by the second term (II). Notice that the phase introduced by the time derivative of the correlation function in (II) is crucial for the successful backprojection into the slowness model that is accomplished by the first term (I). In this term, first ${\bf B}_{\overline{x}}'$ projects the traces of each individual beam into the space of the global array, then the convolution with the recorded data ${\bf {P}_{D} }'$ time shifts the correlation function by the time delay of the events. Finally, the adjoint of the operator ${\partial{\widetilde{{P}} }}/{\partial {s}}$ backprojects the perturbation in the wavefields at the receiver array into the slowness model.

The expression of the gradient of the global objective function 7 with respect to slowness is similarly derived by combining the derivatives in equation 23 with the derivatives in equations 18-19 and is the following three-terms expression:

$${ \nabla{{J_{\rm Global}}}= }$$

$$- \underbrace { {\frac{\partial{\widetilde{{P}} }}{\partial {s}}}... ...};{s}_{0}\right)\frac{\partial {\theta}}{\partial {\boldsymbol \mu}} }_{\rm II}$$

$$\underbrace { \frac {\frac {\partial {J_{\rm Global}}} {{\boldsym... ...x}} {\bf B}_{\overline{x}} C\left({\tau};{s}\right)\right\rangle }. }_{\rm III}$$ (A-25)

The structure of equation 25 is similar to the structure of equation 24 and the terms have similar explanations. The only important difference is that in term II the chain ${\boldsymbol \Sigma}_{\overline{x}}{\bf S}_{\overline{x}}$ performs the stack over the local arrays and the assemblage of the stacked traces into the global array, whereas its adjoint in term I spreads the stacked traces back into the local arrays reforming the local beams.

Wave-equation tomography by beam focusing