Wave-equation tomography by beam focusing

Derivatives with respect to moveout parameters

The computation of the derivatives of 3 with respect to each vector of local-moveout parameters is easily evaluated using the following expression:

$$\frac {\partial {J_{\rm Local}}} {{\boldsymbol \mu}_{\overline{x}... ...rline{x}}\right) ,{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) \right] .$$ (A-12)

The linear operator ${\frac {\partial \mathcal M_{\overline{x}}} {\partial {\boldsymbol \mu}_{\overline{x}}}}$ has the dimensions $\left(N_{\Delta {x_{g}}}N_{{\tau}}\times {N_{{\mu}}}_{{\overline{x}}}\right)$ and is given by

$${\frac {\partial \mathcal M_{\overline{x}}} {\partial {\boldsymbo... ...t) \right] \frac{\partial {\theta}}{\partial {\boldsymbol \mu}_{\overline{x}}},$$ (A-13)

where $\stackrel{.}{C}\left({\tau};{s}_{0}\right) = C\left({\tau}\right)\left[ {\wide... ...}} }\left({t};{s}_{0}\right) , {\stackrel{.}{{P}}}_{D}\left({t}\right) \right]$ , with ${\stackrel{.}{{P}}}_{D}$ being the time derivative of the recorded-data traces. For the choice of moveout parameters expressed in equation 6 we have $\partial {\theta}/\partial {\boldsymbol \mu}_{\overline{x}}=\partial {\theta}/\partial {\mu_C}= \Delta {x_{g}}^2$ .

Similarly, the evaluation of the derivatives of 7 with respect to each shift parameter ${\boldsymbol \mu}$ is easily carried out by the following:

$$\frac {\partial {J_{\rm Global}}} {\partial {\boldsymbol \mu}} = ... ...rline{x}}\right) ,{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) \right] ,$$ (A-14)

where the linear operator ${\frac {\partial \mathcal M} {\partial {\boldsymbol \mu}}}$ is given by

$${\frac {\partial \mathcal M} {\partial {\boldsymbol \mu}} } = \ma... ...}\right) \right] \right\} \frac{\partial {\theta}}{\partial {\boldsymbol \mu}}.$$ (A-15)

When the moveout parameters are simple trace-by-trace phase shifts, as defined in equation 10, it results that ${\partial {\theta}}/{\partial {\boldsymbol \mu}}=1$ .

On a practical note, the preceding expressions look more daunting than they are in practice. They greatly simplify in the important case when the gradient is evaluated for $\bar{{\boldsymbol \mu}}_{\overline{x}}=0$ and $\bar{{\boldsymbol \mu}}=0$ . This simplifying condition is actually always fulfilled unless the optimization algorithm includes inner iterations for fitting the moveout parameters using a linearized approach. Under these conditions, equations 12 and 13 become, respectively,

$$\left. \frac {\partial {J_{\rm Local}}} {{\boldsymbol \mu}_{\over... ... {\bf S}_{\overline{x}} {{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) },$$ (A-16)

and

$$\frac {\partial \mathcal M_{\overline{x}}} {\partial {\boldsymbol... ...}\right) } \frac{\partial {\theta}}{\partial {\boldsymbol \mu}_{\overline{x}}}.$$ (A-17)

Similarly, equations 14 and 15 become

$$\left. \frac {\partial {J_{\rm Global}}} {\partial {\boldsymbol \... ... {\bf S}_{\overline{x}} {{\bf B}_{\overline{x}} C\left({\tau};{s}_{0}\right) },$$ (A-18)

and

$$\frac {\partial \mathcal M} {\partial {\boldsymbol \mu}} = {\bold... ...t({\tau};{s}_{0}\right) } \frac{\partial {\theta}}{\partial {\boldsymbol \mu}}.$$ (A-19)

Wave-equation tomography by beam focusing