Wave-equation tomography by beam focusing

Details of gradient computation

In this appendix I present the analytical development needed to derive equations 22-23 from equations 20-21.

Equation 20 can be rewritten as

$${ \left. \frac {\partial {\boldsymbol \mu}_{\overline{x}}} {\part... ...ol \mu}_{\overline{x}}\right) } {\partial {\boldsymbol \mu}_{\overline{x}}} } }$$

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

where,

$$\frac {\partial^2\mathcal M_{\overline{x}}\left[ {\theta}\left({\... ...ft({\tau};{s}_{0}\right) \right]} {\partial {\boldsymbol \mu}_{\overline{x}}^2} = \mathcal M_{\overline{x}}\left[ {\theta}\left({\boldsymbol \mu}_{... ... \right]\frac{\partial^2 {\theta}}{\partial {\boldsymbol \mu}_{\overline{x}}^2}$$

$$+ \mathcal M_{\overline{x}}\left[ {\theta}\left({\boldsymbol \mu}_{... ...ht) \right]\frac{\partial {\theta}}{\partial {\boldsymbol \mu}_{\overline{x}}},$$

and in which $\stackrel{..}{C}\left({\tau}\right) = C\left({\tau}\right)\left[ {\widetilde{{P}} }\left({t}\right) , {\stackrel{..}{{P}}}_{D}\left({t}\right) \right]$ . Given the moveout parametrization expressed in 6, ${\partial^2 {\theta}}/{\partial {\boldsymbol \mu}_{\overline{x}}^2}=0$ and the previous expression simplifies into the following:

$$\frac {\partial^2 \mathcal M_{\overline{x}}\left[ {\theta}\left({... ...t) \right] \frac{\partial {\theta}}{\partial {\boldsymbol \mu}_{\overline{x}}}.$$

Consequently, the general expression for the gradient of the local moveout parameters with respect to the slowness model is:

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

When $\bar{{\boldsymbol \mu}}_{\overline{x}}=0$ , the general expression further simplifies into:

$$\left. \frac {\partial {\boldsymbol \mu}_{\overline{x}}} {\partia... ...line{x}}} , {{\bf B}_{\overline{x}} C\left({\tau};{s}\right) } \right\rangle }.$$ (A-1)

Similar derivation can be developed for the derivative of the global moveout parameters with respect to slowness. Equation 21 can be rewritten as:

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

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

where,

$${ \frac {\partial^2 { \mathcal M\left\{ {\theta}\left({\boldsymbo... ...ft({\tau};{s}_{0}\right) \right] \right\} }} {\partial {\boldsymbol \mu}^2} = }$$

$$\mathcal M\left\{ {\theta}\left({\boldsymbol \mu}\right) ,{\bolds... ...ight) \right]\right\}\frac{\partial^2 {\theta}}{\partial {\boldsymbol \mu}^2} +$$

$$\mathcal M\left\{ {\theta}\left({\boldsymbol \mu}\right) ,{\bolds... ...{0}\right) \right]\right\}\frac{\partial {\theta}}{\partial {\boldsymbol \mu}}.$$

Given the moveout parametrization in expressed in 10, ${\partial^2 {\theta}}/{\partial {\boldsymbol \mu}^2}=0$ and the previous expression simplifies into:

$${ \frac {\partial^2 { \mathcal M\left\{ {\theta}\left({\boldsymbo... ...ft({\tau};{s}_{0}\right) \right] \right\} }} {\partial {\boldsymbol \mu}^2} = }$$

$$\mathcal M\left\{ {\theta}\left({\boldsymbol \mu}\right) ,{\bolds... ...{0}\right) \right]\right\}\frac{\partial {\theta}}{\partial {\boldsymbol \mu}}.$$

The general expression for the gradient of the global moveout parameters with respect to the slowness model is:

$${ \left. \frac {\partial {\boldsymbol \mu}} {\partial {s}} \right\vert _{{\mu}={\bar{{\mu}}}}= }$$

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

When $\bar{{\boldsymbol \mu}}_{\overline{x}}=0$ and $\bar{{\boldsymbol \mu}}=0$ the general expression further simplifies into:

$$\left. \frac {\partial {\boldsymbol \mu}} {\partial {s}} \right\v... ...{\overline{x}} {\bf B}_{\overline{x}} C\left({\tau};{s}\right) \right\rangle }.$$ (A-2)

${}_{}$


${}_{}$

Wave-equation tomography by beam focusing