Modeling, migration, and inversion in the generalized source and receiver domain

APPENDIX C

This appendix derives the objective function in the simultaneously encoded source and receiver domain. We start with the objective function in the source and receiver domain as follows:

$$J(m({\bf x})) = \frac{1}{2}\sum_{\omega}\sum_{{\bf x}_s}\sum_{{\b... ...(d({\bf x}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf x}_s,\omega))\vert^2.$$ (C-1)

If we follow a discussion similar to those in Appendices A and B, we obtain the general expression of the inverse transform of the simultaneous encoding:

$$w({\bf x}_r,{\bf x}_s,\omega)d({\bf x}_r,{\bf x}_s,\omega) = \ver... ...\omega) \alpha'({\bf x}_s,{\bf p}_s,\omega) \beta'({\bf x}_r,{\bf p}_r,\omega).$$ (C-2)

Substituting Equation C-2 into C-1 yields:

$$J(m({\bf x})) = \frac{1}{2}\sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r}\vert c\v... ...pha'({\bf x}_s,{\bf p}_s,\omega)\beta'({\bf x}_r,{\bf p}_r,\omega)\right\vert^2$$

$$= \frac{1}{2}\sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r}\vert c\v... ...ha'({\bf x}_s,{\bf p}_s,\omega)\beta'({\bf x}_r,{\bf p}_r,\omega)\right)'\times$$

$$\left( \sum_{{\bf p}'_s}\sum_{{\bf p}'_r}r({\bf p}'_r,{\bf p}'_s,... ...)\alpha'({\bf x}_s,{\bf p}'_s,\omega)\beta'({\bf x}_r,{\bf p}'_r,\omega)\right)$$

$$= \frac{1}{2}\sum_{\omega} \vert c\vert^8 \sum_{{\bf p}_s} \sum_{{\... ...bf p}'_r} r'({\bf p}_r,{\bf p}_s ,\omega)r({\bf p}'_r,{\bf p}'_s,\omega) \times$$

$$\sum_{{\bf x}_s} \sum_{{\bf x}_r} \alpha({\bf x}_s,{\bf p}_s,\ome... ...omega) \alpha'({\bf x}_s,{\bf p}'_s,\omega)\beta'({\bf x}_r,{\bf p}'_r,\omega),$$ (C-3)

where $r({\bf p}_r,{\bf p}_s,\omega)$ is defined to be the residual in the encoded source and receiver domain:

$$r({\bf p}_r,{\bf p}_s,\omega) = d({\bf p}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf p}_r,{\bf p}_s,\omega).$$ (C-4)

For plane-wave phase encoding, with sampling dense enough in ${\bf x}_s$ and ${\bf x}_r$, the inner-most summations become Dirac delta functions:

$$\sum_{{\bf x}_s} \sum_{{\bf x}_r} \alpha({\bf x}_s,{\bf p}_s,\ome... ...\omega) \alpha'({\bf x}_s,{\bf p}'_s,\omega)\beta'({\bf x}_r,{\bf p}'_r,\omega)$$

$$\approx \frac{1}{\vert\omega\vert^4}\delta({{{\bf p}'_r-{\bf p}_r}}) \delta({{{\bf p}'_s-{\bf p}_s}}).$$ (C-5)

For random phase encoding, we can also approximately have

$$\sum_{{\bf x}_s} \sum_{{\bf x}_r} \alpha({\bf x}_s,{\bf p}_s,\ome... ...\omega) \alpha'({\bf x}_s,{\bf p}'_s,\omega)\beta'({\bf x}_r,{\bf p}'_r,\omega)$$

$$\approx \delta({{{\bf p}'_r-{\bf p}_r}}) \delta({{{\bf p}'_s-{\bf p}_s}}).$$ (C-6)

Therefore the data-misfit function in the encoded source and receiver domain is

$$J(m({\bf x})) \approx \frac{1}{2}\sum_{\omega} \vert c\vert^4 \su... ...t d({\bf p}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf p}_r,{\bf p}_s,\omega)\vert^2.$$ (C-7)

Modeling, migration, and inversion in the generalized source and receiver domain