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

APPENDIX B

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

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

Similar to the discussion in Appendix A, the general form of the inverse transform of receiver phase encoding can be written as follows:

$$w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) = \vert c\ver... ...sum_{{\bf p}_r}d({\bf p}_r,{\bf x}_s,\omega)\beta'({\bf x}_r,{\bf p}_r,\omega),$$ (B-2)

where for plane-wave phase encoding, $c=\omega$ and $\beta({\bf x}_r,{\bf p}_r,\omega)=e^{i\omega{\bf p}_r{\bf x}_r}$; for random phase encoding, $c=1$ and $\beta({\bf x}_r,{\bf p}_r,\omega) = e^{i\gamma({\bf x}_r,{\bf p}_r,\omega)}$. Substituting Equation B-2 into B-1 yields:

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

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

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

$$= \frac{1}{2}\sum_{\omega} \sum_{{\bf x}_s}\vert c\vert^4 \sum_{{\b... ...bf p}'_r} r'({\bf p}_r,{\bf x}_s ,\omega) r({\bf p}'_r,{\bf x}_s,\omega) \times$$

$$\sum_{{\bf x}_r} \beta({\bf x}_r,{\bf p}_r,\omega)\beta'({\bf x}_r,{\bf p}'_r,\omega),$$ (B-3)

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

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

Similar to the discussion in Appendix A, the inner-most summation in Equation B-3 is approximately a Dirac delta function under certain conditions. Therefore, the data-misfit function in the encoded receiver domain reads as follows:

$$J(m({\bf x})) \approx \frac{1}{2}\sum_{\omega} \vert c\vert^2 \su... ...t d({\bf p}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf p}_r,{\bf x}_s,\omega)\vert^2.$$ (B-5)

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