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

APPENDIX A

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

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

From Equation 8, we can also get the inverse phase-encoding transform. For source plane-wave phase encoding, since the forward operator is unitary, the inverse transform can be written as follows:

$$w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) = \vert\omega... ...^2 \sum_{{\bf p}_s}d({\bf x}_r,{\bf p}_s,\omega)e^{-i\omega{\bf p}_s{\bf x}_s}.$$ (A-2)

For random phase encoding, a similar result can also be obtained, because different realizations of random sequences should be approximately orthogonal, provided that those random sequences are "sufficiently" random; thus we have

$$w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) = \sum_{{\bf p}_s}d({\bf x}_r,{\bf p}_s,\omega)e^{-i\gamma({\bf x}_s,{\bf p}_s,\omega)}.$$ (A-3)

Therefore, we can use a more general form to express the inverse phase-encoding transform:

$$w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) = \vert c\ver... ...um_{{\bf p}_s}d({\bf x}_r,{\bf p}_s,\omega)\alpha'({\bf x}_s,{\bf p}_s,\omega),$$ (A-4)

where for plane-wave phase encoding, $c=\omega$ and $\alpha({\bf x}_s,{\bf p}_s,\omega)=e^{i\omega{\bf p}_s{\bf x}_s}$; for random phase encoding, $c=1$ and $\alpha({\bf x}_s,{\bf p}_s,\omega)=e^{i\gamma({\bf x}_s,{\bf p}_s,\omega)}$.

Substituting Equation A-4 into A-1 yields:

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

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

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

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

$$\sum_{{\bf x}_s} \alpha({\bf x}_s,{\bf p}_s,\omega) \alpha'({\bf x}_s,{\bf p}'_s,\omega),$$ (A-5)

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

$$r({\bf x}_r,{\bf p}_s,\omega) = d({\bf x}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf p}_s,\omega).$$ (A-6)

For plane-wave phase encoding, if the ${\bf x}_s$ is sampled densely enough,

$$\sum_{{\bf x}_s} \alpha({\bf x}_s,{\bf p}_s,\omega) \alpha'({\bf ... ...\bf x}_s} \approx \frac{1}{\vert\omega\vert^2}\delta({{{\bf p}'_s-{\bf p}_s}}).$$ (A-7)

For random phase encoding, the following property also holds as long as the random sequences are "sufficiently" random:

$$\sum_{{\bf x}_s} \alpha({\bf x}_s,{\bf p}_s,\omega) \alpha'({\bf ... ...-\gamma({\bf x}_s,{\bf p}_s,\omega))} \approx \delta({{{\bf p}'_s-{\bf p}_s}}).$$ (A-8)

Substituting Equation A-7 or A-8 into A-5, we get the data-misfit function in the encoded source domain:

$$J(m({\bf x})) \approx \frac{1}{2}\sum_{\omega} \vert c\vert^2 \su... ...t d({\bf x}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf p}_s,\omega)\vert^2.$$ (A-9)

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