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

encoded receivers

Let us define the encoding transform along the ${\bf x}_r$ coordinate of the surface data as follows:

$$d({\bf p}_r,{\bf x}_s,\omega) = \sum_{{\bf x}_r} w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) \beta({\bf x}_r,{\bf p}_r,\omega),$$ (A-17)

where $\beta({\bf x}_r,{\bf p}_r,\omega)$ is the receiver phase-encoding function. Equation 17 integrates along the vertical dashed lines shown in Figure 1 for each source location ${\bf x}_s$ and transforms the surface data from $({\bf x}_s,{\bf x}_r)$ domain into the $({\bf x}_s,{\bf p}_r)$ domain. The receiver phase-encoding function is defined similar to the source phase-encoding function discussed in the previous section. For receiver plane-wave phase encoding,

$$\beta({\bf x}_r,{\bf p}_r,\omega)=e^{i\omega{\bf p}_r{\bf x}_r},$$ (A-18)

where ${\bf p}_r$ is the ray parameter for the receiver plane-waves. For receiver random phase encoding,

$$\beta({\bf x}_r,{\bf p}_r,\omega)=e^{i\gamma({\bf x}_r,{\bf p}_r,\omega)},$$ (A-19)

where $\gamma({\bf x}_r,{\bf p}_r,\omega)$ is the ${\bf p}_r$th random realization. Substituting Equation 1 into 17, we get the forward modeling equation in the receiver plane-wave domain:

$$d({\bf p}_r,{\bf x}_s,\omega) = \sum_{\bf x} G({\bf x},{\bf x}_s,\omega) G({\bf x},{\bf p}_r,\omega;{\bf x}_s) m({\bf x}),$$ (A-20)

where $G({\bf x},{\bf p}_r,\omega;{\bf x}_s)$ is the encoded receiver Green's function defined as follows:

$$G({\bf x},{\bf p}_r,\omega;{\bf x}_s) = \sum_{{\bf x}_r} w({\bf x}_r,{\bf x}_s)G({\bf x},{\bf x}_r,\omega) \beta({\bf x}_r,{\bf p}_r,\omega).$$ (A-21)

Also note that $G({\bf x},{\bf p}_r,\omega;{\bf x}_s)$ depends on ${\bf x}_s$ because of the acquisition mask matrix inside the summation. It should only integrate the grey segment for each vertical dashed line shown in Figure 1.

We minimize the following objective function in the encoded receiver domain (see Appendix B for derivation):

$$J(m({\bf x})) = \sum_{\omega}\vert c\vert^2 \sum_{{\bf p}_r} \sum... ...t d({\bf p}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf p}_r,{\bf x}_s,\omega)\vert^2,$$ (A-22)

The gradient of the objective function in Equation 22 gives the following migration formula in the encoded receiver domain:

$$\nabla J({\bf x}) = \Re \left( \sum_{\omega}\vert c\vert^2\sum_{{... ...r} G'({\bf x},{\bf p}_r,\omega;{\bf x}_s)r({\bf p}_r,{\bf x}_s,\omega) \right),$$ (A-23)

where the residual $r({\bf p}_r,{\bf x}_s,\omega)$ is defined as follows:

$$r({\bf p}_r,{\bf x}_s,\omega) = d({\bf p}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf p}_r,{\bf x}_s,\omega).$$ (A-24)

The Hessian in the encoded receiver domain is then obtained by taking the second derivative of the objective function:

$$H({\bf x},{\bf y}) = \Re \left( \sum_{\omega}\vert c\vert^2 \sum_{{\bf x}_s} G({\bf x},{\bf x}_s,\omega) G'({\bf y},{\bf x}_s,\omega) \right. \times$$

$$\left. \sum_{{\bf p}_r} G({\bf x},{\bf p}_r,\omega;{\bf x}_s) G'({\bf y},{\bf p}_r,\omega;{\bf x}_s) \right).$$ (A-25)

We can rewrite Equation 25 as follows

$$H({\bf x},{\bf y}) = \sum_{{\bf p}_r} H({\bf x},{\bf y},{\bf p}_r),$$ (A-26)

where

$$H({\bf x},{\bf y},{\bf p}_r) = \Re \left( \sum_{\omega}\vert c\vert^2 \sum_{{\bf x}_s} G({\bf x},{\bf x}_s,\omega) G'({\bf y},{\bf x}_s,\omega) \right. \times$$

$$\left. G({\bf x},{\bf p}_r,\omega;{\bf x}_s) G'({\bf y},{\bf p}_r,\omega;{\bf x}_s) {\frac{}{}}^{\frac{}{}}_{\frac{}{}} \right)$$

$$= \Re \left( \sum_{\omega}\vert c\vert^2 \sum_{{\bf x}_s} G({\bf x},{\bf x}_s,\omega) G'({\bf y},{\bf x}_s,\omega) \right. \times$$

$$\left( \sum_{{\bf x}_r } w({\bf x}_r,{\bf x}_s) G ({\bf x},{\bf x}_r ,\omega)\beta({\bf x}_r,{\bf p}_r,\omega) \right) \times$$

$$\left. \left( \sum_{{\bf x}'_r} w({\bf x}_r,{\bf x}_s)G({\bf y},{\bf x}'_r,\omega)\beta({\bf x}'_r,{\bf p}_r,\omega) \right)' \right).$$ (A-27)

Equation 27 is equivalent to Equations $9$ and B-$1$ in Tang (2008), which are called the receiver-side encoded Hessian. As I show here, the receiver-side encoded Hessian is the same as the Hessian in the encoded receiver domain; both of them are derived from the same forward modeling equation defined in Equation 20.

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