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

encoded sources

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

$$d({\bf x}_r,{\bf p}_s,\omega) = \sum_{{\bf x}_s} w({\bf x}_r,{\bf x}_s)d({\bf x}_r,{\bf x}_s,\omega) \alpha({\bf x}_s,{\bf p}_s,\omega),$$ (A-8)

where $\alpha({\bf x}_s,{\bf p}_s,\omega)$ is the source phase-encoding function. Equation 8 integrates along the horizontal dashed lines shown in Figure 1 for each receiver location ${\bf x}_r$ and transforms the surface data from $({\bf x}_s,{\bf x}_r)$ domain into the $({\bf p}_s,{\bf x}_r)$ domain. For the plane-wave phase encoding, the encoding function is:

$$\alpha({\bf x}_s,{\bf p}_s,\omega)=e^{i\omega{\bf p}_s{\bf x}_s},$$ (A-9)

where ${\bf p}_s$ is defined to be the ray parameter of the source plane waves. For random phase encoding, the encoding function is

$$\alpha({\bf x}_s,{\bf p}_s,\omega)=e^{i\gamma({\bf x}_s,{\bf p}_s,\omega)},$$ (A-10)

where $\gamma({\bf x}_s,{\bf p}_s,\omega)$ is a random sequence in ${\bf x}_s$ and $\omega$, and ${\bf p}_s$ defines the index of different realizations of the random sequence.

Substituting Equation 1 into 8, rearranging the order of summation, we get the forward modeling equation in the encoded source domain:

$$d({\bf x}_r,{\bf p}_s,\omega) = \sum_{\bf x} G({\bf x},{\bf x}_r,\omega) G({\bf x},{\bf p}_s,\omega;{\bf x}_r) m({\bf x}),$$ (A-11)

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

$$G({\bf x},{\bf p}_s,\omega;{\bf x}_r) = \sum_{{\bf x}_s} w({\bf x}_r,{\bf x}_s)G({\bf x},{\bf x}_s,\omega)\alpha({\bf x}_s,{\bf p}_s,\omega).$$ (A-12)

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

As derived in Appendix A, the objective function in the encoded source domain can be written as follows:

$$J(m({\bf x})) = \frac{1}{2}\sum_{\omega} \vert c\vert^2 \sum_{{\b... ...t d({\bf x}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf p}_s,\omega)\vert^2,$$ (A-13)

where $c=\omega$ for plane-wave phase encoding, and $c=1$ for random phase encoding.

The gradient of the objective function in Equation 13 gives the following migration formula in the encoded source domain:

$$\nabla J({\bf x}) = \Re \left( \sum_{\omega} \vert c\vert^2 \sum_... ...{{\bf x}_r} G'({\bf x},{\bf x}_r,\omega) r({\bf x}_r,{\bf p}_s,\omega) \right),$$ (A-14)

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

$$r({\bf x}_r,{\bf p}_s,\omega) = d({\bf x}_r,{\bf p}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf p}_s,\omega).$$ (A-15)

It is easy to see that Equation 14 defines the phase-encoding migration (Romero et al., 2000; Liu et al., 2006). By taking the second-order derivatives of the objective function defined in Equation 13 with respect to the model paramters, we obtain the Hessian in the encoded source domain:

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

$$\left. \sum_{{\bf p}_s} G ({\bf x},{\bf p}_s,\omega;{\bf x}_r) G'({\bf y},{\bf p}_s,\omega;{\bf x}_r) \right).$$ (A-16)

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