Subsurface-offset Hessian

The prestack migration image (subsurface offset domain) for a group of shots positioned at ${\bf x}_s=(x_s,y_s,0)$ and a group of receivers positioned at ${\bf x}_r=(x_r,y_r,0)$ can be given by the adjoint of a linear operator ${\bf L}$ acting on the data-space ${\bf d}({\bf x}_s,{\bf x}_r;\omega)$ as

$$\begin{eqnarray} {\bf m}({\bf x},{\bf h}) &=& {\bf L}'{\bf d}({\bf x}_s,{\bf x}_... ...\sideset{}{'}\sum_{{\bf x}}{\bf d}({\bf x}_s,{\bf x}_r;\omega), \end{eqnarray}$$ (1)

where ${\bf G}({\bf x},{\bf x}_s;\omega)$ and ${\bf G}({\bf x},{\bf x}_r;\omega)$ are respectively the Green's functions from the shot position ${\bf x}_s$ and from the receiver position ${\bf x}_r$ to a point in the model space ${\bf x}$, and ${\bf h}=(h_x,h_y)$ is the subsurface offset. The symbols $\sideset{}{'}\sum_{{\bf h}}$ and $\sideset{}{'}\sum_{{\bf x}}$ are spray (adjoint of the sum) operators in the subsurface offset and model space dimensions, respectively. The Green's functions are computed by means of the one-way wave-equation.

The synthetic data can be modeled (as the adjoint of equation 1) by the linear operator ${\bf L}$ acting on the model space ${\bf m}({\bf x},{\bf h})$ with ${\bf x}=(x,y,z)$ and ${\bf h}=(h_x,h_y)$

$$\begin{eqnarray} {\bf d}({\bf x}_s,{\bf x}_r;\omega)&=&{\bf L}{\bf m}({\bf x},{\... ...\bf x}_s}\sideset{}{'}\sum_{\omega} {\bf m}({\bf x},{\bf h}), \end{eqnarray}$$ (2)

where the symbols $\sideset{}{'}\sum_{{\bf x}_r}$,$\sideset{}{'}\sum_{{\bf x}_s}$, and $\sideset{}{'}\sum_{\omega}$ are spray operators in the shot, receiver, and frequency dimensions, respectively.

The quadratic cost function is

$$\begin{eqnarray} S({\bf m}) &=& \frac{1}{2} \sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r} \Vert {\bf d} - {\bf d}_{obs} \Vert^2, \end{eqnarray}$$ (3)

and its second derivative with respect to the model parameters ${\bf m}({\bf x},{\bf h})$ and ${\bf m}({\bf x'},{\bf h'})$ is the subsurface offset Hessian:

$$\begin{eqnarray} {\bf H}({\bf x,h};{\bf x',h'})&=&\sum_{\omega} \sum_{{\bf x}_s... ...f x}_r;\omega) {\bf G}({\bf x'-h'},{\bf x}_r;\omega). \nonumber \end{eqnarray}$$

The next subsection shows how to use the subsurface-offset domain Hessian to compute the angle Hessian following the Fomel (2004) approach.