Subsurface-offset Hessian

Valenciano et al. (2006) define the zero subsurface-offset domain Hessian by using the adjoint of the zero subsurface-offset domain migration as the modeling operator ${\bf L}$. Then the zero-subsurface-offset inverse image can be estimated as the solution of a non-stationary least-squares filtering problem, using an iterative inversion algorithm Valenciano et al. (2006).

The subsurface-offset Hessian was defined by Valenciano and Biondi (2006). The definition can be summarized as follows.

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

$$\begin{eqnarray} {\bf m}({\bf x},{\bf h}) &=& {\bf L}'{\bf d}({\bf x}_s,{\bf x}_... ...\sideset{}{'}\sum_{{\bf x}}{\bf d}({\bf x}_s,{\bf x}_r;\omega), \end{eqnarray}$$ (4)

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 ${\bf x}$, ${\bf f}(\omega)$ is the source wavelet, and ${\bf h}=(h_x,h_y)$ is the subsurface-offset. The symbols $\sideset{}{'}\sum_{{\bf h}}$ and $\sideset{}{'}\sum_{{\bf x}}$ are spray operators (adjoint of the sum) in the subsurface-offset and physical space dimensions ${\bf x}=(x,y,z)$, 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 4) by the linear operator ${\bf L}$ acting on the model space ${\bf m}({\bf x},{\bf h})$

$$\begin{eqnarray} {\bf d}({\bf x}_s,{\bf x}_r;\omega)&=&{\bf L}{\bf m}({\bf x},{\... ...\bf x}_s}\sideset{}{'}\sum_{\omega} {\bf m}({\bf x},{\bf h}), \end{eqnarray}$$ (5)

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 second derivative of the quadratic cost function with respect to the model parameters is the subsurface-offset Hessian:

$$\begin{eqnarray} {\bf H}({\bf x,h};{\bf x',h'})=\sum_{\omega}{\bf f}(\omega)^2 ... ... x+h},{\bf x}_r;\omega) {\bf G}({\bf x'+h'},{\bf x}_r;\omega), \end{eqnarray}$$ (6)

where $({\bf x',h')}$ are the off-diagonal terms of the Hessian matrix.

An approximation to the full subsurface-offset Hessian involves computing only the off-diagonal terms at close to the diagonal Valenciano and Biondi (2006).

$$\begin{eqnarray} {\bf H}({\bf x,h};{\bf x+a,h'})=\sum_{\omega}{\bf f}(\omega)^2 ... ...x+h},{\bf x}_r;\omega) {\bf G}({\bf x+a+h'},{\bf x}_r;\omega), \end{eqnarray}$$ (7)

where ${\bf a}=(a_x,a_y,a_z)$ are the off-diagonal coefficients. The impact of this approximation will be evaluated in the following sections.