Angle-domain Hessian

Sava and Fomel (2003) define an image space transformation from subsurface offset to reflection and azimuth angle as:

$$\begin{eqnarray} {\bf m}({\bf x},{\Theta}) &=& {\bf T'} (\Theta,{\bf h}){\bf m}({\bf x},{\bf h}), \end{eqnarray}$$ (12)

where ${\Theta}=(\theta,\alpha)$ are the reflection and the azimuth angles, and ${\bf T'} (\Theta,{\bf h})$ is the adjoint of the angle-to-offset transformation operator (slant stack).

Substituting the prestack migration image (subsurface offset domain) in equation 7 into equation 12 we obtain the expression for the prestack migration image in the angle-domain that follows:

$$\begin{eqnarray} {\bf m}({\bf x},{\Theta})&=& {\bf T'} (\Theta,{\bf h}) {\bf L}'... ...{'}\sum_{{\bf x}}{\bf d}({\bf x}_s,{\bf x}_r;\omega). \nonumber \end{eqnarray}$$ (13)

The synthetic data can be modeled (as the adjoint of equation 14) by the chain of linear operator ${\bf L}$ and the angle-to-offset transformation operator acting on the model space,

$$\begin{eqnarray} {\bf d}({\bf x}_s,{\bf x}_r;\omega)&=&{\bf L}{\bf T} (\Theta,{\... ...m_{\omega} {\bf T} (\Theta,{\bf h}){\bf m}({\bf x},{\Theta}), \end{eqnarray}$$ (14)

The quadratic cost function is

$$\begin{eqnarray} S({\bf m}) &=& \frac{1}{2} \sum_{\omega}\sum_{{\bf x}_s}\sum_{{... ...\left[ {\bf d}({\bf x}_s,{\bf x}_r;\omega)-{\bf d}_{obs} \right], \end{eqnarray}$$ (15)

while its first derivative with respect to the model parameters ${\bf m}({\bf x},\Theta)$ is

$$\begin{eqnarray} \frac{\partial{S({\bf m})}}{\partial{{\bf m}({\bf x},\Theta)}}=... ...) {\bf G}({\bf x+h},{\bf x}_s;\omega){\bf T} (\Theta,{\bf h}) \} \end{eqnarray}$$ (16)

and its second derivative with respect to the model parameters ${\bf m}({\bf x},\Theta)$ and ${\bf m}({\bf x'},\Theta')$ is the angle-domain Hessian

$$\begin{eqnarray} {\bf H}({\bf x,\Theta};{\bf x',\Theta'})&=& \frac{\partial^2{S... ...bf h}){\bf H}({\bf x,h};{\bf x',h'}) {\bf T}(\Theta',{\bf h'}). \end{eqnarray}$$ (17)