Angle-domain Hessian

Fomel (2004) define an image space transformation from subsurface offset to reflection and azimuth angle as:

$$\begin{eqnarray} {\bf m}({\bf x},{\Theta}) &=& {\bf T'} (\Theta,{\bf h}){\bf m}({\bf x},{\bf h}), \end{eqnarray}$$ (4)

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 1 into equation 4 we obtain the expression for the prestack migration image in the angle-domain that follows:

$$\begin{eqnarray} {\bf m}({\bf x},{\Theta})&=& {\bf T'} (\Theta,{\bf h}) {\bf L}'{\bf d}({\bf x}_s,{\bf x}_r;\omega), \end{eqnarray}$$ (5)

The synthetic data can be modeled (as the adjoint of equation 5) by the chain of linear operator ${\bf L}$ and the angle-to-offset transformation operator acting on the model space,

$$\begin{eqnarray} {\bf d}({\bf x}_s,{\bf x}_r;\omega)&=&{\bf L}{\bf T} (\Theta,{\bf h}){\bf m}({\bf x},{\Theta}), \end{eqnarray}$$ (6)

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

$$\begin{eqnarray} {\bf H}({\bf x,\Theta};{\bf x',\Theta'})&=& {\bf T'} (\Theta,{\bf h}){\bf H}({\bf x,h};{\bf x',h'}) {\bf T}(\Theta',{\bf h'}). \end{eqnarray}$$ (7)