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 . 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 and a group of receivers positioned at can be given by the adjoint of a linear operator acting on the data-space as
The synthetic data can be modeled (as the adjoint of equation 4) by the linear operator acting on the model space
The second derivative of the quadratic cost function with respect to the model parameters is the subsurface-offset Hessian:
An approximation to the full subsurface-offset Hessian involves computing only the off-diagonal terms at close to the diagonal Valenciano and Biondi (2006).