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

(4) |

The synthetic data can be modeled (as the adjoint of equation 4) by the linear operator acting on the model space

(5) |

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

(6) |

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

(7) |

5/6/2007