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## 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 . 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)
where and are respectively the Green's functions from the shot position and from the receiver position to a point in the model space , is the source wavelet, and is the subsurface-offset. The symbols and are spray operators (adjoint of the sum) in the subsurface-offset and physical space dimensions , 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 acting on the model space
 (5)
where the symbols ,, and 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:
 (6)
where 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).
 (7)
where are the off-diagonal coefficients. The impact of this approximation will be evaluated in the following sections.

Next: Data fitting goal Up: Linear least-squares inversion Previous: Linear least-squares inversion
Stanford Exploration Project
5/6/2007