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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 dataspace as
 

 (1) 
where and are respectively the Green's functions from the shot position and from the receiver position to a point in the model space , and is the subsurface offset. The symbols and are spray (adjoint of the sum) operators in the subsurface offset and model space dimensions, respectively.
The Green's functions are computed by means of the oneway waveequation.
The synthetic data can be modeled (as the adjoint of equation 1) by the linear operator acting on the model space with and
 

 (2) 
where the symbols ,, and are spray operators in the shot, receiver, and frequency dimensions, respectively.
The quadratic cost function is
 
(3) 
and its second derivative with respect to the model parameters and is the subsurface offset Hessian:
The next subsection shows how to use the subsurfaceoffset domain Hessian to compute the angle Hessian following the Fomel (2004) approach.
Next: Angledomain Hessian
Up: Expanding Hessian dimensionality
Previous: Expanding Hessian dimensionality
Stanford Exploration Project
1/16/2007