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Sava and Fomel (2003) define an image space transformation from subsurface offset to reflection and azimuth angle as:
 
(12) 
where are the reflection and the azimuth angles, and is the adjoint of the angletooffset transformation operator (slant stack).
Substituting the prestack migration image (subsurface offset domain) in equation 7 into equation 12 we obtain the expression for the prestack migration image in the angledomain that follows:
 
(13) 
 
The synthetic data can be modeled (as the adjoint of equation 14) by the chain of linear operator and the angletooffset transformation operator acting on the model space,
 

 (14) 
The quadratic cost function is
 

 (15) 
while its first derivative with respect to the model parameters is
 

 (16) 
and its second derivative with respect to the model parameters and is the angledomain Hessian
 

 
 
 
 
 (17) 
Next: Explicit vs. implicit Hessian
Up: Expanding Hessian dimensionality
Previous: Subsurfaceoffset Hessian
Stanford Exploration Project
10/31/2005