Expanding Hessian dimensionality

Valenciano et al. (2005b) define the zero subsurface-offset domain Hessian by using the adjoint of the zero subsurface-offset domain migration as the modeling operator ${\bf L}$. 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. (2005a,b).

From the results reported by Prucha et al. (2000), Kuehl and Sacchi (2001), and Valenciano et al. (2005a), regularization in the reflection angle dimension is necessary to stabilize the wave-equation inversion problem. That is why Valenciano and Biondi (2005) defined the wave-equation angle-domain Hessian from the subsurface offset wave-equation Hessian via an angle-to-offset transformation Fomel (2004); Sava and Fomel (2003).

The next subsections summarize the theory presented in Valenciano et al. (2005b).

 

• Subsurface-offset Hessian
• Angle-domain Hessian