Wave-Equation Angle-Domain Hessian

Abstract

A regularization in the reflection angle dimension (and, more generally in the reflection and azimuth angles) is necessary to stabilize the wave-equation inversion problem. The angle-domain Hessian can be computed from the subsurface-offset Hessian by an offsetto-angle transformation. This transformation can be done in the image space following the Sava and Fomel (2003) approach. To perform the inversion, the angle-domain Hessian matrix can be used explicitly, or implicitly as a chain of the offset-to-angle operator and the subsurface offset Hessian matrix.

Least-squares migration with angle gathers

“…Expanding Hessian dimensionality to the angle domain Valenciano and Biondi (2006) defined the Hessian matrix in the prestack image domain as a chain of operators from the subsurface offset h = (hx,hy) to the reflection and azimuth angle Θ = (θ,α): ) where the operator T defines the transformation from reflection and azimuth angle to subsurface offset (Sava and Fomel, 2003). The approach of Valenciano and Biondi (2006) can be applied to any prestack volume where angle gathers are produced from direct binning using Poynting Vectors (Yoon and Marfurt, 2006), or extended imaging conditions (Sava and Fomel, 2005).…”