Regularizing over offset

There is a notable drawback from the approach described above. The operator $\bf S$ can be quite costly, We are doing nx*ny AMO transforms for every output (hx,hy). If we are only interested in a common azimuth dataset $\rm hy=0$, the cost is acceptable as long as ny is fairly small. If we want any cross line offset output offset the cost isn't acceptable. In addition $\bf S$ is a modified version (because of the AMO transform) of a small 2-D box car filter. If you desire additional smoothness in the in-line offset direction (to suppress amplitude variations) we must try a different approach.

Biondi and Vlad (2001) proposed reducing the dimensionality of the problem by ignoring the azimuth direction. They added a smoothness constraint to the problem by applying a Leaky derivative $\bf D$operator between AMO transformed (t,cmpx,cmpy) cubes setting up the minimization,  

$$Q(\bf m) = \vert\vert\bf d- \bf L\bf m\vert\vert^2 + \epsilon^2 \vert\vert\bf D \bf m\vert\vert^2 ,$$ (7)

where $\epsilon$ controls the weighting between the two goals. They preconditioned the problem with the inverse of $\bf D$,leaky integration between AMO transformed cubes $\bf C$ and applied the same Hessian approximation to obtain the approximation  

$$\bf m= \bf C \bf W \bf C' \bf L' \bf d ,$$ (8)

where $\bf C$ is the inverse of $\bf D$and

$$\bf W^{-1} = {\rm diag} \left[ \bf C' \bf L' \bf L\bf C {\bf 1} +\epsilon^2 \right] .$$ (9)

Clapp (2003b) noted that ignoring the azimuth removed some of the advantage of using the AMO operator and suggested that $\bf C$ should be applying polynomial division with a 2-D filter operating in the the (hx,hy) plane. For this exercise I chose a small helical derivative for my 2-D filter Claerbout (1999).

 

• Implementation
• Results