The regularization operator A is designed to shape the model to conform to some expectations of its characteristics, particularly its covariance Prucha et al. (2000); Tarantola (1986). When using downward continuation migration as the linear imaging operator in the least-squares inversion problem, it is known that poor illumination will cause the resulting model to have gaps in events along the offset ray parameter (or the equivalent reflection angle) axis Prucha et al. (2000). When attempting to image the subsurface where we know illumination is poor, it is reasonable to design A to regularize amplitudes along the offset ray parameter (ph) axis.
To create a simple regularization operator, I begin with the assumption that the velocity model used by the imaging operator is correct and therefore there is no moveout along the ph axis. Therefore, A can act horizontally along the ph axis, penalizing sudden changes in amplitude. This is called ``geophysical'' regularization. Essentially, A is a derivative operator. It will act most forcefully on very large, sudden amplitude changes along the ph axis. This should compensate for the sudden holes caused by poor illumination without changing AVA trends that generally vary smoothly Shuey (1985).
Previous experiments using this geophysical regularization where illumination is poor have shown that it helps to ``heal'' the gaps along the ph axis Prucha et al. (2000). While this is encouraging, the question of how much the geophysical regularization affects any genuine amplitude variation with offset ray parameter or angle (AVA) remains open. To answer this question, I have designed a simple test with known AVA and no illumination problems.