The main problem stems from computing the Laplacian on a grid that is too sparse in places (close to spatial aliasing) or whose step size varies too quickly. Previous attempts Dellinger and Muir (1986) as well as the results of this work show that abrupt variations in the mesh size lead to numerical artifacts under the form of reflections off the irregularities in the grid. Such spurious reflections do not appear when the mesh step varies smoothly, or when the variation is less than half the grid step size. Therefore, two straightforward SMM applications are:

1. Migrating reflection data acquired from a platform moving with a nonconstant velocity (i.e., seismic acquisition ship which was not able to maintain constant speed; GPR vehicle that had to accelerate or decelerate; radar-bearing aircraft that encounters various air currents). In all these cases, the velocity variation is small enough that it should lead to a smoothly-varying grid.

2. Migrating reflection data acquired on a
*heated atomic lattice* - a regular grid whose nodes have been
displaced with small (known) amounts from the periodic positions.

In neither of these cases was the data irregular enough to justify the cost of full-fledged regularization, which involves interpolating the whole dataset. Most often the fact that the mesh is not really regular is simply overlooked. Applying SMM to such datasets will surely increase the quality of the image. The implementation is simple, consisting (at least for the 2D zero-offset case that has been implemented, and for its prestack extension) of simply replacing the (1,-2,1) coefficients of the second derivative in the differencing star with sets of three precomputed geometry-dependent values.

6/8/2002