There are five basic ways in which spurious reflections can be
eliminated: 1) Letting the mesh points drift across the *x* axis as
we downward continue. This approach, suggested by
Dellinger and Muir (1986), would regularize the spatial grid and
address the actual cause of the problem. 2) Planting interpolated
traces at strategic locations. 3) Solving for a wave equation that
incorporates the grid irregularity (*dx* as a function of *x*, or by
applying a warping transform). 4) Filtering them out based on the
fact that they are localized and highly coherent and their dip is
opposite from that of local geologic dip. 5) Avoiding the problem
altogether by using a numerical method for downward continuation that
handles irregular data better than the finite difference method.

Method 1 seems to be the most elegant and efficient. For all methods, the biggest problem is posed by large gaps in the data coverage which would still need to be filled in with interpolated traces. A minimum trace density, related to the minimum spatial wavelength present in the data, must be maintained. The recent advances in interpolation methods of nonuniformly sampled data Aldroubi and Grochenig (2001) can be instrumental in this respect. Even with interpolation, the number of fill-in traces required will be smaller for an unconstrained mesh than that in the case of a Cartesian mesh. This is due to the Cartesian lattices not being the best at filling space; the same area can be covered by fewer traces placed on a quasi-regular triangular mesh. This at least should offset the burden of node number bookkeeping for an unstructured mesh. Large coverage gaps can be covered using the boundary element method. This method was created with the specific goal of not having to deal with very large numbers of elements inside a domain - its elements are only on the border of the domain. Another solution may be presented by the finite element method Marfurt (1984), which naturally handles unstructured meshes.

6/8/2002