Many geophysical algorithms can be cast in the form of a line integral in the time-space domain. The most commonly used are NMO-Stack, DMO , slant stack and Kirchhoff migration. The integral may be written in several ways, as an integral over angle, over slowness, over space, over time, or over arc length. Since seismic data is collected at discrete spatial locations, these integrals are most often implemented as a space integral (although this is not the case when the time coordinate of the curve is a multi-valued function of space). The integrals are derived in a continuous space but when they are applied to seismic data they are implemented in a discretely sampled space.
There are many ways to approximate the continuous integral. One of the most common is to sample the curve evenly in space and sum a single sample for each spatial location. Hale 1991 demonstrated that this simple approach can produce operator aliasing when the operator has steep dips. He suggested the use of equal time sampling to overcome this problem. However, equal time sampling will not sample all the available information when the dips are flat.
I propose a simple way of looking at different strategies for performing the integration and their use on a seismic dataset. The line integral should be just that, an integral along the line! The problem then becomes one of interpolating the sampled data from its discrete locations to a continuous function along the line, and integrating the result. All of the interpolation/integration strategies can be coded as a weighted summation of one or more time samples at each trace location. The anti-aliasing Kirchhoff migration proposed by Claerbout Claerbout (1992) is also in the form of a weighted summation of samples on each trace. I discuss the relationship between his methods and the methods I propose here.