Three-dimensional kinematic migration produces a “best-fit” migrated surface, in three-dimensions, of any chosen reflector. The data are the travel times of all observed reflections and diffractions produced by the reflector. Source/receiver configurations are arbitrary, but should be arranged to provide good spatial sampling of the reflector. Each observation contributes an ellipsoid containing all possible reflection points. From this family of ellipsoids, the optimal reflector surface (the envelope of the family) is estimated by use of a statistical imaging condition. The “best-fit” position and shape of the reflector surface is obtained by defining a regular (x, y) grid over the region of interest and estimating the reflector depth beneath each grid point from the distribution of ellipsoids that are present there. The imaging criterion is implemented at each grid point by convolution of a Gaussian with the vector of ellipsoid depths. The width of the Gaussian is chosen to correspond to the scale of the features one wishes to resolve in the image. For each grid point, the migrated image is located at the depth at which the convolution is a maximum.
The algorithm is applied to both synthetic and real data. The synthetic data are constructed by ray tracing in a known structure. The real data are from a seismic survey at the Nevada Test Site; here, a reflector is imaged and interpreted as the surface of a high velocity Paleozoic dolomite that is overlain by low-velocity tuffs.