The velocity‐depth ambiguity in seismic traveltime data

Abstract

An observed disturbance in seismic traveltimes to a reflector can be caused either by an anomalous velocity zone between the surface and the reflector or by a structural variation in the reflector itself. This velocity‐depth ambiguity is formulated in terms of linear estimation theory. Such a formulation allows integration of various published results on velocity‐depth ambiguity and suggests improved methods of stabilizing the solution of a depth‐conversion problem. By solving a relatively simple problem that is amenable to analysis—a single reflector beneath an overburden with a variable velocity—the following conclusions can be drawn: 1) The velocity‐depth ambiguity is caused by traveltime errors and can be quantitatively related to those errors by closed‐form expressions if the velocities do not vary laterally (or vary very slowly). Among other things, those expressions show that for small spread lengths (shorter than haft the depth) the errors in velocity and depth are inversely proportional to the square of the spread length. Errors can thus be reduced more effectively at small spread lengths by increasing the maximum offset rather than by including more offsets. 2) Laterally varying velocities can be estimated accurately at all but isolated points in their spatial frequency spectrum, called “wavelengths of maximum ambiguity.” If these ambiguous wavelengths are stabilized by damping them rather than by more traditional lateral smoothing techniques, structural or velocity features smaller than a spread length need not be smeared laterally. 3) A deep velocity anomaly is estimated with lower accuracy than is a shallow one. The theory presented here is a complement to more general methods of velocity inversion, such as tomography, which can be used to solve very complex problems beyond the scope of this analysis.