THE VELOCITY-DEPTH AMBIGUITY IN SEISMIC TRAVEL-TIME DATA

An observed disturbance in seismic traveltimes to a reflector can be c aused 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 t heory. Such a formulation allows integration of various published resu lts on velocity-depth ambiguity and suggests improved methods of stabi lizing the solution of a depth-conversion problem. By solving a relati vely simple problem that is amenable to analysis-a single reflector be neath 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-fo rm expressions if the velocities do not vary laterally (or vary very s lowly). Among other things, those expressions show that for small spre ad lengths (shorter than half the depth) the errors in velocity and de pth are inversely proportional to the square of the spread length. Err ors can thus be reduced more effectively at small spread lengths by in creasing the maximum offset rather than by including more offsets. 2) Laterally varying velocities can be estimated accurately at all but is olated points in their spatial frequency spectrum, called ''wavelength s of maximum ambiguity.'' If these ambiguous wavelengths are stabilize d by damping them rather than by more traditional lateral smoothing te chniques, structural or velocity features smaller than a spread length need not be smeared laterally. 3) A deep velocity anomaly is estimate d with lower accuracy than is a shallow one. The theory presented here is a complement to more general methods of velocity inversion, such a s tomography, which can be used to solve very complex problems beyond the scope of this analysis.