Velocity-depth ambiguity of reflection traveltimes

Abstract The conversion of time horizons to depth is fundamental to exploration geophysics. The interval velocity used in the conversion is often estimated from the stacking velocity, assuming that each layer's interval velocity is homogeneous. However, even for one laterally inhomogeneous layer above a flat reflector the stacking velocity can swing violently about its average and conventional methods of velocity estimation fail. I show that violent swings in the stacking velocity are a symptom of a long-wavelength ambiguity between the burial depth to an interface and interval velocity. Lateral variations in seismic velocity with a spatial wavelength of about 2.7 D, where D is the depth to the reflecting horizon, cannot be unambiguously resolved from traveltime measurements. The spatial wavelength of this ambiguous component varies from 2.57 D, for very small source-receiver separations, to 2.86 D for source-receiver separations equal to D. Spectral components of the stacking velocity at wavelengths shorter than this ambiguous value are amplified in size and reversed in polarity relative to the interval velocity. A practical inverse filter that corrects for these distortions produces an interval velocity that is almost totally lacking in low-frequency components, giving a very distorted picture of the interval velocity.Since the wavelength of total ambiguity changes with offset, a complete description of the velocity and depth fields can, in theory, be extracted from a combination of multiple-offset traveltime measurements. However, the wavelength of total ambiguity is such a weak function of source-receiver separation that multiple offset processing, in practice, does little to resolve the ambiguity.In fact, the Rayleigh resolution limit implies that three or more offset measurements are more effective than two only if the seismic-line length is at least 20 D. In a series of numerical experiments with the line set to 100 D and a spatial noise level of .01% in each channel I used a two-channel Wiener filter to successfully extract the full-band response for a simultaneous step change in velocity and in depth. The method fails for lines shorter than 20 D because of the transients that arise when the data are shorter than the filter. Stability was achieved by increasing the noise level to 1% in the design of the Wiener filter, but low spatial frequencies were lost and the estimated velocity-depth model was distorted.If the results of this single flat-layer analysis apply to practical situations, the velocity-depth ambiguity may continue to plague exploration seismologists for some time to come.