Analysis of the spectral hole in velocity versus depth resolution for reflection traveltimes with limited aperture

It is difficult to resolve the ambiguity between velocity and reflector depth using reflection traveltimes when the aperture is small, as is common for deep reflectors. For velocity perturbations that are independent of the vertical variable, there is an even stronger velocity-versus-depth ambiguity at a horizontal wavelength of 2.5 times the reflector depth. We give a theoretical explanation of why this spectral hole occurs. When the maximum offset is small, there are velocity and reflector depth perturbations that cause almost cancelling traveltime perturbations; the net traveltime perturbations are second order in offset, making resolution between velocity and depth difficult at all wavelengths. But for the particular wavelength 2π/6 ≈ 2.565 times the reflector depth, an extra term in the Taylor expansion for traveltime near zero offset vanishes; the net traveltime perturbations are fourth order in offset. Thus velocity-versus-depth resolution degrades much sooner at this wavelength as the maximum offs...