Resolution limits for crosswell migration and traveltime tomography

Equations are derived for the point-scatterer response of the crosswell migration and traveltime tomography operators. These formulas are used to estimate the limits of spatial resolution in reflection migration images and traveltime tomograms. In particular, for a crosswell geometry with borehole length L, well offset 2(xo), source wavelength lambda, and a centred point scatterer I show the following. (1) The vertical resolution Delta(z)(mig) of the reflection migration image is equal to 2 lambda(xo)/L under the far-field approximation. Under the Fresnel approximation, Delta(z)(mig) approximate to 2 lambda x(o)/L - 2 lambda(2)/L, which says that images become better resolved with an increase in aperture and a decrease in wavelength and well offset. (2) The horizontal resolution Delta(x)(mig) of the migration image is equal to 16 lambda x(o)(2)/L(2). The lateral resolution of the migrated image is worse than the vertical resolution by a factor of 8x(o)/L (where x(o)/L > 1 under the far-field approximation). (3) The vertical resolution Delta(z)(tomo) of the traveltime velocity tomogram is proportional root lambda x(o). This estimate agrees with that of a previous study. However, the tomographic image of the slowness perturbation behaves as a non-local cosine function along the depth axis, whereas the migration image behaves as a localized squared sine function in the depth coordinate. This is consistent with the empirical observation that interface boundaries are more sharply resolved by migration than by traveltime tomography. (4) The horizontal resolution of the slowness image in a traveltime tomogram is equal to (4x(o)/L)root 3x(o) lambda/4, a factor more than 4x(o)/L worse than the vertical resolution. (5) For N-s sources and N-g geophones, the dynamic range of the migrated image is proportional to NsNg. The dynamic range of the slowness tomogram is proportional to root NgNs. Many of the estimates for the resolution limits have simple geometrical interpretations. For example, the minimum vertical (horizontal) resolution in a migrated reflectivity section corresponds to the minimum vertical (horizontal) stretch that a migrated wavelet undergoes in going from the time domain to the depth domain. In addition, the minimum vertical (horizontal) resolution in a traveltime tomogram corresponds to the minimum vertical (horizontal) width of the wavepath at the scatterer location.