RESOLUTION LIMITS FOR CROSSWELL MIGRATION AND TRAVEL-TIME TOMOGRAPHY

Equations are derived for the point-scatterer response of the crosswel l migration and traveltime tomography operators. These formulas are us ed to estimate the limits of spatial resolution in reflection migratio n images and traveltime tomograms. In particular, for a crosswell geom etry with borehole length L, well offset 2(xo), source wavelength lamb da, and a centred point scatterer I show the following. (1) The vertic al resolution Delta(z)(mig) of the reflection migration image is equal to 2 lambda(xo)/L under the far-held approximation. Under the Fresnel approximation, Delta(z)(mig) approximate to 2 lambda x(o)/L - 2 lambd a(2)/L, which says that images become better resolved with an increase in aperture and a decrease in wavelength and well offset. (2) The hor izontal 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 roo t lambda x(o). This estimate agrees with that of a previous study. How ever, 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 coordi nate. 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 t raveltime 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 sour ces and N-g geophones, the dynamic range of the migrated image is prop ortional to NsNg. The dynamic range of the slowness tomogram is propor tional to root NgNs. Many of the estimates for the resolution limits h ave simple geometrical interpretations. For example, the minimum verti cal (horizontal) resolution in a migrated reflectivity section corresp onds to the minimum vertical (horizontal) stretch that a migrated wave let undergoes in going from the time domain to the depth domain. In ad dition, the minimum vertical (horizontal) resolution in a traveltime t omogram corresponds to the minimum vertical (horizontal) width of the wavepath at the scatterer location.