<title>Resolution limits for Crosswell migration and travel-time tomography</title>

Abstract

I derive the formulas for the seismic response of a line scatterer for the crosswell migration and traveltime tomography operators. These formulas are used to estimate the limits of spatial resolution in reflectivity images obtained from migration and slowness images reconstructed from traveltime tomography. In particular, for a crosswell geometry with borehole length L, well offset 2x<SUB>o</SUB>, source wavelength (lambda) , and a centered line scatterer I show that: (1) The vertical resolution (Delta) <SUB>z</SUB><SUP>mig</SUP> of the migration image is equal to 2(lambda) x<SUB>o</SUB>/L under the far-field approximation. (2) The horizontal resolution (Delta) <SUB>x</SUB><SUP>mig</SUP> of the migration image is equal to 16(lambda) x<SUP>2</SUP><SUB>o</SUB>/L<SUP>2</SUP>. The lateral resolution of the migrated image is worse than the vertical resolution by the factor 8x<SUB>o</SUB>/L (where x<SUB>o</SUB>/L &gt; 1 under the far-field approximation). (3) For inverting traveltimes associated with a localized slowness perturbation midway between the wells, the vertical resolution (Delta) <SUB>x</SUB><SUP>tomo</SUP> of the slowness tomogram is proportional to (root)(lambda) x<SUB>o</SUB>. This estimate agrees with that of a previous study. (4) The horizontal resolution of the slowness image in a traveltime tomogram is equal to [4x<SUB>o</SUB>/L](root)3x<SUB>o</SUB>(lambda) /4, a factor 4(root)3x<SUB>o</SUB>/L worse than the vertical resolution. (5) For N<SUB>s</SUB> and N<SUB>g</SUB> geophones, the dynamic range of the migrated image is proportional to N<SUB>s</SUB>N<SUB>g</SUB>. The dynamic range of the slowness tomogram is proportional to (root)N<SUB>g</SUB>N<SUB>s</SUB>.