Characterization of resolution and uniqueness in crosswell direct‐arrival traveltime tomography using the Fourier projection slice theorem

Abstract

The process of acquiring a crosswell seismic direct‐arrival traveltime data set can be approximated by a series of truncated plane‐wave projections through an interwell slowness field. Using this approximation, the resolution and uniqueness of crosswell direct‐arrival traveltime tomograms can be characterized by invoking the Fourier projection slice theorem, which states that a plane‐wave projection through an object constitutes a slice of the object’s spatial spectrum. The limited vertical aperture of a crosswell survey introduces a small amount of nonuniqueness into the reconstructed tomogram by truncating the plane‐wave projection. By contrast, the limitations on angular aperture have a significant effect on resolution. The reconstructed tomogram is smeared primarily along the limiting projection angles, with the amount of smearing dependent upon the well spacing and the angular aperture. The amount of smearing was found to be inversely proportional to tan Δϕ, where Δϕ is the angular aperture illuminating a sector of the interwell plane. Consequently, the amount of smearing can be large where the angular aperture becomes small, such as at the top and bottom of the tomogram. For interwell sectors illuminated by large angular apertures, Fresnel zone effects will generally be the limiting factor in crosswell tomogram resolution. However, in some circumstances, angular aperture effects may control the tomogram resolution. The effects of angular aperture and direct‐arrival Fresnel zones produce tomograms with spatial resolution that is dependent upon the well spacing. This study indicates that direct‐arrival traveltime tomography will not usually produce tomograms with substantially greater resolution than surface seismic techniques for normal oil and gas well spacings.