Uniqueness of Reflector Depths and Characterization of the Slowness Null Space in Linearized Seismic Reflection Tomography

Abstract

Seismic reflection tomography attempts to match traveltimes obtained from surface seismic data with the corresponding traveltimes of rays traced through a model of the subsurface. Traveltimes along rays from surface shot locations down to reflecting interfaces and then back up to surface receiver locations are used; the goal is to determine both the position of the reflectors and the slowness field above the reflectors. Seismic reflection tomography is closely related to the inversion of a limited-angle Radon transform. The author formulates a continuum version of reflection tomography analogous to the discrete version appropriate for much of surface seismic data; the continuum version models a finite cable length with positive minimum offset, where the shot and receiver spacing are small compared with the cable length. It is proven that in a continuum formulation of reflection tomography with infinite horizontal extent, linearized about a constant background, the reflector depth perturbations are uniquely determined by the traveltime perturbations. Moreover, the slowness perturbations in the null space of this linearized problem can be completely characterized: they are polynomials in the horizontal variable with coefficients which are functions of the depth variable satisfying integral orthogonality constraints.