In ray theoretic traveltime tomography, the solution of a linear system of equations is the heart of the problem. By solving this linear system, variations in the traveltimes are transformed into variations in the model parameters. This transformation from variations in the data to variations in the model depends on the properties of the matrix that describes the linear system and singular value decomposition (SVD) is the tool to study such properties.
SVD has been applied in the past to study the structure of the matrices involved in tomographic traveltime inversion problems (White, 1989; Bregman et al., 1989; Pratt and Chapman, 1990). In these studies the results of the SVD have been only partially reported because no reference has been made to the properties of the singular vectors in data space.
In this paper, I present the complete results of the SVD of the matrices that result after the following three types of parametrization: (a) in regions of constant slowness (McMechan, 1983), (b) in natural pixels (Michelena and Harris, 1991) and (c) in homogeneous anisotropic regions with elliptical velocity dependencies (Michelena and Muir, 1991).
By studying the structure of the corresponding matrices from the SVD results, it is possible to gain some insight on how iterative techniques such as conjugate gradients resolve the data and the model when used to solve tomographic inversions.