Optimizing model parameterization in 2D linearized seismic traveltime tomography

Abstract

In seismic traveltime tomography, a set of linearized equations is solved for the unknown slowness perturbations. The matrix of this set of equations (the tomographic matrix) is usually ill-conditioned, because the model space contains more details than can be resolved using the available data. The condition of the tomographic matrix can be characterized by its singular value spectrum: the larger the normalized singular values and the greater the rank of the matrix, the smaller the null space and the better the condition of the inversion problem. The structure of the tomographic matrix depends on the source–receiver configuration and the model parameterization. Since the source–receiver geometry is often fixed (e.g. in earthquake tomography), conditioning can only be influenced through the model parameterization, i.e. the structure of the model space. In this paper, we demonstrate a method for finding an optimal, irregular triangular cell parameterization that best suits the raypath geometry. We define a practical cost function, whose minimization is equivalent to the minimization of the null space of the tomographic matrix. Since the cost function depends on the model discretization in a highly nonlinear manner, a simulated annealing algorithm is used to find the optimal parameterization. We show that the cost value for the optimal triangulated model is about two times smaller than that for the regular gridded model with the same dimension, resulting in more accurate and reliable inversion results. The method is demonstrated through some cross–borehole tomographic examples with given acquisition geometries.