Tomographic traveltime inversion using natural pixels

Abstract

The problem consists of determining the unknown coefficients an from the measured traveltimes. Once these coefficients have been calculated, the computation of the sum (1) is straightforward.The representation (1) has two important degrees of freedom that influence decisively the kind of results obtained. These are the number (M) and kind of functions I3n(r) to be used. The most common choice for the functions f3n(r) is orthogonal cells (square or cubic pixels), and in that case the coefficients an represent the slowness within each cell (MeMeehan, 1983;Ivansson, 1985). Although this is the most popular basis function for estimating the slowness model, others have been suggested recently. Harlan (1989) defines the velocity function as a sum of smooth basis functions (Gaussians), and Van Trier (1988) defines the functions f3n(r) as cubic B-splines multiplied by functions that reproduce the expected structure of the model. The number of functions M is also arbitrary but is usually small to avoid having to solve a huge system of equations.The kind and number of functions used for expanding the slowness model determine many of the general features of the final image. With the same data set it is possible to obtain different results just because different parameterizations have been used. However, the goal is to obtain a reconstructed model free from these artifacts of parameterization, which means the selection of the basis function is critical in the inversion process and thus should be considered more carefully, as described below.There are no general criteria for deciding which representation is the best, although some may have clear advantages for solving specific problems. Our selection of the basis function will be based on minimization of the expression that estimates the norm of the null space of the problem ABSTRACT Traditionally in the problem of tomographic traveltime inversion, the model is divided into a number of rectangular cells of constant slowness. Inversion consists of finding these constant values using the measured traveltimes. The inversion process can demand a large computational effort if a high-resolution result is desired.We show how to use a different kind of parameterization of the model based on beam propagation paths. This parameterization is obtained within the framework of reconstruction in Hilbert spaces by minimizing the error between the true model and the estimated model. The traveltimes are interpreted as the projections of the slowness along the beampaths. Although the actual beampaths are described by complicated spatial functions, we simplify the computations by approximating these functions with functions of constant width and height, i.e., "fat" rays, which collectively form a basis set of natural pixels.With a simple numerical example we demonstrate that the main advantage of this parameterization. compared with the traditional decomposition of the model in rectangular pixels, is that 2-D reconstructed images of similar quality can be obtained with considerably less computat...

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