Illustrating the trade-off between velocity and reflector position in travel-time inversion by using a two-dimensional subspace

Abstract

Inversion of surface seismic reflection travel times for both the velocity distribution above a reflecting horizon, and the reflector position, is known to suffer from problems. These include non-uniqueness, poor resolution and ambiguity between velocity and reflector position. Often this inverse problem is treated by using a minimization technique, where the objective is to minimize the difference between modelled and observed travel times. Gradient-based methods such as Steepest Descent and Conjugate Gradients often are used to produce a model that best fits the observed data in some sense. These techniques use local gradient information of the objective function that measures the goodness-of-fit between observed and modelled travel times. One way of avoiding problems associated with differences in physical dimensions in gradient-based optimization methods is to use a Petrov-Galerkin, or Subspace, method (Saad and Schultz, 1985; Kennett and Williamson, 1988), in which the gradient vector is split into two orthogonal component vectors, producing a two-dimensional subspace. The first component is the gradient vector associated with the slowness parameters, while the second is the gradient vector associated with the reflector position parameters. After making this decomposition, the minimum is sought in the subspace spanned by these two vectors rather than searching for the minimum along the combined gradient direction alone. By producing contours of the objective function within the two-dimensional subspace described above, insight can be gained into some of the problems faced by travel-time inversion. The trade-off between velocity and reflector position is indicated by the trend of the contours. Non-unique solutions are indicated by the fact that a broad range of earth models can produce an equally good fit to the data. They also highlight the need for reasonable starting models, which often are not available in practice, as these significantly influence the final model. In this paper, brief details of Petrov-Galerkin methods for optimization are given. In addition, objective-function contours, within the two-dimensional subspace described above, are produced for a synthetic travel-time inversion problem.