Nonlinear Traveltime Tomography

By regularizing the tomography problem in such a manner, we obtain a stable, unique solution of the nonlinear tomography problem for transmission surveys, including the cases of completely surrounding ray coverage, VSP surveying in laterally homogeneous media, and cross-borehole experiments with accompanying slowness logs. We demonstrate stability by showing that the discrete solutions approach continuum solutions at fine discretization. We also show that the same solution can be obtained from different starting models. This implies that physically meaningful reconstructions can be made in the absence of good prior information.

Computationally, we solve the nonlinear inversion by way of the Conjugate Gradient method, employing fast and efficient traveltime/ray modeling to refine the propagation paths in keeping with model changes. We show that this method may be straightforwardly implemented on a parallel computer to solve large tomography problems almost interactively. The implementation involves decomposing source positions among the parallel processors and performing survey-wide traveltime modeling at the same cost as serially computing traveltimes from a single source. ``Eikonal'' modeling is used to propagate first-arrival traveltimes from the source to all receivers simultaneously. On medium-grained (100-1000 processor) parallel computers, we can apply the method readily to typical field surveys. The same inversion algorithm may also be applied to the linearized inversion problem with improved performance. As the nonlinear inversion converges toward a solution, the algorithm behaves more and more like linearized inversion.

While optimality of the reconstruction may be cast in terms of how well the data are fit, suitability of the solution is better judged by way of uncertainty analysis. We perform uncertainty analysis to quantify the resolution and variance at various positions in the model. We appeal to Backus and Gilbert's notions of the spread-vs.-variance trade-off, but within a Bayesian framework whereby both variance and resolution are inferred from the posterior covariance associated with our reconstruction. For computational efficiency we calculate posterior covariance by linearizing about the reconstructed model obtained from the nonlinear inversion. We compute the posterior covariance using a Monte Carlo algorithm in which many realizations of perturbed data and model roughnesses are inverted. Finally we demonstrate, for field data examples as well as numerical problems, that the method gives resolution and variance information that behaves correctly with regard to the ray coverage and the character of the associated reconstruction.

We apply nonlinear traveltime to two data sets: a multiple offset VSP survey shot near Larderello, Italy, and a cross-well survey conducted by MIT in the Michigan Basin. In the first case, we show that the uncertainty analysis may be applied to determine the validity of information obtained in a tomogram. In the case of the Michigan survey, we demonstrate the robustness of the inversion in reconstructing subsurface structure from slowness log and first arrival traveltime data in the absence of a good prior model. We apply the uncertainty analysis to characterize the resolution and variance of these results as well.