Non-Linear Reflection Tomography

Abstract

Reflection tomography, the determination of velocity distribution and reflector position from reflection travel-time data, is a very non-linear inverse problem. Unlike in transmission tomography, ray paths have to be iteratively updated, because travel-time variations cannot be computed by integration of slowness along the original unperturbed ray paths. From a study of parameter sensitivity we conclude that in seismic reflection experiments the vertical variation of slowness inside layers is poorly resolved from travel-time data. For this reason, in each layer the slowness was modelled with functions varying only in the horizontal direction. A B-spline representation is adopted for lateral velocity heterogeneity and interfaces. These splines are local and well adapted for tomography because the spline parameters have a geometrical interpretation and they may be explicitly used as unknowns in the inverse problem. For each iteration, and for every source-receiver pair, two-point ray tracing was performed by paraxial ray tracing, and the inverse problem was solved by iterative least-squares. A priori data, necessary to stabilize the inverse problem, were introduced by a penalty function approach. This method is equivalent to using a priori convariance matrices, but it has a simpler physical interpretation and is faster to use. Damping was used to improve the convergence. The method was first tested in the inversion of synthetic data. These synthetic examples illustrate the limitations of reflection tomography: non-linearity effects, poor vertical resolution of the velocity, and decrease of the resolution with the ratio of maximum offset to interface depth. Finally, we inverted a data set from the Paris Basin. The inversion method reduces the residual norm to 6 ms, which is less than the expected error on the data.