Seismic Velocity Field Estimation — Strategies for Large-scale Nonlinear Inverse Problems

The estimation of the seismic velocity field in two or three dimensions, by modelling the travel times of particular seismic phases or by matching observed and computed seismograms, represents a large-scale nonlinear inverse problem. The solution can be obtained by determining the minimum of a misfit function between observations and theoretical predictions, subject to some regularisation conditions on the behaviour of the model parameters. The minimisation can be achieved without the inversion of large matrices by using a search scheme based on the local properties of the misfit function. At each step in the iterative process, a subspace of a small number of directions is constructed in model space and then the minimum sought in a quadratic approximation on this set. At least two directions are required for rapid convergence. This approach is very suitable when the model parameters are of different types, since partitioning by parameter class avoids dependence on scaling.

If the model is to remain close to a reference then the regularisation term is particularly important and different types of a priori information (e.g. geological) can be introduced via the character of this term. When fit-to-data is emphasised there is the chance of finding features suppressed in a more conservative approach, but at the risk of introducing spurious detail.