Bayesian first arrival travel time tomography by interacting MCMC
Résumé
First arrival travel time tomography aims at estimating the velocity field of the subsurface. The resulting velocity field is then commonly used as a starting point for further seismological, mineralogical or tectonic analysis in a wide range of applications such as geothermal energy, volcanoes studies,... The estimated velocity field is obtained through inverse modeling by minimizing an objective function that compares observed and modeled travel times. This step is usually performed by steepest descent optimization algorithms. The major flaw of such optimization schemes, beyond the possibility of staying stuck in a local minima, is that they do not account for the multiple possible solutions of the inverse problem at stake. Therefore, they are unable to assess the uncertainties linked to the solution. In a Bayesian perspective however, the inverse problem can be seen as a conditional simulation problem, that can be solved using iterative algorithms. Indeed, Markov chains Monte-Carlo (MCMC) methods are known to produce samples of virtually any distribution. They have already been widely used in the resolution of non-linear inverse problems where no analytical expression for the forward relation between data and model parameters is available, and where linearization is unsuccessful. In Bayesian inversion, the total number of simulations we can afford is highly related to the computational cost of the forward model. Although fast algorithms have been recently developed for computing first arrival travel time of seismic waves, the complete browsing of the posterior distribution is hardly performed at final time, especially when it is high dimensional and/or multimodal. In the latter case, the chain may stay stuck in one of the modes. One way to improve the mixing properties of classical single MCMC is by making interact several Markov chains at different temperatures. These methods can make efficient use of large CPU clusters, without increasing the global computational cost with respect to classical MCMC and are therefore particularly suited for Bayesian inversion. The approach is illustrated on two geophysical case studies.