Local three-dimensional earthquake tomography by trans-dimensional Monte Carlo sampling

Local earthquake tomography is a non-linear and non-unique inverse problem that uses event arrival times to solve for the spatial distribution of elastic properties. The typical approach is to apply iterative linearization and derive a preferred solution, but such solutions are biased by a number of subjective choices: the starting model that is iteratively adjusted, the degree of regularization used to obtain a smooth solution, and the assumed noise level in the arrival time data. These subjective choices also affect the estimation of the uncertainties in the inverted parameters. The method presented here is developed in a Bayesian framework where a priori information and measurements are combined to define a posterior probability density of the parameters of interest: elastic properties in a subsurface 3-D model, hypocentre coordinates and noise level in the data.We apply a trans-dimensional Markov chain Monte Carlo algorithm that asymptotically samples the posterior distribution of the investigated parameters. This approach allows us to overcome the issues raised above. First, starting a number of sampling chains from random samples of the prior probability distribution lessens the dependence of the solution from the starting point. Secondly, the number of elastic parameters in the 3-D subsurface model is one of the unknowns in the inversion, and the parsimony of Bayesian inference ensures that the degree of detail in the solution is controlled by the information in the data, given realistic assumptions for the error statistics. Finally, the noise level in the data, which controls the uncertainties of the solution, is also one of the inverted parameters, providing a first-order estimate of the data errors. We apply our method to both synthetic and field arrival time data. The synthetic data inversion successfully recovers velocity anomalies, hypocentre coordinates and the level of noise in the data. The Bayesian inversion of field measurements gives results comparable to those obtained independently by linearized inversion, reconstructing the geometry of the main seismic velocity anomalies. The quantification of the posterior uncertainties, a crucial output of Bayesian inversion, allows for visualizing regions where elastic properties are closely constrained by the data and is used here to directly compare our results to the ones obtained with the linearized inversion. In the case we examined the results of two inversion techniques are not significantly different.