Stochastic sampling method for Full Moment Tensor inversion and uncertainty estimation

We develop efficient stochastic samplers for full moment tensor and point force inversion and uncertainty estimation, within the framework of the open-source software package MTUQ (moment tensor uncertainty quantification). Accurate determination of full moment tensor and point force parameters is necessary for the characterization of natural and anthropogenic seismic sources, and it is also an important component of seismic tomography and nuclear monitoring. We investigate the use of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) and the Hamiltonian Monte Carlo (HMC) samplers to improve efficiency upon standard grid search approaches to cover model parameter space. The CMA-ES method has been shown to be efficient on a variety of misfit functionals, both in the context of nonlinear and non convex optimization. In each new generation of this evolutionary algorithm, the parameter space is sampled by a multivariate Gaussian distribution, whose mean, covariance matrix, and step size are deterministically adjusted according to the ranked n best-fitting samples. In the case of convex quadratic functions, these adjustments amount to learning the second-order information of the misfit function, and emulates quasi-newton methods. We have applied CMA-ES to both moment tensor and single force source estimation problems. CMA-ES converges quickly to a solution and saves computation by concentrating sampling in the low-misfit region, resulting in lower computational overhead. The HMC method relied on the gradient of the misfit function and has been successfully applied to seismic source problems. The misfit gradient information is efficiently computed from Strain Green's Function databases which are obtained by saving the strain wavefield in the vicinity of the source region as a result of unit vector forces injected at target stations. Both methods outperformed the conventional grid search method for both synthetic and field data, and they were able to obtain comparable uncertainty estimates at several orders of magnitudes lower than the cost of the grid search method. These sampling methods provide a natural framework for tackling complex inversions, such as joint moment tensor and source location inversions, which are challenging for grid search methods.