10:45 AM - 11:00 AM
[S13-2-02] A Bayesian hierarchical model for a seismic source inversion
A source inversion analysis using spatio-temporal displacement field data is able to be formulated as a discrete linear inverse problem when the Green function and the source fault are known. However, it is difficult to calculate an accurate Green function. Previous studies approximated the effects of the uncertainty of the Green function by introducing a new correlated or uncorrelated error term, which is added to data. The approximation fails to capture important characteristics such as peak shift and heavy tails of the likelihood function under the uncertainty of the Green function.
We propose a hierarchical Bayes model for a multi-data analysis with the multi-time-window finite fault parameterization. In the model, a Green function is treated as a realization of a random variable G. The marginalization of the hyperparameters, which control the prior distribution of the model parameters and the observation errors, is approximated by plugging in of the maximum a posteriori hyperparameters given G. The marginal likelihood function for G is approximated by the Laplace approximation. The marginalization of G is approximated by a Monte-Carlo integration method.
We applied the method to synthetic data. We set a 1-D velocity structure at the source region. We drew thousands of velocity structure samples from the prior distribution of the velocity structure, and then calculated the corresponding Green functions. For reference, we also conducted a conventional inversion, which used only a reference velocity structure. We found that the conventional inversion result suffers from artifacts especially at the later shallow part of the rupture process, while the result with the proposed method does not suffer from the artifact. Note that the mitigation of the artifact was not possible with the simple mean-of-the-posterior-mean approach. We also found increase of the variance of the posterior distribution of the potency due to the marginalization of G.
We propose a hierarchical Bayes model for a multi-data analysis with the multi-time-window finite fault parameterization. In the model, a Green function is treated as a realization of a random variable G. The marginalization of the hyperparameters, which control the prior distribution of the model parameters and the observation errors, is approximated by plugging in of the maximum a posteriori hyperparameters given G. The marginal likelihood function for G is approximated by the Laplace approximation. The marginalization of G is approximated by a Monte-Carlo integration method.
We applied the method to synthetic data. We set a 1-D velocity structure at the source region. We drew thousands of velocity structure samples from the prior distribution of the velocity structure, and then calculated the corresponding Green functions. For reference, we also conducted a conventional inversion, which used only a reference velocity structure. We found that the conventional inversion result suffers from artifacts especially at the later shallow part of the rupture process, while the result with the proposed method does not suffer from the artifact. Note that the mitigation of the artifact was not possible with the simple mean-of-the-posterior-mean approach. We also found increase of the variance of the posterior distribution of the potency due to the marginalization of G.