Research Outputs
Characterizing the spatial correlation of coseismic slip distributions: a data driven Bayesian approach
The spatial correlation of coseismic slip is a necessary input for generating stochastic seismic rupture models, which are commonly used in seismic and tsunami hazard assessments. To date, the spatial correlation of individual earthquakes is characterized using finite fault models by finding the combination of parameters of a von KĂ¡rmĂ¡n autocorrelation function that best fits the observed autocorrelation function of the finite fault model. However, because a priori spatial correlation conditions (i.e. not in the data) are generally applied in finite fault model generation, the results obtained using this method may be biased. Additionally, robust uncertainty estimates for spatial correlations of coseismic slip are generally not performed. Considering these limitations in the classic method, here, a method is developed based on a Bayesian formulation of Finite Fault Inversion (FFI) with positivity constraints. This method allows for characterizing the spatial correlation of coseismic slip and its uncertainties for an earthquake by using samples of coseismic slip from a posterior probability density function (PDF). Furthermore, a Bayesian model selection criterion called Akaike Bayesian Information Criterion (ABIC) is applied to objectively choose between different prior spatial correlation schemes before computing the posterior, to reduce subjectivity due to this prior condition. The ABIC is calculated using an approximate analytical expression of Bayesian evidence. The method is applied to simulated P waves, demonstrating that model selection allows for objectively estimating the most suitable prior spatial correlation scheme in FFI. Additionally, the target spatial correlation of coseismic slip is accurately recovered using samples from the posterior PDF, as well as their uncertainties. Moreover, in the simulated experiment, it is shown that a non-robust choice of the prior spatial correlation scheme can significantly bias the estimated spatial correlations of coseismic slip. We apply our method to observed P waves from the 2015, Illapel earthquake ($M_{\rm w} = 8.3$), finding that the spatial correlation of coseismic slip of this earthquake is better described by a von KĂ¡rmĂ¡n ACF, with mean correlation lengths of around 47 km and Hurst parameter of 0.58. We conclude that using our method reduces biases associated with prior spatial correlation conditions and allows for robust estimation of spatial correlations of coseismic slip and their uncertainties.
Efficient Bayesian uncertainty estimation in linear finite fault inversion with positivity constraints by employing a log-normal prior
Obtaining slip distributions for earthquakes results in an ill-posed inverse problem. While this implies that only limited and uncertain information can be recovered from the data, inferences are typically made based only on a single regularized model. Here, we develop an inversion approach that can quantify uncertainties in a Bayesian probabilistic framework for the finite fault inversion (FFI) problem. The approach is suitably efficient for rapid source characterization and includes positivity constraints for model parameters, a common practice in FFI, via coordinate transformation to logarithmic space. The resulting inverse problem is nonlinear and the most probable solution can be obtained by iterative linearization. In addition, model uncertainties are quantified by approximating the posterior probability distribution by a Gaussian distribution in logarithmic space. This procedure is straightforward since an analytic expression for the Hessian of the objective function is obtained. In addition to positivity, we apply smoothness regularization to the model in logarithmic space. Simulations based on surface wave data show that smoothing in logarithmic space penalizes abrupt slip changes less than smoothing in linear space. Even so, the main slip features of models that are smooth in linear space are recovered well with logarithmic smoothing. Our synthetic experiments also show that, for the data set we consider, uncertainty is low at the shallow portion of the fault and increases with depth. In addition, a simulation with a large station azimuthal gap of 180° significantly increases the slip uncertainties. Further, the marginal posterior probabilities obtained from our approximate method are compared with numerical Markov Chain Monte Carlo sampling. We conclude that the Gaussian approximation is reasonable and meaningful inferences can be obtained from it. Finally, we apply the new approach to observed surface wave records from the great Illapel earthquake (Chile, 2015, Mw = 8.3). The location and amplitude of our inferred peak slip is consistent with other published solutions but the spatial slip distribution is more compact, likely because of the logarithmic regularization. We also find a minor slip patch downdip, mainly in an oblique direction, which is poorly resolved compared to the main slip patch and may be an artefact. We conclude that quantifying uncertainties of finite slip models is crucial for their meaningful interpretation, and therefore rapid uncertainty quantification can be critical if such models are to be used for emergency response.