Accounting for theory errors with empirical Bayesian noise models in nonlinear centroid moment tensor estimation

Centroid moment tensor (CMT) parameters can be estimated from seismic waveforms. Since these data indirectly observe the deformation process, CMTs are inferred as solutions to inverse problems which are generally underdetermined and require significant assumptions, including assumptions about data noise. Broadly speaking, we consider noise to include both theory and measurement errors, where theory errors are due to assumptions in the inverse problem and measurement errors are caused by the measurement process. While data errors are routinely included in parameter estimation for full CMTs, less attention has been paid to theory errors related to velocity-model uncertainties and how these affect the resulting moment-tensor (MT) uncertainties. Therefore, rigorous uncertainty quantification for CMTs may require theory-error estimation which becomes a problem of specifying noise models. Various noise models have been proposed, and these rely on several assumptions. All approaches quantify theory errors by estimating the covariance matrix of data residuals. However, this estimation can be based on explicit modelling, empirical estimation and/or ignore or include covariances. We quantitatively compare several approaches by presenting parameter and uncertainty estimates in nonlinear full CMT estimation for several simulated data sets and regional field data of the M-1 4.4, 2015 June 13 Fox Creek, Canada, event. While our main focus is at regional distances, the tested approaches are general and implemented for arbitrary source model choice. These include known or unknown centroid locations, full MTs, deviatoric MTs and double-couple MTs. We demonstrate that velocity-model uncertainties can profoundly affect parameter estimation and that their inclusion leads to more realistic parameter uncertainty quantification. However, not all approaches perform equally well. Including theory errors by estimating non-stationary (non-Toeplitz) error covariance matrices via iterative schemes during Monte Carlo sampling performs best and is computationally most efficient. In general, including velocity-model uncertainties is most important in cases where velocity structure is poorly known.

Automated summary

This study focuses on estimating centroid moment tensor (CMT) parameters from seismic waveforms, highlighting the importance of incorporating theory errors, especially related to velocity-model uncertainties, in parameter estimation to improve the realism of parameter uncertainty quantification. Various noise models and estimation approaches are compared, with the iterative non-stationary error covariance matrix estimation method showing the best performance and computational efficiency. The inclusion of velocity-model uncertainties is particularly crucial in cases where the velocity structure is poorly known.