@article{VasyuraBathkeDettmerDuttaetal.2021,
  author    = {Vasyura-Bathke, Hannes and Dettmer, Jan and Dutta, Rishabh and Mai, Paul Martin and J{\´o}nsson, Sigurj{\´o}n},
  title     = {Accounting for theory errors with empirical Bayesian noise models in nonlinear centroid moment tensor estimation},
  series = {Geophysical journal international / the Royal Astronomical Society, the Deutsche Geophysikalische Gesellschaft and the European Geophysical Society},
  volume    = {225},
  journal   = {Geophysical journal international / the Royal Astronomical Society, the Deutsche Geophysikalische Gesellschaft and the European Geophysical Society},
  number    = {2},
  publisher = {Oxford University Press},
  address   = {Oxford},
  issn      = {0956-540X},
  doi       = {10.1093/gji/ggab034},
  pages     = {1412 -- 1431},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}
@article{DuttaJonssonVasyuraBathke2021,
  author    = {Dutta, Rishabh and J{\´o}nsson, Sigurj{\´o}n and Vasyura-Bathke, Hannes},
  title     = {Simultaneous Bayesian estimation of non-planar fault geometry and spatially-variable slip},
  series = {JGR / AGU, American Geophysical Union : Solid earth},
  volume    = {126},
  journal   = {JGR / AGU, American Geophysical Union : Solid earth},
  number    = {7},
  publisher = {Wiley},
  address   = {Hoboken, NJ},
  issn      = {2169-9313},
  doi       = {10.1029/2020JB020441},
  pages     = {28},
  year      = {2021},
  abstract  = {Large earthquakes are usually modeled with simple planar fault surfaces or a combination of several planar fault segments. However, in general, earthquakes occur on faults that are non-planar and exhibit significant geometrical variations in both the along-strike and down-dip directions at all spatial scales. Mapping of surface fault ruptures and high-resolution geodetic observations are increasingly revealing complex fault geometries near the surface and accurate locations of aftershocks often indicate geometrical complexities at depth. With better geodetic data and observations of fault ruptures, more details of complex fault geometries can be estimated resulting in more realistic fault models of large earthquakes. To address this topic, we here parametrize non-planar fault geometries with a set of polynomial parameters that allow for both along-strike and down-dip variations in the fault geometry. Our methodology uses Bayesian inference to estimate the non-planar fault parameters from geodetic data, yielding an ensemble of plausible models that characterize the uncertainties of the non-planar fault geometry and the fault slip. The method is demonstrated using synthetic tests considering slip spatially distributed on a single continuous finite non-planar fault surface with varying dip and strike angles both in the down-dip and along-strike directions. The results show that fault-slip estimations can be biased when a simple planar fault geometry is assumed in presence of significant non-planar geometrical variations. Our method can help to model earthquake fault sources in a more realistic way and may be extended to include multiple non-planar fault segments or other geometrical fault complexities.},
  language  = {en}
}