Outlier-insensitive Bayesian inference for linear inverse problems (OutIBI) with applications to space geodetic data

Abstract

SUMMARY Inverse problems play a central role in data analysis across the fields of science. Many techniques and algorithms provide parameter estimation including the best-fitting model and the parameters statistics. Here, we concern ourselves with the robustness of parameter estimation under constraints, with the focus on assimilation of noisy data with potential outliers, a situation all too familiar in Earth science, particularly in analysis of remote-sensing data. We assume a linear, or linearized, forward model relating the model parameters to multiple data sets with a priori unknown uncertainties that are left to be characterized. This is relevant for global navigation satellite system and synthetic aperture radar data that involve intricate processing for which uncertainty estimation is not available. The model is constrained by additional equalities and inequalities resulting from the physics of the problem, but the weights of equalities are unknown. We formulate the problem from a Bayesian perspective with non-informative priors. The posterior distribution of the model parameters, weights and outliers conditioned on the observations are then inferred via Gibbs sampling. We demonstrate the practical utility of the method based on a set of challenging inverse problems with both synthetic and real space-geodetic data associated with earthquakes and nuclear explosions. We provide the associated computer codes and expect the approach to be of practical interest for a wide range of applications.

Mantle flow distribution beneath the California margin

“…Inversion method and resolution. Imaging of localized and distributed deformation can be cast as an inverse problem (see Methods section) where plastic strain-rate in volume elements and slip velocity on surface elements are related to surface GPS velocity vectors by Green's functions [52][53][54][55][56] . The formulation enforces that all the strain tensor components derive from a single-valued velocity field with conservation of angular and linear momentum in a half space 49,50 .…”

Bayesian Inversion for a Stress‐Driven Model of Afterslip and Viscoelastic Relaxation: Method and Application to Postseismic Deformation Following the 2011 MW 9.0 Tohoku‐Oki Earthquake

“…is satisfied for all i = 1, …, M. An alternative choice of the prior PDF that represents a priori ignorance of the model parameters is the Jeffreys prior (Hang et al, 2020;Jeffreys, 1939). Here we arbitrarily employ the uniform prior PDF (Equation 13) because we find that the choice of either the uniform or Jeffreys prior have a negligible influence on the results presented in this paper.…”

“…It is also well known that kinematic slip inversions are highly nonunique because of their large degrees of freedom and the limited spatial and temporal coverage of observations. Another kinematic approach is to simultaneously invert for afterslip and viscoelastic flow (Hang et al., 2020; Moore et al., 2017; Qiu et al., 2018; Tang et al., 2019; Tsang et al., 2016; Weiss et al., 2019) using the Green's functions for distributed anelastic deformation (Barbot, 2018b; Barbot et al., 2017).…”

Static Source Properties of Slow and Fast Earthquakes

“…Inversion of geodetic data for the spatial distribution of slip on a fault is also subject to fundamental limitations, notably due to the St. Venant principle that implies a decreasing resolution with increasing distance between source and observations. However, the deployment of increasingly large and dense geodetic observatories, the development of better analytic standards in inverse theory (Aster et al, 2012; Funning et al, 2014; Fukahata & Wright, 2008; Hang et al, 2020; Nocquet, 2018; Yabuki & Matsu'ura, 1992), and the joint inversion of complementary data sets, both geodetic and seismological, has increased the accuracy of slip distributions (Atzori & Antonioli, 2011; Amey et al, 2018; Barbot et al, 2013; Duputel et al, 2014; DeVries et al, 2017; Evans & Meade, 2012; Gombert et al, 2017, 2018; McGuire & Segall, 2003; Minson et al, 2014; Sathiakumar et al, 2017). For example, the large uncertainties associated with shallow slip near the trench during the 2011 Mw = 9.1 Tohoku, Japan, earthquake were largely reduced by considering tsunami data (e.g., Bletery et al, 2014; Jiang & Simons, 2016; Yamazaki et al, 2011).…”