Non-linearity in Bayesian 1-D magnetotelluric inversion

Abstract

S U M M A R YThis paper applies a Bayesian approach to examine non-linearity for the 1-D magnetotelluric (MT) inverse problem. In a Bayesian formulation the posterior probability density (PPD), which combines data and prior information, is interpreted in terms of parameter estimates and uncertainties, which requires optimizing and integrating the PPD. Much work on 1-D MT inversion has been based on (approximate) linearized solutions, but more recently fully non-linear (numerical) approaches have been applied. This paper directly compares results of linearized and non-linear uncertainty estimation for 1-D MT inversion; to do so, advanced methods for both approaches are applied. In the non-linear formulation used here, numerical optimization is carried out using an adaptive-hybrid algorithm. Numerical integration applies Metropolis-Hastings sampling, rotated to a principal-component parameter space for efficient sampling of correlated parameters, and employing non-unity sampling temperatures to ensure global sampling. Since appropriate model parametrizations are generally not known a priori, both under-and overparametrized approaches are considered. For underparametrization, the Bayesian information criterion is applied to determine the number of layers consistent with the resolving power of the data. For overparametrization, prior information is included which favours simple structure in a manner similar to regularized inversion. The data variance and/or trade-off parameter regulating data and prior information are treated in several ways, including applying fixed optimal estimates (an empirical Bayesian approach) or including them as hyperparameters in the sampling (hierarchical Bayesian). The latter approach has the benefit of accounting for the uncertainty in the hyperparameters in estimating model parameter uncertainties. Non-linear and linearized inversion results are compared for synthetic test cases and for the measured COPROD1 MT data by considering marginal probability distributions and marginal profiles. In some cases, important differences are indicated, including poorer sensitivity to thin and/or low-conductivity layers for linearized inversion, and multimodal PPDs which cannot be addressed within a linearized approach.

Thermoelectric properties of β-As, Sb and Bi monolayers

“…The observably enhanced effects on Seebeck coefficient can be observed in monolayers WX 2 (X=S, Se and Te), and detrimental effects in MX 2 (M=Zr, Hf, Mo and Pt; X=S, Se and Te). For bulk materials, the detrimental effects on Seebeck coefficient caused by SOC also can be found in Mg 2 X (X = Si, Ge, Sn) 50,51 and half-Heusler ANiB (A = Ti, Hf, Sc, Y; B = Sn, Sb, Bi) 52 . Therefore, including SOC is very important for electronic transport coefficients of β-As, Sb and Bi monolayers.…”

Trans‐dimensional Bayesian inversion of controlled‐source electromagnetic data in the German North Sea

“…sophisticated inversion techniques or additional structural constraints to estimate model parameters. For example, smoothness regularization has been included by Minsley (2011), and the regularization weight has been sampled rather than specified by Rosas-Carbajal et al (2013) and Guo et al (2011). Linearized 1-D inversions for both time-and frequency-domain data have been carried out extensively and successfully in the past (e.g., Schwalenberg et al 2010;Scholl 2010;Key 2009).…”

“…Bayesian inversion and regularization have been combined in past work by defining the prior accordingly and carefully adjusting the acceptance criteria. For example, smoothness regularization has been included by Minsley (2011), and the regularization weight has been sampled rather than specified by Rosas-Carbajal et al (2013) and Guo et al (2011). Other implementations of Bayesian algorithms for marine CSEM in fixed dimensions either constrain the subsurface resistivity layering using depths inferred from seismic data such as hydrocarbon reservoir depth (Chen et al 2007), severely constrain the prior parameter widths (Buland and Kolbjørnsen 2012), or use the Bayesian information criterion to estimate the most probable number of sub-seafloor layers, which is held fixed in the inversion (Gehrmann et al 2015).…”

Transdimensional change-point modeling as a tool to investigate uncertainty in applied geophysical inference: An example using borehole geophysical logs

“…In contrast, much of the research into demonstration algorithms addresses the nonuniqueness of models constrained by the available data as the dominant theme (Stoffa and Sen, 1991;Sen and Stoffa, 1992;Yamanaka and Ishida, 1996;Sambridge, 1999). Some algorithms also enable the investigation of uncertainty as part of the exploration of the parameter space Guo et al, 2011;Bodin et al, 2012b).…”