Use of Machine Learning to Estimate Statistics of the Posterior Distribution in Probabilistic Inverse Problems—An Application to Airborne EM Data

Abstract

A key challenge in geoscience is that of combining different kinds of geo-information into one geo-model, typically describing the subsurface. This information can be direct information about geological processes, and spatial variability, or it can be indirect information from measurements of properties related to the subsurface, such as geophysical data. Ideally, when such a geo-model has been established, one should be able to quantify information about specific features related to the geo-model, consistent with all information.This integration of geo-information is typically solved using inverse problem theory (Menke, 2012;Tarantola & Valette, 1982a). Fast deterministic methods exist and have been widely used. For such methods, the goal is to obtain one optimal model, such as the simplest possible model, consistent with available information, typically in the form of observed data (Constable et al., 1987;Menke, 2012;Tikhonov, 1963). In practice, in part due to noise on data and model nonlinearities and imperfections, infinitely many models exist that will be consistent with data, and the deterministic approach can in general not account properly for such uncertainty.Probabilistic inversion methods can, in principle, take into account arbitrarily complex information, and integrate the information into one consistent model, as given by the posterior probability distribution. A full analytic expression of the posterior distribution is rarely possible. Instead, sampling methods can be used to generate a

Probabilistic Petrophysical Reconstruction of Danta's Alpine Peatland via Electromagnetic Induction Data

“…The forward modeling tools used in electromagnetic inverse problems are, nowadays, fully nonlinear and, despite 3D forward options could be available (Cox et al., 2012; Oldenburg et al., 2020), the use of 1D approximations is still the standard in the industry and in the academia (Bai et al., 2020; Dzikunoo et al., 2020; Hansen & Finlay, 2022; Ley‐Cooper et al., 2015; Viezzoli et al., 2008). The reason is mainly related to the computational costs of 3D modeling (Cai et al., 2022; Guillemoteau et al., 2017; Kara & Farquharson, 2023).…”

Probabilistic petrophysical reconstruction of Danta's Alpine peatland via electromagnetic induction data

“…The forward modelling tools used in electromagnetic inverse problems are, nowadays, fully nonlinear and, despite 3D forward options could be available (Cox et al, 2012;Oldenburg et al, 2020), the use of 1D approximations is still the standard in the industry and in the academia (Viezzoli et al, 2008;Ley-Cooper et al, 2015;Dzikunoo et al, 2020;Bai et al 2020;Hansen & Finlay, 2022). The reason is mainly related to the computational costs of 3D modelling (Guillemoteau et al, 2017;Cai et al, 2022;Kara & Farquharson, 2023).…”

Bayesian Inversion, Uncertainty Analysis and Interrogation Using Boosting Variational Inference

“…Since then variational methods have been applied to a variety of problems including travel time tomography (Levy, Laloy, & Linde, 2022; X. Zhang & Curtis, 2020a; X. Zhao et al., 2021), seismic denoising (Siahkoohi et al., 2021, 2023), seismic amplitude inversion (Zidan et al., 2022), earthquake hypocenter inversion (Smith et al., 2022), slip distribution inversion (Sun et al., 2023), full waveform inversion in 2D (Urozayev et al., 2022; W. Wang et al., 2023; X. Zhang & Curtis, 2020b) and in 3D (Lomas et al., 2023; X. Zhang et al., 2023), and survey or experimental design (Strutz & Curtis, 2024). In addition, various types of neural networks produce probabilistic outputs and can be considered variational methods (Bishop, 1994), and these have been applied to subsurface imaging problems for more than two decades (Bloem et al., 2023; Cao et al., 2020; Devilee et al., 1999; de Wit et al., 2013; Earp & Curtis, 2020; Earp et al., 2020; Hansen & Finlay, 2022; Käufl et al., 2014, 2016; Lubo‐Robles et al., 2021; Meier et al., 2007a, 2007b; A. K. Ray & Biswal, 2010; Shahraeeni & Curtis, 2011; Shahraeeni et al., 2012; Siahkoohi et al., 2022; X. Zhang & Curtis, 2021b; X. Zhao et al., 2021). Interestingly, X. Zhang et al.…”