Hierarchical Bayesian Inversion of Global Variables and Large‐Scale Spatial Fields

Wang, Lijing; Kitanidis, Peter K.; Caers, Jef

doi:10.1029/2021wr031610

Cited by 9 publications

(7 citation statements)

References 68 publications

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“…Recently, Wang et al (2022) proposed an hierarchical Bayesian inversion approach targeting first so-called global variables (such as hyperparameters but also physical variables) and then estimating the posterior of the whole field. For the estimation of the global variable's posterior in a non-linear setting, Wang et al (2022) apply a machine-learning based approach and train a neural network to output the global variables given a data realization followed by kernel density estimation of the results. Such a method relies on the ability to estimate the hyperparameters by brute-force prior sampling and subsequent comparison of the resulting data with the true measurements.…”

Section: Discussionmentioning

confidence: 99%

“…Zhao and Luo (2021) applied an iterative approach based on principal components which is updating biased or unknown hyperparameters while solving a non-linear inversion problem. Recently, Wang et al (2022) proposed an hierarchical Bayesian inversion targeting first global variables (such as hyperparameters but also physical variables) and later the posterior of the whole field (referred to as spatial variables). Note that none of these studies focus on inferring the hyperparameters only.…”

Section: Introductionmentioning

confidence: 99%

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Inference of (geostatistical) hyperparameters with the correlated pseudo-marginal method

Friedli¹,

Linde²,

Ginsbourger

et al. 2022

Preprint

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<p>We consider non-linear Bayesian inversion problems to infer the (geostatistical) hyperparameters of a random field describing (hydro)geological or geophysical properties by inversion of hydrogeological or geophysical data. This problem is of particular importance in the non-ergodic setting as no analytical upscaling relationships exist linking the data (resulting from a specific field realization) to the hyperparameters specifying the spatial distribution of the underlying random field (e.g., mean, standard deviation, and integral scales). Jointly inferring the hyperparameters and the "true" realization of the field (typically involving many thousands of unknowns) brings important computational challenges, such that in practice, simplifying model assumptions (such as homogeneity or ergodicity) are made. To prevent the errors resulting from such simplified assumptions while circumventing the burden of high-dimensional full inversions, we use a pseudo-marginal Metropolis-Hastings algorithm that treats the random field as a latent variable. In this random effect model, the intractable likelihood of observing the hyperparameters given the data is estimated by Monte Carlo averaging over realizations of the random field. To increase the efficiency of the method, low-variance approximations of the likelihood ratio are ensured by correlating the samples used in the proposed and current steps of the Markov chain and by using importance sampling. We assess the performance of this correlated pseudo-marginal method to the problem of inferring the hyperparameters of fracture aperture fields using borehole ground-penetrating radar (GPR) reflection data. We demonstrate that the correlated pseudo-marginal method bypasses the computational challenges of a very high-dimensional target space while avoiding the strong bias and too low uncertainty ranges obtained when employing simplified model assumptions. These advantages also apply when using the posterior of the hyperparameters describing the aperture field to predict its effective hydraulic transmissivity.</p>

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inference of (geostatistical) hyperparameters with the correlated pseudo-marginal method

Friedli¹,

Linde²,

Ginsbourger

et al. 2022

Preprint

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show abstract

“…After 𝑓 S ! "# and 𝑓 S $+#, are sampled, the densities can be estimated using a variety of methods, including kernel density estimation [35], mixture density networks [36], or normalizing flows [21,34].…”

Section: Dimension Reduction Density Estimation and Sample Weightingmentioning

confidence: 99%

Formulating and Solving the Data-Consistent Geophysical Inverse Problem for Subsurface Modeling Applications

Miltenberger¹,

Wang²,

Mukerji³

et al. 2023

Preprint

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Today, most probabilistic geophysical inverse problems are formulated using one of two methods: (1) conditional probability and Bayes’ Theorem, or (2) Tarantola’s theory of intersecting probability densities. More recently, a third inverse problem formulation based on pushforward probability measures was proposed, termed “data-consistent inversion”. Many practical problems can be cast into any of these three formulations, but the choice of formulation may change the posterior. To help geophysicists understand how the formulation affects the posterior, we present a mathematical comparison of the three formulations accompanied by simple physical example to illustrate the difference between posteriors from each method. The distinguishing feature of the data-consistent formulation is that it yields a posterior that maps onto the measurement distribution, whereas the traditional formulations do not. Then we propose a flexible Sequential Monte Carlo algorithm for solving practical subsurface modeling problems conditioned to geophysical data. This algorithm uses density estimation techniques to fit a generative probabilistic model to the posterior. The algorithm is demonstrated using a synthetic gravity inversion example. The results are then compared to results from the classical Bayesian formulation solved using Markov Chain Monte Carlo. The comparison shows that even though the difference between the data-consistent posterior and Bayesian posterior can be small in practice, the lack of conditional probabilities in the data-consistent formulation can make it simpler to fit generative probabilistic models to the solution of the inverse problem.

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“…However, PCA may not be able to capture non-linear structures in the data. It has been extensively applied across fields and decades, including research on water resources, e.g., Diaz et al (1968); Haan and Allen (1972); Keating et al (2010); Saad and Turgeon (1988); Tripathi and Govindaraju (2008); Wang et al (2022b); Zhao et al (2022); Zhao and Luo (2020).…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

Machine Learning for Bayesian Experimental Design in the Subsurface

Thibaut¹

2023

Preprint

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Hierarchical Bayesian Inversion of Global Variables and Large‐Scale Spatial Fields

Cited by 9 publications

References 68 publications

Inference of (geostatistical) hyperparameters with the correlated pseudo-marginal method

Inference of (geostatistical) hyperparameters with the correlated pseudo-marginal method

Formulating and Solving the Data-Consistent Geophysical Inverse Problem for Subsurface Modeling Applications

Machine Learning for Bayesian Experimental Design in the Subsurface

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