Bayesian Inversion of Multi‐Gaussian Log‐Conductivity Fields With Uncertain Hyperparameters: An Extension of Preconditioned Crank‐Nicolson Markov Chain Monte Carlo With Parallel Tempering

Abstract

The characterization of hydraulic properties of aquifers and soils is essential to better predict water flow in the subsurface and the transport of heat or solutes. Typically, not enough direct data (e.g., hydraulic conductivity) are available to characterize the heterogeneous subsurface. Thus, additional indirect data (e.g., hydraulic heads) are important for improving characterization and, in turn, predictions by subsurface flow and transport models. In the geostatistical context, the resulting inverse problem for subsurface problems is typically underdetermined. Therefore, approaches were developed which limited the number of independent parameters to be estimated, either by defining a limited number of zones with constant parameters (Carrera & Neuman, 1986) or parameterizing the spatially variable parameter field by a geostatistical function with a few unknown parameters (Kitanidis & Vomvoris, 1983). Later, methods were formulated to estimate a series of equally likely solutions to the groundwater inverse problem, either by an ensemble-based variational data assimilation approach (Gómez-Hernández et al., 1997) or by a sequential data assimilation approach for an ensemble of random parameter fields (Chen & Zhang, 2006). In summary, the combination of regularization and casting the problem in a stochastic framework, helped to tackle groundwater inverse problems.Bayesian inversion has been widely used for model parameter inference (

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

“…This formulation which considers both uncertainty in global and spatial variables and solves inversion problem hierarchically is known as hierarchical Bayes in statistics (Banerjee et al., 2003; Gelman et al., 1995; Kitanidis, 1995; Malinverno & Briggs, 2004). Previous formulations (Laloy et al., 2015; Xiao et al., 2021) for joint global and spatial inverse problems emphasize geostatistical variables $${\boldsymbol{\theta }}_{\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$$ where $$\boldsymbol{\theta }={\boldsymbol{\theta }}_{\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$$. Kitanidis (1995) mentions other uncertain physical variables $${\boldsymbol{\theta }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$$ and also focuses more on $${\boldsymbol{\theta }}_{\mathrm{g}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}$$ in the quasi‐linear geostatistical theory.…”

“…This formulation which considers both uncertainty in global and spatial variables and solves inversion problem hierarchically is known as hierarchical Bayes in statistics (Banerjee et al, 2003;Gelman et al, 1995;Kitanidis, 1995;Malinverno & Briggs, 2004). Previous formulations (Laloy et al, 2015;Xiao et al, 2021) for joint global and spatial inverse problems emphasize geostatistical variables 𝐴𝐴 𝜽𝜽gstat where 𝐴𝐴 𝜽𝜽 = 𝜽𝜽gstat . Kitanidis (1995) mentions other uncertain physical variables 𝐴𝐴 𝜽𝜽phys and also focuses more on 𝐴𝐴 𝜽𝜽gstat in the quasi-linear geostatistical theory.…”

“…In this paper, we denote those non‐spatial variables as global variables. Global variables include both geostatistical variables, also known as structural parameters and hyperparameters (Emerick, 2016; Kitanidis, 1995; Xiao et al., 2021), and non‐geostatistical physical global variables such as boundary and initial conditions. Even with the simplest linear forward model, the relationship between geostatistical variables and data variables are non‐linear.…”

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

“…One way to infer hyperparameters in the non-ergodic setting by MCMC methods is to parameterize the field by hyperparameters and white noise to describe the local properties (as e.g. in Laloy et al, 2015, Hunziker et al, 2017and Xiao et al, 2021. The corresponding full inversion problem involves typically many thousands of parameters, for which either an efficient MH proposal scheme has to be designed (e.g., Xiao et al, 2021) or dimensionality reduction arguments have to be invoked (e.g., Laloy et al, 2015, Rubin et al, 2010.…”

“…in Laloy et al, 2015, Hunziker et al, 2017and Xiao et al, 2021. The corresponding full inversion problem involves typically many thousands of parameters, for which either an efficient MH proposal scheme has to be designed (e.g., Xiao et al, 2021) or dimensionality reduction arguments have to be invoked (e.g., Laloy et al, 2015, Rubin et al, 2010. While the first approach is very challenging (curse of dimensionality, e.g., Robert et al, 2018), the second approach may lead to biased estimates (Laloy et al, 2015).…”