Geostatistical inversion with quantified uncertainty for nonlinear problems requires techniques for providing conditional realizations of the random field of interest. Many first‐order second‐moment methods are being developed in this field, yet almost impossible to critically test them against high‐accuracy reference solutions in high‐dimensional and nonlinear problems. Our goal is to provide a high‐accuracy reference solution algorithm. Preconditioned Crank‐Nicolson Markov chain Monte Carlo (pCN‐MCMC) has been proven to be more efficient in the inversion of multi‐Gaussian random fields than traditional MCMC methods; however, it still has to take a long chain to converge to the stationary target distribution. Parallel tempering aims to sample by communicating between multiple parallel Markov chains at different temperatures. In this paper, we develop a new algorithm called pCN‐PT. It combines the parallel tempering technique with pCN‐MCMC to make the sampling more efficient, and hence converge to a stationary distribution faster. To demonstrate the high‐accuracy reference character, we test the accuracy and efficiency of pCN‐PT for estimating a multi‐Gaussian log‐hydraulic conductivity field with a relative high variance in three different problems: (1) in a high‐dimensional, linear problem; (2) in a high‐dimensional, nonlinear problem and with only few measurements; and (3) in a high‐dimensional, nonlinear problem with sufficient measurements. This allows testing against (1) analytical solutions (kriging), (2) rejection sampling, and (3) pCN‐MCMC in multiple, independent runs, respectively. The results demonstrate that pCN‐PT is an asymptotically exact conditional sampler and is more efficient than pCN‐MCMC in geostatistical inversion problems.
Models are used to predict and/or investigate and explain phenomena in nature. Often, many hypotheses exist for these two tasks. Naturally, the question arises, which of the competing modeling approaches predicts or explains nature best. Bayesian model selection (BMS, e.g., Wasserman, 2000) is a statistical method that uses observed data to select between competing models. BMS is settled in a rigorous probabilistic framework and follows the scheme of Bayesian updating: A prior belief about the plausibility of each candidate model is updated to a posterior model weight in the light of measured data (i.e., the probability of the model to have generated the data, given the model set). Posterior model weights are then used as a basis for Bayesian model ranking, selection, or averaging (BMA, Hoeting et al., 1999).To help with the interpretation of posterior model weights, the so-called model confusion matrix (MCM) has been introduced by Schöniger, Illman, et al. (2015). It reveals whether a lack of confidence in model choice is due to similarity between the candidate models or due to weakly informative data. The MCM is a purely synthetic analysis that can be used as a scale of reference for model weights obtained from real data. Schäfer Rodrigues Silva et al. (2020) have recently extended the MCM analysis to identify the best surrogate model from a set of candidates to replace an expensive full-complexity model in stochastic analysis.Technically, the Bayesian updating procedure requires calculating the so-called Bayesian model evidence (BME). BME is the likelihood of a model to have generated the data, integrated over its whole parameter space and all involved probability distributions. While the likelihood accounts for uncertainty in measured data, the integration considers parameter uncertainty, and potentially also uncertainty in model drivers or boundary conditions. In some cases, the integration even accounts for statistical representations of model errors (Leube et al., 2012;Nowak et al., 2012), which is perceived by many studies to be part of the likelihood.
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 (
Optimal design of groundwater monitoring networks is challenging due to (1) conflicting objectives for assessing the performance of candidate monitoring networks, (2) uncertainty in system dynamics and hydrogeological context, and (3) the large decision space of possible monitoring‐well positions (also termed the search space). The immensity of the search space poses a significant challenge for modern multiobjective optimization tools. This study introduces two approaches that improve the efficiency and effectiveness of evolutionary multiobjective optimization tools when solving monitoring design problems. We show how a careful mathematical representation of the monitoring design search space and reductions of possible monitoring‐well positions enhance the solution and attainment of decision‐relevant multiobjective trade‐offs in monitoring quality. We demonstrate the value of our improved representation and reduction techniques on a three‐objective monitoring network design problem focused on urban source water protection (termed the U_Protect benchmarking problem). U_Protect abstracts a real‐world case study within an urban drinking‐water well catchment, including inaccessible and restricted areas for monitoring‐well installation, and random heterogeneities in the conductivity field. Our representation and reduction methods significantly enhance the effectiveness, efficiency, and reliability of the optimization. Our proposed framework shifts focus to the most impactful monitoring design decisions while also enhancing decision makers understanding of key performance trade‐offs. In combination, our proposed representation and reduction techniques have significant promise for enhancing the size and the scope of combinatorial monitoring problems that can be explored.
Hydraulically induced fracturing is widely used in practice for several exploitation techniques. The chosen macroscopic model combines a phase‐field approach to fractures with the Theory of Porous Media (TPM) to describe dynamic hydraulic fracturing processes in fully‐saturated porous materials. In this regard, the solid's state of damage shows a diffuse transition zone between the broken and unbroken domain. Rocks or soils in grown nature are generally inhomogeneous with material imperfections on the microscale, such that modelling homogeneous porous material may oversimplify the behaviour of the solid and fluid phases in the fracturing process. Therefore, material imperfections and inhomogeneities in the porous structure are considered through the definition of location‐dependent material parameters. In this contribution, a deterministic approach to account for predefined imperfection areas as well as statistical fields of geomechanical properties is proposed. Representative numerical simulations show the impact of solid skeleton heterogeneities in porous media on the fracturing characteristics, e. g. the crack path.
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