Nested sampling algorithm for subsurface flow model selection, uncertainty quantification, and nonlinear calibration

Abstract

[1] Calibration of subsurface flow models is an essential step for managing ground water aquifers, designing of contaminant remediation plans, and maximizing recovery from hydrocarbon reservoirs. We investigate an efficient sampling algorithm known as nested sampling (NS), which can simultaneously sample the posterior distribution for uncertainty quantification, and estimate the Bayesian evidence for model selection. Model selection statistics, such as the Bayesian evidence, are needed to choose or assign different weights to different models of different levels of complexities. In this work, we report the first successful application of nested sampling for calibration of several nonlinear subsurface flow problems. The estimated Bayesian evidence by the NS algorithm is used to weight different parameterizations of the subsurface flow models (prior model selection). The results of the numerical evaluation implicitly enforced Occam's razor where simpler models with fewer number of parameters are favored over complex models. The proper level of model complexity was automatically determined based on the information content of the calibration data and the data mismatch of the calibrated model.

Finding the right balance between groundwater model complexity and experimental effort via Bayesian model selection

“…In groundwater modeling, it has been applied to choose between different parameterizations of aquifer heterogeneity, e.g. by Ye et al (2004), Tsai and Li (2008), Rojas et al (2008), Morales-Casique et al (2010), Seifert et al (2012), and Elsheikh et al (2013), to name only a few selected examples. Refsgaard et al (2012) provide a review of strategies, including BMA, to address geological uncertainty in groundwater flow and transport modeling.…”

“…In the context of groundwater model selection and calibration, finding a balance between performance and complexity is of great interest (e.g., Yeh and Yoon, 1981;Fienen et al, 2009;Elsheikh et al, 2013). BMA is ideally suited to guide this search, because it implicitly honors the principle of parsimony or ''Occam's razor'' (Jeffreys, 1939;Gull, 1988).…”

Making Steppingstones out of Stumbling Blocks: A Bayesian Model Evidence Estimator with Application to Groundwater Transport Model Selection

“…To improve on the above limitations of semi-analytical solutions, numerical estimation of the BME with Monte Carlo based methods has become a fundamental computational problem in Bayesian statistics. Investigating the use of Monte Carlo methods to estimate BME has recently gained research interest in hydrology [21,24,26,45,48,[50][51][52]. These studies leverage on the significant effort in different branches of science to develop robust BME numerical estimators with the aim of reducing the estimation bias and increasing the computational efficiency.…”

“…This can be carried out through the semi-empirical method of nested sampling [69] and the path sampling method of thermodynamic integration [22,28,70]. The nested sampling method that is popular in astronomy was introduced to the hydrology community by Elsheikh et al [50], and the thermodynamic integration method that is popular in phylogeny was introduced to the hydrology community by Schoups and Vrugt [45] and Liu et al [24]. Nested sampling avoids sampling the full prior by converting the multidimensional integration of Equation (2) into a one-dimensional integral by relating the likelihood to the prior mass (i.e., integration of prior within a region).…”

Groundwater Flow Determination Using an Interval Parameter Perturbation Method

“…A crucial step of the Bayesian theory is the sampling algorithm. There are several commonly used sampling algorithms, such as the Metropolis-Hastings algorithm, the Gibbs algorithm and the adaptive Metropolis algorithm [30][31][32][33][34][35][36][37][38][39][40]. Bayesian theory can obtain the posterior probability density function of hydrogeological elements, but the premise is that the distribution of hydrogeological parameters is known a priori.…”