Role of model selection criteria in geostatistical inverse estimation of statistical data‐ and model‐parameters

Abstract

[1] We analyze theoretically the ability of model quality (sometimes termed information or discrimination) criteria such as the negative log likelihood NLL, Bayesian criteria BIC and KIC and information theoretic criteria AIC, AICc, and HIC to estimate (1) the parameter vector h of the variogram of hydraulic log conductivity (Y ¼ ln K), and (2) statistical parameters 2 hE and 2 YE proportional to head and log conductivity measurement error variances, respectively, in the context of geostatistical groundwater flow inversion. Our analysis extends the work of Hernandez et al. (2003Hernandez et al. ( , 2006 and Riva et al. (2009), who developed nonlinear stochastic inverse algorithms that allow conditioning estimates of steady state and transient hydraulic heads, fluxes and their associated uncertainty on information about conductivity and head data collected in a randomly heterogeneous confined aquifer. Their algorithms are based on recursive numerical approximations of exact nonlocal conditional equations describing the mean and (co)variance of groundwater flow. Log conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete ''pilot points.'' Optionally, the maximum likelihood function on which the inverse estimation of Y at pilot points is based may include a regularization term reflecting prior information about Y. The relative weight ¼ YE , as well as h are evaluated separately from other model parameters to avoid bias and instability. This evaluation is done on the basis of criteria such as NLL, KIC, BIC, HIC, AIC, and AICc. We demonstrate theoretically that, whereas all these six criteria make it possible to estimate 2 hE , KIC alone allows one to estimate validly h and 2 YE (and thus ). We illustrate this discriminatory power of KIC numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two-dimensional steady state groundwater flow problem. We find that whereas

Data assimilation and parameter estimation via ensemble Kalman filter coupled with stochastic moment equations of transient groundwater flow

“…[] with the ensemble Kalman filter (EnKF) obviates the need for computationally intensive (a) batch inverse solution of the MEs in the manner of Riva et al . [] and (b) MC simulation commonly associated with EnKF.…”

“…This in turn might obviate the need to estimate variogram parameters jointly with Y , as done in the context of steady state and batch transient geostatistical inversion of MEs by Riva et al . []. In contrast, adopting an incorrect variogram model caused the quality of E Y , V Y , and correlations between estimated and true Y values to deteriorate at all times.…”

Estimation of Single-Metal and Competitive Sorption Isotherms through Maximum Likelihood and Model Quality Criteria

“…While the relative merits of KIC and AIC^ within the context of applications to natural systems are still studied and debated extensively (e.g., Foglia et al, 2007;Ye et al, 2008;Li, 2008, 2010;Li and Tsai, 2009;Riva et al, 2011), our results suggest that in the case of an unweighted calibration it is not possible to select one model to interpret the observed data at the expense of the others. On the other hand, we observe that adopting a weighted calibration for our experiments leads to a clear identification of a best performing model.…”

Multi-Scale Assessment of Prediction Uncertainty in Coupled Reactive Transport Models Conducted at the Florida State University

“…While the method is developed for serial data with temporal correlation, it can be adapted for data with spatial correlation using geostatistical theories. When both spatial and temporal correlations exist, one may first characterize them separately and then aggregate them in the manner of and Riva et al [2011]. Different from the two-stage method of Seber and Wild [2003, p. 279] and Yeh [1984] in groundwater modeling, the iterative process does not need to invoke any assumption on the order of the AR(p) model.…”

“…This has been motivated by a growing recognition that environmental systems are open and complex, rendering them prone to multiple conceptualizations and mathematical descriptions, regardless of the quantity and quality of available data and knowledge [Beven, 2002;Bredehoeft, 2003Bredehoeft, , 2005]. Multimodel analysis has become popular for quantification of model uncertainty [Burnham and Anderson, 2002;Ye et al, , 2008aYe et al, , 2008bYe et al, , 2010bYe et al, , 2010cMarshall et al, 2005;Beven, 2006;Ajami et al, 2007;Vrugt and Robinson, 2007;Li, 2008a, 2008b;Wohling and Vrugt, 2008;Rojas et al, 2008Rojas et al, , 2009Winter and Nychka, 2010;Riva et al, 2011;Nowak et al, 2012;Seifert et al, 2012;Rings et al, 2012;Parrish et al, 2012;Dai et al, 2012]. In multimodel analysis, rather than choosing a single model, modeling predictions and associated uncertainty from multiple competing models are aggregated, typically in a model averaging process.…”