Parameter estimation by ensemble Kalman filters with transformed data: Approach and application to hydraulic tomography

Abstract

[1] Ensemble Kalman filters (EnKFs) are a successful tool for estimating state variables in atmospheric and oceanic sciences. Recent research has prepared the EnKF for parameter estimation in groundwater applications. EnKFs are optimal in the sense of Bayesian updating only if all involved variables are multivariate Gaussian. Subsurface flow and transport state variables, however, generally do not show Gaussian dependence on hydraulic log conductivity and among each other, even if log conductivity is multi-Gaussian. To improve EnKFs in this context, we apply nonlinear, monotonic transformations to the observed states, rendering them Gaussian (Gaussian anamorphosis, GA). Similar ideas have recently been presented by Béal et al. (2010) in the context of state estimation. Our work transfers and adapts this methodology to parameter estimation. Additionally, we address the treatment of measurement errors in the transformation and provide several multivariate analysis tools to evaluate the expected usefulness of GA beforehand. For illustration, we present a first-time application of an EnKF to parameter estimation from 3-D hydraulic tomography in multi-Gaussian log conductivity fields. Results show that (1) GA achieves an implicit pseudolinearization of drawdown data as a function of log conductivity and (2) this makes both parameter identification and prediction of flow and transport more accurate. Combining EnKFs with GA yields a computationally efficient tool for nonlinear inversion of data with improved accuracy. This is an attractive benefit, given that linearization-free methods such as particle filters are computationally extremely demanding.

Estimation of heterogeneous aquifer parameters using centralized and decentralized fusion of hydraulic tomography data

“…In HT studies, hydraulic head transient data have been inverted using different algorithms, such as the sequential successive linear estimator (SSLE) (Yeh and Liu, 2000), the quasi-linear approach (Kitanidis, 1995;Liu and Kitanidis, 2011), the Bayesian maximum a posteriori (MAP) approach (Castagna and Bellin, 2009), and the ensemble Kalman filter (EnKF) (Schöniger et al, 2012).…”

“…Therefore, the ensemble mean of the posterior ensemble provides an unbiased estimate of the system parameters. The EnKF offers several other advantages, such as computational efficiency (Franssen and Kinzelbach, 2009), avoiding sensitivity computations, such as those required by the SSLE (Yeh and Liu, 2000), and improved accuracy when using ensemble-based covariance estimations instead of sensitivity-based covariance estimations (Schöniger et al, 2012).…”

Joint inference of groundwater–recharge and hydraulic–conductivity fields from head data using the ensemble Kalman filter

“…Most parameter estimations used 2-D models, as these are conceptually simpler, faster, and easier to constrain and display. However, EnKF has also successfully been applied to infer 3-D hydraulic-conductivity fields (e.g., Camporese et al, 2011;Schöniger et al, 2012). …”

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

“…[2] Calibrating subsurface models and accounting for parameter uncertainty have received increasing attention in recent studies Neuman, 1986a, 1986b;Kitanidis, 1997;Efendiev et al, 2005;Dostert et al, 2006;Vrugt et al, 2005;Jiang and Woodbury, 2006;Ronayne et al, 2008;Fu and Gomez-Hernandez, 2009;Schöniger et al, 2012]. These techniques can be classified into Bayesian methods based on Markov chain Monte Carlo (MCMC) [Oliver et al, 1997;Efendiev et al, 2005;Elsheikh et al, 2012], gradient-based optimization methods [McLaughlin and Townley, 1996;Carrera et al, 2005;Altaf et al, 2013], stochastic search algorithms [Li and Reynolds, 2011;Elsheikh et al, 2013aElsheikh et al, , 2013bElsheikh et al, , 2013cElsheikh et al, , 2013d, and ensemble Kalman filter methods [Moradkhani et al, 2005;Naevdal et al, 2005;Oliver et al, 2008;Luo et al, 2012;Elsheikh et al, 2013e].…”