A Filtering Approach For Hydrological Process Monitoring With Approximation Error Method

A. Lehikoinen1, S. Finsterle2, J. M. J. Huttunen1, A. Voutilainen1, M. B. Kowalsky2, J. P. Kaipio1

1Department of Physics, University of Kuopio, P.O.Box 1627, FIN-70211, Kuopio, Finland, Email: Jari.Kaipio@uku.fi

2Earth Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 90-1116, Berkeley, California 94720, USA.

ABSTRACT

We present a new methodology for imaging the evolution of electrically conductive fluids in porous media. The inversion problem is formulated as a state estimation problem, which is based on an observation-evolution model. The state estimation problem is solved using Bayesian filtering techniques, such as extended Kalman filter and iterated extended Kalman filter algorithms. The synthetic study we consider involves the imaging of time-varying distributions of water saturation in porous media using electrical resistance tomography (ERT). The complete electrode model with Archie’s law relating water saturation to electrical conductivity is used as the observation model. The evolution model simulates flow through unsaturated porous media. We propose to account for approximation errors in the evolution model by constructing a statistical model for the discrepancy between the realizations of the unknown heterogeneous permeability field and a homogeneous permeability field used in the solution of the inverse problem, and by including this information in the calculation of the posterior probability density of the estimated system state. The proposed method provides improved estimates of water saturation distribution compared to traditional reconstruction schemes that rely on conventional stabilization methods (e.g., using a smoothness prior) and compared to Bayesian filtering approaches that do not incorporate the approximation error method. Moreover, the approximation error method allows for the use of a simplified and computationally efficient evolution model in the state estimation scheme. In addition, the non-stationary inversion methods presented here for unsaturated flow through porous media may be extended for imaging and estimating parameters of dynamical processes using a variety of geophysical methods.

Keywords electrical resistance tomography, non-stationary inverse problem, extended Kalman filter, nonstationary approximation error method