Adaptive multi-scale parameterization for one-dimensional flow in unsaturated porous media Mohamed Hayek, Franc ¸ois Lehmann, Philippe Ackerer * Institut de Me ´canique des Fluides et des Solides, Universite ´ Louis Pasteur de Strasbourg – CNRS – UMR 7507, 2 rue Boussingault, F-67000 Strasbourg, France Received 12 January 2007; received in revised form 27 June 2007; accepted 27 June 2007 Available online 6 July 2007 Abstract In the analysis of the unsaturated zone, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the porous media. Adaptative multi-scale parameterization consists in solving the problem through successive approximations by refining the parameter at the next finer scale all over the domain and stopping the process when the refinement does not induce significant decrease of the objective function any more. In this context, the refinement indicators algorithm provides an adap- tive parameterization technique that opens the degrees of freedom in an iterative way driven at first order by the model to locate the discontinuities of the sought parameters. We present a refinement indicators algorithm for adaptive multi-scale parameterization that is applicable to the estimation of multi-dimensional hydraulic parameters in unsaturated soil water flow. Numerical examples are pre- sented which show the efficiency of the algorithm in case of noisy data and missing data. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Unsaturated flow; Parameter estimation; Inverse method; Refinement indicator; Parameterization 1. Introduction Numerical models which describe the movement of water and chemicals into and through the unsaturated zone have become essential tools for quantifying and integrating transport processes in the unsaturated soil zone. The appli- cation of these models to field-scale flow and transport problems relies heavily on the quality of the model param- eters, especially the unsaturated hydraulic properties. Direct parameter measurements in heterogeneous soils are very difficult and scale-dependent. An alternative approach for estimating hydraulic parameters at the field- scale is inverse modelling which uses, for example, non-lin- ear regression methods to estimate the effective hydraulic parameters. This method minimizes the difference between the predicted and measured hydraulic variables, such as pressure head h, water content h or flow rate. A number of laboratory and field applications [2,17,19,29,34,36,39] have shown the potential of inverse techniques for improv- ing the design and analysis of vadose-zone flow and trans- port experiments. A serious problem encountered in the estimation of hydraulic parameters stems from over-parameterization. Indeed, since experimental measurements are expensive, the set of data is usually insufficient to estimate the values of the parameters in each cell of the computational mesh. The resolution of this problem is done by parameteriza- tion. The most popular one consists in defining sub- domains (zones) in which the parameters are assumed to be constant. Therefore, the number of parameters to iden- tify can be considerably reduced but the number and the shape of the zones have to be known a priori. In multi-scale parameterization, the number of sub-domains is increased during the inverse procedure and the process is stopped when the refinement does not induce a significant decrease of the difference between simulated and measured data [1,8,13,24]. In adaptative multi-scale estimation, the zones 0309-1708/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2007.06.009 * Corresponding author. E-mail address: ackerer@imfs.u-strasbg.fr (P. Ackerer). www.elsevier.com/locate/advwatres Available online at www.sciencedirect.com Advances in Water Resources 31 (2008) 28–43