Conceptual and computational aspects of the mixing cell method to determine groundwater recharge components

In a work by Adar et al., the mixing cell method to determine spatial recharge distributions with the use of environmental isotopes and geochemical species was formulated as a quadratic programming problem, i.e. with a quadratic objective function subject to linear constraints. In this paper the problem is critically reviewed and an alternative formulation is given in terms of general linear regression theory, bringing it in line with the statistical framework developed by Wagner and Gorelick. For the implementation of the alternative solution method the Singular Value Decomposition (SVD) algorithm is suggested which provides information about the existence of a unique solution, makes it possible to deal with ill-conditioned problems and finally also provides the variances and covariances of the calculated flow parameters. The theory of this paper and the SVD algorithm are illustrated with two simple examples, one of which is derived from the eastern Botswana situation.