A methodology to compute mixing ratios with uncertain end‐members

Mixing calculations involve computing the ratios in which two or more end‐members are mixed in a sample. Mixing calculations are useful for a number of tasks in hydrology, such as hydrograph separation, water or solute mass balances, and identification of groundwater recharge sources. Most methods available for computing mixing ratios are based on assuming that end‐member concentrations are perfectly known, which is rarely the case. Often, end‐members cannot be sampled, and their concentrations vary in time and space. Still, much information about them is contained in the mixtures. To take advantage of this information, we present here a maximum likelihood method to estimate mixing ratios, while acknowledging uncertainty in end‐member concentrations. Maximizing the likelihood of concentration measurements with respect to both mixing ratios and end‐member concentrations leads to a general constrained optimization problem. An algorithm for solving this problem is presented and applied to two synthetic examples of water mixing problems. Results allow us to conclude that the method outperforms traditional approaches, such as least squares or linear mixing, in the computation of mixing ratios. The method also yields improved estimates of end‐member concentrations, thus enlarging the potential of mixing calculations. The method requires defining the reliability of measurements, but results are quite robust with respect to the assumed standard deviations. A nice feature of the method is that it allows for improving the quality of computations by increasing the number of samples and/or analyzed species.