Estimation of single-metal and competitive sorption isotherm through Maximum likelihood and model quality criteria

Metal sorption of single and binary (competitive) systems for several soils is analyzed to assess the ability of alternative isotherm models to interpret experimental observations. The analysis is performed within a Maximum Likelihood framework and on the basis of model identifi cation (sometimes termed “quality” or “information”) criteria. These methodologies allow the assessment of the measurement error variance in the parameter estimation process and the uncertainty arising from the use of alternative (conceptual-mathematical) models. We first analyze Cu and Zn sorption in two Israeli soils, Bet Dagan and Yatir, which are slightly alkaline but with substantially different sorption capacities and perform an extensive set of batch experiments in single and binary systems. We then analyze the data set published by Liao and Selim (2009) where Ni and Cd sorption was studied in three different (one neutral and two acidic) soils. Single component data from both sets of experiments are interpreted on the basis of the Langmuir, Freundlich, and Redlich–Peterson (RP) models. The family of binary systems results is analyzed in light of the Sheindorf–Rebhun–Sheintuch (SRS) model, the modified RP model, and the modified and extended Langmuir models. All of the considered models are expressed in terms of initial and equilibrium concentrations, two variables that are measured independently. Maximum Likelihood and model identification criteria (such as Bayesian criteria BIC and KIC, and information theoretic criteria AIC, AICc, and HIC) are employed to (a) estimate model parameters, (b) rank alternative models, and (c) estimate the relative degree of likelihood of each model by means of a weight, or posterior probability. We show that modeling observation error variance either as a constant or as a function of concentration does not significantly affect parameter estimates for a given model. These different representations of measurement error variance impact the ranking of alternative models based on posterior probability weights. The weights associated with different models can be very similar when a uniform measurement error variance is considered, so that it is difficult to clearly identify a single best model.