We analyze theoretically the ability of model quality criteria such as negative log likelihood, Bayesian criteria BIC and KIC and information theoretic criteria AIC, AICc and HIC to estimate (a) the parameter vector of the variogram of hydraulic log conductivity (Y), and (b) statistical parameters proportional to head and log conductivity measurement error variances, respectively, in the context of geostatistical groundwater flow inversion. We demonstrate that only the Bayesian criterion KIC is suitable for this purpose. We illustrate this discriminatory power of KIC numerically by using a differential evolution genetic search algorithm to minimize it in the context of a two-dimensional steady state groundwater flow problem.