A robust objective function for calibration of groundwater models in light of deficiencies of model structure and observations

Groundwater models require parameter optimization based on the minimization of objective functions describing, for example, the residual between observed and simulated groundwater head. At larger scales, constraining these models requires large datasets of groundwater head observations. These observations are typically only available from databases comprised of varying quality data from a variety of sources and will be associated with unknown observational uncertainty. At the same time the model structure, especially the hydrogeological description, will inevitably be a simplification of the complex natural system. As a result, calibration of groundwater models often results in parameter compensation for model structural deficiency, or can be affected by observation errors. This problem can be amplified by the application of common squared error-based performance criteria, which are most sensitive to the largest errors. In our context, we assume that the residuals that remain large during the optimization process likely do so because of either model structural error or observation error. Based on this assumption it is desirable to design an objective function that is less sensitive to these large residuals of low probability, and instead favours the majority of observations that can fit the given model structure and likely are free of large observation errors. We suggest a Continuous Ranked Probability Score (CRPS) based objective function that limits the influence of large residuals in the optimization process as the metric puts more emphasis on the position of the residual along the cumulative distribution function than on the magnitude of the residual. The CRPS-based objective function was applied in the calibration of regional-scale coupled surface-groundwater models and compared to conventional objective functions based on mean of squared, absolute and root errors, using synthetic as well as real observations. The optimization tests illustrated that the novel CRPS-based objective function successfully limited the dominance of large residuals in the optimization process and consistently reduced overall bias.