The CRPS – used as a robust objective function for groundwater model calibration in light of observation and model structural uncertainty

The Continuous Ranked Probability Score (CRPS) is a popular evaluation tool for probabilistic forecasts. We suggest using it, outside its original scope, as an objective function in the calibration of large-scale groundwater models, due to its robustness to large residuals in the calibration data.

Groundwater models commonly require their parameters to be estimated in an optimization where some objective function measuring the model’s performance is to be minimized. Many performance metrics are squared error-based, which are known to be sensitive to large values or outliers. Consequently, an optimization algorithm using squared error-based metrics will focus on reducing the very largest residuals of the model. In many cases, for example when working with large-scale groundwater models in combination with calibration data from large datasets of groundwater heads with varying and unknown quality, there are two issues with that focus on the largest residuals: Such outliers are often i) related to observational uncertainty or ii) model structural uncertainty and model scale. Hence, fitting groundwater models to such deficiencies can be undesired, and calibration often results in parameter compensation for such deficiencies.

Therefore, we suggest the use of a CRPS-based objective function that is less sensitive to (the few) large residuals, and instead is more sensitive to fitting the majority of observations with least bias. We apply the novel CRPS-based objective function to the calibration of large-scale coupled surface-groundwater models and compare to conventional squared error-based objective functions. These calibration tests show that the CRPS-based objective function successfully limits the influence of the largest residuals and reduces overall bias. Moreover, it allows for better identification of areas where the model fails to simulate groundwater heads appropriately (e.g. due to model structural errors), that is, where model structure should be investigated.

Many real-world large-scale hydrological models face similar optimizations problems related to uncertain model structures and large, uncertain calibration datasets where observation uncertainty is hard to quantify. The CRPS-based objective function is an attempt to practically address the shortcomings of squared error minimization in model optimization, and is expected to also be of relevance outside our context of groundwater models.