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

Abstract

Abstract. 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, due to the size of the inverse problem. 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. This problem can be amplified by the application of common squared error-based performance criteria, which are most sensitive to the largest errors. 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. 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 two regional scale coupled surface-groundwater models and compared to calibrations using conventional sum of absolute and squared errors. 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. Furthermore, it highlighted areas in the model where the structural model should be revisited.

Using a Groundwater Adjusted Water Balance Approach and Copulas to Evaluate Spatial Patterns and Dependence Structures in Remote Sensing Derived Evapotranspiration Products

“…The model structure is based on a comprehensive hydro-stratigraphic model for Denmark and parameterized through calibration against objective functions based on 305 discharge stations and 25,981 hydraulic head observation wells for the period 2000-2010 (for the current application run for the period 2002-2014). The main objective functions are Kling-Gupta efficiency (KGE) [70] and percent bias (Pbias) for streamflow, and continuous ranked probability score (CRPS) values for groundwater heads [53]. Details on the model data, construction, calibration, and validation can be found in [71], while calibration statistics will be presented in brevity in the Result section.…”

“…In the literature, ET datasets from the remote-sensing and surface water balance approaches have been compared and studied (e.g., [7,8,53,54]). For example, [17,20] compared water balance ET estimates to ET estimates based on hydrological modeling and RS methods over central Europe and Canada, respectively.…”

Calibration of subseasonal sea‐ice forecasts using ensemble model output statistics and observational uncertainty

“…Typically, F obs (x) is represented by a Heaviside step function, H(x − y); that is, the observation is assumed to be known perfectly. Recent work, however, has demonstrated that model fitting through CRPS minimization is more robust when observational uncertainty is represented explicitly (Schneider et al, 2020). Given this, and the significant uncertainty in daily passive microwave SIC estimates (e.g., Ivanova et al, 2015), we assess the benefits of including the observational uncertainty provided with OSI-SAF SIC in the representation of F obs (x).…”

Hunting for Information in Streamflow Signatures to Improve Modelled Drainage

“…The mean errors per model grid then were weighted based on the number of observations-with a weight of 1 for cells with one observation, a weight of 2 for cells with two to nine observations, a weight of 3 for cells with ten to 99 observations, and a weight of 5 for cells with 100 or more observations. The final objective function for groundwater heads was formulated using the Continuous Ranked Probability Score (CRPS), as described by [61]. The CRPS-based approach was chosen because conventionally used squared-errorbased objective functions are particularly sensitive to large residuals.…”