Model smoothing strategies to remove microscale discontinuities and spurious secondary optima in objective functions in hydrological calibration

Environmental processes frequently exhibit threshold-type behavior, e. g., the initiation of fluxes such as snowmelt, recharge, and quick flow. Incorporating such thresholds into hydrological models introduces discontinuities into the objective functions used in model calibration, making parameter estimation unnecessarily more difficult. Moreover, this study shows that model thresholds can produce spurious multimodality in least squares objective functions even if the underlying model is near linear in its parameters. In contrast, smoothing the model with respect to its parameters and inputs yields differentiable objective functions and, in some cases, can also improve its macroscale characteristics by removing spurious secondary optima. This simplifies model calibration and sensitivity analysis by reducing the complexity of objective functions and permitting the use of powerful derivative-based analysis methods such as Newton-type optimization and Hessian-based uncertainty assessment. This paper details smoothing strategies for several classes of thresholds and discontinuities commonly found in hydrological models, including step and angle discontinuities in the constitutive functions and flux constraints arising from conservation laws in the governing differential equations. The smoothing algorithms and their parameters are selected to ensure infinite differentiability of the model and its objective functions while preserving the macroscale behavior of the original governing equations. The improvements in the structure of the model and its objective functions are illustrated empirically for a degree-day-based snow model. The smoothing techniques are general and can be applied to other models with thresholds.