Leveraging Probabilistic Parameter Estimation to Address Data Scarcity in Lake Hydrodynamic Modeling

Hydrodynamic lake models simulate water temperature at various depths using physics-based equations. These models rely on parameters representing system properties that are not directly measurable, and are generally adjusted through deterministic calibration (DC) methods to find a single parameter set that aligns simulated temperatures with observed data. However, DC overlooks the inherent prediction uncertainties arising from (1) inadequate process representation in the model, (2) measurement errors in hydrometeorological inputs, and (3) errors in water temperature observations used for calibration. Additionally, temperature observations in many lakes and reservoirs are often restricted to a single depth with frequent gaps, necessitating synthetic gap-filling techniques like interpolation, which increase uncertainty and compromise predictive accuracy.

This study explores the potential of probabilistic parameter estimation (PE), which evaluates the uncertainty around likely parameter values, to address these limitations and improve prediction performance in a lake hydrodynamic model. Using the Generalized Likelihood Uncertainty Estimation (GLUE) method, we quantify uncertainty in model predictions by identifying and aggregating acceptable parameter sets that meet predefined performance criteria conditioned on observed data. Unlike DC, GLUE emphasizes the range of plausible outcomes rather than a single optimal solution. We also propose a method to eliminate the subjectivity in selecting the performance criteria.

We apply this approach to the General Lake Model (GLM), a state-of-the-art 1D vertical hydrodynamic model, using Lake Mendota (WI, USA) as a case study. We use 8 years of hourly seasonal observations, including water temperature measurements at 1-meter intervals from the surface to a depth of 20 meters. Our analysis investigates the impact of data gaps and synthetic gap-filling on prediction accuracy and uncertainty. We systematically compare PE and DC to determine which method better handles data scarcity and improves predictive accuracy. Furthermore, we assess whether PE with multi-depth profile observations provide better predictions than single-depth observations and identify the optimal location for single-depth calibration, focusing on the surface mixed layer (SML), metalimnion, and hypolimnion.

Our results reveal that non-calibrated GLM tends to predict better in the SML than in the hypolimnion and PE becomes increasingly necessary as prediction depth increases. Strikingly, single-depth hypolimnion observations yield more accurate prediction uncertainty bounds across the water column and reduce overfitting compared to profile observations. In contrast, including observations from the SML and metalimnion weakens prediction performance at greater depths. Additionally, synthetic gap-filling in observational data degrades prediction accuracy and amplifies uncertainty. Furthermore, PE consistently outperforms DC in predictive accuracy, especially in deeper waters, and proves more robust under conditions of limited data availability.

These results offer practical insights into instrumentation, data collection and calibration strategies for lake hydrodynamic modeling. They underscore the value of probabilistic approaches like GLUE for robust model development and provide guidance for addressing similar challenges in other aquatic systems.