Bayesian Inference in Physically-based Vadose Zone Modeling: The Good, The Bad and The Ugly

Mechanistic models, grounded in the Richards and advection-dispersion equations, provide a comprehensive theoretical framework for describing hydrological processes and solute transport in the vadose zone. Due to the limited transferability of laboratory estimates to field conditions, model parameters are often inversely estimated from transient field observations, making calibration an increasingly common practice in vadose zone modeling. The inescapable necessity to include some form of uncertainty assessment has led to the rise of Bayesian inference as the preferred tool for probabilistic calibration. By combining prior information with observations and model predictions, Bayesian inference enables the estimation of parameter posterior distributions, verification of model adequacy, and assessment of the model’s predictive uncertainty (The Good). Nevertheless, its application to mechanistic vadose zone models poses multiple challenges, among which the curse of dimensionality is likely the most critical (The Bad). We demonstrate that the performance of state-of-the-art Markov Chain Monte Carlo (MCMC) methods deteriorates even for moderately high-dimensional inverse problems, due to the shrinking of the typical set and improper spatiotemporal discretizations of the vadose zone domain during Monte Carlo runs, with both issues being exacerbated under model misspecification. Although using the gradient of the posterior density could mitigate the former problem, it is often rendered impractical due to numerical challenges. While these issues are generally manageable in low-dimensional settings, Bayesian inference remains hindered when applied in combination with computationally intensive vadose zone models (e.g., 2D/3D models, reactive solute transport) (The Ugly). We demonstrate that surrogate-based models can alleviate this problem, though their training and validation are not without difficulties. Based on these findings, we draw some conclusions and propose possible future directions for uncertainty assessment in physically based vadose zone modeling.