Parameter estimation, uncertainty analysis and data worth assessment of soil water and heat transport models using an Iterative Ensemble Smoother

Mechanistic soil water and heat transport models are important tools for quantifying and predicting exchange fluxes between the atmosphere, vegetation, soils, and aquifers. For instance, these models are essential for assessing how soils help mitigate flood and heatwave risks, particularly in urban environments. Mechanistic soil water and heat transport models rely on parameters characterising the soils' hydraulic and thermal properties. The estimation of these parameters through inverse modelling and the quantification of associated uncertainties is challenging due to the non-linear nature of the processes and computational demands of the simulations. However, recent advances in inverse modelling algorithms such as Iterative Ensemble Smoothers (IES) allow us to handle highly parameterised, non-linear models while keeping the number of model runs relatively small (~1000).  These advances not only keep the computational demand for inverse modelling and parameter estimation tractable, but in addition allow for the quantitative assessment of data worth of available or planned observations, which can be used to increase the efficiency of experimental designs. In this work, we tested a Levenberg-Marquardt form of IES, a relatively novel method increasingly used in reservoir and groundwater modelling, with a mechanistic soil water and heat transport model. We firstly generated reference values of soil moisture, temperature and fluxes using three synthetic models representing different characteristic soil profiles with contrasting parameter values. We simulated a calibration period representing our planned field experiments, consisting of two infiltration tests with warm and cold water, and a prediction period including a heat wave and an extreme rainfall event. Secondly, we used the IES algorithm to history-match the model-generated “observations” of soil moisture and temperature at six depths for the calibration period. We finally evaluated the algorithm's ability to estimate the reference parameters, as well as predict soil moisture, temperature, and fluxes. The results show that the posterior distributions obtained with the IES algorithm are consistent with the reference values for all parameters and predictions considered. Furthermore, the relatively small number of runs required (< 10,000) allowed us to perform parameter estimation and uncertainty quantification across different experimental scenarios, thereby quantifying their data worth and optimising our experimental design. The synthetic approaches formed the basis for simulating water and heat transport in three real-world urban soils and assessing the worth of temperature measurements in tracing water and heat fluxes. IES algorithms have strong potential to become standard tools for vadose zone modelling, and the insights gained from our study offer a solid foundation for their effective application.