Exploring soil moisture dynamics and variability across scales and geological settings using gaussian mixture-long short-term memory networks

Soil moisture is a key variable for a range of hydrological and ecological processes, yet capturing its small-scale variability and preferential flow phenomena remains challenging. Recent advancements in deep learning have demonstrated potential in predicting hydrological variables, but conventional data-driven models often struggle to represent small-scale variability effectively. In this study, we integrate Long-Short Term Memory (LSTMs) networks and Gaussian Mixture Models (GMMs) to simulate soil moisture dynamics while explicitly quantifying its associated variability. Unlike deterministic approaches, our probabilistic framework accounts for nonlinear relationships between inputs and outputs while modeling the inherent small-scale variability in soil moisture. We apply this methodology to a comprehensive in-situ soil moisture dataset from the Attert experimental basin, where the experimental design incorporates three replicated soil moisture profiles at each location and depth within a 5-meter radius. These replications are fundamental to our probabilistic framework: they provide direct, co-located observations of the natural spread in soil moisture under identical boundary conditions, allowing the model to learn the statistical structure of small-scale variability. This design enables disentangling sensor noise from genuine spatial heterogeneity and provides an empirical basis for training models that capture both temporal dynamics and local-scale variability.. Our results demonstrate that the proposed model reproduces soil moisture dynamics across multiple depths and scales, achieving an average Kling-Gupta Efficiency (KGE) of 0.52, Rank Correlation of 0.72, and Root Mean Squared Error of 0.036 m³m⁻³, while also capturing the key aspects of small-scale variability and sensor uncertainty. Furthermore, the modeled distributions offer new insights into the spatiotemporal structure of soil moisture and underscore the value of probabilistic modeling in hydrological approaches. By explicitly incorporating small-scale variability into the modeling process, our approach enhances both the interpretability and reliability of soil moisture predictions. While LSTMs effectively capture temporal dynamics, our findings underscore the necessity of incorporating variability quantification to improve model accuracy and generalization. This study highlights the potential of probabilistic deep learning frameworks in hydrological modeling and supports their broader application for improved soil moisture estimation and variability assessment.