A Memory-Based, non-Markovian, Linear Integro-Differential Equation for Root-Zone Soil Moisture

Soil-moisture memory (SMM) regulates the evolution of drought, hydrological predictability, and land–atmosphere coupling, yet many conventional diagnostic metrics simplify this complex phenomenon into a sole memory timescale. In this paper, we introduce a unified observation-driven framework—a scale-aware Linear Integro-Differential Equation (LIDE) for root zone soil moisture—to infer the complete distributed memory kernel that effectively models soil-moisture dynamics. When applied to multi-year in situ observations from energy-limited, water-limited, and intermediate hydro-climatic regimes, LIDE reveals a rich hierarchy of memory structures that conventional e-folding autocorrelation or hybrid deterministic-stochastic metrics are unable to capture. Application of LIDE in examined sites revealed a fast-memory timescale from ∼3–32 days, a short-term slow-memory timescale from 13 to 39 days, an intermediate slow-memory from ∼115–127 days, a long-term slow-memory from ∼218–541 days, and a theoretical saturation timescale from ~9 to 15 years. LIDE also provides additional quantitative information about memory strength, as assessed by actual memory capacity (Q_∞), which is not available through conventional persistence analyses, with Q_∞ being relatively constant over the examined sites (1.12–1.24 days⁻²) despite large hydro-climatic contrasts among sites. Applying LIDE on hourly, daily, and monthly data reveals that high-frequency data provides information on sub-daily fast memory timescales (~6 hours at the intermediate site, namely Schöneseiffen in Germany), as well as an additional very short slow-memory timescale (~14 hours at Schöneseiffen) that is not observable in daily or monthly data. The integrated kernel also accounts for the oscillatory saturation dynamics associated with soil-moisture reemergence, making it possible to retrieve this process from observations for the first time. Collectively, these results place LIDE as a state-of-the-art and state-of-the-practice approach in diagnosing multiscale memory of the soil moisture that is physically interpretable and scalable and can greatly advance drought sciences, ecohydrology, and land-surface modeling.