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

Abstract. 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 – which quantifies the accumulation of memory at different timescales, e.g., fast memory (τ_F) and slow memory with very-short-term (τ_VSS), short-term (τ_SS), mid-term (τ_MS), and long-term (τ_LS) components as well as an additional memory saturation timescale (τ_Sat). A helper function, namely Logit–Piecewise Memory Segmentation (LPMS) method, is also developed which automates the timescales detection. When applied to lysimeter-based in-situ daily-based observations from three different hydro-climatic regimes in Germany lasting for 2013 to 2018, LIDE reveals a τ_F timescale from ∼3–32 days and τ_SS, τ_MS, and τ_LS timescales from ∼13–39, ∼115–127, and ∼218–541 days, respectively, and a theoretical τ_Sat timescale from ∼9–15 years, while the τ_VSS remained undetectable. On top of the multi-timescales’ quantification, LIDE also provides additional quantitative information about memory strength, as assessed by actual memory capacity (Κ_Sat), which is not available through conventional diagnostic metrics; with Κ_Sat being relatively constant over the examined sites (1.12–1.24 days^-1). The integrated kernel also allows to retrieve the oscillatory saturation dynamics associated with soil-moisture reemergence from observations for the first time. Applying LIDE to hourly, daily, and monthly data reveals its scale-aware nature, whereas when applied to hourly data, it provides additional timescales (e.g., sub-daily τ_F and τ_VSS timescales), while when applied to coarser data, it smooths them out. Collectively, obtained results place LIDE as a state-of-the-art and state-of-the-practice approach in quantifying SMM characteristics that are physically interpretable and scalable and can greatly advance drought sciences, ecohydrology and land-surface modeling.