| 04 Feb 2026
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 (τF) and slow memory with very-short-term (τVSS), short-term (τSS), mid-term (τMS), and long-term (τLS) components as well as an additional memory saturation timescale (τSat). 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 τF timescale from ∼3–32 days and τSS, τMS, and τLS timescales from ∼13–39, ∼115–127, and ∼218–541 days, respectively, and a theoretical τSat timescale from ∼9–15 years, while the τVSS remained undetectable. On top of the multi-timescales’ quantification, LIDE also provides additional quantitative information about memory strength, as assessed by actual memory capacity (ΚSat), which is not available through conventional diagnostic metrics; with ΚSat 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 τF and τVSS 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.
General Comments
This manuscript presents a significant advancement in the quantification of soil moisture memory (SMM) by introducing the Linear Integro-Differential Equation (LIDE) framework. By utilizing the Mori-Zwanzig formalism, the author moves beyond the sole memory timescale limitation of conventional diagnostic metrics. The introduction of the Logit-Piecewise Memory Segmentation (LPMS) method provides an automated tool for identifying multi-scale memory regimes. The paper is well-structured, mathematically rigorous, and physically grounded under two contrasting hydro-climatic data. Unlike Markovian models, LIDE separates the instantaneous decay (via the frequency coefficient l) from the distributed memory via the kernel function, K(t). This allows for a more nuanced characterization of "fast" vs. "slow" memory. The finding that fast memory (t_F) is significantly shorter in energy-limited sites (~7.6 days) compared to water-limited sites (~32.4 days) is physically consistent with the system's ability to dissipate anomalies through evapotranspiration. The article is worth publication in HESS, however, could improve consistently if the author addresses the following concerns:
Minor comments
Please, number all lines
Line 87: “function”
Lines 87-88: the sentence construction is unclear. Rephrase
Lines 84-96: In Section 2.1 the author provides two methods to describe the temporal variation of soil water content in a layered soil profile. The first method is based on the water flux simulation across the soil profile using the Richards equation. In such circumstances the numerical solution of the Richards equation occurs in the soil profile that is discretized in N adjoining elements. Then, the soil water content on the soil surface, in the root zone (i.e., at the depth of 30 cm) and in the deep layer (i.e., at the depth of 100 cm) can be recorded because the Richards equation is solved in every element of the soil profile.
From line 96 the author presents the second method to get soil water content at the desired soil depth. I think the text needs to be clarified because Equations (3,4,5) can replace and (crudely) approximate the Richards equation (under which assumptions??), then we simplify to theta_root. Is that correct?
Line 110: why is theta_root usually set to zero at t0?
Eq. 8: is the l in Eq. 8 the same l appearing in Equations (3,4,5)?
Line 121: I am not familiar with Gottwald et al. (2016), but the LIDE is a crude approximation of the Richards equation which is the benchmark equation in soil hydrology in most cases (assuming preferential flow is negligible etc.). Or the author refers to another equation? Please clarify in the reply and in the text.
Line 131: “… and determine its parameters..”
Please add a Nomenclature Table in the Appendix by declaring all symbols used in the article with their meaning and units
Line 196: replace “frequencies of days or longer” with “daily temporal resolution or coarser”
Figure quality can improve consistently. Harmonize fontsize. For instance in Figure 3 the figure titles and panels a1, a2, etc. are larger than the other characters (axis labels, axis titles, equations and so on)
Conclusion
This study is of high quality and provides a "state-of-the-practice" approach for land-surface modeling and drought science. I recommend it for publication after addressing the points raised in this review.