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

Rahmati, Mehdi

doi:10.5194/egusphere-2026-549

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https://doi.org/10.5194/egusphere-2026-549

Preprints

04 Feb 2026

| 04 Feb 2026

Status: this preprint is open for discussion and under review for Hydrology and Earth System Sciences (HESS).

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

Mehdi Rahmati

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.

Received: 29 Jan 2026 – Discussion started: 04 Feb 2026

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RC1:
'Revision of Manuscript hess-2026-549', Anonymous Referee #1, 14 Feb 2026 reply
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:
The paper is too long and heavy to read. It should be consistently shortened. Some parts can be moved to Supplemental Material.

While the manuscript is technically sound, several sections would benefit from a thorough linguistic review to improve flow, correct minor grammatical inconsistencies, and ensure idiomatic accuracy in complex descriptions. I reported a few inconsistencies in the Minor Comments section, but I invite the author to thoroughly revise the draft. There are a few instances where plural subjects are paired with singular verbs, such as "several other metrics is also introduced" or "convolutions of forcings... is". Some sentences are quite long and contain multiple nested clauses, which can obscure the core message. A native speaker might suggest more varied phrasing to keep the reader engaged.

I suggest to set a schematic figure in Section 2.1 or 2.2 or 2.4 (please, move Fig. 1 into one of these Sections and expand this Figure by adding a clear schematics) to help the reader fully understand the methodology indicating the key equations, the parameters into play, the parameters to be fitted, the expected model output and so on. Maybe another schematic figure could be added in Section 3.2 or 3.3 to help the reader understand on how experimental data were used in this study.

Sections 2 (Introduction) and 3 (Materials and Methods) are too fragmented. The author should re-organize the various sub-sections by stating crystal-clear titles. First of all, Section 2 has the same title of Section 1! Please, revise thoroughly for sake of clarity.

I suggest to separate Results from Discussion. Please, shorten Results as much as possible and move minor findings into Supplemental Materials. A new Discussion section is of paramount importance. The authors here should highlight the novel aspect of this study (there is novelty), how the author filled the knowledge gaps. Why is SMM so important for the soil community? SMM is used as a critical diagnostic metric to characterize the temporal behavior of soil moisture across different hydro-climatic regimes. SMM serves as an indicator of a system's ability to retain information about past trajectories. SMM identifies how quickly a soil system responds to perturbations, such as a one-week decay period in energy-limited sites versus a four-week period in water-limited sites. The framework introduces actual memory capacity, which quantifies the total influence of past moisture states, allowing for a quantitative comparison of memory strength between different soil types or climates. Different memory timescales are linked to specific soil-related processes, such as topsoil hydraulic equilibration, root-zone storage buffering, and deep vadose-zone leakage. Indicators like the t50 (halfway point of memory accumulation) reveal that intermediate sites may build up memory in 48 days, while water-limited sites require nearly a year. SMM regulates the evolution and persistence of droughts. Long-term memory (seasons to years) is essential for understanding multi-year drought legacies and recovery. Can the author expand this discussion? What are the limitations of the study? What are the future implications? The author probably already wrote these parts in the Results and Discussion but I missed them in the text that is too long and heavy. Some parts can be taken from the Conclusions where the author inserted important discussions with several citations. The conclusions should not contain citations but briefly report the most important findings (possible in bullets points). I invite the author to consistently shorten up the Conclusions.

The model predicts a saturation timescale (t_Sat) of 9–15 years. Given that the study uses a 6-year dataset, the author should clarify the uncertainty associated with extrapolating the saturation window beyond the length of the observations.

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.

Reply
Citation: https://doi.org/10.5194/egusphere-2026-549-RC1
- AC1:
  'Reply on RC1', Mehdi Rahmati, 17 Mar 2026 reply
  Response to Reviewer #1 comments on paper (egusphere-2026-549) entitled “A Memory-Based, non-Markovian, Linear Integro-Differential Equation for Root-Zone Soil Moisture” submitted to Hydrology and Earth System Sciences (HESS)
  I would like to thank the reviewer for his/her constructive and helpful evaluation of this manuscript. What I am particularly happy with is reading that the reviewer understands the novelty of the LIDE framework and its potential to account for multi-scale memory dynamics hidden when using traditional, single-timescale persistence quantities. I greatly value your feedback, and I agree that reformatting the manuscript in a way you suggested will highly improve the clarity, usefulness, and accessibility of the information for the hydrological community.
  Here are the structural and content revisions I am going to make:
  Structural Reorganization: One of the major issues creating fragmentation in my draft is that subsections under the introduction are logically not progressing from motivation to literature review and then theoretical foundation. In addition, I will split the Results and Discussion into separate sections. This will enable a concise presentation of results followed by a vigorous discussion section focused on different aspects as suggested by the reviewer.
  
  Expanded Visual Representations: As recommended, I will also move Figure 1 to the Methods section and expand its content with a clear conceptual schematic representation of the LIDE methodology. This diagram will outline the main equations, parameters, and results of interest to the reader as a guide for visualizing how the framework works. Further, I will add a second schematic showing how lysimeter data is decoded and passed to the LIDE framework.
  
  Streamlining for Readability: To help reduce length and improve readability, I will shift extended mathematical derivations and auxiliary findings to an appendix or supplementary material. Furthermore, a complete nomenclature table will also be provided in the appendix.
  
  Methodology Clarification: I will clarify much better in the new version how the LIDE approximation relates to Richards’ equation and what assumptions are involved. I will also address the uncertainty on the saturation timescale (τ_sat) when it is set outside the region of observational fit (6 yr).
  
  Refining the Narrative: I will conduct a thorough linguistic review to make sure everything flows perfectly and correct grammatical inconsistencies and ensure the text is concise and accessible. The conclusions will also be distilled into a bulleted key finding list, in response to your suggestion.
  
  I am quite confident that these revisions will preserve the paper's mathematical rigor while clarifying a diagnostic relationship of the LIDE framework for the broad audience. I am now preparing a point-by-point response to cover each specific technical and editorial issue, and I hope to submit a much-improved version of this manuscript in the near future.
  
  Reply
  
  Citation: https://doi.org/10.5194/egusphere-2026-549-AC1
RC2:
'Comment on egusphere-2026-549', Anonymous Referee #2, 11 Mar 2026 reply
General comments
The manuscript presents a framework for quantifying soil moisture memory (SMM) using a non-Markovian Linear Integro-Differential Equation (LIDE) approach. The motivation is clearly stated, as conventional persistence metrics often reduce soil moisture memory to a single characteristic timescale may not adequately capture the complex and multi-scale nature of soil moisture dynamics. The author derives the governing equation using the Mori-Zwanzig projection formalism and interprets soil moisture dynamics as a system with memory effects represented through a kernel function that accounts for the influence of past states. This formulation separates instantaneous responses from distributed memory effects and allows the characterization of multiple memory regimes. The framework is applied to lysimeter observations from sites in Germany, where the analysis reveals a hierarchy of soil moisture memory timescales ranging from days to several years. The topic is relevant and of high importance, since soil moisture memory plays an important role for hydrological predictability, drought persistence, and land-climate interactions. The theoretical development is sound and the results are clear. Overall, the manuscript addresses an interesting problem and the proposed framework provides a useful perspective for analyzing soil moisture persistence. However, several aspects of the manuscript would benefit from clarification and revision before publication.
Comments
The overall structure of the manuscript could be improved to enhance clarity and readability. The manuscript currently contains two sections titled Introduction, which should be revised to ensure a clearer progression of ideas. In addition, the extensive mathematical explanations would be more appropriately placed in the Methods section and could be shortened or partially moved to supplementary material to improve readability. The Results and Discussion should be presented as separate sections, allowing results to be clearly separated from their interpretation that would also make novelty and importance more highlighted.

The manuscript introduces several memory metrics, including fast memory, cumulative memory, and saturation memory. While these metrics are mathematically defined, their hydrological interpretation remains somewhat unclear. A clearer conceptual explanation of how these regimes relate to known soil and vegetation processes such as infiltration, drainage, evapotranspiration, and deeper soil storage would improve the accessibility of the results.

The empirical evaluation is based on a small number of lysimeter observations in Germany. Although lysimeters provide high-quality measurements, the limited spatial coverage raises questions regarding the generality of the conclusions. Soil moisture memory strongly depends on factors such as soil texture, vegetation type, rooting depth, and climatic conditions. A discussion of how the identified memory regimes might vary under different hydro-climatic settings would strengthen the manuscript and help place the results in a broader context.

Results and discussion section presents various different findings. The reader has problems identifying their importance and significance. I believe a large benefit would be to explicitly articulate the main goals in the introduction section and discuss those in a separate discussion section. That would help the reader grasp the main message of the paper that wants to be delivered. Minor/less important findings can be put in a separate section in discussion, provided in the supplementary material, or due to the length of the documents just left out to make the reader able to focus on the more important and novel outcomes of the research.

Minor comments
Lines 295–320: The phrasing in this section lacks clarity and flow and should be revised.

Due to the large amount of information presented in the manuscript, some of the extended mathematical derivations and secondary findings could be moved to an Appendix or Supplementary Material to improve readability and maintain focus on the main results.

Reply
Citation: https://doi.org/10.5194/egusphere-2026-549-RC2
- AC2:
  'Reply on RC2', Mehdi Rahmati, 17 Mar 2026 reply
  Response to Reviewer #2 comments on paper (eagusphere-2026-549) entitled “A Memory-Based, non-Markovian, Linear Integro-Differential Equation for Root-Zone Soil Moisture” submitted to Hydrology and Earth System Sciences (HESS)
  I appreciate the reviewer for his/her thoughtful and constructive review of this manuscript. I am happy that the reviewer acknowledged the significance of the adopted Linear Integro-Differential Equation (LIDE) approach to capture multi-scale properties of soil moisture memory, and I appreciate specific, actionable guidelines for improving the clarity and impact of this work.
  I agree with the reviewer’s suggestions to improve the writing structure of the manuscript. More specifically, I intend to address these points as follows:
  Redefining the Content: I will combine all parts currently listed as "Introduction" into a single, cohesive narrative. In addition, I will restructure the manuscript to clearly distinguish between the Methods, Results, and Discussion sections. Doing so will enable the key findings and their hydrological significance to be highlighted more clearly, as recommended.
  
  Improving the Accessibility of the Metrics: I realize that the hydrological meaning of the derived memory metrics—fast, slow, saturation—should be more intuitive. Nonetheless, I am going to aim towards providing a more obvious conceptual mapping from the defined mathematical terms to physical processes (infiltration, drainage, and evapotranspiration) in a later version of the paper to better assist the reader in interpreting the results.
  
  Contextualizing the Results and Future Scope: I understand the interest of the reviewer in the generality of these findings beyond German lysimeter sites. Although a global-scale analysis of soil moisture memory drivers will not be conducted in the current paper (in particular, to maintain the focus and length of this manuscript), it is an important objective of a follow-on study underway. In that ongoing effort, we are extending the LIDE framework to a broader (global-scale) array of hydroclimatic environments in order to systematically assess the relative contributions of various soil textures, vegetation phonologies, and rooting depths on the identified memory regimes. In the Discussion, I will point out that this ongoing work is far better equipped to systematically address these variances, and I will offer a brief synthesis of likely drivers of memory variability, which can benchmark the current results as a methodologically robust comparison point for these future, broader analyses.
  
  Focusing the Narrative: I will take the reviewer’s advice to clarify how the results are presented. I will simplify the mathematical derivation in the main body of the paper by moving secondary mathematics and less interesting results into an appendix or supplementary material so that I can keep it focused on novel, impactful research outcomes.
  
  I believe these revisions will greatly improve both the readability and perceived impact of the framework. I am currently working on a point-by-point response in order to respond to all comments in detail, and I will ensure that suggested structural and conceptual changes will appear in the revised draft.
  
  Reply
  
  Citation: https://doi.org/10.5194/egusphere-2026-549-AC2
RC3:
'Comment on egusphere-2026-549', Anonymous Referee #3, 16 Mar 2026 reply

Dear author,
The paper adds little to the available toolbox for time series analysis. The Introduction does not discuss competing methods in sufficient detail. The theory section (of which the first part is labeled Introduction) is not well written, and leaves out essential information (see annotated manuscript).
The test of the model only comprises a calibration, followed by an excessively long presentation of details of this calibration. The amount of conjecture show how poor the model is in explaining anything.
The discussion leaves the relevance of the work unclear.
Below are my main comments. Detailed comments are in the attached file.
________________________________________________________________________________
You introduce you model equations without explaining at which spatial and temporal scales you model operates, although the introduction gives some hints.
You have a root zone, but no clear representation of the vegetation, apart from a root water uptake term that is no longer there when you present your model equations.
In section 2.1, you offer a rationale for your approach by suggesting that the conventional alternative is to solve Richards' equation for three coupled layers in the unsaturated zone. As I explain in the detailed comment there, you incorrectly describe how Richards' solvers (mainly Hydrus-1D and SWAP) are used. You need to come up with something better.
Equations 3-5 coalesce external forcings and noise in the same variable. Why is this not a terrible ideas? If all the forcing terms are in epsilon, you have no way of separating the effect of the forcing from that of the noise, for instance. The terms in these equations are not adequately explained. I offer a few possibilities in the detailed comments, that come with their own problems.
In the case of infiltration, you cannot switch from conductivity-limited to capacity-limited infiltration. In the case of root water uptake, you can neither include the effect of water stress, nor the effect on root water uptake capability and potential transpiration of previous stress events. These limitations apply even more clearly if you place root water uptake in the linear loss term (with lambda). With these limitations, what potential use do you envision for your model?
The surface layer is minute, so it will have a very small characteristic time. Also, when saturated, it does not operate as a linear reservoir during periods with infiltration (or it does, but for a reservoir with a characteristic time equal to zero), because it will pass on infiltration to the root zone instantaneously. This creates several problems, as indicated in the detailed comments.
Overall, the very limited capability of the model to respond to forcings in a way that reflects its current status (liquid flow or vapor flow in the surface layer, water stress and effects of previous stressors on the vegetation in the root zone, dynamics of he groundwater table) limit the contribution of this work rather dramatically, especially since there are plenty of models that incorporate this.
The paper uses data o the target variable only, fits a sigmoidal function to a quantity that is related to the temporal autocorrelation of that variable on three data sets, and then elaborately describes the result of this fitting exercise. There is no validation, and hence no evaluation of the predictive quality of the model. But because only the target variable was measured, the model does not have a predictive mode. This is simply a descriptive model, essentially an autocorrelation analysis. The author appears top have published something akin to a semivariogram for the root zone water content time series, but nothing more. I do not see why that should warrant a publication, nor do I see the relevance of this.
These weaknesses are sufficient to recommend rejection, unfortunately. I therefore did not review the remaining sections in detail.
You seem to use the term 'memory' for temporal autocorrelation, which is a much clearer term. When you use that term, you may benefit from studying the literature on spatial and temporal autocorrelation. Later in the paper you go into temporal autocorrelation, but not to benefit from it, but mainly to criticize how it has been deployed so far in a limited number of papers. A more extensive review would have strengthened the Introduction, but possibly would have pointed you to useful alternatives to you approach..
You claim that the sum of parameters alpha and beta quantify the total memory capacity. I believe that beta represents zero memory, and alpha maximum memory. Perfect memory arises if alpha = 0: in that case, a value at an arbitrary time can be predicted error-free from a previous value of the same variable at the shortest lag-time (presumably zero). If alpha is larger then zero, the target variable has a memory-free component,.i.e., here is some random variation that generates a base level of noise. Later in the paper you appear to acknowledge this, contradicting this part of the text.
If the notation in a paper is elaborate, an appendix with the names, notation, dimensions, and meaning of all variables is useful.
The differential operator 'd' is an operator, not a variable. It should not be in italics.
The notation in unnecessarily frivolous. Limit yourself to numbers for footnotes, and Latin and Greek alphabets in equations if possible.

Reply

Citation: https://doi.org/10.5194/egusphere-2026-549-RC3
- AC3: 'Reply on RC3', Mehdi Rahmati, 17 Mar 2026 reply
  
  Response to Reviewer #3 comments on paper (egusphere-2026-549) entitled “A Memory-Based, non-Markovian, Linear Integro-Differential Equation for Root-Zone Soil Moisture” submitted to Hydrology and Earth System Sciences (HESS)
  I would like to thank the reviewer for his/her time and specific comments on the manuscript. Although I value the attempt to engage with this work, I have some concerns regarding the reviewer’s understanding of the fundamental scope of the paper. The reviewer critiques this work mostly as a predictive model; however, the proposed framework is constructed as a diagnostic, data-driven framework, specifically for observation-based model diagnostics—a Linear Integro-Differential Equation (LIDE)—that would support identification and quantification of the multi-scale memory effects inherent to root-zone soil moisture.
  If indeed the LIDE were to be used as a fully predictive hydrological model, I agree with the reviewer that our choice of vegetation dynamics, explicit use of Richards' equation coupling, and independent validation datasets will be crucial (to my point, however, this is not currently necessary). Indeed, the current work lays down the diagnostic foundation specifically to enable such future advancements. Furthermore, imposing these “predictive-model” constraints onto the existing “diagnostic framework” conflates our goal of extracting and quantifying (non-Markovian) memory features.
  This is most certainly the case with respect to independent validation, where indeed full time series are needed to numerically solve the integrodifferential equation and characterize the memory kernel in accordance with LIDE. Because this is a diagnostic derivation based on the historical time trajectory of the system, rather than a forward-predictive simulation, the standard “calibration/validation” split does not apply to this methodology.
  In addition, I would like to highlight that we do provide a specialized section in the manuscript where the LIDE framework is contrasted against some of the prior existing approaches—such as e-folding-based methods or hybrid models. These comparisons show that existing approaches represent local aspects of memory in soils, while the LIDE is a general, multi-scale quantity that neither of these simpler metrics captures.
  Currently, I am writing a point-by-point response to the reviewer's comments. I hope to address the reviewer's concerns, where they are valid, and provide justification for those that I consider not valid. I welcome constructive dialogue that highlights the diagnostic contribution of this study while outlining the necessary steps for future predictive applications.
  
  Reply
  
  Citation: https://doi.org/10.5194/egusphere-2026-549-AC3

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Short summary

Soil moisture memory impacts drought onset, weather forecast and land–atmosphere interactions while it is often characterized by one or two memory time-scales. We developed an observation-driven approach to uncover it across many short and long time scales. Based on measurements in Germany, we demonstrate that soil moisture exhibits the memory from days up to years and have a limit value where it is saturated. This helps understanding the drought persistence and land-surface response.


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