<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing DTD v3.0 20080202//EN" "https://jats.nlm.nih.gov/nlm-dtd/publishing/3.0/journalpublishing3.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" specific-use="SMUR" dtd-version="3.0" xml:lang="en">
<front>
<journal-meta>
<journal-id journal-id-type="publisher">EGUsphere</journal-id>
<journal-title-group>
<journal-title>EGUsphere</journal-title>
<abbrev-journal-title abbrev-type="publisher">EGUsphere</abbrev-journal-title>
<abbrev-journal-title abbrev-type="nlm-ta">EGUsphere</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub"></issn>
<publisher><publisher-name>Copernicus Publications</publisher-name>
<publisher-loc>Göttingen, Germany</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.5194/egusphere-2026-549</article-id>
<title-group>
<article-title>A Memory-Based, non-Markovian, Linear Integro-Differential Equation for Root-Zone Soil Moisture</article-title>
</title-group>
<contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Rahmati</surname>
<given-names>Mehdi</given-names>
<ext-link>https://orcid.org/0000-0001-5547-6442</ext-link>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
</contrib-group><aff id="aff1">
<label>1</label>
<addr-line>Institute of Bio- and Geosciences: Agrosphere (IBG-3), Forschungszentrum Jülich, Germany</addr-line>
</aff>
<pub-date pub-type="epub">
<day>04</day>
<month>02</month>
<year>2026</year>
</pub-date>
<volume>2026</volume>
<fpage>1</fpage>
<lpage>40</lpage>
<permissions>
<copyright-statement>Copyright: &#x000a9; 2026 Mehdi Rahmati</copyright-statement>
<copyright-year>2026</copyright-year>
<license license-type="open-access">
<license-p>This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit <ext-link ext-link-type="uri"  xlink:href="https://creativecommons.org/licenses/by/4.0/">https://creativecommons.org/licenses/by/4.0/</ext-link></license-p>
</license>
</permissions>
<self-uri xlink:href="https://egusphere.copernicus.org/preprints/2026/egusphere-2026-549/">This article is available from https://egusphere.copernicus.org/preprints/2026/egusphere-2026-549/</self-uri>
<self-uri xlink:href="https://egusphere.copernicus.org/preprints/2026/egusphere-2026-549/egusphere-2026-549.pdf">The full text article is available as a PDF file from https://egusphere.copernicus.org/preprints/2026/egusphere-2026-549/egusphere-2026-549.pdf</self-uri>
<abstract>
<p>Soil-moisture memory (SMM) regulates the evolution of drought, hydrological predictability, and land&amp;ndash;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 &amp;ndash; a scale-aware Linear Integro-Differential Equation (LIDE) for root-zone soil moisture &amp;ndash; which quantifies the accumulation of memory at different timescales, e.g., fast memory (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;F&lt;/sub&gt;) and slow memory with very-short-term (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;VSS&lt;/sub&gt;), short-term (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;SS&lt;/sub&gt;), mid-term (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;MS&lt;/sub&gt;), and long-term (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;LS&lt;/sub&gt;) components as well as an additional memory saturation timescale (&lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;Sat&lt;/sub&gt;). A helper function, namely Logit&amp;ndash;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 &amp;tau;&lt;sub&gt;F&lt;/sub&gt; timescale from &amp;sim;3&amp;ndash;32 days and &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;SS&lt;/sub&gt;, &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;MS&lt;/sub&gt;, and &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;LS&lt;/sub&gt; timescales from &amp;sim;13&amp;ndash;39, &amp;sim;115&amp;ndash;127, and &amp;sim;218&amp;ndash;541 days, respectively, and a theoretical &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;Sat&lt;/sub&gt; timescale from &amp;sim;9&amp;ndash;15 years, while the &amp;tau;&lt;sub&gt;VSS&lt;/sub&gt; remained undetectable. On top of the multi-timescales&amp;rsquo; quantification, LIDE also provides additional quantitative information about memory strength, as assessed by actual memory capacity (&lt;em&gt;&amp;Kappa;&lt;/em&gt;&lt;sub&gt;Sat&lt;/sub&gt;), which is not available through conventional diagnostic metrics; with &lt;em&gt;&amp;Kappa;&lt;/em&gt;&lt;sub&gt;Sat&lt;/sub&gt; being relatively constant over the examined sites (1.12&amp;ndash;1.24 days&lt;sup&gt;-1&lt;/sup&gt;). 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 &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;F&lt;/sub&gt; and &lt;em&gt;&amp;tau;&lt;/em&gt;&lt;sub&gt;VSS&lt;/sub&gt; 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.</p>
</abstract>
<counts><page-count count="40"/></counts>
</article-meta>
</front>
<body/>
<back>
</back>
</article>