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<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Remote Sens.</journal-id>
<journal-title-group>
<journal-title>Frontiers in Remote Sensing</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Remote Sens.</abbrev-journal-title>
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<issn pub-type="epub">2673-6187</issn>
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<publisher-name>Frontiers Media S.A.</publisher-name>
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<article-meta>
<article-id pub-id-type="publisher-id">1718353</article-id>
<article-id pub-id-type="doi">10.3389/frsen.2025.1718353</article-id>
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<article-categories>
<subj-group subj-group-type="heading">
<subject>Original Research</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>ASCAT soil moisture retrieval using deep learning: a focus on localization strategy</article-title>
<alt-title alt-title-type="left-running-head">Dinh</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/frsen.2025.1718353">10.3389/frsen.2025.1718353</ext-link>
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<name>
<surname>Dinh</surname>
<given-names>Lan Anh</given-names>
</name>
<xref ref-type="aff" rid="aff1"/>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
<xref ref-type="author-notes" rid="fn001">
<sup>&#x2020;</sup>
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<uri xlink:href="https://loop.frontiersin.org/people/1352816"/>
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<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Data curation" vocab-term-identifier="https://credit.niso.org/contributor-roles/data-curation/">Data curation</role>
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<aff id="aff1">
<institution>Estellus</institution>, <city>Paris</city>, <country country="FR">France</country>
</aff>
<author-notes>
<corresp id="c001">
<label>&#x2a;</label>Correspondence: Lan Anh Dinh, <email xlink:href="mailto:lan-anh.dinh@obspm.fr">lan-anh.dinh@obspm.fr</email>
</corresp>
<fn fn-type="present-address" id="fn001">
<label>&#x2020;</label>
<p>
<bold>Present addresses:</bold> Lan Anh Dinh, LIRA, Observatoire de Paris, Universit&#xe9; PSL, Sorbonne Universit&#xe9;, CNRS, Paris, France</p>
</fn>
</author-notes>
<pub-date publication-format="electronic" date-type="pub" iso-8601-date="2026-01-23">
<day>23</day>
<month>01</month>
<year>2026</year>
</pub-date>
<pub-date publication-format="electronic" date-type="collection">
<year>2025</year>
</pub-date>
<volume>6</volume>
<elocation-id>1718353</elocation-id>
<history>
<date date-type="received">
<day>03</day>
<month>10</month>
<year>2025</year>
</date>
<date date-type="rev-recd">
<day>11</day>
<month>12</month>
<year>2025</year>
</date>
<date date-type="accepted">
<day>15</day>
<month>12</month>
<year>2025</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2026 Dinh.</copyright-statement>
<copyright-year>2026</copyright-year>
<copyright-holder>Dinh</copyright-holder>
<license>
<ali:license_ref start_date="2026-01-23">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License (CC BY)</ext-link>. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</license-p>
</license>
</permissions>
<abstract>
<p>This study investigates the intercomparison of daily soil moisture (SM) retrieval from ASCAT (Advanced SCATterometer) observations using machine learning. The exploitation of spatial structure through convolutional neural networks (CNNs) is shown to significantly enhance retrieval performance compared to a standard multilayer perceptron (MLP), with spatial correlation with the target ERA5 SM increasing from 0.55 to 0.91 and temporal correlation from 0.61 to 0.73. Incorporating &#x201c;localization&#x201d; (i.e., a strategy to adjust the neural network (NN) behavior to local conditions) into the model is a key factor for improving retrieval quality, resulting in more accurate SM estimates, reduced regional biases, improved temporal dynamics, and more realistic representations of extreme SM events. Our NN-based retrievals show strong agreement with <italic>in situ</italic> SM measurements, achieving temporal correlations of 0.60 and 0.68 for the MLP and CNN models, respectively, in the contiguous United States (CONUS) during 2019. These findings underscore the critical role of spatial learning and localization in SM retrieval from remote sensing data such as ASCAT.</p>
</abstract>
<kwd-group>
<kwd>ASCAT</kwd>
<kwd>deep learning</kwd>
<kwd>localization</kwd>
<kwd>neural network</kwd>
<kwd>soil moisture retrieval</kwd>
</kwd-group>
<funding-group>
<funding-statement>The author(s) declared that financial support was received for this work and/or its publication. The CERISE project (grant agreement no. 101082139) is funded by the European Union.</funding-statement>
</funding-group>
<counts>
<fig-count count="9"/>
<table-count count="2"/>
<equation-count count="3"/>
<ref-count count="62"/>
<page-count count="18"/>
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<custom-meta-group>
<custom-meta>
<meta-name>section-at-acceptance</meta-name>
<meta-value>Microwave Remote Sensing</meta-value>
</custom-meta>
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</front>
<body>
<sec sec-type="intro" id="s1">
<label>1</label>
<title>Introduction</title>
<p>Soil moisture (SM) is a vital component of the Earth system, regulating the exchange of water and energy between the land surface and the atmosphere. It serves as a critical factor in numerous applications, including agricultural monitoring, drought and flood forecasting, water resource management, and climate prediction (<xref ref-type="bibr" rid="B51">Seneviratne et al., 2010</xref>; <xref ref-type="bibr" rid="B39">Ochsner et al., 2013</xref>; <xref ref-type="bibr" rid="B57">Vereecken et al., 2015</xref>). Recognizing its importance, extensive efforts have been made to obtain SM estimates across different temporal and spatial scales. These methods include ground-based <italic>in situ</italic> measurements (<xref ref-type="bibr" rid="B16">Dorigo et al., 2013</xref>), satellite-based retrievals (<xref ref-type="bibr" rid="B35">Liu et al., 2012</xref>; <xref ref-type="bibr" rid="B6">Aires et al., 2021b</xref>), and land surface model simulations (<xref ref-type="bibr" rid="B23">Guo et al., 2006</xref>). While each method has its strengths, they also face limitations related to coverage, representativeness, and accuracy.</p>
<p>In the last two decades, data-driven methods, particularly neural networks (NNs), have shown strong potential for SM retrieval (<xref ref-type="bibr" rid="B2">Aires et al., 2001</xref>; <xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>; <xref ref-type="bibr" rid="B46">Rodr&#xed;guez-Fern&#xe1;ndez et al., 2015</xref>; <xref ref-type="bibr" rid="B6">Aires et al., 2021b</xref>). Multilayer perceptrons (MLPs) provide a robust and interpretable pixel-scale baseline, aligning with current operational retrieval frameworks (<xref ref-type="bibr" rid="B47">Rodr&#xed;guez-Fern&#xe1;ndez et al., 2019</xref>; <xref ref-type="bibr" rid="B6">Aires et al., 2021b</xref>). More advanced convolutional neural networks (CNNs) extend these models by exploiting the satellite data at the image scale. Unlike physically based inversion methods, these approaches are purely data-driven: they learn statistical relationships between remote sensing observations and soil moisture states, enabling flexible, scalable, and potentially more accurate retrievals (<xref ref-type="bibr" rid="B62">Zhang et al., 2025</xref>). However, their performance is highly dependent on the choice of input data. The inclusion of relevant predictor variables, whether physical (e.g., brightness temperature, vegetation indices, land surface temperature) or geographical (e.g., land cover, soil texture, topography), directly influences retrieval accuracy and generalization.</p>
<p>Previous retrieval studies have improved physical realism by integrating auxiliary variables to constrain the radiative transfer models better. For instance, <xref ref-type="bibr" rid="B2">Aires et al. (2001)</xref> included surface emissivities along with microwave satellite observations to improve surface temperature retrieval, effectively accounting for local variations such as humidity and vegetation. Due to the lack of reliable, real-time emissivity products at that time, first-guess emissivities from climatologies were used to provide a seasonal and location-specific constraint. In another example, as SM influences various satellite observations (<xref ref-type="bibr" rid="B42">Prigent et al., 2005</xref>), early efforts such as <xref ref-type="bibr" rid="B4">Aires et al. (2005)</xref> developed global NN retrieval models that leveraged synergies across visible/infrared, active, and passive microwave sensors. This global approach has continued in later studies, such as in <xref ref-type="bibr" rid="B32">Kolassa et al. (2016)</xref>, and has evolved to include a broader range of inputs. More recently, <xref ref-type="bibr" rid="B22">Greifeneder et al. (2021)</xref> integrated data from a diverse collection of datasets (i.e., synthetic aperture radar, Landsat, MODIS, land cover, and other sources), with a total of 62 input features for a global NN-based SM retrieval. The growing trend in NN-based SM retrievals has been toward using increasingly large and diverse input sets. While this may help capture complex environmental interactions, it can sometimes obscure the role of localized factors. To address this, some studies have begun to explicitly introduce geo-localization information to enhance localization in NN models (<xref ref-type="bibr" rid="B38">Mahara and Rishe, 2023</xref>; <xref ref-type="bibr" rid="B53">Singh and Gaurav, 2023</xref>; <xref ref-type="bibr" rid="B26">Han et al., 2023</xref>). Incorporating both physical (i.e., dynamic) and geographical (i.e., static) inputs has thus become a common strategy.</p>
<p>In this study, we evaluate the performance of NN-based models for SM retrieval from the Advanced SCATterometer (ASCAT) observations, with a particular focus on the role of &#x201c;localization&#x201d;&#x2014;&#x2013;the incorporation of region-specific information to allow NN models to adapt more effectively to local conditions (<xref ref-type="bibr" rid="B13">Boucher et al., 2023</xref>; <xref ref-type="bibr" rid="B40">Pellet et al., 2025</xref>). We compare MLPs and CNN2 with varying levels of localization, ranging from simple models based only on physical or geographical inputs to highly localized approaches such as pixel-wise MLPs and locally connected CNNs. Other modern architectures, such as convolutional neural network-long short-term memory (ConvLSTM) models, which account for both spatial and temporal dependencies, have demonstrated strong performance in forecasting SM (<xref ref-type="bibr" rid="B60">Wang et al., 2024</xref>; <xref ref-type="bibr" rid="B44">Rabiei et al., 2025b</xref>; <xref ref-type="bibr" rid="B43">Rabiei et al., 2025a</xref>). However, as this study focuses on retrieval rather than forecasting, such models were excluded from our analysis. Nonetheless, temporal modeling could also offer potential benefits for retrieval. Further extensions will therefore investigate temporal architectures that are better suited for retrieval tasks than ConvLSTM, such as transformers or embedding-based methods. We assess here the ability of MLPs and CNNs to retrieve daily SM and capture extreme values (minimum and maximum) using a 2019 validation dataset. Model performance is evaluated against <italic>in situ</italic> measurements from the International Soil Moisture Network (ISMN; <xref ref-type="bibr" rid="B16">Dorigo et al., 2013</xref>; <xref ref-type="bibr" rid="B17">Dorigo et al., 2021</xref>) and compared with the ASCAT-derived SM product. The analysis is focused on the contiguous United States (CONUS; 125<inline-formula id="inf1">
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</inline-formula>N), a region characterized by dense <italic>in situ</italic> measurements and pronounced spatial variability (<xref ref-type="bibr" rid="B10">Bernhardt et al., 2018</xref>). While CONUS provides an ideal testbed for assessing model skill, the methodology is inherently generalizable and applicable to other well-instrumented regions, such as Europe.</p>
<p>In the following, <xref ref-type="sec" rid="s2">Section 2</xref> describes the datasets employed in this study and presents preliminary information content analysis between the target SM and the other considered predictors. <xref ref-type="sec" rid="s3">Section 3</xref> details the NN-based architectures, localization strategies, and evaluation metrics. <xref ref-type="sec" rid="s4">Section 4</xref> assesses model performance at the daily scale and investigates the effects of localization. <xref ref-type="sec" rid="s5">Section 5</xref> focuses on extreme SM retrievals, comparing localized and non-localized NN-based models. <xref ref-type="sec" rid="s6">Section 6</xref> reports the evaluation against <italic>in situ</italic> measurements. Finally, <xref ref-type="sec" rid="s7">Section 7</xref> summarizes the findings, highlights practical implications, and discusses the potential of deep learning and localization approaches for advancing remote sensing&#x2013;based SM retrievals.</p>
</sec>
<sec id="s2">
<label>2</label>
<title>Datasets</title>
<sec id="s2-1">
<label>2.1</label>
<title>ERA5 database</title>
<p>We employed the ERA5 reanalysis dataset (i.e., the fifth generation of ECMWF (European Centre for Medium-Range Weather Forecasts) reanalysis) (<xref ref-type="bibr" rid="B29">Hersbach et al., 2023</xref>), which provides consistent land&#x2013;atmosphere variables at hourly temporal resolution and 0.25<inline-formula id="inf5">
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<mml:mrow>
<mml:mo>&#xb0;</mml:mo>
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</inline-formula> spatial resolution, based on the ECMWF coupled assimilation system. Here, we chose ERA5 SM as the training target since it currently provides one of the most accurate and spatially coherent large-scale SM datasets available (<xref ref-type="bibr" rid="B55">Soci et al., 2024</xref>). This choice is a well-established practice in previous developments of machine-learning SM retrievals (<xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>; <xref ref-type="bibr" rid="B47">Rodr&#xed;guez-Fern&#xe1;ndez et al., 2019</xref>; <xref ref-type="bibr" rid="B6">Aires et al., 2021b</xref>; <xref ref-type="bibr" rid="B40">Pellet et al., 2025</xref>).</p>
<p>The following ERA5 variables relevant to SM dynamics were considered in this study:</p>
<p>
<italic>Soil moisture (SM)</italic> &#x2013; Volumetric SM (<inline-formula id="inf6">
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<p>
<italic>Soil temperature (ST)</italic> &#x2013; ST is important for retrieving SM due to its thermal properties (<xref ref-type="bibr" rid="B14">Campbell, 1985</xref>), which indirectly relate ST and soil water content. We used the topmost layer (0&#x2013;7&#xa0;cm), which is most relevant to ASCAT observations.</p>
<p>
<italic>Leaf area index (LAI)</italic> &#x2013; LAI was included as a vegetation descriptor due to its impact on the backscatter coefficient in microwave remote sensing (<xref ref-type="bibr" rid="B42">Prigent et al., 2005</xref>; <xref ref-type="bibr" rid="B41">Petchiappan et al., 2022</xref>).</p>
<p>
<italic>Antecedent precipitation evaporation index (APEI)</italic> &#x2013; APEI is a well-established predictor of SM, representing the balance between precipitation and evaporation integrated over a decaying temporal window (<xref ref-type="bibr" rid="B52">Shaw et al., 1997</xref>; <xref ref-type="bibr" rid="B26">Han et al., 2023</xref>). It is defined as:<disp-formula id="e1">
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</inline-formula> &#x3d; 33, as this configuration demonstrated a strong correlation with ERA5 SM in the preliminary analysis (see <xref ref-type="sec" rid="s2-5">Section 2.5</xref>). Although these values were presented as globally optimized parameters, they have been validated against a long-term <italic>in situ</italic> SM network across the United States from the United States Climate Reference Network (USCRN). We also conducted additional tests (not shown), exploring alternative parameter combinations, but did not observe any significant improvements. Therefore, we have decided to use the same combinations of <inline-formula id="inf16">
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</inline-formula> as in <xref ref-type="bibr" rid="B26">Han et al. (2023)</xref>, which are considered appropriate for our study region.</p>
<p>
<italic>Amplitude of the diurnal surface temperature cycle (dT)</italic> &#x2013; The diurnal temperature amplitude is closely linked to soil thermal inertia and, consequently, to SM content (<xref ref-type="bibr" rid="B42">Prigent et al., 2005</xref>). Therefore, alongside the aforementioned variables, dT serves as an auxiliary predictor in our SM retrieval framework. We computed dT as the difference between the daily maximum and minimum hourly ERA5 surface temperatures.</p>
<p>The original hourly ERA5 variables were aggregated to a daily scale. Precipitation and evaporation were converted to cumulative daily values, while all other variables were averaged over a 24-h period.</p>
</sec>
<sec id="s2-2">
<label>2.2</label>
<title>ASCAT database</title>
<p>ASCAT is a C-band radar onboard the Metop-A, B, and C satellites, providing near-global coverage every 12&#xa0;h from a sun-synchronous orbit at 817&#xa0;km altitude. We used the Metop ASCAT Climate Data Record (CDR, H120 v7), which offers consistent SM-related products at a resampled 12.5&#xa0;km resolution (from native 25&#x2013;34&#xa0;km, depending on swath position) (<xref ref-type="bibr" rid="B24">H SAF, 2021a</xref>; <xref ref-type="bibr" rid="B25">H SAF, 2021b</xref>). Two ASCAT variables were employed:</p>
<p>
<italic>Backscatter</italic> <inline-formula id="inf18">
<mml:math id="m19">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> &#x2013; Normalized radar backscatter (<inline-formula id="inf19">
<mml:math id="m20">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, in dB) was adjusted to a 40<inline-formula id="inf20">
<mml:math id="m21">
<mml:mrow>
<mml:mo>&#xb0;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> incidence angle. Backscatter is highly sensitive to soil&#x2019;s dielectric properties and thus widely used in SM retrieval (<xref ref-type="bibr" rid="B58">Wagner et al., 1999</xref>; <xref ref-type="bibr" rid="B19">El Hajj et al., 2016</xref>).</p>
<p>
<italic>ASCAT soil moisture (SM)</italic> &#x2013; Relative SM (%) derived by rescaling <inline-formula id="inf21">
<mml:math id="m22">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> between dry and wet reference conditions (<xref ref-type="bibr" rid="B25">H SAF, 2021b</xref>). To ensure comparability with ERA5 and retrieval results, we converted ASCAT relative SM into volumetric units (<inline-formula id="inf22">
<mml:math id="m23">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>) using soil porosity data from the Global Land Data Assimilation System [GLDAS; <xref ref-type="bibr" rid="B45">Rodell et al. (2004)</xref>], following the approach of <xref ref-type="bibr" rid="B49">Saxton and Rawls (2006)</xref>. This variable is hereafter referred to as the H120 SM product.</p>
<p>Both <inline-formula id="inf23">
<mml:math id="m24">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and H120 SM data were collocated with ERA5 variables by regridding to 0.25<inline-formula id="inf24">
<mml:math id="m25">
<mml:mrow>
<mml:mo>&#xb0;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> resolution and aggregating to daily values. The analysis covers a 4-year period (2016&#x2013;2019).</p>
</sec>
<sec id="s2-3">
<label>2.3</label>
<title>
<italic>In situ</italic> data from the international soil moisture network</title>
<p>
<italic>In situ</italic> SM measurements from the International Soil Moisture Network [ISMN; <xref ref-type="bibr" rid="B16">Dorigo et al. (2013)</xref>; <xref ref-type="bibr" rid="B17">Dorigo et al. (2021)</xref>] were used to assess retrieval performance in 2019. The ISMN provides a globally accessible archive of harmonized SM measurements, established through international collaboration and supported by the European Space Agency (ESA) funding and community contributions. This database serves as an essential reference for validating the retrieval of SM products (<xref ref-type="bibr" rid="B46">Rodr&#xed;guez-Fern&#xe1;ndez et al., 2015</xref>; <xref ref-type="bibr" rid="B8">Batchu et al., 2023</xref>). For the CONUS, we selected data from three main networks, including: (1) the Soil Climate Analysis Network [SCAN; <xref ref-type="bibr" rid="B50">Schaefer et al. (2007)</xref>], (2) the SNOwpack TELemetry network [SNOTEL; <xref ref-type="bibr" rid="B33">Leavesley et al. (2010)</xref>], and (3) the United States Climate Reference Network [USCRN; <xref ref-type="bibr" rid="B9">Bell et al. (2013)</xref>]. We used measurements from the top layer (0&#x2013;7&#xa0;cm) to match the depth sensitivity of ASCAT and ERA5. The measurements, originally reported at hourly intervals in volumetric units (<inline-formula id="inf25">
<mml:math id="m26">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), were aggregated into daily averages. Stations with more than 100 missing records were excluded.</p>
</sec>
<sec id="s2-4">
<label>2.4</label>
<title>Soil texture and land-cover data</title>
<p>To further evaluate how retrieval performance varies across different environmental conditions, we incorporated soil texture and land-cover information into our analysis. Soil texture classes at the surface level (0&#xa0;cm) were obtained from the United States Department of Agriculture (USDA) soil texture classification system provided by <xref ref-type="bibr" rid="B28">Hengl (2018)</xref>. Land-cover information was derived from the Annual National Land Cover Database (NLCD) Collection 1 Land Cover 2019 product (<xref ref-type="bibr" rid="B56">U.S. Geological Survey, 2024</xref>).</p>
<p>We extracted the local soil texture and dominant land-cover type associated with each ISMN site based on its geographic coordinates. These ancillary datasets allow us to stratify the evaluation of SM retrievals across major soil texture classes and land-cover categories for all selected <italic>in situ</italic> sites across CONUS during 2019.</p>
</sec>
<sec id="s2-5">
<label>2.5</label>
<title>Preliminary information content analysis</title>
<p>To assess the spatial and temporal dependencies among variables (<xref ref-type="sec" rid="s2-2">Sections 2.2</xref>, <xref ref-type="sec" rid="s2-1">2.1</xref>), we performed a sensitivity analysis using ERA5 and ASCAT data aggregated at the daily scale across all pixels in the study domain from 2016 to 2019. Geographic coordinates (latitude and longitude) were also included. <xref ref-type="fig" rid="F1">Figure 1</xref> summarizes the results as correlation matrices, reporting total, temporal, and spatial correlations. Here, the &#x201c;spatial correlation&#x201d; denotes the temporal average of spatial correlation coefficients across the domain, while the &#x201c;temporal correlation&#x201d; corresponds to the spatial average of temporal correlations computed at each grid point.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Correlations between all the considered variables&#x2014;ERA5 soil moisture (SM), ASCAT backscatter <inline-formula id="inf26">
<mml:math id="m27">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, soil temperature (ST), leaf area index (LAI), antecedent precipitation evaporation index (APEI), amplitude of the diurnal cycle of surface temperature (dT), latitude (lat), longitude (lon)&#x2014;based on pooled data across all pixels in the study domain for the 2016&#x2013;2019 period. The panels show total correlations (left), temporal correlations (middle), and spatial correlations (right) among variables.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g001.tif">
<alt-text content-type="machine-generated">Correlation matrices for total, temporal, and spatial correlations. Each matrix displays variables SM, &#x3C3;&#x2084;&#x2080;, ST, LAI, APEI, dT, lat, and lon, with color gradients indicating correlation strength from -1.0 (blue) to 1.0 (red).</alt-text>
</graphic>
</fig>
<p>The analysis reveals that ASCAT <inline-formula id="inf27">
<mml:math id="m28">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> backscatter and APEI exhibit moderate positive total correlations with ERA5 SM (0.39 and 0.45, respectively), whereas ST and dT show moderate and strong negative correlations (&#x2212;0.37 and &#x2212;0.67). Longitude also correlates positively with SM (0.38), suggesting the presence of large-scale geographic gradients. These results highlight dT, APEI, <inline-formula id="inf28">
<mml:math id="m29">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and longitude as particularly influential predictors in terms of their overall and spatial association with SM. On the other hand, temporal correlations further emphasize the relevance of ST (&#x2212;0.44) and LAI (&#x2212;0.30), underscoring their role in capturing day-to-day SM variability. As these correlations aggregate all locations and seasons, they offer a general overview of variable interactions. Still, examining total, spatial, and temporal correlations provides a comprehensive understanding of variable dynamics and helps inform predictor selection. There is a trade-off between capturing temporal <italic>versus</italic> spatial dynamics, which should be balanced when designing retrieval models.</p>
<p>Importantly, correlations also reflect confounding influences. For example, <inline-formula id="inf29">
<mml:math id="m30">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> correlates with LAI (0.40), indicating sensitivity to vegetation as well as SM. This mixed signal complicates large-scale retrieval, reinforcing the need to incorporate auxiliary (e.g., ERA5) and ancillary (independent from ERA5) data to improve SM estimates (<xref ref-type="bibr" rid="B5">Aires et al., 2021a</xref>).</p>
<p>Finally, the link between auxiliary variables and geographic coordinates also offers insights into spatial patterns&#x2014;for example, <inline-formula id="inf30">
<mml:math id="m31">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and longitude (0.36), or ST and latitude (&#x2212;0.49). Recognizing such spatial and temporal dependencies should help in better designing models and selecting inputs, taking into account the underlying correlations that could either support or limit retrieval. These impacts will be discussed further in the following sections.</p>
</sec>
</sec>
<sec id="s3">
<label>3</label>
<title>Retrieval methods and localization strategies</title>
<sec id="s3-1">
<label>3.1</label>
<title>Neural network models</title>
<p>NN retrieval models have been extensively applied in the field of remote sensing (<xref ref-type="bibr" rid="B20">Go&#xef;ta et al., 1994</xref>; <xref ref-type="bibr" rid="B7">Atkinson and Tatnall, 1997</xref>; <xref ref-type="bibr" rid="B37">Maggiori et al., 2017</xref>). In this study, we investigated two widely used NN architectures&#x2014;MLPs and CNNs&#x2014;for the SM retrieval.</p>
<sec id="s3-1-1">
<label>3.1.1</label>
<title>Multilayer perceptrons</title>
<p>An MLP, also known as a feedforward NN, consists of multiple layers of neurons (perceptrons) organized sequentially (<xref ref-type="bibr" rid="B48">Rumelhart et al., 1986</xref>). Each neuron performs a weighted summation of its inputs, followed by the application of a nonlinear activation function. In a fully connected architecture, every neuron in a given layer is connected to all neurons in the preceding layer. The output of neuron <inline-formula id="inf31">
<mml:math id="m32">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is computed as: <inline-formula id="inf32">
<mml:math id="m33">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>y</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>f</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mrow>
<mml:mo>&#x2211;</mml:mo>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mi>N</mml:mi>
</mml:mrow>
</mml:msubsup>
<mml:msub>
<mml:mrow>
<mml:mi>w</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
<mml:msub>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf33">
<mml:math id="m34">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf34">
<mml:math id="m35">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mo>&#x3d;</mml:mo>
<mml:mtext>&#x2009;</mml:mtext>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mo>&#x2026;</mml:mo>
<mml:mo>,</mml:mo>
<mml:mi>N</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> are the inputs to neuron <inline-formula id="inf35">
<mml:math id="m36">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf36">
<mml:math id="m37">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>w</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> are the synaptic weights associated with each input, <inline-formula id="inf37">
<mml:math id="m38">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the bias term, and <inline-formula id="inf38">
<mml:math id="m39">
<mml:mrow>
<mml:mi>f</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> denotes an activation function (<xref ref-type="bibr" rid="B11">Bishop, 1995</xref>). Typically, neurons in the hidden layers (i.e., all layers except input and output layers) use a sigmoid activation (f(x) &#x3d; <inline-formula id="inf39">
<mml:math id="m40">
<mml:mrow>
<mml:mfrac>
<mml:mrow>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>), while neurons in the output layer employ the linear activation function (f(x) &#x3d; x).</p>
</sec>
<sec id="s3-1-2">
<label>3.1.2</label>
<title>Convolutional neural networks</title>
<p>Unlike MLPs, which process inputs as flattened vectors (e.g., a single pixel or feature at a time), CNNs are specifically designed for grid-like data such as images. A CNN processes inputs as multidimensional tensors, e.g., <inline-formula id="inf40">
<mml:math id="m41">
<mml:mrow>
<mml:mi mathvariant="bold">X</mml:mi>
<mml:mo>&#x2208;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="bold">H</mml:mi>
<mml:mo>&#xd7;</mml:mo>
<mml:mi mathvariant="bold">W</mml:mi>
<mml:mo>&#xd7;</mml:mo>
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> where <inline-formula id="inf41">
<mml:math id="m42">
<mml:mrow>
<mml:mi mathvariant="bold">H</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf42">
<mml:math id="m43">
<mml:mrow>
<mml:mi mathvariant="bold">W</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> denote the spatial dimensions (image height and width) and <inline-formula id="inf43">
<mml:math id="m44">
<mml:mrow>
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the number of channels (or input depth). In our case, <inline-formula id="inf44">
<mml:math id="m45">
<mml:mrow>
<mml:mi mathvariant="bold">H</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 100 and <inline-formula id="inf45">
<mml:math id="m46">
<mml:mrow>
<mml:mi mathvariant="bold">W</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 200 pixels covering the CONUS domain, while <inline-formula id="inf46">
<mml:math id="m47">
<mml:mrow>
<mml:mi mathvariant="bold">C</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> ranges from two to seven, including <inline-formula id="inf47">
<mml:math id="m48">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and other auxiliary variables.</p>
<p>As suggested by the term <italic>convolutional</italic>, CNNs consist of multiple convolutional layers that transform the input tensor <inline-formula id="inf48">
<mml:math id="m49">
<mml:mrow>
<mml:mi mathvariant="bold">X</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> into successively higher-level feature representations. Each convolutional layer applies a set of learnable kernels (or filters) across the input. As the kernel slides over the tensor, it computes localized weighted sums that collectively form a feature map. Mathematically, this can be expressed as:<disp-formula id="e2">
<mml:math id="m50">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mi mathvariant="bold">W</mml:mi>
<mml:mo>&#x2217;</mml:mo>
<mml:mi mathvariant="bold">X</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>b</mml:mi>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>where <inline-formula id="inf49">
<mml:math id="m51">
<mml:mrow>
<mml:mi mathvariant="bold">W</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> denotes the convolutional kernel, <inline-formula id="inf50">
<mml:math id="m52">
<mml:mrow>
<mml:mo>&#x2217;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> represents the convolution operation, <inline-formula id="inf51">
<mml:math id="m53">
<mml:mrow>
<mml:mi mathvariant="bold">X</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the local input, b is the bias term, and <inline-formula id="inf52">
<mml:math id="m54">
<mml:mrow>
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<mml:mrow>
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</mml:mrow>
<mml:mrow>
<mml:mo>&#x2032;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> is the output feature. Unlike element-wise multiplications in MLPs, CNN kernels exploit spatial locality, making them effective for capturing spatial patterns. Typically, multiple kernels are used in each layer to extract diverse features.</p>
<p>The behavior of a convolutional layer is influenced by several architectural parameters:<list list-type="bullet">
<list-item>
<p>
<italic>Padding</italic> extends the input tensor by adding extra pixels (commonly zeros) around the borders, which allows control over the spatial size of the output. Padding is often used to preserve the input dimensions after convolution.</p>
</list-item>
<list-item>
<p>
<italic>Stride</italic> specifies the step size with which the kernel slides across the input. A stride greater than one reduces the output resolution.</p>
</list-item>
<list-item>
<p>
<italic>Dilation</italic> expands the convolutional kernel (also known as the receptive field) without increasing the number of parameters by involving pixel skipping with a given rate (typically 2 or 3), allowing the model to capture long-range dependencies efficiently (<xref ref-type="bibr" rid="B61">Yu and Koltun, 2016</xref>).</p>
</list-item>
</list>
</p>
<p>After convolution, activation functions are applied element-wise to introduce nonlinearity. Common activation functions include:<list list-type="bullet">
<list-item>
<p>
<italic>Rectified Linear Unit (ReLu)</italic>, <inline-formula id="inf53">
<mml:math id="m55">
<mml:mrow>
<mml:mi mathvariant="normal">R</mml:mi>
<mml:mi mathvariant="normal">e</mml:mi>
<mml:mi mathvariant="normal">L</mml:mi>
<mml:mi mathvariant="normal">U</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:mi>z</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>max</mml:mi>
<mml:mrow>
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<mml:mrow>
<mml:mn>0</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>z</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, is widely used due to its simplicity and effectiveness in mitigating the vanishing gradient problem (<xref ref-type="bibr" rid="B21">Goodfellow et al., 2016</xref>).</p>
</list-item>
<list-item>
<p>
<italic>Tanh</italic>, <inline-formula id="inf54">
<mml:math id="m56">
<mml:mrow>
<mml:mi>tanh</mml:mi>
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
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<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>&#x2212;</mml:mo>
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</mml:mrow>
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<mml:mi>x</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi>x</mml:mi>
</mml:mrow>
</mml:msup>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
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<mml:mi>x</mml:mi>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:mfrac>
</mml:mrow>
</mml:math>
</inline-formula>, maps inputs to (&#x2212;1,1), centered around zero, but can still suffer from vanishing gradients.</p>
</list-item>
<list-item>
<p>
<italic>Softmax</italic> converts a vector of outputs into a probability distribution, typically applied only in the final layer for multi-class classification.</p>
</list-item>
</list>
</p>
<p>Beyond the core convolutional layers, a CNN model can include additional specialized layers or processing steps that enhance its ability to learn the input&#x2013;target relationship:<list list-type="bullet">
<list-item>
<p>
<italic>Dropout layer</italic> temporarily deactivates a fraction of neurons during each training iteration, introducing randomness into the learning process and helping to mitigate overfitting. This technique promotes more uniform learning across neurons by preventing reliance on any single neuron. During inference, all neurons are active, but their outputs are scaled to account for the dropout applied during training.</p>
</list-item>
<list-item>
<p>
<italic>Batch normalization layer</italic> normalizes the input of each layer for all training batches. This process helps stabilize and accelerate training by reducing internal covariate shift across all layers.</p>
</list-item>
</list>
</p>
</sec>
</sec>
<sec id="s3-2">
<label>3.2</label>
<title>Localization approaches</title>
<p>NNs are statistical models typically trained to minimize global errors over large datasets. While effective on average, their performance can degrade in regions with complex surface conditions&#x2014;such as areas with dense vegetation or significant topographic variability&#x2014;leading to localized regional biases. These biases are often not due to insufficient data samples but rather a lack of input features that adequately represent the local surface conditions.</p>
<p>To address this challenge, the concept of &#x201c;localization&#x201d; has been introduced (<xref ref-type="bibr" rid="B13">Boucher et al., 2023</xref>; <xref ref-type="bibr" rid="B40">Pellet et al., 2025</xref>). Localization refers to strategies that enable NNs to adapt their behavior to local conditions. This can be achieved in several ways: by augmenting input features (e.g., adding physical or geographical variables) or by modifying the model architecture to account for spatial heterogeneity. In the following, we discuss several localization strategies for the two types of NN architectures considered: MLPs and CNNs.</p>
<sec id="s3-2-1">
<label>3.2.1</label>
<title>For multilayer perceptrons</title>
<p>MLPs are commonly applied in remote sensing as pixel-wise regressors, where a single model is trained over all spatial locations. This imposes a compromise solution that does not account for regional variability in land surface characteristics, often leading to regional biases. To mitigate these issues, several localization strategies have been proposed (<xref ref-type="bibr" rid="B13">Boucher et al., 2023</xref>):<list list-type="bullet">
<list-item>
<p>
<italic>Physical variable augmentation</italic>: Inputs, such as soil ST or vegetation indices (e.g., LAI), are introduced to better represent the surface conditions. This approach enhances the model&#x2019;s physical realism by better constraining the inverse radiative transfer and allows for more generalized models with global applicability (<xref ref-type="bibr" rid="B2">Aires et al., 2001</xref>; <xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>; <xref ref-type="bibr" rid="B31">Kolassa et al., 2013</xref>).</p>
</list-item>
<list-item>
<p>
<italic>Geographic coordinate addition</italic>: Spatial static information such as latitude and longitude [or other coordinate-derived variables (<xref ref-type="bibr" rid="B38">Mahara and Rishe, 2023</xref>)] is introduced as proxy features to localize the model. This strategy helps partition the input space and simplifies the underlying relationships by implicitly capturing location-dependent processes. While less physically interpretable (latitude and longitude do not intervene in a radiative transfer model linking satellite observations and variables to retrieve), this statistical localization can be effective in improving regional accuracy in a global model (<xref ref-type="bibr" rid="B36">Madadikhaljan and Schmitt, 2025</xref>) by modulating the NN behavior in several parts of the spatial domain.</p>
</list-item>
</list>
</p>
<p>The choice between these strategies depends on the intended application. For example, purely statistical approaches [e.g., CDF-matching at the site level (<xref ref-type="bibr" rid="B18">Drusch et al., 2005</xref>)] are commonly used in assimilation frameworks, where improving dynamics may outweigh physical interpretability. However, it is important to note that even physically motivated inputs may be exploited by the NN in a purely statistical way&#x2014;leveraging correlations rather than causal relationships. This is evident when inspecting Jacobians (i.e., sensitivities) of trained models, which may show unrealistic patterns. Addressing this challenge may require the integration of physical constraints during training (<xref ref-type="bibr" rid="B1">Aires et al., 1999</xref>; <xref ref-type="bibr" rid="B3">Aires et al., 2004</xref>) or the development of hybrid models where physical knowledge provides structural constraints and NNs learn the unknown parameters.</p>
<p>In practice, especially when the information content of the observations is not enough to fully characterize the variable to monitor, some form of localization is necessary to reduce regional biases. One can use only physical variables, but incorporating both physical and geographical inputs has become a common strategy (<xref ref-type="bibr" rid="B53">Singh and Gaurav, 2023</xref>; <xref ref-type="bibr" rid="B26">Han et al., 2023</xref>).<list list-type="bullet">
<list-item>
<p>
<italic>Pixel-scale modeling</italic>: At the most localized level, fully independent MLPs can be trained per pixel, allowing the model to specialize in the local observation-to-variable relationships. This &#x201c;pixel-specific&#x201d; modeling simplifies the retrieval task but sacrifices generalization and scalability. There is also a risk of overfitting due to the smaller sample size per model. Such a strategy may be appropriate in cases where auxiliary physical variables are unavailable or unreliable. It is worth noting that approaches such as CDF matching (<xref ref-type="bibr" rid="B5">Aires et al., 2021a</xref>) or a linear regression (<xref ref-type="bibr" rid="B59">Wagner et al., 2013</xref>) of an SM index to a physical SM range (e.g., from ECMWF) are also examples of pixel-specific models.</p>
</list-item>
</list>
</p>
</sec>
<sec id="s3-2-2">
<label>3.2.2</label>
<title>For convolutional neural networks</title>
<p>
<list list-type="bullet">
<list-item>
<p>
<italic>Grid information</italic>: CNNs can inherently incorporate spatial information through convolutional layers that apply filters or kernels across the input images. When trained on a fixed domain (e.g., CONUS), the pixel location in the image is directly equivalent to the geographic coordinates information previously cited for MLPs. As such, traditional CNNs already embed a form of localization. However, standard CNNs rely on &#x201c;weight sharing&#x201d; (<xref ref-type="bibr" rid="B34">Lecun et al., 1998</xref>), where the same set of filters is applied across all spatial locations. This can limit their capacity to capture region-specific behaviors in large, heterogeneous domains. To address this, CNNs typically use multiple stacked convolutional layers to develop hierarchical features that generalize spatial variations. Nonetheless, this approach may require a large number of filters to approximate diverse land surface conditions.</p>
</list-item>
<list-item>
<p>
<italic>Local connection</italic>: the locally connected layers were first proposed by <xref ref-type="bibr" rid="B15">Chen et al. (2015)</xref> and later adopted in remote sensing [e.g., in <xref ref-type="bibr" rid="B13">Boucher et al. (2023)</xref>; <xref ref-type="bibr" rid="B40">Pellet et al. (2025)</xref>]. These layers are convolutional layers but do not share weights across space. Instead, each spatial region has its own filter set, allowing the network to learn location-specific patterns. This architectural approach is analogous to the pixel-specific strategy used in MLPs, while benefiting the local neighborhood (within the kernel window) to enhance retrieval for the central pixel.</p>
</list-item>
</list>
</p>
<p>To comprehensively assess the impact of localization strategies, we explored a set of model architectures with varying input configurations for SM retrieval. The following section introduces these models and the experimental setups used to investigate how different forms of localization influence the spatial and temporal accuracy of the retrievals.</p>
</sec>
</sec>
<sec id="s3-3">
<label>3.3</label>
<title>Proposed model architectures</title>
<p>Various model architectures can be exploited to retrieve SM.</p>
<p>
<italic>Standard MLP</italic>: A simple MLP model consists of a single hidden layer with 10 neurons with sigmoid activation functions, followed by a linear output layer. The input can be limited to only <inline-formula id="inf55">
<mml:math id="m57">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and/or incorporates auxiliary inputs, including physical variables and geographic coordinates. Different architectures are explored, denoted as <inline-formula id="inf56">
<mml:math id="m58">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf57">
<mml:math id="m59">
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the number of physical variables and <inline-formula id="inf58">
<mml:math id="m60">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> the number of geographic variables.</p>
<p>
<italic>Localized MLP (MLP-l)</italic>: A pixel-specific MLP where, instead of pooling all pixels together, a separate MLP is trained for each individual pixel, allowing highly localized predictions. We chose to test here <inline-formula id="inf59">
<mml:math id="m61">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l with three physical inputs: <inline-formula id="inf60">
<mml:math id="m62">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, ST, and LAI, without including any geographic input, as a single model for each pixel.</p>
<p>
<italic>Standard CNN</italic>: The proposed CNN consists of multiple convolutional layers, followed by a ReLU activation function. The architecture employs dilated convolutions to expand the receptive field without reducing spatial resolution. The network begins with a standard convolutional layer of <inline-formula id="inf61">
<mml:math id="m63">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf62">
<mml:math id="m64">
<mml:mrow>
<mml:mo>&#xd7;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 16 kernels to extract low-level spatial features, followed by three dilated convolutional layers (<inline-formula id="inf63">
<mml:math id="m65">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf64">
<mml:math id="m66">
<mml:mrow>
<mml:mo>&#xd7;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> D, where D corresponds to 32, 64, and 32 kernels) with rates of 1, 2, and 1, respectively, to capture broader contextual information. An additional standard convolutional layer (<inline-formula id="inf65">
<mml:math id="m67">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf66">
<mml:math id="m68">
<mml:mrow>
<mml:mo>&#xd7;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> 16 kernels) is included before the final output layer. Batch normalization is applied after each layer to enhance convergence, and dropout (30%) is introduced to mitigate overfitting. &#x201c;Same&#x201d; padding is considered here to preserve input dimensions. The input consists of a 2D image of <inline-formula id="inf67">
<mml:math id="m69">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and/or additional auxiliary inputs. Similar to the standard MLP, the notation <inline-formula id="inf68">
<mml:math id="m70">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> indicates a CNN with <inline-formula id="inf69">
<mml:math id="m71">
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> physical variables and <inline-formula id="inf70">
<mml:math id="m72">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> geographic variables (although the latter case is not really meaningful).</p>
<p>
<italic>Localized CNN (CNN-l)</italic>: A localized variant that considers a locally connected convolutional layer, allowing the filter to learn unique weights for different spatial locations. The architecture includes a zero-padding to preserve the spatial dimensions, followed by a single locally connected <inline-formula id="inf71">
<mml:math id="m73">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> layer to capture local-specific patterns, without weight sharing. We considered in this case a <inline-formula id="inf72">
<mml:math id="m74">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l model with three inputs, including three images of <inline-formula id="inf73">
<mml:math id="m75">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, ST, and LAI.</p>
<p>All models use consistent hyperparameters: Adam optimizer (with an initial learning rate of 0.001), Glorot Uniform initialization with a fixed seed, and early stopping after 10 epochs of no validation loss improvement (maximum of 200 epochs), restoring best weights. The only variation is in the batch size, which depends on the model type: MLPs use a batch size of 64, while CNNs use 16. For the localized models, MLP-l uses a batch size of 16, whereas CNN-l uses 8. Since the dataset covers the entire CONUS domain, computationally intensive optimization strategies (e.g., Bayesian tuning) were infeasible, consistent with observations in previous large-sample remote sensing studies (<xref ref-type="bibr" rid="B43">Rabiei et al., 2025a</xref>). As a result, a structured manual tuning strategy is adopted here: key hyperparameters were first selected within standard ranges reported in the literature (e.g., <xref ref-type="bibr" rid="B40">Pellet et al., 2025</xref>; <xref ref-type="bibr" rid="B43">Rabiei et al., 2025a</xref>) and then iteratively refined based on validation performance. Further adjustments around the final configurations produced only marginal improvements, indicating that the selected architecture is adequate for capturing SM retrieval accuracy.</p>
<p>In all the experiments, the overall dataset was split sequentially rather than spatially: the 3-year 2016&#x2013;2018 dataset was divided into 80% for training and 20% for validation, while 2019 was reserved for testing and performance evaluation. This approach ensures that the model is evaluated on a truly unseen period, thus providing a realistic assessment of generalization ability. Other temporal combinations (e.g., using 2016 as the test year) are also possible, provided the test set remains independent of the training period. A Monte Carlo-like test using four different test years, followed by averaging the test rates for each year, will be evaluated in upcoming work.</p>
<p>The ascending and descending satellite orbits were trained separately.</p>
</sec>
<sec id="s3-4">
<label>3.4</label>
<title>Evaluation metrics</title>
<p>To evaluate the performance of various SM retrieval models, we used several statistical evaluation metrics, including: Pearson&#x2019;s correlation coefficient (r, unitless), root mean square error (RMSE, <inline-formula id="inf74">
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<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), and bias (Bias, <inline-formula id="inf75">
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</inline-formula>/m<sup>3</sup>). The standard deviation (Std, <inline-formula id="inf76">
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<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
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</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>) of the difference between two datasets is also reported.</p>
<p>In addition, we also evaluated the dampening and inflating effects (<xref ref-type="bibr" rid="B12">Boucher and Aires, 2023</xref>) by comparing the range of values in the predicted SM dataset <inline-formula id="inf77">
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<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">pred</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x3c;</mml:mo>
<mml:mspace width="0.3333em"/>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">ref</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, indicating a reduced range in the predictions. Conversely, an inflating effect is observed when <inline-formula id="inf80">
<mml:math id="m82">
<mml:mrow>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">pred</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x3e;</mml:mo>
<mml:mspace width="0.3333em"/>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">ref</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. These effects are quantified as:<disp-formula id="equ1">
<mml:math id="m83">
<mml:mrow>
<mml:mi>E</mml:mi>
<mml:mfenced open="(" close=")">
<mml:mrow>
<mml:mi>%</mml:mi>
</mml:mrow>
</mml:mfenced>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mfrac>
<mml:mrow>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">pred</mml:mi>
</mml:mrow>
</mml:msub>
<mml:mo>&#x2212;</mml:mo>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">ref</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mrow>
<mml:mi>R</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>n</mml:mi>
<mml:mi>g</mml:mi>
<mml:msub>
<mml:mrow>
<mml:mi>e</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">ref</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:mfrac>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
</disp-formula>
</p>
<p>Negative values indicate dampening, while positive values indicate inflation. The range quantifies the difference between the average SM of the upper and lower 10% quantiles at each pixel (<xref ref-type="bibr" rid="B12">Boucher and Aires, 2023</xref>).</p>
</sec>
</sec>
<sec id="s4">
<label>4</label>
<title>Analysis of daily soil moisture retrievals</title>
<sec id="s4-1">
<label>4.1</label>
<title>Overall performance of MLPs and CNNs</title>
<p>As described in <xref ref-type="sec" rid="s3-3">Section 3.3</xref>, we investigated various model configurations based on MLP and CNN architectures, incorporating different combinations of physical and geographical input variables for both MLPs and CNNs. These configurations were designed to assess the models&#x2019; sensitivity to varying degrees of localization and input diversity. <xref ref-type="table" rid="T1">Table 1</xref> summarizes the evaluation metrics for all MLP and CNN retrieval models, applied to the 2019 testing dataset, covering eight scenarios for each model type. Detailed results, including spatial maps and time series plots, are presented in the following for selected representative models.</p>
<table-wrap id="T1" position="float">
<label>TABLE 1</label>
<caption>
<p>Performance of MLP and CNN retrieval models over the 2019 validation dataset, for different scenarios with different combinations of input parameters: <italic>Physical variables</italic>: ASCAT backscatter <inline-formula id="inf81">
<mml:math id="m84">
<mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>, soil temperature (ST), leaf area index (LAI), antecedent precipitation evaporation index (APEI), amplitude of the diurnal cycle of surface temperature (dT); <italic>Geographic coordinates</italic>: latitude (lat), longitude (long).</p>
</caption>
<table>
<thead valign="top">
<tr>
<th rowspan="2" align="center">Scenarios</th>
<th colspan="7" align="center">Input parameters</th>
<th colspan="6" align="center">Statistical metrics</th>
</tr>
<tr>
<th align="center">
<inline-formula id="inf82">
<mml:math id="m85">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="center">ST</th>
<th align="center">LAI</th>
<th align="center">APEI</th>
<th align="center">dT</th>
<th align="center">lat</th>
<th align="center">lon</th>
<th align="center">
<inline-formula id="inf83">
<mml:math id="m86">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tot</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="center">
<inline-formula id="inf84">
<mml:math id="m87">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="center">
<inline-formula id="inf85">
<mml:math id="m88">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf86">
<mml:math id="m89">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>)</th>
<th align="center">
<inline-formula id="inf87">
<mml:math id="m90">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">spa</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="center">
<inline-formula id="inf88">
<mml:math id="m91">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">spa</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf89">
<mml:math id="m92">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>)</th>
<th align="center">Dampening/Inflating (%)</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td align="center">MLP<sub>0,2</sub>
</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.44</td>
<td align="center">-</td>
<td align="center">0.109</td>
<td align="center">0.44</td>
<td align="center">0.115</td>
<td align="center">&#x2212;100</td>
</tr>
<tr>
<td align="center">MLP<sub>1,0</sub>
</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">0.37</td>
<td align="center">0.24</td>
<td align="center">0.110</td>
<td align="center">0.41</td>
<td align="center">0.118</td>
<td align="center">&#x2212;74</td>
</tr>
<tr>
<td align="center">MLP<sub>1,2</sub>
</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.51</td>
<td align="center">0.24</td>
<td align="center">0.104</td>
<td align="center">0.54</td>
<td align="center">0.110</td>
<td align="center">&#x2212;78</td>
</tr>
<tr>
<td align="center">MLP<sub>3,0</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">
<bold>0.59</bold>
</td>
<td align="center">
<bold>0.61</bold>
</td>
<td align="center">
<bold>0.096</bold>
</td>
<td align="center">
<bold>0.55</bold>
</td>
<td align="center">
<bold>0.103</bold>
</td>
<td align="center">
<bold>&#x2212;45</bold>
</td>
</tr>
<tr>
<td align="center">MLP<sub>3,2</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.64</td>
<td align="center">0.60</td>
<td align="center">0.092</td>
<td align="center">0.60</td>
<td align="center">0.098</td>
<td align="center">&#x2212;53</td>
</tr>
<tr>
<td align="center">MLP<sub>5,0</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">0.74</td>
<td align="center">0.52</td>
<td align="center">0.091</td>
<td align="center">0.71</td>
<td align="center">0.086</td>
<td align="center">&#x2212;25</td>
</tr>
<tr>
<td align="center">MLP<sub>5,2</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.74</td>
<td align="center">0.53</td>
<td align="center">0.081</td>
<td align="center">0.71</td>
<td align="center">0.085</td>
<td align="center">&#x2212;25</td>
</tr>
<tr>
<td align="center">MLP<sub>3,0</sub>-l</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">
<bold>0.92</bold>
</td>
<td align="center">
<bold>0.69</bold>
</td>
<td align="center">
<bold>0.047</bold>
</td>
<td align="center">
<bold>0.92</bold>
</td>
<td align="center">
<bold>0.048</bold>
</td>
<td align="center">
<bold>&#x2212;26</bold>
</td>
</tr>
<tr>
<td align="center">CNN<sub>0,2</sub>
</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.64</td>
<td align="center">-</td>
<td align="center">0.063</td>
<td align="center">0.65</td>
<td align="center">0.074</td>
<td align="center">&#x2212;100</td>
</tr>
<tr>
<td align="center">CNN<sub>1,0</sub>
</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">0.85</td>
<td align="center">0.50</td>
<td align="center">0.065</td>
<td align="center">0.85</td>
<td align="center">0.067</td>
<td align="center">&#x2212;29</td>
</tr>
<tr>
<td align="center">CNN<sub>1,2</sub>
</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.87</td>
<td align="center">0.52</td>
<td align="center">0.062</td>
<td align="center">0.87</td>
<td align="center">0.064</td>
<td align="center">&#x2212;39</td>
</tr>
<tr>
<td align="center">CNN<sub>3,0</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">
<bold>0.91</bold>
</td>
<td align="center">
<bold>0.73</bold>
</td>
<td align="center">
<bold>0.055</bold>
</td>
<td align="center">
<bold>0.91</bold>
</td>
<td align="center">
<bold>0.057</bold>
</td>
<td align="center">
<bold>&#x2212;36</bold>
</td>
</tr>
<tr>
<td align="center">CNN<sub>3,2</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.92</td>
<td align="center">0.73</td>
<td align="center">0.050</td>
<td align="center">0.91</td>
<td align="center">0.052</td>
<td align="center">&#x2212;32</td>
</tr>
<tr>
<td align="center">CNN<sub>5,0</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">0.91</td>
<td align="center">0.72</td>
<td align="center">0.058</td>
<td align="center">0.91</td>
<td align="center">0.059</td>
<td align="center">&#x2212;44</td>
</tr>
<tr>
<td align="center">CNN<sub>5,2</sub>
</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">0.90</td>
<td align="center">0.71</td>
<td align="center">0.055</td>
<td align="center">0.89</td>
<td align="center">0.057</td>
<td align="center">&#x2212;30</td>
</tr>
<tr>
<td align="center">CNN<sub>3,0</sub>-l</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="center">x</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="left">&#x200b;</td>
<td align="center">
<bold>0.93</bold>
</td>
<td align="center">
<bold>0.69</bold>
</td>
<td align="center">
<bold>0.048</bold>
</td>
<td align="center">
<bold>0.92</bold>
</td>
<td align="center">
<bold>0.049</bold>
</td>
<td align="center">
<bold>&#x2212;32</bold>
</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn>
<p>For example, <inline-formula id="inf90">
<mml:math id="m93">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the MLP model with inputs including <inline-formula id="inf91">
<mml:math id="m94">
<mml:mrow>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> physical variables and <inline-formula id="inf92">
<mml:math id="m95">
<mml:mrow>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> geographical variables; <inline-formula id="inf93">
<mml:math id="m96">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi>a</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>b</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l is the localized version. The statistical metrics include: total correlation (<inline-formula id="inf94">
<mml:math id="m97">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tot</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>); temporal correlation (<inline-formula id="inf95">
<mml:math id="m98">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) and RMSE (<inline-formula id="inf96">
<mml:math id="m99">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf97">
<mml:math id="m100">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>); spatial correlation (<inline-formula id="inf98">
<mml:math id="m101">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">spa</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) and RMSE (<inline-formula id="inf99">
<mml:math id="m102">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">spa</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf100">
<mml:math id="m103">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>); and dampening/inflating effects (%).</p>
</fn>
<fn>
<p>Bold values highlight the performance of the four representative models that are examined in the following sections.</p>
</fn>
</table-wrap-foot>
</table-wrap>
<p>The performance of MLP models improves with the addition of more auxiliary features. For example, as shown in <xref ref-type="table" rid="T1">Table 1</xref>, the addition of ST and LAI data results in a significant increase in total correlation, rising from 0.37 in <inline-formula id="inf101">
<mml:math id="m104">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>&#x2014;which uses only the ASCAT <inline-formula id="inf102">
<mml:math id="m105">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> backscatter as an input&#x2014;to 0.59 in <inline-formula id="inf103">
<mml:math id="m106">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. Corresponding improvements are also observed in temporal correlation (increasing from 0.24 to 0.61) and spatial correlation (increasing from 0.41 to 0.55). Overall, model performance continues to enhance with the inclusion of additional features such as APEI, dT, and geographic coordinates. This trend is further illustrated in <xref ref-type="fig" rid="F2">Figure 2</xref>, which shows SM retrievals from different models for the ascending orbit on 20 January 2019. Although <inline-formula id="inf104">
<mml:math id="m107">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> captures some SM patterns in the southeastern part, <inline-formula id="inf105">
<mml:math id="m108">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> yields improved retrievals that, while not perfect, are closer to the ERA5 reference data. This enhancement can be attributed to the MLP model&#x2019;s ability to treat all input features as independent, thereby effectively integrating additional information.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>From top to bottom: the soil moisture (SM) retrievals from different MLP (left) and CNN (right) models, and the target ERA5 SM on the ascending orbit of 20 January 2019.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g002.tif">
<alt-text content-type="machine-generated">Six maps of the United States display soil moisture data in cubic meters per cubic meter (m&#xB3;/m&#xB3;) using color gradients from 0.0 to 0.5. The top row compares MLP and CNN estimates with mostly uniform color. Subsequent rows show more detailed predictions with visible white areas indicating missing data. The final map, labeled ERA5, serves as the reference. Each map varies in color intensity, representing different soil moisture levels.</alt-text>
</graphic>
</fig>
<p>On the other hand, CNNs exhibit a different behavior. While performance improves from <inline-formula id="inf106">
<mml:math id="m109">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> to <inline-formula id="inf107">
<mml:math id="m110">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> or <inline-formula id="inf108">
<mml:math id="m111">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, the model achieves a high total correlation of 0.91 using only three physical inputs (e.g., <inline-formula id="inf109">
<mml:math id="m112">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, ST, LAI), suggesting it was already near its capacity. Adding more features results in only marginal gains and, in some instances, slight performance degradation. This phenomenon can be attributed to the architectural characteristics of CNNs, which are designed to extract spatial hierarchies and local patterns through convolutional layers. As shown in <xref ref-type="fig" rid="F2">Figure 2</xref>, even a single <inline-formula id="inf110">
<mml:math id="m113">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> input allows <inline-formula id="inf111">
<mml:math id="m114">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> to effectively capture meaningful SM patterns. For example, it identifies very dry conditions (indicated in red) over the Rocky Mountains and the southwestern part, while showing wetter conditions (in blue) across the southeastern region&#x2013;these patterns closely resemble the ERA5 reference data for 20 January 2019. Therefore, the <inline-formula id="inf112">
<mml:math id="m115">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model is recommended here.</p>
<p>CNN consistently outperforms the equivalent MLP models, as evidenced by the statistical metrics reported in <xref ref-type="table" rid="T1">Table 1</xref>. This superiority is further illustrated in <xref ref-type="fig" rid="F3">Figure 3</xref>, which compares the spatial correlation scores of various MLP and CNN models. Across all configurations, CNNs demonstrate higher spatial correlation than classical MLP models. This improvement can be attributed to the CNNs&#x2019; ability to exploit spatial patterns, which is particularly advantageous when such patterns are strongly expressed in the target variable, such as SM. However, both model types still have difficulty capturing abrupt events (i.e., sudden SM peaks). For example, during a pronounced wetting event in the eastern United States around late May 2019, all models showed a reduced spatial correlation compared to the surrounding periods (see <xref ref-type="fig" rid="F3">Figure 3</xref>).</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Comparison of spatial correlations over the 2019 validation dataset for six MLP and CNN models (as detailed in <xref ref-type="table" rid="T1">Table 1</xref>), relative to the target ERA5 soil moisture (SM). The left y-axis shows the spatial correlation scores for each model. The right y-axis displays the average ERA5 SM over the western United States (all pixels from 125&#xb0; W to 97.5&#xb0; W) and eastern United States (all pixels from 97.5&#xb0; W to 70&#xb0; W).</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g003.tif">
<alt-text content-type="machine-generated">Line graph showing spatial correlation versus time from January 2019 to January 2020. Multiple lines represent different models and data sources, with varying spatial correlations. Legends include MLP and CNN models, and ERA5 SM for West and East regions. Y-axis on the left shows spatial correlation, right shows soil moisture in cubic meters per cubic meter.</alt-text>
</graphic>
</fig>
<p>It is also noted that when increasing the localization: <inline-formula id="inf117">
<mml:math id="m120">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x2192;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf118">
<mml:math id="m121">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,2</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x2192;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf119">
<mml:math id="m122">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>5,0</mml:mn>
</mml:mrow>
</mml:msub>
<mml:mspace width="0.3333em"/>
<mml:mo>&#x2192;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> <inline-formula id="inf120">
<mml:math id="m123">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>5,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> the total correlation increases (from 0.59 to 0.74) at the detriment to the temporal correlation (from 0.61 to 0.53). This means that the localization (by adding geographic or auxiliary features) allows to constrain better the overall SM pattern, but the SM dynamics are degraded locally. This is a true issue because even if we need a correct spatial pattern of SM, the SM dynamics are an essential piece of information, for instance, for assimilation. There is clearly a balance that needs to be done between a better spatial pattern, reducing local biases, and a satisfactory local dynamics. We strongly suggest that any new product should be evaluated using these two diagnostics.</p>
</sec>
<sec id="s4-2">
<label>4.2</label>
<title>Impacts of the localization information</title>
<p>We examined a simplified case where only geographic coordinates (latitude and longitude) were used as input variables. Although not practical for real-world SM retrieval, this setup highlights the role of spatial information in model performance. The results showed non-negligible total correlations: just latitude and longitude information provides a total correlation of 0.44 for <inline-formula id="inf121">
<mml:math id="m124">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>0,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. Incorporating coordinate information significantly enhances spatial correlation, as demonstrated by an increase from 0.37 in the <inline-formula id="inf122">
<mml:math id="m125">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model (model with only <inline-formula id="inf123">
<mml:math id="m126">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> as an input) to 0.51 in the <inline-formula id="inf124">
<mml:math id="m127">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model (MLP with <inline-formula id="inf125">
<mml:math id="m128">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, latitude, and longitude as inputs). In contrast, the temporal correlation does not change and is equal to 0.24. This addition of the geographic features only improves the spatial correlation, reducing the local biases, without increasing the temporal correlations. This is natural, as latitude and longitude are static.</p>
<p>On the other hand, substituting physical variables (e.g., in the <inline-formula id="inf126">
<mml:math id="m129">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model with inputs including <inline-formula id="inf127">
<mml:math id="m130">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, ST, and LAI) instead of using coordinate information (as done in the <inline-formula id="inf128">
<mml:math id="m131">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model with three inputs: <inline-formula id="inf129">
<mml:math id="m132">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, latitude, and longitude) enhances both temporal and spatial correlations. These physical features carry inherent spatiotemporal information, allowing the model to better capture variations across both dimensions. This shows that physical variables are more SM information-carrying features than just latitude and longitude.</p>
<p>While coordinate information benefits MLPs, it has minimal impact on CNNs. For example, in the <inline-formula id="inf130">
<mml:math id="m133">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>1,2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> model, which incorporates <inline-formula id="inf131">
<mml:math id="m134">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, latitude, and longitude as inputs, all correlation metrics (total, temporal, and spatial) increase by less than 0.02 compared to a standard CNN model that uses only <inline-formula id="inf132">
<mml:math id="m135">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mn>40</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> as an input feature. This limited improvement occurs because the grid position of the pixels in the input image is exploited directly by the CNN as latitude and longitude information when trained on a fixed domain as CONUS. We will not comment further on the <inline-formula id="inf133">
<mml:math id="m136">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mtext>_</mml:mtext>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> configurations.</p>
<p>It is not surprising that the two more extreme localization cases&#x2014;<inline-formula id="inf134">
<mml:math id="m137">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l and <inline-formula id="inf135">
<mml:math id="m138">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l&#x2014;outperform all other experiments, as presented in <xref ref-type="table" rid="T1">Table 1</xref> with high total correlations of 0.92 and 0.93, respectively. An example of <xref ref-type="fig" rid="F2">Figure 2</xref> also shows that the <inline-formula id="inf136">
<mml:math id="m139">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l and <inline-formula id="inf137">
<mml:math id="m140">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l retrieved SM patterns more closely align with the ERA5 target for the ascending orbit on 20 January 2019, compared to other classical models. Both architectures demonstrate similar spatial and temporal performance, as illustrated in <xref ref-type="fig" rid="F3">Figures 3</xref>, <xref ref-type="fig" rid="F4">4</xref>, respectively, and they far surpass classical NN models such as <inline-formula id="inf138">
<mml:math id="m141">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf139">
<mml:math id="m142">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. The local bias in these localized models is notably reduced, with near-zero bias errors and very low standard deviations observed in most regions of the study area (see <xref ref-type="fig" rid="F4">Figure 4</xref>).</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Comparison of retrieval statistics, namely, temporal correlation (left), bias (middle), and standard deviation (right), over the 2019 validation dataset; from the top to bottom: for the <inline-formula id="inf140">
<mml:math id="m143">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf141">
<mml:math id="m144">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf142">
<mml:math id="m145">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l, and <inline-formula id="inf143">
<mml:math id="m146">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l models, compared to the target ERA5 SM.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g004.tif">
<alt-text content-type="machine-generated">Four rows of maps display correlation, bias, and standard deviation across the United States for different models, labeled MLP and CNN. Each row represents a model with correlation values ranging from 0.61 to 0.73. The middle maps show bias with values near zero, and the right maps depict the standard deviation, mostly around 0.045 cubic meters per cubic meter. Color gradients highlight regional variations.</alt-text>
</graphic>
</fig>
<p>Despite these promising results, it is important to acknowledge the limitations of such localization strategies. By design, these models primarily learn localized statistical relationships, which may limit their ability to generalize or represent broader physical processes. Consequently, they may inadvertently propagate inherent biases in the target dataset. Furthermore, the reliability of the models is highly dependent on the quality of training data at each location; areas with sparse observations may experience instability, leading to issues such as overfitting or underfitting.</p>
</sec>
</sec>
<sec id="s5">
<label>5</label>
<title>Evaluation of extreme cases</title>
<sec id="s5-1">
<label>5.1</label>
<title>Minimum and maximum soil moisture retrievals</title>
<p>We further assessed the models&#x2019; ability to capture SM extremes, both dry and wet conditions, by comparing their retrievals of minimum and maximum values. <xref ref-type="fig" rid="F5">Figure 5</xref> shows the corresponding maps derived from several NN-based models alongside the ERA5 reference. The results from the ASCAT product (i.e., H120) are also presented for comparison purposes.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Minimum soil moisture (SM) (left); maximum SM (middle); dampening (in magenta) and inflating (in green) effects (right), from top to the bottom, for the: H120 product, <inline-formula id="inf144">
<mml:math id="m147">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf145">
<mml:math id="m148">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf146">
<mml:math id="m149">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l, <inline-formula id="inf147">
<mml:math id="m150">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l, and ERA5, during year 2019.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g005.tif">
<alt-text content-type="machine-generated">Maps compare soil moisture data across the United States using different models, labeled as H1 2 0, M L P 3 comma 0, C N N 3 comma 0, M L P 3 comma O dash I, C N N 3 comma O dash I, and E R A 5. Each model shows columns for minimum, maximum, and dampening/inflating percentages. Color gradients indicate variations in soil moisture levels, with scales at the bottom ranging from red to blue for moisture content and green to violet for percentage change.</alt-text>
</graphic>
</fig>
<p>Both the non-localized models, <inline-formula id="inf148">
<mml:math id="m151">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf149">
<mml:math id="m152">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, tend to underestimate SM extremes. Notably, they struggle to represent dry conditions in the western region and high SM values in the eastern part, because these are anomalies compared to standard cases.</p>
<p>In contrast, the localized models, <inline-formula id="inf150">
<mml:math id="m153">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l and <inline-formula id="inf151">
<mml:math id="m154">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l, demonstrate improved estimates of extreme values. Localization enables the models to adapt more effectively to regional variability, allowing them to detect local anomalies with higher accuracy. The enhanced ability of localized models to capture extreme wet and dry events highlights their potential value for targeted applications such as drought monitoring or flood forecasting.</p>
</sec>
<sec id="s5-2">
<label>5.2</label>
<title>Dampening and inflating effects</title>
<p>In general, statistical models (e.g., NNs) tend to dampen the range of values, as regression-based models typically capture a smaller portion of the total variability than the original data (<xref ref-type="bibr" rid="B27">Hastie et al., 2009</xref>; <xref ref-type="bibr" rid="B54">Skafte et al., 2019</xref>). However, under certain conditions&#x2014;particularly when training data from different locations are pooled&#x2014;regression models can also exhibit inflation (<xref ref-type="bibr" rid="B12">Boucher and Aires, 2023</xref>). Inflating effects are more commonly associated with non-statistical models.</p>
<p>As illustrated in the third column of <xref ref-type="fig" rid="F5">Figure 5</xref>, the ASCAT H120 product shows a widespread inflating effect (depicted in green), largely due to an excessive number of pixels with a minimum SM of zero. It is important to note that ASCAT SM is not trained on ERA5; it is more &#x201c;independent&#x201d; than the NN-based results presented here. In contrast, our NN-based retrievals (MLPs and CNNs) rarely show inflation. These models mainly exhibit dampening (shown in magenta), consistent with their statistical nature. Among the NN-based models, the localized versions (<inline-formula id="inf152">
<mml:math id="m155">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l and <inline-formula id="inf153">
<mml:math id="m156">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>-l) show a reduced dampening effect, as also seen in <xref ref-type="table" rid="T1">Table 1</xref>, indicating better characterization of SM extremes. By tailoring the model to regional characteristics, localization improves the representation of both minima and maxima, further enhancing retrieval realism.</p>
</sec>
</sec>
<sec id="s6">
<label>6</label>
<title>Evaluation of daily soil moisture retrievals against <italic>in situ</italic> measurements</title>
<p>In this section, we evaluated the performance of our daily NN-based SM retrievals using the <italic>in situ</italic> measurements from the ISMN dataset during the 2019 validation period at 638 sites in the study area. For this assessment, we selected two representative models&#x2014;<inline-formula id="inf154">
<mml:math id="m157">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf155">
<mml:math id="m158">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>&#x2014;and compared them alongside the ERA5 and ASCAT H120 SM products against measurements. Each ISMN site corresponds to a specific geographic point, while the SM retrievals are provided at a 0.25<inline-formula id="inf156">
<mml:math id="m159">
<mml:mrow>
<mml:mo>&#xb0;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> resolution. Therefore, for each site, we selected the nearest pixel cell to represent the corresponding model data for comparison. We first present time series examples from randomly selected sites, followed by a comprehensive statistical evaluation across all 638 locations.</p>
<sec id="s6-1">
<label>6.1</label>
<title>Time series analysis</title>
<p>
<xref ref-type="fig" rid="F6">Figure 6</xref> presents examples of daily SM time series in 2019 from six randomly selected <italic>in situ</italic> sites, along with the correlation scores between model outputs and ground measurements. Among these six examples, the ASCAT H120 product demonstrates a relatively good correlation with the <italic>in situ</italic> data; however, it exhibits a pronounced bias in absolute values, as shown by the consistently offset range (orange lines) in <xref ref-type="fig" rid="F6">Figure 6</xref>.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Soil moisture (SM) time series in 2019 for six <italic>in situ</italic> sites, including <italic>in situ</italic> measurements (ISMN) and ASCAT retrievals: H120 product, <inline-formula id="inf157">
<mml:math id="m160">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf158">
<mml:math id="m161">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and ERA5. The temporal correlation (<inline-formula id="inf159">
<mml:math id="m162">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, denoted as r in each panel for brevity) between <italic>in situ</italic> and SM retrieval data is presented in each panel.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g006.tif">
<alt-text content-type="machine-generated">Multiple line graphs depicting soil moisture content (SM in cubic meters per cubic meter) over time from January 2019 to January 2020 at six different sites. Each graph includes four data series: ISMN, H120, MLP3,0, CNN3,0, and ERA5, differentiated by color. Correlation coefficients for each dataset are provided above each graph. The graphs display variations in soil moisture across different locations and models.</alt-text>
</graphic>
</fig>
<p>On the other hand, the other three retrievals&#x2014;our two NN-based models and ERA5&#x2014;show more reasonable SM ranges and better capture observed temporal dynamics. ERA5 demonstrates the highest agreement with ISMN data, with correlation coefficients exceeding 0.80 across all six sites, consistent with its assimilation of extensive <italic>in situ</italic> and satellite observations (<xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>). <inline-formula id="inf160">
<mml:math id="m163">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> also performs well, closely following the ERA5 time series. This is consistent with expectations, given that ERA5 SMs served as the training target for the models. Interestingly, at one of the six randomly selected sites (last panel of <xref ref-type="fig" rid="F6">Figure 6</xref>), <inline-formula id="inf161">
<mml:math id="m164">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> surpasses ERA5, with a correlation of 0.83 against the <italic>in situ</italic> data. In most cases, <inline-formula id="inf162">
<mml:math id="m165">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> outperforms <inline-formula id="inf163">
<mml:math id="m166">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, further highlighting the advantages of convolutional architectures in capturing spatial and temporal patterns of SM.</p>
</sec>
<sec id="s6-2">
<label>6.2</label>
<title>Statistical comparison</title>
<p>
<xref ref-type="table" rid="T2">Table 2</xref> presents the statistical performance of four SM retrievals (H120 product, <inline-formula id="inf164">
<mml:math id="m167">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf165">
<mml:math id="m168">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and ERA5) evaluated against <italic>in situ</italic> measurements from 638 sites during the 2019 validation period. To complement this tabular summary, <xref ref-type="fig" rid="F7">Figure 7A</xref> illustrates the distributions or the probability density functions (PDFs) of three temporal metrics: correlation (<inline-formula id="inf166">
<mml:math id="m169">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>), bias (<inline-formula id="inf167">
<mml:math id="m170">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf168">
<mml:math id="m171">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), and standard deviation (<inline-formula id="inf169">
<mml:math id="m172">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf170">
<mml:math id="m173">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>).</p>
<table-wrap id="T2" position="float">
<label>TABLE 2</label>
<caption>
<p>Performance metrics of soil moisture (SM) retrievals compared to the ISMN <italic>in situ</italic> measurements across 638 sites for the year 2019.</p>
</caption>
<table>
<thead valign="top">
<tr>
<th colspan="2" align="center">SM estimates</th>
<th align="center">
<inline-formula id="inf171">
<mml:math id="m174">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</th>
<th align="center">
<inline-formula id="inf172">
<mml:math id="m175">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf173">
<mml:math id="m176">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>)</th>
<th align="center">
<inline-formula id="inf174">
<mml:math id="m177">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<inline-formula id="inf175">
<mml:math id="m178">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>)</th>
<th align="center">
<inline-formula id="inf176">
<mml:math id="m179">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<inline-formula id="inf177">
<mml:math id="m180">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>)</th>
</tr>
</thead>
<tbody valign="top">
<tr>
<td rowspan="3" align="center">ASCAT retrievals</td>
<td align="center">H120</td>
<td align="center">0.61</td>
<td align="center">0.108</td>
<td align="center">0.024</td>
<td align="center">0.080</td>
</tr>
<tr>
<td align="center">
<inline-formula id="inf178">
<mml:math id="m181">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">0.60</td>
<td align="center">0.136</td>
<td align="center">0.108</td>
<td align="center">0.068</td>
</tr>
<tr>
<td align="center">
<inline-formula id="inf179">
<mml:math id="m182">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>
</td>
<td align="center">0.68</td>
<td align="center">0.107</td>
<td align="center">0.067</td>
<td align="center">0.061</td>
</tr>
<tr>
<td colspan="2" align="center">ERA5</td>
<td align="center">0.75</td>
<td align="center">0.106</td>
<td align="center">0.061</td>
<td align="center">0.060</td>
</tr>
</tbody>
</table>
<table-wrap-foot>
<fn>
<p>Metrics include temporal correlation (<inline-formula id="inf180">
<mml:math id="m183">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>), root mean square error (<inline-formula id="inf181">
<mml:math id="m184">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf182">
<mml:math id="m185">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), bias (<inline-formula id="inf183">
<mml:math id="m186">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf184">
<mml:math id="m187">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), and standard deviation (<inline-formula id="inf185">
<mml:math id="m188">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf186">
<mml:math id="m189">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), reported as the median values across all sites. The evaluated SM retrievals include the ASCAT-derived retrievals: H120 product, <inline-formula id="inf187">
<mml:math id="m190">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf188">
<mml:math id="m191">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>; and ERA5 reanalysis.</p>
</fn>
</table-wrap-foot>
</table-wrap>
<fig id="F7" position="float">
<label>FIGURE 7</label>
<caption>
<p>
<bold>(a)</bold> Probability density functions (PDFs) of temporal correlation (<inline-formula id="inf189">
<mml:math id="m192">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>), bias (<inline-formula id="inf190">
<mml:math id="m193">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf191">
<mml:math id="m194">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>) and standard deviation (<inline-formula id="inf192">
<mml:math id="m195">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf193">
<mml:math id="m196">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>) for four SM products (ASCAT-derived retrievals: H120, <inline-formula id="inf194">
<mml:math id="m197">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf195">
<mml:math id="m198">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and ERA5), evaluated against 638 <italic>in situ</italic> sites during 2019. <bold>(b)</bold> Map of temporal correlation (<inline-formula id="inf196">
<mml:math id="m199">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) between <inline-formula id="inf197">
<mml:math id="m200">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <italic>in situ</italic> measurements, along with the corresponding improvement map showing the correlation difference between <inline-formula id="inf198">
<mml:math id="m201">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf199">
<mml:math id="m202">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<inline-formula id="inf200">
<mml:math id="m203">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">&#x394;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>r</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; <inline-formula id="inf201">
<mml:math id="m204">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf202">
<mml:math id="m205">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) - <inline-formula id="inf203">
<mml:math id="m206">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf204">
<mml:math id="m207">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>)) from <inline-formula id="inf205">
<mml:math id="m208">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. Red (gray) indicates an improvement (decrease) in correlation.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g007.tif">
<alt-text content-type="machine-generated">(a) Three density plots displaying data comparisons for variables \( r_{\text{tem}} \), \(\text{Bias}_{\text{tem}}\), and \(\text{Std}_{\text{tem}}\) in various colors: H120 (yellow), MLP\(_{3,0}\) (red), ERA5 (black), and CNN\(_{3,0}\) (blue). (b) Two US maps showing \( r_{\text{tem}} \) distribution with colored dots indicating values from 0.0 to 1.0 and a map showing improvement from MLP\(_{3,0}\) to CNN\(_{3,0}\) with color gradation from -0.4 to 0.4.</alt-text>
</graphic>
</fig>
<p>Across 638 sites, the ACAST H120 SM product demonstrates a moderate median temporal correlation (<inline-formula id="inf206">
<mml:math id="m209">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.61), which is comparable to the other three evaluated SM products. Although it achieves the lowest median bias (<inline-formula id="inf207">
<mml:math id="m210">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.024 <inline-formula id="inf208">
<mml:math id="m211">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>) among all products, its standard deviation is relatively higher (<inline-formula id="inf209">
<mml:math id="m212">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.080 <inline-formula id="inf210">
<mml:math id="m213">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>), suggesting greater variability compared to the alternatives. Additionally, <xref ref-type="fig" rid="F7">Figure 7A</xref> shows that the H120 product exhibits a longer negative tail in its bias distribution, and its standard deviation values extend beyond the typical range observed in the other retrieval methods.</p>
<p>In contrast, ERA5 performs best among all retrievals, with the highest median correlation (<inline-formula id="inf211">
<mml:math id="m214">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.75) and low errors (<inline-formula id="inf212">
<mml:math id="m215">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>RMSE</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.106, <inline-formula id="inf213">
<mml:math id="m216">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Bias</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.061, <inline-formula id="inf214">
<mml:math id="m217">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>Std</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; 0.060 <inline-formula id="inf215">
<mml:math id="m218">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mtext>m</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>/m<sup>3</sup>). This superior performance is attributed to its reanalysis nature, which combines model data with observations, thereby enhancing consistency with ground-based measurements (<xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>). Our NN-based retrievals, <inline-formula id="inf216">
<mml:math id="m219">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf217">
<mml:math id="m220">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, also show competitive performance. In particular, <inline-formula id="inf218">
<mml:math id="m221">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> achieves a temporal correlation of 0.68, closely approaching ERA5, while <inline-formula id="inf219">
<mml:math id="m222">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> records a lower correlation of 0.60. Notably, the bias and standard deviation distributions of <inline-formula id="inf220">
<mml:math id="m223">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> retrieval closely resemble those of ERA5 (as presented in <xref ref-type="fig" rid="F7">Figure 7a</xref>), indicating that it provides more stable and reliable predictions than the other ASCAT-derived products.</p>
<p>
<xref ref-type="fig" rid="F7">Figure 7b</xref> provides spatial insights by mapping the temporal correlation (<inline-formula id="inf221">
<mml:math id="m224">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) of <inline-formula id="inf222">
<mml:math id="m225">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> with the <italic>in situ</italic> measurements over 638 considered sites. Lower correlation values, indicated by yellow markers, are primarily found in the Rocky Mountains. A corresponding improvement map highlights the change in temporal correlation between <inline-formula id="inf223">
<mml:math id="m226">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf224">
<mml:math id="m227">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> (<inline-formula id="inf225">
<mml:math id="m228">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi mathvariant="normal">&#x394;</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mi>r</mml:mi>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> &#x3d; <inline-formula id="inf226">
<mml:math id="m229">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf227">
<mml:math id="m230">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) - <inline-formula id="inf228">
<mml:math id="m231">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>r</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mi mathvariant="italic">tem</mml:mi>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>(<inline-formula id="inf229">
<mml:math id="m232">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>)). The observed increases in correlation (shown in red) over the mountainous areas underscore the ability of CNN architectures to better capture spatial variability, outperforming MLPs in complex environmental conditions, such as those presented by varied topography and heterogeneous mountain landscapes.</p>
</sec>
<sec id="s6-3">
<label>6.3</label>
<title>Effects of soil texture and land-cover types</title>
<p>To further investigate environmental controls on model performance, we evaluated SM retrievals within individual soil texture groups and land-cover classes against <italic>in situ</italic> SM measurements. <xref ref-type="fig" rid="F8">Figures 8</xref>, <xref ref-type="fig" rid="F9">9</xref> present the comparison metrics for the major soil texture classes and dominant land-cover types, respectively. Overall, ASCAT H120 exhibits smaller biases but with higher standard deviations than both NN-based retrievals and ERA5 across most soil texture and land-cover categories, likely due to the retrieval&#x2019;s calibration to local conditions (<xref ref-type="bibr" rid="B6">Aires et al., 2021b</xref>). Between two NN models, <inline-formula id="inf230">
<mml:math id="m233">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> consistently outperforms <inline-formula id="inf231">
<mml:math id="m234">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, showing higher correlations and lower RMSE, bias, and standard deviation across all classes. Across soil texture groups (<xref ref-type="fig" rid="F8">Figure 8</xref>), <inline-formula id="inf232">
<mml:math id="m235">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, although trained on ERA5, demonstrates better performance compared to ERA5, particularly for clay loam and silty clay loam in terms of correlation, and for most texture classes in terms of RMSE. Across land-cover types (<xref ref-type="fig" rid="F9">Figure 9</xref>), <inline-formula id="inf233">
<mml:math id="m236">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> also outperforms ERA5 in correlation over mixed forest, and in RMSE over evergreen and deciduous forests, cropland, grassland, pasture, and developed areas.</p>
<fig id="F8" position="float">
<label>FIGURE 8</label>
<caption>
<p>Boxplots of performance metrics for soil moisture (SM) retrievals (i.e., ASCAT-derived retrievals: H120 product, <inline-formula id="inf234">
<mml:math id="m237">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>MLP</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf235">
<mml:math id="m238">
<mml:mrow>
<mml:msub>
<mml:mrow>
<mml:mtext>CNN</mml:mtext>
</mml:mrow>
<mml:mrow>
<mml:mn>3,0</mml:mn>
</mml:mrow>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, and ERA5 reanalysis) compared with ISMN measurements, stratified by soil texture class (clay loam, silty clay loam, sandy clay loam, loam, silty loam, sandy loam, loam sandy). The number of sites within each class is shown in parentheses. Soil texture types represented by only a few sites (e.g., sand, silt, clay) are omitted.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g008.tif">
<alt-text content-type="machine-generated">Box plot grid comparing four models (H120, MLP&#x2083;,&#x2080;, CNN&#x2083;,&#x2080;, ERA5) for different soil types: Clay Loam, Silty Clay Loam, Sandy Clay Loam, Loam, Silty Loam, Sandy Loam, and Loam Sandy. Metrics shown include r&#x2091;&#x209C;&#x2091;&#x2098;, RMSE&#x2091;&#x209C;&#x2091;&#x2098;, Bias&#x2091;&#x209C;&#x2091;&#x2098;, and Std&#x2091;&#x209C;&#x2091;&#x2098;. Each plot visualizes performance variations across models and metrics.</alt-text>
</graphic>
</fig>
<fig id="F9" position="float">
<label>FIGURE 9</label>
<caption>
<p>Same as <xref ref-type="fig" rid="F8">Figure 8</xref>, but stratified by major land-cover categories including forests (evergreen, deciduous, mixed), croplands, crops, shrubs, grasslands, pasture, and developed areas. Land-cover types represented by only a few sites (e.g., wetlands, barren) are omitted.</p>
</caption>
<graphic xlink:href="frsen-06-1718353-g009.tif">
<alt-text content-type="machine-generated">Box plots comparing four models: H120, MLP3,0, CNN3,0, and ERA5 across eight land cover types: Evergreen, Deciduous, Mixed, Crop, Shrub, Grassland, Pasture, and Developed. Metrics include correlation coefficient (r), root mean square error (RMSE), bias, and standard deviation (Std). Each plot shows model performance variability across different vegetation types with sample sizes given in parentheses.</alt-text>
</graphic>
</fig>
</sec>
</sec>
<sec id="s7">
<label>7</label>
<title>Conclusions and perspectives</title>
<p>This study presented a comprehensive evaluation of various NN-based models (MLPs and CNNs) for daily SM retrieval, with a particular focus on the role of localization. Overall, NN-based models demonstrate strong performance in capturing both temporal and spatial dynamics of SM, showing high correlations with the target ERA5 SM as well as <italic>in situ</italic> measurements.</p>
<p>Among the two types of models evaluated, CNNs more effectively exploit spatial patterns and outperform MLPs. Our investigation of localization strategies suggests that physical variables should be preferred over purely geographical ones:<list list-type="bullet">
<list-item>
<p>Latitude and longitude, while useful for capturing the overall structure, do not fully convey finer details. Specifically, they tend to smooth variations even when the variable displays spatial heterogeneity, as observed in surface variables such as SM.</p>
</list-item>
<list-item>
<p>Physical variables, particularly those linked to radiative transfer processes, preserve spatial heterogeneity and promote more physically consistent retrievals. Moreover, when CNNs are used, coordinate information can be implicitly encoded in input imagery, especially in localized CNN configurations.</p>
</list-item>
</list>
</p>
<p>Localized NN-based models (MLP-l and CNN-l) show improved performance in retrieving extreme values, highlighting their potential for specialized applications such as drought monitoring or flood forecasting. However, caution is needed concerning the risks of overfitting&#x2014;especially for pixel-specific models like MLP-l&#x2014;due to the limited sample size available for each localized model.</p>
<p>The study highlights several methodological considerations that are relevant to future SM retrieval research. ERA5 remains one of the most reliable large-scale reference datasets for SM, as it integrates physical modeling with the assimilation of diverse observational inputs (i.e., multiple satellite products, various precipitation datasets, and <italic>in situ</italic> measurements). As a result, NN-based SM retrievals cannot be expected to surpass ERA5 skill globally. It is nevertheless important to acknowledge that ERA5 exhibits local-dependent biases, and NNs trained on ERA5 inevitably inherit these biases. However, our results suggest that localization can mitigate some of these local errors. In certain cases, our retrieval (based on ASCAT backscatter and auxiliary variables) aligns more closely with <italic>in situ</italic> measurements than ERA5. Moreover, the performance improvements were observed across different soil textures and land-cover types. This improvement can arise from various factors: for example, the models may extract observation signal components not fully exploited in the reanalysis, correct specific local biases present in ERA5, or better capture high-frequency variability at certain sites. This finding aligns with previous work (<xref ref-type="bibr" rid="B47">Rodr&#xed;guez-Fern&#xe1;ndez et al., 2019</xref>).</p>
<p>Overall, although a single-sensor retrieval cannot comprehensively outperform ERA5, our findings demonstrate that deep learning can extract high-quality SM information from ASCAT alone. This contributes to the advancement of independent satellite-based SM retrievals, which remain essential for climate analysis, model evaluation, long-term consistency assessments, and data assimilation (<xref ref-type="bibr" rid="B4">Aires et al., 2005</xref>). Furthermore, in many practical applications, the retrieved SM will ultimately be assimilated into a land surface model whose climatology is already based on ERA-type reanalyses. This further supports the use of ERA5 as an appropriate and consistent training target for the NN retrieval.</p>
<p>For future works, these deep learning approaches can be extended to different regions or even to a global-scale analysis. This can ultimately support the development of a long-term SM record by leveraging observations from multiple satellite sensors. However, integrating data across different sensors poses notable challenges, including inter-sensor calibration and varying retrieval uncertainties. Addressing these complexities is essential for generating a coherent multi-sensor SM product. Incorporating <italic>in situ</italic> measurements into training (e.g., via regional fine-tuning) can be a promising approach, though technically challenging due to scale mismatches between point and satellite measurements. Furthermore, deep learning techniques have the potential not only to exploit spatial patterns, as demonstrated in this study using CNNs, but also to utilize temporal information. Therefore, we plan to explore the possibility of testing transformers and embedding representation methods to perform retrieval across a full sequence of images, rather than just a single image at a specific time. Another direction is the potential for forecasting, rather than just retrieval. Specifically, we might focus on short- and medium-term SM forecasting using ConvLSTM models (<xref ref-type="bibr" rid="B60">Wang et al., 2024</xref>; <xref ref-type="bibr" rid="B43">Rabiei et al., 2025a</xref>). Given that SM layers in the ground exhibit strong dependencies tied to soil texture and geology, our retrieval methods could also aim to estimate moisture levels in deeper layers beyond the surface layer we considered here.</p>
<p>The methods proposed in this study are general and hold potential for a wide range of applications beyond SM retrieval. For example, localized CNN approaches could be used to retrieve surface water extent at high spatial resolution. Moreover, these methods can also be applied in forward mode, using surface parameters such as land surface temperature, vegetation cover, and surface roughness to estimate land surface emissivities or even satellite-observed brightness temperatures. Such a forward model could be integrated into data assimilation frameworks to support numerical weather prediction systems. Finally, although this study focused on methodological development, at 0.25<inline-formula id="inf236">
<mml:math id="m239">
<mml:mrow>
<mml:mo>&#xb0;</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> and daily resolution, producing higher-resolution near-real-time SM maps would enhance real-world practical applications such as precision agriculture (for optimal watering), hydrological modeling, fire security during dry periods, and identifying areas prone to flooding (<xref ref-type="bibr" rid="B30">Huang et al., 2025</xref>). This improvement can be achieved by using higher-resolution sensors (e.g., Sentinel) or by directly downscaling the obtained SM fields within a transformer-based retrieval scheme.</p>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s8">
<title>Data availability statement</title>
<p>Publicly available datasets were analyzed in this study. This data can be found here: The ASCAT data supporting this study can be obtained from the Metop ASCAT SSM CDR (<xref ref-type="bibr" rid="B25">H SAF, 2021b</xref>), <ext-link ext-link-type="uri" xlink:href="https://hsaf.meteoam.it/">https://hsaf.meteoam.it/</ext-link> (last access: 22 January 2025). Porosity data (<xref ref-type="bibr" rid="B45">Rodell et al., 2004</xref>) are from <ext-link ext-link-type="uri" xlink:href="https://ldas.gsfc.nasa.gov/gldas/soils">https://ldas.gsfc.nasa.gov/gldas/soils</ext-link> (last access: 31 May 2025). The ERA5 reanalysis dataset can be downloaded from <ext-link ext-link-type="uri" xlink:href="https://cds.climate.copernicus.eu">https://cds.climate.copernicus.eu</ext-link> (Hersbach et al., 2023) (last access: 15 March 2025). Finally, the International Soil Moisture Network data are available at <ext-link ext-link-type="uri" xlink:href="https://ismn.earth/en/data/">https://ismn.earth/en/data/</ext-link> (<xref ref-type="bibr" rid="B17">Dorigo et al., 2021</xref>) (last access: 3 March 2025).</p>
</sec>
<sec sec-type="author-contributions" id="s9">
<title>Author contributions</title>
<p>LD: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Software, Validation, Visualization, Writing &#x2013; original draft, Writing &#x2013; review and editing.</p>
</sec>
<ack>
<title>Acknowledgements</title>
<p>The author thanks Filipe Aires (LIRA) and Victor Pellet (LMD) for valuable discussions and for their comments and suggestions on the original draft of the manuscript. She also thanks Patricia de Rosnay and Peter Weston (ECMWF) for interesting discussions.</p>
</ack>
<sec sec-type="COI-statement" id="s11">
<title>Conflict of interest</title>
<p>Author LD was employed by Estellus.</p>
</sec>
<sec sec-type="ai-statement" id="s12">
<title>Generative AI statement</title>
<p>The author(s) declared that generative AI was used in the creation of this manuscript. Generative AI tools, including Grammarly and ChatGPT, were used solely for language editing and refinement. All ideas, interpretations, and original text were created by the author. AI assistance was limited to enhancing clarity, grammar, and style, without contributing to the conceptual or intellectual content of the work.</p>
<p>Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.</p>
</sec>
<sec sec-type="disclaimer" id="s13">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<sec sec-type="disclaimer" id="s14">
<title>Author disclaimer</title>
<p>Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the Commission. Neither the European Union nor the granting authority can be held responsible for them.</p>
</sec>
<fn-group>
<fn fn-type="custom" custom-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/1299904/overview">Amen Al-Yaari</ext-link>, Universit&#xe9; Paris-Sorbonne, France</p>
</fn>
<fn fn-type="custom" custom-type="reviewed-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/2809560/overview">Xiangzhuo Liu</ext-link>, INRA Centre Provence-Alpes-C&#xf4;te d&#x2019;Azur, France</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/3241169/overview">Saman Rabiei</ext-link>, University of Florida, United States</p>
</fn>
</fn-group>
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