{"paper_id":"40f99771-a539-4a49-a137-ced9b1e56768","body_text":"Soil Moisture Evapotranspiration (SMET): A Low-Cost Method for Determining Seasonal Crop Water Demand | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Soil Moisture Evapotranspiration (SMET): A Low-Cost Method for Determining Seasonal Crop Water Demand Masoumeh Hashemi, Tejinder Singh, Oliver Hargreaves, Alfonso Torres-Rua, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7124239/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Evapotranspiration (ET) is an essential component of water balances that can be estimated using different tools, but heterogeneous landscapes and experimental designs limit their applicability. This study aims to develop a cost-effective, simplified method for estimating seasonal crop ET using soil moisture sensors along the root zone profile and reference ET data from local weather stations. The study was conducted at two sites in Utah, USA to develop and validate a new empirical soil-moisture-based evapotranspiration (SMET) model that estimates seasonal ET based on the relationship between reference ET (ET r ), changes in soil moisture, and actual evapotranspiration (ET a ). The initial model was constructed using detailed flux measurements from Modena, Utah, where an Eddy Covariance (EC) flux tower was installed. The model was validated with EC tower measurements at one of the research sites in Vernal, Utah showing a Root Mean Square Error (RMSE) and Root Relative Squared Error (RRSE) of 1.26 mm day − 1 and 0.65, respectively, which demonstrated high accuracy. This approach offers a practical solution for farmers, water managers, and policymakers to estimate crop water consumption, supporting better water resource management in arid and semi-arid regions. Actual Evapotranspiration Eddy Covariance flux tower Soil Moisture Sensor Agricultural Water Management Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Practical Application Knowing how much water crops use is critical for farmers, especially in dry regions like Utah. Many current methods to measure this water use, called evapotranspiration, are expensive, complex, and not practical for most farms. This study introduces a new, simple method called SMET that helps estimate how much water crops actually use during the growing season. It uses common tools that many farmers already have: soil moisture sensors and local weather station data. The SMET method was tested on alfalfa fields in Utah and showed strong agreement with advanced measurement systems, proving that it is both reliable and cost-effective. Farmers, land managers, and researchers can use this method to better understand how much water crops need and to improve irrigation decisions. The model also works well in small research plots where advanced tools like eddy covariance towers or lysimeters can’t be used. By making water use estimates easier and more affordable, SMET can help support sustainable agriculture in water-scarce regions. 1. Introduction Water is an indispensable natural resource essential for sustaining life and ecosystems. However, its availability is limited by climate-induced extremes like drought and over-extraction (Hashemi et al. 2019; Scanlon et al. 2023 ). In regions where economies are predominantly agrarian, the agricultural sector accounts for approximately 90% of total water consumption, with industrial use representing only 3% (Fao n.d.). In arid regions of the US, like Utah, irrigation represents nearly 80% of diverted water consumption (Maughan et al. 2015 ). The predominance of agriculture in the US and global water use necessitates a detailed assessment of water use in water-scarce regions. Efficient use of water in arid regions in the context of irrigation remains the utmost priority for land managers. Irrigation efficiency in crop fields depends on the agricultural water balance accounting for the inflow [Precipitation (P) + Irrigation (I) + Capillary Rise (CR)]; outflow [evapotranspiration (ET) + deep percolation (DP) + runoff (RO)] and changes in the soil water storage (Hashemi and Sepaskhah 2020 ). Of these components, ET measurements are critical in influencing the crop's irrigation water requirement. ET is quantified as the amount of water lost from the crop root zone through evaporation from the soil and transpiration by existing vegetation. Accurate and effective estimation of ET remains a persistent challenge in hydrological and agricultural research. Conventional methodologies, including eddy covariance (EC) systems, lysimeter-based measurements, and remote sensing techniques, are constrained by certain limitations (Peddinti and Kisekka 2025 ). The EC system and lysimeter provide high-resolution ET measurements but are expensive, labor-intensive, and require technical expertise for operation. Furthermore, the EC system is well-suited for larger spatial footprints limiting its applicability in small agricultural research experiments. Remote sensing imagery tools developed based on data from these methods are also constrained by spatial and temporal resolution limiting their use (Abdelmajeed and Juszczak 2024 ). These limitations highlight the need for scalable, low-cost ET estimation tools for land managers. Soil moisture sensors are widely used for irrigation scheduling in agricultural fields. Soil moisture sensors, combined with climatic data, can be effective tools for irrigation scheduling that improves water use efficiency under diverse environmental and agronomic conditions (Mathewos Boltana et al. 2023 ). The effectiveness of soil moisture sensors for irrigation scheduling is supported by many studies. For example, using a capacitance-based soil moisture sensor controller reduced irrigation water use by 55% for green bell peppers (Capsicum annum) (Zotarelli et al. 2011 ). In the mid-southern US, on-farm research using soil tensiometer sensors indicates that − 80 to -100 cbar reduces water usage by 40% in maize ( Zea mays ) and soybean ( Glycine max ) while improving water use efficiency by up to 51% (Bryant et al. 2023 ). Several different types of soil moisture sensors have been used for irrigation scheduling. A recent review on soil water sensors for irrigation scheduling highlights that 33% of studies used electrical resistivity (ER) sensors, followed by 21%, 16%, and 13% use for frequency domain reflectometry (FDR), neutron probes, and time domain reflectometry (TDR), along with another 17% that combined tensiometer and TDR in the US (Datta and Taghvaeian 2023 ). Although soil moisture sensors are widely used for irrigation scheduling their potential for estimating actual evapotranspiration (ET a ) has not been fully investigated. Using these readily available sensors that are already installed in some land managers' fields or research plots to estimate ET a could enhance the spatial resolution of in-field estimations. It may also serve as cost-effective ground validation for satellite-based ET a models. The objective of this study was to develop a cost-effective methodology using soil moisture data collected by farmers and land managers, along with reference evapotranspiration (ET r ) from nearby weather stations, to estimate ET a . This tool may provide a simple and cost-effective tool for estimating ETa compared to more cost or labor-prohibitive Bowen Ratio Energy Balance (BREB) method (Bowen 1926 ), the micrometeorological technique for vertical turbulent fluxes of gases (Baldocchi 2003 ), surface energy balance system (Allen et al. 2007 ; Bastiaanssen et al. 1998 ; Norman et al. 1995 ; Su 2002 ). 2. Materials and methods 2.1. Model description Actual evapotranspiration (ET a ) refers to water loss due to evaporation and transpiration under existing environmental and management conditions, encompassing both standard and non-standard scenarios (DeJonge 2024 ). ET a provides essential information that can help farmers and researchers make informed decisions about agricultural management by allowing them to evaluate the efficiency of their irrigation systems to help them optimize water use and improve sustainability. Various methodologies have been developed to measure ET a , but many of them are limited by spatial and temporal resolution. This study considers two primary methods for quantifying crop water use, the Water Budget (WB) and the Crop Coefficient Method (CCM) aims to integrate the relation between WB and CCM, and synthesizes these methods to determine ET a (Fig. 1 ). A case study using EC flux tower data as a ground truth for ET a was employed to build and refine the model. A two-stage evaluation by calibration and validation was performed to assess the use of crop water at different locations. A detailed explanation of each step is presented in the following section. 2.1.1. Interpreting ET a , Crop Coefficient Method, and Water Budget Relationships 2.1.1.1. Water Budget (WB) \\(\\:{ET}_{a}=I+P+G-{\\Delta\\:}{\\theta\\:}-RO-DP\\) (1) The WB method is based on the principle of mass conservation, where the change in storage is equal to the net difference between all input and output flows (Fig. 2 ). This approach can be explained using Eq. 1,(Wan et al. 2015 ). Where, I is the amount of the irrigation (mm), P is the amount of the precipitation (mm), G is the groundwater contribution (mm), Δθ is the change in soil water storage in the root zone (mm), RO is the amount of runoff (mm), and DP is the amount of deep percolation (mm), which can be used to estimate the residual term of ET a . Although a simple model in theory, the WB is challenging in practice due to the difficulty in accurately measuring deep percolation (Walker and Skogerboe 1987 ) and groundwater contributions, and to a lesser extent, irrigation and runoff. Due to difficulties in accurately estimating each parameter, a simplified model to estimate ET a was used with certain assumptions. The water balance, assuming that RO, G, and DP are negligible (Assumption 1: \\(\\:RO=G=DP\\approx\\:0\\) ), allows for ET a to be calculated as a function of P, I, and Δθ (Eq. 2 ). $$\\:{\\Delta\\:}\\theta\\:=P+I-ETa$$ 2 These assumptions are made to simplify the ET a estimation, but under actual field conditions, the magnitude of each parameter varies with the irrigation system. By considering applying the function (Eq. 2 ) in the regions where precipitation is infrequent and light during the growing season, P will be considered negligible compared to the irrigation amount (Assumption 2: \\(\\:P\\ll\\:I\\to\\:P+I\\approx\\:I\\) ). This assumption can be easily verified as a posteriori using local meteorological data, which further simplifies the water balance: $$\\:{\\Delta\\:}\\theta\\:=I-ETa$$ 3 Finally, as precise irrigation data are not always available, the algorithms developed in this analysis are not designed to produce an estimate of ET a during irrigation days (Assumption 3: \\(\\:I=0\\) ). This approach helps mitigate error due to Assumption 1, since the greatest RO and DP occur during irrigation. After integrating all these assumptions, the ET a in the water budget is simplified to depend solely on the change in soil moisture: $$\\:ETa=-{\\Delta\\:}\\theta\\:$$ 4 In brief, a simplified version of the water balance works under assumptions of no irrigation or precipitation (I = P = 0), well-drained soils (RO = DP = 0), and a water table well below the root zone (G = 0). This approach assumes ET a is represented by the negative change in soil moisture (Eq. 4 ). These assumptions are influenced by soil characteristics (e.g., water retention, soil texture) and water movement following irrigation events. When RO, G, and DP are significant, the only conclusion that can be drawn without further verification is that ET a is a function of change in soil moisture (Eq. 5). \\(\\:{ET}_{a}=f\\left(-{\\Delta\\:}\\theta\\:\\right)\\) (5) 2.1.2. Crop Coefficient Method The crop coefficient method utilizes a reference value for ET a that represents the atmospheric demand (i.e., either ET r or ET o ) multiplied by the crop coefficient (K c ), i.e., specific to the crop, growth stage, and climatic condition (e.g., wind and relative humidity), to estimate ET a (Allen et al. 1998 ). $$\\:{ET}_{a}={K}_{c}\\times\\:{ET}_{r}$$ 6 $$\\:{ET}_{a}=f\\left({ET}_{r}\\right)$$ 7 In this method, it is assumed that the soil moisture is at field capacity, which means that evapotranspiration is governed by climatic conditions and is considered potential evapotranspiration. Consequently, the crop's water needs follow the pattern of atmospheric demand, and its water use is similar to that of reference crops [e.g., alfalfa (Medicago sativa)]. Therefore, the following relationship can be expressed: 2.2. Proposed methodology Equations 5 and 7 demonstrate how ET a is a complex phenomenon that can be modeled as part of the water cycle, but also as a biophysical process that responds to environmental factors. Solving equations for ET a within the complete water balance requires the use of advanced modeling software such as HYDRUS 1-D, which uses linear finite elements to numerically solve the Richards equation for saturated-unsaturated water flow and Fickian-based advection-dispersion equations for both heat and solute transport (Šimůnek et al., 2008). The use of this type of complex software requires training and is not feasible for most land managers outside of research settings. The method proposed in this paper attempts to combine and simplify Equations 5 and 7 by expressing ET a as a function of only the changes in soil water depletion (Δθ) and the atmospheric demand (ET r ), which can be easily measured (Eq. 8 ). $$\\:{ET}_{a}=f\\left(\\varDelta\\:\\theta\\:,{\\:ET}_{r}\\right)$$ 8 2.2.1. Developing the equation and calibrating Figure 3 illustrates the process of building and calibrating the proposed model. In the first step, we developed the Soil Moisture-based Evapotranspiration estimation (SMET) model, which estimates ET a using Soil Moisture (SM) change and ET r . To achieve this, soil moisture change is calculated using equations 9 , and ET r is obtained from the nearest weather station. $$\\:{\\varDelta\\:\\theta\\:}_{j,\\:\\:i}=f\\left(x\\right)=\\left\\{\\begin{array}{c}\\frac{\\left({z}_{j}-{z}_{j-1}\\right)}{2}\\times\\:\\left({\\theta\\:}_{j,\\:\\:i-1}-{\\theta\\:}_{j,\\:\\:i}\\right),\\:\\:\\left({\\theta\\:}_{j,\\:\\:i-1}-{\\theta\\:}_{j,\\:\\:i}\\right)<0\\\\\\:0,\\:\\:\\left({\\theta\\:}_{j,\\:\\:i-1}-{\\theta\\:}_{j,\\:\\:i}\\right)\\ge\\:0\\end{array}\\right.\\varDelta\\:\\theta\\:=\\sum\\:_{j=1}^{m}{\\varDelta\\:\\theta\\:}_{j,\\:\\:i}$$ 9 Where, \\(\\:{\\theta\\:}_{j,\\:\\:i-1}\\) is the soil moisture on (i-1) th day at depth \\(\\:{z}_{j-1}\\) , \\(\\:{\\theta\\:}_{j,\\:\\:i}\\) is the soil moisture on day i th at depth \\(\\:{z}_{j-1}\\) , \\(\\:{z}_{j-1}\\) and \\(\\:{z}_{j}\\) ​ are the depth at which the soil moisture sensors are located, \\(\\:{\\varDelta\\:\\theta\\:}_{j,\\:\\:i}\\) represents soil water depletion from the layer between depth \\(\\:{z}_{j}\\) and \\(\\:{z}_{j-1}\\) on i th day, m is the number of the soil moisture sensors, determined based on crop root zoon. If ET r is not measured at the weather station, it can be estimated using equations such as Penman-Monteith (PM), and if sufficient weather data is not available to calculate PM, the Hargreaves equation can be used instead. Following, data from the Eddy Covariance (EC) flux tower is used to calculate actual evapotranspiration, serving as the ground truth. EC flux tower measurements are often used to validate ET a models (Drechsler et al. 2022 ; Franssen et al. 2010; Kisekka et al. 2022 ; Markwitz and Siebicke 2019 ). Flux data does not provide a single value but instead offers an expected ET a range due to EC sensors measuring ET a both from the atmospheric fluxes (ET original ) and from energy fluxes (ET closed ) in the closure of the energy balance (Foken and Foken, 2008 ; Franssen et al., 2010). When it was impractical to use a range, the geometric mean of the original and closed ET a measurements was calculated and considered to be the best approximation of ET a . $$\\:ETa=\\sqrt{{ET}_{original}\\bullet\\:{ET}_{closed}}$$ 10 Subsequently, Eureqa software was used to determine the most accurate model by mapping the dependent variable to the input variables. This software tool is designed to describe datasets in a simplified form using a technique known as symbolic regression. Developed by(Schmidt and Lipson 2009 ), the tool works by generating random equations based on the data through an evolutionary search process that utilizes a genetic algorithm. While the initial guesses may not fit the data well, some equations perform better than others. These better-fitting equations are then used as the foundation for the next round of guesses. The process continues until no further improvement in fit can be achieved. 2.2.2. Validation SMET model The SMET model was initially developed using data from a site equipped with an EC tower and subsequently validated at a second site that also featured an EC tower. Figure 4 illustrates the validation approach. In this process, ET a was estimated using the calibrated SMET model, which depends on ET r and soil moisture changes. Additionally, ET a was measured using the EC flux tower and considered as observational data to assess the model's accuracy. The estimated daily ET a was compared with EC flux tower measurements (ET a ) to calculate residuals and assess the performance of the SMET model. 2.2.3. Statistical analysis As mentioned, the SMET model was initially developed and calibrated using data from one site. The calibrated model was then validated at a different location. During both the calibration and validation process, the model's performance was evaluated using several statistical indices. In this study, Root Mean Square Error (RMSE), Root Relative Squared Error (RRSE), and Pearson correlation coefficient (r) were calculated using Equations 11–13, respectively, to assess model performance at each step. \\(\\:RMSE=\\sqrt{\\frac{{\\sum\\:}_{i=1}^{n}{({ET}_{a(\\:\\text{i},\\:sim)}-{ET}_{a\\:(i,EC\\:tower)\\:})}^{2}}{n}}\\) (11) \\(\\:RRSE=\\sqrt{\\frac{\\sum\\:_{i=1}^{n}{\\left({ET}_{a(\\:\\text{i},\\:sim)}-{ET}_{a\\:(i,\\:EC\\:tower)\\:}\\right)}^{2}}{\\sum\\:_{i=1}^{n}{\\left({ET}_{a\\:(i,\\:EC\\:tower\\:)}-\\stackrel{-}{{ET}_{a\\:\\left(EC\\:tower\\:\\right)}}\\right)}^{2}}}\\) (12) \\(\\:r=\\frac{\\sum\\:_{i=1}^{n}(\\left({ET}_{a\\left(\\:\\text{i},\\:sim\\right)}-\\stackrel{-}{{ET}_{a\\:\\left(sim\\:\\right)}}\\right)\\times\\:\\left({ET}_{a\\:\\left(i,\\:EC\\:tower\\:\\right)}-\\stackrel{-}{{ET}_{a\\:\\left(EC\\:tower\\:\\right)}}\\right))}{\\sqrt{\\sum\\:_{i=1}^{n}{\\left({ET}_{a(\\:\\text{i},\\:sim)}-\\stackrel{-}{{ET}_{a\\:\\left(sim\\:\\right)}}\\right)}^{2}\\times\\:\\sum\\:_{i=1}^{n}{\\left({ET}_{a(\\:\\text{i},\\:\\text{E}\\text{C}\\:\\text{t}\\text{o}\\text{w}\\text{e}\\text{r})}-\\stackrel{-}{{ET}_{a\\:\\left(\\:EC\\:tower\\right)\\:}}\\right)}^{2}}}\\) (13) Where, \\(\\:{ET}_{a\\:(i,\\:EC\\:tower)\\:}\\) is the ET a calculated using EC flux tower data (mm) on i th day, \\(\\:{ET}_{a(\\:\\text{i},\\:sim)}\\) is the ET a simulated using the SMET model on i th day, \\(\\:\\stackrel{-}{{ET}_{a\\:\\left(sim\\:\\right)}}\\) is the average value of simulated ET a using SMET, \\(\\:\\stackrel{-}{{ET}_{a\\:\\left(EC\\:tower\\:\\right)}}\\) is the average value of ET a measured by EC tower, and n is the number of the days since the start of the growing season. 2.3. Study sites The study was conducted at two agricultural fields in Utah, USA, with one site in Vernal, and another in Modena (Fig. 5 and Table 1 ). Utah is classified as a semi-arid region with hot, dry summers and cold winters with most of the annual precipitation occurring during the winter months (Gillies et al. 2009 ). Utah’s mean altitude is 1835 m, mean precipitation is 359 mm, mean temperature is 9.2˚C, and mean solar radiation is 17 MJ/m 2 day (PRISM Climate Group at Oregon State University n.d.). The specific climate parameters for each study site are reported in Table 1 . The Vernal and Modena study sites were each equipped with an EC flux tower, a weather station, and an array of soil moisture sensors in alfalfa under sprinkler irrigation. Soil moisture sensors at Vernal were in close proximity to the EC tower, while at Modena two sets of soil moisture sensors were installed on the north and east side of the field (Fig. 5 ). Table 1 Study site details including elevation, and 30- year normal precipitation, mean air temperature, and solar radiation retrieved from the PRISM climate group at Oregon State University. Study site Vernal, UT, USA Modena, UT, USA Longitude (DD) -109.564270 -113.789074 Latitude (DD) 40.457876 37.747303 Irrigation System Wheel-lines Center pivot Area (Ha) 31.7 51.1 Growing season 2019–2020 2021 Elevation (m) 1703 1596 Precipitation (mm) 261 286 Mean Air Temperature (˚C) 8.1 9.5 Mean Solar Radiation (MJ/m 2 day) 17.2 18.5 Soil Moisture Sensor Locations 1 2 Sensor Depth (cm) 3, 10, 25, and 215 cm 8, 15, 30, 61, 91, 122, and 152 cm 2.3.1. Soil moisture data collection The volumetric water content data at each site were collected using a site-specific array of (Acclima TDR) sensors. According to the manufacturer’s specifications, the reporting accuracy of the sensors is ± 1% for coarse and medium textured soils and ± 2.5% for fine textured soils (Acclima, 2022 ). Details of the installation depths at each site are presented in Fig. 6 . The data were collected every 15 minutes and the reading from midnight of each day was used for analysis. 2.3.2. Model development at the Vernal site Raw soil moisture (SM) data were collected near the EC flux tower (Fig. 5 ) using seven TDR sensors at various depths (Fig. 6 A). At the Vernal site, sensor depths were chosen to represent the root zone of alfalfa, which extends down to ~ 2 m. 2.3.3. Model validation at the Modena site The raw soil moisture data were collected at two locations in the field, East and North (Fig. 5 ) in Modena. The East location had an array of seven TDR soil moisture sensors (Fig. 6 B) and the North location had an array of six TDR soil moisture sensors. On the north side of the field, the 152 cm deep sensor was not installed due to a hard gravel layer hindering root penetration. At Modena site, sensors were installed in the top-soil layers representing maximum root concentration with the greatest changes in soil moisture. This rationale led to having a close spacing near the surface to monitor soil moisture in the topsoil layers and a greater spacing at deeper depths to monitor soil moisture for the remainder rooting depth, which is expected to vary to a lesser extent. 3. Results and Discussion 3.1. Developing SMET model To develop the SMET model, which is a function of soil moisture depth and ET r , the data from the Vernal site was utilized. Soil moisture depth was calculated after screening the dataset to remove any values outside the expected range or those affected by sensor malfunctions. The data screening procedure is explained in the supplementary materials. After screening, soil moisture depth (Δθ) was calculated using Eq. 9 . By observing the soil moisture data for the Vernal study site (Supplementary Figure S1) irrigation events can be identified as they coincide with rapid increases followed by gradual decreases in volumetric water content. The spikes in soil moisture level caused by irrigation can be observed more easily at shallower soil moisture timeseries (3, 10, and 25 cm deep) and become more subtle with the increase in depth until they become difficult to discern in the deepest soil layer (215 cm deep). After irrigation events were identified, the Eureqa software was then used to identify the appropriate model that maps ET r and soil moisture depth to actual evapotranspiration, which was measured using an EC flux tower. Among the first ten most accurate models provided by Eureqa, the most straightforward equation was selected (Eq. 14 ). $$\\:{ET}_{a}={a\\times\\:ET}_{r}+b\\times\\:{ET}_{r}^{+}-c\\times\\:{{\\Delta\\:}\\theta\\:}^{-}$$ 14 Where, ET a is the daily actual evapotranspiration measured by the EC tower (mm d − 1 ), ET r is the daily reference evapotranspiration (mm d − 1 ), \\(\\:{{\\Delta\\:}{\\theta\\:}}^{-}\\) is the daily soil moisture depletion (negative changes in soil moisture) (mm d − 1 ), and \\(\\:{ET}_{r}^{+}\\) is the daily ET r on days when soil moisture is increasing. Eq. 14 was then split up into a conditional equation based on whether the soil moisture was increasing or decreasing: $$\\:{ET}_{a}=\\left\\{\\begin{array}{cc}a{ET}_{r}-c\\varDelta\\:\\theta\\:&\\:if\\:\\varDelta\\:\\theta\\:<0\\to\\:{{ET}_{r}}^{+}=0\\\\\\:a{ET}_{r}+b{ET}_{r}&\\:if\\:\\varDelta\\:\\theta\\:\\ge\\:0\\to\\:{\\varDelta\\:\\theta\\:}^{-}=0\\end{array}\\right.$$ 15 By simplifying Eq. 15 , the a, b, and c parameters were combined into a single α value, giving the following conditional equation: $$\\:{ET}_{a}=\\left\\{\\begin{array}{cc}\\propto\\:({ET}_{r}-\\varDelta\\:\\theta\\:)&\\:if\\:\\varDelta\\:\\theta\\:<0\\to\\:{{ET}_{r}}^{+}=0\\\\\\:2\\propto\\:{ET}_{r}&\\:if\\:\\varDelta\\:\\theta\\:\\ge\\:0\\to\\:{\\varDelta\\:\\theta\\:}^{-}=0\\end{array}\\right.$$ 16 Where α is a dimensional constant and ET a , ET r and Δθ are in mm/day. Eq. 16 is what will be referred to as the Soil Moisture based EvapoTranspiration estimation (SMET) model. 3.3. SMET model calibration The optimization of the α constant was performed by applying the SMET model (Eq. 16 ) to the combined 2019 and 2020 datasets collected at the research study in Vernal, UT using an initial value of \\(\\:\\alpha\\:=0.30\\) . This initial value for α was selected as an approximate average of the a, b, and c parameters provided by the Eureqa model (Eq. 13). The residuals and the RMSE were then calculated and used to fit Eq. 16 using the Solver function in Excel, yielding an optimized value for α of 0.43. The optimized model was then used to calculate the daily ET estimates and the cumulative monthly and cumulative seasonal values for the 2019 and 2020 growing seasons at Vernal. As shown in Table 2 , SMET estimated ET a , with an accuracy of 1.37 and 1.15 mm during 2019 and 2020, respectively. Table 2 Summary statistics for the daily calibrated SMET estimates for the 2019 and 2020 growing seasons. Year r RMSE (mm) RRSE (mm) 2019 0.76 1.37 0.67 2020 0.81 1.15 0.63 The second verification of the SMET model’s performance involved comparing its monthly and seasonal cumulative ET a values with those obtained from EC flux tower measurements for both 2019 and 2020. As shown in Fig. 7 , the model generally follows the seasonal ET a trend observed in the EC data, capturing the peak evapotranspiration during summer months. Although the SMET model slightly underestimates monthly ET a in several months, particularly in July, where the EC original value is highest, it still aligns well with the overall seasonal totals for both years. In 2020, the agreement is stronger, especially during peak months (June–July 2020), where SMET estimates closely match or even exceed EC values. These results confirm that while some month-to-month discrepancies exist, the model is reliable for seasonal-scale water use estimation. By separating the irrigation from the non-irrigation days, during which soil moisture increases or decreases, respectively (Eq. 16 ), the SMET model can provide some insight into the irrigation treatment in the absence of such data. For the Vernal site, there were 24 and 27 irrigation days for the 2019 and 2020 years, respectively and the ET occurring during those days amounted to approximately 16% and 19% of the total, respectively (Table 3 ). Table 3 Irrigation information inferred from the SMET model for the 2019 and 2020 growing seasons in Vernal, UT, USA Year Total days Irrigation days ET during irrigation days (mm) 2019 203 24 12% 116 16% 2020 213 27 13% 138 19% 3.4. SMET model verification The optimized SMET model, with the same α value obtained from the calibration performed with the Vernal data, was applied to the Modena study site in two locations within the alfalfa field (Fig. 5 ) using the soil moisture and ET r data during the 2021 growing season from April 1st to October 22nd, to calculate cumulative ET a . The field average seasonal ET a calculated by SMET was 1114 mm, which was within the range measured by the EC tower of 1032–1191 mm (Fig. 8 ). These findings indicated that as long as the fundamental conditions (i.e., deep sensor array, crop canopy completely covers the surface) of the SMET model are not violated, it can successfully be applied to sites other than the one used for its calibration to obtain reasonable estimates of ET on a seasonal scale. Similar to the approach used at Vernal study site, the SMET model provided some insight into irrigations applied to the field in Modena. The estimated number of irrigation days for the North and East locations in the field was 61 (30%) and 57 (28%), respectively and the amount of ET a that occurred during irrigation days was 27% and 29%, respectively (Table 4 ). While these results illustrate the model’s potential for evaluating field-scale irrigation practices, it is important to note that the model’s accuracy can be influenced by factors such as irrigation method, monitoring depth, and data continuity. As outlined in the supplementary section, specific considerations must be evaluted, such as ensuring full root zone sensor coverage, applying the model on appropriate temporal scales, and accounting for variations in irrigation system type (e.g., drip vs. flood). These considerations are critical for improving the reliability and broader applicability of the SMET model across diverse agricultural settings. Table 4 SMET model irrigation days estimates for the Modena, UT, USA study site during the 2021 growing season. Site Total days Irrigation days ET a during irrigation days North 204 61 30% 362 mm 27% East 204 57 28% 344 mm 29% 4. Conclusions The proposed SMET model in this study is capable of reliably estimating seasonal ET a with accuracy using daily soil moisture data from sensors and weather data. The model provides a practical tool for estimating water demand under different fields with limited available data. The results were consistent with those measured by the EC tower across the two alfalfa sites, though further calibration is needed for α under different crops and climatic conditions. The simplicity and ease of the implementation of the SMET model make it a valuable tool for land managers, offering an efficient and cost-effective way to estimate seasonal water demands with minimal investment. It could also be highly beneficial for research trials conducted at the plot or strip plot scale, a common practice in the crop and soil science disciplines. Further, many research plots already employ soil moisture sensors to monitor crop water use and needs and SMET could provide a useful tool for estimating ET a at smaller scales than existing ET a instruments or models. The SMET model can be used in areas with a heterogeneous landscape, allowing easy installation of soil moisture sensors, which is impractical with ET a flux tower and lysimeter. Further, it could be an efficient tool to estimate ET a under small agricultural research setting experiments with different treatments, where EC tower might not be able to detect the subtle differences among treatments and installing micro lysimeters would be laborious. Finally, the model can also offer valuable insights when used alongside an EC tower under a heterogeneous environment by providing detailed resolution of water consumption. This approach could be particularly valuable if ET a data is remotely sensed using drones or satellites (e.g. OpenET), enabling estimates of root zone soil moisture. Declarations Data Availability Statement Some or all data, models, or code supporting the findings of this study are available from the corresponding author upon reasonable request. References Abdelmajeed AYA, Juszczak R (2024) Challenges and limitations of remote sensing applications in northern peatlands: present and future prospects. Remote Sens 16(3):591. 10.3390/RS16030591 Acclima (2022) True TDR-315H Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspiration – Guidelines for computing crop water requirements. FAO Irrigation and Drainage Paper 56. Food and Agriculture Organization of the United Nations, Rome Allen RG, Tasumi M, Trezza R (2007) Satellite-based energy balance for mapping evapotranspiration with internalized calibration (METRIC)—Model. Journal of Irrigation and Drainage Engineering 133 (4): 380–394. 10.1061/(ASCE)0733-9437 (2007)133:4(380) Baldocchi DD (2003) Assessing the eddy covariance technique for evaluating carbon dioxide exchange rates of ecosystems: past, present and future. Glob Change Biol 9(4):479–492. 10.1046/j.1365-2486.2003.00629.x Bastiaanssen WGM, Menenti M, Feddes RA, Holtslag AAM (1998) A remote sensing surface energy balance algorithm for land (SEBAL). 1. Formulation. J Hydrol 212–213(1–4):198–212. 10.1016/S0022-1694(98)00253-4 Bowen IS (1926) The ratio of heat losses by conduction and by evaporation from any water surface. Phys Rev 27(6):779. 10.1103/PhysRev.27.779 Bryant CJ, Spencer GD, Gholson DM, Plumblee MT, Dodds DM, Oakley GR, Reynolds DZ, Krutz LJ (2023) Development of a soil moisture sensor-based irrigation scheduling program for the midsouthern United States. Crop Forage Turfgrass Manage 9(1):e20217. 10.1002/CFT2.20217 Datta S, Taghvaeian S (2023) Soil water sensors for irrigation scheduling in the United States: A systematic review of literature. Agric Water Manage 278:108148. 10.1016/j.agwat.2023.108148 DeJonge K (2024) Evapotranspiration terminology: historical evolution and current standardization. AGU Fall Meeting Abstracts H13:S–06 Drechsler K, Fulton A, Kisekka I (2022) Crop coefficients and water use of young almond orchards. Irrig Sci 40(3):379–395 FAO n.d. Food and Agriculture Organization of the United Nations Foken T (2008) The energy balance closure problem: an overview. In: Flux Measurements: Ecological Applications Franssen HJH, Stöckli R, Lehner I, Rotenberg E, Seneviratne SI (2010a) Energy balance closure of eddy-covariance data: a multisite analysis for European FLUXNET stations. Agric For Meteorol 150(12):1553–1567 Franssen HJH, Stöckli R, Lehner I, Rotenberg E, Seneviratne SI (2010b) Energy balance closure of eddy-covariance data: a multisite analysis for European FLUXNET stations. Agric For Meteorol 150(12):1553–1567. 10.1016/j.agrformet.2010.08.005 Gillies RR, Ramsey RD, Banner RE, Baldwin BD, McGinty EL (2009) Climate of Utah. Rangelands Resources of Utah. Utah State University Cooperative Extension Service, Logan, pp 39–45 Hashemi M, Sepaskhah AR (2020) Evaluation of artificial neural network and Penman–Monteith equation for the prediction of barley standard evapotranspiration in a semi-arid region. Theoret Appl Climatol 139(1–2):275–285. 10.1007/s00704-019-02966-x Keller J, Bliesner RD (1990) Sprinkle and Trickle Irrigation. Van Nostrand Reinhold, New York Kisekka I, Peddinti SR, Kustas WP, McElrone AJ, Bambach-Ortiz N, McKee L, Bastiaanssen W (2022) Spatial–temporal modeling of root zone soil moisture dynamics in a vineyard using machine learning and remote sensing. Irrig Sci 40(4):761–777 Markwitz C, Siebicke L (2019) Low-cost eddy covariance: a case study of evapotranspiration over agroforestry in Germany. Atmos Meas Tech 12(9):4677–4696 Mathewos Boltana S, Demelash W, Bekele T, Yisihak U, Lohani TK, Wondemeneh Bekele D, Ukumo TY (2023) Evaluation of irrigation scheduling to maximize tomato production using comparative assessment of soil moisture and evapotranspiration in restricted irrigated regions. Cogent Food Agric 9(1). 10.1080/23311932.2023.2214428 Maughan T, Allen LN, Drost D (2015) Soil moisture measurement and sensors for irrigation management. All Current Norman JM, Kustas WP, Humes KS (1995) Source approach for estimating soil and vegetation energy fluxes in observations of directional radiometric surface temperature. Agric For Meteorol 77(3–4):263–293. 10.1016/0168-1923(95)02265-Y Peddinti SR, Kisekka I (2025) Evaluation of the LI-710 evapotranspiration sensor in comparison to full eddy covariance for monitoring energy fluxes in perennial and annual crops. Agric Water Manage 313:109501. 10.1016/j.agwat.2025.109501 PRISM Climate Group at Oregon State University n.d. 30-year normals. Accessed July 31, 2022. Available at: https://prism.oregonstate.edu/normals/ Scanlon BR, Fakhreddine S, Rateb A, de Graaf I, Famiglietti J, Gleeson T, Grafton RQ, Jobbagy E, Kebede S, Kolusu SR, Konikow LF, Long D, Mekonnen M, Schmied HM, Mukherjee A, MacDonald A, Reedy RC, Shamsudduha M, Simmons CT, Sun A, Taylor RG, Villholth KG, Vörösmarty CJ, Zheng C (2023) Global water resources and the role of groundwater in a resilient water future. Nat Reviews Earth Environ 4(2):87–101. 10.1038/s43017-022-00378-6 Schmidt M, Lipson H (2009) Distilling free-form natural laws from experimental data. Science 324(5923):81–85 Šimůnek J, van Genuchten MTh ŠejnaM (2008) Development and applications of the HYDRUS and STANMOD software packages and related codes. Vadose Zone J 7(2):587–600. 10.2136/vzj2007.0077 Su Z (2002) The Surface Energy Balance System (SEBS) for estimation of turbulent heat fluxes. Hydrol Earth Syst Sci 6(1):85–100. 10.5194/hess-6-85-2002 Walker WR, Skogerboe GV (1987) Surface Irrigation: Theory and Practice. Prentice-Hall, Englewood Cliffs, NJ Wan Z, Zhang K, Xue X, Hong Z, Hong Y, Gourley JJ (2015) Water balance-based actual evapotranspiration reconstruction from ground and satellite observations over the conterminous United States. Water Resour Res 51(8):6485–6499 Zotarelli L, Dukes MD, Scholberg JMS, Femminella K, Muñoz-Carpena R (2011) Irrigation scheduling for green bell peppers using capacitance soil moisture sensors. J Irrig Drain Eng 137(2):73–81. 10.1061/(ASCE)IR.1943-4774.0000281 Additional Declarations The authors declare no competing interests. Supplementary Files Supplementary.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-7124239\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":true,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":485371356,\"identity\":\"0b9d4916-978e-4401-9c11-bad6a17becc1\",\"order_by\":0,\"name\":\"Masoumeh Hashemi\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYBCDBAYJBgZmIIOHHyLATIIWyQZStTAYHCCgRbf9+MPHvDk2eQzSzY8/F1RskzG+kf5MgqHCOrEBhxazMznGxrzb0ooZZI6ZSc84c5vH7EaOmQTDmXTcWg7ksEnzbjuc2CCRYMbM2wbWwibB2HYYt5bzz58BtfwHakn//Jn3320e4xlAhzH+w6PlRoIZUMsBoJYcA2nehts8BkDrJBgb8Gl5Y2w4d1tyYptETpk0z7HbPBJn3hhbJBxLN8btsPSHD95us0vsl0jf/Jmn5rY9f3v6wxsfaqxlcWmBAzYUXgIh5aNgFIyCUTAK8AIAUXxYBdYDKyEAAAAASUVORK5CYII=\",\"orcid\":\"https://orcid.org/0000-0001-8649-5208\",\"institution\":\"Utah State University\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Masoumeh\",\"middleName\":\"\",\"lastName\":\"Hashemi\",\"suffix\":\"\"},{\"id\":485371357,\"identity\":\"78940f7a-590f-487a-88db-263273fe504c\",\"order_by\":1,\"name\":\"Tejinder Singh\",\"email\":\"\",\"orcid\":\"https://orcid.org/0009-0007-8184-8028\",\"institution\":\"Utah State University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Tejinder\",\"middleName\":\"\",\"lastName\":\"Singh\",\"suffix\":\"\"},{\"id\":485373216,\"identity\":\"bb50e578-9511-47ba-88f0-a36134c84980\",\"order_by\":2,\"name\":\"Oliver Hargreaves\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Utah State University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Oliver\",\"middleName\":\"\",\"lastName\":\"Hargreaves\",\"suffix\":\"\"},{\"id\":485373217,\"identity\":\"7f663528-1391-4308-911f-eff4a7bff065\",\"order_by\":3,\"name\":\"Alfonso Torres-Rua\",\"email\":\"\",\"orcid\":\"https://orcid.org/0000-0002-2238-9550\",\"institution\":\"Utah State University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Alfonso\",\"middleName\":\"\",\"lastName\":\"Torres-Rua\",\"suffix\":\"\"},{\"id\":485374119,\"identity\":\"2e06b14a-4cb8-47fa-9a55-e0faad97c5d7\",\"order_by\":4,\"name\":\"Matt Yost\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Utah State University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Matt\",\"middleName\":\"\",\"lastName\":\"Yost\",\"suffix\":\"\"},{\"id\":485374120,\"identity\":\"25cb13f5-b0c3-4718-9f07-7a06dfcde248\",\"order_by\":5,\"name\":\"Lawrence Hipps\",\"email\":\"\",\"orcid\":\"\",\"institution\":\"Utah State University\",\"correspondingAuthor\":false,\"prefix\":\"\",\"firstName\":\"Lawrence\",\"middleName\":\"\",\"lastName\":\"Hipps\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2025-07-14 20:17:13\",\"currentVersionCode\":1,\"declarations\":{\"humanSubjects\":false,\"vertebrateSubjects\":false,\"conflictsOfInterestStatement\":false,\"humanSubjectEthicalGuidelines\":false,\"humanSubjectConsent\":false,\"humanSubjectClinicalTrial\":false,\"humanSubjectCaseReport\":false,\"vertebrateSubjectEthicalGuidelines\":false},\"doi\":\"10.21203/rs.3.rs-7124239/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-7124239/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":86890322,\"identity\":\"541dbb0d-62eb-4306-9359-dec580b1db6f\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:39:02\",\"extension\":\"png\",\"order_by\":1,\"title\":\"Figure 1\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":198389,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eWorkflow of the proposed methodology for estimating actual evapotranspiration (ET\\u003csub\\u003ea\\u003c/sub\\u003e) in a field using soil moisture data and reference evapotranspiration\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure1.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/165189dec72d18d36d103daf.png\"},{\"id\":86889699,\"identity\":\"42e41b8b-3760-4bae-b82d-d322af678130\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:23:02\",\"extension\":\"png\",\"order_by\":2,\"title\":\"Figure 2\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":4045559,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eWater balance model scheme. The components of the water balance that influence changes in soil moisture (Δθ) are represented by the arrows. Arrows pointing toward the plants indicate a process that increases soil moisture, while the arrows pointing away from plants represent a process that decreases soil moisture.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure2.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/a4de16f50749edfd84a90df5.png\"},{\"id\":86889408,\"identity\":\"5a9a1c6b-1d71-4a22-9f2a-862e1fad7737\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:15:02\",\"extension\":\"png\",\"order_by\":3,\"title\":\"Figure 3\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":516601,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eFlowcharts of developing SMET model and calibration\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure3.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/3b8a1c507daf33ad78118067.png\"},{\"id\":86890071,\"identity\":\"634c5a5d-930e-4712-9375-40ece23ace49\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:31:02\",\"extension\":\"png\",\"order_by\":4,\"title\":\"Figure 4\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":252461,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eFlowchart of the validation process for the calibrated SMET model using EC flux tower data\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure4.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/94f792436e97e69daad24b4d.png\"},{\"id\":86889428,\"identity\":\"370042ba-36a2-4214-a6a3-9d8a3b636018\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:15:02\",\"extension\":\"png\",\"order_by\":5,\"title\":\"Figure 5\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":18964956,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eMaps of the two study sites in UT, USA used to develop SMET\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure5.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/8e9f275a7bc2adda564f7a66.png\"},{\"id\":86890323,\"identity\":\"ba0781ff-df8d-4c4b-93e2-0d9311e1dbc3\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:39:02\",\"extension\":\"png\",\"order_by\":6,\"title\":\"Figure 6\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":1428280,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eSoil moisture sensor arrays for the Vernal, UT, USA (A), and Modena, UT sites\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure6.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/0d00abef5b04e729457d2f72.png\"},{\"id\":86890076,\"identity\":\"b79b9063-24b8-4bd9-862d-f9df5ae24fb2\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:31:02\",\"extension\":\"png\",\"order_by\":7,\"title\":\"Figure 7\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":898777,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eComparison of the monthly cumulative values from the Soil Moisture Evapotranspiration (SMET) model and the eddy covariance (EC) tower measurements for the site in Vernal, UT, USA in 2019 (left) and 2020 (right).\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure7.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/c758e713ee68e56b7742c2d6.png\"},{\"id\":86889419,\"identity\":\"686e40fb-323f-4fa4-b881-3e8c3eeca279\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:15:02\",\"extension\":\"png\",\"order_by\":8,\"title\":\"Figure 8\",\"display\":\"\",\"copyAsset\":false,\"role\":\"figure\",\"size\":475967,\"visible\":true,\"origin\":\"\",\"legend\":\"\\u003cp\\u003eSoil Moisture EvapoTranspiration (SMET) model ET\\u003csub\\u003ea\\u003c/sub\\u003e estimates for the 2021 growing season at the East soils moisture sensors station at the Modena, UT, USA site compared to the EC tower closed measurements.\\u003c/p\\u003e\",\"description\":\"\",\"filename\":\"MasoumehhashemiFigure8.png\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/295eee85fe7aff7e70ad0d46.png\"},{\"id\":87466912,\"identity\":\"79bc07e1-26ec-48f7-a228-95795e25ba0e\",\"added_by\":\"auto\",\"created_at\":\"2025-07-24 07:36:31\",\"extension\":\"pdf\",\"order_by\":0,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":26603382,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"manuscript.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/16b5a566-1bdb-48f0-8c15-e3f78872ab96.pdf\"},{\"id\":86890074,\"identity\":\"9a7e429e-6853-4a69-8eee-3a1054d433d3\",\"added_by\":\"auto\",\"created_at\":\"2025-07-16 19:31:02\",\"extension\":\"docx\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"supplement\",\"size\":1030069,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"Supplementary.docx\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-7124239/v1/44182cf4df915c35320b044c.docx\"}],\"financialInterests\":\"The authors declare no competing interests.\",\"formattedTitle\":\"\\u003cp\\u003e\\u003cstrong\\u003eSoil Moisture Evapotranspiration (SMET): A Low-Cost Method for Determining Seasonal Crop Water Demand\\u003c/strong\\u003e\\u003c/p\\u003e\",\"fulltext\":[{\"header\":\"Practical Application\",\"content\":\"\\u003cp\\u003eKnowing how much water crops use is critical for farmers, especially in dry regions like Utah. Many current methods to measure this water use, called evapotranspiration, are expensive, complex, and not practical for most farms. This study introduces a new, simple method called SMET that helps estimate how much water crops actually use during the growing season. It uses common tools that many farmers already have: soil moisture sensors and local weather station data. The SMET method was tested on alfalfa fields in Utah and showed strong agreement with advanced measurement systems, proving that it is both reliable and cost-effective. Farmers, land managers, and researchers can use this method to better understand how much water crops need and to improve irrigation decisions. The model also works well in small research plots where advanced tools like eddy covariance towers or lysimeters can\\u0026rsquo;t be used. By making water use estimates easier and more affordable, SMET can help support sustainable agriculture in water-scarce regions.\\u003c/p\\u003e\"},{\"header\":\"1. Introduction\",\"content\":\"\\u003cp\\u003eWater is an indispensable natural resource essential for sustaining life and ecosystems. However, its availability is limited by climate-induced extremes like drought and over-extraction (Hashemi et al. 2019; Scanlon et al. \\u003cspan citationid=\\\"CR26\\\" class=\\\"CitationRef\\\"\\u003e2023\\u003c/span\\u003e). In regions where economies are predominantly agrarian, the agricultural sector accounts for approximately 90% of total water consumption, with industrial use representing only 3% (Fao n.d.). In arid regions of the US, like Utah, irrigation represents nearly 80% of diverted water consumption (Maughan et al. \\u003cspan citationid=\\\"CR22\\\" class=\\\"CitationRef\\\"\\u003e2015\\u003c/span\\u003e). The predominance of agriculture in the US and global water use necessitates a detailed assessment of water use in water-scarce regions.\\u003c/p\\u003e\\u003cp\\u003eEfficient use of water in arid regions in the context of irrigation remains the utmost priority for land managers. Irrigation efficiency in crop fields depends on the agricultural water balance accounting for the inflow [Precipitation (P)\\u0026thinsp;+\\u0026thinsp;Irrigation (I)\\u0026thinsp;+\\u0026thinsp;Capillary Rise (CR)]; outflow [evapotranspiration (ET)\\u0026thinsp;+\\u0026thinsp;deep percolation (DP)\\u0026thinsp;+\\u0026thinsp;runoff (RO)] and changes in the soil water storage (Hashemi and Sepaskhah \\u003cspan citationid=\\\"CR17\\\" class=\\\"CitationRef\\\"\\u003e2020\\u003c/span\\u003e). Of these components, ET measurements are critical in influencing the crop's irrigation water requirement. ET is quantified as the amount of water lost from the crop root zone through evaporation from the soil and transpiration by existing vegetation. Accurate and effective estimation of ET remains a persistent challenge in hydrological and agricultural research.\\u003c/p\\u003e\\u003cp\\u003eConventional methodologies, including eddy covariance (EC) systems, lysimeter-based measurements, and remote sensing techniques, are constrained by certain limitations (Peddinti and Kisekka \\u003cspan citationid=\\\"CR24\\\" class=\\\"CitationRef\\\"\\u003e2025\\u003c/span\\u003e). The EC system and lysimeter provide high-resolution ET measurements but are expensive, labor-intensive, and require technical expertise for operation. Furthermore, the EC system is well-suited for larger spatial footprints limiting its applicability in small agricultural research experiments. Remote sensing imagery tools developed based on data from these methods are also constrained by spatial and temporal resolution limiting their use (Abdelmajeed and Juszczak \\u003cspan citationid=\\\"CR1\\\" class=\\\"CitationRef\\\"\\u003e2024\\u003c/span\\u003e). These limitations highlight the need for scalable, low-cost ET estimation tools for land managers.\\u003c/p\\u003e\\u003cp\\u003eSoil moisture sensors are widely used for irrigation scheduling in agricultural fields. Soil moisture sensors, combined with climatic data, can be effective tools for irrigation scheduling that improves water use efficiency under diverse environmental and agronomic conditions (Mathewos Boltana et al. \\u003cspan citationid=\\\"CR21\\\" class=\\\"CitationRef\\\"\\u003e2023\\u003c/span\\u003e). The effectiveness of soil moisture sensors for irrigation scheduling is supported by many studies. For example, using a capacitance-based soil moisture sensor controller reduced irrigation water use by 55% for green bell peppers (Capsicum annum) (Zotarelli et al. \\u003cspan citationid=\\\"CR32\\\" class=\\\"CitationRef\\\"\\u003e2011\\u003c/span\\u003e). In the mid-southern US, on-farm research using soil tensiometer sensors indicates that \\u0026minus;\\u0026thinsp;80 to -100 cbar reduces water usage by 40% in maize (\\u003cem\\u003eZea mays\\u003c/em\\u003e) and soybean (\\u003cem\\u003eGlycine max\\u003c/em\\u003e) while improving water use efficiency by up to 51% (Bryant et al. \\u003cspan citationid=\\\"CR8\\\" class=\\\"CitationRef\\\"\\u003e2023\\u003c/span\\u003e). Several different types of soil moisture sensors have been used for irrigation scheduling. A recent review on soil water sensors for irrigation scheduling highlights that 33% of studies used electrical resistivity (ER) sensors, followed by 21%, 16%, and 13% use for frequency domain reflectometry (FDR), neutron probes, and time domain reflectometry (TDR), along with another 17% that combined tensiometer and TDR in the US (Datta and Taghvaeian \\u003cspan citationid=\\\"CR9\\\" class=\\\"CitationRef\\\"\\u003e2023\\u003c/span\\u003e).\\u003c/p\\u003e\\u003cp\\u003eAlthough soil moisture sensors are widely used for irrigation scheduling their potential for estimating actual evapotranspiration (ET\\u003csub\\u003ea\\u003c/sub\\u003e) has not been fully investigated. Using these readily available sensors that are already installed in some land managers' fields or research plots to estimate ET\\u003csub\\u003ea\\u003c/sub\\u003e could enhance the spatial resolution of in-field estimations. It may also serve as cost-effective ground validation for satellite-based ET\\u003csub\\u003ea\\u003c/sub\\u003e models. The objective of this study was to develop a cost-effective methodology using soil moisture data collected by farmers and land managers, along with reference evapotranspiration (ET\\u003csub\\u003er\\u003c/sub\\u003e) from nearby weather stations, to estimate ET\\u003csub\\u003ea\\u003c/sub\\u003e. This tool may provide a simple and cost-effective tool for estimating ETa compared to more cost or labor-prohibitive Bowen Ratio Energy Balance (BREB) method (Bowen \\u003cspan citationid=\\\"CR7\\\" class=\\\"CitationRef\\\"\\u003e1926\\u003c/span\\u003e), the micrometeorological technique for vertical turbulent fluxes of gases (Baldocchi \\u003cspan citationid=\\\"CR5\\\" class=\\\"CitationRef\\\"\\u003e2003\\u003c/span\\u003e), surface energy balance system (Allen et al. \\u003cspan citationid=\\\"CR4\\\" class=\\\"CitationRef\\\"\\u003e2007\\u003c/span\\u003e; Bastiaanssen et al. \\u003cspan citationid=\\\"CR6\\\" class=\\\"CitationRef\\\"\\u003e1998\\u003c/span\\u003e; Norman et al. \\u003cspan citationid=\\\"CR23\\\" class=\\\"CitationRef\\\"\\u003e1995\\u003c/span\\u003e; Su \\u003cspan citationid=\\\"CR29\\\" class=\\\"CitationRef\\\"\\u003e2002\\u003c/span\\u003e).\\u003c/p\\u003e\"},{\"header\":\"2. Materials and methods\",\"content\":\"\\u003cdiv id=\\\"Sec3\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e2.1. \\u003cb\\u003eModel description\\u003c/b\\u003e\\u003c/h2\\u003e\\u003cp\\u003eActual evapotranspiration (ET\\u003csub\\u003ea\\u003c/sub\\u003e) refers to water loss due to evaporation and transpiration under existing environmental and management conditions, encompassing both standard and non-standard scenarios (DeJonge \\u003cspan citationid=\\\"CR10\\\" class=\\\"CitationRef\\\"\\u003e2024\\u003c/span\\u003e). ET\\u003csub\\u003ea\\u003c/sub\\u003e provides essential information that can help farmers and researchers make informed decisions about agricultural management by allowing them to evaluate the efficiency of their irrigation systems to help them optimize water use and improve sustainability. Various methodologies have been developed to measure ET\\u003csub\\u003ea\\u003c/sub\\u003e, but many of them are limited by spatial and temporal resolution.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003cp\\u003eThis study considers two primary methods for quantifying crop water use, the Water Budget (WB) and the Crop Coefficient Method (CCM) aims to integrate the relation between WB and CCM, and synthesizes these methods to determine ET\\u003csub\\u003ea\\u003c/sub\\u003e (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e). A case study using EC flux tower data as a ground truth for ET\\u003csub\\u003ea\\u003c/sub\\u003e was employed to build and refine the model. A two-stage evaluation by calibration and validation was performed to assess the use of crop water at different locations. A detailed explanation of each step is presented in the following section.\\u003c/p\\u003e\\u003cdiv id=\\\"Sec4\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.1.1. Interpreting ET\\u003csub\\u003ea\\u003c/sub\\u003e, Crop Coefficient Method, and Water Budget Relationships\\u003c/h2\\u003e\\u003cdiv id=\\\"Sec5\\\" class=\\\"Section4\\\"\\u003e\\u003ch2\\u003e2.1.1.1. Water Budget (WB)\\u003c/h2\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"No\\\" id=\\\"Taba\\\" border=\\\"1\\\"\\u003e\\u003ccolgroup cols=\\\"2\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e\\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{ET}_{a}=I+P+G-{\\\\Delta\\\\:}{\\\\theta\\\\:}-RO-DP\\\\)\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e(1)\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eThe WB method is based on the principle of mass conservation, where the change in storage is equal to the net difference between all input and output flows (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e). This approach can be explained using Eq.\\u0026nbsp;1,(Wan et al. \\u003cspan citationid=\\\"CR31\\\" class=\\\"CitationRef\\\"\\u003e2015\\u003c/span\\u003e).\\u003c/p\\u003e\\u003cp\\u003eWhere, I is the amount of the irrigation (mm), P is the amount of the precipitation (mm), G is the groundwater contribution (mm), Δθ is the change in soil water storage in the root zone (mm), RO is the amount of runoff (mm), and DP is the amount of deep percolation (mm), which can be used to estimate the residual term of ET\\u003csub\\u003ea\\u003c/sub\\u003e .\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003cp\\u003eAlthough a simple model in theory, the WB is challenging in practice due to the difficulty in accurately measuring deep percolation (Walker and Skogerboe \\u003cspan citationid=\\\"CR30\\\" class=\\\"CitationRef\\\"\\u003e1987\\u003c/span\\u003e) and groundwater contributions, and to a lesser extent, irrigation and runoff. Due to difficulties in accurately estimating each parameter, a simplified model to estimate ET\\u003csub\\u003ea\\u003c/sub\\u003e was used with certain assumptions.\\u003c/p\\u003e\\u003cp\\u003eThe water balance, assuming that RO, G, and DP are negligible (Assumption 1: \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:RO=G=DP\\\\approx\\\\:0\\\\)\\u003c/span\\u003e\\u003c/span\\u003e), allows for ET\\u003csub\\u003ea\\u003c/sub\\u003e to be calculated as a function of P, I, and Δθ (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ1\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e).\\u003cdiv id=\\\"Equ1\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ1\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{\\\\Delta\\\\:}\\\\theta\\\\:=P+I-ETa$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e2\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eThese assumptions are made to simplify the ET\\u003csub\\u003ea\\u003c/sub\\u003e estimation, but under actual field conditions, the magnitude of each parameter varies with the irrigation system. By considering applying the function (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ1\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e) in the regions where precipitation is infrequent and light during the growing season, P will be considered negligible compared to the irrigation amount (Assumption 2: \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:P\\\\ll\\\\:I\\\\to\\\\:P+I\\\\approx\\\\:I\\\\)\\u003c/span\\u003e\\u003c/span\\u003e). This assumption can be easily verified as a posteriori using local meteorological data, which further simplifies the water balance:\\u003cdiv id=\\\"Equ2\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ2\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{\\\\Delta\\\\:}\\\\theta\\\\:=I-ETa$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e3\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eFinally, as precise irrigation data are not always available, the algorithms developed in this analysis are not designed to produce an estimate of ET\\u003csub\\u003ea\\u003c/sub\\u003e during irrigation days (Assumption 3: \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:I=0\\\\)\\u003c/span\\u003e\\u003c/span\\u003e). This approach helps mitigate error due to Assumption 1, since the greatest RO and DP occur during irrigation. After integrating all these assumptions, the ET\\u003csub\\u003ea\\u003c/sub\\u003e in the water budget is simplified to depend solely on the change in soil moisture:\\u003cdiv id=\\\"Equ3\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ3\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:ETa=-{\\\\Delta\\\\:}\\\\theta\\\\:$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e4\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eIn brief, a simplified version of the water balance works under assumptions of no irrigation or precipitation (I\\u0026thinsp;=\\u0026thinsp;P\\u0026thinsp;=\\u0026thinsp;0), well-drained soils (RO\\u0026thinsp;=\\u0026thinsp;DP\\u0026thinsp;=\\u0026thinsp;0), and a water table well below the root zone (G\\u0026thinsp;=\\u0026thinsp;0). This approach assumes ET\\u003csub\\u003ea\\u003c/sub\\u003e is represented by the negative change in soil moisture (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ3\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003e).\\u003c/p\\u003e\\u003cp\\u003eThese assumptions are influenced by soil characteristics (e.g., water retention, soil texture) and water movement following irrigation events. When RO, G, and DP are significant, the only conclusion that can be drawn without further verification is that ET\\u003csub\\u003ea\\u003c/sub\\u003e is a function of change in soil moisture (Eq.\\u0026nbsp;5).\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"No\\\" id=\\\"Tabb\\\" border=\\\"1\\\"\\u003e\\u003ccolgroup cols=\\\"2\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e\\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{ET}_{a}=f\\\\left(-{\\\\Delta\\\\:}\\\\theta\\\\:\\\\right)\\\\)\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e(5)\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec6\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.1.2. \\u003cb\\u003eCrop Coefficient Method\\u003c/b\\u003e\\u003c/h2\\u003e\\u003cp\\u003eThe crop coefficient method utilizes a reference value for ET\\u003csub\\u003ea\\u003c/sub\\u003e that represents the atmospheric demand (i.e., either ET\\u003csub\\u003er\\u003c/sub\\u003e or ET\\u003csub\\u003eo\\u003c/sub\\u003e) multiplied by the crop coefficient (K\\u003csub\\u003ec\\u003c/sub\\u003e), i.e., specific to the crop, growth stage, and climatic condition (e.g., wind and relative humidity), to estimate ET\\u003csub\\u003ea\\u003c/sub\\u003e (Allen et al. \\u003cspan citationid=\\\"CR3\\\" class=\\\"CitationRef\\\"\\u003e1998\\u003c/span\\u003e).\\u003cdiv id=\\\"Equ4\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ4\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}={K}_{c}\\\\times\\\\:{ET}_{r}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e6\\u003c/div\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Equ5\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ5\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}=f\\\\left({ET}_{r}\\\\right)$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e7\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eIn this method, it is assumed that the soil moisture is at field capacity, which means that evapotranspiration is governed by climatic conditions and is considered potential evapotranspiration. Consequently, the crop's water needs follow the pattern of atmospheric demand, and its water use is similar to that of reference crops [e.g., alfalfa (Medicago sativa)]. Therefore, the following relationship can be expressed:\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec7\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e2.2. Proposed methodology\\u003c/h2\\u003e\\u003cp\\u003eEquations\\u0026nbsp;5 and \\u003cspan refid=\\\"Equ5\\\" class=\\\"InternalRef\\\"\\u003e7\\u003c/span\\u003e demonstrate how ET\\u003csub\\u003ea\\u003c/sub\\u003e is a complex phenomenon that can be modeled as part of the water cycle, but also as a biophysical process that responds to environmental factors. Solving equations for ET\\u003csub\\u003ea\\u003c/sub\\u003e within the complete water balance requires the use of advanced modeling software such as HYDRUS 1-D, which uses linear finite elements to numerically solve the Richards equation for saturated-unsaturated water flow and Fickian-based advection-dispersion equations for both heat and solute transport (Šimůnek et al., 2008). The use of this type of complex software requires training and is not feasible for most land managers outside of research settings. The method proposed in this paper attempts to combine and simplify Equations 5 and \\u003cspan refid=\\\"Equ5\\\" class=\\\"InternalRef\\\"\\u003e7\\u003c/span\\u003e by expressing ET\\u003csub\\u003ea\\u003c/sub\\u003e as a function of only the changes in soil water depletion (Δθ) and the atmospheric demand (ET\\u003csub\\u003er\\u003c/sub\\u003e), which can be easily measured (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ6\\\" class=\\\"InternalRef\\\"\\u003e8\\u003c/span\\u003e).\\u003cdiv id=\\\"Equ6\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ6\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}=f\\\\left(\\\\varDelta\\\\:\\\\theta\\\\:,{\\\\:ET}_{r}\\\\right)$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e8\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cdiv id=\\\"Sec8\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.2.1. Developing the equation and calibrating\\u003c/h2\\u003e\\u003cp\\u003eFigure \\u003cspan refid=\\\"Fig3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e illustrates the process of building and calibrating the proposed model. In the first step, we developed the Soil Moisture-based Evapotranspiration estimation (SMET) model, which estimates ET\\u003csub\\u003ea\\u003c/sub\\u003e using Soil Moisture (SM) change and ET\\u003csub\\u003er\\u003c/sub\\u003e. To achieve this, soil moisture change is calculated using equations \\u003cspan refid=\\\"Equ7\\\" class=\\\"InternalRef\\\"\\u003e9\\u003c/span\\u003e, and ET\\u003csub\\u003er\\u003c/sub\\u003e is obtained from the nearest weather station.\\u003cdiv id=\\\"Equ7\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ7\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{\\\\varDelta\\\\:\\\\theta\\\\:}_{j,\\\\:\\\\:i}=f\\\\left(x\\\\right)=\\\\left\\\\{\\\\begin{array}{c}\\\\frac{\\\\left({z}_{j}-{z}_{j-1}\\\\right)}{2}\\\\times\\\\:\\\\left({\\\\theta\\\\:}_{j,\\\\:\\\\:i-1}-{\\\\theta\\\\:}_{j,\\\\:\\\\:i}\\\\right),\\\\:\\\\:\\\\left({\\\\theta\\\\:}_{j,\\\\:\\\\:i-1}-{\\\\theta\\\\:}_{j,\\\\:\\\\:i}\\\\right)\\u0026lt;0\\\\\\\\\\\\:0,\\\\:\\\\:\\\\left({\\\\theta\\\\:}_{j,\\\\:\\\\:i-1}-{\\\\theta\\\\:}_{j,\\\\:\\\\:i}\\\\right)\\\\ge\\\\:0\\\\end{array}\\\\right.\\\\varDelta\\\\:\\\\theta\\\\:=\\\\sum\\\\:_{j=1}^{m}{\\\\varDelta\\\\:\\\\theta\\\\:}_{j,\\\\:\\\\:i}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e9\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eWhere, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{\\\\theta\\\\:}_{j,\\\\:\\\\:i-1}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the soil moisture on (i-1)\\u003csup\\u003eth\\u003c/sup\\u003e day at depth \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j-1}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{\\\\theta\\\\:}_{j,\\\\:\\\\:i}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the soil moisture on day i\\u003csup\\u003eth\\u003c/sup\\u003e at depth \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j-1}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j-1}\\\\)\\u003c/span\\u003e\\u003c/span\\u003eand \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e​ are the depth at which the soil moisture sensors are located, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{\\\\varDelta\\\\:\\\\theta\\\\:}_{j,\\\\:\\\\:i}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e represents soil water depletion from the layer between depth \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e and \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{z}_{j-1}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e on i\\u003csup\\u003eth\\u003c/sup\\u003e day, m is the number of the soil moisture sensors, determined based on crop root zoon.\\u003c/p\\u003e\\u003cp\\u003eIf ET\\u003csub\\u003er\\u003c/sub\\u003e is not measured at the weather station, it can be estimated using equations such as Penman-Monteith (PM), and if sufficient weather data is not available to calculate PM, the Hargreaves equation can be used instead. Following, data from the Eddy Covariance (EC) flux tower is used to calculate actual evapotranspiration, serving as the ground truth.\\u003c/p\\u003e\\u003cp\\u003eEC flux tower measurements are often used to validate ET\\u003csub\\u003ea\\u003c/sub\\u003e models (Drechsler et al. \\u003cspan citationid=\\\"CR11\\\" class=\\\"CitationRef\\\"\\u003e2022\\u003c/span\\u003e; Franssen et al. 2010; Kisekka et al. \\u003cspan citationid=\\\"CR19\\\" class=\\\"CitationRef\\\"\\u003e2022\\u003c/span\\u003e; Markwitz and Siebicke \\u003cspan citationid=\\\"CR20\\\" class=\\\"CitationRef\\\"\\u003e2019\\u003c/span\\u003e). Flux data does not provide a single value but instead offers an expected ET\\u003csub\\u003ea\\u003c/sub\\u003e range due to EC sensors measuring ET\\u003csub\\u003ea\\u003c/sub\\u003e both from the atmospheric fluxes (ET\\u003csub\\u003eoriginal\\u003c/sub\\u003e) and from energy fluxes (ET\\u003csub\\u003eclosed\\u003c/sub\\u003e) in the closure of the energy balance (Foken and Foken, \\u003cspan citationid=\\\"CR13\\\" class=\\\"CitationRef\\\"\\u003e2008\\u003c/span\\u003e; Franssen et al., 2010). When it was impractical to use a range, the geometric mean of the original and closed ET\\u003csub\\u003ea\\u003c/sub\\u003e measurements was calculated and considered to be the best approximation of ET\\u003csub\\u003ea\\u003c/sub\\u003e.\\u003cdiv id=\\\"Equ8\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ8\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:ETa=\\\\sqrt{{ET}_{original}\\\\bullet\\\\:{ET}_{closed}}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e10\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eSubsequently, Eureqa software was used to determine the most accurate model by mapping the dependent variable to the input variables. This software tool is designed to describe datasets in a simplified form using a technique known as symbolic regression. Developed by(Schmidt and Lipson \\u003cspan citationid=\\\"CR27\\\" class=\\\"CitationRef\\\"\\u003e2009\\u003c/span\\u003e), the tool works by generating random equations based on the data through an evolutionary search process that utilizes a genetic algorithm. While the initial guesses may not fit the data well, some equations perform better than others. These better-fitting equations are then used as the foundation for the next round of guesses. The process continues until no further improvement in fit can be achieved.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec9\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.2.2. Validation SMET model\\u003c/h2\\u003e\\u003cp\\u003eThe SMET model was initially developed using data from a site equipped with an EC tower and subsequently validated at a second site that also featured an EC tower. Figure\\u0026nbsp;\\u003cspan refid=\\\"Fig4\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003e illustrates the validation approach. In this process, ET\\u003csub\\u003ea\\u003c/sub\\u003e was estimated using the calibrated SMET model, which depends on ET\\u003csub\\u003er\\u003c/sub\\u003e and soil moisture changes. Additionally, ET\\u003csub\\u003ea\\u003c/sub\\u003e was measured using the EC flux tower and considered as observational data to assess the model's accuracy. The estimated daily ET\\u003csub\\u003ea\\u003c/sub\\u003e was compared with EC flux tower measurements (ET\\u003csub\\u003ea\\u003c/sub\\u003e) to calculate residuals and assess the performance of the SMET model.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec10\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.2.3. Statistical analysis\\u003c/h2\\u003e\\u003cp\\u003eAs mentioned, the SMET model was initially developed and calibrated using data from one site. The calibrated model was then validated at a different location. During both the calibration and validation process, the model's performance was evaluated using several statistical indices. In this study, Root Mean Square Error (RMSE), Root Relative Squared Error (RRSE), and Pearson correlation coefficient (r) were calculated using Equations 11\\u0026ndash;13, respectively, to assess model performance at each step.\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"No\\\" id=\\\"Tabc\\\" border=\\\"1\\\"\\u003e\\u003ccolgroup cols=\\\"2\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003cthead\\u003e\\u003ctr\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e\\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:RMSE=\\\\sqrt{\\\\frac{{\\\\sum\\\\:}_{i=1}^{n}{({ET}_{a(\\\\:\\\\text{i},\\\\:sim)}-{ET}_{a\\\\:(i,EC\\\\:tower)\\\\:})}^{2}}{n}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e(11)\\u003c/p\\u003e\\u003c/th\\u003e\\u003c/tr\\u003e\\u003c/thead\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e\\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:RRSE=\\\\sqrt{\\\\frac{\\\\sum\\\\:_{i=1}^{n}{\\\\left({ET}_{a(\\\\:\\\\text{i},\\\\:sim)}-{ET}_{a\\\\:(i,\\\\:EC\\\\:tower)\\\\:}\\\\right)}^{2}}{\\\\sum\\\\:_{i=1}^{n}{\\\\left({ET}_{a\\\\:(i,\\\\:EC\\\\:tower\\\\:)}-\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(EC\\\\:tower\\\\:\\\\right)}}\\\\right)}^{2}}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e(12)\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e\\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:r=\\\\frac{\\\\sum\\\\:_{i=1}^{n}(\\\\left({ET}_{a\\\\left(\\\\:\\\\text{i},\\\\:sim\\\\right)}-\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(sim\\\\:\\\\right)}}\\\\right)\\\\times\\\\:\\\\left({ET}_{a\\\\:\\\\left(i,\\\\:EC\\\\:tower\\\\:\\\\right)}-\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(EC\\\\:tower\\\\:\\\\right)}}\\\\right))}{\\\\sqrt{\\\\sum\\\\:_{i=1}^{n}{\\\\left({ET}_{a(\\\\:\\\\text{i},\\\\:sim)}-\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(sim\\\\:\\\\right)}}\\\\right)}^{2}\\\\times\\\\:\\\\sum\\\\:_{i=1}^{n}{\\\\left({ET}_{a(\\\\:\\\\text{i},\\\\:\\\\text{E}\\\\text{C}\\\\:\\\\text{t}\\\\text{o}\\\\text{w}\\\\text{e}\\\\text{r})}-\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(\\\\:EC\\\\:tower\\\\right)\\\\:}}\\\\right)}^{2}}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e(13)\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eWhere, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{ET}_{a\\\\:(i,\\\\:EC\\\\:tower)\\\\:}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the ET\\u003csub\\u003ea\\u003c/sub\\u003e calculated using EC flux tower data (mm) on i\\u003csup\\u003eth\\u003c/sup\\u003e day, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{ET}_{a(\\\\:\\\\text{i},\\\\:sim)}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the ET\\u003csub\\u003ea\\u003c/sub\\u003e simulated using the SMET model on i\\u003csup\\u003eth\\u003c/sup\\u003e day, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(sim\\\\:\\\\right)}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the average value of simulated ET\\u003csub\\u003ea\\u003c/sub\\u003e using SMET, \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:\\\\stackrel{-}{{ET}_{a\\\\:\\\\left(EC\\\\:tower\\\\:\\\\right)}}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the average value of ET\\u003csub\\u003ea\\u003c/sub\\u003e measured by EC tower, and n is the number of the days since the start of the growing season.\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec11\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e2.3. Study sites\\u003c/h2\\u003e\\u003cp\\u003eThe study was conducted at two agricultural fields in Utah, USA, with one site in Vernal, and another in Modena (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e and Table\\u0026nbsp;\\u003cspan refid=\\\"Tab1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e). Utah is classified as a semi-arid region with hot, dry summers and cold winters with most of the annual precipitation occurring during the winter months (Gillies et al. \\u003cspan citationid=\\\"CR16\\\" class=\\\"CitationRef\\\"\\u003e2009\\u003c/span\\u003e). Utah\\u0026rsquo;s mean altitude is 1835 m, mean precipitation is 359 mm, mean temperature is 9.2˚C, and mean solar radiation is 17 MJ/m\\u003csup\\u003e2\\u003c/sup\\u003e day (PRISM Climate Group at Oregon State University n.d.). The specific climate parameters for each study site are reported in Table\\u0026nbsp;\\u003cspan refid=\\\"Tab1\\\" class=\\\"InternalRef\\\"\\u003e1\\u003c/span\\u003e.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003cp\\u003eThe Vernal and Modena study sites were each equipped with an EC flux tower, a weather station, and an array of soil moisture sensors in alfalfa under sprinkler irrigation. Soil moisture sensors at Vernal were in close proximity to the EC tower, while at Modena two sets of soil moisture sensors were installed on the north and east side of the field (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e).\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab1\\\" border=\\\"1\\\"\\u003e\\u003ccaption language=\\\"En\\\"\\u003e\\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 1\\u003c/div\\u003e\\u003cdiv class=\\\"CaptionContent\\\"\\u003e\\u003cp\\u003eStudy site details including elevation, and 30- year normal precipitation, mean air temperature, and solar radiation retrieved from the PRISM climate group at Oregon State University.\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/caption\\u003e\\u003ccolgroup cols=\\\"3\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e\\u003cthead\\u003e\\u003ctr\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eStudy site\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003eVernal, UT, USA\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003eModena, UT, USA\\u003c/p\\u003e\\u003c/th\\u003e\\u003c/tr\\u003e\\u003c/thead\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eLongitude (DD)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e-109.564270\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e-113.789074\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eLatitude (DD)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e40.457876\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e37.747303\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eIrrigation System\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003eWheel-lines\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003eCenter pivot\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eArea (Ha)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e31.7\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e51.1\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eGrowing season\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e2019\\u0026ndash;2020\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e2021\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eElevation (m)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e1703\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e1596\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003ePrecipitation (mm)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e261\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e286\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eMean Air Temperature (˚C)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e8.1\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e9.5\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eMean Solar Radiation (MJ/m\\u003csup\\u003e2\\u003c/sup\\u003eday)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e17.2\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e18.5\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eSoil Moisture Sensor Locations\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e1\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e2\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eSensor Depth (cm)\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e3, 10, 25, and 215 cm\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e8, 15, 30, 61, 91, 122, and 152 cm\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cdiv id=\\\"Sec12\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.3.1. Soil moisture data collection\\u003c/h2\\u003e\\u003cp\\u003eThe volumetric water content data at each site were collected using a site-specific array of (Acclima TDR) sensors. According to the manufacturer\\u0026rsquo;s specifications, the reporting accuracy of the sensors is \\u0026plusmn;\\u0026thinsp;1% for coarse and medium textured soils and \\u0026plusmn;\\u0026thinsp;2.5% for fine textured soils (Acclima, \\u003cspan citationid=\\\"CR2\\\" class=\\\"CitationRef\\\"\\u003e2022\\u003c/span\\u003e). Details of the installation depths at each site are presented in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig6\\\" class=\\\"InternalRef\\\"\\u003e6\\u003c/span\\u003e. The data were collected every 15 minutes and the reading from midnight of each day was used for analysis.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec13\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.3.2. Model development at the Vernal site\\u003c/h2\\u003e\\u003cp\\u003eRaw soil moisture (SM) data were collected near the EC flux tower (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e) using seven TDR sensors at various depths (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig6\\\" class=\\\"InternalRef\\\"\\u003e6\\u003c/span\\u003eA). At the Vernal site, sensor depths were chosen to represent the root zone of alfalfa, which extends down to ~\\u0026thinsp;2 m.\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec14\\\" class=\\\"Section3\\\"\\u003e\\u003ch2\\u003e2.3.3. Model validation at the Modena site\\u003c/h2\\u003e\\u003cp\\u003eThe raw soil moisture data were collected at two locations in the field, East and North (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e) in Modena. The East location had an array of seven TDR soil moisture sensors (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig6\\\" class=\\\"InternalRef\\\"\\u003e6\\u003c/span\\u003eB) and the North location had an array of six TDR soil moisture sensors. On the north side of the field, the 152 cm deep sensor was not installed due to a hard gravel layer hindering root penetration. At Modena site, sensors were installed in the top-soil layers representing maximum root concentration with the greatest changes in soil moisture. This rationale led to having a close spacing near the surface to monitor soil moisture in the topsoil layers and a greater spacing at deeper depths to monitor soil moisture for the remainder rooting depth, which is expected to vary to a lesser extent.\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/div\\u003e\"},{\"header\":\"3. Results and Discussion\",\"content\":\"\\u003cdiv id=\\\"Sec16\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e3.1. Developing SMET model\\u003c/h2\\u003e\\u003cp\\u003eTo develop the SMET model, which is a function of soil moisture depth and ET\\u003csub\\u003er\\u003c/sub\\u003e, the data from the Vernal site was utilized. Soil moisture depth was calculated after screening the dataset to remove any values outside the expected range or those affected by sensor malfunctions. The data screening procedure is explained in the supplementary materials. After screening, soil moisture depth (Δθ) was calculated using Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ7\\\" class=\\\"InternalRef\\\"\\u003e9\\u003c/span\\u003e. By observing the soil moisture data for the Vernal study site (Supplementary Figure S1) irrigation events can be identified as they coincide with rapid increases followed by gradual decreases in volumetric water content. The spikes in soil moisture level caused by irrigation can be observed more easily at shallower soil moisture timeseries (3, 10, and 25 cm deep) and become more subtle with the increase in depth until they become difficult to discern in the deepest soil layer (215 cm deep).\\u003c/p\\u003e\\u003cp\\u003eAfter irrigation events were identified, the Eureqa software was then used to identify the appropriate model that maps ET\\u003csub\\u003er\\u003c/sub\\u003e and soil moisture depth to actual evapotranspiration, which was measured using an EC flux tower. Among the first ten most accurate models provided by Eureqa, the most straightforward equation was selected (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ9\\\" class=\\\"InternalRef\\\"\\u003e14\\u003c/span\\u003e).\\u003cdiv id=\\\"Equ9\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ9\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}={a\\\\times\\\\:ET}_{r}+b\\\\times\\\\:{ET}_{r}^{+}-c\\\\times\\\\:{{\\\\Delta\\\\:}\\\\theta\\\\:}^{-}$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e14\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eWhere, ET\\u003csub\\u003ea\\u003c/sub\\u003e is the daily actual evapotranspiration measured by the EC tower (mm d\\u003csup\\u003e\\u0026minus;\\u0026thinsp;1\\u003c/sup\\u003e), ET\\u003csub\\u003er\\u003c/sub\\u003e is the daily reference evapotranspiration (mm d\\u003csup\\u003e\\u0026minus;\\u0026thinsp;1\\u003c/sup\\u003e), \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{{\\\\Delta\\\\:}{\\\\theta\\\\:}}^{-}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the daily soil moisture depletion (negative changes in soil moisture) (mm d\\u003csup\\u003e\\u0026minus;\\u0026thinsp;1\\u003c/sup\\u003e), and \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:{ET}_{r}^{+}\\\\)\\u003c/span\\u003e\\u003c/span\\u003e is the daily ET\\u003csub\\u003er\\u003c/sub\\u003e on days when soil moisture is increasing. Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ9\\\" class=\\\"InternalRef\\\"\\u003e14\\u003c/span\\u003e was then split up into a conditional equation based on whether the soil moisture was increasing or decreasing:\\u003cdiv id=\\\"Equ10\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ10\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}=\\\\left\\\\{\\\\begin{array}{cc}a{ET}_{r}-c\\\\varDelta\\\\:\\\\theta\\\\:\\u0026amp;\\\\:if\\\\:\\\\varDelta\\\\:\\\\theta\\\\:\\u0026lt;0\\\\to\\\\:{{ET}_{r}}^{+}=0\\\\\\\\\\\\:a{ET}_{r}+b{ET}_{r}\\u0026amp;\\\\:if\\\\:\\\\varDelta\\\\:\\\\theta\\\\:\\\\ge\\\\:0\\\\to\\\\:{\\\\varDelta\\\\:\\\\theta\\\\:}^{-}=0\\\\end{array}\\\\right.$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e15\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eBy simplifying Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ10\\\" class=\\\"InternalRef\\\"\\u003e15\\u003c/span\\u003e, the a, b, and c parameters were combined into a single α value, giving the following conditional equation:\\u003cdiv id=\\\"Equ11\\\" class=\\\"Equation\\\"\\u003e\\u003cdiv format=\\\"TEX\\\" class=\\\"mathdisplay\\\" id=\\\"FileID_Equ11\\\" name=\\\"EquationSource\\\"\\u003e\\n$$\\\\:{ET}_{a}=\\\\left\\\\{\\\\begin{array}{cc}\\\\propto\\\\:({ET}_{r}-\\\\varDelta\\\\:\\\\theta\\\\:)\\u0026amp;\\\\:if\\\\:\\\\varDelta\\\\:\\\\theta\\\\:\\u0026lt;0\\\\to\\\\:{{ET}_{r}}^{+}=0\\\\\\\\\\\\:2\\\\propto\\\\:{ET}_{r}\\u0026amp;\\\\:if\\\\:\\\\varDelta\\\\:\\\\theta\\\\:\\\\ge\\\\:0\\\\to\\\\:{\\\\varDelta\\\\:\\\\theta\\\\:}^{-}=0\\\\end{array}\\\\right.$$\\u003c/div\\u003e\\u003cdiv class=\\\"EquationNumber\\\"\\u003e16\\u003c/div\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eWhere α is a dimensional constant and ET\\u003csub\\u003ea\\u003c/sub\\u003e, ET\\u003csub\\u003er\\u003c/sub\\u003e and Δθ are in mm/day. Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ11\\\" class=\\\"InternalRef\\\"\\u003e16\\u003c/span\\u003e is what will be referred to as the Soil Moisture based EvapoTranspiration estimation (SMET) model.\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec17\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e3.3. SMET model calibration\\u003c/h2\\u003e\\u003cp\\u003eThe optimization of the α constant was performed by applying the SMET model (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ11\\\" class=\\\"InternalRef\\\"\\u003e16\\u003c/span\\u003e) to the combined 2019 and 2020 datasets collected at the research study in Vernal, UT using an initial value of \\u003cspan class=\\\"InlineEquation\\\"\\u003e\\u003cspan class=\\\"mathinline\\\"\\u003e\\\\(\\\\:\\\\alpha\\\\:=0.30\\\\)\\u003c/span\\u003e\\u003c/span\\u003e. This initial value for α was selected as an approximate average of the a, b, and c parameters provided by the Eureqa model (Eq.\\u0026nbsp;13). The residuals and the RMSE were then calculated and used to fit Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ11\\\" class=\\\"InternalRef\\\"\\u003e16\\u003c/span\\u003e using the Solver function in Excel, yielding an optimized value for α of 0.43. The optimized model was then used to calculate the daily ET estimates and the cumulative monthly and cumulative seasonal values for the 2019 and 2020 growing seasons at Vernal. As shown in Table\\u0026nbsp;\\u003cspan refid=\\\"Tab2\\\" class=\\\"InternalRef\\\"\\u003e2\\u003c/span\\u003e, SMET estimated ET\\u003csub\\u003ea\\u003c/sub\\u003e, with an accuracy of 1.37 and 1.15 mm during 2019 and 2020, respectively.\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab2\\\" border=\\\"1\\\"\\u003e\\u003ccaption language=\\\"En\\\"\\u003e\\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 2\\u003c/div\\u003e\\u003cdiv class=\\\"CaptionContent\\\"\\u003e\\u003cp\\u003eSummary statistics for the daily calibrated SMET estimates for the 2019 and 2020 growing seasons.\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/caption\\u003e\\u003ccolgroup cols=\\\"4\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e\\u003cthead\\u003e\\u003ctr\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eYear\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003er\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003eRMSE (mm)\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003eRRSE (mm)\\u003c/p\\u003e\\u003c/th\\u003e\\u003c/tr\\u003e\\u003c/thead\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e2019\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e0.76\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e1.37\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e0.67\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e2020\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e0.81\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e1.15\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e0.63\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003cp\\u003eThe second verification of the SMET model\\u0026rsquo;s performance involved comparing its monthly and seasonal cumulative ET\\u003csub\\u003ea\\u003c/sub\\u003e values with those obtained from EC flux tower measurements for both 2019 and 2020. As shown in Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig7\\\" class=\\\"InternalRef\\\"\\u003e7\\u003c/span\\u003e, the model generally follows the seasonal ET\\u003csub\\u003ea\\u003c/sub\\u003e trend observed in the EC data, capturing the peak evapotranspiration during summer months. Although the SMET model slightly underestimates monthly ET\\u003csub\\u003ea\\u003c/sub\\u003e in several months, particularly in July, where the EC original value is highest, it still aligns well with the overall seasonal totals for both years. In 2020, the agreement is stronger, especially during peak months (June\\u0026ndash;July 2020), where SMET estimates closely match or even exceed EC values. These results confirm that while some month-to-month discrepancies exist, the model is reliable for seasonal-scale water use estimation.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003cp\\u003eBy separating the irrigation from the non-irrigation days, during which soil moisture increases or decreases, respectively (Eq.\\u0026nbsp;\\u003cspan refid=\\\"Equ11\\\" class=\\\"InternalRef\\\"\\u003e16\\u003c/span\\u003e), the SMET model can provide some insight into the irrigation treatment in the absence of such data. For the Vernal site, there were 24 and 27 irrigation days for the 2019 and 2020 years, respectively and the ET occurring during those days amounted to approximately 16% and 19% of the total, respectively (Table\\u0026nbsp;\\u003cspan refid=\\\"Tab3\\\" class=\\\"InternalRef\\\"\\u003e3\\u003c/span\\u003e).\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab3\\\" border=\\\"1\\\"\\u003e\\u003ccaption language=\\\"En\\\"\\u003e\\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 3\\u003c/div\\u003e\\u003cdiv class=\\\"CaptionContent\\\"\\u003e\\u003cp\\u003eIrrigation information inferred from the SMET model for the 2019 and 2020 growing seasons in Vernal, UT, USA\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/caption\\u003e\\u003ccolgroup cols=\\\"6\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c5\\\" colnum=\\\"5\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c6\\\" colnum=\\\"6\\\"\\u003e\\u003c/div\\u003e\\u003cthead\\u003e\\u003ctr\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eYear\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003eTotal days\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colspan=\\\"2\\\" nameend=\\\"c4\\\" namest=\\\"c3\\\"\\u003e\\u003cp\\u003eIrrigation days\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colspan=\\\"2\\\" nameend=\\\"c6\\\" namest=\\\"c5\\\"\\u003e\\u003cp\\u003eET during irrigation days (mm)\\u003c/p\\u003e\\u003c/th\\u003e\\u003c/tr\\u003e\\u003c/thead\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e2019\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e203\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e24\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e12%\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e\\u003cp\\u003e116\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e\\u003cp\\u003e16%\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003e2020\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e213\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e27\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e13%\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c5\\\"\\u003e\\u003cp\\u003e138\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e\\u003cp\\u003e19%\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\\u003cdiv id=\\\"Sec18\\\" class=\\\"Section2\\\"\\u003e\\u003ch2\\u003e3.4. SMET model verification\\u003c/h2\\u003e\\u003cp\\u003eThe optimized SMET model, with the same α value obtained from the calibration performed with the Vernal data, was applied to the Modena study site in two locations within the alfalfa field (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig5\\\" class=\\\"InternalRef\\\"\\u003e5\\u003c/span\\u003e) using the soil moisture and ET\\u003csub\\u003er\\u003c/sub\\u003e data during the 2021 growing season from April 1st to October 22nd, to calculate cumulative ET\\u003csub\\u003ea\\u003c/sub\\u003e. The field average seasonal ET\\u003csub\\u003ea\\u003c/sub\\u003e calculated by SMET was 1114 mm, which was within the range measured by the EC tower of 1032\\u0026ndash;1191 mm (Fig.\\u0026nbsp;\\u003cspan refid=\\\"Fig8\\\" class=\\\"InternalRef\\\"\\u003e8\\u003c/span\\u003e). These findings indicated that as long as the fundamental conditions (i.e., deep sensor array, crop canopy completely covers the surface) of the SMET model are not violated, it can successfully be applied to sites other than the one used for its calibration to obtain reasonable estimates of ET on a seasonal scale.\\u003c/p\\u003e\\u003cp\\u003e\\u003c/p\\u003e\\u003cp\\u003eSimilar to the approach used at Vernal study site, the SMET model provided some insight into irrigations applied to the field in Modena. The estimated number of irrigation days for the North and East locations in the field was 61 (30%) and 57 (28%), respectively and the amount of ET\\u003csub\\u003ea\\u003c/sub\\u003e that occurred during irrigation days was 27% and 29%, respectively (Table\\u0026nbsp;\\u003cspan refid=\\\"Tab4\\\" class=\\\"InternalRef\\\"\\u003e4\\u003c/span\\u003e). While these results illustrate the model\\u0026rsquo;s potential for evaluating field-scale irrigation practices, it is important to note that the model\\u0026rsquo;s accuracy can be influenced by factors such as irrigation method, monitoring depth, and data continuity. As outlined in the supplementary section, specific considerations must be evaluted, such as ensuring full root zone sensor coverage, applying the model on appropriate temporal scales, and accounting for variations in irrigation system type (e.g., drip vs. flood). These considerations are critical for improving the reliability and broader applicability of the SMET model across diverse agricultural settings.\\u003c/p\\u003e\\u003cp\\u003e\\u003cdiv class=\\\"gridtable\\\"\\u003e\\u003ctable float=\\\"Yes\\\" id=\\\"Tab4\\\" border=\\\"1\\\"\\u003e\\u003ccaption language=\\\"En\\\"\\u003e\\u003cdiv class=\\\"CaptionNumber\\\"\\u003eTable 4\\u003c/div\\u003e\\u003cdiv class=\\\"CaptionContent\\\"\\u003e\\u003cp\\u003eSMET model irrigation days estimates for the Modena, UT, USA study site during the 2021 growing season.\\u003c/p\\u003e\\u003c/div\\u003e\\u003c/caption\\u003e\\u003ccolgroup cols=\\\"6\\\"\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c1\\\" colnum=\\\"1\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c2\\\" colnum=\\\"2\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"char\\\" char=\\\".\\\" class=\\\"colspec\\\" colname=\\\"c3\\\" colnum=\\\"3\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c4\\\" colnum=\\\"4\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c5\\\" colnum=\\\"5\\\"\\u003e\\u003c/div\\u003e\\u003cdiv align=\\\"left\\\" class=\\\"colspec\\\" colname=\\\"c6\\\" colnum=\\\"6\\\"\\u003e\\u003c/div\\u003e\\u003cthead\\u003e\\u003ctr\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eSite\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003eTotal days\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colspan=\\\"2\\\" nameend=\\\"c4\\\" namest=\\\"c3\\\"\\u003e\\u003cp\\u003eIrrigation days\\u003c/p\\u003e\\u003c/th\\u003e\\u003cth align=\\\"left\\\" colspan=\\\"2\\\" nameend=\\\"c6\\\" namest=\\\"c5\\\"\\u003e\\u003cp\\u003eET\\u003csub\\u003ea\\u003c/sub\\u003e during irrigation days\\u003c/p\\u003e\\u003c/th\\u003e\\u003c/tr\\u003e\\u003c/thead\\u003e\\u003ctbody\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eNorth\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e204\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e61\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e30%\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e\\u003cp\\u003e362 mm\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e\\u003cp\\u003e27%\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003ctr\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c1\\\"\\u003e\\u003cp\\u003eEast\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c2\\\"\\u003e\\u003cp\\u003e204\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"char\\\" char=\\\".\\\" colname=\\\"c3\\\"\\u003e\\u003cp\\u003e57\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c4\\\"\\u003e\\u003cp\\u003e28%\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c5\\\"\\u003e\\u003cp\\u003e344 mm\\u003c/p\\u003e\\u003c/td\\u003e\\u003ctd align=\\\"left\\\" colname=\\\"c6\\\"\\u003e\\u003cp\\u003e29%\\u003c/p\\u003e\\u003c/td\\u003e\\u003c/tr\\u003e\\u003c/tbody\\u003e\\u003c/colgroup\\u003e\\u003c/table\\u003e\\u003c/div\\u003e\\u003c/p\\u003e\\u003c/div\\u003e\"},{\"header\":\"4. Conclusions\",\"content\":\"\\u003cp\\u003eThe proposed SMET model in this study is capable of reliably estimating seasonal ET\\u003csub\\u003ea\\u003c/sub\\u003e with accuracy using daily soil moisture data from sensors and weather data. The model provides a practical tool for estimating water demand under different fields with limited available data. The results were consistent with those measured by the EC tower across the two alfalfa sites, though further calibration is needed for α under different crops and climatic conditions. The simplicity and ease of the implementation of the SMET model make it a valuable tool for land managers, offering an efficient and cost-effective way to estimate seasonal water demands with minimal investment. It could also be highly beneficial for research trials conducted at the plot or strip plot scale, a common practice in the crop and soil science disciplines. Further, many research plots already employ soil moisture sensors to monitor crop water use and needs and SMET could provide a useful tool for estimating ET\\u003csub\\u003ea\\u003c/sub\\u003e at smaller scales than existing ET\\u003csub\\u003ea\\u003c/sub\\u003e instruments or models.\\u003c/p\\u003e\\u003cp\\u003eThe SMET model can be used in areas with a heterogeneous landscape, allowing easy installation of soil moisture sensors, which is impractical with ET\\u003csub\\u003ea\\u003c/sub\\u003e flux tower and lysimeter. Further, it could be an efficient tool to estimate ET\\u003csub\\u003ea\\u003c/sub\\u003e under small agricultural research setting experiments with different treatments, where EC tower might not be able to detect the subtle differences among treatments and installing micro lysimeters would be laborious. Finally, the model can also offer valuable insights when used alongside an EC tower under a heterogeneous environment by providing detailed resolution of water consumption. This approach could be particularly valuable if ET\\u003csub\\u003ea\\u003c/sub\\u003e data is remotely sensed using drones or satellites (e.g. OpenET), enabling estimates of root zone soil moisture.\\u003c/p\\u003e\"},{\"header\":\"Declarations\",\"content\":\"\\u003ch2\\u003eData Availability Statement\\u003c/h2\\u003e\\u003cp\\u003eSome or all data, models, or code supporting the findings of this study are available from the corresponding author upon reasonable request.\\u003c/p\\u003e\"},{\"header\":\"References\",\"content\":\"\\u003col\\u003e\\u003cli\\u003e\\u003cspan\\u003eAbdelmajeed AYA, Juszczak R (2024) Challenges and limitations of remote sensing applications in northern peatlands: present and future prospects. 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Prentice-Hall, Englewood Cliffs, NJ\\u003c/span\\u003e\\u003c/li\\u003e\\u003cli\\u003e\\u003cspan\\u003eWan Z, Zhang K, Xue X, Hong Z, Hong Y, Gourley JJ (2015) Water balance-based actual evapotranspiration reconstruction from ground and satellite observations over the conterminous United States. Water Resour Res 51(8):6485\\u0026ndash;6499\\u003c/span\\u003e\\u003c/li\\u003e\\u003cli\\u003e\\u003cspan\\u003eZotarelli L, Dukes MD, Scholberg JMS, Femminella K, Mu\\u0026ntilde;oz-Carpena R (2011) Irrigation scheduling for green bell peppers using capacitance soil moisture sensors. J Irrig Drain Eng 137(2):73\\u0026ndash;81. \\u003cspan class=\\\"ExternalRef\\\"\\u003e\\u003cspan class=\\\"RefSource\\\"\\u003e10.1061/(ASCE)IR.1943-4774.0000281\\u003c/span\\u003e\\u003cspan address=\\\"10.1061/(ASCE)IR.1943-4774.0000281\\\" targettype=\\\"DOI\\\" class=\\\"RefTarget\\\"\\u003e\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/span\\u003e\\u003c/li\\u003e\\u003c/ol\\u003e\"}],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":true,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":true,\"hideJournal\":true,\"highlight\":\"\",\"institution\":\"Utah State University\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":false,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":false,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true},\"keywords\":\"Actual Evapotranspiration, Eddy Covariance flux tower, Soil Moisture Sensor, Agricultural Water Management\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-7124239/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-7124239/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eEvapotranspiration (ET) is an essential component of water balances that can be estimated using different tools, but heterogeneous landscapes and experimental designs limit their applicability. This study aims to develop a cost-effective, simplified method for estimating seasonal crop ET using soil moisture sensors along the root zone profile and reference ET data from local weather stations. The study was conducted at two sites in Utah, USA to develop and validate a new empirical soil-moisture-based evapotranspiration (SMET) model that estimates seasonal ET based on the relationship between reference ET (ET\\u003csub\\u003er\\u003c/sub\\u003e), changes in soil moisture, and actual evapotranspiration (ET\\u003csub\\u003ea\\u003c/sub\\u003e). The initial model was constructed using detailed flux measurements from Modena, Utah, where an Eddy Covariance (EC) flux tower was installed. The model was validated with EC tower measurements at one of the research sites in Vernal, Utah showing a Root Mean Square Error (RMSE) and Root Relative Squared Error (RRSE) of 1.26 mm day\\u003csup\\u003e\\u0026minus;\\u0026thinsp;1\\u003c/sup\\u003e and 0.65, respectively, which demonstrated high accuracy. This approach offers a practical solution for farmers, water managers, and policymakers to estimate crop water consumption, supporting better water resource management in arid and semi-arid regions.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Soil Moisture Evapotranspiration (SMET): A Low-Cost Method for Determining Seasonal Crop Water Demand\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2025-07-16 19:14:57\",\"doi\":\"10.21203/rs.3.rs-7124239/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true}}],\"origin\":\"\",\"ownerIdentity\":\"e76cb06a-83ee-46b1-afe6-7ebeb3ed3f37\",\"owner\":[],\"postedDate\":\"July 16th, 2025\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2025-07-16T19:14:57+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2025-07-16 19:14:57\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-7124239\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-7124239\",\"identity\":\"rs-7124239\",\"version\":[\"v1\"]},\"buildId\":\"8U1c8b4HqxoKbykW_rLl7\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}