every 5 yr (USGS, 2018). The lack of irrigation water use data sets at high spatio-temporal resolution hampers efficient regional water management, designing effective water policies, and understanding terrestrial water cycle in irrigated croplands (Koch et al., 2020;Kumar et al., 2015).

Estimations of irrigation water use has received significant attention in recent years (Foster et al., 2014;Haddeland et al., 2006;Jalilvand et al., 2019;Massari et al., 2021;Sun et al., 2017;Vörösmarty & Sahagian, 2000;Wisser et al., 2008). The existing literature on estimating irrigation water use is mainly classified into two categories. The first type is to incorporate some simplified soil/plant-based irrigation rules with predefined thresholds in the process-based models (de Rosnay et al., 2003;Haddeland et al., 2006;Lawston et al., 2015;Nie et al., 2022Nie et al., , 2018;;Ozdogan et al., 2010), such as the most widely used maximum allowable depletion (MAD) irrigation scheme with 50% based on soil moisture (Ozdogan et al., 2010). However, irrigation decisions are complicated by many factors, such as farmers' preferences and irrigation infrastructures, thus these frameworks for estimating irrigation water use accompany large uncertainties from farmers' irrigation behavior and process-based models (Foster et al., 2014;Lamb et al., 2021;D. Wang & Cai, 2009).

Second, estimating irrigation water use relies on remote sensing-based soil moisture and/or evapotranspiration (ET) data sets, as irrigation increases soil moisture and ET significantly in the soil-plant-atmosphere-continuum. From the soil moisture perspective, some studies applied data assimilation methods, such as particle batch smoother, for estimating irrigation water use through integrating remotely sensed soil moisture retrievals (such as SMAP) into the process-based models (Abolafia-Rosenzweig et al., 2019;Filippucci et al., 2020;Kumar et al., 2015;Zaussinger et al., 2019). Some studies also used some simple inverted soil water balance models, such as SM2RAIN proposed by Brocca et al. (2013) for rainfall estimation at first and then adapted for quantifying irrigation (Brocca et al., 2018;Jalilvand et al., 2019). However, estimations of irrigation water use based on remotely sensed soil moisture retrievals is strongly connected with the quality of soil moisture data sets as well as some empirical parameters required in the inverted soil water balance models and/or process-based models, such as soil depth and water holding capacity (Jalilvand et al., 2021;Zappa et al., 2022;K. Zhang et al., 2022). Specifically, current satellite-based large-scale soil moisture data sets, which only provide soil moisture information in the shallow soil depth (<0.05 m), are limited by either coarse spatial resolutions (e.g., >10 km in SMOS L3 and SMAP L3 products; Al Bitar et al., 2017;Chan et al., 2016), or coarse temporal resolutions (6-12 days by using Sentinel-1 SAR information in soil moisture retrieval; Das et al., 2019). Some proposed approaches for soil moisture observations with higher spatial and temporal resolution, such as optical approaches based on SWIR and NDVI (Babaeian et al., 2021(Babaeian et al., , 2018;;Sadeghi et al., 2017) and estimating soil moisture via assimilating the remote sensing retrievals (thermal infrared, TIR, and synthetic aperture radar, SAR; Lei et al., 2020), could help for estimating irrigation water use at high spatio-temporal resolution.

Estimations of irrigation water use based on remotely sensed ET observations has emerged in recent years. Most studies obtained net ET difference from dual modeling of ET as irrigation water use. Specifically, dual modeling usually used remote sensing-based models (e.g., the two-source energy balance, TSEB, the Priestly-Taylor JPL, PT-JPL, and the Operational Simplified Surface Energy Balance, SSEB op ) based on surface energy balance with irrigation impact and land surface models (e.g., Noah-MP) and/or hydrological models (e.g., the multiscale Hydrologic Model, mHM, and World-Wide Water, W3) without irrigation impact (Hain et al., 2015;Koch et al., 2020;Romaguera et al., 2012Romaguera et al., , 2014;;D. Wang & Cai, 2007; C. Zhang & Long, 2021). However, the irrigation water use data sets from these studies usually had coarse temporal (monthly, seasonal, and annual) and/ or spatial (1-5 km) resolutions. Some studies estimated irrigation water use through long-term net water balance based on land surface/hydrological models with assimilation of satellite-based information, such as ET, soil moisture, leaf area index (LAI), and land surface temperature (LST; Van Dijk et al., 2018). However, these studies ignored the water unavailable for crop use, such as runoff, drainage, and deep percolation, and also had coarse temporal and/or spatial resolutions. Some researchers estimated irrigation water use using machine learning but required the detailed irrigation water use data sets, such as the in situ pumping records (Wei et al., 2022).

Model-data fusion approaches integrate models and observations to improve the accuracy of model predictions (Gettelman et al., 2022; Y.-P. Wang et al., 2009), which have been applied in many areas, including climate projections using the earth systems (Gettelman et al., 2022), terrestrial carbon cycles tracking (Keenan et al., 2012; Y.-P. Wang et al., 2009), and hydrological cycles simulations (Abrahart & See, 2002;Stampoulis et al., 2019). There are multiple model-data fusion methods, such as data assimilation, Bayesian inference, and model calibration (Gettelman et al., 2022;Guan et al., 2022;Liu & Gupta, 2007; Y.-P. Wang et al., 2009;J. Zhang et al., 2021a).

The application of model-data fusion for estimating irrigation water use can be characterized as an inverse application, analyzing model forcing (i.e., irrigation water use) from observations (e.g., soil moisture and/or ET) with irrigation impacts. This study aims to use the model-data fusion approaches (such as data assimilation) to estimate center-pivot irrigation water use at high spatio-temporal resolution (daily and field scale) based on the process-based model and the remotely sensed ET observations. There are two possible pathways: The first one is to continuously assimilate the remotely sensed ET observations into the process-based models to estimate irrigation timing and amount concurrently. For the second pathway, we assume that the difference between the remotely sensed ET observations with irrigation impacts and ET simulations from ecosys without irrigation impacts may reflect the irrigation signals. Thus, the ET difference can be used to check whether irrigation occurs or not (i.e., irrigation timing), and then data assimilation can be applied to determine irrigation amount.

In this study, we proposed a new model-data fusion framework with two configurations, that is, concurrent (CON) and sequential (SEQ), embedded in the particle filtering for estimating center-pivot irrigation water use at high spatio-temporal resolution (daily and field-scale

The model-data fusion approach integrating a well-calibrated agroecosystem model and remotely sensed ET observations is applied for estimating irrigation water use at high spatio-temporal resolution (daily and field scale, i.e., determined by the field boundary from Common Land Unit, CLU) in this study (Figure 1a). The agroecosystem model is forced with climate and field data, then data assimilation (particle filtering) with two configurations (CONcurrent, CON, and SEQuential, SEQ) is used to update the water fluxes (ET and irrigation) from the agroecosystem model based on the remotely sensed ET observations. Specifically, remotely sensed ET observations with actual irrigation impact are assimilated into the agroecosystem model in real-time at the daily scale through particle filtering to determine when and how much irrigation has already been applied in the past. Irrigation signals are detected from the difference between remotely sensed ET observations with irrigation impact and ET simulations from the agroecosystem model without irrigation impacts.

Two configurations embedded in the data assimilation (particle filtering) for estimating irrigation water use are proposed, including the CONcurrent (CON) and SEQuential (SEQ) (Figures 1b and1c). CON uses data assimilation for each time period to investigate whether irrigation occurs or not at the daily scale and to estimate corresponding daily irrigation water use amount concurrently (Figure 1b). SEQ first determines whether irrigation occurred or not based on ET difference at the daily scale, and then applies data assimilation at the days with the occurring irrigation to estimate daily irrigation water use amount sequentially (Figure 1c). Specifically, particle filtering is selected as the sequential data assimilation approach. The daily irrigation events are set as the particles in the particle filtering, and they are randomly generated from the random distribution with given irrigation ranges [0, β 𝐴𝐴 × I max ] mm (β, needs to be calibrated) at the daily scale (Equation 1). If the weight of the irrigation particle with 0 mm amount is maximum among all the particles (irrigation events with different amounts) on the target day, CON claims that there is no irrigation on that day. Otherwise, irrigation occurs on the target day, and the daily irrigation amount is determined as the weighted average of all the irrigation particles. Different from CON, SEQ first determines whether irrigation occurs or not based on the daily relative difference of ET (the ratio of ET difference to ET simulation) on the target day. Specifically, there is no irrigation event on the target day for the case that the daily relative difference of ET between remotely sensed observation with irrigation impact and model simulation without irrigation impact is smaller than the set threshold (α, needs to be calibrated; Equation 2). However, if the daily relative difference of ET reaches the threshold (α), irrigation is assumed as the reason for the large difference of ET. Thus, irrigation occurs on the target day and daily irrigation amount is the weighted average of all the irrigation particles. In addition, estimating irrigation water use is also constrained by the pumping wells and field area (Equation 3).

Figure 1. Framework of (a) the proposed model-data fusion approach with two configurations (CON and SEQ) for estimating irrigation water use at high spatio-temporal resolution. (b) Concurrent (CON) determines whether irrigation occurs or not and irrigation water use amount concurrently. (c) Sequential (SEQ) first determines whether irrigation occurred or not and then estimates irrigation water use amount sequentially. Three parameters need to be calibrated for CON and SEQ, including threshold for relative difference of ET (α), irrigation duration (dT), and irrigation range (β).

In addition, once irrigation starts, it usually continues for several days (i.e., irrigation duration, dT, needs to be calibrated). The reason is that center-pivot irrigation systems usually take several days to complete an irrigation cycle due to the large irrigated field, and irrigation will be delayed if a rainfall event exceeds certain amounts during the irrigation cycle, such as the 6.5 mm day -1 for 1-day delay, 13 mm day -1 for 2-day delay, and 45.5 mm day -1 for 7-day delay set in J. Gibson et al. (2017). Thus, irrigation duration (dT) is complicated by multiple factors, such as irrigation systems and climate status.

where 𝐴𝐴 𝐴𝐴 𝑛𝑛 𝑡𝑡 is the irrigation particle n at time period t (mm d -1 ); β is the parameter needed to be calibrated to determine the irrigation ranges to generate all the irrigation particles; N is the particle size; ET t,obs is the remotely sensed ET observation with irrigation impact at time period t (mm d -1 ); 𝐴𝐴 ET ′ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is the model simulation of ET without irrigation impact before data assimilation at time period t (mm d -1 ); α is the set threshold of the relative difference of ET; and I max is the maximum allowed irrigation amount (mm d -1 ), which is usually determined by the capacity of pumping wells of surface water and groundwater for irrigation (gallon per minute, GPM) and the field area (S field , acre) (Equation 3, 1 gallon = 1/27,154 acre-inch, 1 inch = 25.4 mm).

Data assimilation, one typical method of model-data fusion, can effectively correct state estimations to reduce uncertainties from process-based models and observations (Moradkhani, 2008;Weerts & El Serafy, 2006;J. Zhang, Cai, et al., 2021). Particle filtering, one of the sequential data assimilation schemes based on Monte Carlo algorithms, is used in this study. The key idea is to determine the final irrigation amount through the weighted average of all the irrigation particles (Weerts & El Serafy, 2006). Specifically, all the daily irrigation particles are generated from random distribution (Equation 1), then are incorporated into the advanced agroecosystem model to obtain ET simulations with the impacts of different irrigation water use amounts. Note that we combine the uncertainties from the process-based model and observations together, rather than quantifying the ET uncertainties from the process-based model and observations separately. The reason is that the true-values of ET are difficult to obtain without eddy covariance techniques, thus it is difficult to quantify the ET uncertainties from remotely sensed observations. The combined ET uncertainties are represented by the Gaussian distribution (bias, μ s-o , and standard deviation, σ s-o ) of ET difference between ET simulations from the calibrated process-based model and remotely sensed ET observations both with irrigation impacts. Then, the probability of all the irrigation particles is determined by the daily ET difference between model simulations with different irrigation particles and remotely sensed observations based on pre-determined Gaussian distribution of ET difference (Equation 4). The ET uncertainties can be reduced through the probability calculation with the pre-determined Gaussian distributions with the parameters (bias, μ s-o , and standard deviation, σ s-o ). The associated weights (w t ) for all the irrigation particles are determined as the normalized probabilities (Equation 5).

As irrigation impact is incorporated into the advanced agroecosystem model through the assimilation of remotely sensed ET observations in real-time at the daily scale, the weights of all the particles are determined to be diverse without weight degeneracy. Thus, the resampling scheme is not adopted in this study to simplify its process. Lastly, the final irrigation amount is estimated as the minimum of maximum allowed irrigation amount (I max ) and estimations of irrigation water use, which is determined based on the weighted average of all the particles with their associated weights considering the irrigation application efficiency (λ in Equation 6). Note that the irrigation application efficiency of the center pivots is set as 85% to account for the irrigation water loss unavailable for crop use (U.S. GAO, 2019). In general, three parameters (threshold for relative difference of ET, α, irrigation duration, dT, and irrigation range, β) are needed to be calibrated for CON and SEQ.

where 𝐴𝐴 ΔET 𝑛𝑛 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is the difference of ET between ET simulations ( 𝐴𝐴 ET 𝑛𝑛 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 , mm d -1 ) from the process-based model with the impacts of irrigation particle n and remotely sensed ET observation (ET t,obs , mm d -1 ) with actual irrigation impacts at time period t (mm d -1 ); the ET difference is assumed to follow the Gaussian distribution with the pre-determined parameters of mean (μ s-o ) and standard deviation (σ s-o );

𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴

is the probability of ET difference of irrigation particle n at time period t; 𝐴𝐴 𝐴𝐴 𝑛𝑛 𝑡𝑡 is the associated weight of the irrigation particle n at time period t; λ is the irrigation application efficiency of the center pivots (set as 85% in this study); 𝐴𝐴 𝐴𝐴 * 𝑡𝑡 is the estimated irrigation amount at time period t (mm d -1 ).

The ecosys model is used in the model-data fusion as the process-based model. Ecosys is an advanced agroecosystem model based on biophysical and biochemical mechanisms, and it uses the multi-layered soil-root-canopy system to track the water, energy, carbon, and nutrient cycles at the hourly scale (Grant, 1995(Grant, , 1997;;Grant et al., 1993). More details about the ecosys model can be found in Appendix A. We will implement the sensitivity analysis and model calibration for the ecosys model before we use it to estimate irrigation water use. Sensitive parameters related to phenology, canopy carbon assimilation, stomatal conductance, and root water uptake of maize and soybean are screened out using the Sobol method. Then, sensitive parameters of maize and soybean are calibrated separately by minimizing the normalized root mean square error (NRMSE) of daily ET and LAI using eddy covariance measurements in eastern Nebraska and satellite-based data sets in western Nebraska (Equation 7). The NRMSE metric is used to avoid the impacts of different magnitudes of ET and LAI. Irrigation impact is considered through incorporating farmer irrigation records and/or the auto-irrigation scheme based on soil moisture. If farmer irrigation records are available at the field level, they can be provided as the forcing in model calibration. Otherwise, the auto-irrigation scheme in the ecosys model with the widely used soil-based MAD-50% with a depth of 0.92 m in the top nine soil layers is applied to incorporate irrigation impact in model calibration (i.e., irrigation is triggered to fill current soil moisture to field capacity when MAD increases above the MAD threshold of 0.5; Malejane et al., 2018). Finally, the planting date within the ranges from USDA NASS weekly Crop Progress Reports in western Nebraska is calibrated individually for each site-year to minimize NRMSE of daily ET and LAI.

where ET t,sim and ET t,obs represent ET of model simulation and observation at time period t (mm d -1 ); LAI t,sim and LAI t,obs represent LAI of model simulation and observation at time period t (m 2 m -2 ); σ(ET obs ) and σ(LAI obs ) are the standard deviations of observed ET and LAI; T and S are the total number of days and irrigated fields, respectively.

Two types of experiments are designed in this study, including synthetic and real experiments using synthetic and real ET observations, respectively. The first type of experiments (i.e., synthetic experiments) use the synthetic ET observations to quantify the impacts of different magnitude of ET uncertainties on the estimation performance of irrigation water use. The performance of estimating irrigation water use through the proposed model-data fusion framework is degraded by the uncertainties of ET difference between observations and simulations from process-based models, which are systematic bias and variability (i.e., standard deviation, treated as noise). The synthetic ET observations are generated by adding random synthetic perturbations (ET pert ) to ET simulations from the process-based model with synthetic irrigation records. The perturbations can be attributed to the uncertainties of ET difference between observations and model simulations both with irrigation impact. Thus, three groups of scenarios (including B, V, and BV) are applied to investigate the impacts on degrading the estimation performance of irrigation water use resulting from bias, variability, and the combinations of bias and variability, separately. The first group of six scenarios (B 1 , B 2 , B 3 , B 4 , B 5 , and B 6 ) are designed with ET perturbations resulting from different magnitudes of bias (1%, 10%, 20%, 30%, 40%, and 50% of ET observations) but no noise.

The second group of six scenarios The second type of experiments (i.e., real experiments) are the real cases for estimating irrigation water use using real ET observations. Ten replicates of real experiments are applied for each site-year to investigate the robustness of the proposed model-data fusion framework for estimating irrigation water use. Irrigation season for each siteyear is determined as the period from the first irrigation day to the last irrigation day based on farmer irrigation records in this study.

Three statistical indexes between estimations of irrigation water use and farmer irrigation records are calculated for each site-year at different temporal scales (daily, weekly, and monthly) for two types of experiments using synthetic and real ET observations, including Pearson correlation coefficient (r, Equation 8), root mean square error (RMSE, Equation 9), and BIAS (Equation 10).

where 𝐴𝐴 𝐴𝐴 * 𝑡𝑡 is the final estimated irrigation amount at time period t (mm d -1 ); I t is the irrigation amount based on farmer irrigation records at time period t (mm d -1 ); 𝐴𝐴 𝐼𝐼 * and 𝐴𝐴 𝐼𝐼 are the averages of estimations of irrigation water use and farmer irrigation records, respectively; T is the total time periods with the unit of day, week, and month.

Two sets of irrigated fields in eastern and western Nebraska, one of the largest irrigated states in the U.S., were selected for this study (Figure 2a). The first set of irrigated fields in eastern Nebraska were two AmeriFlux sites (US-Ne1 and US-Ne2, https://ameriflux.lbl.gov/) with center pivot irrigation systems, which were located at the University of Nebraska-Lincoln's (UNL) Eastern Nebraska Research and Education Center (ENREC) with annual precipitation around 700-800 mm (Figure 2b). US-Ne1 had continuous maize cropping systems, while US-Ne2 had maize-soybean rotation cropping systems. The second set of irrigated fields were 29 fields located in Keith and Deuel counties in western Nebraska, planted with maize or soybean with annual precipitation around 400-500 mm (Figure 2c and Table S1 in Supporting Information S1).

BESS-STAIR ET and LAI data sets at high spatial (30 m) and temporal (daily) resolutions (C. Jiang et al., 2020) during the growing seasons (May-October) in 2015-2016 were used as the satellite-based ET and LAI observations for the second set of irrigated fields in western Nebraska (Table 1). The grid-scale (30 m) ET and LAI data sets were further processed into the field-scale ET and LAI data sets based on the averages of all grids within the field boundary from CLU. Breathing Earth System Simulator (BESS) is a satellite-driven biophysical model, coupling atmosphere and canopy radiative transfer, canopy photosynthesis, and evapotranspiration processes for water, energy, and carbon cycles (C. Jiang & Ryu, 2016). The satellite fusion algorithm, SaTellite dAta IntegRation (STAIR), integrated Landsat data sets with high spatial resolution and MODIS data sets with high temporal resolution to generate daily 30 m resolution surface spectral reflectance under all-sky conditions (Luo et al., 2018(Luo et al., , 2020)). The daily 30 m surface spectral reflectance data was used to drive the BESS model to generate BESS-STAIR ET and LAI data sets at high spatio-temporal resolution (30 m and daily; C. Jiang et al., 2020). Its satisfactory performance has been demonstrated by benchmarking with 12 eddy covariance sites across the U.S. Corn Belt and by constraining process-based models (such as Noah-MP) on croplands (C. Jiang et al., 2020;Yang et al., 2020).

Daily farmer irrigation records at the first set of irrigated fields (two eddy-covariance sites: US-Ne1 and US-Ne2) during the period 2001-2012 were obtained from the Carbon Sequestration Program (CSP) at the UNL ENREC near Mead, Nebraska (http://csp.unl.edu/Public/sites.htm; Table 1). For the second set of irrigated fields in western Nebraska, daily farmer irrigation records during the period from 2015 to 2016 were processed based on the hourly raw data downloaded from AgSense and FieldNET irrigation software provided by the Nebraska Chapter of The Nature Conservancy through the Western Nebraska Irrigation Project (Table 1). The spatially distributed maps of annual and field-scale irrigation water use at the second set of irrigated fields in western Nebraska during the period from 2015 to 2016 were shown in Figure S1 in Supporting Information S1.

Synthetic ET observations for synthetic experiments (Section 2.3) were generated by adding random synthetic perturbations (ET pert ) to ET simulations from the ecosys model with synthetic irrigation records (Table 1). ET perturbations (ET pert ) resulting from different magnitudes of bias (1%-50% of ET observations) were generated for the first group of six scenarios (B 1 , B 2 , B 3 , B 4 , B 5 , and B 6 in Equation 11). ET perturbations (ET pert ) resulting from different magnitudes of variability (i.e., standard deviation, treated as noise, 1%-50% of ET observations) were generated for the second group of six scenarios (V 1 , V 2 , V 3 , V 4 , V 5 , and V 6 in Equation 12). ET perturbations (ET pert ) resulting from different combinations of bias and noise were generated for the third group of six scenarios (BV 1 , BV 2 , BV 3 , BV 4 , BV 5 , and BV 6 in Equation 13), including scenarios for both small bias and noise, small bias but large noise, large bias but small noise, and both large bias and noise. ET simulations from ecosys have already incorporated irrigation impact through farmer irrigation records, which were treated as synthetic irrigation records. Farmer irrigation records for the irrigated fields in eastern Nebraska were obtained from UNL-ENREC's CSP (Table 1).

where ET t,pert denoted ET perturbation at time period t (mm d -1 ), which represented the difference between ET observations and simulations from process-based models both with irrigation impact; B k represented the kth scenario of the first group of synthetic experiments to investigate the impacts of bias on degrading the estimation performance of irrigation water use; V k represented the kth scenario of the second group of synthetic experiments to investigate the impacts of variability (noise) on degrading the estimation performance of irrigation water use; BV k represented the kth scenario of the third group of synthetic experiments to investigate the impacts of the combination of bias and variability (noise) on degrading the estimation performance of irrigation water use; γ k represented the ratio of ET observations for bias for the kth scenario; and ε k represented the ratio of ET observations for variability (noise) for the kth scenario.

The first set of irrigated fields (US-Ne1 and US-Ne2) in Mead, eastern Nebraska had the complete data sets from AmeriFlux during the period 2001-2012, including water (ET), energy (sensible and latent heat), and carbon (gross primary production-GPP and net ecosystem exchange-NEE) fluxes and hourly gap-filled meteorological data (i.e., humidity, downward shortwave radiation, precipitation, air temperature, and wind speed; Table 1). The golden standard data sets of field management and crop growth data, including planting/harvest dates, planting density, fertilization records, and LAI, were obtained from UNL-ENREC's CSP (http://csp.unl.edu/Public/sites. htm).

For the second set of irrigated fields in western Nebraska, the hourly meteorological data (i.e., humidity, downward shortwave radiation, precipitation, air temperature, and wind speed) was obtained from North American Land Data Assimilation System (NLDAS-2). BESS-STAIR ET and LAI data sets were processed into the daily and field-scale ET and LAI observations, while detailed field management and other crop growth data were unavailable. Crop type and field boundary were obtained from Cropland Data Layer and CLU, respectively. The initial planting date of the second set of irrigated fields was obtained from USDA NASS weekly Crop Progress Reports, and crops were harvested on October 31. For fertilizer, 18g N m -2 and 5g P m -2 per year were applied 2 days before planting for corn while no fertilizer was applied for soybean. Other land management practices were set as the same across all the irrigated fields in western Nebraska, including tillage practice (no-tillage) and planting density (8.4 plants m -2 for corn and 37.1 plants m -2 for soybean). In addition, soil properties (field capacity, wilting point, saturated hydraulic conductivity, etc) at two sets of irrigated fields in eastern and western Nebraska with a maximum root-zone depth of 2.0 m were obtained from the gSSURGO data set (Table 1).

The sensitive parameters of the ecosys model for maize and soybean were screened out using the Sobol method, respectively (Figures S2 and S3 in Supporting Information S1). The identified sensitive parameters of the ecosys model for maize and soybean were then calibrated separately in eastern and western Nebraska. For the irrigated AmeriFlux fields (US-Ne1 and US-Ne2) in eastern Nebraska with detailed ground-based management data, the daily and monthly ecosys-simulated ET and LAI of maize and soybean matched very well with the eddy-covariance and ground-based observations (Figures 3a-3d and Figures S4a-S4d in Supporting Information S1). Both R 2 of daily ET of maize and soybean were 0.87, and R 2 of monthly ET of maize and soybean were 0.94 and 0.96, respectively. For LAI, R 2 of maize and soybean at the daily scale were 0.81 and 0.75, respectively; and R 2 of maize and soybean at the monthly scale were 0.83 and 0.91, respectively. Ecosys could also capture the magnitude and seasonal patterns of ET and LAI with high accuracy at US-Ne1 and US-Ne2 (Figure S5 in Supporting Information S1). However, as there were no available ground-based management observations at the 29 irrigated fields in western Nebraska, the planting date for each site-year was calibrated using daily and field-scale satellite-based ET and LAI data sets (Table S1 in Supporting Information S1). The scatter of daily and monthly ecosys-simulated ET and LAI with the satellite-based observations distributed evenly on both sides of the 1-to-1 line at the field level (Figures 3e-3h and Figures S4e-S4h in Supporting Information S1). R 2 of daily ET of maize and soybean were 0.49 and 0.47, respectively, and R 2 of monthly ET of maize and soybean were 0.91 and 0.85, respectively. For LAI, R 2 of maize and soybean at the daily scale were 0.93 and 0.80, respectively; and R 2 of maize and soybean at the monthly scale were 0.94 and 0.83, respectively.

The first type of synthetic experiments (including three groups of scenarios B, V, and BV, more details in Section 2.3) using synthetic ET observations (details in Section 3.2.3) for estimating irrigation water use were conducted at two AmeriFlux fields (US-Ne1 and US-Ne2) in eastern Nebraska during the period from 2001 to 2012. Synthetic ET observations for three groups of scenarios were generated by adding random synthetic perturbations (ET pert ) resulting from bias (scenarios B), noise (scenarios V), and the combination of bias and noise (scenarios BV) to virtual observations, that is, ET simulations from ecosys with synthetic irrigation records (details in Section 3.2.3). Based on the performance comparison among different particle sizes (8, 10, 20, 30, and 40), we chose the particle size (N = 10) in this study via the tradeoff between performance and computation time. Three parameters (threshold for relative difference of ET, α, irrigation range, β, and irrigation duration, dT) of the proposed model-data fusion with CON and SEQ configurations were calibrated manually as 0.02, 3.0, and 1.0, respectively.

Three indexes (r, RMSE, and BIAS) between estimations of irrigation water use and synthetic irrigation records at daily, weekly, and monthly scales under three groups of scenarios (B, V, and BV) were summarized in the boxplots (Figures 4 and5, and Figure S6 in Supporting Information S1). Results indicated that the perturbations resulting from noise had larger impacts on degrading the estimation performance of irrigation water use than those resulting from bias. Pearson correlation coefficients (r) of scenarios V (perturbations resulting from noise) were lower than those of scenarios B (perturbations resulting from bias; Figure 4). For scenarios BV, r of scenarios BV 2 and BV 4 with smaller noise (10%) were higher than those of scenarios BV 3 and BV 5 with larger noise (30%). In addition, r decreased with the perturbations resulting from noise (scenarios V), while there was little difference for scenarios B with different magnitudes of perturbations resulting from bias. The reason was that bias correction embedded in particle filtering (Equation 4) could effectively remove the impacts of perturbations resulting from bias. For another two indexes (RMSE and BIAS), the similar patterns also were indicated by the results in Figure 5 and Figure S6 in Supporting Information S1. RMSE and BIAS of scenarios V (perturbations resulting from noise) were larger than those of scenarios B (perturbations resulting from bias). For scenarios BV, RMSE and BIAS of scenarios BV 2 (bias: 10%, noise: 10%) and BV 4 (bias: 30%, noise: 10%) were lower than those of scenarios BV 3 (bias: 10%, noise: 30%) and BV 5 (bias: 30%, noise: 30%), which had larger noise but same bias. In addition, RMSE and BIAS increased with the perturbations resulting from noise (scenarios V), while there was little difference for scenarios B with different magnitudes of perturbations resulting from bias. For the performance at different temporal scales, the statistical indexes (r and RMSE) between estimations of irrigation water use and farmer irrigation records were different, except BIAS. Particularly, the statistical indexes (r and RMSE) performed the best at the monthly scale but worst at the daily scale, while the statistical index (BIAS) had little difference among different temporal scales (daily, weekly, and monthly).

Taking US-Ne1 (a continuous maize cropping system) in 2004 as an example, CON and SEQ could effectively detect synthetic irrigation records with high accuracy for scenario BV 1 with the combinations of small bias and small noise (1%, 1%) (Figure 6 and Figure S7 in Supporting Information S1). The overlapped irrigation in red bar denoted that the estimations of irrigation water use from CON and SEQ (i.e., irrigation estimations, black bar with hatches) hit farmer irrigation records (blue bar). Daily ET difference between ET simulations from the ecosys model before data assimilation (without irrigation impact, red dashed lines) and ET observations (with irrigation impact, blue solid lines) were calculated on each day in real-time. The daily ET differences were removed by incorporating daily irrigation events, thus daily ET simulations from the ecosys model before data assimilation without irrigation impact on the target day could be corrected as ET simulations after data assimilation (black solid lines) in real-time with irrigation impact for the next day. Estimations of irrigation water use from CON only missed the irrigation event on 22 July 2004 for 1 day (Figure 6), while SEQ missed irrigation events for 6 days (Figure S7 in Supporting Information S1). It indicated that CON performed slightly better than SEQ for scenario BV 1 . Meanwhile, cumulative estimations of irrigation water use was close to cumulative synthetic irrigation records, demonstrating the satisfactory performance of CON and SEQ on estimating irrigation water use at high spatio-temporal resolution (daily and field-scale).

Estimations of irrigation water use using real ET observations was conducted at two sets of irrigated fields in eastern and western Nebraska. Based on the results of the ecosys model calibration (Section 4.1), the best distributions of daily ET difference between ecosys simulations and observations both with irrigation impact for maize and soybean in eastern and western Nebraska were screened out as Gaussian distributions (Figure S8 in Supporting Information S1). The mean and standard deviation of Gaussian distributions were estimated for bias correction (Equation 4) embedded in particle filtering for estimating irrigation water use. Particle size in the real experiments (N = 10) was selected as the same as the synthetic experiments. In addition, three parameters of the proposed model-data fusion framework with CON and SEQ configurations were manually calibrated separately in eastern and western Nebraska, that is, α = 0.05, β = 1.5, dT = 3 in eastern Nebraska and α = 0.05, β = 3.0, dT = 7 in western Nebraska. Irrigation range (β) and irrigation duration (dT) in western Nebraska were larger than those in eastern Nebraska, and this could be explained by climatic differences. Specifically, the climate in western Nebraska (annual precipitation around 450 mm) was much drier than that in eastern Nebraska (annual precipitation around 750 mm), thus requiring more irrigation water use. Based on the calibrated parameters, estimating irrigation water use at high spatio-temporal resolution was conducted by the proposed model-data fusion with CON and SEQ configurations for each site-year in eastern and western Nebraska. The estimations of irrigation water use at the monthly and annual scales matched well with farmer irrigation records for all the site-years in Nebraska (Figure 7). At the monthly scale, Pearson correlation coefficients of CON and SEQ configurations were 0.80 (p-value < 0.001) and 0.81 (p-value < 0.001), respectively, with bias of 0.19 and -0.88 mm/m (Figure 7a). Both bimodal distributions with the peak around 0 and 100 mm/m were found for farmer irrigation records and estimations of irrigation water use. At the annual scale, 95% confidence intervals of estimations of irrigation water use were very close to the 1-to-1 line, with Pearson correlation coefficients of 0.55 (p-value < 0.001) and 0.47 (p-value < 0.001) for CON and SEQ configurations, respectively (Figure 7b). For the performance comparison between CON and SEQ, we found that SEQ performed better than CON in eastern Nebraska, while there was little difference between CON and SEQ in western Nebraska (Figure 8 and Table 2).

For the spatial performance comparison between eastern and western Nebraska, we found that CON and SEQ performed better in eastern Nebraska than those in western Nebraska, that is, higher r and lower RMSE and BIAS in eastern Nebraska (Figure 8 and Table 2). In addition, there was little difference in estimating irrigation water use between fields planted with maize and soybean (Table 2 and Figure S9 in Supporting Information S1). Taking the fields (1013503, 1011702, and 1011960) which grew maize in 2015 in western Nebraska as examples, the model-data fusion with CON and SEQ configurations successfully detected most of the irrigation events at the daily scale (the overlapped irrigation events, red bar) but still with some missing (blue bar) or redundant irrigation events (black bar with hatches) (Figure 9, and Figures S10-S14 in Supporting Information S1). The daily ET difference between model simulations before data assimilation (without irrigation impact, red dashed lines) and daily ET observations (with irrigation impact, blue solid lines) were used for irrigation detection in real time (Figure 9, and Figures S10-S14 in Supporting Information S1). Then, 10 irrigation particles with different amounts in the particle filtering were incorporated into the ecosys model to account for the impacts of irrigation with different amounts in real-time on each day (Figure S15 in Supporting Information S1). The final daily irrigation amount was determined by the weighted average of all the irrigation particles, and it was assimilated into the ecosys model to remove the ET difference in real-time. Thus, daily ET simulations from the ecosys model before data assimilation without irrigation impacts (red dashed lines) could be updated into ET simulations after data assimilation with irrigation impacts (black solid lines) in real-time. In addition, the 95% confidence interval of cumulative estimations of irrigation water use based on 10 replicates were narrow and covered cumulative farmer irrigation records for most time periods (Figure 9, and Figures S10-S14 in Supporting Information S1). All these results demonstrated that the proposed model-data fusion with CON and SEQ configurations were robust and reliable for estimating irrigation water use at high spatio-temporal resolution.

The total deviation from remotely sensed ET and precipitation during the growing seasons (i.e., the estimation of minimum irrigation water use based on water balance (I min = max[ET obs -P obs , 0])) (Irmak et al., 2011;Van Dijk et al., 2018) was compared with the estimations of irrigation water use based on CON and SEQ configurations and actual irrigation records applied by farmers (Figure 10). The boxplots indicated that the estimation of minimum irrigation water use based on water balance (mean: 120.59 and 73.92 mm/yr in eastern and western Nebraska) were much lower than the estimations of irrigation water use using CON (mean: 195.22 Figures 10a and10c). The estimations of irrigation water use based on CON and SEQ configurations were close to the farmer irrigation records in eastern and western Nebraska. It indicated that our proposed model-data fusion framework with CON and SEQ configurations was reliable for the estimations of irrigation water use. In addition, we analyzed the relationships between irrigation water use and precipitation during the growing seasons to check the sensitivity of irrigation water use to climate variability (Figures 10b and10d). Irrigation water use decreased with the increasing precipitation, which were reflected in four cases (r < 0), indicating that irrigation water use was smaller in wet years with higher precipitation, especially in eastern Nebraska (r = -0.60). We also found that the fitted regression lines with 95% confidence intervals of the estimations of irrigation water use based on CON and SEQ were close to those of farmer irrigation records in eastern and western Nebraska, further demonstrating Note. The p-values were represented the symbols (ns meaning p > 0.05; * meaning p ≤ 0.05; ** meaning p ≤ 0.01; *** meaning p ≤ 0.001). There were 24 and 52 site-years in eastern and western Nebraska, respectively. There were 66 and 10 site-years planted with maize and soybean in Nebraska, respectively.

the reliability of the proposed model-data fusion framework with CON and SEQ configurations for estimating irrigation water use.

In addition, the difference between farmer irrigation records and the water balance case could be treated as water loss unavailable for crop use (water loss = max[I obs + P obs -ET obs , 0]) from five potential sources, including runoff, deep percolation, air evaporation, plant interception, and leakage in the irrigation systems, ignoring the change of soil water storage (Irmak et al., 2011;Van Dijk et al., 2018). For the water balance case, we found that about one-third (33.56%) of available water (including precipitation and irrigation) was not used by crops across all the site-years (2015-2016 at 29 fields) located in Keith and Deuel counties in western Nebraska. The unused water fraction (33.56%) in western Nebraska was larger than that during 2001-2012 at two AmeriFlux sites (US-Ne1 and US-Ne2) (17.47%) in eastern Nebraska. It was consistent with previous studies that total irrigation water use in Nebraska could potentially be reduced up to 25%-40% without hurting crop yields (K. E. Gibson et al., 2019). In addition, we found that water loss increased with precipitation (Figures 10b and10d), indicating that farmers applied more excess irrigation with higher precipitation during the growing seasons. All these results suggested that the farmers' irrigation behavior at the research fields in Nebraska might be suboptimal for water saving, especially at the irrigated fields in western Nebraska. Thus, there should be a large opportunity to improve irrigation efficiency in Nebraska by adopting more efficient irrigation scheduling technologies (J. Zhang et al., 2021b;J. Zhang et al., 2021c).

The proposed model-data fusion approach with CON and SEQ configurations for estimating center-pivot irrigation water use at high spatio-temporal resolution (daily and field-scale) is successfully applied in synthetic and real experiments at two sets of irrigated fields in eastern and western Nebraska. The fundamental theory behind this approach to estimate center-pivot irrigation water use is the difference between remotely sensed ET observations with irrigation impacts and ET simulations from the process-based ecosys model without irrigation impacts. The daily irrigation events are estimated by assimilating the remotely sensed ET observations into the ecosys model via particle filtering with multiple irrigation events. Thus, the uncertainties of observed and simulated ET have critical impacts on the efficacy of the proposed model-data fusion approach with CON and SEQ configurations.

If there are relatively small uncertainties in the remotely sensed ET observations and ET simulations from the ecosys model, the ET difference can largely be attributed to the irrigation, and our proposed model-data fusion can effectively detect all the irrigation events, as shown in the synthetic experiment of scenario BV 1 (the combinations of low bias, 1%, and low noise, 1%) in Figure 6. However, the uncertainties in the ET observations and simulations are unavoidable and unquantifiable. In this study, we combine the uncertainties from process-based model and observations together, and attribute the ET uncertainties into two sources (i.e., bias and noise). The ET uncertainties are represented by the Gaussian distribution (bias, μ s-o , and standard deviation, i.e., noise, σ s-o ) of ET difference between the simulations from the calibrated process-based model and remotely sensed observations both with irrigation impact. The synthetic experiments (i.e., three groups of scenarios including bias-B, noise-V, and bias and noise-BV) indicated that the ET uncertainties with noise had larger impacts on degrading the performance of the proposed model-data fusion to estimate irrigation water use than those with bias. The reason is that bias correction embedded in particle filtering could effectively remove the ET uncertainties resulting from bias, while the ET uncertainties resulting from noise is difficult to be removed via particle filtering.

For the ET uncertainties in the real experiments, the noise (i.e., standard deviation) in eastern Nebraska is smaller than that in western Nebraska for maize and soybean (Figure S8 in Supporting Information S1). Thus, the proposed model-data fusion approach with CON and SEQ configurations has better performance on the estimations of irrigation water use in eastern Nebraska with smaller ET uncertainties than that in western Nebraska with larger ET uncertainties. The fundamental reason for smaller ET uncertainties in eastern Nebraska (i.e., smaller noise in eastern Nebraska in Figure S8 in Supporting Information S1) is that the irrigated fields (US-Ne1 and US-Ne2) in eastern Nebraska use the eddy covariance-based ET observations and ET simulations from the ecosys model with the complete field management data sets from UNL-ENREC's CSP, while the irrigated fields in western Nebraska use the satellite-based BESS-STAIR ET observations and ET simulations from the ecosys model without detailed field management data sets. Note that the irrigated fields in western Nebraska are more realistic compared to the irrigated fields (US-Ne1 and US-Ne2) in eastern Nebraska, which have the gold standard of scientific observations. For most irrigated fields, the BESS-STAIR ET data set can be used as ET observations, while the management information is usually unknown due to lack of open access, technological knowledge, or held by private companies, which increases the ET uncertainties for the estimations of irrigation water use.

This study has demonstrated that the ET uncertainties have significant negative impacts on the efficacy of the proposed model-data fusion with CON and SEQ configurations for the estimations of irrigation water use. We will discuss the possible pathways for reducing the uncertainties from both data and model perspectives. From the data perspective, ET observations with higher accuracy could reduce the ET uncertainties and improve the estimations of irrigation water use. For example, the real experiments for estimating irrigation water use using eddy covariance measurements of ET in eastern Nebraska performed better than those using satellite-based BESS-STAIR ET measurements in western Nebraska (Figure 8 and Table 2). The reason was that ET observations from eddy covariance measurements were more accurate than the satellite-based ET observations. It further confirmed the importance of ET observations for estimating irrigation water use based on the model-data fusion framework. To improve remotely sensed ET observations can be possibly achieved by constraining the satellite-based ET models (Courault et al., 2005) with LST data set, as LST reflects the cooling effects due to irrigation and in turn can be used to improve ET observations (J. Zhang et al., 2021a). However, the existing satellite-based LST data sets usually could not fulfill high resolutions in space and time simultaneously; for example, MODIS LST data is at the daily step (snapshot) but at 1 km resolution; Landsat LST resampled data is at 30 m resolution but has 16-day interval; and ECOSTRESS LST data is at 70 m resolution but has irregular revisit time (Cook, 2014;Cook et al., 2014;Li et al., 2021). In addition, existing global satellite LST products generally show large daytime uncertainties (e.g., bias = 2°C and RMSE = 3.5°C) in croplands (Li et al., 2021), which might be propagated to ET observations. More advanced satellite thermal sensors (Wright et al., 2020), multi-satellite LST fusion algorithms (Xu & Cheng, 2021), and data-model fusion algorithms (X. Zhang, Zhou, et al., 2021) will potentially lead to the development of high-quality LST products at high spatio-temporal resolution in the future, ultimately for remotely sensed ET data sets with high accuracy.

The possible pathways from the model perspective (including inputs, process, and parameters) for reducing the uncertainties will be discussed here. (a) More detailed field-scale management data collection (such as planting date, fertilizer records, and irrigation systems) could effectively reduce the uncertainties of model simulations of ET and thus improve the estimations of irrigation water use (Massari et al., 2021) Jiang et al., 2021;Kimm et al., 2020;Wu et al., 2020), could be applied to constrain these parameters in the ecosystem models to obtain reliable crop water use and hydrological simulations.

Estimating irrigation water use at high spatio-temporal resolution in this study can contribute to sustainable regional water management and understanding groundwater depletion for sustainable groundwater management in groundwater-dependent irrigated regions. The proposed model-data fusion framework for estimating irrigation water use can be applied in the irrigated agricultural lands with different crops and/or multiple cropping systems at large-scale. The remotely sensed BESS-STAIR ET, GPP, and LAI data sets at high-spatiotemporal resolutions (daily and field-scale) can be used to constrain the process-based models at large-scale. However, the parameters (ET difference threshold and irrigation duration) need to be calibrated with some necessary information, such as the field-scale irrigation water use at some sample fields or total irrigation water use at large-scale regions.

The application of this model-data fusion framework to estimate irrigation water use can be treated as water footprint estimation in the irrigated lands. Current existing water footprint data sets have coarse spatial and temporal resolutions, and our study can provide water footprint data sets at the daily and field scale. Long-term water footprint data sets can be used to investigate the spatio-temporal variability of irrigation efficiency, thus to evaluate different irrigation conservation programs, such as the Local Enhanced Management Area program in Kansas (Deines et al., 2021) and the widespread adoption of low-energy precision application technologies (McCarthy et al., 2020). Then, potential regions with low irrigation efficiency but water scarcity can be identified to reform regional water policies to regulate farmers' irrigation water use for sustainable water management (Foster et al., 2020;D. Wang & Cai, 2007). In addition, water footprint data sets at high spatio-temporal resolution can be used to understand groundwater pumping and recharge rate patterns for long-term aquifer conservation in groundwater-dependent irrigated regions, such as the High Plains Aquifer (HPA). Agriculture in Nebraska, Kansas, and Texas largely depends on groundwater pumping in the HPA for irrigation, but there is large spatial variability for the groundwater level changes, that is, slight groundwater level declines in northern HPA (Nebraska) while serious groundwater level declines in southern HPA (Kansas and Texas; Nie et al., 2018;Scanlon et al., 2012). Application of our methods in HPA may provide a pathway to investigate the primary causes of the disparities for groundwater level changes and the possible strategies for groundwater sustainability in the HPA.

Estimating irrigation water use at high spatio-temporal resolution can also help farmers to achieve sustainable irrigation. The proposed method in this study can effectively reflect farmers' irrigation behavior, thus its applications at large-scale can be used to understand the spatio-temporal variability of farmers' irrigation practices and to estimate water loss fractions unavailable for crop use (Foster et al., 2019). Water loss fractions (such as around one-third in western Nebraska) can indicate how much space is left for irrigation improvements. Thus, estimating irrigation water use at high spatio-temporal resolution can help to improve farmers' irrigation practices with more science-based irrigation methods in regions with low irrigation efficiencies and high water loss fractions (J. Zhang et al., 2021b;J. Zhang et al., 2021c). In addition, estimations of irrigation water use can compensate for the missing irrigation records if farmers do not provide, then accurate soil water budgets at the field level can be provided for precision irrigation with the estimations of irrigation water use as past water inputs. Furthermore, accurate soil water budgets with the estimations of irrigation water use can also help to estimate ecosystem carbon, energy, and nitrogen budgets, thus for modeling crop growth and precision nitrogen management.

Last but not least, the estimations of irrigation water use (irrigation timing and amount) in this study can help the ecosystem and hydrological models to track terrestrial water and energy cycles in the irrigated regions. Irrigation is the major anthropogenic disturbance of terrestrial water and energy cycles in the irrigated agricultural lands (McDermid et al., 2021), as irrigation performs as water inputs and affects surface energy budget (i.e., energy partitioning between latent and sensible heat fluxes). The estimations of irrigation water use reflecting farmers' irrigation behavior in this study can be used to investigate the direct impacts of irrigation on regional water and energy budgets with land-atmosphere feedbacks. The traditional criterion-based irrigation scheduling (such as maximum allowable depletion, MAD-50%) set in the ecosystem and hydrological models cannot reflect farmers' irrigation decisions (de Rosnay et al., 2003;Haddeland et al., 2006), as farmers' irrigation decisions are complicated due to many factors, such as irrigation infrastructures, farmers' preferences, and climate conditions (Foster et al., 2020;D. Wang & Cai, 2007). The proposed method in this study offers a pathway to estimate irrigation water use at high spatio-temporal resolution reflecting farmers' irrigation behavior.

We have demonstrated that our proposed model-data fusion framework with CON and SEQ configurations is effective for estimating center-pivot irrigation water use at the daily and field scale. The irrigation signals are detected by the difference between remotely sensed ET observations with irrigation impacts and ET simulations from the agroecosystem (ecosys) model without irrigation impacts. The uncertainties of remotely sensed ET observations and ET simulations from the ecosys model have critical impacts on the efficacy of the proposed model-data fusion approach with CON and SEQ configurations. We conclude two major findings based on two types of experiments using synthetic and real ET observations in Nebraska: (a) Three groups of scenarios (bias-B, noise-V, and bias and noise-BV) for the synthetic experiments indicated that, among two sources of ET uncertainties (bias and noise), noise had larger impacts on degrading the performance of the proposed model-data fusion to estimate irrigation water use than those with bias. Systematic bias can be removed by bias correction embedded in the particle filtering. (b) The proposed model-data fusion framework with CON and SEQ configurations offered a pathway to estimate center-pivot irrigation water use at high spatio-temporal resolution. The estimations of irrigation water use at the monthly and annual scales matched well with farmer irrigation records, with RMSE less than 30 mm/m and 80 mm/yr, absolute BIAS less than 1 mm/m and 6 mm/yr, and r around 0.80 and 0.50, respectively. Although detecting daily irrigation records was very challenging, our method still gave a good performance with RMSE, BIAS, and r around 2.90, 0.03, and 0.4 mm/d, respectively. The proposed model-data fusion framework for estimating irrigation water use at high spatio-temporal resolution could contribute to regional water management, sustainable irrigation, and better tracking terrestrial water and energy cycles.