Abdelhakim Amazirh
email: abdelhakim.amazirh@um6p.ma

Salah Er-Raki

Nitu Ojha

Houssaine El

Vincent Bouras

Olivier Rivalland

Abdelghani Merlin

Chehbouni

El Houssaine Bouras

Vincent Rivalland

Merlin C Abdelghani Olivier

Dr Amazirh Abdelhakim

Assimilation of SMAP disaggregated soil moisture and Landsat land surface temperature to improve FAO-56 estimates of ET in semi-arid regions

Keywords: Evapotranspiration, Data assimilation, FAO-dualKc, Soil moisture, Land surface temperature

come

Introduction

Accurate estimation of surface evapotranspiration (ET) is crucial to understand land-surface interaction processes. Additionally, ET is the dominant factor in the water cycle, especially in arid and semi-arid regions. About 85 to 90 % of precipitation returns to the atmosphere through ET [START_REF] Rosenberg | Microclimate. The Biological Environment[END_REF]. Moreover, ET estimates have an important role in monitoring drought [START_REF] Bhattarai | An automated multi-model evapotranspiration mapping framework using remotely sensed and reanalysis data[END_REF][START_REF] Gerhards | Challenges and future perspectives of multi-/Hyperspectral thermal infrared remote sensing for crop water-stress detection: A review[END_REF] and other extreme climatic events [START_REF] Littell | A review of the relationships between drought and forest fire in the United States[END_REF][START_REF] Molden | Improving agricultural water productivity: Between optimism and caution[END_REF], as well as water resource management [START_REF] Madugundu | Performance of the METRIC model in estimating evapotranspiration fluxes over an irrigated field in Saudi Arabia using Landsat-8 images[END_REF][START_REF] Tasumi | Estimating evapotranspiration using METRIC model and Landsat data for better understandings of regional hydrology in the western Urmia Lake Basin[END_REF]. In this case the objective is to have accurate ET estimates that allow computing irrigation requirements. Precise actual crop evapotranspiration (ETc act) estimates are the key for the quantification of crop water requirements, which allow the optimization of irrigation especially in semi-arid and arid areas that suffer from a pronounced shortage of water [START_REF] Allen | Evapotranspiration information reporting: I. Factors governing measurement accuracy[END_REF][START_REF] French | Remote sensing of evapotranspiration over cotton using the TSEB and METRIC energy balance models[END_REF]. Several techniques were developed and used to monitor ETc act at different space-time scales. At plot or field scale, we found the eddy correlation system [START_REF] Allen | Evapotranspiration information reporting: I. Factors governing measurement accuracy[END_REF][START_REF] Baldocchi | Measuring biosphere-atmosphere exchanges of biologically related gases with micrometeorological methods[END_REF], while at a larger scale, the scintillometer can provide ETc act over a transect up to 5 km [START_REF] Ezzahar | The use of scintillometry for validating aggregation schemes over heterogeneous grids[END_REF][START_REF] Kohsiek | An Extra Large Aperture Scintillometer For Long Range Applications[END_REF]. Other approaches for measuring ETc act were discussed in [START_REF] Alfieri | A Brief Overview of Approaches for Measuring Evapotranspiration[END_REF] and [START_REF] Er-Raki | Micrometeorology tools for measuring evapotranspiration from the leaf to the region[END_REF].

Concerning ETc act predictions, several models with different degrees of complexity have been developed during the past 30 years [START_REF] Li | A review of current methodologies for regional Evapotranspiration estimation from remotely sensed data[END_REF][START_REF] Seguin | Using midday surface temperature to estimate daily evaporation from satellite thermal IR data[END_REF]. Note that,crop ET (ETc) refer to optimal, well-watered conditions and pristine cultivated crops, i.e. cropped in conditions without any stress conditions and may be estimated with models while under field conditions actual ET is estimated with models because crops are generally subject to water (and/or other stress conditions). Some of these models use remotely sensed data such as surface albedo, Leaf Area Index (LAI), Normalized Difference Vegetation Index (NDVI), Land Surface Temperature (LST) [START_REF] Allen | Evapotranspiration information reporting: I. Factors governing measurement accuracy[END_REF][START_REF] Granger | Satellite-derived estimates of evapotranspiration in the Gediz basin[END_REF][START_REF] Kharrou | Assessment of Equity and Adequacy of Water Delivery in Irrigation Systems Using Remote Sensing-Based Indicators in Semi-Arid Region, Morocco[END_REF][START_REF] Kustas | Use of remote sensing for evapotranspiration monitoring over land surfaces[END_REF][START_REF] Li | A review of current methodologies for regional Evapotranspiration estimation from remotely sensed data[END_REF] and soil moisture [START_REF] Elfarkh | Integrating thermal stress indexes within Shuttleworth-Wallace model for evapotranspiration mapping over a complex surface[END_REF][START_REF] Gokmen | Integration of soil moisture in SEBS for improving evapotranspiration estimation under water stress conditions[END_REF][START_REF] Walker | Regional evapotranspiration estimates using the relative soil moisture ratio derived from SMAP products[END_REF]. ETc act component can be predicted based on water balance at a daily time scale. Among this family of models, the FAO dual crop coefficient model (FAO-dualKc, Allen et al., 1998) is the most common and operational model to retrieve ETc act, which was extensively used for modeling water consumption and growth of plants (e.g., [START_REF] Alberto | Actual evapotranspiration and dual crop coefficients for dry-seeded rice and hybrid maize grown with overhead sprinkler irrigation[END_REF][START_REF] Barker | Evaluation of variable rate irrigation using a remote-sensing-based model[END_REF][START_REF] Ko | Determination of growth-stage-specific crop coefficients (Kc) of cotton and wheat[END_REF][START_REF] Liu | A consolidated evaluation of the FAO-56 dual crop coefficient approach using the lysimeter data in the North China Plain[END_REF]. The SIMDualKc software [START_REF] Paço | Evapotranspiration and crop coefficients for a super intensive olive orchard. An application of SIMDualKc and METRIC models using ground and satellite observations[END_REF][START_REF] Paço | Evapotranspiration and crop coefficients for a super intensive olive orchard. An application of SIMDualKc and METRIC models using ground and satellite observations[END_REF]Pereira et al., 2020;Rosa et al., 2012bRosa et al., , 2012a) ) was set up based on FAO-dualKc to simulate ETc act and its components. In contrast with other models based on energy balance theory, the FAO-56 model requires limited input parameters. Several studies used the FAO model to retrieve ETc act over various crop types where wheat is the most studied crop among small grain cereals (e.g. [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF][START_REF] Drerup | Evapotranspiration of winter wheat estimated with the FAO 56 approach and NDVI measurements in a temperate humid climate of NW Europe[END_REF][START_REF] French | Satellite-based NDVI crop coefficients and evapotranspiration with eddy covariance validation for multiple durum wheat fields in the US Southwest[END_REF][START_REF] Rafi | Partitioning evapotranspiration of a drip-irrigated wheat crop: Inter-comparing eddy covariance-, sap flow-, lysimeter-and FAO-based methods[END_REF]. Recently, Pereira et al. (2021) provides a review of the studies using the FAO-dualKc over wheat and other crop types. The FAO-dualKc approach allows for the partitioning of ETc act into soil evaporation (Es) and plant transpiration (Tc act) by separating the crop coefficients (Kc act) into a basal crop (Kcb act) and soil evaporation (Ke) coefficients.

Despite the operationality of the FAO-dualKc, several studies stated that FAO-dualKc tends to under/overestimate soil evaporation (Es) at the beginning or the end of the season when the surface is under bare soil conditions [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF][START_REF] Boulet | Evapotranspiration and Evaporation/Transpiration Retrieval Using Dual-Source Surface Energy Balance Models Integrating VIS/NIR/TIR Data with Satellite Surface Soil Moisture Information[END_REF][START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF]. In this context, many papers deal with the use of soil moisture (SM) data to retrieve soil evaporation under bare soil conditions [START_REF] Amazirh | Retrieving surface soil moisture at high spatio-temporal resolution from a synergy between Sentinel-1 radar and Landsat thermal data: A study case over bare soil[END_REF][START_REF] Zribi | Soil surface moisture estimation over a semi-arid region using ENVISAT ASAR radar data for soil evaporation evaluation[END_REF]. Moreover, the crop Tc act estimates is based on the root zone water balance reflecting the variation of the root zone soil moisture (RZSM). Therefore, using RZSM measurements will improve the Tc act estimates. The root zone soil moisture is a state variable that is quite difficult to estimate from meteorological data alone or remote sensing data. For this reason, as an attempt to improve ETc act estimates, [START_REF] Kalma | Estimating land surface evaporation: A review of methods using remotely sensed surface temperature data[END_REF] and [START_REF] Li | A review of current methodologies for regional Evapotranspiration estimation from remotely sensed data[END_REF] used LST data to estimate water stress indices. More recently, LST data were used to retrieve the stress coefficient (Ks) used by the FAO approach [START_REF] Dejonge | Comparison of canopy temperature-based water stress indices for maize[END_REF][START_REF] Ihuoma | Recent advances in crop water stress detection[END_REF][START_REF] Kullberg | Evaluation of thermal remote sensing indices to estimate crop evapotranspiration coefficients[END_REF].

Other techniques attempt to combine complementary information from hydrological models and observed data for mapping ETc act. This technique is named data assimilation [START_REF] Kumar | A land surface data assimilation framework using the land information system: Description and applications[END_REF][START_REF] Mclaughlin | Computational issues for large-scale land surface data assimilation problems[END_REF]. The data assimilation technique is based on minimizing the mismatch between the model outputs and observations, by making the optimal use of all available information. We distinguish between two categories of data assimilation schemes: sequential assimilation like Ensemble Kalman Filter and optimal interpolation. The former technique has been widely used in hydrology [START_REF] Chen | An evapotranspiration assimilation method based on ensemble kalman filter and À trous wavelet[END_REF][START_REF] Moradkhani | Uncertainty assessment of hydrologic model states and parameters: Sequential data assimilation using the particle filter[END_REF][START_REF] Wang | Assimilation of soil moisture in LPJ-DGVM[END_REF][START_REF] Xie | Data assimilation for distributed hydrological catchment modeling via ensemble Kalman filter[END_REF]. The latter is variational assimilation (e.g., 4-dimensional variational assimilation [START_REF] Caparrini | Estimation of surface turbulent fluxes through assimilation of radiometric surface temperature sequences[END_REF][START_REF] Courtier | Important literature on the use of adjoint, variational methods and the Kalman filter in meteorology[END_REF]). Several research works used data assimilation techniques to estimate ETc act at different levels [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF][START_REF] Merlin | Assimilation of Disaggregated Microwave Soil Moisture into a Hydrologic Model Using Coarse-Scale Meteorological Data[END_REF][START_REF] Neale | Soil water content estimation using a remote sensing based hybrid evapotranspiration modeling approach[END_REF][START_REF] Pipunic | Assimilation of remotely sensed data for improved latent and sensible heat flux prediction: A comparative synthetic study[END_REF]. Recently, many research works used the variational data assimilation technique to update and improve the ETc act estimates and its components (e.g. [START_REF] Abdolghafoorian | LIDA: A Land Integrated Data Assimilation Framework for Mapping Land Surface Heat and Evaporative Fluxes by Assimilating Space-Borne Soil Moisture and Land Surface Temperature[END_REF][START_REF] Tajfar | Estimation of surface heat fluxes via variational assimilation of land surface temperature, air temperature and specific humidity into a coupled land surface-atmospheric boundary layer model[END_REF][START_REF] Xu | Mapping regional turbulent heat fluxes via variational assimilation of land surface temperature data from polar orbiting satellites[END_REF][START_REF] Xu | Partitioning evapotranspiration into soil evaporation and canopy transpiration via a two-source variational data assimilation system[END_REF]. For the sequential data assimilation technique, Kalman Filter (KF) and its variants are the widely used techniques for the ETc act retrieval due to their simple conceptual formulation and relative ease of implementation compared to other techniques [START_REF] Bateni | Surface heat flux estimation with the ensemble Kalman smoother: Joint estimation of state and parameters[END_REF][START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF][START_REF] He | Mapping Regional Turbulent Heat Fluxes via Assimilation of MODIS Land Surface Temperature Data into an Ensemble Kalman Smoother Framework[END_REF][START_REF] Xu | Estimating turbulent fluxes through assimilation of geostationary operational environmental satellites data using ensemble Kalman filter[END_REF]. KF provides a sequential, unbiased, and minimum error variance estimate under the assumption of known statistics of system and measurement errors for a linear problem. The KF provided a comparable result with other popular assimilation methods, despite its simple conceptual formulation which did not require any integrations backward in time compared to other variational assimilation techniques (4DVAR, 3DVAR…). However, KF does not account for nonlinearities in hydrological systems, for this, a variant of KF (e.g. extended Kalman filters) could solve some nonlinear problems. Even that, KF affords very good results compared to other complicated models such as the variational method [START_REF] Tian | An ensemble-based explicit four-dimensional variational assimilation method[END_REF]. In addition, KF requires few restrictive assumptions which make it operational. Therefore, KF seems to be suitable for hydrological linear problems such as modeling ETc act. Data assimilation technique seems to be a promising tool to improve the ETc act estimates when combining with FAO model. [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF] have shown that assimilating LST data into FAO-56 single crop model improves the overall ETc act estimates. [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF] in their paper reach a satisfactory result after assimilating LST data in FAO single crop coefficient with RMSE values dropping down from 0.69 to 0.46 mm/day compared to open-loop FAO, which corresponds to an improvement of 40% when assimilating LST into the FAO56 approach. In the same vein, the objective of this paper is to use data assimilation in conjunction with FAO-dualKc methods to improve ETc act estimates.

We first assimilate time domain reflectometry (TDR) near-surface SM (0-5cm) observations into FAO-dualKc to improve the soil Es estimates. Then, TDR-SM and infrared thermal radiometer (IRT) derived LST data were jointly assimilated to improve soil Es and wheat crop Tc act, respectively. A sequential assimilation scheme was applied to update the Es and Tc act estimates. Secondly, Landsat 7/8 derived-LST and disaggregated Soil Moisture Active Passive (SMAP)-SM data at 100 m resolution were used in conjunction with a sequential assimilation approach over two wheat fields named B123 and F16 near Marrakech city in central Morocco.

Sites and data description

Sites description

The present work was conducted over the Tensift region in central Morocco (Figure 1). This area is known as a semi-arid area with low and irregular precipitation against high potential ETo. Numerous studies have been performed in this region since 2002-2003 to address various problems regarding the use of water, understanding the integrated hydrological functioning of the Tensift semi-arid basin, as well as improving our knowledge of fundamental hydrological processes [START_REF] Amazirh | Modified Penman-Monteith equation for monitoring evapotranspiration of wheat crop: Relationship between the surface resistance and remotely sensed stress index[END_REF][START_REF] Chehbouni | An integrated modelling and remote sensing approach for hydrological study in arid and semi-arid regions: the SUDMED Programme[END_REF][START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF]. The Tensift basin receiving about 250 mm/year of rainfall, while the reference evapotranspiration (ETo) is about 1600 mm/year, according to the FAO-56 model [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF]. Two study fields have been monitored during 2002-2003 and 2015-2016 wheat agricultural seasons named by B123 and F16, respectively. Both fields were irrigated by flood gravitation irrigation systems. Flowmeters were used to collect precisely the irrigation volumes of the two irrigated fields. Irrigation was applied every 1 to 3 weeks from December 2015 to April 2016 during the 2015-2016 season.

The 4 ha F16 field was irrigated 8 times with a volume of 64 mm each, while the B123 field was irrigated 4 times with a volume of 24 mm each. These fields are located in an irrigated perimeter named by R3 situated about 40 km east of the city of Marrakech (31°40′9.46″N, 7°35′45.64″W). The B123 and F16 fields occupied an area of approximately 4 ha each (0.85 km * 4.15 km) involving 7 pixels. The surrounding two fields were also cultivated with wheat crops and were similar to the same irrigation system. Both fields were known for high clay contents (47%) and 18% of sand fraction. The sowing dates were the 13 th December 2015 and 14 th January 2003 for the F16 and B123 irrigated sites, respectively [START_REF] Amazirh | Modified Penman-Monteith equation for monitoring evapotranspiration of wheat crop: Relationship between the surface resistance and remotely sensed stress index[END_REF][START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF].

Experimental data set

Weather station

During the investigated agricultural seasons, an experiment has been set up for each site. An automatic weather station was set up, over an alfalfa cover near the studied wheat fields, to provide continuous climatological data at a sub-hourly time step, including air temperature (Ta,HMP60,Vaisala,Oyj,Helsinki,Finland), relative humidity (rha, HMP60, Vaisala, Oyj, Helsinki, Finland), solar radiation (Rg, pyranometer, LI-COR LI-200X, Lincoln, NE, USA), wind velocity (ua, anemometer, R.M. Young 3002, Traverse City, USA) and precipitation (P, automatic rain gauge, Texas Electronics TE525 MM, USA). Reference evapotranspiration (ET0) was calculated on a daily basis using the Penman-Monteith equation for short canopies [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF].

Eddy covariance stations

Each field was equipped with an eddy covariance station (EC) consisting of a 3D sonic anemometer (CSAT3, Campbell Scientific Ltd. Logan, UT, USA) that measures the three components of wind speed and a Krypton hygrometer (KH21, Campbell Scientific Ltd. Logan, UT), providing continuous measurements of energy fluxes, vertical sensible heat (Hobs) and latent heat (LEobs) fluxes which is equivalent to ETc act obs. The net radiometer (Kipp and Zonen CNR4, Campbell Sci, Delft, The Netherlands) measured the surface net radiation (Rn) at 2.6 m height and the soil heat flux (G) was measured at 5 cm depth by using two heat flux plates (HPF01, Campbell Sci, Logan, USA) over the wheat crop fields. An average output of the two plates was used. The two plates were buried with thermocouples at two depths to account for heat storage. Note that, the EC systems were installed at a height of 2.6 m, and the maximum height of wheat was 0.74 m and 0.90 m for B123 (2002-2003) and F16 (2015-2016) sites, respectively. Therefore, the EC system was at least 1.70 m above the canopy. The choice of the installation height actually resulted in a compromise between two constraints: 1) sampling within the field for a range of wind directions and 2) measuring above the roughness layer all along the agricultural season. The relative height of 1.70 m above the canopy ensures that the EC is in the constant flux layer, which is located approximately 1.5-2 canopy heights above the soil surface [START_REF] Burba | Eddy Covariance Method-for Scientific[END_REF]. The two energy fluxes were extracted from KH21 hygrometer measurements, which quantify the fluctuations of atmospheric water vapor and temperature.

The collected raw EC data at 20 Hz are processed in the laboratory using the EC-pack software developed by the Meteorology and Air Quality Group, Wageningen University (available for download at http://www.met.wau.nl/). The flux tower ETc act is calculated as a multiplication of the air density by the fluctuation of the mean density of water vapor in the air, the fluctuation of the mean vertical wind velocity, and the latent heat of vaporization. Before using such an energy flux data set, a verification of the reliability and the quality of these measurements is needed. The energy balance closure is a way to assess that, by comparing the surface available energy (Rn-G) to the sum of turbulent fluxes (Hobs+ LEobs). For the F16 field, the energy budget closure was achieved in [START_REF] Amazirh | Modified Penman-Monteith equation for monitoring evapotranspiration of wheat crop: Relationship between the surface resistance and remotely sensed stress index[END_REF]. The ratio of the turbulent energy flux to available energy (EBR) was 78%, with a strong coefficient of determination (R 2 =0.91), while for B123 field, the EBR is equal to 80% with an R 2 equal to 0.92 [START_REF] Er-Raki | Derived crop coefficients for winter wheat using different reference evapotranspiration estimates methods[END_REF]. The slope of the regression forced through the origin was about 1.30 and 1.22 for F16 and B123 fields, respectively. The closure error is close to 10 % for both field which is in the acceptable ranges (from 10% to 30%) reported in [START_REF] Twine | Correcting eddy-covariance flux underestimates over a grassland[END_REF]. The obtained results are relatively good and the closure of the energy balance is relatively well verified in comparison with other studies [START_REF] Elfarkh | Integrating thermal stress indexes within Shuttleworth-Wallace model for evapotranspiration mapping over a complex surface[END_REF][START_REF] Testi | Evapotranspiration of a young irrigated olive orchard in southern Spain[END_REF][START_REF] Twine | Correcting eddy-covariance flux underestimates over a grassland[END_REF]. To correct some small uncertainties of non-closure of the energy balance, the Bowen ratio was used to recompute H and LE by forcing the energy balance closure with the measured values of Bowen ratio. The observed data were carefully processed and rigorously screened, some gaps were observed in the collected data, these data were excluded.

In situ soil moisture and land surface temperature measurements

Soil moisture data were also collected over both sites at different depths [START_REF] Ko | Determination of growth-stage-specific crop coefficients (Kc) of cotton and wheat[END_REF]10,20,30,50,70 cm), at a half-hourly time step, using time-domain reflectometer (TDR) probes buried under the eddy covariance system for each field. Only one point was selected under the eddy covariance stations for each field. This is due to fact that the fields are irrigated with flooding system which implies a homogeneous fraction of wetted areas, where all the pixels have the same percentage of irrigation water, which means uniform soil moisture within the site. Also, the wheat crop covers the field uniformly (uniform distribution of seeds), which means that the shaded area within the field is uniform. This homogeneity all over the field could explain the use of one location instead of several locations. Note that, SM was measured up to 0.7 m, but only data from 0.05 to 0.5 m were used in this study. This is due to, the measured rooting depths which are 0.52 and 0.50 m for F16 and B123 fields [START_REF] Er-Raki | Combining FAO-56 model and ground-based remote sensing to estimate water consumptions of wheat crops in a semi-arid region[END_REF]. The collected SM data were calibrated using the gravimetric technique by taking soil samples at each depth. SM at the surface was used in this study to improve soil Es through assimilation technic.

In addition, brightness surface temperature (Tbs) was measured by an Apogee 8-14 μm thermal radiometer sensor (SI-121, Apogee IRTS-P's, Inc., Logan, Utah, USA) in each field, set up at a 2 m height looking at nadir. The Tbs is corrected from sensor errors using the sensor body temperature, then was converted to surface temperature as in [START_REF] Tardy | A Software Tool for Atmospheric Correction and Surface Temperature Estimation of Landsat Infrared Thermal Data[END_REF], using the approach proposed by [START_REF] Olioso | Estimating the water budget components of irrigated crops: Combining the FAO-56 dual crop coefficient with surface temperature and vegetation index data[END_REF]. The thermal data were corrected from atmospheric effects using the spectral atmospheric transmission and the atmospheric downwelling and upwelling radiances. The atmospheric parameters are obtained from an atmospheric radiative transfer model (MODTRAN, Berk et al., 2005). Finally, LST was estimated by inverting the simplified Plank's law. The derived spectral atmospheric parameters and the surface emissivity permit the conversion of the at-sensor radiance into the top of canopy radiance exempt from atmospheric effects. The corrected temperature was used to improve Tc act by updating the stress coefficient (Ks).

Remote sensing data

In this work, Landsat-7 (L7) and Landsat-8 (L8) data were used to provide land surface temperature (LST) while SMAP data were used to extract surface soil moisture (SM) Table (1).

Only data over the F16 field were derived because SMAP sensor was launched in 2015 and B123 field was set in 2002.

Landsat data

L7 ETM+ and L8 OLI-TIRS were used in this study. The surface reflectance and thermal radiance (TIR) were downloaded from the USGS website (https://earthexplorer.usgs.gov/). L7 provides one thermal band (band 6) while the L8 provides two thermal bands (bands 10 and 11) but only band 11 was considered which is recommended to be used regarding the quality. TIR data were freely acquired with a spatial resolution of 60 and 100 m for L7 and L8, respectively.

The revisit time for each sensor is 16 days with an 8-day offset between the two overpasses, which offers TIR data every 8 days. 13 cloud-free images were collected in the 2015-2016 agricultural season over F16 study field. As mentioned in section 2.1, the F16 field occupied an area of approximately 4 ha (0.85 km × 4.15 km) involving 7 pixels. However, a selection of the pixel where the eddy covariance system was installed has been performed to overcome the edge problem, which can involve the pixels of the surrounded fields. Even if an average over the field was made, it would not impact the results because the surrounding fields are also cultivated with wheat crop with the same irrigation system (gravity) and at the same time.

L7 and L8 have different resolutions, a rescaling approach is needed. Since the spatial resolution of L7 thermal data (60 m) is much finer than spatial resolution of L8 thermal data (100 m), the 60 m TIR data were resampled linearly to 100 m resolution which is the lowest spatial resolution between L7 and L8. The 13 TIR data were then preprocessed to extract the LST. TIR data were converted to LST data after applying atmospheric correction and correction of surface emissivity. The followed steps are described in [START_REF] Tardy | A Software Tool for Atmospheric Correction and Surface Temperature Estimation of Landsat Infrared Thermal Data[END_REF] and the same processing chain was used in [START_REF] Amazirh | Including Sentinel-1 radar data to improve the disaggregation of MODIS land surface temperature data[END_REF][START_REF] Amazirh | Retrieving surface soil moisture at high spatio-temporal resolution from a synergy between Sentinel-1 radar and Landsat thermal data: A study case over bare soil[END_REF]. The different correction steps defined in [START_REF] Tardy | A Software Tool for Atmospheric Correction and Surface Temperature Estimation of Landsat Infrared Thermal Data[END_REF] allow converting the Landsat digital number to top of atmosphere radiance to top of canopy radiance then, to the physical LST at the surface. An atmospheric correction of the thermal infrared bands' data was firstly carried out using the MODTRAN atmospheric radiative transfer model software. The atmospheric profile composition (vertical air temperature and water content) needed as input to the model has been obtained from the ECMWF European Reanalysis (ERA) Interim product [START_REF] Dee | The ERA -Interim reanalysis: Configuration and performance of the data assimilation system[END_REF]. As the second step, the at-sensor radiance was converted into surface radiance using the estimated surface emissivity. Then the LST was obtained by inverting Plank's law.

SMAP

SMAP mission was launched in January 2015. This satellite was launched by the National Aeronautics and Space Administration (NASA) with an L-band passive sensor onboard [START_REF] Entekhabi | SMAP Handbook-Soil Moisture Active Passive: Mapping Soil Moisture and Freezs/Thaw from Space[END_REF]. SMAP satellite [START_REF] Entekhabi | SMAP Handbook-Soil Moisture Active Passive: Mapping Soil Moisture and Freezs/Thaw from Space[END_REF] is the first L-band mission that combines both radiometer (passive) and radar (active) data dedicated to the study of soil moisture at a range of resolutions from 3 km (active) to 36 km (passive) with a revisit cycle of 2-3 days. It orbits at an altitude of 658 km and provides SM from both descending and ascending overpasses. Despite the high frequency of SMAP data acquisition, the spatial resolution is very low which limits their application, especially to agricultural fields. To fill the gap, many disaggregation methodologies were developed to provide SM at high spatial resolution [START_REF] Abbaszadeh | Downscaling SMAP Radiometer Soil Moisture Over the CONUS Using an Ensemble Learning Method[END_REF]Peng et al., 2017). [START_REF] Ojha | Stepwise Disaggregation of SMAP Soil Moisture at 100 m Resolution Using Landsat-7/8 Data and a Varying Intermediate Resolution[END_REF] (2018) during the 2015-2016 season. The disaggregation procedure was tested on 7 dates at Landsat overpass against in situ SM. The evaluation of disaggregated SM was done based on statistical results in terms of the correlation coefficient (R), the slope of the linear regression, absolute mean bias, and root mean square difference (RMSD). The results show good results in terms of R which is ranged between 0.6 and 0.9 with a relatively low absolute bias lower than 0.03 m 3 /m 3 . This approach developed by [START_REF] Ojha | Stepwise Disaggregation of SMAP Soil Moisture at 100 m Resolution Using Landsat-7/8 Data and a Varying Intermediate Resolution[END_REF], was applied during this work to 36 km-SMAP resolution data products, and the disaggregated data set was used in this study. In this study, only 9 available dates were selected on the Landsat overpass, due to in situ data gaps.

Another issue is that the evaluation was assessed from December 27, 2015 to January June 01, 2016 (winter wheat growth season) which limit the number of satellite overpasses. In addition, the disaggregated SM products include data gaps, which on some dates unfortunately covered the wheat site. + VNIR stands for Visible-near infrared. × TIR stands for Thermal infrared. B stands for Band.

Methodologies

FAO-56: Dual crop coefficient

FAO-56 [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF] was extensively used to retrieve ET and irrigation scheduling. This is due to its operationality and its simplicity of implementation. The FAO dual crop coefficient (FAO-dualKc) comes to improve the ET estimates by partitioning ET into soil Es and plant Tc using reduction evaporation coefficient (Ke) and basal crop coefficient (Kcb), respectively, from standard evapotranspiration (ET0). Under well-watered condition ETc is written as:

ET c = (K cb + K e )ET 0 (1)
ET0 being the ET rate over a well-watered crop land covered by a short green, grass-like crop (reference ET), depending only on atmospheric conditions (ASCE-EWRI, 2004). Equation ( 1)

depends only on atmospheric conditions and is adapted to non-hydric stressed crops (standard conditions). To take into account the crop water stress and to catch the real ET (ETc act), a water stress coefficient (Ks) was introduced into Equation ( 1) and it becomes:

ET c act = ET FAO = K cb K s ET 0 + K e ET 0 = T c act + E s (2)
Where ET c act ≡ ETFAO is the actual crop evapotranspiration under non-standard conditions. Also, the stages durations are changing from one year to another due to its dependence on the thermal accumulation. The Kcb parameter was decomposed into Kcb ini, Kcb mid, and Kcb end. The Kcb dev is the Kcb during the development stage and is estimated using linear interpolation between Kcb ini, and Kcb mid. The Kcb was interpolated at the daily scale between the start and end value (Kcb ini, Kcb mid, Kcb end) of each phenological phase as in the FAO-56 documentation.

Kcb values were adjusted in each growing stage using climatic data. Ke is the evaporation coefficient which is calculated as:

K e = min{[K r (K c max -K cb )], [f ew K c max ]} (3)
where Kc max is the maximum value of crop coefficient (following rain or irrigation), ranged from 1.05 to 1.30 and estimated using meteorological data [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF]. few is the fraction of exposed and wetted soil from which most Es occurs. It is calculated as a function of fraction cover derived from NDVI as suggested by [START_REF] Er-Raki | Combining FAO-56 model and ground-based remote sensing to estimate water consumptions of wheat crops in a semi-arid region[END_REF]. Kr is the reduction coefficient of evaporation. This coefficient depends on the amount of water that can be depleted by Es during a complete drying cycle (TEW) and the cumulative depth of water depleted at the end of day i-1 (De,i-1). It is calculated as:

K r,FAO = TEW-D e,i-1 TEW-REW ; De,i-1>REW (4) K r,FAO = 1 ; De,i-1<REW
where REW is the readily evaporated water, which is the maximum depth of water that can be easily evaporated without restriction. REW (mm) depends on soil properties. FAO provides REW value for each soil texture [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF] ranging from 5 to 12 mm. To estimate the maximum amount of water that can be evaporated (TEW), the top layer depth that can be evaporated (Ze), the soil moisture at field capacity (SM FC ), and the soil moisture at wilting point (SM WP ) are required. TEW is calculated as:

TEW = 1000(SM FC -0.5SM WP )Z e (5)
SM FC and SM WP are reported for each soil texture classification in FAO-56.

A daily water balance is required to estimate the cumulative depth of Es (depletion) from the soil surface layer (De). A simplification of the water balance equation was established, where runoff flux is ignored when working on a flat area.

D e,i = D e,i-1 -P i -(

I i f w ) + E s,i f ew + DP e,i (6)
where Pi is precipitation input (mm) on day i and Ii is irrigation measured for each site (mm) on day i, Es,i is soil evaporation (mm) on day i, calculated as Ke × ET 0 [START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage[END_REF][START_REF] Bos | Water Requirements for Irrigation and the Environment[END_REF] and fw is the fraction of soil surface wetted by irrigation. In case of irrigation event or precipitation more than 3 mm/day, fw is taken equal to 1, otherwise, fw is set to zero. In this study, fields are irrigated by flood-irrigation technique, therefore, fw set to 1 during irrigation event. The DPe,i is deep percolation loss (downward drainage) from the topsoil layer on day i (mm). Deep percolation occurs when soil moisture exceeds field capacity following heavy rain or irrigation. The DPe,i equation assumes that water content is at field capacity so that De,i set to zero and the Es process switch off on the day of a complete wetting event. From equation ( 6), DPe,i then is calculated as follow:

DP e,i = P i + (

I i f w ) -D e,i-1 (7)
The procedure to estimate the transpiration reduction factor Ks [0-1] is similar to Kr. Ks is calculated based on a daily computation of the water balance for the root-zone layer Zr (m) which is written as follow:

K s,FAO = TAW-D r TAW-RAW = TAW-D r (1-p)TAW (8)
When Dr > RAW the stress is presumed to start, and Ks is calculated using Equation (8).

Conversely, when Dr ≤ RAW then Ks is equal to 1.

Dr is the root zone depletion (mm), TAW is total available water in the root zone [mm], RAW is readily available soil water of the root zone (p*TAW) and p is the depletion fraction for no stress which is a fraction of TAW that a crop can extract from the root zone without causing water stress. The TAW is linked to the root depth and to the difference between the water content at field capacity and wilting point as expressed in Equation ( 9):

TAW = 1000(SM FC -SM WP )Z r (9)

Assimilation approaches

Through assimilation, we seek to improve first daily soil Es by integrating observed and remotely sensed SM, then integrating observed and remotely sensed LST into the Tc act rate.

Figure 2 presents an illustration of how the correction is performed using sequential assimilation to improve state variables using observations to correct the model. The state variables in our case are the soil Es and the plant Tc act. The used sequential assimilation approach is based on generating an ensemble of perturbations, to obtain the forecast error covariance information required by the standard Kalman filter update equation.

Figure 2: Sequential data assimilation principle to obtain corrected value (analysis) by

updating model using observation [START_REF] Stanev | Methods of data assimilation[END_REF].

Two assimilation methods were used, the first one, by assimilating SM into FAO-dualKc soil Es component and the second, by assimilating jointly SM and LST into FAO-dualKc soil Es and wheat Tc act. Both approaches are detailed below.

Method 1: SM only

As a first step, SM was assimilated into the FAO model to improve the Es component estimates.

The data assimilation scheme is used to correct the depletion depth De used by the FAO-dualKc by using first, in situ and then remote sensing SM at field scale integrated into Ke equation.

Data assimilation is based on the use of an optimal and simple linear scheme to sequentially assimilate Ke as proposed by [START_REF] Schuurmans | Assimilation of remotely sensed latent heat flux in a distributed hydrological model[END_REF] and used by [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF]. This method has been chosen due to its simplicity and to keep FAO method operational. The assimilated Ke is written as: K e,New = K e,FAO + 𝐊 𝐞𝐯𝐚𝐩 (K e,SM -K e,FAO )

where Kevap is the Kalman gain that minimizes the analyzed error covariance, calculated as:

𝐊 𝐞𝐯𝐚𝐩 =
K s,New = K s,FAO + 𝐊 𝐬𝐭𝐫𝐞𝐬𝐬 (K s,LST -K s,FAO ) (13)
where the 𝐊 𝐬𝐭𝐫𝐞𝐬𝐬 is the Kalman gain calculated as equation:

𝐊 𝐬𝐭𝐫𝐞𝐬𝐬 = σ Ks,FAO 2 σ Ks,FAO 2 +σ Ks,LST 2 (14)
Ks,FAO is the stress coefficient calculated by water balance in the root zone layer, While Ks,LST is estimated as a proxy of LST, where the LST is normalized by its maximal and minimal values as in [START_REF] Idso | Normalizing the stress-degree-day parameter for environmental variability[END_REF].

K s,LST = LST proxy = (LST-T a ) max -(LST-T a ) (LST-T a ) max -(LST-T a ) min (15)
where LST -Ta is the difference between surface temperature and air temperature; (LST-Ta)min is the lower limit of (LST -Ta) of a surface which is transpiring at the potential rate, and

(LST-Ta)max the expected differential in the case of a non-transpiring crop.

An overview of the methodologies using remote sensing data is represented in the flowcharts, Figure 3. Standard FAO-dualKc is also presented in (a).

Results and discussions

Remote sensing data analysis

Landsat against in situ LST

As an important step for investigating the use of LST data for the assimilation approach, it is essential to investigate the relevance between the Landsat LST data and in situ measurements.

In this section a comparison between the 100 m Landsat LST and in situ TIR measurements was performed. In Figure 4, Landsat LST is plotted against in situ LST data in order to check the reliability of the remote sensed Landsat LST. The comparison between both temperatures showed good agreement, with a determination coefficient equal to 0.95 and an error (RMSE) equal to 2.16 °C. The results are slightly consistent with the one obtained in [START_REF] Amazirh | Retrieving surface soil moisture at high spatio-temporal resolution from a synergy between Sentinel-1 radar and Landsat thermal data: A study case over bare soil[END_REF], where they found an error close to 3 °C, over a field near the study field, by applying the same algorithm used in this work. Landsat overestimates slightly LST. This is due to the difference in the spatial extent of remotely sensed and in situ observations, where the Apogee radiometer was installed at a 2-m height, which limits the spatial representativeness of its measurements. The observed underestimation is quite similar for both L7 and L8 compared to in situ measurements. An RMSE of 2.33 °C and MBE of -1.30 °C were obtained when comparing L-7 and in situ measurements, while when comparing L8 against in situ LST, an RMSE of 1.98 °C and MBE of -1.18 °C were obtained.

This allows, to combine the both satellites to benefit from the temporal resolution of both satellites.

TDR against disaggregated-SMAP soil moisture

Before using the 100 m resolution SMAP disaggregated SM (SMDISPATCH), it is important to check their reliability and quality. To this end, an evaluation against in situ SM over the F16 study field was performed. The comparison results are illustrated in Figure 5. that could lead to differences between the disaggregated and in situ SM is that the ground SM sensors are buried at a depth of 5 cm, while the penetration of the L-band wave varies between 2 and 5 cm depending on soil conditions (eg. soil texture).

In situ evaluation of the assimilation approaches

As an assessment of the assimilation methodologies, this procedure was tested over both B123 and F16 sites. In situ SM and LST data are available over both sites. The first step was to test the effect of near-surface SM assimilation alone into FAO-dualKc Es component. The second step was to assess the performance of coupled assimilation of both SM and LST variables into FAO-dualKc ETc act. Assimilating LST alone into the FAO model was also tested over both sites. Results (results not shown) indicate that assimilating LST only, slightly improves the ETc act estimations (an error difference of 0.92 mm/day) and an R 2 equal to 0.64 with and an MBE less than 0.57 m/day for the F16 site. The slightly improvement of ETc act estimates using LST only is due to the fact that LST data is representing the soil and wheat temperatures and a partition of LST is needed to incorporate LST data into FAO model. Otherwise, assimilating LST data into FAO single approach coefficient could improve the ETc act results significantly as found in [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF].

Figure 6 presents the daily temporal comparison of the ETc act estimates using the assimilation approaches and the observed ETc act. The statistical results of the assimilation approaches and the standard FAO-dualKc are presented in Table 2. The ETc act estimates are evaluated by comparing them with the measured ETc act by the Eddy covariance system from both wheat sites (B123 and F16). Ground-based SM and LST data are available at a 30 min time scale, which allows us to choose the number of assimilation dates without restrictions. In our case, an 8-day frequency was used, which corresponds to the Landsat revisit time-frequency over our study sites. First only SM data are assimilated (SM-only) then SM and LST couple are assimilated. From Table 2 the RMSE between measured and estimated ETc act using standard FAO-dualKc (no assimilation) for the F16 and B123 sites are, 0.98 and 1.01 mm/day, respectively. As seen in Figure 6, without assimilation, standard FAO-dualKc severely underestimates ETc act (see the MBE values in Table 2). The assimilation of surface soil moisture alone is significantly improving the estimate of soil Es component, which ameliorates ETc act estimates at the field scale. Without and with SM assimilation, the RMSE decreases from 0.98 and 1.01 to 0.65 mm/day and 0.73 mm/day, and the R 2 increases from 0.63 and 0.72 to 0.85 and 0.77 for F16

and B123, respectively. The classical FAO-dualKc is based on an estimated SM from a simplified water balance budget (Equation 6), where some water fluxes were neglected in our case (runoff, deep percolation, and capillarity rise…). Where runoff flux takes place above a certain threshold value of soil moisture present in the soil (Manabe, 1969), which can generate some uncertainties. The underestimation of ETc act is observed at the beginning of the season, during the germination and emergence stage of growing crops and at the end (senescent stage) of the agricultural season, due to the limitation of the standard FAO-56 model in the estimation of the Es component where water is lost mainly by Es.

These results are in accordance with several works [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF][START_REF] Er-Raki | Combining FAO-56 model and ground-based remote sensing to estimate water consumptions of wheat crops in a semi-arid region[END_REF][START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF][START_REF] Rafi | Partitioning evapotranspiration of a drip-irrigated wheat crop: Inter-comparing eddy covariance-, sap flow-, lysimeter-and FAO-based methods[END_REF]. Realistic dynamics of SM allow reflecting correctly the water budget closure to catch the variability of soil Es. The better results obtained when assimilating SM into FAO-dualKc confirm this improvement where the integration of SM data allows better control of the model variation, and by assimilation, we tend to adjust soil water content and calibrating FAO-dualKc model. In the classical FAO, the evaporation coefficient Ke is linked mainly to soil texture based on a standard formulation using SM data (ad-hoc relationship with soil texture). This could provide a limitation to some extent in the Es estimation over bare soil conditions (under/overestimation). By assimilating SM, we tend to correct this issue by adjusting the soil evaporation coefficient, based on a relationship between SM and Es process taking into account soil types and surface conditions as in [START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF][START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF].

Results of the coupled assimilation of SM and LST into FAO-dualKc for both sites are also given in Table 2. As seen from Table 2, combining LST and SM assimilation schemes provides good results in comparison with standard FAO-dualKc. The RMSE and the bias are reduced from 0.98 mm/day and -0.47 mm/day to 0.55 mm/day and -0.14 mm/day while R 2 increases from 0.63 to 0.87 using FAO-dualKc without assimilation and with assimilation of both SM and LST-proxy conjointly, for F16 site, respectively. These results reveal that the best statistical parameters are provided by jointly assimilating LSTproxy and SM data. Looking at the results, for B123 site, it is noticed that the best results are provided when assimilating SM and LST couple with an RMSE of 0.71 mm/day and MBE equal to 0.03 mm/day. While before assimilation, the RMSE and MBE values for the ETc act were 1.01 and -0.44 mm/day for the B123 field, respectively. In the case of assimilating SM and LST couple, the effectiveness of the assimilation process is mainly dominated by SM assimilation. Overall, the assimilation approaches provide better results than the standard FAO-56. For F16 sites as an example, the main discrepancies between ETc act estimates before and after assimilation can be observed during the development (between December 2015 to January 31, 2016) and late (from 10 May to the end of the season) stages due to great differences in Ke estimates and thus in Es. Late in the season, a difference is observed between Es estimates before and after the assimilation procedure. In the standard FAO which is based on daily water balance, the water in the surface evaporable layer is fully depleted means Ke,FAO ~ 0, and Es≈0.. Whereas, when assimilating SM, we tend to update the Ke, and then the evaporation increases to catch better the EC observations.

The increase in Es can be explained by an increase of the sun-exposed soil due to the reduction of vegetation and the capillary rise from the root zone, which can be detected from the SM assimilated into the Es estimates. Moreover, the underestimation of Es by FAO-56 could be explained by the definition of the depth of Ze, where a 5 cm SM was used in this study to be consistent with the remote sensing depth. As FAO suggested, Ze should be ranging from 0.10-0.15 cm based on the soil texture and this could affect the final result of Es estimation as shown previously in [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF].

Assimilating remote sensing data into FAO-dualKc

To evaluate the spatial extensibility of the assimilation scheme, the daily ETc act over the selected F16 field during 2015-2016 was simulated at a spatial resolution of 100 m. Only F16

was chosen for the spatial analysis due to the availability of thermal data at high spatio-temporal resolution as Landsat 7 and 8 data as well as SM data derived from SMAP. In this case, SM data was derived from SMAP data product, which were disaggregated to 100 m [START_REF] Ojha | Stepwise Disaggregation of SMAP Soil Moisture at 100 m Resolution Using Landsat-7/8 Data and a Varying Intermediate Resolution[END_REF], and LST data were derived from Landsat-7 and Landsat-8 data. As the in situ evaluation, two assimilation techniques were tested: SM only and assimilating jointly SM and LSTproxy couple into FAO model. Figures 7 and8 present the scatter and time series plots of the obtained results using the assimilation techniques, respectively. From the obtained results in Figures 7 and8, the assimilation approaches tend to improve the ETc act estimates either using the proposed assimilation approaches. In the first case, the SMDISPATCH products are only assimilated into FAO-dualKc. Figures 7 and8 compare the results of the assimilation procedure with the one from the open-loop ETFAO (no assimilation, Figure 7a and Figure 8 'red line'). When assimilating the SMAP based disaggregated SMDISPATCH (Figure 7b and Figure 8 'green line'), the ETc act estimates is improved where the error is reduced compared to ETFAO from 0.98 mm/day to 0.75 mm/day, as well as the bias decreased from -0.47 mm/day to -0.06 mm/day. For the other case, when coupled assimilation of SMDISPATCH and LST through using the normalized LST proxy, the RMSE is equal to 0.73 mm/day with a bias that doesn't exceed 0.15 mm/day. Comparing both assimilation approaches, the lowest RMSE is assessed by assimilating both SM and LST into FAO model (Figure 7c and Figure 8 'blue line'). Without assimilation (ETFAO), FAO-dualKc severely underestimates ETc act during the senescence period (Figure 7a), where both assimilation experiments have slightly similar accuracy in ETc act estimates. This is seen especially at the beginning and the end of the wheat growing season (Figure 8), where the SM improves the soil Es when soil is under bare soil conditions.

We note that the assimilation using in situ data provided better and more accurate results than using remote sensing ones. This is due first to the data, where in situ data are the truth observations while the remotely sensed ones are estimated from model (Dispatch for SM) or corrected (MODTRAN for LST) which generate a cumulated error. In this study a proxy of LST was used, therefore the observed error of 2.16 °C when comparing Landsat LST and in situ LST may less impact the ETc act results since a relative value of LST data (between 0-1) was used. Then the frequency of the in situ perturbations (observations) is higher than the remote sensing data (cloud effects and satellite revisit time). This could also impact the results, where the higher assimilation frequency (sequence), the better the ETc act estimates are, as found As seen from Figure 9, Ke ranges from 0 to 1.2 for both cases. Nine available dates are used for the SM assimilation when SMAP SMDISPATCH data are available. For dry conditions (absence of irrigation or precipitation, e.g. from the 15 th to 31 st of January), an increase in the surface depletion leads to a decrease of both Ke and an increase of Es surface resistance, which induces water stress at the surface. While, after an irrigation or rainfall, soil evaporates at a potential rate when the soil resistance is decreasing, which leads to an increase of Ke. From Figure 9, the daily pattern of Ke,FAO, and Ke,New are similar and responds perfectly to water supply (precipitation or irrigation). Despite that, the assimilated Ke,New is more physical which means it increased and decreased progressively, especially during the dryness period. This clearly appears at the 22 nd and 11 th of January, where the Ke,FAO decreased sharply and suddenly than the Ke,New that decreased gradually and slowly. It can be concluded that SM assimilation tends to adjust the duration of the dryness phenomena when the water content in the upper soil becomes limiting, taken into account the soil properties as shown in [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF].

Especially in stage 2 when soil moisture is the limiting factor, the Ke,New decreases progressively depending on the soil properties, this because of the response and the soil drying time which is well taken into account in the Ke,New approach. For the coupled assimilation, the Ke,New was taken to update the Es component, while the introduction of LST data will update the stress coefficient Ks. Figure 10 shows the time variation comparison between the estimated Ks by FAO-56 model with assimilation (Ks,New) and without assimilation (Ks,FAO). The Ks,SM (black diamonds) and Ks,T (pink diamonds ) are presented indicating the SM and LST assimilation dates, respectively.

Looking at Figure 10, it can be seen that the Ks,FAO, and Ks,New follow the same variation and in some cases are superposed. Ks reached the value 1 when the Dr is lower than RAW, these cases appeared when the water is supplied by rain or irrigation (Figure 10). When the water supply is insufficient or ET0 is very high, the root zone depletion increased, and the Ks value dropped below the value 1, which generated wheat stress. The removal of water by ETc act leads to activation and an increasing in wheat stomatal resistance as well as an increasing in soil water depletion, and this induced water stress conditions. Ks,New simulates stress conditions in a more pronounced way than the classical FAO-dualKc. Both Ks before and after assimilation responded to water supply and reach the value 1. The main discrepancies between both methods are observed when Ks dropped below 1, where the Ks,FAO abruptly decreased while Ks,New decreased progressively.

As seen, the flood-irrigated field (F16) was irrigated every 1 to 3 weeks depending on precipitation and a sum of 8 irrigations were observed. Generally, the field was well irrigated and under minimal stress. This makes quantifying the impact of assimilating LST into the FAO-ETc act estimation difficult. However, the stress periods were detected when assimilating LST into Ks.

Conclusions

The investigated work deals with improving actual crop evapotranspiration (ETc act) estimates using data assimilation techniques. The idea was to improve the accuracy of the FAO-dualKc ET components through assimilating remote sensing data. Surface soil moisture (SM) data is assimilated in order to update soil evaporation and land surface temperature LST assimilation, the ETc act estimates are slightly improved compared to assimilating only SM data. However, some uncertainties were observed, and this could be due to the sensing depth of SM data. The assimilation technique has demonstrated the potential of inferring valuable information from remotely sensed land surface data (SM and LST) for evaluating distributed water balance models, given that forcing data (including irrigation) are accurately known. The proposed study showed its capability to retrieve ETc act at field scale, a spatialization of ETc act is needed. This could be done by using LST and SM products at high resolution (Amazirh et al., 2019;[START_REF] Ojha | Stepwise Disaggregation of SMAP Soil Moisture at 100 m Resolution Using Landsat-7/8 Data and a Varying Intermediate Resolution[END_REF]. Recently, [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF] proposed a new Es formulation by including the soil texture information into the reduction coefficient (Kr). The proposed formulation demonstrates its better performance than the classical FAO. Combining the assimilation of SM into Es formulation developed in [START_REF] Amazirh | Implementing a new texture-based soil evaporation reduction coefficient in the FAO dual crop coefficient method[END_REF] could provide more accurate results by improving the Es component.

Figure 1 :

 1 Figure 1: Location of the study area R3 (red rectangle). Study fields are illustrated with blue rectangles.

  used DISPATCH (DISaggregation based on Physical And Theoretical scale Change) method to disaggregate 36 km-SMAP SM resolution to 100 m resolution. All the followed disaggregation steps were described in Ojha et al. (2019). Three steps were followed to downscale the 36 km SMAP SM to 100 m. The 36 km SM is first disaggregated to 1 km resolution using MODIS LST and NDVI (DISPATCH-1 km). The 1 km resolution disaggregated data is next aggregated to the intermediate spatial resolution, and then further disaggregated to 100 m resolution using Landsat-derived LST and NDVI (DISPATCH-100 m). The disaggregation approaches are based on thermal-derived soil evaporative efficiency (SEE) which is a normalization of LST data by its maximum and minimum values derived from the energy balance model. The disaggregated SMDISPATCH products were evaluated over 22 irrigated fields in the same used area. The in situ SM measurements have been collected and calibrated as in Amazirh et al.

*

  SMAP data were disaggregated from 36 km to 100 m resolution.

  Kcb, Ke, and Ks are basal crop Tc act coefficient under non-standard conditions, soil evaporation, and water stress coefficients, respectively. The Kcb values are defined for each crop growth stage. The wheat growth stages are distinct by their lengths (initial (Lini), development (Ldev), mid-season (Lmid), and late (Llate)). Lengths of growth stages are usually calculated based on the fractional vegetation cover (fc) derived from normalized difference vegetation index (NDVI) as in Er-raki et al. (2008), Er-Raki et al. (2007), Olivera-Guerra et al. (2018) and Rafi et al. (2019). During the initial stage, Kcb named Kcb ini, mid-season named Kcb mid, and at the end of the growing season, Kcb named Kcb end where the values were taken from Allen et al. (1998). Local calibration of the FAO model has been stressed in this work, by adjusting the default values of lengths of growth stages reported in FAO based on the different equations provided in FAO-56 for adjusting the Kcb values. Lengths of growth stages significantly differ from the FAO-56 default values, especially for the duration of phenological phases. However, the calibrated lengths of growth stages were in accordance with the NDVI measurements carried out on the site. Note that, the lengths of the growth stage reported in FAO-56 are informative values which can only be used in locations with no data.

Figure 3 :

 3 Figure 3: Schematic diagrams presenting an overview of the main inputs, models and outputs of the assimilation of soil moisture into FAO-dualKc (a) and SM & LST conjointly (b).

Figure 4 :

 4 Figure 4: Land surface temperature derived from Landsat-7 (L7) and -8 (L8) versus in situ (In situ LST) measurements for F16 field. Line 1:1 (dotted blue line) and regression line (black solid line) were also presented.

Figure 5 :

 5 Figure 5: Disaggregated SMAP-soil moisture (SMDISPATCH) against in situ soil moisture (In situ SM) over the F16 field. Line 1:1 (dotted blue line) and regression line (black solid line) were also presented.

Figure 6 :

 6 Figure 6: Daily temporal comparison between FAO-dualKc and observed ETc act without assimilation (ETFAO) and when assimilating SM-only and when assimilating SM and LSTproxy couple for B123 (top) and F16 (bottom) sites.

Figure 7 :

 7 Figure 7: Scatter plot comparison between FAO-dualKc and observed ETc act (ETobs) without assimilation 'ETFAO' (a), when assimilating only soil moisture 'SM-only' (b) and conjointly soil moisture and land surface temperature 'SM and LST' (c).

Figure 8 :

 8 Figure 8: Daily temporal comparison between FAO-dualKc and observed ETc act (ETobs) without assimilation and when assimilating SM-only (SMNew (SM-only)) and conjointly SM and LST-proxy (SMNew (SM + LSTproxy)). The ETSM is presented also (yellow hexagram) indicating the assimilation dates.

in

  [START_REF] Er-Raki | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF] and[START_REF] Fu | Analysis of influence of observation operator on sequential data assimilation through soil temperature simulation with common land model[END_REF]. The assimilation of SM allows the updating of the surface depletion (De), thus controlling the water budget at the surface then updating the Ke coefficient of evaporation. By integrating the LST information, this allows the updating of the root zone depletion Dr, which controls the temporal course of the root zone water budget, which leads to correcting the Ks values. The idea of assimilating LST into the stress coefficient is to improve the transpiration component. Both sites were well irrigated, and the stress appears just in some periods during the investigated study. The real impact of LST on the water balance at the root zone could be seen clearly in the case of a field that undergoes several stress periods when irrigation is deliberately cut. In this study, just on some dates that the field is undergoing stress period and by assimilating LST into FAO model we succeeded to capture the stress periods. To better observe the difference on the assimilated Ke,SM and the surface water balance based Ke,FAO, Figure9illustrates the temporal course comparison between the estimated Ke by FAO-dualKc model without and with SM assimilation during 2015-2016 growing season for F16 field.

Figure 9 :

 9 Figure 9: Comparison between the FAO-dualKc evaporation coefficient Ke,FAO before assimilation (red line) and the updated one (Ke,New) after assimilation (blue line). The Ke,SM is

Figure 10 :

 10 Figure 10: Comparison between the FAO-dualKc stress coefficient Ks,FAO before assimilation (red line) and the updated one (Ks,New) after assimilating LSTproxy and SM couple (blue line).

  (LST) is assimilated to update the crop wheat transpiration (Tc act). In situ SM and LST data are firstly assimilated at the point scale. Then SMAP disaggregated SM and Landsat derived LST was used at field scale. The investigated techniques were tested over two different fields in central Morocco, during the 2002-2003 and 2015-2016 wheat growing seasons, respectively. Our results point out that by assimilating only the SM data, we yield a very satisfying estimation of ETc act. The modelled ETc act tracks successfully the ETc act observation. By coupling SM and

Table 1

 1 lists the characteristics and the selected dates of L7/8 and SMAP overpasses used in this work.

Table 1 :

 1 Technical characteristics of satellite products and the selected dates collected in clear sky conditions over the study site.

Method 2: Combined assimilation of SM and Land Surface Temperature FAO

  e,FAO and K e,New are the estimated Ke by the FAO-dualKc approach before and after data assimilation, respectively. While K e,SM is the Ke estimated from SM observations derived from in situ or SMAP disaggregated data. The Ke,SM is calculated as Equation (3) except that the De equation is forced by SM data instead of using the water balance model at the surface. The De equation using SM information is written as follow:

	𝜎 𝐾𝑒,𝐹𝐴𝑂 2	and 𝜎 𝐾𝑒,𝑆𝑀 2	are the variances of predicted errors on Ke estimates from FAO-dualKc
	model and SM observations, respectively.
			D e = 1000(SM FC -SM 𝑜𝑏𝑠 )Z e	(12)
	3.2.2.
			2 σ Ke,FAO σ Ke,FAO 2 +σ Ke,SM 2	(11)

K -dualKc model is based on water balance model to estimate Ks. In order to improve wheat crop Tc act estimates, surface temperature can be used to reflect the crop water status by calculating stress index for the root-zone layer. The used assimilation scheme is the same used for assimilating SM (sequential assimilation):

Table 2 :

 2 Statistical analysis of ETc act simulated by the FAO-dualKc before assimilation and after assimilations for both wheat fields.

	Statistics	R 2		RMSE (mm/day)	MBE (mm/day)
	Fields	B123	F16	B123	F16	B123	F16
	Methods
	ETFAO (no assimilation)	0.72 0.63	1.01	0.98	-0.44 -0.47
	SM-only	0.77 0.85	0.73	0.65	-0.14 -0.32
	SM+LSTproxy	0.77 0.87	0.71	0.55	0.03 -0.14

NOTE. ETFAO (no assimilation

): open loop FAO model. SM-only: assimilating only soil moisture data into FAO model. SM+LSTproxy: Assimilating conjointly soil moisture and Land surface temperature into FAO model.

Acknowledgements

This study was conducted within the Center of remote sensing applications (https://crsa.um6p.ma/), at the Mohammed VI university-Morocco and funded by OCP (AS No 71). This work was also partly funded by the European Commission Horizon 2020 Programme for Research and Innovation (H2020) in the context of the Marie Sklodowska-Curie Research and Innovation Staff Exchange (RISE) action (REC project, grant agreement no: 645642), followed by ACCWA project, grant agreement no. 823965). The in situ data set was provided by the Joint International Laboratory TREMA (http://trema.ucam.ac.ma). PRIMA IDEWA and ALTOS projects are also acknowledged.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.