Abdelhakim Amazirh
email: abdelhakim.amazirh@um6p.ma

Olivier Merlin

Salah Er-Raki

Elhoussaine Bouras

Abdelghani Chehbouni

Dr Amazirh Abdelhakim

Implementing a new texture

Keywords: FAO-2Kc, soil evaporation, soil texture, soil moisture, evapotranspiration

Introduction

Surface evapotranspiration (ET) is an important flux in water and energy exchange processes at the interface between land surfaces and the atmosphere. Soil evaporation (E) is one of the main components of ET beside plant transpiration. E accounts for a substantial part of ET from the exposed soil surface in growing crops, where its spatial distribution plays an important role in various fields such as hydrology, meteorology and agronomy. In addition, E is the only hydrological flow that connects both water (through soil moisture) and energy (through land surface temperature) balances. About 50 % to 70 % of annual precipitation is consumed by E to the atmosphere which is considered as a water loss and not used for crop productivity [START_REF] Harrold | Transpiration Evaluation of Corn Grown on a Plastic-Covered Lysimeter 1[END_REF][START_REF] Peters | Relative Magnitude of Evaporation and Transpiration 1[END_REF][START_REF] Wallace | Increasing agricultural water use efficiency to meet future food production[END_REF].

Partitioning ET is essential in modeling land atmosphere interactions and vegetation water uptake and it is crucial in the monitoring of plant water uptake and water stress (Er-raki et al., 2010;[START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF][START_REF] Porporato | Plants in water-controlled ecosystems: Active role in hydrologic processes and responce to water stress IV. Discussion of real cases[END_REF]Rafi et al., 2019). In agricultural crops, water is mainly lost by E during the germination and emergence stage of growing crops. Then, through the crop growth and development, the surface is covered; therefore, E impacts less the water change in the soil while the transpiration becomes the main process and dominates ET. In arid and semi-arid regions including irrigated areas, E is the main outward flux and influences the soil water budget, which is very vital for agriculture [START_REF] Suleiman | Modeling Soil Water Redistribution during Second-Stage Evaporation[END_REF]. [START_REF] Schlesinger | Transpiration in the global water cycle[END_REF] reported that the E contribution into ET ranges from 20 % to 40 %, and it is an essential boundary condition between soil and atmosphere.

It is well known that the E process occurs in two main stages. The first stage is characterized by a constant rate of E which is also known as the energy-limited stage. It corresponds to when the evaporation occurs at a maximum rate meaning that it is limited only by the external meteorological (radiation, wind speed and relative humidity) conditions. This stage continues as long as the soil underneath has enough water storage. The second stage, also named the falling rate stage, corresponds to when the surface dries so that the evaporation becomes limited by soil moisture ( and by the soil hydraulic properties that determine the transfer of liquid and vaporized water to the surface. In this stage the soil resistance to water diffusion increases until reaches its minimal value [START_REF] Chanzy | Significance of Soil Surface Moisture With Respect to Daily Bare Soil Evaporation[END_REF]. The rate of evaporation and the duration of both stages vary with soil texture. Therefore, more water can be evaporated from saturated clayey soil than from saturated sandy soils because more water is stored in a clayey soil than in a sandy soil.

Different techniques have been used to measure E. Among them, we cite weighing lysimetry [START_REF] Balwinder-Singh | The effect of rice straw mulch on evapotranspiration, transpiration and soil evaporation of irrigated wheat in Punjab, India[END_REF][START_REF] Boast | A "Micro-Lysimeter" Method for Determining Evaporation from Bare Soil: Description and Laboratory Evaluation[END_REF][START_REF] Leuning | Rainfall interception and evaporation from soil below a wheat canopy[END_REF], isotopy [START_REF] Aouade | Combining stable isotopes, Eddy Covariance system and meteorological measurements for partitioning evapotranspiration, of winter wheat, into soil evaporation and plant transpiration in a semi-arid region[END_REF][START_REF] Zhang | Evapotranspiration and its partitioning in an irrigated winter wheat field: A combined isotopic and micrometeorologic approach[END_REF] and portable evaporation chamber [START_REF] Luo | Portable canopy chamber measurements of evapotranspiration in corn, soybean, and reconstructed prairie[END_REF][START_REF] Raz-Yaseef | Effects of spatial variations in soil evaporation caused by tree shading on water flux partitioning in a semi-arid pine forest[END_REF]. Recently, Rafi et al. (2019) intercompared different methods based on eddy covariance, sap flow and weighing lysimetry measurements to assess E. The above techniques provided E measurements at a crop field scale.

The FAO dual crop coefficient model (FAO-2Kc, [START_REF] Allen | Using the FAO-56 dual crop coefficient method over an irrigated region as part of an evapotranspiration intercomparison study[END_REF] is one of the most widely used models to estimate crop ET due to its simplicity and operationality, and it has been applied in standard and beyond standard conditions [START_REF] Suleiman | Evaluation of FAO-56 crop coefficient procedures for deficit irrigation management of cotton in a humid climate[END_REF]. The FAO-2Kc consists in splitting the crop coefficient K c into two separate coefficients, one for crop transpiration, called the basal crop coefficient (K cb ) and one for soil evaporation (K e ). FAO-2Kc model requires limited input parameters and it has been used to simulate ET of crops like wheat (Drerup et al., 2017;Er-raki et al., 2010;Er-Raki et al., 2007;[START_REF] Jin | Validation of two Huanjing-1A/B satellite-based FAO-56 models for estimating winter wheat crop evapotranspiration during mid-season[END_REF], olive [START_REF] Chehbouni | Improvement of FAO-56 method for olive orchards through sequential assimilation of thermal infrared-based estimates of ET[END_REF]Er-Raki et al., 2010), citrus [START_REF] Rallo | Using field measurements and FAO-56 model to assess the eco-physiological response of citrus orchards under regulated deficit irrigation[END_REF], table grapes [START_REF] Rodriguez | Determination of crop evapotranspiration of table grapes in a semi-arid region of Northwest Mexico using multi-spectral vegetation index[END_REF], sugar beet [START_REF] Anderson | Assessing FAO-56 dual crop coefficients using eddy covariance flux partitioning[END_REF][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] crops grown under different climates [START_REF] Ayyoub | A simple and alternative approach based on reference evapotranspiration and leaf area index for estimating tree transpiration in semi-arid regions[END_REF][START_REF] Debnath | Sensitivity Analysis of FAO-56 Penman-Monteith Method for Different Agro-ecological Regions of India[END_REF].

Despite the numerous applications of FAO-2Kc, many studies reported that the E component is not well quantified. In some studies, the FAO-2Kc method overestimates E at the start of the season and underestimates it at the end of the season [START_REF] Mutziger | Comparison of measured and FAO-56 modeled evaporation from bare soil[END_REF][START_REF] Parlange | physical basis for time series model of soil water content[END_REF][START_REF] Torres | Bare soil evaporation under high evaporation demand: a proposed modification to the FAO-56 model[END_REF]. Rafi et al., (2019) and [START_REF] Olivera-Guerra | Estimating the water budget components of irrigated crops: Combining the FAO-56 dual crop coefficient with surface temperature and vegetation index data[END_REF] found an underestimation of E at the start and the end of the wheat growing season. This is potentially due to the classical K e or more specifically to the reduction coefficient K r of FAO-2Kc which is based on an ad hoc relationship with texture via the at field capacity [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF]. Moreover, [START_REF] Phillips | Using ARM Observations to Evaluate Climate Model Simulations of Land-Atmosphere Coupling on the U.S. Southern Great Plains[END_REF] stated that most land-surface models used to estimate E cannot differentiate soil texture and this is due to the difficulty to incorporate the soil texture effect on the E rate. [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] showed that the nonlinear relationship between E rate and surface over bare soils varied systematically with soil texture across several well-instrumented sites. Based on this work, a new K r formula was proposed based on the soil texture information using a new pedo-transfer function that physically links K r to texture (K r,text ). The new pedo-transfer function was developed in [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] using a meta-analysis approach and was recently corroborated by a physically-based modelling approach relying on soil hydrodynamical properties to simulate capillary flow [START_REF] Lehmann | Soil Texture Effects on Surface Resistance to Bare-Soil Evaporation[END_REF]). This gives independent insight and additional confidence on the robustness of the proposed pedo-transfer function.

The aim of the study is therefore to assess the usefulness of the pedo-transfer function of [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] within the FAO formalism using -for the first time -36 contrasted sites.

First, the new (K r,text ) and classical (K r,FAO ) formulations are tested over 33 bare (or approximately bare) soil sites with different soil textures and compared against E measurements. In this case, in situ soil moisture data are used as forcing (offline validation), by considering the evaporation module separately from the water budget model, meaning that the soil moisture forcing of Kr is measured. Note that among the 33 bare sites used to evaluate the FAO evaporation estimates in the offline mode, 32 sites belong to the calibration data set of [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] while the remaining site (MOBou) is used in this paper for the evaluation. Then, as a complementary assessment of the new evaporation formulation, crop ET was retrieved during the entire agricultural season. In this case, the simulated soil moisture (θ FAO ) is used as forcing to K r,FAO and K r,text (online validation) by considering the evaporation module coupled to the water budget model, meaning that the soil moisture forcing of Kr is simulated by the FAO water budget model. Both formulations of soil evaporation were evaluated in terms of total ET estimates over three wheat sites located in Morocco near Marrakech. Note that the 3 wheat sites (B123, EC1, EC2) was used in this paper to validate the online implementation of the soil evaporation reduction coefficient in the FAO model. All the three sites do not belong to the calibration data set of [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF].

Sites and data description

In this study, a significant number of sites (33 bare soil and 3 wheat sites) have been used and divided into two categories depending on the desired objectives: i) the 33 bare soil sites have been used for comparing K r,FAO and K r,text under bare soil conditions only, and ii) the 3 wheat plots have been used to evaluate the new evaporation module when coupled to the crop water budget during the entire growing season including all phenological stages of wheat. A description of all the sites is presented below. More details about all the 32 bare soil sites but MOBou can be found in [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF].

The studied 33 bare soil sites were differing in climatic conditions and soil textures (Table 1).

These sites are situated in 13 countries all over the world among which 8 are located over uncropped lands, containing the natural lands including sand desert (NIHAP), savanna fallow (NISav) and degraded land (NIDeg), native grass (AUStu), grass for silage or hay (USDk1), short grass following fireforest (USFwf), and sparse shrub (USMo1 and USMo7). The remaining 25 sites are located over agricultural fields where an identification of bare soil periods is performed. In this study, ''bare soil'' period is defined as a time period when the plant transpiration is either negligible or small compared to E. Hence the term ''bare soil'' includes both actual bare soil conditions, and soils partially covered by mulch, crop residue, or sparse vegetation. The data were collected from several national and international flux station networks, intensive field measurements and extracted during bare or quasi bare soil periods by [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF]. Table 1 indicates the localization of the different 33 bare soil sites, where most sites are located in Europe continent. In Table 1 , the clay and sand fractions are presented. The sand fraction ranges from 0.05-0.92 and clay fraction from 0.02-0.56. Such a wide observation range of sand and clay fraction allows for testing the methodologies over different soils. Additional soil physical properties are also given in Table 1 such as, the volumetric soil moisture at saturation ( , soil moisture at field capacity and soil moisture at wilting point ( ) as well as the readily evaporable water (REW). These parameter values are estimated with pedotransfer functions based on clay and sand fractions [START_REF] Brisson | A semi-empirical model of bare soil evaporation for crop simulation models[END_REF][START_REF] Cosby | A Statistical Exploration of the Relationships of Soil Moisture Characteristics to the Physical Properties of Soils[END_REF][START_REF] Noilhan | The ISBA land surface parameterisation scheme[END_REF]. In addition to soil texture, continuous measurements of energy fluxes had been monitored in each site such as net radiation (Rn), soil conduction (G) and sensible and latent heat fluxes (H) and (LE), respectively. Measured LE is corrected using the Bowen ratio [START_REF] Twine | Correcting eddy-covariance flux underestimates over a grassland[END_REF]. Note that, in some sites LE is missing. In this case, LE is estimated as the residual of the energy balance equation.

The three wheat crops used for evaluating E estimations by simulating total ET are located close to Marrakech, Morocco (Figure 1). This region is characterized by a semi-arid to arid climate with high reference ET and low precipitation. Agricultural areas are generally covered by winter wheat crops. The three winter wheat fields are comprised of one flood irrigation plot (B123) of 4 ha located in an irrigated agricultural zone (called R3) and two drip-irrigated fields, both being 1.5 ha and situated in Chichaoua region. Both regions are located in the Haouz plain. The area undergoes contrasted atmospheric conditions all over the year with high temperature and low rainfall (less than 250 mm/ year) during summer and low temperature and irregular rainfall from autumn to spring with a high evaporative demand (1600 mm/year). Wheat is the dominating crop (50 %) of the R3 zone, which is known for its heterogeneity and contains different types of land cover (alfalfa, wheat, olive, orange and al., 2018;[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][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] Khabba | The SudMed Program and the Joint International Laboratory TREMA: A Decade of Water Transfer Study in the Soil-Plant-Atmosphere System over Irrigated Crops in Semi-Arid Area[END_REF][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][START_REF] Olivera-Guerra | Irrigation retrieval from Landsat optical/thermal data integrated into a crop water balance model: A case study over winter wheat fields in a semi-arid region[END_REF]. The Chichaoua site has been monitored since 2016.

Both wheat sites in Chichaoua were sown in November 2016, with the same drip irrigation system. One of the plots (EC1) had been irrigated according to FAO crop water requirements and the other plot (EC2) had undergone stress in controlled conditions. Additional details and information about the study sites can be found 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], [START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF] and Rafi et al. (2019).

During the investigated agricultural seasons, all the three sites were monitored by an eddy covariance system, equipped with different sensors providing continuous measurements of energy and water fluxes exchanged between soil, vegetation and atmosphere. The installed hygrometer provided continuous measurements of H and LE fluxes. Additional instruments were installed in the tower providing extra measurements such as a Sonic 3D anemometer designed to measure the wind speed over the 3 orthogonal axes. R n was measured by the net radiometer Kipp and Zonen CNR4. Half-hourly or sub-hourly measurements of classical meteorological data composed of air temperature T a , wind speed u a , incoming solar radiation R g and air relative humidity rh a were collected at a reference height (2 m). Soil moisture data were also measured by Time Domain

Reflectometry sensors over the 3 sites at different soil depths (5, 10, 15, 30, 100 cm). In this work two surface layers were used: 5 and 10 cm. A mean value between the two layers has been used to assess soil moisture within the 0-10 cm soil column. For the sites where the soil moisture at 10 cm is unavailable and only soil moisture at 5 and 15 cm are available, a linearly interpolation as in [START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF] was used to estimate the integrated soil moisture from 0 to 10 cm depth.

Methodologies

Dual crop coefficient method FAO-56

Allen et al. (1998) describe with details the theoretical background to assess crop ET. The FAO-56 method is based on FAO-24 (Doorembos and Pruitt, 1975), where ET is estimated based on the crop coefficient (K c ) and the reference evapotranspiration (ET 0 ). [START_REF] Allen | Using the FAO-56 dual crop coefficient method over an irrigated region as part of an evapotranspiration intercomparison study[END_REF] and Allen et al. (2005bAllen et al. ( , 2005a[START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage paper 56[END_REF] incorporate the dual crop coefficient, splitting K c into crop and bare soil contributions. Using the dual crop coefficient method allows for partitioning ET into bare soil evaporation and plant transpiration beyond standard conditions using two coefficients: the soil water evaporation coefficient (K e ) to describe E from the soil surface and the basal crop coefficient (K cb ) to describe plant transpiration (Equation 1). The daily variation in the crop coefficient is a function of the wetness of the soil surface and plant development. The crop ET under well-watered conditions is written as [START_REF] Steduto | Aquacrop-the FAO crop model to simulate yield response to water: I. concepts and underlying principles[END_REF]:

(1)

with ET c being the maximal crop ET, ET 0 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. An adjustment is made to obtain the actual estimates of the real crop ET a by introducing the water stress coefficient K s into the Equation ( 1), and it becomes:

(

In this paper we aim to improve the representation of E in Equation (2). Under bare soil conditions, crop transpiration is equal to zero (K cb =0). E from Equation ( 3) is expressed as:

(3) Soil evaporative coefficient (K e ) is a function of the amount of water in the soil, soil properties and the exposed and wetted soil fraction to solar radiation where most E occurs (f ew ). K e is defined using Eq. 4:

(4)

where few = min (1-fc, fw) with fw being the fraction of the soil surface wetted by irrigation or rainfall (fw is 1 in case rainfall) and fc the fraction of surface covered by vegetation, and where K cmax is the maximum value (ranging from 1.05 to 1.30) of crop coefficient K c following rain or irrigation. It represents the upper limit on E and transpiration that reflects the natural constraints on available energy. For bare soil, K cb is again equal to 0 and K cmax is equal to 1.2 (Allen et al., 2005b[START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage paper 56[END_REF]. K e formula becomes:

(5) with K r being the reduction coefficient, which depends on the cumulative depth of water depleted (D e ) and the amount of water that can be depleted by E during a complete drying cycle (TEW). Following wetting by precipitation or irrigation, K r,FAO can be as high as 1.

When the soil surface is dry, K r,FAO is small and can even reach zero. E is presumed to take place in two stages and the reduction coefficient K r,FAO is expressed differently for both stages The reduction coefficient decreases and becomes smaller than 1 in proportion to the amount of water present in the surface soil layer. K r,FAO of Equation ( 5) is written as:

; D e >REW (6)

; D e <REW with REW (mm) depending on soil texture. FAO-56 provides a table for each soil type. REW normally ranges from 5 to 12 mm and is highest for medium and fine textured soils [START_REF] Ritchie | Model [or PredictingE vaporatio[nr oma Row Crop with Incomplete Cover[END_REF][START_REF] Ritchie | Soil and weather inputs for the IBSNAT crop models[END_REF]. D e is the cumulative depletion from the soil surface layer by evaporation (mm). The estimation of D e requires a daily water balance model of the topsoil layer. Otherwise soil moisture ( information can be used directly in the D e calculation as follows.

(7) TEW (mm) is the maximum depth of water evaporated from the surface layer:

(8

)
where Ze is the effective depth of soil in which the atmosphere can extract water by E.

, and REW are reported for each site in table 1 depending on the soil texture classification. For Ze, FAO-56 recommended a value of 10 cm for coarse soils and 15 cm for fine textured soils [START_REF] Hanks | Modeling crop response to irrigation in relation to soils, climate and salinity[END_REF][START_REF] Ritchie | Model [or PredictingE vaporatio[nr oma Row Crop with Incomplete Cover[END_REF][START_REF] Wright | New evapotranspiration crop coefficients[END_REF]. The values of TEW, Z e and REW are important factors that control and influence the E process.

Step calculation of the classical Kr FAO evaporation model is presented in the flowchart (Figure 3).

New texture-based approach

The FAO-56 classical E method (Kr FAO ) is sensitive to the E depth. Thus, the FAO-2Kc E equation uses a given soil thickness. In the FAO-2Kc, they recommended a value of Z e of 0.1 to 0.15 m which is suitable for the water balance model. where is the surface soil moisture, is taken equal to soil moisture at saturation by contrast of the work of [START_REF] Jacquemin | Sensitivity study and validation of a land surface parameterization using the HAPEX-MOBILHY data set[END_REF] and [START_REF] Lee | Estimating the soil surface specific humidity[END_REF], and P is a semi-empirical parameter depending on the soil moisture sensing depth and soil proprieties (texture). [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] proposed an empirical Pedo-transfer (PTF) equation of the critical soil moisture at which the actual E rate is half the potential E rate. In practice was correlated with soil texture (clay and sand content) with a good correlation coefficient (0.6 for sand fraction and 0.8 for clay fraction). A negative correlation was found between and sand fraction while it increases as a function of clay fraction. To benefit from the complementary information on soil water retention capacity, a multilinear regression of with both sand and clay fractions has been established named as the texture-based PTF. In this study, the PTF of [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF], is used to check out its utility within the FAO formalism. This formulation is tested over 33 study sites to evaluate the FAO evaporation estimates, with 32 sites belonging to the calibration data set of [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] and the remaining site (MOBou) used as an independent data set for the evaluation of E estimates.

Then, the three additional independent sites (EC1, EC2, B123) are used to assess wheat surface evapotranspiration. All the three sites were not used in the calibration process in [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF].

is then written as:

(10)

The advantage of the Eq 9, is that it can be calibrated using one single parameter P as shown in [START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF]. Indeed, the formulation presented in Merlin et al. ( 2011) is a parsimonious formulation well adapted for parameterization with texture information. Hence, the idea is to parametrize simply presented in Equation ( 10) as a function of the single parameter P in which K r,text is approached linearly at the mid-value (K r,text = 0.5).

Therefore, P parameter can be deduced at by inverting Eq 9:

where K r,text = 0.5 (11)

The flowchart for E calculation using the new approach is also illustrated in Figure 3.

Results and discussions

Figure 4 plots the texture-based simulated Kr (K r,text ) as a function of soil moisture at 0-5 cm depth for all bare studied sites. Looking at the Figure 4, it can be observed that the K r,textrelationship shows two different shape curves: The S-shaped curve and the Г-shaped curve.

The shape curves depend on the P value which is different for each site due to the soil texture information. For the 33 bare soil sites, the P values ranged from 0.24 to 2.46 from sandy ( to clayey ( soils, respectively. For fine textured soils K r,text increased rapidly with the increase of while K r,text increased slowlier for coarse texture soils. The same behavior was observed with physically-based simulated Kr in [START_REF] Chanzy | Significance of Soil Surface Moisture With Respect to Daily Bare Soil Evaporation[END_REF].

For P value higher than 0.5 the K r,text increases rapidly with . In this case the derivative of Kr with respect to is equal to zero at and when reaches the saturation ( ) K r,text equals 1. Contrarily, for the sites with a P value smaller than 0.5, the slope at tends to infinity. The scatter plot between observed and predicted E using the based-texture K r (Figure 5) appears less dispersed around the 1:1 line compared to the classical Kr FAO . Also, the bias is almost systematically reduced with the new texture-based formulation. A linear correlation is generally observed, which reveals that the new approach could be effective and successfully used to retrieve soil evaporation under different soil and climate conditions.

Although the statistical analyses provided relevant information about the effectiveness of the new developed model, statistical summary from Figure 6 revealed that the new K r formalism has the capability to predict soil evaporation more accurately than the classical Kr FAO . When comparing the R 2 , RMSE, MBE, slope and intercept for each site, one can observe that the performance of the new K r based on soil texture information is better than the classical representation of K r . By using the new formalism, an improvement of R In some study sites, even if the new approach still provides better results, a relatively poor performance may occur for both approaches, (eg. USIb1, FRAur and CHOe2 sites). The relatively poor results obtained at those sites are not due to the model itself but might be due to the collected true measurements. As indicated in [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] most of the study sites were not in a fully bare soil condition, as these sites were cropped and a selection of the periods when soil is quasi bare was investigated, assuming no crop (no transpiration) over the sites. This assumption could be a source of errors and could degrade the data quality. In addition, soil depth could impact the results and could generate some uncertainties. To assess the impact of the soil depth on the proposed approach, a sensitivity analysis is carried out by using soil moisture at 0-10 cm depth. Overly, the new textural based Kr provides more accurate results than the K r,FAO . This is due to its physical basis, given that the K r,text is implemented in a phenomenological model based on observational data, while the K r,FAO was mainly built on ad hoc assumptions. [START_REF] Lehmann | Soil Texture Effects on Surface Resistance to Bare-Soil Evaporation[END_REF] showed the importance of soil texture on the variation of surface evaporation as a function of soil moisture content.

Moreover, as cited in [START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF] and [START_REF] Wetzel | Evapotranspiration from nonuniform surfaces: a first approach for short-term numerical weather prediction[END_REF], the nonlinear relationship between soil evaporation and soil moisture varied systematically with soil properties. In addition, the critical soil moisture content where the evaporation rate drops below potential rate, was related to soil texture through the field capacity [START_REF] Budyko | Heat balance of the Earth's surface[END_REF][START_REF] Manabe | Climate and the Ocean Circulation I. The Atmospheric Circulation and the Hydrology of the Earth's Surface[END_REF]. Through the new K r,text , we succeeded in introducing the effect of soil texture on E rate, which is not taken into account properly in other land surface models [START_REF] Phillips | Using ARM Observations to Evaluate Climate Model Simulations of Land-Atmosphere Coupling on the U.S. Southern Great Plains[END_REF].

Sensitivity of Kr formulations to the thickness of surface soil layer

In this study the E estimates is assessed using soil moisture data at shallow depths near the soil surface (0-5 cm and 0-10 cm depending on the site). As a prospect to integrate K r,text in the FAO-2Kc (next sub-section), which uses a surface soil thickness (Z e ) of either 10 or 15 cm, we need to assess the applicability of the K r,text to the 0-15 cm soil layer. This is more crucial as all soil evaporation formulations are sensitive to the thickness of the surface soil layer [START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF]. Therefore, a sensitivity analysis has been made to quantify the impact of active soil evaporation depth to E as well as the integration of the new K r,text into the FAO-2Kc formulation. This was made for both approaches using 0-10 cm (in replacement of the 0-5 cm) collected soil moisture. Note that, only the sites where soil moisture data at 10 cm were available have been used for the sensitivity analysis. Figure 7 presents the simulated soil E using texture-based Kr text and classical Kr FAO compared to in-situ measurements for the 0-10 cm soil moisture depth. Due to the non-availability of soil moisture at 10 cm depth for all sites, eleven sites were selected to test the new approach where soil moisture data at 10 cm depth were available. Figure 7 shows the simulated evaporation using both classical Kr FAO and new model Kr text compared to in situ measurements for six sites with different soil textures and climate conditions: clayey (MOBou, MEYaq), silty loam (BELon, FRGri, DESeh) and sandy loam (IECa1). Figure 8 presents the performance of the approaches over all the eleven study sites. Looking at the results in Figures 7 and8, generally the new textural-based model captures well the variation of evaporation measurements. Mostly the new model provides a more accurate prediction of E all over the 11 study sites. The new model shows its capabilities to solve the common overestimation problem that occurs especially under arid and semi-arid conditions [START_REF] Michel | The WACMOS-ET project -Part 1: Tower-scale evaluation of four remotesensing-based evapotranspiration algorithms[END_REF] as shown at sites MOBou, BELon and DESeh. The new model yielded lower RMSEs over almost all sites in comparison with classical Kr FAO , with a value of 0.94 mm.day -1 and 1.70 mm.day -1 , respectively, at BELon site as an example.

The statistical results are presented in Figure 8, showing the performance of both approaches (classic Kr FAO and new texture-based Kr text ) using soil moisture data at 0-5 cm and 0-10 cm depth. In same sites, it appears from the statistical results that the performance of the new approach slightly decreases while the classical Kr FAO slightly improves providing better results by using soil moisture at 0-10 cm instead of 0-5 cm depths. However, the new approach still outperformed the Kr FAO approach regardless of the sensing depth (0-5 or 0-10 cm) of the soil moisture measurements. The error between Kr FAO and in-situ evaporation measurements is slightly reduced when using soil moisture at 0-10 cm instead of 0-5 cm depths, from 0.85 to 0.60 mm.day -1 while by using new model the error is slightly increased from 0.58 to 0.75 mm.day -1 over FRGri site. The improvement of the performance of Kr FAO approach by using soil moisture at 0-10 cm instead of soil moisture at 0-5cm, is due to the fact that the evaporation process can occur from deeper layers than the 0-5 cm layer. This is especially occurred in arid and semi-arid areas in which the FAO-2Kc recommends the use of a layer of 0-15 cm. Similarly, the new formulation of the evaporation reduction factor is intrinsically dependent on the soil moisture measurements depth. It is reminded that the used formulation has been developed and calibrated using a 0-5 cm soil layer. As stated by [START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF], the P value is an increasing function of the soil depth for a given soil moisture value. This could be responsible for the slight underestimation of new model when using the 0-10 cm layer. However, the Kr FAO formulation is generally based on moisture data from 0 to 15 cm [START_REF] Allen | Using the FAO-56 dual crop coefficient method over an irrigated region as part of an evapotranspiration intercomparison study[END_REF]. The sensitivity analysis shows that the new approach generally outperformed the classical Kr FAO at either 5cm or 10 cm soil depths.

Coupling the K r,text model with the FAO crop water budget model: application to wheat cropped fields

As mentioned above, the main objective is to improve the soil evaporation estimates through using a formulation of Kr based on soil texture information and soil moisture estimates. The point is that retrieving soil evaporation is more relevant over bare soils and sparse areas.

However, over cropped area, retrieving surface evapotranspiration is more important. For this reason, as a source of validating of our approach, we tested the new textural-based Kr approach over the cropped fields defined before: B123 (2002[START_REF] Komatsu | Toward a robust phenomenological expression of evaporation efficiency for unsaturated soil surfaces[END_REF]), EC1 and EC2 (2016-2017). Two cases were tested for each approach, where the soil evaporation component is forced by i) in-situ soil moisture , using the classical FAO-2Kc (E FAO + ) or the new approach (E New + ) and ii) simulated soil moisture from the water balance ( FAO ), for classical FAO-2Kc and new approach termed in the plots as (E FAO + FAO ) and (E New + FAO ), respectively.

In order to observe the response of the evaporation coefficients to water supply, Figure 9 shows the temporal variation of Kr for FAO-2Kc (K r,FAO ) and new textural-based Kr (K r,text ) approaches for the cropped fields. From the Figure 9, it can be observed that at the initial stage of crop growth, the K r,FAO values are high and reach its maximal value [START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF]. This is due to the small fraction of soil surface covered by wheat crop. In addition, following rain or irrigation both Kr increase, while the Kr is small and can reach zero in the drier periods when the soil resistance to evaporation increases.

Overall, K r,text detects soil stress periods and responds well to the water supplied by precipitation or/and irrigation. After an irrigation or precipitation event both Kr increase with different amplitudes. The new textural-based Kr appears to be more physical than the K r,FAO .

It consistently increases following the supplied water amounts and then gradually decreases. This is not the case for K r,FAO , which increases and decreases rapidly. This is due to the response and the soil drying time which is well taken into account in the K r,text approach.

Especially in stage 2 when soil moisture is the limiting factor, the K r,text decreases progressively for each soil type depending on the soil properties.

An increase of Kr comes with an increase of the evaporation rate after a water supply by irrigation or/and rain. Consequently, soil evaporation losses are considerable. Figure 10 presents a daily evolution comparison of the measured ET (ET obs ) with simulated ET using standard FAO-2Kc (E FAO + FAO ) and new approach (E New + FAO ). Soil evaporation and wheat transpiration were also presented in the same figure for the three studied fields. According to figure 10, we notice that the new approach follows correctly the measured ET.

As it is seen, the new formulation of E helps catch the variation of surface ET especially at the beginning and the end of the season when ET is dominated by E and when transpiration is relatively small. This behavior of the classical FAO-2Kc was also reported in Rafi et al. (2019), where the classical FAO was found to underestimate E leading to an underestimation of ET especially during the senescence period. While in Olivera-Guerra et al. ( 2018) an overestimation issue was observed using thermal-derived E estimates. Using soil moisture and texture information lead to a correction of E in the initial and late stage. To evaluate visually and quantitatively the used approaches, the predicted ET were plotted against the observed ET (ET EC ) and are presented in Figure 11 for the three wheat crop fields. From the statistical results presented in Figure 11, the new method provides the best results in term of accuracy and stability over the three wheat crop fields. The ET is estimated with an RMSE equal to 0.71 and 0.66 mm/day and an R 2 equal to 0.78 and 0.83 by using the classical FAO-2Kc and the proposed method for B123 field, respectively. An improvement is observed when using the new method. This is especially seen for small values of ET where the estimations are less scattered around the 1:1 line.

Statistics of the studied four cases are shown and reported in Figure 12. Using in situ soil moisture or simulated from water balance in the new model gives almost the same results.

A slight difference is observed in the statistical results in terms of RMSE and R 2 . The E New + shows the best results compared to the other methods with the lowest RMSE (0.55 mm/day) and the highest R 2 of about 0.84 in the B123 field. The reason behind the good results when in-situ soil moisture was used is the fact that soil moisture is directly measured using TDR sensors while the simulated soil moisture is estimated using water balance equation where errors can accumulate from each water balance components. However, the classical approach shows good results compared to new approach in the EC2 field, although the difference between both approaches is smaller.

Overly, results showed that the new textural approach for E provided more accurate results in terms of ET regardless of the source (direct measurements and model simulations) of soil moisture estimates.

Conclusion

This work aims to improve the representation of the soil evaporation (E) component of the total evapotranspiration (ET). In the classical FAO-2Kc, the evaporation reduction coefficient Kr is estimated based on an ad-hoc relationship with soil texture. This standard formulation provides some limitation to quantify the E process. Rafi et al. (2019) and Olivera-Guerra et al.

(2018) reported that the standard FAO-2Kc overestimates or underestimates E in the beginning or/and the end of the agricultural season.

A new E formulation is developed by including the soil texture information into Kr. The idea here is to adapt the Kr representation to the nonlinear relationship between soil moisture and E in a range of soil texture types and surface conditions as in Merlin et al. (2016a[START_REF] Merlin | A phenomenological model of soil evaporative e ffi ciency using surface soil moisture and temperature data[END_REF]. By including the new formulation into the FAO-2Kc the typical drying process is modified. The new evaporation formulation was tested over 33 bare soil sites distributed all over the world.

The sites are different in term of soil texture and meteorological conditions. Soil moisture measurements at 0-5cm and 0-10 cm were used to further assess the impact of soil measurement depths on the E rate. FAO-2Kc and the new texture-based approach were evaluated by comparing them against in situ E measurements. From the obtained results, the new approach provides the best results in term of accuracy and robustness. The new model provided solid basis for describing the soil texture impact on the E reduction coefficient.

However, some uncertainties were observed, and this could be due to the water diffusion flux which is not taken into account in the FAO-2Kc model and represents the soil as a simple reservoir. Further, the new evaporation model was evaluated over a 3 monitored wheat cropped fields, where evapotranspiration is retrieved. The simulated evapotranspiration was compared to eddy covariance measurements installed in the 3 sites. A comparison of the 4 cases is investigated by using both standard FAO-2Kc and the new approach. Each model is forced by soil moisture either measured in situ or simulated by the water balance model.

Results showed that the texture-based approach provides the best results with low errors over the studied sites using field-scale measurements data.

To cope with some limitations of FAO-2Kc model, remote sensing data can be assimilated [START_REF] Amazirh | Improving FAO-56 estimates of ET in semi-arid region through combined assimilation of SMAP based disaggregated soil moisture and Landsat Surface Temperature[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]. However, the use of remote sensing data in the FAO-2Kc would require a match between the soil layers represented by the model and the sensing depth of spaceborne observations. The point is that the sensed thickness of remote sensing data rarely exceeds 0-5 cm. In this vein, coupling the FAO-2Kc with remote sensing data will be an asset for minimizing modeling errors, and for better constraining the 'often unavailable over most irrigated area' irrigation input data. To do this, changing the depth of the top-soil layer represented by the FAO-2Kc model to fit with the near-surface soil moisture derived from remote sensing is needed.

Figure 1 :

 1 Figure 1: Study areas where the coupled approach (K r,text -FAO-2Kc) is evaluated.

(Figure 2 )

 2 : energy is the limiting factor of E during stage I (constant drying rate) and soil moisture is the limiting factor during stage II (falling drying rate). Stage I begins from D e =0 and holds until D e is equal to the readily evaporable water (REW) and where the soil moisture equals to a given critical soil moisture (value when soil begins to resist to evaporation). In this stage, the soil surface remains wet and the evaporation rate is controlled by the energy available at the soil surface. Stage I holds until water cannot be transported to near surface as fast as the rate to supply the atmospheric demand. Here stage II begins (D e exceeds REW).

Figure 2 :

 2 Figure 2: K r,FAO as function of De for classical FAO[START_REF] Allen | Crop evapotranspiration -Guidelines for computing crop water requirements -FAO Irrigation and drainage paper 56[END_REF]

  [START_REF] Jacquemin | Sensitivity study and validation of a land surface parameterization using the HAPEX-MOBILHY data set[END_REF],[START_REF] Lee | Estimating the soil surface specific humidity[END_REF] and[START_REF] Noilhan | A Simple Parameterization of Land Surface Processes for Meteorological Models[END_REF] proposed a formulation of soil evaporative efficiency based on surface soil moisture and soil moisture at the field capacity in two specific soil thickness layers.Chanzy and Bruckler, (1993), Komatsu. (2003) and[START_REF] Merlin | Modeling soil evaporation efficiency in a range of soil and atmospheric conditions using ameta-analysis approach[END_REF][START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF] found a nonlinear behavior of soil evaporative efficiency as a function of soil moisture.[START_REF] Merlin | An analytical model of evaporation efficiency for unsaturated soil surfaces with an arbitrary thickness[END_REF] proposed an expression of the soil evaporative efficiency for various soil thicknesses based on soil moisture and soil texture information. The reduction coefficient K r based on soil texture (K r,text ) is written as:(9)

Figure 3 :

 3 Figure 3: Flowchart presenting an overview of the main input and output data of the classical Kr FAO (Em-dashed red rectangle) and the new textural-based approach Kr text (Em-dashed blue rectangle) to estimate soil evaporation (E).

Figure 4 :

 4 Figure 4: Texture-based K r simulated by the new model as function of in situ surface soil moisture for the different 33 used sites. The very clayey soil and very sandy soil in our data set correspond to P values equal to 2.46 and 0.24, respectively.

Figure 5 :

 5 Figure 5: Evaporation estimates from new model (E New ) and FAO-2Kc (E FAO ) using soil moisture at 5-cm depth compared to measured evaporation.

Figure 6 :

 6 Figure 6: Bar graph of R 2 , RMSE, MBE, intercept and slope of the linear regression between simulated and observed E using the Kr text (black) and Kr FAO (grey) for each site.

Figure 7 :

 7 Figure 7: Evaporation estimates from new model (E New ) and Kr FAO (E FAO ) using soil moisture at 10 cm depth compared to measured evaporation.

Figure 8 :

 8 Figure 8: Bar graph of the statistical parameters (R 2 , RMSE, MBE), the intercept and slope of the linear regression between simulated and observed E using the new texture-based (black '5cm', grey '10 cm') and Kr FAO (Brown '5cm', orange '10 cm') for each site. 5 cm and 10 cm are the soil depths.

Figure 9 :

 9 Figure 9: Time variation of the K r estimated from FAO-2Kc (K r,FAO ) and new textural based Kr (K r,text ) for the studied fields.

Figure 10 :

 10 Figure 10: Time series comparison of daily plant transpiration (T), soil evaporation (E), ET 0 as well as the ET observed by EC (ET obs ) and simulated by FAO-2Kc (left) and new model (right) over the study fields.

Figure 11 :

 11 Figure 11: Scatterplot of simulated ET-FAO using the FAO-2Kc (left plot) and the new model forced by simulated soil moisture ( FAO ) (right plot) versus Eddy covariance ET (ET EC ) over the B123, EC1 and EC2 study fields.

Figure 12 :

 12 Figure 12: Bar graph of R 2 , RMSE, MBE, intercept and slope of the linear regression between simulated and observed ET using the 4 approaches.

Table 1 :

 1 Study sites and their soil physical properties.

	Sites	Sand (%)	Clay
			(%)	(m 3

/m 3 ) (m 3 /m 3 ) (m 3 /m 3 ) REW (mm) Country

	AUStu	34.3	14.5	0.4079	0.205	0.335	9.5	Australia
	BELon	7.5	20	0.4566	0.205	0.335	9.5	Belgium
	CHOe2	9.5	43	0.4940	0.23	0.36	10	Switzerland
	DEGeb	9.5	30	0.467	0.205	0.335	9.5	Germany
	DEKli	21.5	55.7	0.4798	0.22	0.36	10	Germany

Acknowledgements

This study was conducted within the Center of remote sensing applications, at the Mohammed VI university-Morocco and was 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), the European Fluxes Database Cluster (http://www.europe-fluxdata.eu), the AmeriFlux sites, OzFlux site, long term observatories (AMMA, HOBE and SudMed), and the short term intensive field campaigns (EFEDA, ReSeDa, Yaqui'08, IHOP, Monsoon'90 and SGP'97). The French Agence Nationale de la Recherche (MIXMOD-E project, ANR-13-JS06-0003-01) is also acknowledged for extracting and pre-processing the bare soil evaporation database.

Environmental Processes 2,[689][690][691][692][693][694][695][696][697][698][699][700][701][702][703][704].1007/s40710-015-0107-1 Diarra, A., Jarlan, L., Er-Raki, S., Le Page, M., Aouade, G., Tavernier, A., Boulet, G., Ezzahar, J., Merlin, O., Khabba, S., 2017