Application of SEBAL and Ts/VI Trapezoid Models for Estimating Actual Evapotranspiration in the Algerian Semi-Arid Environment to Improve Agricultural Water Management

Accurate spatio-temporal estimation of evapotranspiration (ET) and surface energy fluxes is crucial for many agroenvironmental applications, including the determination of water balance, irrigation scheduling, agro-ecological zoning, simulation of global changes in land use and forecasting crop yields. Remote sensing based energy balance models are presently most suitable for estimating ET at both temporal and spatial scales. This study presents an intercomparison of ET maps over the Habra plain in western Algeria obtained with two different models: Ts/VI trapezoid (Surface temperature/Vegetation Index Trapezoid Model) and SEBAL (Surface Energy Balance Algorithm for Land). Ts/VI trapezoid is the most used model, due to its simplicity, ease of use, few data input requirements and relatively high accuracy. It allows estimating ET directly by using the Priestley-Taylor equation. Whereas SEBAL allows estimating ET as the residual term of the energy balance equation, by using a rather complex hot and cold pixel based contextual approach to internally calibrate sensible heat flux through an iterative approach. The data set consists of four Landsat-8 OLI/TIRS images acquired on 2018-2019 and some ground measurements. In conclusion, the results show that SEBAL and Ts/VI trapezoid models provide comparable outputs and suggest that both the two models are suitable approaches for ET mapping over agricultural areas where ground measurements are scarce or difficult to collect.


Introduction
The spatio-temporal estimation of evapotranspiration (ET) from agricultural regions is important for agriculture water management, especially in arid and semiarid regions where water deficiency is becoming a major constraint for economic development (Yang et al., 2019). Conventional methods (such as Bowen ratio system, eddy covariance system, or weighting lysimeters) that use point measurements to estimate ET are representative only of local areas and cannot be extended to large areas because of heterogeneity of landscape (Liu and Xu, 2019). Remote sensing-based energy balance models are presently best alternatives for estimating ET at both field and regional scales where ground measurements are not feasible and reliable estimates of ET are needed (Yang et al., 2019;Zhang et al., 2016).
The ET is a major component of the terrestrial hydrological cycle, nearly two-thirds of precipitation over land is returned back to the atmosphere by ET (Brutsaert, 2013). This proportion may be higher in dry regions, such as the Mediterranean basin (Boulet et al., 2020). Inaccurate estimates of ET in these regions can cause large errors in the hydrological components prediction such as runoff and recharge, and in the associated water balance and water resources availability (Zhang et al., 2017).
Remote sensing technology facilitates the calculation of albedo, vegetation indexes and surface temperature, which are necessary to remote sensing-based energy balance models for scaling up ET and surface energy fluxes to larger spatial and longer temporal scales (Chen and Liu, 2020). To this end, major effort has been devoted over the past three decades to improve remote sensingbased methods that provide spatially distributed surface fluxes maps using airborne and satellite data (Nehal et al., 2017). These methods can be classified into three main categories: a) Methods using simple empirical relationships relating daily ET to an instantaneous surface temperature measurement (Trezza et al., 2013;Huang et al., 2019); 2) Methods using deterministic relationships based on more complex models such as Soil-Vegetation-Atmosphere Transfer models (SVAT) (Olioso et al., 2005;Chebbi et al., 2018;Bigeard et al., 2019); 3) Methods based on ET estimation as the residual term of the energy balance equation, which can be divided into two categories: 1) Single-source models, such as SEBAL (Surface Energy Balance Algorithm for Land) (Bastiaanssen et al., 1998;Bastiaanssen et al., 2005), METRIC (Mapping Evapotranspiration with Internalized Calibration) Ramírez-Cuesta et al., 2020), T s /VI trapezoid model (Jiang and Islam, 2001;Stisen et al., 2008;Zhu et al., 2017), SEBS (Surface Energy Balance System) (Su, 2002;Chen et al., 2019), S-SEBI (Simplified Surface Energy Balance Index) (Roerink et al, 2000;Allies et al., 2020), that do not distinguish between soil evaporation and transpiration. Their simplicity has made the single-source models widely used; 2) dual-source models, such as TSEB (Two Source Energy Balance) (Kustas et al., 2018), ALEXI or Dis-ALEXI  and SPARSE (Soil Plant Atmosphere and Remote Sensing Evapotranspiration)  that discriminates the soil and vegetation component.
This study evaluates the performance of T s /VI trapezoid and SEBAL models for ET estimation by using four images acquired by Landsat-8 OLI/TIRS. This later provides updated images at 30-m resolution every 16 days. The estimates of these models are then compared with ground observations on barley and artichoke crops using the Bowen Ratio Energy Balance (BREB) method. T s /VI trapezoid model allows estimating ET directly by using the Priestley-Taylor equation, without the need to calculate the sensible heat flux. It requires few input data. It is based on a graphical method in which the extreme values of surface temperature are deduced from the scatterplot between vegetation index (NDVI) and surface temperature, and then the Priestley-Taylor parameter is calculated (Stisen et al., 2008). However SEBAL follows an energy balance approach, where the latent heat flux (corresponding to the energetic equivalent of ET) is estimated as the residual term when net radiation, sensible and soil heat fluxes are known, by using a rather complex hot and cold pixel based contextual approach to internally calibrate sensible heat flux through an iterative approach. SEBAL may not be as applicable as other models for ET spatialization over agricultural areas where ground information is scarce or difficult to collect (Khaldi et al., 2011). Moreover, SEBAL has the particularity of using a calibration procedure to compensate for temperature and albedo errors without the need for a complex atmospheric correction (Bastiaanssen et al., 2005).
The goal of this study is to use SEBAL and T s /VI trapezoid models for estimating the actual ET over the agricultural plain of Habra, a semi-arid region with heterogeneous surface conditions in northwestern Algeria, where ground data are scarce or difficult to collect. The region contains extremes in surface albedo, vegetation cover and surface temperature, due to a varied land cover types including irrigated agriculture, rainfed agriculture, bare soil and livestock grazing. The irrigated agriculture in the area is now facing a significant threat, due to limited water supplies, climate change and an increase in extreme events (Benzater et al., 2021), which appeals us to develop water-saving irrigation for sustainable water use. To this end, a remote sensing approach is required to be routinely applied as a tool for providing both historical and near-real time ET and surface energy fluxes for performing a better management of the agricultural water resources of the area.

Study area description
The study area corresponds to the agricultural plain of Habra, which houses the irrigation perimeter of Mohammadia. It is located in northwestern Algeria (Oran) between longitudes 0°6'44" W and 0°6'33" E and latitude 35°32'45" N and 35°41'33" N. It covers an area of 328 km 2 (Fig. 1).
The selected area is part of the great interior plain of Macta, which is the receptacle of the second watershed of Algeria by its area (14500 km 2 ) and only communicates with the Mediterranean Sea by a narrow channel (Benzater et al., 2019). The average altitude is about 40 m.
The soils in the plain of Habra are sedimentary formation with variable texture intake alluvial and alluviocolluvial. They are distributed in the plain into entities more or less uniform and regular. Soil salinity is between 8 and 16 mS/cm at depth of more than 50 cm with low rate of leaching.
The climate in the study region is Mediterranean semi-arid with mild winter. Two main periods characterized this region, a rainy period during the months of autumn (September, October and November), winter (December, January and February) and early spring (March and April), and a dry and hot period during the months of summer (June, July and August). The absolute minimum air temperature during winter down to 6 to 8°C. Summer is usually dry and warm. The absolute maximum air temperature is equal to 42°C. The average annual rainfall for the period 1970-2011 is about 450 mm (Elouissi et al., 2017).
The rural areas with rich soils are suitable for agriculture but where the soils are poor, livestock grazing (rangelands) is dominant. Irrigated agriculture is dominant in the study area and the main crops include fruits mainly citrus and garden crops such as artichoke. Rainfed agriculture occupies a small part and the main crops include cereals mainly barley (Fig. 2).
The irrigation perimeter of Mohammadia is relatively old, dating back to 1940. It covers an area equipped for irrigation (irrigable area) of about 19630 ha (the total area is about 21210 ha), predominantly arboreal and particularly citrus. It is considered the most important in Algeria northwestern. Despite this, few studies on the management of its irrigation water were performed (Tazekrit et al., 2017). Irrigation water is conveyed to the irrigation perimeter through a classical irrigation network from three dam reservoirs, namely Fergoug, Bouhanifia and Ouizert, which operate in cascade. This water knows significant losses during transfer, due to illegal consumption along the stream. The irrigation network is of the gravity type, with open channels of trapezoidal or circular shape totaling a length of 256 km (Zerkaoui et al., 2017). Its management is provided by the national office for irrigation and drainage in consultation with the association of irrigators. So far, this network is the only way to supply the irrigation water to the majority of irrigated crops because groundwater in the area is brackish and cannot be used to irrigate citrus orchards (Tazekrit et al., 2017).
In the early 1960s, the irrigation perimeter received a total water volume of about 80 million m 3 /year. This volume has been constantly decreasing since 1977 and the irrigation quotas allocated for all irrigable area did not exceed 32 million m 3 /year. This reduction is due to the lack of water resources caused by recurrent drought plaguing Algeria's western region for two decades  and by allocating water as a priority for domestic use (Sutton and Zaimeche, 1992).Over the past decade, the allocated irrigation water volumes are still weak and have little exceeded 26 million m 3 . Such a situation of water resources cannot meet the survival needs of orchards, which has led to the decline of several hundred hectares of trees and the reduction of irrigated area to about 50% (Zerkaoui, 2017).

The used data
Remote sensing data used in this work consists of four Landsat-8 OLI/TIRS images acquired during 2018 and 2019 (Table 1). They are freely downloadable https:// earthexplorer.usgs.gov.
Remote sensing data are supplemented by ground measurements which were performed on two points located in the experimental site of the National Institute of Plant Protection in Mohammadia city. The first point was located on barley crop and the second on artichoke crop (Fig. 1). These measurements were intended for the daily monitoring of energy fluxes at the soil-plant-atmosphere continuum. They correspond to the radiometric surface temperature, the reflected radiation and the three components of surface energy balance, i.e. soil heat flux, sensible heat flux and latent heat flux. Incoming shortwave (solar) and thermal radiations were measured on the meteorological station of Mohammadia (ONM), located in the study area (Fig. 1) using a pyranometer and a pyrgeometer, respectively. The meteorological station also provided measurements on the reference variables which are air temperature, air humidity, wind speed, air pressure, sunshine duration and daily potential ET (Table 2). On the experimental device, installed on the plots of barley and artichoke, the albedo was obtained from the ratio of the reflected radiation (measured by an Apogee pyranometer ) and the incoming shortwave radiation. The net radiation is determined from the radiative balance equation, depending on the albedo, the incoming shortwave and thermal radiations and the surface emission which is deduced from the radiometric surface temperature. The latter is measured by an Apogee infrared radiometer (IRTS-P model). The soil heat flux is measured  using three Hukseflux conductive flux plates (HFP01SC model) installed at 5 cm depth in the soil. Sensible and latent heat fluxes were computed from measurements at two levels (0.5 and 2.5 m above the surface) of air temperature and relative humidity using the BREB technique.

Surface Energy Balance Algorithm for Land (SEBAL) model
The following presents a brief description of SE-BAL, with further details found in Bastiaanssen et al. (1998) and Bastiaanssen et al. (2005). SEBAL is a remote sensing processing model for estimating instantaneous terms of the energy balance equation, including ET flux, at the satellite overpass time.
For the determination of ET, SEBAL estimates the latent heat flux (λE) as the residual term of the energy balance equation which describes the energy exchange between the land surface and the atmosphere: where Rn is the net radiation at the surface (W/m 2 ), H is the sensible heat flux (W/m 2 ), G is the soil heat flux (W/m 2 ) and λE is the latent heat flux (energy consumed by ET, W/m 2 ) ( Table 3). Net radiation (Rn) estimation is quite similar for both models; it is calculated according to: where Rg is the incoming shortwave radiation, partly reflected depending on the albedo r 0 , L ↓ and L ↑ are the incoming and the emitted outgoing longwave radiations (W/m 2 ), respectively. Mapping the net radiation (Rn) requires evaluation of 1) the incoming shortwave radiation (Rg) which is obtained from the weather observations. It allowed us to estimate the atmospheric transmittance (which represents the capacity of the atmosphere to transmit solar radiation), 2) the outgoing longwave radiations (L ↑ ), obtained by the expression of Stephan-Boltzmann, using surface temperature and surface emissivity and 3) the incoming longwave radiation (L ↓ ), using air temperature and atmosphere emissivity (ε a ). This latter is calculated depending on atmospheric transmittance (τ) following the expression (Bastiaanssen et al., 1998): The soil heat flux (G) cannot be directly determined from satellite sensors and requires an empirical formulation. It has been estimated as a fraction of Rn with the coefficient a function of surface temperature, albedo and vegetation index (Bastiaanssen et al., 2000): The sensible heat flux (H) is expressed as a function of the near-surface air temperature difference (T aero -T a ) as follows: where ρ is air density (kg/m 3 ), Cp is air specific heat at constant pressure (J/kg/K), r ah is the aerodynamic resis- tance to heat transfer, T aero is the aerodynamic temperature (K) and T a is the air temperature (K). In satellite remote sensing applications, the radiometric surface temperature (T 0 ) retrieval is often used instead of the aerodynamic temperature (T aero ) in Eq. (5) (Kustas et al., 1989). In SEBAL, the sensible heat flux (H) is estimated by using the following expression: where dT is the near-surface temperature difference between two near surface heights z 1 = 0.1 m and z 2 = 2 m above the canopy layer, and r ah is the aerodynamic resistance to heat transport between these levels (s/m). dT is used in Eq. (6) because of the difficulty in estimating surface temperature T 0 accurately from satellite due to uncertainty in atmospheric attenuation or contamination and radiometric calibration of the sensor .
In addition, T 0 , as measured by satellite (i.e., radiometric or kinetic temperature) can deviate from the "aerodynamic" temperature that drives the heat transfer process by several degrees (Qualls and Brutsaert, 1996). In this model, the difference dT between the two near surface heights 0.1 and 2 m is approximated by a simple linear function: The coefficients a and b in Eq. (7) are empirically determined using the properties of pixels in extreme water conditions (hot/cold and dry/wet). These pixels are identified on the image by analyzing the vegetation index and the surface temperature relationship according to the trapezoid method . The dry pixels are indicated at bare soils (NDVI values close to zero) having high surface temperature. However, the wet pixels are indicated at fully vegetation (NDVI > 0.7) having low surface temperature. The thresholds of low and high temperatures are defined as the equilibrium surface temperatures resulting from the energy balance for well-watered dense vegetation and dry bare soil, respectively .
With the identification of wet and dry pixels, we can estimate H wet and H dry from the energy balance equation as follows: A dry pixel is characterized by a zero latent heat flux (λE dry = 0), which means, the overall available energy (Rn-G) dry is partitioned into sensible heat flux. For a wet pixel, the latent heat flux (λE wet ) in SEBAL is assumed to be equal to the hourly grass reference ET (ETr), by using Penman-Monteith equation (Allen et al., 1998), multiplied by an empirical coefficient of 1.05. The choice of this coefficient is primarily dictated by assumption that a wet pixel (fully covered by vegetation) usually has an ET value of 5% larger than ETr (Gavilán et al., 2019). By inverting the Eq. (6), H wet (or H dry ) value allows deducing the temperature difference dT wet (or dT dry ) between the two near surface heights 0.1 and 2 m. The coefficients a and b of Eq. (7) are determined by fitting a line using dT and T 0 values of both dry and wet pixels in the image.
In SEBAL, r ah is calculated between z 1 and z 2 using a wind speed extrapolated from some blending height above the surface (∼200 m, a height of 200 m was used in this study) and an iterative procedure for correcting atmospheric stabilities to heat and momentum transfer, based on the Monin-Obukhov's similarity theory. For the first estimation of H, the atmospheric condition is assumed to be neutral, and r ah is calculated as: where k is von Karman's constant (= 0.41) and u* is friction velocity (m/s), calculated using the logarithmic wind profile for neutral atmospheric condition for the first iteration as: where u 200 is wind speed at 200 m (corresponding to blending height) and z om is roughness length for momentum transport (m). In subsequent iterations, the Monin-Obukhov length (L) is first calculated to examine the stability conditions of the atmosphere: where g is the acceleration due to gravity (= 9.81 m/s 2 ). Then, the corrected values for r ah and u* for each iteration are calculated from Eqs. (13) and (14), respectively, as follows: where ψ h (z 2 ) and ψ h (z 1 ) are stability functions for heat transport at heights z 2 and z 1 , respectively, and ψ m (200) is the stability function for momentum transport at blending height (=200m). Both ψ h and ψ m are functions of the Monin-Obukhov length, and are computed following Allen et al. (2007). The corrected values of r ah and u* are then used to recalculate H until the value of r ah stabilizes.
Once H is estimated, λE is computed using Eq. (1). This last step leads mapping the latent heat flux. This should help in the interpretation of a surface behaviour with respect to water stress (Bastiaanssen et al., 1998). It is therefore preferable for an easier interpretation to deduce moisture indicators such as the evaporative fraction (EF), the surface resistance to evaporation (r s ) and the Priestley-Taylor parameter (α) The flowchart presented in Fig. 3 summarizes the methodology used in SEBAL model for mapping ET from satellite data and ground measurements.

Surface temperature/Vegetation Index trapezoid (Ts/VI trapezoid) model
The relationship between surface temperature (T 0 , or temperature difference dT S ) and vegetation index (the nor-malized difference vegetation index (NDVI)) can be used to describe the evaporative capacity of the land surface assuming that T 0 varies for a given vegetation index mainly depending on the availability of soil moisture rather that differences in atmospheric forcing over a relatively flat area. This relationship has been widely used to obtain information about the energy fluxes or soil moisture of the land surface (Jiang and Islam, 2001;Stisen et al., 2008;Bai et al., 2019). Detailed discussions on the T 0 -NDVI relation are found in e.g. Sandholt et al. (2002), Hu et al. (2019), Carlson and Petropoulos (2019) and Carlson (2007). Scatterplots between remotely sensed T 0 and NDVI often results in a trapezoidal/triangular shape.
The prerequisite for estimating evaporative fraction (EF) and ET from the T s /VI trapezoid is to determine accurately the lower edge of this space (wet edge) which is characterized by saturated surface soil water content with maximum ET and the upper edge (dry edge) of the scatter plot representing lower limit of surface soil moisture content with limited ET and higher limit of surface temperature for a given NDVI (Sandholt et al., 2002). The T s /VI trapezoid method should ideally be applied over smaller regions and those with little topographic variation (Zhu et al., 2017). The four points of the trapezoid corresponds to extreme conditions of surface in terms of surface temperature and NDVI, which allow deducing the extreme values of surface temperature and NDVI (Moran et al., 1994;Hu et al., 2019).
The Priestley-Taylor formulation with fully remotely sensed data proposed by Jiang and Islam (2001) and further improved and validated by Stisen et al. (2008) representatively based on the interpretations of the T s /VI trapezoid model, has been employed to estimate ET using the following equation: Where Rn and G are obtained using Eqs. (2) and (4) respectively (both models use the same equations for estimating net radiation and soil heat flux), γ is the psychrometric constant (≈0.66 mbar/K) and Δ is the slope of the saturation vapor curve at air temperatures (T a ), calculated with: Equation (15) is a modified version of the Priestley-Taylor equation in the case of unsaturated surfaces by the introduction of the parameter φ which represents the so-called Priestley-Taylor parameter, and which accounts for aerodynamic and canopy resistances, and is slightly different from the original Priestley-Taylor's parameter α (≈1.26). This parameter depends on surface moisture conditions . It is defined globally to range from φ min = 0, for a dry bare soil, to φ max = (∆+γ)/∆, for a saturated or well vegetated surface (Stisen et al., 2008). As illustrated in Fig. 4, φ min is assigned to a pixel with minimum NDVI and maximum temperature; φ max is assigned to pixels with maximum NDVI. φ can be expressed as a function of the evaporative fraction (EF) as follows: where EF is defined as the ratio of ET or latent heat flux (λE) to available energy (Rn-G): The parameter φ is estimated following the approach proposed by Jiang and Islam (2001) using three steps (Stisen et al., 2008): In the first step, surface moisture condition is estimated by interpolating the surface temperature between the wet and dry edges. The dry and wet edges are experimentally or theoretically identified by determining the surface temperature in the trapezoid corners (Fig. 5) using surface energy balance equation and boundary conditions represented by the surface resistances values for each moisture condition for soil and vegetation. In this study, we set a surface resistance of 10 s/m for wet vegetation cover, 400 s/m for dry vegetation cover, 0 for wet bare soil and ∞ for dry bare soil (Boegh et al., 2002).
In the second step, we estimate φ i,min which represents the minimum value of φ for a given fraction cover (f c ) value, as: where φ i,min is the value of the Priestley-Taylor parameter at the dry edge for a given value of NDVI i , φ max is the value of φ at the wet edge (φ max = (∆+γ)/∆).
The fraction cover (f c ) is expressed as (Stisen et al., 2008): where NDVI min and NDVI max are the minimum and maximum observed vegetation index values, corresponding respectively to bare soil and fully vegetated surfaces, defining the extremes of the trapezoid. The third step is to interpolate φ between φ i,min and φ i,max within each NDVI class between the lowest temperature (T s,i,min ) at wet edge and highest temperature (T s,i,max ) at dry edge. The linear interpolation of φ i with temperature leads to normalization of surface temperature and is given as: where T 0,i,min is the lowest surface temperature at the wet edge for a given NDVI and T 0,i,max is the highest temperature at the dry edge for a given NDVI.

Results and discussion
Both SEBAL and T s /VI trapezoid models are developed in C++ code. In the framework of this study, the modeling of the energy balance equation by SEBAL and T s /VI trapezoid models allows us to show that the surface parameters, obtained from the satellite scanned spectral radiance in the optical and thermal infrared ranges, namely the albedo, the vegetation index and the surface temperature lead to determine the SEBAL-based latent heat flux (λE) as the residual term of the energy balance equation and the T s /VI trapezoid-based λE directly by using the Priestley-Taylor equation. To evaluate the performance of the energy balance results from these two models, four statistical metrics were used: root mean square error (RMSE), mean absolute percent difference (MAPD), Bias and determination coefficient (R 2 ) (Timmermans et al., 2007). Note that the validation of models outputs with an energy fluxes measurement system (such as the Bowen ratio system) is often used to assess the performance of energy balance models. However, this type of point-based comparison does not guarantee that the model provides consistent fluxes over the whole range of surface conditions that may exist in a landscape (Choi et al., 2009;Khaldi et al., 2011).

Surface temperature
The surface temperature (T 0 ) is indirectly related to the latent heat flux (λE) through the energy balance equation. It provides important information on surface moisture conditions that is useful to support agricultural water management at spatial scale. The analysis of the correlation between the satellite-derived surface temperature (T 0,Sat ) and the ground-measured surface temperature (T 0,Ground ), obtained by an Apogee infrared radiometer (IRTS-P), showed reliable results with R 2 of 0.93 (Fig. 6). However, T 0,Sat in our case is slightly underestimated compared to the T 0,Ground as evidenced by the RMSE of 3.11°C (corresponding to MAPD of 10.47%) and a Bias of about −2°C. The recorded temperature errors (3.6 to 18.5%) are relatively close to the values reported in previous studies with RMSE approximately ranging from 2.9 to 4.2°C (Timmermans et al., 2007;Santos et al., 2020;Consoli and Vanella, 2014;Zou et al., 2018). For example, a slightly similar RMSE and Bias (4.2 and −3.4°C, respectively) were detected by Madugundu et al. (2017) in comparing T 0,Sat against T 0,Ground over an arid irrigated field in Saudi Arabia by using Landsat-8 data. In another study over the Texas High Plains, Chávez et al. (2009) reported remote sensing errors of 1.9 and 11.1% in estimating T 0,Sat (compared to T 0,Ground ) for sorghum and corn fields, respectively.
The analysis of the correlation between T 0 and λE for DOY 2 indicates significantly better agreement between these two variables, with R 2 of 0.93 and 0.97 for SEBAL and T s /VI trapezoid, respectively (Fig. 7). Moreover, the NDVI and albedo parameters, even if they offer interesting additional information in the interpretation of thermal infrared data (Carlson, 2007), are less significant in discriminating surface moisture conditions (Khaldi et al., 2011).
Surface temperature variations in the study area are shown in Table 4. Higher values correspond to pixels where bare soils (dry pixels) are dominant, while low values are associated with irrigated dense vegetation (wet pixels). Similarly, the average surface temperature over dry pixels is higher than over wet pixels (Table 4).

Net Radiation
Higher values of net radiation (Rn) are recorded over dense vegetation cover (NDVI > 0.7) having optimum water supply conditions. Bare soils (NDVI < 0.

Soil heat flux
For the image acquired on autumn (DOY 271), the average soil heat flux (G) is 37.7 W/m 2 . However, the average of G is higher (

Sensible heat flux
In the studies of ET estimation through the energy balance equation, the sensible heat flux evaluation is the most delicate for residual models such as SEBAL . To reduce errors due to this flux, SEBAL used an approach based on the Monin-Oubukhov similarity theory in the atmospheric boundary layer. In fact, the surface boundary layer modeling allows mapping the sensible heat flux which is obtained by estimating two key parameters of the energy balance regulation, depending on the surface type and its thermodynamic properties which are the aerodynamic resistance to heat transfer (r ah ) and the surface-air temperatures difference.
We note also on Table 4 that for dry pixels (bare soil and urban) the aerodynamic resistance to heat transfer (r ah ) is low (19.6, 21.3, 16.3 and 18.7 s/m for the DOY 271, 2, 75, 130, respectively), causing the release of sensible heat to the atmosphere. This is justified by high differences between surface and air temperatures. However, for wet pixels (freshly irrigated plots) r ah values are high (18.4,24.6,19.2 and 26.5 s/m for the DOY 271,2,75,130,respectively) because the available energy (Rn -G) is mainly consumed by ET. This differentiation of the sensible heat flux for dry and wet pixels is caused by the surface water status and its influence on the energy partition between the latent and sensible heat. Specifically, wet surfaces are individualized by low H values while high H values are assigned to dry areas (Table 4). Compared to field BREB observations, SEBAL-estimated sensible heat flux (H) agrees relatively well, yielding a RMSE and Bias values of 55.4 and 21.6 W/m 2 , respectively. These results indicate that SEBAL slightly overestimated H flux by a relative Bias of around 17%. Therefore, they still considered in reasonable agreement with the Bowen ratio measurements, having ∼20% uncertainty (Kustas and Norman, 1999;Liu and Xu, 2019;Timmermans et al., 2007), and are in good agreement with the values reported by the previous studies of Tang and Li, (2015), Santos et al. (2020), Wagle et al. (2017) and Ochege et al. (2019), indicating RMSE values of 46.4, 82.8, 72 and 48.29 W/m 2 , respectively. However, the T s /VI trapezoid model exhibited that it has significantly greater difficulty in estimating H flux (deduced as H = Rn -G -λE), producing a RMSE of 97.3 W/m 2 and Bias of 78.5 W/m 2 , which indicates an overestimation of H by a relative Bias of about 37%. The errors on T s /VI trapezoid model estimates are slightly similar to those detected by Tang and Li (2015), with RMSE and Bias values were 119 and 98 W/ m 2 , respectively. This indicates that the T s /VI trapezoid model is more sensitive to the estimated available energy (Rn-G) than SEBAL, and is because SEBAL is designed to compensate for the available energy and H-flux Bias, by using a contextual image-based calibration approach (Bastiaanssen et al., 1998).

Latent heat flux
In the irrigation scheme, computed water allocations can be updated using the information obtained by some irrigation performance indicators, such as uniformity and adequacy (Bastiaanssen et al., 1996). The latent heat flux (λE, corresponding to the energy consumed by ET) can be used to express the uniformity of crop water use, which is related to the equity of irrigation water distribution, or to detect the adequacy of regional water management, through the evaporative fraction (EF) which is mainly deduced from λE (Eq. (18)).
The latent heat flux is generally high for dense canopy and low for dry bare soils having high surface temperatures, low net radiations and high sensible heat fluxes. Table 5 summarizes the results of energy fluxes and moisture indicators obtained from the two models (SEBAL and T s /VI trapezoid) for different land use categories. It shows that high values of ET (λE) are observed over the irrigated areas with dense vegetation, while low values are over the bare soils, corresponding to high values of albedo. This allows emphasizing that the spatial distribution of SEBAL and T s /VI trapezoid-derived ET is relatively well correlated to the water regimes of the different land use units.
Overall, both SEBAL and T s /VI trapezoid models reproduce the estimated latent heat fluxes fairly well, compared to field BREB observations. The SEBAL-estimated latent heat flux values shows a slight discrepancy and better performance than T s /VI trapezoid, with RMSE of 49.1 W/m 2 (corresponding to MAPD of 18.2%), a Bias of −15.1 W/m 2 (relative Bias = −5.8%) and R 2 of 0.77 (Fig. 8a). These statistics are in good agreement with those obtained on Caatinga biome site (RMSE = 82.8 W/m 2 and Bias = −18 W/m 2 ) by Santos et al. (2020), on the SMA-CEX site (RMSE = 57 W/m 2 and Bias = −29 W/m 2 ) by Choi et al. (2009), and on sub-humid grassland (Southern Great Plains '97) and semi-arid rangeland (Monsoon '90) (RMSE = 49 W/m 2 and Bias = −13 W/m 2 ) by Timmer-mans et al. (2007). However, the T s /VI trapezoid model tends to significantly underestimate latent heat flux with a Bias of −39.2 W/m 2 (corresponding to −15% in relative value), RMSE of 52.5 W/m 2 (MAPD=20.7%) and a determination coefficient (R 2 ) of 0.81 (Fig. 8b). This significant underestimation of T s /VI trapezoid-derived λE flux was also demonstrated by Choi et al. (2009), Tang andLi (2015) and Lian and Huang (2016). Figure 9 shows the spatial distributions of the latent heat fluxes (λE) derived from the two models. Although the same anchor pixels (hot and cold) were selected with the two models, it was observed that the SEBAL model predicted a slightly higher λE than the T s /VI trapezoid model (Fig. 10), especially for DOY 130 (10 May 2019), as shown in Fig.10d. These spatial patterns are likely due to different approaches used to estimate H and λE by the two models. The SEBAL calculates H using a singlesource temperature gradient technique for heat transport, accounting for stability effects based on the Monin-Obukhov theory, and λE is computed as a residual of the energy balance equation, while the T s /VI trapezoid model calculates λE directly from Eq. (10), but H is calculated as a residual of the energy balance equation. Based on the identified spatial patterns for H and λE, the SEBAL computational scheme seems to be more physically comprehensive considering the stability for the aerodynamic resistance of heat transport (Wagle et al., 2017). The comparison of the latent heat flux estimates with SEBAL and T s /VI trapezoid models is shown in Fig.10. In general, a good agreement with determination coefficients (R 2 ) of 0.95, 0.95, 0.96 and 0.91 is shown for the DOY 271, 2, 75 and 130 respectively, with RMSE of 33.6, 14.1, 25.9 and 56.3 W/m 2 , respectively (Fig.10). The result of this comparison leads to the conclusion that the two models provide close outputs and suggests that both models can be considered as operational approaches for monitoring ET and surface water status over agricultural areas having limited amount of ground information. We can also note that the simple contextual methods, such as T s /VI tra-pezoid model, have the potential to provide ET estimations consistent with the rather complex physically-based SEBAL model, as indicated by Lian and Huang (2016). However, ET estimations from the T s /VI trapezoid model are quite sensitive to derived (Rn-G), T 0 and EF, while SEBAL can compensate for errors in estimated (Rn-G) and T 0 by using a hot and cold pixel based contextual approach to internally calibrate sensible heat flux through an iterative approach.
Different moisture indicators, such as evaporative fraction (EF) and surface resistance to evaporation (r s ), that can provide direct information on the stress condition  of crops, are computed from the latent heat flux (λE). In Figs. 11 and 12, we represent and compare the spatial distributions of the evaporative fraction obtained by SEBAL and T s /VI trapezoid model for the DOY 2. This comparison shows good agreement, with R 2 of 0.92 and RMSE of 0.074 (-), and justifies that both models produce nearly the same output.
Another way to validate our results is to analyze the frequency distribution of the surface resistance to evaporation (Fig. 13). Bastiaanssen et al. (1996) showed that for most crops covering fully soil (NDVI > 0.65), this resistance vary between 10 and 300 s/m with generally  peaking in the class of 30 to 80 s/m. There is a general consensus that the surface resistance for crops which cover the soil entirely lies in approximately the same range López-Urrea, 2019, Bougeault et al., 1991). The results shown in Fig. 13 are approximately consistent with this indication.

Conclusion
Different models have been developed to estimate ET from remote sensing data. In this paper, SEBAL and T s /VI trapezoid models were applied using Landsat-8 OLI/TIRS data over the Habra plain (western Algeria), a semiarid region with heterogeneous surface conditions, to estimate actual ET. The models outputs were compared with field observations using BREB method, to identify the most appropriate model. A significant discrepancy between remote sensing and ground estimates of latent heat flux is shown, with RMSE value of 49.1W/m 2 and 52.5 W/m 2 for SEBAL and T s /VI trapezoid respectively. That means 18.2% and 20.7% in relative terms and a determination coefficient  (R 2 ) of 0.77 and 0.81 respectively. These results are ascribed to errors committed in estimating the net radiation, soil heat flux and the sensible heat flux. These differences can be explained by the inaccuracies on the intermediate variables such as surface emissivity, soil heat flux, roughness length and air temperature and by excess in the net radiation occurred in the irrigated plots of this semi-arid region, due to the advection effects.
The results presented above confirm the possibilities offered by the Landsat-8 OLI/TIRS satellite data to solve the energy balance equation and to be routinely applied as a tool for providing both historical and near-real time ET for performing a better management of the agricultural water resources of the area. Despite the recorded inaccuracies, the obtained results show that both SEBAL and T s /VI trapezoid models provide close outputs and suggest that these models are suitable approaches to monitor actual ET and surface water status over agricultural areas where ground information is scarce or difficult to collect.