Introduction

One of the key economic and social challenges in developing nations, particularly those in barren and semiarid regions, is water shortage (Wu et al. 2018). Iran, with much of its land in arid and semiarid zones, relies heavily on water and irrigation for agriculture (Karandish and Hoekstra 2017). As the global population is projected to reach 7.8 billion by 2025, there will be heightened pressure on food security in developing nations, home to over 80 percent of this increase. Consequently, these countries will confront water scarcity in meeting agricultural, industrial, and urban demands (Nazari et al. 2013). Thus, accurate water demand forecasting is crucial for agriculture (Gonçalves et al. 2022). ET estimation is a method that enhances water management, leading to improved water efficiency (Venancio 2019).

Numerous methods have been developed and studied using meteorological data to calculate ET in various climatic and geographic conditions. These methods range from simple empirical relationships to those with a complex physical basis. Most rely on point-to-point measurements, limiting their applicability to local areas and making them unsuitable for large basins due to the dynamic and regional nature of ET (Volk et al. 2023). Remote sensing technology and geographic information systems are commonly used for estimating ET, especially in extensive areas (Yao 2019). By utilizing satellite imagery, which contains valuable information for estimating actual ET, researchers have developed methods to reduce reliance on ground data. Satellite images provide data on different surface properties across various bands of visible, thermal, and infrared spectra, enabling the estimation of real ET (Zhou et al. 2014).

RS is the sole technology capable of extracting parameters like surface temperature, albedo coefficient, and plant index in an efficient and environmentally friendly manner, while also being cost-effective (Campos et al. 2018). GEE stands out as a top-notch web-based RS system that has greatly simplified numerous satellite image processing tasks in recent years (Gorelick et al. 2017). The primary objective in estimating ET is to boost agricultural output with reduced water usage, thereby decreasing the agricultural water footprint and reallocating water to other purposes, particularly environmental needs. Currently, the agricultural sector, through global research initiatives like the Global Research on Water and Food Challenges led by the International Agricultural Research Advisory Group, aims to utilize excess water across various economic sectors to not only enhance productivity but also improve community livelihoods and environmental conditions. Consequently, these programs strive to enhance agricultural production per unit of water, starting from plant level and extending to irrigation systems and watersheds.

Recent years have seen valuable research on estimating ET under changing climate conditions worldwide. Gonçalves et al. (2022) examined RS ET determination using GeeSEBAL for sugarcane irrigation management in Brazil. Landsat 8 (OLI) and Landsat 7 (ETM+) satellite images were employed in the study, revealing energy balance components computed by GeeSEBAL to be acceptable with R2 = 0.97. Kayser et al. (2022) explored GeeSEBAL's automatic calibration uncertainty for estimating ET in subtropical humid climates, validating ET estimates with air circulation covariance measurements. Processing 132 cloudless Landsat images yielded an RMSE value of 0.91 mm/day and R2 of 0.82. Sun et al. (2022) assessed changes in water use efficiency and its correlation with land use changes using three different data inputs in China's Yellow River. The paper proposed a framework for real ET processing and WUE calculation through the maximum entropy generation method and GEE environment. By applying this framework and utilizing the three data inputs in GEE, ET and real WUE results were computed from 2001 to 2020. The study demonstrated that this system, combined with the maximum entropy production method, presents new opportunities for enhancing water resources management.

This study aimed to introduce and evaluate new methods for estimating ET using satellite data in the Kermanshah plain lands. The research employed Google Earth Reference ET Engine products and the SEBAL energy balance algorithm to calculate actual ET via the GeeSEBAL model on GEE. To evaluate global and national initiatives in the field of ET, OpenET and FAO Wapor were utilized. Landsat satellite images at a 30 × 30-meter resolution from 2013 to 2022, corresponding to the wheat cultivation period in Kermanshah plain, were used to estimate wheat plant ET. The ET time series was generated by utilizing advanced weather data reanalysis from ERA5. The expected outcome of this study is an improvement in water resource management through sustainable methods, offering managers precise information derived from ET data access.

Materials and methods

The study area

Kermanshah province, situated in the western part of the country, relies heavily on agriculture for its economy. The Kermanshah plain, spanning 28797 hectares, is positioned between 46˚45΄ and 47˚5΄ east longitude and 34˚23΄ to 34˚35΄ north latitude (Fig. 1). Every year almost 109,000 hectares of Kermanshah farms are cultivated with wheat. The average elevation of the plain is 1330 meters above sea level, with an average annual temperature of 14.1 degrees Celsius and a total rainfall of 458 mm. Kermanshah is divided into 14 counties and shares borders with Kurdistan province to the north, Lorestan and Ilam provinces to the south, Hamedan province to the east, and Iraq to the west, with over 330 km of common border. The region experiences most of its precipitation from humid Mediterranean fronts, resulting in rain and hail during autumn and spring, and snow during winter. Wheat is the primary agricultural product in the region, covering the largest portion of agricultural land.

Fig. 1
figure 1

Location of the study area in Iran and Kermanshah province

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Lysimeter specification

Studies of wheat ET in 2013 and 2014 were carried out at the Research Farm of Water Resources Engineering Department located on the campus of Faculty of Agriculture and Natural Resources of Razi University of Kermanshah, which has a longitude of 47° 9" east and latitude of 34° 21" north and is located at an altitude of 1319m above sea level. During the study, meteorological data were collected daily from a fully automated meteorological station located 50 m from the lysimetric station site.

Six drainable lysimeters with diameter of 1.20 and depth of 1.20 m were used for this research and wheat was planted on them. The soil texture used in lysimeters was silty clay and its moisture content in the range of soil agronomic capacity was 24% by weight and its specific bulk density was 1.3 g/cm3.

Sampling was started with the beginning of the growing season. In each record, the values of irrigation depth, rainfall, drainage, and changes in soil moisture reserve were recorded and using water balance equation, the actual ET and ET of wheat plants were determined.

The irrigation cycle was selected in a way that the lowest stress was applied to the plant. To measure ET by drained lysimeter for a given period of time, the water balance of the soil was used as Eq. 1.

$$ {\text{ETc }} = I + P - D \pm \Delta S $$
(1)

where ETc = wheat ET (mm); I= irrigation depth (mm); P= precipitation (mm); D= depth of drainage water (mm); ΔS = changes in soil moisture in the given period (mm).

During irrigation, the value of moisture content in soil profile was measured using TDR equipment and installed the corresponding sensors at depths of 10, 30, 60, and 80 cm inside of the drainable lysimeters. Excess water extracted from lysimeters that was drained using underground pipes into tanks in the underground access chamber adjacent to the lysimeters was measured by a calibration container (Fig. 2).

Fig. 2
figure 2

Location of drainage lysimeters in the agricultural campus of Razi University to calculate the ET of wheat plants

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Satellite images used

The satellite images used in the study include Landsat 8 images taken at 16-day intervals during the wheat growing season from 2013 to 2022, ensuring that there are no clouds covering the area. The specific date and time of the satellite imaging are detailed in Table 1. Kermanshah synoptic station served as the reference station (Fig. 3). Essential meteorological parameters for the study encompass solar radiation, relative humidity, wind speed, and temperature on satellite transit days. GeeSEBAL, a novel tool for automated ET estimation, employs SEBAL via GEE. The study integrated GIS and GEE.

Table 1 Satellite imagery used in the research
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Fig. 3
figure 3

Location of the study area, synoptic and lysimetric stations in GEE

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SEBAL algorithm

The immediate calculation of transpiration evaporation (λET) for satellite transit time is determined by the net radiation flux (Rn), sensible heat flux (G), and soil heat flux (H) values per pixel as per the equation by Jiang et al. (2015).

$$ \lambda {\text{ET}} = {\text{Rn}} - G - H $$
(2)

Net radiation (Rn)

The net radiation flux at the Earth's surface is calculated using the balance equation of the diffusion flux from the atmosphere to the surface. The amount of net radiation on the Earth's surface is determined by the following equation.

$$ {\text{Rn}} = \left( {1 - \alpha } \right){\text{Rs}} \downarrow + {\text{RL}} \downarrow - {\text{RL}} \uparrow - \left( {1 - \varepsilon_{0} } \right){\text{RL}} \downarrow $$
(3)

where ⍺ Albedo, Rs↓ incoming shortwave radiation (0.3–μm) (W/m2), RL↓ incoming longwave radiation (3–100 μm) (W/m2), RL↑ longwave output (W/m2), ε0 surface emissivities.

Calculation of albedo surface (α)

Albedo represents the ratio of reflected solar radiation to the total solar radiation received on Earth's surface. This value is influenced by the characteristics, composition, and intensity of solar radiation. The surface albedo for individual pixels in satellite imagery is determined using Eq. 4 (Silva et al. 2018).

$$ \alpha = \frac{{\alpha_{{{\text{toa}}}} - \alpha_{{\text{path - radiance}}} }}{{\tau_{b}^{2} }} $$
(4)

In the above equation, \({\tau }_{b}\) is the atmospheric transmission coefficient and \({\alpha }_{\text{toa}}\) is the albedo in suitable atmospheric conditions, which is calculated from Eq. 5. Also, \({\alpha }_{\text{path}-\text{radiance}}\) is the amount of deviant radiation albedo, the average solar radiation in all bands that are reflected to the sensor through the atmosphere before reaching the earth, is a value between 0.025 and 0.04.

$$ \alpha_{{{\text{toa}}}} = \sum \left( {W_{\lambda } \times \rho_{\lambda } } \right) $$
(5)

In Eq. 4, \({\rho }_{\lambda }\) is the reflectivity rate for band λ and \({\rho }_{\lambda }\) is the weight coefficient for λ, which is calculated as Eq 6.

$$ {\text{W}}_{\lambda } = \frac{{{\text{ESUN}}_{\lambda } }}{{\sum {\text{ESUN}}_{\lambda } }} $$
(6)

According to the above relationship, ESUNλ is the average exo-atmospheric brightness for the λ band, whose value is different for each band and a constant value for each Satellite image sensor (Richter et al. 2009). The coefficients of each reflective band of Landsat 8 images for albedo calculation are shown in Table 2.

Table 2 Coefficients of reflective bands of Landsat 8 images
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Normalized difference vegetation index (NDVI)

Vegetation indicators were enhanced with the introduction of the initial satellites designed for monitoring and assessing vegetation cover (Huete and Glenn 2011). This index relies on vegetation quantity and health. NDVI is computed using infrared and near-infrared bands satellite data.

$$ {\text{NDVI}} = \frac{{\rho_{{{\text{NIR}}}} - \rho_{R} }}{{\rho_{{{\text{NIR}}}} + \rho_{R} }} $$
(7)

In which ⍴NIR is the reflectivity recorded in the near-infrared band and ⍴R is the reflectivity recorded in the infrared band. In different sensors, the bands of infrared and near-infrared wavelengths are different. In Table 3, the infrared and near-infrared bands in the OLI sensor are shown.

Table 3 Infrared and near-infrared bands in OLI sensor
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Soil adjusted vegetation index (SAVI)

The SAVI index is an indicator to eliminate the effects of background soil on the NDVI index.

$$ {\text{SAVI}} = \frac{{\rho_{{{\text{NIR}}}} - \rho_{R} }}{{\rho_{{{\text{NIR}}}} + \rho_{R} + L}}\left( {1 + L} \right) $$
(8)

In which L is a factor for eliminating the effects of vegetation that ranges from zero for dense vegetation to 1 for low density.

Leaf area index (LAI)

LAI is the ratio of the total area of the plant leaves to its canopy. Different relationships have been developed to obtain leaf area index, one of these methods is as follows:

$$ {\text{LAI}} = \frac{{ - \ln \left( {\frac{{0/69 - {\text{SAVI}}}}{0/59}} \right)}}{0/91} $$
(9)

Calculation of incoming short-wavelength radiation (R s↓)

Incoming short-wavelength solar radiation is the sunlight that reaches Earth's surface directly or diffusely under clear sky conditions. This value is influenced by atmospheric factors, time, and location, and can be determined using relationships (10) to (12).

$$ R_{{\text{S}}} \downarrow = G_{{{\text{sc}}}} \times {\text{COS}}\theta \times {\text{dr}} \times \tau_{{{\text{sw}}}} $$
(10)
$$ {\text{dr}} = 1 + 0/0033 \times {\text{COS}}\left( {\left( {{\text{DOY}} \times 2\pi } \right)/365} \right) $$
(11)
$$ \tau_{{{\text{sw}}}} = 0/75 + 2 \times 10 - 5 \times Z $$
(12)

The solar constant GSC is 1367 (Wm-2). The angle θ is 90°-β, where β represents the altitude angle of the Sun. The term dr squared inversely represents the relative distance of the Earth to the Sun, which is derivable from the Day of Year (DOY) and τ_sw, the atmospheric permittivity. The latter is determined using the Z zone average height, which is better calculated using the DEM instead of the area's average height for more precise outcomes. RS↓ can vary between 200 and 1000 depending on the specific time and location of the image.

Calculation of incoming longwave radiation (R L↓)

The incoming longwave radiation is the thermal flux entering the atmosphere to the surface of the earth in terms of W/m2 calculated by the Stephen–Boltzmann equation as (13).

$$ R_{{\text{L}}} \downarrow = \varepsilon_{{\text{a}}} \sigma T_{{\text{a}}}^{4} $$
(13)

In relation to (13), εa is the atmospheric emission coefficient, σ is the Stephen–Boltzmann constant, and Ta is the near-surface air temperature in degrees Kelvin. In this context, the cold pixel temperature can be considered as the air temperature near the surface. To calculate the emission coefficient, the following empirical relation can be utilized.

$$ \varepsilon_{{\text{a}}} = 0/85 \times \left( { - \ln \tau_{{{\text{sw}}}} } \right)^{0/09} $$
(14)

In relation to (14), τsw is the atmospheric transit coefficient. The incoming long wavelength radiation has values ranging from 200 to 600 (Wm-2).

Calculation of outgoing longwave radiation (R L↑)

The outgoing longwave radiation is calculated according to Stephen–Boltzmann's law and using the (15) relation.

$$ R_{{\text{L}}} \uparrow = \varepsilon_{0} \times \sigma \times T_{{\text{S}}}^{4} $$
(15)

In relation to (15), RL↑ longwave radiation, \({\varepsilon }_{0}\) power that represents thermal radiation behavior in the range of 6–14 microns and TS (Kelvin) surface temperature.

Earth surface temperature (T s)

Surface temperature is one of the main parameters for calculating SEBAL algorithm.

Surface temperature of Landsat Satellite 8

In Landsat 8, thermal bands 10 or 11 are used to calculate surface temperature. In this study, the following relationship is used to estimate the surface temperature.

$$ T_{{\text{S}}} = \frac{K2}{{\ln \left( {\frac{{\varepsilon_{{{\text{NB}}}} K1}}{{{\text{RC}}}} + 1} \right)}} $$
(16)

In the above relationship, RC is a surface corrected thermal radiance for bands 10 or 11 Landsat 8, values of K1 and K2 are constant values equal to 774.89 and 1321.08, respectively, and εNB emission coefficient in low bands of thermal width which can be calculated with \({\varepsilon }_{0}\) according to the below table.

Surface emission (εNB)

Surface emission is the ratio of heat energy radiated by the surface to the heat energy radiated by the black body at the same temperature. The surface emissivity for the earth's surface coverage, established through empirical relations, is presented in Table 4 (Allen et al. 2005).

Table 4 Surface emissions based on land cover
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Hot and cold pixels

The SEBAL method uses two square pixels of indicators to determine the constant boundary conditions in the energy balance equation. One of these pixels, called the cold pixel, is from a well-watered and well-watered area, the ground surface temperature in this pixel is close to the air temperature and the evaporation and sweating equivalent to ET is referenced. The second pixel, called the warm pixel, is dry and vegetation-free agricultural land where ET is considered zero.

Soil heat flux (G)

Soil heat flux is the amount of heat transfer within the soil and vegetation due to molecular conductivity. The following equation can be used to calculate it (de Lima et al. 2021).

$$ \frac{G}{{{\text{Rn}}}} = \frac{{T_{{\text{s}}} }}{\alpha } \times \left( {0/0032 \times \alpha + 0/0062 \times \alpha^{2} } \right) \times \left( {1 - 0/978 \times {\text{NDVI}}^{4} } \right) $$
(17)

In this regard, Ts is surface temperature, surface albedo α, NDVI vegetation index, and solar net radiation Rn. The above relationship has acceptable performance in agricultural lands. In SEBAL, the ratio of G/R_n for clear, deep water, and snow is equal to 0.5.

This ratio is estimated using the values of net radiation flux, heat flux, and soil heat flux obtained at the time of imaging (Ferreira et al. 2013).

$$ \Delta_{{{\text{ins}}}} = \frac{{R_{n} - G - H}}{{R_{n} - G}} $$
(18)

Heat flux (H)

The last parameter in the energy balance equation is the H (the permeable heat flux). Thermal heat flux is the amount of heat loss to air by convection and molecular conductivity due to the difference in temperature. The most important and difficult part of SEBAL's algorithm is to calculate the heat flux based on heat transfer and can be calculated through the following equation (Wang et al. 2020).

$$ H = \frac{{\rho \times Cp \times d_{T} }}{{r_{{{\text{ah}}}} }} $$
(19)

In the above relationship, ρ represents the air density (Kg/m3), Cp is the specific heat of air (1004J/kg/k), dT denotes the temperature difference T1-T2 at heights Z1-Z2 in Kelvin, and rah stands for the aerodynamic resistance for heat transfer (m/s). The challenge in computation arises from the presence of two unknowns, rah and dT, in Eq 19, making it a complex problem to solve. To simplify the calculations and address this issue, the method involves utilizing two hot and cold pixels as discussed earlier to derive dependable estimates for H.

To account for the impact of buoyancy resulting from surface heating, SEBAL incorporates the Monin–Obukhov theory through an iterative procedure. When computing the sensible heat flux, it is crucial to account for atmospheric stability conditions, particularly under dry circumstances. The atmosphere can exist in three stable states: unstable, neutral, and stable. The stability or instability of the air is determined by the Monin–Obukhov length, where an atmosphere is considered unstable if L<0, neutral if L=0, and stable if L>0. The Monin–Obukhov length is calculated using the following equation (Sun et al. 2011).

$$ L = \frac{{\rho {\text{C}}pU*^{3} T_{{\text{s}}} }}{KgH} $$
(20)

In SEBAL algorithm, we first obtain the initial values of U* and rah in the meteorological station by assuming neutral atmospheric conditions:

$$ r_{{{\text{ah}}}} = \frac{{\ln \left( {\frac{{Z_{2} }}{{Z_{1} }}} \right)}}{{U_{*} \times K}} $$
(21)
$$ U_{*} = \frac{{KU_{x} }}{{\ln \frac{{Z_{x} }}{{Z_{{{\text{om}}}} }}}} $$
(22)
$$ Z_{{{\text{om}}}} = 0/12 \times h $$
(23)
$$ U_{200} = U_{*} \frac{{\ln \left( {\frac{200}{{Z_{{{\text{om}}}} }}} \right)}}{K} $$
(24)

In the relationship between Z1 and Z2, elevations above displacement are zero when Z1 equals vegetation height and Z2 is slightly higher. The SEBAL algorithm uses 0.1 m and 2 m for these values. K is the Van Carmen constant at 0.41. U* represents frictional velocity indicating turbulence fluctuations in air (m/s), \({Z}_{\text{om}}\) is momentum roughness, Ux is wind speed at Zx height, and H is average vegetation height in meters. Initially, Zom is calculated by averaging vegetation height near the weather station. Then, the target altitude with average wind speed at that level is determined, yielding U* for the weather station using the mentioned formulas. Subsequently, rah for the weather station is computed. With U* and rah known, U200 for the weather station is derived, followed by calculating rah for the entire satellite image. To find the total image \({r}_{\text{ah}}\), \({Z}_{\text{om}}\) for the whole satellite image is necessary. If a land use map is available, \({Z}_{\text{om}}\) per pixel can be obtained from a provided Table 5.

Table 5 Zom values for types of ground surfaces
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For agricultural areas and if there is no map available, the value of \({Z}_{\text{om}}\) or as a function of the leaf area index is calculated:

$$ Z_{{{\text{om}}}} = 0/018LAI $$
(25)

Or by using albedo and NDVI from the following relationship.

$$ Z_{{{\text{om}}}} = \exp \left[ {a\left( {\frac{{{\text{NDVI}}}}{\alpha }} \right) + b} \right] $$
(26)

The coefficients of a and b are based on the sampling of the area can be estimated.

In the next step, we need to get the dT value for the points of the image so that we can reach the final goal of calculating the heat flux. In the previous steps, we selected hot and cold pixels and took their main information. In cold pixels, it is assumed that all the energy is spent on evapotranspiration. On the other hand, in the warm pixel that is without vegetation or water, we assume that the rate of ET is zero. The choice of these two pixels requires skill and practice, and the quality of ET calculations depends on the precise selection of these two pixels. By specifying hot and cold pixels, the difference in air temperature and aerodynamic temperature can also be estimated. Thus, between the surface temperature (Ts) of two hot and cold pixels and the difference in temperature \({\text{dT}}_{\text{cold}}\) and \({\text{dT}}_{\text{hot}}\), they establish a regression relationship (dT=b+aTs) and by finding the regression coefficients (a), (b) and specifying the total dT, they estimate the value of H with \({r}_{\text{ah}}\). dT for two hot and cold pixels is obtained through the following relationships.

$$ {\text{dT}}_{{{\text{cold}}}} = \frac{{H_{{{\text{cold}}}} \times r_{{\text{ah - cold}}} }}{{\rho \times C_{p} }} $$
(27)
$$ {\text{dT}}_{{{\text{hot}}}} = \frac{{H_{{{\text{hot}}}} \times r_{{\text{ah - hot}}} }}{{\rho \times C_{p} }} $$
(28)

In hot and cold pixels, the heat flux is obtained from the following relationships.

$$ H_{{{\text{hot}}}} = {\text{Rn}} - G $$
(29)
$$ H_{{{\text{cold}}}} = {\text{Rn}} - G - K\lambda {\text{ET}}_{{{\text{ref}}}} $$
(30)

In which \({\lambda \text{ET}}_{\text{ref}}\) is equal to hourly reference ET and is estimated through the FAO Penman–Monteith relation. The K-factor per cold pixel of the plant creates a high sensitivity in the SEBAL model, and its determination requires the coverage information in the cold pixel coordinates. On the other hand, the air temperature for pixels is calculated from the following Eq 31.

$$ T_{{{\text{air}}}} = T_{s} - d_{T} $$
(31)

By calculating the air temperature for each pixel, the air density can be corrected and the initial H value for each pixel can be calculated by inserting into the equation. By specifying the initial perceptible flux, it is necessary to calculate the length of Monin–Obukhov described earlier. Then, we get the modified values U and \({\text{r}}_{\text{ah}}\) using the following relations.

$$ U_{*} = \frac{{U_{200} K}}{{\ln \left( {\frac{200}{{Z_{om} }}} \right) - \emptyset_{m} \left( {200{\text{m}}} \right)}} $$
(32)
$$ r_{ah} = \frac{{\ln \left( {\frac{{z_{2} }}{{z_{1} }}} \right) - \emptyset_{{hz_{2} }} + \emptyset_{{hz_{1} }} }}{{KU_{*} }} $$
(33)

In these relationships, ∅m (200 m), stability correction for momentum transfer at 200 m and ∅h (z2) and ∅h (z1), respectively, are stability corrections for heat transfer at 2 and 0.1 m, respectively.

If L<0:

$$ \emptyset_{m} \left( {200{\text{m}}} \right) = 2\ln \left( {\frac{{1 + X_{{200{\text{m}}}} }}{2}} \right) + \ln \left( {\frac{{1 + X^{2} \left( {200{\text{m}}} \right)}}{2}} \right) - 2{\text{ARCTAN}}\left( {X_{{200{\text{m}}}} } \right) + 0/5\pi $$
(34)
$$ \emptyset_{h} \left( {2{\text{m}}} \right) = 2\ln \left( {\frac{{1 + X^{2} \left( {2{\text{m}}} \right)}}{2}} \right) $$
(35)
$$ \emptyset_{h} \left( {0/1{\text{m}}} \right) = 2\ln \left( {\frac{{1 + X^{2} \left( {0/1{\text{m}}} \right)}}{2}} \right) $$
(36)

If L>0:

$$ \emptyset_{m} \left( {200{\text{m}}} \right) = - 5\left( \frac{2}{L} \right) $$
(37)
$$ \emptyset_{h} \left( {2{\text{m}}} \right) = - 5\left( \frac{2}{L} \right) $$
(38)
$$ \emptyset_{h} \left( {200{\text{m}}} \right) = - 5\left( {\frac{0/1}{L}} \right) $$
(39)

If L = 0:

$$ \emptyset_{m} ,\emptyset_{h} = 0 $$
(40)

In these relations, the values x200, x2, and x 0/1 are obtained from the following relations:

$$ X\left( {200{\text{m}}} \right) = \left( {1 - 16\frac{200}{L}} \right)^{0/25} $$
(41)
$$ X\left( {0/1{\text{m}}} \right) = \left( {1 - 16\frac{0/1}{L}} \right)^{0/25} $$
(42)

Calculation of ET

Finally, to calculate ET in mm/day, we use the following formula:

$$ {\text{ET}}_{{{\text{inst}}}} = 3600\frac{{\lambda {\text{ET}}}}{\lambda } $$
(43)

In this formula, λ is the latent heat of vaporization, which is 2450 J/g. If we want to calculate the amount of ET in mm/day, we use the following equation:

$$ {\text{ET}}_{r} F = \frac{{{\text{ET}}_{{{\text{inst}}}} }}{{{\text{ET}}_{r} }} $$
(44)

Reference ET is calculated using available meteorological data, using FAO Penman–Monteith and ETinst, calculated hourly ET in the previous step. The amount of daily ET is obtained from the following relationship:

$$ {\text{ET}}_{24} = {\text{ET}}_{r} 24 \times {\text{ET}}_{r} F $$
(45)

ETr24 is the reference ET throughout the day. Finally, ET maps can be prepared and stored using the above relationships (Silva et al. 2018).

GeeSEBAL

The GeeSEBAL algorithm was created within the GEE using JavaScript and implements the SEBAL algorithm. The emergence of the GEE has significantly transformed satellite image processing, accelerating data processing and calculation, thereby enhancing the efficiency of RS in earth sciences. Improved computational speed, automation, and development of user-friendly algorithms are key attributes of this system. This system enables rapid access to large satellite image datasets without the need for extensive downloads or complex processes in RS software (Gorelick et al. 2017). This study utilized the ERA5 reanalysis dataset for meteorological inputs (Munoz-Sabater et al. 2021). Figure 4 illustrates the calculation stages of wheat ET using the GeeSEBAL model.

Fig. 4
figure 4

Steps to calculate ET using GeeSEBAL algorithm.

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Daily real ET (mm d-1) estimated by SEBAL is calculated using the following equation (Bastiaanssen 2000).

$$ {\text{ET}} = 0.035 \times \left( {\frac{{{\text{LE}}}}{{{\text{Rn}} - G}}} \right)\left[ {\left( {1 - \alpha } \right){\text{Rs}} - 110\tau 24h} \right] $$
(46)

where LE, latent heat flux (W m-2); rn, Net radiation (W m-2); G, soil heat flux (W m-2); α, surface albedo; Rs, daily global solar radiation (W m-2); τ24h, the average daily atmospheric transmittance.

In this algorithm, a linear equation (dT= a+bTsup) is assumed between air temperature (dT) and surface temperature. Here, 'a' and 'b' represent calibration coefficients. In the cold pixel, H=0, and in the warm pixel, LE=0 (Oliveira et al. 2014). The SEBAL algorithm, developed in GEE, is implemented using JavaScript exclusively for Landsat8 images. GEE provides free access with a vast and expanding user base, access to all Landsat images, direct management of time series stacks, and ease of parallel processing to quicken calculations. GEE is an integrated environment designed for petabyte-scale scientific analysis and visualization of spatial datasets. The system offers a vast data list analyzed by thousands of computers (Xiong et al. 2017). Users can request new catalogs to be added to the public catalog, upload their data via the REST interface using command-line or browser-based tools, and share with other users or groups as desired. Users can access public data as well as their private data using the operator library provided by the API in GEE and analyze them. These operators are part of a large parallel processing system that automatically distributes calculations and ensures high throughput (Patel et al. 2015).

Model validation

In 2013 and 2014, the GeeSEBAL model's accuracy was assessed by comparing the results with lysimeter data using metrics such as R2, RMSE, MAE, NSE, and NRMSE that are presented in Table 6. RMSE values indicate errors in statistical methods. Lower RMSE values signify higher model accuracy, with zero indicating perfect estimation. Absolute error ranges from zero to infinity, always positive; closer to zero implies better performance. NSE ranges from -∞ to 1, with 0 to 1 considered acceptable and 1 being optimal. Correlation coefficient values range from 0 to 1; closer to 1 indicates better performance. Variable estimate accuracy is classified into four categories based on NRMSE values: NRMSE<0.2, 0.2≤NRMSE<0.3, 0.3≤NRMSE<0.5, and NRMSE≥0.5 (Li et al. 2013).

Table 6 Evaluation criteria of indicators
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In which Si is the estimated value, \(\overline{\text{S} }\) is the mean of the estimated value, Oi is the observed value, \(\overline{\text{O} }\) is the mean of the observed value, and N is the number of data.

Results and discussion

The study aimed to compare GeeSEBAL model results for estimating ET of wheat plant with data from ground (lysimeter). GEE coding was used to calculate actual ET in the study area. The basin shapefile has been uploaded to the GEE environment, and the GeeSEBAL code results, including a histogram, ET colorful map, and ET values, are displayed in Fig. 5.

Fig. 5
figure 5

How to code and extract the ET map in the GEE environment

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Based on data from a 10-year research period, the satellite passed over the study area a total of 33 times without clouds between 2013 and 2022. Reference ET in this study was computed using established products in GEE, utilizing the FAO Penman–Monteith relationship. In order to validate the results of GeeSEBAL algorithm, 2013 and 2014 lysimetry results were selected and finally the ET values were calculated for other years as well. The average ET values derived from the GeeSEBAL algorithm for each satellite image captured for 2013 and 2014 are detailed in Table 7.

Table 7 Reference and daily wheat ET values obtained from the GeeSEBAL algorithm for the years 2013 and 2014
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Considering that the goal was to compare the model with the lysimeter, the years 2013 and 2014 were chosen to determine the NDVI and LST values. The subsequent images display the minimum and maximum values of NDVI and LST recorded in the region on various dates. These visual representations allow for the adjustment of hot and cold pixel values in the region (Fig. 6).

Fig. 6
figure 6

Trends of NDVI and TS changes calculated in 2013–2014 using the GeeSEBAL model

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In Figs. 7 and 8, dated 01/06/2013 and 20/06/2014, the ET in Kermanshah plain, as per the GeeSEBAL algorithm results, shows an increase. This rise is attributed to the growth in crop vegetation, air temperature, and subsequently ET in these regions. Analysis of the maps indicates that well-irrigated areas exhibit the highest ET rates. Moving away from agricultural lands toward areas with sparse vegetation leads to a significant decrease in the ET rate. Figure 9 illustrates the reference and daily ET values recorded by lysimeter and calculated by the GeeSEBAL algorithm during satellite overpass days in the Kermanshah study area. The average ET for both methods in 2013 and 2014 stood at 13.61 and 13.95 mm/day for GeeSEBAL and lysimeter, respectively.

Fig. 7
figure 7

Instantaneous ET calculated in 2013–2014 using GeeSEBAL model

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Fig. 8
figure 8

Map of actual ET for the wheat growing season in Kermanshah plain.

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Fig. 9
figure 9

Reference and daily ET measured by lysimeter and estimated by GeeSEBAL algorithm

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In Table 8, the statistical indices for 2013 and 2014 are presented. The correlation coefficients between the measured and estimated ET values using lysimeter and GeeSEBAL are notably high at 0.98 and 0.99, respectively. This suggests a strong alignment of the GeeSEBAL method in determining the ET of wheat. The RMSE values between lysimeter and GeeSEBAL data are 0.42 and 0.55, indicating good agreement. Additionally, NSE and MAE coefficients stand at 0.95, 0.96, 0.40, and 0.42, respectively. With NRMSE values of 0.07 and 0.04 for reference and daily ET, the effectiveness of the GeeSEBAL algorithm in the Kermanshah plain region is well demonstrated. Leveraging the benefits of the GEE environment, it is advisable to employ the GeeSEBAL method for estimating wheat plant transpiration and reference ET in this area.

Table 8 The performance evaluation results of GeeSEBAL algorithm based on lysimeter data in 2013–2014
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Given that GeeSEBAL effectively fitted the values of wheat ET for the years 2013 and 2014, it is possible to rely on its results for future years as well (Table 9).

Table 9 Reference and daily wheat ET values obtained from the GeeSEBAL algorithm for the years 2015-2022
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Changes in ETo are crucial for agricultural water management, irrigation system design, and planning. The linear regression trend shows a consistent increase in daily ET from 2013 to 2022 as depicted in Fig. 10.

Fig. 10
figure 10

Daily ET changes of wheat from 2013 to 2022 using GeeSEBAL algorithm

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Tabari et al. (2011) showed a positive trend in 70% of the stations in western Iran using the Mann–Kendall test and Sen's slope estimator, and in 75% of stations using linear regression for annual ETo series in the region, which is in line with the findings of this study.

Due to the increase in the rate of ET, the maps of actual ET, Trends of NDVI and TS changes, and instantaneous ET have been extracted for the year 2022 in Figs. 11, 12, and 13.

Fig. 11
figure 11

Maps of actual ET for the wheat growing season for 2022 in Kermanshah plain.

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Fig. 12
figure 12

Trends of NDVI and TS changes calculated in 2022 using the GeeSEBAL model

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Fig. 13
figure 13

Instantaneous ET calculated in 2022 using GeeSEBAL model

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Discussion

Few studies have explored the ET levels using the GeeSEBAL model and Landsat 8 images, making it challenging to contrast the findings of this study with those of other researchers. Santos et al. (2021) introduced the GeeSEBAL code tool, noting its versatility in Earth science applications. Comparisons between EC and estimated data from GeeSEBAL demonstrated the model's effectiveness across various vegetation types and ecosystems, yielding consistently accurate results (RMSD < 0.7 mm/day in most instances). Thus, the tool provides a swift and precise method for estimating ET. In a study by Gonçalves et al. (2022) in a major sugarcane production area, daily ET from GeeSEBAL was compared with EC data, resulting in R2 and RMSE values of 0.97 and 0.46, respectively. Kayser et al. (2022) utilized Eddy covariance data from five flow towers in southern Brazil to validate ET estimates. Processing 132 cloud-free Landsat images with adjusted parameters yielded a RMSE of 0.91 and an R2 of 0.82. Andrade et al. (2024) similarly demonstrated the utility of GeeSEBAL-MODIS for tracking climate change and human-induced effects on ET. This model paves the path for precise global monitoring and enhances progress in worldwide water resource management. In addition, studies that have compared SEBAL results with lysimeter results for wheat plants will be reviewed. Bala et al. (2016) evaluated and validated ET using SEBAL algorithm and lysimetric data in wheat fields in India. Results from this study revealed that the RMSE of crop-growing period was 0.51 mmd−1 for ETSEBAL. NRMSE (0.21) and R2 (0.91) tests indicated that model prediction is significant, and model can be effectively used for the estimation of ET from SEBAL as input of RS datasets. Rawat et al. (2017) determined wheat ET using the SEBAL model. The results demonstrated that the SEBAL-based ET conformed with lysimeter method with R2 value of 0.91. Asadi and Kamran (2022) compared the SEBAL, METRIC, and ALARM algorithms for estimating actual ET of wheat in the Parsabad–Moghan region during the years 2016–2019. The results indicated that the SEBAL algorithm had the lowest RMSE of 0.633 and the highest R2 of 0.93 compared to the lysimetric results. Zoratipour et al. (2023) estimated the daily ET of wheat using two algorithms (SEBS) and (SEBAL) in the central province of Khuzestan during 2019–2020. According to the results, both SEBAL and SEBS algorithms showed the highest compatibility with lysimeter data (R2 = 0.92 and 0.96, RMSE = 2.15 and 1.53 mm/day, respectively). Given that the exponential coefficient (R2=0.99) for determining wheat ET in the GeeSEBAL model has been higher than the results of the SABAL in various studies, it can be concluded that this model is very important for determining ET and can be a suitable replacement for the SABAL algorithm. Also, the difference of this method from other methods is that it can estimate actual ET for a large area such as a plain without the need for land cover data, while other methods like lysimeters simulate or calculate ET at a point or farm level.

Conclusion

Various algorithms and techniques for calculating ET have been suggested, with remote sensing algorithms and satellite imagery playing an increasingly important role. These methods have consistently demonstrated results, particularly when contrasted with the limitations of using lysimeters for estimating ET (e.g., time consumption, high costs, and limited data generalization on a large scale). Employing RS approaches, like the GeeSEBAL model, represents a scientific and cost-efficient method for computing ET at a regional level. Recent progress in RS holds promise for estimating actual ET across extensive agricultural regions. This investigation was carried out in the Kermanshah plain utilizing the open-source SEBAL framework and GEE through the API. A comparison of the ET values derived from the GeeSEBAL algorithm and the lysimeter in the study zone uncovers a strong correlation between the results obtained from both techniques. The R2 values for reference and actual ET were found to be 0.98 and 0.99, respectively. Furthermore, statistical evaluations employing RMSE, MAE, NSE, and NRMSE metrics revealed a slight quantitative variance in the mean ET data between the two methodologies. Given GeeSEBAL's autonomy from land measurements as inputs, this model possesses worldwide applicability for field and regional inquiries into water and energy balance, as well as water resource management in areas lacking climate data. Ultimately, harnessing global meteorological networks and observational datasets with advanced methods for estimating ET in cloud computing environments, like the OpenET initiative, will bolster agricultural water management on a global scale.