1 Introduction

As the competition for the finite water resources on earth increases due to growth in population and affluence, agriculture is faced with intensifying pressure to improve the efficiency of water used for food production (Hsiao et al. 2007; Yao et al. 2018; Mason 2019). Therefore, in increasing population growth, rapid development in agriculture and growing scarcity of water resources, much of the studies have focused on water tension management (Lalehzari et al. 2020), water delivery scheduling (Omidzade et al. 2020), groundwater level monitoring (Fry et al. 2020), efficient allocation of water resources (Yan et al. 2020), and resolving conflicts between water consumers (Lalehzari and Kerachian 2020; Li and Kinzelbach 2020). Irrigation agriculture is one of the large consumers of water in arid and semi-arid regions which its allocation affects directly the yield production, food security, and system efficiency (Fallah-Mehdipour et al. 2013; Sun et al. 2017; Li and Kinzelbach 2020). The sustainable management of water resources in agriculture is often accompanied by the development of simulation models based on the crop growth patterns or estimating the irrigation water requirement (Yan et al. 2018; Sang et al. 2019; Varzi et al. 2019).

Evapotranspiration is an important factor in irrigation water planning and crop water tension monitoring (Li and Kinzelbach 2020). However, direct measurement of actual evapotranspiration is difficult and, at best, mostly provides point values (Akbari et al. 2007; Reyes-Acosta and Lubczynski 2011; Balugani et al. 2017). Actual evapotranspiration (ETa) of plants has been considered as a main factor of decision system in different studies (Parsinejad et al. 2013; Li and Kinzelbach 2020), which is estimated by various procedures such as remote sensing (Mahan et al. 2012; Colak et al. 2015; Veysi et al. 2017), crop water stress index (Lalehzari et al. 2016) and soil moisture balance (Vedula et al. 2005; Balugani et al. 2016; Maroufpoor et al. 2019; Lalehzari and Kerachian 2020).

At present, satellite remote sensing data can be used to estimate actual and potential evapotranspiration in real time at regional scale (Chen et al. 2005; Ines et al. 2006). Moreover, predicting the evapotranspiration based on the satellite images would be helpful to reduce uncertainty in the simulated parameters of the agricultural system and soil water budget (Irmak and Kamble 2009). The main advantages of remote sensing application in agriculture are the high level of covered area and obtainable data without extensive monitoring networks (Chemin et al. 2004; Akbari et al. 2007). Various algorithms have been developed to utilize the satellite observations for quantifying evapotranspiration by different researchers (Allen et al. 2007; Irmak and Kamble 2009; Veysi et al. 2017).

Water shortage in Iran requires the development and evaluation of different decision-making strategies appropriate for irrigation planning to improve production, reduce water consumption and make optimal use of water. Due to the impact of soil water balance on evaporation, transpiration, and water use efficiency, the soil water balance (SWB) was simulated for 13 crops in Shahrekord plain, Iran. Crop growth modeling and soil moisture balance are provided using real-time programing in MATLAB software. Furthermore, SEBAL technique is incorporated as an efficient tool to estimate the actual evapotranspiration in the studied crops for considering the accuracy of predicted results.

2 Materials and Methods

2.1 Soil Water Balance (SWB)

The main structure of the modeling process to estimate the actual evapotranspiration has been summarized in Fig. 1. In the first step, soil water balance (SWB) was formulated in daily time steps by transpiration (Tr), evaporation (E), deep percolation (DP), irrigation (I), and rainfall (R) components, and the water stored in root zone. Therefore, the soil moisture balance in the root zone could be defined as following equation:

$$\Delta S = I + R {-} E {-} Tr {-} DP$$
(1)

where ∆S = the change of the moisture storage in the root zone (mm). As shown in Fig. 1, simulating the crop growth was carried out to achieve the daily values of canopy cover and transpiration (Tr).

$$Tr = Kc \times Ksw \times Kcs \times ET_{o}$$
(2)

where Kc = the crop transpiration coefficient that is addressed as a proportional ratio to the fraction of the soil surface covered by canopy cover, Ksw = the soil water stress coefficient which is smaller than 1, Kcs = the cold stress coefficient which becomes smaller than 1 when there are not enough growing degrees in the day.

Fig. 1
figure 1

Schematic of simulation–optimization model

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The evaporation (E) in Eq. (1) was estimated by multiplying the daily potential evapotranspiration with three reduction factors (Raes et al. 2017).

$$E = Kr \times \left( {\sqrt[4]{{\frac{{RAW_{E} + 1}}{20}}}} \right) \times \left( {1 - Kc} \right) \times ET_{o}$$
(3)

where Kr = the evaporation reduction coefficient which becomes \(0 \le Kr \le 1\); REW = readily allowable water for evaporation (mm) and (1 − Kc) could be defined as soil evaporation coefficient. In Eq. (3), Kr reduces soil evaporation rate when insufficient moisture is available in the root zone to the evaporative demand.

To determine the calibrated values of effective rainfall, a daily soil water balance incorporating crop transpiration, evaporation, irrigation, rainfall, and the storage capacity of the root zone was used. Hence, by considering the daily potential evapotranspiration and rainfall, the daily effective rainfall (RE) was obtained by the following empirical equation:

$$R_{E} = \left( {0.8433*R^{0.6453} - 0.03324} \right) \times 10^{{\left( {0.0332*ETo} \right)}}$$
(4)

Therefore, actual evapotranspiration (ETa) in each day of growing season was obtained based on the simulated canopy cover, water tensions and soil water balance (SWB).

$$ETa = Tr + E$$
(5)

2.2 Satellite Data Processing

For dealing with the necessity of considering the critical growth stages of a crop, four critical times take into consideration including: I: initial canopy cover at the time of 90% crop emergence; II: time to maximum canopy cover; III: start of canopy senescence; and IV: time to maturity. Twenty-four days used for satellite imagery for each crop according to the four above-mentioned points are shown in Fig. 2.

Fig. 2
figure 2

Time points for estimating the actual evapotranspiration using SEBAL

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The Level 1B moderate resolution imaging spectroradiometer, MODIS, images of the Shahrekord plain are acquired to estimate daily evapotranspiration that covered the 2018–2019 growing season at the experimental site. Table 1 shows the details of MODIS products used in the study. The MOD11 L2 data comprised of two thermal bands with a 1- km resolution and were used to estimate surface temperature and emissivity. Extraction of the binary file was performed for two visible (bands 1 and 2), five short-wave infrared (bands 3, 4, 5, 6, and 7) and two thermal (bands 31 and 32) bands.

Table 1 Details of MODIS products
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The original MODIS data were provided in hierarchical data format (HDF). The HDF is a multi-object file format for sharing scientific data in multi-platform distributed environments (Irmak and Kamble 2009). ArcGIS 10.2, ENVI 5, and ERDAS 9.1 were used to analyze the downloaded images. Individual images of each band were created for each day by converting their corresponding HDF files. MODIS land surface temperature (LST) product was downscaled to 250 m using a cubic convolution technique to be consistent with MODIS visible and near-infrared (MOD09) data (Irmak and Kamble, 2009).

SEBAL is based on the energy balance where sensible heat flux is related to temperature. SEBAL requires a number of computational steps to calculate the actual ET, as well as other energy exchanges between land and atmosphere, with the following input data (Bastiaanssen 2000): visible, near-infrared, and thermal infrared bands from MODIS and measurements of air temperature, relative humidity, wind speed, and solar radiation. Under the absence of advection, the energy balance in SEBAL is calculated at an instant time t for each satellite overpass by the following equation (Irmak and Kamble 2009):

$$\lambda ET = R_{n} - G - H$$
(6)

where \(\lambda ET\) is the latent heat flux which is the energy necessary to vaporize water (W m−2); Rn is the net radiation (W m−2), G is the soil heat flux (W m−2) and H is the sensible heat flux (W m−2). The net radiation flux (Rn) is calculated using the surface radiation balance:

$$R_{n} = \left( {1 - \alpha } \right)R_{s \downarrow } + R_{l \downarrow } - R_{l \uparrow } - \left( {1 - \varepsilon_{0} } \right)R_{l \downarrow }$$
(7)

where \(R_{s \downarrow }\) is incoming shortwave, \(R_{l \downarrow }\) is incoming longwave, and \(R_{l \uparrow }\) is outgoing longwave radiation fluxes (W m−2), \(\varepsilon_{0}\) is the surface emissivity, and \(\alpha\) is the surface albedo. G Index as a conduction flux through the soil matrix is driven by a thermal gradient in the uppermost topsoil. Heat flow varies with the state of the vegetation covering the soil that is influencing the light interception by the soil surface. SEBAL computes the ratio G/Rn near midday as:

$$\frac{G}{{R_{n} }} = \frac{{T_{s} }}{\alpha }\left( {0.0038\alpha + 0.0074\alpha^{2} } \right)\left( {1 - 0.98NDVI^{4} } \right)$$
(8)

where Ts is the satellite-derived surface temperature (°C), and NDVI is the normalized difference vegetation index (Eq. 8). The NDVI is the ratio of the differences in reflectivities for the near-infrared band (NIR) and the red band (RED) to their sum:

$${\text{NDVI}} = \frac{{{\text{NIR}} - {\text{RED}}}}{{{\text{NIR}} + {\text{RED}}}}$$
(9)

The sensible heat flux (H) is a convection flux through the atmosphere layers coming from the surface skin boundary layer with the topmost soil/vegetation layer:

$$H = \frac{{\rho_{\text{air}} C_{p} {\text{d}}T}}{{r_{aH} }}$$
(10)

where \(\rho_{\text{air}}\) is the density of moist air (kg m−3), Cp is specific heat of air (1004 J kg−1 K−1), dT is the vertical temperature difference between two heights (z0 = 0.1 m, and zs = 2 m) (K), and raH is the aerodynamic resistance to heat transfer (s m−1). The dT is determined from the hot and cold pixels with assumed values of H (Bastiaanssen 2000). Selection of representative hot and cold pixels is extremely important for the accuracy of the ET estimations with SEBAL or with other models originated from SEBAL because the model is calibrated based on the parameters from the two well-known locations (hot and cold pixels) for the entire image. The energy necessary to vaporize water (\(\lambda ET\)) under given atmospheric conditions is calculated as a residual of the energy balance components in Eq. (5). The instantaneous evaporative fraction (R) is used to express the ratio of the actual to the crop evaporative demand when the atmospheric moisture conditions are in equilibrium with the soil moisture conditions. Daily actual evapotranspiration (ET24) is calculated from the daily averaged net radiation, Rn24. The SEBAL model ignores G for time scales of 1 day or longer. Thus, net available energy (Rn - G) reduces to net radiation (Rn) at daily timescales. T24 is computed as:

$$ET_{24} = 86400\frac{\lambda ET}{\lambda }\frac{{R_{n24} }}{{R_{n} }}$$
(11)

where Rn24 is the 24-h averaged net radiation (W m−2), k is the latent heat of vaporization (J kg−1), and ET24 is daily actual ET (cm day−1).

2.3 Study Area

The case study was conducted in Shahrekord plain during 2018–2019 season. Shahrekord plain located in 32° 07″ N to 32° 35″ N latitude and 50° 38″ E to 51° 10″ E longitude in Chaharmahal and Bakhtiari province, Iran (Fig. 3). The average annual precipitation is only 120 mm, which corresponds to semiarid conditions. Water shortage and agricultural demands are the main reasons for developing the decision-making framework for estimating the crop water requirement in this area. (Tabatabaei et al. 2010; Lalehzari et al. 2010; Fakharinia et al. 2012; Lalehzari et al. 2013, 2014; Lalehzari and Tabatabaei 2015).

Fig. 3
figure 3

The study area in Chaharmahal and Bakhtiari province, Iran

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3 Results and Discussion

The common crops selected to predict the canopy cover and transpiration in study area are summarized in Table 2. Moreover, the canopy cover used for estimating the transpiration in critical points is presented (Table 2). Canopy cover has the considerable role on the calculation of evaporation and transpiration. Evaporation values in the first period of growing season from sowing time to emergence completion are the major fraction of ETa.

Table 2 Canopy cover in four critical stages of growing season
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The actual evapotranspiration values calculated by SWB and SEBAL are compared in the Fig. 4. As shown in the figure and the scatter diagrams (together with calculated coefficient of determination), the accuracy of the SWB technique is satisfactory for predicting ETa, because the recently obtained results are close to the previously obtained by SEBAL.

Fig. 4
figure 4

Comparison between calculated ETa using SEBAL and SWB methods

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Alfalfa is a plant that has different growth periods and structures to simulate growth and calculate water requirements. Therefore, in this study, it was evaluated separately by applying special restrictions. Figure 5 shows a comparison between water requirements at ten points of the alfalfa growing season. Results showed that at the beginning and end of the growing season, the two methods were more compatible with each other. Moreover, it seems to have been more accurate in periods when the role of evaporation outweighed transpiration.

Fig. 5
figure 5

Calculated actual evapotranspiration of alfalfa using SEBAL and SWB methods

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The accuracy of the proposed methods could be evaluated by root mean square error, RMSE, and coefficient of residual mass, CRM (Lalehzari and Boroomand-Nasab 2017).

$${\text{RMSE}} = \pi \sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {ETa_{\text{SEBAL}} - ETa_{\text{SWB}} } \right)_{i}^{2} }}{n}}$$
(12)
$${\text{CRM}} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {ETa_{\text{SEBAL}} - ETa_{\text{SWB}} } \right)_{i}^{{}} }}{{\mathop \sum \nolimits_{i = 1}^{n} ETa_{\text{SWB}} }}$$
(13)

where ETaSEBAL and ETaSWB are the calculated ETa using SEBAL and SWB, respectively, and n is the number of output data. Table 3 shows a summary of the evaluation results for crops and satellite images.

Table 3 Comparison with the soil water balance and SEBAL algorithm for estimating the ETa
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The RMSE has a minimum value of 0, with a better agreement close to 0. The coefficient of residual mass, CRM, indicates the underestimation and overestimation amounts by negative and positive values, respectively. According to the table, daily ETa predicted based on the SEBAL method are underestimated for barley, cucumber, chickpea, and alfalfa and overestimated for wheat, potato, onion, lentil, bean, and maize and are close to the calculated values of SWB for tomato, colza, and sugar beet.

4 Summary and Conclusion

In this study, the remote sensing images as a data source for agricultural water modeling and soil water balance components were considered to estimate the actual evapotranspiration. SEBAL technique was applied to characterize the surface energy balance for predicting the evapotranspiration, and then, the derived ETa was compared with results obtained by a calibrated framework of soil, water, and crop relationships. Preliminary results obtained by applying SWB and SEBAL show a good agreement between the both estimated evapotranspiration fluxes. According to the coefficient of residual mass index, actual evapotranspiration in daily time steps predicted based on the SEBAL method is underestimated for barley, cucumber, chickpea, and alfalfa and overestimated for wheat, potato, onion, lentil, bean, and maize and are close to the calculated values of SWB for tomato, colza, and sugar beet.

Due to the fact that the SWB method is based on step-by-step mathematical calculations, it is more efficient in computer models and optimization than remote sensing. Furthermore, the time required to estimate SWB transpiration is much less than other experimental and physical methods.