Evaluation of Hargreaves–Samani and Valiantzas Solar Radiation Models for FAO-56 Reference Evapotranspiration Estimation in Salem (Valapadi), Tamil Nadu, India

Reference evapotranspiration (ET₀) represents the atmospheric demand for water from a standardized reference surface and is a critical parameter for irrigation scheduling, crop water requirement estimation, and agricultural water management. Accurate estimation of ET₀ plays a vital role in optimizing water-use efficiency and improving agricultural productivity, particularly in regions with limited water resources. The FAO-56 Penman–Monteith method is widely accepted as the standard method for estimating reference evapotranspiration because it integrates the physical processes governing evapotranspiration and has been validated across a wide range of climatic conditions worldwide (Allen et al., 1998).

Although the FAO-56 Penman–Monteith method provides reliable estimates of evapotranspiration, its application requires a complete set of meteorological parameters, including solar radiation, air temperature, relative humidity, and wind speed. In many agricultural regions, especially in developing countries, such comprehensive meteorological data are not always available due to limited weather-monitoring infrastructure and the absence of reliable radiation-measuring instruments. Solar radiation observations are particularly scarce because radiation sensors require specialized equipment and regular maintenance. Consequently, the lack of solar radiation data limits the application of the FAO-56 Penman–Monteith method in many agricultural areas (Samani, 2000; Djaman et al., 2016).

This challenge is evident in several parts of Tamil Nadu, India, where agricultural meteorological stations often record basic parameters such as temperature and humidity but lack consistent measurements of solar radiation. Since solar radiation is one of the most important drivers of evapotranspiration, its absence introduces significant uncertainty in estimating evapotranspiration and in irrigation planning. To address this limitation, several simplified empirical approaches have been developed to estimate solar radiation using more readily available climatic variables such as air temperature and relative humidity.

One of the most widely used simplified approaches is the Hargreaves–Samani model, which estimates solar radiation based on the difference between daily maximum and minimum air temperatures (Hargreaves & Samani, 1982). Because this model requires only temperature data, it has been extensively applied in regions where radiation measurements are unavailable. However, the accuracy of the Hargreaves–Samani model may vary with local climatic conditions, particularly in humid or tropical regions, where atmospheric moisture influences the solar radiation reaching the surface (Samani et al., 2011).

To improve the performance of temperature-based radiation estimation models, Valiantzas proposed several simplified formulations of evapotranspiration and radiation estimation models using limited meteorological data (Valiantzas, 2006, 2013a, 2013b, 2013c, 2013d). These approaches were developed to improve evapotranspiration estimation in situations where complete meteorological data are not available. Later studies further refined these models and demonstrated their applicability under different climatic conditions (Valiantzas, 2015; Valiantzas, 2018b). In particular, Valiantzas (2017) proposed a modification of the Hargreaves–Samani model that incorporates relative humidity as an additional parameter. By accounting for atmospheric moisture conditions, the modified model aims to improve the accuracy of radiation estimates, especially in climates where humidity significantly influences solar radiation attenuation.

Previous studies have shown that simplified evapotranspiration estimation methods based on limited meteorological data can provide reasonable approximations of the FAO-56 Penman–Monteith method when applied under appropriate climatic conditions (Djaman et al., 2016; Valipour, 2014). Several comparative studies have also demonstrated that simplified Penman-based and temperature-based models can produce acceptable ET₀ estimates under data-limited conditions (Valiantzas, 2012, 2014a, 2014b; Kisi, 2014). These findings indicate that simplified radiation and evapotranspiration models can serve as practical alternatives in regions lacking complete meteorological observations.

Evaluating the performance of these simplified radiation models is therefore important for determining their applicability in regions with incomplete meteorological datasets. Many studies conducted in different climatic regions have demonstrated that temperature-based and humidity-based radiation models can provide acceptable evapotranspiration estimates when radiation measurements are unavailable, although their accuracy often depends on regional climatic characteristics and local calibration (Kisi, 2014; Valiantzas, 2018b).

Recent studies conducted in Tamil Nadu have also evaluated the applicability of simplified radiation estimation models that use temperature and humidity as inputs. For example, a study in Coimbatore demonstrated that both the Hargreaves–Samani and Valiantzas models provide reliable estimates of solar radiation and FAO-56 Penman–Monteith evapotranspiration when meteorological data are limited (Rashwin et al., 2025).

Similarly, under semi-arid conditions in India, the performance of the Hargreaves–Samani and Valiantzas models was assessed using long-term meteorological data, confirming that these models can effectively estimate solar radiation using temperature and humidity parameters (Ramachandran et al., 2022).

These findings highlight the potential of simplified radiation models to support evapotranspiration estimation and irrigation planning in regions where direct solar radiation measurements are unavailable.

The present study focuses on Valapadi in the Salem district of Tamil Nadu, India (11.6686° N latitude and 78.4067° E longitude), located at an elevation of approximately 305 m above mean sea level. The region experiences a tropical climate characterized by relatively high temperatures and moderate humidity throughout the year. Agriculture in the Salem region relies heavily on effective water management, making accurate estimates of evapotranspiration essential for irrigation planning and sustainable agricultural practices. However, the limited availability of solar radiation measurements in the region restricts the application of the standard FAO-56 Penman–Monteith method.

Therefore, there is a need to evaluate alternative approaches to estimating solar radiation that can provide reliable evapotranspiration estimates with limited meteorological data. This study aims to assess the applicability of simplified radiation estimation models for evapotranspiration calculation in Valapadi, Salem. By comparing evapotranspiration estimates obtained using simplified radiation models with those derived from measured solar radiation within the FAO-56 Penman–Monteith framework, the study seeks to determine whether these models can serve as reliable alternatives for evapotranspiration estimation in data-limited environments.

Meteorological data required for estimating reference evapotranspiration using the FAO-56 Penman–Monteith (FAO-56 PM) method were obtained from the NASA POWER database for 25 years, from January 2000 to December 2025, corresponding to the geographical location of Valapadi in Salem district, Tamil Nadu, India. The study area is located at approximately 11.6686° N latitude and 78.4067° E longitude, with an elevation of 305 meters above mean sea level (MSL). The dataset includes key meteorological parameters such as maximum and minimum air temperatures, relative humidity, and solar radiation, which are required for estimating reference evapotranspiration and evaluating simplified solar radiation models.

Estimation of Solar Radiation

 Hargreaves-Samani R_S formula based on T (HS R_S [T])

Hargreaves and Samani (1982) recommended a simplified equation to estimate solar radiation:

                                                        (1)

Where,

R_A is extra-terrestrial radiation [MJ m-2 day-1].

T_max is the maximum temperature [°C]; and

T_min is the minimum temperature [°C].

The radiation (R_A) for each day of the year and at different latitudes can be estimated from the solar constant, the solar declination, and the time of the year using the following equation.

   (2)

Where,

R_A is extraterrestrial radiation [MJ m-2 day-1],

G_SC is solar constant = 0.0820 MJ m-2 min-1,

d_r is the inverse relative distance Earth-Sun,

 is sunset hour angle [rad],

φ is latitude [rad] and

δ is solar declination [rad].

Valiantzas’ R_S formula based on T and RH (Val Rs [T & RH])

            Hargreaves’ model for estimating the solar radiation as a function of RH (Hargreaves and Allen, 2003) is given as follows:

                                                 (3)

Where x is an empirical exponent.

By combining the relevant equations, solar radiation (Rs) can be expressed as a function of air temperature and relative humidity. Valiantzas (2017) estimated the regression coefficients via a calibration procedure using a global climate dataset. The dataset included monthly meteorological observations from the FAO CLIMWAT database (Smith, 1993), representing climatic conditions across several countries and a wide range of environments. Based on this analysis, Valiantzas proposed a simplified empirical formulation that estimates solar radiation using temperature and relative humidity data. This modified version of the Hargreaves–Samani model incorporates both temperature and humidity variables, thereby improving the estimation of solar radiation under varying atmospheric conditions.

                            (4)

In Eq. (4), a value of 1.001 was used instead of 1 to avoid a numerical singularity that occurs when the ratio RH/100 equals 1. Hence, the final form of Valiantzas R_S equation is given as:

                          (5)

Estimation of Reference Evapotranspiration

To calculate this daily water demand, the FAO-56 framework relies on the following comprehensive mathematical relationship:

                       (6)

Where,

ET₀ is reference evapotranspiration [mm day-1],

R_n is net radiation at the crop surface [MJ m-2 day-1],

G is soil heat flux density [MJ m-2 day-1],

T is the mean daily air temperature [°C],

U is wind speed at 2 m height [m s-1],

 is saturation vapor pressure [kPa],

 is actual vapor pressure [kPa],

 is saturation vapor pressure deficit [kPa],

 Δ is the slope of the vapor pressure curve [kPa °C-1],

γ is the psychrometric constant [kPa °C-1]

Comparison of methods

Comparisons were made between the measured solar radiation and the solar radiation derived using the HS Rs [T] model and the Val Rs [T & RH] models separately. In addition, a comparison was made between the reference evapotranspiration estimated from the FAO56-PM equation, using measured Rs values and solar radiation derived from the HS Rs [T] model and the Val Rs [T & RH] models, applied separately

In this study, a comparison is made using simple error analysis and linear regression.

                                            (7)

Where,

S is the regression coefficient (slope of the line),

Y is the reference measured values of R_S, and

X = corresponding estimates of R_S by the comparison formula.

The indices used in the error analysis are the standard error estimate (SEE) and the long-term average ratio (rt).

              (8)

                                  (9)

Where,

Y_i is the reference measured value at i^th data point.

X_i is the corresponding estimate by the comparison formula.

n is the total number of observations.

X_av and Y_av are the long-term average values of the tested models and the reference values, respectively.

Evaluation of Solar Radiation Estimation Models

The primary objective of this study was to evaluate the performance of two empirical models, namely the Hargreaves–Samani (HS) model and the Valiantzas (VAL) model, for estimating solar radiation (Rs) under the climatic conditions of Valapadi, Salem district, Tamil Nadu, India. These models were evaluated using readily available meteorological parameters obtained from the NASA POWER dataset. The performance of these models was assessed by comparing estimated solar radiation with measured data and by examining their influence on FAO-56 Penman–Monteith reference evapotranspiration (ET₀) estimates.

Solar Radiation Estimation

The mean solar radiation values indicate that both the Hargreaves–Samani (HS) model (23.40 MJ m⁻² day⁻¹) and the Valiantzas (VAL) model (22.27 MJ m⁻² day⁻¹) slightly underestimate the observed solar radiation (23.48 MJ m⁻² day⁻¹). The HS model shows only a small underestimation, whereas the VAL model exhibits a larger deviation from the measured values.

The variability of solar radiation estimates was analyzed using the coefficient of variation (CV%). The measured solar radiation showed a CV of 8.06%, while the HS model produced a CV of 8.38%, indicating that the HS estimates closely follow the variability pattern of the measured data. In contrast, the VAL model exhibited a higher CV of 11.44%, indicating greater variability in the radiation estimates it produced.

The higher variability observed in the VAL model may be attributed to the inclusion of relative humidity as an additional parameter in the radiation estimation process. Since relative humidity varies seasonally and responds to atmospheric moisture conditions, it introduces additional fluctuations in the solar radiation estimates, thereby increasing the overall variability of the model output.

Reference Evapotranspiration Estimation

The mean reference evapotranspiration (ET₀) values calculated using the FAO-56 Penman–Monteith method with different radiation inputs are also very similar. The mean ET₀ obtained using measured solar radiation is 5.78 mm day⁻¹, while ET₀ estimated using HS radiation is 5.79 mm day⁻¹, indicating a very close agreement between the two methods. The ET₀ calculated using VAL radiation is slightly lower at 5.58 mm day⁻¹, suggesting a minor underestimation.

The variability of ET₀ estimates also reflects the variability in the radiation inputs. The coefficient of variation for ET₀ calculated using measured radiation is 14.85%, while HS-based ET₀ shows a slightly lower CV of 14.45%, indicating stable evapotranspiration estimates. In contrast, the VAL-based ET₀ exhibits a higher CV of 17.24%, consistent with the greater variability observed in VAL solar radiation estimates.

Overall, the results indicate that the Hargreaves–Samani model provides solar radiation estimates that closely track the variability of measured radiation, yielding evapotranspiration estimates nearly identical to those obtained from measured data. The Valiantzas model, while slightly underestimating radiation and evapotranspiration values, captures greater variability due to the influence of relative humidity. These findings suggest that both models can be used as alternative approaches to estimating evapotranspiration in regions where measured solar radiation data are limited, although the HS model shows slightly better agreement with observed values in the Valapadi region.

Table 1. Summary of Solar Radiation and Reference Evapotranspiration

Table 2.  Statistics of Solar Radiation and Reference Evapotranspiration

Figure 1: Mean Monthly Maximum Temperature for the study area

Figure 2: Mean Monthly Minimum Temperature for the study area

Figure 3: Mean Monthly Relative Humidity for the study area

Figure 4: Mean Monthly Measured Solar Radiation for the study area

Seasonal Pattern of Measured Solar Radiation

The measured solar radiation in the Valapadi region shows a clear seasonal variation throughout the year, influenced primarily by solar geometry, cloud cover, and monsoonal weather conditions. The annual mean measured solar radiation during the study period was 23.48 MJ m⁻² day⁻¹, with values ranging from 19.30 MJ m⁻² day⁻¹ to 26.05 MJ m⁻² day⁻¹.

Higher solar radiation levels are typically observed during the summer months (March–June) when the region experiences clear skies and high solar elevation angles. During this period, daily radiation values frequently range between 24.5 and 26.0 MJ m⁻² day⁻¹, corresponding closely with the upper quartile value of 24.80 MJ m⁻² day⁻¹ and the observed maximum of 26.05 MJ m⁻² day⁻¹.

In contrast, solar radiation decreases during the monsoon season (July–October) due to increased cloud cover, atmospheric moisture, and rainfall associated with southwest and northeast monsoon systems. During this period, radiation values generally fall within the range of 21.5 to 23.0 MJ m⁻² day⁻¹, with a lower quartile of 21.78 MJ m⁻² day⁻¹.

During the winter season (November–February), solar radiation gradually increases again as cloud cover decreases and atmospheric conditions stabilize. Radiation values during this period generally range from 23.0 to 24.5 MJ m⁻² day⁻¹, close to the median of 24.46 MJ m⁻² day⁻¹.

Overall, the seasonal distribution of measured solar radiation in Valapadi shows higher radiation in summer, moderate levels in winter, and relatively lower radiation during the monsoon period. This seasonal variability directly influences evapotranspiration rates and highlights the importance of accurate solar radiation estimation for reliable evapotranspiration modeling in the region.

Daily and Monthly Variation of Reference Evapotranspiration (ET₀)

The daily variation in reference evapotranspiration (ET₀) in the Valapadi region reflects seasonal changes in solar radiation, air temperature, and atmospheric moisture. Based on the FAO-56 Penman–Monteith method using measured solar radiation, the average daily ET₀ during the study period was 5.78 mm day⁻¹, with values ranging from 3.77 mm day⁻¹ to 6.91 mm day⁻¹.

The distribution of ET₀ values indicates moderate variability throughout the year, with a standard deviation of 0.86 mm day⁻¹. The interquartile statistics show that 25% of the ET₀ values were below 4.93 mm day⁻¹, and 75% were below 6.51 mm day⁻¹, indicating that most daily evapotranspiration values fall within this range. The median ET₀ value of 6.16 mm day⁻¹ indicates that evapotranspiration demand is relatively high for a significant portion of the year, owing to the warm climatic conditions in the Salem region.

The seasonal variation of ET₀ follows the typical tropical climatic pattern observed in Tamil Nadu. Higher evapotranspiration rates are generally observed during the summer months (March–June) when air temperature and solar radiation are at their peak. During this period, ET₀ values frequently approach the upper range of 6.5 to 6.9 mm day⁻¹, reflecting high atmospheric evaporative demand.

During the monsoon season (July–October), ET₀ values tend to decrease due to increased cloud cover, higher humidity levels, and reduced solar radiation. As a result, evapotranspiration values typically fall within the range of 5.0 to 5.8 mm day⁻¹ during this period.

The winter season (November–February) shows the lowest evapotranspiration values due to relatively lower temperatures and reduced solar radiation. During this period, ET₀ values may decrease to approximately 3.8–5.0 mm day⁻¹, which corresponds closely with the observed minimum value of 3.77 mm day⁻¹.

The monthly ET₀ pattern shows a gradual increase from winter to summer, reaching maximum levels during the pre-monsoon period, followed by a decline during the monsoon months. This seasonal trend is consistent with the observed variations in solar radiation and temperature in the Valapadi region. Overall, the results indicate that evapotranspiration demand remains relatively high throughout the year, emphasizing the importance of accurate ET₀ estimation for irrigation management and water resource planning in the region.

Comparison of Measured Solar Radiation with Hargreaves–Samani and Valiantzas Models

The performance of the Hargreaves–Samani (HS) and Valiantzas (VAL) models for estimating solar radiation (Rs) was evaluated by comparing estimated values with measured solar radiation data for the Valapadi region. The comparison was carried out using statistical indicators including the coefficient of determination (R²), standard error of estimate (SEE), and ratio statistic (rt).

The measured solar radiation during the study period had a mean of 23.48 MJ m⁻² day⁻¹, a standard deviation of 1.89 MJ m⁻² day⁻¹, and values ranging from 19.30 to 26.05 MJ m⁻² day⁻¹. The HS model produced a mean radiation estimate of 23.40 MJ m⁻² day⁻¹, which is very close to the measured mean value, indicating a minimal mean deviation of approximately 0.34%. The VAL model produced a slightly lower mean radiation value of 22.27 MJ m⁻² day⁻¹, corresponding to an underestimation of approximately 5.15% compared with the measured radiation.

In terms of variability, the HS model exhibited a standard deviation of 1.96 MJ m⁻² day⁻¹, comparable to that observed in the measured radiation data. In contrast, the VAL model showed higher variability, with a standard deviation of 2.55 MJ m⁻² day⁻¹, indicating greater sensitivity to meteorological inputs, such as relative humidity, included in the model formulation.

The statistical performance indicators further highlight differences in model performance. The comparison between measured radiation and HS-estimated radiation produced a coefficient of determination (R²) of 0.3697, indicating a relatively weak linear correlation between the two datasets. The standard error of estimate (SEE) for the HS model was 1.71 MJ m⁻² day⁻¹, indicating a moderate deviation of the estimated values from the observed radiation. However, the ratio statistic (rt) of 0.9964 indicates that the HS model reproduces the overall magnitude of solar radiation very closely to the measured values.

In contrast, the Valiantzas model demonstrated a much stronger relationship with measured radiation, with an R² of 0.8874, indicating a high degree of correlation between estimated and observed radiation. The SEE value of 1.57 MJ m⁻² day⁻¹ is slightly lower than that of the HS model, suggesting improved predictive accuracy. The rt value of 0.9483 indicates a small systematic underestimation of solar radiation by the VAL model.

Overall, the results indicate that while the Hargreaves–Samani model reproduces the mean magnitude of solar radiation very closely, it shows weaker correlation with measured radiation values. In contrast, the Valiantzas model demonstrates a significantly stronger correlation with measured radiation and lower prediction error, although it slightly underestimates the mean radiation levels. These findings suggest that including relative humidity in the Valiantzas model improves its ability to capture variability in solar radiation under the climatic conditions of the Valapadi region.

                                                  5a)                                                                                    5b)

Figure 5a – 5b: Comparison of Daily Solar Radiation Estimated by Hargreaves-Samani and Valiantzas’ Model Versus Measured Solar Radiation for the study area

FAO Reference Evapotranspiration (ET₀) Using Measured, HS, and Valiantzas Radiation

Reference evapotranspiration (ET₀) for the study area was estimated using the FAO-56 Penman–Monteith method with three different solar radiation inputs: measured solar radiation (Rs), Hargreaves–Samani estimated radiation (HS Rs), and Valiantzas estimated radiation (Val Rs). The ET₀ estimates were compared to evaluate the impact of radiation estimation models on evapotranspiration calculations.

The mean ET₀ calculated using measured solar radiation was 5.78 mm day⁻¹, while the ET₀ estimated using HS-derived radiation was 5.79 mm day⁻¹, indicating an almost identical average evapotranspiration estimate between the two approaches. In contrast, the ET₀ estimated using Valiantzas radiation was slightly lower at 5.58 mm day⁻¹, indicating a small underestimation relative to the measured radiation-based ET₀.

The variability of ET₀ estimates was evaluated using the standard deviation and coefficient of variation (CV). The ET₀ derived from measured radiation exhibited a standard deviation of 0.86 mm day⁻¹ and a CV of 14.85%, indicating moderate variability in evapotranspiration demand. The HS-based ET₀ showed a similar variability with a standard deviation of 0.84 mm day⁻¹ and a CV of 14.45%, suggesting that the HS model closely reproduces the variability observed in the measured radiation-based ET₀ estimates. The VAL-based ET₀ exhibited slightly higher variability, with a standard deviation of 0.96 mm day⁻¹ and a CV of 17.24%, reflecting the greater variability introduced by radiation estimates from the Valiantzas model.

Statistical comparison further confirms the strong agreement between the evapotranspiration estimates derived from measured and model-estimated radiation. The relationship between measured radiation-based ET₀ and HS radiation-based ET₀ showed a coefficient of determination (R²) of 0.8936, indicating a strong correlation between the two datasets. The standard error of estimate (SEE) was 0.2818 mm day⁻¹, and the ratio statistic (rt) was 1.0014, indicating that the HS model reproduces ET₀ values almost identical to those obtained using measured radiation.

Similarly, the comparison between measured radiation-based ET₀ and VAL radiation-based ET₀ showed an even stronger relationship with an R² value of 0.9821, indicating excellent agreement between the two datasets. The SEE value was 0.2593 mm day⁻¹, and the rt value of 0.9648 indicates a slight underestimation of ET₀ when using Valiantzas radiation.

Figure 6: Comparison of FAO-56 ET₀ calculated using measured, HS, and Valiantzas radiation in the study area.

Overall, the results demonstrate that both the Hargreaves–Samanian and Valiantzas radiation models can be effectively used within the FAO-56 Penman–Monteith framework for estimating reference evapotranspiration in the Valapadi region. While the HS model reproduces ET₀ magnitudes very closely to those obtained using measured radiation, the Valiantzas model shows stronger statistical correlation with measured ET₀ values, indicating its ability to better capture radiation variability under the climatic conditions of the study area.

7a)                                                                   7b)

Figure 7a – 7b: Comparison of daily FAO ET₀ estimated by substituting measured and Solar Radiation obtained from the Hargreaves–Samani’s Model and Valiantzas’ Model

This study evaluated the performance of the Hargreaves–Samani (HS) and Valiantzas (VAL) solar radiation estimation models for estimating reference evapotranspiration using the FAO-56 Penman–Monteith (PM) method in the Valapadi region of Salem district, Tamil Nadu. The analysis was conducted using 25 years of meteorological data (2000–2025) obtained from the NASA POWER database.

The results indicate that both HS and VAL models can reasonably estimate solar radiation under the climatic conditions of the study area. The HS model produced a mean solar radiation of 23.40 MJ m⁻² day⁻¹, which is very close to the measured mean of 23.48 MJ m⁻² day⁻¹, whereas the VAL model slightly underestimated radiation, with a mean of 22.27 MJ m⁻² day⁻¹. Statistical evaluation showed that the VAL model had a stronger correlation with measured radiation (R² = 0.887), whereas the HS model exhibited a lower correlation (R² = 0.369) but reproduced the overall magnitude of radiation more closely.

When incorporated into the FAO-56 Penman–Monteith equation, both radiation models produced evapotranspiration estimates comparable to those obtained using measured radiation. The mean ET₀ calculated using measured radiation was 5.78 mm day⁻¹, while HS-based ET₀ was 5.79 mm day⁻¹, and VAL-based ET₀ was 5.58 mm day⁻¹. The HS model showed very close agreement with measured ET₀ values (R² = 0.894, rt ≈ 1.001), indicating minimal bias. The VAL model demonstrated an even stronger statistical relationship with measured ET₀ (R² = 0.982) but showed a slight tendency to underestimate evapotranspiration.

Overall, the findings suggest that both HS and VAL radiation estimation models can serve as effective alternatives for evapotranspiration estimation when measured solar radiation data are unavailable. However, the Hargreaves–Samani model provides ET₀ estimates that are very close to measured values, whereas the Valiantzas model captures radiation variability more effectively by including relative humidity. These results highlight the practical applicability of simplified radiation models for estimating evapotranspiration in data-limited agricultural regions such as Valapadi, Salem, thereby supporting improved irrigation planning and water resource management.

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