Calibration of the Hargreaves–Samani method for the calculation of reference evapotranspiration in different Köppen climate classes

Evapotranspiration (ET) from the Earth's surface is the driver of the global atmospheric water cycle. Accurate irrigation water requirement estimations are crucial for efficient management and planning of water resources and therefore the estimation of reference evapotranspiration (ETo) is important for irrigation management. The knowledge of monthly ET is of great importance in the study and management of present and future water resources and for solving many theoretical problems in the field of hydrology, climatology and geographical ecology. As an agroclimatic index, long-term monthly means of ETo have been widely used to assess the effect of water in agronomy, irrigation planning as well as climate change impact.

Many estimation methods based on meteorological variables have been developed. Numerous formulations, classified as radiation based, temperature based, pan-evaporation based, mass-transfer based as well as combinations of those have been developed (Jensen et al. 1990).

The FAO-56 Penman–Monteith (hereafter FAO-56) equation is recommended as the standard model for estimating ETo (Allen et al. 1998) in all climates. Many researchers have confirmed that FAO-56 is superior to other estimation methods (Hargreaves & Allen 2003; Berengena & Gavilán 2005; Sentelhas et al. 2010; Tabari & Talaee 2011; Rahimikhoob et al. 2012; Maestre-Valero et al. 2013; Tabari et al. 2013; Valiantzas 2013). Due to the absence of experimental records, data-driven models are often calibrated to calculate FAO-56 ETo targets. This procedure is also adopted for calibrating more conventional empirical approaches (Marti et al. 2015).

FAO-56 is of the best quality since it models evaporation based on physics. This is possible only if daily minimum and maximum temperature, humidity, wind speed and sunshine duration are known. However, observations of these variables are often not available. This is the major obstacle to using FAO-56 widely.

Among many other methods, the Hargreaves–Samani (HS) equation, which employs only maximum and minimum temperature data, can be used to approximate ETo. This empirical model is one of the most commonly used. It is recommended by Allen et al. (1998) and again by Allen (2006) as the best approximation to FAO-56. As an empirical method, HS involves empirical coefficients. While HS can be applied with some standard values of these coefficients, several authors (Allen et al. 1998) recommended calibrating HS with respect to FAO-56 at locations with comparable climate.

Samani (2000) found that factors other than solar radiation, wind speed and humidity can influence local observations of the difference in maximum and minimum temperature and thus the results of HS. These factors include latitude, elevation, topography, storm pattern, advection and proximity to large water bodies. At low latitudes, for example, the diurnal temperature range can become very small and, consequently, HS can significantly underestimate both solar radiation and ET.

Several attempts to improve the accuracy of HS have been made, mainly by using wind speed, elevation, precipitation, distance to coastline, air humidity or solar radiation data (Shahidian et al. 2013). However, it is not desirable to use these climatic variables directly in the HS equation because this would take away the ease of implementation and simplicity of the model.

HS modifications in the literature include modifying the original calibration coefficients by a linear regression with or without a constant term. Some authors include more variables that could be fit to observations or FAO-56 (Allen et al. 1998; Droogers & Allen 2002; Trajkovic 2007; Tabari & Talaee 2011; Razzaghi & Sepaskhah 2012; Mohawesh & Talozi 2012; Ngongondo et al. 2013; Raziei & Pereira 2013; Heydari & Heydari 2014). According to Alexandris et al. (2006), HS is often unable to capture the effect of some important climatic variables. Some authors have found overestimations of ET by HS under humid environments and underestimation under windy conditions, compared to the Penman–Monteith equation (Allen et al. 1998; Droogers & Allen 2002). Hargreaves & Allen (2003) have shown that the HS ET estimation differences are caused primarily by impacts of local humidity on the computed vapor pressure deficit.

HS performed better in semi-arid and arid regions (López-Urrea et al. 2006), while it performed poorly in humid climates (Yoder et al. 2005; Wang et al. 2014). In this regard, the performance of the HS model and its calibration have been widely assessed in different climatic zones. The study was carried out using a monthly timescale and FAO-56 benchmarks.

The main objective of the present study is the evaluation of the HS global performance for all climate zones and the fit of a calibrated Hargreaves equation (HSc) for different climate conditions. The goal is to provide optimized versions of the Hargreaves formulation for each individual Köppen class as an alternative to FAO-56 when only maximum and minimum daily temperatures are available.

The models are applied using the climate database from the Agromet Group of United Nations Food and Agriculture Organization (FAO). The data set of long-term monthly means can be downloaded together with the interpolation tool New_LocClim from http://www.fao.org/nr/climpag/. Monthly mean values are available for 4,368 stations worldwide. Table 1 shows how the stations are distributed among the continents.

This data set is considered to be an excellent source of information to compare different ET estimates for different climate zones.

Climate classifications are introduced to organize large amounts of climatic information and provide a concise description. A widely accepted system of climate classification is summarized by Köppen (1936). Recently updated world maps are available (Kottek et al. 2006; Peel et al. 2007). Although introduced in the nineteenth century, the Köppen classification was published in substantially its present form in 1918. It is still the most widely used since it reflects natural vegetation as an expression of climate. This climate–vegetation association provides a relation between a multivariate description of climate and an easily visualized landscape.

The Köppen classification identifies five main groups of climates: tropical climates (A), dry climates (B), temperate climates (C), boreal climates (D) and polar climates (E). All these main groups are further divided by Köppen in order to better classify sub-climates. In this paper we use the Köppen classes as provided in Table 2.

Legates & McCabe (1999) suggested using the coefficient of efficiency for evaluating the goodness of fit. C_eff ranges from minus infinity to 1.0 with larger values indicating better agreement. The coefficient of efficiency is recommended by several authors to evaluate the performance of a model. Moriasi et al. (2007) classified model performance as very good if C_eff >0.75, good if 0.65 < C_eff ≤0.75, satisfactory if 0.50 < C_eff ≤0.65 and poor if C_eff ≤0.50.

In this work the coefficient of efficiency is used in conjunction with the RMSE and MAE. The RMSE is one of the commonly used error indices. Willmott & Matsuura (2005) indicate that MAE is the most natural measure of average error magnitude.

While RMSE, MAE and C_eff provide information on the quality of ET_HSc, BIC (Schwarz 1978; Beal 2007) is especially useful for model selection since it penalizes models with a higher degree of freedom, i.e., more free coefficients. Usually models with the lowest BIC are preferred, according to the parsimony principle.

Köppen climate class, ETo by FAO-56, HS and the five versions of ET_HSc are calculated for the 4,368 stations of the FAO Agromet global climate database. All stations are grouped by the 12 Köppen climate classes considered. The five calibrated ET_HSc equations are optimized with respect to FAO-56 for each Köppen climate class separately.

The results of the best fit of the different ET_HSc versions are provided in Table 3. It shows the RMSE, the MAE, the coefficient of efficiency (C_eff) and the BIC value and the best estimates of k₁, k₂, k₃ for different Köppen climate zones and each ET_HSc.

Both MAE and RMSE are lowest for ET_HS5 (sometimes together with ET_HS3 or ET_HS4). Improvements compared to the original HS method in MAE and RMSE are in the order of 20 to 50% for all three Köppen A classes. For other climate classes improvements are in the order of 10 to 40%.

In the ET_HS5 model, the highest errors are observed in the dry B classes with RMSE (ET_HS5) = 1.211 mm day⁻¹ for climate class BW (desert) followed by the class BS with RMSE (ET_HS5) = 0.881 mm day⁻¹. The lowest errors are observed in the class Ds with RMSE (ET_HS5) = 0.327 mm day⁻¹ followed by the E climates with RMSE (ET_HS5) = 0.378 mm day⁻¹.

Table 3 also presents the coefficients of the calibrated Hargreaves method by Köppen class. It can be seen that the regression coefficients vary widely by Köppen class indicating the importance of optimized coefficients.

The best estimate of the exponent k₂ depends strongly on the climate class. Its original value in the Hargreaves approach is 0.5. If all data are used the best fit is k₂ = 0.495 confirming that 0.5 is a good choice in general. However, for A climates much lower best-fit values are obtained, ranging from 0.074 for Af over 0.143 for As to 0.333 for Aw climates. This reflects the relatively weak dependence of ET on the daily temperature range (T_x – T_n) in tropical climates. The good performance of ET_HS3 in these cases indicates that the inappropriately high value of k₂ = 0.5 in the original Hargreaves method is the major source of the relatively large errors for A climates. For E and Df climates best estimates of k₂ reach 0.592 and 0.659, respectively, and are therefore considerably higher than the original value of 0.5.

Although BIC values do not indicate that ET_HS5 is superior to simpler models in each climate class, we suggest this model for all climate classes with the best-fit coefficient values obtained from our analysis. Aside from providing the lowest estimation error (RMSE and MAE) and the best C_eff, this allows for one generalized formulation for all climate classes.

It should be taken into account, however, that this study is based on the analysis of long-term monthly means. It is important to point out that the results encountered and conclusions drawn might only be valid for planning in general, whereas they might be unsuitable for monitoring applications.

The results show that HS deviates significantly from FAO-56 even for climatic values and that these deviations are climate-class specific. For some purposes that require long-term ET estimates the differences between models may not be significant to justify the extra effort. In tropical climates the differences between modified models are higher and the daily temperature range is not an efficient indicator of ETo in terms of C_eff. In these regions further research should be undertaken for evaluating the validity of other simple methods as suggested by De Pauw (2008).

Long-term monthly means of ET for 4,368 stations worldwide are estimated by the Penman–Monteith model (FAO-56) and several variations of the HS method. The comparison of the results by climate class showed that simple modifications of the HS model with climate-class specific coefficient values lead to significant improvements compared with the un-calibrated HS method. The addition of a constant term yields an unbiased estimator. Calibrating all the coefficients in HS and introducing a new constant term transforms HS into a highly efficient equation for estimating FAO-56 when only temperature data are available. This allows for a wide range of applications.

Considering the problems associated with the availability of meteorological data in the world, the HS temperature-based model is recommended as the most simple and practical method for estimating FAO-56 in the literature. Results obtained from the comparisons of ET estimates by the HS equation and its modifications against FAO-56 throughout different Köppen climate classes showed high variability even for long-term monthly means. For all but tropical climates the results indicate that monthly ET values estimated by the HS method are a suitable approximation of FAO-56 for agro-climatological studies.

In A climates the reduction of estimation errors due to calibration of the HS method is largest. This is most pronounced in the Af climate where MAE is reduced from its original value of 0.663 to 0.335 mm day⁻¹ for ET_HS5. However, the low values of C_eff indicate the need for the search of other simple methods for the estimation of ET in tropical climates.

This study strongly supports the use of the calibrated Hargreaves equation ET_HS5 at different climatic conditions in the case when only maximum and minimum temperature data are available.