Modifying Hargreaves–Samani equation with meteorological variables for estimation of reference evapotranspiration in Turkey

Evapotranspiration is the summation of evaporation from wet soil plus transpiration from the leaves of plants existing on that soil, expressed in depth of water in mm over a certain area in daily, weekly, monthly, or yearly periods. Because it is needed for calculation of the irrigation water requirement, accurate estimation of evapotranspiration is vitally important due to the steadily increasing demand for food.

In compliance with the suggestion by Allen et al. (1998), Trajkovic (2005), Kisi (2008), Cobaner (2011, 2013), Ngongondo et al. (2013), and Shiri et al. (2014) used the linear regression method for calibrations of such empirical formulas at some specific local geographic locations. Gocic & Trajkovic (2011) proposed the FAO-56 PM and Hargreaves models based on web services using Davis weather station data. Their web services either directly provide the measured data or suggest methods to estimate values of the meteorological variables required by the FAO-56 PM and Hargreaves formulas. Patel et al. (2014) calibrated and validated the empirical Hargreaves coefficient and the exponent coefficient in the HS equation with a fuzzy-logic-based approach for diverse climate locations of India. They reported that the modified HS equation resulted in improved ET₀ estimates, and the location-specific calibration methodology presented in their paper increased the accuracy of the HS equation. Marti et al. (2015b) applied a gene expression programming (GEP)-based approach for estimating ET₀, considering lysimetric records taken at two stations in Spain which actually measured ET₀ along with other pertinent meteorological variables over the period 2007–2012, and reported that the locally trained, what they called GEP4 and GEP6 models, estimated ET₀ even better than FAO-56 PM. Shiri et al. (2015) also applied a GEP-based model for ET₀ estimation in daily periods in Iran using relevant data having a record length of nine years measured at eight coastal and 21 inland stations in Iran. Similarly to many relevant studies, Shiri et al. (2015) also took the FAO-56 PM results as true values and assessed a few different versions of their GEP-based model by various comparison statistics like determination coefficient and mean absolute relative error. Shiri et al. (2015) noted usually underestimations at coastal stations and overestimations at inland stations by the original HS equation, and they reported that with locally calibrated coefficients the modified HS equation was better. For the GEP-based model, Shiri et al. (2015) developed different equations involving fairly long analytical expressions individually pertaining to each single station. According to Marti et al. (2015a) however, application of such data-driven models ‘requires the implementation of specific software, and the obtained models can generally not be expressed in straightforward simple equations’ like that of HS.

Originally, the HS equation was developed for semiarid environments (Hargreaves & Samani 1985). Some studies showed that the HS equation overestimated ET₀ in warm humid areas, such as the southeastern USA (Lu et al. 2005) and India (Kashyap & Panda 2001). Therefore, numerous attempts were made to improve the estimation capability of the HS equation, some of which are already mentioned above (e.g., Martinez-Cob & Tejero-Juste 2004; Ravazzani et al. 2012; Shahidian et al. 2012; Hosseinzadeh Talaee 2014). Jensen et al. (1997) noted a significant correlation between the wind speed and the slope of the PM/HS regression with a determination coefficient: R² = 0.88. Droogers & Allen (2002), after having collected and analyzed pertinent weather data in many meteorological stations all over the world, stated that by adding the precipitation as an extra term in it the modified HS equation yielded better ET₀ estimates in monthly periods than its original form. Fooladmand et al. (2008) further modified the HS equation, which was analytically extended by Droogers & Allen (2002) by inclusion of the monthly precipitation term, for some stations in the south of Iran.

Calculation of suitable magnitudes for the coefficients of the modified HS equation for estimation of monthly mean ET₀ for seven geographical regions of Turkey has been the main objective of this study. Seven distinct climate regions are spread throughout Anatolia and the Balkans part of Turkey. The Marmara Region in the northeast has a climate similar to that in the Balkans, which has precipitation throughout the year with cold winters. The Aegean Region in the west has a mild climate similar to that of Mediterranean shores at coastal areas, but has a land climate towards the inner parts. The Mediterranean Region in the south has a typical mild climate similar to that in Greece and Italy, which receives considerable orographic precipitation year-round. Its upper boundary is delineated by the peaks of the Taurus Mountain ranges, which mostly extend parallel to the shoreline. The Southeastern Anatolia Region has hot and dry summers and cold winters receiving less than overall average precipitation. The Eastern Anatolia Region is a mountainous area covered by extensions of both the Taurus and the Black Sea Mountains, and it has hot but short summers, long winters with harshly low temperatures with a lot of snow. The Black Sea Region receives much higher precipitation than the average of Turkey, which is mostly orographic type, and it has a somewhat similar climate to the Mediterranean Region. The Inner Anatolia Region, the seventh region, receives much lower precipitation than the average of Turkey because the Black Sea Mountain ranges, mostly running parallel to the shoreline of the Black Sea in the north, and the Taurus Mountain ranges, mostly running parallel to the shoreline of the Mediterranean Sea in the south, prevent passage of moisture-laden clouds inwards. In this region the continental climate prevails, with hot and mostly dry summers and cold winters.

Various agricultural activities, including citrus fruit crops, banana plantations, pistachio plantations, tea plantations, cereal crops, sugar beets, etc. in all these seven regions of Turkey take place at an ever-increasing rate. Some large-scale irrigation projects in Turkey, like the Southeastern Anatolia Project (known as GAP in Turkey), have already been completed and are operational, while construction of some others is still continuing at a rapid pace (see www.dsi.gov.tr). Turkey is said to be one of the few self-sufficient countries from an agricultural products’ standpoint. In recent years, various crops have been exported to many countries. In short, accurate estimation of irrigation water requirements in those seven regions, and hence of the evapotranspiration from cultivated lands, has become a significantly important issue for Turkey.

The purpose of the studies summarized in the above paragraphs is calculation of suitable magnitudes for the coefficients of the modified HS equation pertinent to the local conditions of a specific geographical area. All these calibrated HS equations are for the regions whose data are used, and they cannot be valid at different meteorological conditions, such as the geographical regions of Turkey. Hence, the main objective of this study is to improve the ET₀ estimation capability of the HS equation by determining the magnitudes of its coefficients most suitable to each one of the seven geographical regions in Turkey, using the relevant meteorological data measured at the weather stations throughout Turkey so as to match the ET₀ value given by it to that of the FAO-56 PM equation as much as possible. Moreover, as summarized above, most studies on a similar theme attempted to calibrate the K_H coefficient while leaving the other two intact, keeping their original magnitudes assigned by Hargreaves and Samani. In our study, we will extend the calibration of these three parameters in all possible combinations, namely: first, modifying only one of the three while keeping the other two at their original values; second, modifying any two as the third one is at its original value; and third, modifying all three coefficients to enable the HS equation to estimate the ET₀ values with better accuracy anywhere in any one of the seven regions of Turkey. Further, by incorporating the wind speed as an extra predictor for those meteorology stations where reliable wind speed data are available, we will search for the possibility of further improvement in its capability for ET₀ estimation. The contribution of the wind speed for betterment of the HS equation will be taken into account by multiplying the value given by the HS equation by an extra coefficient, which will be related to the wind speed by linear regression, an approach not practiced before by any similar studies.

Due to the lack of experimental ET₀ measurements, the numbers given by the FAO-56 PM equation have been accepted as the true values, and this equation has been used for calibrating the modified versions of the HS equations (e.g., Allen et al. 1998; Almorox & Grieser 2016). Step-by-step calculation of the FAO-56 PM equation (Equation (1) here) is done in this study as summarized below.

Alternatively, some researchers calibrated the K_H coefficient while leaving the others at their original values, and just a few studies calibrated all three coefficients (K_H, K_T, and e_H) of the HS equation specific to a geographical location (e.g., Trajkovic 2007; Adeboye et al. 2009; Sepaskhah & Razzaghi 2009; Maestre-Valero et al. 2013; Lee 2015). In our study, the K_H, e_H, and K_T coefficients of the HS equation are adjusted in all possible combinations with the help of the Solver menu of Microsoft Excel using the monthly average values of all the pertinent meteorological variables recorded at 275 stations over the period 1975–2010 and assuming the ET₀ values computed by the FAO-56 PM equation were the true ET₀ values, which is a common practice pursued by many studies as summarized in the Introduction. Excel Solver is part of a broader system of commands called Simulation Solver in Excel, and it is used to determine the most suitable form of the analytical expression chosen in a window known as target cell. The Solver utilizes the data in another cell pertaining to the chosen analytical model input by the user. Constraints for the coefficients of the analytical expression can also be specified [Excel–Help]. The Microsoft Office Excel Solver tool was developed by Lasdon & Waren (1978) using the generalized reduced gradient (GRG2) algorithm, which is an improved state of the simplex method for non-linear programming (Lasdon et al. 1978, 1996). The GRG2 algorithm is explained in Lasdon et al. (1978) and Ecker & ve Kupferschmid (1991).

The averages for each one of the seven regions of salient statistical characteristics of the meteorological data used are given in Tables 2 and 3. Adjusted K_H, e_H, and K_T coefficients of the modified HS equations and the performance measures (MAE, mm day^–1; MARE, %; and RMSE, mm day^–1) of the original and modified HS equations for all combinations are presented in the Appendix, Tables A1–A7 (available with the online version of this paper). The values of the performance statistics for, first, the original HS equation, next, for the best combination with all three calibrated coefficients (combination 7), for the ones including the wind speed (combinations 8 and 9), and for the one given by Almorox & Grieser (2016) (combination 10) are summarized in Tables 4 and 5. As seen in these tables, first of all, the original HS equation estimates the ET₀s in all seven regions with fairly high MARE values (about 20–25%). For combinations which modified the magnitudes of K_H, e_H, and K_T coefficients (combinations 1–7), the best combination having the smallest RMSE statistic is the last one (combination 7), which allows all the three coefficients to assume different magnitudes from those of the original HS equation. RMSE is known as the standard error of the estimates in the regression, which is a major criterion of superiority in most similar studies. MARE values in our study were mostly in parallel with those of RMSE also. Calibration of the original and modified HS equations by inclusion of the wind speed (WS-HS and WS-MHS, combinations 8 and 9, respectively) enhanced the accuracy of the HS equation. The best estimations are obtained from the ninth combination, which allows calibration of all three coefficients of the HS equation along with inclusion of the wind speed (WS-MHS) (Equation (15)). The second best estimation results are obtained from the original HS equation with the wind speed data included (WS-HS) for the coastal regions. The HS equation with three calibrated coefficients but not including the WS gives the second best estimation for the inner regions of Turkey. It is also concluded from Tables 4 and 5 that calibration of coefficients of the HS equation reduces the MARE statistics by an average of 5%. In addition to that, calibration of the modified HS equation with inclusion of the wind speed decreases the MARE statistics almost 10%.

Here, the three coefficients of the original HS equation are modified for distinct climate regions of Turkey, which provide better ET₀ estimates. The results indicate that the best combination for modification of the HS equation, which allows all the three coefficients to assume different magnitudes from those of the original HS equation, provides the smallest values of RMSE statistics of those of the other coefficient adjusting combinations except for the Black Sea Region, where the best estimation is obtained by adjusting the other two coefficients while keeping the original magnitude for the e_H coefficient. With the purpose of further enhancing its accuracy, the wind speed is introduced as another explanatory variable to the HS equation by Equation (15) here, which has improved the estimation capability of the HS equation considerably.

When the results of all the combinations are compared, for all seven regions of Turkey, the best estimations are obtained from the ninth combination, which is the HS equation having all the three coefficients adjusted, which also has the wind speed as an extra explanatory variable. The effect of the wind speed is taken into account as a second step using Equation (15), which multiplies the ET₀ given by the modified HS equation by another coefficient, computed by a linear equation whose two coefficients are determined beforehand by regression. The second best estimation results are obtained by the original HS equation to which the wind speed is added for the coastal regions. Although the wind speed has similar statistical parameters for all regions, the results show that the coastal regions are more sensitive to the wind speed for estimation of ET₀ than the inner regions (Inner Anatolia, Southeastern Anatolia, and Eastern Anatolia regions) at which the continental climate prevails. The results indicate that the modified HS equation reduces the MAE and RMSE statistics for each region in Turkey more than the original HS equation. Calibration of coefficients of the HS equation reduces the MARE statistics on an average of 5%. In addition to that, calibration of the modified HS equation with wind speed data decreases the MARE statistics by almost 10%. Suitable magnitudes of the three coefficients of the HS equation can be computed by a similar analysis in other regions of the world as well. It should be kept in mind that the validity of the empirical equations is legitimate only in the regions where they are developed and within the ranges of the available data.

Although we have taken the results provided by the FAO-56 PM equation as the benchmark values throughout our study, it should still be borne in mind that this is also a calculation model, which may be inadequate for some cases. Also, some of the terms of the FAO-56 PM equation need to be calculated themselves, which may include some degree of uncertainty. It would be therefore much more reliable to attempt to improve either the HS equation or any other model based on actual lysimeter measurements naturally.

This work was supported by Research Fund of the Erciyes University, Project Number FBA-2014-4875. The authors wish to thank the Turkish State Meteorological Service (MGM) for the supply of the mean monthly values of maximum, minimum, and average temperature data.