ارزیابی و واسنجی معادله ترونت‌وایت برای تخمین تبخیر تعرق در اقلیم باد خیز مطالعه موردی منطقه سیستان

A B S T R A C T
There are many methods for calculating evapotranspiration that require a lot of data, but a few require only air temperature. One of these methods is Trontwait. The Sistan region in the southeast of Iran is one of the regions that is unique in Iran due to the 120 days of winds. The purpose of this research is: 1) to evaluate 6 different Trontwait methods available in the sources compared to the Fau-Penman-Monteith method and 2) to modify the equation for the windy region of Sistan. The results showed that the original Trontwhite equation underestimates the amount of evaporation-transpiration. Among the existing methods, the use of coefficient k=0.72 had the best results. In order to recalibrate the Trunthwaite equation, the effective temperature coefficient of the equation (k) must be modified. The results showed that the optimal value of k varies between 0.755 and 1.04. The annual average value of the root mean square error (RMSE), according to the variable k values, was equal to 0.14 mm per day. Also, by minimizing the square of the error, we considered the k value to be 0.802 as a constant, and the RMSE value was equal to 1.19 mm per day. It can be concluded that after correcting the Trontwhite equation, it can be used in the Sistan region
Extended Abstract
Introduction
Evapotranspiration is one of the most important components of the hydrologic cycle in the nature and its exact determination is essential for water balance studies, irrigation and water resources management. One of the most accurate methods of estimating ETo is in different climates of FAO Penman-Monteith equation (PMF-56) the accuracy of this method for calculating evapotranspiration in all over the world has been proven successfully. PMF-56 method as a standard method requires a large meteorological data such as air temperature, relative humidity, wind speed at a height of 2 meters and solar radiation. Provide exact data on all areas is not and also if it is possible it is not very reliable. Some of the evapotranspiration methods require only air temperature for ETo estimation. One of these methods is Thornthwaite. The Sistan region in the southeast of Iran is one of the regions that is unique in Iran due to the 120-day winds and high day-night temperature changes. The purpose of this research is to evaluate the different existing methods of the Thornthwaite equation (6 methods) and adjustment the Thornthwaite equation for the Sistan region.
 Methodology
Six Thornthwaite approaches were used in this study are described below:
1) The Thornthwaite method (Thornthwaite 1948) is a temperature-based method for the estimation of ET0 as a function of the average monthly temperature.
2) Camargo et al. (1999) improved the performance of the Thornthwaite method using an effective temperature ( ) instead of the mean temperature ( ) and of the daily temperature amplitude:
(1)
 where k = calibration coefficient. Camargo et al. (1999) found that k = 0.72 is the best value for estimating monthly ET₀.
3) Pereira and Pruitt (2004) recommended k = 0.69 for estimating daily ET₀.
4) Trajkovic (2005a) expressed the equation reference evapotranspiration where Thornthwaite equation depend on maximum possible duration of sunshine and mean air temperature in the i-th month.
5) The data from Serbian stations Palic, Belgrade, and Nis were used to calibrate the Thornthwaite equation (Trajkovic 2005b):
(2)
6) Bautista et al. (2009) calibrated the Thornthwaite equation by changing the value of the corresponding constant p = 16.
In this research, in addition to evaluating different methods and determining the best method, the value of coefficient k for Sistan region is recalibrated and its best value is stated.
 Results and Discussion
The results based on FAO-56 PM equation show that amount of daily evapotranspiration varied from about 21 mm per day in July to 0.7 mm day^-1 in the beginning of December.The annual average value of evapotranspiration according to the FAO-56 PM equation is 8.21 mm day^-1.
Six Thornthwaite approaches were used in this study were compared to full FAO-56 PM equation. The statistical summary including RMSE (mm day^-1), MBE (mm day^-1) and r ( ) for Sistan location is presented in Table 1. The value of RMSE in original Thornthwaite equation was equal to 4.15 mm day^-1, the value of MBE was equal to -3.92 mm day^-1, and the value of r was equal to 0.52, which indicates Thornthwaite equation was very poor in estimating ET0 and greatly underestimated PM values. The method of Camargo et al. (1999) with RMSE value of 3.14 mm day^-1 and MBE value of -2.49 mm day^-1 and r value of 0.7 had the best results among all methods. Generally, the accuracy of this method is low compared to the PMF-56 method. Among the studied approaches, only the method of Bautista et al (2009) overestimates the amount of evapotranspiration. The average amount of annual evaporation in this approache is equal to 11.84, which has been overestimated by about 44%.
The best monthly values of k (optimal value) was obtained using the trial and error method and minimizing the value. The results show that the value of coefficient k varies from 0.755 in March to 1.04 in October and its average value is 0.825. By using the solver option and based on the lowest value of the root mean square error between Thornthwaite evapotranspiration and the FAO-56 PM method, the value of k coefficient was also obtained, which was equal to 0.802. In the following, by using three k coefficient values, including optimal variable coefficient ( ), average variable coefficient ( ) and obtained from the solver option ( ), the evapotranspiration value were calculated and evaluated. Table (2) shows the statistical indices of the evapotranspiration value calculated with the adjustment Thornthwaite equation based on the coefficient k.
The results showed that the use of variable  coefficient with RMSE equal to 0.14 mm day^-1, MBE -0.04 mm day^-1 and r equal to 0.99 had the best results. It can be concluded that usingof  and   we will reach satisfactory results in calculating evapotranspiration in Sistan region.
 Conclusion
The results of this study showed that the 60% of the amount of evapotranspiration in the Sistan region occurs in the four months of June to September when the winds blow for 120 days. The total evapotranspiration in the three months of December, January and February is equal to 8% of the annual evapotranspiration value. The results showed that the six existing methods of estimating evapotranspiration with Thornthwaite method have low accuracy, so its value should be recalibrated. Hence, based on this study, reference evapotranspiration can be easily calculated for the windy region of Sistan with the available metorological data and the calibrated Thornthwaite equation. In developing countries where good quality data are relatively scarce, using such simple methods may be beneficial for the farmers and local water organizations.
 Funding
There is no funding support.
 Authors’ Contribution
All of the authors approved thecontent of the manuscript and agreed on all aspects of the work.
 Conflict of Interest
Authors declared no conflict of interest.
 Acknowledgments
We are grateful to all the scientific consultants of this paper.