Evaluating Methods of Estimation and Modelling Spatial Distribution of Evapotranspiration in the Middle Heihe River Basin, China

Seven models commonly used to estimate the daily reference evapotranspiration (ET0) were evaluated in the middle Heihe River Basin of the arid northwestern part of China. The objectives of the study are to choose the appropriate model for estimating the areal distribution of ET0 and to explain the spatial–temporal distribution of the same through GIS in the study area. The results indicated that the FAO-Penman model is the best way to estimate ET0; its RMSE ranged from 1.11 to 1.70 mm, and r 2 from 0.59 to 0.93. The spatial variations of ET0 are higher in the western part than in the middleeastern part of the study area. The temporal variations of daily differences in ET0 rates are mainly due to the differences in irradiance (Rn) and to daily differences in the vapor pressure deficit (D). The spatially modeled ET0 results (r 2 = 0.88) are in agreement with the corresponding data in situ on the 15th of each month.


INTRODUCTION
Increasing food production to ensure food security is a great challenge for the ever growing population of the arid northwestern part of China in the forthcoming decades. Most of the earlier studies have focused on estimation of crop water demand which is essentially governed by crop evapotranspiration (ET) [1][2][3][4][5][6] . Crop ET is a function of the reference crop evapotranspiration (ET 0 ) and crop coefficient (K c ). Over the past 50 years, many methods, namely FAO-Penman, Penman, 1982-Kimberly-Penman, FAO-corrected-Penman, Penman-Monteith, Blaney-Criddle, Priestley-Taylor, FAO-Radiation, Hargreaves, and FAO-Blaney-Criddle, have been developed for estimating ET 0 . Some statistical relationships between ET 0 and temperature and precipitation have been used [7][8] . These statistical relationships are specific to points or to zones where the stations were located. Therefore, it is desirable to have one method that estimates the reference crop evapotranspiration consistently well on the regional scale. Studies on spatial and temporal variations of ET 0 are limited, and the spatial and temporal variations of ET 0 on a regional scale are important for irrigation planning and related issues. Accordingly, there is a need for modeling ET 0 through GIS to determine the spatially distributed estimation of ET 0 .
Irrigation is an obvious option for improving crop production in the study area. Major investments have been made in irrigation over the past 30 years by diverting surface water and extracting groundwater. As water resources becoming scarce, the region faces a serious water deficiency. The sustainability of irrigated agriculture depends primarily on the efficient management of irrigation water, which is governed by ET 0 . In this study, the main objective is to evaluate the methods commonly used to estimate reference crop evapotranspiration and use the most suitable one to estimate the temporal and spatial distribution of reference crop evapotranspiration.

Study area:
The middle Heihe River Basin (17.000 km 2 ) is located in northwestern China between 96˚42 -102˚00 E and 37˚41 -42˚42 N (Fig. 1). Sandwiched between the southern Qilian Mountains and the northern Mazong Mountains, the area is like a corridor trending from northwest to southeast. The elevation ranges from 1300 to 2500 m. The climate is characteristically arid because the area is situated in the inner part of the Asia-Europe continent. The mean annual precipitation varies from 250 mm in the southern mountainous area to less than 100 mm in the northern highland. Two characteristics of the precipitation deserve mention. First, the inter-annual variability in the precipitation is as high as 80%. Second, over 60% of the precipitation falls between June and August. Figure 2 shows the pattern of rainfall over the year at the Zhangye meteorological station (a representative meteorological station in the study area). The mean annual air temperature is 8°C at the lower (northern) part of the basin and decreases to 2.1°C at the higher (southern) part of the basin. Zonal soil types in the area include gray-brown desert soil, gray desert soil, and mountain gray cinnamony soil. Azonal soil types consist of irrigation-warping soil, saline soil, and brown sand soil, which are embedded within the zonal soil types. Most common vegetation encountered in the area is temperate dwarf shrub and sub-shrub desert vegetation dominated by Chenopodiaceae, Zygophyllaceae, Ephedranceae, Asteraceae, Poaceae, and Leguminosae. Under the influence of the water resource distribution and human activities, there are crops and afforested areas distributed on the piedmont lower alluvial fan and fluvial plain in the area.  Description of models: The definition of reference crop evapotranspiration (ET 0 ) has been updated by Allen et al [9] . The only factors affecting ET 0 are climatic parameters. Consequently, ET 0 is a climatic parameter and can be computed from weather data. It expresses the evaporating power of the atmosphere at a specific location and time of the year and does not consider the crop characteristics and soil factors. A range of more or less empirical methods have been developed over the last 50 years by various scientists and specialists worldwide to estimate ET 0 from different climatic variables [10] . The main climatological methods considered in the study are FAO-Penman, FAO-Radiation, Hargreaves, Penman, Penman-Monteith, Priestley-Taylor, and Blaney-Criddle. These methods are showed in Table 1. The equations listed in Table 1 include two important variables, i.e. the net radiation (R n ) and the soil heat flux (G). The net radiation can be expressed as [10] : Where: S 0 is the extraterrestrial radiation (MJ m -2 per day) e d is the vapor pressure (kPa) Eq. (2) FAO-Penman Eq. (4) Priestley-Taylor ( ) Eq. (5) FAO-Radiation  [15,20] Eq. (6) Hargreaves Eq. (7) Blaney-Criddle the Stefan-Boltzmann constant (4.903×10 -9 MJ m -2 K-4 per day) T is the air temperature (K) r is the reflection coefficient (0.23) n is the bright sunshine hours per day (h) N is total day length (h). A part of the net radiation is converted into sensible heat to warm up the soil and is thus not available for evaporation. The radiation used to heat the soil (i.e., sensible heat) depends on the vegetation coverage and the physical properties of the soil (e.g., soil bulk density and soil depth). With dense vegetation, little radiation reaches the ground, and the heat storage in the soil can be often neglected [11] . To estimate the change in soil heat content for a given period, the following equation can be used: Where: c s is the soil heat capacity, 1.215 MJ m -3 °C -1 [12] d s is the estimated effective soil depth. The effective soil depth is typically 0.5 m in the study area T 2 is the temperature at the end of the period considered (°C) T 1 is the temperature at the beginning of the period considered (°C) ∆t is the length of the period (days)

RESULTS AND DISCUSSION
Net radiation and soil heat flux: The net daily radiation is the fundamental variable for the simulation of evapotranspiration. However, direct measurements are not available for the study area except for Linze station, so simple radiation models become an effective alternative in estimating net radiation through observed meteorological data. The observed net daily radiation data are available for June to September 2003 at Linze station. By comparing the observed and estimated net radiation (Fig. 3), we found that Eq. (8) had a high accuracy in the study area. The soil heat flux was relatively small compared with R n and was underestimated by Eq. (9), when compared with the measurement. In this study, the fitness of two series of data was not good. However, the soil heat flux is not sensitive to evapotranspiration. Selection of models: Because these models were developed in different climatic settings, their performance in a new climatic setting may vary. We used meteorological data from 2003 from five representative stations (Table 2) in the middle Heihe River Basin to calculate ET 0 . The estimated ET 0 was validated by pan evaporation data because pans provide a measure of the combined effect of radiation, wind temperature, and humidity on the evaporation from an open water surface. Pans also respond in a similar fashion to the same climatic factors affecting ET 0 [24] . Through a comparison, the best-performing model was then chosen to estimate the spatial distribution of ET 0 in the study area.
There are two criteria commonly used for a relative performance study of the estimation methods. One is the root mean square error (RMSE) criterion, used to compare the ET 0 observed at a weather station in situ with the ET 0 estimated by various climatological methods. RMSE provides a good measure of how closely two independent data sets match [25] . The other is the coefficient of determination (r 2 ). The r 2 and RMSE values, as well as the slope and intercept obtained by the seven methods are shown in Table 2. The seven models are radiation-based (e.g. Priestley-Taylor and FAO-Radiation), temperature-based (Hargeaves and Blaney-Criddle), or combination-based (e.g. Penman, FAO-Penman, and Penman-Monteith). Their performance at the five stations changes greatly (Fig. 4). The Priestley-Taylor model did not perform well (r 2 = 0.53-0.86), with the greatest RMSE (1.46-1.80 mm). The FAO-Penman model, in which ET 0 is a function of the net solar radiation, wind speed, and vapor pressure deficit performed extremely well at the five stations, with r 2 ranging from 0.65 to 0.93 and RMSE from 1.06 to 1.47 mm. The Hargreaves, Penman, FAO-Radiation, Blaney-Criddle, and Penman-Monteith models also did well. These methods were ranked based on the average RMSE at the five stations (   As stated earlier, the purpose of evaluating these commonly used models is to choose the best-performing model to estimate the spatial distribution of ET 0 . Our comparison of their performances demonstrates that the FAO-Penman model has the best performance, with the average r 2 being the highest (0.77) and the average root mean square error (RMSE) the lowest (1.23 mm per day). Figure 5 shows the significant correlation between the estimated and observed data at five stations. The best performance of the FAO-Penman model and the availability of the parameters required by the FAO-Penman model at all the meteorological stations in the study area made this model the best choice.

Spatially distributed modeling of ET 0 :
The FAO-Penman model requires four parameters: (1) the net solar radiation (R n , MJ m -2 per day); (2) the slope of the saturation vapor pressure-temperature curve ( ); (3) the wind speed (U z , m s -1 ); and (4) the vapor pressure deficit (D, kPa). They can be calculated using models or as observed in meteorological stations.
1. Spatial interpolation of parameters required by the model The meaning of the four parameters mentioned above and their detailed calculations are explained in Table 4. The spatial distribution of the air temperature (T), the actual vapor pressure (e d ), the bright sunshine hours per day (n), and the wind speed (U z ) were interpolated using the Kriging method. Spatially distributed altitude data (Z) and latitude data ( ) were obtained from DEM data (1:100,000). After the basal variables were spatialized (T, e d , n, U z , , and Z), the other variables could be brought to the regional scale using the equations in Table 4. 2. Spatial distribution of ET 0 Finally, ET 0 was spatially estimated using the FAO-Penman model, described in Table 1 (Eq. (2)). Since the purpose of modeling the spatial distribution of ET 0 in the middle Heihe River Basin is to provide a baseline (or plane) for spatially mapping the water demand of crops, we focus here on the spatial distribution of ET 0 during the ecologically meaningful time period, i.e., the growing season approximately from May to October. Figure 6 showed the temporalspatial distribution of ET 0 on the representative date each month (15th). Spatially, the higher ET 0 values are found in the western part of the Heihe River, and the lower ET 0 values are found in the middle-eastern part, where the temperatures are lower because of the altitude and where it is cloudier. Figure 6 also showed that the ET 0 value has a temporal variation (during the 5 days), the highest mean ET 0 value, 8.78 mm, appearing on 15th May and the lowest mean ET 0 value, 3.38 mm, being seen on 15th October.
To test the spatially modeled ET 0 results, we compared the ET 0 values observed by the station on the 15th of each month at the six representative stations (Gaotai, Zhangye, Shandan, Jiuquan, Linze, and Minle) with the spatially modeled results for the corresponding  time at the six stations (Fig. 7). The modeled ET 0 values are in agreement with those measured in situ (the correlation coefficient is quite high, r 2 = 0.88), giving us confidence in the spatially-modeled ET 0 results. the FAO-Penman model at all the meteorological stations in the study area made this model the best choice. 2. Our comparison shows that the net daily radiation, the fundamental variable for the simulation of evapotranspiration, was calculated with high accuracy in the study area. The modeled soil heat flux was not in agreement with that observed at the Linze station. However, the soil heat flux is not sensitive to evapotranspiration. 3. The modeled ET 0 value has a temporal and spatial distribution. Spatially, the higher ET 0 values are found in the western part of the Heihe River, and lower ET 0 values appear in the middle-eastern part, where the temperatures, depending on the altitude, are lower and where it is cloudier. Temporally (during the 5 days), the highest mean ET 0 value, 8.78 mm, is seen on the 15th May, and the lowest mean ET 0 value, 3.38 mm, is seen on the 15th October. 4. The spatially modeled ET 0 results were compared with the station-observed ET 0 on the 15th of each month at the six representative stations (Gaotai, Zhangye, Shandan, Jiuquan, Linze, and Minle). The modeled ET 0 values during the simulated period are in agreement with those measured in situ (the correlation coefficient r 2 = 0.88), giving us confidence in the spatially modeled ET 0 results.