Applicability of 12 PET estimation methods in different climate regions in China

Potential evapotranspiration (PET) is a comprehensive factor that characterises climate change, and considering the numerous methods to calculate PET, it is difficult to objectively select a method according to the requirements. In this study, the applicability of 12 commonly used PET estimation methods in China was studied. Based on temperature and humidity, China is divided into 11 temperature zones (TZ) and 5 arid and humid regions (AHRs). The study used the FAO Penman– Monteith (P-M) method as the standard, and the applicability of the 12 methods was analysed using four factors: correlation, annual mean values, seasonal distribution, and parameter characteristics. The results show that the radiation-based methods have the best monthly correlation with the P-M method, the temperature-based methods are second best, and mass-transfer-based methods perform the worst. Among these, the P-T method is the best, and the Hamon method is the worst. The Kharrufa and Abtew methods have the better applicability in higher TZs, whereas the Harg method has the least applicability. The seasonal distribution of radiation-based methods (excluding the Jensen method) in the different AHRs and different TZs is better than that of temperature-based and mass-transfer-based methods. According to the evaluation results of all factors, the Rohwer, P-T, and Mark methods are recommended when the data conditions are not conducive for the P-M method.


INTRODUCTION
Due to the increasing impact of human activities in recent decades, the global climate has undergone drastic changes (Cong et  China, due to the large differences in geographical environments, the differences in climate change trends are more observable (Zhang & Cong ). For example, although the overall temperature in China shows an increasing trend, a downward trend exists in the northeast region.
The overall wind speed shows a downward trend; however, the change in wind speed in the southeast area of the Tibetan Plateau is not evident, whereas in other areas, an increasing wind speed trend has been observed (Huang et al. ). The water cycle is an important component of the climate system, and climate change impacts on the water cycle elements will inevitably lead to the temporal and spatial changes of regional or basin water resources (Yang et al. ). The potential evapotranspiration (PET) plays an important climatic role in the water cycle and is a comprehensive meteorological factor. It is a key variable in the methods or models that predict climate change impacts on water resources, and simulation accuracy directly affects the prediction rationality (Douglas et al. ). Climate change directly or indirectly affects the extent and spatio-temporal distribution of PET (Wang et al. ; Dinpashoh et al. ), which necessitates an estimation study of PET in the context of climate change.
Currently, approximately 50 methods or models are used worldwide to estimate PET, which can be divided into the following four categories: (1) temperature-based methods, (2) mass-transfer-based methods, (3) radiationbased methods, and (4) comprehensive methods (Singh & Xu ). These methods are derived experimentally or theoretically, with different test conditions, required input data, and applicability in different regions.
The most internationally accepted method for estimating PET is the FAO Penman-Monteith (FAO P-M) formula. It is internationally recognised as the most effective method to estimate PET and has been shown to provide accurate estimations under different conditions (Allen et al. ). The P-M method was used to calculate PET in China, and the results confirmed its suitability in China (Liu et al. ). However, this formula requires a large amount of meteorological input data, such as maximum, minimum, and mean air temperatures, wind speed, relative humidity, and solar radiation. In practical applications, because the data are limited or the accuracy of the data is not sufficient, especially in remote areas, PET calculations are limited and cannot be estimated (Er-Raki et al. ).
Simple empirical methods require less data. For example, the Linacre method only requires temperature, altitude, and latitude data; the Abtew method only needs total solar radiation data. These methods can accurately estimate PET and are widely used (Ye et al. ). Determining the applicability of the PET equations, the variety of data types required, and the extensive expertise required to properly use the various equations makes it difficult to select the appropriate method for further research on practical applications (Xu & Singh ). Therefore, it is necessary to study the applicability of various PET estimation methods under different conditions and provide the basis for the selection of PET for further research.
PET applicability analysis began in the 1990s, with a large-scale systematic study focussing on dozens of commonly used methods for estimating PET, such as Priestley-Taylor (P-T), Thornthwaite (Th), Blaney-Criddle (BC), Hargreaves (Harg), Jensen-Haise (JH), P-T, Penman, and Turc (Tu). Practically, fewer than ten methods for estimating PET were typically used in most studies (Table 1 only nine TZs with stations are discussed, and the specific distribution is shown in Figure 1 and Table 2.

Model comparison and statistical error analysis
The correlation coefficient is the linear correlation degree between study variables, and the root-mean-square error is a measure of the deviation between the observed and true values. In this study, the correlation coefficient and rootmean-square error were used to measure the correlation and deviation between 12 PET methods and the P-M method.
where R 2 is the correlation coefficient (dimensionless), PETPM i is the calculated value of the P-M formula (mm); PET i,m is the calculated value of the m method (12 methods total, mm), PET i,m is the average value of the P-M formula (mm), and RMSE is the root-mean-square error (dimensionless).

PET methods
A total of 13 methods for estimating PET were used, including the benchmark method FAO P-M formula and four temperature-based, two mass-transfer-based, and six radiation-based methods ( Table 3).
The standard method used in this study for estimating PET is the FAO Penman-Monteith formula (P-M formula) recommended by the United Nations Food and Agriculture Organization. This method is based on the principle of energy balance and aerodynamics, and it has a complete theoretical basis, high precision, and high applicability among many PET estimation formulas; therefore, it can be used as a benchmark method for the applicability analysis of other methods.
where E P is the PET (mm/d); (E P ) units are the same in the subsequent formulas; G is the soil heat flux density (MJ/ (m 2 d)); R n is the net radiation on the surface of the crop (MJ/(m 2 d)); γ is the dry-wet constant (kPa/ C); T a is the daily average temperature ( C) at a height of 2 m; e s is the saturated vapour pressure (kPa); e a is the actual vapour pressure (kPa); Δ is the slope of the temperature-saturated water pressure curve T (kPa/ C); and U 2 is the wind speed of 2 m (m/s). When the G value is small, it is neglected in the process.

RESULTS AND DISCUSSION
This study analyses the applicability of 12 PET methods in different arid and humid zones and TZs in China using four factors: correlation analysis, annual mean values, intra-seasonal distribution, and parameter adjustments.

Correlation analysis
The monthly correlation coefficients of all the methods were above 0.89 (Table 4) k is the reactant influence coefficient; p is the percentage of daytime hours as a percentage of the daytime hours; T d is the dew point temperature ( C); h is the site elevation (m); A is the latitude ( ), where the site is located, it is the daytime duration (h); Pt is the saturated water vapour density term; Rs is the short-wave radiation (MJ/(m 2 d)); in the Doorenbos & Pruitt (1977) formula, α ¼ 1:066 À 0:13 × 10 À2 RH þ 0:045U d À 0:20 × 10 À3 RH × U d À 0:135 × 10 À4 RH 2 À 0:11 × 10 À2 U 2 d ; Tx is the temperature constant, this study considers À3 C; removes the selected parameters that need to be adjusted (experience coefficients are not explained), and other symbols have the same meaning as formula 3. were close to those obtained by the FAO P-M method (Table 5). Before the adjustment, the value calculated by the P-T method was closest to the value calculated by the  (Table 6), which is consistent with the negative correlation between the FAO P-M formula and humidity. The increase in the humid region (5-HR) value relative to the subhumid region (4-SHR) is primarily due to higher temperatures in the humid region compared with the subhumid region; therefore, the PET value increased slightly with an increase in humidity. Among the different TZs (Table 6), PET increased with an increase in temperature in the plateau (TZ1-3), the temperate zone (TZ4-5), and the tropical zone (TZ6-9).

Seasonal distribution
The PET calculated by the FAO P-M formula is evenly dis-

Parameter adjustment feature
The parameter adjustment values using different methods are listed in Table 7. In addition to the Hamon method

Correlation analysis
The correlation coefficients of the different methods cal-   methods showed that these methods are more suitable in the arid areas than in the humid areas. Among the methods, the six methods based on radiation showed a little change in each AHR, without conspicuous low.
The monthly correlation coefficients between these six methods and the FAO P-M method are greater than 0.94, indicating well the applicability of these six methods in different AHRs.

Annual mean values
With an increase in humidity, the annual mean changes in evapotranspiration calculated using BC, Linacre, Kharrufa, Hamon, Rohwer, P-T, Mak, and Door methods decrease gradually, which indicates that the range of PET values to be adjusted by these methods in wetter areas is less. This also indicates that these methods are more applicable in the humid areas. However, the Penman is more suitable for the application in the arid areas. The annual mean change in evapotranspiration calculated by the Abtew method first increases and then decreases, indicating that it is more suitable for the arid or humid areas. In contrast, the Harg method is applicable for boundaries between the arid and humid areas ( Figure 5).

Seasonal spatial distribution
The distribution of the P-M method increased with the increase in wetness throughout the year, and the proportion of PET decreased in summer and increased significantly in winter, especially in the humid zone (5-HR). The PET in winter accounted for approximately 13%. In the different AHRs ( Figure 6, 1-EAR), the mass-transfer-based methods are significantly better than the temperature-based methods, and the seasonal distribution of the calculated values is similar to that of the FAO P-M method, especially in the extreme arid region (1-EAR) and the arid region (2-AR) ( Figure 6).
The seasonal distribution of temperature-based methods is better in humid regions than in arid regions, which is most clearly demonstrated by the Kharrufa method ( Figure 6).
The seasonal distribution of mass-transfer-based methods performs lower than that in arid regions. However, it performs more effectively than the temperature-based method, indicating that the seasonal distribution of mass-transferbased methods is more applicable in arid regions versus in humid regions and is better performing than the temperature-based methods in all AHRs. Among the six methods, the Rohwer method has the greatest applicability in AHRs,   Table 10.

Parameter adjustment feature
The adjustment values of the parameters using different methods based on temperature and mass-transfer in different AHRs are shown in Table 8. The standard deviation of the BC method parameters needs to be adjusted in different arid or humid regions. The standard deviation is only 3.36%; therefore, the parameter adjustment range of the BC method does not change with the changes in humidity. The parameter ranges, which need to be adjusted, in Linacre, Kharrufa, P-T, Rohwer, Door, and Mak methods, decrease with increasing humidity. The six methods are more suitable in humid regions; the Penman method exhibits the contrary.
As the humidity increases, the range of parameter adjustment is gradually increased; therefore, the applicability of the method is more effective in arid regions.
For the adjustment value of parameters using different methods based on radiation in different AHRs (Table 9), the Abtew method gradually increases the range of parameters from AHRs to the arid and humid boundary region, thereby representing a mountain peak shape. However, the range of the Harg method parameter adjustment decreases between the AHRs and the arid and humid boundary region, presenting the shape of a valley. These lows are almost identical to the distribution of the PET variation shown in Figure 5.

Correlation analysis
Because some of the sites were in the plateau areas (1-3), the law of monthly correlation between the various methods and the FAO P-M method cannot be fully reflected; therefore, the monthly correlation of the Linacre method suddenly drops in the 2-3 region. As shown in Figure 7(a), in the regions with lower and higher temperatures, the monthly correlation coefficients between the various methods and the FAO P-M method are not high, and the monthly correlation is greater when temperatures are optimal. Therefore, the temperaturebased and mass-transfer-based methods are more suitable  for these TZs. In Figure 7(b), the monthly correlation coefficients between the various radiation-based methods and the FAO P-M method are mostly above 0.94, and the variation in different TZs is small, indicating that these methods have better applicability in each TZ.

Annual mean values
The annual mean of PET in different TZs using different methods is shown in Figure 8. With increasing temperature, the annual mean variation values calculated using the Kharrufa and Abtew methods gradually decrease; that is, the magnitude of the PET that needs to be adjusted gradually decreases, indicating the excellent application of the two methods in higher TZs. However, the Harg method performs well at lower temperatures. The annual mean variation value of PET calculated by BC, Hamon, and Penman is independent of temperature, and the applicability of these methods is the same in each TZ. The applicability of the Door method in the plateau zones (TZ1-3) is poor and is better in other zones (TZ4-9).

Seasonal spatial distribution
The FAO P-M method had the same regularity in the TZs     Table 10.

Parameter adjustment feature
The adjustment values of the parameters using different methods based on temperature, mass-transfer, and radiation in different TZs are presented in Tables 11 and 12

CONCLUSIONS
Many methods exist for estimating PET, and this study examined four temperature-based methods, two mass-transfer-based methods, and six radiation-based methods, while