Potential evapotranspiration (PET), the maximum evapotranspiration rate under unlimited water supply, reflects the capacity for transpiration flow and plant primary production. Numerous models have been developed to quantify PET, but there are still large uncertainties in PET estimations. In this study, the authors conducted spatially explicit estimations of daily PET from 1981 to 2010 for eight different land-cover types on the Tibetan Plateau by applying three types of PET models including a combination model (Penman–Monteith), a radiation-based model (Priestley–Taylor), and a temperature-based model (Thornthwaite). This study found that the PET estimated by Thornthwaite model (PETT) was lower than those estimated by Priestley–Taylor (PETPT) and Penman–Monteith models (PETPM). Penman–Monteith model gave the highest estimates of PET on annual and daily scales. The mean annual PET for the whole plateau estimated by these three models varied from 675.1 to 700.5 mm yr−1, and daily PET varied from 1.33 to 1.92 mm day−1. The spatial pattern of PETT did not agree with the PETPT and PETPM, while the latter two agreed well with each other. Because of different model structures and dominant meteorological drivers, the interannual variability of PET varied significantly among the models. PETPT and PETPM showed a transition around 1993 since the dominant meteorological drivers were different before and after 1993. These disagreements among different models suggested that PET models with different algorithms should be used with caution. This study provided a validation to assist those undertaking PET estimations on the Tibetan Plateau.
Potential evapotranspiration (PET), a function of available energy, vapor pressure gradient, and vegetation type, indicates an upper limit to water losses by evapotranspiration (Thornthwaite 1948; Penman 1948). In well-watered circumstances, PET can reflect the capacity of ecosystems for transpiration flow and primary production (Fisher et al. 2011). Moreover, as an important input to a variety of aridity indices, drought indices, and climate and hydrologic models (Sentelhas et al. 2010; Currie 1991), PET has drawn increasing attention in the past decades. Different PET models behave disparately, and the simulated PET values greatly affect the results of climate and hydrological models. Therefore, it is important to assess the uncertainties of different PET models.
The choice of PET models depends not only on the important potential controls of the study system but also on what data are available to run the PET model. There are more than 50 PET models varying from temperature-based to radiation-based to physically based process models. Previous studies found that intercomparisons of PET estimates have revealed large uncertainties because of different assumptions and input data involved in PET models. For example, Fisher et al. (2011) compared three different types of PET models and found that the temperature-based model estimated 20%–30% less than the radiation-based and combination models. By contrasting three temperature-based and three radiation-based PET models, Lu et al. (2005) found that PET values were significantly different and greater deviations were found among the temperature-based models. Moreover, Douglas et al. (2009) demonstrated that annually aggregated Priestley–Taylor (PETPT) and Turc models outperformed the Penman–Monteith (PM; PETPM) model, and Priestley–Taylor performance appeared to be superior to the other two models on a daily scale. It is therefore important to identify the deviations among PET models, since different PET models give widely inconsistent values at particular locations, as demonstrated in previous studies (Fisher et al. 2005).
This study focuses on the three most commonly used models, and each of them belongs to a different category. The first one is the well-known Penman–Monteith model, which is a combination model taking into consideration energetic drivers (i.e., solar radiation and air temperature) and atmospheric drivers (i.e., vapor pressure deficit and surface wind speed). PET estimates of the Penman–Monteith model are commonly used as the standard values and adopted to evaluate other PET models (Lu et al. 2005). However, the main drawback of the Penman–Monteith model is its strict inputs. The second category is the temperature-based models, which are widely used because of their simple structures, especially when only temperature data are available (Xu and Singh 2001). The Thornthwaite (1948) model, a typical temperature-based model, has one particular advantage for ecological application—the inclusion of daylight hours, which are implicitly related to radiation and the relative ability to photosynthesize (Fisher et al. 2011). Another simpler, but effective, PET model was developed by Priestley and Taylor (1972) for a well-watered surface by introducing a unitless constant to the Penman equation. The Priestley–Taylor model has been recognized as one of the most popular radiation-based models (Xu and Singh 2000).
The Tibetan Plateau, which is rich in lakes, rivers, glaciers, and wetlands, is an ideal area for investigating the interactions between hydrological cycles and climatic change because the ecosystems remain relatively undisturbed and the South Asian Monsoon develops wide ranges of temperature and moisture gradients across the plateau (Li and Fang 1999). The spatial and temporal variability of PET on the Tibetan Plateau have especially been of concern in climatological and hydrological studies in recent decades. However, most of the previous studies were conducted by adopting the Penman–Monteith model to estimate PET and identify the dominant meteorological factors that drive PET on the Tibetan Plateau (Zhang et al. 2009; Chen et al. 2006; Wang et al. 2013; Liang et al. 2013; Yin et al. 2010). Because of inaccessibility, the complex terrain, sparse meteorological stations, and high elevation (>4500 m) of the Tibetan Plateau, not all meteorological variables required by the Penman–Monteith model are completely observed in some regions. The applicability of alternative PET models has not yet been assessed on the Tibetan Plateau. Therefore, it is necessary to assess the uncertainties associated with alternative PET models, especially when the meteorological data are incompletely observed.
In this study, we extend previous studies by 1) comparing the performance of three PET models on daily, seasonal, and annual scales; 2) assessing the differences in PET values for different land-cover types on the Tibetan Plateau during 1981–2010; 3) investigating the dominant meteorological variables that determine the interannual variability of PET; and 4) providing a validation for choosing a PET model for use on the Tibetan Plateau.
2. Materials and methods
2.1. Study site
The Tibetan Plateau (25°–40°N, 75°–105°E), the highest (average elevation > 4000 m) and largest (>2.5 million km2) highland in the world, is located in western China. Most areas of the plateau are characterized with an arid–semiarid climate. The mean annual precipitation is 473 mm, ranging from 50 to 1000 mm. The mean annual temperature is 3.8°C, ranging from −15° to 10°C. The spatial patterns of precipitation and temperature result in an increase in aridity from southeast to northwest of the Tibetan Plateau. More than 60% of the plateau is alpine grasslands, which includes alpine meadows, alpine steppes, and alpine desert.
2.2. Data sources
High-quality daily in situ meteorological datasets from 1981 to 2010, including maximum air temperature Tmax (°C), minimum air temperature Tmin (°C), relative humidity RH (%), wind speed at 10-m height U (m s−1), sunshine duration S (h), and precipitation P (mm) from 80 out of 109 meteorological stations (all 80 meteorological stations are built before 1981, the locations never change during the study period, and the stations have more than 95% of the recorded data) over the Tibetan Plateau were downloaded from China Meteorological Administration (CMA) website (http://data.cma.cn/site/index.html). The altitudes of all 80 stations varied between 505 and 4670 m. Station density in high-altitude regions (western of plateau) is much more sparse than in the lowlands (eastern of plateau; e.g., only two stations are available in the west; Figure 1). Missing data were estimated from average values of other years observed at the same station. The 80 sites covered a variety of land-cover types: needleleaf forest (5), broadleaf forest (1), shrub (10), desert (3), alpine steppe (16), alpine meadow (19), alpine cushion vegetation (22), and others (4). Annual, seasonal, and monthly average values of PET were calculated from the daily values.
2.3. Calculation of PET
2.3.1. Penman–Monteith model
The Penman–Monteith model (Penman 1948; Monteith 1965) is a physically based method that is widely used for PET estimation. In 1998, the Food and Agricultural Organization (FAO) modified the Penman–Monteith model by defining a hypothetical reference grass with an assumed height of 0.12 m, a fixed surface resistance of 70 s m−1, and an albedo of 0.23. The assumptions can be applied to the Tibetan Plateau, where more than 60% is alpine grasslands (Song et al. 2017). The Penman–Monteith model is expressed as (Allen et al. 1998)
where PETPM is the potential evapotranspiration (mm day−1), Δ is the slope of the saturated vapor pressure curve (kPa °C−1), Rn is the net radiation (MJ m−2 day−1), G is the soil heat flux density (MJ m−2 day−1), es is the saturation vapor pressure (kPa), ea is the actual vapor pressure (kPa), (es − ea) is the saturation vapor pressure deficit (VPD; kPa) at temperature T, γ is the psychrometric constant (kPa °C−1), and U2 is the wind speed at 2-m height (m s−1). A full description of the Penman–Monteith model is available in appendix A.
2.3.2. Priestley–Taylor model
The Priestley–Taylor equation (Priestley and Taylor 1972) is a simplification of the original Penman method, where the aerodynamic term is replaced by an empirical coefficient, known as the Priestley–Taylor parameter α. The model takes the following form:
where λ is the latent heat of vaporization (MJ kg−1), and α is an empirical constant. The terms Rn, G, Δ, and γ are as defined for the Penman–Monteith model.
Instead of setting α as an empirical constant (α = 1.26), we calculated α using the equation recommended by Er-Raki et al. (2010), since previous studies have shown that the appropriate value for the parameter α varied considerably from humid to arid regions. For example, McNaughton and Black (1973) suggested that α should be 1.05 in a humid region. However, the value of 1.74 had been suggested for an arid region by Steiner et al. (1991). The α equation in Er-Raki et al. (2010) was calibrated by minimizing the RMSE between the PET computed by Priestley–Taylor model and FAO-PM model in a semiarid region. Er-Raki et al. (2010) concluded that the calibration of α could be applied in many areas (arid, humid):
It is noted that α can vary from 0.93 to 2.33, depending on the climatic conditions.
2.3.3. Thornthwaite model
The PET in the Thornthwaite model (PETT; Thornthwaite 1948) is a function of the mean air temperature Tmean (°C), climatological normal annual temperature Tn, and photoperiod (maximum number of sunshine duration S):
where PETT is the standard, 30-day evapotranspiration [mm (30 day)−1], considering S = 12 h; and I and α are thermal indices, calculated by
Finally, PETT,day, in millimeters per day, is estimated by the following expression:
To adjust the original Thornthwaite model to arid and very humid conditions, Camargo et al. (1999) replaced the Tmean by the effective temperature Tef, which is used in this study:
2.4. Spatiotemporal analyses of PET
2.4.1. Spatial interpolation
2.4.2. Trend analyses
To detect the magnitude of the overall trends in meteorological factors and PET, the linear regression method was used, which is the most frequently used method for quantifying long-term trends in hydroclimatic variables (Wang et al. 2013).
The significance of a trend was assessed by the Mann–Kendall test (Mann 1945; Kendall 1975), which is a nonparametric method used commonly for time series (Sneyers 1990). The Mann–Kendall test is popular because it is distribution free, robust against outliers, and has a higher power-to-test tendency than many other commonly used tests. The equation below is used to estimate whether a series has a significant trend:
where N is the series length and V is the number of the dual observed values in a series. In this study, p < 0.05 is regarded as a significant trend. A positive M indicates an ascending trend and vice versa. If a series has a significant trend, |M| > Mp/2 = 1.96.
The segmented regression, which simultaneously detects both shift trends and step changes (Shao and Campbell 2002; Shao et al. 2010) as compared with other detection regression, was employed to investigate the segmented change patterns in meteorological and PET variables. All the statistical analyses involved in this paper were implemented with MATLAB 10.0.
3.1. Performance of different PET models at mean annual scales
At an annual time scale, PET estimated by three models are all very similar: the PETPM and PETPT differ by only 7.2 mm, and PETT and PETPM differ by 25.5 mm. However, there is large spatial variability of PET (Figure 2). Specifically, a west-to-east declining gradient of mean annual PET is obvious for both PETPM and PETPT estimates. The mean annual PETPM is 700.5 ± 143.2 mm (ranging from 474.0 to 1112.5 mm) followed by PETPT, which provides a lower mean annual PET of 693.3 ± 108.8 mm (ranging from 505.7 to 1052.1 mm). The mean annual PETT (675.1 ± 98.4 mm yr−1) is slightly lower than those calculated by the other two models, with its lowest value of 477.4 mm yr−1 in the middle of the Tibetan Plateau and highest value of 1004.3 mm yr−1 in the southeast.
From spring to autumn, the spatial distribution of PET values estimated by different models is similar to that of mean annual PET (Figure 3). For the whole region, the maximum PET occurs in summer with values ranging from 252.7 ± 36.3 (PETT) to 308.3 ± 58.5 mm (PETPT). The minimum PET occurs in winter with values ranging from 35.8 ± 20.2 (PETPT) to 80.3 ± 32.2 mm (PETT). Our results show that the standard deviations are largest in summer and smallest in winter.
At site levels, the temporal changes of annual PET show a mixed pattern of both upward and downward trends for different parts of the Tibetan Plateau from 1981 to 2010 (Figure 4b). At an annual scale, increasing trends in annual PET are observed for 43% of all meteorological stations with PETPM, 69% with PETPT, and 95% with PETT, wherein annual PET shows significant changing trends (p < 0.05) at 18%, 23%, and 80% of all stations, corresponding to PETPM, PETPT, and PETT, respectively. The stations with significant negative trends estimated by PETPM are mainly distributed at the northern, southwestern, and southeastern corners of the plateau. Significant increases in annual PETPM mainly occur east of the plateau. Similar to PETPM, the stations with significant increasing trends of PETPT are mainly distributed in the eastern plateau.
Meanwhile, Figure 4a illustrates the interannual trends of PET for the entire plateau. It shows that the plateau is characterized with the different interannual trends by different models. First, the interannual PETPM and PETPT trends both have a transition period, in which PET decreases from 1981 through 1993 and reverses after 1994. Specifically, PETPM has decreased by 3.4 mm yr−1 (p < 0.05) during 1981–93, while PETPM has increased by 2.3 mm yr−1 (p < 0.1) since 1994. PETPT decreases significantly (p < 0.05) by 2.0 mm yr−1 and increases insignificantly by 1.0 mm yr−1. Second, as opposed to the results estimated by the other two models, PETT demonstrates a slightly increasing trend (0.5 mm yr−1, p = 0.9) before 1993 and shows a strong increasing trend after 1994 (1.9 mm yr−1, p < 0.01).
3.2. Performance of different PET models at daily time scales
The PET values estimated by the three models show a strong seasonality: mean daily PET averaged over 80 stations gradually increases from spring to summer and then decreases from autumn to winter (Figure 5). The PETPM ranges from 0.60 ± 0.12 mm day−1 in winter to 3.21 ± 0.42 mm day−1 in summer with a mean daily PETPM of 1.92 ± 0.28 mm. The long-term, annual-mean, daily PETPT is 1.90 ± 0.30 mm day−1 with an annual maximum of 3.78 ± 0.62 mm day−1 and minimum of 0.16 ± 0.02 mm day−1. When compared with PETPM and PETPT, PETT has the lowest value. From 1981 to 2010, PETT ranges from 0.45 ± 0.15 to 2.89 ± 3.17 mm day−1 with a mean daily value of 1.33 ± 0.97 mm day−1.
With respect to the 1:1 line (where the results from the two models would equal each other), the PETPT provides the best agreement with PETPM (Figure 6). The coefficient of determination (R2 = 0.98, p < 0.001) and the slope are close to 1, with acceptable RMSE of 0.12 mm day−1. When compared with PETPM, PETT performs less well with R2 of 0.80 (p < 0.001) and RMSE of 0.3 mm day−1.
3.3. PET for different land-cover types
Our results show that, for eight land-cover types, the three PET models perform similarly for the majority of land-cover types but diverge greatly for some land-cover types (Figure 7). For example, PETPM and PETPT differ by only 0.3% at shrub sites and 0.4% at broadleaf forest sites. PETT for desert and alpine steppe are lower than PETPM by 11% and 12%. However, PETT for broadleaf forest is higher than PETPT and PETPM by 18%. Furthermore, the results of these model estimations indicate that the broadleaf forest has the highest PET varying from 817.2 to 1004.3 mm yr−1, while alpine meadow has the lowest PET, ranging from 589.2 to 648.5 mm yr−1.
Interannual variabilities of PET for different land-cover types are different among the three models (Table 1). For example, PETT for all land-cover types significantly increased for the last 30 years, and the largest increase is found in broadleaf forest (2.79 mm yr−1). However, most land-cover types show decreasing trends in PETPM. Only desert, alpine steppe, and “others” show increases in PETPM with the trends ranging from 0.38 to 2.66 mm yr−1. Similar with PETT, six of the eight land-cover types demonstrate slightly increasing patterns during 1981–2010 in PETPT.
3.4. Differences in meteorological regulations of the PET models
The meteorological factors, which control temporal variations of PET, include dynamic factors such as wind speed U; thermodynamic factors such as temperature Tmean, radiation Rn, and sunshine duration S; and water factors such as precipitation P and relative humidity RH (Q. Liu et al. 2010). The interannual trends of meteorological factors are investigated to explain the detected PET trends (Figure 8). The results show that Rn (R2 = 0.20, p < 0.05), RH (R2 = 0.12, p < 0.1), U (R2 = 0.81, p < 0.001), and S (R2 = 0.25, p < 0.05) decrease as a whole during 1981–2010. Nevertheless, Tmean (R2 = 0.76, p < 0.001) and P (R2 = 0.25, p < 0.05) increase from 1981 to 2010. Moreover, according to the temporal variation patterns of PET, we divide the trends of meteorological factors into two stages. Although there are the same increasing or decreasing trends in Rn, Tmean, U, S, and P between two stages, the slopes of trends are totally different. For example, U decreases at a rate of 0.02 m s−1 yr−1 during the first period (1981–93) and then decreases at a rate of 0.004 m s−1 yr−1 during the second period (1994–2010). Different from other meteorological factors, a slightly increasing trend in RH (R2 = 0.10, p = 0.3) is found during the period of 1981–93, while an obvious decreasing trend (R2 = 0.62, p < 0.001) can be found during the period of 1994–2010.
To identify the controlling meteorological factors driving interannual variations of PET and the relative contributions of different factors, we explore the correlations between PET and meteorological factors. The correlation analyses are conducted for two periods (Table 2). The results illustrate that the decreasing U (R2 = 0.72, p < 0.01) is most correlated with the decrease of PETPM during 1981–93. However, the most dominating factor for the increasing PETPM is RH (R2 = 0.73, p < 0.01), followed by Tmean (R2 = 0.48, p < 0.05) in the second period (1994–2010). PETPT is driven by S (R2 = 0.72, p < 0.01) during 1981–93, while RH (R2 = 0.49, p < 0.05) leads to the increasing PETPT during 1994–2010. As opposed to the other two models, the increasing Tmean dominates the increasing PETT in two stages (R2 = 0.47 and 0.89, respectively, p < 0.01).
4.1. From daily to annual scale
In this study, we used meteorological data to estimate PET with different models and investigated PET variations on daily, seasonal, and annual scales. Although the three models demonstrated similar PET values on average annual and daily scales, the spatial distribution and interannual trends of estimated PET revealed considerable inconsistence among different parts of the Tibetan Plateau.
For both mean daily and annual scales, the lowest PETT value of the three models was similar to previous PET intercomparison studies. For example, Hulme et al. (1996) showed that the value of PETT in Africa was generally lower than PETPM and PETPT. Similarly, Chen et al. (2006) demonstrated that values from PETT were lower than PETPM on the Tibetan Plateau. Fisher et al. (2011) compared three different types of PET models for 11 land-cover types at global scale, showing that PETT estimated 20%–30% less than PETPT and PETPW. Furthermore, Lu et al. (2005) showed that PETT for the United States was the lowest among six PET models.
In addition, the PETT did not agree well with the other two models either in regional variations (Figure 2) or among different land-cover types (Figure 7). The mean annual bias indicated that PETT overestimated PET over monsoon-affected areas (forests in the southeastern plateau) where climate was relatively humid, whereas it gave an underestimation over arid areas in the west of the Tibetan Plateau. The reason for this was that the Thornthwaite model assumed vegetation cover and depended heavily on the accuracy of temperature without considering net radiation function. For instance, erroneously low temperature can lead to excessively low PET in the Thornthwaite model.
Our results showed PETPT agreed well with PETPM not only for temporal variations (at both mean daily and annual scales) but also for spatial variations except for the deserts and alpine steppe (Figures 2, 3). The reason why PETPM was higher than PETPT in these regions (low Rn and relatively high VPD) was that the VPD-driven Penman–Monteith model gave higher PET than that from the Priestley–Taylor model without considering VPD.
Overall, much care must be taken when selecting the appropriate model for estimating PET for a particular region. Generally, a temperature-based model should be avoided if data are available to run the other models or if the vegetation cover is low. If only data for a temperature-based model are available, the authors should consider the calibration for the results according to our results on daily and annual scales. In addition, although the radiation-based model showed the closest results with combination model, it should be noted that the combination model will probably give lower PET estimates in alpine meadow and alpine cushion vegetation on the Tibetan Plateau.
4.2. Impact of meteorological factors
Differences in the interannual variation of PET are regulated by different model-dominated meteorological factors; Tmean, as a surrogate for atmospheric demand, is a critical meteorological driver for the temperature-based models (Fisher et al. 2011). It can be inferred from the Thornthwaite model that the increase in PETT on the plateau was mainly due to the rising Tmean from 1981 to 2010. This increasing trend in PETT was more obvious after 1994 when Tmean started to increase sharply (Figure 8).
In our study, we confirmed the general pattern of decreasing trends in PETPM across the Tibetan Plateau as identified by previous studies (Chen et al. 2006; Zhang et al. 2009; Wang et al. 2013; Liang et al. 2013; Yin et al. 2010; Figure 4a). Similar with previous studies, the decreasing wind speed, which was caused by the decrease in Asian monsoon strength, was the primary factor for the interannual pattern of PETPM before 1993 (Yin et al. 2010; Wang et al. 2013). Some research pointed out that urbanization (e.g., tall buildings) may cause the wind speed reduction. However, the anthropogenic impacts on the plateau are far less serious than most regions of the world, and the impact of urbanization on wind speed might be negligible (Q. Liu et al. 2010). During the past two decades, the decreasing RH and increasing air temperature, which offset the negative effects of wind speed, resulted in increasing PETPM. This result can be explained by the fact that the increasing Tmean caused a simultaneous increase in both VPD and outward longwave radiation. The increase in VPD accelerated the water transfer, while the increase in longwave radiation decreased Rn (Figure 8). PETPM increased when VPD-induced increase in PETPM was higher than the amount of decrease in PETPM caused by the increase of longwave radiation under low air temperature conditions.
Our results demonstrated that the decreasing S followed by Rn (calculated by S) were the dominant factors to drive the interannual PETPT trend before 1993. However, after 1994, the contributions of both S and Rn were less than those of RH (Table 2), which demonstrated that PETPT was thermodynamic limited before 1993 and turned to be water limited after 1994. It should be noted that, as opposed to the interannual pattern of PETT, the mean annual PETPT and PETPM experienced a concave pattern with decreases from 1980 to 1993, and this reversed after 1994. Previous studies also documented that the 1990s marked a transition period in which the trend in PET showed an increasing pattern (Wang et al. 2013; Zhang et al. 2009). We also compared our results with the pan evaporation ETpan measured by previous studies (the meteorological records used in our study did not include the ETpan) and found that ETpan have similar patterns. For example, M. Liu et al. (2010) demonstrated that the nationwide average ETpan data first decreased during 1960–92 and then increased after 1992 because of seriously rising air temperature. Similar results were also reported by Wang et al. (2013).
The magnitude of interannual PET variations will differ because of different meteorological drivers. Disparate combinations of meteorological factors may induce diverse PET trends under the influence of the same meteorological variables. Generally, if only data for a temperature-based model are available for the interannual PET estimation, it is necessary to consider the changing points identified from our analyses.
In this study, we compared three PET models, including a radiation-based model (Priestley–Taylor model), a temperature-based model (Thornthwaite model), and a combination model (Penman–Monteith model), based on daily data from 80 meteorological stations on the Tibetan Plateau from 1981 to 2010. The results showed that the Penman–Monteith and Priestley–Taylor models demonstrated similar estimations at annual and daily scales. The Thornthwaite estimates were lower than the other two models. Regionally, PET estimated by Penman–Monteith and Priestley–Taylor models showed similar spatial patterns with a declining gradient from west to east. PET estimated by Thornthwaite showed a significant increasing trend from 1981 to 2010, but the other two models revealed transitions around 1993. The differences in model structures and their dominant variables were the major reasons explaining the differences among their interannual PET variations. Overall, it is important to understand that most PET models have been developed for use in specific studies and that they were constructed to be most appropriate for use in the climates of the locations where they were developed. Our study suggests that careful consideration and verification are needed before applying a PET model in hydrological and ecological studies.
This study was supported by the Natural Science Foundation of Fujian Province, China (2017J01069); the National Natural Science Foundation of China (31600368); the Fundamental Research Funds for the Central Universities of China (20720160109); the National Key Research and Development Program of China (2017YFC0506100); the NASA Land-Cover and Land-Use Change Program (NASA-NNX09AI26G); and the NSF Division of Information and Intelligent Systems (NSF-1028291).
Appendix A: A Full Description of the Penman–Monteith Model
The parameters of Equation (1) were determined as follows below.
Net radiation Rn was computed by equations recommended by Yin et al. (2008), which was calibrated by observed solar radiation of 80 meteorological stations over China and measurements of net longwave radiation on the Tibetan Plateau:
where σ is the Stefan–Boltzmann constant (4.903 × 10−9 MJ K−4 m−2 day−1); Tmax,k and Tmin,k are the maximum and minimum temperatures in kelvins, respectively; s is actual sunshine duration (h); S is potential sunshine hours; and Rso is clear-sky solar radiation. The calculation of Rso followed the procedure outlined in Allen et al. (1998).
Soil heat flux G was calculated using mean temperature with a simple formula:
where Tmean,i is the mean temperature of the i day, and Tmean,i−1 is the mean temperature of the previous day.
The parameters Δ and γ were computed as
where Cp is the specific heat of moist air (kJ kg−1 °C−1), P is atmospheric pressure (kPa), λ is the latent heat of vaporization (MJ kg−1), and h is altitude (m).
To convert wind speed data obtained at 10 m to the standard height of 2 m,
The es and ea were estimated with mean daily relative humidity RHmean and temperature:
Appendix B: The Mean and Square Deviation of PET Values Estimated by Three Models
Potential evapotranspiration values are given by Penman–Monteith, Priestley–Taylor, and Thornthwaite models in Table B1.