In many places around the world, panevaporation has been detected to decrease with the increase in temperature, which is known as the “panevaporation paradox.” An example of the paradox was found in the Haihe River Basin from 1957 to 2001. To explain the mechanism of the paradox, an approach to quantify the contributions of climate factors to the panevaporation trend has been proposed, in which the individual contribution was defined as the product of the partial derivative and slope of the trend for the concerned variables. Four variables, including temperature, wind speed, solar radiation, and vapor pressure, were selected based on the Penman-Monteith method to assess their individual contribution to the panevaporation trend. The results showed that an increase in temperature resulted in the increase of panevaporation, but this effect had been offset by an increase in vapor pressure and decrease in wind speed and solar radiation. Wind speed was the dominant factor contributing to panevaporation decreases in the Haihe River Basin.
 The explanations from different researchers seem inconsistent with each other because of the differences in the study areas and the approaches used. However, they imply that the decrease in panevaporation may not be due to a single factor, but rather, to an integrated effect of radiation, wind, temperature, humidity, and so on. It is therefore important for us to quantify the contributions of the climate factors to the long-term panevaporation trend. The purpose of this article is to propose an approach to quantify the contributions to panevaporation trends to help explain the panevaporation paradox; the Penman-Monteith equation is used to determine the climate factors that may contribute to panevaporation trends. The contributions of each climate factor are derived according to the perfect differential of the Penman-Monteith equation, but first, the long-term trends of panevaporation, reference evapotranspiration, and the climate factors concerned will be detected.
2.1. Trend Test
 The rank-based nonparametric Mann-Kendall statistical test [Mann, 1945; Kendall, 1975] has been commonly used for trend detection [Yue and Wang, 2002; Zheng et al., 2007] because of its robustness for nonnormally distributed and censored data, which are frequently encountered in hydroclimatic time series. This method defines the test statistic Z as
where n is the data record length and xi and xj are the sequential data values. The function sgn(x) is defined as
Equation (3) gives the standard deviation of S with correction for ties in the data, with ei denoting the number of ties of extent i. The upward or downward trend in the data is statistically significant if ∣Z∣ > u1−α/2, where u1−α/2 is the (1 − α/2) quantile of the standard normal distribution [Kendall, 1975] and when α = 0.05 and u1−α/2 = 1.96. A positive Z indicates an increasing trend in the time series, and a negative Z indicates a decreasing trend.
 An alternative method widely used to detect the trend of a time series x is the linear regression approach against time t. For the linear regression function (i.e., x = a + bt), we have dx/dt = b, in which the slope b can be considered an indicator describing the trend of the variable concerned and can be estimated as
2.2. Estimation of Potential Evapotranspiration
 In literature, there are many methods available for potential evapotranspiration estimation, among which the Penman-Monteith method is widely accepted. In this study, the Penman-Monteith method introduced by the Food and Agriculture Organization (FAO) is used to estimate the so-called reference evapotranspiration (ETref), where the land cover is regarded as 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 [Allen et al., 1998]. The FAO method to estimate reference evapotranspiration can be expressed as
where ETref is reference evapotranspiration (millimeters per day), Rn is net radiation at the reference surface (megajoules per meter squared per day), G is soil heat flux density (megajoules per meter squared per day), Tmean is daily mean temperature (degrees Celsius), U2 is the wind speed at 2 m height (meters per second), VPs is saturated vapor pressure (kilopascals), VP is actual vapor pressure (kilopascals), Δ is the slope of the vapor pressure curve versus temperature (kilopascals per degree Celsius), and γ is the psychrometric constant (kilopascals per degree Celsius). Rn represents the difference between incoming net shortwave radiation (Rns) and outgoing net longwave radiation (Rnl). Rns is estimated from surface solar radiation (Rs):
where λ (= 0.23) is the albedo of the reference grassland, alfalfa. Rs is estimated as
where S is the actual duration of sunshine (in hours), N is the maximum possible duration of sunshine or daylight hours (S/N is thus the relative sunshine duration), and Ra is the extraterrestrial radiation intensity (megajoules per meter squared per day). The coefficients as (= 0.25) and bs (= 0.50) were estimated from measured solar radiation and sunshine hours at the eight radiation stations that have solar radiation records (Figure 1).
 Pans are also widely used to estimate potential evapotranspiration. As mentioned by Allen et al. , several factors produce significant differences in water loss from pans and from a cropped surface such as differences in reflection of solar radiation, turbulence, temperature, and humidity of the air directly above the respective surfaces. Notwithstanding the difference between panevaporation and the reference evapotranspiration of cropped surfaces, panevaporation is related to the reference evapotranspiration by a regression function:
where Kp and Kc are regression coefficients and Epan is panevaporation (millimeters per day).
2.3. Sensitive Coefficient
 To identify the attributions of long-term evaporation change, a mathematically defined sensitivity coefficient is used to evaluate the sensitivity of reference evapotranspiration related to climate variables [McCuen, 1974]:
where xi is the ith climate variable and S(xi) is the sensitivity coefficient of reference evapotranspiration related to xi. The sensitivity coefficient was first adopted by McCuen  and is now widely used [Coleman and DeCoursey, 1976; Beven, 1979; Rana and Katerji, 1998; Qiu et al., 1998; Hupet and Vanclooster, 2001; Gong et al., 2006]. Essentially, a positive or negative sensitivity coefficient of a meteorological variable indicates that ETref will increase or decrease as the meteorological variable increases. The larger the sensitivity coefficient, the larger effect a given variable has on ETref. A sensitivity coefficient of 0.1 for a variable would mean that a 10% increase of that variable may increase ETref by 1%, while all other variables remain unchanged [Gong et al., 2006].
2.4. Contribution Assessment
 Mathematically, for the function y = f(x1, x2,…), the variation of the dependent variable y can be expressed by a differential equation as
where xi is the ith independent variable and f′i = ∂f/∂xi. Moreover, as y varies with time t, we can rewrite equation (11) as
If we let TRy = dy/dt and TRi = dxi/dt be the long-term trend in y and xi, then equation (12) can be rewritten as
If TRy and TRi are estimated as the slope of the linear regression for y and xi against time t given in equation (5), C(xi) can then be estimated as the contribution of xi to the long-term trend in y, which is exactly the product of the partial derivative and long-term trend in xi.
 Therefore, following the equation (equation (6)) for reference evapotranspiration estimation, we have, approximately,
or, written briefly,
where TRref is the long-term trend in ETref and C(Rs), C(Tmean), C(U2), and C(VP) are the contributions to the long-term trend in ETref due to the change in Rs, Tmean, U2, and VP, respectively. It should be noted that only four meteorological variables, including Rs, Tmean, U2, and VP, are selected in equation (14) and equation (15) because Tmean is in good relationship with Tmax and Tmin and can represent the effect of change in the air temperature. Moreover, the variables considered in equation (14) and equation (15) should be independent of each other to ensure that each factor represents its individual contribution. In such a case, vapor pressure (VP) instead of vapor pressure deficit (VPD) is considered here because VPD is the difference between VPs and VP, where VPs is a function of temperature. Soil heat flux density is also a function of temperature but is neglected because of its rather small long-term change, while δ represents the systemic error in the estimation of the contributions using equation (14) and equation (15).
 For the panevaporation Epan, according to equation (9), we have dEpan/dt = Kp × dETref/dt; therefore, equation (15) can be rewritten to estimate the contribution to long-term trend in Epan as
or, simplified, as
where TRpan is the long-term trend in Epan and can be estimated by equation (5), and C′(Rs), C′(Tmean), C′(U2) and C′(VP) are individual contributions to the long-term trends in Epan due to a change in Rs, Tmean, U2, and VP, respectively; ɛ is the error item. Furthermore, the individual proportional contribution of climate factors to the long-term trend in Epan can be estimated as
where x may be Rs, Tmean, U2, or VP, and Cpan is the estimated total contribution to the panevaporation trend.
3. Study Area and Data
 Located in northern China, the Haihe River Basin (HRB; 112°E∼120°E and 35°N∼43°N) has a total area of more than 318 × 103 km2, including two megacities, Beijing and Tianjin. The HRB is one of the most developed areas in China, with a population accounting for about 10% of the nation's total population. Climatically, the HRB belongs to the East Asian monsoon region. The annual mean temperature varies from 8°C to 12°C, while annual precipitation and evaporation is about 539 mm and 470 mm, respectively; relative humidity varies from 50% to 70%.
 In this study, a data set of 45 national meteorological stations over the HRB is available for the period 1957–2001 from the National Climatic Centre of the China Meteorological Administration, of which eight stations have solar radiation records (Figure 1). The data set includes daily observations of maximum, minimum, and average air temperatures (Tmax, Tmin, Tmean) at 2 m height, wind speed measured at 10 m height, vapor pressure (VP) at 2 m height, sunshine duration, and panevaporation (Epan). Epan was measured using a metal pan, 20 cm in diameter and 10 cm high, installed 70 cm above the ground. To estimate the reference evapotranspiration, the measured wind speed was transferred to wind speed at 2 m height (U2) by the wind profile relationship introduced by Allen et al. . For the basin as a whole, the values of the variables concerned were obtained by the kriging interpolating method of ArcGIS based on station observations.
4. Results and Discussions
4.1. Correlation Between ETref and Epan
Figure 2 shows that the value for Epan was in good agreement with that for ETref. The R2 between annual Epan and ETref was about 0.91 for all 45 stations over the HRB (Figure 2a), while it was about 0.95 for the whole basin (Figure 2b). In addition, the regression functions between Epan and ETref for different seasons listed in Table 1 show that R2 values were above 0.93, with pan coefficient (Kp) varying from 1.981 to 2.776. The close relationship between Epan and ETref suggests that reference evapotranspiration can be a good estimation of panevaporation in the HRB if regression coefficients are known. This may not be surprising as Epan measures the integrated effect of radiation, wind, temperature, and vapor pressure on evaporation from an open-water surface and reflects evapotranspiration ability in certain environments [Xu et al., 2006]. Therefore, it was reasonable to use equation (16) to estimate contributions to the long-term trend in Epan.
Table 1. Regression Equations Between Epan and ETref for the Whole Haihe River Basin
Epan = 2.737 × ETref − 302.4
Epan = 2.776 × ETref − 492.3
Epan = 2.539 × ETref − 165.2
Epan = 1.981 × ETref − 32.1
Epan = 2.738 × ETref − 1086.4
4.2. Trend of Potential Evapotranspiration
Figure 3 shows the trends of annual Epan and ETref from 1957 to 2001, using the Mann-Kendall method, for all stations in the HRB. Spatially, one may notice that there was a close correlation between the trend of Epan and ETref. As shown in Figures 3a and 3b, except for two stations (53487 and 53588), which show an increasing trend in both Epan and ETref, most stations show a decreasing trend. In particular, 25 stations show a significant decreasing trend at the level of α = 0.05, mainly situated at the southeastern part of the HRB. The decreasing trend of Epan and ETref in the hilly region, however, was not statistically significant. The slope of the Epan trend varied from −17.6 to 6.7 mm yr−2, and that of ETref ranged from −5.7 to 2.2 mm yr−2. Again, the slope of the Epan trend was in good agreement with that of ETref with R2 = 0.91 (Figure 3c), indicating the efficiency of panevaporation estimation using the Penman-Monteith method introduced by Allen et al. .
 For the basin as a whole, the results of the trend test show that both annual Epan and ETref decreased significantly (α = 0.05) during the period from 1957 to 2001. Annual Epan decreased at 4.91 mm yr−1 and annual ETref decreased at 1.77 mm yr−1 (Figure 4). The decreasing rate is less than that in the western United States (6.3 mm yr−1) [Peterson et al., 1995] but larger than that in the eastern United States (3.2 mm yr−1) [Peterson et al., 1995], Australia (4.0 mm yr−1) [Roderick and Farquhar, 2004], the Tibetan Plateau (4.57 mm yr−1) [Zhang et al., 2007], and the Yangtze River Basin (3.09 mm yr−1) [Xu et al., 2006] in China. It is noted that there is a depression of Epan and ETref around 1964, possibly because the year 1964 was an exceptional rain year, implying lower sunshine durations and higher vapor pressures, which could lead to lower evapotranspiration rates.
 Seasonally, as shown in Figure 4, in the spring and summer, Epan significantly decreased, with reduction rates of −2.02 mm yr−1 and −1.95 mm yr−1, respectively (Table 2), while the decreasing trend in Epan in the autumn and winter seasons was not significant, with Mann-Kendall statistics of Z less than 1.96 (Table 2). The result was almost the same as that of ETref (Table 2), implying that the decrease of annual potential evapotranspiration was mainly due to the decrease in the spring and summer seasons.
Table 2. Trends in Epan, ETref, and Climate Variables of the Basin as a Whole
 According to the Penman-Monteith method described in equation (6), Tmax, Tmin, Tmean, VP, U2, and Rs are the meteorological variables that determine the estimation of reference evapotranspiration. As shown in Figure 5, significantly, Tmax, Tmin, and Tmean increased at most stations of the HRB, while U2 and Rs decreased at more than 75% of the 45 stations. VP showed an increasing trend at almost all stations, of which 22 stations had trends that increased significantly and only 1 station showed a significant decreasing trend (Figure 5).
 For the basin as a whole, as shown in Figure 6 and Table 2, Tmax, Tmin, and Tmean increased with a change rate of 0.017°C yr−2, 0.033°C yr−2, and 0.025°C yr−2, respectively, while U2 and Rs decreased considerably at the rate of −0.014 ms−1 yr−2 and −0.023 MJ m−2 d−1 yr−2, respectively. However, the increasing trend of VP was not statistically significant at the level of α = 0.05 but was significant at the level of α = 0.1. The increasing trends in Tmax, Tmin, and Tmean would have led to an increasing trend in Epan and ETref, while the increasing trend in VP and decreasing trend in U2 and Rs would have resulted in decreasing Epan and ETref.
4.4. Sensitivity of ETref
 As mentioned earlier, partial derivatives (equation (11)) and sensitivity coefficients (equation (10)) are two essential and useful indicators for showing the impacts of meteorological factors on the change of potential evapotranspiration. According to equation (10) and the Penman-Monteith method in equation (6), the annual and seasonal means of daily partial derivatives and sensitivity coefficients for Tmean, VP, U2, and Rs were calculated at the basin scale.
 Annually, for the basin as a whole, the mean partial derivatives of ETref in relation to Tmean, VP, U2, and Rs were 0.15, −1.66, 0.41, and 0.065, respectively (Table 3). The positive f′(Tmean), f′(U2), and f′(Rs) and negative f′(VP) indicated that ETref decreased with the decrease of Tmean, U2, and Rs but increased with the reduction of VP. In addition, the absolute value of f′(VP) was the largest in comparison with f′(Tmean), f′(U2), and f′(Rs), indicating that ETref was more sensitive to VP than to Rs, U2, and Tmean. As shown in Table 3, the annual mean sensitivity coefficient of ETref with respect to Tmean, VP, U2, and Rs was 0.43, −0.51, 0.27, and 0.34, respectively. Similar to those of partial derivatives, sensitivity coefficients were positive for U2, Rs, and Tmean but negative for VP, suggesting that a 10% increase of VP could result in a 5.1% decrease of ETref, while a 10% increase of Tmean, U2, and Rs could lead to a 4.3%, 2.7%, and 3.4% increase in ETref, respectively. Evidently, ETref was again shown to be most sensitive to vapor pressure.
Table 3. Sensitivities of ETref in Relation to Wind Speed, Vapor Pressure, Solar Radiation, and Mean Temperature
 It is noted that both partial derivatives and sensitivity coefficients were not constant, but varied with the meteorological condition. Intraannually, as shown in Table 3, f′(U2) held the largest value in spring but the smallest in winter. However, f′(Tmean) and f′(Rs) were the largest values in summer but the smallest in winter; f′(VP) was negative for all seasons, with a maximum absolute value in winter and a minimum absolute value in summer. The seasonal variation of the sensitivity coefficient was not the same as that of partial derivatives. As shown in Table 3, S(U2) was the largest in winter but the smallest in summer, indicating that ETref in the winter season was more sensitive to U2 than in other seasons. On the contrary, both S(RS) and S(Tmean) had the largest value in summer but the smallest in winter, while S(VP) had the largest absolute value in winter but the smallest in summer. One may notice that in the winter season, when Tmean was below zero, the sensitivity coefficient of ETref in relation to Tmean was negative, suggesting that a 10% increase of temperature in winter could lead to a 4.4% increase of ETref in that season.
Figure 7 shows the interannual variation of annual partial derivatives and sensitivity coefficients. Apparently, the partial derivatives of ETref in relation to both U2 and Rs showed an increasing trend from 1957 to 2001, which indicated that ETref was becoming more sensitive to solar radiation and wind speed but less sensitive to vapor pressure. However, f′(Tmean) and the absolute value of f′(VP) tended to decrease, implying that ETref was becoming less sensitive to Tmean and VP; f′(Tmean) was larger than the long-term average since the 1970s, while f′(VP) was larger than the long-term average since the 1980s. Being different to partial derivatives, the sensitivity coefficients did not show any trend during the period 1957–2001 (Figure 7b), which means that the ratio between the proportional change of meteorological variables and proportional change of ETref remained quite stable.
4.5. Contributions to Epan Trend
 According to equation (16), the contributions of climate factors to long-term trends in ETpan can be estimated with the known partial derivatives and the trends of the climate factors concerned. As shown in Figure 8, the annual and seasonal panevaporation trends calculated using equation (16) of the 45 stations fit well with those detected from the observed panevaporation (TRpan). The largest absolute error between seasonal TRpan and Cpan was 1.8 mm yr−1 in the summer at station 54906, with a relative error of −22.6%. Annually, the largest absolute error between TRpan and Cpan was 3.9 mm yr−1 at station 54714, with a relative error of −26.0%. Especially for seasonal panevaporation or stations with a smaller slope of trend (i.e., ∣TRpan∣ < 10 mm yr−2), there was better agreement between the observed and calculated panevaporation trends. For the basin as a whole, Table 4 shows that the errors between TRpan and Cpan in the spring, summer, autumn, and winter were 0.09, 0.14, −0.02, and −0.02 mm yr−1, respectively, with relative errors of −4.48%, −7.18%, 3.03%, and 8.00%. Annually, the absolute error was 0.32 mm yr−1 and the relative error was −6.52%. The results indicate the effectiveness of the approach proposed in this study, considering the uncertainty of the observed data and the systemic error in estimating panevaporation using the Penman-Monteith method.
Table 4. Contributions and Proportional Contributions of Climate Factors to the Trends in Epana
TRpan is the long-term trend of observed Epan; Cpan is the estimated trend of Epan equaling the sum of C'(U2), C'(VP), C'(Rs), and C'(Tmean); while ρsum is the sum of the proportional contributions ρ(U2), ρ(VP), ρ(Rs), and ρ(Tmean).
Table 4 shows that except for Tmean, the other three factors VP, U2, and Rs all had negative contributions to the long-term trend of Epan, which means that the increase in temperature had resulted in the increase in panevaporation. However, this effect of temperature has been offset by the increase of vapor pressure and decrease of wind speed and solar radiation.
 Annually, the increasing Tmean led to a 3.64 mm yr−1 increase of Epan; meanwhile, the change of U2, Rs, and VP led to the decrease of Epan at the rate of −5.48 mm yr−1, −1.47 mm yr−1, and −1.28 mm yr−1, respectively. The proportional contributions of Tmean, VP, U2, and Rs to the long-term trend in annual Epan were −79.3%, 27.89%, 119.39%, and 32.03%, respectively, where the decrease in U2 was shown to be the dominant factor for the decrease in Epan. Figure 9 shows the contributions and proportional contributions of the four climate variables to long-term trends in annual panevaporation at each station. In 31 out of 45 stations, U2 was the dominant factor for the change in Epan, while Tmean, Rs, and VP played the most important role in the change of Epan at 5, 6, and 3 stations, respectively. The results mean that the decreasing trend in U2 was the most crucial factor for the decreasing trend detected in Epan in the HRB, followed by Rs and VP.
 Seasonally, as shown in Table 4, the total contributions of wind speed, vapor pressure, solar radiation, and temperature to panevaporation trends in spring and summer were larger than in autumn and winter, which well explained the decreasing trend of Epan in spring and summer. Except for the summer season, the absolute contributions of wind speed were largest in the other three seasons, which covered the effect of increasing temperature. For instance, in the winter season, the increase of temperature may have resulted in a 1.09 mm yr−1 increase of Epan, but the decrease of wind speed could have led to a 1.31 mm yr−1 decrease of Epan. It should also be noted that the contributions of VP and Rs to Epan trends in spring and summer were both larger than in autumn and winter, indicating stronger changes of VP and Rs in spring and summer.
4.6. Explanation of Panevaporation Paradox
 The panevaporation paradox has underlined the confusing situation in which global warming is accompanied by the decrease of panevaporation. In this study, however, we found that increasing temperature indeed led to the increase of panevaporation, but this effect was offset by changes in other climate factors. The decreasing wind speed and solar radiation and the increasing vapor pressure resulted in the decrease of panevaporation in the HRB.
 Wind speed was shown to be the dominant factor contributing to the decrease of panevaporation in the HRB. Such results were also found in the United States [Hobbins, 2004], the Tibetan Plateau of China [Zhang et al., 2007], and Australia [Rayner, 2007]. However, in other regions, other factors, such as solar radiation or vapor pressure, may play a more important role in the decrease of panevaporation. Even in the HRB, according to Table 3 and Figure 9, the dominant factor in the panevaporation trend has shown spatial and temporal differences, despite wind speed being the crucial factor at most stations (68.9%) and most of the time (spring, autumn, and winter). We should be aware that the contribution of climate factors to long-term trends of panevaporation is the product of partial derivatives and the slope of the trend, as described in equation (14). Therefore, the largest contribution may not result from the most sensitive factors. For instance, although vapor pressure was the most sensitive variable for potential evapotranspiration, its contribution to the long-term trend of potential evapotranspiration could not be the largest because of its relatively small change.
 Uncertainties existed in the estimation of the contribution of climate factors using equation (16). First, the uncertainty may be due to the correlation between reference evapotranspiration and panevaporation. Apparently, the better agreement between reference evapotranspiration and panevaporation indicates a more reliable equation (16). Second, for simplification, not all variables related to potential evaporation were considered in equation (16), which could be the reason for the systemic errors. Third, the uncertainty may come from the determination of the partial derivatives. As mentioned earlier, the partial derivatives of each climate factor were not constant but varied during the period 1957–2001. The mean partial derivatives used in this study may result in the uncertainty in estimating the sensitivity coefficients and the contributions of the climate variables concerned.
 In many places around the world in the last half century, pan evapotranspiration has been detected to decrease as temperature increases, a phenomenon known as the “panevaporation paradox.” The panevaporation paradox was detected in the HRB for the period 1957–2001, with results showing that panevaporation has decreased significantly, especially in the spring and summer seasons, while maximum temperature, minimum temperature, and mean temperature increased. Concurrently, wind speed and solar radiation decreased but vapor pressure increased.
 To explain the mechanism of the panevaporation paradox in the HRB, an approach to quantify the individual contributions of climate variables to panevaporation trends has been proposed. Four climate factors, including temperature, wind, vapor pressure, and solar radiation, were selected, according to the Penman-Monteith method. The contribution of a climate factor to the panevaporation trends was defined as the product of the partial derivative and the slope of the trend for the concerned factor. The increasing mean temperature in the HRB could have resulted in the increase of panevaporation; however, this effect has been offset by a decrease of wind speed, a decrease of solar radiation, and an increase of vapor pressure in this region. Wind speed was the dominant factor in decreasing panevaporation in the HRB.
 This research was supported by the Chinese National Key Program 973 project (2006CB403407), Key Projects in the National Science and Technology Pillar Program (2007BAC03A11), and the Natural Science Foundation of China (40601015). We appreciate the suggestions of the two anonymous reviewers and the help of Zhang Minghua of UC Davis in the improvement of this article.