Abstract

A total differential equation was proposed to assess the driving factors for the spatial variations in potential evapotranspiration (ET0). Using China's Loess Plateau as an example study area, three transects with distinct ET0 gradients in space, i.e., northwest–east, northwest–south and northwest–southwest, were chosen to sample spatially varied ET0 and four climatic variables (solar radiation, actual vapor pressure, wind speed, and mean temperature) at an interval of 10 km. Considered an independent variable, the distance was differentiated to quantify the contribution of each climatic variable to the spatial ET0 variations along each transect. A significant decrease in solar radiation and an increase in actual vapor pressure were identified as the dominant impact factors that led to a decreased ET0 in the northwest–east and northwest–south directions, respectively. As another key contributor, the decreasing wind speed induced a decreasing trend in ET0 from northwest to southwest. The above results implied that the dominant factor(s) for the spatial variations in ET0 differed among the regions. Therefore, the total differential equation is a powerful approach to determine the driving factors and to quantify their individual contribution to the spatial variations in ET0.

INTRODUCTION

As one of the major components of the hydrologic cycle, evapotranspiration (ET) drives energy and water exchanges among the hydrosphere, atmosphere, and biosphere (Wang et al. 2007). An accurate estimate of ET is important for improving the efficient use of water resources and for planning future water resource management (Gavilan & Castillo-Llanque 2009). However, direct measurement of ET is time-consuming and expensive. Thus, potential evapotranspiration (ET0) has been widely used in water resource studies and management practices (Li et al. 2014) because it can be directly estimated by meteorological data (Allen et al. 1998).

In recent decades, global air temperature has become increasingly significant. However, ET0 and pan evaporation have decreased worldwide over the past decades, known as the ‘evaporation paradox’ phenomenon (Brutsaert & Parlange 1998). Subsequently, attribution analysis has been widely conducted to understand this phenomenon. Three main approaches including the linear stepwise multivariate regression, detrended method, and sensitivity analysis method are commonly used. The linear stepwise multivariate regression method generally considers the major climatic factors as independent variables; a higher correction associated with ET0 for a certain variable suggests that the variable has a larger impact on the ET0 variations (Liu et al. 2004, 2015; Shan et al. 2015). The steps of the detrended method for quantifying the contributions of a certain climatic variable to the trend of ET0 include the following: (1) removing the trend from a time-series climatic variable to make it stationary; (2) recalculating ET0 by using this detrended variable and other climatic variables; (3) the differences between the original ET0 values and the recalculated ones indicate the influence of this variable imposed upon ET0 (Xu et al. 2006; Liu & Yang 2010). Compared with the former two methods, the total differential method has solid theories, because it calculates the contributions of major variables based on the total differential equation. Specifically, the ET0 change induced by a certain variable is evaluated by multiplying the long-term trend of the variable with its partial derivative. Further, many studies have demonstrated that the attribution errors induced by this method are relatively small and the sum of the contributions of each variable fit well with the actual trend of ET0 (Roderick et al. 2007; Zheng et al. 2009; Liu et al. 2011; Ning et al. 2016).

However, the above three approaches have rarely been applied to conduct quantitative attribution analysis of the spatial ET0 variations. Current studies are more likely to focus on qualitative analysis to solve the most related climatic variables that contribute to the spatial distribution of ET0; for example, only by qualitatively analyzing the spatial patterns of ET0 and related climatic variables, did Li et al. (2012) detect the main cause of the lowest ET0 values in the southwest region and the highest ET0 values in the northwest region. Furthermore, methods such as clustering or zonality analysis have also been used to assist the attribution study over the spatial variations of ET0 and other climatic variables. Based on clustering analysis, Guangdong Province in China was divided into four parts by He et al. (2015) to calculate the correlation between pan evaporation and certain climatic variables in order to detect the underlying driving factors that cause pan evaporation changes in subregions. The same method was also used to study the spatiotemporal variations of pan evaporation over all of China by Zhang et al. (2015). Even though clustering analysis can be regarded as a quantitative method operated on the spatial variations of ET0, it is still a temporal analysis that essentially explores the attributes of ET0 changes in subregions. The zonality analysis method, building the multiple regression equations for climatic variables with longitude, latitude, and altitude information, can indicate how the climatic variables vary spatially with these geographical factors (Li et al. 2016a; Sun & Zhang 2016). Nevertheless, the problem with this method is that it ignores the interaction among the climatic variables.

Therefore, in this study, we extended the total differential method based on the FAO 56 Penman–Monteith formula to conduct an attribution analysis of the spatial ET0 variations over the Loess Plateau, China. In addition, the attribution results were compared with those of the multiple regression analysis. Three main directions with distinct ET0 gradients were selected to operate the attribution analysis.

DATA AND METHODS

Study area

The Loess Plateau is located in the upper and middle reaches of the Yellow River in North China (Figure 1), covering a total area of 6.4 × 105 km2. Most areas are dominated by the semi-arid and subhumid climate, with decreased precipitation along the southeast–northwest direction, ranging from 200 to 750 mm. Characterized by highly erodible wind-deposited loess soil, sparse vegetation, but heavy rainstorms in the summer, the plateau faces the severe problems of erosion and sedimentation. Several soil and water conservation practices have been implemented since the 1950s. Other human activities, such as coal mining and urbanization, have also seriously altered the local landscapes and hydrological cycle.

Figure 1

Location of the Loess Plateau and the spatial distribution of the meteorological stations used in this study.

Figure 1

Location of the Loess Plateau and the spatial distribution of the meteorological stations used in this study.

Data collection

Daily meteorological data composed of daily maximum and minimum temperature at an elevation of 2 m, atmospheric pressure, wind speed at an elevation of 10 m, mean relative humidity, and sunshine duration were used to calculate the daily ET0. They were obtained from the China Meteorological Administration (http://data.cma.cn/); the records were collected from 96 stations during the period of 1960–2013. There are a few missing data points that were replenished based on the regression relationship with their neighboring station records. DEM (digital elevation model) data were provided by Geospatial Data Cloud of China.

Calculation of potential evapotranspiration

The Penman–Monteith formula, recommended by the FAO (Food and Agriculture Organization) in 1998, was often used to calculate the daily reference evapotranspiration (mm d−1). Reference evapotranspiration is a special case of potential evapotranspiration, because it also describes the maximum evaporative capacity from the surface, and even the surface is defined as a hypothetical grass reference crop with specific characteristics. Thus, reference evapotranspiration was used to represent the potential evapotranspiration in this study, and its specific calculation was as follows: 
formula
(1)
where Rn is the net radiation at the crop surface (MJ m−2 d−1); G is soil heat flux density (MJ m−2 d−1); Tmean is the mean daily air temperature measured at an elevation of 2 m (°C), which is defined as the mean of the daily maximum (Tmax) and minimum temperatures (Tmin); U2 is the wind speed at an elevation of 2 m (m s−1); es is the saturation vapor pressure (kPa); ea is the actual vapor pressure (kPa); Δ is the slope of the vapor pressure curve (kPa °C−1); and is the psychrometric constant (kPa °C−1).

Spatial interpolation

First, the mean annual ET0 of each station was calculated. Then, the cokriging method was chosen to spatially interpolate the mean annual ET0, U2, Rs, Tmean, and ea at a resolution of 1 km. Elevation information extracted from the DEM data was used as a correction factor. The correlation between climatic variables and elevation were described by the crossover-mutation function as: 
formula
(2)
where z(x) and Zui are the climate data for the unknown point and station i, respectively; y(x) is the elevation data; n is the number of stations; my and mz are the mean elevation and climatic variable values, respectively; and and are the weights.

The root mean square error (RMSE) and the relative error (RE) were employed to assess the performance of the cokriging interpolation method (Table 1). The results showed that the cokriging method that considers the elevation information can improve the accuracy of the interpolation for all variables.

Table 1

The cross-validation tests for the mean annual ET0 and the four climatic factors spatially interpolated by the cokriging method

  ET0 (mm) U2 (m/s) Rs (MJ/m2Tmean (°C) ea (kPa) 
Not considering elevation RMSE 9.79 0.22 0.35 1.16 0.012 
RE (%) 10.3 5.5 1.7 12.1 4.2 
Considering elevation RMSE 5.82 0.18 0.28 0.33 0.007 
RE (%) 8.2 4.9 1.3 3.6 2.8 
  ET0 (mm) U2 (m/s) Rs (MJ/m2Tmean (°C) ea (kPa) 
Not considering elevation RMSE 9.79 0.22 0.35 1.16 0.012 
RE (%) 10.3 5.5 1.7 12.1 4.2 
Considering elevation RMSE 5.82 0.18 0.28 0.33 0.007 
RE (%) 8.2 4.9 1.3 3.6 2.8 

Transect sampling for spatial ET0 variations

Considering the spatial distribution of ET0, three transects along different directions were chosen to conduct the attribution analysis of the spatial variation of ET0. The ET0 samples were taken at an interval distance of 10 km along each transect. Then, the mean ET0 at each point with a 5-km buffering area was extracted to represent the spatial ET0 variations along the transects. Similar operations were performed on the four variables of U2, Rs, Tmean, and ea. Subsequently, the spatial variation sequences for ET0 and those for the four climatic variables were obtained along the three transects.

Attribution analysis for the spatial ET0 variations

Total differential method

Following Equation (1), ET0 is a function of U2, Rs, Tmean, and ea; thus, Equation (1) can be rewritten as: 
formula
(3)
The sensitivity of ET0 to a particular independent variable x can be calculated as: 
formula
(4)
The contribution of each variable to the spatial change in ET0 can be reformulated as: 
formula
(5)
where l represents the distance (10 km) of every sample point from its origin.
Combined with Equation (1), Equation (5) can be expressed as: 
formula
(6)
Equation (6) can be simplified as: 
formula
(7)
where is the variation in ET0 for a certain transect with a unit of mm/10 km; C_(U2), C_(Rs), C_(Tmean), and C_(ea) are the contributions of U2, Rs, Tmean, and ea to the spatial ET0 changes, respectively. Furthermore, the relative contributions of each factor to the spatial change in ET0 for each transect can be calculated as: 
formula
(8)

Multiple regression analysis

To prove the attribution results of the total differential method, a multiple stepwise regression method was also applied to identify the primary and the leading climatic variables. Using SPSS, a multiple regression analysis was carried out by considering the spatial series of ET0 as a dependent variable and the four climatic variables (U2, Rs, Tmean, and ea) as independent variables. Specifically, the standard regression coefficient served as the basis to determine the contribution of each climatic variable to the ET0 variation. The larger the standard regression coefficient is, the larger the contribution.

RESULTS

Spatial variations in the mean annual ET0 and in the other four climatic variables

Figure 2 shows the spatial distribution of mean annual ET0 over the Loess Plateau for the period 1960–2013. The mean annual ET0 was 987.3 mm, ranging from 700 to 1,226 mm. ET0 distinctly decreased in three directions, i.e., from the northwest to east, south, and southwest of the Loess Plateau. To perform the quantitative attribution analysis for the spatial variations in ET0 over the Loess Plateau, three transects were set along these three directions, namely, northwest–east, northwest–south, and northwest–southwest. High ET0 values (more than 1,050 mm/a) were found in the Ordos Plateau and the Ningxia Plain to the northwest and in Sanmenxia to the southeast. The lowest ET0 values (less than 900 mm/a) were discovered in the west of Liupan Mountain, the southwest and the north of Wutai Mountain, and the northeast of Lvliang Mountain.

Figure 2

Spatial distribution of the mean annual ET0 over the Loess Plateau for the period 1960–2013.

Figure 2

Spatial distribution of the mean annual ET0 over the Loess Plateau for the period 1960–2013.

The spatial distributions of U2, Rs, Tmean, and ea over the Loess Plateau are shown in Figure 3. The wind speed was not substantially varied in space. Except for the regions to the west and to the east, the U2 values in most of the regions are in the range of 1.5–2.5 m/s. Solar radiation decreased from northwest to southeast, with the highest Rs values (more than 16.5 MJ /m2) found in the northwest and the lowest values (less than 15.0 MJ /m2) found in the southeast. The spatial distribution of the mean temperature was generally in line with the changes in the actual vapor pressure, decreasing from southeast to northwest.

Figure 3

Spatial distribution of mean annual wind speed, solar radiation, air temperature, and actual vapor pressure over the Loess Plateau for the period 1960–2013.

Figure 3

Spatial distribution of mean annual wind speed, solar radiation, air temperature, and actual vapor pressure over the Loess Plateau for the period 1960–2013.

Spatial changes in the sensitivity of ET0 to the four climatic variables

The mean annual sensitivity coefficients for wind speed (S_U2), solar radiation (S_Rs), mean temperature (S_Tmean), and actual vapor pressure (S_ea) were calculated for each station, and then they were interpolated to show the spatial pattern (Figure 4). S_U2 increased from southeast to northwest in the range of 0.13–0.28, which means a 10% increase in U2 would result in a larger change in ET0 in the northwest than in the southeast. Conversely, S_Rs decreased from southeast to northwest in the range of 0.25–0.50. The highest values of S_Tmean were found in the southeast (more than 0.6), and then it decreased from southeast to southwest, northwest, and north. The absolute value of S_ea decreased from south to north ranging from −0.75 to −0.34.

Figure 4

Spatial distribution of the mean annual sensitivity coefficients of ET0 to wind speed, solar radiation, air temperature, and actual vapor pressure over the Loess Plateau for the period 1960–2013.

Figure 4

Spatial distribution of the mean annual sensitivity coefficients of ET0 to wind speed, solar radiation, air temperature, and actual vapor pressure over the Loess Plateau for the period 1960–2013.

The mean annual sensitivity coefficients for the whole Loess Plateau were the largest for ea (−0.51), intermediate for Rs and Tmean (0.38 and 0.38), and the smallest for U2 (0.20). These results indicated that a 10% decrease in Rs, Tmean, and U2 would result in a 3.8%, 3.8%, and 2.0% decrease in ET0, respectively, while a 10% decrease in ea would result in a 5.1% increase in ET0.

ET0 attribution analysis along three transects

Along the northwest–east transect

Figure 2 shows that there was a substantial decrease in ET0 from the Ertuoke Banner in the northwest of the Loess Plateau to Fanzhi County in the east. Figure 5(a) presents how ET0 varied with different sample points along the northwest–east transect. Except for the relatively high values captured around Shuozhou County in Shanxi Province, ET0 exhibited a clear longitudinal zonality with the change rate of −4.9 mm/10 km from northwest to east. In the same direction, U2 and ea increased and Rs and Tmean decreased. Except for the U2, trends in the other three climatic variables all were significant (p < 0.01) (Table 2).

Table 2

Trends of the four climatic factors and their (relative) contributions to the trend in ET0 along the northwest-east and northwest–south transects over the Loess Plateau

  ET0 U2 Rs Tmean ea Ɛ(%) 
Northwest to east Slope −4.9** 0.006 −0.031** −0.033* 0.002**  
Contributions (mm/10 km)  −0.27 −2.65 −0.80 −1.32 −0.12 
Relative contributions (%)  5.3 52.6 15.8 26.3 −2.4 
Northwest to south Slope −4.5** −0.012** −0.039** 0.072** 0.008**  
Contributions (mm/10 km)  −1.50 0.13 7.97 −10.95 0.10 
Relative contributions (%)  34.4 −3.04 −183.2 251.9 2.3 
  ET0 U2 Rs Tmean ea Ɛ(%) 
Northwest to east Slope −4.9** 0.006 −0.031** −0.033* 0.002**  
Contributions (mm/10 km)  −0.27 −2.65 −0.80 −1.32 −0.12 
Relative contributions (%)  5.3 52.6 15.8 26.3 −2.4 
Northwest to south Slope −4.5** −0.012** −0.039** 0.072** 0.008**  
Contributions (mm/10 km)  −1.50 0.13 7.97 −10.95 0.10 
Relative contributions (%)  34.4 −3.04 −183.2 251.9 2.3 

Note: * and **suggest that the trend of a given variable is significant at the level of p= 0.05 and p= 0.01 by the F test, respectively; Ɛ is the error between the slope of ET0 and the sum of contributions of each climatic factor to the change in ET0. (The same meanings apply to Tables 3 and 4.)

Figure 5

Spatial variations in ET0 along the (a) northwest–east, (b) northwest–south, and (c) northwest–southwest directions.

Figure 5

Spatial variations in ET0 along the (a) northwest–east, (b) northwest–south, and (c) northwest–southwest directions.

Equations (5)–(9) were used to quantify the contribution associated with the four individual factors to the spatial variation in ET0. The results implied that the downward trend in the mean annual ET0 along the direction of 39°N was potentially contributed by a decrease in Rs and an increase in ea, with the contribution values identified as −2.65 and −1.32 mm/10 km, equal to 52.6% and 26.3%, respectively. Lower impacts were calculated for U2 (5.3%) and Tmean (15.8%) on the decreased ET0 along the northwest–east transect. The attribution results of the multiple regression analysis showed that the contribution of Rs to the ET variation along this transect was largest with a stand regression coefficient of 0.55 (p < 0.01), followed by ea, Tmean, and U2, which agrees with the results of the total differential method (Table 3). However, the multiple regression analysis could not calculate the specific contribution rate of each variable.

Table 3

Standard regression coefficients between the changes in ET0 and the climatic variables for three transects

Transects U2 Rs Tmean ea 
Northwest to east −0.35** 0.55** 0.45** −0.52** 
Northwest to south 0.37** −0.03 1.70** −2.27** 
Northwest to southwest 0.47** 0.39** 0.28** 0.22 
Transects U2 Rs Tmean ea 
Northwest to east −0.35** 0.55** 0.45** −0.52** 
Northwest to south 0.37** −0.03 1.70** −2.27** 
Northwest to southwest 0.47** 0.39** 0.28** 0.22 

Along the northwest–south transect

The ET0 variation from northwest to south across the Loess Plateau is shown in Figure 5(b). This transect is at the longitude of 108°E. ET0 decreased overall from northwest to south at the rate of −4.5 mm/10 km. Along this transect, U2 and Rs decreased significantly (p < 0.01), and their contributions to the decreased ET0 were calculated as −1.50 and 0.13 mm/10 km, equal to the relative contributions of 34.4% and −3.04%, respectively. In contrast, Tmean and ea showed an increasing trend (significance test: p < 0.01); their contributions to the decreased ET0 were identified as 7.97 and −10.95 mm/10 km, which are equivalent to a relative contribution of −183.2% and 251.9%, respectively. In summary, even though an increasing Tmean could result in an increase in ET0, such a trend might be offset by the function of ea and U2. The attribution results of the multiple regression also proved that ea was the dominant factor that influenced the ET0 variation along the northwest–south transect.

It is worth noting that ET0 strongly decreased from Dingbian County to Wuqi County in the Shaanxi Province at the rate of 15.8 mm/10 km along the northwest–south transect. The decreasing U2 made the maximum contribution to the decreased ET0 (58.8%), followed by increasing Tmean and ea (23.2% and 15.7%), and the contribution of decreasing Rs was minimal.

Along the northwest–southwest transect

ET0 also decreased significantly from northwest to southwest at the rate of −4.0 mm/10 km (Figure 5(c)). Except for ea, the other three variables all exhibited a decreasing trend, while the four climatic variables all decreased ET0 in this direction. The maximum relative contribution to the decreased ET0 along this transect was U2, with the values computed as 46.6%, followed by Rs (37.7%) and ea (15.4%), compared with the minimum percentage of Tmean (Table 4). The order of importance of the climatic variables to the ET0 variation along this transect detected by the sensitivity method agrees with that by the multiple regression method.

Table 4

Trends of the four climatic factors and their (relative) contributions to the trend in ET0 along the northwest–southwest transects over the Loess Plateau

  ET0 U2 Rs Tmean ea Ɛ (%) 
Ertuoke-Lingwu Slope −2.9** −0.021** 0.004* 0.116** 0.011**  
Contributions (mm/10 km)  −5.71 0.65 4.85 −2.72 −0.01 
Relative contributions (%)  194.6 −22.2 −165.3 92.9 −0.4 
Lingwu-Zhongning Slope 10.6** 0.043** 0.020** 0.049** −0.001  
Contributions (mm/10 km)  8.99 1.56 −0.14 0.18 −0.003 
Relative contributions (%)  84.9 14.8 −1.3 1.7 −0.03 
Zhongning-Dongxiang Slope −9.4** −0.032** −0.043** −0.106** −0.001*  
Contributions (mm/10 km)  −2.30 −6.06 −0.89 −0.14 −0.01 
Relative contributions (%)  24.5 64.6 9.5 1.5 −0.05 
Slope −4.0** −0.017** −0.018** −0.001 0.002**  
Ertuoke-Dongxiang Contributions (mm/10 km)  −1.86 −1.51 −0.01 −0.62 0.02 
Relative contributions (%)  46.6 37.7 0.31 15.4 0.5 
  ET0 U2 Rs Tmean ea Ɛ (%) 
Ertuoke-Lingwu Slope −2.9** −0.021** 0.004* 0.116** 0.011**  
Contributions (mm/10 km)  −5.71 0.65 4.85 −2.72 −0.01 
Relative contributions (%)  194.6 −22.2 −165.3 92.9 −0.4 
Lingwu-Zhongning Slope 10.6** 0.043** 0.020** 0.049** −0.001  
Contributions (mm/10 km)  8.99 1.56 −0.14 0.18 −0.003 
Relative contributions (%)  84.9 14.8 −1.3 1.7 −0.03 
Zhongning-Dongxiang Slope −9.4** −0.032** −0.043** −0.106** −0.001*  
Contributions (mm/10 km)  −2.30 −6.06 −0.89 −0.14 −0.01 
Relative contributions (%)  24.5 64.6 9.5 1.5 −0.05 
Slope −4.0** −0.017** −0.018** −0.001 0.002**  
Ertuoke-Dongxiang Contributions (mm/10 km)  −1.86 −1.51 −0.01 −0.62 0.02 
Relative contributions (%)  46.6 37.7 0.31 15.4 0.5 

Even though ET0 showed an overall decreasing trend along this direction, it displayed the following regional differences: (1) from Ertuoke Banner to Lingwu County in Ningxia Province, ET0 decreased only at the rate of −2.9 mm/10 km; (2) from Lingwu to Zhongning County in Ningxia Province, ET0 increased up to 1,170 mm/a by a rate of 10.6 mm/10 km; (3) from Zhongning to Dongxiang County in Ningxia Province, ET0 decreased at a rate of −9.4 mm/10 km to only 870 mm per year. U2 seems to be the dominant factor for the decreasing trend in ET0 in the first two regions, with 194.6% and 84.9% in the relative contribution; however, U2 decreased in the first region but increased in the second region. In the third region, the impacts of U2 on ET0 were degraded to only 24.5%; ET0 seems to be strongly associated with Rs (64.6%).

DISCUSSION

The possible impacts of topography and vegetation change on the spatial pattern of ET0

Due to little precipitation and cloud coverage in the northwest of the Loess Plateau, where the region is dominated by a semi-arid climate (Qian 1991), the actual vapor pressure tends to be low but high in solar radiation (see Figure 3). This collectively contributes to the high ET0 in this region. However, the southeastern part of the Loess Plateau is affected by a semi-humid continental monsoon climate, which is jointly controlled by summer monsoon and westerlies (Liang et al. 2015; Yan et al. 2016); thus, the local precipitation is richer than other regions of the Loess Plateau. As a consequence, the high actual vapor pressure and low solar radiation in the southeast result in relatively low values of ET0. Combining the relatively low elevation in the southeast region with a valley terrain (Wu et al. 1982), the temperature in this area is high, and it makes a large positive contribution to ET0. Furthermore, there are several abrupt points of ET0 on the three transects in our study (see Figure 4); they are highly related to the variation in terrain. For example, from the Ordos Plateau in the northwest to the loess hilly-gully region in the south (Figure 1), the terrain became more complex and the land surface elevation roughly decreased, which may explain the decrease in wind speed in the loess hilly-gully region (i.e., from Dingbian County to Wuqi County in Figure 5(b)), and finally resulted in a rapid decrease in ET0. To further quantify the impacts of topography on the spatial change in ET0, the correlation between the mean annual ET0 and elevation for 96 stations was calculated. The calculation results showed that the determining coefficient was 0.25 (p < 0.01), indicating that regional variation in topography would significantly impact the spatial pattern of ET0 (Figure 6(a)).

Figure 6

The relationship between mean annual ET0 and elevation; NDVI for 96 stations.

Figure 6

The relationship between mean annual ET0 and elevation; NDVI for 96 stations.

Moreover, the change in vegetation cover is another potential impact on ET0. It was reported that the vegetation cover has been effectively improved in the loess hilly-gully region due to the implementation of the ‘Grain to Green Project’ since 1999 (Zhang et al. 2013; Sun et al. 2015), which may increase the surface roughness, decrease the wind speed (Vautard et al. 2010; Cowie et al. 2013) and, finally, impact ET0. With the Global Inventory Modeling Studies 3 g database (GIMMS3 g), the mean annual NDVI was calculated from 1982 to 2012, which can reflect the vegetation condition of the whole Loess Plateau. Then, the NDVI value for each station was extracted and its relationship with ET0 was evaluated. The results showed that ET0 was significantly related to NDVI (p < 0.05), suggesting that the improvement of the vegetation condition would result in the ET0 change (Figure 6(b)). However, the revegetation project was not carried out for the whole Loess Plateau, and it mainly focused on the hilly-gully region (Xin et al. 2008). Thus, the heterogeneity of human interference would influence the spatial pattern of ET0.

The possible impacts of the spatial interpolation accuracy on the attribution results

There is no denying that the spatial interpolation accuracy of ET0 and the four related climatic variables could impact the final imposed uncertainty of our attribution analysis. Here, we will try to discuss this uncertainty. The spatial distribution of these variables generally follows patterns similar to those of previous studies (Li et al. 2012; Li et al. 2016b). Thus, it can be determined that the interpolation error in this study is acceptable and it will not influence the global spatial distribution characteristics of climatic variables. Following the same steps of interpolating mean annual values of ET0 and the four climatic variables, the annual values of these five variables from 1961 to 2012 were first interpolated. Then, their values at each corresponding station were extracted. Finally, the attribution analysis of the extracted ET0 change was conducted and the results were compared with that of the actual ET0 for each station (Figure 7). The results showed that the contributions of the interpolated U2, Rs, Tmean, and ea variables to ET0 were similar to those of the four actual variables to ET0, with a high determination coefficient R2 (0.85, 0.88, 0.91, and 0.80). Thus, it can be concluded that the influence of the spatial interpolation error on the attribution result was acceptable.

Figure 7

Comparison between the contributions of the interpolated U2, Rs, Tmean, and ea variables to ET0 and those of the actual four variables to ET0 at 96 stations.

Figure 7

Comparison between the contributions of the interpolated U2, Rs, Tmean, and ea variables to ET0 and those of the actual four variables to ET0 at 96 stations.

Other uncertainties

Based on the FAO 56 Penman–Monteith formula, this study used the total differential equation method to quantify the contributions of each impact factor to the spatial ET0 variation across the Loess Plateau. Although this method is effective in presenting the contribution of a certain climatic variable to the spatial variation in ET0, errors exist between the calculated and observed total ET0 changes. The errors exist in all three transects (see Ɛ values in Tables 13). The maximum Ɛ appeared in the direction of 39°N, accounting for 2.4% of the observed ET0 change. Similar phenomena have been detected in the attribution analysis for the temporal variation of ET0 (Zheng et al. 2009; Liu et al. 2011, 2013; Feng et al. 2014).

Except for the impact of the interpolation accuracy, several other aspects can also potentially contribute to the errors. First, only a limited number of climatic variables (four in this study) are considered in such kinds of studies; therefore, the ET0 changes calculated from the limited climatic variables cannot represent those from the climate (Zheng et al. 2009). In addition, the errors could come from the assumption of the total differential equation that the climatic variables are independent from each other, which is not true in reality; for example, the increasing temperature would result in a decrease in actual vapor pressure (Liu et al. 2011; Liu & Zhang 2013). Furthermore, the method used for contribution analysis is actually a Taylor expansion. Considering only the first-order approximation to evaluate their contributions would underestimate the calculated total changes in ET0.

CONCLUSIONS

In this study, the total differential equation was applied to analyze the contribution of four climatic factors to spatial variation in ET0 over the Loess Plateau. First, the annual ET0 records collected from 96 stations in the study area during 1960–2013 were calculated using the FAO 56 Penman–Monteith equation. Second, the annual sensitivity coefficients of ET0 to U2, Rs, Tmean, and ea were calculated for each station. Then, the mean annual ET0, four climatic variables and sensitivity coefficients were mapped using the cokriging interpolation method to show their spatial patterns. Three transects with distinguishable ET0 gradients were selected and sampled with an interval of 10 km for maps of ET0 and the four climatic variables. Finally, using distance as an independent variable, the contributions of each climatic variable to the spatial ET0 variations along the three transects were assessed by the total differential equation method. In addition, the attribution results were compared with those of the multiple regression analysis. The analysis results showed the following: the mean annual ET0 roughly decreased from northwest to east, south, and southwest over the Loess Plateau. The mean annual sensitivity coefficients for the whole Loess Plateau were largest for ea, intermediate for Rs and Tmean, and smallest for U2. The dominant factors that caused ET0 to decrease from northwest to east and to southwest were the decreasing solar radiation and increasing actual vapor pressure, respectively. As another key contributor, the decreasing wind speed induced a decreasing trend in ET0 from northwest to southwest. Except for the three directions given in our study, the total differential equation method can also be used for any direction. Compared with the result of the multiple regression analysis, the total differential method can not only obtain the dominant factor but also accurately evaluate the contribution rate of each climatic factor.

Current studies are more likely to focus on qualitative analysis to determine the most related climatic variables that contribute to the spatial distribution of ET0. This study provides a quantitative method to conduct the attribution analysis of the spatial ET0 variations. In addition, this method can also conduct an attribution analysis of the spatial variation in other climatic variables, which should have a clear functional relation with its impact factors.

ACKNOWLEDGEMENTS

This study was supported by the National Key Research and Development Program of China (No. 2016YFC0501602), the Opening Fund of State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau (A314021402-1804), the National Natural Science Foundation of China (No. 41571036), and the Public Welfare Industry (Meteorological) Research Project of China (No. GYHY201506001).

REFERENCES

REFERENCES
Allen
R. G.
,
Pereira
L. S.
,
Raes
D.
&
Smith
M.
1998
Crop Evapotranspiration: Guidelines for Computing Crop Water Requirements
.
Irrigation and Drainage Paper 56
,
Food and Agriculture Organization
,
Rome
.
Brutsaert
W.
&
Parlange
M.
1998
Hydrologic cycle explains the evaporation paradox
.
Nature
396
(
6706
),
30
30
.
doi:10.1038/23845
.
Cowie
S. M.
,
Knippertz
P.
&
Marsham
J. H.
2013
Are vegetation-related roughness changes the cause of the recent decrease in dust emission from the Sahel?
Geophysical Research Letters
40
(
9
),
1868
1872
.
doi:10.1002/grl.50273
.
Feng
J.
,
Yan
D. H.
,
Li
C. Z.
,
Yu
F. L.
&
Zhang
C.
2014
Assessing the impact of climatic factors on potential evapotranspiration in droughts in North China
.
Quaternary International
336
(
12
),
6
12
.
doi:10.1016/j.quaint.2013.06.011
.
Gavilan
P.
&
Castillo-Llanque
F.
2009
Estimating reference evapotranspiration with atmometers in a semiarid environment
.
Agricultural Water Management
96
(
3
),
465
472
.
doi:10.1016/j.agwat.2008.09.011
.
He
Y.
,
Lin
K.
,
Chen
X.
,
Ye
C.
&
Cheng
L.
2015
Classification-based spatiotemporal variations of pan evaporation across the Guangdong Province, South China
.
Water Resources Management
29
,
901
912
.
doi:10.1007/s11269-014-0850-5
.
Li
Z.
,
Zheng
F. L.
&
Liu
W. Z.
2012
Spatiotemporal characteristics of reference evapotranspiration during 1961–2009 and its projected changes during 2011–2099 on the Loess Plateau of China
.
Agricultural and Forest Meteorology
154
,
147
155
.
doi:10.1016/j.agrformet.2011.10.019
.
Li
Z.
,
Chen
Y. N.
,
Yang
J.
&
Wang
Y.
2014
Potential evapotranspiration and its attribution over the past 50 years in the arid region of Northwest China
.
Hydrological Processes
28
(
3
),
1025
1031
.
doi:10.1002/hyp.9643
.
Li
Q.
,
Yang
M. X.
,
Wan
G. N.
&
Wang
X. J.
2016a
Spatial and temporal precipitation variability in the source region of the Yellow River
.
Environmental Earth Sciences
75
(
7
),
594
.
doi:10.1007/s12665-016-5583-8
.
Li
Y.
,
Liang
K.
,
Bai
P.
,
Feng
A.
,
Liu
L.
&
Dong
G.
2016b
The spatiotemporal variation of reference evapotranspiration and the contribution of its climatic factors in the Loess Plateau, China
.
Environmental Earth Sciences
75
(
4
),
1
14
.
doi:10.1007/s12665-015-5208-7
.
Liang
W.
,
Bai
D.
,
Wang
F. Y.
,
Fu
B. J.
,
Yan
J. P.
,
Wang
S.
,
Yang
Y.
,
Long
D.
&
Feng
M.
2015
Quantifying the impacts of climate change and ecological restoration on streamflow changes based on a Budyko hydrological model in China's Loess Plateau
.
Water Resources Research
51
(
8
),
6500
6519
.
doi:10.1002/2014WR016589
.
Liu
Q. A.
&
Yang
Z. F.
2010
Quantitative estimation of the impact of climate change on actual evapotranspiration in the Yellow River Basin, China
.
Journal of Hydrology
395
(
3–4
),
226
234
.
doi:10.1016/j.jhydrol.2010.10.031
.
Liu
B. H.
,
Xu
M.
,
Henderson
M.
&
Gong
W. G.
2004
A spatial analysis of pan evaporation trends in China, 1955–2000
.
Journal of Geophysical Research-Atmospheres
109
(
D15
),
D15102
.
doi:10.1029/2004JD004511
.
Liu
X. M.
,
Luo
Y. Z.
,
Zhang
D.
,
Zhang
M. H.
&
Liu
C. M.
2011
Recent changes in pan-evaporation dynamics in China
.
Geophysical Research Letters
38
,
L13404
.
doi:10.1029/2011GL047929
.
Liu
X.
,
Zhang
D.
,
Luo
Y.
&
Liu
C.
2013
Spatial and temporal changes in aridity index in northwest China: 1960 to 2010
.
Theoretical and Applied Climatology
112
(
1–2
),
307
316
.
doi:10.1007/s00704-012-0734-7
.
Liu
Y.
,
Liu
B. C.
,
Yang
X. J.
,
Bai
W.
&
Wang
J.
2015
Relationships between drought disasters and crop production during ENSO episodes across the North China Plain
.
Regional Environmental Change
15
(
8
),
1689
1701
.
doi:10.1007/s10113-014-0723-8
.
Ning
T.
,
Li
Z.
,
Liu
W.
&
Han
X.
2016
Evolution of potential evapotranspiration in the northern Loess Plateau of China: recent trends and climatic drivers
.
International Journal of Climatology
36
(
12
),
4019
4028
.
doi:10.1002/joc.4611
.
Qian
L.
1991
The Climate on the Loess Plateau
.
China Meteorological Press
,
Beijing
(in Chinese)
.
Roderick
M. L.
,
Rotstayn
L. D.
,
Farquhar
G. D.
&
Hobbins
M. T.
2007
On the attribution of changing pan evaporation
.
Geophysical Research Letters
34
,
L17403
.
doi:10.1029/2007GL031166
.
Shan
N.
,
Shi
Z.
,
Yang
X.
,
Zhang
X.
,
Guo
H.
,
Zhang
B.
&
Zhang
Z.
2015
Trends in potential evapotranspiration from 1960 to 2013 for a desertification-prone region of China
.
International Journal of Climatology
36
(
10
),
3434
3445
.
doi: 10.1002/joc.4566
.
Sun
R. H.
&
Zhang
B. P.
2016
Topographic effects on spatial pattern of surface air temperature in complex mountain environment
.
Environmental Earth Sciences
75
(
7
),
621
.
doi:10.1007/s12665-016-5448-1
.
Sun
W. Y.
,
Song
X. Y.
,
Mu
X. M.
,
Gao
P.
,
Wang
F.
&
Zhao
G. J.
2015
Spatiotemporal vegetation cover variations associated with climate change and ecological restoration in the Loess Plateau
.
Agricultural and Forest Meteorology
209
,
87
99
.
doi: 10.1016/j.agrformet.2015.05.002
.
Vautard
R.
,
Cattiaux
J.
,
Yiou
P.
,
Thepaut
J.-N.
&
Ciais
P.
2010
Northern Hemisphere atmospheric stilling partly attributed to an increase in surface roughness
.
Nature Geoscience
3
(
11
),
756
761
.
doi:10.1038/NGEO979
.
Wang
K.
,
Wang
P.
,
Li
Z.
,
Cribb
M.
&
Sparrow
M.
2007
A simple method to estimate actual evapotranspiration from a combination of net radiation, vegetation index, and temperature
.
Journal of Geophysical Research-Atmospheres
112
(
D15
),
D15107
.
doi:10.1029/2006JD008351
.
Wu
X.
,
Lv
G.
&
Tang
J.
1982
The hydrological characteristics of Sanmanxia-Huayuankou region of Yellow River
.
Journal of China Hydrology
25
(
4
),
53
57
(in Chinese)
.
Xin
Z. B.
,
Xu
J. X.
&
Zheng
W.
2008
Spatiotemporal variations of vegetation cover on the Chinese Loess Plateau (1981–2006): impacts of climate changes and human activities
.
Science in China Series D-Earth Sciences
51
(
1
),
67
78
.
doi: 10.1007/s11430-007-0137-2
.
Xu
C. Y.
,
Gong
L. B.
,
Jiang
T.
,
Chen
D. L.
&
Singh
V. P.
2006
Analysis of spatial distribution and temporal trend of reference evapotranspiration and pan evaporation in Changjiang (Yangtze River) catchment
.
Journal of Hydrology
327
(
1–2
),
81
93
.
doi:10.1016/j.jhydrol.2005.11.029
.
Yan
M.
,
Li
F.
,
He
L.
,
Lv
M.
&
Chen
D.
2016
Effects of summer monsoon and other atmospheric circulation factors on periodicities of runoff in the Middle Huanghe River during 1919–2010
.
Scientia Geographica Sinica
36
(
2
),
917
925
(in Chinese)
.
Zhang
B. Q.
,
Wu
P. T.
,
Zhao
X. N.
,
Wang
Y. B.
&
Gao
X. D.
2013
Changes in vegetation condition in areas with different gradients (1980–2010) on the Loess Plateau, China
.
Environmental Earth Sciences
68
(
8
),
2427
2438
.
doi:10.1007/s12665-012-1927-1
.
Zhang
Q.
,
Qi
T. Y.
,
Li
J. F.
,
Singh
V. P.
&
Wang
Z. Z.
2015
Spatiotemporal variations of pan evaporation in China during 1960–2005: changing patterns and causes
.
International Journal of Climatology
35
(
6
),
903
912
.
doi:10.1002/joc.4025
.
Zheng
H. X.
,
Liu
X. M.
,
Liu
C. M.
,
Dai
X. Q.
&
Zhu
R. R.
2009
Assessing contributions to panevaporation trends in Haihe River Basin, China
.
Journal of Geophysical Research
114
,
D24105
.
doi:10.1029/2009JD012203
.