Abstract

Application of complementary relationship (CR) approaches using only routine meteorological data is a very convenient method of estimating actual evapotranspiration (ETa). Reanalysis datasets and remote sensing data provide good tools to overcome the difficulties in obtaining observation data. This study of the Hotan River Basin (HRB) in northwest China serves as a prime example for estimation of ETa during 2006–2014 by using the modified generalized CR. Based on comparison and analysis, the maximum potential evaporation calculated by the Penman-based equation was adopted. The estimated ETa rates were verified using a regional water balance method at annual time scales because of the limited available data. The calibration parameter was calibrated based on the elevation and underlying surface types. The mean annual ETa ranged from 2.3 mm to 800 mm during 2006–2014. ETa rates in the plains regions were higher than those in the mountainous regions. Most of ETa was concentrated in the months of May to September. A water deficit occurred in the middle and lower regions, while a water surplus occurred in the upper regions. This study not only provided a new concept for calibration, but also a potential solution for different underlying surfaces and time scales.

INTRODUCTION

Actual evapotranspiration (ETa) is a key variable in the energy, water, and carbon cycles of terrestrial ecosystems (Vinukollu et al. 2011; Ma et al. 2015a; Crago et al. 2016; Aminzadeh & Or 2017; Szilagyi et al. 2017). Accurate estimates of ETa are needed to address a wide range of problems such as global climate change, water balance computation, agricultural management, and hydrological modeling (Xu & Singh 2005; Crago et al. 2016). Evapotranspiration is the major water consumption pathway in the water budget, accounting for more than 60% of annual precipitation on a global scale (Oki & Kanae 2006) and more than 90% in extreme arid areas (Glenn et al. 2007). ETa exhibits strong heterogeneity because of the wide spatial variability of precipitation, ground conditions, and vegetation types (Mallick et al. 2015). The harsh natural conditions in extreme cold and arid areas make it difficult to obtain sufficient hydrological and meteorological data in situ. Despite the urgent need for it, estimating ETa rates and analyzing the temporal and spatial distribution features of ETa with limited observation data in extremely cold and arid basins is a tremendous challenge.

Although ETa can be directly observed by the eddy covariance method, weighing lysimeter method, energy balance Bowen ratio method, or the scintillometer method (Wang & Dickinson 2012; Ma et al. 2015b), these all require sufficient manpower, materials, and financial resources. Several approaches to the estimation of ETa have been developed to date, some of which were derived from the land surface energy balance equation using remotely sensed surface temperatures (e.g., Bastiaanssen et al. 1998; Su 2002; Mariotto et al. 2011), while others, such as soil-vegetation-atmosphere transfer models, are physically based methods (e.g., Haverd et al. 2011) and some are derived from the Penman (1948) or Priestley & Taylor (1972) equations (e.g., Monteith 1965; Allen et al. 1998; Monteith & Unsworth 2013). The complementary relationship (CR) between ETa and potential evaporation (ETp) was first proposed by Bouchet (1963). Several models based on the CR have been suggested during the past several decades, such as the advection-aridity (AA) model developed by Brutsaert & Stricker (1979), the complementary relationship areal evapotranspiration (CRAE) model of Morton (1983), the Granger model described by Granger & Gray (1989), as well as some nonlinear normalized CR models (Han et al. 2012, 2014; Brutsaert 2015; Crago et al. 2016; Szilagyi et al. 2016, 2017). The CR models are intellectually appealing since they only use routine meteorological data and bypass detailed soil moisture, land cover type, and terrain and vegetation information (Xu & Singh 2005; Ma et al. 2015b; Szilagyi et al. 2016). The CR approaches have been widely applied over different underlying conditions, spatial scales, and temporal scales (e.g., Xu & Singh 2005; Huntington et al. 2011; Wang et al. 2011; Han et al. 2012, 2014; Crago & Qualls 2013; Mallick et al. 2015; Hobbins et al. 2016; McEvoy et al. 2016). However, one criticism of the CR is the lack of definitive derivation and physical constraints (Crago et al. 2016; Szilagyi et al. 2016), although some studies have attempted to rectify this (Szilagyi 2007; Han et al. 2012, 2014; Brutsaert 2015; Aminzadeh et al. 2016; Crago et al. 2016; Szilagyi et al. 2016, 2017). In addition, there is a strong argument between symmetric and asymmetric CR under different settings (Szilagyi 2007; Jaksa et al. 2013; Ma et al. 2015b; Crago et al. 2016). Han et al. (2012) proposed boundary conditions (BCs) in the CR formulation and a new nonlinear function satisfying these BCs. Through comparison of the nonlinear CR developed by Han et al. (2012), the CRAE, and the Penman–Monteith approaches, Ma et al. (2015b) suggested that the nonlinear CR performed well with limited observation data for local calibration of the model parameter values. Moreover, they found that this CR model had an obvious advantage in both theory and practice in that it did not require users to select a symmetric or asymmetric CR relationship. Brutsaert (2015) reformulated the CR as a general polynomial based on strictly physical considerations. Szilagyi et al. (2016) compared the original version of the generalized CR of Brutsaert (2015) with a revised version in which the BCs-(ii) of Brutsaert (2015) was replaced. Furthermore, Crago et al. (2016) elaborated why these BCs-(ii) needed to be replaced, and introduced a new concept, the maximum ETp (denoted by Epads in the literature), to normalize ETp as xmin. They also proposed a new version of the CR based on rescaling, which not only reduced scatter of ETa estimates, but led to formation of a proposed self-adjusting CR model. Szilagyi et al. (2017) redefined the maximum ETp as an invariable result of adiabatic drying under near-neutral atmospheric conditions (denoted by Epmax in the literature). Epads was calculated by the mass-transfer-based method, while Epmax was calculated by the Penman-based equation (Ma & Zhang 2017). How to define the maximum ETp could be the key to the application of the modified Brutsaert (2015) generalized CR model by Crago et al. (2016) and Szilagyi et al. (2017).

In recent years, the rapid development of satellite remote sensing has led to remote sensing data providing great convenience for spatio-temporal samplings (Singh et al. 2008; Liu et al. 2009). However, reanalysis-based methods (Mesinger et al. 2006; Yang et al. 2010; Lavers et al. 2012) have unique advantages, such as less sensitivity to cloud cover, longer temporal coverage, and better representation of actual conditions for spatially distributed data (Szilagyi et al. 2017). The China Meteorological Forcing Dataset is a reanalysis dataset of surface meteorological and environmental factors developed by the Institute of Tibetan Plateau Research, Chinese Academy of Sciences. Using Princeton reanalysis forcing data, Global Land Data Assimilation System data, Global Energy and Water Exchanges-Surface Radiation Budget data, and Tropical Rainfall Measuring Mission data (3B42) as the background, this dataset also combines China Meteorological Administration station data from 1979 to 2015. The spatial resolution of this dataset is 0.1° and its temporal resolution is 3 h (He & Yang 2011).

The Hotan River Basin (HRB) is located in the hinterland of the Taklimakan Desert of northwest China (the largest desert in China) and at the north side of the Kunlun Mountains (part of the Tibetan Plateau). There are limited observation data available for extreme cold and arid areas of the river basin. In this study, the modified generalized CR framework of Crago et al. (2016), with the maximum ETp defined by Crago et al. (2016) and Szilagyi et al. (2017), was applied to estimate the daily ETa rates in the HRB from 2006 to 2013. The China Meteorological Forcing Dataset was used to calculate the CR components and analyze regional water balance, while Moderate Resolution Imaging Spectroradiometer (MODIS) data were used to calculate the net solar radiation (Rn). The daily precipitation data and monthly runoff data were collected to calculate a regional water budget. The objectives of this study were: (i) to analyze the improvement effects of different modified methods and select an appropriate definition of the maximum ETp according to conditions in the study basin; (ii) to calibrate the local parameter values of the Priestley–Taylor coefficient and the adjustable parameter to estimate daily ETa rates and validate the annual ETa by using a regional water balance; and (iii) to evaluate the temporal and spatial distribution variations of ETa and the water budget in the HRB during 2006–2014.

MODEL AND PARAMETERIZATION

Background of the CR

The CR theory proposed by Bouchet (1963) can be described as follows (Brutsaert & Parlange 1998): 
formula
(1)
where ETw is the wet-environment evaporation, and b is a proportionality parameter. The CR postulates that opposite changes exist between ETa and ETp, meaning that when soil moisture decreases and the available energy remains constant, the energy would have been consumed by ETa, while when soil moisture is saturated, ETa = ETp = ETw. Note that the symmetric CR postulated by Bouchet (1963) implies b equals unity (Aminzadeh & Or 2017). The AA model or modified AA models with b = 1 have achieved good performance (Xu & Singh 2005; Hobbins et al. 2001; Ma et al. 2015b; Szilagyi 2015). Nevertheless, under many conditions reported in the literature, b is considerably larger than 1 (Aminzadeh & Or 2017), and is even thought to be variable (Granger & Gray 1989; Szilagyi 2007; Han et al. 2012).
In the most widely used AA model, ETp should be estimated using the Penman (1948) equation and ETw should be estimated using the Priestley–Taylor (1972) equations: 
formula
(2)
 
formula
(3)
where is the slope of the saturation vapor pressure curve at the air temperature (kPa °C−1); is the psychometric constant (kPa °C−1); Rn is the net radiation (MJ m−2 day−1); G is the soil heat flux (MJ m−2 day−1); is the latent heat of vaporization (MJ kg−1); es is the saturation vapor pressure at air temperature (kPa); ea is the actual vapor pressure (kPa); (es−ea) is the saturation vapor pressure deficit (kPa); and is the coefficient with a default value of approximately 1.26, which depends mainly on underlying surface conditions (Brutsaert 2015). In the present study, the values of were recalculated based on underlying surface conditions. Additionally, f (U) is a function of the mean wind speed at a reference level (mm day−1 kPa−1), which is either theoretically or empirically derived. The Penman's (1948) empirical linear f (U) is commonly used (Brutsaert 1982; Qualls & Gultekin 1997; Xu & Singh 2005): 
formula
(4)
where U2 is the wind speed (m s−1) at a height of 2 m.
As demonstrated by Hobbins et al. (2001) and Ma et al. (2015b), Equation (4) should be replaced with a more appropriate method to accurately estimate ETp. At daily or longer time scales, atmospheric stability is generally considered to be neutral (Ma et al. 2015b). Brutsaert & Stricker (1979) suggested that the f (U) should be derived through the Monin–Obukhov similarity theory as follows: 
formula
(5)
where t = 86,400 s; k = 0.4 is the von Karman constant; is the density of air (kg m−3); Pa is the air pressure (kPa); z is the measurement height of wind speed and humidity (2 m in this study); d is the displacement height (m); and zom and zov are the roughness lengths of momentum and water vapor (m), respectively. zom and d are correlated with the effective vegetation height, h (Guo & Shen 2015). For trees, zom= 0.075 h and d = 0.78 h (Guo & Shen 2015); for cropland and grass, zom= 0.123 h and d = 0.67 h (Allen et al. 1998); for urban and barren land, zom= 0.004 m (Wang et al. 1988; Han et al. 2012). Finally, zov can typically be expressed as zov= 0.14 zom (Ryu et al. 2008; Ma et al. 2015b).
Szilagyi & Jozsa (2008) suggested that in Equation (3) should be evaluated at the wet environment air temperature (Tw) rather than air temperature (Ta). This modification is necessary to estimate Tw in arid or semiarid regions because of the large difference between Tw and Ta (Huntington et al. 2011; Szilagyi 2014). Tw is generally unknown for water-limited conditions, but can be approximated by the wet environment surface temperature (Tws) (Huntington et al. 2011). Szilagyi & Jozsa (2008) recommended an implicit equation for Tws based on the Bowen ratio (Bo) for a small wet patch: 
formula
(6)
where e0 (Tws) is the saturated vapor pressure at Tws (K). Tws can be solved through iterations. Normally, for a small wet surface, Tws is typically lower than Ta; thus, Tws calculated by Equation (6) can be taken for Tw. If Tws > Ta, Tw should be replaced by Ta (Huntington et al. 2011; Szilagyi 2014; Ma et al. 2015b).
Actual vapor pressure has a good correlation with dew point temperature, relative humidity, specific humidity, dry bulb temperature, or wet bulb temperature. ea can be approximated as follows: 
formula
(7)
where q is the specific humidity (kg kg−1).

Utilizing the Surface Energy Balance System method of Su (2002), Rn was calculated using remote sensing data as an instantaneous value. The daily Rn values were estimated by implementing the sinusoidal model given by Bisht et al. (2005). The other climate-related parameters, G,, , , , , and , were calculated using the method recommended by the Food and Agriculture Organization (Allen et al. 1998).

Generalized CR and theoretical development

The dimensionless form of the AA model with ETp is formed from combining Equation (1) with Equations (2) and (3): 
formula
(8)
Equation (8) generates the dimensionless variables and . Values of y and x are between 0 and 1. Equation (8) becomes 
formula
(9)
Brutsaert (2015) imposed four BCs by setting physical constraints to develop a fourth-order polynomial relationship between y and x that was inspired by Han et al. (2012). The four BCs are as follows: (i) y = 1 at x = 1; (ii) y = 0 at x = 0; (iii) dy/dx = 1 at x = 1; and (iv) dy/dx = 0 at x = 0. However, Szilagyi et al. (2016) recommended that BCs-ii should be replaced by y = 0 at x = xmin. To calculate xmin, Crago et al. (2016) introduced the maximum ETp (hereinafter ETpmax) and recommended that ETpmax be obtained from a small wet surface according to the aerodynamic mass transfer equation, denoted by ETpds in the present study: 
formula
(10)
where refers to saturated specific humidity at Tws. The specific humidity, q = 0 at height z. In the present study, z = 2 m. Substituting Equation (5) into Equation (10), one obtains the following: 
formula
(11)
where is the saturation vapor pressure at Tws.
As an alternative, could also be obtained from the Penman equation with ea = 0, and the temperature for a dry surface (denoted by Tdry), denoted by in the present study, can be obtained according to the definition by Szilagyi et al. (2017): 
formula
(12)
where is the slope of the saturation vapor pressure curve at Tdry, and Tdry is defined for adiabatic changes as . Tdry is described in more detail by Szilagyi (2014) and Szilagyi et al. (2017).
Once has been determined by employing Equation (11) or Equation (12), xmin can be obtained using the formula given by Crago et al. (2016): 
formula
(13)
Crago et al. (2016) suggested that x = xmin is the lower boundary of the CR, so the CR could be formulated by rescaling x: 
formula
(14)
Considering the special geographical situation, such as the significant elevation change or the different underlying conditions, some flexibility in CR formulations may be needed (Crago et al. 2016). As recommended by Crago et al. (2016), a general polynomial function can be sought in the form of 
formula
(15)
We might specify four BCs inspired by Brutsaert's (2015): (i) y = 1 at X = 1; (ii) y = 0 at X = 0; (iii) dy/dX = 1 at X = 1; (iv) dy/dX = at X = 0. Equation (15) becomes 
formula
(16)
containing only one adjustable parameter . When = 1, Equation (16) reduces to as used by Crago et al. (2016). When = 0, Equation (16) reduces to as used by Szilagyi et al. (2017).

STUDY AREA AND DATA

Study area

The HRB is located in northwest China (77.40°E–81.59°E, 34.84°N–40.44°N), covering an area of 88,753 km2 (draining to the Xiaota hydrological station) (Figure 1). The HRB is one of the main headwaters of the Tarim River, which is the largest inland river in China. The altitude of the HRB varies between 1,014 m and 6,858 m above sea level (asl) (Figure 1). The source of the Hotan River is located at the north side of the Kunlun Mountains, and the lower reaches pass through the hinterland of the Taklimakan Desert. The mountainous regions (above 2,000 m asl) are covered with ice and snow, alpine grasslands, and bare rock (Figure 2), and are characterized by a frigid arid climate (Zheng et al. 2013). The plains regions (below 2,000 m asl) are mainly covered with croplands, desert steppes, and desert (Figure 2). The oasis-desert systems are characterized by a warm temperate arid climate (Zheng et al. 2013). Because of the colossal irrigation system in the HRB, drainage collects downstream of the irrigated area, where an oasis-desert transition area is formed. The Kalakashi River and the Yulongkashi River are two branches of the Hotan River that cross the Hotan oasis. There are only two meteorological stations in the Hotan oases and five hydrological stations on the Hotan River (Figure 1). All of these stations are at less than 2,000 m asl, and there are no stations in the mountainous regions or desert regions of the HRB. Two mountain-pass hydrological stations, Wuluwati and Tonguziluoke, are at around 2,000 m asl. The Aigeliya and Tuzhiluke hydrological stations are at the convergence of the two branches. The Xiaota hydrological station is at 319 km downstream of the convergence, which is the single basin export to the Tarim River (Zhao et al. 2009; Lyu et al. 2015).

Figure 1

Location of the study basin and the hydrological and meteorological stations. The HRB is divided into three parts (the upper, middle, and lower regions) according to the location of hydrological stations.

Figure 1

Location of the study basin and the hydrological and meteorological stations. The HRB is divided into three parts (the upper, middle, and lower regions) according to the location of hydrological stations.

Figure 2

Land use/cover in the HRB in 2011.

Figure 2

Land use/cover in the HRB in 2011.

Data description

In the present study, 0.1° grid-based meteorological data of the China Meteorological Forcing Dataset in the HRB during 2006–2014 were used, including mean daily air temperature, mean specific humidity, mean wind speed, precipitation, and 3-hourly mean air temperature. As mentioned in the Introduction, the China Meteorological Forcing Dataset was produced by merging a variety of data sources (He & Yang 2011). This study used 3-hourly mean temperature from this dataset to facilitate the preparation of daily maximum and minimum temperatures. Note that the wind speed in this dataset was at a height of 10 m (U10); therefore, U10 values were transformed to U2 values via U2=0.748 U10 (Allen et al. 1998). We also used MODIS data at a spatial resolution of 1 km, including MOD11A2 (emissivity, land surface temperature), MCD43B3 (albedo), MCD15A2 (leaf area index), and MCD12Q1 (land cover type). Additionally, digital elevation model (DEM) data of approximately 90 m resolution were collected. The forcing datasets and DEM data were used with the cubic convolution resampling technique to attain the 1 km spatial resolution matching the remotely sensed data. MODIS data collected at 8-day time steps was linearly interpolated to obtain daily values. Spatio-temporal gap filling of missing data during cloudy periods was accomplished through compositing time series data (Cleugh et al. 2007). Moreover, the monthly runoff data from the five hydrological stations were collected to analyze regional water balance.

RESULTS AND ANALYSES

Testing different versions of key parameters

In the above description, was replaced by , and in Equation (3) was evaluated at Tw rather than at Ta. In this section, the improvement effects of different modified methods were analyzed. In addition, the two definitions of ETpmax of Crago et al. (2016) and Szilagyi et al. (2017) described above, were compared and selected according to conditions in the study basin. Analysis of annual precipitation data from 2006 to 2014 revealed that the precipitation in 2011 was in the median value. Therefore, 2011 was chosen as an example to represent the climate of the study region.

As shown in Figure 3(a) and 3(b), the spatial distribution of the daily mean wind function derived from Equation (4) was compared with that derived from Equation (5) . The mean value of (3.11 mm day−1 kPa−1) was smaller than the mean value of (4.54 mm day−1 kPa−1). The spatial distribution of the absolute percentage difference (APD) between and (denoted by APD_f (U)), as well as the spatial distribution of the APD between ETp derived from and that derived from (denoted by APD_ETp) can be seen in Figure 3(c) and 3(d). The mean values of APD_f (U) and APD_ETp were 0.32 and 0.23, respectively, with larger values in the plains regions than in the mountainous regions. Moreover, the values of APD_ETp in desert regions were larger than those in oasis regions. These results indicate that the correction of the wind function had a more significant reduced-ETp effect in the plains regions than in the mountainous regions, while the correction of the wind function to the estimate of ETp had the most significant influence in desert regions of the plains regions, followed by influence in oasis regions of the plains regions; meanwhile, it had the least influence in the mountainous regions.

Figure 3

Spatial distribution of (a) the daily mean wind function derived from Equation (4) () and (b) the daily mean wind function derived from Equation (5) (); (c) the APD between and (denoted by APD_f (U)) and (d) the APD between ETp derived from Equation (4) and that derived from Equation (5) (denoted by APD_ETp) in 2011 as an example.

Figure 3

Spatial distribution of (a) the daily mean wind function derived from Equation (4) () and (b) the daily mean wind function derived from Equation (5) (); (c) the APD between and (denoted by APD_f (U)) and (d) the APD between ETp derived from Equation (4) and that derived from Equation (5) (denoted by APD_ETp) in 2011 as an example.

Figure 4(a) and 4(b) display the spatial distribution of the daily mean Ta and the daily mean Tw, respectively. The spatial distribution of the mean absolute error (MAE) between Ta and Tw (denoted by MAE (TaTw)), and the MAE between ETrad derived at Ta and that derived at Tw (denoted by MAE (ETrad)) can be seen in Figure 4(c) and 4(d). The values of MAE (TaTw) in the plains regions were larger than those in the mountainous regions, while those in oasis-desert transition regions were larger than those values in desert and/or oasis regions. The spatial distribution of MAE (ETrad) was similar to the distribution of MAE (TaTw). The correction of Tw to the estimate of ETrad had a relatively small influence, with the maximum MAE (ETrad) value of 32.87 mm (about 3.5% of the ETrad derived at Ta).

Figure 4

Spatial distribution of (a) the daily mean air temperature (Ta) and (b) the daily mean wet environment air temperature (Tw); (c) the MAE between Ta and Tw (denoted by MAE (TaTw)) and (d) the MAE between ETrad derived at Ta and that derived at Tw (denoted by MAE (ETrad)). The year 2011 is used as an example.

Figure 4

Spatial distribution of (a) the daily mean air temperature (Ta) and (b) the daily mean wet environment air temperature (Tw); (c) the MAE between Ta and Tw (denoted by MAE (TaTw)) and (d) the MAE between ETrad derived at Ta and that derived at Tw (denoted by MAE (ETrad)). The year 2011 is used as an example.

Figure 5(a) and 5(b) show the spatial distribution of the annual ETpds estimated from Equation (11) and the annual ETpdry estimated from Equation (12), respectively. Figure 5(c) and 5(d) display the spatial distribution of the difference between ETpds and ETp (denoted by ETpdsETp) and the difference between ETpdry and ETp (denoted by ETpdryETp), respectively. The values of ETpdsETp in the plains regions were greater than 0, while those in the mountainous regions were less than 0. These results indicated that ETpds was less than ETp in the mountainous regions. According to the basic assumptions of Crago et al. (2016), , and the defined by Crago et al. (2016), i.e., ETpds, was not applicable in the study basin, especially in the mountainous regions. However, the values of ETpdryETp were greater than 0 in the whole basin, with the minimum value of 170.2 mm yr−1 in the mountainous regions and the maximum value of 660.2 mm yr−1 near Xiaota station in the plains regions. Therefore, defined by Szilagyi et al. (2017), i.e., ETpdry, should be adopted in the present study.

Figure 5

Spatial distribution of (a) annual ETpds estimated from Equation (11) and (b) annual ETpdry estimated from Equation (12); (c) the difference between ETpds and ETp (denoted by ETpdsETp) and (d) the difference between ETpdry and ETp (denoted by ETpdryETp). The year 2011 is used as an example.

Figure 5

Spatial distribution of (a) annual ETpds estimated from Equation (11) and (b) annual ETpdry estimated from Equation (12); (c) the difference between ETpds and ETp (denoted by ETpdsETp) and (d) the difference between ETpdry and ETp (denoted by ETpdryETp). The year 2011 is used as an example.

Calibration and validation

Priestley & Taylor (1972) and subsequent studies have suggested a value of the Priestley–Taylor coefficient in Equation (3) near 1.26 with a narrow range (e.g., Eichinger et al. 1996; Crago et al. 2010). Hobbins et al. (2001) obtained a value of = 1.3177 using data from 92 basins of the USA. Brutsaert (2005) suggested that for saturated surfaces typically falls between 1.20 and 1.30. Xu & Singh (2005) determined values for three study regions at 1.18, 1.04, and 1.00. Yang et al. (2008) obtained an average = 1.17 with a range of 0.87–1.48 from 108 basins of China, whereas Gao et al. (2012) suggested falls between 1 and 1.23 for nine sub-basins of the Haihe River Basin. Similar seasonal variability for in the range of 1.1–1.4 was reported for the Asian monsoon (Yang et al. 2013). Brutsaert (2015) and Ma et al. (2015b) used near 1.13. The value was also reported to be a function of the water content of drying soil surfaces, reaching values as low as 0.25 for relatively dry soil surfaces (Aminzadeh & Or 2014). To accommodate the observed range of , we assume . Szilagyi et al. (2017) proposed a novel approach to obtain the value of that applied to a calibration-free formulation of the CR. However, their approach to obtain the value of is unsuitable in extreme cold and arid basins. Due to the lack of measured data, a value of cannot be accurately determined. However, the fact that ETwETp can estimate the possible maximum value of , thus, for the HRB. We assume that a fixed value of is applicable in the whole basin, even though we agree with the view that changes over time and underlying surface. Hence, we assume that for the HRB in the present study. In addition, we assume that ETa = 0 when Ta is less than the dew point temperature.

The adjustable parameter in Equation (16) was treated as the only calibration parameter. Note that one can deduce from Equation (16) that ETa rates increase monotonously as the value increases. Because measured data for the HRB in extreme cold and arid areas were limited, the calculated results were not verified at daily and monthly time scales. Therefore, the calibration of and the validation of ETa used the regional water balance method at annual time scales.

ETa estimated from the water balance equation (ETwb) was used as the reference at annual time scales: 
formula
(17)
where P is the annual precipitation; R is the annual runoff; and denotes the water storage change, which includes groundwater, glacial water, and soil moisture. was assumed to be 0 at the annual time scale in the present study because the accurate water storage change data were almost impossible to obtain in the HRB, there was not much groundwater extracted for irrigation, and the change of water storage was small in the calibration period (just one year in the present study). Considering the severe spatio-temporal heterogeneity between precipitation and ETa, the HRB was divided into upper, middle, and lower regions to calibrate the coefficient and verify ETa rates according to the location of hydrological stations (Figure 1). The year 2011 was chosen to determine the optimal values of because it was neither too wet nor too dry, while other years were used as the validation period.

Calibration was conducted as follows. First, the coefficient in desert or urban regions of the plains regions was determined since precipitation does not produce runoff and is almost completely evaporated (Ma et al. 2014). Second, the coefficient in vegetated areas of the plains regions was determined according to the regional water balance method. Third, the coefficient in rock bare regions of the mountainous regions was determined. Considering that the daily maximum temperature was less than 0 °C above 5,000 m asl during most of the year, ETa rates above 5,000 m asl were quite low. As mentioned above, ETa rates increased monotonously with increasing ; therefore, we assumed that the values of decreased monotonously with increasing elevation. For conveniently calibrating, elevation of the mountainous regions was divided into three parts, and we assumed that the relationship between and elevation was linear in each elevation interval. Note that the relationship between and elevation might be actually nonlinear. Finally, the coefficient in vegetated areas of the mountainous regions was determined using a method similar to that employed for the rock bare regions. The optimal values of used for Equation (16) were found (Table 1) by minimizing the sum of absolute errors (MSAE) between ETwb and ETa. The optimal values of were applied to estimate ETa in the validation period, and the results are listed in Table 2. Note that Zhao et al. (2009) obtained mean annual evapotranspiration (198.78 mm) from year 1954 to 2003 in middle regions. In this study, the mean annual evapotranspiration in middle regions was 209.2 mm which is very close to the results of Zhao et al. (2009).

Table 1

Values of optimized parameter in the upper, middle, and lower regions

Elevation (m asl) Vegetated regions
 
Unvegetated regions
 
ETp (mm) P (mm)  ETp (mm) P (mm)  
<1,200 1,287.17 81.56 −3.2 1,239.99 83.09 −7 
1,200–2,000 1,560.45 51.89 0.6 1,343.87 82.04 −4.2 
2,000–3,000 1,278.33 161.3 100 − 50*(Elevation/1,000) 1,258.84 148.99 82 − 44*(Elevation/1,000) 
3,000–4,000 1,256.35 182.85 400 − 150*(Elevation/1,000) 1,168.83 188.92 700 − 250*(Elevation/1,000) 
>4,000 1,156.5 196.94 400 − 150*(Elevation/1,000) 1,006.5 196.15 1,300 − 400*(Elevation/1,000) 
Elevation (m asl) Vegetated regions
 
Unvegetated regions
 
ETp (mm) P (mm)  ETp (mm) P (mm)  
<1,200 1,287.17 81.56 −3.2 1,239.99 83.09 −7 
1,200–2,000 1,560.45 51.89 0.6 1,343.87 82.04 −4.2 
2,000–3,000 1,278.33 161.3 100 − 50*(Elevation/1,000) 1,258.84 148.99 82 − 44*(Elevation/1,000) 
3,000–4,000 1,256.35 182.85 400 − 150*(Elevation/1,000) 1,168.83 188.92 700 − 250*(Elevation/1,000) 
>4,000 1,156.5 196.94 400 − 150*(Elevation/1,000) 1,006.5 196.15 1,300 − 400*(Elevation/1,000) 
Table 2

Annual precipitation (P), runoff (R), ETwb, ETa, and MSAE in the upper, middle, and lower regions during 2006–2014

Year Upper regions
 
Middle regions
 
Lower regions
 
 
P R ETwb ETa P R ETwb ETa P R ETwb ETa MSAE 
2011 183.74 137.44 46.30 46.39 104.95 −104.43 209.38 209.95 82.93 −61.14 144.07 144.80 1.39 
2006 225.59 163.72 61.87 48.79 155.87 −94.42 250.30 233.43 78.54 −58.48 137.01 119.13 −47.83 
2007 156.60 117.01 39.59 34.44 79.89 −98.46 178.36 155.87 59.58 −44.53 104.11 93.30 −38.45 
2008 186.33 121.89 64.44 46.42 83.66 −109.21 192.87 166.52 70.15 −33.60 103.75 121.07 −27.05 
2009 123.17 98.29 24.88 38.71 54.80 −87.39 142.19 141.58 67.85 −33.16 101.02 91.45 3.65 
2010 237.17 182.98 54.19 53.55 202.07 −104.12 306.18 257.54 149.00 −96.30 245.30 156.11 −138.47 
2012 204.86 162.99 41.87 62.12 105.38 −108.69 214.07 298.03 88.04 −55.03 143.07 182.51 143.65 
2013 247.13 201.91 45.22 46.06 113.08 −126.02 239.10 220.79 114.85 −46.82 161.67 151.49 −27.65 
2014 194.09 134.69 59.40 48.20 82.86 −99.11 181.96 199.86 70.53 −41.16 111.69 135.04 30.05 
Mean 196.87 147.94 48.93 47.25 109.70 −103.43 213.13 209.20 87.32 −51.14 138.45 131.26 −12.8 
Year Upper regions
 
Middle regions
 
Lower regions
 
 
P R ETwb ETa P R ETwb ETa P R ETwb ETa MSAE 
2011 183.74 137.44 46.30 46.39 104.95 −104.43 209.38 209.95 82.93 −61.14 144.07 144.80 1.39 
2006 225.59 163.72 61.87 48.79 155.87 −94.42 250.30 233.43 78.54 −58.48 137.01 119.13 −47.83 
2007 156.60 117.01 39.59 34.44 79.89 −98.46 178.36 155.87 59.58 −44.53 104.11 93.30 −38.45 
2008 186.33 121.89 64.44 46.42 83.66 −109.21 192.87 166.52 70.15 −33.60 103.75 121.07 −27.05 
2009 123.17 98.29 24.88 38.71 54.80 −87.39 142.19 141.58 67.85 −33.16 101.02 91.45 3.65 
2010 237.17 182.98 54.19 53.55 202.07 −104.12 306.18 257.54 149.00 −96.30 245.30 156.11 −138.47 
2012 204.86 162.99 41.87 62.12 105.38 −108.69 214.07 298.03 88.04 −55.03 143.07 182.51 143.65 
2013 247.13 201.91 45.22 46.06 113.08 −126.02 239.10 220.79 114.85 −46.82 161.67 151.49 −27.65 
2014 194.09 134.69 59.40 48.20 82.86 −99.11 181.96 199.86 70.53 −41.16 111.69 135.04 30.05 
Mean 196.87 147.94 48.93 47.25 109.70 −103.43 213.13 209.20 87.32 −51.14 138.45 131.26 −12.8 

Note that R refers to a ratio of the difference in runoff volume observed between the downstream and upstream hydrological stations to region area; and Mean refers to the mean annual value during the validation period (all in mm yr−1).

In general, the mean annual ETa estimated using the optimal values of matched well with the mean annual ETwb, which indicates that the MSAE in the validation period was −12.8 mm yr−1. However, this does not necessarily indicate that the annual MSAE was also small. Indeed, there was a certain deviation between the annual ETa and annual ETwb with the maximum MSAE value of 143.65 mm in 2010.

Spatial and temporal variations in precipitation, ETa, and water budget

Figure 6 shows the spatial distribution of annual ETa, and Figure 7 shows the spatial distribution of mean annual precipitation, ETa, and PETa difference during 2006–2014 in the HRB. The mean annual precipitation ranged from 46.0 mm to 342.0 mm, while the mean annual ETa ranged from 2.3 mm to 800 mm. The higher precipitation values were in the mountainous regions, indicating that the mean annual precipitation was higher at high elevation. ETa rates in the plains regions were higher than those in the mountainous regions, with the highest values being observed in croplands and shrublands along the riverside. Perhaps the irrigation or groundwater plays a key role in this issue. Relatively lower ETa occurred in ice and snow regions, as well as in barren land of the mountainous regions. The difference (PETa) reflects water surplus or deficits (Figure 7(c)). The mean annual PETa difference in the mountainous regions was positive (except in alpine grasslands), while it was negative in the plains regions. These findings indicate that a great deal of precipitation and runoff dissipated in the form of ETa in the plains regions.

Figure 6

Spatial distribution of annual ETa rates during 2006–2014 in the HRB (all in mm yr−1).

Figure 6

Spatial distribution of annual ETa rates during 2006–2014 in the HRB (all in mm yr−1).

Figure 7

Spatial distribution of mean annual (a) precipitation (P), (b) ETa, and (c) PETa difference during 2006–2014 in the HRB (all in mm yr−1).

Figure 7

Spatial distribution of mean annual (a) precipitation (P), (b) ETa, and (c) PETa difference during 2006–2014 in the HRB (all in mm yr−1).

Figure 8 shows the monthly mean P, ETa, and PETa difference from 2006 to 2014 in the upper, middle, and lower regions. As shown in Figure 8(a), most of the P was concentrated in the months of May to September, accounting for 65% (128 mm) of annual P in the upper regions, 61% (66 mm) in the middle regions, and 70% (61 mm) in the lower regions. As shown in Figure 8(b), the variations in ETa in different regions were similar, following a single peak normal distribution. Most of ETa was concentrated in the months of May to September, accounting for 89% of the mean annual ETa (42 mm) in the upper regions, 76% (159 mm) in the middle regions, and 80% (105 mm) in the lower regions. The inter-annual changes in ETa in winter months were lower than in summer months. As shown in Figure 8(c), a water deficit occurred in the middle and lower regions, and a water surplus occurred in the upper regions. Because the inter-annual changes in P were relatively large, the inter-annual changes in water budget were relatively large as well. Water deficits occurred during most months throughout the year in the middle regions (except January and February). In the lower regions water deficits occurred during May to September. From October to April, water budget was a basic equilibrium, because the river was drying up during these months. Additionally, the annual deficit was estimated to be 100 mm in the middle regions and 44 mm in the lower regions. Furthermore, the amount of available water was sufficient to satisfy ETa in most months in the upper regions, where the mean annual water surplus was 150 mm.

Figure 8

Monthly mean (a) precipitation (P), (b) ETa, and (c) PETa difference from 2006 to 2014 in the upper, middle, and lower regions. The bars indicate the maximum and minimum values of P, ETa, and PETa difference for the land cover types.

Figure 8

Monthly mean (a) precipitation (P), (b) ETa, and (c) PETa difference from 2006 to 2014 in the upper, middle, and lower regions. The bars indicate the maximum and minimum values of P, ETa, and PETa difference for the land cover types.

DISCUSSION

There is great debate regarding whether proportionality parameter b is constant. Huntington et al. (2011) applied a modified AA model with b = 1 to estimate monthly and annual ETa from arid shrublands of the southwestern USA and obtained quite satisfactory prediction accuracy. Ma et al. (2015b) applied a modified CR-based AA model to obtain daily ETa in the alpine steppe of the Tibetan Plateau and believed that a symmetric CR (b = 1) contradicted previous research that used default parameter values to claim an asymmetric CR in arid and semiarid regions of the Tibetan Plateau. Sugita et al. (2001) demonstrated that b equals unity only when the underlying surface is smooth enough and soil moisture is sufficient. Furthermore, Ma et al. (2015b) suggested that a strictly symmetric CR is difficult to achieve because the actual wet surface is too small or too large, or there is the influence of some additional heat transfer though the bottom or side of pans. Kahler & Brutsaert (2006) also suggested that the asymmetry with ETp given by pan evaporation was due to the effect of a significant bottom or sidewall effect of pans. However, Szilagyi (2007) and Brutsaert (2015) demonstrated that the asymmetry is the inherent nature rather than the cases. In the present study, Equation (14), proposed by Crago et al. (2016), was used to estimate the ETa. The present CR model is a self-adjusting CR since b could be obtained for each daily and each grid value as b= (1 xmin)/xmin (Crago et al. 2016). The CR is only linear (symmetric) when xmin= 0.5. The definition of variable X defined by Crago et al. (2016) is essentially the same as the definition of Szilagyi et al. (2017). However, the relationship between y and x or X is actually still unspecified. Brutsaert (2015) and Crago et al. (2016) recommended the simple polynomial to add some flexibility in CR formulations, so that its parameters can be calibrated easily. Hence, Equation (16) was adopted in this study.

Two definitions of ETpmax reported by Crago et al. (2016) and Szilagyi et al. (2017) were contrasted above to select an appropriate definition according to conditions in the study basin. The ETpmax defined by Szilagyi et al. (2017) was chosen because of the physically based limits, i.e., . We found that ETpETpds > 0 where the elevation is more than approximately 2,300 m asl (not displayed). Because TwsTa in the mountainous regions, one obtains the following: 
formula
(18)
ETpds defined by Crago et al. (2016) is not applicable where ETpETpds > 0, i.e., ((RnG)/(λ))−f(U) es > (γ/Δ)f(U)ea. In the present study, ETpds was unsuitable in the mountainous regions. Applicability of ETpds depends on the combined effect of Rn, wind speed, Ta, and elevation. Ma & Zhang (2017) thought that the reason of inapplicability of ETpds is mainly due to small wind speed and large available energy. This conclusion is essentially consistent with comments of Ma & Zhang (2017) since the comments had assumed time-invariant Tws and landscape in setting the numerical experiment. Although the values of the wind speed in the mountainous regions are higher than these values in the plains regions, as shown in Figure 3, the available energy and roughness length should play roles in impacting the relationship between ETp and ETpds as well. That is, when ETpds is defined, wind speed is not the only factor that needs to be considered.

In the current study, the HRB was divided into two terrain conditions: the plains regions (below 2,000 m asl) and the mountainous regions (above 2,000 m asl), while the HRB was divided into two underlying surface conditions: vegetated regions and unvegetated regions. Hence, to precisely describe the spatial distribution of ETa, the of each condition was recalculated according to the regional water balance and the basic evaporation laws, as mentioned above (Table 1). The underlying surface of the mountainous regions changed with elevation; therefore, a linear function of elevation was used to illustrate the change of in the mountainous regions. Note that calibrate varies significantly. This may show that this CR method might not be used in extreme cold and arid areas.

Obviously, estimated ETa has some uncertainties. For example, extreme cold and arid areas often have precipitation saved in the form of snow and ice until they melt in subsequent years. This means that has a greater effect at smaller time scales, but this should not be assumed to be 0. However, in the present study, it had to be assumed to be 0, as explained above. Moreover, over the past few years, the data obtained from the Gravity Recovery and Climate Experiment (GRACE) have notably improved our understanding of water storage change at monthly time scales in large basins globally (Scanlon et al. 2015; Save et al. 2016). However, the area of the HRB makes it unsuitable for analysis because it uses GRACE data. Secondly, there are uncertainties regarding the function relations of and elevation in the mountainous regions. A linear function was selected since it was easy to understand and be calibrated; however, this does not mean that linear function is the best relationship between and elevation. In addition, the values of changed with elevation rather than time. The values of were also reported to be a function of surface meteorological and environmental factors (Aminzadeh & Or 2014). Obviously, the setting of values of can make a significant difference in the calibration parameter . As noted by Szilagyi et al. (2017), any CR-based method is quite sensitive to the parameter value of . Finally, the reanalysis dataset and runoff data still had significant uncertainties in extreme cold and arid areas because of the extremely sparse meteorological stations and harsh natural conditions. Although ETa estimated in the mountainous regions remains uncertain and could not be verified at daily and monthly time scales, this study not only provides a new concept for calibration, but also a potential solution for different underlying surfaces and time scales.

CONCLUSIONS

In this study, we used the generalized CR to estimate ETa rates during 2006–2014 at daily time scales in extreme cold and arid areas. The HRB served as a prime example. The China Meteorological Forcing Dataset and MODIS data were used to calculate CR components and to analyze regional water balance. Comparison and analysis revealed that correction of the wind function had more significant effects than correction of Tw. , defined by Crago et al. (2016), was not applicable in the study basin, especially in the mountainous regions. Hence, we adopted , defined by Szilagyi et al. (2017). Since available data in the HRB were limited, the validity and accuracy of the calculated ETa were verified using a regional water balance method at annual time scales. The coefficient was calibrated based on the elevation and underlying surface types. Mean annual ETa estimated using optimal values of matched the mean annual ETwb well, while there was deviation between annual ETa and annual ETwb. Based on the model estimations, spatial and temporal distributions of ETa and the water budget in the basin from 2006 to 2014 could offer a reference at monthly or even daily time scales. The mean annual ETa ranged from 2.3 mm to 800 mm during 2006–2014. ETa rates in the plains regions were higher than those in the mountainous regions, with the highest values being observed in croplands and shrublands along the riverside. Most of ETa was concentrated in the months of May to September, accounting for 89% of the mean annual ETa (42 mm) in the upper regions, 76% (159 mm) in the middle regions, and 80% (105 mm) in the lower regions. A water deficit occurred in the middle and lower regions, and a water surplus occurred in the upper regions. The annual deficit was estimated to be 100 mm in the middle regions and 44 mm in the lower regions. The amount of available water was sufficient to satisfy ETa in most months in the upper regions, where the mean annual water surplus was 150 mm.

If meteorological stations and in situ flux stations could be established in the mountainous and desert regions in the future, the accuracy of estimated ETa and water management efficiency could be improved. Moreover, a model-independent calculation of considering underlying surface conditions and environmental conditions has important implications for a calibration-free CR model. That CR model could be easily applied to all kinds of conditions, including extreme cold and arid areas. However, additional studies should be conducted to further improve the model.

ACKNOWLEDGEMENTS

The forcing datasets used in this study were developed by the Data Assimilation and Modeling Center for Tibetan Multi-spheres, Institute of Tibetan Plateau Research, and Chinese Academy of Sciences (http://westdc.westgis.ac.cn/data/). MODIS data and DEM data were provided by the Geospatial Data Cloud site, Computer Network Information Center, and Chinese Academy of Sciences (http://www.gscloud.cn). This work was supported by the National Natural Science Foundation of China (51509202 and 51679185). We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript. We are grateful to the four anonymous reviewers and the editor whose valuable comments and suggestions led to important revisions.

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