Actual daily evapotranspiration (ET_{a}) was measured in two weighing micro-lysimeters, from 1 July 2009 to 30 June 2011 on an alpine meadow in the Qilian Mountains, northwest China. The findings showed that the mean daily ET_{a} in the unfrozen and frozen periods was 2.0 mm and 0.2 mm, respectively. The dominant factor affecting ET_{a} in the unfrozen period was net radiation (*RN*), whereas those in the frozen period were soil surface temperature (*T _{s}*) and air temperature (

*T*). The mean value of the daily crop coefficient (

_{a}*K*) was 0.82 in the unfrozen period, and 0.19 in the frozen period. Regression analysis of

_{c}*K*and the environmental variables indicated that

_{c}*T*,

_{a}*T*, relative humidity and soil surface water content (

_{s}*SWC*), rather than

*RN*, were the major factors influencing

*K*throughout the whole measurement period and in the unfrozen period, although no clear correlative relationships were found between

_{c}*K*and meteorological factors in the frozen period. Three daily empirical ET

_{c}_{a}models for an alpine meadow were developed, also utilizing the FAO-56 Penman–Monteith approach to estimate reference evapotranspiration. They all exhibited good performances in the unfrozen period, but none were found suitable for the frozen period.

## INTRODUCTION

The majority of inland rivers in China, including the Heihe River, the Shiyang River, and the Shule River, originate from the Qilian Mountains, northwestern China, a huge mountain range on the northeastern Qinghai–Tibet Plateau (Dong *et al.* 2015). As a main water source conservation zone of northwest China (Sun & Liu 2013), the Qilian Mountains play a critical role in regional sustainable development, especially for the oasis belt in the arid Hexi Corridor (Dong *et al.* 2015). Under the influence of global warming, the rate of warming in high mountain regions is amplified with elevation (Mountain Research Initiative EDW Working Group 2015), and the temperature on the north slope of Qilian Mountains has increased by 0.29 °C/10a in the past 50 years (Li *et al.* 2012). As in other cold high mountain regions, warming has caused a series of changes in hydrological processes in the Qilian Mountains, including changes in glacier melt (Wu *et al.* 2015), seasonal snow cover (Bourque & Mir 2012), alpine permafrost distribution (Zhao *et al.* 2012), and runoff (Li *et al.* 2012), which have influenced the water resources of the mountains and the middle and downstreams area of inland rivers. However, evapotranspiration (ET), an integral part of the water cycle under cold high mountain conditions (Herrnegger *et al.* 2012), is still incompletely understood in the Qilian Mountains because of sparse observations and a lack of measured data (Kang *et al.* 2008).

In fact, measuring ET in cold high mountain regions poses a number of challenges. The low temperatures test the endurance of fieldworkers and their instruments, and the distance of these areas from large urban centers raises the cost of logistics and reduces the availability of measured data (Woo 2008). The Bowen ratio–energy balance and eddy correlation methods are widely used to obtain actual evapotranspiration (ET_{a}) in a diversity of sites, but in cold alpine meadow regions, they are both time- and apparatus-consuming and applicable only when a number of requirements are fulfilled, and their assumption of spatial homogeneity is also frequently violated (Pauwels & Samson 2006). Micro-lysimeters offer a relatively cheap, robust and practical alternative for measuring ET_{a}, particularly for the high mountains (Zhang *et al.* 2003; Yang *et al.* 2013), where it is too expensive and difficult to build the large-scale weighing lysimeters. Although there are some limitations to the use of micro-lysimeters to estimate ET_{a}, such as the influence on ET_{a} of lysimeter length and surface soil moisture content, they have long been used for this purpose, and their reliability and modifications have been discussed by Lascano & van Bavel (1986) and Zhang *et al.* (2003). In sum, their low cost and ease of application renders micro-lysimeters appropriate for measuring ET_{a} on alpine meadows.

Owing to the difficulties of direct ET measurement in cold high mountain conditions, estimated methods provide an acceptable way of obtaining ET. As an analytical approach, the FAO-56 publication on crop ET_{a} (Allen *et al.* 1998) is the most widely used method worldwide (e.g., Sumner & Jacobs 2005; Pauwels & Samson 2006; Ngongondo *et al.* 2013; Reddy 2015). Ma *et al.* (2015b) proposed a complementary relationship method to simulate the ET in high-altitude regions. The FAO-56 method is commonly called the two-step approach, because it calculates the reference evapotranspiration (ET_{0}) first, and is then empirically adjusted using the crop coefficients (*K _{c}*) (Allen

*et al.*1998). Although the FAO-56 method has attracted criticism, it is a convenient way to estimate ET

_{a}(Lhomme

*et al.*2015; Ma

*et al.*2015c). However, as an important adjustment ratio,

*K*depends on the type and varieties of crops, crop height, leaf characteristics, soil properties, climate conditions, irrigation methods, and so on (Reddy 2015). Due to data limitations and for calculation simplification, Allen

_{c}*et al.*(1998) suggest that

*K*can be assumed to be a constant for a given vegetation growth stage, involving initial, crop development, mid-season and late season. Following the guidelines for computing ET

_{c}_{a}(Allen

*et al.*1998),

*K*can be simply estimated as a function of the green-leaf area index (

_{c}*LAI*) (Sumner & Jacobs 2005), a regression function of air temperature (

*T*), soil water content, relative humidity (

_{a}*RH*), and

*RN*(Zhou & Zhou 2009; Yang & Zhou 2011), or a polynomial function of sowing day (Reddy 2015). However, unlike the low altitude regions, because of a lack of ET

_{a}measurements, there are few reports and/or discussions on

*K*in high altitude mountainous regions.

_{c}The aims of this paper were to: (1) measure actual daily ET_{a} by micro-lysimeters, compare it with daily ET_{0} from the FAO-56 Penman–Monteith (FPM) approach method; (2) investigate the factors influencing ET_{a} and *K _{c}*; and (3) develop a

*K*and daily ET

_{c}_{a}model on the basis of FPM ET

_{0}estimates for the application on an alpine meadow in the Qilian Mountains, northwest China.

## STUDY SITE AND EXPERIMENTAL SETUP

The experiment was conducted at the Qilian Alpine Ecology & Hydrology Research Station (38°16′N, 99°52′E, altitude 2,980 m), which is located in the headwaters region of Heihe River in the Qilian Mountains (Figure 1). A standard meteorological observation field was built at the station on 20 June 2009, and an Environmental Information System (ENVIS, IMKO Micromodu1technik GmbH, Germany), was also installed to record half-hourly measurement of soil temperature (T8, IMKO, Germany) and water contents at seven depths (0 cm for temperature/2 cm for water content, 20 cm, 40 cm, 60 cm, 80 cm, 120 cm and 160 cm, TRIME–EZ, IMKO, Germany), and a number of other meteorological variables, including solar radiation and reflected radiation (CMA6, Kipp & Zonen, the Netherlands), *RN* (Type 8110, Ph. Schenk GmbH Wien, Austria), precipitation (RG50, Seba Hydrometrie GmbH, Germany), soil heat flow (0.05 m, HFT-3, Campbell Scientific Ltd., USA), *T _{a}* and

*RH*(HMP45D, Vaisala, Finland) at 1.5 m and 2.5 m, wind speed (

*WS*) and direction (RITA Gray, IISA, Siggelkow Geratebau GmbH, Germany). According to meteorological data from the observation field and ENVIS, from 1 July 2009 to 30 June 2011, the mean

*T*was approximately 1.5 °C in the observed period, with January and July averages of −8.6 °C and 14.5 °C, respectively. The mean annual

_{a}*RH*and

*RN*was 54.8% and 80.7 W m

^{−2}, respectively. The mean annual precipitation over the two observation years was 446.9 mm, 90% of which occurred in the plant growing season (May to September). The research site was classified as seasonal frozen soil region, and the maximum frozen soil depth (

*FSD*) was about 240 cm (Figure 2). The dominant species in the area were

*Kobresia capillifolia*(Decne.) C.B. Clarke and

*Carex moorcroftii*(Falc. ex Boott)

*.*The canopy cover was above 98%, and the vegetation root depth was nearly 35 cm.

Two lysimeters (labeled A and B) used in this study, of metal material, filled with a block of natural soil of the same size and shape as the drum, were installed by setting a cylindrical container into the soil at the level of the natural surface. Drawing from the useful research in similar localities (Zhang *et al.* 2003; Yang *et al.* 2013), the two lysimeters chosen for this work were both 40 cm in depth and 31.5 cm in diameter, to accommodate the depth of the majority of vegetation roots. The lysimeters were installed in the standard meteorological observation field, which was located in a flat pasture with representative vegetation and soil. Each was weighed daily at 20:00 (China Standard Time), with a balance of 2 g in precision, which corresponds to 0.026 mm. There are some limitations of micro-lysimeters in obtaining ET_{a} in an alpine environment. The lysimeter container could block the deep drainage, and the lysimeter material could influence the soil heat transfer. In this study, the designed length of the lysimeters was long enough to reduce the influences from deep soil water flow, and the metal material could transfer the heat well between soils in and out of the lysimeters. There have been some successful field experiments with similar lysimeters in an alpine environment (Yang *et al.* 2013), so it was reasonable and acceptable to use the designed lysimeters to obtain ET_{a} at the research site. The micro-lysimeters were installed on 25 June 2009, with measurements taken from 1 July 2009 to 30 June 2011, to obtain the daily ET_{a} readings for the two consecutive years. Owing to the fact that soil freezes during the cold seasons in high-altitude mountainous regions, the measurement period was divided into unfrozen and frozen periods based on the measured soil surface temperature (*T _{s}*). Days on which

*T*fell below 0 °C constituted the frozen period, and all other days were the unfrozen period.

_{s}## METHODS

### Measurement method

_{a}to be calculated using the following equation: where ET

_{a}is the measured ET by the lysimeter method (mm day

^{−1});

*ΔW*is the weight difference measured by the lysimeter (g day

^{−1}); is the water density (1 g cm

^{−3});

*S*is the surface area of the soil within it (cm

^{2});

*P*is precipitation (mm day

^{−1}); and

*I*is infiltration (g day

^{−1}), measured by weighing daily, the same as the lysimeter.

### FPM

_{0}(Allen

*et al.*1998) can be expressed as: where ET

_{0}is the reference ET estimated by FPM (mm day

^{−1});

*R*is the

_{n}*RN*at the grass surface (MJ m

^{−2}day

^{−1});

*G*is the soil heat flux (MJ m

^{−2}day

^{−1});

*T*is the average air temperature at 2 m height (°C);

_{a}*u*is the

_{2}*WS*at 2 m height (m s

^{−1});

*e*is the saturation vapor pressure (kPa);

_{s}*e*is the actual vapor pressure (kPa);

_{a}*Δ*is the slope of the saturated water vapor pressure curve (kPa °C

^{−1}); and

*γ*is the psychrometric constant (kPa °C

^{−1}).

### Crop coefficient

*K*), as an adjustment ratio, can be expressed as: where

_{c}*K*is the crop coefficient, ET

_{c}_{a}is from Equation (1), and ET

_{0}is from Equation (2).

### Statistical analysis

_{a}modeling performance. ET

_{a}values measured by the lysimeters and computed through developed methods were compared by using a series of statistics. Errors were calculated as: where RMSE, MAD,

*R*, and

*R*are the root mean square error (mm day

^{2}^{−1}), mean absolute deviation (mm day

^{−1}), Pearson correlation coefficients, and coefficient of determination;

*n*is the number of observation days;

*E*are the estimated ET values (mm day

_{i}^{−1});

*M*are ET values measured by the lysimeters (mm day

_{i}^{−1}). The value of Pearson correlation coefficients is between 1 and −1 inclusively, where 1 is total positive correlation, 0 is no correlation, and −1 is total negative correlation. The regression functions and regression fitting lines in this study were directly provided by Origin 8.0 software.

## RESULTS AND DISCUSSION

### Determination of ET_{a} by lysimeters

Two micro-lysimeters were used to obtain an accurate determination of ET_{a} in an alpine meadow in the Qilian Mountains from 1 July 2009 to 30 June 2011. Figure 3 shows a comparison between the ET_{a} values recorded by lysimeter A and B. The values of the two lysimeters were in a very close range, and the average of the two lysimeters was taken as the actual daily ET on the research site. The total ET_{a} over the two years was 887.8 mm, 1.2 mm per day, accounting for 98.0% of precipitation during the experimental period. In the first year (from 1 July 2009 to 30 June 2010), the sum of ET_{a} was 451.5 mm, accounting for 99.1% of precipitation in the corresponding period. The figures for the second year (from 1 July 2010 to 30 June 2011) were 436.3 mm and 99.6%, respectively. The ratio of ET_{a} and precipitation showed that ET accounted for the largest loss of water cycle at the research site; another study reported a similar conclusion in a high altitude region (Ma *et al.* 2015a).

Obviously, seasonal variation in ET_{a} was observed at the research site (Figure 4), and the average daily ET_{a} in spring (March–May), summer (June–August), autumn (September–November), and winter (December–February) was 0.7 mm, 3.1 mm, 1.0 mm, and 0.1 mm, respectively, and the daily reference ET_{0} in four seasons was 2.2 mm, 2.9 mm, 2.0 mm, and 0.8 mm, respectively. The complicated and changeable weather in the alpine mountainous area led to short-term variations of the daily ET_{a}. The ET_{a} values in the frozen period were very small due to the low temperatures and frozen soil with little liquid water; and the bulk of ET_{a} occurred in the unfrozen period. The average daily ET_{a} in the frozen and unfrozen periods was 0.2 mm and 2.0 mm, respectively, and the reference ET_{0} was 1.0 mm and 2.5 mm, respectively, in the two consecutive measured years. In the first year, the sum of ET_{a} in the unfrozen period (200 days) was 418.0 mm, accounting for 92.6% of ET_{a} in that year, whereas it was 411.7 mm in the second year (208 days), accounting for 94.4% of ET_{a} in that year.

### Controlling factors affecting ET_{a}

The principal meteorological parameters affecting ET_{a} are radiation, *T _{a}*,

*RH*, and

*WS*(Allen

*et al.*1998). Water in the near surface soil is the source of ET

_{a}.

*T*was used to distinguish the frozen and unfrozen periods in this paper. Therefore, the influential meteorological factors discussed to control ET

_{s}_{a}in this paper were

*RN*(W m

^{−2}),

*T*(°C),

_{a}*T*(°C),

_{s}*RH*(%),

*WS*(m s

^{−1}), and

*SWC*(%).

The Pearson correlation coefficients between ET_{a} and the meteorological factors were computed at daily scale (Table 1), with *T _{s}*,

*T*, and

_{a}*RN*found to be the most important factors in controlling ET

_{a}throughout the whole measurement period, followed by

*SWC*and

*RH*. As most of the precipitation took place in the warm season on the research site, the

*SWC*was close to saturation most of the time during the summer months (minimum of 22.5% and maximum of 35.5%, Figure 2). The land surface was wet or nearly wet in the unfrozen period, and the strong relationship between ET

_{a}and

*RN*indicated that the dominant factor affecting ET

_{a}in this period was radiation, followed by

*T*and

_{s}*T*(Table 1). In the frozen period, in contrast, the plants grew very slowly and the frozen soil had very little liquid water content, roughly about 6% (Figure 2). The Pearson coefficients between ET

_{a}_{a}and the meteorological parameters showed that ET

_{a}was dominated by

*T*and

_{s}*T*in the frozen period, with

_{a}*RN*and

*SWC*exerting a similarly sized effect on ET

_{a}.

*RH*influenced ET

_{a}over the entire measurement period, but its separate effects in the frozen and unfrozen periods could not be distinguished.

*WS*was not a major control factor on ET

_{a}in either period at the research sites.

Period | Factor | ET_{a} | RN | T _{a} | T _{s} | RH | WS | SWC |
---|---|---|---|---|---|---|---|---|

W | ET_{a} | 1 | ||||||

RN | 0.730^{**} | 1 | ||||||

T _{a} | 0.736^{**} | 0.578^{**} | 1 | |||||

T _{s} | 0.752^{**} | 0.561^{**} | 0.971^{**} | 1 | ||||

RH | 0.430^{**} | 0.122^{**} | 0.608^{**} | 0.688^{**} | 1 | |||

WS | − 0.165^{**} | 0.052 | − 0.061 | − 0.092^{*} | − 0.174^{**} | 1 | ||

SWC | 0.617^{**} | 0.382^{**} | 0.785^{**} | 0.846^{**} | 0.691^{**} | − 0.171^{**} | 1 | |

U | ET_{a} | 1 | ||||||

RN | 0.688^{**} | 1 | ||||||

T _{a} | 0.601^{**} | 0.296^{**} | 1 | |||||

T _{s} | 0.651^{**} | 0.216^{**} | 0.898^{**} | 1 | ||||

RH | 0.059 | − 0.387^{**} | 0.231^{**} | 0.467^{**} | 1 | |||

WS | − 0.209^{**} | 0.042 | − 0.155^{**} | − 0.271^{**} | − 0.304^{**} | 1 | ||

SWC | 0.275^{**} | − 0.128^{**} | 0.160^{**} | 0.411^{**} | 0.552^{**} | − 0.330^{**} | 1 | |

F | ET_{a} | 1 | ||||||

RN | 0.408^{**} | 1 | ||||||

T _{a} | 0.487^{**} | 0.304^{**} | 1 | |||||

T _{s} | 0.572^{**} | 0.329^{**} | 0.824^{**} | 1 | ||||

RH | 0.139^{*} | − 0.134^{*} | 0.007 | 0.157^{**} | 1 | |||

WS | 0.161^{**} | 0.238^{**} | 0.245^{**} | 0.280^{**} | 0.077 | 1 | ||

SWC | 0.407^{**} | 0.035 | 0.445^{**} | 0.570^{**} | 0.122^{*} | 0.030 | 1 |

Period | Factor | ET_{a} | RN | T _{a} | T _{s} | RH | WS | SWC |
---|---|---|---|---|---|---|---|---|

W | ET_{a} | 1 | ||||||

RN | 0.730^{**} | 1 | ||||||

T _{a} | 0.736^{**} | 0.578^{**} | 1 | |||||

T _{s} | 0.752^{**} | 0.561^{**} | 0.971^{**} | 1 | ||||

RH | 0.430^{**} | 0.122^{**} | 0.608^{**} | 0.688^{**} | 1 | |||

WS | − 0.165^{**} | 0.052 | − 0.061 | − 0.092^{*} | − 0.174^{**} | 1 | ||

SWC | 0.617^{**} | 0.382^{**} | 0.785^{**} | 0.846^{**} | 0.691^{**} | − 0.171^{**} | 1 | |

U | ET_{a} | 1 | ||||||

RN | 0.688^{**} | 1 | ||||||

T _{a} | 0.601^{**} | 0.296^{**} | 1 | |||||

T _{s} | 0.651^{**} | 0.216^{**} | 0.898^{**} | 1 | ||||

RH | 0.059 | − 0.387^{**} | 0.231^{**} | 0.467^{**} | 1 | |||

WS | − 0.209^{**} | 0.042 | − 0.155^{**} | − 0.271^{**} | − 0.304^{**} | 1 | ||

SWC | 0.275^{**} | − 0.128^{**} | 0.160^{**} | 0.411^{**} | 0.552^{**} | − 0.330^{**} | 1 | |

F | ET_{a} | 1 | ||||||

RN | 0.408^{**} | 1 | ||||||

T _{a} | 0.487^{**} | 0.304^{**} | 1 | |||||

T _{s} | 0.572^{**} | 0.329^{**} | 0.824^{**} | 1 | ||||

RH | 0.139^{*} | − 0.134^{*} | 0.007 | 0.157^{**} | 1 | |||

WS | 0.161^{**} | 0.238^{**} | 0.245^{**} | 0.280^{**} | 0.077 | 1 | ||

SWC | 0.407^{**} | 0.035 | 0.445^{**} | 0.570^{**} | 0.122^{*} | 0.030 | 1 |

W: whole measured period, *n* = 730; U: unfrozen period, *n* = 408; F: frozen period, *n* = 322.

^{**}*P* < 0.01; ^{*}*P* < 0.05.

The controlling factors that affect ET_{a} over different underlying surfaces in different climate regions are different; climate background, plant and soil type, and human interference can all be dominant factors. Over wet temperate grassland in Japan, *RN*, the atmospheric vapor pressure deficit, canopy surface conductance, and *LAI* all had an appreciable effect on ET_{a} measured via the eddy covariance technique (Li *et al.* 2005). Ryu *et al.* (2008) measured ET_{a} directly with the eddy covariance technique on a grassland in the Mediterranean climate zone of California, and found that monthly ET_{a} scaled negatively with solar radiation and was restrained by precipitation in the water-limited period; in the energy-limited period, on the other hand, the majority of ET_{a} scaled positively with solar radiation. In the temperate desert steppe in Inner Mongolia, China, ET_{a} measured via the eddy covariance method during the growing season showed *SWC* to be the dominant influential factor (Yang & Zhou 2011). In a study carried out in seven cropping zones in the Indus Basin in Pakistan, monthly ET_{a} estimated by the surface energy balance system was significantly controlled by mean *T _{a}* and rainfall, among several other climatological variables (e.g.,

*RN*, sunshine hours, and

*WS*) (Liaqat

*et al.*2015).

*K*_{c} and its influence factors

_{c}

In the current study, *K _{c}*, calculated based on Equation (3), displayed obvious daily and seasonal variations (Figure 5). The values of

*K*were low in the frozen period, exhibited a gradual rise from the frozen to the unfrozen period, and reached a peak in summertime, and then fell off as the frozen period approached. The daily

_{c}*K*values ranged from 0.07 to 2.47 in the unfrozen period, with a mean value of 0.82, and from 0.01 to 0.79 in the frozen period, with a mean value of 0.19. Due to differences in climate background and plant type,

_{c}*K*values in this study varied to a greater extent than those in studies carried out in other regions.

_{c}*K*for reed marsh areas in the Liaohe Delta, Northeast China, ranged from 0.2 to 1.4, with a mean value of 0.53 in 2005 (Zhou & Zhou 2009). Migliaccio & Shoemaker (2014) reported monthly

_{c}*K*values ranging from 0.62 to 0.92 for non-irrigated bahiagrass in subtropical south Florida, from May 2010 to January 2013.

_{c}It has long been known that climate exerts a major influence on *K _{c}*, which varies due to many meteorological factors, such as cloudiness, radiation,

*WS*, temperature, and so on (Zhou & Zhou 2009). In this study, regression analysis was conducted between

*K*and a number of environmental variables, namely,

_{c}*T*,

_{a}*T*,

_{s}*RN*,

*RH*, and

*SWC*. Figure 6 and Table 2 show that

*T*,

_{a}*T*,

_{s}*RH*, and

*SWC*, but not

*RN*, were the major factors influencing

*K*throughout the whole measurement period and the unfrozen period. However, there were no clear correlative relationships between

_{c}*K*and any of the meteorological factors in the frozen period. During that period, very little liquid water was available in the soil for evaporation, and there was almost no transpiration because of extremely limited plant activity. The FPM method, which was developed to calculate ET

_{c}_{0}with actively green grass growing and adequately watered (Allen

*et al.*1998), may not be suitable for the meteorological conditions that prevail in the frozen period on high altitude mountainous regions. This may explain the lack of clear statistical relationship between

*K*and the meteorological factors in the frozen period.

_{c}Variables | Whole period | Unfrozen period | Frozen period |
---|---|---|---|

T _{a} | y = 0.0008x^{2} + 0.0328x + 0.4321; R² = 0.5264 | y = 0.0007x^{2} + 0.0233x + 0.5477; R² = 0.1573 | y = 0.0002x^{2} + 0.0064x + 0.2271; R² = 0.0125 |

T _{s} | y = 0.0015x^{2} + 0.0308x + 0.3029; R² = 0.6552 | y = 0.0026x^{2} + 0.0096x + 0.3844; R² = 0.3688 | y = 0.0017x^{2} + 0.0330x + 0.3071; R² = 0.1204 |

RN | y = 0.0278x + 0.3451; R² = 0.1229 | y = −0.00001x + 0.8164; R² = 2E-06 | y = −0.0025x + 0.1977; R² = 0.0050 |

RH | y = 0.0334e^{0.0434x}; R² = 0.4451 | y = 0.1143e^{0.0282x}; R² = 0.3398 | y = 0.0738e^{0.0155x}; R² = 0.0410 |

SWC | y = 0.0898e^{0.0754x}; R² = 0.575 | y = 0.1323e^{0.0639x}; R² = 0.4015 | y = 0.1059e^{0.0373x}; R² = 0.0670 |

DOY | y = −2E-07x^{3} + 6E-05x^{2} − 0.0019x + 0.0946; R² = 0.5997 | y = −1E-07x^{3} − 4E-07x^{2} + 0.0179x − 1.5438; R² = 0.4248 | y = −3E-08x^{3} + 1E − 05x^{2} + 0.0001x + 0.1311; R² = 0.1448 |

Variables | Whole period | Unfrozen period | Frozen period |
---|---|---|---|

T _{a} | y = 0.0008x^{2} + 0.0328x + 0.4321; R² = 0.5264 | y = 0.0007x^{2} + 0.0233x + 0.5477; R² = 0.1573 | y = 0.0002x^{2} + 0.0064x + 0.2271; R² = 0.0125 |

T _{s} | y = 0.0015x^{2} + 0.0308x + 0.3029; R² = 0.6552 | y = 0.0026x^{2} + 0.0096x + 0.3844; R² = 0.3688 | y = 0.0017x^{2} + 0.0330x + 0.3071; R² = 0.1204 |

RN | y = 0.0278x + 0.3451; R² = 0.1229 | y = −0.00001x + 0.8164; R² = 2E-06 | y = −0.0025x + 0.1977; R² = 0.0050 |

RH | y = 0.0334e^{0.0434x}; R² = 0.4451 | y = 0.1143e^{0.0282x}; R² = 0.3398 | y = 0.0738e^{0.0155x}; R² = 0.0410 |

SWC | y = 0.0898e^{0.0754x}; R² = 0.575 | y = 0.1323e^{0.0639x}; R² = 0.4015 | y = 0.1059e^{0.0373x}; R² = 0.0670 |

DOY | y = −2E-07x^{3} + 6E-05x^{2} − 0.0019x + 0.0946; R² = 0.5997 | y = −1E-07x^{3} − 4E-07x^{2} + 0.0179x − 1.5438; R² = 0.4248 | y = −3E-08x^{3} + 1E − 05x^{2} + 0.0001x + 0.1311; R² = 0.1448 |

*K*_{c} and ET_{a} models

_{c}

*K*model have been conducted. Sumner & Jacobs (2005) reported that

_{c}*K*in a non-irrigated pasture site in Florida, USA, was a simple function of

_{c}*LAI*. Zhou & Zhou (2009) reported that the daily

*K*in a reed marsh in the Liaohe Delta, Northeast China, could be a function of

_{c}*RN*,

*T*, and

_{a}*RH*. As there were no continuous

*LAI*measurement data on the research site, and

*T*,

_{a}*T*,

_{s}*RH*, and

*SWC*were the major factors influencing

*K*over the entire measurement period, the estimated daily

_{c}*K*can be obtained from the equation empirically from the correlations and regression analysis between

_{c}*K*and its major environmental controls: where

_{c}*K*is estimated

_{ce}*K*;

_{c}*T*and

_{a}*T*are the air and soil surface temperature (°C);

_{s}*RH*is relative humidity (%); and

*SWC*is soil surface liquid water content (%); and

*a*,

*b*,

*c*,

*d*,

*e*,

*f*, and

*g*are the parameters estimated through the regression.

*K*was derived as a polynomial function of the number of days after sowing. Based on the good relationship between

_{c}*K*and day of year (

_{c}*DOY*) over the entire measurement period on the research site (Figure 6, Table 2), the estimated daily

*K*can be given by the following equation: where

_{c}*D*is

*DOY*; and

*a*,

*b*,

*c*, and

*d*are parameters estimated through the regression.

*K*models (Equations (7)–(9)), and the data from the second year (from 1 July 2010 to 30 June 2011) were then used to validate the ET models. Based on daily meteorological data and ET

_{c}_{a}in the first year, three empirical daily ET models for alpine meadow in the Qilian Mountains were developed: where ET

_{ae}is estimated ET

_{a}(mm day

^{−1}), and ET

_{0}is the reference ET estimated by Equation (2) (mm day

^{−1}).

The three models were tested using ET_{a} data collected during the measurement period. The slope of regression line (0.87) between the estimated (Equation (10)) and measured ET was close to 1, with *R ^{2}* of 0.89, RMSE of 0.49 mm day

^{−1}, and MAD of 0.32 mm day

^{−1}in the calibration period, whereas the slope was 1.08, with

*R*of 0.83, RMSE of 0.79 mm day

^{2}^{−1}, and MAD of 0.46 mm day

^{−1}in the validation period. Equation (11) based on

*T*,

_{s}*RH*, and

*SWC*achieved better performance, with the slope of the regression line of 0.94,

*R*of 0.88, RMSE of 0.55 mm day

^{2}^{−1}, and MAD of 0.36 mm day

^{−1}over the entire validation period. A possible reason for this result was that

*T*, rather than

_{s}*T*, was the key indicator parameter of the energy process on soil surface, and exerted a strong influence on both

_{a}*K*and ET. Based on the relationship between

_{c}*K*and

_{c}*DOY*, Equation (12), although it displayed poorer performance than the two other models over the entire validation period, with a slope of 1.17,

*R*of 0.80, RMSE of 1.28 mm day

^{2}^{−1}, and MAD of 0.96 mm day

^{−1}, could be used when data were limited because of its minimal input needs.

Scatter plots (Figure 7) and statistical analysis (Table 3) of the estimated and measured ET show that the three empirical daily ET models were able to calculate ET well for the unfrozen period, but were found to be unsuitable for the frozen period. The *R ^{2}* of estimated ET from Equations (10)–(12) against ET

_{a}in the frozen validation period were 0.25, 0.25, and 0.23, respectively. The values of RMSE were 0.13 mm day

^{−1}, 0.17 mm day

^{−1}, 0.66 mm day

^{−1}, respectively, and MAD were 0.10 mm day

^{−1}, 0.13 mm day

^{−1}, and 0.47 mm day

^{−1}, respectively, whereas the mean daily ET

_{a}in the frozen period was 0.2 mm day

^{−1}. The assumptions of

*K*and ET models, built to estimate crop ET with active green grass growth and adequate water in the low-altitude regions, might not be suitable for the actual growing and meteorological conditions in a high-altitude mountainous region during the winter months in which the ground was frozen. ET

_{c}_{a}in the frozen period was very low, and potential measurement errors would be relatively large. The measurement errors might be the other reason for poor performances of ET

_{a}models in the frozen period. Building a simple empirical daily

*K*or ET model using meteorological variables might prove difficult in periods of freeze, because there was no clear relationship between those variables and

_{c}*K*in the frozen period.

_{c}Model | Period | Slope | Intercept | R ^{2} | RMSE (mm day^{−1}) | MAD (mm day^{−1}) |
---|---|---|---|---|---|---|

Equation (10) | W | 0.87/1.08 | 0.16/0.21 | 0.89/0.83 | 0.49/0.79 | 0.32/0.46 |

U | 0.79/0.98 | 0.43/0.51 | 0.82/0.73 | 0.64/1.04 | 0.49/0.73 | |

F | 0.73/0.36 | 0.06/0.16 | 0.37/0.25 | 0.16/0.13 | 0.10/0.10 | |

Equation (11) | W | 0.89/0.94 | 0.14/0.27 | 0.90/0.88 | 0.45/0.55 | 0.29/0.36 |

U | 0.82/0.85 | 0.39/0.58 | 0.84/0.81 | 0.60/0.72 | 0.46/0.55 | |

F | 0.74/0.55 | 0.06/0.16 | 0.51/0.25 | 0.13/0.17 | 0.09/0.13 | |

Equation (12) | W | 1.08/1.17 | 0.47/0.72 | 0.88/0.80 | 0.82/1.28 | 0.60/0.96 |

U | 0.94/0.99 | 0.94/1.28 | 0.83/0.71 | 1.02/1.60 | 0.84/1.33 | |

F | 1.74/1.90 | 0.13/0.32 | 0.39/0.23 | 0.48/0.66 | 0.31/0.47 |

Model | Period | Slope | Intercept | R ^{2} | RMSE (mm day^{−1}) | MAD (mm day^{−1}) |
---|---|---|---|---|---|---|

Equation (10) | W | 0.87/1.08 | 0.16/0.21 | 0.89/0.83 | 0.49/0.79 | 0.32/0.46 |

U | 0.79/0.98 | 0.43/0.51 | 0.82/0.73 | 0.64/1.04 | 0.49/0.73 | |

F | 0.73/0.36 | 0.06/0.16 | 0.37/0.25 | 0.16/0.13 | 0.10/0.10 | |

Equation (11) | W | 0.89/0.94 | 0.14/0.27 | 0.90/0.88 | 0.45/0.55 | 0.29/0.36 |

U | 0.82/0.85 | 0.39/0.58 | 0.84/0.81 | 0.60/0.72 | 0.46/0.55 | |

F | 0.74/0.55 | 0.06/0.16 | 0.51/0.25 | 0.13/0.17 | 0.09/0.13 | |

Equation (12) | W | 1.08/1.17 | 0.47/0.72 | 0.88/0.80 | 0.82/1.28 | 0.60/0.96 |

U | 0.94/0.99 | 0.94/1.28 | 0.83/0.71 | 1.02/1.60 | 0.84/1.33 | |

F | 1.74/1.90 | 0.13/0.32 | 0.39/0.23 | 0.48/0.66 | 0.31/0.47 |

W: whole measurement period; U: unfrozen period; F: frozen period^{a}.

^{a}ET_{ae} = Slope × ET_{a} + Intercept.

## CONCLUSIONS

On an alpine meadow site located in the high Qilian Mountains, the total ET_{a} measured from micro-lysimeters, from 1 July 2009 to 30 June 2011, was 887.8 mm, 1.2 mm per day. The total ET_{a} was 451.5 mm in the first year (from 1 July 2009 to 30 June 2010), and 436.3 mm in the second. The ratio of ET to precipitation in the measurement period showed that ET was the main cause of water loss at the research site. Seasonal variations were found in the measured ET_{a} in the high-mountain regions, with the mean daily ET_{a} in the unfrozen and frozen periods measuring 2.0 and 0.2 per day, respectively. The dominant factor affecting ET_{a} in the unfrozen period was *RN*, whereas it was *T _{s}* and

*T*in the frozen period.

_{a}The crop coefficient (*K _{c}*) for the alpine meadow in the high mountains showed obvious daily and seasonal variations, with daily

*K*ranging from 0.07 to 2.47 in the unfrozen period, with a mean value of 0.82, and from 0.01 to 0.79 in the frozen period, with a mean value of 0.19.

_{c}*T*,

_{a}*T*,

_{s}*RH*, and

*SWC*, but not

*RN*, were the major factors of influence for

*K*over the whole measurement period and during the unfrozen period, but there were no clear correlative relationships between

_{c}*K*and these meteorological factors in the frozen period.

_{c}Three daily *K _{c}* models were constructed. The first one was a function of

*T*,

_{a}*RH*, and

*SWC*, while the second one was a function of

*T*,

_{s}*RH*, and

*SWC*. The third

*K*model was a polynomial function of

_{c}*DOY*. Then, three empirical daily ET

_{a}models derived from the FPM method were developed on the basis of those

*K*models. The results indicated that the ET

_{c}_{a}model based on

*T*,

_{s}*RH*, and

*SWC*offered the best performance, with the slope of the regression line of 0.94,

*R*of 0.88, RMSE of 0.55 mm day

^{2}^{−1}, and MAD of 0.36 mm day

^{−1}for the whole validation period, and slope,

*R*, RMSE, and MAD was 0.85, 0.81, 0.72, mm day

^{2}^{−1}and 0.55 mm day

^{−1}, respectively for the unfrozen validation period; and 0.55, 0.25, 0.17 mm day

^{−1}, and 0.13 mm day

^{−1}, respectively, for the frozen validation period. The three empirical daily ET

_{a}models performed well in calculating ET

_{a}at the research site during the unfrozen period, but were not suitable for the frozen period. The probable reason was that

*K*and ET

_{c}_{0}from the FPM method provided inappropriate assumption for the actual plant growing and meteorological conditions in the high-altitude mountainous regions during the frozen period.

It is difficult to achieve a true understanding of ET in high mountain areas, because direct measurements of ET_{a} in those regions are not easily obtained. The results from this study will advance the understanding of actual daily ET_{a} and the difference in control factors between frozen and unfrozen periods. The three empirical daily ET_{a} models provided in this paper can be used for accurate quantitative estimates of the sensitivity of alpine meadow to climate change in high-altitude mountainous regions.

## ACKNOWLEDGEMENTS

This work was carried out with financial support from the National Basic Research Program of China (2013CBA01806) and the National Natural Sciences Foundation of China (41401041). We thank the four reviewers and the editor for their constructive comments that led to improvements in the manuscript.