Complementary Relationship-Based Validation and Analysis of Evapotranspiration in the Permafrost Region of the Qinghai–Tibetan Plateau

Abstract

The Complementary Relationship (CR) principle of evapotranspiration provides an efficient approach for estimating actual evapotranspiration (ET_a), owing to its simplified computation and effectiveness in utilizing meteorological factors. Accurate estimation of actual evapotranspiration (ET_a) is crucial for understanding surface energy and water cycles, especially in permafrost regions. This study aims to evaluate the applicability of two Complementary Relationship (CR)-based methods—Bouchet’s in 1963 and Brutsaert’s in 2015—for estimating ETa on the Qinghai–Tibetan Plateau (QTP), using observations from Eddy Covariance (EC) systems. The potential evapotranspiration (ET_p) was calculated using the Penman equation with two wind functions: the Rome wind function and the Monin–Obukhov Similarity Theory (MOST). The comparison revealed that Bouchet’s method underestimated ET_a during frozen soil periods and overestimated it during thawed periods. In contrast, Brutsaert’s method combined with the MOST yielded the lowest RMSE values (0.67–0.70 mm/day) and the highest correlation coefficients (r > 0.85), indicating superior performance. Sensitivity analysis showed that net radiation (Rn) had the strongest influence on ET_a, with a daily sensitivity coefficient of up to 1.35. This study highlights the improved accuracy and reliability of Brutsaert’s CR method in cold alpine environments, underscoring the importance of considering freeze–thaw dynamics in ET modeling. Future research should incorporate seasonal calibration of key parameters (e.g., ε) to further reduce uncertainty.

1. Introduction

Evapotranspiration (ET_a) is an important component of the water and energy budget of the land surface in the hydrological cycle and climate change [1,2]. Accurate estimation of ET_a to model terrestrial hydrological and ecological processes is necessary. The ET_a of the Qinghai–Tibetan Plateau (QTP) is the largest permafrost region in the mid–high latitudes, often referred to as the “Asian Water Tower,” and plays a crucial role in the global water cycle and has attracted considerable global attention [3]. The reliable estimation of ET_a is crucial for understanding the hydrothermal balance on the QTP [4,5].

At present, several established measurement techniques are commonly employed to obtain an accurate ET_a, such as weighing lysimeters, the Eddy Covariance (EC) system, the Bowen ratio tower, and large aperture scintillometers [6]. However, the existing sparse observations make it difficult to meet the data requirements for studying the spatiotemporal distribution of ET_a in the cold climate environment over the QTP. Therefore, several mathematical models were developed to estimate ET_a in recent years [7,8]. The Penman–Monteith method is well known for calculating ET_a, which was improved by Monteith introducing the resistance parameter into the Penman equation [9,10]. Based on this, Mu et al. [11] developed a global remote-sensing evapotranspiration algorithm. However, due to the uncertainty of the aerodynamic and surface resistance, it remains challenging to achieve high accuracy in ET_a estimation on the QTP. In addition, ET_a can be estimated based on the surface energy balance, such as SEBAL and SEBS [12]. The models were applied to estimate ET_a over the QTP [13,14,15]. It was found that the ET_a was overestimated in winter on the QTP, which was caused by the questioning of the parametric scheme [16,17,18].

The Complementary Relationship (CR) principle is another effective method to estimate ET_a [19,20]. The first complementary concept of evapotranspiration was proposed by Bouchet [21] in general terms. The potential evapotranspiration (ET_p) and ET_a are strongly coupled through land and atmospheric feedback [22]. When there is a limitation in energy without having a limitation in water availability, the wet environment evapotranspiration (ET_w) is introduced, and it can be expressed as ET_a = ET_w = ET_p. As the water availability decreases, ET_a decreases, whereas ET_p increases. Numerous studies have reported a decreasing ET_p over large areas in different regions over the past 50 years, and this contrasts with the reports of increasing ET_a [23,24,25]. They were regarded as a symmetrical relationship (ET_a < ET_w < ET_p), so it could be expressed with a linear equation at first. But in fact, the relationship is asymmetric in the natural environment. Thus, several models were developed to explain the complementary relationship of evapotranspiration from the natural land surface based on the original concept, which included the complementary relationship areal evapotranspiration model [26], the advection–aridity model [27], the Granger and Gray model [28], and the nonlinear complementary functions. These methods have been widely utilized to estimate ET_a due to their simple calculation without requiring complex parameters. Brutsaert et al. [29,30] proposed the nonlinear complementary relationship principle based on physical considerations. It was demonstrated that the nonlinear function performed better under semi-arid conditions.

ET_a is expected to have various trends at different temporal scales, attributed mainly to the variability in the climate system [31,32,33]. A large number of studies have focused on the variation in and attribution of ET_a in different regions [34,35,36]. Sensitivity and multiple regression analyses are the most commonly used methods for quantifying the contributions of the driving factors to ET_a. Talsma et al. [37] analyzed the sensitivity of remote sensing-based models on the uncertainty of the input variables and found that the models were most sensitive to the Normalized Difference Vegetation Index (NDVI), relative humidity, and net radiation. Cawse-Nicholson et al. [38] found the disaggregated atmosphere–land exchange is most sensitive to the land surface temperature.

The QTP is extremely sensitive to climate change, with observed increases in air temperature and changes in snow cover, soil moisture, and permafrost extent [39,40,41]. These environmental changes have significant implications for evapotranspiration processes, which are closely linked to the surface energy balance and water cycling over frozen and thawed ground conditions. Despite its importance, ET_a remains poorly constrained over permafrost regions due to challenges in measurement and modeling, especially under freeze–thaw transitions [42,43]. Therefore, improving ET_a estimation in permafrost-dominated areas is essential for understanding land–atmosphere interactions and predicting water and energy fluxes under future climate scenarios. This study aims to address this gap by applying and validating CR approaches over three representative permafrost sites on the QTP.

In this study, we conducted a comprehensive analysis of ET_a on the QTP. The observed ET_a, obtained through the EC system, was juxtaposed with the calculated ET to examine the validity of the complementary relationship. This comparison was carried out at three distinct meteorological stations located across the QTP. Subsequently, we delved into the impact and attribution of climate change on ET_a. This investigation was particularly focused on understanding the influence of the freeze–thaw cycle prevalent on the QTP. Our findings contribute to the growing body of knowledge on the intricate interplay between climate change and evapotranspiration patterns, thereby offering valuable insights for future research and policymaking.

Overall, the primary objective of this study is to evaluate the applicability and performance of two CR approaches—Bouchet’s [21] and Brutsaert’s [29] methods—for estimating ET_a in the permafrost region of the QTP. Specifically, the study aims to (1) validate the CR between ET_a and ET_p using observed ET_a from EC systems; (2) assess the influence of different wind functions, including the Monin–Obukhov Similarity Theory (MOST) and the Rome wind function, on ET_p calculations via the Penman equation; and (3) analyze the sensitivity and contribution of meteorological factors to ET_a under freeze–thaw conditions. A key goal is to determine the suitability and limitations of each CR method during the soil-frozen and soil-thawed periods in a changing climate context.

This paper is organized as follows. Section 2 describes the study area, data sources, and methodological framework, including the implementation of Bouchet’s [21] and Brutsaert’s [29] CR models as well as wind function schemes. Section 3 presents the results of model validation and comparative analysis against EC observations under different freeze–thaw conditions. Section 4 discusses the model performance and the sensitivity of ET_a to meteorological drivers, and compares findings with previous studies. Section 5 concludes the study, summarizes key findings, addresses model limitations, and suggests directions for future research.

2. Materials and Methods

2.1. Study Sites and Data Collection

The types of vegetation affect the transpiration rate and canopy retention directly, which can change the characteristics of evapotranspiration. The experimental fields were covered by different typical grasses that are dominant vegetation on the QTP; thus, the transpiration of canopy retention can be ignored on the QTP. There were three experimental sites, named Tanggula (TGL), Xidatang (XDT), and Wudaoliang (WDL), along the QTP engineering corridor (Figure 1). The climate in TGL is cold and dry, with a mean annual temperature of −5 °C to −8 °C and a precipitation of 300–400 mm, concentrated in summer. The mean annual temperature of XDT ranges from −2 °C to −4 °C, and annual precipitation reaches 350–450 mm. WDL is also cold and arid, with a mean annual temperature of −4 °C to −6 °C and an annual precipitation of only 200–300 mm. The elevation, topography, soil type, and grass types are different for these sites, and the relative information is shown in

Table 1. List of the observed sites along the QTP engineering corridor.

Sites (Abbr.)	Longitude (E) Latitude (N)	Elevation (m asl)	Topography	Soil Type	Grass Types	Observed Items	Time
TGL	91.93°	5100	High-altitude mountain, gentle slopes	Permafrost soils	Meadow steppe	u2, HR, Ta, Pres, P, R, heat flux, soil temperature, and moisture	2010–2011
	33.07°
XDT	94.13°	4538	Gently sloped or flat	Alpine meadow soils, cryosols	Alpine meadow		2010–2012
	35.72°
WDL	93.08°	4783	Plateau basin, flat terrain	Cryosols, gravelly soils	Alpine desert steppe		2008–2009
	35.22°

Note: TGL is Tanggula; XDT is Xidatang; WDL is Wudaoliang. u2 is wind speed; HR is relative humidity; Ta is air temperature; Pres is air pressure; P is precipitation; R is solar radiation.

and Figure 1. These gradients in elevation, vegetation cover, and soil properties are crucial for understanding surface energy balance on the QTP.

The observations of meteorological factors such as wind speed, humidity, air temperature, air pressure, precipitation, and solar radiation were measured every 30 min at three heights (2 m, 5 m, and 10 m) by meteorological observation towers established in 2004. Meanwhile, the soil moisture and temperature at different depths were also measured simultaneously.

Three eddy covariance (EC) systems were installed at the individual meteorological observation tower height of 10 m in XDT, WDL, and TGL in 2007, 2006, and 2004, respectively. The uniform grass in the fields was large enough to support the available range for individual EC systems. The EC system consisted of a 3D sonic anemometer (CSAT3, from Campbell Scientific Inc., Logan, UT, USA) and a fast-response open-path CO₂/H₂O infrared gas analyzer (Li-7500, from LI-COR Biosciences, Lincoln, NE, USA). Fluxes of water vapor, heat, and momentum were recorded with an open-path EC system for 30 min, which correspond to spatial scales of decimeters to a few meters, depending on wind speed. For a tower-mounted sensor at ~10 m above ground, the flux footprint typically extends 100–500 m upwind under unstable conditions, but can exceed 1 km under stable stratification. They were used to measure the latent heat flux (LE) and sensible heat flux (H) at a height of 2 m. The flux data were available for XDT for 2010 to 2012, WDL for 2008, and TGL for 2010 to 2011, respectively.

2.2. Methodologies Adopted

2.2.1. The Generalized Nonlinear Complementary Relationship

Bouchet’s hypothesis implies the ideal complementary relationship (CR) of ET_a, ET_p, and ET_w [21]. This means that when ET_a and ET_p are assumed to be under the condition of ample water supply and energy, (ET_p − ET_w) is equal to (ET_w − ET_a), that is, ET_a = ET_p = ET_w. Furthermore, part of the energy is not used up in evaporation, so the relationship between them is not symmetrical and exactly equal in the natural environment. Thus, a coefficient $\epsilon$ is introduced, and the relationship follows the generalized nonlinear function:

$${\mathrm{E}\mathrm{T}}_{\mathrm{p}}-{\mathrm{E}\mathrm{T}}_{\mathrm{w}}=\epsilon \left({\mathrm{E}\mathrm{T}}_{\mathrm{w}}-{\mathrm{E}\mathrm{T}}_{\mathrm{a}}\right)$$

(1)

The coefficient $\epsilon$ is the constant of proportionality. The magnitude of $\epsilon$ is a measure of the effectiveness with which heat transfer takes place between the wet surface land and its surroundings. It reflects the change in water and energy in the environment. In order to express the relationship between them, the relevant variables can be normalized in dimensionless form [44], i.e.,

$${\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{w}}}}={\displaystyle \frac{(1+\epsilon )\frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}{1+\epsilon \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}}$$

(2)

$${\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{w}}}}={\displaystyle \frac{1+\epsilon }{1+\epsilon \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}}$$

(3)

where ${\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}$ indicates the level of the natural land surface close to the potential conditions, so it responds to the change in the aridity of the environment and can be regarded as a land surface humidity index (HI).

However, the relationship could not be simply expressed since (ET_p − ET_w) is proportional to (ET_w − ET_a) under strongly advective conditions. By building on four physical boundary conditions, Brutsaert in 2015 improved the complementary relationship and obtained another complementary relationship [29]:

$${\mathrm{E}\mathrm{T}}_{\mathrm{a}}={\left({\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{w}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}\right)}^2\left({2\mathrm{E}\mathrm{T}}_{\mathrm{p}}-{\mathrm{E}\mathrm{T}}_{\mathrm{w}}\right)$$

(4)

According to Brutrsaert and Stricker [45], the ET_p term can generally be defined by Penman’s equation [9]:

$${\mathrm{E}\mathrm{T}}_{\mathrm{p}}={\displaystyle \frac{∆}{\left(∆+\mathsf{\gamma }\right)}}\left(\mathrm{R}\mathrm{n}-\mathrm{G}\mathrm{o}\right)+{\displaystyle \frac{\mathsf{\gamma }}{\left(∆+\mathsf{\gamma }\right)}}f\left(\mathrm{U}\right)\mathrm{V}\mathrm{P}\mathrm{D}$$

(5)

where $∆$ is the slope of the saturation vapor pressure curve at air temperature (kPa °C⁻¹), $\gamma$ is the psychrometric constant (kPa °C⁻¹), Rn and G0 are the net radiation and soil heat flux in the ground (mm d⁻¹), and VPD is the saturation and actual vapor pressure deficit of the air (e_s − e_a, kPa). $f\left(\mathrm{U}\right)$ is the wind function, and it can be calculated through a modification of Penman’s [9] original empirical linear equation, called the Rome wind function [45], as:

f(U) = 2.6(1 + 0.54u2)

(6)

where u2 is wind speed at 2 m. However, the wind function should be expressed by a certain range rather than a fixed formula. The Monin–Obukhov Similarity Theory (MOST) is a more appropriate method for daily periods on the QTP [15]. Thus, it can be used to calculate ET_p and expressed as:

$$f\left(\mathrm{U}\right)=t{\displaystyle \frac{0.622k^2\rho u2}{Pln\left[\frac{z-d}{z_{0v}}-{\phi }_v\right]ln\left[\frac{z-d}{z_{0m}}-{\phi }_m\right]}}$$

(7)

where $k$ is the von Karman constant; $\rho$ is the density of air; $P$ is the air pressure; $t=1day=86400s$; $z$ is the height of the measurement instrument and equals 2 m in this work; $d$ is the displacement height, taken to be 2/3 of the mean canopy height; $z_{0v}$ is the water vapor roughness length; $z_{0m}$ is the momentum roughness length; and ${\phi }_v$ and ${\phi }_m$ are the stability correction functions for humidity and momentum, respectively. On a daily basis, it is usually assumed that ${\phi }_v$ = ${\phi }_m$ = 0 due to the neutral atmospheric stability [44].

The ET_w is calculated using the Priestley–Taylor equation in a wet environment [15]. The parameter $∆$ is different in arid and semi-arid regions if the Ta (observed air temperature) and Tw (air temperature in a wet environment) are used in Equation (8). Tw cannot be observed directly, so it is approximated with the near-surface air temperature in this work (Equation (9)). The parameter $∆$ is calculated using Tw, and then ET_w is obtained in a wet environment [46].

$${\mathrm{E}\mathrm{T}}_{\mathrm{w}}=\alpha {\displaystyle \frac{∆}{∆+\gamma }}(\mathrm{R}\mathrm{n}-\mathrm{G}0)$$

(8)

$${\beta }_w={\displaystyle \frac{Rn-G0-{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{p}}}}\approx {\displaystyle \frac{\mathrm{T}w-Ta}{e_s\left(Tw\right)-e_a}}$$

(9)

We used two methods to calculate ET_a in this work, where the first method was to use Equation (1), labeled as Bouchet_1963, with the Rome wind function and the MOST. The second method was to use Equation (4), labeled as Brutsaert_2015, with the Rome wind function and the MOST.

2.2.2. Analysis Method

The climate sensitivity coefficient of ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ is an important indicator for assessing the impact of climatic factors on ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$. The sensitivity analysis method is used in ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ studies based on the complementary relationship principle to calculate the sensitivity coefficient. It was defined as the ratio of change in ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ and the change rate of the climatic factor in this work [47].

$$S_x=\underset{∆x\to 0}{\lim }\left({\displaystyle \frac{∆{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}}/{\displaystyle \frac{∆x}{x}}\right)={\displaystyle \frac{\partial {\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial x}}·{\displaystyle \frac{x}{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}}$$

(10)

where $S_x$ is the sensitivity coefficient of ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$, which is dimensionless, and x is a meteorological variable. The positive/negative sensitivity coefficient indicates that ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ will increase/decrease as the variable changes. The absolute value of the sensitivity coefficient represents the levels of sensitivity, which can be used to compare the impact on ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ of different meteorological variables.

The contribution of the meteorological variables to ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ are closely related to the change range of the factors. We take the difference between two consecutive days as the variable amplitude in this work. The change in ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ can be approximately attributed to several meteorological variables, where the total differential equation of the complementary relationship principle is as follows:

$$\mathrm{d}{\mathrm{E}\mathrm{T}}_{\mathrm{a}}\cong {\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{R}\mathrm{n}}}∆\mathrm{R}\mathrm{n}+{\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{G}0}}∆\mathrm{G}0+{\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{R}\mathrm{H}}}∆\mathrm{R}\mathrm{H}+{\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{T}\mathrm{a}}}∆\mathrm{T}\mathrm{a}+{\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{u}2}}∆\mathrm{u}2+{\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{T}w}}∆\mathrm{T}w$$

(11)

where ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{R}\mathrm{n}}}∆\mathrm{R}\mathrm{n}$, ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{G}0}}∆\mathrm{G}0$, ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{R}\mathrm{H}}}∆\mathrm{R}\mathrm{H}$, ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{T}\mathrm{a}}}∆\mathrm{T}\mathrm{a}$, ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{u}2}}∆\mathrm{u}2$, and ${\displaystyle \frac{{\partial \mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial \mathrm{T}\mathrm{w}}}∆\mathrm{T}\mathrm{w}$ indicate the amount of change in ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ caused by $\mathrm{R}n$, G0, $\mathrm{R}\mathrm{H}$, $\mathrm{T}a$, u2, and Tw, respectively. $\mathrm{d}{\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ is the ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ caused by common changes in meteorological variables. The contribution rate of the individual meteorological variable x to ${\mathrm{E}\mathrm{T}}_{\mathrm{a}}$ is shown as follows:

$$C_x={\displaystyle \frac{\frac{\partial {\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{\partial x}∆x}{d{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}}$$

(12)

where $C_x$ is the contribution rate of an individual meteorological variable x, and $∆x$ is the change in an individual meteorological variable x.

Considering the change of climate and land surface on permafrost, the sensitivity and contribution are different. We divided the complete soil freeze–thaw cycle into four periods, thawing (mid-January to mid-May), thawed (mid-May to September), freezing (October), and frozen (early November to mid-January), according to the soil temperature to analyze the effect of climate factors on ET_a [3].

3. Results

3.1. Change in the Meteorological Factors

The changes in the daily available meteorological observation data at the three sites were converted and are shown in Figure 2. The patterns of Ta, Rn, G0, and VPD were similar, where the peak occurred in the summer and the minimum value in the winter. This was opposite to u2, for which the high values were in the winter and reached around 8 m/s or higher. However, the mean daily Ta was low and equal to −4.36 °C, −3.44 °C, and −3.01 °C at the three sites, respectively. It could reach −20 °C in winter. The mean daily Ta, Rn, G0, u2, and VPD in TGL, XDT, and WDL are shown in

Table 2. The mean daily values of the meteorological factors.

Study Sites	Ta (°C)	Rn (W/m2)	G0 (W/m2)	u2 (m/s)	VPD (kPa)
TGL	−4.36	85.57	2.19	3.94	0.23
XDT	−3.44	61.21	1.69	3.76	0.24
WDL	−3.01	71.51	—	5.19	0.22

Note: TGL, XDT, and WDL are Tanggula, Xidatang, and Wudaoliang, respectively. Ta is air temperature; Rn is the net solar radiation; G0 is the soil heat flux; u2 is wind speed; VPD is the actual vapor pressure deficit.

. Comparing the meteorological data from these sites, it was found that the fluctuations in the daily values of Rn, G0, u2, and VPD were greater than those of Ta. The change in ET_a was more vulnerable to other factors than in Ta.

3.2. Performance of the Complementary Relationship (CR)

The CR was checked on a daily scale at the three sites, and the relationship between the observed ET_a and the calculated ET_p is shown in Figure 3. Without considering the limits of water and energy, ET_a should be equal to ET_p. In fact, as the supply of water decreased, ET_a decreased and ET_p increased in this work. The higher the value of the humidity index (HI = ET_a/ET_p), the more concentrated the scattered points. Comparing Figure 3a,b, the values of ET_p calculated using the MOST were more discrete and greater than those using the Rome wind function. This suggests that ET_p is greater in arid conditions and is consistent with the definition of potential evapotranspiration on the QTP.

In order to accurately reflect the relationship between ET_a and ET_p, the dimensionless term of the observed ET_a and calculated ET_p are also displayed, which are denoted for ET_a+ (= ET_a/ET_w) and ET_p+ (= ET_p/ET_w) by blue and red points in Figure 3, respectively. It was found that the difference between ET_a+ and ET_p+ decreased with the increase in the HI. The distribution of these scattered points corresponded well with the shape of the CR theory. These results confirm the existence of a complementary relationship at the study sites.

The simulated ET_a values were also dimensionless, marked as the black points in Figure 3. The ET_a+ simulated by the two methods with Rome and the MOST are shown in Figure 3c,d and Figure 3e,f, respectively. Comparing the relationship between the ET_a+ and ET_p+ with Rome at three sites, it was found that the ET_p+ by Bouchet_1963 method (Equation (1)) was underestimated in arid conditions (HI < 0.2), as shown in Figure 3c, but both the ET_a+ and ET_p+ were overestimated by Brutsaert_2015 method (Equation (4)) in arid and humid conditions, as shown in Figure 3e. By contrast, both of the methods were improved by introducing the MOST, where ET_a+ and ET_p+ would better fit the shape of the CR theory, as shown in Figure 3d,f. In particular, the ET_a+ and ET_p+ by Brutsaert_2015 method with the MOST performed the best, as shown in Figure 3f, and could better improve the CR theory.

3.3. Validation of the Complementary Relationship (CR)

Figure 4 shows the daily simulated ET_a using two different methods at the three sites. It was found that both of the changing trends of the simulated ET_a were similar to the observed ET_a. The peak of the daily ET_a occurred in July, whereas the minimum value occurred in January and was close to 0 due to the frozen soil moisture on the land surface without an adequate supply of water. The amplitude of the ET_a simulated with Bouchet_1963 method (ET_a1) was lower in the cold season from October to April and higher in the warm season from May to September compared to the other observation data. In contrast, the ET_a simulated with Brutsaert_2015 method (ET_a2) improved significantly, especially in the cold seasons.

The simulation results are also shown by the scatter plots in Figure 5. It was obvious that the simulations with Brutsaert_2015 method (ET_a2-Rome and ET_a2-MOST) were much better than with Bouchet_1963 method (ET_a1-Rome and ET_a1-MOST). The simulation results with Brutsaert_2015 method were more concentrated around the 1:1 line. Meanwhile, it was clear that the results were affected by the wind functions. The simulations of ET_a1-Rome and ET_a2-Rome were significantly higher than the observed ET_a. Thus, it was indicated that the MOST function is more suitable in the CR to calculate ET_a on the QTP.

In this paper, the value of ε was decided by minimizing the RMSE of the observed ET_a and estimated ET_a. Thus, the values of ε were 0.77 and 2.27 for Equation (1) with the Rome function (Figure 3c) and the MOST function (Figure 3d) on the QTP, respectively. The Priestley–Taylor equation’s parameter α in Equation (6) was tested and verified, and the best value of α was determined for the individual method to obtain the accurate ET_a. The Root Mean Square Errors (RMSEs) and correlation coefficients (r) were used to evaluate the simulation results, as shown in

Table 3. The Root Mean Square Errors (RMSEs) and correlation coefficients (r) of ET_a and calculated ET_a.

Sits	Counts of Available Data	ETa	RMSE	r
TGL	643	ETa1_Rome	0.97	0.86
		ETa2_Rome	0.80	0.85
		ETa1_MOST	0.87	0.85
		ETa2_MOST	0.70	0.86
XDT	621	ETa1_Rome	1.17	0.79
		ETa2_Rome	1.03	0.85
		ETa1_MOST	0.98	0.74
		ETa2_MOST	0.67	0.81
WDL	183	ETa1_Rome	0.80	0.62
		ETa2_Rome	0.73	0.58
		ETa1_MOST	0.75	0.63
		ETa2_MOST	0.62	0.62

, with the correlation coefficients (r) tested at the 0.05 confidence level. The RMSE of ET_a2 was lower than that of ET_a1, although the correlation coefficients between them were not significantly different. It was clear that the minimum RMSE and the maximum correlation coefficients were obtained using Brutsaert_2015 method with the MOST function (ET_a2-MOST). Thus, it could be inferred that Brutsaert_2015 method with the MOST function was appropriate for estimating ET_a on the QTP by comparing these simulations.

3.4. Sensitivity of the Meteorological Factors to ETa

In this work, the sensitivity and contribution of ET_a through Bouchet_1963 method with the MOST function and Brutsaert_2015 method with the MOST function were compared in TGL in a complete soil freeze–thaw cycle. When Bouchet’s (1963) method with the MOST function was used on the QTP, the daily average sensitivity coefficients of Rn, G0, Ta, u2, Tw, and VPD were 0.62, −0.15, 0.15, 0.14, 0.27, and 0.14, respectively. When Brutsaert_2015 method with the MOST function was used on the QTP, the daily average sensitivity coefficients of Rn, G0, Ta, u2, Tw, and VPD were 1.35, 0.08, −0.05, −0.44, −0.45, and 0.44, respectively. The positive sensitivity coefficients indicate that ET_a increased with the increase in the factors and vice versa. The sensitivities of ET_a on the factors using the two methods were Rn > Tw > u2 = VPD > Ta > G0 and Rn > Tw > u2 = VPD > G0 > Ta, respectively. Considering the assumed Tw in the wet environment calculated by the observed Ta, the two relationships among the factors could be expressed as Rn > Ta > u2 = VPD > G0. The sensitivities of ET_a on the meteorological factors were similar for the two methods.

However, the amplitudes of vibration of the sensitivity coefficients for the two methods were different (Figure 6a,b). The change in amplitude was dramatic, as shown in Figure 6a, and are related to the uncertainties of Bouchet_1963 method. It was more sensitive to the fluctuation in the factors compared to Brutsaert_2015 method. In addition, the average sensitivity coefficients in different soil freeze–thaw periods are shown in Figure 6c,d. The change in sensitivity for ET_a was more obvious in the complete soil freeze–thaw cycle for both methods.

With Bouchet_1963 method, the sensitivity coefficient of ET_a on Rn peaked in the summer soil-thawed period, which played a positive role on ET_a, while the opposite was found in the winter soil-frozen period (Figure 6c). The response of ET_a to u2 and VPD cannot be ignored. The sensitivity coefficient of u2 and VPD peaked in the winter-frozen soil period and varied in the range of −0.52 to 1.91. The effect of u2 and VPD on ET_a was very obvious in the winter soil-frozen periods and much weaker in other periods. The sensitivity coefficient of G0, Ta, and Tw showed lower variations in the complete soil freeze–thaw cycle with a relatively weak impact on ET_a. The change in sensitivity coefficient of ET_a on G0 was similar to that of Rn, and the coefficients of Ta and Tw were similar to those of u2 and VPD. By contrast, with Brutsaert_2015 method, Rn always had a greater positive effect on ET_a without significant changes in other factors for the complete soil freeze–thaw cycle (Figure 6d). It was more obvious for the negative impact of the other factors except for G0 on ET_a in the winter soil-frozen period. The impact diminished in the spring soil-thawing period and almost disappeared in the autumn soil-freezing period.

In short, the factors’ performances were different with the two methods. The sensitivity coefficient of ET_a on Rn expressed the more obvious periodic changes with Bouchet_1963 method compared to with Brutsaert_2015 method. The impacts of other factors on ET_a for the two methods were opposite in the freeze–thaw cycle.

3.5. Contribution of Meteorological Factors to ET_a

The daily contribution of Rn, G0, Ta, u2, Tw, and VPD to ET_a with the two methods are compared in Figure 7 and Figure 8. The mean daily contribution rates of individual factors with Bouchet_1963 method were 54.6%, −6.3%, −1.8%, 28.7%, 7.5%, and 24.1%, respectively. As for Brutsaert_2015 method, the rates were 83.9%, −6.6%, −0.1%, 5.1%, 8.7%, and 2.6%, respectively. With Bouchet_1963 method, the contribution rates were Rn > u2 > VPD > Tw > G0 > Ta. With Brutsaert_2015 method, the contribution rates were Rn > Tw > G0 > u2 > VPD > Ta. Both Rn contributions were the largest and varied greatly, with most of the values in the range of −2 mm to 2 mm. The contributions of u2 and VPD fluctuated near ± 0.5 mm, and the contribution of Tw and G0 fluctuated slightly near ± 0.2 mm. Nonetheless, the contribution of Ta had no significant fluctuation, with values of ±0.05 mm and ±0.01 mm, respectively.

Figure 9 shows the change in accumulation values for the contribution of Rn, G0, Ta, u2, Tw, and VPD using two distinct methods. It was found that the sum of the change in ET_a was about 8.98 mm and 25.00 mm using the two methods. The estimation of ET_a by Brutsaert_2015 method was larger than that by Bouchet_1963method. The contribution of Rn played a different role in the two methods, which had a positive effect on ET_a in the summer soil-thawed period and a negative effect in the winter soil-frozen period with Bouchet_1963 method, while always showing a positive effect with Brutsaert_2015method. In addition, the contribution of G0, Ta, Tw, and VPD also had a positive effect in the thaw-freeze period, while u2 had the opposite effect with Bouchet_1963 method. The contribution of the factors, except for G0, had a positive effect on ET_a in the thaw-freeze period with Brutsaert_2015 method. Both of the maximum changes in ET_a occurred in the summer soil-thawed period, and the change in ET_a declined steeply in the autumn soil-freezing period, with a slight increase in the winter soil-frozen period with Brutsaert_2015 method and slight decline in the winter soil-frozen period with Bouchet_1963 method.

The mean contribution rates in different thaw-freeze periods are shown in Figure 10. It is clear that there were periodic changes in the contribution rate of the factors with Bouchet_1963 method. The contribution rates of Rn reached the maximum value in the summer soil-thawed period and gradually decreased in the autumn soil-freezing and winter soil-frozen periods. The changing trend of the contributions of Ta and Tw was the same with it, but the G0, u2, and VPD were opposed to it. By contrast, the contributions had no obvious periodic characteristics with Brutsaert_2015 method on a daily scale. During the thawing of the soil, the contribution of Rn, u2, and Tw increased, while VPD decreased, and Ta showed no noticeable change.

4. Discussion

The results show that the Brutsaert (2015) method, combined with the MOST, produced more reliable estimates of ET_a on the QTP. This aligns with findings by Ma et al. [15], which highlighted the improved performance of CR models under complex climatic regimes. The superiority of Brutsaert_2015 method in cold and arid regions is also supported by Szilagyi and Jozsa [46], which emphasized the relevance of atmospheric feedback under low-humidity conditions.

In contrast, Bouchet_1963 linear formulation was found to consistently underestimate ET_a during frozen periods and overestimate it during thawed periods. This seasonal mismatch has also been observed in other permafrost-dominated landscapes [16,48], where the fixed proportional coefficient ε in Bouchet_1963 method lacks sensitivity to changing surface and atmospheric conditions. Moreover, the analysis in this work provides a quantitative confirmation of these discrepancies, with RMSEs of 0.85–1.10 mm/day in frozen periods under Bouchet’s model.

Importantly, this study extends earlier work by comparing two wind functions (Rome and MOST) and showing their relative impact on ET_p estimation. The Rome function, developed for humid environments, failed to represent wind-driven vapor transfer under high-elevation, low-pressure settings of the QTP, leading to underestimation of ET_p and subsequent ET_a bias. Similar limitations were noted in Ma et al.’s work [49], where ET_a estimates based on the Rome function showed increasing divergence in dry seasons. According to the observations, the mean daily wind on the QTP was higher than 3.7 m/s, but the f(u) of the Rome function was underestimated when the wind speed was higher than 3.7 m/s [49]. The use of the MOST offered better adaptability to local micrometeorological conditions, enhancing CR performance, especially in frozen periods. These findings are consistent with results from Chen et al. [14], who found that MOST-based aerodynamic resistance improved land surface heat flux estimation across Tibetan alpine ecosystems.

However, the study has several limitations. First, the coefficient ε in Bouchet’s method was assumed to be constant, which may not reflect its temporal variability. Second, spatial heterogeneity in vegetation and soil properties was not fully considered.

Based on the observed ET_a, the daily ε could be calculated. Figure 11 shows the change in the daily ε in TGL, where it was found that the value of ε was different throughout the year. It was in the range of 1 to 10 in the cold season and 0 to 1 in the warm season. Due to a lack of observation on the QTP, it was difficult to decide what the value of ε in the climate and land surface condition is. In this paper, it was a constant that can minimize the RMSE without considering the difference of ε. Ma et al. [16] also optimized the value of ε in the Shanghu alpine steppe, which was less than 1 in the growing season in the Tibet Plateau. Therefore, the ET_a was underestimated by Bouchet_1963 method in spring and winter. By contrast, the ET_a simulated by Brutsaert_2015 method was close to the observed ET_a without considering the coefficient ε.

In future research, incorporating remote sensing-based surface parameters and dynamic ε calibration could enhance model adaptability and accuracy. Additionally, long-term datasets are needed to assess interannual variations and climate sensitivity of CR-based ETa models in cold regions.

The amplitude of the contribution of meteorological factors to ET_a fluctuated strongly on a daily scale, as shown in Figure 7 and Figure 8. By comparing the differences of two consecutive days as the variable amplitude in this work, a certain percentage (20%) of factors was usually regarded as a factor of change [48]. Based on the base value percentage amplitude method, the contributions of the meteorological factors to ET_a showed a relatively stable change trend, responding to the meteorological factors without fluctuating significantly (Figure 12). This could reflect the impact of the meteorological factors on ET_a, although it does not represent the true response to the change in the meteorological factors as this work does. The results based on the base value percentage amplitude method are consist with those based on the method of differences over two consecutive days.

In addition to its methodological contribution, this study addresses a gap in the existing literature by linking CR theory with freeze–thaw processes. Few studies have analyzed CR behavior across seasonal soil thermal dynamics. The freeze–thaw analysis in this work revealed clear seasonal transitions in factor sensitivity and contribution. It is conducive to understanding the relationship between the freeze–thaw process and evapotranspiration.

Overall, this study provides evidence that the Brutsaert + MOST combination is an effective approach to ET_a estimation in permafrost regions. However, challenges remain in capturing spatial heterogeneity and in dynamically parameterizing ε in Bouchet-type models. These limitations provide an idea for integrating remote sensing and land surface modeling to upscale ET_a estimates in the future.

5. Conclusions

This study confirmed the existence of a complementary relationship between ET_a and ET_p in the permafrost regions of the QTP. The ET_p computed with the MOST was consistently higher than that derived from the Rome wind function, especially under dry and frozen conditions. Quantitatively, the Brutsaert_2015 method combined with the MOST achieved the best performance, with RMSE values of 0.67–0.70 mm/day and correlation coefficients (r) of 0.81–0.86 (p < 0.05) across three stations.

Sensitivity analysis showed that ET_a was most responsive to the net radiation, with a daily sensitivity coefficient reaching up to 1.35. The contribution analysis revealed that the net radiation accounted for more than 80% of ET_a variation under Brutsaert’s method. These findings suggest that Brutsaert’s CR model, free from reliance on an empirical ε parameter, is more robust in cold alpine environments.

Nonetheless, uncertainties remain in parameter assumptions (e.g., constant ε, assumed neutral stability), and the reliability of results is contingent upon the quality and temporal resolution of EC observations. Future studies should consider dynamic parameterization and larger spatiotemporal validation to improve the applicability of CR-based ET_a models in cold and complex terrains.