The Influence of Meteorological Variables on Reference Evapotranspiration Based on the FAO P-M Model—A Case Study of the Taohe River Basin, NW China

Abstract

To explore the mechanisms driving variation in ET₀ (reference evapotranspiration) in an arid inland region of Northwest China, daily meteorological data from 1960 to 2019 from 19 meteorological stations in the Taohe River basin and its surrounding areas were used to analyze the temporal and spatial distributions of ET₀ and meteorological variables. Various qualitative and quantitative analysis methods were used to reveal the correlation between ET₀ and meteorological variables. The degree of sensitivity of ET₀ variations to meteorological variables and the contribution from each meteorological variable were clarified, and the mechanisms driving variation in ET₀ were fully revealed. These are the results: (1) ET₀ in the Taohe River basin presented a significant upward trend with a linear change rate of 0.93 mm/a, and a sudden change occurred in 1994. The spatial variation in ET₀ ranged from 779.8 to 927.6 mm/a, with low values in the upper and middle reaches and high values in the lower reaches. The ET₀ at 14 stations (73.68% of the total) was significantly increased (p < 0.05), and that at 5 stations (26.32% of the total) was not significantly increased (p > 0.05). (2) RH, R_n, and u₂ did not change significantly, while T_max and T_min showed a significant upward trend. (3) R_n is a meteorological variable closely related to variations in ET₀, and is the most sensitive variable for variations in ET₀, followed by T_max and u₂. (4) T_max is the meteorological variable that contributes most to the variation in ET₀ (30.98%), followed by T_min (29.11%), u₂ (6.57%), R_n (2.22%), and RH (0.05%). The research results provide a scientific basis for the rational and efficient utilization of water resources and the maintenance of ecosystem health.

1. Introduction

As an important link in the process of the hydrologic cycle, evapotranspiration promotes the continuous flux and conversion of water and energy in the three surface layers, the pedosphere, biosphere, and atmosphere [1], and constitutes a central link in the climate system. Variations in evapotranspiration have a strong impact on hydrological processes [2], and this is a popular field of multidisciplinary research in the context of climate change [3]. Reference evapotranspiration (ET₀) represents the evapotranspiration of a hypothetical grass reference crop with specific characteristics [4], which significantly affects the availability of water resources [5]. It is often used to estimate crop evapotranspiration [6,7] and develop effective irrigation plans to ensure agricultural productivity [8]. The multitype drought index considers ET₀ as an important input variable, which is the key to accurately monitoring and alerting to drought conditions [9]. In the context of global warming, increasing variations in ET₀ aggravate the frequency, intensity, and duration of drought [10]. Therefore, the study of ET₀ is of great significance for ensuring the water use efficiency of ecosystems, for efficiently utilizing water resources, and for monitoring the water and heat status of the surface [1,8]. Owing to the interactions among atmosphere–vegetation–surface systems, various factors such as meteorological conditions and underlying surface characteristics affect the variation in ET₀ [11,12]. According to the FAO P-M model, ET₀ is calculated by temperature, relative humidity, wind speed, sunshine duration, and geographical location [13,14]. Any change in these variables may trigger a change in ET₀, and the interaction among various meteorological factors is relatively complex [15,16]. In the context of global warming, it is helpful to quantitatively study the spatiotemporal variation in ET₀ and the interaction between meteorological factors, to explore the sensitivity of ET₀ to variation in different meteorological factors and to identify the influence of various climate variables to reveal the response mechanism of hydrological processes to climate change [17].

Some valuable research has been conducted on ET₀ estimation and its spatiotemporal variation, driving mechanisms, and influence [2,18,19,20,21]. Dinpashoh et al. [22] used the Mann–Kendall method and Sen’s estimator method to analyze monotonic trends in ET₀ in the Lake Urmieh basin. Di Nunno et al. [23] compared the reference evapotranspiration predicted by machine learning algorithms with the evapotranspiration calculated for two climate scenarios, RCP 4.5 and RCP 8.5, and predicted future trends in ET₀. Jerin et al. [24] used a modified Mann–Kendall test, wavelet analysis, and kriging models to study the spatiotemporal variation in ET₀ and used linear regression analysis and partial correlation coefficients to identify the factors controlling these variations. Fu et al. [25] constructed multiple linear regression equations for temperature, precipitation, sunshine duration, wind speed, and relative humidity with terrestrial evapotranspiration (ET) to determine climatic factors on ET. Jiang et al. [26] obtained the contribution of climate variables to ET₀ variation by multiplying the sensitivity coefficients of individual climate variables by their relative change. The relevant studies used statistical analysis, the Sobol method, sensitivity analysis, and contribution rate analysis together with a variety of analytical methods that are applicable for this research [15,26,27]. Considering the numerous factors affecting the variation in ET₀ and the complex interactions among meteorological factors, the combined application of multiple methods is very important to reveal the qualitative and quantitative relationship between ET₀ and meteorological factors. In this paper, various methods, such as correlation coefficient, grey correlation degree, path analysis, elasticity coefficient, and contribution rate calculations, are exploratively combined in a way that has rarely been previously performed. As a transitional basin between the Qinghai–Tibet Plateau and Loess Plateau and with its heterogeneity of climate and surface conditions, the Taohe River basin in China is one of the most sensitive regions to climate change [28], but little research has been done on ET₀ in the Taohe River basin [2,18,19,20,21,28]. By integrating qualitative analysis and quantitative analysis, the internal relationship and the degree of correlation between ET₀ variations and meteorological factors are analyzed in a more comprehensive and objective way. The key leading factors are identified, and the sensitivity of ET₀ variations to meteorological variables and the contribution from each meteorological variable are determined to reveal the mechanisms influencing ET₀ variations. This study can provide a new idea to reveal the mechanisms influencing ET₀ variations by combining qualitative and quantitative methods to predict more scientifically the impact of ET₀ variations on hydrological processes under climate change, and complement the regional studies on ET₀ in the Taohe River basin.

This research has three purposes: (1) Use the multivariate statistical method and Thiessen polygon weighted method to analyze the variations and change point in ET₀ and meteorological variables in a basin and use the spatial interpolation method to determine the spatial distribution of ET₀. (2) Based on qualitative analysis methods such as the correlation coefficient, grey correlation degree and path analysis, reveal close correlation between ET₀ variations and meteorological variables. (3) Using quantitative analysis methods such as the elasticity coefficient and contribution rate calculation, explore the sensitivity of ET₀ variations to meteorological variables and the contribution from each meteorological variable.

2. Materials and Methods

2.1. Study Area

The Taohe River is a first-level tributary on the right bank of the upper Yellow River. Originating from Xiqingshan, Qinghai Province, it flows into the Liujiaxia Reservoir in the upper main stream of the Yellow River, with a total length of 673 km and a watershed area of 25,527 km². Its geographical location is 101°36’~104°20’ E, 34°06’~36°01’ N (Figure 1). The upper reaches are above the Xiabagou hydrological station, the middle reaches extend from the Xiabagou hydrological station to Lijiacun hydrological station, and the lower reaches are below the Lijiacun hydrological station to Hekou hydrological station. The multiyear average annual temperature increases from 1 °C in the upper reaches to 9 °C in the lower reaches, while the multiyear average annual precipitation decreases from 650 to 300 mm [29]. The upper and middle reaches are cold and humid, while the lower reaches are warm and arid [28]. The basin contains abundant water resources with large amounts of runoff in the Gansu section of the Yellow River basin. The land spans three major geomorphic units: the plateau region in the upper reaches, the alpine valley region in the middle reaches, and the loess hilly–gully region in the lower reaches. The upper and middle reaches are 2000 m above sea level and have good vegetation coverage, while the lower reaches have low elevation, poor vegetation coverage, and severe soil and water erosion. The water and sand are from different sources, with water originating from the upper reaches and sand from the lower reaches [30].

2.2. Data Source

The daily meteorological data from 1960.1.1 to 2019.12.31 (21,915 days and 175,320 data points) were obtained from the China Meteorological Data Network (http://data.cma.cn, accessed on 12 June 2022); these included the daily mean temperature and maximum/minimum temperature, daily mean relative humidity, daily precipitation, daily mean wind speed at 10 m above ground, and sunshine duration from 19 meteorological stations in and around the basin. It is the daily values dataset containing basic meteorological elements from national ground-based meteorological stations in China (V3.0). The upper reaches in the plateau region and the middle reaches in the alpine valley region, with high altitude and complex topographic conditions, have fewer meteorological monitoring stations than the lower reaches in the loess hilly areas, which have relatively more ground monitoring stations. If only 8 stations within the basin were chosen to study ET₀, representing the whole basin with area 25,527 km², the accuracy of ET₀ in the Taohe River basin would be difficult to guarantee, so 19 meteorological stations were chosen in and around the basin. The observation data were produced by the National Meteorological Information Center after strict quality control and are widely used in scientific research [15,19,26]. The meteorological stations with less than 1% missing daily data were selected and the missing data points were interpolated based on a regression relationship with data from nearby stations. The seasons are divided into spring from March to May, summer from June to August, autumn from September to November, and winter from December to February of the following year.

2.3. Methods

2.3.1. Calculation of Reference Evapotranspiration (ET₀)

The FAO Penman-Monteith formula recommended by the United Nations Food and Agriculture Organization (FAO) [31,32,33,34] is used to calculate reference evapotranspiration. The FAO P-M method is based on energy balance and aerodynamic principles, and considers comprehensively the influence of temperature, radiation, wind speed, humidity, and regional location (altitude and latitude), which has been proven to have high calculation accuracy in different regions and under different climatic conditions [35,36]. The FAO P-M equation used to estimate ET₀ can be expressed as [31,32,33]:

$$\mathrm{E}{\mathrm{T}}_0=\frac{0.408\mathrm{\Delta }\left(R_{\mathrm{n}}-G\right)+\gamma \frac{900}{T+273}{u}_2\left(e_s-e_a\right)}{\mathsf{\Delta }+\gamma \left(1+0.34u_2\right)}$$

(1)

where ET₀ is the daily reference evapotranspiration (mm⋅day⁻¹); Δ is the slope vapor pressure curve (kPa⋅°C⁻¹); R_n is the net radiation (MJ⋅m⁻²⋅day⁻¹); G is the soil heat flux density (MJ⋅m⁻²⋅day⁻¹), which is ignored on the daily scale and therefore assumed to be zero [31]; γ is the psychrometric constant (kPa⋅°C⁻¹); u₂ is the wind speed at a height of 2 m (m⋅s⁻¹); T is the average air temperature (°C); e_s is the saturation vapor pressure (kPa); e_a is the actual vapor pressure (kPa); and (e_s − e_a) is the saturation vapor pressure deficit (kPa).

Allen et al. [31] proposed a formula for wind speed conversion at different elevations, converting wind speed at a height of 10 m into wind speed at a height of 2 m, as shown in Equation (2):

$$u_2=u_z\frac{4.87}{\log (67.8z-5.42)}$$

(2)

where u₂ is the wind speed at a height of 2 m (m⋅s⁻¹); u_z is the wind speed at a height of z m (m⋅s⁻¹); and z is the height of the anemometer above the ground. The expression for solar net radiation R_n is [31]:

$$\begin{array}{l}{R}_{\mathrm{n}}=R_{\mathrm{n}\mathrm{s}}-R_{\mathrm{n}\mathrm{l}}\\ R_{\mathrm{n}\mathrm{s}}=\left(1-\alpha \right)\left[a_s+b_s\left(\frac{n}{N}\right)\right]R_a\end{array}$$

(3)

where R_n is the net radiation (MJ⋅m⁻²⋅day⁻¹); R_ns is net shortwave radiation (MJ⋅m⁻²⋅day⁻¹); R_nl is net longwave radiation (MJ⋅m⁻²⋅day⁻¹) [31]; R_a is extraterrestrial radiation (MJ⋅m⁻²⋅day⁻¹); and the albedo α is 0.23. According to Zhu Changhan [37], the values of a_s and b_s in the Northwest region are 0.281 and 0.441, respectively. n is actual sunshine duration (h) and N is the daylight hours (h).

2.3.2. Calculation of the Elasticity Coefficient and Contribution Rate

ET₀ is a function of R_n, u₂, T_max, T_min, and RH in the FAO P-M model, which can be written as ET₀ = f(R_n,u₂,T_max,T_min,RH), and the total derivative of ET₀ [38] can be written as:

$$d\mathrm{E}{\mathrm{T}}_0\approx \frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial R_{\mathrm{n}}}d{R}_{\mathrm{n}}+\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial u_2}d{u}_2+\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial T_{\max }}d{T}_{\max }+\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial T_{\min }}d{T}_{\min }+\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial \mathrm{R}\mathrm{H}}d\mathrm{R}\mathrm{H}$$

(4)

According to the definition by Schaake [39], the elasticity coefficient of evapotranspiration to the variation in climate factors is expressed by the ratio of the change in evapotranspiration to the rate of change in climate factors. The formula is shown in Equation (5):

$${\epsilon }_x=\underset{\mathrm{\Delta }x/x\to 0}{\lim }\left(\frac{\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_0/\mathrm{E}{\mathrm{T}}_0}{\mathrm{\Delta }x/x}\right)=\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial x}\cdot \frac{x}{\mathrm{E}{\mathrm{T}}_0}$$

(5)

where ε_x is the elasticity coefficient of meteorological Factor x, and ∂ET₀/∂x is obtained by partial differentiation. The elasticity coefficients for each climate element are shown as follows [39]:

$${\epsilon }_{R_{\mathrm{n}}}=\frac{\partial E{T}_0}{\partial R_{\mathrm{n}}}\frac{R_{\mathrm{n}}}{\mathrm{E}{\mathrm{T}}_0},\hspace{1em}{\epsilon }_{u_2}=\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial u_2}\frac{u_2}{\mathrm{E}{\mathrm{T}}_0},\hspace{1em}{\epsilon }_{T_{\max }}=\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial T_{\max }}\frac{T_{\max }}{\mathrm{E}{\mathrm{T}}_0},\hspace{1em}{\epsilon }_{T_{\min }}=\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial T_{\min }}\frac{T_{\min }}{\mathrm{E}{\mathrm{T}}_0},\hspace{1em}{\epsilon }_{\mathrm{R}\mathrm{H}}=\frac{\partial \mathrm{E}{\mathrm{T}}_0}{\partial \mathrm{R}\mathrm{H}}\frac{\mathrm{R}\mathrm{H}}{\mathrm{E}{\mathrm{T}}_0}$$

(6)

Equation (7) [38] can be obtained by substituting Equation (6) into Equation (4), and the expression is:

$$\frac{d\mathrm{E}{\mathrm{T}}_0}{\mathrm{E}{\mathrm{T}}_0}\approx {\epsilon }_{R_{\mathrm{n}}}\frac{d{R}_{\mathrm{n}}}{R_{\mathrm{n}}}+{\epsilon }_{u_2}\frac{d{u}_2}{u_2}+{\epsilon }_{T_{\max }}\frac{d{T}_{\max }}{T_{\max }}+{\epsilon }_{T_{\min }}\frac{d{T}_{\min }}{T_{\min }}+{\epsilon }_{\mathrm{R}\mathrm{H}}\frac{d\mathrm{R}\mathrm{H}}{\mathrm{R}\mathrm{H}}$$

(7)

According to the concept of total differentiation, the variation in ET₀ can be considered to be caused by the variation in various climate factors; the degree of influence on the variation in climate factors (such as temperature, wind speed, humidity, etc.) on ET₀ is [38]:

$$\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_0=\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_{0R\mathrm{n}}+{\mathrm{\Delta }\mathrm{E}\mathrm{T}}_{{0u}_2}+{\mathrm{\Delta }\mathrm{E}\mathrm{T}}_{{0T}_{\max }}+{\mathrm{\Delta }ET}_{{0T}_{\min }}+{\mathrm{\Delta }ET}_{0\mathrm{R}\mathrm{H}}$$

(8)

$$\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_0={\epsilon }_{R_{\mathrm{n}}}\frac{\mathrm{\Delta }{R}_{\mathrm{n}}}{R_{\mathrm{n}}}\mathrm{E}{\mathrm{T}}_0+{\epsilon }_{u_2}\frac{\mathrm{\Delta }{u}_2}{u_2}\mathrm{E}{\mathrm{T}}_0+{\epsilon }_{T_{\max }}\frac{\mathrm{\Delta }{T}_{\max }}{T_{\max }}\mathrm{E}{\mathrm{T}}_0+{\epsilon }_{T_{\min }}\frac{\mathrm{\Delta }{T}_{\min }}{T_{\min }}\mathrm{E}{\mathrm{T}}_0+{\epsilon }_{\mathrm{R}\mathrm{H}}\frac{\mathrm{\Delta }\mathrm{R}\mathrm{H}}{\mathrm{R}\mathrm{H}}\mathrm{E}{\mathrm{T}}_0$$

(9)

The contribution rate of meteorological Factor x to ET₀ variation is shown in Equation (10) [38]:

$$\delta \mathrm{E}{\mathrm{T}}_{0x}=\frac{\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_{0x}}{\mathrm{\Delta }\mathrm{E}{\mathrm{T}}_0}\times 100\%$$

(10)

where δET_0x is the contribution rate of meteorological Factor x to ET₀ variation (%); ∆ET_0x is the contribution of meteorological Factor x to ET₀ variation; and ∆ET₀ is the variation in ET₀.

2.3.3. Trend and Change Point Analysis

In this paper, the linear regression method and the Mann–Kendall nonparametric statistical test method were used to analyze the statistical variations in time series for ET₀ and meteorological variables within a certain confidence interval [40]. Combining the Mann–Kendall test with the Pettitt test [41] and cumulative anomaly curve [42], the change points in the time series were analyzed. The Mann–Kendall test method is suitable and simple for hydrometeorological data that do not follow a normal distribution [43].

2.3.4. Correlation Analysis

According to the monthly time series of ET₀ and meteorological variables, the Pearson correlation coefficient and Spearman correlation coefficient were calculated [44,45]. Based on the grey system theory, ET₀ and meteorological variables are regarded as a grey system, in which ET₀ is used as the reference series and meteorological variables are used as the comparison series. The greater the weighted correlation value is, the higher the degree of similarity of geometric shapes of curves and the closer the connection [46]. Path analysis is used to analyze complex linear relationships among multiple variables, which is an extension of regression analysis. By determining the direct and indirect effects of independent variables on dependent variables, the interaction between variables is analyzed to provide a reliable basis for statistical decision-making [47].

3. Results and Discussion

3.1. Temporal and Spatial Variation Characteristics of ET₀

3.1.1. Interannual Variation in ET₀

ET₀ was calculated for each station based on meteorological data during 1960–2019 from 19 meteorological stations in and around the Taohe River basin. Site ET₀ was weighted based on the Thiessen polygon to obtain the basin ET₀ [48]. The interannual variation in the basin ET₀ is shown in Figure 2a. The time series of ET₀ showed a significant upward trend over 60 years, with a linear change rate of 0.93 mm/a, and the Mann–Kendall (M-K) statistic Z was 4.31, reaching a significance level of 0.05. The annual mean value of ET₀ was 819.46 mm, and the coefficient of variation was 3.59%. The maximum value during 1960–2019 was 884.52 mm in 2016, the minimum value during 1960–2019 was 760.15 mm in 1989, and the extreme value ratio was 1.16. The M-K test is shown in Figure 2b. The ET₀ time series jumped in 1994, which is consistent with the cumulative anomaly curve shown in Figure 2c. ET₀ decreased from 1960 to 1993 and increased from 1994 to 2019, when the mean value of ET₀ increased from 803.39 to 840.49 mm, with an increase of 4.62%. The ET₀ time series reached a significance level of 0.05 in 2000 and showed a significant increase.

3.1.2. Spatial Distribution of Average Annual Value and Annual Change Rate of ET₀

Kriging spatial interpolation was carried out based on the annual ET₀ for 60 years at 19 stations in and around the basin. The spatial distribution of the average annual value of ET₀ is shown in Figure 3a. The range of the basin ET₀ was 779.8–927.6 mm, and ET₀ presents an increasing variation from upstream to downstream. The upper and middle reaches of the basin were low-value areas with ET₀ ranging from 779.8 to 829.0 mm. The lower reaches were high-value areas with ET₀ ranging from 829.0 to 927.6 mm. The difference in the spatial distribution of ET₀ was affected by the regional geographic environment and climatic conditions. The basin spans three geomorphic units from the upper reaches to the lower reaches, namely, the plateau, alpine canyon area, and loess hilly area. The upper and middle reaches are cold and humid, while the lower reaches are warm and arid. The spatial distribution of the linear change rate in ET₀ is shown in Figure 3b. The change rate in ET₀ in the basin was between −0.48 and 2.95 mm/a. The change rate in the upstream and downstream was lower, which was between −0.48 and 1.20 mm/a. The change rate in the middle reaches was higher, which was between 1.20 and 2.95 mm/a. The ET₀ at 14 stations (73.68%) was significantly increased (p < 0.05). ET₀ at 5 stations (26.32%) was not significantly increased (p > 0.05), and they were distributed in areas with low change rates around the upstream and downstream basins, namely, Weiyuan Station, Linxia Station, Yongjing Station, Guanghe Station, and Henan Station. Therefore, ET₀ in the whole basin increased significantly except for Guanghe station.

3.1.3. Spatial Distribution of ET₀ over Four Seasons

The average annual values from 19 stations during four seasons were used for spatial interpolation. The spatial distribution of the average annual values of ET₀ for the four seasons is shown in Figure 4. The average annual values of ET₀ in spring, summer, autumn, and winter were 251.93, 325.04, 153.92, and 88.51 mm, respectively. The order was summer, spring, autumn and winter, with the largest value in summer and the smallest value in winter, and the value in summer was 3.67 times that in winter. In both spring and summer, ET₀ increased from the upper reaches to the lower reaches. The upper and middle reaches showed low values, ranging from 235.8 to 266.1 mm in spring and from 297.2 to 346.9 mm in summer, which increased by approximately 61.4 to 80.8 mm in summer compared with that in spring. The lower reaches showed high values, ranging from 266.1 to 296.3 mm in spring and from 346.9 to 396.6 mm in summer, which increased about 80.8~100.3 mm more in summer than in spring. In both autumn and winter, ET₀ decreased from south to north, with high values in the upper and middle reaches and low values in the lower reaches, ranging from 145.9 to 166.5 mm in autumn and from 73.0 to 104.6 mm in winter, which decreased by approximately 61.9 to 72.9 mm in winter compared with that in autumn. In general, the distributions in autumn and winter were similar, with the upper and middle reaches showing greater values and the lower reaches showing lower values. Since ET₀ in spring and summer accounted for 70.41% of the whole year, the spatial distribution of ET₀ in spring and summer was similar to the distribution of annual ET₀, with the upper and middle reaches showing lower values and the lower reaches showing larger values.

3.2. Variation Characteristics of Meteorological Factors Related to ET₀

The FAO P-M model can be written as ET₀ = f (R_n,u₂,T_max,T_min,RH), where R_n, u₂, T_max, T_min, and RH are the independent variables and ET₀ is the dependent variable. Meteorological data from 1960 to 2019 from 19 meteorological stations and the FAO P-M formula were used to calculate R_n and u₂. The basin R_n, u₂, RH, T_max, and T_min were obtained by weighting the Thiessen polygon. The interannual variation trends in these five variables are shown in Figure 5. In Figure 5, linear regression was used to reveal variations in meteorological variables over time, identifying trends and linear change rates in meteorological data from 1960–2019. RH did not change significantly (Z = −1.57), with an annual mean value of 65.19%. Compared with the annual average value of 65.76% during 1960–1993 according to the period separation shown in Figure 2, RH decreased annually by 1.33% during 1994–2019 (64.43%). R_n did not change significantly (Z = −0.48), with an average annual value of 3068.66 MJ⋅m⁻²⋅d⁻¹. Comparing the value of R_n during 1960–1993 (3066.80 MJ⋅m⁻²⋅d⁻¹) with that during 1994–2019 (3071.08 MJ⋅m⁻²⋅d⁻¹), R_n increased annually by 4.27 MJ⋅m⁻²⋅d⁻¹ during 1960–2019. u₂ did not change significantly (Z = −0.89), with an annual mean value of 1.27 m/s. Compared with the annual average value of 1.29 m/s during 1960–1993, u₂ showed an annual decrease of 0.03 m/s during 1994–2019 (1.26 m/s). T_max showed a significant upward trend (Z = 5.84), with an average annual value of 12.34 ℃, and the average annual value of T_max during 1960–1993 (11.82 ℃) was 1.21 ℃ lower than that during 1994–2019 (13.03 ℃). T_min showed a significant upward trend (Z = 7.65), with an annual average of −1.33 ℃. Compared with the annual average value of −1.82 ℃ during 1960–1993, T_min during 1994–2019 (−0.70 ℃) increased by 1.12 ℃. In conclusion, RH, R_n, and u₂ did not change significantly, while T_max and T_min showed a significant upward trend.

3.3. Qualitative Correlation Analysis between ET₀ and Meteorological Variables

The FAO P-M model shows that ET₀ is a function of R_n, u₂, T_max, T_min, and RH, and there is a certain correlation between the variables R_n, u₂, T_max, T_min, RH, and the dependent variable ET₀. The correlation coefficient, grey correlation degree, regression relationship curve, and path analysis were used to study the correlation between ET₀ and meteorological variables and to qualitatively reveal the statistical correlation degree between ET₀ and R_n, u₂, T_max, T_min, and RH.

3.3.1. Correlation Coefficient and Grey Correlation Degree between ET₀ and Meteorological Variables

The monthly time series of ET₀, R_n, u₂, T_max, T_min, and RH were used to calculate the correlation coefficient and grey correlation degree between ET₀ and each meteorological variable. ET₀ and R_n are monthly values, and u₂, T_max, T_min, and RH are monthly daily mean values. The calculation results are shown in

Table 1. Correlation between ET₀ and meteorological variables.

Correlation Coefficient	Pearson						Spearman						Grey Relational Degree
	ET0	RH	Rn	Tmax	Tmin	u2	ET0	RH	Rn	Tmax	Tmin	u2
ET0	1	0.490 **	0.989 **	0.936 **	0.892 **	0.447 **	1	0.489 **	0.982 **	0.941 **	0.890 **	0.486 **	1
RH		1	0.493 **	0.671 **	0.786 **	−0.169 **		1	0.490 **	0.67 4 **	0.788 **	−0.171 **	0.614
Rn			1	0.901 **	0.869 **	0.495 **			1	0.896 **	0.851 **	0.499 **	0.887
Tmax				1	0.977 **	0.245 **				1	0.974**	0.247 **	0.804
Tmin					1	0.211 **					1	0.198 **	0.736
u2						1						1	0.625

Note: ** 0.01 significant level.

. The order for the correlation coefficients between meteorological variables and ET₀ is R_n, T_max, T_min, RH, u₂, and the order for the grey correlation degree is R_n, T_max, T_min, u₂, RH. ET₀ had high consistency with the time series for R_n, T_max, and T_min, with Pearson’s coefficients of 0.989, 0.936, and 0.892; Spearman’s coefficients of 0.982, 0.941, and 0.890, and grey correlation degrees of 0.887, 0.804, and 0.736. R_n was strongly correlated with T_max and T_min, with Pearson’s coefficients of 0.901 and 0.869, and Spearman’s coefficients of 0.896 and 0.851, respectively. T_max was strongly correlated with T_min, with a Pearson’s coefficient of 0.977 and a Spearman’s coefficient of 0.974.

3.3.2. Regression Analysis between ET₀ and Meteorological Variables

The regression curves between ET₀ and R_n, u₂, T_max, T_min, and RH are shown in Figure 6. In Figure 6, linear regression was used to reveal the correlation between ET₀ and the meteorological variables. The more the data points are concentrated on a particular line and the closer R² is to 1.0, the stronger the correlations between ET₀ and the meteorological variables. ET₀ is closely related to R_n, T_max, and T_min. The linear regression of ET₀ and R_n is the most significant, and the determination coefficient R² is 0.978 (Figure 6a). Second, the determination coefficients R² between ET₀ and T_max and T_min are 0.876 (Figure 6b) and 0.795 (Figure 6c), respectively. There is no obvious regression relationship between ET₀ and RH or u₂. The regression relationship between ET₀ and various meteorological variables is consistent with the results of the correlation coefficient and grey correlation degree noted in Section 3.3.1, which shows that ET₀ has a close statistical correlation with the time series for R_n, T_max, and T_min, and poor consistency with the time series for RH and u₂.

3.3.3. Path Analysis of ET₀ by Meteorological Variables

Path analysis was conducted on the five meteorological variables R_n, u₂, T_max, T_min, and RH to determine the direct and indirect effects (effects through other factors) on ET₀. The path analysis results are shown in

Table 2. Path analysis of ET₀ by meteorological factors.

Meteorological Factors	Path Coefficient (Direct Effect)	Indirect Path Coefficient (Indirect Effect)						Simple Correlation Coefficient
		Rn	Tmax	RH	u2	Tmin	Σ
Rn	0.743		0.311	−0.057	−0.013	0.005	0.246	0.989
Tmax	0.345	0.670		−0.078	−0.006	0.006	0.591	0.936
RH	−0.116	0.366	0.232		0.004	0.005	0.607	0.490
u2	−0.026	0.368	0.084	0.020		0.001	0.473	0.447
Tmin	0.006	0.646	0.337	−0.091	−0.005		0.886	0.892

Notes: R_n, T_max, RH, u₂, and T_min represent net radiation, daily maximum temperature, relative humidity, wind speed at 2 m, and daily minimum temperature, respectively.

. The path coefficient reflects the direct effect of various meteorological variables on ET₀, and the order is R_n > T_max > RH > u₂ > T_min. R_n has the largest direct effect, while T_min has the smallest effect. The path coefficients of RH and u₂ are negative, indicating that RH, u₂, and ET₀ change in opposite, while R_n, T_max, T_min, and ET₀ change in the same direction. The indirect path coefficient reflects the indirect effect of each meteorological variable, and the order is T_min > RH > T_max > u₂ > R_n. The indirect effect of T_min is the largest, and the indirect effect of R_n is the smallest. The simple correlation coefficient comprehensively reflects the direct and indirect effects of each meteorological variable, and the order is R_n > T_max > T_min > RH > u₂, which is the same as the calculation result for the Pearson’s coefficient. R_n is a meteorological variable closely related to the variation in ET₀.

3.4. Quantitative Correlation Analysis between ET₀ and Meteorological Variables

3.4.1. Sensitivity of ET₀ to Variations in Meteorological Variables

Based on the FAO P-M model, the elasticity coefficients for the five meteorological variables at 19 meteorological stations were obtained by solving differential equations (Equations (4)–(6)). The inverse distance weight method was used to carry out spatial interpolation with the elasticity coefficients for each meteorological variable. The spatial distribution of the R_n, u₂, T_max, T_min, and RH elasticity coefficients is shown in Figure 7. The R_n (E-R_n) elasticity coefficient varies from 0.622 to 0.806, that is, if R_n increases by 10%, then ET₀ increases by 6.22%~8.06%. E-R_n increases from upstream to downstream, and the high value area is located in the middle and lower reaches, with E-R_n 0.765~0.806. The T_max (E-T_max) elasticity coefficient varies from 0.113 to 0.162, that is, if T_max increases by 10%, then ET₀ increases by 1.13%~1.62%. The spatial distributions of E-T_max and E-R_n are similar, and the high value range for E-T_max and E-R_n is between 0.153 and 0.162. The u₂ (E-u₂) elasticity coefficient ranges from −0.141 to −0.077, that is, if u₂ decreases by 10%, then ET₀ increases by 0.77%~1.41%. The upper and middle reaches in the basin are high value areas, ranging from −0.141 to −0.125. The T_min (E-T_min) elasticity coefficient varies from −0.064 to 0.034, and ET₀ varies from 0% to 0.64% if T_min increases by 10%. The RH (E-RH) elasticity coefficient varies from −0.0015 to −0.0008, that is, if RH decreases by 10%, then ET₀ increases by 0.008%~0.015%. The middle reaches in the basin are high value areas. In conclusion, the order in ET₀ elasticity coefficients caused by variations in meteorological variables is R_n, T_max, u₂, T_min, and RH. The Thiessen polygon weighted method was used to obtain the annual mean value for the basin. The maximum R_n elasticity coefficient was 0.722, followed by T_max (0.143), u₂ (−0.110), and T_min (−0.016), and the minimum value for RH was −0.001. ET₀ was the most sensitive to variations in R_n, followed by T_max and u₂. ET₀ was least sensitive to variations in T_min and RH.

3.4.2. Contribution Rate of Meteorological Variables to ET₀ Variation

The contribution rate of meteorological variables to the variation in ET₀ is shown in

Table 3. Contribution rate of meteorological variables to ET₀.

Meteorological Factors	Elasticity Coefficient	Average Annual Value	Average Annual Change	Annual Change in ET0/mm	Contribution Rate/%
Rn	0.722	3068.66 MJ⋅m−2⋅d−1	4.27 MJ⋅m−2⋅d−1	0.82	2.22
Tmax	0.143	12.34 °C	1.21 °C	11.49	30.98
Tmin	−0.016	−1.33 °C	1.12 °C	10.80	29.11
u2	−0.110	1.27 m/s	−0.03 m/s	2.44	6.57
RH	−0.001	65.19%	−1.33%	0.02	0.05
ET0	-	819.46 mm	37.10 mm	-	-

Notes: R_n, T_max, RH, u₂, and T_min represent net radiation, daily maximum temperature, relative humidity, wind speed at 2 m, and daily minimum temperature, respectively.

. The order for the contribution rate is T_max > T_min > u₂ > R_n > RH. As the ET₀ time series jumped in 1994, the period 1960–1993 is defined as the base period and the period 1994–2019 as the change period, which are shown in Figure 2b,c in Section 3.3.1. Compared with the base period (1960–1993), the average annual increase in ET₀ was 37.10 mm during the change period (1994–2019). The contribution rates of each meteorological variable to the variation in ET₀ were as follows: T_max 11.49 mm (30.98%), T_min 10.80 mm (29.11%), u₂ 2.44 mm (6.57%), R_n 0.82 mm (2.22%), and RH 0.02 mm (0.05%). T_max is the most important meteorological variable that promotes the increase in ET₀, followed by T_min, u₂, and R_n, and RH has the least influence. In the elasticity coefficient and contribution rate calculation, only RH, R_n, u₂, T_max, and T_min were considered, and the contributions of mean temperature, altitude, and latitude were not considered. In the process of solving the differential equations for the elasticity coefficient, R_n was a compound variable deduced from T_max and T_min. As R_n was taken as an independent variable, the influence of T_max and T_min on R_n was not considered. The total contribution from the five variables was 68.93%.

4. Discussion

Qualitative and quantitative methods are combined in a progressive and complementary relationship. The linear trend method and the Mann–Kendall test were used to analyze trends in the time series for ET₀ and meteorological variables. The Mann–Kendall test, Pettitt test, and cumulative anomaly curve were used to analyze the change points in the time series and the results were validated against each other to increase reliability. Correlation coefficients, grey correlation degree, regression curves, and pass analysis qualitatively revealed complex correlations between ET₀ and meteorological variables. Differential solution equations for meteorological variables affecting ET₀ were constructed on the basis of the FAO P-M model to identify the sensitivity of ET₀ to variations in meteorological variables and the contribution from each meteorological variable. The exploratory application of multiple methods, with qualitative and quantitative methods complementing each other, provides new theoretical ideas and analytical bases to fully reveal the influencing mechanisms of ET₀ variations.

ET₀ in the Taohe basin showed a significant increase, with a linear change rate of 0.93 mm/a, and a sudden change occurring in 1994. The results are consistent with the previous research conclusion [32] that ET₀ in the temperate continental zone and temperate monsoon zone of China showed an obvious upward trend from 1985 to 2015. The spatial distribution of ET₀ ranged from 779.8 to 927.6 mm, increasing from upstream to downstream, with low values in the upper and middle reaches and high values in the lower reaches. It is consistent with the conclusion [28] that “ET₀ gradually increases from upstream to downstream areas”. RH, R_n, and u₂ did not change significantly, while T_max and T_min showed a significant increase. Fan et al. [32] found that the maximum and minimum temperatures have increased significantly during 1956 to 2015 in China, which is generally consistent with the conclusion of this study.

The regression relationship curve is consistent with the results for the correlation coefficient. ET₀ is closely correlated with R_n, T_max, and T_min, but has poor consistency with the time series for RH and u₂. ET₀ was the most sensitive to variations in R_n, followed by T_max and u₂, and ET₀ is the least sensitive to the variation in T_min and RH. Yang et al. [28] showed that R_n was the most sensitive factor affecting ET₀ in the Taohe River basin from 1981 to 2010, which is consistent with the conclusion of this study. Compared with the base period (1960–1993), the order for the contribution of meteorological variables to ET₀ variation during the change period (1994–2019) was T_max, T_min, u₂, R_n, and RH. T_max was the most important meteorological variable (30.98%) that promoted an increase in ET₀, followed by T_min (29.11%), u₂ (6.57%), and R_n (2.22%); RH (0.05%) had the least influence. This is consistent with the conclusion of previous studies that “the dominant factor is the increase in ET₀ due to air temperature in the Taohe River basin from 1981 to 2010” [28]. Although ET₀ is closely related to R_n and is most sensitive to variations in R_n, the contribution of R_n is only 2.22%, which is related to whether R_n has a significant magnitude of change from 1960 to 2019, based on Equation (9). The contribution is influenced by both the sensitivity coefficient and the magnitude of change in the meteorological variables. In the elasticity coefficient and contribution rate calculations, only RH, R_n, u₂, T_max, and T_min were considered, and the contributions from other meteorological variables were not considered. Therefore, not all climatic and geographical environment conditions influencing ET₀ variations have been taken into account and the total contribution from the above five variables was 68.93%.

R_n is a compound variable deduced from T_max and T_min, but it was taken as an independent variable in the process of solving the differential equations for the elasticity coefficient, which may affect the calculation results. The estimation of ET₀ is available based on FAO P-M model on the basis of a large amount of meteorological data, but its implementation is still inconvenient due to its requirement for a large amount of meteorological data. Therefore, machine learning forecasting algorithms [33,49] for ET₀ estimation are highly desirable.

5. Conclusions

Based on the FAO P-M model, we selected ET₀ and five meteorological variables in the formula ET₀ = f(R_n,u₂,T_max,T_min,RH), and analyzed the spatiotemporal distribution of ET₀ and meteorological variables. Qualitative and quantitative analysis methods were used to reveal the correlation between ET₀ and various meteorological variables to clarify the sensitivity of ET₀ to variations in meteorological variables and the contribution from each meteorological variable. The main conclusions are as follows:

(1)

ET₀ in the Taohe basin showed a significant increase, with a linear change rate of 0.93 mm/a. The spatial distribution of ET₀ ranged from 779.8 to 927.6 mm, increasing from upstream to downstream. The ET₀ at 14 stations (73.68%) was significantly increased (p < 0.05), and that at 5 stations (26.32%) was not significantly increased (p > 0.05).

(2)

RH, R_n, and u₂ did not change significantly, while T_max and T_min showed a significant increase. The annual average values of RH, R_n, u₂, T_max, and T_min were 65.19%, 3068.66 MJ⋅m⁻²⋅d⁻¹, 1.27 m/s, 12.34 °C, and −1.33 °C, while the annual average change was ﹣1.33%, + 4.27 MJ⋅m⁻²⋅d⁻¹, −0.03 m/s, +1.21 °C, and +1.12 °C.

(3)

ET₀ is closely correlated with R_n, T_max, and T_min, but has poor consistency with the time series for RH and u₂. ET₀ was the most sensitive to variations in R_n, followed by T_max and u₂, and ET₀ is the least sensitive to the variation in T_min and RH.

(4)

Compared with the base period (1960–1993), the order for the contribution of meteorological variables to ET₀ increase during the change period (1994–2019) was T_max (30.98%), followed by T_min (29.11%), u₂ (6.57%), and R_n (2.22%), and RH (0.05%), and the total contribution from these five variables was 68.93%.