Common Calibration of Solar Radiation and Net Longwave Radiation Is the Key to Accurately Estimating Reference Crop Evapotranspiration over the Tibetan Plateau

Abstract

Reference crop evapotranspiration (ET₀) is crucial for water management. Although the FAO56-PM method is widely used to estimate ET₀, its input variables of solar radiation (R_s) and net longwave radiation (R_nl) are not readily available. Currently, mere calibration of the formula for R_s is assumed to be effective in improving FAO56-PM’s performance. To test this hypothesis, all input variables for FAO56-PM were measured in Lhasa, Linzhi, and Bange over the Tibetan Plateau (TP) to assess how different calculation methods affect ET₀ estimates. Compared to the original FAO56-PM, calibration of both R_s and R_nl formulas yielded the best model performance. Mere calibration of the formula for R_nl notably improved ET₀ estimation accuracy, while mere calibration of the formula for R_s reduced its accuracy. The general calibration model was slightly less effective than calibration of both R_s and R_nl formulas but obviously outperformed the original FAO56-PM. This model showed that ET₀ increased from east to west, ranging from 569.4 mm/year to 1118.5 mm/year. Trend analysis indicated rapid increases in ET₀ in the eastern region and significant decreases in the western region of the TP over recent decades. The findings are useful for the regional application of FAO56-PM to achieve sustainable development in Tibet.

1. Introduction

Evapotranspiration (ET) plays a crucial role in the hydrological cycle and water balance [1,2,3]. Accurate ET estimation is vital, serving as a key element in hydrological [4,5,6] and crop growth models [7,8,9]. Additionally, precise ET data is essential for developing optimal strategies in agricultural water management [10,11,12], and it is particularly important for crafting policies that promote sustainable agriculture in water-scarce areas such as the Tibetan Plateau [13].

ET can be measured directly using instruments like lysimeters or eddy covariance measurements [14,15], but these methods are laborious, time-consuming, and costly [16]. Currently, ET is typically estimated using the two-step approach recommended by the Food and Agriculture Organization (FAO). This method calculates ET as the product of reference crop evapotranspiration (ET₀) and the crop coefficient [13,17,18]. Therefore, accurate estimation of ET₀ is important for reliable ET calculations.

ET₀ is defined as the evapotranspiration rate from a surface of green grass with a uniform height of 8–15 cm. The grass is assumed to be actively growing, fully shading the ground, and free from water stress [19]. Numerous models have been developed to estimate ET₀ [20], which can be categorized into four main types [21]: combination-based models [17,22,23], radiation-based models [24,25,26], temperature-based models [27,28,29], and aerodynamic models [30,31,32]. Among these, the FAO-56 Penman–Monteith model [17] (FAO56-PM for short) is considered the most reliable for ET₀ estimation. It is often used as the standard for evaluating other models [21,33,34], and is applied globally [35,36,37].

Application of the FAO-56 PM method is often limited by its required input variables, which include maximum and minimum temperatures, water vapor pressure, wind speed, and radiation-related variables [17]. Among these, maximum and minimum temperatures, water vapor pressure, and wind speed are readily available at most weather stations (for example, all 2400 weather stations in China routinely measure these variables). However, radiation-related variables such as solar radiation (R_s) and net longwave radiation (R_nl) are measured at only a very few stations (for instance, R_s is measured at approximately 140 stations in China, while R_nl is only recorded in specific radiation observation experiments). To address this, the Angstrom formula is recommended for estimating R_s. The Allen formula, based on the Stefan–Boltzmann law, is used for estimating R_nl in the FAO-56 PM method [17].

The introduction of these two formulas has caused confusion regarding the standard ET₀ by FAO-56 PM. Some scientists believe that the standard ET₀ should be calculated strictly using the recommended formulas and default coefficients [21,33,35]. Conversely, many scientists believe that R_s, estimated using the recommended Angstrom formula and default coefficients, may not be reliable. They suggest that R_s should first be estimated using a calibrated Angstrom formula or other methods, and then the standard ET₀ by FAO-56 PM can be obtained using the estimated R_s. In this process, R_nl should be calculated strictly according to the recommended Allen formula and default coefficients [38,39,40].

However, Allen et al. [17] stated that “where measurements of incoming and outgoing short and longwave radiation during bright sunny and overcast hours are available, calibration of the coefficients in Equation (39) can be carried out” ([17], p. 52. Here, Equation (39) refers to the formula for net longwave radiation). This statement indicates clearly that all input variables, including both R_s and R_nl, should be measurements rather than estimates. In other words, the standard ET₀ by FAO-56 PM refers to the ET₀ calculated with all measured input variables, including measured R_s and R_nl. Using the FAO-56 PM with the recommended formulas and default coefficients should only be considered as a last resort when no measurements are available, and the results cannot be regarded as the standard ET₀ by FAO-56 PM. The formulas for both R_s and R_nl should be calibrated simultaneously whenever measurements are available [17].

Based on the arguments above, errors in ET₀ estimation will arise from both R_s and R_nl, if no calibration of the recommended formulas is performed. It is generally assumed that calibrating R_s would partly reduce these errors, a topic that has recently gained attention from researchers [41,42,43,44]. However, R_nl can also play an important role in energy balance in certain regions [45,46,47], especially in the alpine regions like the Tibetan Plateau, where high R_nl was identified [48]. In such climates, it is reasonable to hypothesize that errors in ET₀ estimation might decrease if the errors from the recommended formulas have the same sign for both R_s and R_nl to result in offsetting effects (i.e., both are overestimated or underestimated simultaneously). In this situation, studies focusing solely on calibrating R_s may not reduce, but rather increase, the errors in ET₀ estimation. Simultaneous calibration of both R_s and R_nl is the only effective method for accurate ET₀ estimation. However, up to now, few studies have addressed the simultaneous calibration of both R_s and R_nl, particularly in alpine regions like the Tibetan Plateau.

The Tibetan Plateau (TP), located in southwestern China, is the largest plateau in the country and the highest globally. Highland barley serves as the staple food for the Tibetan population, yet its cultivation is constrained by local water conditions [49]. Accurate data on evapotranspiration is crucial for the local government to develop effective irrigation plans. Despite having around 40 weather stations scattered over the TP, routine measurements of R_s are only conducted at Lhasa and Linzhi stations, while R_nl is not routinely measured at any station. However, R_nl has been recorded in Lhasa and Linzhi during experiments in recent years, and measurements of both R_s and R_nl were taken from 2014 to 2015 at Bange during the Third Scientific Expedition over the Tibetan Plateau.

In this study, for the first time, supported by complete data required for FAO56-PM, including measurements of both R_s and R_nl, we calibrated the recommended formulas for R_s and R_nl in FAO56-PM over the TP, aiming to (1) analyze errors in ET₀ arising from the recommended formulas for R_s and R_nl; (2) assess the significance of mere calibration of R_s for ET₀ estimation; and (3) comprehensively calibrate both R_s and R_nl for accurate ET₀ estimation over the TP. Ultimately, a general model was developed for regional application of FAO56-PM, which was used to calculate the spatial distribution and temporal trend of ET₀, in order to support sustainable agricultural development in this region.

2. Materials and Methods

2.1. Description of the Study Region

Situated in southwestern China, the TP is the world’s highest plateau, known for its complex terrain [50]. Approximately 40 weather stations are sparsely distributed across the region (Figure 1). These stations routinely measure basic meteorological variables such as sunshine duration, maximum and minimum temperatures, vapor pressure, and wind speed on a daily basis. Radiation-related variables, including both R_s and R_nl, are measured only during specific experiments or scientific expeditions at three stations: Lhasa, Linzhi, and Bange. The comprehensive data from these three stations were used for the calibration of FAO56-PM. The detailed information on the experimental sites and the weather stations is provided in

Table 1. Geographic characteristics for the experimental sites and weather stations.

Station	Latitude/N	Longitude/E	Altitude/m	Experimental Period/Recording Time
Lhasa	29.67	91.13	3648.9	20 October 1993 to 31 December 2020
Linzhi	29.59	94.25	2991.8	1 September 2015 to 31 December 2020
Bange	31.38	90.02	4700.0	1 August 2014 to 31 July 2015
Shiquanhe	32.50	80.08	4278.6	1 January 1991 to 31 December 2020
Gaize	32.15	84.42	4414.9	1 January 1991 to 31 December 2020
Anduo	32.35	91.10	4800.0	1 January 1991 to 31 December 2020
Nanqu	31.48	92.07	4507.0	1 January 1991 to 31 December 2020
Pulan	30.28	81.25	3900.0	1 January 1991 to 31 December 2020
Shenzha	30.95	88.63	4672.0	1 January 1991 to 31 December 2020
Dangxiong	30.48	91.10	4200.0	1 January 1991 to 31 December 2020
Lazi	29.08	87.60	4000.0	1 January 1991 to 31 December 2020
Nanmulin	29.68	89.10	4000.1	1 January 1991 to 31 December 2020
Rikaze	29.25	88.88	3836.0	1 January 1991 to 31 December 2020
Nimu	29.43	90.17	3809.4	1 January 1991 to 31 December 2020
Gongga	29.30	90.98	3555.3	1 January 1991 to 31 December 2020
Qongjie	29.03	91.68	3740.0	1 January 1991 to 31 December 2020
Zedang	29.25	91.77	3551.7	1 January 1991 to 31 December 2020
Nielamu	28.18	85.97	3810.0	1 January 1991 to 31 December 2020
Dingri	28.63	87.08	4300.0	1 January 1991 to 31 December 2020
Jiangzi	28.92	89.60	4040.0	1 January 1991 to 31 December 2020
Langkazi	28.97	90.40	4431.7	1 January 1991 to 31 December 2020
Cuona	27.98	91.95	4280.3	1 January 1991 to 31 December 2020
Longzi	28.42	92.47	3860.0	1 January 1991 to 31 December 2020
Pali	27.73	89.08	4300.0	1 January 1991 to 31 December 2020
Suoxian	31.88	93.78	4022.8	1 January 1991 to 31 December 2020
Biru	31.48	93.78	3940.0	1 January 1991 to 31 December 2020
Dingqing	31.42	95.60	3873.1	1 January 1991 to 31 December 2020
Leiwuqi	31.22	96.60	3810.0	1 January 1991 to 31 December 2020
Changdu	31.15	97.17	3306.0	1 January 1991 to 31 December 2020
Jiali	30.67	93.28	4488.8	1 January 1991 to 31 December 2020
Luolong	30.75	95.83	3640.0	1 January 1991 to 31 December 2020
Bomi	29.87	95.77	2736.0	1 January 1991 to 31 December 2020
Basu	30.05	96.92	3260.0	1 January 1991 to 31 December 2020
Jiacha	29.15	92.58	3260.0	1 January 1991 to 31 December 2020
Milin	29.22	94.22	2950.0	1 January 1991 to 31 December 2020
Zuogong	29.67	97.83	3780.0	1 January 1991 to 31 December 2020
Mangkang	29.68	98.60	3870.0	1 January 1991 to 31 December 2020
Chayu	28.65	97.47	2327.6	1 January 1991 to 31 December 2020

.

2.2. Data Collection

2.2.1. Data on Basic Meteorological Variables

The basic meteorological variables, such as sunshine duration, maximum and minimum temperatures, vapor pressure, and wind speed, have been recorded daily at all weather stations in China. These data were compiled by the National Meteorological Information Center (NMIC), a division of the China Meteorological Administration (CMA). NMIC has developed the Tianqin platform, featuring a stringent data quality control system to maintain measurement accuracy. For this study, NMIC provided daily meteorological data from 1991 to 2020. The abnormal records accounted for approximately 0.75%, and were removed from the data sets in this study.

2.2.2. Data on Radiation-Related Variables

Daily R_s has been measured routinely at Lhasa and Linzhi since 1961. R_nl has been measured at Lhasa from 20 October 1993 to 31 December 2020, and at Linzhi from 1 September 2015 to 31 December 2020. Both R_s and R_nl were measured simultaneously at Bange from 1 August 2014 to 31 July 2015, during the Third Scientific Expedition over the Tibetan Plateau (TSETP).

2.2.3. Datasets Used for This Study

Two datasets were created for the comprehensive calibration and regional application of FAO56-PM. For comprehensive calibration, the data from R_nl measurements for Lhasa, Linzhi, and Bange were used. These measurements, along with the corresponding R_s and basic meteorological variables, were combined into a complete dataset for FAO56-PM comprehensive calibration. The coefficients were calibrated using data on odd-numbered days in this dataset. For regional application, basic meteorological variables from 1991 to 2020 were compiled into a dataset for all stations across the TP.

2.3. Description of FAO56-PM

The widely used FAO56-PM is expressed as [17]:

$${\mathrm{E}\mathrm{T}}_0={\displaystyle \frac{0.408∆\left(R_n-G\right)+\left(900\gamma u_2\right(e_s-e_a)/T+273)}{∆+\gamma (1+0.34u_2)}}$$

(1)

where ET₀ represents the reference crop evapotranspiration (mm d⁻¹); Δ is the slope of saturated vapor pressure-temperature curve (kPa °C⁻¹); R_n is the net radiation (MJ m⁻² d⁻¹); G is the soil heat flux (MJ m⁻² d⁻¹); e_s and e_a are the saturated and actual vapor pressure (kPa), respectively; γ is the psychrometric constant (kPa °C⁻¹); T is mean air temperature (°C); u₂ is wind speed at 2 m height (m s⁻¹). For standardization, Allen et al. [17] defined T as the mean value of the daily maximum (T_max) and minimum temperature (T_min), rather than the average of the hourly temperature measurements.

$$T={\displaystyle \frac{T_{max}+T_{min}}{2}}$$

(2)

The slope of the saturated vapor pressure-temperature curve Δ can be expressed as:

$$∆={\displaystyle \frac{2504\exp \left(17.27T/(T+237.3)\right)}{{\left(T+237.3\right)}^2}}$$

(3)

At the daily scale, the soil heat flux G can be neglected [17], so G = 0 in the Equation (1). The psychrometric constant γ is provided by:

$$\gamma =0.665\times {10}^{-3}p$$

(4)

where p is atmospheric pressure (kPa), calculated as follows [17]:

$$p=101.3{\left({\displaystyle \frac{293-0.0065H}{293}}\right)}^{5.26}$$

(5)

where H is the altitude (m) of the study site. Wind speed at 2 m (u₂) can be calculated from the wind speed measured at a height of z m (u_z) using the following equation:

$$u_2={\displaystyle \frac{4.87}{\mathrm{l}\mathrm{n}(67.8z-5.42)}}{u}_z$$

(6)

The saturated vapor pressure e_s is the average of e_s (T_max) and e_s (T_min), calculated as:

$$e_s\left(t\right)=0.6108exp\left({\displaystyle \frac{17.27t}{t+237.3}}\right)$$

(7)

where t denotes either T_max or T_min.

For most weather stations in China, all input variables except R_n are readily available for calculating ET₀. The net radiation R_n is given by:

$$R_n=\left(1-\alpha \right)\times R_s-R_{nl}$$

(8)

where R_n, R_s and R_nl represent net radiation, solar radiation (or downward shortwave radiation), and net longwave radiation, respectively. The coefficient α denotes albedo, and Allen et al. [17] set α to 0.23 for the reference crop.

If both R_s and R_nl are available, the standard ET₀ can be calculated directly using FAO56-PM. However, obtaining R_s and R_ln, particularly R_nl, is often difficult at most weather stations. Therefore, various methods exist to obtain R_s and R_nl, leading to different calculation methods for FAO56-PM as follows.

2.4. Calculation Methods

2.4.1. AllenM Method

If the input variables for both R_s and R_nl are measurements, the ET₀ calculated using the FAO56-PM method is based on all measured input variables. In this case, the calculated ET₀ is referred to as the standard value for FAO56-PM, and this method is defined as AllenM.

2.4.2. AllenO Method

If the measurements of both R_s and R_nl are not available, Allen et al. [17] recommended two empirical formulas for estimating R_s and R_nl. R_s is typically estimated using the Angstrom formula (Hereafter referred to as F_s):

$${\displaystyle \frac{R_s}{R_0}}=a+b\times {\displaystyle \frac{S}{S_0}}$$

(9)

where R₀ represents extraterrestrial radiation, S is actual sunshine hours, and S₀ is potential sunshine hours. In the absence of measured data for calibration, Allen et al. [17] suggested default values of 0.250 for coefficient a and 0.500 for coefficient b, respectively. The FAO56-PM model for calculating net longwave radiation is formulated as follows (Hereafter referred to as F_nl):

$$R_{nl}=\sigma \left({\displaystyle \frac{{T_{max,k}}^4+{T_{min,k}}^4}{2}}\right)\left(c-d\sqrt{e_a}\right)\left(e{\displaystyle \frac{R_s}{R_{s0}}}-f\right)$$

(10)

where σ represents the Stefan–Boltzmann constant (4.895 × 10⁻⁶ MJ m⁻² d⁻¹ K⁻⁴); T_max,k and T_min,k denote the absolute maximum and minimum temperature in Kelvin, respectively. R_s stands for the solar radiation, while R_s0 indicates the calculated clear-sky solar radiation, typically derived from the Angstrom model under full sunshine conditions, where S = S₀. The coefficients 0.34, 0.14, 1.35 and 0.35 are the default values for the coefficients of c, d, e and f, respectively, as recommended by Allen et al. [17]. This method is actually the original FAO56-PM, which is defined as AllenO.

2.4.3. AllenS Method

Considering the reliability of R_s, some scientists have focused on calibrating F_s in an effort to improve the accuracy of the FAO56-PM model. Mere calibration of F_s is defined as AllenS.

2.4.4. AllenL Method

In contrast to AllenS, mere calibration of F_nl without calibration of F_s is defined as AllenL.

2.4.5. AllenC Method

In contrast to AllenS and AllenL, comprehensive calibration of both F_s and F_nl is defined as AllenC.

2.4.6. AllenR Method

The calibrated coefficients for F_s and F_nl are site-specific, necessitating local calibration using measurements of both R_s and R_nl. This requirement makes it challenging, if not impossible, to apply AllenC on a regional scale. So, we developed a general method allows for the easy derivation of coefficients using geographic or basic meteorological parameters, facilitating its regional application over the TP.

Based on our previous study on estimating regional distribution of solar radiation over the TP [51], the coefficients for F_s over the TP can be calculated from the following functions.

$$\left(a+b\right)=0.106\times \ln \left(H\right)-0.060$$

(11)

$$b=0.373\times {\displaystyle \frac{1}{\mu }}+0.483$$

(12)

where µ is the average daily water vapor pressure (hPa). Based on all measured R_nl at Lhasa, Linzhi and Bange, a set of coefficients for F_nl can be obtained by calibrating F_nl against these measurements. This method can be easily applied at a regional scale and is referred to as AllenR.

2.5. Model Evaluation

The performance of these models was evaluated using root mean square error (RMSE), which measures random error; mean absolute percentage error (MAPE), which measures the error relative to the standard value; mean bias error (MBE), which measures systematic error; and coefficient of determination (R²), which measures the model fitting effect [52].

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{(C_i-M_i)}^2}$$

(13)

$$MAPE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|{\displaystyle \frac{C_i-M_i}{M_i}}\right|\times 100\mathrm{\%}$$

(14)

$$MBE={\displaystyle \frac{1}{n}}\sum _{i=1}^n(C_i-M_i)$$

(15)

$${ R}^2={\displaystyle \frac{{\left[{\sum }_{i=1}^n\left(M_i-\overline{M}\right)\left(C_i-\overline{C}\right)\right]}^2}{{\sum }_{i=1}^n{\left(M_i-\overline{M}\right)}^2{\sum }_{i=1}^n{\left(C_i-\overline{C}\right)}^2}}$$

(16)

where M_i is the measured value; C_i is the calculated value; $\overline{M}$ and $\overline{C}$ are the average of the measured and calculated values, respectively, and n is the total sample number.

3. Results

3.1. Errors in Estimated ET₀ Resulting from Recommended F_s and F_nl

First, the standard ET₀ was calculated under the AllenM method, i.e., all of the input variables were measurements. Errors in the estimated ET₀, arising from recommended F_s (or F_nl), were identified by replacing the measured R_s (or R_nl) with the values calculated using the recommended F_s (or F_nl).

3.1.1. Errors in Estimated ET₀ Resulting from Recommended F_s

First, R_s was calculated using F_s. Then, the estimated R_s, along with all other measured data, including the measured R_nl, was used to drive the FAO56-PM. It is evident that the recommended F_s performed well in estimating R_s at all three stations, with R² values of 0.742, 0.816, and 0.860 for Lhasa, Linzhi, and Bange, respectively (Figure 2). The negative values of MBE at all stations indicated that the recommended F_s, with its default coefficients, underestimated R_s over the TP. These errors in estimated R_s led to errors in estimated R_n, with negative MBE values of −0.728, −1.136, and −2.126 at Lhasa, Linzhi, and Bange, respectively (Figure 2). The errors in estimated R_n were further transmitted to errors in estimated ET₀, with negative MBE values of −0.124, −0.176, and −0.344 at Lhasa, Linzhi, and Bange, respectively (Figure 2). Overall, the recommended F_s with its default coefficients underestimated R_s at all stations, resulting in underestimated R_n and ET₀ accordingly.

3.1.2. Errors in Estimated ET₀ Resulting from Recommended F_nl

First, R_nl was estimated using F_nl with its default coefficients. Then, the estimated R_nl, along with all other measured data, including the measured R_s, was used to drive the FAO56-PM model. The recommended formula for R_nl performed poorly, with R² values of 0.294, 0.551, and 0.654 for Lhasa, Linzhi, and Bange, respectively (Figure 3). F_nl significantly underestimated R_nl at all stations, with MBE values of −4.464, −3.649, and −3.974 for Lhasa, Linzhi, and Bange, respectively. As R_n was calculated as the difference between (1 − α)R_s and R_nl, i.e., (0.77R_s − R_nl), and R_s was the measured data, the underestimated R_nl led to an overestimated R_n, which in turn resulted in an overestimated ET₀ (Figure 3).

3.1.3. Errors in Estimated ET₀ Resulting from Both Recommended F_s and F_nl

First, R_s and R_nl were calculated using F_s and F_nl, respectively. Then, these estimated values, along with other measured variables, were used to drive the FAO56-PM. This approach is essentially the AllenO method, which is the original FAO56-PM with its recommended formulas and default coefficients for estimating R_s and R_nl.

The errors in R_n (Figure 4) were due to both the estimated R_s (Figure 2) and the estimated R_nl (Figure 3). R_s was underestimated by F_s (Figure 2), while R_nl was even more significantly underestimated by F_nl (Figure 3). The larger negative MBE for R_nl was partly offset by the smaller negative MBE for R_s, resulting in a smaller MBE for R_n (Figure 4). Overall, F_nl caused a greater underestimation of R_nl compared to the underestimation of R_s by F_s. When both R_s and R_nl were estimated, the errors were offset partially, rather than accumulating, thus reducing the errors in R_n and increasing the accuracy in ET₀.

3.2. Calibration of FAO56-PM

The original FAO56-PM, AllenO method, produced errors in the estimated ET₀ due to its recommended F_s and F_nl with the default coefficients (Figure 4). Therefore, calibrating F_s and F_nl in AllenO became crucial for enhancing the performance of FAO56-PM.

3.2.1. Mere Calibration of the Recommended F_s

Calibration of only F_s was referred to as AllenS, which involved determining the coefficients a and b for F_s, as shown in

Table 2. Coefficients in the empirical formulas of F_s and F_nl under different calculation methods.

Method	Site	Fs		Fnl
		a	b	c	d	e	f
AllenO	All stations	0.250	0.500	0.340	0.140	1.350	0.350
	Lhasa	0.280	0.500	0.340	0.140	1.350	0.350
AllenS	Linzhi	0.304	0.471	0.340	0.140	1.350	0.350
	Bange	0.304	0.563	0.340	0.140	1.350	0.350
	Lhasa	0.250	0.500	0.630	0.296	0.673	−0.149
AllenL	Linzhi	0.250	0.500	0.280	0.076	0.997	−0.324
	Bange	0.250	0.500 500	0.846	0.345	0.507	−0.054
	Lhasa	0.280	0.500	0.630	0.296	0.673	−0.149
AllenC	Linzhi	0.304	0.471	0.280	0.076	0.997	−0.324
	Bange	0.304	0.563	0.846	0.345	0.507	−0.054
	Lhasa	0.253	0.556	0.582	0.264	0.721	−0.149
AllenR	Linzhi	0.258	0.530	0.582	0.264	0.721	−0.149
	Bange	0.238	0.598	0.582	0.264	0.721	−0.149

. Compared to the original F_s (Figure 2), the calibrated F_s provided better estimates of R_s, evidenced by higher R² values (Figure 5). Additionally, the calibrated F_s showed no significant discrepancies between measured and estimated R_s, with acceptable MBE values of 0.027, −0.101 and 0.206 for Lhasa, Linzhi and Bange, respectively (Figure 5). However, the original F_nl’s tendency to underestimate R_nl could not be compensated by the accurate estimation of R_s, resulting in a larger absolute MBE in R_n compared to AllenO (Figure 5 vs. Figure 4). In summary, while AllenS improved the accuracy of R_s estimation, it diminished the error offset between F_s and F_nl, thereby increasing errors in estimating R_n and ET₀. Consequently, despite the effort put into the calibration process, AllenS performed worse than AllenO.

3.2.2. Mere Calibration of the Recommended F_nl

Calibration of only F_nl was referred to as AllenL. Table 2 shows the calibrated coefficients of c, d, e, and f for F_nl under AllenL. The calibrated F_nl outperformed the original F_nl, as demonstrated in Figure 2 and Figure 6, with higher R² values of 0.319, 0.577, and 0.701, respectively. Compared to the original F_nl, the calibrated version significantly reduced systematic error in the estimated R_nl, achieving lower absolute MBE values of 0.016, 0.009, and 0.041 for Lhasa, Linzhi and Bange, respectively. After calibration of F_nl, errors in R_n were primarily due to F_s. Error analysis above indicated that the original F_nl was the main source of errors in R_n estimation, whereas F_s contributed comparatively minor errors (Figure 2 and Figure 3). Consequently, AllenL performed significantly better than AllenO. For AllenL, the system errors in R_n were notably small, with MBE values of −0.744, −1.145, and −2.167 for Lhasa, Linzhi and Bange, respectively. This enhanced accuracy in estimating R_n also improved the estimation of ET₀ at all stations (Figure 6).

3.2.3. Comprehensive Calibration of Both F_s and F_nl

Only calibration of F_s or F_nl led to high accuracy in estimation of R_s or R_nl, respectively (Figure 5 and Figure 6). Thus, the comprehensive calibration of both F_s and F_nl, referred to as AllenC, resulted in small systemic errors in both the estimated R_s and the estimated R_nl (Figure 5 and Figure 6), leading to high accuracy in estimating R_n (Figure 7). Compared to AllenS and AllenL, AllenC produced smaller absolute MBE values for R_n at all stations (Figure 7). This high accuracy in R_n estimation also led to precise ET₀ estimations, with MBE values of −0.004 at Lhasa, −0.011 at Linzhi, and 0.019 at Bange, respectively (Figure 7). In summary, AllenC significantly improved the accuracy of R_n and ET₀ estimations compared to AllenO, AllenS, and AllenL methods.

3.3. General Method Developed for Regional Application of FAO56-PM over the TP

AllenC performed well in estimating R_n and ET₀, but the calibrated coefficients for F_s and F_nl were site-specific, necessitating local calibration using measurements of both R_s and R_nl. In contrast, the coefficients of F_s and F_nl can be easily obtained by the general method of AllenR.

The coefficients for F_s and F_nl were calculated under the AllenR method, as shown in Table 2, and used to estimate R_s and R_nl (Figure 8). Compared to the errors in estimated R_s and R_nl using the original F_s and F_nl (Figure 2 and Figure 3), the errors using AllenR’s coefficients decreased significantly (Figure 8), indicating that AllenR outperformed the original FAO56-PM (AllenO) for estimating R_s and R_nl. This high accuracy in estimating R_s and R_nl contributed to the high accuracy in estimating R_n, which in turn led to accurate ET₀ estimates (Figure 8). Naturally, the estimated R_s by AllenR was slightly less accurate than that by AllenS with local calibration, and the estimated R_nl by AllenR was also slightly less accurate than that by AllenL with local calibration (Figure 8 vs Figure 5 and Figure 6), which resulted in slightly larger errors in R_n and ET₀ compared to the AllenC method. However, AllenR is still considered the most suitable method for regional ET₀ estimation over the TP due to its higher accuracy than AllenO and broader applicability compared to AllenC.

3.4. Spatial Distribution of ET0 over the TP

Meteorological data from 40 weather stations across the TP were used to estimate the regional ET₀ using the AllenR method. From 1991 to 2020, the average annual ET₀ ranged from 569 to 1118 mm, increasing from east to west, with some high values centered in Basu, Gongga, and Lazi (Figure 9). Within a year, the high monthly values predominantly occurred from April to September (Figure 9), coinciding with the growth period of highland barley in most areas of the TP.

The estimated ET₀ using AllenR was set as the base value (V_b), while estimates from other calibration methods were set as the compared values (V_c). The relative error in ET₀ due to other methods was calculated as (V_c − V_b)/V_b × 100%. According to AllenO’s calculations, ET₀ was significantly overestimated, with the relative error decreasing from east to west, ranging from 12.6% to 29.6% (Figure 10a). The current popular method, AllenS, only calibration of F_s, produced a similar distribution pattern for the relative error, but the error increased, ranging from 18.8% to 36.2% (Figure 10b). Conversely, calibrating F_nl alone led to an underestimation of ET₀, with the relative error ranging from −12.7% to −0.3%. The most significant underestimation occurred in southeastern Tibet, decreasing from southeast to northwest (Figure 10c).

3.5. Trend Analysis of ET₀ over the TP

Trend analyses of ET₀ at Lhasa from 1994 to 2020 were conducted using various calibration methods (Figure 11). When both F_s and F_nl were locally calibrated, AllenC outperformed all other methods. Calibrating only F_nl, the AllenL method performed worse than AllenC but significantly better than AllenO. The widely used AllenS method, which calibrates only F_s, performed even worse than AllenO. In contrast, AllenR performed slightly worse than AllenC but better than all other methods (Figure 11). Interestingly, the trends (slopes of regression lines) showed minimal differences among the calibration methods, with values of 9.1 mm/year for AllenM, 7.6 mm/year for AllenO, 7.5 mm/year for AllenS, 7.2 mm/year for AllenL, 7.2 mm/year for AllenC, and 7.3 mm/year for AllenR, respectively (Figure 11). Overall, different calibration methods led to significant variations in estimated ET₀ but had only a minor impact on trend values.

The spatial distribution of trends in ET₀ from 1991 to 2000 was analyzed using the AllenR method. In the eastern and central parts of the TP, ET₀ increased rapidly compared to other areas. Conversely, ET₀ decreased in the western part of the TP over the past three decades (Figure 12a). Overall, ET₀ showed increasing trends with an average value of 2.8 mm/year, ranging from −2.2 to 10.4 mm/year across the TP.

The AllenO method resulted in overestimated trends in ET₀ in most regions of the TP (Figure 12b). The absolute error, calculated as (V_c − V_b), averaged 0.51 mm/year, with values ranging from −0.4 to 1.6 mm/year (Figure 12b). AllenS produced a similar distribution of absolute error, ranging from −0.6 to 1.6 mm/year, with the largest errors occurring in the southeastern TP (Figure 12c). Conversely, AllenL led to a distinctly different distribution pattern of absolute error (Figure 12d). Trends in ET₀ were underestimated in eastern TP but overestimated in western TP, with values ranging from −0.3 to 0.7 mm/year and an average of 0.1 mm/year across the TP.

4. Discussion

Various calibration methods led to discrepancies in the estimation of ET₀. The widely used AllenS method, which involves only calibration of F_s, actually increased, rather than decreased, errors in the estimated ET₀ over Tibet. These findings and their implications are discussed below.

4.1. High Rs and Rnl over the TP

The TP is known as the Third Pole of the world because of its high elevation [53,54,55]. The energy balance over the TP is notably distinct from that of the surrounding regions [56,57,58].

Radiation over the TP is significantly higher than that in surrounding regions [59,60,61], due to its high elevation and clear skies. The average daily radiation showed notably high values at Lhasa, Linzhi, and Bange (Figure 13), with mean values of 19.96 MJ m⁻² d⁻¹, 15.59 MJ m⁻² d⁻¹ and 20.61 MJ m⁻² d⁻¹, respectively. These radiation values are much higher than those measured at other stations with similar latitudes [62,63,64,65,66].

R_nl represents the difference between outgoing longwave radiation and incoming longwave radiation. Outgoing longwave radiation over the TP is slightly lower than that in surrounding areas due to the region’s lower temperature caused by high elevation [67]. In contrast, incoming longwave radiation is significantly lower due to clear sky conditions and low humidity in this highland area [68]. Consequently, R_nl over the TP is much higher than that in surrounding regions [69], which was also manifested in our study (Figure 13). The average R_nl values were 9.41, 6.62, and 8.17 MJ m⁻² d⁻¹ at Lhasa, Linzhi, and Bange, respectively (Figure 13), which are much higher than those in plains at similar latitudes [48,70,71]. Therefore, R_nl plays a crucial role in R_n over the TP [69], indicating that calibrating F_nl is essential for accurately estimating R_n and ET₀ in this highland region.

4.2. Offsetting Effects Between Errors in Estimated R_s and R_nl

The original F_s and F_nl models were developed using data primarily collected from flat regions [17]. In contrast to the flat regions, the transmissions of solar radiation are approximately 10–15% greater over the plateau than those at sea-level stations because of its high altitude, thin air and good air transparency [72]. Therefore, the coefficients of b over the TP are usually greater than those in the other regions [41,73,74]. Thus, using F_s without calibration results in an underestimation of R_s over the TP [60], a finding corroborated in this study (Figure 2). Similarly, applying F_nl directly leads to an even greater underestimation of R_nl (Figure 3). In the FAO56-PM model, R_n is calculated as 0.77R_s minus R_nl. Consequently, the larger underestimation of R_nl partially offsets the smaller underestimation of R_s, leading to an overestimation of R_n at each station (Figure 4). The widely used AllenS method, which involves mere calibration of F_s, reduces this offsetting effect and further increases the overestimation of R_n (Figure 5). In contrast, mere calibration of F_nl (AllenL method) means that the primary errors in R_n estimation will mainly stem from F_s. Compared to F_nl, F_s contributes relatively smaller errors to R_n (Figure 2 and Figure 3), resulting in higher accuracy in estimating R_n and ET₀ (Figure 6). Comprehensive calibration of both F_s and F_nl (AllenC method) yields very small errors in estimating both R_s and R_nl, leading to the best model performance in estimating R_n and ET₀ (Figure 7). This offsetting effect can be intuitively observed in the errors of estimated R_s, R_nl, R_n and ET₀ under different calculation methods (Figure 14).

4.3. Spatial Distribution and Temporal Trend of ET₀ over the TP

According to AllenR, the spatial distribution of ET₀ across TP increases from east to west, with notable high centers at Basu, Gongge, and Lazi (Figure 9). Previous studies using the original FAO56-PM (AllenO method) reported a very similar pattern [13]. Additionally, calibrating only F_s (AllenS) also produced a distribution pattern comparable to those of AllenR and AllenO [75]. However, compared to AllenR, AllenO resulted in a relative error range of 12.6% to 29.6%, while AllenS had an even higher error range of 18.8% to 36.2% over the TP (Figure 10a,b). These large estimation errors in ET₀ could lead to impractical water management policies implemented by local governments [39].

Trend analysis by AllenR shows that ET₀ has increased rapidly in the eastern part and decreased significantly in the western part of the TP (Figure 10a). This distribution pattern is similar to findings by AllenO [13] and AllenS [38,75,76], suggesting that different calibration methods of FAO56-PM may significantly affect the estimated ET₀ values but have little impact on the trends in recent decades (see Figure 11). Therefore, we cautiously envisage that calibration methods have minimal influence on the results of ET₀ trend analysis. Consequently, previous trend analyses by AllenO or AllenS may still be reliable [38,75,76], despite large errors in the estimated ET₀ values [13].

4.4. Uncertainties and Future Studies

ET₀ estimated by FAO56-PM is typically considered as the standard value for evaluating the performance of other empirical models in comparative studies (e.g., [21,33,34]). However, this study reveals that ET₀ calculated using FAO56-PM is not a fixed value but varies depending on different calculation methods. Consequently, the standard ET₀ used in previous studies may lead to confusion when assessing model performance. For instance, if the ET₀ values are 1.0 mm/day, 2.0 mm/day, and 3.0 mm/day for AllenO, AllenS, and AllenM, respectively, and empirical models A, B, and C produce ET₀ values of 1.0 mm/day, 2.0 mm/day, and 3.0 mm/day, respectively, model C would actually perform the best among the three models. However, if ET₀ from AllenO or AllenS is used as the standard value (e.g., [33,34]), the evaluation results could differ significantly. Therefore, the standard ET₀ used in past studies might lead to unreliable conclusions about model performance. Future research should reevaluate these findings.

In recent years, uncertainty analysis of ET₀ under climate change scenarios has become a significant research focus [77,78,79]. Typically, ET₀ calculated using the FAO56-PM method serves as a benchmark for assessing uncertainties from Global Climate Models (GCMs) and climate change scenarios [80,81,82]. However, as discussed above, ET₀ derived from FAO56-PM can vary depending on the calculation method used. When calculated using the original FAO56-PM (AllenO), the maximum relative error in ET₀ estimation can reach up to 29.6% (Figure 10a). Consequently, the benchmark itself is uncertain, and this uncertainty has not been accounted for in current analyses [80,81,82]. To mitigate this issue, the benchmark for ET₀ should be determined using FAO56-PM with comprehensive calibration of both F_s and F_nl. Additionally, ET₀ plays a crucial role in various drought indices [83,84,85], hydrological models [86,87,88], and crop growth models [7,8,9]. The outcomes of these indices or models may be affected by different calculation methods for FAO56-PM, a factor that should be addressed in future research. Furthermore, we observed that FAO56-PM is often briefly mentioned in related literature without details on the description of the data for R_s and R_nl (e.g., [80,82]). Given that the estimated ET₀ is significantly influenced by the calculation methods for R_s and R_nl, we strongly recommend that all authors clearly specify the methods used for calculating R_s and R_nl in future studies involving FAO56-PM.

As revealed in this study, R_nl plays a very important role in estimating R_n, a finding also identified in other studies [45,46,89]. However, compared to the accurate estimation of R_s by the calibrated F_s (Figure 3), F_nl performed worse in estimating R_nl over the TP in this study, even with calibrated coefficients (Figure 4). Our findings are consistent with previous reports [47,90], which indicated that R_nl was also underestimated in other locations. Therefore, simultaneous calibration of both R_s and R_nl based on local conditions is crucial for accurately estimating R_n and ET₀ [89,91,92]. In addition, Kofronova et al. [90] used different empirical models to estimate R_nl, showing that empirical models for F_nl exhibited large RMSE values with calibrated coefficients, which agrees well with our findings. Therefore, new methods should be developed for accurate estimation of R_nl to improve the model performance of FAO56-PM in future studies, incorporating current popular approaches based on various machine learning models [41,93,94,95]. Additionally, though the FAO56-PM is considered the most reliable method for calculating ET₀, direct validation of the FAO56-PM against observed measurements from lysimeters and similar instruments is also crucial. This should be an important scientific task for future work over the Tibetan Plateau.

5. Conclusions

Both R_s and R_nl values are significantly higher over the TP compared to surrounding regions. The formulas for R_s and R_nl recommended in the original FAO56-PM led to underestimations of both values, with R_nl being more significantly underestimated than R_s.

Comprehensive calibration of the formulas for both R_s and R_nl led to the best performance of the FAO56-PM. The general calibration model, using coefficients derived from readily available geographical and basic meteorological parameters, was slightly less effective than local calibration of the formulas for both R_s and R_nl, but obviously outperformed the original FAO56-PM, demonstrating significant regional applicability over the TP. Compared to the original FAO56-PM, mere calibration of the formula for R_nl notably improves accuracy in estimating ET₀, whereas only calibration of the formula for R_s reduces, rather than improves, the accuracy due to the diminished offsetting effect between underestimated R_s and R_nl.

The general model calculated that ET₀ increased from east to west, ranging from 569.4 mm/year to 1118.5 mm/year. Other calculation methods showed a similar spatial distribution pattern, with relative errors in ET₀ ranging from 12.6% to 29.6% under the original FAO56-PM. Trend analysis revealed that ET₀ increased rapidly in the eastern region while decreasing significantly in the western region of the TP over recent decades. Interestingly, although different calculation methods affect the estimated values of ET₀, they have minimal impact on the trend analysis results.