A Performance Evaluation of Nine Potential Evapotranspiration Methods Against the FAO-56 Penman–Monteith Benchmark at the Broadleaf Forest of Taxiarchis in Northern Greece ^†

Abstract

Potential evapotranspiration (PET) is a critical component of the water cycle, driving plants’ growth and survival. This study focused on estimating the daily potential evapotranspiration (PET) in a forest site in Northern Greece and assessing the performance of nine empirical PET estimation methods. These methods, categorized into mass-transfer, temperature-based, and radiation-based models, were compared against the widely used FAO-56 Penman–Monteith benchmark. The results highlight significant seasonal and monthly variations in vegetation water requirements. Among the methods tested, radiation-based models, particularly the Makkink equation, outperformed the others, followed by the Turc and Priestley–Taylor models. Temperature-based methods showed moderate performance and could serve as viable alternatives in forests with limited data availability, though local calibration is advisable.

1. Introduction

Potential evapotranspiration (PET) plays a crucial role in understanding plants’ water needs and is vital for various assessments in hydrology, biometeorology, and environmental management. Despite its importance, direct measurement of PET is challenging and requires costly equipment and continuous maintenance.

Most existing PET estimation methods rely on data from urban or agricultural areas [1,2], with limited testing in forest environments [3,4], urban forests, or green spaces [5]. This scarcity may stem from the difficulty of installing and maintaining meteorological stations in forested areas.

The Penman–Monteith method, recommended by the FAO [6], is widely regarded as the most accurate method but has high data requirements. Consequently, alternative models with varied scientific approaches and reduced data needs have been proposed. Continuous testing of these models across different environments is essential, as performance can vary significantly [7,8,9]. Local calibration may also be necessary [10].

This study aimed to estimate the PET in a mountainous Mediterranean forest in Northern Greece using the Penman–Monteith (FAO-56 PM) method as a benchmark. Additionally, it evaluated nine widely used empirical methods, based on mass transfer, temperature, and radiation, to identify suitable alternatives for environments with limited data availability.

2. Study Site, Data, and Methods

This study was conducted in the university forest site of Taxiarchis, located on Mt. Cholomontas in Northern Greece. Managed by the Aristotle University of Thessaloniki since 1934, this coppice oak forest spans approximately 5800 hectares, with elevations ranging from 320 to 1165 m above sea level. Part of the forest is designated as a Natura 2000 site (GR1270001). The region experiences a sub-humid Mediterranean climate, characterized by brief droughts, hot summers, and mild winters.

An automated micrometeorological station, equipped with high-precision sensors, was installed within the forest at an altitude of 860 m above sea level (40.43° N, 23.50° E) by the Laboratory of Mountainous Water Management and Control of the Aristotle University of Thessaloniki. Daily meteorological data from 1 July 2012 to 31 August 2019 were utilized for this study. Detailed information about the site can be found in the studies by Stefanidis and Alexandridis [11] and Stefanidis et al. [12].

This study employed empirical models for estimating potential evapotranspiration (PET), categorized into mass-transfer, temperature-based, and radiation-based methods. Three widely used models per category were selected and compared against the PET estimates derived from the FAO-56 Penman–Monteith model (considered the benchmark method). The equations for these models utilize basic meteorological data, with the analytical expressions detailed in

Table 1. Potential evapotranspiration (PET) empirical methods per category.

PET Method/Category	Equation *	Eq. No.	References
Benchmark model
FAO56-PM	$\mathrm{P}\mathrm{E}\mathrm{T}={\displaystyle \frac{0.408∆\left({\mathrm{R}}_{\mathrm{n}}-\mathrm{G}\right)+\mathsf{\gamma }\frac{900}{\mathrm{T}+273} \mathrm{u} ({\mathrm{e}}_{\mathrm{s}}-{\mathrm{e}}_{\mathrm{a}})}{∆+\mathsf{\gamma }(1+0.34 \mathrm{u})}}$	(1)	[6]
Mass-transfer models
Dalton, 1801	$\mathrm{P}\mathrm{E}\mathrm{T}=\left(3.648+0.7223\mathrm{u}\right)({\mathrm{e}}_{\mathrm{s}}-{\mathrm{e}}_{\mathrm{a}})$	(2)	[15]
Penman, 1948	$\mathrm{P}\mathrm{E}\mathrm{T}=\left(2.625+1.3812 \mathrm{u}\right)({\mathrm{e}}_{\mathrm{s}}-{\mathrm{e}}_{\mathrm{a}})$	(3)	[16]
Szász, 1973	$\mathrm{P}\mathrm{E}\mathrm{T}=0.00536 {\left(\mathrm{T}+21\right)}^2{\left(1+{\displaystyle \frac{\mathrm{R}\mathrm{H}}{100}}\right)}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\left(0.0519 \mathrm{u}+0.905\right)$	(4)	[17]
Temperature-based models
Thornthwaite, 1948	${\mathrm{P}\mathrm{E}\mathrm{T}=16\left({\displaystyle \frac{10 \mathrm{T}}{\mathrm{I}}}\right)}^{\mathrm{a}}{\displaystyle \frac{\mathrm{N}}{360}}$	(5)	[18,19]
Hargreaves and Samani, 1985	$\mathrm{P}\mathrm{E}\mathrm{T}=0.0023{\left({\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{T}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}^{0.5}\left(\mathrm{T}+17.8\right) {\mathrm{R}}_{\mathrm{a}}$	(6)	[20]
Droogers and Allen, 2002	$\mathrm{P}\mathrm{E}\mathrm{T}=0.0013{\left({\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{T}}_{\mathrm{m}\mathrm{i}\mathrm{n}}-0.0123\mathrm{P}\mathrm{R}\right)}^{0.76}\left(\mathrm{T}+17\right) {\mathrm{R}}_{\mathrm{a}}$	(7)	[21]
Radiation-based models
Makkink, 1957	$\mathrm{P}\mathrm{E}\mathrm{T}=0.61{\displaystyle \frac{\mathsf{\Delta }}{\mathsf{\Delta }+\mathsf{\gamma }}}{\displaystyle \frac{{\mathrm{R}}_{\mathrm{s}}}{\mathsf{\lambda }}}-0.12$	(8)	[22]
Turc, 1961	$\mathrm{P}\mathrm{E}\mathrm{T}=\left\{\begin{array}{c}0.013 \left({23.9 \mathrm{R}}_{\mathrm{s}}+50\right){\displaystyle \frac{\mathrm{T}}{\mathrm{T}+15}} , \mathrm{for} \mathrm{RH}>50\\ 0.013 \left({23.9 \mathrm{R}}_{\mathrm{s}}+50\right){\displaystyle \frac{\mathrm{T}}{\mathrm{T}+15}} \left(1+{\displaystyle \frac{50-\mathrm{R}\mathrm{H}}{70}}\right), \mathrm{for} \mathrm{RH}\le 50\end{array}\right.$	(9)	[23]
Priestley and Taylor, 1972	$\mathrm{P}\mathrm{E}\mathrm{T}=1.26{\displaystyle \frac{\mathsf{\Delta }}{\mathsf{\Delta }+\mathsf{\gamma }}}{\displaystyle \frac{{\mathrm{R}}_{\mathrm{n}}-\mathrm{G}}{\mathsf{\lambda }}}$	(10)	[24]

* where T, T_max, and T_min are the daily mean, maximum, and minimum air temperatures in °C; RH is the relative humidity in %; Rs and Rn are the global solar and the net radiation fluxes in MJ m⁻² day⁻¹; G is the soil heat flux in MJ m⁻² day⁻¹ (G ≈ 0); Δ is the slope of the vapor pressure curve (kPa °C⁻¹); γ is the psychrometric constant (kPa °C⁻¹); e_s and e_a are the saturation and actual vapor pressures in kPa; u is the windspeed at a height of 2 m in m/s; λ = 2.501−0.002361 Τ in MJ kg⁻¹; R_a is the extraterrestrial radiation in mm day⁻¹; a = 6.75 × 10⁻⁷ I³ − 7.71 × 10⁻⁵ I² +1.7912 × 10⁻² I + 0.49239; $\mathrm{I}=\sum _1^{12}{\left(0.2 \mathrm{T}\right)}^{1.514}$; N represents the maximum sunshine daily hours; and PR is the monthly precipitation in mm.

. Further information regarding parameter estimation can be found in the studies by Allen et al. [6] and Proutsos et al. [5,13,14].

A total number of 2289 daily values was analyzed, and the results were assessed through several statistical indices, including the slope a and the offset b of the linear regression line y = ax + b, the coefficient of determination R², the mean bias error (MBE), the variance of the differences distribution sd², the mean absolute error (MAE), the root-mean-square error (RMSE), and the index of agreement (d). Based on the above indices, the tested models were ranked by applying the standardization procedure suggested by Aschonitis et al. [25] and also described by Proutsos et al. [5], and the standardized ranking performance index (sRPI) was calculated for each method.

3. Results and Discussion

The FAO-56 PM model was used to calculate the daily PET values for the mountainous forest site of Taxiarchis. The distribution of the PET confirmed its expected seasonal variability typical of Mediterranean climates [1,5]. On an annual basis, the PET averaged 2.308 mm/d (sd = 1.554), varying seasonally from 0.746 mm/d (sd = 0.422) in winter to 4.097 mm/d (sd = 1.080) in summer, with intermediate values in spring (2.479 mm/d; sd = 1.138) and autumn (1.625 mm/d; sd = 0.981), based on the data of the period 2012–2019. The monthly PET values showed even greater variation, ranging from 0.629 mm/d (sd = 0.430) in December to 4.382 mm/d (sd = 1.111) in July.

The vegetation’s water needs, indicated by the PET estimated by the FAO-56 PM model, varied notably compared with the other methods. Figure 1 illustrates the comparison of the nine PET methods against the benchmark, along with the correlation statistics.

Table 2. Mean values of PET, regression line (y = ax + b) statistics (slope a, intercept b, and coefficient of determination R²), mean bias error (MBE), root-mean-square error (RMSE), mean absolute error (MAE), standard deviation square (sd²), and index of agreement d for the nine tested empirical PET modes compared with the FAO56-PM benchmark method in the forest site of Taxiarchis. Shadowed cells indicate the best value for each statistical index.

PET Category/Method	N	Mean	a	b	MBE	RMSE	MAE	sd2	d	R2	sRPI
Benchmark method
FAO56-PM	2289	2.308
Mass-transfer methods
Dalton, 1801	2098	3.105	1.26	−0.01	0.63	1.50	0.98	1.85	0.85	0.68	0.126
Penman, 1948	2098	2.706	1.15	−0.14	0.23	1.31	0.89	1.67	0.91	0.65	0.411
Szász, 1973	2081	2.854	1.13	0.04	0.36	1.26	0.94	1.46	0.89	0.67	0.374
Temperature-based methods
Thornthwaite, 1948	2289	2.099	0.90	0.02	−0.21	0.82	0.63	0.63	0.95	0.76	0.662
Hargreaves and Samani, 1985	2289	2.696	0.93	0.54	0.39	0.77	0.60	0.44	0.92	0.83	0.542
Droogers and Allen, 2002	2277	2.489	0.96	0.26	0.17	0.74	0.55	0.52	0.96	0.81	0.730
Radiation-based methods
Makkink, 1957	2268	2.120	0.91	0.01	−0.21	0.49	0.37	0.19	0.97	0.92	0.934
Turc, 1961	2289	2.482	1.09	−0.03	0.17	0.58	0.43	0.31	0.97	0.91	0.879
Priestley and Taylor, 1972	2289	2.553	1.10	0.01	0.24	0.56	0.44	0.25	0.98	0.93	0.872

displays the regression line coefficients and various statistical indices used to assess the models’ performances. Figure 1 illustrates the comparison of the nine PET methods against the benchmark, along with the correlation statistics.

The assessment of the nine empirical PET methods reveals that radiation-based equations generally provide more accurate estimates compared with other methods, consistent with the findings from previous studies [4,5,7]. Specifically, at our site, the Makkink equation showed the best performance for the RMSE (0.49), MAE (0.37), and sd² (0.19) and the closest average PET value (2.120 mm/d) to the benchmark method (2.308 mm/d), outperforming all other methods. Similarly, the Priestley and Taylor PET performed best for three statistical indices: offset b of the regression line (0.01), d (0.98), and R² (0.93). Among the temperature-based methods, Droogers and Allen’s method exhibited the best slope a value (0.96) and the best MBE (0.17), while all mass-transfer models had overall less good statistics compared with all other categories. In general, temperature-based methods offer a viable alternative for PET estimation in sites with limited data availability. This is also supported by the findings of Bourletsikas et al. [3], who tested 24 PET methods in a central Greek forest. The combined evaluation of all statistical indices yielded the sRPI score per method, depicted in Table 2, providing an overview of the model’s overall performance.

Radiation-based methods outperformed temperature-based and mass-transfer methods, yielding more accurate PET estimates with sRPI values ranging from 0.916 to 0.945, followed by temperature-based methods (sRPI: 0.600 to 0.778) and mass-transfer methods (sRPI: 0.137 to 0.382). The Makkink equation stood out as the best method for the forest site (sRPI = 0.945), while the Dalton equation performed the poorest (sRPI = 0.137).

The average annual PET varied between overestimations of +8% to +35% for most methods, except Thornthwaite’s and Makkink’s, which were slightly underestimated by −9% and −8%, respectively. Four of the nine methods (Thornthwaite’s, Droogers and Allen’s, Makkink’s, and Turc’s methods) closely matched the average FAO-56 PM PET, differing by an absolute value of less than 10%.

Performance varied across seasons. The Makkink method consistently underestimated PET, producing slightly smaller seasonal averages (−6% to −10%), while the Thornthwaite method underestimated in winter (−40%) and spring (−32%) but only slightly overestimated in summer (+3%) and autumn (+9%), resulting in an annual −9% underestimation.

Dalton’s equation performed well only in spring (+5%), indicating unsuitability for other seasons. Penman’s method showed good performance in spring (−7%) and summer (+9%) but highly overestimated in winter (+69%) and autumn (+70%). Szász’s model performed well in spring but overestimated in other seasons. Hargreaves and Samani’s method consistently overestimated in all seasons (+17% annually), highlighting the need for local calibration.

Turc’s method performed well across all seasons, except for autumn when it overestimated the PET. Priestley and Taylor’s method provided accurate estimates in autumn but overestimated in spring and summer and underestimated in winter. This model’s performance aligned with previous findings, ranking first among 48 temperature-based methods in urban environments [5].

4. Conclusions

Potential evapotranspiration (PET) is crucial for understanding forest water needs, particularly in the Mediterranean climate. However, studies on PET in forests are limited due to sparse data from forest meteorological stations. In this study, we utilized accurate meteorological data from a Northern Greek forest site spanning 2012–2020 to estimate the daily PET using the Penman–Monteith method (FAO-56 PM). We analyzed 2632 daily values and tested nine empirical PET estimation methods, categorized into mass-transfer, temperature-based, and radiation-based models, against the FAO-56 PM benchmark. The results showed an average daily PET of 2.279 mm/d (sd = 1.542), with significant seasonal and monthly variability due to the Mediterranean climate. Radiation-based methods outperformed temperature-based and mass-transfer methods, with the Makkink equation exhibiting the best performance across all seasons. Temperature-based models showed a moderate performance and are recommended with local calibration.