# Parametric Modelling of Potential Evapotranspiration: A Global Survey

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## Abstract

**:**

## 1. Introduction

_{0}, which refers to the evapotranspiration from a standardized vegetated surface (i.e., actively growing and completely shading grass of 0.12 m height, surface resistance 70 s · m

^{−1}, and albedo = 0.23). The globally accepted method for consistent estimation of PET is the Penman-Monteith (herein referred to as PM) equation, as formalized by the Food and Agriculture Organization (FAO), which is physically-based, and is therefore used as standard for comparisons with other, more simple approaches [6]. The major drawback for the generalized application of the PM method worldwide is the need of simultaneous measurements of four meteorological variables (air temperature, wind speed, relative humidity, and net radiation or, alternatively, sunshine duration), at the desirable spatial and temporal resolution.

_{0}). The meteorological inputs and ET

_{0}data are retrieved by the FAO CLIMWAT database that provides monthly climatic information at 4300 locations worldwide. A preliminary analysis of these data allowed explaining the major drivers of PET over the globe and across seasons. We perform extended analysis of the model inputs and outputs, including the production of global maps of optimized model parameters and associated performance metrics, as well as comparisons with a widely known formula by Hargreaves and Samani [26]. Finally, we use the interpolated values of the optimized parameter values to validate the predictive capacity of the model against detailed meteorological data, in terms of monthly time series, at several stations worldwide. The results are very encouraging, since even with the use of abstract climatic information for its calibration, the model generally ensures very reliable PET estimations. However, we have detected few cases where the model systematically fails to reproduce the reference PET, particularly across tropical areas. Except for these specific areas, the parameter estimations through the derived maps can be directly employed within the proposed formula, at both point and regional scales.

## 2. Theoretical Background

#### 2.1. The Penman-Monteith Equation

^{−1}); R

_{n}is the net incoming daily radiation at the vegetated surface (MJ · m

^{−2}· d

^{−1}); G is the soil heat flux (MJ · m

^{−2}· d

^{−1}); ρ

_{a}is the mean air density at constant pressure (kg · m

^{−3}); c

_{a}is the specific heat of the air (MJ · kg

^{−1}· °C

^{−1}); r

_{a}is an aerodynamic or atmospheric resistance to water vapour transport for neutral conditions of stability (s · m

^{−1}); r

_{s}is a surface resistance term (s · m

^{−1}); v

_{a}* − v

_{a}is the vapour pressure deficit of the air (kPa), defined as the difference between the saturation vapour pressure v

_{a}* and the actual vapour pressure v

_{a}; λ is the latent heat of vaporization (MJ · kg

^{−1}); Δ is the slope of the saturation vapour pressure curve at specific air temperature (kPa · °C

^{−1}); and, γ is the psychrometric constant (kPa · °C

^{−1}). Given that the typical time scale of the PM equation is daily, all of the associated fluxes are expressed in daily or mean daily units.

_{n}, as the sole energy term to be assessed; the latter is defined as the difference between incoming and outgoing radiation of short and long wavelengths.

_{a}*, and hence a lower vapour pressure difference v

_{a}* – v

_{a}, and lower reference evapotranspiration estimates [30].

#### 2.2. The Parametric Formula

_{a}(kJ · m

^{−2}) is the extraterrestrial radiation, T (°C) is the mean air temperature, and a (kg · kJ

^{−1}), b (kg · m

^{−2}), and c (°C

^{−1}) are model parameters that should be inferred through calibration, against “reference” PET data, either modelled or measured. We remark that from a macroscopic point-of-view, the above parameterization has some physical correspondence to the PM equation, since the product a R

_{a}represents the overall energy term (i.e., incoming minus outgoing solar radiation), parameter b represents the missing aerodynamic term, while quantity (1 − c T) is an approximation of the denominator term of the PM formula [24].

_{a}, and temperature, T, and thus it belongs to the so-called radiation-based approaches. The extraterrestrial radiation, defined as the solar radiation received at the top of the Earth’s atmosphere on a horizontal surface, is an astronomic variable, given by:

_{sc}is the solar constant, with typical value 82 kJ · m

^{−2}· min

^{−1}, d

_{r}is the inverse relative distance of the Earth from the Sun, ω

_{s}(rad) is the sunset hour angle, φ is the latitude (rad), and δ is the solar declination (rad). Variables d

_{r}and δ are periodic functions of time, while ω

_{s}is function of latitude and time. For details on computing the above astronomic variables, the reader may refer to the literature (e.g., [3]).

_{min}) and maximum temperature (T

_{max}) values are available.

#### 2.3. Modified Parametric Model

## 3. The CLIMWAT Database: Preliminary Analysis

#### 3.1. Database Overview

^{−1}), daily sunshine duration (h), daily solar radiation (MJ · m

^{-2}), monthly rainfall, gross and effective (mm), as well as mean monthly ET

_{0}estimations through the Penman-Monteith formula.

#### 3.2. Which Meteorological Drivers Explain Mean Annual PET over the Globe?

_{0}) data against the four meteorological variables that are embedded in the Penman-Monteith equation, i.e., solar radiation, mean temperature estimated from Equation (4), relative humidity and wind speed, at the annual scale, and fitted the most suitable regression model.

_{0}over the globe is highly correlated with mean annual solar radiation and temperature, particularly when considering power-type or exponential regression functions. As expected, mean annual ET

_{0}is negatively correlated with mean relative humidity, while it seems uncorrelated to wind speed. It is worth mentioning that as the solar radiation and temperature increase, the variance of ET

_{0}increases significantly. Therefore, in order to reduce heteroscedasticity effects, it is essential considering at least two explanatory variables in the context of empirical PET modelling.

_{0}against mean annual sunshine duration and annual extraterrestrial radiation, which are typically used instead of solar radiation, in PET estimations (given that solar radiation observations are generally sparse, due to the cost of associated equipment, i.e., pyranometers, radiometers or solarimeters). Surprisingly, the mean annual sunshine duration is slightly less correlated with mean annual ET

_{0}than extraterrestrial radiation, although the former is expected to be better estimator of the actual solar energy received in the Earth’s surface. This is a very important conclusion that confirms the suitability of radiation-based approaches, using both temperature and extraterrestrial radiation as explanatory variables of PET. However, it is essential remarking that the overall driver of PET and temperature as well is net solar radiation, which is a portion of the extraterrestrial one. Furthermore, the correlation between PET and temperature is so much significant only at coarse time scales, such as the annual one, while its correlation with the solar radiation remains significant, at all temporal scales [40].

#### 3.3. How Well Do Extraterrestrial Radiation and Temperature Explain the Seasonal Patterns of PET?

_{a}, and temperature, T. In general, a loop-type shape exists between the mean monthly PET and the two aforementioned variables, due to the influence of thermal inertia, causing a delay in temperature changes against solar radiation changes across seasons. Apparently, due to the loop-shape relationship, the two pairs of variables are expected to be linearly correlated; actually, the more elongate the loop, the higher should be the correlation. In Figure 3, we plotted the relationships between monthly extraterrestrial radiation vs. mean monthly ET

_{0}and mean monthly temperature vs. ET

_{0}, at five characteristic stations in Australia, exhibiting different hydroclimatic conditions, which confirm the above hypothesis. However, there are also cases where the shapes of T − ET

_{0}and R

_{a}− ET

_{0}loops are irregular (nonconvex), thus resulting in very low, even negative, correlations. Such examples are shown in Figure 4, involving another set of stations in Australia.

_{0}over the globe, we formulated the linear regression models of mean monthly ET

_{0}against the two variables and calculated the coefficient of determination, r

^{2}(i.e., square of Pearson correlation coefficient), at the full sample of 4300 CLIMWAT stations. Table 1 summarizes the results, by means of number of stations corresponding to ranges of r

^{2}, from 0–10% up to 90–100%. It is shown that ET

_{0}exhibits very high linear correlation, by means of r

^{2}values greater than 0.90 against both extraterrestrial radiation and temperature at only 642 out of 4300 stations (14.9%). This percentage rises up to 49.7% (2135 stations) is we consider a wider acceptable range for r

^{2}, i.e., upper than 0.80.

_{0}ensures r

^{2}greater than 0.90, while at the same stations, the correlation between ET

_{0}and R

_{a}is negligible (r

^{2}< 0.10). The opposite case, i.e., very high correlation of PET with R

_{a}, while very low with T appears only once, thus it is statistically negligible. In this vein, we can consider a linear regression model between mean monthly T and ET

_{0}as benchmark to evaluate the performance of any other empirical model, which parameters are identified through calibration.

## 4. Model Calibration

#### 4.1. Evaluation Criteria

_{0}) we used the following statistical criteria:

- The coefficient of determination, most commonly referred to as efficiency or Nash-Sutcliffe efficiency:$$\mathrm{NSE}=1-\frac{{{\displaystyle \sum}}_{t=1}^{T}{\left({\mathrm{PET}}_{\mathrm{mod}}^{t}-{\mathrm{PET}}_{\mathrm{obs}}^{t}\right)}^{2}}{{{\displaystyle \sum}}_{t=1}^{T}{\left({\mathrm{PET}}_{\mathrm{obs}}^{t}-{\overline{\mathrm{PET}}}_{\mathrm{obs}}\right)}^{2}}$$
- The mean absolute error:$$\mathrm{MAE}=\frac{{{\displaystyle \sum}}_{t=1}^{T}|{\mathrm{PET}}_{\mathrm{obs}}^{t}-{\mathrm{PET}}_{\mathrm{mod}}^{t}|}{T}$$
- The relative bias:$$\mathrm{BIAS}=\frac{{{\displaystyle \sum}}_{t=1}^{T}({\mathrm{PET}}_{\mathrm{mod}}^{t}-{\mathrm{PET}}_{\mathrm{obs}}^{t})}{{{\displaystyle \sum}}_{t=1}^{T}({\mathrm{PET}}_{\mathrm{obs}}^{t})}$$

_{0}value, estimated by the PM formula at time step t, ${\mathrm{PET}}_{\mathrm{mod}}^{t}$ is the modeled value at time step t, ${\overline{\mathrm{PET}}}_{\mathrm{obs}}$ is the monthly average value of the reference PET, and T is total number of time steps (in the particular case, T equals the number of months, i.e., 12).

_{a}and T against ET

_{0}, we only consider values greater than 0.70 as satisfying, whereas positive values less than 0.50 are only marginally accepted. On the other hand, negative NSE values are definitely unacceptable, since they indicate that the mean observed value is a better predictor than the simulated value. The mean absolute error and the bias are quite similar metrics, quantifying in absolute (i.e., mm/month) and relative (%) terms the deviation of the mean modelled ET

_{0}from the corresponding mean reference value, ${\overline{Q}}_{\mathrm{obs}}$.

#### 4.2. Optimization Procedure

_{0}, and using NSE as the objective function to maximize against parameters a′ and c′. Within calibration, we have considered a quite extended feasible space, by allowing a′ and c′ to vary within ranges [–0.02, 0.02] and [–5.0, 5.0], respectively. The global search was carried out with the evolutionary annealing-simplex algorithm, which is a heuristic technique that has been proved very effective on locating global optima in highly nonlinear spaces [49,50].

#### 4.3. Assessment against Linear Regression Estimations

_{a}. In Figure 5 we contrast the ranges of coefficients of determination, r

^{2}, achieved by the two linear regression models and the nonlinear parametric model, for the entire sample of 4300 stations. The parametric model ensures very satisfying efficiency (NSE > 0.90) in 58.8% of stations, while only 32.8% and 35.9% of stations exhibit such good performance, considering the linear regression models against T and R

_{a}, respectively. In 2562 stations (59.6%), the parametric approach outperforms both regression models, while in 1327 stations (30.9%) it outperforms at least one model. Only in 411 stations (9.6%) the two benchmarks achieve a higher r

^{2}than the parametric approach. We remark that in linear regression theory, r

^{2}is mathematically equivalent to efficiency, which is the most widely used goodness-of-fitting measure for evaluating nonlinear models. However, while the coefficient of determination of a nonlinear model can take any value from −∞ to unity, in linear regression this metric is by definition non-negative (r

^{2}). Moreover, linear regression models are by definition unbiased, given that the least-square line is forced to pass through the observed mean.

_{0}through the parameterization implemented in Equation (5), because of the irregular relationship of ET

_{0}vs. the two explanatory variables, or due to the influence of additional meteorological drivers (relative humidity and wind speed) as rationalized in Section 3.3.

#### 4.4. Assessment against Hargreaves-Samani Estimations

_{0}estimations across stations. This bias is actually embedded in the coefficients that are embedded in Equation (9), which have been estimated on the basis of specific climatic regime, which cannot be representative of any conditions worldwide. On the other hand, Equation (5) with calibrated parameters ensures very satisfactory performance in an extended part of the station sample, since the model is adapted to local climatic conditions.

## 5. Assessment of Model Performance across Geographical Zones

#### 5.1. Final Data Sample

#### 5.2. ResidualsAnalysis for Stations with Negative NSE

_{0}due to energy or water limitations mainly in the tropical zone, as shown in Figure 6.

#### 5.3. Evaluation of Model Performance across Geographical Zones

## 6. Spatial Analysis and Model Validation

#### 6.1. Spatial Interpolation of Optimized Parameters

_{0}given the observed y values at sampled locations S

_{i}in the following manner:

_{0}is a linear combination of the weights (λ

_{i}) and observed y values in S

_{i}, where λ

_{i}is defined as:

_{0i}between S

_{0}and S

_{i}with a power α, and the denominator is the sum of all inverse-distance weights for all locations i (in the particular case, all stations exhibiting positive efficiency).

#### 6.2. Spatial Distribution of Parameters

#### 6.3. Model Validation

_{0}predictions provided by the parametric formula (Equation (5)), using interpolated parameters against PM estimates in a number of independent stations. In particular, we considered two validations sets, a “local” and a “global” one. The former comprises 37 stations across California, for which monthly meteorological time series are available from the California Irrigation Management Information System [60]. The “global” set comprises 17 stations from countries with different hydroclimatic regimes (Spain, Germany, Ireland, Greece, Iran, and Australia), for which we obtained full time series of the required meteorological data, at the monthly scale, form various data sources.

_{0}with significant accuracy, thus exhibiting an average efficiency up to 0.855, and an average bias of only −0.07. Except for three stations (Bishop, Castroville, De Laveaga), the NSE exceeds 0.70, while in 17 out of 37 stations it exceeds 0.90. This indicates an almost perfect performance, particularly when taking into account that the model has been calibrated using abstract (i.e., mean monthly) meteorological information over the entire globe, while the validation set comprises detailed data, both in terms of spatial extent and temporal resolution. Similarly satisfying are the outcomes from the global validation set, which are summarized in Table 8 (average efficiency 0.852, average bias 0.02), thus confirming the model predictive capacity across different climates.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Food and Agriculture Organization (FAO CLIMWAT) hydrometeorological network (dark areas indicate high altitudes).

**Figure 2.**Scatter plot of mean annuals of (

**a**) solar radiation, (

**b**) temperature, (

**c**) relative humidity, (

**d**) wind speed, (

**e**) sunshine duration, (

**f**) extraterrestrial radiation vs. mean annual ET

_{0}.

**Figure 3.**Scatter plots of monthly extraterrestrial radiation, R

_{a}, vs. mean monthly ET

_{0}(

**a**) and mean monthly temperature, T, vs. ET

_{0}(

**b**) at five stations in Australia, exhibiting loop-type patterns.

**Figure 4.**Scatter plots of monthly extraterrestrial radiation, R

_{a}, vs. mean monthly ET

_{0}(

**a**) and mean monthly temperature, T, vs. ET

_{0}(

**b**) at five stations in Australia, exhibiting irregular patterns.

**Figure 5.**Ranges of coefficient of determination for the linear regression functions of monthly reference PET against R

_{a}and T, and the nonlinear parametric model.

**Figure 7.**Normal probability plot of the empirical distribution function of the mode residuals using Weibull plotting positions against normal distribution function N (0, 0.7), for stations with negative NSE.

**Figure 12.**Scatter plot of optimized parameters through the final data sample of 4088 stations exhibiting positive NSE values.

**Table 1.**Ranges of coefficient of determination, r

^{2}, between monthly ET

_{0}and the two explanatory variables, R

_{a}and T, across the full sample of 4300 CLIMWAT stations.

T vs. ET_{0} | R_{a} vs. ET_{0} | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0–10% | 10–20% | 20–30% | 30–40% | 40–50% | 50–60% | 60–70% | 70–80% | 80–90% | 90–100% | Total | |

0–10% | 55 | 17 | 9 | 12 | 8 | 9 | 8 | 7 | 4 | 1 | 130 |

10–20% | 38 | 11 | 7 | 4 | 11 | 8 | 3 | 3 | 5 | 3 | 93 |

20–30% | 33 | 16 | 13 | 13 | 5 | 7 | 8 | 10 | 9 | 4 | 118 |

30–40% | 36 | 14 | 24 | 10 | 7 | 5 | 12 | 12 | 8 | 15 | 143 |

40–50% | 29 | 14 | 17 | 18 | 22 | 17 | 19 | 13 | 13 | 18 | 180 |

50–60% | 34 | 10 | 17 | 16 | 17 | 28 | 26 | 21 | 26 | 31 | 226 |

60–70% | 30 | 10 | 23 | 15 | 21 | 30 | 37 | 30 | 31 | 52 | 279 |

70–80% | 45 | 11 | 15 | 19 | 28 | 20 | 44 | 48 | 77 | 135 | 442 |

80–90% | 69 | 14 | 14 | 10 | 17 | 34 | 38 | 78 | 362 | 643 | 1279 |

90–100% | 42 | 6 | 6 | 5 | 9 | 30 | 35 | 147 | 488 | 642 | 1410 |

Total | 411 | 123 | 145 | 122 | 145 | 188 | 230 | 369 | 1023 | 1544 | 4300 |

Quartiles | Hargreaves-Samani | Parametric |
---|---|---|

Minimum value | −327.204 | −5.997 |

1 | −5.834 | 0.721 |

2 | −0.971 | 0.947 |

3 | 0.245 | 0.984 |

4 | 0.980 | 0.999 |

**Table 3.**Altitude distribution (%) of the calibration set of CLIMWAT stations (4088 stations, in total).

Region | Altitude | |||
---|---|---|---|---|

<500 m | 500–1000 m | 1000–1500 m | >1500 m | |

Africa | 53.6 | 14.5 | 16.2 | 15.7 |

Oceania | 90.9 | 6.7 | 0.9 | 1.5 |

Eurasia | 75.8 | 12.9 | 6.9 | 4.4 |

N. America | 68.1 | 14 | 7.7 | 10.2 |

S. America | 65.4 | 15.9 | 5.3 | 13.4 |

Total | 69.9 | 13.3 | 8.3 | 8.5 |

Region | 1.0–0.9 | 0.9–0.8 | 0.8–0.7 | 0.7–0.6 | 0.6–0.5 | <0.5 |
---|---|---|---|---|---|---|

Africa | 34 | 16 | 12 | 9 | 7 | 22 |

Oceania | 67 | 10 | 7 | 4 | 1 | 11 |

Eurasia | 68 | 12 | 7 | 4 | 3 | 6 |

N. America | 65 | 15 | 5 | 3 | 2 | 10 |

S. America | 54 | 12 | 10 | 7 | 6 | 11 |

Region | 0–2 mm | 2–4 mm | 4–6 mm | 6–8 mm | 8–10 mm | >10 mm |
---|---|---|---|---|---|---|

Africa | 36 | 36 | 15 | 6 | 3 | 4 |

Oceania | 52 | 36 | 9 | 3 | 0 | 0 |

Eurasia | 39 | 37 | 17 | 5 | 1 | 1 |

N. America | 40 | 39 | 17 | 3 | 1 | 0 |

S. America | 69 | 26 | 4 | 1 | 0 | 0 |

Region | −0.122–0.000 | 0.000–0.001 | 0.001–0.062 |
---|---|---|---|

Africa | 65 | 14 | 21 |

Oceania | 38 | 12 | 50 |

Eurasia | 72 | 5 | 23 |

N. America | 68 | 13 | 19 |

S. America | 55 | 15 | 30 |

a/a | Station | Validation Period | NSE | MAE (mm) | BIAS |
---|---|---|---|---|---|

1 | Five Points | June 1982–June 2013 | 0.880 | 20.4 | −0.09 |

2 | Davis | October 1982–June 2013 | 0.857 | 13.8 | −0.01 |

3 | Firebaugh/Telles | October 1982–June 2013 | 0.897 | 16.8 | −0.09 |

4 | Gerber | October 1982–June 2013 | 0.896 | 17.9 | −0.10 |

5 | Durham | October 1982–June 2013 | 0.870 | 19.7 | −0.14 |

6 | Carmino | November 1982–June 2013 | 0.952 | 11.3 | −0.01 |

7 | Stratford | November 1982–June 2013 | 0.913 | 17.2 | −0.06 |

8 | Castroville | December 1982–June 2013 | 0.442 | 23.7 | −0.23 |

9 | Kettleman | December 1982–June 2013 | 0.903 | 18.8 | −0.10 |

10 | Bishop | March 1983–June 2013 | 0.475 | 16.5 | 0.03 |

11 | Parlier | June 1983–June 2013 | 0.858 | 22.1 | −0.16 |

12 | McArthur | December 1983–June 2013 | 0.940 | 11.5 | 0.01 |

13 | U.C. Riverside | June 1985–June 2013 | 0.858 | 13.2 | 0.08 |

14 | Brentwood | May 1986–October 2006 | 0.930 | 13.2 | −0.06 |

15 | San Luis Obispo | May 1986–June 2013 | 0.856 | 12.0 | −0.08 |

16 | Blackwells corner | May 1987–June 2013 | 0.939 | 13.7 | −0.05 |

17 | Los Banos | June 1988–June 2013 | 0.926 | 14.0 | −0.06 |

18 | Buntingville | May 1986–June 2013 | 0.953 | 11.1 | 0.03 |

19 | Temecula | December 1986–June 2013 | 0.769 | 12.9 | 0.02 |

20 | Santa Ynez | December 1986–June 2013 | 0.842 | 13.6 | −0.10 |

21 | Seeley | June 1987–June 2013 | 0.845 | 18.4 | 0.03 |

22 | Manteca | December 1987–June 2013 | 0.796 | 25.2 | −0.10 |

23 | Modesto | October 1987–June 2013 | 0.922 | 14.7 | −0.06 |

24 | Irvine | November 1987–June 2013 | 0.803 | 13.2 | −0.10 |

25 | Oakville | October 1989–June 2013 | 0.930 | 13.3 | −0.10 |

26 | Pomona | April 1989–June 2013 | 0.701 | 19.0 | −0.15 |

27 | Fresno State | November 1988–June 2013 | 0.906 | 18.4 | −0.12 |

28 | Santa Rosa | January 1990–June 2013 | 0.894 | 11.5 | −0.09 |

29 | Browns Valley | May 1989–June 2013 | 0.856 | 22.3 | −0.16 |

30 | Lindcove | June 1989–June 2013 | 0.782 | 31.0 | −0.22 |

31 | Alturas | May 1989–June 2013 | 0.916 | 10.4 | −0.02 |

32 | Cuyama | October 1989–June 2013 | 0.950 | 11.5 | 0.05 |

33 | Tulelake FS | May 1989–June 2013 | 0.922 | 11.9 | 0.05 |

34 | Windsor | January 1991–June 2013 | 0.905 | 11.4 | −0.09 |

35 | De Laveaga | October 1990–June 2013 | 0.676 | 21.8 | −0.19 |

36 | Westlands | May 1992–June 2013 | 0.932 | 15.0 | −0.03 |

37 | Sanel Valley | February 1991–June 2013 | 0.939 | 11.0 | −0.02 |

Average | 0.855 | 16.0 | −0.07 |

a/a | Station | Country | Validation Period | NSE | MAE (mm) | BIAS |
---|---|---|---|---|---|---|

1 | Aachen | Germany | January 1951–May 2011 | 0.955 | 6.8 | 0.06 |

2 | Bremen | Germany | January 1951–May 2011 | 0.954 | 5.5 | 0.03 |

3 | Alicante | Spain | January 1980–September 2010 | 0.916 | 11.1 | 0.00 |

4 | Badajoz | Spain | January 1961–May 2005 | 0.921 | 13.0 | −0.09 |

5 | Valencia | Spain | September 1954–August 1964 | 0.893 | 10.0 | −0.06 |

6 | Zaragoza | Spain | February 1974–January 1996 | 0.953 | 10.8 | −0.01 |

7 | Herakleion | Greece | January 1968–December 1989 | 0.947 | 10.2 | −0.00 |

8 | Kerkyra | Greece | January 1968–December 1989 | 0.936 | 9.8 | −0.09 |

9 | Kavala | Greece | January 1968–December 1989 | 0.835 | 13.5 | 0.04 |

10 | Limnos | Greece | January 1968–December 1989 | 0.762 | 24.3 | 0.12 |

11 | Athens | Greece | January 1968–December 1989 | 0.924 | 13.6 | 0.03 |

12 | Melbourne | Australia | January 2009–January 2016 | 0.752 | 18.5 | 0.17 |

13 | Dublin | Ireland | January 2013–June 2016 | 0.870 | 5.1 | −0.09 |

14 | Bandar-Anzali | Iran | January 1990–December 2005 | 0.875 | 13.9 | −0.16 |

15 | Ramsar | Iran | January 1990–December 2005 | 0.788 | 16.2 | 0.15 |

16 | Khorram-Abad | Iran | January 1990–December 2005 | 0.400 | 38.3 | 0.37 |

17 | Kashan | Iran | January 1990–December 2005 | 0.804 | 19.6 | −0.13 |

Average | 0.852 | 14.1 | 0.02 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tegos, A.; Malamos, N.; Efstratiadis, A.; Tsoukalas, I.; Karanasios, A.; Koutsoyiannis, D.
Parametric Modelling of Potential Evapotranspiration: A Global Survey. *Water* **2017**, *9*, 795.
https://doi.org/10.3390/w9100795

**AMA Style**

Tegos A, Malamos N, Efstratiadis A, Tsoukalas I, Karanasios A, Koutsoyiannis D.
Parametric Modelling of Potential Evapotranspiration: A Global Survey. *Water*. 2017; 9(10):795.
https://doi.org/10.3390/w9100795

**Chicago/Turabian Style**

Tegos, Aristoteles, Nikolaos Malamos, Andreas Efstratiadis, Ioannis Tsoukalas, Alexandros Karanasios, and Demetris Koutsoyiannis.
2017. "Parametric Modelling of Potential Evapotranspiration: A Global Survey" *Water* 9, no. 10: 795.
https://doi.org/10.3390/w9100795