Theor Appl Climatol DOI 10.1007/s00704-016-1996-2

ORIGINAL PAPER

Worldwide assessment of the Penman–Monteith temperature approach for the estimation of monthly reference evapotranspiration Javier Almorox 1

&

Alfonso Senatore 2 & Victor H. Quej 1 & Giuseppe Mendicino 2

Received: 17 June 2016 / Accepted: 31 October 2016 # Springer-Verlag Wien 2016

Abstract When not all the meteorological data needed for estimating reference evapotranspiration ETo are available, a Penman–Monteith temperature (PMT) equation can be adopted using only measured maximum and minimum air temperature data. The performance of the PMT method is evaluated and compared with the Hargreaves–Samani (HS) equation using the measured long-term monthly data of the FAO global climatic dataset New LocClim. The objective is to evaluate the quality of the PMT method for different climates as represented by the Köppen classification calculated on a monthly time scale. Estimated PMT and HS values are compared with FAO-56 Penman–Monteith ETo values through several statistical performance indices. For the full dataset, the approximated PMT expressions using air temperature alone produce better results than the uncalibrated HS method, and the performance of the PMT method is even more improved adopting corrections depending on the climate class for the estimation of the solar radiation, especially in the tropical climate class.

1 Introduction Reference evapotranspiration (ETo) is a pivotal component of the hydrological cycle and an important parameter in

* Javier Almorox [email protected]

1

Departamento de Producción Agraria, ETSIAAB, Universidad Politécnica de Madrid, UPM, Avd. Puerta de Hierro, 2, Madrid 28040, Spain

2

Department of Environmental and Chemical Engineering, University of Calabria, via P. Bucci 42B, 87036 Rende, Cosenza, Italy

climatology, hydrology, agricultural water management, meteorology, and environmental research. ETo is a crucial input for many climatological, hydrological, and geo-botanical models and estimation of sensitivity to climatic change (Paparrizos et al. 2016). Allen et al. (1998) defined ETo as Bthe rate of evapotranspiration from a hypothetical crop.^ The most widely accepted standard method for ETo estimation from meteorological and climatological data is the FAO-56 Penman–Monteith (PM) formulation (Allen et al. 1998). However, this equation cannot be used in many cases, since its required meteorological input is lacking, at least partially. Specifically, often in the developing countries available meteorological datasets are inaccurate, and sparse, especially concerning global radiation or sunshine hours, water vapor pressure or relative humidity, and wind speed (Almorox et al. 2015). Conversely, numerous meteorological stations around the world collect high-quality air temperature data. The search for solutions for calculating ETo in the absence of full data has led to the development of a variety of models (Rojas and Sheffield 2013; Pereira et al. 2015), including the use of soft-computing techniques: multiple regression analyses, artificial neural networks (ANNs), genetic algorithms (GAs), fuzzy logic (FL), adaptive neuro-fuzzy inference systems (ANFIS), and support vector machines (SVMs) (Rahimikhoob 2010; Gocić et al. 2015). These algorithms are not easily implementable and generally have been calibrated with ETo using the PM equation and may not translate properly in time and space. Image processing from satellites is another way which makes it possible to fill the lack of reference evapotranspiration information (Pereira et al. 2015). In fact, remote sensing has been used in combination with interpolation of ground meteorological observations (e.g., Senatore et al. 2015). Despite the introduction of increasingly sophisticated methods of estimation and advanced remote sensing

J. Almorox et al.

measurement techniques, ETo temperature-based models are very useful in data-poor areas of the world. Air temperature is the most widely registered climatic parameter needed for ETo calculation (Mendicino and Senatore 2013). Hence, air temperature models are very widely used in the literature. Allen et al. (1998) in the FAO-56 guidelines for PM computation include two alternative approaches requiring temperature data only, i.e., (1) a temperature-based model such as the empirical Hargreaves–Samani model (referred to as HS) (Hargreaves and Samani 1985), or (2) methods to estimate weather datasets lacking. The second approach is called herein the PM temperature (herein referred to as PMT) method. Preferences have been frequently directed toward the HS formulation that, as an empirical method, requires empirical coefficients. Recent research has focused on finding robust temperature-based models for estimating ETo based on the HS method (e.g., Vanderlinden et al. 2004; Trajkovic 2007; Ravazzani et al. 2012; Mendicino and Senatore 2013; Martí et al. 2015). Other authors recommended calibrating HS with respect to PM at locations with comparable climate (Almorox and Grieser 2016). The PMT formulation, when applied using only air temperature data, is based on the physics from the PM formulation. This formulation retains many of the philosophy and principles of the physically based combination model PM, considering a combination of energy balance and aerodynamic principles (Pereira et al. 2015). The PMT application requires a more complex calculation than temperature-based models. Numerous researches evaluated the accuracy of the PMT formulation by comparing it with results of PM and with other ETo temperature-based models, mainly HS. Popova et al. (2006) concluded that when only air temperature is observed, the PMT method performs better than the HS model. Jabloun and Sahli (2008) proposed procedures to estimate ETo with missing climate data for different Tunisian locations and revealed that the difference between ETo obtained from full and limited database is small. López Moreno et al. (2009) concluded for a mountainous site in Spain that the estimate of the missing PM parameters offers a more accurate estimation of ETo than the HS formula. Gocic and Trajkovic (2010) suggested that the PM-reduced set ETo estimates should be recommended when only limited data is available in the Sacramento valley region (California). Raziei and Pereira (2013) concluded that the HS and PMT methods appropriately estimate ETo for all climatic regions of Iran and that the performance of the PMT model is further improved when minimum air temperature is adjusted for estimation of dewpoint Tdew. Todorovic et al. (2013) showed that the performance of HS and PMT models is different according to the climate conditions. Vangelis et al. (2013) observed that the PMT FAO Penman–Monteith temperature methods performed better than HS, Thornthwaite, and Blaney–Criddle methods. Córdova et al. (2015) found that

using estimated wind speed data has no major effect on the calculated ETo but that if solar radiation data are estimated, ETo calculations may be erroneous in the high Andes of southern Ecuador. The objectives of this study are (1) to estimate the errors that can arise at the monthly scale if climatological and meteorological data are not available and (2) to compare at the monthly scale the performance of the PMT model using minimum climatic data and the HS model with respect to PM ETo values. Both the analyses are performed for the first time at a global scale, using a complete world dataset under different climate conditions. In the next sections, materials and methods used in the study are introduced, with a specific focus on the estimation methods for missing meteorological data leading to the formulation of the PMT equation, and then the performances of the various methods applied to the global monthly dataset are shown and discussed.

2 Materials and methods 2.1 Climate database and Köppen climate classification The empirical equations are applied using the New LocClim climate database from the Agromet Group of United Nations Food and Agriculture Organization (FAO) (www.fao. org/nr/climpag/), including long-term average climatological data of 4367 stations (1042 from Africa, 1026 from America, 1254 from Asia, 763 from Europe, and 282 from Oceania). The Köppen climate classification system (Köppen 1936) proved to be very useful for investigating the performances of the temperature-based models with respect to PM (Almorox et al. 2015) for different climate zones. This is a widely accepted system for climate classification. The Köppen classification identifies five main groups of climates, and these main groups are further sub-divided by Köppen in order to better classify sub-climates. In this paper, we utilize the following Köppen classes: A tropical climates (Af: rainforest, As: savannah with dry summer, Aw: savannah with dry winter), B arid climates (BS: steppe, BW: desert), C temperate climates (Cs: dry summer, Cw: dry winter, Cf: without dry season); D cold climates (Ds: dry summer, Dw: dry winter, Df: without dry season); and E polar climates. Figure 1 shows the number of stations of the FAO climate database for each of the subclimate types considered. Data are available for 12 calendar months per station. Hence, the analysis is based on 52,404 data pairs. 2.2 Penman–Monteith FAO-56 equation The Penman–Monteith FAO-56 (PM) method is the standard procedure for accurate estimation of ETo and is recommended

Worldwide assessment of the Penman–Monteith temperature approach Fig. 1 Number of stations in the New LocClim climate database per Köppen climate class

by FAO due to its physically based characteristics that allow to incorporate aerodynamic and physiological parameters (Allen et al. 1998; Pereira et al. 2015). The PM formulation is the following:

ETo ¼

900 u2 ðes −ea Þ T mean þ 273 Δ þ γ ð1 þ 0:34u2 Þ

0:408ΔðRn −GÞ þ γ

ð1Þ

where ETo is the reference crop evapotranspiration for short crop (grass, mm day −1). Units for the 0.408 coefficient are m2 mm MJ−1; Rn is the calculated net radiation at the crop surface (MJ m−2 day−1), given by the difference between net shortwave radiation Rns (which is equal to (1 − α)·Rs, with Rs (MJ m−2 day−1) the incoming solar radiation and α = 0.23 the albedo) and outgoing net longwave radiation (Rnl); and G is the soil heat flux density (MJ m−2 day−1). For monthly periods, it is not 0 and it is computed from air temperature of previous and next months; Tmean is the daily mean of air temperature (°C); u2 is the mean wind speed at 2 m height (m s−1); 900 is a coefficient value for short grass reference (K mm s3 Mg−1 day−1); 0.34 is a denominator constant for short grass reference (these values change with reference type and calculation time step) (s m−1); es is the saturation vapor pressure (kPa); ea is the actual vapor pressure (kPa), which can be determined from dewpoint temperature or from psychrometric readings; Δ is the slope of the saturation vapor pressure function (kPa °C−1); and γ is the psychrometric constant (kPa °C−1). The main obstacle to using the PM method widely is the required meteorological and climatological data (Allen et al. 1998). The PM formulation requires measurements of solar radiation (Rs), actual vapor pressure (ea), wind speed (u2), and maximum and minimum air temperatures (Tmax and Tmin, respectively).

Rs is the primary driver of evapotranspiration; however, solar radiation and sunshine duration are an infrequently measured input. Where global solar radiation and insolation data are not available, the subtraction between the maximum and minimum air temperatures can be used (Hargreaves and Samani 1982). If air relative humidity data is lacking, an estimate of ea can be made by assuming a relationship between minimum air temperature and dewpoint. Allen et al. (1998) proposed that the estimated Tdew value from Tmin requires a correction. Finally, wind speed is one of the least available parameters among the ones needed for estimating ETo. The common alternative is to use Bdefault^ wind speed data, which can be estimated as the average wind speed. When wind data are lacking, the average value of u2 = 2 m s−1 (Allen et al. 1998) can be used in the PM equation. The procedures for estimating missing global radiation, air humidity, and wind speed data are given in the paragraphs that follow.

2.3 Performance assessment The performances of the different models are measured using the determination coefficient (R2), the root mean square error (RMSE), and the mean absolute error (MAE). hX n 

 i2 P −P −O O i avg i avg i¼1 R2 ¼ X n  2 X n  2 Pi −Pavg Oi −Oavg i¼1 i¼1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1Xn RMSE ¼ ðPi −Oi Þ2 i¼1 n 1 Xn MAE ¼ ðjPi −Oi jÞ i¼1 n

ð2Þ

ð3Þ ð4Þ

where n is the number of data, P is the predicted (modeled) value, Pavg is the average of the predicted values, O is the

J. Almorox et al.

measured value, and Oavg is the average of the measured values. Smaller values of RMSE and MAE imply higher accuracy in the modeling, while the larger R2 values indicate a close coupling between the estimated and measured data. 2.4 Estimating missing solar radiation: the Hargreaves and Samani model Sunshine duration is a good proxy for accurate estimates of global solar radiation. However, also sunshine data are lacking in most of the climatological stations. As an alternative, models based on geographical coordinates and air temperatures are attractive and viable options (Almorox et al. 2013). When solar radiation, cloudiness, and hours of sun measurements are not available, the Hargreaves–Samani formulation (HSrad) for the estimation of Rs is proposed (Hargreaves and Samani 1982): h i HSrad ¼ Ra  k1  ðT max −T min Þ0:5 ð5Þ where Ra (MJ m−2 day−1) is the extraterrestrial solar radiation, Tmax (°C) is the maximum air temperature, Tmin

(°C) is the minimum air temperature, and k1 (°C−0.5) is an empirical coefficient (k1 = 0.16 for interior regions and k 1 = 0.19 for coastal regions, according to Hargreaves 1994). The HSrad model performs satisfactorily and was recommended for situations when only air temperature is available (Allen et al. 1998). However, the calibration of k1 and 0.5 exponent values for different climate conditions is an accepted approach to accomplish error estimation from the HSrad model (Hargreaves and Allen 2003): HSrad ¼ Ra k 1 ðT max −T min Þk 2

ð6Þ

where HSrad is calculated with calibrated coefficient k1 and exponent k2. Table 1 shows the calculated values of both coefficients estimated for the different Köppen climate types over the globe and shows the calibrated value of coefficient k1 when the k2 exponent is set equal to 0.5. Once Rs is estimated for a given location, net radiation Rn is calculated as the difference between upward and downward radiation fluxes. The outgoing net longwave radiation is

" # #  ðT max þ 273:15Þ4 þ ðT min þ 273:15Þ4  k1⋅ðT max −T min Þk2 0:5 −0:35 Rn1 ¼ σ ⋅ 0:34−0:14⋅ea ⋅ 1:35⋅ 2 ð0; 75 þ z⋅2⋅10−5 Þ "

where σ is the Stefan–Boltzmann constant (4.903 × 10−9 MJ K−4 m−2 day−2); the coefficients 0.34, 0.14, 1.35, −0.35, and 0.75 are recommended in regions where no actual global radiation or hours of sun data are available. The expression 0.75 + 2z × 10−5 is used when calibrated values for the Ba^ and Bb^ Ångström–Prescott

ð7Þ

coefficients are not available (z is the elevation, in meters, and 0.75 is the sum of a + b, a = 0.25 and b = 0.50; when Ångström–Prescott calibrated values are available, the clear sky solar radiation is Rso = (a + b)·Ra and the expression is replaced by the sum of a + b in the denominator) (Allen et al. 1998).

Values of k1 and k2 Hargreaves and Samani coefficients for the different Köppen climate types

Table 1 k1

k2

MAE MJ m−2 day−1

RMSE MJ m−2 day−1

R2

Af As Aw BS

0.475 0.455 0.338 0.348

0.022 0.050 0.187 0.186

1.4488 1.5406 1.8508 1.7059

1.8875 1.8998 2.3359 2.2249

0.335 0247 0.223 0.831

BW Cf Cs Cw Df Ds Dw E

0.364 0.304 0.319 0.233 0.197 0.152 0.211 0.304

0.183 0.194 0.201 0.330 0.313 0.482 0.305 0.204

1.4735 1.4572 1.7391 1.7439 1.9988 1.8130 2.3461 1.6587

1.9466 1.8820 2.3319 2.1751 2.4899 2.6492 2.7438 2.2938

0.852 0.898 0.867 0.621 0.855 0.889 0.782 0.885

k1

k2

MAE MJ m−2 day−1

RMSE MJ m−2 day−1

R2

Af As Aw BS

0.166 0.159 0.162 0.157

0.5 0.5 0.5 0.5

2.200 2.257 2.210 2.062

2.829 2.983 2.877 2.849

0.146 0.134 0.269 0.723

BW Cf Cs Cw Df Ds Dw E

0.158 0.148 0.152 0.153 0.125 0.145 0.128 0.147

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

1.894 1.623 1.959 1.848 2.009 1.815 2.512 2.179

2.636 2.329 2.748 2.356 2.533 2.643 3.009 3.008

0.729 0.844 0.815 0.556 0.849 0.889 0.738 0.801

Worldwide assessment of the Penman–Monteith temperature approach

2.5 Estimating missing air humidity Actual vapor pressure (ea) is defined as   17:27  T dew * ea ¼ eðT dew Þ ¼ 0:6108 exp T dew þ 237:3

ð8Þ

When humidity data are lacking, ea can be estimated by assuming that dewpoint temperature (Tdew) is near the minimum air temperature (Tmin). Although Tmin is a good estimation of Tdew, in general Tmin is often higher than Tdew in arid climates: to amend this problem, the estimated value for Tdew from Tmin may require correction. Then, considering different climate zones, corrections of Tdew are defined in Table 2 (Todorovic et al. 2013; Raziei and Pereira 2013). 2.6 Estimating missing wind speed data

"" h

i 0:77⋅Ra⋅k 1 ⋅ðT max −T min Þk 2 −

" 1:35⋅

PMTrad ¼

2.7 Temperature Penman–Monteith equation The PMT method uses as input only air temperature for the estimation of ETo, whereas wind speed, global solar radiation, and humidity are estimated. The complete expression of the PMT equation is the following one: PMT ¼ PMTrad þ PMTaero

If wind speed data are missing or cannot be calculated, data can be estimated importing wind data from a nearby

0:408⋅Δ⋅

station with similar mesoclimate, or using average estimates of wind speed. In local applications, the common alternative is to use a default wind speed. For the objectives of the study, in this paper we use the Bclimatic default^ value of wind speed fixed to 2 m s−1; this climatic default value was obtained as the average of the stations in the different climates (Allen et al. 1998; Todorovic et al. 2013).

ð9Þ

where

! !## # 4 4   0:5 ðT k 1 ⋅ðT max −T min Þk 2 max þ 237:15Þ þ ðT min þ 273:15Þ 17:27T dew −0:35 ⋅σ⋅ 0:34−0:14⋅ 0:6108⋅exp ⋅ −G T dew þ237:3 2 0:75 þ 2⋅z⋅10−5 Δ þ γ ð1 þ 0:34u2 Þ

ð10Þ

PM T aero

 

esðT max Þ þ esðT min Þ 900⋅u2 −esðT dew Þ ⋅ γ 2 T mean þ 273 ¼ Δ þ γ ð1 þ 0:34u2 Þ

PMT is the reference crop evapotranspiration for short crop (grass) when PM is applied using only measured temperature data (mm day−1), PMTrad is the radiation term of the PMT (mm day−1), and PMTaero is the aerodynamic component (mm day−1). Table 3 shows the six versions of the PMT method used for comparison in this work, together with their ID (from M1 to M6). Specifically, in models from M1 to M3, Tdew was simply calculated reducing by 2 °C Tmin, while in models from M4 to M6 the method described in Table 2 was used. Concerning the parameter k1 of Eq. (6), it was estimated by linear regression using local data in all models except M3 and M6, where typical constant Table 2

Correction of Tdew estimates from Tmin

Climate zones

Corrected Tdew (°C)

Hyper arid Arid Semi-arid Dry sub-humid Moist sub-humid. Humid

Tdew = Tmin − 4 Tdew = Tmin − 2 Tdew = Tmin − 1 Tdew = Tmin − 1 Tdew = Tmin

ð11Þ

values were adopted. Finally, also the parameter k2 of Eq. (6) was estimated for models M1 and M4, while the constant value of 0.5 was retained for the other models. When k1 and k2 were Table 3 HS ID M0 PMT ID M1 M2 M3 M4 M5 M6

HS model and the different PMT model versions used in this study

Equation ETo = 0.408·KET·(Tmean + 17.8)·(Tmax − Tmin)0.5 Ra Tdew Tmin − 2 °C Tmin − 2 °C Tmin − 2 °C Razieia Razieia Razieia

HSrad k1 Estimated Estimated 0.17 Estimated Estimated 0.17

HSrad k2 Estimated 0.5 0.5 Estimated 0.5 0.5

Tdew was calculated reducing by 2 °C Tmin in M1–M3 and using the method described in Table 2 in M4–M6. The parameter k1 of Eq. (6) was estimated by linear regression using local data in all models except M3 and M6. The parameter k2 of Eq. (6) was estimated for M1 and M4. The default value of wind speed is fixed to 2 m s−1 a

Computed in Table 2 (Raziei and Pereira 2013)

J. Almorox et al.

estimated, their values were achieved considering a single calibrated value for each Köppen class which minimizes the errors.

Table 4 Performance measures (R2, RMSE, mm day−1, MAE, mm day−1) of HS and PMT as compared to PM (FAO-56) evapotranspiration R2

RMSE (mm day−1)

MAE (mm day−1)

M0 M1

0.010 0.631

0.575 0.355

0.451 0.281

M2

0.275

0.498

0.396

M3 M4

0.276 0.631

0.497 0.355

0.395 0.281

M5

0.275

0.498

0.396

M6 As

0.276

0.497

0.395

M0

0.104

0.710

0.523

M1

0.591

0.484

0.368

M2 M3

0.309 0.315

0.629 0.627

0.476 0.475

M4 M5

0.595 0.311

0.482 0.628

0.366 0.476

M6 Aw M0

0.317

0.626

0.474

0.406

0.739

0.551

M1 M2 M3 M4 M5 M6

0.673 0.562 0.566 0.676 0.564 0.569

0.543 0.630 0.626 0.541 0.628 0.624

0.410 0.480 0.478 0.408 0.479 0.476

BS M0 M1

0.765 0.854

0.893 0.706

0.664 0.508

M2 M3 M4 M5 M6 BW M0 M1 M2 M3 M4 M5 M6 Cf M0 M1 M2 M3

0.833 0.833 0.857 0.836 0.836

0.755 0.755 0.698 0.749 0.748

0.550 0.552 0.502 0.545 0.547

0.749 0.830 0.815 0.816 0.829 0.814 0.815

1229 1015 1058 1057 1016 1061 1059

0.929 0.739 0.780 0.778 0.739 0.780 0.779

0.851 0.920 0.912 0.906

0.540 0.399 0.419 0.434

0.403 0.298 0.310 0.326

M4 M5

0.921 0.913

0.397 0.418

0.297 0.310

2.8 Hargreaves–Samani model Af

In the comparison analysis, also the Hargreaves and Samani HS model is considered (Hargreaves and Samani 1985): ET0 ¼ 0:408 K ET ðT mean þ 17:8ÞðT max −T min Þ0:5 Ra

ð12Þ

where KET is an empirical coefficient, initially set to 0.0023 but later on recalibrated in different ways. Hereafter, in the comparison analysis, the Hargreaves–Samani model, where the KET coefficient is calibrated for each climate class in the way proposed by Almorox and Grieser (2016), is referred to as the M0 model (Table 3).

3 Results and discussion PM by FAO-56, HS, and the six versions of PMT are calculated for the 4367 stations of the New LocClim climate database. All stations are grouped by the 12 Köppen climate classes considered, and the equations are calibrated by regression with respect to PM for each Köppen climate class separately. Table 4 shows the statistical summary of the comparison between the PMT and HS models and the standardized reference PM method for the different Köppen climate classes. Figures 2 and 3 show the comparison of the models for the 12 climate classes using monthly data from the 4367 stations over the globe, while Figs. 4 and 5 show the comparison for the five main groups of Köppen climates. Considering all data, the R2, RMSE, and MAE differences for the PMT methods are practically negligible, the scatter of the results varying from 0.863 (M3) to 0.884 (M4). HS formulation provides poorer estimates than PMT, with a higher scatter of results (R2 = 0.785) and significantly higher RMSE (0.846 mm day−1) and MAE (0.6 mm day−1). Models M4 and M1 (which uses Tdew = Tmin − 2) are the best with all performance indices, which are very similar (R2 = 0.884 and 0.881, RMSE = 0.62–0.63, and MAE = 0.43–0.44, respectively for M4 and M1). Model M0 provides the worst results among all methods in the 12 Köppen climates. Net radiation is an important and specific variable to determine ETo; the radiation model needs to be calibrated according to climatological conditions (Yin et al. 2008). In the HSrad formulation, the calibration of k1 and k2 values for different climate conditions is an accepted approach to accomplish error estimation from the HS model. As shown in Table 1, the k1 empirical radiation adjustment coefficient (with k2 = 0.5) in this work differs from 0.125 to 0.166 (M2 and M5 models), against an original value of 0.17 (M3 and M6 models). In M1 and M4, the two coefficients are estimated, the k1 empirical

Worldwide assessment of the Penman–Monteith temperature approach Table 4 (continued)

M6

Table 4 (continued)

R2

RMSE (mm day−1)

MAE (mm day−1)

0.907

0.432

0.325

R2

RMSE (mm day−1)

MAE (mm day−1)

M6

0.918

0.298

0.240

Cs M0

0.857

0.557

0.423

All data M0

0.785

0.846

0.600

M1

0.925

0.413

0.311

M1

0.881

0.632

0.438

M2 M3

0.902 0.900

0.470 0.475

0.355 0.361

M2 M3

0.864 0.863

0.676 0.679

0.474 0.479

M4 M5

0.926 0.903

0.410 0.469

0.309 0.354

M4 M5

0.884 0.866

0.624 0.670

0.433 0.471

M6 Cw

0.901

0.474

0.360

M6

0.865

0.672

0.475

M0 M1

0.715 0.801

0.645 0.533

0.476 0.385

M2

0.785

0.554

0.408

M3 M4

0.788 0.803

0.549 0.530

0.402 0.383

M5 M6 Df

0.786 0.790

0.551 0.547

0.406 0.400

M0 M1 M2 M3 M4 M5

0.905 0.943 0.943 0.938 0.943 0.944

0.474 0.364 0.363 0.380 0.363 0.362

0.329 0.238 0.237 0.261 0.238 0.236

M6 Ds

0.939

0.378

0.260

M0

0.964

0.330

0.224

M1 M2 M3 M4 M5 M6 Dw M0 M1 M2 M3 M4 M5 M6 E M0 M1

0.980 0.980 0.981 0.980 0.980 0.982

0.243 0.243 0.234 0.242 0.242 0.233

0.163 0.163 0.156 0.162 0.162 0.155

0.875 0.941 0.934 0.930 0.941 0.934 0.931

0.534 0.366 0.387 0.399 0.364 0.385 0.396

0.408 0.254 0.266 0.276 0.253 0.265 0.274

0.832 0.941

0.436 0.251

0.346 0.197

M2 M3 M4 M5

0.924 0.918 0.941 0.924

0.286 0.298 0.286 0.286

0.226 0.240 0.226 0.226

adjustment coefficient differs from climate to climate from 0.152 to 0.475, and the k2 differs from 0.022 to 0.482 (Table 1). The comparison shows that simple M1 and M4 modifications of the HSrad model with climate-class-specific parameter values lead to some improvements of the PMT model. Results achieved throughout different Köppen climate classes show high variability. However, in tropical A climates, the reduction of estimation errors due to calibration of the HSrad solar radiation model and PMT application is the largest. Model M0 is for all the Köppen climates the worst model and shows the worst results for the tropical climates, specifically in terms of scatter. Results show that M0 deviates significantly from ETo, and these deviations are climate-class-specific. However, at locations where no global radiation data or no sunshine data are available for specific estimation of empirical coefficients, the use of nominal values of 0.17 and 0.5 appears to be well suited; the M3 and M6 PMT models perform better than the M0 model. Despite the relatively poor results obtained with the M0 model, the performance of the optimized values of the k1 coefficient of the HSrad formulation still shows the potentiality of the simple HS model. If station-by-station (and not class-byclass) optimization of the KET coefficient is considered for

Fig. 2 R2 of models for the 12 Köppen climates classes

J. Almorox et al. Fig. 3 MAE of models for the 12 Köppen climates classes

Eq. (12), the overall mean MAE reduces to 0.237 mm day−1 (0.218, 0.292, 0.213, 0.177, and 0.219 mm day−1, respectively for classes A, B, C, D, and E), lower than the best results achieved with the different PMT models. Figure 6 shows the global distribution of the station-by-station calibrated KET coefficient with the HS model, along with the frequency distribution of the values. The mean value of the KET coefficient over the whole dataset is 0.0022, in agreement with literature, but ranges between a minimum value of 0.0009 (a Cf station in Chile) to a maximum of 0.0048 (an Aw station in Brazil), with a standard deviation of about 0.0004. From Fig. 6 some KET patterns emerge, like, e.g., averagely high values in the desert areas of Australia, Northern Africa, and Northern America or averagely low values in both humid tropical areas (South America and Africa) and cold highest latitudes in the northern hemisphere. Even though these patterns already provide some information for a regional estimate of the KET coefficient, further research is needed for finding and evaluating a fitting global regionalization strategy for the HS equation. Fig. 4 R2 of models per main groups of Köppen climate classification

While the potential of a regionalized HS equation is acknowledged, the results of our work demonstrate however that the PMT-based approach significantly improves current global methods based on various versions of the calibrated HS equation (Almorox and Grieser 2016). The RMSE values found in this study range from 0.23 to 1.01 mm day−1 with 0.62 mm day−1 for all locations. The statistical errors are similar to others achieved by other authors using simplified forms of the PM equation (RMSE 0.41–0.80 mm day−1, Jabloun and Sahli 2008; RMSE lower (90%) than 0.70 mm day−1, Raziei and Pereira 2013). In the calibrated model, they observed the highest errors in the dry B classes with RMSE = 1.211 mm day −1 for climate class BW (1.01 mm day−1 for M1 model) followed by the class BS with RMSE = 0.881 mm day−1 (0.698 mm day−1 for M4 model). The lowest errors were observed in the class Ds with RMSE = 0.327 mm day−1 (0.234 mm day−1 for M3 model) followed by the E climates with RMSE = 0.378 mm day−1 (0.251 mm day−1 for M1 model). In A tropical climates, the lowest

Worldwide assessment of the Penman–Monteith temperature approach Fig. 5 MAE of models per main groups of Köppen climate classification

RMSE occurred in climate class Af with 0.425 mm day−1 (0.355 mm day−1 for M4 model), followed by As with 0.594 mm day−1 (0.482 mm day−1 for M4 model) and Aw with 0.704 mm day−1 (0.541 mm day−1 for M4 model). In this study, it is underlined that the best evaluation of ETo is given by the PMT equation. The PMT method provides lower errors of the ETo estimate and improves the estimation of HS ETo temperature-based models. Furthermore, results confirm that when only air temperature data are available, the PMT method can be used, with today’s common software, for example a spreadsheet program, as an adequate rigorous formulation, because it retains the physical philosophy of the PM method. Results of this work show that the accuracy of the PM approximation based on the PMT model is reasonably high, with an overall MAE value in any case lower than 0.5 mm day−1. PMT achieves much higher R2 values under single climate Köppen classes than those reported by the M0 method in this paper or than those reported by Shahidian et al. (2012) while RMSE and MAE are lower than the values obtained in Almorox and Grieser (2016). Figure 7 shows a

comparison of the spatial distributions of MAEs calculated on each station with both models M0 and M4, highlighting the better performance of the latter model almost everywhere, even though some regions where M0 works better than M4 are also evident (e.g., central and eastern USA or southeastern coast of Australia). Since the PMT method depends only on temperature data, which can be easily spatially interpolated, it is currently used for drawing regional and global maps (e.g., FAO 2012). From this point of view, the detailed comparison with observed climatological data performed in this study provides useful information about the actual reliability of these maps, with their strengths and weaknesses. The PMT method is not a perfect replacement for the PM model, but more accurate approximations may be developed by applying the methodology of this work rather than other temperature-based simple equations. The PMT model might be rather convenient, from a practical point of view, since climatological studies are often limited by the incompleteness of meteorological and climatic dataset. The PMT calculation process could be improved with new procedures for

Fig. 6 Left: global distribution of the station-by-station calibrated KET coefficient with the HS model. Right: associated frequency distribution

J. Almorox et al.

Fig. 7 Spatial distributions of the MAEs calculated on each station with models M0 (upper map) and M4 (bottom map)

estimating missing weather values with the use of published or available values for the location-specific conditions or by means of remote sensing estimates. This study strongly supports the use of the PMT formulation and recommends the calibration of the radiation model according to local climatic conditions.

4 Conclusions With an analysis performed for the first time at the global scale, this study has shown that PMT estimates of ETo are

more in agreement with PM than the HS estimates. The PMT estimates are accurate and provide lower errors. The PMT model, using only measured temperature data, retains the philosophy of the full data PM equation. When only air temperature data are available, empirical models for estimating solar radiation, Tdew, and wind speed are viable options for agroclimatic and hydrologic applications. The study shows that using appropriate empirical HS calibration coefficients for estimating solar radiation results in improved accuracy in ETo estimates. The adjustment of Tmin for estimating Tdew in the PMT method has shown to enhance the performance of the method in A, B,

Worldwide assessment of the Penman–Monteith temperature approach

and C climates; the dewpoint temperature estimation can be further improved with the use of published or available values for the location-specific conditions. In places where no wind data are available, the average value of 2 m s−1 (Allen et al. 1998) can be used for ETo estimates. The use of mean values for wind speed in the PMT model, calculated for the study area, could also improve the estimate. This work shows that the performance of PMT and HS (M0) models may vary significantly according to the climate under consideration, ranging from tropical to polar. In tropical climates, the performances of the PMT approach and the M0 method are very variable, although M1 and M4 show the best performance. In other climates, the differences between M0 and PMT methods are smaller in terms of performance indices, but again PMT shows for every climate the best performance. In the case of missing weather data or data that cannot be calculated, we recommend that missing climatic data are estimated and the PMT method for the calculation of ETo is adopted.

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