Computing FAO56 reference grass evapotranspiration PM-ETo from temperature with focus on solar radiation

Agricultural Water Management 215 (2019) 86–102

Contents lists available at ScienceDirect

Agricultural Water Management journal homepage: www.elsevier.com/locate/agwat

Computing FAO56 reference grass evapotranspiration PM-ETo from temperature with focus on solar radiation

T

P. Paredes , L.S. Pereira ⁎

Centro de Investigação em Agronomia, Alimentos, Ambiente e Paisagem (LEAF), Instituto Superior de Agronomia, Universidade de Lisboa, Tapada da Ajuda, 1349-017, Lisboa, Portugal

ARTICLE INFO

ABSTRACT

Keywords: PM-ETo with reduced datasets (PMT) Dew-point temperature Default wind speed Solar radiation Radiation adjustment coefficient kRs Predictive equations for kRs

The computation of the reference crop evapotranspiration (ETo) using the FAO Penman-Monteith equation (PMETo) requires data on maximum and minimum air temperatures (Tmax, Tmin), vapour pressure deficit (VPD), solar radiation (Rs) and wind speed at 2 m height (u2). However, those data are often not available, or data sets may be incomplete or have questionable quality. Various procedures were proposed in FAO56 to overcome these limitations and an abundant literature has been and is being produced relative to alternative computational methods. Studies applied to a variety of climates, from hyper-arid to humid, have demonstrated that improved methods to compute PM-ETo from temperature only (PMT approach) have appropriate accuracy. These methods refer to estimating: (i) the dew point temperature (Tdew) from Tmin or, in case of humid climates, from the mean temperature, Tmean; (ii) Rs from the temperature difference (TD = Tmax-Tmin); and (iii) u2 using default global or regional values. Greater difficulties refer to the need for locally calibrating the radiation adjustment coefficient (kRs) used with the Rs equation. Therefore, considering that calibrated kRs values were made available by past studies for a large number of locations and diverse climates, the current study developed and tested simple computational approaches relating locally calibrated kRs with various observed weather variables – TD, relative humidity (RH) and average u2. The equations were developed using CLIMWAT monthly full-data relative to all the Mediterranean countries. The equations refer to all available data, or to data grouped as hyper-arid and arid, semi-arid, dry and moist sub-humid, and humid climates. To test those kRs equations, ETo computed from temperature and using the predicted kRs values were compared with ETo computed with full data sets of the same Mediterranean locations and of Iran, Inner Mongolia, Portugal and Bolivia. RMSE average values result then small, ranging from 0.34 to 0.54 mm day−1, therefore not very far from values obtained when a trial and error procedure was used for all the same locations, from 0.27 to 0.46 mm day−1. These indicators allow to propose the use of kRs obtained from the predictive equations instead of locally calibrated kRs values, which greatly eases computations and may largely favour the use of the PMT approach.

1. Introduction The reference evapotranspiration (ETo, Allen et al., 1998), also named potential ET in non-agricultural studies, is of main importance for computing crop and vegetation evapotranspiration, and for characterizing the local climate. Its accurate calculation is crucial in applications to crop water management, irrigation planning, basins water balance, climate characterization and climate change studies as recently reviewed (Pereira et al., 2015; Pereira, 2017). Grass reference ETo was defined through a FAO experts’ consultation (Smith et al., 1991) by parameterizing the Penman-Monteith equation (Monteith, 1965) for a cool season grass, so resulting in the Penman-Monteith ETo (PM-ETo). Grass reference ETo is defined as the rate of ⁎

evapotranspiration from a hypothetical reference crop with an assumed crop height h = 0.12 m, a fixed daily canopy resistance rs = 70 s m−1, and an albedo of 0.23, closely resembling the evapotranspiration from an extensive surface of green grass of uniform height, actively growing, completely shading the ground and not short of water (Allen et al., 1998). As analysed by Pereira et al. (1999), this definition provides for a clear distinction relative to the Penman’s potential ET (Penman, 1948) and the equilibrium evaporation (Slatyer and McIlroy, 1961), which allows assuming a well-defined relationship between ETo and the actual vegetation ET based upon the aerodynamic and surface resistances of the reference crop and of the considered vegetation surface, thus supporting the concept and use of crop coefficients. The computation of ETo using the FAO Penman-Monteith equation

Corresponding author. E-mail addresses: [email protected] (P. Paredes), [email protected] (L.S. Pereira).

https://doi.org/10.1016/j.agwat.2018.12.014 Received 20 August 2018; Received in revised form 22 November 2018; Accepted 5 December 2018 0378-3774/ © 2018 Elsevier B.V. All rights reserved.

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

(PM-ETo) requires full data sets on maximum and minimum air temperatures (Tmax, Tmin), relative humidity (RH) or dew point temperature (Tdew), solar radiation (Rs) and wind speed at 2 m height (u2). However, those data are often not available, or data sets may be incomplete or have questionable quality, thus requiring the adoption of alternative computational procedures such as those proposed in FAO56 (Allen et al., 1998) to overcome those limitations. In addition, very numerous alternative computational methods are dealt in an abundant literature as quoted in the literature cited below. However, the assumed definition implies that meteorological observations should be performed over an extensive green grass field, healthy and not short of water, commonly designated as reference weather station site. The PM-ETo equation is derived from the Penman-Monteith combination equation parameterized for the grass reference crop as referred above (Allen et al., 1998):

ETo =

0.408

(Rn G) + +

900 u (e Tmean + 273 2 s

(1 + 0.34 u2 )

et al. (2015) were likely performed for reference sites. The study by Cai et al. (2007) reported good results of using Eq. (2) except for a site located in a hyper-arid region. Nevertheless, the majority of worldwide weather data are likely reported from non-reference sites, and the adoption of Tdew = Tmin may then cause less accuracy of ETo estimates. In addition, data quality may play a main role in the accuracy of ETo estimates (Allen et al., 2011). The need for temperature correction was discussed by various authors, particularly in FAO56 (Allen et al., 1998), thus doing Tdew = Tmin - aT, often proposing a correction factor aT = 2 or 3 °C. The software proposed by Annandale et al. (2002) took this requirement into consideration and left to the users the selection of that correction factor. Recently, Upreti and Ojha (2017), adopted seasonal correction factors varying from 0 °C during the wet season to 3 °C during the summer months. Alternative computations have been proposed such as the equation developed by Hubbard et al. (2003), where Tdew is computed from Tmin, Tmean and the temperature difference TD = Tmax-Tmin. Differently, Lobit et al. (2018a) computed VPD using Eq. (2) but adopted different parameterizations when the site was considered humid or dry. In an application to the CLIMWAT data base referring to 555 stations in the Mediterranean countries, Todorovic et al. (2013) found that the correction factor aT should vary with the site aridity index (AI, UNEP, 1997), ratio between the long term averages of the annual precipitation and the annual potential climatic ET (PET, Thornthwaite, 1948). Thus, aT depends on the aridity of the site, being larger for hyper-arid locations and null for moist sub-humid sites. Differently from more common approaches, Todorovic et al. (2013) also proposed to compute Tdew from Tmean using a correction factor aD (ºC) for the humid sites, where daily Tdew is expected to be above Tmin, thus Tdew = Tmean – aD. This approach was confirmed in applications to Iran (Raziei and Pereira, 2013a), Inner Mongolia (Ren et al., 2016a) and the Azores Islands (Paredes et al., 2018a). With the purpose of computing the PM-ETo equation in absence of radiation and/or sunshine duration data, Allen (1997) and Allen et al. (1998) proposed to estimate Rs using the predictive Rs equation of Hargreaves and Samani (1982, 1985), herein referred HS-Rs, which expresses Rs as a linear function of the square root of TD = Tmax-Tmin:

ea ) (1)

where ETo is the daily grass reference ET (mm day−1), Rn is the net radiation at the crop surface (MJ m-2 day−1), G is the soil heat flux density (MJ m-2 day−1), Tmean is the mean daily air temperature at 2 m height (°C), u2 is the daily wind speed at 2 m height (m s−1), es is the saturation vapour pressure at 2 m height (kPa), ea is the actual vapour pressure at 2 m height (kPa), es-ea or VPD is the saturation vapour pressure deficit (kPa), Δ is the slope vapour pressure curve (kPa °C−1), and γ is the psychrometric constant (kPa °C−1). G = 0 for daily computations because the daily soil heat flux beneath the grass reference surface is very small (Allen et al., 1998). The hourly or fractions computation of ETo is described by Allen et al. (2006), who also described the ASCE PM-ETr equation that represents alfalfa reference ET. While maximum and minimum temperature (Tmax and Tmin, ºC) are commonly observed worldwide, the other required weather variables are often not available with good quality and their acquisition may be quite expensive. To cope with conditions of reduced data sets, methods were proposed by Allen et al. (1998) for estimation of the missing variables Rs, ea and/or u2. Overall, these methods consist of the so called Penman-Monteith temperature approach (PMT). When relative humidity data or psychrometric observations are missing, Allen et al. (1998) recommended computing ea assuming that the dew point temperature (Tdew, ºC) could be acceptably estimated by Tmin, thus resulting that actual vapour pressure is estimated as:

ea = eo (Tdew ) = 0.611 exp

17.27 Tmin Tmin + 237.3

R s = k R s (Tmax

(3)

Tmin )0.5R a −0.5

where kRs is an empirical radiation adjustment coefficient (°C ) and Ra is the extraterrestrial radiation (MJ m2 day-1). This Eq. (3) is part of the Hargreaves and Samani equation (HS-ETo), which may be written as

ETo = 0.0135/ kRs (Tmax

(2)

Tmin )0.5R a (Tmean + 17.8)

(4a)

or as

This approach holds when applied to a reference site since the PMETo definition implies the consideration of an extensive and actively growing grass crop completely shading the ground and not short of water. Moreover, Allen (1996), Allen et al. (1998) and Temesgen et al. (1999) have shown that this approach (Eq. (2)) does not hold when observations are performed in non-reference sites, i.e., when weather observations are affected by aridity and/or local advection, which cause that Tmin > Tdew. Kimball et al. (1997) observed that: (i) Tmin exceeds Tdew in arid sites leading to average daily differences of 0.8 to 1.2 kPa between ea computed from Tmin and Tdew; (ii) in sites with semi-arid climate, there is seasonality in the differences between Tmin and Tdew resulting in differences between respective ea values varying from 0.1 to 0.6 kPa for winter and summer months, respectively; (iii) smaller differences between Tmin and Tdew occurred in other less arid climates, e.g., humid continental, and humid subtropical conditions, with average daily differences of less than 0.3 kPa between ea computed from Tmin and Tdew. In addition, in humid locations Tdew often remains above Tmin resulting that Eq. (2) does not apply adequately. Numerous applications assuming Tdew = Tmin are reported in literature. Applications described by Liu and Pereira (2001), Pereira et al. (2003), Popova et al. (2006), Jabloun and Sahli (2008), and Córdoba

ETo = CHS (Tmax

Tmin )0.5R a (Tmean + 17.8)

(4b)

where Tmean is mean daily air temperature (°C), λ is the latent heat of vaporization commonly assumed equal to 2.45 MJ kg−1, 0.0135 is a factor for conversion of units from the American to the International system, and 17.8 is an empirical parameter proposed by Hargreaves et al. (1985). When written with a Hargreaves coefficient (CHS = 0.0135/λ kRs ) as per Eq. 4b, the HS-ETo equation is not directly expressed as influenced by the radiation adjustment coefficient kRs; however, calibrating CHS also indirectly calibrates kRs. Following the recommendation by Allen et al. (1998), Eq. 4 is commonly used as alternative to PM-ETo (Eq. (1)) when only reduced data sets are available. Numerous studies focused on both equations 3 and 4, particularly on the accuracy of Eq. (3), i.e., the solar energy term of Eq. 4, comparatively with other Rs estimation equations. Despite many authors keep using the default kRs values 0.16 and 0.19 °C−0.5 for respectively inland and coastal sites (Allen et al., 1998), a great deal of research has provided for more accurate estimation of kRs, directly or through estimating CHS, for a variety of environments. This coefficient kRs relates with the atmospheric conditions that influence the availability of incoming solar radiation at crop and natural 87

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

surfaces, so the availability of energy for evaporation. Allen (1997) suggested estimating kRs as a function of elevation to account for the effect of the thickness of the atmosphere on the volumetric heat capacity of the atmosphere, thus making: kRs = kro (Pz/Po)0.5

showing good performances. Almorox et al. (2011) applied various Rs models to several locations in central Spain and found that HS-Rs performed best; however, authors developed a new model where a saturated vapour pressure (es) term was used in addition to TD. Later, Almorox et al. (2013) reported that performances of two versions of the HS-Rs equation were similar to that of BC-Rs. In a study performed in Mexico (Quej et al., 2017), the best ranking was for BC-Rs but HS-Rs ranked high. Contrasting, in a later study applied to Mexico (Lobit et al., 2018b), none of those models was considered satisfactorily and a new model was developed modified from BC-Rs, also using TD. In a study applied to USA (Woli and Paz, 2011) comparing 16 Rs models, a modified version of the HS-Rs equation ranked in the mid of the list, while higher performances were assigned to models that, in addition to TD, included daily rainfall as predictor. A model comparison performed for a set of Iranian stations (Jahani et al., 2017) has shown that HS-Rs models performed well, mainly when elevation was considered (Eq. (6)); however, they ranked below various purposefully developed TD polynomial equations, however of local interest only. The performance of various forms of HS-Rs models were also good but not the best as reported for other studies in Iran (Besharat et al., 2013) and Canada (Aladenola and Madramootoo, 2014). Summarizing, the comparative studies of Rs prediction models do not provide for a clear ranking of models because they are conceptually different and are applied to a variety of sites having diverse environmental conditions, different data availability and diverse modelling objectives. Various comparisons did not focus ET applications but local or regional solar energy studies, therefore having requirements different of those where Rs is used to compute ETo. Studies have shown that higher prediction performances require somewhat complex parameterization, so contradicting the simplicity of the original HS-Rs Eq. (3). That simplicity was a main reason why HS-Rs was proposed to compute the PMETo when only reduced data sets are available (Allen et al., 1998). The referred studies also show that despite the HS-Rs Eq. (3) may not be the best estimator of Rs, it certainly is highly valuable to provide for computing ETo when solar radiation or sunshine duration are not observed. Numerous applications have demonstrated the appropriateness of the HS-Rs model for estimating Rs for PM-ETo computations with reduced data sets. Among the first results are those for China by Liu and Pereira (2001) and Pereira et al. (2003), for Bulgaria (Popova et al., 2006) and for Tunisia (Jabloun and Sahli, 2008). These studies used simple tests to get the best kRs values, which varied in a limited range. The recommendation by Allen (1997) to adopt self-calibration had no followers because it is quite demanding. Bandyopadhyay et al. (2008) used also a demanding calibration procedure with back estimation of Rs to estimate the kRs coefficients for a large set of weather stations in India. Trial and error procedures for the calibration of kRs were later used by Todorovic et al. (2013), Raziei and Pereira (2013a), Ren al. (2016a) and Paredes et al. (2018a), which aimed at minimizing the differences between PM-ETo computed with full data or with Rs estimated with HS-Rs. The available studies allow to understand that kRs varies spatially not only with site elevation but also with the distance to large water bodies as earlier discussed by Hargreaves and Samani (1982, 1985), Allen (1997) and Hargreaves and Allen (2003). The effect of the proximity to large water bodies, particularly the sea, may refer to the peculiar behaviour of both wind and air moisture in coastal areas which were supposed to increase kRs relative to inland areas. In addition, recent studies (Raziei and Pereira, 2013a; Ren et al., 2016a; Paredes et al., 2018a) reported that kRs increases with the aridity of the site and with wind speed. Reduced data sets often do not include wind speed data. Under these circumstances, Allen et al. (1998) proposed the use of the world average wind speed value u2 = 2 m s−1 as the default estimator of wind speed when related data are missing. When average local or regional

(5)

where Pz and Po are the mean atmospheric pressure (kPa) at the site and at the sea level respectively, and kro is the original value of kRs, 0.16 or 0.17 °C−0.5 after Samani (2000). Later, with the same purpose of taking account for effects of reduced atmospheric thickness on Rs, Annandale et al. (2002) made kRs directly dependent of the elevation z (m) of the site: kRs = kro (1 + 2.7 10−5 z)

(6)

Samani (2000), to ease the use of Eq. 4 without requiring local calibration of kRs, adopted a polynomial relationship with the long term average temperature difference TDavg (ºC) characterizing the site since it may be considered as an indicator of the energy available at the site: kRs = 0.00185 TDavg2 - 0.0433 TDavg + 0.4023

(7)

Similarly, Mendicino and Senatore (2013) developed polynomial relationships with TDavg to predict CHS for various regions of Italy. Differently, Vanderlinden et al. (2004) proposed a linear estimator of CHS using the ratio of daily averages Tavg/TDavg as predictor. Adapting this approach, Thepadia and Martinez (2012) developed a linear kRs prediction equation using data of humid sites: kRs = a Tavg/TDavg + b

(8)

Later, Martí et al. (2015) assessed various polynomial equations to estimate the CHS from various weather and geographical site characteristics in eastern Spain. The HS-Rs models where kRs is replaced by a predictive equation as those described above are often named after the respective authors; however, the radiation prediction model is the same, the HS-Rs model (Eq. (3)), and what changes is the parameterization of kRs. That way of renaming the HS-Rs model due to differences in parameterization results confuse users, as the varied nomenclature and terminology used is also confusing. Various models to estimate Rs from temperature were developed in addition to that of Eq. (3). They have been the object of diverse comparative studies; however, these studies do not provide for a definitive ranking of models. On the one hand, model results depend upon the way how they were parameterized and, on the other hand, the ranking depends on the selected goodness of fit indicators and on the objectives of modelling, e.g., energy assessments, ET estimation or crop yield predictions. Rankings obtained from comparative studies result therefore difficult to interpret. An early study applied to Argentina (Meza and Varas, 2000) shows a slight advantage of HS-Rs over the BC-Rs model of Bristow and Campbell (1984). Chen et al. (2004) considered negatively the HS-Rs model and developed a logarithmic TD equation for replacing HS-Rs in China. A study applied to Central Europe (Trnka et al., 2007) compared various Rs models in terms of their impacts on the performance of crop yield models and has shown that simple TD models ranked below various models that used other variables in addition to TD. Comparative studies by Abraha and Savage (2008), applied to selected weather stations across the world, have shown that the HS-Rs model ranked high among various models, including first ranked in some cases. A study applied to selected Chinese stations (Liu et al., 2009) has shown that HS-Rs ranked high but preference was assigned to the BC-Rs model due to its accuracy and because parameters could be estimated from weather variables, so not requiring calibration using observation data. That same BC-Rs model was also reported with high ranking in a comparative study relative to four countries in four continents (Moradi et al., 2014), but with two different versions of HS-Rs

88

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

wind speed data are available they could be used alternatively following results reported by Popova et al. (2006), Jabloun and Sahli (2008) and Paredes et al. (2018a). Nevertheless, as discussed by Allen et al. (1998), impacts of wind speed are generally not large except for areas where aridity dominates. The use of the above referred approaches for estimating the parameters of the PM-ETo equation allows computing ETo just replacing Rs, ea or u2 by their respective estimators when related weather variables are missing, or using temperature data only when more variables are lacking. The above quoted bibliography includes adequate examples of computing ETo with the PMT approach. Considering the discussion above on procedures to compute Rs, ea and u2 with reduced data sets, and observing that there is a limited use of those computational procedures despite various studies show that

monthly mean climatic data from over 5000 meteorological stations worldwide. The weather data include monthly mean data for maximum and minimum temperature (Tmax and Tmin, ºC), solar radiation (Rs, MJ m−2 day-1), mean relative humidity (RH, %) and wind speed (u2, km day-1). In addition, the database also includes monthly mean data on total and effective precipitation (P and Pe, mm) and on monthly mean daily grass reference evapotranspiration (ETo, mm day-1) computed with the PM-ETo equation (Eq. (1)). The CLIMWAT database has been used in several evapotranspiration studies, namely those reported by Allen (1996, 1997), Temesgen et al. (1999), and Droogers and Allen (2002). Data used in this study are about the same as previously used by Todorovic et al. (2013) and refer to the Mediterranean countries with a total number of 588 weather stations (Fig. 1) hereafter called MedClimwat.

Fig. 1. Spatial distribution of the weather stations of Med-Climwat throughout the climate zones defined with the UNEP aridity index (adapted from Todorovic et al., 2013).

All stations are distributed into six climate zones defined according to the UNEP aridity index AI (UNEP, 1997), with AI consisting of the ratio between the long term mean of annual precipitation (P, mm) and annual potential climatic evapotranspiration (PETTH, mm, Thornthwaite, 1948) The distribution of weather stations into the various AI climate zones is represented in Fig. 1. The respective spatial distribution shows that most humid and sub-humid climates are located in the northern Mediterranean countries, semiarid climates mostly occur nearby the Mediterranean Sea, while arid and hyper-arid climates dominate in the southern Mediterranean countries. Fig. 1 also shows that a good number of stations are located by the coast. Overall, the selected weather stations are well distributed throughout the Mediterranean region and refer to a large spectrum of environmental conditions as summarized in Table 2 where ranges of ETo and elevation are presented in relation to the various climate zones. kRs values of the selected Mediterranean stations range from 0.11 to 0.28 °C−0.5. Weather data of Iran used in the current study corresponds to those previously handled by Raziei and Pereira (2013a). They consist of monthly averages of precipitation (mm), Tmax and Tmin (°C), RH (%),

related calculations are often accurately performed, the objectives of this study consist of: a) revising the use of available procedures for estimating Tdew from temperature, wind speed from local or default values and solar radiation from the temperature difference; b) developing multiple regression equations for the empirical radiation adjustment coefficient kRs to be used in assessing Rs from temperature, and c) testing such equations for various regions and climates as a contribution to define a consolidated method for computing the PM-ETo from temperature when only reduced datasets are available. 2. Materials and methods 2.1. Data Various data sets consisting of daily and monthly weather data were used in the present study. Main data are from the CLIMWAT database (Smith, 1993), which version 2.0 consists of observed long-term 89

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Table 1 Distribution of weather stations of the various datasets per climate zones with indication of the respective aridity index (UNEP, 1997). Climate zones and aridity index

Hyper-arid (< 0.05) Arid (0.05 to 0.20) Semi-arid (0.20 to 0.50) Dry Sub-humid (0.50 to 0.65) Moist Sub-humid (0.65 to 1.00) Humid (> 1.00) Total

Number of weather stations per climate zone Med-Climwat*

Iran*

Inner Mongolia*

Portugal**

Bolivia**

Total

34 49 89 90 191 135 588

1 12 14 4 1 5 37

3 10 27 3 4 – 47

0 0 1 8 20 27 56

0 0 0 4 0 0 4

38 71 131 109 216 167 732

* monthly data. ** daily data. Table 2 Basic characteristics of the sites included in the datasets used: ranges of elevation (m) and ETo (mm day−1) per climate zone. Datasets

Ranges of elevations and ETo per climate zones Humid

Ranges of elevations (m) Med-Climwat 1 to 1916 Iran −26 to −18 Inner Mongolia – Portugal 28 to 1020 Bolivia – Ranges of ETo (mm day−1) Med-Climwat 0.2 to 7.9 Iran 0.5 to 5.7 Inner Mongolia – Portugal 0.2 to 11.2 Bolivia –

Moist sub-humid

Dry sub-humid

Semi-arid

Arid

Hyper-arid

3 to 1775 Up to 1337 286 to 997 5 to 693 –

0 to 1667 13 to 1373 306 to 581 8 to 250 3789 to 3950

−200 to 1585 143 to 2048 241 to 1490 Up to 190 –

−276 to 1140 7 to 1754 965 to 1561 – –

−13 to 1378 489 to 1237 940 to 966 – –

0.1 to 8.2 0.8 to 9.1 0.01 to 5.4 0.2 to 10.0 –

0.3 to 8.8 0.5 to 8.6 0.01 to 6.1 0.2 to 10.9 1.2 to 6.3

0.4 to 9.4 0.4 to 11.2 0.03 to 7.9 0.3 to 10.2 –

0.8 to 10.8 0.5 to 13.7 0.1 to 9.6 – –

1.6 to 11.3 0.9 to 10.9 0.1 to 12.5 – –

sunshine duration (n, hour) and wind speed (u2, m s−1) relative to 37 Iranian synoptic stations. The majority of the selected stations had nearly complete data records for the period 1971–2005. Their data passed careful quality control by the Iranian meteorological organization and were further analysed by the authors of the study for possible non-homogeneities resulting that the wind datasets at 7 stations were corrected against neighbouring stations following the proposed approaches by Allen et al. (1998). The distribution of the selected stations in terms of climate zones defined by the aridity index AI (UNEP, 1997) is summarized in Table 1. AI varies from humid and sub-humid near the Caspian Sea in north-eastern Iran, dry-sub-humid and semi-arid in the northern and western regions, to arid and hyper-arid in central and southern Iran. The ranges of ETo and elevation are given in Table 2 in relation to the referred climate zones. A spatial analysis relative to the weather variables used to compute ETo and of ETo itself, as well as relative to their seasonality, is reported by Raziei and Pereira (2013b). The kRs values used in the current study ranged from 0.11 to 0.21 °C−0.5, with larger values generally assigned to arid and hyper-arid sites (Raziei and Pereira, 2013a). Data of Inner Mongolia used herein refer to those adopted in the study reported by Ren et al. (2016a). They consist of monthly precipitation (mm), Tmax and Tmin (°C), RH (%), sunshine duration (n, hour) and u2 (m s−1) from 47 surface meteorological stations referring to the period of 1981 to 2012. Data were provided by the China Meteorological Organization and have been submitted to quality control previously to be used. The selected weather stations are well distributed throughout Inner Mongolia. The aridity index AI varies from moist subhumid in the East to hyper-arid in the West. Coherently, ETo increases from the eastern to the western Inner Mongolia. The distribution of weather stations per climate zone is shown in Table 1, while the ranges of ETo and elevation are presented in Table 2. kRs ranges from 0.13 to 0.24 °C−0.5, with higher values for the sites having arid and hyper-arid

climates (Ren et al., 2016a). A trend analysis and a related spatial analysis of weather variables and ETo, including when computed by the PMT approach, are presented by Ren et al. (2016b). Daily data relative to 4 weather stations in the Bolivian Altiplano (Chipana et al., 2010) were also used. Data refer to precipitation (mm), Tmax and Tmin (°C), RH (%), u2 (m s−1), and sunshine duration (n, hour) relative to the period 2002-2006. Sites elevation range from 3790 to 3950 m. All stations were classified as dry sub-humid climates (Table 1). The range of ETo is given in Table 2. kRs varied from 0.13 to 0.14 °C−0.5. Daily weather data of Portugal consisted of two datasets relative to the Azores Archipelago and to Continental Portugal. The Azores dataset used in the present study refer to 19 weather stations located in eight islands (Paredes et al., 2018a, b). Data included precipitation (mm), Tmax and Tmin (°C), RH (%), u2 (m s−1), and solar radiation (Rs, W m−2) or sunshine duration (n, hour). All data were collected above green grass. Daily data refers to periods whose lengths varied between 283 and 5478 days. A data quality control was performed according to recommendations by Allen et al. (1998). The yearly average precipitation varies with longitude, from 730 mm year−1 in Santa Maria, the eastmost island, to 1666 mm year−1 in Flores, the west-most island. The weather variables largely vary within the months throughout the year. Air humidity is generally high, with median RH close to 80%. The dominant winds are from SW, with high moisture; winds are often strong, mainly in winter, and show a great variability within months. The variability of solar radiation is high due to effects of variable cloudiness. ETo is greater in summer months and does not change much from an island to another. For all stations, AI ranged from 2.0 to 8.6, thus indicating that a humid climate prevails in all islands. However, kRs varied in a large range, from 0.14 to 0.24 °C−0.5 (Paredes et al., 2018a). The dataset relative to Continental Portugal consists of 37 weather 90

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

stations, 27 from the Portuguese Meteorological Institute (IPMA) and 10 from the Irrigation Operative Center (COTR) and refer to periods of various lengths, from 2004 to 13,111 days. Most of these stations were used in a study on application of ERA-Interim reanalysis data to compute ETo (Paredes et al., 2018c). Data refers to precipitation (mm), Tmax and Tmin (oC), RH (%), u2 (m s−1), and Rs (MJ m-2 day−1) or n (hour). The quality of observed data was previously assessed using the techniques described in FAO56 (Allen et al., 1998). The distribution of weather stations per climate zone is also shown in Table 1. Ranges of ETo and station elevations are summarized in Table 2. kRs ranged from 0.11 to 0.26 °C-0.5.

0.50 recommended by Allen et al. (1998) were used except for the high elevation sites in the Bolivian Altiplano, where calibrated values were adopted (Chipana et al., 2010). Net long wave radiation (Rnl), resulting from the balance between the outgoing long wave radiation emitted by the vegetation and the soil (Rlu↑) and the down-coming long wave radiation from the atmosphere (Rld↓), is:

Rnl =

When full weather data sets were available, the PM-ETo Eq. (1) was computed using the standard methods described in FAO56 (Allen et al., 1998). Eq. (1) was used as standard for assessing the performance of the alternative approaches used to compute ETo with reduced data sets. The saturation vapour pressure (es, kPa) was computed from observed Tmax and Tmin as:

eo (Tmax) + eo (Tmin ) 2

f = ac

(9)

17.27 T T+237.3

+

(10)

50 e o (Tmax )

= 0.34

ea =

RHmax 100

(11)

+ e o (Tmax ) 2

RHmin 100

u2 = uz

(13)

- Tmonth,

i-1)

(20)

4.87 ln ( 67.8 z

5.42 )

(21)

The need to correct Tmin to accurately estimating Tdew in arid areas was early well identified. Following the studies by Todorovic et al. (2013), Raziei and Pereira (2013a), Ren et al. (2016a) and Paredes et al. (2018a), a clear set of temperature corrections was adopted. The latter are defined according to the local aridity index (UNEP, 1997):

(14)

n Ra N

i+1

2.3. Computation of ETo with reduced data sets (PMT approach)

where Rs is the incoming solar radiation and α is the albedo, or canopy reflection coefficient, fixed to 0.23 as for the definition of the grass reference crop. When Rs was not measured, it was calculated from the observed duration of sunshine hours with the Angström (1924) equation:

R s = a s + bs

(19)

0.14 ea

thus from the temperatures of the following and past months, i+1 and i-1 (Allen et al., 1998). To adjust wind speed data, uz obtained by instruments placed at the height zm > 2 m, a logarithmic wind speed profile (Allen et al., 1998) was used to convert uz into u2:

(12)

Rns represents the balance between incoming and reflected solar radiation

Rns = (1- ) R s

(18)

z) R a

Gmonth, i = 0.07 (Tmonth,

thus taking into consideration Tmax and Tmin for increased accuracy (Allen et al., 1998). Net radiation (Rn) was computed as the algebraic sum of the net short wave radiation (Rns) and the net long wave radiation (Rnl), with all radiation units expressed in MJ m−2 day-1:

Rn = Rns + Rnl

5

where ea is the actual vapour pressure of the atmosphere (kPa). The coefficients a1 = 0.34 and b1 = -0.14 are recommended for average atmospheric conditions (Allen et al., 1998). When daily weather data were used, the soil heat flux density (G, MJ m−2 day-1) was assumed null; differently, for monthly calculations G was computed as

For some Portuguese stations, where maximum and minimum RH were available (RHmax and RHmin, %), ea was computed as

e o (Tmin )

(17)

where 0.75 = as + bs (Eq. (15)). This equation is valid for z < 6000 m and low air turbidity. The net emissivity of the surface (ε’) represents the difference from the emissivity of the vegetation and the soil to the effective emissivity of the atmosphere, and is computed as:

RHmean 50 e o (Tmin )

Rs + bc Rso

R so = (0.75+2 × 10

The actual vapour pressure (ea, kPa), when only mean relative humidity (RHmean, %) data were available, as for most data used, was computed as:

ea =

(16)

where the FAO 56 recommended coefficients are ac ≈ 1.35 and bc ≈ -0.35, with ac + bc ≈ 1.0. Daily Rso was computed as a function of site elevation z (m):

where eº(T), the saturation vapour pressure (kPa) for air temperature T (ºC), is given as:

eo (T) = 0.6108 exp

4 4 T Kx + T Kn 2

where f is the cloudiness factor (-), ε’ is the net emissivity of the surface (-), σ is the Stefan-Boltzmann constant = 4.90 × 10-9 MJ m−2 K-4 day-1, and TKx and TKn are respectively Tmax and Tmin expressed in Kelvin (K). The cloudiness factor f is a linear function of the ratio Rs/Rso between actual incoming radiation and clear-sky radiation Rso:

2.2. Computation of ETo with full data sets

es =

f

AI =

P PETTH

(22)

ratio between the long term means of the annual precipitation (P) and the annual potential climatic evapotranspiration (PETTH, Thornthwaite, 1948). AI is computed for the largest number of yearly data available for the site. PETTH may be computed as described by Pereira and Pruitt (2004); in addition, a software to compute PETTH has been proposed by Tigkas et al. (2015). The best approaches for computing Tdew following the above referred studies consist of

(15)

where n is the actual daily duration of sunshine (hour), N is the maximum possible daily duration of sunshine in the same day (hour), n/N is the relative daily sunshine duration (-), Ra is extra-terrestrial radiation (MJ m−2 day-1), as is the coefficient expressing the fraction of extraterrestrial radiation reaching the earth on overcast days (when n = 0), and as+bs is the fraction of Ra reaching the earth on open-sky days (when n = N). Ra and N are computed as a function of the day in the year and the latitude of the site; the default values as = 0.25 and bs =

Tdew = Tmin - aT if AI < 1.00 91

(23)

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

where aT assumes one of the following values depending upon the local AI:

one equation was selected. Using the same predictors, multiple regression equations were also developed for the stations grouped by aridity index into four groups relative to the sites classified as hyperarid and arid, semi-arid, dry and moist sub-humid, and humid. The regression equations were developed with data relative to 555 weather stations from the Med-Climwat data set. 33 outliers were identified, and consequently excluded, using their common definition (Tukey, 1977): any data point more than 1.5 interquartile ranges below the first quartile or above the third quartile. The selected global and AI grouped multiple regression equations were tested with data relative to all datasets described before, thus for a total of 699 sites.

a) Hyper-arid locations, AI < 0.05, aT = 4 °C b) Arid locations, 0.05 ≤ AI < 0.20, aT = 2 °C c) Semiarid locations and dry sub-humid locations, 0.20 ≤ AI < 0.65, aT = 1 °C d) Moist sub-humid locations, 0.65 ≤ AI < 1.00, aT = 0 °C Differently, for humid climates: Tdew = Tmean – aD if AI ≥ 1.00

(24)

with aD = 2 °C for this application. Replacing those Tdew values in the actual vapour pressure equation (Eq. (10)), it results, respectively, the following actual vapour pressure equations to be used when observations of air moisture are lacking, to be used for AI < 1.00

ea = eo (Tdew ) = 0.611 exp

17.27 (Tmin aT) (Tmin aT) + 237.3

(25)

17.27 (Tmean aD) (Tmean aD) + 237.3

(26)

2.4. Accuracy indicators The accuracy of various ETo PMT computation approaches was assessed by comparing ETo PMT results with PM-ETo computed for the same sites. With this purpose, several statistical indicators were used: (1) The regression coefficient (b0) of the regression forced to the origin (FTO, Eisenhauer, 2003) between PM-ETo computed with full data sets, Oi (x), and ETo PMT computed with the PMT approach, Pi (y). A value of b0 = 1.0 indicates that the fitted line is y = x, thus Oi and Pi are similar. A value of b0 > 1.0 suggests overestimation and b0 < 1.0 underestimation. (2) The determination coefficient (R2) of the ordinary least squares (OLS) regression between Oi and Pi. where a R2 close to 1.0 indicates that most of the variation of the observed values is explained by the estimation approach used. However, a high value of R2 is, in itself, insufficient to state that there is good overall agreement between observed and estimated values. (3) The root mean square error (RMSE), which measures the overall differences between observed (Oi) and predicted (Pi) values

or when AI ≥ 1.00

ea = eo (Tdew ) = 0.611 exp

The estimation of wind speed in absence of measurements was performed using the default value of u2 = 2 m s−1. This was the only viable procedure considering the variety of data used belonging to diverse databases. The use of estimated local values revealed not possible. Nevertheless, former studies confirmed that for areas where climate is mild the impact of wind speed on ETo is quite limited. Differently, for arid and windy sites, the use of under-estimated u2 values, e.g., adopting the referred default value, may lead to underestimation of the computed ETo. However, the number of these situations was very reduced in the datasets used in the current study. When radiation or sunshine duration measurements are not available, the PMT method estimates the incoming solar radiation from the HS-Rs Eq. (3), as discussed before. That discussion has shown that the HS-Rs method may not be the best but it is certainly appropriate to compute Rs in absence of observations of radiation or sunshine duration when aiming at estimating ETo. In the current study, the kRs values in Eq. (3) were computed for every site using a trial and error procedure for searching the kRs value that minimizes the difference between ETo computed with full data sets (PM-ETo) and with the PMT approach (ETo PMT), when adopting the Tdew and u2 estimation procedures described above. Considering the studies by Samani (2000), Thepadia and Martinez (2012), Mendicino and Senatore (2013) and Martí et al. (2015), among others, all the computed kRs relative to the Med-Climwat dataset were used in a regression analysis to assess their relationships with various climatic and geographical variables characteristics of the respective sites. The variables used as predictors include averages of TD, Tmean, the ratio Tmean/TD, u2, RH, as well as the aridity index AI and the elevation (z, m), which characterize each site. Regressions were performed either considering those variables alone or in combination with one or two others. All possible single and multiple linear regressions between locally calibrated kRs and those predictors were tested and the regression equations were compared based on several statistical indicators including: the multiple coefficient of correlation, the coefficient of determination (R2), the adjusted R2, the F values in the goodness-of-fit tests and their significance p-values. In addition, for each predictor the regression coefficient, the standard error, the confidence interval and the p-value of the zero null hypothesis were analysed. The multiple regression equations were developed for all MedClimwat sites together and, after analyzing their statistical indicators,

n

RMSE =

i= 1

(Pi

Oi )2

n

0.5

(27)

which should be as small as possible. It has the same units of the variable under analysis. (4) The Nash and Sutcliffe (1970) modelling efficiency (EF, non-dimensional) that provides an indication of the relative magnitude of the mean square error (MSE = RMSE2) relative to the observed data variance: M

EF = 1.0

i= 1 M i= 1

(Oi

Pi )2

(Oi

¯ )2 O

(28)

The maximum value EF = 1.0 can only be achieved if there is a perfect match between all observed (Oi) and predicted (Pi) values, thus when RMSE = 0, R2 = 1.0 and b0 = 1. The closer the values of EF are to 1.0, the better estimators of PM-ETo are the ETo PMT values. As discussed by Legates and McCabe Jr. (1999), negative values of EF indicate that MSE is larger than the observed data variance meaning that the estimation approach is less appropriate for the intended prediction. 3. Results and discussion 3.1. Relating the radiation adjustment coefficient kRs with site weather variables The use of PMT was revised for all datasets. The accuracy indicators relative to ETo-PMT when the radiation coefficient kRs was estimated by trial and error applied to each location are given in Table 3. The

92

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Table 3 Accuracy indicators relative to the computation of ETo-PMT when the radiation coefficient kRs was estimated by trial and error applied to every site of all datasets.

Humid Moist sub-humid Dry sub-humid Semi-arid Arid Hyper-arid All data

Number of observations

b0 (range)

80851 77448 54656 27708 11500 2382 254545

0.94 0.95 0.97 0.96 0.96 0.98 0.94

to to to to to to to

1.03 1.03 1.03 1.04 1.03 1.02 1.04

R2 Mean (s.d.)

RMSE (mm day−1) Mean (s.d.)

EF Mean (s.d.)

0.95 0.98 0.96 0.98 0.97 0.99 0.97

0.27 0.31 0.36 0.41 0.46 0.45 0.35

0.94 0.98 0.95 0.97 0.96 0.99 0.97

( ± 0.11) ( ± 0.03) ( ± 0.09) ( ± 0.02) ( ± 0.05) ( ± 0.02) ( ± 0.07)

( ± 0.20) ( ± 0.16) ( ± 0.17) ( ± 0.16) ( ± 0.22) ( ± 0.20) ( ± 0.19)

( ± 0.12) ( ± 0.04) ( ± 0.12) ( ± 0.03) ( ± 0.06) ( ± 0.03) ( ± 0.08)

s.d – standard deviation.

• (T/TD) (°C), the long term average of the ratios T /(T -T ) • u (m s ), the long term average of the wind speed at 2 m height • RH (%), the long term average of RH • the aridity index AI (Eq. (22)). • the site elevation z (m).

Table 4 Selected statistical indicators of various single and multiple linear regressions between kRs and various predictor variables relative to all sites of the MedClimwat dataset. Predictor variables

avg

2 avg

avg

Statistical indicators Multiple R

Adjusted R2

F value

Significance of F

(T/TD) TD u2 Tmean AI z RH

0.63 0.59 0.57 0.21 0.21 0.19 0.09

0.40 0.35 0.32 0.04 0.04 0.04 0.006

390 315 280 28.3 27.4 21.5 4.8

6.6E-67 9.0E-57 1.4E-51 1.5E-7 2.3E-7 4.3E-6 0.029

TD, u2 TD, RH (T/TD), RH TD, AI TD, z RH, u2

0.74 0.74 0.65 0.60 0.59 0.58

0.54 0.54 0.41 0.36 0.35 0.33

345 345 208 166 157 147

9.7E-100 1.2E-99 5.9E-69 6.4E-48 1.99E-55 2.0E-52

TD, u2, RH u2, RH, (T/TD) AI, TD, RH TD, z, RH AI, TD, u2 TD, z, u2

0.83 0.75 0.74 0.74 0.74 0.74

0.69 0.56 0.54 0.54 0.54 0.54

422 254 231 231 230 230

3.4E-142 3.9E-105 7.8E-99 7.9E-99 1.8 E-98 1.4E-98

avg

kRs = 0.365

min

mean

0.010 TDavg + 0.019 u2 avg

0.002 RHavg

(29)

with all 3 variables being long term averages as defined before. That Eq. (29) refers to a wide panoply of climates, with AI ranging from 0, in various desert sites in Libya and Egypt, to 3.1 in a humid site of Turkey. Therefore, the Med-Climwat sites were grouped into four climate zones and kRs equations relative to the same predictors were developed for those climatic zones resulting in the following equations: a) Humid climate (AI ≥ 1.0)

kRs = 0.519

(°C), the long term average of the temperature differences

0.010 TDavg + 0.019 u2 avg

0.003 RHavg

(30a)

b) Moist and dry sub-humid climates (0.50 ≤ AI < 1.0)

max-Tmin avg

max

The linear regression equations relating kRs with the above defined variables were developed for all sites of the Med-Climwat dataset considering the predictor variables isolated, paired and grouped by 3 as presented in Table 4. Results are presented in a decreasing order of the statistical indicators characterizing each regression for the three cases relative to the number of predictors. Statistical indicators show that best results were achieved when considering three predictors. When only one predictor is considered, the best indicators are (T/TD)avg, TDavg and u2 avg. The first two could be expected considering the common Eqs. (7) and (8). Contrarily to expected (Eqs. (5) and (6)) elevation has quite low effects on the prediction. When two variables are considered together, the best statistical indicators refer to the pairs TDavg - u2 avg and TDavg - RHavg, These results confirm the importance of wind speed, already identified when single variables were considered, and introduce the relative humidity as a variable influencing kRs. The best results (Table 4) were for a combination of three variables TDavg, u2 avg, and RHavg -, which were already identified as best predictors when just one or two variables were considered. Statistical indicators of the linear regression ranked first are definitely better than those of the second ranked equation, which had in common two of the considered variables. Particularly important is the fact that the first ranked equation has an adjusted R2 of 0.69 vs. an adjusted R2 of 0.56, of the second one. In other words, the equation kRs = f(TDavg, u2 avg, RHavg) is able to explain 69% of the variance of kRs while the second one, kRs = f((T/TD)avg, u2 avg, RHavg), explains only 56% of the same variance. That first equation was therefore selected and writes as follows:

regression coefficient b0 varies little around its target value 1.0 and does not show tendency for over- or under-estimation. Moreover, it is associated with very high R2, which indicates that most of the variation of the observed values is explained by the PMT approach used. The RMSE are quite small, with averages varying from 0.27 mm day−1 in case of humid climates to 0.46 mm day−1 for arid climates. All EF values are high, thus indicating that mean square errors are much lower than the variance of observations. These results, particularly RMSE, compare favourably with those in literature, e.g. studies recently performed by Ren et al. (2016a), Almorox et al. (2018), and Paredes et al. (2018b), as well as when assessing the accuracy of ETo computation using reanalysis weather data (Martins et al., 2017; Paredes et al., 2018c). The good accuracy results in Table 3 are certainly related with the accuracy of estimating kRs using a trial and error procedure for every site. These conditions favour to assess the relationships between kRs and main climate variables observed, which may influence its value. As referred before, the considered variables are:

• TD T •T

mean

−1

kRs = 0.396

(°C), the long term average of the mean air temperatures Tmean

93

0.011 TDavg + 0.019 u2 avg

0.002 RHavg

(30b)

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Table 5 Statistical indicators of the selected multiple linear regressions using average TD, u2 and RH as predictors of kRs for all Med-Climwat sites (global equation) and when sites are grouped per climate zones according the Aridity Index AI (climate zones equations). Indicators

Global equation

Number of sites Regression coefficients

β0 β1 β2 β3

Multiple R R2 Adjusted R2 F value Significance of F

Climate zones equations Humid

Moist and dry sub-humid

Semi-arid

Hyper-arid and arid

555 0.365 −0.010 0.019 −0.002

124 0.519 −0.010 0.019 −0.003

263 0.396 −0.011 0.019 −0.002

92 0.388 −0.010 0.022 −0.002

76 0.217 −0.004 0.035 −0.001

0.83 0.70 0.69 421.5 3.4E-142

0.91 0.82 0.82 182.5 3.75E-44

0.88 0.78 0.78 300.4 5.34E-84

0.88 0.77 0.77 100.0 3.01E-28

0.87 0.75 0.74 72.0 1.27E-21

Standard error

Intercept TD u2 RH

0.0104 0.0004 0.0011 0.0001

0.2123 0.0097 0.0020 0.0002

0.0143 0.0005 0.0015 0.0001

0.023 0.001 0.002 0.0003

0.023 0.001 0.003 0.0002

p-value

Intercept TD u2 RH

9.8E-143 5.82E-97 1.12E-54 1.64E-50

5.24E-48 2.67E-19 1.86E-15 1.92E-31

3.74E-79 1.01E-53 3.74E-28 8.73E-29

1.83E-29 6.79E-21 2.94E-14 9.58E-13

1.48E-14 0.0002 6.93E-20 1.04E-07

Confidence interval

Intercept TD u2 RH

0.344, 0.385 −0.011, −0.009 0.017,0.022 −0.002,−0.001

0.477, 0.561 −0.012, −0.009 0.015, 0.023 −0.004, −0.003

0.368, 0.424 −0.012,−0.009 0.016, 0.022 −0.002, −0.002

0.343, 0.434 −0.011, −0.008 0.018,0.027 −0.003,−0.002

0.172, 0.262 −0.006, −0.002 0.030, 0.041 −0.001, −0.0007

c) Semi-arid climates (0.20 ≤ AI < 0.50)

kRs = 0.388

0.010 TDavg + 0.022 u2 avg

0.002 RHavg

coastal areas, where wind is often higher than inland (Allen, 1997; Allen et al., 1998), as well as in studies by Raziei and Pereira (2013a), Ren et al. (2016a) and Paredes et al. (2018a). Finally, considering that RHavg has a negative regression coefficient, that variable may represent the influence of cloudiness and air moisture, which is in agreement with studies referring that the consideration of rainfall or the saturation vapour pressure may favour the results of the HS-Rs model (Hargreaves and Samani, 1985; Droogers and Allen, 2002; Almorox et al., 2011; Woli and Paz, 2011). However, these interpretations have to be considered with care since in a multi-regression all variables play complementary, not individual roles.

(30c)

d) Arid and hyper-arid climates (AI < 0.20)

kRs = 0.217

0.004 TDavg + 0.035 u2 avg

0.001 RHavg

(30d)

The statistical results relative to all 5 equations, the global one (29) and those relative to the climate zones (30a through 30d)), are presented in Table 5. Results show a strong linear relationship between the dependent variable (kRs) and the independent variables (TDavg, u2 avg, RHavg) with multiple coefficient of correlation values higher than 0.83, R2 > 0.70, and adjusted R2 above 0.69. These indicators show that most of the variance of kRs is explained by the linear multiple regression equations, particularly when sites were grouped by climate zone. This could be expected because grouping the sites according to their climate features reduces the variability of the climate variables influencing the computation of kRs. In addition, the overall goodness-of-fit F test gives p-values close to zero, so rejecting the null hypothesis. Likewise, the pvalues for the regression coefficient tests for all variables are almost null. Moreover, the confidence intervals for the regression coefficients do not include the null value, thus assuring that the contribution of the associated predictors are statistically significant. The good results obtained in terms of statistical indicators may be explained by the combined role that the selected variables may play in terms of influencing kRs values. TDavg has a negative regression coefficient which may refer to the fact that when TDavg is high more energy is lost as long wave radiation (see Eq. (16) through (19)). The wind speed average is likely associated with clearness of the atmosphere at the site, which depends upon the wind transport of air moisture away. This fact was noticed when earlier studies defined a larger kRs for

3.2. Using the locally predicted kRs to compute ETo for reduced data sets The good statistical indicators associated with the developed global and climate zones Eqs. (29) and (30) led to a main question: could these equations be used predictively? To answer to this question, ETo PMT were computed using the kRs produced by those Eqs. (29) and (30) and the respective accuracy was assessed against the corresponding PM-ETo. Results of regressing ETo PMT against PM-ETo are shown in Figs. 2 and 3 for all sites of respectively the Med-Climwat dataset and of the Iran, Inner Mongolia, Portugal and Bolivian datasets. Regressions were performed after grouping sites by climate zones. Results in Figs. 2 and 3 are different in terms of R2 because, on the one hand, the Med-Climwat dataset was previously used to derive the kRs regression equations and, on the other hand, data are of different nature, with data of Med-Climwat consisting of observed long-term monthly mean climatic data while data of other datasets are observed monthly or daily weather data. Therefore, dispersion of points around the regression line is much smaller when the Med-Climwat was used. Analyzing Fig. 2, it is apparent that the use of the global predictive

94

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Fig. 2. Comparing ETo PMT with PM-ETo for Med-Climwat monthly data when the kRs is locally calibrated (1st column), derived from the global predictive Eq. (29) (2nd column), or derived from the climate zones predictive equations 30, with data grouped for humid, moist sub-humid, dry sub-humid, semi-arid, arid and hyperarid climates.

95

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Fig. 3. Comparing ETo PMT with PM-ETo for Iran (monthly data), Inner Mongolia (monthly data), Portugal (daily data) and Bolivia (daily data) when the kRs is locally calibrated (1st column), derived from the global predictive Eq. (29) (2nd column), or derived from the climate zones predictive equations 30, with data grouped for humid, moist sub-humid, dry sub-humid, semi-arid, arid and hyper-arid climates.

96

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Table 6 Computing ETo from temperature (ETo-PMT) when the radiation coefficient kRs was estimated from the global predictive equation. Accuracy indicators relative to all climate zones resulting from comparing ETo-PMT with PM-ETo computed with full data sets. b0 range kRs predicted with the global, Eq. (29) Humid (n = 80,851) Moist sub-humid (n = 77,448) Dry sub-humid (n = 54,656) Semi-arid (n = 27,708) Arid (n = 11,500) Hyper-arid (n = 2,382) All data

0.78 0.76 0.90 0.87 0.84 0.83 0.76

to to to to to to to

1.18 1.27 1.18 1.16 1.39 1.18 1.39

kRs predicted with equations 30 for sites grouped per climate zones Humid (n = 80,851) 0.80 to 1.20 Moist sub-humid (n = 77,448) 0.75 to 1.23 Dry sub-humid (n = 54,656) 0.89 to 1.17 Semi-arid (n = 27,708) 0.88 to 1.20 Arid (n = 11,500) 0.85 to 1.21 Hyper-arid (n = 2,382) 0.88 to 1.09 All data 0.75 to 1.23

R2 mean (s.d)

RMSE (mm day−1) mean (s.d)

EF mean (s.d)

0.95 0.98 0.96 0.98 0.97 0.99 0.97

( ± 0.11) ( ± 0.03) ( ± 0.09) ( ± 0.02) ( ± 0.05) ( ± 0.02) ( ± 0.07)

0.40 0.38 0.42 0.46 0.58 0.62 0.44

( ± 0.23) ( ± 0.17) ( ± 0.17) ( ± 0.18) ( ± 0.29) ( ± 0.23) ( ± 0.21)

0.93 0.97 0.95 0.97 0.95 0.98 0.95

( ± 0.16) ( ± 0.04) ( ± 0.14) ( ± 0.04) ( ± 0.10) ( ± 0.04) ( ± 0.10)

0.95 0.98 0.96 0.98 0.97 0.99 0.97

( ± 0.11) ( ± 0.03) ( ± 0.09) ( ± 0.02) ( ± 0.06) ( ± 0.02) ( ± 0.07)

0.34 0.38 0.41 0.47 0.54 0.54 0.41

( ± 0.21) ( ± 0.18) ( ± 0.17) ( ± 0.18) ( ± 0.24) ( ± 0.20) ( ± 0.20)

0.93 0.97 0.95 0.96 0.95 0.98 0.96

( ± 0.16) ( ± 0.04) ( ± 0.13) ( ± 0.05) ( ± 0.08) ( ± 0.03) ( ± 0.10)

equation leads to a decrease of b0 for the humid climates, i.e., a tendency for under-estimating ETo PMT, but no tendencies were detected for the other climates. When using the climate zones predictive Eqs. (30a) to (30d), no tendencies are detected (Fig. 2), which indicates an advantage of these climate focused equations. The R2 values are similar for all three PMT approaches (Fig. 2) except for the hyper-arid sites, with R2 slightly decreasing when the global predictive equation is used. This increased dispersion around the regression line refers to the higher values of ETo, when aridity is highest. Again, differences relative to the case when kRs is obtained using the climate zones predictive equations are small. Results in Fig. 3 relative to the datasets consisting of daily and monthly observed data from Iran, Inner Mongolia, Portugal and Bolivia are quite similar to those analyzed above despite R2 are small then those relative to Med-Climwat, particularly in case of humid climates. However, regression coefficients b0 are generally close to 1.0 except for hyper-arid climates, where there is a trend for under-estimation of ETo PMT, mainly when the global predictive equation is used. Results for all datasets relative to using PMT with kRs obtained from the global and the climate zones predictive Eqs. (29) and 30 are given in Table 6. The regression coefficients b0 show a wider range when the global predictive equation is used, with highest difference for hyperarid sites, where that range changes from [0.83, 1.18] to [0.88, 1.09]. Changes in R2 are very small. The modelling efficiency EF is generally high and differences between using kRs from the global or the climate zones (Eqs. (29) and 30) are negligible. Differently, there are clear improvements in RMSE, particularly for the humid and the hyper-arid climates, when the climate zones predictive equations are used, with RMSE decreasing from 0.40 to 0.34 mm day−1 in case of humid climates and from 0.62 to 0.54 mm day−1 in hyper-arid climates. The results analyzed above, particularly relative to RMSE, compare well with those reported in literature, e.g. studies recently performed (Thepadia and Martinez, 2012; Ren et al., 2016a; Almorox et al., 2018; Paredes et al., 2018b), as well as referring to the accuracy of ETo computed with reanalysis weather data (Martins et al., 2017; Paredes et al., 2018c). A more detailed analysis of the behavior of the accuracy indicators b0 and RMSE relative to using predictively the global or the climate zones predictive Eqs. (29) and 30 to obtain kRs for computing ETo PMT is presented in Fig. 4. It compares the frequency distribution of b0 and RMSE when those equations are used for humid, moist and dry sub-

humid, semiarid, and arid and hyper-arid climates. Relative to b0, it is evident that using the global equation the frequency of under-estimation cases is higher, particularly when comparing with humid climates. Relative to RMSE, differences in frequencies between using global or climate zones kRs predictive equations are small, but larger RMSE are less frequent when using equations 30. However, results are not definitely in favor of the climate zones predictive equations. A test was performed comparing the adoption of local/regional wind speed data vs. default u2 = 2 m s−1 in case of hyper-arid windy sites of Inner Mongolia. Results in Fig. 5 evidence that regression coefficients became closer to 1.0 when adopting a local/regional long term estimate of u2 comparatively to the common default value. In other words, the detected under-estimate of ETo PMT was overcome when using local wind estimates. However, this improvement was not evident for other cases (data not shown). Following results analysed above for a wide set of weather data relative to diverse climates and time steps calculations, it was possible to define an improved methodology to compute ETo with reduced data sets, commonly referred as PMT, which is easy to implement in Excel. It is summarized in Fig. 6 and described in the following; 1st step: Determination of the local aridity index (AI, Eq. (22)) using the long term averages of the annual precipitation and of the PETTH; the latter computed from long term averages of observed Tmax and Tmin. 2nd step: Depending on AI, computing the actual vapour pressure (ea) as: a) If AI < 1.0, using Tmin observed and aT ranging from 0 ºC to 4 ºC selected according to AI

ea = 0.611 exp

17.27 (Tmin aT) (Tmin aT) + 237.3

(25bis)

b) If AI ≥ 1.0, using observed Tmean = (Tmax+Tmin)/2 and aD = 2 ºC

ea = 0.611 exp

17.27 (Tmean aD) (Tmean aD) + 237.3

(26bis)

rd

3 step: Using the observed Tmax and Tmin, calculation of the saturated vapour pressure (es)

es =

97

eo (Tmax) + eo (Tmin ) 2

(9bis)

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Fig. 4. Frequency distribution of b0 and RMSE indicators when estimating ETo PMT with kRs obtained from the (i) global predictive equation ( ) and from (ii) climate zones predictive equations ( ) for various climates grouped as (a) humid, (b) moist and dry sub-humid, (c) semi-arid, and (d) arid and hyper-arid, applied to Iran, Inner Mongolia, Portugal and Bolivia datasets.

4th step: with results of previous steps, calculation of the saturation vapour pressure deficit (VPD = es-ea). 5th step: Selection of the best estimator for wind speed u2, either a local or regional value, or a default u2 value (e.g., u2 = 2 m s−1). 6th step: Calculate TDavg from the long term observed TD = Tmax Tmin. 7th step: Using the long term averages of ea and es, compute RHavg

RHavg = 100

8th step: From wind speed observations in a nearby weather station, or using a u2 value characterizing wind speed at regional or local level, or even adopting a default value, define an average value u2 avg 9th step: Estimate the local kRs value after selecting one of the following equations: i) the global equation

ea avg es avg

(31)

kRs = 0.365

98

0.010 TDavg + 0.019 u2 avg

0.002 RHavg

(29bis)

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Fig. 5. Comparing ETo PMT with for PM-ETo hyper-arid windy sites of Inner Mongolia (monthly data) when using different approaches for estimation of the kRs coefficient using a) default u2 = 2 m s−1, and b) local long term estimates of u2 values.

ii) the climate zone equations according to the AI determined in the 1st step:

4. Conclusions The PMT approach does not receive the preference of users due to its computational burden to estimate Tdew aiming at computing the actual vapor pressure, thus VPD, and to search the best empirical radiation coefficient kRs through a trial and error procedure for estimating Rs from temperature. If searching for an appropriate estimation of Tdew is relatively easy, it is not the case for kRs. However, various polynomial expressions have been proposed in the literature for estimating kRs from weather variables characterizing the sites under study but their use predictively has rarely been assessed, e.g., by Thepadia and Martinez (2012). Moreover, related literature never reported about the accuracy of ETo estimates when kRs is derived from a predictive equation directly or through the estimation of the Hargreaves coefficient (CHS). Advancing from past research, this study demonstrates that Tdew and kRs may be estimated for any site when a few variables characterizing the climate of the site are available, so making it possible to ease the computation of ETo with reduced data sets, at limit when these consist of Tmax and Tmin only. Results of this study confirmed the adequateness of computing Tdew = Tmin – aT, with aT varying from 0 to 4 °C when the climate varies from sub-humid to hyper-arid, and Tdew = Tmean – 2 in case of humid climates. This approach allows to overcome influences of site aridity and local advection on the observed temperature, i.e., allows to correct effects of non-reference weather sites when computing ETo. Relative to the Rs estimation from temperature, the study allowed to conclude that, following previous studies with the Hargreaves coefficient (Eq. 4), it was possible to define predictive equations for kRs from weather variables characteristic of the site or the region. These

Humid climate (AI ≥ 1.0)

kRs = 0.519

0.010 TDavg + 0.019 u2 avg

0.003 RHavg

(30a bis)

Moist and dry sub-humid climates (0.50 ≤ AI < 1.0)

kRs = 0.396

0.011 TDavg + 0.019 u2 avg

0.002 RHavg

(30b bis)

Semi-arid climates (0.20 ≤ AI < 0.50)

kRs = 0.388

0.010 TDavg + 0.022 u2 avg

0.002 RHavg

(30c bis)

Arid and hyper-arid climates (AI < 0.20)

kRs = 0.217

0.004 TDavg + 0.035 u2 avg

0.001 RHavg

(30d bis)

10th step: using the estimated kRs and observed temperature data, compute

R s = k R s (Tmax

(3bis)

Tmin)0.5R a

11th step: Using observed temperature data and the estimated VPD, u2 and Rs, compute ETo

ETo =

0.408

(Rn G) + +

900 u (e Tmean + 273 2 s

ea )

(1 + 0.34 u2 )

(1bis)

where Rn is computed from Rs with Eq. (13).

99

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

Fig. 6. Flux diagram of the PMT approach with estimation of Tdew from Tmin or Tmean depending on the Aridity Index (AI), using a default or average wind speed value and computing the incoming solar radiation with the HS-Rs equation with kRs derived from a predictive Eq. (29) or 30 depending upon the climate through AI.

equations may explain a large fraction of the variance of kRs considering all climates together or grouped according to AI. Moreover, the test of these equations for all sites provided for good statistical indicators when comparing ETo computed with the PMT approach using the kRs predictive equations comparatively to the ETo computed with full data sets. Better results were obtained when kRs predictive equations referred to the climate zones. The PMT approach resulted easy to compute

and related errors have shown not to be much larger than those resulting from computing kRs through a trial and error procedure, which is much more demanding. These results are of great importance for extensively using the PMT approach when radiation is not available, as well as when data sets include only Tmax and Tmin. Further research is required to compare the simplified PMT method with other approaches applicable to reduced data sets such as the use of 100

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira

reanalysis products (Martins et al., 2017; Paredes et al., 2018c) and the use of satellite data (Mokhtari et al., 2018; Trigo et al., 2018). Innovative approaches are likely of main interest when to be used in real time.

for the Northern Great Plains using maximum and minimum temperature. Agron. J. 95, 323–328. Jabloun, M., Sahli, A., 2008. Evaluation of FAO-56 methodology for estimating reference evapotranspiration using limited climatic data: application to Tunisia. Agric. Water Manage. 95, 707–715. Jahani, B., Dinpashoh, Y., Nafchi, A.R., 2017. Evaluation and development of empirical models for estimating daily solar radiation. Renew. Sust. Energ. Rev. 73, 878–891. Kimball, J.S., Running, S.W., Nemani, R., 1997. An improved method for estimating surface humidity from daily minimum temperature Agric. For. Meteorol. 85, 87–98. Legates, D.R., McCabe Jr, G.J., 1999. Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour. Res. 35, 233–241. Liu, Y., Pereira, L.S., 2001. Calculation methods for reference evapotranspiration with limited weather data. J. Hydraul. Eng. 3, 11–17 in Chinese. Liu, X., Mei, X., Li, Y., Wang, Q., Jensen, J.R., Zhang, Y., Porter, J.R., 2009. Evaluation of temperature-based global solar radiation models in China. Agric. Forest Meteorol. 149, 1433–1446. Lobit, P., López Pérez, L., Lhomme, J.P., Gómez Tagle, A., 2018a. Retrieving air humidity, global solar radiation, and reference evapotranspiration from daily temperatures: development and validation of new methods for Mexico. Part I: humidity. Theor. Appl. Climatol. 133, 751–762. Lobit, P., López Pérez, L., Lhomme, J.P., 2018b. Retrieving air humidity, global solar radiation, and reference evapotranspiration from daily temperatures: development and validation of new methods for Mexico. Part II: radiation. Theor. Appl. Climatol. 133, 799–810. Martí, P., Zarzo, M., Vanderlinden, K., Girona, J., 2015. Parametric expressions for the adjusted Hargreaves coefficient in Eastern Spain. J. Hydrol. (Amst) 529, 1713–1724. Martins, D.S., Paredes, P., Raziei, T., Pires, C., Cadima, J., Pereira, L.S., 2017. Assessing reference evapotranspiration estimation from reanalysis weather products. An application to the Iberian Peninsula. Int. J. Climatol. 37, 2378–2397. Mendicino, G., Senatore, A., 2013. Regionalization of the Hargreaves coefficient for the assessment of distributed reference evapotranspiration in southern Italy. J. Irrig. Drain. Eng. 139, 349–362. Meza, F., Varas, E., 2000. Estimation of mean monthly solar global radiation as a function of temperature. Agric. Forest Meteorol. 100, 231–241. Mokhtari, A., Noory, H., Vazifedoust, M., 2018. Performance of different surface incoming solar radiation models and their impacts on reference evapotranspiration. Water Resour. Manage. 32, 3053–3070. Monteith, J.L., 1965. Evaporation and Environment. The State and Movement of Water in Living Organisms, 19th Symp. of Soc. Exp. Biol. Cambridge University Press, Cambridge, pp. pp. 205–234. Moradi, I., Mueller, R., Perez, R., 2014. Retrieving daily global solar radiation from routine climate variables. Theor. Appl. Climatol. 116, 661–669. Nash, J.E., Sutcliffe, J.V., 1970. River flow forecasting through conceptual models: part 1. A discussion of principles. J. Hydrol. (Amst) 10, 282–290. Paredes, P., Fontes, J.C., Azevedo, E.B., Pereira, L.S., 2018a. Daily reference crop evapotranspiration with reduced data sets in the humid environments of Azores islands using estimates of actual vapour pressure, solar radiation and wind speed. Theor. Appl. Climatol. 134, 1115–1133. Paredes, P., Fontes, J.C., Azevedo, E.B., Pereira, L.S., 2018b. Daily reference crop evapotranspiration in the humid environments of Azores islands using reduced data sets. Accuracy of FAO PM temperature and Hargreaves-Samani methods. Theor. Appl. Climatol. 134, 595–611. Paredes, P., Martins, D., Pereira, L.S., Cadima, J., Pires, C., 2018c. Accuracy of daily estimation of grass reference evapotranspiration using ERAInterim reanalysis products with assessment of alternative bias correction schemes. Agric. Water Manage. 210, 340–353. Penman, H.L., 1948. Natural evaporation from open water, bare soil and grass. Proc. Royal Soc., Ser A, London 193, 120–145. Pereira, L.S., 2017. Water, agriculture and food: challenges and issues. Water Resour. Manage. 31, 2985–2999. Pereira, A., Pruitt, W., 2004. Adaptation of the Thornthwaite scheme for estimating daily reference evapotranspiration. Agric. Water Manage 66, 251–257. Pereira, L.S., Perrier, A., Allen, R.G., Alves, I., 1999. Evapotranspiration: review of concepts and future trends. J. Irrig. Drain. Eng. 125 (2), 45–51. Pereira, L.S., Cai, L.G., Hann, M.J., 2003. Farm water and soil management for improved water use in the North China Plain. Irrig. Drain. 52 (4), 299–317. Pereira, L.S., Allen, R.G., Smith, M., Raes, D., 2015. Crop evapotranspiration estimation with FAO56: past and future. Agric. Water Manage. 147, 4–20. Popova, Z., Kercheva, M., Pereira, L.S., 2006. Validation of the FAO methodology for computing ETo with missing climatic data. Application to South Bulgaria. Irrig. Drain. 55 (2), 201–215. Quej, V.H., Almorox, J., Ibrakhimo, M., Saito, L., 2017. Estimating daily global solar radiation by day of the year in six cities located in the Yucatán Peninsula. Mexico. J. Clean. Prod. 141, 75–82. Raziei, T., Pereira, L.S., 2013a. Estimation of ETo with Hargreaves-Samani and FAO-PM temperature methods for a wide range of climates in Iran. Agric. Water Manage. 121, 1–18. Raziei, T., Pereira, L.S., 2013b. Spatial variability analysis of reference evapotranspiration in Iran utilizing fine resolution gridded datasets. Agric. Water Manage. 126, 104–118. Ren, X., Qu, Z., Martins, D.S., Paredes, P., Pereira, L.S., 2016a. Daily reference evapotranspiration for hyper-arid to moist sub-humid climates in Inner Mongolia, china: I. Assessing temperature methods and spatial variability. Water Resour. Manage. 30, 3769–3791. Ren, X., Martins, D.S., Qu, Z., Paredes, P., Pereira, L.S., 2016b. Daily reference evapotranspiration for hyper-arid to moist sub-humid climates in Inner Mongolia, china: II. Trends of ETo and weather variables and related spatial patterns. Water Resour

Acknowledgements The support of the Fundação para a Ciência e a Tecnologia, Portugal, through the Post-Doc grant SFRH/BPD/102478/2014 to the first author and the grant attributed to research unit LEAF (UID/AGR/ 04129/2013) are acknowledged. References Abraha, M.G., Savage, M.J., 2008. Comparison of estimates of daily solar radiation from air temperature range for application in crop simulations. Agric. Forest Meteorol. 148, 401–416. Aladenola, O.O., Madramootoo, C.A., 2014. Evaluation of solar radiation estimation methods for reference evapotranspiration estimation in Canada. Theor. Appl. Climatol. 118, 377–385. Allen, R.G., 1996. Assessing integrity of weather data for reference evapotranspiration estimation. J. Irrig. Drain. Eng. 122 (2), 97–106. Allen, R.G., 1997. Self-calibrating method for estimating solar radiation from air temperature. J. Hydrol. Eng. 2 (2), 56–67. Allen, R.G., Pereira, L.S., Raes, D., Smith, M., 1998. Crop Evapotranspiration. Guidelines for Computing Crop Water Requirements. Irrigation and Drainage Paper 56. FAO Rome, 300 p. Allen, R.G., Pruitt, W.O., Wright, J.L., Howell, T.A., Ventura, F., Snyder, R., Itenfisu, D., Steduto, P., Berengena, J., Baselga Yrisarry, J., Smith, M., Pereira, L.S., Raes, D., Perrier, A., Alves, I., Walter, I., Elliott, R., 2006. A recommendation on standardized surface resistance for hourly calculation of reference ETo by the FAO56 Penman–Monteith method. Agric. Water Manage. 81, 1–22. Allen, R.G., Pereira, L.S., Howell, T.A., Jensen, M.E., 2011. Evapotranspiration information reporting: I. Factors governing measurement accuracy. Agric. Water Manage. 98 (6), 899–920. Almorox, J., Hontoria, C., Benito, M., 2011. Models for obtaining daily global solar radiation with measured air temperature data in Madrid (Spain). Acs Appl. Energy Mater. 88, 1703–1709. Almorox, J., Bocco, M., Willington, E., 2013. Estimation of daily global solar radiation from measured temperatures at Cañadade Luque, Córdoba. Argentina. Renew. Energ. 60, 382–387. Almorox, J., Senatore, A., Quej, V.H., Mendicino, G., 2018. Worldwide assessment of the Penman–Monteith temperature approach for the estimation of monthly reference evapotranspiration. Theor. Appl. Climatol. 131, 693–703. Angström, A., 1924. Solar and terrestrial radiation. Q. J. R. Meteorol. Soc. 50, 121–125. Annandale, J.G., Jovanovic, N.Z., Benadé, N., Allen, R.G., 2002. Software for missing data error analysis of Penman–Monteith reference evapotranspiration. Irrig. Sci. 21, 57–67. Bandyopadhyay, A., Bhadra, A., Raghuwanshi, N.S., Singh, R., 2008. Estimation of monthly solar radiation from measured air temperature extremes. Agric. For. Meteorol. 148, 1707–1718. Besharat, F., Dehghan, A.A., Faghih, A.R., 2013. Empirical models for estimating global solar radiation: a review and case study. Renew. Sust. Energ. Rev. 21, 798–821. Bristow, K.L., Campbell, G.S., 1984. On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric. For. Meteorol. 31, 159–166. Cai, J., Liu, Y., Lei, T., Pereira, L.S., 2007. Estimating reference evapotranspiration with the FAO Penman–Monteith equation using daily weather forecast messages. Agric. For. Meteorol. 145, 22–35. Chen, R., Ersi, K., Yang, J., Lu, S., Zhao, W., 2004. Validation of five global radiation models with measured daily data in China. Energ. Convers. Manage. 45, 1759–1769. Chipana, R., Yujra, R., Paredes, P., Pereira, L.S., 2010. Determinación de la evapotranspiración de referencia con datos limitados en zonas de altura del Altiplano Boliviano aplicando la metodología de la FAO y la ecuación de Hargreaves-Samani. I Congreso. Boliviano De Riego Y Drenaje, Fac. Agronomia. Universidad Mayor de San Andrés, La Paz paper CIDAB-S613-C6. Córdoba, M., Carrillo-Rojas, G., Crespo, P., Wilcox, B., Célleri, R., 2015. Evaluation of the Penman-Monteith (FAO 56 PM) method for calculating reference evapotranspiration using limited data. Application to the wet páramo of Southern Ecuador. Mt. Res. Dev. 35 (3), 230–239. Droogers, P., Allen, R.G., 2002. Estimating reference evapotranspiration under inaccurate data conditions. Irrig. Drain. Syst. Eng. 16, 33–45. Eisenhauer, J.G., 2003. Regression through the origin. Teach. Stat. 25 (3), 76–80. Hargreaves, G.H., Allen, R.G., 2003. History and evaluation of Hargreaves evapotranspiration equation. J. Irrig. Drain. Eng. 129, 53–63. Hargreaves, G.H., Samani, Z.A., 1982. Estimating potential evapotranspiration. J. Irrig. Drain. Eng. 108, 225–230. Hargreaves, G.H., Samani, Z.A., 1985. Reference crop evapotranspiration from temperature. Appl. Eng. Agric. 1 (2), 96–99. Hargreaves, G.L., Hargreaves, G.H., Riley, J.P., 1985. Irrigation water requirements for SENEGAL River basin. J. Irrig. Drain. Eng. 111, 265–275. Hubbard, K.G., Mahmood, R., Carlson, C., 2003. Estimating daily dew point temperature

101

Agricultural Water Management 215 (2019) 86–102

P. Paredes, L.S. Pereira Manage. 30, 3793–3814. Samani, Z., 2000. Estimating solar radiation and evapotranspiration using minimum climatological data. J. Irrig. Drain. Eng. 126, 265–267. Slatyer, R.O., McIlroy, I.C., 1961. Evaporation and the Principles of Its Measurement. in: Practical Micrometeorology. CSIRO (Australia) and UNESCO, Paris. Smith, M., 1993. CLIMWAT for CROPWAT: A climatic database for irrigation planning and management. FAO Irrigation and Drainage Paper No. 49, Rome. (CLIMWAT 2.0). http://www.fao.org/land-water/databases-and-software/climwat-for-cropwat/en/. Smith, M., Allen, R.G., Monteith, J., Perrier, A., Pereira, L.S., Segeren, A., 1991. Report of the Expert Consultation on Procedures for Revision of FAO Guidelines for Prediction of Crop Water Requirements. UN-FAO, Rome, Italy 54p. Temesgen, B., Allen, R.G., Jensen, D.T., 1999. Adjusting temperature parameters to reflect well-watered conditions. J. Irrig. Drain. Eng. 125 (1), 26–33. Thepadia, M., Martinez, C.J., 2012. Regional calibration of solar radiation and reference evapotranspiration estimates with minimal data in Florida. J. Irrig. Drain. Eng. 138 (2), 111–119. Thornthwaite, C.W., 1948. An approach toward a rational classification of climate. Geogr. Rev. (Oxf) 38, 55–94. Tigkas, D., Vangelis, H., Tsakiris, G., 2015. DrinC: a software for drought analysis based on drought indices. Earth Sci. Inf. 8, 697–709.

Todorovic, M., Karic, B., Pereira, L.S., 2013. Reference evapotranspiration estimate with limited weather data across a range of Mediterranean climates. J. Hydrol. (Amst) 481, 166–176. Trigo, I.F., de Bruin, H., Beyrich, F., Bosveld, F.C., Gavilán, P., Groh, J., López-Urrea, R., 2018. Validation of reference evapotranspiration from Meteosat Second Generation (MSG) observations. Agric. Forest Meteorol. 259, 271–285. Trnka, M., Eitzinger, J., Kapler, P., Dubrovský, M., Semerádová, D., Žalud, Z., Formayer, H., 2007. Effect of estimated daily global solar radiation data on the results of crop growth models. Sensors 7, 2330–2362. Tukey, J.W., 1977. Exploratory Data Analysis. Addison-Wesley, Reading, MA. UNEP, 1997. World Atlas of Desertification, 2nd edn. United Nations Environment Programme, Arnold, London, pp. 182 p. Upreti, H., Ojha, C.S.P., 2017. Estimation of relative humidity and dew point temperature using limited meteorological data. J. Irrig. Drain Eng. 143 (9), 05017005. Vanderlinden, K., Giráldez, J.V., Van Meirvenne, M., 2004. Assessing reference evapotranspiration by the Hargreaves method in Southern Spain. J. Irrig. Drain. Eng. 130, 184–191. Woli, P., Paz, J.O., 2011. Evaluation of various methods for estimating global solar radiation in the Southeastern United States. J. Appl. Meteorol. Clim. 51, 972–985.

102