1 Introduction

The reference crop evapotranspiration (ETo), often named potential ET in hydrologic and climatologic studies, is a main factor for characterizing the local climate and for computing crop and vegetation evapotranspiration. Its accurate estimation is relevant in several domains such as crop water management, irrigation planning and management, hydrologic and water balance studies, climate characterization, and climate change analysis as revised by Pereira et al. (2015) and Pereira (2017). This is the case of the Azores islands, where climate studies with model CIELO (Azevedo et al. 1999) would benefit if complemented with ETo information when only reduced data sets are available as discussed below in Section 2.1.

Grass reference ETo was defined by FAO after parameterizing the Penman-Monteith equation (Monteith 1965) for a cool season grass (Smith et al. 1991). ETo was then 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 r s = 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). Later studies confirmed the appropriateness of using a daily fixed surface resistance of 70 s m−1 (Lecina et al. 2003; Steduto et al. 2003). This definition allowed a clear distinction relative to the Penman’s potential ET (Penman 1948) and the equilibrium evaporation (Slatyer and McIlroy 1961), thus assuming a well-defined relationship between ETo and actual ET of a vegetation surface (Pereira et al. 1999).This relationship bases upon the aerodynamic and surface resistances of the reference grass crop and of the considered vegetation, thus theoretically supporting the concept and use of crop coefficients (Pereira et al. 1999). The Penman-Monteith ETo (PM-ETo) is therefore defined by an equation derived from the Penman-Monteith combination equation when parameterized as referred above (Allen et al. 1998). Daily ETo (mm day−1) is therefore given as

$$ {\mathsf{ET}}_{\mathsf{o}}=\frac{\mathsf{0.408}\Delta \left({R}_{\mathsf{n}}-G\right)+\gamma\;\frac{\mathsf{900}}{T_{\mathsf{mean}}+\mathsf{273}}{u}_{\mathsf{2}}\left({e}_{\mathsf{s}}-{e}_{\mathsf{a}}\right)}{\Delta +\gamma\;\left(\mathsf{1}+\mathsf{0.34}\;{u}_{\mathsf{2}}\right)} $$
(1)

where R n 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), T mean is the mean daily air temperature at 2 m height (°C), u 2 is the wind speed at 2 m height (m s−1), e s is the saturation vapor pressure at 2 m height (kPa), e a is the actual vapor pressure at 2 m height (kPa), VPD or e s−e a is the saturation vapor pressure deficit (kPa), Δ is the slope vapor pressure curve (kPa °C−1), and γ is the psychrometric constant (kPa °C−1). For daily computations, G equals zero as the magnitude of daily soil heat flux beneath the grass reference surface is very small (Allen et al. 1998). Computations therefore require observed data on maximum and minimum temperature (T max and T min), solar radiation (R s) or sunshine duration (n), air relative humidity (RH) or psychrometric data, and wind speed (u 2). While T max and T min records are commonly well observed at weather station networks in most countries, the other variables are often not easily available with good quality, or are available for only short periods of time, and/or data sets have frequent gaps; in addition, their acquisition may be very expensive.

To face conditions of incomplete/reduced data sets, methods were proposed by Allen et al. (1998) for estimation of the missing variables R s, e a, and/or u 2. When relative humidity data or psychrometric observations are missing, Allen et al. (1998), following a previous assessment by Jensen et al. (1997), recommended to estimate actual vapor pressure (e a) assuming that T min could be an acceptable estimate of dew point temperature (T dew):

$$ {\mathit{\mathsf{e}}}_{\mathsf{a}}={\mathit{\mathsf{e}}}^{\mathsf{o}}\left({\mathit{\mathsf{T}}}_{\mathsf{dew}}\right)=\mathsf{0.611}\;\mathsf{\exp}\left[\frac{\mathsf{17.27}\;{\mathit{\mathsf{T}}}_{\mathsf{min}}}{{\mathit{\mathsf{T}}}_{\mathsf{min}}+\mathsf{237.3}}\right] $$
(2)

Nevertheless, Allen et al. (1998) referred the need to correct weather data measured in weather stations not respecting the assumption that collected data refer to vertical fluxes of heat and water vapor originated by an extended green grass vegetated surface (Allen 1996). In addition, Kimball et al. (1997) observed that (a) T min exceeds T dew in arid sites leading to average daily differences of 0.8 to 1.2 kPa between e a computed from T min and T dew; (b) in sites with semiarid climate there, is seasonality in the differences between T min and T dew resulting seasonal differences between respective e a values, varying from 0.1 to 0.6 kPa for winter and summer months, respectively; (c) smaller differences between T min and T dew occurred in other less arid climates, e.g., coastal Mediterranean, humid continental, and humid subtropical conditions, with average daily differences of less than 0.3 kPa between e a computed from T min and T dew. The need to correct T min to estimate T dew in arid areas was well identified (Temesgen et al. 1999) but, for a decade, literature has been reporting that users were assuming T dew = T min. This may be due to the fact that, with few exceptions, researchers were focusing on alternative ET equations or numerical methods that do not require computation of e a, or because Allen et al. (1998) did not propose a well-defined procedure but to perform corrections to T min after results of data analysis. Later, studies by Todorovic et al. (2013), Raziei and Pereira (2013) and Ren et al. (2016) developed a clear set of corrections for T min relative to various arid climatic conditions and corrections for the mean daily temperature for estimating T dew in sub-humid and humid climates where T dew is likely above T min. These corrections were set according to the UNEP aridity index (AI; UNEP 1997), which is defined by the ratio between the mean annual precipitation (P) and the mean annual potential climatic evapotranspiration (PETTH, Thornthwaite 1948).

Corrections for aridity consist of

$$ {T}_{\mathrm{dew}}={T}_{\mathrm{min}}-{\mathrm{a}}_{\mathrm{T}} $$
(3)

where the correction factor a T varies with the climate aridity of the station:

  1. a)

    Hyper-arid locations, AI < 0.05, a T = 4 °C,

  2. b)

    Arid locations, AI from 0.05 to 0.20, a T = 2 °C,

  3. c)

    Semiarid locations, AI from 0.20 to 0.50, a T = 1 °C,

  4. d)

    Dry sub-humid locations, AI from 0.50 to 0.65, a T = 1 °C.

Considering the relations for T dew in moist air proposed by Lawrence (2005), for moist sub-humid and humid climates and when the mean temperature is very low, T dew was empirically approximated (Todorovic et al. 2013) by

$$ {T}_{\mathrm{d}\mathrm{ew}}={T}_{\mathrm{mean}}-{\mathrm{a}}_{\mathrm{d}} $$
(4)

where a d = 2 °C when 0.8 < P/PETTH < 1.0 and a d = 1 °C if P/PETTH > 1.0. However, later studies (Ren et al. 2016) have shown that a d varies with factors other than aridity and, depending upon the accuracy requirements, its calibration is needed. It results the equation

$$ {\mathit{\mathsf{e}}}_{\mathsf{a}}={\mathit{\mathsf{e}}}^{\mathsf{o}}\left({T}_{\mathsf{d}\mathsf{ew}}\right)=\mathsf{0.611}\;\mathsf{\exp}\left[\frac{\mathsf{17.27}\;\left({T}_{\mathsf{mean}}-{\mathit{\mathsf{a}}}_{\mathsf{d}}\right)}{\left({T}_{\mathsf{mean}}-{\mathit{\mathsf{a}}}_{\mathsf{d}}\right)+\mathsf{237.3}}\right] $$
(5)

where T dew is replaced by its value in Eq. 4.

Allen (1997) and Allen et al. (1998) proposed to estimate R s for using with the PM-ETo equation (Eq. 1) adopting the R s predictive equation by Hargreaves and Samani (1982) that expresses R s as a linear function of the square root of the temperature difference T max−T min:

$$ {\mathit{\mathsf{R}}}_{\mathsf{s}}={\mathsf{k}}_{{\mathit{\mathsf{R}}}_{\mathsf{s}}}{\left({T}_{\mathsf{max}}-{T}_{\mathsf{min}}\right)}^{\mathsf{0.5}}{R}_{\mathsf{a}} $$
(6)

where k Rs is an empirical radiation adjustment coefficient (°C−0.5) and R a is the extraterrestrial radiation (MJ m2 day−1). Comparative studies by Abraha and Savage (2008), Bandyopadhyay et al. (2008) and Aladenola and Madramootoo (2014) reported good performance of the temperature difference Eq. 6 relative to other models.

The empirical coefficient k Rs was originally considered to range 0.16 to 0.19 °C−0.5, respectively for “interior” or “coastal” regions (Allen 1997; Allen et al. 1998).However, k Rs is supposed to vary with altitude, thus reflecting the air pressure changes as for the volumetric heat capacity of the atmosphere (Allen 1997), which increases linearly with the volumetric mass density and thus the atmospheric pressure (Turbet et al. 2017). Thus, k Rs would vary spatially not only with site elevation but also with the distance to sea as earlier discussed by Allen et al. (1998) and is compatible with influences of air moisture on the volumetric heat capacity of the atmosphere. In addition, k Rs was observed to increase with the aridity of the site and with wind speed (Raziei and Pereira 2013; Ren et al. 2016). Other factors may also influence k Rs. Samani (2000) developed a quadratic relationship between k Rs and the temperature difference T max−T min, but this approach does not contribute to explain the spatial variability of k Rs. The effect of altitude was considered in the relationship between k Rs and the square root of the ratio between the atmospheric pressure at the location and at sea level (Allen 1997). Annandale et al. (2002) also developed a decreasing relationship between k Rs and elevation to account for the effects of reduced atmospheric thickness on R s. Therefore, for island locations, where climate is always influenced by the sea proximity, it is required to calibrate k Rs and not assume the above referred range of its values, respectively 0.16 to 0.19 °C−0.5 for “interior” or “coastal” regions.

In the original version of the Hargreaves-Samani (HS) equation (Hargreaves and Samani, 1982), a bulk constant term of 0.023 is used, which is known as the Hargreaves coefficient and corresponds to the product 0.0135 k Rs, with k Rs = 0.17 °C−0.5, and 0.0135 representing a units’ conversion constant. Various researchers calibrated that coefficient, i.e., inexplicitly calibrated k Rs, using a two-parameter temperature function (Vanderlinden et al. 2004; Mendicino and Senatore 2013; Martí et al. 2015). However, results by these authors show that using two-parameter temperature function to calibrate the Hargreaves coefficient produced quite different parameters for various locations having similar climate in the Mediterranean area, which makes it difficult to interpret results and does not allow to identify trends for these parameters. A function with a single parameter focusing on the Hargreaves coefficient have shown that given the variety of factors influencing R s and T max−T min, there is the need for searching the best value for k Rs at each location.

Allen et al. (1998) proposed the use of the world average wind speed value u 2 = 2 m s−1 as the default estimator of wind speed when related data are missing. When average local wind speed data are available, they were used alternatively following the assessments by Popova et al. (2006) and Jabloun and Sahli (2008).

The use of the referred approaches for estimating the parameters of the PM-ETo equation allows computing ETo just replacing R s, e a, or u 2 by their respective estimators when related weather variables are missing, or using temperature data only when more variables are missing. Considering the discussion above on procedures to compute ETo with reduced data sets when estimating just the missing variables, as well as the lack of appropriate information on estimating ETo and related missing variables in humid island environments, the objective of this study consists of assessing the performance of temperature-based procedures to estimate actual vapor pressure (Eqs. 2 through 5), solar radiation (Eq. 6), and wind speed when used in PM-ETo computations in humid island environments. The assessment of ETo computed with only temperature data is the object of a companion paper (Paredes et al. 2017).

2 Materials and methods

2.1 Study area and data

The archipelago of Azores, of volcanic origin, comprises nine islands and is located in the North Atlantic at latitudes 36° 45′ N to 39° 43′ N and longitudes 24° 45′ W to 31° 17′ W (Fig. 1). The east-most island, Santa Maria, is located approximately 1400 km from mainland Europe, and the west-most island, Flores, lies 1900 km from the North American continent (Fig. 1). Biogeographically, the archipelago of Azores belongs to Macaronesia, which also comprises the archipelagos of Madeira, Canaries, and Cape Verde (Cropper and Hanna 2014), which are drier or much drier than Azores.

Fig. 1
figure 1

The Azores archipelago and location of islands under study (adapted from Elias et al. 2016)

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Daily data used in the present study was collected in various weather stations in eight islands whose locations, basic climate characteristics, and size of data sets are described in Tables 1 and 2. Data included precipitation (P), maximum and minimum air temperatures (T max and T min, °C), relative humidity (RH, %), wind speed (u 2, m s−1), and solar radiation (R s, W m−2) or sunshine duration (n, h). All data were collected above green grass, including those collected in airports. The basic information about variability of daily collected data by month is given in Fig. 2 for eight selected stations, which refer to seven out of the nine islands and two contrasting environments in Terceira island.

Table 1 Weather stations characterization and data range of the weather data series
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Table 2 Weather stations data available and respective institutions in charge
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Fig. 2
figure 2figure 2

Box-and-whisker plot showing the lower quartile (Q1), the median (Q2), and the upper quartile (Q3) as well as the minimum and maximum values relative to the main weather variables for selected locations. Data relative to each weather station refer to the periods of observation indicated in Table 1

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The Azorean climate is strongly influenced by its location in the middle of the North Atlantic Ocean. During most of the year (September to March), the Azores region is frequently crossed by the North Atlantic storm-track, the main path of rain-producing weather systems, while during late Spring and Summer, the Azores climate is influenced by the Azores anticyclone (Santos et al. 2004). Therefore, the yearly average precipitation varies with longitude, from 730 mm year−1 in Santa Maria, the east-most island, to 1666 mm year−1 in Flores, the west-most island (Barceló and Nunes 2012).

As depicted in Fig. 2, the weather variables vary much, not only within the year but also within the months, which is a characteristic of the islands’ climate environment as referred above. Precipitation is more abundant during November to January and varies much within each month, with elevation and longitude. Air humidity is generally high, with median RH values close to 80% and RHmin near 60%. The monthly average temperature varies little along the year, varies much within a month, and the cooler months are generally January and February and the hottest one is August. Frosts are rare below 600 m of altitude. The dominant winds are from SW, with high moisture; winds are quite strong and show a great variability within months mainly in winter. Solar radiation is highest generally by July and smaller by December; its variability is very high mainly in summer months. Therefore, the reference evapotranspiration ETo is greater in summer months when the variability within months is quite high. According to the Köppen-Geiger classification (Kottek et al. 2006), the climate in Faial and Flores is Cfb, fully humid with warm summer (Table 1); in Pico, Terceira, and the western part of S. Miguel (Lagoa do Fogo, Furnas, and Tronqueira) is Csb, humid with dry and warm summer, and in Graciosa, S. Jorge, Santa Maria, and the eastern part of S. Miguel (Chã de Macela, Ponta Delgada, Santana, and Sete Cidades) is Csa, humid with a dry and hot summer. In all locations, the aridity ratio P/PETTH is much higher than 1.0, from 2.0 at Santa Cruz da Graciosa to 8.6 at S. Caetano (Table 1), indicating a very humid climate (UNEP, 1997).

Several studies have been performed relative to the climate of the Azores islands using the CIELO model (Azevedo et al. 1999), which is a physically based model that simulates the transformations experienced by an humid air mass ascending from the windward side, starting from the sea level up to the central mountains of the islands, and then descending from the opposite side. During the ascending path, the air column’s temperature decreases and the air saturates, with the consequent condensation of water vapor that precipitates in favorable physical conditions; on the other side of the island, the descending air mass decreases humidity and increases temperature as for the foehn effect. The model was validated to the Terceira Island (Azevedo et al. 1999) and tested in the S. Miguel Island (Santos et al. 2004; Miranda et al. 2006). Model results consist of the spatial distribution of all simulated climatic variables on the island territory, thus CIELO was applied to all islands to support ecological studies, e.g., vegetation distribution (Elias et al. 2016). Because solar radiation is not simulated, it is important that, using the findings of the current study, both R s and ETo could be estimated using the reduced data sets created by CIELO. Using CIELO in climate change studies (Santos et al. 2004; Miranda et al. 2006) it was concluded that main impacts of global warming will be on the intra-annual precipitation distribution, with wetter winters and other seasons becoming drier, which will impact the Azores water resources and their availability for vegetation and crops.

2.2 Reference evapotranspiration computations

The PM-ETo Eq. (1) corresponds to definition of the grass reference ETo. Computations require the adoption of the standard methods proposed by Allen et al. (1998) for computing the various parameters of the equation. As analyzed by Nandagiri and Kovoor (2005), results of the PM-ETo Eq. (1) using different computational approaches for the respective parameters differ from those obtained when standard computation procedures are adopted. Equation (1) is used herein as standard for assessing the performance of the alternative approaches to compute ETo with reduced data sets.

Following Allen et al. (1998), the standard computation of the vapor pressure deficit (VPD, kPa), which is the difference between the saturation vapor pressure (e s, kPa) and the actual vapor pressure (e a, kPa), was performed with e s given as

$$ {\mathsf{e}}_{\mathsf{s}}=\frac{{\mathsf{e}}^{\mathsf{o}}\left({T}_{\mathsf{max}}\right)+{\mathsf{e}}^{\mathsf{o}}\left({T}_{\mathsf{min}}\right)}{\mathsf{2}} $$
(7)

where e o(T max) and e o(T min) are the saturation vapor pressure at respectively the maximum and minimum daily temperatures (kPa), and e a was computed as

$$ {e}_{\mathrm{a}}=\frac{e^{\mathrm{o}}\left({T}_{\mathrm{min}}\right)\frac{RH_{\mathrm{max}}}{100}+{e}^{\mathrm{o}}\left({T}_{\mathrm{max}}\right)\frac{RH_{\mathrm{min}}}{100}}{2} $$
(8)

in locations where observations of maximum and minimum relative humidity, RHmax, and RHmin (%) were available (Table 2), or as

$$ {\mathit{\mathsf{e}}}_{\mathsf{a}}=\frac{RH_{\mathsf{mean}}}{\mathsf{100}}\left[\frac{{\mathit{\mathsf{e}}}^{\mathsf{o}}\left({T}_{\mathsf{max}}\right)+{\mathit{\mathsf{e}}}^{\mathsf{o}}\left({T}_{\mathsf{min}}\right)}{\mathsf{2}}\right] $$
(9)

when only the mean daily RH (RHmean, %) was available (Table 2).When RH data are missing, given the humid environments of the islands, the approach described with Eq. 5 was adopted. Its results were compared with those obtained with the standard Eqs. 8 or 9 depending upon data observed. T dew was firstly derived from observed e a as

$$ {T}_{\mathsf{dew}}=\frac{\mathsf{116.91}+\mathsf{237.3}\;\mathsf{\ln}\left({\mathit{\mathsf{e}}}_{\mathsf{a}}\right)}{\mathsf{16.78}-\mathsf{\ln}\left({\mathit{\mathsf{e}}}_{\mathsf{a}}\right)} $$
(10)

and the derived T dew was compared with T mean to derive the adjustment factors a d to be used with Eq. 4. To improve accuracy of ETo estimations, the values of a d were derived for the semesters October–March (Winter) and April–September (Summer) seasons, which relate with the intra-annual variation of the precipitation and temperature with winter being the period having most of precipitation and lower temperature (Fig. 2).

Solar radiation, R s, was observed in most weather stations (Table 2) and sunshine duration was observed only in Lajes. For the latter, R s was calculated as (Allen et al. 1998):

$$ {R}_{\mathrm{s}}=\left({a}_{\mathrm{s}}+{b}_{\mathrm{s}}\frac{n}{N}\right)\kern0.22em {R}_{\mathrm{a}} $$
(11)

where R s is shortwave solar radiation (MJ m−2 day−1), n is the actual sunshine duration (hour), N is maximum possible sunshine duration or daylight hours (hour), n/N is relative sunshine duration (−), R a is extraterrestrial radiation (MJ m−2 day−1), a s is the fraction of extraterrestrial radiation reaching the earth on overcast days (n = 0), and a s + b s is the fraction of extraterrestrial radiation reaching the earth on clear-sky days (n = N). The default parameters a s = 0.25 and b s = 0.50 were used as recommended by Allen et al. (1998).

The calibration of the radiation adjustment factor k Rs (Eq. 6) was performed using a trial and error procedure through an iteratively adjusting the k Rs value when comparing R s computed with Eq. 6 with R s observations. For every location, this procedure was applied seasonally as for the calibration of the factor a d as referred above.

To adjust wind speed data obtained from instruments placed above the standard height of 2 m, a logarithmic wind speed profile (Allen et al. 1998) was used:

$$ {u}_2={u}_z\kern0.62em \frac{4.87}{\ln\;\left(67.8\kern1.12em z\kern1.12em -\kern1.12em 5.42\right)} $$
(12)

where u 2 is wind speed at 2 m above ground surface (m s−1), u z is the measured wind speed at z m height (m s−1), and z is height of measurement above ground surface (m).When wind speed data are missing, u 2 was estimated through the local average of wind speed and adopting the default u 2def = 2 m s−1 or 3 m s−1, the latter in case of weather measurements over grass in airports, where wind speed is typically high (Fig. 2).

For all three cases described above, ETo computed with the estimated missing variables e a using T dew estimates, ETo Tdew, R s using the temperature difference method, ETo TD, and u 2 with the yearly average wind speed or the default u 2, respectively u 2avg and u 2def, thus the ETo uavg and ETo udef were assessed for accuracy against the PM-ETo.

2.3 Accuracy indicators

The accuracy of ETo computations using alternative approaches for estimating e a, R s, and u 2, was assessed by comparing their results with those of the PM-ETo equation using full data sets. In addition, the estimators of e a using estimated T dew values (e a Tdew) and of R s using the calibrated k Rs and the temperature difference equation (R s TD) were compared with e a and R s obtained from observations. Following previous applications to ETo studies, accuracy was measured with several statistical indicators as described by Martins et al. (2017):

  1. 1.

    The regression coefficient (b 0) of the regression forced to the origin (FTO, Eisenhauer, 2003) between daily PM-ETo computed with observed data, O i (x), and daily ETo computed with estimated variables, P i (y). A value of b 0 = 1.0 indicates that the fitted line is y = x, so that O i and P i are similar. A value of b 0 > 1.0 suggests over-estimation and b 0 < 1.0 under-estimation.

  2. 2.

    The determination coefficient (R 2) of the ordinary least squares (OLS) regression between O i and P i. The higher the value of R 2, the more variation of the PM-ETo values is explained by the simplified computation approach; however, a high value of R 2 is, in itself, insufficient to state that there is good overall agreement between observed and estimated values.

  3. 3.

    The root mean square error (RMSE, mm day−1) that measures overall discrepancies between observed and estimated values, hence the smaller it is the better is the accuracy.

  4. 4.

    The percent bias (PBIAS, %), which is simply a normalized difference between the means of both sets O i and P i, and as a bias indicator measures the average tendency of the simulated data to be larger or smaller than their corresponding observations.

  5. 5.

    The Nash and Sutcliffe (1970) modeling 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 (Legates and McCabe 1999), i.e., compares “noise” with “information” (Moriasi et al. 2007). EF is estimated as

$$ EF=1-\frac{\sum_{i=1}^M{\left({O}_i-{P}_{\mathsf{i}}\right)}^{\mathsf{2}}}{\sum_{\mathit{\mathsf{i}}=\mathsf{1}}^{\mathit{\mathsf{M}}}{\left({\mathit{\mathsf{O}}}_{\mathsf{i}}-\overline{\mathit{\mathsf{O}}}\right)}^{\mathsf{2}}} $$
(13)

The maximum value EF = 1.0 can only be achieved if there is a perfect match between all observed (O i) and predicted (P i) values, thus with RMSE = 0, R 2 = 1.0, and b 0 = 1. The closer the values of EF are to 1.0, the more “noise” is negligible relative to the “information,” implying that alternative-based values of ETo are good estimators of PM-ETo values. Negative values of EF indicate that MSE is larger than the observed data variance meaning that it would be better to use the mean of observed values rather than the predicted values P i as Legates and McCabe (1999) correctly stressed.

As shown by Martins et al. (2017), the joint assessment of this set of indicators provides a good evaluation of the quality of prediction of weather variables and ETo when using the referred approaches to estimate the missing weather variables. Scatter plots and the regression lines relating O i and P i were also analyzed for every variable set. Eight examples of graphical results are presented to support discussion. They were selected in seven islands and covering various environmental conditions.

3 Results and discussion

3.1 Estimation of ETo in the absence of RH observations

In the absence of RH data, the actual vapor pressure (e a) may be estimated using T min as estimator of T dew (Eq. 2). However, in humid climates T dew > T min and is better related with T mean, i.e., T dew = T mean − a d (Eq. 4), which requires appropriate calibration of a d. Computations using T dew = T min revealed acceptable accuracy but a tendency for under-estimation of ETo in most locations (results not shown). Differently, using a calibrated adjustment factor a d, the regression between (T mean − a d) and T dew has shown regression coefficients ranging from 0.96 to 1.00, coefficients of determination R 2 > 0.81, small RMSE < 1.69 °C, PBIAS in the interval − 1.2 to 2.4%, and EF > 0.81. Examples of scatter plots representing that regression for the eight selected locations (Fig. 3) support the assumption that a linear relation describes well the strong relationship between (T mean − a d) and T dew. It may be observed that, however, (T mean − a d) tends to over-estimate T dew when air temperature is low but not for medium and higher temperatures.

Fig. 3
figure 3

Relationships between (T mean − a d) and T dew for eight selected locations in various islands and having different environments. Included the FTO regression equation and the OLS determination coefficient R 2; also depicted the 1:1 line (- - -)

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A regression analysis was also used to compare e a computed from the estimated T dew, e a Tdew, with e a computed from RH observations. Results for the referred eight selected locations (Fig. 4) show that respective results are very close statistically, with b 0 ranging from 0.95 to 0.98, and R 2 > 0.82, nevertheless with a slight under-estimation for the high e a values and over-estimation for the low ones. In addition, results for all locations present small RMSE < 0.18 kPa, PBIAS in the interval − 0.6 to 3.5%, and EF > 0.82 indicate that the estimation of e a from T dew = T mean − a d is a very good approach for islands’ humid environments.

Fig. 4
figure 4

Comparing e a Tdew, computed from (T mean − a d) and e a, computed with observed data, for eight selected locations in various islands and having different environments. Included the FTO regression equation and the OLS determination coefficient R 2; also depicted the 1:1 line (- - -)

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Testing T dew estimations were performed by comparing ETo Tdew, computed with e a Tdew, with PM-ETo computed with observed full data sets. Results in Table 3 show that a d values ranged from 1.5 to 4 °C and that for most locations, a d values were the same for both seasons; when seasonal differences were observed, the corresponding a d differences were small (0.5 °C),with the lower values for the more humid winter. Higher a d values were more frequent for western locations, where rainfall and climate humidity are higher, and lower values are often for locations of medium to high elevation, also exposed to the wind. However, data available are insufficient to derive a proper relationship between a d values and selected characteristic of the stations.

Table 3 Calibrated values for the temperature adjusting factor a d and accuracy indicators of ETo estimations in the absence of RH data using T dew = T mean − a d (Eq. 4)
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The accuracy of daily ETo Tdew estimations are presented in Table 3 and Fig. 5. Results show to be very good considering the goodness of results for estimating T dew as well as e a Tdew (Figs. 3 and 4). No tendency for under- or over-estimation were observed in 70% of stations, where b 0 values are in the interval 0.99–1.01 and all b 0 values range from 0.98 to 1.02. Consequently, PBIAS values are more often within the interval of − 5 to 5%, with only three locations presenting an under-estimation bias ranging between − 7 and − 5%. Large R 2 values (R 2 > 0.90) are observed in 55% of locations and R 2 > 0.80 was for 95% of cases, thus meaning that ETo Tdew well explains most of the variation of the PM-ETo results computed with full data sets. In addition, RMSE are generally small, ranging from 0.15 to 0.53 mm day−1 with 65% of stations with RMSE < 0.40 mm day−1. Such small RMSE values indicate that the discrepancies between ETo Tdew and PM-ETo are small. The EF values were very high, with EF > 0.90 in 50% of the weather stations and EF > 0.80 in 95% of the locations, thus meaning that MSE are much lower than the variance of PM-ETo results. Hence, using T dew = T mean− a d with a d values listed in Table 2 to estimate actual vapor pressure is appropriate to overcome problems due to missing RH data or when they have no adequate quality.

Fig. 5
figure 5

Frequency distribution of the accuracy indicators relative to the computation of EToTdew when the actual vapor pressure computed from T dew = T mean − a d

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The scatter plots in Fig. 6 not only highlight the very good relationships between ETo Tdew and PM-ETo but also allow to perceive which data shows slight inequalities between both computational approaches: slight over-estimations occur for the smaller values of ETo, and negligible under-estimations occur for the high ETo values. This behavior is well explained by the relationships between e a Tdew and e a analyzed before (Fig. 4).

Fig. 6
figure 6

Comparing EToTdew, computed with the estimator e a Tdew estimated from (T dew = T mean − a d) and PM-ETo computed with full data sets, for eight selected locations in various islands and having different environments. Included the FTO and the OLS regression equations and the OLS determination coefficient R 2; the 1:1 line (- - -) and the OLS line (- . - . -) are depicted

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Few studies report on the accuracy of ETo estimations in the absence of RH, and less are available for humid climates but different of that of Azores islands. Better indicators were reported using T dew = T min for sub-humid climates, e.g., by Liu and Pereira (2001) and Pereira et al. (2003) for North China Plain, and Popova et al. (2006) for Thrace, Bulgaria, as well as by Trajkovic and Kolakovic (2009) for Serbia, and Sentelhas et al. (2010) and Aladenola and Madramootoo (2014) for humid sites of Canada. However, it is not possible to properly compare with climatic conditions as those in Azores where intra-annual and monthly variability of air masses flowing over the islands causes a great variability of climate variables (Fig. 2).

3.2 Estimation of ETo when solar radiation data is not available

As previously stated, when R s observations are not available, estimations of ETo may be performed using the Hargreaves temperature difference equation (Eq. 6) with a locally calibrated radiation adjustment coefficient k Rs, thus resulting in the ETo TD equation.

Results in Table 3 show that k Rs values ranged from 0.14 to 0.25 °C−0.5with lower k Rs values in locations at medium and high altitude. Because the coefficient k Rs is supposed to reflect the volumetric heat capacity of the atmosphere (Allen 1997), therefore, lower k Rs values are to be expected with elevation. As discussed by Turbet et al. (2017), the volumetric heat capacity of the atmosphere increases linearly with the volumetric mass density and thus with the atmospheric pressure. For that reason, Annandale et al. (2002) developed a decreasing relationship between k Rs and elevation to account for the effects of reduced atmospheric thickness. The occurrence of winds transporting humid air masses may also contribute to decrease the heat capacity of the atmosphere, as already described with model CIELO (Azevedo et al. 1999). Nevertheless, the sample size is small and does not allow to develop a statistically significant relationship between k Rs and relevant station characteristics.

Results of the calibration of k Rs show that in 45% of locations, different k Rs values where found for the winter and summer seasons, with lower k Rs in winter (Table 4). Seasonal differences are likely due to the fact that for those locations, the volumetric heat capacity of the atmosphere is higher in summer since the frequent cloud cover may limit it during winter. Computations for a period covering both seasons were performed assigning different k Rs to the winter and summer periods.

Table 4 Accuracy of ETo estimation in the absence of R s observations using temperature difference (Eq. 6) and the calibrated k Rs value
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Results relative to R s estimations with Eq. 6 resulted in a trend for under-estimation with RMSE ranging 3.5 to 5.5 MJ m−2 day−1. Selected examples of regression of R s TD with observed R s are presented in Fig. 7. No study is available for conditions similar to those in Azores and few applications for humid to sub-humid conditions are available in literature. Results of our study are in the range of those reported by Aladenola and Madramootoo (2014) and Kwon and Choi (2011); better results were reported by Almorox et al. (2016) and Lyra et al. (2016) when calibrating k Rs.

Fig. 7
figure 7

Selected examples comparing the estimated solar radiation R s TD computed with the temperature difference and calibrated k Rs (Eq. 6) with and R s from observations. Included the OLS regression equation and the respective R 2; also depicted the 1:1 line (- - -) and the OLS line (─)

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Accuracy indicators (Table 4, Fig. 8) show that in 50% of locations, b 0 ranged from 0.99 to 1.00 and that in the other 50% of cases, b 0 was in the range of 0.96 to 0.98, hence meaning that ETo TD and PM-ETo are statistically very close, however with ETo TD slightly underestimating PM-ETo. It resulted that the PBIAS values are generally negative but small, with PBIAS not exceeding − 7%, generally in the interval − 5 to 5%. Higher PBIAS were observed in locations with medium to high altitude, which may relate with the high cloudiness of stations and combined impacts of winds carrying humid air masses. R 2 values were generally close to 1.0, with R 2 > 0.80 in 80% of cases, thus meaning that most of variation of PM-ETo can be explained with the ETo TD approach. RMSE are quite small with 95% of the locations having RMSE < 0.50 mm day−1. The EF values are high, ranging 0.70 to 0.91, with EF > 0.80 in 80% of locations, indicating that the mean square error is much lower than the variance of ETo computed with observed full data sets.

Fig. 8
figure 8

Frequency distribution of the accuracy indicators relative to the computation of ETo using R s TD derived from the temperature difference and a calibrated radiation adjustment coefficient k Rs

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The scatter plots in Fig. 9 highlight the very good relationships between ETo TD and PM-ETo, despite the less good accuracy in R s TD estimations (Fig. 7) but also allow perceiving that a slight over-estimation occur for the higher values of ETo, and in most locations a negligible under-estimations occur for the low ETo values. This behavior is well explained by the relationships between R s TD and observed R s as previously analyzed (Fig. 7).

Fig. 9
figure 9

Selected examples of comparing ETo TD, computed with the temperature difference equation and adopting calibrated k Rs. The FTO and the OLS regression equations and the OLS coefficient of determination R 2 are included; the 1:1 line (- - -) and the OLS line (- . - . -) are depicted

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The results of the present study are difficult to compare with other studies where only R s estimates from the temperature difference is used to compute ETo in humid island environments. Examples from humid climates, but different from the Azores environment, are available. Smaller RMSE were observed by Popova et al. (2006) and Aladenola and Madramootoo (2014) using a constant value for k Rs. Pereira et al. (2003), Sentelhas et al. (2010), Kwon and Choi (2011), and Lyra et al. (2016) reported higher RMSE values than the current study. No other studies were found for islands with humid climate.

3.3 ETo estimation in the absence of wind speed data

Two approaches were used, the first adopting the local annual average wind speed (u 2avg) as u 2 estimator, and the second using a default value of 2 m s−1 (u 2def) corresponding to the world average wind speed, adjusted for 3 m s−1in case of weather stations located in airports because these show high average values because they are exposed to most of wind directions. In the Azores archipelago, there is a high spatial variability of the average wind speed (Table 5), even within the same island, which relates with the elevation and exposure to the dominant winds. For example, São Caetano and Santa Bárbara weather stations are located at high altitude (Table 1) but have small wind because they are leeward located in the mountains. Furnas and Granja stations are located in old volcanic craters and therefore less exposed to wind. Differently, Ribeirinha and Fontinhas stations are located on the top of small mountains exposed to winds and therefore show high u 2. For all study cases, u 2 is determined by the exposure to winds.

The accuracy of ETo uavg and ETo udef estimates, i.e., using respectively u 2avg and u 2def estimation procedures, is very high (Table 5, Fig. 10). ETo uavg shows no tendency for under- or over-estimation in most locations, with b 0 ranging 0.99 to 1.01 in 90% of cases and PBIAS ranging from − 2.5 to 2.5% in most locations. R 2 values are very high (R 2 > 0.94) for all cases and RMSE values were small in all locations, below 0.28 mm day−1. EF values were very high, with EF > 0.94 in all locations. Good results were also obtained for ETo udef, where 95% of cases have b 0 ranging from 0.98 to 1.02 and all cases with R 2 > 0.93. RMSE was inferior to 0.36 mm day−1 at all locations and EF > 0.90 for all cases. Results of accuracy indicators show that results of ETo uavg and ETo udef are close to those of PM-ETo, the variation of the latter is well explained by both ETo approaches, the bias of estimation are small and vary locally, and high EF values were observed for both approaches indicating that the mean square errors were much smaller than the variance of PM-ETo computed with full data sets. It is concluded that when u 2 data are not available, the most accurate procedure for computing ETo consists of using u 2avg data, but these data are difficult to get for locations other than those used in this study. Nevertheless, the assessed accuracy of ETo udef was very good, close to the former; hence, the most useful approach consists in using u 2def in ETo estimations.

Table 5 Accuracy of ETo estimations in the absence of wind speed data using the local annual average u 2avg and the default u 2def = 2 m s−1 (adjusted in case of weather stations located on grasslands in airports)
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Fig. 10
figure 10

Frequency distribution of the “goodness-of-fit” indicators relative to the computation of ETo in the absence of wind speed using the local annual average, u 2avg () and the default value u 2def = 2 m s−1 (or 3 m s−1 when adjusted for weather stations located on grasslands in airports) ()

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Results in Fig. 11 show that dispersion of values around the regression line is higher for the windy locations, e.g., Ribeirinha, and, inversely, is small when the weather station is not exposed to the dominant high winds, e.g., S. Bárbara, with consequent impacts on R 2 values. For all cases, the regression coefficient is close to 1.0.

Fig. 11
figure 11

Comparing ETo udef computed with the default wind speed with PM-ETo using full data sets. The FTO regression equations and the OLS coefficient of determination R 2 are included; the 1:1 line is also depicted (- - -)

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There are no results available for weather conditions similar to Azores. Various studies relative to humid or sub-humid climates using u 2 = 2 m s−1 report RMSE closer to ours (Sentelhas et al. 2010) or higher (Popova et al. 2006; Jabloun and Sahli 2008; Kwon and Choi 2011).

4 Conclusions

The approaches proposed by Allen et al. (1998) tested in the present study for ETo estimation when observed data is missing yielded good results. The performance of the tested approaches in Azores island environments are closely linked with the very humid climate and are influenced by longitude, altitude, and exposure to dominant winds.

In absence of RH observations, the actual vapor pressure e a could be very accurately estimated, with negligible over- or under-estimation when deriving T dew from T mean − a d, i.e., using a calibrated T dew adjustment coefficient. That coefficient is larger in western islands, often is different in winter and summer, and is influenced by rainfall, elevation, and wind.

The use of a Hargreaves-based root squared temperature difference equation with a calibrated radiation adjustment coefficient k Rs revealed a very good option to estimate both short-wave solar radiation, R s TD, and EToTD when R s observations are not available. k Rs values were influenced by wind and by altitude of the locations, with lower values found in medium and high altitude locations. Despite tendencies for under-estimation of high R s values and over-estimation of low ones, low estimation errors were obtained for R s, with quite small over- or under-estimation errors of ETo. Results reflect the high variability of R s within the year and within the months due to frequent cloud cover, which highly reduces R s in many rainy days. Nevertheless, errors of estimate were relatively small. It was found that k Rs varies with altitude and wind, which influence the heat capacity of the air.

When observations of the wind speed are lacking, two options revealed appropriate to estimate u 2 to be used in the PM equation. The first is the use of the local annual average of u 2 and the second is the use of a default value, 2 m s−1, which is majored for 3 m s−1 in case of weather stations located at airports because of the inherent exposure to all wind directions. Both cases, i.e., the use of the annual average of u 2 or a default value, yield very good accuracy indicators, with RMSE ranging 0.07–0.28 mm day−1and EF > 0.90; however, it is recommended the use of the selected default values because they provide highly accurate estimates.

Results of this study provide for easy parameterization of the PM equation for reduced data sets in Azores archipelago. In particular, results of this study shall be used together with model CIELO to complement the respective spatial distribution of weather variables, which do not include R s or ETo while both may be computed using the variables yielded by the model.

The procedures tested herein are of great importance for humid and cold areas where temperature methods perform contradictorily (Trajkovic and Kolakovic 2009; Tabari 2010; Todorovic et al. 2013). Results obtained in the current study show that procedures herein assessed may be used for other cold humid locations, namely considering that the adjustment coefficients for both T dew and R s TD estimations should be locally calibrated.