Introduction

Evapotranspiration (ET) is the simultaneous process of transfer of water to the atmosphere by transpiration and evaporation in a soil–plant system. ET is an important parameter for climatological and hydrological studies, as well as for irrigation planning and management (Sentelhas et al. 2010). Furthermore, it is necessary to quantify ET for work dealing with water resource management or environmental studies. ET quantification frequently must be preceded by the determination of reference evapotranspiration (ETo) (Lopez–Urrea et al. 2006). Reference evapotranspiration has been defined as the rate of evapotranspiration from an extensive grassed area of 8–15 cm tall, uniform, actively growing, completely shading the ground and with adequate water (Doorenbos and Pruitt 1977). Subsequently, Allen et al. (1998) elaborated on the concept of ETo, by referring it to an ideal 12 cm high crop with a fixed surface resistance of 70 sm−1 and an albedo of 0.23.

Accurate estimation of ETo in irrigated lands is necessary for improving the planning and efficient use of water resources. Application of lysimeters is the most common method for estimating ETo. Unfortunately, lysimeters are unsuitable for monitoring evapotranspiration as compared to direct climate-based measurement at weather stations. This is not only due to their cost and complexity, but also because the limited area of a typical weather station enclosure does not provide sufficient fetch from a representative surface for these measurements to be meaningful (Sentelhas et al. 2010). In practice, ETo can be either estimated using available climatic data from a weather station or derived from the pan observation multiplied by a conversion factor (K pan) (Xing et al. 2008). Numerous equations, classified as temperature-based, radiation-based, pan evaporation-based, mass transfer-based and combination-type, have been developed for estimating ETo, but their performances in different environments vary (Gocic and Trajkovic 2010). The Penman–Monteith FAO 56 (PMF-56) model, which is recommended as the sole method for determining ETo, has been reported to be able to provide consistent ETo values in many regions and climates (Allen et al. 2005, 2006), and it has long been accepted worldwide as a good ETo estimator when compared with others methods (Cai et al. 2007). It is now widely used by agronomists, irrigation engineers and other scientists in field practice and research (Alexandris et al. 2006). The main shortcoming of the PMF-56 equation is that it requires numerous weather data that are not always available for many locations. This is especially true in developing countries where reliable weather data sets of radiation, relative humidity and wind speed are limited (Gocic and Trajkovic 2010; Tabari and Hosseinzadeh Talaee 2011). Furthermore, the installation and maintenance of weather station equipment can be expensive and complicated (Sentelhas et al. 2010).

The application of ETo equations with fewer meteorological parameters requirements is recommended under situations where more complete weather data is lacking. However, before these equations can be used to estimate ETo for a given region, they must be evaluated against either lysimeter measurements or the PMF-56 standard model. Although many studies have been conducted for evaluation of ETo equations under relatively low humidity conditions (semi-arid) throughout the world (e.g., Jensen et al. 1990, 1997; Kashyap and Panda 2001; Irmak et al. 2002, 2003a, b; Grismer et al. 2002; Yoder et al. 2004; Chen et al. 2005; Temesgen et al. 2005; Alkaeed et al. 2006; Trajkovic 2007; Landeras et al. 2008; Xing et al. 2008; Ali and Shui 2009; Trajkovic and Kolakovic 2009; Sentelhas et al. 2010), little such work has been carried out in humid climates of Iran. DehghaniSanij et al. (2004) assessed the estimates of ETo obtained using the Penman, Penman–Monteith, Wright–Penman, Blaney–Criddle, Radiation balance and Hargreaves models against experimentally determined values in a semi-arid environment. The results indicated that the Penman–Monteith model produced the most reliable estimates compared to lysimeter data. Sabziparvar et al. (2010) examined pan evaporation-based equations for estimating ETo in cold semi-arid and warm arid climates. They found that the Orang and Snyder models were the best models for estimation of ETo in cold semi-arid and warm arid environments, respectively. Tabari (2010) evaluated four ETo models with small weather data requirements (Makkink, Turc, Priestley–Taylor and Hargreaves) in four climates. The results showed that the Turc model was the best-suited model in estimating ETo for cold humid and arid climates. In addition, the Hargreaves model was the most precise model under warm humid and semi-arid climatic conditions. Sabziparvar and Tabari (2010) prepared the spatially distributed maps of ETo in the arid and semi-arid regions using the Hargreaves model. The estimated total monthly ETo revealed a significant variation during the growing seasons (April–September) so that the study region experienced the highest and lowest monthly ETo values of 250 and 80 mm in July and April, respectively.

To our knowledge, there are no reports of studies that have been conducted to evaluate the performance of mass transfer-based methods in Iran. In this study, 29 commonly used ETo equations that belonged to four groups: (1) pan evaporation-based methods, (2) temperature-based methods, (3) radiation-based methods, and (4) mass transfer-based methods were evaluated against the PMF-56 standard model; and the best and worst equations of each category were determined using climatic data from the Rasht station located in a humid climate near Rash, Iran. In addition, two radiation-based methods for estimating ETo were derived using air temperature and solar radiation data based on the PMF-56 model as a reference. A cross-comparison of the best equations from each group was also conducted. The assessed methods were: FAO-24 pan table, Cuenca, Allen and Pruitt, Snyder, Modified Snyder, Pereira, Orang, FAO-56 (pan evaporation-based), Thornthwaite, four new types of Hargreaves equation reported by Droogers and Allen (2002) and Trajkovic (2007), Blaney–Criddle and Schendel (temperature-based), Jensen–Haise, Ritchie, McGuinness and Bordne, Irmak and two equations developed here (radiation-based), Dalton, Trabert, Meyer, Rohwer, Penman, Albrecht, Romanenko, Brockamp and Wenner, WMO and Mahringer (mass transfer-based).

Materials and methods

Data set

The data set used in this study was obtained from Rasht station in northern Iran. The station is located between the coast of the Caspian Sea and the slopes of the Alborz mountain (37°15′N, 49°36′E; −6.9 m a.s.l.). Rash city has a mild humid climate with plenty of annual rainfall and is known as the “City of Rain” around Iran. Rasht receives about 1,000–1,400 mm of annual precipitation in the form of rain. The wettest months are October (215 mm) and November (186 mm), respectively. Long-term (41 years) climate data of the experimental area identified January as the coldest month, with a mean temperature of 6.8°C, whereas the hottest month is July, with a mean temperature of 25.2°C. The amount of humidity is truly high throughout the year. The average annual relative humidity is 82%, with an average of 86% during October, November and December, and 75% during July. The average wind speed is 1.3 m/s with an average of 1.6 m/s in January and February, and 1 m/s in July. Climatic variables including mean, maximum and minimum air temperatures, relative humidity, dew point temperature, water vapor pressure, wind speed, atmospheric pressure, precipitation, solar radiation and sunshine hours for the period 1965–2005 and Class A pan evaporation for the period 1993–2005 (period of record) were obtained from IRIMO (2007). The monthly means of the primary climate parameters are summarized in Table 1.

Table 1 Monthly means of the main climatic variables at Rasht station during 1965–2005
Full size table

Evapotranspiration estimation methods

The FAO Penman–Monteith method for calculating ETo can be expressed as (Allen et al. 1998):

$$ {\text{ET}}_{o} = \frac{{0.408\Updelta (R_{\text{n}} - G) + \gamma \frac{900}{{T_{\text{a}} + 273}}U_{2} (e_{\text{s}} - e_{\text{a}} )}}{{\Updelta + \gamma (1 + 0.34U_{2} )}} $$
(1)

where ETo is the reference crop evapotranspiration (mm day−1), R n is the net radiation (MJ m−2 day−1), G is the soil heat flux (MJ m−2 day−1), \( \gamma \) is the psychrometric constant (kPa °C−1), es is the saturation vapor pressure (kPa), e a is the actual vapor pressure (kPa), and \( \Updelta \) is the slope of the saturation vapor pressure–temperature curve (kPa °C−1), T a is the average daily air temperature (°C), and U 2 is the mean daily wind speed at 2 m (m s−1). The computation of all data required for calculating ETo followed the method and procedure given in Chapter 3 of FAO-56 (Allen et al. 1998).

The soil heat flux for monthly periods was estimated as

$$ G = 0.14(T_{{{\text{month}}2}} - T_{{{\text{month}}1}} ) $$
(2)

where T month2 is the temperature at the end of the period in °C, T month1 is the temperature at the beginning of the period in °C, 0.14 is the soil heat capacity coefficient at effective soil depth, typically at 2 m (Allen et al. 1998). Furthermore, the solar radiation gaps were filled using the Angstrom equation (Allen et al. 1998).

$$ R_{s} = \left( {a_{s} + b_{s} \frac{n}{N}} \right)R_{a} $$
(3)

where R a is the extraterrestrial radiation (MJm−2 day−1), n is the actual duration of sunshine (h), N is the maximum possible duration of sunshine or daylight hours (h), a s is the regression constant, expressing 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 days (n = N).

Pan evaporation-based ETo equations

In many areas, the necessary meteorological data are lacking, and simpler techniques such as pan evaporation-based methods are required. Class A pan evaporation (E pan) data are used for estimating ETo (Eq. 4) throughout the world because of the simplicity of technique, low cost and ease of application in determining crop water requirements for irrigation scheduling (Singh 1989; Stanhill 2002).

$$ {\text{ET}}_{o} = E_{\text{pan}} \cdot K_{\text{pan}} $$
(4)

where K pan is pan coefficient. In this study, eight methods were applied for estimating ETo at the humid location. Cuenca (1989):

$$ \begin{gathered} K_{\text{pan}} = 0.475 - (0.245 \times 10^{ - 3} U_{2} ) + (0.516 \times 10^{ - 2} {\text{RH}}) + (0.118 \times 10^{ - 2} F) - \hfill \\ (0.16 \times 10^{ - 4} {\text{RH}}^{2} ) - (0.101 \times 10^{ - 5} F^{2} ) - (0.8 \times 10^{ - 8} {\text{RH}}^{2} U_{2} ) - (0.1 \times 10^{ - 7} {\text{RH}}^{2} F) \hfill \\ \end{gathered} $$
(5)

Allen and Pruitt (1991):

$$ \begin{gathered} K_{\text{pan}} = 0.108 - (3.31 \times 10^{ - 4} \,U_{2} ) + (0.0422\,{ \ln }(F)) + \hfill \\ (0.1434\,{ \ln }({\text{RH}})) - [6.31 \times 10^{ - 4} (({ \ln }(F))^{2} \,{ \ln }({\text{RH}}))] \hfill \\ \end{gathered} $$
(6)

Snyder (1992):

$$ K_{\text{pan}} = 0.482 + [0.24\,{ \ln }(F)] - (3.76 \times 10^{ - 4} \,U_{2} ) + (0.0045\,{\text{RH}}) $$
(7)

Modified Snyder:

$$ K_{\text{pan}} = 0.5321 - (3 \times 10^{ - 4} \,U_{2} ) + (0.0249\,{ \ln }(F)) + (0.0025\,{\text{RH}}) $$
(8)

Pereira (Pereira et al. 1995):

$$ K_{\text{pan}} = 0.85 \times (\Updelta + \gamma )/[\Updelta + \gamma (1 + 0.33\,U_{2} )] $$
(9)

Orang (1998):

$$ K_{\text{pan}} = 0.51206 - (0.000321 \cdot U_{2} ) + (0.002889 \cdot {\text{RH}}) + (0.03188 \cdot { \ln }(F)) - (0.000107 \cdot {\text{RH}} \cdot { \ln }(F)) $$
(10)

FAO-56 (Allen et al. 1998):

$$ K_{\text{pan}} = 0.108 - 0.0286\,U_{2} + 0.0422\,{ \ln }(F) + 0.1434\,{ \ln }({\text{RH}}) - 0.000631\,[{ \ln }(F)]^{2} { \ln }({\text{RH}}). $$
(11)

In the above pan evaporation-based equations, U 2 is the mean daily wind speed measured at 2 m height (km day−1), RH is the mean daily relative humidity (%), F is the upwind fetch distance of low-growing vegetation (m), ∆ is the slope of the vapor pressure curve (kPa °C−1) and γ is the psychrometric constant (kPa °C−1). In the FAO-56 pan equation, U 2 is in m s−1. In addition to the above mentioned equation, the K pan values obtained from the FAO-24 pan table (Doorenbos and Pruitt 1977) were also evaluated.

Temperature-based ETo equations

The temperature-based ETo models are some of the earliest methods for estimating ET (Xu and Singh 2002). According to Jensen et al. (1990), the relation of ET to air temperature dated back to the 1920s. In this study, seven temperature-based methods were used. In the following equations, T a, T max and T min are the mean, maximum and minimum air temperatures, respectively (oC), RH is the relative humidity (%) and R a is the extraterrestrial radiation (MJ m−2 day−1).

Thornthwaite (1948):

$$ {\text{ET}}_{o} = 16\left( {10\frac{{T_{\text{a}} }}{I}} \right)^{a} $$
(12)
$$ I = \sum\limits_{n = 1}^{12} {(0.2T_{\text{a}} )^{1.514} } $$
(13)
$$ a = 6.75 \times 10^{ - 7} I^{3} - 7.71 \times 10^{ - 5} I^{2} + 1.7912 \times 10^{ - 2} I + 0.49239. $$
(14)

Where ETo is in mm month−1. I is a thermal index imposed by the local normal climatic temperature regime, and the exponent a is a function of I. In order to convert the estimates from a standard monthly (mm month−1) to a daily time scale (mm day−1), the following correction factor (C) was used:

$$ C = \frac{N}{360}. $$
(15)

Where N is the photoperiod (h) for a given day.Blaney and Criddle (1950):

$$ {\text{ET}}_{o} = a + b[P(0.46T_{\text{a}} + 8.13)] $$
(16)

where ETo is in mm day−1, P is the mean annual percentage of daytime hours that can be obtained from Doorenbos and Pruitt (1977), and a and b are the parameters of the equation. The a and b coefficients were computed using regression equations developed by Allen and Pruitt (1991). Schendel (1967):

$$ {\text{ET}}_{o} = 16 \cdot \frac{{T_{\text{a}} }}{\text{RH}} $$
(17)

where ETo is in mm day−1. Droogers and Allen (2002) reported three new types of the Hargreaves equation (Hargreaves and Samani 1985) as follows:

$$ {\text{ET}}_{o} = 0.408 \times 0.0030 \times (T_{\text{a}} + 20) \times (T_{\max } - T_{\min } )^{0.4} \times R_{a} $$
(18)
$$ {\text{ET}}_{o} = 0.408 \times 0.0025 \times (T_{\text{a}} + 16.8) \times (T_{\max } - T_{\min } )^{0.5} \times R_{a} $$
(19)
$$ {\text{ET}}_{o} = 0.408 \times 0.0013 \times (T_{\text{a}} + 17) \times (T_{\max } - T_{\min } - 0.0123P)^{0.76} \times R_{a} $$
(20)

where ETo is in mm day−1 and P is monthly rainfall (mm). The coefficient of 0.408 is for converting MJ m−2 day−1 into mm day−1 (Allen et al. 1998). The Eqs. 18, 19 and 20 are defined hereafter as Hargreaves-M1, Hargreaves-M2 and Hargreaves-M3, respectively. Trajkovic (2007) adjusted the Hargreaves equation for the humid climate of Western Balkans region (hereafter as Hargreaves-M4) as follows:

$$ {\text{ET}}_{o} = 0.408 \times 0.0023 \times (T_{\text{a}} + 17.8) \times (T_{\max } - T_{\min } )^{0.424} \times R_{a}. $$
(21)

Radiation-based ETo equations

Four commonly used radiation-based equations including Jensen–Haise, Ritchie, McGuinness and Bordne and Irmak were evaluated and compared in this study. Selection of the equations was carried out by taking into account the equations (Makkink 1957; Turc 1961, Priestley and Taylor 1972) used in the previous study carried out in the region (Tabari 2010). In the following equations, T a, ∆, γ and R n have the same meaning as those defined in the PMF-56 model, R s is the solar radiation, T max and T min are the maximum and minimum air temperatures, respectively and λ is the latent heat.

Jensen and Haise (1963):

$$ {\text{ET}}_{o} = \frac{{C_{\text{T}} (T_{\text{a}} - T_{x} ) \times R_{\text{s}} }}{\lambda } $$
(22)

where ETo is in mm day−1, λ is in cal gr−1, R s is in mm day−1, C T (temperature constant) = 0.025, and T x  = −3 when T a is in degrees Celsius. These coefficients were considered to be constant for a given area (Xu and Singh 2000).

McGuinness and Bordne (1972):

$$ {\text{ET}}_{o} = \left\{ {(0.0082 \times T_{\text{a}} - 0.19)\left( {\frac{{R_{\text{s}} }}{1500}} \right)} \right\} \times 2.54 $$
(23)

where ETo is in cm day−1 for a monthly period, T a is in degrees Fahrenheit, R s is in cal/cm2/day. Ritchie (1972) method as described by Jones and Ritchie (1990):

$$ {\text{ET}}_{o} = \alpha_{1} [3.87 \times 10^{ - 3} \cdot R_{\text{s}} (0.6T_{\max } + 0.4T_{\min } + 29)] $$
(24)

where T max and T min are in °C and the ETo units are the same as those of R s. When

$$ 5 < T_{ \max } 35^\circ {\text{C}}\quad \alpha = 1.1 $$
$$ T_{ \max } > 35^\circ {\text{C}}\quad \alpha = 1.1 + 0.05 \cdot (T_{\max } - 35) $$
(25)
$$ T_{ \max } < 5^\circ {\text{C}}\quad \alpha = 0.1\,{\text{exp[}}0.18 \cdot (T_{ \max } + 20 ) ] $$
(26)

Irmak (Irmak et al. 2003b):

$$ {\text{ET}}_{o} = - 0.611 + 0.149 \times R_{\text{s}} + 0.079 \times T_{\text{a}} $$
(27)

where the units of ETo, R s and T a are same as those defined in the PMF-56 model.

Similar to the study of Irmak et al. (2003b), two radiation-based equations were developed in this study using multiple linear regressions. In the multiple linear regressions, the PMF-56 ETo values were used as the dependent variable and T max and T min or T a and R s were the independent variables. The developed radiation-based equations are as follows:

$$ {\text{ET}}_{o} = - 0.642 + 0.174\,R_{\text{s}} + 0.0353\,T_{\text{a}} $$
(28)
$$ {\text{ET}}_{o} = - 0.478 + 0.156\,R_{\text{s}} - 0.0112\,T_{ \max } + 0.0733\,T_{ \min } $$
(29)

where ETo, R s, T a, T max and T min have the same meaning as before, ETo is in mm day−1. It should be noted that 65% of the data (1965–1990) were used for development of the equations and the rest of data (1991–2005) were applied for validation.

Mass transfer-based ETo equations

The mass transfer-based methods utilize the concept of eddy transfer of water vapor from an evaporating surface to the atmosphere. All such methods are fundamentally based on Dalton’s gas law. The mass transfer-based methods give satisfactory results in many cases and normally use easily measurable variables and have simple model forms (Singh and Xu 1997). Ten mass transfer-based equations were used in this study.

Dalton (1802):

$$ {\text{ET}}_{o} = (0.3648 + 0.07223u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(30)

Trabert (1896):

$$ {\text{ET}}_{o} = 0.3075 \cdot \sqrt u \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(31)

Meyer (1926):

$$ {\text{ET}}_{o} = (0.375 + 0.05026u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(32)

Rohwer (1931):

$$ {\text{ET}}_{o} = 0.44(1 + 0.27u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(33)

Penman (1948):

$$ {\text{ET}}_{o} = 0.35(1 + 0.98/100u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(34)

Albrecht (1950):

$$ {\text{ET}}_{o} = (0.1005 + 0.297u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(35)

Romanenko (1961):

$$ {\text{ET}}_{o} = 0.0018(T_{\text{a}} + 25)^{2} \cdot (100 - {\text{RH}}) $$
(36)

Brockamp and Wenner (1963):

$$ {\text{ET}}_{o} = 0.543 \cdot u^{0.456} \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(37)

WMO (1966):

$$ {\text{ET}}_{o} = (0.1298 + 0.0934u) \cdot (e_{\text{s}} - e_{\text{a}} ) $$
(38)

Mahringer (1970):

$$ {\text{ET}}_{o} = 0.15072 \cdot \sqrt {3.6u} \cdot (e_{\text{s}} - e_{\text{a}} ). $$
(39)

In the above equations, e s and e a are the saturation and actual vapor pressure, respectively, u is the wind speed, RH is the relative humidity (%) and T a is the mean air temperature (oC). e s and e a are in hPa in all the equations except Rohwer and Penman models, e s and e a are in mmHg in Rohwer and Penman models, u is in m s−1 in all the equations except Penman model, u is in miles day−1 in Penman model, ETo is in mm day−1 in all the equations except Romanenko model where ETo is in cm month−1.

Evaluation criteria

In this study, the root mean square error (RMSE), percentage error of estimate (PE), mean bias error (MBE) and coefficient of determination (R 2) were used for the evaluation of the simplified ETo equations. The RMSE, PE, MBE and R 2 are defined as:

$$ {\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {(P_{i} - O_{i} )^{2} } }}{n}} $$
(40)
$$ {\text{PE}} = \left| {\frac{{\bar{P} - \bar{O}}}{{\bar{O}}}} \right| \times 100\% $$
(41)
$$ {\text{MBE}} = \frac{{\sum\nolimits_{i = 1}^{n} {(P_{i} - O_{i} )} }}{n} $$
(42)
$$ R^{2} = \frac{{\left[ {\sum\nolimits_{i = 1}^{n} {(P_{i} - \bar{P})(O_{i} - \bar{O})} } \right]^{2} }}{{\sum\nolimits_{i = 1}^{n} {(P_{i} - \bar{P})^{2} } \sum\nolimits_{i = 1}^{n} {(O_{i} - \bar{O})^{2} } }} $$
(43)

where P i and O i are the predicted and observed values, respectively; \( \bar{P} \) and \( \bar{O} \) are the average of P i and O i , and n is the total number of data.

Results and discussion

Pan evaporation-based ETo equations

First, we calculated K pan values using the pan evaporation-based methods and then evaluated their relative performance with respect to PMF-56 ETo estimates in the study area. The comparisons of calculated mean monthly K pan values using the pan evaporation-based methods are given in Fig. 1. In the K pan calculations, the upwind fetch of low-growing vegetation (F) was taken as 1,000 m since the weather station was surrounded by irrigated agricultural crops. The highest K pan values were obtained by the Snyder and Cuenca equations, respectively. The K pan values generated by the Snyder equation varied from 0.99 in November to 0.89 in July, with an average of 0.97. Moreover, the K pan values calculated from Cuenca equation ranged from 0.91 in November to 0.88 in July, with an average of 0.89. The K pan values determined by the Snyder and Cuenca equations in this study are higher than those reported by Irmak et al. (2002) who obtained average K pan values of 0.93 and 0.85 by the Snyder and Cuenca equations at a humid location in Florida, USA. This is due to the higher relative humidity at Rasht station (82%) as compared with that at the Green Acres Agricultural Research Center weather station in Florida (73%). The average K pan values generated by Allen and Pruitt, Orang, Modified Snyder, FAO-24 pan table, FAO-56 pan and Pereira methods were 0.89, 0.86, 0.86, 0.83, 0.82 and 0.73, respectively.

Fig. 1
figure 1

Mean monthly K pan obtained from the pan evaporation-based methods

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The mean monthly of ETo values calculated from the PMF-56 model and the pan evaporation-based methods were plotted in Fig. 2. As shown, all of the pan evaporation-based methods underestimated PMF-56 ETo at the Rasht study site. The underestimation of ETo values by the pan evaporation-based equations was also found in the United States (Grismer et al. 2002) and Canada (Xing et al. 2008). The Snyder equation provided the least underestimate average of 0.11 mm/day, while the Pereira equation yielded the greatest underestimate average of 0.67 mm/day (Table 2). The ETo calculated by the Snyder equation best matched the ETo estimates by the PMF-56 equation with the lowest errors rates (RMSE = 0.53 mm/day and PE = 4.91%). Xing et al. (2008) evaluated the Snyder and Cuenca equations to estimate ETo in Maritime region of Canada and found that the Snyder equation generally performed better than the Cuenca equation. According to the results (Table 2), the Allen and Pruitt equation can be selected as the second best method with the R 2 value of 0.88, the RMSE value of 0.56 mm/day and an underestimation of 11.25%. Overall performances suggest that the FAO-24 pan table and Cuenca methods can be more reliable than the Orang, Modified Snyder, FAO-56 pan and Pereira methods for estimating ETo for the study area. Grismer et al. (2002) found that pan evaporation-based estimates of ETo using both K pan tables and equations were generally within an error of approximately 10% for humid regions of California.

Fig. 2
figure 2

Comparison of 13-year mean monthly ETo calculated from the PMF-56 model and the pan evaporation-based methods

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Table 2 Statistical performance of the pan evaporation-based methods versus the PMF-56 model for estimating monthly ETo during the study period (1993–2005)
Full size table

Temperature-based ETo equations

Table 3 summarizes the results of the application of the temperature-based methods for the Rasht humid site, when compared with the full-data PMF-56 method. Consideration of all the results from the analysis indicated that the Blaney–Criddle equation had the best performance (R 2 = 99, RMSE = 0.33 mm/day and PE = 1.17%) among the temperature-based methods, followed by the Hargreaves-M4 (R 2 = 95, RMSE = 0.34 mm/day and PE = 7.87%) and Thornthwaite equations (R 2 = 82, RMSE = 0.64 mm/day and PE = 10.30%). Good performance of the Blaney–Criddle equation may stem from its original development for humid areas where the advective effect is usually negligible and has been reported by several researchers (Irmak et al. 2003b; Ali and Shui 2009). The Blaney–Criddle and Hargreaves-M4 equations overestimated PMF-56 ETo by 0.03 and 0.182 mm/day, respectively, while the Thornthwaite equation underestimated it by 0.24 mm/day (Fig. 3). Jensen et al. (1990), Alkaeed et al. (2006), Trajkovic and Kolakovic (2009) and Sentelhas et al. (2010) found that the Thornthwaite equation underestimated ETo in relation to the PMF-56 method at humid locations.

Table 3 Statistical performance of the temperature-based methods versus the PMF-56 model for estimating monthly ETo during the study period (1965–2005)
Full size table
Fig. 3
figure 3

Comparison of 41-year mean monthly ETo calculated from the PMF-56 model and the temperature-based methods

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The Hargreaves-M1, Hargreaves-M2 and Hargreaves-M3 equations performed relatively well with a R 2 higher than 0.90. The results indicated that the new version of the Hargreaves equation that contains the rainfall parameter provided closer ETo estimates than the other new types of the Hargreaves equation developed by Droogers and Allen (2002). In addition, the performance of the Hargreaves-M3 model was better than that (RMSE = 0.70 mm/day and MBE = −0.62 mm/day) for the original Hargreaves equation reported by Tabari (2010) at Rasht station. The overestimation of the Hargreaves-M1, Hargreaves-M2 and Hargreaves-M3 equations varied from 0.32 mm/day (14.21%) to 0.96 mm/day (41.57%). The overestimation of the Hargreaves equation under humid conditions were found by Jensen et al. (1997); Kashyap and Panda (2001); Yoder et al. (2004); Trajkovic (2007) and Landeras et al. (2008). Furthermore, according to Temesgen et al. (2005), higher wind speed combined with lower humidity resulted in lower values of Hargreaves ETo compared to PMF-56 ETo. Also, lower wind speed combined with higher humidity resulted in higher values of Hargreaves ETo compared to PMF-56 ETo. This is probably due to the lack of explicit wind speed and humidity terms in the Hargreaves equation. The Schendel equation was not a suitable method for estimation of ETo at the humid location due to the high overestimations (37.32%) it presented, with a RMSE of more than 1 mm day−1.

Radiation-based ETo equations

The results of the statistical analysis of the radiation-based methods versus the PMF-56 model are given in Table 4. As listed, good coefficients of determination were obtained for all the radiation-based equations, with values greater than 0.93. The derived equations (Eqs. 28, 29), Irmak and Ritchie models were the best options to estimate ETo in the study area. Eq. 29 slightly overestimated PMF-56 ETo by 0.22% with a R 2 value of 0.98 and RMSE of 0.18 mm/day (Fig. 4). Equation 28 had a lower R 2 (0.94) and higher error (RMSE = 0.26 mm/day, MBE = −0.02 mm/day and PE = 0.26%) than Eq. 29 for the study site. It means that the inclusion of maximum and minimum air temperatures instead of mean air temperature resulted in better ETo estimates. The Irmak model overestimated PMF-56 ETo by 18.10% with a R 2 value of 0.93 and RMSE of 0.54 mm/day. The overestimation of the Irmak equation was also reported by Irmak et al. (2003b) under humid conditions of Florida. The Ritchie equation overestimated ETo as compared to the PMF-56 model (MBE = −0.50 mm/day), with a R 2 value of 0.98 and RMSE of 0.57 mm/day. The Ritchie equation is a modification of the Priestley–Taylor equation. A slightly better ETo estimates (R 2 = 0.98, RMSE = 0.44 mm/day and MBE = −0.25 mm/day) were obtained by the Priestley–Taylor model (Tabari 2010) compared with the Ritchie equation at the Rasht station. The Jensen–Haise and McGuinness and Bordne models demonstrated the worst performances among the radiation-based methods with the RMSE of 1.18 and 1.87 mm/day, respectively. The poor performance of the Jensen–Haise equation obtained in this study is in good agreement with the results found in humid climates of Serbia (Trajkovic and Kolakovic 2009) and Florida (Irmak et al. 2003a, b). The Jensen–Haise and McGuinness and Bordne models greatly overestimated PMF-56 ETo by 30.24 and 59.79%, respectively. Analyses by Jensen et al. (1990) showed the Jensen–Haise equation had a tendency to overestimate ETo in humid climates.

Table 4 Statistical performance of the radiation-based methods versus the PMF-56 model for estimating monthly ETo during the study period (1965–2005)
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Fig. 4
figure 4

Comparison of 41-year mean monthly ETo calculated from the PMF-56 model and the radiation-based methods

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Mass transfer-based ETo equations

Table 5 summarizes the results from comparing the ten evaluated mass transfer-based estimates to that from the PMF-56 model. According to the MBE values, all of the mass transfer-based equations underestimated PMF-56 ETo except Rohwer, Albrecht and Brockamp and Wenner. The Romanenko (R 2 = 0.92, RMSE = 0.66 mm/day and PE = 11.99%), Dalton (R 2 = 0.81, RMSE = 0.79 mm/day and PE = 13.92%) and Meyer (R 2 = 0.84, RMSE = 0.80 mm/day and PE = 14.36%) equations yielded the best ETo estimations as compared to that from the PMF-56 method. Furthermore, the Rohwer and Penman equations provided satisfactory estimations of ETo in the study area. The WMO, Mahringer and Trabert equations with average underestimations of 44.41, 31.18 and 25.99% and the Brockamp and Wenner with an average overestimation of 26.09% showed the worst performances among the mass transfer-based methods for estimating ETo in the humid area. The mean monthly ETo estimated by the mass transfer-based methods and the PMF-56 model is plotted in Fig. 5.

Table 5 Statistical performance of the mass transfer-based methods versus the PMF-56 model for estimating monthly ETo during the study period (1965–2005)
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Fig. 5
figure 5

Comparison of 41-year mean monthly ETo calculated from the PMF-56 model and the mass transfer-based methods

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Cross-comparison of the ETo methods

According to the RMSE values, the 10 best methods were selected among the 31 considered ETo methods (Fig. 6). Equation 29 (radiation-based) ranked first with a RMSE of 0.18 mm/day. Equation 28 (radiation-based) ranked second with a RMSE of 0.26 mm/day. The temperature-based Blaney–Criddle and Hargreaves-M4 equations can be considered as the third and fourth best methods with RMSE values of 0.33 and 0.34 mm/day, respectively. The fifth was the Snyder radiation-based equation with a RMSE of 0.53 mm/day. The Irmak, Ritchie, Allen and Pruitt, FAO-24 pan table and Cuenca methods ranked sixth place to tenth, respectively. In general, the comparative results showed that the mass transfer-based equations had the worst performances among the ETo methods evaluated. The radiation-based and temperature-based models were the best-suited equations for the humid climate. Furthermore, the pan evaporation-based methods performed well in the study area, indicating that the pan measurement simulates the change in all relevant climatic conditions fairly well. This may not be surprising as pan evaporation provides an integrated measurement of the effects of solar radiation, wind speed, air temperature and relative humidity (Chen et al. 2005). To evaluate the best ETo equations obtained, the Eqs. 29, 28, Blaney–Criddle and Hargreaves-M4 were tested at another humid site (Bandar–Anzali). The equations with the R 2 values higher than 0.94 and the RMSE values lower than 0.7 mm/day presented the good performances at Bandar–Anzali station (Table 6).

Fig. 6
figure 6

The RMSE values for the 10 best methods among the 31 considered ETo methods

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Table 6 Performance evaluation of the four best ETo equations at Bandar–Anzali station
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Summary and conclusions

In this study, 29 commonly used ETo equations that developed from four different approaches (1) pan evaporation-based, (2) temperature-based, (3) radiation-based, and (4) mass transfer-based were tested against the PMF-56 standard model. The best and worst equations of each group were determined using climatic data from Rasht station located in a humid climate of northern Iran. In addition, two radiation-based methods for estimating ETo were derived using air temperature and solar radiation data based on the PMF-56 model as a reference. The results indicated that all of the pan evaporation-based methods had a tendency to underestimate PMF-56 ETo. Similarly, the majority of the mass transfer-based equations underestimated PMF-56 ETo in the humid environment. Among the pan evaporation-based methods, the ETo calculated by the Snyder equation best matched the ETo estimates from the PMF-56 equation with the lowest errors rates (RMSE = 0.53 mm/day and PE = 4.91%). The Romanenko (R 2 = 0.92, RMSE = 0.66 mm/day and PE = 11.99%), Dalton (R 2 = 0.81, RMSE = 0.79 mm/day and PE = 13.92%) and Meyer (R 2 = 0.84, RMSE = 0.80 mm/day and PE = 14.36%) equations gave the best ETo estimations among the mass transfer-based methods.

In contrast with the pan evaporation-based and mass transfer-based methods, the temperature-based and radiation-based equations overestimated PMF-56 ETo. The analysis also showed that the Blaney–Criddle equation had the best performance (R 2 = 99, RMSE = 0.33 mm/day and PE = 1.17%) among the temperature-based methods, followed by the Hargreaves-M4 (R 2 = 95, RMSE = 0.34 mm/day and PE = 7.87%). Furthermore, the ETo values estimated by the two radiation-based equations developed in this study were superior to the corresponding values obtained from the existing radiation-based methods. Comparison of the 31 considered ETo methods showed that the two developed radiation-based equations yielded ETo values most similar to those from the PMF-56 model, and the Blaney–Criddle, Hargreaves-M4, Snyder, Irmak, Ritchie, Allen and Pruitt, FAO-24 pan table and Cuenca methods were the third to tenth best methods, respectively. In general, the comparative results showed that the mass transfer-based equations had the worst performances, while the radiation-based and temperature-based models were the best-suited equations for estimating ETo in this humid climate of Iran. Considering the unavailability of full weather data for applying the PMF-56 model for estimation of ETo in many regions of the world, especially in developing countries, the results will be useful for choosing the simpler ETo methods in humid climates. Such comprehensive studies as that conducted here are recommended for evaluation of the simpler ETo methods in other climatic conditions.