The method can be described in following way:

PMET ref = 0.408 ⋅ Δ ⋅ ( R n − G ) + γ ( 900 T k + 273 ) ⋅ u 2 ⋅ ( e s − e a ) Δ + γ ⋅ ( 1 + 0.34 ⋅ u 2 )

where ET_ref is the reference evapotranspiration (mm/day), 0.408 MJ·m⁻²·day⁻¹ converting the latent heat at 20˚C into mm/day value, Δ slope of vapour pressure curve (kPa·˚C⁻¹), R_n net radiation at crop surface (MJ·m²·day⁻¹), G soil heat flux (MJ·m⁻²·day⁻¹), that can be neglected if the soil is totally shaded by grass surface, γ psychometric constant (kPa·˚C⁻¹), considering the altitude of Hungarian meteorological stations it equals 0.066, T_mean the mean air temperature, u₂ wind speed at 2 m height e_s – e_a vapour pressure deficit (kPa). The elements of FAO Penman-Monteith equation were determined with the help of methods applied type by Allen et al. [4] and Walter et al. [3]. This method was evaluated on the data measured by lysimeter in Szarvas experimental station in Hungary. Calculated data by Penman-Monteith method and measured data by Lysimeter in Szarvas were in good agreement Varga-Haszonits et al. [5].

The slope (Δ) of the curve of temperature-saturation vapour pressure:

Δ = 2503 ( 17.27 T mean T mean + 237.3 ) ( T mean + 237.3 ) 2

The T_mean mean temperature calculated from maximum and minimum temperatures is:

T mean = T max + T min 2

Radiation balance (R_n):

R n = ( 1 − α ) R g l − R n l

whereα commonly used crop albedo (0.23);

R_gl global radiation determined by Angström formula;

R g l = ( 0.25 + 0.50 n N ) R a

R_a extraterrestrial radiation can be estimated from solar radiation data:

R a = 24 ( 60 ) Π G s c d r [ ω s sin ( φ ) sin ( δ ) + cos ( φ ) cos δ sin ω s ]

where R_a is the extraterrestrial radiation (MJ·m⁻²·day⁻¹);

G_sc is the solar constant (4.92 MJ·m⁻²·hour⁻¹);

d_r inverse relative distance Earth-Sun;

ω_s sunset hour angle (rad), φ s latitude (rad) and δ solar declination (rad);

R_nl net outgoing longwave radiation as can be described as

R n l = σ ( ( T max , K ) 4 + ( T min , K ) 4 2 ) ( 0.34 − 0.14 e a ) ( 1.35 R g l R g l 0 − 0.35 )

where R_nl net outgoing long wave radiation (MI·m⁻²·day⁻¹);

σ Stefan-Boltzmann constant (4.903 × 10⁻⁹ MJ·K⁻⁴·m⁻²·day⁻¹);

T_max,K and T_min,K a max. and min. temperature in Kelvin;

e_a actual vapour pressure (kPa), R_gl global radiation (MJ·m⁻²·day⁻¹);

R_gl0 (MJ·m⁻²·day⁻¹) clear sky solar radiation max possible (MJ·m⁻²·day⁻¹).

The saturation deficit, which is the difference of saturation vapour pressure (e_s) and actual vapour pressure (e_a), determines the aerodynamic vapour transportation.

To estimate saturation vapour pressure, Allen et al. [4]:

e s = e 0 ( T max ) + e 0 ( T min ) 2

The e⁰(T_max) and e⁰(T_min) can be estimated based on the equation below if we use values off aT_max and T_min:

e 0 ( T ) = 0.6108 exp ( 17.27 T T + 237.3 )

Actual vapour pressure can be estimated with the dewpoint temperature (T_dew):

e a = 0.6108 exp ( 17.27 T dew T dew + 237.3 )

If there are no dewpoint temperature data available, we can calculate the actual vapour pressure values from the minimum temperature values under the subhumid climatic conditions of Hungary with reasonable accuracy (Varga-Haszonits) [6].

Wind speed is measured at different heights, not at 2 meter height, where the other meteorological elements are measured. Wind speed measured at different heights can be adjusted with the help of logarithmic wind profile to 2 m above ground level (Campbell and Norma) [7].

The Penman-Monteith formula requires wind speed data measured at 2 m above ground level to determine other elements at the same height, and can be adjusted according to the following equation (Allen et al.) [4]:

u 2 = u z ⋅ 4.87 ln ( 67.8 ⋅ z − 5.42 )

where u₂ is the wind speed at 2 m height (m/s), u_z is the wind speed at z height (m/s), and z is the height of measurement (m).

The method suggests an average wind speed of 2 m/s in regions with low constant wind speed data if there are no wind speed data available (Allen et al.) [4], then the Penman-Monteith formula can be described as:

PMET cw = 0.408 ⋅ Δ ⋅ ( R n − G ) + γ ( 900 T k + 273 ) ⋅ 2 ⋅ ( e s − e a ) Δ + γ ⋅ ( 1 + 0.34 ⋅ 2 )

where PMET_cw (cw = constant wind speed).

When there are no wind speed data available, Penman-Monteith formula can be expressed exclusively using radiation, temperature and vapour pressure data:

MET rtv = 0.408 ⋅ Δ ⋅ ( R n − G ) + γ ( 900 T k + 273 ) ⋅ ( e s − e a ) Δ + γ

where PMET_rtv (rtv = radiation, temperature, vapour pressure) is the reference evapotranspiration calculated from the radiation, temperature and vapour pressure data.

The relationship between FAO-PMET and the climatic factors was determined using linear regression. It is well known, that the higher the value of the determination coefficient of the linear relationship (r²), the more the measured climatic factor explains the changes in evapotranspiration. Furthermore, the form of the line y = a +bx shows how closely the calculated relationship approximates the original one. Namely, if b is close to 1 and a to 0, then the result gives a close approach, but if b = 1 and a = 0, then y = x. In this case, it may be necessary to examine only those factors that most intensively influence the evaporation. It needs to take into consideration, however, Wilks [8] revealed the ambiguity of these expectations. Namely, the regression relationship quantitatively expresses the nature and closeness of the relationship between two variables, but does not give information regarding causality. Nevertheless in the agro- and hydroclimatology the effecting factors and the dependent variables can be satisfactorily distinguished. The regression relationship between the climatic factors affecting evaporation and the FAO PMET_ref values was analysed in a way where except the selected factors all the other values were kept constant.

To analyse the effect of wind speed at 2 m above ground level and under Hungary’s climatic conditions, the wind speed data of the 14 meteorological stations

are presented in Table 2. Table 2 shows that the average wind speed values of the months do not reach 3 m/s and they fall under the value of 1.5 m/s for only some months. Data vary around 2 m/s. It is, therefore, reasonable to examine what values we get if we apply the Penman-Monteith formula and calculate with 2 m/s average wind speed. Allen et al. [4] also advise to calculate reference evapotranspiration with a constant 2 m/s value if there are no wind speed data available, Allen et al. [4].

The Penman-Monteith formula was calculated in original form and in modified form using 2 m/s wind speed data for 14 meteorological stations for the time period between 1976-2000. The relationship between the results of the two methods was expressed in a form of regression equations. The result is shown in Table 3.

Table 3 shows a very high correlation between the two formulas, If the form and relationship of the linear equation are expressed as y = bx + a and b = 1 and a = 0, then x = y; Therefore the received result is more accurate if b is much closer to 1 and a to 0. In Table 3, the value of a can be regarded as 0 and b shows that the received value is a few tenths lower than the original Penman-Monteith formula.

We compared the results of calculations without wind speed data with the results received with the application of the original formula (Table 4).

Table 4 shows if we apply PMET_rtv formula to estimate the reference ET with

the application of PMET_ref then the coefficient of b changes between 0.92 and 0.96, which indicates a good approach. The line’s a constant of 0.03 and 0.16 mm shows differences of only 0 and 0.2 mm.

If the amount of irrigation water need has to be calculated, then it is very important to know the difference between the values of the original reference evapotranspiration and the values of evapotranspiration estimated with modified formulas. From a practical perspective, it is very useful to know not only the value of error but its frequency as well. Therefore when assessing the usefulness of the modified formulas we looked at the frequency of errors and chose the method which produced the smallest errors with the highest frequency.

Table 5 shows the frequency of errors in the case of formulas estimated at 2 m constant wind speed. It can be seen from the data of Table 5 that differences close to 0 occur mostly at the measuring sites. Zero is the most frequent difference at most sites. There are only 4 of the measuring sites where the most frequent values fall into the interval of 0.1. Differences between −0.2 and +0.2 mm are the most frequent in relation to all measuring sites. Table 5 also shows differences of −0.1 and +0.1 mm, which have the most frequent occurrence. They show rates of 70% or higher at all measuring sites. Their frequency of occurrence is the lowest in Miskolc, where it is exactly 70% and the highest rate is in Mosonmagyaróvár, with 95%. As a result, it may be concluded that considering wind speed conditions in Hungary PMET_cw formula can be applied to estimate reference evapotranspiration in regions where there are no wind speed data available. Another approach was where wind speed data was completely excluded. Table 4 shows this formula which produced very good results, so we compared this formula of PMET_rtv with the results of the original PMET_ref formula (Table 6).

The most frequent estimation errors fall into the interval of −0.1 and +0.2.

The estimation error of +0.2 mm is also relatively frequent. These results fit well with the results achieved with other alternative formulas in humid and sub-humid regions (Fisher and Pringle) [9].