TPenMonteith

1 Summary

1.1 Short Info

 Unit for calculating transpiration, evaporation, and interception using standard weather and plant data,
 based on the Penman-Monteith approach and related literature.
 

1.2 Author

 Henning Kage & Agronomy Group, University of Kiel
 

1.3 Timestamp

 First edited: 6.10.89
 Last edited: 02.08.25
 

1.4 Key References

• van Bavel, C.H.M. (1966). Potential evaporation: The Combination Concept and its Experimental Verification. Water Resour.Res. (1966) 2, 455-467
• Duynisveld, W.H.M. (1983). Entwicklung von Simulationsmodellen fuer den Transport von geloesten Stoffen in wasserungesaettigten Boeden and Lockersedimenten. Texte Umweltbundesamt (1983) 17, 197 Seiten
• Monteith, J.L. (1973). Principles of Environmental Physics. Edward Arnold, London, 1973, 241 Seiten
• Groot, J.J.R. (1987). Simulation of nitrogen balance in a system of winter wheat and soil. Simulation Reports CABO-TT, Wageningen 1987
• Loebmeier, F.J. (1983). Agrarmeteorologisches Model zur Berechnung der aktuellen Verdunstung (AMBAV). Beitraege zur Agrarmeteorologie Nr. 7/83

1.5 Classes

The unit UPenMonteith defines the following classes:

TPenMonteith

1.6 Descendence

The class TPenMonteith is derived from: TPlantRelatedSubMod <- TSubmodel <- TGraphicControl <- TControl <- TComponent <- TPersistent <- TObject

2 Submodel objects

Dynamic system model consists according to the Forrester approach of state variable, parameters, variables and external driving forces. This is implemented in Hume by dynamic lists, which contents is given in the following sections.

2.1 State variables

The class TPenMonteith has 1 following state variable(s).

State variable	Units	InitialValue	Description
int_stor	[mm]	0	intercepted water on canopy

TRUE

2.2 Parameters

The class TPenMonteith has 7 following parameter(s).

Parameter	Units	Value	Description
CiThreshold	[ppm]	380	threshold for CO2 impact on rc0
Elev	[m]	50	Höhe über NN [m]
exk_GlobRad	[-]	0.5	extiction coefficient for global radiation
measure_height	[m]	2	Measurement height [m]
rc0	[s.m-1]	50	Stomatawiderstand bei guter Wasserversorgung
relRc0Inc_CO2	[(s.m-1)/ppm]	0.3878	mediates the relative CO2 impact on rc0 estimated from: Elevated CO2 effects on canopy and soil water flux parameters… Burkart et al. 2010
SIC	[mm.m-2.m-2]	0.15	specific interception capacity

TRUE

2.3 Variables

The class TPenMonteith has 21 following variable(s).

kable(df.var,  escape = FALSE)

Variable	Units	Description
CO2pp	[ppm]	external CO2 concentration
CO2TransDiff	[mm/d]	CO2 induced reduction of pot_trans
ET0	[]	reference evapotranspiration short grass (FAO)
interception	[mm/d]	daily interception rate
k_GlobRad	[-]	actual extinction coefficient for global radiation, can be from parameter or from external crop model
NetRad	[W.m-2]	net radiation
NetRain	[mm/d]	rain - interception
P	[mbar]	air pressure
pETP	[]	potential evporation
pETP_ambient	[]	potential evapotranspiration ohne CO2 Einfluss
pot_Evapo_ambient	[mm/d]
PotEvap	[mm/d]	potential soil evaporation rate
PotTrans	[mm/d]	potential plant transpiration
potTrans_ambient	[mm/d]	potential transpiration under ambient CO2
ra	[s/m]	aerodynamic resistance
rc	[s/m]	canopy resistance
rc_ambient	[s/m]	canopy resistance under abient CO2
rc0_ambient	[s.m-1]	rc0 value without CO2 effect
rc0_Var	[s.m-1]	rc0 value as used for calculation (from parameter or plant model)
relCO2TransDiff	[-]	rel. CO2 induced reduction of pot_trans
VapPress	[mbar]	saturated vapour pressure

TRUE

2.4 External variables

The class TPenMonteith has 8 following external variable(s).

External variable	Units	Description	Source
CropHeight		crop height	NA
CO2pp		external CO2 concentration	NA
Rad_Int		gobal radiation in [W.m-2]	NA
LAI		leaf area index	NA
rain		rainfall rate	NA
Sat_def		Sättigungsdefizit [hPa]	NA
TMPM		average daily temperature	NA
Wind		wind speed	NA

TRUE

2.5 Options

The class TPenMonteith has 5 following option(s).

Option	Units	Description
ContOutput	NA	Output every time step?
FinalOutput	NA	Output of final values in separate file?
optCO2	NA	Option for including effects of elevated CO2 in calculations
optExCO2	NA	Option for using external supplied CO2 concentration in calculations
ra_Option	NA	Option for ra(u, CropHeight)-function

TRUE

3 Further source code documentation

The following section includes information extracted from the XML documentation of the class TPenMonteith. It includes constants, enumerations and functions.

3.1 Enumerations

Within the unit (2) enumeration types are defined :

Table 1: Enumerations of unit UPenMonteith

Name	Summary	Elements
TSource	Source of extinction coefficient / rc0	fromParameter, fromPlantModel
T_ra_Funct	Options for ra calculation	PenmanMonteith, ThomOliver

TRUE

The enumeration types are used to internally define options for different process formulation. By setting an TOption object

3.2 Constants

The following internal constants are defined within the Unit

Table 2: Constants of unit UPenMonteith

Name	Type	Summary	Value
l_h_v_water	Extended	latent heat for water evaporation at 10 °C in [J/Kg]	2.477e+06
MW_ratio	Comp	ratio of the molecular weight of water to the molecular weight of dry air	0.622
c_p	Comp	specific heat of air at constant pressure [J/kg/K]	1005

TRUE

3.3 Functions

paged_table(output$fun_df)

Table 3: Functions of class TPenMonteith

TRUE

The procedures defined in the class TPenMonteith are:

paged_table(output$proc_df)

Table 4: Functions of class TPenMonteith

TRUE

4 Scientific Background

Evaporation of a plant canopy depends like dry matter production on the amount of intercepted radiant energy but also on the vapour pressure deficit of the ambient air. One of the most fundamental methods to calculate the evaporation rate of a closed plant canopy is the Penman-Monteith equation (Monteith, 1973):

\lambda E=\frac{s{{R}_{n}}+{{\rho }_{a}}{{c}_{p}}\frac{{{e}_{s}}-{{e}_{a}}}{{{r}_{a}}}}{s+\gamma \left( 1+\frac{{{r}_{c}}}{{{r}_{a}}} \right)} \tag{1} with evaporation E (kg^.m^-2.s^-1), latent heat of vaporisation of water \lambda (J^.kg^-1), net radiation R_n (W^.m^-2), total resistance of the pathway between the evaporating sites and the bulk air r_a (s^.m^-1), the canopy resistance r_c (s^.m^-1), the vapour pressure deficit between the evaporating sites and the bulk air e_s-e_a (hPa), the slope of the vapour saturation curve s (hPa^.K^-1), the volumetric heat capacity of dry air c_p (J^.kg^-1), the density of dry air \rho_a (kg^.m^-3) and the psychrometer constant \gamma (hPa^.K^-1).

In order to calculate the evaporation rate, the Penman-Monteith equation requires the net radiation, the vapor pressure deficit, the canopy resistance, the aerodynamic resistance.

The net radiation is the difference between the incoming shortwave radiation and the reflected shortwave radiation plus the incoming longwave radiation minus the outgoing longwave radiation and the soil heat flux. The vapour pressure deficit is the difference between the saturation vapour pressure and the actual vapour pressure. The canopy resistance is the resistance to water vapour transfer from the crop surface to the atmosphere. The aerodynamic resistance is the resistance to water vapour transfer from the leaf surface to the atmosphere.

4.0.1 Net radiation

The net radiation R_n [W^.m^-2] is computed from the global radiation, GR [W^.m^-2], using an empirical regression equation derived from measurements of global and net radiation over grass at the experimental field ‘Ruthe’ (Figure 1):

{{R}_{n}}=0.6494\cdot GR-18.417 \tag{2}

The FAO-56 method calculates the net radiation as the difference between the incoming shortwave radiation and the reflected shortwave radiation plus the incoming longwave radiation minus the outgoing longwave radiation and the soil heat flux (Carmona et al. (2017), Allen et al. (1998)):

R_n = (1 - \alpha) \cdot R_s + R_l - G \tag{3}

where \alpha is the albedo, R_s is the incoming shortwave radiation, R_l is the incoming longwave radiation and G is the soil heat flux. The albedo is the fraction of the incoming shortwave radiation that is reflected by the surface. The soil heat flux is the heat flux from the soil to the atmosphere. The soil heat flux is usually small compared to the other components of the net radiation and is often neglected.

The net energy flux leaving the earth’s surface is, however, less than that emitted and given by the Stefan-Boltzmann law due to the absorption and downward radiation from the sky. Water vapour, clouds, carbon dioxide and dust are absorbers and emitters of longwave radiation. Their concentrations should be known when assessing the net outgoing flux. As humidity and cloudiness play an important role, the Stefan-Boltzmann law is corrected by these two factors when estimating - the net outgoing flux of longwave radiation. It is thereby assumed that the concentrations of the other absorbers are constant.

The equation for estimating net radiation in the FAO Penman-Monteith method (daily time resolution”), R_{nrefcrop}, is expressed as:

\begin{aligned} & R_{nrefcrop}=R_{ns}+R_{nl}=R_s\downarrow \left(1-\alpha \right )+R_{nl}=R_{ns}+R_{nl} \\ & =R_s\downarrow \left(1-\alpha \right )+\left( {R}_{l\downarrow 0}-{R}_{l\uparrow}\right) \end{aligned} \tag{4}

The FAO-56 method suggests a semi empirical approach for the calculation of the net radiation for a reference crop surface:

R_{nl}=-\sigma\left[\frac{T^4_{max,K}+T^4_{min,K}}{2} \right] \left( a_1+b_1\sqrt{e_a} \right) \left( a_c \frac{R_s\downarrow}{R_{s0}\downarrow}+b_c \right) \tag{5}

where α = 0.23 is the surface albedo, σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴) or 4.903 10^-9 [MJ m^-2 K^-4] , T_max,K and T_min,K are the daily maximum and minimum air temperatures [°K] at screen height (2 m), respectively, e_a is the water vapor pressure (kPa), R_{s \downarrow 0} is clear-sky solar radiation [MJ m⁻² d^-1], and a₁, b₁, a_c, and b_c are calibration coefficients (dimensionless). An average of the maximum air temperature to the fourth power and the minimum air temperature to the fourth power is commonly used in the Stefan-Boltzmann equation for 24-hour time steps.The term (a_l + b_l\cdot\sqrt{e_a}) expresses the correction for air humidity, and will be smaller if the humidity increases. The effect of cloudiness is expressed by (a_c\cdot (R_s↓/R_{s0}↓) + b_c). The term becomes smaller if the cloudiness increases and hence R_s↓ decreases (Allen et al. (1998)). For the application of the FAO Penman–Monteith method, values of a₁ = 0.34, b₁ = −0.14, a_c = 1.35, and b_c = −0.35 have been recommended (Allen et al. (1998)), using data from an arid climate for its calibration (Wright and Jensen 1972). However, if measurements of incoming and outgoing short and longwave radiation are available, calibration of these coefficients should be carried out for each climatic condition. For example, Jensen et al. (1990) found the best results with a_c and b_c values of [1.2, −0.2], [1.1, −0.1], and [1.0, 0.0] for arid, semiarid, and humid areas, respectively (Kjaersgaard et al. (2009)).

The clear sky shortwave radiation is the shortwave radiation that would be received if the sky were clear. The clear sky shortwave radiation can be calculated from the solar constant, the solar declination, the solar hour angle and the extraterrestrial radiation.

Figure 1: Net radiation as a function of global radiation from hourly measurements at the experimental station Ruthe in 1995

The comparison of daily measured global radiation, and empirical regression equation derived from measurements of global and net radiation over grass at the experimental field ‘Ruthe’ together with the net radiation calculated by Eq. Eq. 5 is shown in Figure 2. It is obvious that Eq. Eq. 5 overestimates net radiation which is in line with the findings of Kjaersgaard et al. (2009) for a site in Denmark.

For sake of simplicity, the empirical regression equation derived from measurements of global and net radiation over grass at the experimental field ‘Ruthe’ is used in the class TPenMonteith.

Figure 2: Net radiation as a function of global radiation from daily measurements at the experimental station Ruthe in 1995

4.0.2 Vapour pressure deficit

The vapour pressure deficit is the difference between the saturation vapour pressure and the actual vapour pressure. The saturation vapour pressure is the maximum amount of water vapour that the air can hold at a given temperature. The actual vapour pressure can be calculated from the relative humidity and the saturation vapour pressure. The relative humidity is the ratio of the actual vapour pressure to the saturation vapour pressure. The vapour pressure deficit is the difference between the saturation vapour pressure and the actual vapour pressure. The vapour pressure deficit is a measure of the drying power of the air. The greater the vapour pressure deficit, the greater the drying power of the air.

The saturation vapour pressure e_s in [hPa] can be calculated from the temperature [°C] using the Tetens-equation:

e_s = 6.1078 \cdot 10^{ \frac{7.5 \cdot T }{T + 237.3 } } \tag{6}

Please note the units used for e_s in TPenMonteith are for historical reasons in [hPa].

The saturation deficit [hPa] then is calculated from the relative humidity as:

e_{s}-e_{a} = e_s - \frac{RH}{100} \cdot e_s \tag{7}

Figure 3: Vapour pressure values for 100 and 60% relative humidity and vapour pressure deficit as a function of temperature from the Tetens-equation

4.0.3 Slope of saturation vapour pressure curve

The slope of the saturation vapour pressure curve s [hPa^.K^-1] is calculated as the derivative of the saturation vapour pressure with respect to temperature:

s = 239.0 \cdot 17.4 \cdot \frac{e_s} {(Temp + 239.0)^2} \tag{8}

4.0.4 Air pressure

The pressure of the air [hPa] can be calculated from the elevation [m] and the temperature [°C] using the barometric formula:

P = 1013.0 \cdot e^{\left(\frac{-0.034 \cdot Elev } {Temp + 273}\right)} \tag{9}

Note that the class TPenMonteith uses the pressure in [hPa] for historical reasons.

4.0.5 CO_2 concentration

The CO_2 concentration [ppm] can be supplied as an external variable or calculated from the date using a quadratic regression equation derived from measurements of CO_2 concentration at the Mauna Loa Observatory:

CO_2 = 45549.96 -47.00505 \cdot year + 0.01220744 \cdot year^2 \tag{10}

where Year is the year of the measurement.

4.0.6 Density of dry air

The density of dry air \rho_a [kg^.m^-3] can be calculated from the pressure of the air [hPa], the temperature [°C] and the gas constant for dry air [J^.kg^-1.K^-1] using the ideal gas law:

\rho_a = 1.2917 - 0.00434 \cdot T \tag{11}

where T is the temperature [°C].

4.0.7 Psychrometer constant

\gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } \tag{12}

where c_p is the specific heat of air at constant pressure [J^.kg^-1.^K-1], P is the pressure of the air [hPa], \lambda is the latent heat of vaporisation of water [J^.kg^-1] and MW_{ratio} is the ratio of the molecular weight of water to the molecular weight of dry air.

At an air pressure of 1013 hPa and a temperature of 10°C the psychrometer constant is 0.661 [hPa K^-1].

4.0.8 Aerodynamic resistance

The aerodynamic resistance depends on wind speed and the height of the vegetation. Under so called stable conditions (canopy temperature equals air temperature) a logarithmic wind profile can be found, i.e. wind speed is linearly related to the logarithm of the height. From the measurement height (z) the wind speed (u) decreases down to zero at a height z₀ which can be found by linear extrapolation on the semi-logarithmic plot of height vs wind speed. The term z₀ is also called the aerodynamic roughness length. It can be approximated by 0.13^.h, where h is the crop height [m].

Monteith and Unsworth (1990) are giving the following equation for r_a [s^.m^-1]:

{{r}_{a}}=\frac{1}{{{k}^{2}}u}\left( \ln {{\left( \frac{z}{{{z}_{0}}} \right)}^{2}} \right) \tag{13}

where k is the von Karman constant (0.41) and u is the wind speed at height z.

It is always important to consider the measurement height of the wind speed. Wind speed is measured at different heights above ground even within a specific network.

This equation, however, has the problem that when wind speed gets very low, values of r_a get unrealistically high. In fact as U \rightarrow 0, r_a \rightarrow \infty and (T_c - T_a) \rightarrow \infty, an unrealistic result. Windspeeds < 1 m s^-1 frequently occur during the course of canopy temperature measurements, a fact that must be accounted for in the calculation of r. A further complication becomes evident at the lower windspeeds when the measurement of windspeed is considered. The stall speed of many anemometers is relatively large (i.e., 0.5 m s^-1), which means that a value of U=0 can be recorded although the air may be moving above the canopy. Even without measurable horizontal wind, air movement takes place within the canopy because free convective conditions exist which results in buoyancy driven, energy containing eddies.

Thom and Oliver (1977) suggested a semi-empirical approach for overcoming this problem:

{{\text{r}}_{\text{a}}}\text{=}\frac{\text{4.72 ln}{{\left( \frac{\text{z-d}}{{{\text{z}}_{\text{0}}}} \right)}^{\text{2}}}}{\text{1 + 0.54u}} \tag{14}

where d is the zero plane displacement height, i.e. the height to which plant canopies increase the plane of zero wind speed. The displacement height is according to Monteith (1973) 0.63^.h, were h is the crop height [m].

Within the TPenMoneith class there is an option to calculate the aerodynamic resistance using either the Penman-Monteith equation or the Thom-Oliver equation.

4.0.8.1 FAO-56 method for aerodynamic resistance

The FAO-56 method differs in some details from the above approach.

For a wide range of crops the zero plane displacement height, d [m], and the roughness length governing momentum transfer, z_om [m], can be estimated from the crop height h [m] by the following equations:

d = \frac{2}{3} h \tag{15}

z_{om} = 0.123 h \tag{16}

The roughness length governing transfer of heat and vapour, zoh [m], can be approximated by:

z_{oh} = 0.1 z_{om} \tag{17}

Assuming a constant crop height of 0.12 m and a standardized height for wind speed, temperature and humidity at 2 m (z_m = z_h = 2 m), the aerodynamic resistance r_a [s m^-1] for the grass reference surface becomes

r_a = \frac{4.72 \ln \left(\frac{2-0.12}{0.123 \cdot 0.12}\right)^2}{1+0.54 \cdot 1} \approx 70 [s \cdot m^{-1}] \tag{18}

4.0.9 Canopy resistance

The second resistance considered in the Penman-Monteith equation is the canopy resistance. The canopy resistance r_c is in principle the value of the stomatal resistance r_s divided by the “active” leaf area index, assuming that the leaves act independently as emitters of water vapour:

{{r}_{c}}\approx \frac{r_s}{LAI};\,LAI<< \tag{19}

For higher values of leaf area index, however, additional leaves do not decrease the canopy resistance as the first leaves do, i.e. local increase of the vapour pressure of the air by existing leaves reduces the additional transpiration rate of additional leaves.

The canopy resistance is then calculated as:

We use an equation proposed by Stockle (personal communication) for the canopy resistance:

rc = \begin{cases} rc_0, & \text{if } LAI < 1.0 \\[8pt] \frac{rc_0}{LAI}, & \text{if } 1.0 \leq LAI < 2.0 \\[8pt] \frac{rc_0}{2} - \left(\frac{rc_0}{2} - \frac{rc_0}{3}\right) \cdot \frac{LAI - 2}{4}, & \text{if } 2.0 \leq LAI < 6.0 \\[8pt] \frac{rc_0}{3}, & \text{if } rc \text{ remains undefined} \\[8pt] \end{cases} \tag{20}

For a well watered crop the value of r_s is at around 50 [s.m^{-1}]. The resulting values of r_c plotted against LAI look like shown in Figure 4.

4.0.9.1 FAO-56 method for canopy resistance

The FAO reference evapotranspiration model (FAO-56, https://www.fao.org/4/x0490e/x0490e06.htm) suggests a simple empirical model for the canopy resistance as a function of the leaf area index.

r_{surf}=\frac{r_s}{LAI_{active}} \tag{21}

where r_{surf} is the surface resistance, r_s is the stomatal resistance and LAI_{active} is the active leaf area index. The active leaf area index is the leaf area index that is actually transpiring. This value is for the FAO-56 model calculated as 0.5 times the total leaf area index.

THE FAO-56 method further assumes a stomatal resistance, r_s, of a single leaf of 100 s m^-1 under well-watered conditions and postulates for clipped grass a general equation for LAI:

LAI = 24 \cdot h \tag{22}

where h is the crop height [m].

By assuming a crop height of 0.12 m, the surface resistance, r_c [s^.m-1^^], for the grass reference surface becomes Eq. 23:

r_c = \frac{100}{0.5 \cdot 24 \cdot 0.12} \approx 70 [s \cdot m^{-1}] \tag{23}

4.0.9.2 CO2-effect on canopy resistance

The relative difference of the CO2 concentration in the sub stomatal cavity, CO₂pp, to the CO₂ concentration in the ambient air, CO₂a, is calculated as:

rel \Delta C_i = \frac{(CO_2pp- C_iThreshold)}{C_ithreshold} \tag{24}

The canopy resistance is then calculated as:

rc_0 = Plant_rc0 \cdot \left(1+rel \Delta C_i \dot relR_{c0_{IncCO2}} \right ) \tag{25}

Figure 4: Canopy resistance rc as a function of leaf area index according to the empirical model of Stockle

The component offers the option to import the r_c0 value from a connected plant component. This is controlled by the option ‘frc0Opt’, which has to be set to ‘fromPlantModel’ in order to do so.

where Plant_rc0 is the canopy resistance of the plant model, relR_c0Inc_{CO2} is the relative increase of the canopy resistance per unit increase of the CO₂ concentration in the sub stomatal cavity and C_iThreshold is the CO₂ concentration in the sub stomatal cavity at which the canopy resistance is increased by 50%.

4.1 Dividing potential Evapotranspiration into components

Potential evaporation is the sum potential transpiration, T_p, potential evaporation, E_p, and of interception evaporation, I:

E{{T}_{p}}={{T}_{p}}+{{E}_{p}}+I \tag{26} Interception evaporation kg.m-2.d-1 is assumed to take place from a storage pool, IP, (kg^.m^-2) situated on the surface of the canopy. The capacity of this storage pool, CIP, kg^.m^-2 is calculated from a specific interception capacity, SIC, (kg^.m^2.m^-2) and the leaf area index LAI m^2.m^-2.

CIP=SIC\cdot LAI \tag{27}

Interception is the minimum of the sum of the maximum possible change of the interception pool and the precipitation rate, Pr, kg^.m^-2.d^-1 and the potential Evapotranspiration:

I=\min \left( {}^{IP}/{}_{dt}+\min \left( \Pr ,\,CIP-IP \right),\,\,E{{T}_{p}} \right) \tag{28}

The change of the storage pool is calculated from the minimum of the precipitation rate and the actual unused capacity of the precipitation pool, CIP-IP:

\frac{dIP}{dt}=\min \left( \Pr ,\,CIP-IP \right)-I \tag{29}

The potential evaporation rate is determined by the fraction of radiation energy which is reaching the soil surface, R_ns. This fraction can be calculated using a analogue of the Lambert-Beer law:

{{R}_{ns}}={{R}_{n}}\cdot {{e}^{-{{k}_{G}}\cdot LAI}} \tag{30}

Were k_G is the extinction coefficient for global radiation, which has a default value of 0.5. It can be estimated from the extinction coefficient for photosynthetic active radiation by dividint it by 1.35.

The potential evaporation then gets:

{{E}_{p}}=\left( E{{T}_{p}}-I \right)\cdot \frac{{{R}_{ns}}}{{{R}_{n}}} \tag{31}

Actual evaporation was determined from an empirical function using potential evaporation and the water potential in 10 cm depth as input parameters (Beese et al. (1978)). The potential transpiration is calculated as the remaining part of the potential evapo-transpiration after subtracting potential evaporation and interception and taking also into account an empirical crop resistance.

References

Allen, R.G., Pereira L.S., Raes D., Smith M., 1998. Crop evapotranspiration, Irrigation and drainage paper. FAO, Rome.

Beese, F., R.R. Van Der Ploeg, Richter, W., 1978. Der wasserhaushalt einer löß-parabraunerde unter winterweizen und brache. Computermodelle und ihre experimentelle verifizierung. Z. Acker und Pflanzenbau 146, 1–19.

Carmona, F., Rivas, R., Kruse, E., 2017. Estimating daily net radiation in the FAO penman–monteith method. Theoretical and Applied Climatology 129, 89–95. https://doi.org/10.1007/s00704-016-1761-6

Jensen, M.E., Burman R.D., Allen, R.G., 1990. Evapotranspiration and irrigation water requirements, ASCE manual. American Society of Civil Engineers, New York.

Kjaersgaard, J.H., Cuenca, R.H., Martínez-Cob, A., Gavilán, P., Plauborg, F., Mollerup, M., Hansen, S., 2009. Comparison of the performance of net radiation calculation models. Theoretical and Applied Climatology 98, 57–66. https://doi.org/10.1007/s00704-008-0091-8

Monteith, J.L., 1973. Principles of environmental physics. Edward Arnold, London pp.241.

Monteith, J.L., Unsworth, M.H., 1990. Principles of environmental physics. Edward Arnold, London.

Thom, A.S., Oliver, H.R., 1977. On penman’s equation for estimating regional evaporation. Journal of the Royal Meteorological Society 103, 345–357.