Meteorological factors determining ET
Atmospheric parameters
Air temperature
Air humidity
Radiation
Wind speed
Climatic data acquisition
Estimating missing climatic data
Minimum data requirements
The methods for calculating evapotranspiration from meteorological data require various climatological and physical parameters. Some of the data are measured directly in weather stations. Other parameters are related to commonly measured data and can be derived with the help of a direct or empirical relationship. This chapter discusses the source, measurement and computation of all data required for the calculation of the reference evapotranspiration by means of the FAO PenmanMonteith method. Different examples illustrate the various calculation procedures. Appropriate procedures for estimating missing data are also provided.
Meteorological data can be expressed in several units. Conversion factors between various units and standard S. I. units are given in Annex 1. Climatic parameters, calculated by means of the equations presented in this chapter are tabulated and displayed for different meteorological conditions in Annex 2. Only the standardized relationships are presented in this chapter. The background of certain relationships and more information about certain procedures are given in Annex 3. Annexes 4, 5 and 6 list procedures for the statistical analysis, assessment, correction and completion of partial or missing weather data.
The meteorological factors determining evapotranspiration are weather parameters which provide energy for vaporization and remove water vapour from the evaporating surface. The principal weather parameters to consider are presented below.
The evapotranspiration process is determined by the amount of energy available to vaporize water. Solar radiation is the largest energy source and is able to change large quantities of liquid water into water vapour. The potential amount of radiation that can reach the evaporating surface is determined by its location and time of the year. Due to differences in the position of the sun, the potential radiation differs at various latitudes and in different seasons. The actual solar radiation reaching the evaporating surface depends on the turbidity of the atmosphere and the presence of clouds which reflect and absorb major parts of the radiation. When assessing the effect of solar radiation on evapotranspiration, one should also bear in mind that not all available energy is used to vaporize water. Part of the solar energy is used to heat up the atmosphere and the soil profile.
The solar radiation absorbed by the atmosphere and the heat emitted by the earth increase the air temperature. The sensible heat of the surrounding air transfers energy to the crop and exerts as such a controlling influence on the rate of evapotranspiration. In sunny, warm weather the loss of water by evapotranspiration is greater than in cloudy and cool weather.
FIGURE 10. Illustration of the effect of wind speed on evapotranspiration in hotdry and humidwarm weather conditions
While the energy supply from the sun and surrounding air is the main driving force for the vaporization of water, the difference between the water vapour pressure at the evapotranspiring surface and the surrounding air is the determining factor for the vapour removal. Wellwatered fields in hot dry arid regions consume large amounts of water due to the abundance of energy and the desiccating power of the atmosphere. In humid tropical regions, notwithstanding the high energy input, the high humidity of the air will reduce the evapotranspiration demand. In such an environment, the air is already close to saturation, so that less additional water can be stored and hence the evapotranspiration rate is lower than in arid regions.
The process of vapour removal depends to a large extent on wind and air turbulence which transfers large quantities of air over the evaporating surface. When vaporizing water, the air above the evaporating surface becomes gradually saturated with water vapour. If this air is not continuously replaced with drier air, the driving force for water vapour removal and the evapotranspiration rate decreases.
The combined effect of climatic factors affecting evapotranspiration is illustrated in Figure 10 for two different climatic conditions. The evapotranspiration demand is high in hot dry weather due to the dryness of the air and the amount of energy available as direct solar radiation and latent heat. Under these circumstances, much water vapour can be stored in the air while wind may promote the transport of water allowing more water vapour to be taken up. On the other hand, under humid weather conditions, the high humidity of the air and the presence of clouds cause the evapotranspiration rate to be lower. The effect on evapotranspiration of increasing wind speeds for the two different climatic conditions is illustrated by the slope of the curves in Figure 10. The drier the atmosphere, the larger the effect on ET and the greater the slope of the curve. For humid conditions, the wind can only replace saturated air with slightly less saturated air and remove heat energy. Consequently, the wind speed affects the evapotranspiration rate to a far lesser extent than under arid conditions where small variations in wind speed may result in larger variations in the evapotranspiration rate.
Atmospheric pressure (P)
Latent heat of vaporization (l)
Psychrometric constant (g)
Several relationships are available to express climatic parameters. The effect of the principal weather parameters on evapotranspiration can be assessed with the help of these equations. Some of the relationships require parameters which express a specific characteristic of the atmosphere. Before studying the four principal weather parameters, some atmospheric parameters will be discussed.
The atmospheric pressure, P, is the pressure exerted by the weight of the earth's atmosphere. Evaporation at high altitudes is promoted due to low atmospheric pressure as expressed in the psychrometric constant. The effect is, however, small and in the calculation procedures, the average value for a location is sufficient. A simplification of the ideal gas law, assuming 20°C for a standard atmosphere, can be employed to calculate P:
_{} (7)
where
P atmospheric pressure [kPa],
z elevation above sea level [m],
Values for atmospheric pressure as a function of altitude are given in Annex 2 (Table 2.1).
The latent heat of vaporization, l, expresses the energy required to change a unit mass of water from liquid to water vapour in a constant pressure and constant temperature process. The value of the latent heat varies as a function of temperature. At a high temperature, less energy will be required than at lower temperatures. As l varies only slightly over normal temperature ranges a single value of 2.45 MJ kg^{1} is taken in the simplification of the FAO PenmanMonteith equation. This is the latent heat for an air temperature of about 20°C.
The psychrometric constant, g, is given by:
_{} (8)
where
g psychrometric constant [kPa °C^{1}],
P atmospheric pressure [kPa],
l latent heat of vaporization, 2.45 [MJ kg^{1}],
c_{p} specific heat at constant pressure, 1.013 10^{3} [MJ kg^{1} °C^{1}],
e ratio molecular weight of water vapour/dry air = 0.622.
The specific heat at constant pressure is the amount of energy required to increase the temperature of a unit mass of air by one degree at constant pressure. Its value depends on the composition of the air, i.e., on its humidity. For average atmospheric conditions a value c_{p} = 1.013 10^{3} MJ kg^{1} °C^{1} can be used. As an average atmospheric pressure is used for each location (Equation 7), the psychrometric constant is kept constant for each location. Values for the psychrometric constant as a function of altitude are given in Annex 2 (Table 2.2).
EXAMPLE 2. Determination of atmospheric parameters.
Determine the atmospheric pressure and the psychrometric constant at an elevation of 1800 m.  
With: 
z = 
1800 
m 
From Eq. 7: 
P = 101.3 [(293  (0.0065) 1800)/293]^{5.26} = 
81.8 
kPa 
From Eq. 8: 
g = 0.665 10^{3} (81.8) = 
0.054 
kPa °C^{1} 
The average value of the atmospheric pressure is 81.8 kPa. 
Agrometeorology is concerned with the air temperature near the level of the crop canopy. In traditional and modem automatic weather stations the air temperature is measured inside shelters (Stevenson screens or ventilated radiation shields) placed in line with World Meteorological Organization (WMO) standards at 2 m above the ground. The shelters are designed to protect the instruments from direct exposure to solar heating. The louvered construction allows free air movement around the instruments. Air temperature is measured with thermometers, thermistors or thermocouples mounted in the shelter. Minimum and maximum thermometers record the minimum and maximum air temperature over a 24hour period. Thermographs plot the instantaneous temperature over a day or week. Electronic weather stations often sample air temperature each minute and report hourly averages in addition to 24hour maximum and minimum values.
Due to the nonlinearity of humidity data required in the FAO PenmanMonteith equation, the vapour pressure for a certain period should be computed as the mean between the vapour pressure at the daily maximum and minimum air temperatures of that period. The daily maximum air temperature (T_{max}) and daily minimum air temperature (T_{min}) are, respectively, the maximum and minimum air temperature observed during the 24hour period, beginning at midnight. T_{max} and T_{min} for longer periods such as weeks, 10day's or months are obtained by dividing the sum of the respective daily values by the number of days in the period. The mean daily air temperature (T_{mean}) is only employed in the FAO PenmanMonteith equation to calculate the slope of the saturation vapour pressure curves (D) and the impact of mean air density (P_{a}) as the effect of temperature variations on the value of the climatic parameter is small in these cases. For standardization, T_{mean} for 24hour periods is defined as the mean of the daily maximum (T_{max}) and minimum temperatures (T_{min}) rather than as the average of hourly temperature measurements.
_{} (9)
The temperature is given in degrees Celsius (°C) or Fahrenheit (°F). The conversion table is given in Annex 1. In some calculation procedures, temperature is required in Kelvin (K), which can be obtained by adding 273.16 to the temperature expressed in degrees Celsius (in practice K = °C + 273.16). The Kelvin and Celsius scale have the same scale interval.
The water content of the air can be expressed in several ways. In agrometeorology, vapour pressure, dewpoint temperature and relative humidity are common expressions to indicate air humidity.
Vapour pressure
Water vapour is a gas and its pressure contributes to the total atmospheric pressure. The amount of water in the air is related directly to the partial pressure exerted by the water vapour in the air and is therefore a direct measure of the air water content.
In standard S. I. units, pressure is no longer expressed in centimetre of water, millimetre of mercury, bars, atmosphere, etc., but in pascals (Pa). Conversion factors between various units and Pa are given in Annex 1. As a pascal refers to a relatively small force (1 newton) applied on a relatively large surface (1 m^{2}), multiples of the basic unit are often used. In this handbook, vapour pressure is expressed in kilopascals (kPa = 1000 Pa).
When air is enclosed above an evaporating water surface, an equilibrium is reached between the water molecules escaping and returning to the water reservoir. At that moment, the air is said to be saturated since it cannot store any extra water molecules. The corresponding pressure is called the saturation vapour pressure (e°(T)). The number of water molecules that can be stored in the air depends on the temperature (T). The higher the air temperature, the higher the storage capacity, the higher its saturation vapour pressure (Figure 11).
As can be seen from Figure 11, the slope of the curve changes exponentially with temperature. At low temperatures, the slope is small and varies only slightly as the temperature rises. At high temperatures, the slope is large and small changes in T result in large changes in slope. The slope of the saturation vapour pressure curve, D, is an important parameter in describing vaporization and is required in the equations for calculating ET_{o} from climatic data.
FIGURE 11. Saturation vapour pressure shown as a function of temperature: e°(T) curve
FIGURE 12. Variation of the relative humidity over 24 hours for a constant actual vapour pressure of 2.4 kPa
The actual vapour pressure (e_{a}) is the vapour pressure exerted by the water in the air. When the air is not saturated, the actual vapour pressure will be lower than the saturation vapour pressure. The difference between the saturation and actual vapour pressure is called the vapour pressure deficit or saturation deficit and is an accurate indicator of the actual evaporative capacity of the air.
Dewpoint temperature
The dewpoint temperature is the temperature to which the air needs to be cooled to make the air saturated. The actual vapour pressure of the air is the saturation vapour pressure at the dewpoint temperature, The drier the air, the larger the difference between the air temperature and dewpoint temperature.
Relative humidity
The relative humidity (RH) expresses the degree of saturation of the air as a ratio of the actual (e_{a}) to the saturation (e°(T)) vapour pressure at the same temperature (T):
_{} (10)
Relative humidity is the ratio between the amount of water the ambient air actually holds and the amount it could hold at the same temperature. It is dimensionless and is commonly given as a percentage. Although the actual vapour pressure might be relatively constant throughout the day, the relative humidity fluctuates between a maximum near sunrise and a minimum around early afternoon (Figure 12). The variation of the relative humidity is the result of the fact that the saturation vapour pressure is determined by the air temperature. As the temperature changes during the day, the relative humidity also changes substantially.
It is not possible to directly measure the actual vapour pressure. The vapour pressure is commonly derived from relative humidity or dewpoint temperature.
Relative humidity is measured directly with hygrometers. The measurement is based on the nature of some material such as hair, which changes its length in response to changes in air humidity, or using a capacitance plate, where the electric capacitance changes with RH. Vapour pressure can be measured indirectly with psychrometers which measure the temperature difference between two thermometers, the socalled dry and wet bulb thermometers. The dry bulb thermometer measures the temperature of the air. The bulb of the wet bulb thermometer is covered with a constantly saturated wick. Evaporation of water from the wick, requiring energy, lowers the temperature of the thermometer. The drier the air, the larger the evaporative cooling and the larger the temperature drop. The difference between the dry and wet bulb temperatures is called the wet bulb depression and is a measure of the air humidity.
The dewpoint temperature is measured with dewpoint meters. The underlying principle of some types of apparatus is the cooling of the ambient air until dew formation occurs. The corresponding temperature is the dewpoint temperature.
Relative humidity and dewpoint temperature data are notoriously plagued by measurement errors. Measurement error is common for both older hygrothermograph types of instruments and for the more modem electronic instruments. These instruments are described in Annex 5. Great care should be made to assess the accuracy and integrity of RH and dewpoint data. The user is encouraged to always compare computed dewpoint temperatures to daily minimum air temperatures, as described at the end of this chapter and in Annexes 5 and 6. Frequently, it is better to utilize a dewpoint temperature that is predicted from daily minimum air temperature, rather than to use unreliable relative humidity measurements. The user is encouraged to utilize good judgement in this area.
Mean saturation vapour pressure (e_{s})
As saturation vapour pressure is related to air temperature, it can be calculated from the air temperature. The relationship is expressed by:
_{} (11)
where
e°(T) saturation vapour pressure at the air temperature T [kPa],
T air temperature [°C],
exp[..] 2.7183 (base of natural logarithm) raised to the power [..].
Values of saturation vapour pressure as a function of air temperature are given in Annex 2 (Table 2.3). Due to the nonlinearity of the above equation, the mean saturation vapour pressure for a day, week, decade or month should be computed as the mean between the saturation vapour pressure at the mean daily maximum and minimum air temperatures for that period:
_{} (12)
Using mean air temperature instead of daily minimum and maximum temperatures results in lower estimates for the mean saturation vapour pressure. The corresponding vapour pressure deficit (a parameter expressing the evaporating power of the atmosphere) will also be smaller and the result will be some underestimation of the reference crop evapotranspiration. Therefore, the mean saturation vapour pressure should be calculated as the mean between the saturation vapour pressure at both the daily maximum and minimum air temperature.
EXAMPLE 3. Determination of mean saturation vapour pressure
The daily maximum and minimum air temperature are respectively 24.5 and 15°C.  
From Eq. 11 
e°(T_{max}) = 0.6108 exp[17.27(24.5)/(24.5 + 237.3)] 
3.075 
kPa 
From Eq. 11 
e°(T_{min}) = 0.6108 exp[17.27(15)/(15 + 237.3)] 
1.705 
kPa 
From Eq. 12 
e_{s} = (3.075 + 1.705)/2 
2.39 
kPa 

Note that for temperature 19.75°C (which is T_{mean}). e°(T) = 
2.30 
kPa 
The mean saturation vapour pressure is 2.39 kPa. 
Slope of saturation vapour pressure curve (D )
For the calculation of evapotranspiration, the slope of the relationship between saturation vapour pressure and temperature, D, is required. The slope of the curve (Figure 11) at a given temperature is given by.
_{} (13)
where
D slope of saturation vapour pressure curve at air temperature T [kPa °C^{1}],
T air temperature [°C],
exp[..] 2.7183 (base of natural logarithm) raised to the power [..].
Values of slope D for different air temperatures are given in Annex 2 (Table 2.4). In the FAO PenmanMonteith equation, where D occurs in the numerator and denominator, the slope of the vapour pressure curve is calculated using mean air temperature (Equation 9).
Actual vapour pressure (e_{a}) derived from dewpoint temperature
As the dewpoint temperature is the temperature to which the air needs to be cooled to make the air saturated, the actual vapour pressure (e_{a}) is the saturation vapour pressure at the dewpoint temperature (T_{dew}) [°C], or:
_{} (14)
Actual vapour pressure (e_{a}) derived from psychrometric data
The actual vapour pressure can be determined from the difference between the dry and wet bulb temperatures, the socalled wet bulb depression. The relationship is expressed by the following equation:
e_{a} = e° (T_{wet})  g _{psy} (T_{dry}  T_{wet}) (15)
where
e_{a} actual vapour pressure [kPa],
e°(T_{wet}) saturation vapour pressure at wet bulb temperature [kPa],
g _{psy} psychrometric constant of the instrument [kPa °C^{1}],
T_{dry}T_{wet} wet bulb depression, with T_{dry} the dry bulb and T_{wet} the wet bulb temperature [°C].
The psychrometric constant of the instrument is given by:
g _{psy} = a_{psy} P (16)
where a_{psy} is a coefficient depending on the type of ventilation of the wet bulb [°C^{1}], and P is the atmospheric pressure [kPa]. The coefficient a_{psy} depends mainly on the design of the psychrometer and rate of ventilation around the wet bulb. The following values are used:
a_{psy} = 
0.000662 
for ventilated (Asmann type) psychrometers, with an air movement of some 5 m/s, 

0.000800 
for natural ventilated psychrometers (about 1 m/s), 

0.001200 
for nonventilated psychrometers installed indoors. 
EXAMPLE 4. Determination of actual vapour pressure from psychrometric readings
Determine the vapour pressure from the readings of an aspirated psychrometer in a location at an elevation of 1200 m. The temperatures measured by the dry and wet bulb thermometers are 25.6 and 19.5°C respectively.  
From Eq. 7 (Table 2.1), at: 
z= 
1200 
m 
Then: 
P= 
87.9 
kPa 
From Eq. 11 (Table 2.3), for 
T_{wet} = 
19.5 
°C 
Then: 
e°(T_{wet}) = 
2.267 
kPa 
Ventilated psychrometer 
a_{psy} = 
0.000662 
°C^{1} 
From Eq. 15: 
e_{a} = 2.267  0.000662 (87.9) (25.6  19.5) = 
1.91 
kPa 
The actual vapour pressure is 1.91 kPa. 
Actual vapour pressure (e_{a}) derived from relative humidity data
The actual vapour pressure can also be calculated from the relative humidity. Depending on the availability of the humidity data, different equations should be used.
· For RH_{max} and RH_{min}:
_{} (17)
where
e_{a} actual vapour pressure [kPa],
e°(T_{min}) saturation vapour pressure at daily minimum temperature [kPa],
e°(T_{max}) saturation vapour pressure at daily maximum temperature [kPa],
RH_{max} maximum relative humidity [%],
RH_{min} minimum relative humidity [%].
For periods of a week, ten days or a month, RH_{max} and RH_{min} are obtained by dividing the sum of the daily values by the number of days in that period.
· For RH_{max}:
When using equipment where errors in estimating RH_{min} can be large, or when RH data integrity are in doubt, then one should use only RH_{max}:
_{} (18)
· For RH_{mean}:
In the absence of RH_{max} and RH_{min}, another equation can be used to estimate e_{a}:
_{} (19)
where RH_{mean} is the mean relative humidity, defined as the average between RH_{max} and RH_{min}. However, Equation 19 is less desirable than are Equations 17 or 18.
EXAMPLE 5. Determination of actual vapour pressure from relative humidity
Given the following daily minimum and maximum air temperature and the corresponding relative humidity data: T_{min} = 18°C and RH_{max} = 82% Determine the actual vapour pressure.  
From Eq. 11 (Table 2.3), at: 
T_{min} = 
18 
°C 
Then: 
e°(T_{min}) = 
2.064 
kPa 
From Eq. 11 (Table 2.3), at: 
T_{max} = 
25 
°C 
Then: 
e°(T_{max}) = 
3.168 
kPa 
From Eq. 17: 
e_{a} = [2.064 (82/100) + 3.168 (54/100)] = 
1.70 
kPa 
Note that when using Eq. 19: 
e_{a }= 
1.78 
kPa 
Vapour pressure deficit (e_{s}  e_{a})
The vapour pressure deficit is the difference between the saturation (e_{s}) and actual vapour pressure (e_{a}) for a given time period. For time periods such as a week, ten days or a month e_{s} is computed from Equation 12 using the T_{max} and T_{min} averaged over the time period and similarly the e_{a} is computed with one of the equations 4 to 19, using average measurements over the period. As stated above, using mean air temperature and not T_{max} and T_{min} in Equation 12 results in a lower estimate of e_{s}, thus in a lower vapour pressure deficit and hence an underestimation of the ET_{o} (see Box 7). When desired, e_{s} and e_{a} for long time periods cal also be calculated as averages of values computed for each day of the period.
EXAMPLE 6. Determination of vapour pressure deficit
Determine the vapour pressure deficit with the data of the previous example (Example 5).  
From Example 5: 
e°(T_{min}) = 
2.064 
kPa 

e°(T_{max}) = 
3.168 
kPa 

e_{a} = 
1.70 
kPa 

e_{s}  e_{a} = (2.064 + 3.168)/21.70 = 
0.91 
kPa 
The vapour pressure deficit is 0.91 kPa. 
BOX 7. Calculation sheet for vapour pressure deficit (e_{s}  e_{a})  
Saturation vapour pressure: e_{s} (Eq. 11 or Table 2.3)  
T_{max} 

°C 
_{} 

kPa  
T_{min} 

°C 
_{} 

kPa  
saturation vapour pressure e_{s} = [e°(T_{max}) + e°(T_{min})]/2 Eq. 12 

kPa  
Actual vapour pressure: e_{a}  
1. e_{a} derived from dewpoint temperature (Eq. 14 or Table 2.3)  
T_{dew} 

°C 
_{} 

kPa  
OR 2. e_{a} derived from maximum and minimum relative humidity  
RH_{max} 

% 
_{} 

kPa  
RH_{min} 

% 
_{} 

kPa  
e_{a} = [e°(T_{min}) RH_{max}/100 + e°(T_{max}) RH_{min}/100]/2 Eq. 17 

kPa  
OR 3. e_{a} derived from maximum relative humidity (errors in RH_{min})  
RH_{max} 

% 
e_{a} = e°(T_{min}) RH_{max}/100 Eq. 18 

kPa  
OR 4. e_{a} derived from mean relative humidity (less recommended)  
RH_{mean} 

% 
e_{a} = e_{s} (RH_{mean})/100 Eq. 19 

kPa  
Vapour pressure deficit: (e_{s}  e_{a}) 

kPa 
Extraterrestrial radiation (R_{a})
The radiation striking a surface perpendicular to the sun's rays at the top of the earth's atmosphere, called the solar constant, is about 0.082 MJ m^{2} min^{1}. The local intensity of radiation is, however, determined by the angle between the direction of the sun's rays and the normal to the surface of the atmosphere. This angle will change during the day and will be different at different latitudes and in different seasons. The solar radiation received at the top of the earth's atmosphere on a horizontal surface is called the extraterrestrial (solar) radiation, R_{a}.
If the sun is directly overhead, the angle of incidence is zero and the extraterrestrial radiation is 0.0820 MJ m^{2} min^{1}. As seasons change, the position of the sun, the length of the day and, hence, R_{a} change as well. Extraterrestrial radiation is thus a function of latitude, date and time of day. Daily values of R_{a} throughout the year for different latitudes are plotted in Figure 13.
FIGURE 13. Annual variation in extraterrestrial radiation (R_{a}) at the equator, 20 and 40° north and south
Solar or shortwave radiation (R_{s})
As the radiation penetrates the atmosphere, some of the radiation is scattered, reflected or absorbed by the atmospheric gases, clouds and dust. The amount of radiation reaching a horizontal plane is known as the solar radiation, R_{s}. Because the sun emits energy by means of electromagnetic waves characterized by short wavelengths, solar radiation is also referred to as shortwave radiation.
FIGURE 14. Annual variation of the daylight hours (N) at the equator, 20 and 40° north and south
For a cloudless day, R_{s} is roughly 75% of extraterrestrial radiation. On a cloudy day, the radiation is scattered in the atmosphere, but even with extremely dense cloud cover, about 25% of the extraterrestrial radiation may still reach the earth's surface mainly as diffuse sky radiation. Solar radiation is also known as global radiation, meaning that it is the sum of direct shortwave radiation from the sun and diffuse sky radiation from all upward angles.
Relative shortwave radiation (R_{s}/R_{so})
The relative shortwave radiation is the ratio of the solar radiation (R_{s}) to the clearsky solar radiation (R_{so}). R_{s} is the solar radiation that actually reaches the earth's surface in a given period, while R_{so} is the solar radiation that would reach the same surface during the same period but under cloudless conditions.
The relative shortwave radiation is a way to express the cloudiness of the atmosphere; the cloudier the sky the smaller the ratio. The ratio varies between about 0.33 (dense cloud cover) and 1 (clear sky). In the absence of a direct measurement of R_{n}, the relative shortwave radiation is used in the computation of the net longwave radiation.
Relative sunshine duration (n/N)
The relative sunshine duration is another ratio that expresses the cloudiness of the atmosphere. It is the ratio of the actual duration of sunshine, n, to the maximum possible duration of sunshine or daylight hours N. In the absence of any clouds, the actual duration of sunshine is equal to the daylight hours (n = N) and the ratio is one, while on cloudy days n and consequently the ratio may be zero. In the absence of a direct measurement of R_{s}, the relative sunshine duration, n/N, is often used to derive solar radiation from extraterrestrial radiation.
As with extraterrestrial radiation, the day length N depends on the position of the sun and is hence a function of latitude and date. Daily values of N throughout the year for different latitudes are plotted in Figure 14.
Albedo (a) and net solar radiation (R_{ns})
A considerable amount of solar radiation reaching the earth's surface is reflected. The fraction, a, of the solar radiation reflected by the surface is known as the albedo. The albedo is highly variable for different surfaces and for the angle of incidence or slope of the ground surface. It may be as large as 0.95 for freshly fallen snow and as small as 0.05 for a wet bare soil. A green vegetation cover has an albedo of about 0.200.25. For the green grass reference crop, a is assumed to have a value of 0.23.
The net solar radiation, R_{ns}, is the fraction of the solar radiation R_{s} that is not reflected from the surface. Its value is (1a)R_{s}.
Net longwave radiation (R_{nl})
The solar radiation absorbed by the earth is converted to heat energy. By several processes, including emission of radiation, the earth loses this energy. The earth, which is at a much lower temperature than the sun, emits radiative energy with wavelengths longer than those from the sun. Therefore, the terrestrial radiation is referred to as longwave radiation. The emitted longwave radiation (R_{l, up}) is absorbed by the atmosphere or is lost into space. The longwave radiation received by the atmosphere (R_{l, down}) increases its temperature and, as a consequence, the atmosphere radiates energy of its own, as illustrated in Figure 15. Part of the radiation finds it way back to the earth's surface. Consequently, the earth's surface both emits and receives longwave radiation. The difference between outgoing and incoming longwave radiation is called the net longwave radiation, R_{nl}. As the outgoing longwave radiation is almost always greater than me incoming longwave radiation, R_{nl} represents an energy loss.
Net radiation (R_{n})
The net radiation, R_{n}, is the difference between incoming and outgoing radiation of both short and long wavelengths. It is the balance between the energy absorbed, reflected and emitted by the earth's surface or the difference between the incoming net shortwave (R_{ns}) and the net outgoing longwave (R_{nl}) radiation (Figure 15). R_{n} is normally positive during the daytime and negative during the nighttime. The total daily value for R_{n} is almost always positive over a period of 24 hours, except in extreme conditions at high latitudes.
Soil heat flux (G)
In making estimates of evapotranspiration, all terms of the energy balance (Equation 1) should be considered. The soil heat flux, G, is the energy that is utilized in heating the soil. G is positive when the soil is warming and negative when the soil is cooling. Although the soil heat flux is small compared to R_{n} and may often be ignored, the amount of energy gained or lost by the soil in this process should theoretically be subtracted or added to R_{n} when estimating evapotranspiration.
The standard unit used in this handbook to express energy received on a unit surface per unit time is megajoules per square metre per day (MJ m^{2} day^{1}). In meteorological bulletins other units might be used or radiation might even be expressed in units no longer accepted as standard S. I. units, such as calories cm^{2} day^{1}.
FIGURE 15. Various components of radiation
In the FAO PenmanMonteith equation (Equation 6), radiation expressed in MJ m^{2} day^{1} is converted (Box 8) to equivalent evaporation in mm day^{1} by using a conversion factor equal to the inverse of the latent heat heat of vaporization (1/l = 0.408):
equivalent evaporation [mm day^{1}] = 0.408 x Radiation [MJ m^{2} day^{1}] (20)
BOX 8. Conversion from energy values to equivalent evaporation The conversion from energy values to depths of water or vice versa is given by: _{} where l latent heat of vaporization [MJ kg^{1}], By using a single value of 2.45 MJ kg^{1} for l (see section on atmospheric parameters and Annex 3) and multiplying the above equation by 1000 to obtain mm: _{} 
Common units used to express energy received on a unit surface per unit time, and conversion factors are summarized in Table 3.
TABLE 3. Conversion factors for radiation

multiplier to obtain energy received on a unit surface per unit time 
equivalent evaporation 

MJ m^{2} day^{1} 
J cm^{2} day^{1} 
cal cm^{2} day^{1} 
W m^{2} 
mm day^{1} 

1 MJ m^{2} day^{1} 
1 
100 
23.9 
11.6 
0.408 
1 cal cm^{2} day^{1} 
4.1868 10^{2} 
4.1868 
1 
0.485 
0.0171 
1 W m^{2} 
0.0864 
8,64 
2.06 
1 
0.035 
1 mm day^{1} 
2.45 
245 
58.5 
28.4 
1 
Solar radiation can be measured with pyranometers, radiometers or solarimeters. The instruments contain a sensor installed on a horizontal surface that measures the intensity of the total solar radiation, i.e., both direct and diffuse radiation from cloudy conditions. The sensor is often protected and kept in a dry atmosphere by a glass dome that should be regularly wiped clean.
Net longwave and net shortwave radiation can be measured by recording the difference in output between sensors facing upward and downward. In a net radiometer, the glass domes are replaced by polyethylene domes that have a transmission range for both shortwave and longwave radiation.
Where pyranometers are not available, solar radiation is usually estimated from the duration of bright sunshine. The actual duration of sunshine, n, is measured with a CampbellStokes sunshine recorder. This instrument records periods of bright sunshine by using a glass globe that acts as a lens. The sun rays are concentrated at a focal point that burns a hole in a specially treated card mounted concentrically with the sphere. The movement of the sun changes the focal point throughout the day and a trace is drawn on the card. If the sun is obscured, the trace is interrupted. The hours of bright sunshine are indicated by the lengths of the line segments.
The quantity of heat conducted into the soil, G, can be measured with systems of soil heat flux plates and thermocouples or thermisters.
Extraterrestrial radiation for daily periods (R_{a})
The extraterrestrial radiation, R_{a}, for each day of the year and for different latitudes can be estimated from the solar constant, the solar declination and the time of the year by:
_{} (21)
where
R_{a} extraterrestrial radiation [MJ m^{2} day^{1}],
G_{sc} solar constant = 0.0820 MJ m^{2} min^{1},
d_{r} inverse relative distance EarthSun (Equation 23),
w _{s} sunset hour angle (Equation 25 or 26) [rad],
j latitude [rad] (Equation 22),
d solar decimation (Equation 24) [rad].
R_{a} is expressed in the above equation in MJ m^{2} day^{1}. The corresponding equivalent evaporation in mm day^{1} is obtained by multiplying R_{a} by 0.408 (Equation 20). The latitude, j, expressed in radians is positive for the northern hemisphere and negative for the southern hemisphere (Example 7). The conversion from decimal degrees to radians is given by:
_{} (22)
EXAMPLE 7. Conversion of latitude in degrees and minutes to radians
Express the latitudes of Bangkok (Thailand) at 13°44'N and Rio de Janeiro (Brazil) at 22°54'S in radians.  
Latitude 
Bangkok (northern hemisphere) 
Rio de Janeiro (southern hemisphere) 
degrees & minutes 
13°44'N 
22°54'S 
decimal degrees 
13 + 44/60=13.73 
(22) + (54/60) =  22.90 
radians 
(p /180) 13.73 = + 0.240 
(p /180) (22.90) =  0.400 
The latitudes of Bangkok and Rio de Janeiro are respectively +0.240 and 0.400 radians. 
The inverse relative distance EarthSun, d_{r}, and the solar declination, d, are given by:
_{} (23)_{} (24)
where J is the number of the day in the year between 1 (1 January) and 365 or 366 (31 December). Values for J for all days of the year and an equation for estimating J are given in Annex 2 (Table 2.5).
The sunset hour angle, w _{s}, is given by:
w _{s} = arccos [tan (j) tan (d)] (25)
As the arccos function is not available in all computer languages, the sunset hour angle can also be computed using the arctan function:
_{} (26)
where
X = 1  [tan(j)]^{2} [tan(d)]^{2} (27)
and X = 0.00001 if X £ 0
Values for R_{a} for different latitudes are given in Annex 2 (Table 2.6). These values represent R_{a} on the 15^{th} day of each month. These values deviate from values that are averaged over each day of the month by less than 1% for all latitudes during nonfrozen periods and are included for simplicity of calculation. These values deviate slightly from the values in the Smithsonian Tables. For the winter months in latitudes greater than 55° (N or S), the equations for R_{a} have limited validity. Reference should be made to the Smithsonian Tables to assess possible deviations.
EXAMPLE 8. Determination of extraterrestrial radiation
Determine the extraterrestrial radiation (R_{a}) for 3 September at 20°S.  
From Eq. 22 
20°S or j = (p /180) (20) = (the value is negative for the southern hemisphere) 
0.35 
rad 
From Table 2.5: 
The number of day in the year, J = 
246 
days 
From Eq. 23 
d_{r} = 1 + 0.033 cos(2p (246)/365) = 
0.985 
rad 
From Eq. 24 
d = 0.409 sin(2p (246)/365  1.39) = 
0.120 
rad 
From Eq. 25: 
w _{s} = arccos[tan(0.35)tan(0.120)] = 
1.527 
rad 
Then: 
sin(j)sin(d) = 
0.041 
 
and: 
cos(j)cos(d) = 
0.933 
 
From Eq. 21 
R_{a} = 24(60)/p (0.0820)(0.985)[1.527(0.041) + 0.933 sin(1.527)] = 
32.2 
MJ m^{2} d^{1} 
From Eq. 20 
expressed as equivalent evaporation = 0.408 (32.2) = 
13.1 
mm/day 
The extraterrestrial radiation is 32.2 MJ m^{2} day^{1}. 
Extraterrestrial radiation for hourly or shorter periods (R_{a})
For hourly or shorter periods the solar time angle at the beginning and end of the period should be considered when calculating R_{a}:
_{} (28)
where
R_{a} extraterrestrial radiation in the hour (or shorter) period [MJ m^{2} hour^{1}],
G_{sc} solar constant = 0.0820 MJ m^{2} min^{1},
d_{r} inverse relative distance EarthSun (Equation 23),
d solar declination [rad] (Equation 24),
j latitude [rad] (Equation 22),
w _{1} solar time angle at beginning of period [rad] (Equation 29),
w _{2} solar time angle at end of period [rad] (Equation 30).
The solar time angles at the beginning and end of the period are given by:
_{} (29)_{} (30)
where
w solar time angle at midpoint of hourly or shorter period [rad],
t_{1} length of the calculation period [hour]: i.e., 1 for hourly period or 0.5 for a 30minute period.
The solar time angle at midpoint of the period is:
_{} (31)
where
t standard clock time at the midpoint of the period [hour]. For example for a period between 14.00 and 15.00 hours, t = 14.5,L_{z} longitude of the centre of the local time zone [degrees west of Greenwich]. For example, L_{z} = 75, 90, 105 and 120° for the Eastern, Central, Rocky Mountain and Pacific time zones (United States) and L_{z} = 0° for Greenwich, 330° for Cairo (Egypt), and 255° for Bangkok (Thailand),
L_{m} longitude of the measurement site [degrees west of Greenwich],
S_{c} seasonal correction for solar time [hour].
Of course, w < w _{s} or w > w _{s} from Equation 31 indicates that the sun is below the horizon so that, by definition, R_{a} is zero.
The seasonal correction for solar time is:
S_{c} = 0.1645 sin(2 b)  0.1255 cos(b)  0.025 sin(b) (32)_{} (33)
where J is the number of the day in the year.
Daylight hours (N)
The daylight hours, N, are given by:
_{} (34)
where w _{s} is the sunset hour angle in radians given by Equation 25 or 26. Mean values for N (15^{th} day of each month) for different latitudes are given in Annex 2, Table 2.7.
EXAMPLE 9. Determination of daylight hours
Determine the daylight hours (N) for 3 September at 20°S.  
From Example 8: 
w _{s} = arccos[tan(0.35)tan(0.120)] = 
1.527 
rad 
From Eq. 34: 
N= 24/p (1.527) = 
11.7 
hour 
The number of daylight hours is 11.7 hours. 
BOX 9. Calculation sheet for extraterrestrial radiation (R_{a}) and daylight hours (N)  
Latitude  
Degrees and minutes are + positive for northern hemisphere  
Degrees 

° 
> 

°  
Minutes 

¢ 
/60> 

°  
Decimal degrees = Sum(degrees + minutes/60) 

°  
j = p /180*[decimal degrees] Eq. 22 

rad  
Day of the year  
Day 


 
Month 


J Table 2.5 (Annex 2) 

 
d_{r} = 1 + 0.033 cos(2p J/365) Eq. 23 

 
d = 0.409 sin(2p J/3651.39) Eq. 24 

rad  
sin(j)sin(d) 

 
cos(j)cos(d) 

 
w _{s} = arccos[tan(j)tan(d)] Eq. 25 

rad  
(24(60)/p G_{sc} 
37.59 
MJ m^{2} day^{1}  
Extraterrestrial radiation: R_{a}  
_{} Eq. 21 

MJ m^{2} day^{1}  
Daylight hours: N  
_{} Eq. 34 

hour/day 
Solar radiation (R_{s})
If the solar radiation, R_{s}, is not measured, it can be calculated with the Angstrom formula which relates solar radiation to extraterrestrial radiation and relative sunshine duration:
_{} (35)
where
R_{s} solar or shortwave radiation [MJ m^{2} day^{1}],n actual duration of sunshine [hour],
N maximum possible duration of sunshine or daylight hours [hour],
n/N relative sunshine duration [],
R_{a} extraterrestrial radiation [MJ m^{2} day^{1}],
a_{s} regression constant, expressing the fraction of extraterrestrial radiation reaching the earth on overcast days (n = 0),
a_{s}+b_{s} fraction of extraterrestrial radiation reaching the earth on clear days (n = N).
R_{s} is expressed in the above equation in MJ m^{2} day^{1}. The corresponding equivalent evaporation in mm day^{1} is obtained by multiplying R_{s} by 0.408 (Equation 20). Depending on atmospheric conditions (humidity, dust) and solar declination (latitude and month), the Angstrom values a_{s} and b_{s} will vary. Where no actual solar radiation data are available and no calibration has been carried out for improved a_{s} and b_{s} parameters, the values a_{s} = 0.25 and b_{s} = 0.50 are recommended.
The extraterrestrial radiation, R_{a}, and the daylight hours or maximum possible duration of sunshine, N, are given by Equations 21 and 34. Values for R_{a} and N for different latitudes are also listed in Annex 2 (Tables 2.6 and 2.7). The actual duration of sunshine, n, is recorded with a Campbell Stokes sunshine recorder.
EXAMPLE 10. Determination of solar radiation from measured duration of sunshine
In Rio de Janeiro (Brazil) at a latitude of 22°54'S, 220 hours of sunshine were recorded in May. Determine the solar radiation. 

From Eq. 22: 
latitude = 22°54'S = 22.90°S or p /180(22.90) = 
0.40 
rad 
From Table 2.5: 
for 15 May, the day in the year (J) = 
135 
 
From Eq. 21 or Table 2.6: 
R_{a} = 
25.1 
MJ m^{2} day^{1} 
From Eq. 34 or Table 2.7 
N = 
10.9 
hours day^{1} 
n = 220 hours/31 days = 
7.1 
hours day^{1} 

From Eq. 35: 
R_{s} = [0.25 + 0.50 (7.1/10.9)] R_{a} = 0.58 R_{a} = 0.58 (25.1) = 
14.5 
MJ m^{2} day^{1} 
From Eq. 20: 
expressed as equivalent evaporation = 0.408(14.5) = 
5.9 
mm/day 
The estimated solar radiation is 14.5 MJ m^{2} day^{1}. 
Clearsky solar radiation (R_{so})
The calculation of the clearsky radiation, R_{so}, when n = N, is required for computing net longwave radiation.
· For near sea level or when calibrated values for a_{s} and b_{s} are available:
R_{so} = (a_{s}+b_{s})R_{a} (36)
where
R_{so} clearsky solar radiation [MJ m^{2} day^{1}],
a_{s}+b_{s} fraction of extraterrestrial radiation reaching the earth on clearsky days (n = N).
· When calibrated values for a_{s} and b_{s} are not available:
R_{so} = (0.75 + 2 l0^{5}z)R_{a} (37)
where
z station elevation above sea level [m].
Other more complex estimates for R_{so}, which include turbidity and water vapour effects, are discussed in Annex 3 (Equations 3.14 to 20).
Net solar or net shortwave radiation (R_{ns})
The net shortwave radiation resulting from the balance between incoming and reflected solar radiation is given by:
R_{ns} = (1a)R_{s} (38)
where
R_{ns} net solar or shortwave radiation [MJ m^{2} day^{1}],a albedo or canopy reflection coefficient, which is 0.23 for the hypothetical grass reference crop [dimensionless],
R_{s} the incoming solar radiation [MJ m^{2} day^{1}].
R_{ns} is expressed in the above equation in MJ m^{2} day^{1}.
Net longwave radiation (R_{nl})
The rate of longwave energy emission is proportional to the absolute temperature of the surface raised to the fourth power. This relation is expressed quantitatively by the StefanBoltzmann law. The net energy flux leaving the earth's surface is, however, less than that emitted and given by the StefanBoltzmann 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 StefanBoltzmann 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:
_{} (39)
where
R_{nl} net outgoing longwave radiation [MJ m^{2} day^{1}],
s StefanBoltzmann constant [4.903 10^{9} MJ K^{4} m^{2} day^{1}],
T_{max, K} maximum absolute temperature during the 24hour period [K = °C + 273.16],
T_{min, K} minimum absolute temperature during the 24hour period [K = °C + 273.16],
e_{a} actual vapour pressure [kPa],
R_{s}/R_{so} relative shortwave radiation (limited to £ 1.0),
R_{s} measured or calculated. (Equation 35) solar radiation [MJ m^{2} day^{1}],
R_{so} calculated (Equation 36 or 37) clearsky radiation [MJ m^{2} day^{1}].
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 StefanBoltzmann equation for 24hour time steps. The term (0.340.14Ö e_{a}) expresses the correction for air humidity, and will be smaller if the humidity increases. The effect of cloudiness is expressed by (1.35 R_{s}/R_{so}  0.35). The term becomes smaller if the cloudiness increases and hence R_{s} decreases. The smaller the correction terms, the smaller the net outgoing flux of longwave radiation. Note that the R_{s}/R_{so} term in Equation 39 must be limited so that R_{s}/R_{so} £ 1.0.
Where measurements of incoming and outgoing short and longwave radiation during bright sunny and overcast hours are available, calibration of the coefficients in Equation 39 can be carried out.
Annex 2 (Table 2.8) lists values for _{} for different air temperatures.
EXAMPLE 11. Determination of net longwave radiation
In Rio de Janeiro (Brazil) at a latitude of 22°54'S (= 22.70°), 220 hours of bright sunshine, a mean monthly daily maximum and minimum air temperature of 25.1 and 19.1°C and a vapour pressure of 2.1 kPa were recorded in May. Determine the net longwave radiation.  
From Example 10: 
R_{s} = 
14.5 
MJ m^{2} day^{1} 
From Eq. 36: 
R_{so} = 0.75 R_{a} = 0.75. 25.1 = 
18.8 
MJ m^{2} day^{1} 
From Table 2.8 or for: 
s = 
4.903 10^{9} 
MJ K^{4} m^{2} day^{1} 
Then: 
T_{max} = 25.1°C = 
298.3 
K 
and: 
_{} 
38.8 
MJ m^{2} day^{1} 
and: 
T_{min} = 19.1°C = 
292.3 
K 
and: 
_{} 
35.8 
MJ m^{2} day^{1} 
and: 
e_{a} = 
2.1 
kPa 
and: 
0.340.14 Ö e_{a} = 
0.14 
 
and: 
R_{s}/R_{so} = (14.5)/(18.8) 
0.77 
 
 
1.35(0.77)  0.35 = 
0.69 
 
From Eq. 39: 
R_{nl} = [(38.7 + 35.7)/2] (0.14) (0.69) = 
3.5 
MJ m^{2} day^{1} 
From Eq. 20: 
expressed as equivalent evaporation = 0.408 (3.5) = 
1.4 
mm/day 
The net longwave radiation is 3.5 MJ m^{2} day^{1}. 
Net radiation (R_{n})
The net radiation (R_{n}) is the difference between the incoming net shortwave radiation (R_{ns}) and the outgoing net longwave radiation (R_{nl}):
R_{n} = R_{ns}  R_{nl} (40)
EXAMPLE 12. Determination of net radiation
Determine the net radiation in Rio de Janeiro in May with the data from previous examples.  
From Example 10: 
R_{s} = 
14.5 
MJ m^{2} day^{1} 
From Eq. 39: 
R_{ns} = (1  0.23) R_{s} = 
11.1 
MJ m^{2} day^{1} 
From Example 11: 
R_{nl} = 
3.5 
MJ m^{2} day^{1} 
From Eq. 40: 
R_{n} = 11.13.5 = 
7.6 
MJ m^{2} day^{1} 
From Eq. 20: 
expressed as equivalent evaporation = 0.408 (7.7) = 
3.1 
mm/day 
The net radiation is 7.6 MJ m^{2} day^{1}. 
BOX 10. Calculation sheet for net radiation (R_{n})  
Latitude 

° 



Day 


R_{a} (Box 9 or Table 2.6) 

MJ m^{2} d^{1} 
Month 


N (Box 9 or Table 2.7) 

hours 
n 

hours 
(in absence of R_{s}) n/N 


Net solar radiation: R_{ns}  
If n is measured instead of R_{s}: 

 
R_{s} = (0.25+0.50 n/N) R_{a} Eq. 35 

MJ m^{2} d^{1}  
R_{so} = [0.75 + 2 (Altitude)/100000] R_{a} Eq. 37 

MJ m^{2} d^{1}  
R_{s}/R_{so} (£ 1.0) 

 
R_{ns} = 0.77 R_{s} Eq. 38 

MJ m^{2} d^{1}  
Net longwave radiation: R_{nl}  
T_{max} 

°C 
T_{max, K} = T_{max} + 273.16 

K 
T_{min} 

°C 
T_{min, K} = T_{min} + 273.16 

K 



_{} (Table 2.8) 

MJ m^{2} d^{1} 



_{} (Table 2.8) 

MJ m^{2} d^{1} 
_{} 

MJ m^{2} d^{1}  
e_{a} 

kPa 
(0.340.14Ö e_{a}) 


R_{s}/R_{so} 


(1.35 R_{s}/R_{so}  0.35) 


_{} Eq. 39 

MJ m^{2} d^{1}  
Net radiation: R_{n}  
R_{n} = R_{ns}  R_{nl} Eq. 40 

MJ m^{2} d^{1} 
Soil heat flux (G)
Complex models are available to describe soil heat flux. Because soil heat flux is small compared to R_{n}, particularly when the surface is covered by vegetation and calculation time steps are 24 hours or longer, a simple calculation procedure is presented here for long time steps, based on the idea that the soil temperature follows air temperature:
_{} (41)
where
G soil heat flux [MJ m^{2} day^{1}],
c_{s} soil heat capacity [MJ m^{3} °C^{1}],
T_{i} air temperature at time i [°C],
T_{i1} air temperature at time i1 [°C],
D t length of time interval [day],
D z effective soil depth [m].
As the soil temperature lags air temperature, the average temperature for a period should be considered when assessing me daily soil heat flux, i.e., D t should exceed one day. The depth of penetration of the temperature wave is determined by the length of the time interval. The effective soil depth, D z, is only 0.100.20 m for a time interval of one or a few days but might be 2 m or more for monthly periods. The soil heat capacity is related to its mineral composition and water content.
· For day and tenday periods:
As the magnitude of the day or tenday soil heat flux beneath the grass reference surface is relatively small, it may be ignored and thus:
G_{day} » 0 (42)
· For monthly periods:
When assuming a constant soil heat capacity of 2.1 MJ m^{3} °C^{1} and an appropriate soil depth, Equation 41 can be used to derive G for monthly periods:
G_{month, i} = 0.07 (T_{month, i+1}  T_{month, i1}) (43)
or, if T_{month, i+1} is unknown:
G_{month, i} = 0.14 (T_{month, i}  T_{month, i1}) (44)
where
T_{month, i} mean air temperature of month i [°C],
T_{month, i1} mean air temperature of previous month [°C],
T_{month, i+1} mean air temperature of next month [°C].
· For hourly or shorter periods:
For hourly (or shorter) calculations, G beneath a dense cover of grass does not correlate well with air temperature. Hourly G can be approximated during daylight periods as:
G_{hr} = 0.1 R_{n} (45)
and during nighttime periods as:
G_{hr} = 0.5 R_{n} (46)
Where the soil is warming, the soil heat flux G is positive. The amount of energy required for this process is subtracted from R_{n} when estimating evapotranspiration.
EXAMPLE 13. Determination of soil heat flux for monthly periods
Determine the soil heat flux in April in Algiers (Algeria) when the soil is warming. The mean monthly temperatures of March, April and May are 14.1, 16.1, and 18.8°C respectively.  
From Eq. 43 
for the month of April: 
0.33 
MJ m^{2} day^{1} 
From Eq. 20 
expressed as equivalent evaporation = 0.408(0.33) = 
0.13 
mm/day 
The soil heat flux is 0.33 MJ m^{2} day^{1}. 
Wind is characterized by its direction and velocity. Wind direction refers to the direction from which the wind is blowing. For the computation of evapotranspiration, wind speed is the relevant variable. As wind speed at a given location varies with time, it is necessary to express it as an average over a given time interval. Wind speed is given in metres per second (m s^{1}) or kilometres per day (km day^{1}).
Wind speed is measured with anemometers. The anemometers commonly used in weather stations are composed of cups or propellers which are turned by the force of the wind. By counting the number of revolutions over a given time period, the average wind speed over the measuring period is computed.
Wind speeds measured at different heights above the soil surface are different. Surface friction tends to slow down wind passing over it. Wind speed is slowest at the surface and increases with height. For this reason anemometers are placed at a chosen standard height, i.e., 10 m in meteorology and 2 or 3 m in agrometeorology. For the calculation of evapotranspiration, wind speed measured at 2 m above the surface is required. To adjust wind speed data obtained from instruments placed at elevations other than the standard height of 2m, a logarithmic wind speed profile may be used for measurements above a short grassed surface:
_{} (47)
where
u_{2} wind speed at 2 m above ground surface [m s^{1}],
u_{z} measured wind speed at z m above ground surface [m s^{1}],
z height of measurement above ground surface [m].
The corresponding multipliers or conversion factors are given in Annex 2 (Table 2.9) and are plotted in Figure 16.
FIGURE 16. Conversion factor to convert wind speed measured at a certain height above ground level to wind speed at the standard height (2 m)
EXAMPLE 14. Adjusting wind speed data to standard height
Determine the wind speed at the standard height of 2 m, from a measured wind speed of 3.2 m/s at 10 m above the soil surface.  
For: 
u_{z} = 
3.2 
m/s 
And: 
z = 
10 
m 
Then: 
Conversion factor = 4.87/ln (67.8 (10)  5.42) = 
0.75 
 
From Eq. 47: 
u_{2} = 3.2 (0.75) = 
2.4 
m/s 
The wind speed at 2 m above the soil surface is 2.4 m/s. 
Meteorological data are recorded at various types of weather stations. Agrometeorological stations are sited in cropped areas where instruments are exposed to atmospheric conditions similar to those for the surrounding crops. In these stations, air temperature and humidity, wind speed and sunshine duration are typically measured at 2 m above an extensive surface of grass or short crop. Where needed and feasible, the cover of the station is irrigated. Guidelines for the establishment and maintenance of agrometeorological stations are given in the FAO Irrigation and Drainage Paper No. 27. This handbook also describes the different types of instruments, their installation and reliability.
Data collected at stations other than agrometeorological stations require a careful analysis of their validity before their use. For example, in aeronautic stations, data relevant for aviation are measured. As airports are often situated near urban conditions, temperatures may be higher than those found in rural agricultural areas. Wind speed is commonly measured at 10 m height above the ground surface.
The country's national meteorological service should be contacted for information on the climatic data collected at various types of weather stations in the country. National services commonly publish meteorological bulletins listing processed climatic data from the various stations.
The annexes list procedures for the statistical analysis, assessment, correction and completion of partial or missing weather data:
Annex 4: Statistical analysis of weather data sets;
Annex 5: Measuring and assessing integrity of weather data;
Annex 6: Correction of weather data observed at nonreference sites for computing ET_{o}.
Starting in 1984, FAO has published mean monthly agroclimatic data from 2300 stations in the FAO Plant Production and Protection Series. Several volumes exist:
No. 22: 
Volume 1: data for Africa, countries north of the equator (1984), 

Volume 2: data for Africa, countries south of the equator (1984); 
No. 24: 
Agroclimatic data for Latin America and the Caribbean (1985); 
No. 25: 
Volume 1: Agroclimatic data for Asia (AJ) (1987), 

Volume 2: Agroclimatic data for Asia (KZ) (1987). 
CLIMWAT for CROPWAT (FAO Irrigation and Drainage Paper No. 46) contains monthly data from 3 262 climatic stations contained on five separate diskettes. The stations are grouped by country and by continent. Monthly averages of maximum and minimum temperatures, mean relative humidity, wind speed, sunshine hours, radiation data as well as rainfall and ET_{o} calculated with the FAO PenmanMonteith method are listed on the diskettes for mean longterm conditions.
FAOCLIM provides a user friendly interface on compact disc to the agroclimatic database of the Agrometeorology Group in FAO. The data presented are an extension of the previously published FAO Plant Production and Protection Series and the number of stations has been increased from 2300 to about 19000, with an improved world wide coverage. However, values for all principal weather parameters are not available for all stations. Many contain air temperature and precipitation only.
These databases can be consulted in order to verify the consistency of the actual database or to estimate missing climatic parameters. However, they should only be used for preliminary studies as they contain mean monthly data only. FAOCLIM provides monthly time series for only a few stations. The information in these databases should never replace actual data.
Other electronic databases for portions of the globe have been published by the International Water Management Institute (IWMI). These databases include daily and monthly air temperature, precipitation and ET_{o} predicted using the Hargreaves ET_{o} equation that is based on differences between daily maximum and minimum air temperature.
Estimating missing humidity data
Estimating missing radiation data
Missing wind speed data
The assessment of the reference evapotranspiration ET_{o} with the PenmanMonteith method is developed in Chapter 4. The calculation requires mean daily, tenday or monthly maximum and minimum air temperature (T_{max} and T_{min}), actual vapour pressure (e_{a}), net radiation (R_{n}) and wind speed measured at 2 m (u_{2}). If some of the required weather data are missing or cannot be calculated, it is strongly recommended that the user estimate the missing climatic data with one of the following procedures and use the FAO PenmanMonteith method for the calculation of ET_{o}. The use of an alternative ET_{o} calculation procedure, requiring only limited meteorological parameters, is less recommended. Procedures to estimate missing humidity, radiation and wind speed data are given in this section.
Where humidity data are lacking or are of questionable quality, an estimate of actual vapour pressure, e_{a}, can be obtained by assuming that dewpoint temperature (T_{dew}) is near the daily minimum temperature (T_{min}). This statement implicitly assumes that at sunrise, when the air temperature is close to T_{min}, that the air is nearly saturated with water vapour and the relative humidity is nearly 100%. If T_{min} is used to represent T_{dew} then:
_{} (48)
The relationship T_{dew} » T_{min} holds for locations where the cover crop of the station is well watered. However, particularly for arid regions, the air might not be saturated when its temperature is at its minimum. Hence, T_{min} might be greater than T_{dew} and a further calibration may be required to estimate dewpoint temperatures. In these situations, "T_{min}" in the above equation may be better approximated by subtracting 23 °C from T_{min}. Appropriate correction procedures are given in Annex 6. In humid and subhumid climates, T_{min} and T_{dew} measured in early morning may be less than T_{dew} measured during the daytime because of condensation of dew during the night. After sunrise, evaporation of the dew will once again humidify the air and will increase the value measured for T_{dew} during the daytime. This phenomenon is demonstrated in Figure 5.4 of Annex 5. However, it is standard practice in 24hour calculations of ET_{o} to use T_{dew} measured or calculated during early morning.
The estimate for e_{a} from T_{min} should be checked. When the prediction by Equation 48 is validated for a region, it can be used for daily estimates of e_{a}.
Net radiation measuring devices, requiring professional control, have rarely been installed in agrometeorological stations. In the absence of a direct measurement, longwave and net radiation can be derived from more commonly observed weather parameters, i.e., solar radiation or sunshine hours, air temperature and vapour pressure. Where solar radiation is not measured, it can perhaps be estimated from measured hours of bright sunshine. However, where daily sunshine hours (n) are not available, solar radiation data cannot be computed with the calculation procedures previously presented. This section presents various methods to estimate solar radiation data with an alternative methodology.
Solar Radiation data from a nearby weather station
This method relies on the fact that for the same month and often for the same day, the variables affecting incoming solar radiation, R_{s}, and sunshine duration, n, are similar throughout a given region. This implies that: (i) the size of the region is small; (ii) the air masses governing rainfall and cloudiness are nearly identical within parts of the region; and (iii) the physiography of the region is almost homogenous. Differences in relief should be negligible as they strongly influence the movement of air masses. Under such conditions, radiation data observed at nearby stations can be used.
Caution should be used when applying this method to mountainous and coastal areas where differences in exposure and altitude could be important or where rainfall is variable due to convective conditions. Moreover, data from a station located nearby but situated on the other side of a mountain may not be transferable as conditions governing radiation are different. The user should observe climatic conditions in both locations and obtain information from local persons concerning general differences in cloud cover and type.
Where the northsouth distance to a weather station within the same homogeneous region exceeds 50 km so that the value for R_{a} changes, the R_{s} measurement should be adjusted using the ratio of the solar to extraterrestrial radiation, R_{s}/R_{a}:
_{} (49)
where
R_{s, reg} solar radiation at the regional location [MJ m^{2} day^{1}],
R_{a, reg} extraterrestrial radiation at the regional location [MJ m^{2} day^{1}].
Once the solar radiation has been derived from the radiation data of a nearby station, the net longwave radiation (Equation 39) and the net radiation (Equation 40) can be calculated.
The estimation method of Equation 49 is recommended for monthly calculations of ET_{o}. If using the method for daily estimates of ET_{o}, a more careful analysis of weather data in the importing and exporting meteorological stations has to be performed to verify whether both stations are in the same homogeneous climatic region and are close enough to experience similar conditions within the same day. The analysis should include the comparison of daily weather data from both stations, particularly the maximum and minimum air temperature and humidity. In fact, similar cloudiness and sunshine durations are related to similarities in temperature and humidity trends.
Generally, daily calculations of ET_{o} with estimated radiation data are justified when utilized as a sum or an average over a severalday period. This is the case for the computation of the mean evapotranspiration demand between successive irrigations or when planning irrigation schedules. Under these conditions, the relative error for one day often counterbalances the error for another day of the averaging period. Daily estimates should not be utilized as true daily estimates but only in averages over the period under consideration.
Solar Radiation data derived from air temperature differences
The difference between the maximum and minimum air temperature is related to the degree of cloud cover in a location. Clearsky conditions result in high temperatures during the day (T_{max},) because the atmosphere is transparent to the incoming solar radiation and in low temperatures during the night (T_{min}) because less outgoing longwave radiation is absorbed by the atmosphere. On the other hand, in overcast conditions, T_{max} is relatively smaller because a significant part of the incoming solar radiation never reaches the earth's surface and is absorbed and reflected by the clouds. Similarly, T_{min} will be relatively higher as the cloud cover acts as a blanket and decreases the net outgoing longwave radiation. Therefore, the difference between the maximum and minimum air temperature (T_{max}  T_{min}) can be used as an indicator of the fraction of extraterrestrial radiation that reaches the earth's surface. This principle has been utilized by Hargreaves and Samani to develop estimates of ET_{o} using only air temperature data.
The Hargreaves' radiation formula, adjusted and validated at several weather stations in a variety of climate conditions, becomes:
_{} (50)
where
R_{a} extraterrestrial radiation [MJ m^{2} d^{1}],
T_{max} maximum air temperature [°C],
T_{min} minimum air temperature [°C],
k_{Rs} adjustment coefficient (0.16.. 0.19) [°C^{0.5}].
The square root of the temperature difference is closely related to the existing daily solar radiation in a given location. The adjustment coefficient k_{Rs} is empirical and differs for 'interior' or 'coastal' regions:
· for 'interior' locations, where land mass dominates and air masses are not strongly influenced by a large water body, k_{Rs} @ 0.16;· for 'coastal' locations, situated on or adjacent to the coast of a large land mass and where air masses are influenced by a nearby water body, k_{Rs} @ 0.19.
The relationship between R_{s}/R_{a} and the temperature difference is plotted in Figure 17 for interior and coastal locations. The fraction of extraterrestrial radiation that reaches the earth's surface, R_{s}/R_{a}, ranges from about 0.25 on a day with dense cloud cover to about 0.75 on a cloudless day with clear sky. R_{s} predicted by Equation 50 should be limited to £ R_{so} from Equation 36 or 37.
FIGURE 17. Relationship between the fraction of extraterrestrial radiation that reaches the earth's surface, R_{s}/R_{a}, and the air temperature difference T_{max}  T_{min} for interior (k_{Rs} = 0.16) and coastal (k_{Rs} = 0.19) regions
The temperature difference method is recommended for locations where it is not appropriate to import radiation data from a regional station, either because homogeneous climate conditions do not occur, or because data for the region are lacking. For island conditions, the methodology of Equation 50 is not appropriate due to moderating effects of the surrounding water body.
Caution is required when daily computations of ET_{o} are needed. The advice given for Equation 49 fully applies. It is recommended that daily estimates of ET_{o} that are based on estimated R_{s} be summed or averaged over a severalday period, such as a week, decade or month to reduce prediction error.
EXAMPLE 15. Determination of solar radiation from temperature data
Determine the solar radiation from the temperature data of July in Lyon (France) at a latitude of 45°43'N and at 200 m above sea level. In July, the mean monthly maximum and minimum air temperatures are 26.6 and 14.8°C respectively.  

latitude = 45°43' = +45.72" decimal degrees = 
0.80 
radian 
From Table 2.5: 
The day of the year for 15 July is 
196 
 
From Eq. 21 or Annex 2 Table 2.6): 
R_{a} = 
40.6 
MJ m^{2} day^{1} 
From Eq. 50 (same latitude): 
R_{s} = 0.16 [Ö (26.614.8)] R_{a} = 0.55 (40.6) = 
22.3 
MJ m^{2} day^{1} 
From Eq. 20 (same latitude): 
equivalent evaporation = 0.408 (22.3) = 
9.1 
mm/day 
In July, the estimated solar radiation, R_{s}, is 22.3 MJ m^{2} day^{1} 
EXAMPLE 16. Determination of net radiation in the absence of radiation data
Calculate the net radiation for Bangkok (13°44'N) by using T_{max}, and T_{min}. The station is located at the coast at 2 m above sea level. In April, the monthly average of the daily maximum temperature, daily minimum temperature and daily vapour pressure are 34.8°C, 25.6°C and 2.85 kPa respectively. 

For Latitude 13°44'N = +13.73° decimal degrees = 0.24 radian and for 15 April, J = 105: 

From Eq. 21 or Table 2.6. 
R_{a} = 
38.1 
MJ m^{2} day^{1} 
(in coastal location) k_{Rs} = 
0.19 


(T_{max}  T_{min}) = (34.8  25.6) = 
9.2°C 
°C 

From Eq. 50: 
R_{s} = 0.19Ö (9.2)R_{a} 
21.9 
MJ m^{2} day^{1} 
From Eq. 36: 
R_{so} = 0.75 R_{a} 
28.5 
MJ m^{2} day^{1} 
From Eq. 38: 
R_{ns} = 0.77R_{s} 
16.9 
MJ m^{2} day^{1} 

s = 
4.903 10^{9} 
MJ K^{4} m^{2} day^{1} 
T_{max} = 
34.8 
°C 

_{} 
44.1 
MJ m^{2} day^{1} 


T_{min} 
25.6 
°C 
_{} 
39.1 
MJ m^{2} day^{1} 

_{} 
41.6 
MJ m^{2} day^{1} 

For: 
e_{a} = 2.85 kPa 
2.85 
kPa 
(0.340.14Ö e_{a}) = 
0.10 
 

For: 
R_{s}/R_{so} = 
0.77 
 
Then: 
(1.35(R_{s}/R_{so})  0.35)= 
0.69 
 
From Eq. 39: 
R_{nl} = 41.6(0.10)0.69 = 
3.0 
MJ m^{2} day^{1} 
From Eq. 40: 
R_{n} = (16.92.9) = 
13.9 
MJ m^{2} day^{1} 
From Eq. 20: 
equivalent evaporation = 0.408 (13.9) = 
5.7 
mm/day 
The estimated net radiation is 13.9 MJ m^{2} day^{1}. 
Empirical methodology for island locations
For island locations, where the land mass has a width perpendicular to the coastline of 20 km or less, the air masses influencing the atmospheric conditions are dominated by the adjacent water body in all directions. The temperature method is not appropriate for this situation. Where radiation data from another location on the island are not available, a first estimate of the monthly solar average can be obtained from the empirical relation:
R_{s} = 0.7 R_{a}  b (51)
where
R_{s} solar radiation [MJ m^{2} day^{1}],
R_{a} extraterrestrial radiation [MJ m^{2} day^{1}],
b empirical constant, equal to 4 MJ m^{2} day^{1}.
This relationship is only applicable for low altitudes (from 0 to 100 m). The empirical constant represents the fact that in island locations some clouds are usually present, thus making the mean solar radiation 4 MJ m^{2} day^{1} below the nearly clear sky envelope (0.7 R_{a}). Local adjustment of the empirical constant may improve the estimation.
The method is only appropriate for monthly calculations. The constant relation between R_{s} and R_{a} does not yield accurate daily estimates.
Wind speed data from a nearby weather station
Importing wind speed data from a nearby station, as for radiation data, relies on the fact that the air flow above a 'homogeneous' region may have relatively large variations through the course of a day but small variations when referring to longer periods or the total for the day. Data from a nearby station may be imported where air masses are of the same origin or where the same fronts govern air flows in the region and where the relief is similar.
When importing wind speed data from another station, the regional climate, trends in variation of other meteorological parameters and relief should be compared. Strong winds are often associated with low relative humidity and light winds are common with high relative humidity. Thus, trends in variation of daily maximum and minimum relative humidities should be similar in both locations. In mountainous areas, data should not necessarily be imported from the nearest station but from nearby stations with similar elevation and exposure to the dominant winds. The paired stations may even vary from one season to another, depending on the dominant winds.
Imported wind speed data can be used when making monthly estimates of evapotranspiration. Daily calculations are justified when utilized as a sum or average over a severalday period, such as a week or decade.
Empirical estimates of monthly wind speed
As the variation in wind speed average over monthly periods is relatively small and fluctuates around average values, monthly values of wind speed may be estimated. The 'average' wind speed estimates may be selected from information available for the regional climate, but should take seasonal changes into account. General values are suggested in Table 4.
TABLE 4. General classes of monthly wind speed data
Description 
mean monthly wind speed at 2 m 
light wind 
...£ 1.0 m/s 
light to moderate wind 
1  3 m/s 
moderate to strong wind 
3  5 m/s 
strong wind 
... ³ 5.0 m/s 
Where no wind data are available within the region, a value of 2 m/s can be used as a temporary estimate. This value is the average over 2000 weather stations around the globe.
In general, wind speed at 2 m, u_{2}, should be limited to about u_{2} ³ 0.5 m/s when used in the ET_{o} equation (Equation 6). This is necessary to account for the effects of boundary layer instability and buoyancy of air in promoting exchange of vapour at the surface when air is calm. This effect occurs when the wind speed is small and buoyancy of warm air induces air exchange at the surface. Limiting u_{2} ³ 0.5 m/s in the ET_{o} equation improves the estimation accuracy under the conditions of very low wind speed.
An alternative equation for ET_{o} when weather data are missing
This section has shown how solar radiation, vapour pressure and wind data can be estimated when missing. Many of the suggested procedures rely upon maximum and minimum air temperature measurements. Unfortunately, there is no dependable way to estimate air temperature when it is missing. Therefore it is suggested that maximum and minimum daily air temperature data are the minimum data requirements necessary to apply the FAO PenmanMonteith method.
When solar radiation data, relative humidity data and/or wind speed data are missing, they should be estimated using the procedures presented in this section. As an alternative, ET_{o} can be estimated using the Hargreaves ET_{o} equation where:
ET_{o} = 0.0023(T_{mean} + 17.8)(T_{max}  T_{min})^{0.5} R_{a} (52)
where all parameters have been previously defined. Units for both ET_{o} and R_{a} in Equation 52 are mm day^{1}. Equation 52 should be verified in each new region by comparing with estimates by the FAO PenmanMonteith equation (Equation 6) at weather stations where solar radiation, air temperature, humidity, and wind speed are measured. If necessary, Equation 52 can be calibrated on a monthly or annual basis by determining empirical coefficients where ET_{o} = a + b ET_{o} Eq. 52, where the "Eq. 52" subscript refers to ET_{o} predicted using Equation 52. The coefficients a and b can be determined by regression analyses or by visual fitting. In general, estimating solar radiation, vapor pressure and wind speed as described in Equations 48 to 51 and Table 4 and then utilizing these estimates in Equation 6 (the FAO PenmanMonteith equation) will provide somewhat more accurate estimates as compared to estimating ET_{o} directly using Equation 52. This is due to the ability of the estimation equations to incorporate general climatic characteristics such as high or low wind speed or high or low relative humidity into the ET_{o} estimate made using Equation 6.
Equation 52 has a tendency to underpredict under high wind conditions (u_{2} > 3 m/s) and to overpredict under conditions of high relative humidity.