There are different methods to estimate or measure the ET and the PET. The precision and reliability vary from one method to another, some provide only an approximation.
Each technique has been developed with the available climatological data to estimate the ET. The direct measurements of PET are expensive and are only used for local calibration of a given method using climatological data. The most frequently used techniques are: hydrological method or water balance method, climatic methods, and micro meteorological methods. Many investigators have modified the equations that are already established. For example, one may find modification of Blaney–Criddle formula, Hargreaves–Samani, Class A pan evaporation and so forth. Allen [1] inves-tigated 13 variations of the Penman equation. He found that the Penman–Monteith formula was most precise. Modified equations are actually recommended by the FAO and the USDA––Soil Conservation Service. Most of investigators agree that Penman, Class A pan evaporation, Blaney–Criddle, and Hargreaves–Samani equations, can be trusted. High precision can be obtained with local calibration of a given method. Ev-ery researcher has its preferred formula that may give good results. Hargreaves and Samani [7] presented their formulae as to be simplest and practical. I can add that, “There is no evidence of a superior method.” Allen and Pruitt [2] presented the FAO modified Blaney–Criddle method, which involves relatively easy calculations and give precise estimates of PET (when it is calibrated for local conditions). Every researcher has preference. However each formula, depending where it was evaluated, may or may not result in the first or the last place.
Hydrologic Method or Water Balance
This technique employs periodic determination of rainfall, irrigation, drainage, and soil moisture data. The hydrologic method uses water balance equation:
PI + SW – RO – D – ET = 0 (1) where: PI = Precipitation and/or irrigation.
RO = Runoff.
D = Deep percolation.
SW = Change in the soil moisture, and ET = Evapotranspiration.
In equation (1), every variable can be measured with precision with the lysimeters (Figure 2.2 a, b, c). The ET can be calculated as residual, knowing values of all other parameters (Figure 2.3).
Figure 2.2a. A typical lysimeter with its components in Australia.
At Universiy of California, Davis.
Figure 2.2b. A field lysimeter test facility.
Figure 2.2c. Plastic tube lysimeter.
Figure 2.3. Water balance method.
Climatic Methods
Using weather data [3], numerous equations have been proposed. Also, numerous modifications have been made to the available formulae for application to a particular region.
Penman
The Penman formula was presented in 1948. It employs net radiation, air temperature, wind velocity, and deficit in the vapor pressure. He gave the following equation:
PET =Rn/a + b Ea
c + b (2)
where: PET = Daily potential evapotranspiration, mm/day.
C = Slope of saturated air vapor pressure curve, mb/°C.
Rn = Net radiation, cal/cm2 day.
a = Latent heat of vaporization of water = [59.59 – 0.055 T], cal/cm2 – mm = 58 cal/cm2 – mm at 29°C.
Ea = 0.263 [(ea – ed) * (0.5 + 0.0062 * u2)] (2a) Ea = Average vapor pressure of air, mb = (emax – emin)/2.
ed = Vapor pressure of air at minimum air temperature, mb.
u2 = Wind velocity a height of 2 meters, km/day. b = Psychrometric constant = 0.66, mb/°C.
T = [(Tmax – Tmin)/2], in degrees °C.
(emax – emin) = Difference between maximum and minimum vapor pressure of air vapor, mb.
(Tmax – Tmin) = Difference between maximum and minimum daily temperature, °C.
Penman modified by Monteith [14]
After modification, the resultant equation is as follows:
LE = – s (Rn – S) + Pa * Cp (es – ea)/r
[(s + b) * (ra + rc)]/ra (3) where: LE = Latent heat flow.
Rn = Net radiation.
S = Soil heat flow.
Cp = Air specific energy at constant pressure.
s = Slope of saturated vapor pressure of air, at air average temperature of wet bulb thermometer.
Pa = Density of humid air.
es = Saturated vapor pressure of water.
ea = Partial water vapor pressure of air.
ra = Air resistance.
rc = Leaf resistance.
b = Psychrometric constant.
This method has been successfully used to estimate the ET of a crop. The Penman-Monteith equation is limited to research work (experimentation) since the ra and rc data are not always available.
Penman modified by Doorenbos and Pruitt
PET = c * [W*Rn + (1 – W) * F(u) * (ea – ed)] (4) where: PET = Potential evapotranspiration, mm/day.
W = A factor related to temperature and elevation.
Rn = Net radiation, mm/day.
F(u) = Wind related function.
(ea – ed) = Difference between the saturated vapor pressure of air at average tem-perature and vapor pressure of air, mb.
c = A correction factor.
The Penman formula is not popular, because it needs data that is not available at majority of the weather stations. Estimations of PET using Penman formula cannot be complex. The equation contains too many components, which should be measured or estimated, when data is not available.
Thornthwaite
This method uses monthly average temperature and the length of the day.
PET = 16 Ld [10 T/I]a (5)
where: PET = Estimated evapotranspiration for 30 days, mm.
Ld = Hours of the day divided by 12.
A = (6.75 x 10-7 I3) – (7.71 x 105 I2) + 0.01792 I + 0.49239 (5a) T = Average monthly temperature, °C.
I = i1 + i2 + . . . + i12, where, i = [Tm/5] x 1.514 (5b) The Thornthwaite method underestimates PET during the summer when maxi-mum radiation of the year occurs. Besides, the application of equation to short periods of time can lead to an error. During short periods, the average temperature is not an adequate measure of the received radiation [14]. During long terms, the temperature and the ET are similar functions of the net radiation. These are related, when the long periods are considered.
Blaney–Criddle
The original Blaney–Criddle equation was developed to predict the consumptive use of PET in arid climates. This formula uses percentage of monthly sunshine hours and monthly average temperature.
PET = Km F (6)
where: PET = Monthly potential evapotranspiration, mm.
Km = Empirically derived coefficient for the Blaney–Criddle method.
F = Monthly ET factor = [25.4 * PD * (1.8 T + 32)]/100 (6a) T = Monthly average temperature, °C.
PD = Monthly percentage daily sunshine hour.
This method is easy to use and the necessary data are available. It has been widely used in the Western United States with accurate results, but not the same in Florida, where ET is underestimated during summer months.
Blaney–Criddle modified by FAO [4]
PET = C * P * [0.46 * T + 8] (7)
where: PET = Potential evapotranspiration, mm/day.
T = Monthly average temperature.
P = Percentage daily sunshine hours, Table 2.1.
C = Correction factor, which depends on the relative humidity, light hours, and wind.
Doorenbos and Pruitt [4, 5] recommended individual calculation for each month.
They indicated that it may be necessary to increase its value for high elevations.
Table 2.1. Percentage average daily sunshine hours (P) based on annual day light hour for different latitudes.
Latitude, degrees January February March April May June
North
South* July August Sept. October Nov. Dec.
60 0.15 0.20 0.26 0.32 0.38 0.41
Latitude, degrees
July August Sept. October Nov. Dec.
North
South* January February March April May June
60 0.40 0.34 0.28 0.22 0.17 0.13
*Southern latitudes have six months of difference as shown in Table 2.1.
Blaney and Criddle modified by Shih
PET = 25.4 * K * [MRs * (1.8 T + 32)]/[TMRs] (8) where: PET = Monthly potential evapotranspiration, mm.
K = Coefficient for this modified method.
MRs = Monthly solar radiation, cal/cm2. T = Monthly average temperature, °C.
TMRs = Sum of monthly solar radiation during the year, cal/cm2. Jensen–Haise
The Jensen–Haise equation [9] resulted from about 3000 measurements of the ET taken in the Western Regions of the United States for a 35 years period. It is an empiri-cal equation.
PET = Rs (0.025 * T + 0.08) (9)
where: PET = Potential evapotranspiration, mm/day.
Rs = Daily total solar radiation, mm of water.
T = Air average temperature, °C.
This method seriously underestimates ET under conditions of high movements of atmospheric air masses. However, it gives reliable results for calm atmospheres.
Table 2.1. (Continued)
Stephens–Stewart
Stephens–Stewart utilized solar radiation data. It is similar to the original Jensen–
Haise method [9]. The equation is as follow:
PET = 0.01476 * [(T + 4.905) * MRs]/b (10) where: PET = Monthly potential evapotranspiration, mm.
T = Monthly average temperature, °C.
MRs = Monthly solar radiation, cal/cm2.
b = Latent vaporization energy of water = [59.59 – 0.055 Tm], cal/cm2-mm.
= 58 cal/cm2-mm at 29°C.
Pan evaporation
Class A pan is commonly used instrument to measure evaporation. The evaporation pan (Figure 2.4) integrates the climate factors and has proven to give accurate estima-tions of PET. It requires a good service, maintenance, and management. Table 2.2 gives class A pan coefficients under different conditions [Doorenbos and Pruitt, 4 and 5]. The relationship between PET and pan evaporation can be expressed as:
PET = Kp * PE (11)
where: PET = Potential evapotranspiration, mm/day.
Kp = Pan coefficient.
PE = Class A pan evaporation.
View of climatological station Figure 2.4. A typical class A pan.
Table 2.2. Pan coefficient (KP) for the class A pan evaporation under different conditions.
Class A Condition A Condition B*
Pan Pan surrounded by grass Pan surrounded by dry uncovered soil
Average of HR% Low Medium High Low Medium High
40 40–70 70 40 40–70 70
Wind** Distance from Distance from
km/day the green crop, the dry fallow,
m m
Light 0 0.55 0.55 0.75 0 0.70 0.80 0.85
175 10 0.65 0.75 0.85 10 0.60 0.70 0.80
100 9.70 0.80 0.85 100 0.55 0.65 0.75
1000 0.75 0.85 0.85 1000 0.50 0.60 0.70
Moderate 0 0.50 0.60 0.65 0 0.65 0.75 0.80
175–425 10 0.60 0.70 0.75 10 0.55 0.65* 0.70
100 0.65 0.75 0.80 100 0.50 0.60 0.65
1000 0.70 0.80 0.80 1000 0.45 0.55 0.60
Strong 0 0.45 0.50 0.60 0 0.60 0.65 0.70
425–700 10 0.55 0.60 0.65 10 0.50 0.55 0.65
100 0.60 0.65 0.70 100 0.45 0.45 0.60
1000 0.65 0.70 0.75 1000 0.40 0.45 0.55
Very 0 0.40 0.45 0.50 0 0.50 0.60 0.65
Strong 10 0.45 0.55 0.60 10 0.45 0.50 0.55
100 0.50 0.60 0.65 100 0.40 0.45 0.50
1000 0.55 0.60 0.65 1000 0.35 0.40 0.45
*For areas of extensive uncovered and not developed agricultural soils.
Reduce values of KP by 20% under hot wind conditions and by 5–10% for moderate wind conditions, temperature and humidity.
**Total wind movement in km/day.
Hargreaves method
Hargreaves method uses a minimum of climatic data. The formula is as bellow:
PET = MF * (1.8 T + 32) * CH (12)
where: PET = Potential evapotranspiration, mm/month.
MF = Monthly factor depending on the latitude.
T = Monthly average temperature, °C.
CH = Correction factor for the relative humidity (RH) = To be used for RH > 64%
= 0.166 [(100 – HR)]1/2 (12a)
The Hargreaves original formula for the PET was based on radiation and tempera-ture as given below:
PET = [(0.0135 * RS)] * [T + 17.8] (13) where: RS = Solar radiation, mm/day.
T = Average temperature, °C.
To estimate solar radiation (RS) using extraterrestrial radiation (RA), Hargreaves and Samani [7, 8] formulated the following equation:
RS = Krs * RA * TD0.50 (13a) where: T = Average temperature, °C.
RS = Solar radiation.
RA = Extraterrestrial radiation.
Krs = Calibration coefficient.
TD = Difference between maximum and minimum temperatures.
Hargreaves and Samani modified method
Finally after several years of calibration, equation (13) was modified as follows:
PET = 0.0023 Ra * [T + 17.8] * (TD)0.50 (14) where: PET = Potential evapotranspiration, mm/day.
Ra = Extraterrestrial radiation, mm/day.
T = Average temperature, °C.
TD = Difference between maximum and minimum temperatures, °C.
This equation requires only maximum and minimum temperature data. This data is normally available. This formula is precise and reliable.
Linacre method
The Linacre equation is as follow:
PET =700 Tm/[100 – La] + 15 [T – Td]
[80 – T] (15)
where: PET = Potential evapotranspiration, mm.
Tm = (Ta + 0.0062 * Z) (15a)
Z = Elevation, m.
T = Average temperature, °C.
La = Latitude, degrees.
Td = Daily average temperature, °C.
The variations in PET values by this formula are 0.3 mm/day based annual data and 1.7 mm/day based on daily data.
Makkink method
This formula provides good results in humid and cold climates, and in arid regions.
Makkink used radiation measurements to develop a following regression equation:
PET = Rs * [s/(a + b)] + 0.12 (16) where: PET = Potential evapotranspiration, mm/day.
Rs = Total daily solar radiation, mm/day.
b = Psychrometric constant.
s = Slope of saturated vapor pressure curve at average air temperature.
Radiation method
Doorenbos and Pruitt [4] presented following radiation equation, which is a modified Makkink formula [16]:
PET = c * (W * Rs) (17)
where: PET = Potential evapotranspiration for the considered period, mm/day.
Rs = Solar radiation, mm/day.
W = Correction factor related to temperature and elevation.
C = Correction factor, which depends on the average humidity and average wind speed.
This method was employed in the Equator zone, in small islands and in high lati-tudes. Solar radiation maps provide the necessary data for the formula.
Regression method
The simple lineal regression equation is given as follow:
PET = [a * Rs] + b (18)
where: PET = Potential evapotranspiration, mm/day.
a and b = Empirical constants (regression coefficients), which depend on the loca-tion and season.
Rs = Solar radiation, mm/day.
This regression method is simple and easy to use. However, it is not frequently used because of highly empirical nature.
Priestly–Taylor method
In the absence of atmospheric air mass movement, Priestly and Taylor showed that the PET is directly related to evaporation equilibrium:
PET = A [s/(S + B)] [(Rn + S)] (19) where: PET = Potential evapotranspiration, mm/day.
A = Empirically derived constant.
s = Slope of saturated vapor pressure curve, at average air temperature.
B = Psychrometric constant.
Rn = Net radiation, mm/day.
This method is of semi-empirical in nature. It is reliable in humid zones, and is not adequate in arid regions. Table 2.3 shows advantages and disadvantages of different methods of estimation of potential evapotranspiration.
Table 2.3. Advantages and disadvantages of different methods to estimate PET.
Method Advantages Disadvantages
1. Penman Easy to apply. Underestimates ET under high movement condi-tions of atmospheric air masses.
2. Penman (FAO) Provide satisfactory results. The formula contains many components, which may result complex calculations.
3. Water balance Easy to process the data and integrate with the observa-tions.
Low precision on the daily measures and difficult to obtain the ET when it is raining.
4. Thornthwaite Is reliable for long terms. Underestimate the ET during the summer. Is not precise for short terms.
5. Blaney–Criddle Easy to use, the data is
usu-ally available. The crop coefficient depends a lot on the climate.
6. Blaney–Criddle
(FAO) The given crop coefficient
depend less on the climate. In high elevations, coasts, and small islands there is no relation between temperature and solar radiation.
7. Stephens–
Stewart Is reliable on the western side of the United States (where it was developed).
Need to be evaluated in other locations.
8. Jensen–Haise Is reliable under calm
atmo-spheric conditions. Underestimates ET under conditions of high move-ment of atmospheric air masses.
9. Evaporation
pan Integrate all climatological
factors. Evaporation continues during the night in the pan, which affects the PET estimates.
10. Hargreaves Requires a minimum of
cli-matological data. Underestimates PET on the coasts and under high movements of air masses.
11. Hargreaves and
Samani Requires only maximum
and minimum temperature data.
Needs to be evaluated in many locations for its ac-ceptance.
12. Radiation Is reliable in Equator, small
islands and high altitudes. Monthly estimates are often necessary outside Equator.
13. Makkink Good for humid and cold
climates. It is not reliable in arid regions.
14. Linacre Is precise on annual basis. Precision decreases on daily base.
15. Priestly–Taylor Reliable on humid areas. Not adequate for arid zones.