FAO Irrigation and Drainage Paper

No. 56



Crop
Evapotranspiration

(guidelines for computing crop water requirements)







by
Richard G. ALLEN
Utah State University
Logan, Utah, U.S.A.

Luis S. PEREIRA
Instituto Superior de Agronomia
Lisbon, Portugal

Dirk RAES
Katholieke Universiteit Leuven
Leuven, Belgium

Martin SMITH
FAO, Water Resources, Development and Management Service
Rome, Italy

Crop evapotranspiration





iii

Preface




This publication presents an updated procedure for calculating reference and crop
evapotranspiration from meteorological data and crop coefficients. The procedure, first presented
in the FAO Irrigation and Drainage Paper No. 24 'Crop Water Requirements', is termed the 'Kc
ETo' approach, whereby the effect of the climate on crop water requirements is given by the
reference evapotranspiration ETo and the effect of the crop by the crop coefficient Kc. Other
procedures developed in FAO Irrigation and Drainage Paper No. 24 such as the estimation of
dependable and effective rainfall, the calculation of irrigation requirements and the design of
irrigation schedules are not presented in this publication but will be the subject of later papers in
the series.

Since the publication of FAO Irrigation and Drainage Paper No. 24 in 1977, advances in research
and more accurate assessment of crop water use have revealed the need to update the FAO
methodologies for calculating ETo. The FAO Penman method was found to frequently
overestimate ETo while the other FAO recommended equations, namely the radiation, the Blaney-
Criddle, and the pan evaporation methods, showed variable adherence to the grass reference crop
evapotranspiration.

In May 1990, FAO organized a consultation of experts and researchers in collaboration with the
International Commission for Irrigation and Drainage and with the World Meteorological
Organization, to review the FAO methodologies on crop water requirements and to advise on the
revision and update of procedures.

The panel of experts recommended the adoption of the Penman-Monteith combination method as a
new standard for reference evapotranspiration and advised on procedures for calculating the
various parameters. The FAO Penman-Monteith method was developed by defining the reference
crop as a hypothetical crop with an assumed height of 0.12 m, with a surface resistance of 70 s m-1
and an albedo of 0.23, closely resembling the evaporation from an extensive surface of green grass
of uniform height, actively growing and adequately watered. The method overcomes the
shortcomings of the previous FAO Penman method and provides values that are more consistent
with actual crop water use data worldwide. Furthermore, recommendations have been developed
using the FAO Penman-Monteith method with limited climatic data, thereby largely eliminating
the need for any other reference evapotranspiration methods and creating a consistent and
transparent basis for a globally valid standard for crop water requirement calculations.

The FAO Penman-Monteith method uses standard climatic data that can be easily measured or
derived from commonly measured data. All calculation procedures have been standardized
according to the available weather data and the time scale of computation. The calculation methods,
as well as the procedures for estimating missing climatic data, are presented in this publication.

In the 'Kc-ETo' approach, differences in the crop canopy and aerodynamic resistance relative to the
reference crop are accounted for within the crop coefficient. The Kc coefficient serves as an
aggregation of the physical and physiological differences between crops. Two calculation methods

iv
to derive crop evapotranspiration from ETo are presented. The first approach integrates the
relationships between evapotranspiration of the crop and the reference surface into a single Kc
coefficient. In the second approach, Kc is split into two factors that separately describe the
evaporation (Ke) and transpiration (Kcb) components. The selection of the Kc approach depends
on the purpose of the calculation and the time step on which the calculations are to be executed.

The final chapters present procedures that can be used to make adjustments to crop coefficients to
account for deviations from standard conditions, such as water and salinity stress, low plant
density, environmental factors and management practices.

Examples demonstrate the various calculation procedures throughout the publication. Most of the
computations, namely all those required for the reference evapotranspiration and the single crop
coefficient approach, can be performed using a pocket calculator, calculation sheets and the
numerous tables given in the publication. The user may also build computer algorithms, either
using a spreadsheet or any programming language.

These guidelines are intended to provide guidance to project managers, consultants, irrigation
engineers, hydrologists, agronomists, meteorologists and students for the calculation of reference
and crop evapotranspiration. They can be used for computing crop water requirements for both
irrigated and rainfed agriculture, and for computing water consumption by agricultural and natural
vegetation.

Crop evapotranspiration





v
Acknowledgements





These guidelines constitute the efforts of eight years of deliberations and consultations by the
authors, who together formed the working group to pursue the recommendations of the FAO expert
consultation that was held in May 1990 in Rome. The consultation was organized to review the then
current FAO guidelines to determine Crop Water Requirements, published in 1977 as FAO
Irrigation and Drainage paper No. 24 (FAO-24) and authored by J. Doorenbos and W. Pruitt. The
conceptual framework for the revised methodologies introduced in this publication came forth out
of the advice of the group of eminent experts congregated in the 1990 meetings and who have
importantly contributed to the development of the further studies conducted in the framework of
the publication. Members of the 1990 FAO expert consultation included Dr P. Fleming of
Australia, Dr A. Perrier of France, Drs L. Cavazza and L. Tombesi from Italy, Drs R. Feddes and
J. Doorenbos of the Netherlands, Dr L.S. Pereira of Portugal, Drs J.L. Monteith and H. Gunston
from the United Kingdom, Drs R. Allen, M. Jensen and W.O. Pruitt of USA, Dr D. Rijks from the
World Meteorological Organization and various staff of FAO .

Many other experts and persons from different organizations and institutes have provided, in
varying degrees and at different stages, important advice and contributions. Special
acknowledgements for this are due in particular to Prof. W.O. Pruitt (retired) of the University of
California, Davis and J. Doorenbos of FAO (retired) who set the standard and template for this
work in the predecessor FAO-24, and to Prof. J.L. Monteith whose unique work set the scientific
basis for the ETo review. Prof. Pruitt, despite his emeritus status, has consistently contributed in
making essential data available and in advising on critical concepts. Dr James L. Wright of the
USDA, Kimberly, Idaho, further contributed in providing data from the precision lysimeter for
several crops. Further important contributions or reviews at critical stages of the publication were
received from Drs M. Jensen, G. Hargreaves and C. Stockle of USA, Dr B. Itier of France, and
various members of technical working groups of the International Commission on Irrigation and
Drainage (ICID) and the American Societies of Civil and Agricultural Engneers.

The authors thank their respective institutions, Utah State University, Instituto Superior de
Agronomia of Lisbon, Katholieke Universiteit Leuven and FAO for the generous support of faculty
time and staff services during the development of this publication.

The authors wish to express their gratitude to Mr H. Wolter, Director of the Land and Water
Development Division for his encouragement in the preparation of the guidelines and to FAO
colleagues and others who have reviewed the document and made valuable comments.

Special thanks are due to Ms Chrissi Redfern for her patience and valuable assistance in the
preparation and formatting of the text. Mr Julian Plummer further contributed in editing the final
document.

Crop evapotranspiration





vii
Contents



Page
1. INTRODUCTION TO EVAPOTRANSPIRATION 1
Evapotranspiration process 1
Evaporation 1
Transpiration 3
Evapotranspiration 3
Units 3
Factors affecting evapotranspiration 5
Weather parameters 5
Crop factors 5
Management and environmental conditions 5
Evapotranspiration concepts 7
Reference crop evapotranspiration (ETo) 7
Crop evapotranspiration under standard conditions (ETc) 7
Crop evapotranspiration under non-standard conditions (ETc adj) 9
Determining evapotranspiration 9
ET measurement 9
ET computed from meteorological data 13
ET estimated from pan evaporation 13


PART A. REFERENCE EVAPOTRANSPIRATION (ETO) 15
2. FAO PENMAN-MONTEITH EQUATION 17
Need for a standard ETo method 17
Formulation of the Penman-Monteith equation 18
Penman-Monteith equation 18
Aerodynamic resistance (ra) 20
(Bulk) surface resistance (rs) 20
Reference surface 23
FAO Penman-Monteith equation 23
Equation 23
Data 25
Missing climatic data 27
3. METEOROLOGICAL DATA 29
Meteorological factors determining ET 29
Solar radiation 29
Air temperature 29
Air humidity 30
Wind speed 30
Atmospheric parameters 31
Atmospheric pressure (P) 31

viii
Page

Latent heat of vaporization (λ) 31
Psychrometric constant (γ) 31
Air humidity 33
Concepts 33
Measurement 35
Calculation procedures 36
Radiation 41
Concepts 41
Units 43
Measurement 45
Calculation procedures 45
Wind speed 55
Measurement 55
Wind profile relationship 55
Climatic data acquisition 57
Weather stations 57
Agroclimatic monthly databases 57
Estimating missing climatic data 58
Estimating missing humidity data 58
Estimating missing radiation data 59
Missing wind speed data 63
Minimum data requirements 64
An alternative equation for ETo when weather data are missing 64

4. DETERMINATION OF ETO 65
Penman-Monteith equation 65
Calculation procedure 66
ETo calculated with different time steps 66
Calculation procedures with missing data 76
Pan evaporation method 78
Pan evaporation 78
Pan coefficient (Kp) 79


PART B. CROP EVAPOTRANSPIRATION UNDER STANDARD CONDITIONS 87

5. INTRODUCTION TO CROP EVAPOTRANSPIRATION (ETC) 89
Calculation procedures 89
Direct calculation 89
Crop coefficient approach 90
Factors determining the crop coefficient 91
Crop type 91
Climate 91
Soil evaporation 93
Crop growth stages 95

Crop evapotranspiration





ix
Page

Crop evapotranspiration (ETc) 97
Single and dual crop coefficient approaches 98
Crop coefficient curve 99
Flow chart of the calculations 101
6. ETC - SINGLE CROP COEFFICIENT (KC) 103
Length of growth stages 103
Crop coefficients 109
Tabulated Kc values 109
Crop coefficient at the initial stage (Kc ini) 114
Crop coefficient at the mid-season stage (Kc mid) 121
Crop coefficient at the end of the late season stage (Kc end) 125
Construction of the Kc curve 127
Annual crops 127
Kc curves for forage crops 127
Fruit trees 129
Calculating ETc 129
Graphical determination of Kc 129
Numerical determination of Kc 132
Alfalfa-based crop coefficients 133
Transferability of previous Kc values 134

7. ETC - DUAL CROP COEFFICIENT (KC = KCB + KE) 135
Transpiration component (Kcb ETo) 135
Basal crop coefficient (Kcb) 135
Determination of daily Kcb values 141
Evaporation component (Ke ETo) 142
Calculation procedure 142
Upper limit Kc max 143
Soil evaporation reduction coefficient (Kr) 144
Exposed and wetted soil fraction (few) 147
Daily calculation of Ke 151
Calculating ETc 156


PART C. CROP EVAPOTRANSPIRATION UNDER NON-STANDARD CONDITIONS 159
8. ETC UNDER SOIL WATER STRESS CONDITIONS 161
Soil water availability 161
Total available water (TAW) 161
Readily available water (RAW) 162
Water stress coefficient (Ks) 167
Soil water balance 169
Forecasting or allocating irrigations 171
Effects of soil salinity 174
Yield-salinity relationship 175
Yield-moisture stress relationship 176

xii
List of figures



1. Schematic representation of a stoma 2
2. The partitioning of evapotranspiration into evaporation and transpiration over the
growing period for an annual field crop 2
3. Factors affecting evapotranspiration with reference to related ET concepts 4
4. Reference (ETo), crop evapotranspiration under standard (ETc) and non-standard
conditions (ETc adj) 6
5. Schematic presentation of the diurnal variation of the components of the energy
balance above a well-watered transpiring surface on a cloudless day 10
6. Soil water balance of the root zone 12
7. Simplified representation of the (bulk) surface and aerodynamic resistances for
water vapour flow 19
8. Typical presentation of the variation in the green Leaf Area Index over the
growing season for a maize crop 22
9. Characteristics of the hypothetical reference crop 24
10. Illustration of the effect of wind speed on evapotranspiration in hot-dry and humid-
warm weather conditions 30
11. Saturation vapour pressure shown as a function of temperature: e°(T) curve 34
12. Variation of the relative humidity over 24 hours for a constant actual vapour
pressure of 2.4 kPa 34
13. Annual variation in extraterrestrial radiation (Ra) at the equator, 20 and 40° north
and south 41
14. Annual variation of the daylight hours (N) at the equator, 20 and 40° north and
south 42
15. Various components of radiation 44
16. Conversion factor to convert wind speed measured at a certain height above
ground level to wind speed at the standard height (2 m) 56
17. Relationship between the fraction of extraterrestrial radiation that reaches the
earth's surface, Rs/Ra, and the air temperature difference Tmax - Tmin for
interior (kRs = 0.16) and coastal (kRs = 0.19) regions 61
18. ETo computed by CROPWAT 69
19. Two cases of evaporation pan siting and their environment 79
20. Typical Kc for different types of full grown crops 92
21. Extreme ranges expected in Kc for full grown crops as climate and weather change 92
22. The effect of evaporation on Kc. The horizontal line represents Kc when the soil
surface is kept continuously wet. The curved line corresponds to Kc when the
soil surface is kept dry but the crop receives sufficient water to sustain full
transpiration 94
23. Crop growth stages for different types of crops 94

Crop evapotranspiration





xiii
Page
24. Typical ranges expected in Kc for the four growth stages 97
25. Generalized crop coefficient curve for the single crop coefficient approach 100
26. Crop coefficient curves showing the basal Kcb (thick line), soil evaporation Ke)
(thin line) and the corresponding single Kc = Kcb + Ke curve (dashed line) 100
27. General procedure for calculating ETc 102
28. Variation in the length of the growing period of rice (cultivar: Jaya) sown during
various months of the year at different locations along the Senegal River (Africa) 109
29. Average Kc ini as related to the level of ETo and the interval between irrigations
and/or significant rain during the initial growth stage for all soil types when
wetting events are light (about 10 mm per event) 117
30. Average Kc ini as related to the level of ETo and the interval between irrigations
greater than or equal to 40 mm per wetting event, during the initial growth stage
for: a) coarse textured soils; b) medium and fine textured soils 118
31. Partial wetting by irrigation120
32. Adjustment (additive) to the Kc mid values from Table 12 for different crop
heights and mean daily wind speeds (u2) for different humidity conditions 122
33. Ranges expected for Kc end 126
34. Crop coefficient curve 126
35. Constructed curve for Kc for alfalfa hay in southern Idaho, the United States using
values from Tables 11 and 12 and adjusted using Equations 62 and 65 128
36. Kc curve and ten-day values for Kc and ETc derived from the graph for the dry
bean crop example (Box 15) 132
37. Constructed basal crop coefficient (Kcb) curve for a dry bean crop (Example 28)
using growth stage lengths of 25, 25, 30 and 20 days 142
38. Soil evaporation reduction coefficient, Kr 145
39. Determination of variable few as a function of the fraction of ground surface
coverage (fc) and the fraction of the surface wetted (fw) 148
40. Water balance of the topsoil layer 152
41. Depletion factor for different levels of crop evapotranspiration 166
42. Water stress coefficient, Ks 167
43. Water balance of the root zone 169
44. The effect of soil salinity on the water stress coefficient Ks 181
45. Different situations of intercropping 198
46. Kc curves for small areas of vegetation under the oasis effect as a function of the
width of the expanse of vegetation for conditions where RHmin = 30%, u2 =
2 m/s, vegetation height (h) = 2 m and LAI = 3 203
47. Mean evapotranspiration during non-growing, winter periods at Kimberly, Idaho,
measured using precision weighing lysimeters 210

Crop evapotranspiration





xvii
Page
31. Determination of the evapotranspiration from a bare soil 146
32. Calculation of the crop coefficient (Kcb + Ke) under sprinkler irrigation 150
33. Calculation of the crop coefficient (Kcb + Ke) under furrow irrigation 151
34. Calculation of the crop coefficient (Kcb + Ke) under drip irrigation 151
35. Estimation of crop evapotranspiration with the dual crop coefficient approach 154
36. Determination of readily available soil water for various crops and soil types 166
37. Effect of water stress on crop evapotranspiration 168
38. Irrigation scheduling to avoid crop water stress 172
39. Effect of soil salinity on crop evapotranspiration 182
40. First approximation of the crop coefficient for the mid-season stage for sparse
vegetation 185
41. Estimation of mid-season crop coefficient 190
42. Estimation of mid-season crop coefficient for reduced ground cover 191
43. Estimation of Kcb mid from ground cover with reduction for stomatal control 192
44. approximate estimation of Ks from crop yield data 194
45. Effects of surface mulch 197
46. Intercropped maize and beans 200
47. Overlapping vegetation 201

xviii
List of equations




Page
1. Energy balance equation 11
2. Soil water balance 12
3. Penman-Monteith form of the combination equation 19
4. Aerodynamic resistance (ra) 20
5. (Bulk) surface resistance (rs) 21
6. FAO Penman-Monteith equation for daily, ten-day and monthly time steps 24
7. Atmospheric pressure (P) 31
8. Psychrometric constant (γ) 32
9. Mean air temperature (Tmean) 33
10. Relative humidity (RH) 35
11. Saturation vapour pressure as a function of temperature (e°(T)) 36
12. Saturation vapour pressure (es) 36
13. Slope e°(T) curve (∆) 37
14. Actual vapour pressure derived from dewpoint temperature (ea) 37
15. Actual vapour pressure derived from psychrometric data (ea) 37
16. Psychrometric constant of the (psychrometric) instrument (γpsy) 37
17. Actual vapour pressure derived from RHmax and RHmin (ea) 38
18. Actual vapour pressure derived from RHmax (ea) 39
19. Actual vapour pressure derived from RHmean (ea) 39
20. Conversion form energy to equivalent evaporation 44
21. Extraterrestrial radiation for daily periods (Ra) 46
22. Conversion from decimal degrees to radians 46
23. Inverse relative distance Earth-Sun (dr) 46
24. Solar declination (δ) 46
25. Sunset hour angle - arccos function (ωs) 46
26. Sunset hour angle - arctan function (ωs) 47
27. Parameter X of Equation 26 47
28. Extraterrestrial radiation for hourly or shorter periods (Ra) 47
29. Solar time angle at the beginning of the period (ω1) 48

Crop evapotranspiration





xix
Page
30. Solar time angle at the end of the period (ω2) 48
31. Solar time angle at midpoint of the period (ω) 48
32. Seasonal correction for solar time (Sc) 48
33. Parameter b of Equation 32 48
34. Daylight hours (N) 48
35. Solar radiation (Rs) 50
36. Clear-sky radiation near sea level (Rso) 51
37. Clear-sky radiation at higher elevations (Rso) 51
38. Net solar or net shortwave radiation (Rns) 51
39. Net longwave radiation (Rnl) 52
40. Net radiation (Rn) 53
41. Soil heat flux (G) 54
42. Soil heat flux for day and ten-day periods (Gday) 54
43. Soil heat flux for monthly periods (Gmonth) 54
44. Soil heat flux for monthly periods if Tmonth,i+1 is unknown (Gmonth) 54
45. Soil heat flux for hourly or shorter periods during daytime (Ghr) 55
46. Soil heat flux for hourly or shorter periods during nighttime (Ghr) 55
47. Adjustment of wind speed to standard height (u2) 56
48. Estimating actual vapour pressure from Tmin (ea) 58
49. Importing solar radiation from a nearby weather station (Rs) 59
50. Estimating solar radiation from temperature differences (Hargreaves’ formula) 60
51. Estimating solar radiation for island locations (Rs) 62
52. 1985 Hargreaves reference evapotranspiration equation 64
53. FAO Penman-Monteith equation for hourly time step 74
54. Actual vapour pressure for hourly time step 74
55. Deriving ETo from pan evaporation 79
56. Crop evapotranspiration (ETc) 90
57. Dual crop coefficient 98
58. Crop evapotranspiration - single crop coefficient (ETc) 103
59. Interpolation for infiltration depths between 10 and 40 mm 117
60. Adjustment of Kc ini for partial wetting by irrigation 119
61. Irrigation depth for the part of the surface that is wetted (Iw) 119
62. Climatic adjustment for Kc mid 121

Crop evapotranspiration





xxi
95. Adjustment coefficient (from LAI) 185
96. Adjustment coefficient (from fc) 185
97. K(cb mid) adj from Leaf Area Index 186
98. K(cb mid) adj from effective ground cover 187
99. Kcb full for agricultural crops 189
100. Kcb full for natural vegetation 189
101. Kcb h for full cover vegetation 189
102. Adjustment for stomatal control (Fr) 191
103. Water stress coefficient (Ks) estimated from yield response to water function 194
104. Crop coefficient estimate for intercropped field (Kc field) 199
105. Crop coefficient estimate for windbreaks (Kc) 203

xxii
List of principal symbols and acronyms



apsy coefficient of psychrometer [°C-1]
as fraction of extraterrestrial radiation reaching the earth on an overcast day [-]
as+bs fraction of extraterrestrial radiation reaching the earth on a clear day [-]
cp specific heat [MJ kg-1 °C-1]
cs soil heat capacity [MJ m-3 °C-1]
CR capillary rise [mm day-1]
De cumulative depth of evaporation (depletion) from the soil surface layer [mm]
Dr cumulative depth of evapotranspiration (depletion) from the root zone [mm]
d zero plane displacement height [m]
dr inverse relative distance Earth-Sun [-]
DP deep percolation [mm]
DPe deep percolation from the evaporation layer [mm]
E evaporation [mm day-1]
Epan pan evaporation [mm day-1]
e°(T) saturation vapour pressure at air temperature T [kPa]
es saturation vapour pressure for a given time period [kPa]
ea actual vapour pressure [kPa]
es- ea saturation vapour pressure deficit
ECe electrical conductivity of the saturation extract of the soil [dS m-1]
ECe, threshold electrical conductivity of the saturation extract of the soil above which yield
begins to decrease [dS m-1]
ET evapotranspiration [mm day-1]
ETo reference crop evapotranspiration [mm day-1]
ETc crop evapotranspiration under standard conditions [mm day-1]
ETc adj crop evapotranspiration under non-standard conditions [mm day-1]
exp[x] 2.7183 (base of natural logarithm) raised to the power x
Fr resistance correction factor [-]

Crop evapotranspiration





xxiii
fc fraction of soil surface covered by vegetation (as observed from overhead) [-]
fc eff effective fraction of soil surface covered by vegetation [-]
1-fc exposed soil fraction [-]
fw fraction of soil surface wetted by rain or irrigation [-]
few fraction of soil that is both exposed and wetted (from which most evaporation
occurs) [-]
G soil heat flux [MJ m-2 day-1]
Gday soil heat flux for day and ten-day periods [MJ m-2 day-1]
Ghr soil heat flux for hourly or shorter periods [MJ m-2 hour-1]
Gmonth soil heat flux for monthly periods [MJ m-2 day-1]
Gsc solar constant [0.0820 MJ m-2 min-1]
H sensible heat [MJ m-2 day-1]
HWR height to width ratio
h crop height [m]
I irrigation depth [mm]
Iw irrigation depth for that part of the surface wetted [mm]
J number of day in the year [-]
Kc crop coefficient [-]
Kc ini crop coefficient during the initial growth stage [-]
Kc mid crop coefficient during the mid-season growth stage [-]
Kc end crop coefficient at end of the late season growth stage [-]
Kc max maximum value of crop coefficient (following rain or irrigation) [-]
Kc min minimum value of crop coefficient (dry soil with no ground cover) [-]
Kcb basal crop coefficient [-]
Kcb full basal crop coefficient during mid-season (at peak plant size or height) for
vegetation with full ground cover of LAI > 3 [-]
Kcb ini basal crop coefficient during the initial growth stage [-]
Kcb mid basal crop coefficient during the mid-season growth stage [-]
Kcb end basal crop coefficient at end of the late season growth stage [-]
Ke soil evaporation coefficient [-]
Kp pan coefficient [-]
Kr soil evaporation reduction coefficient [-]

Crop evapotranspiration





xxv
REW readily evaporable water (i.e., maximum depth of water that can be evaporated
from the soil surface layer without restriction during stage 1) [mm]
RH relative humidity [%]
RHhr average hourly relative humidity
RHmax daily maximum relative humidity [%]
RHmean daily mean relative humidity [%]
RHmin daily minimum relative humidity [%]
RO surface runoff [mm]
Sc seasonal correction factor for solar time [hour]
SF subsurface flow [mm]
T air temperature [°C]
TK air temperature [K]
TKv virtual air temperature [K]
Tdew dewpoint temperature [°C]
Tdry temperature of dry bulb [°C]
Tmax daily maximum air temperature [°C]
Tmax,K daily maximum air temperature [K]
Tmean daily mean air temperature [°C]
Tmin daily minimum air temperature [°C]
Tmin,K daily minimum air temperature [K]
Twet temperature of wet bulb [°C]
TAW total available soil water of the root zone [mm]
TEW total evaporable water (i.e., maximum depth of water that can be evaporated from
the soil surface layer)[mm]
t time [hour]
u2 wind speed at 2 m above ground surface [m s-1]
uz wind speed at z m above ground surface [m s-1]
W soil water content [mm]
Ya actual yield of the crop [kg ha-1]
Ym maximum (expected) yield of the crop in absence of environment or water stresses
[kg ha-1]
Ze depth of surface soil layer subjected to drying by evaporation [m]

xxvi
Zr rooting depth [m]
z elevation, height above sea level [m]
zh height of humidity measurements [m]
zm height of wind measurements [m]
zom roughness length governing momentum transfer [m]
zoh roughness length governing heat and vapour transfer [m]
α albedo [-]
γ psychrometric constant [kPa °C-1]
γpsy psychrometric constant of instrument [kPa °C-1]
∆ slope of saturation vapour pressure curve [kPa °C-1]
∆SW variation in soil water content [mm]
∆t length of time interval [day]
∆z effective soil depth [m]
δ solar declination [rad]
ε ratio molecular weight of water vapour/dry air (= 0.622)
η mean angle of the sun above the horizon
θ soil water content [m3(water) m-3(soil)]
θFC soil water content at field capacity [m3(water) m-3(soil)]
θt threshold soil water content below which transpiration is reduced due to water
stress [m3(water) m-3(soil)]
θWP soil water content at wilting point [m3(water) m-3(soil)]
λ latent heat of vaporization [MJ kg-1]
λET latent heat flux [MJ m-2 day-1]
ρa mean air density [kg m-3]
ρw density of water [kg m-3]
σ Stefan-Boltzmann constant [4.903 10-9 MJ K-4 m-2 day-1]
ϕ latitude [rad]
ω solar time angle at midpoint of hourly or shorter period [rad]
ω1 solar time angle at beginning of hourly or shorter period [rad]
ω2 solar time angle at end of hourly or shorter period [rad]
ωs sunset hour angle [rad]

Introduction to evapotranspiration





2


FIGURE 1
Schematic representation of a stoma

water vapour
cuticula
epidermal
cells
mesophyll
cells
intercellular
space
Atmosphere
wate
r
Leaf
FIGURE 2
The partitioning of evapotranspiration into evaporation and transpiration over the growing period
for an annual field crop

0%
20%
40%
60%
80%
100%
time
pa
rti
tio
nin
g o
f e
va
po
tra
ns
pir
ati
on
lea
f a
re
a i
nd
ex
(L
AI
)
evaporation
transpiration
sowing harvest
crop
soil
L A I

Crop evapotranspiration





3
Transpiration
Transpiration consists of the vaporization of liquid water contained in plant tissues and the
vapour removal to the atmosphere. Crops predominately lose their water through stomata.
These are small openings on the plant leaf through which gases and water vapour pass
(Figure 1). The water, together with some nutrients, is taken up by the roots and transported
through the plant. The vaporization occurs within the leaf, namely in the intercellular spaces,
and the vapour exchange with the atmosphere is controlled by the stomatal aperture. Nearly
all water taken up is lost by transpiration and only a tiny fraction is used within the plant.

Transpiration, like direct evaporation, depends on the energy supply, vapour pressure
gradient and wind. Hence, radiation, air temperature, air humidity and wind terms should be
considered when assessing transpiration. The soil water content and the ability of the soil to
conduct water to the roots also determine the transpiration rate, as do waterlogging and soil
water salinity. The transpiration rate is also influenced by crop characteristics, environmental
aspects and cultivation practices. Different kinds of plants may have different transpiration
rates. Not only the type of crop, but also the crop development, environment and
management should be considered when assessing transpiration.

Evapotranspiration (ET)
Evaporation and transpiration occur simultaneously and there is no easy way of distinguishing
between the two processes. Apart from the water availability in the topsoil, the evaporation
from a cropped soil is mainly determined by the fraction of the solar radiation reaching the
soil surface. This fraction decreases over the growing period as the crop develops and the
crop canopy shades more and more of the ground area. When the crop is small, water is
predominately lost by soil evaporation, but once the crop is well developed and completely
covers the soil, transpiration becomes the main process. In Figure 2 the partitioning of
evapotranspiration into evaporation and transpiration is plotted in correspondence to leaf area
per unit surface of soil below it. At sowing nearly 100% of ET comes from evaporation,
while at full crop cover more than 90% of ET comes from transpiration.


UNITS
The evapotranspiration rate is normally expressed in millimetres (mm) per unit time. The rate
expresses the amount of water lost from a cropped surface in units of water depth. The time
unit can be an hour, day, decade, month or even an entire growing period or year.

As one hectare has a surface of 10 000 m2 and 1 mm is equal to 0.001 m, a loss of 1
mm of water corresponds to a loss of 10 m3 of water per hectare. In other words, 1 mm day-1
is equivalent to 10 m3 ha-1 day-1.

Water depths can also be expressed in terms of energy received per unit area. The
energy refers to the energy or heat required to vaporize free water. This energy, known as
the latent heat of vaporization (λ), is a function of the water temperature. For example, at
20°C, λ is about 2.45 MJ kg-1. In other words, 2.45 MJ are needed to vaporize 1 kg or 0.001
m3 of water. Hence, an energy input of 2.45 MJ per m2 is able to vaporize 0.001 m or 1 mm
of water, and therefore 1 mm of water is equivalent to 2.45 MJ m-2. The evapotranspiration
rate expressed in units of MJ m-2 day-1 is represented by λET, the latent heat flux.

Crop evapotranspiration





5
FACTORS AFFECTING EVAPOTRANSPIRATION

Weather parameters, crop characteristics, management and environmental aspects are factors
affecting evaporation and transpiration. The related ET concepts presented in Figure 3 are
discussed in the section on evapotranspiration concepts.

Weather parameters

The principal weather parameters affecting evapotranspiration are radiation, air temperature,
humidity and wind speed. Several procedures have been developed to assess the evaporation
rate from these parameters. The evaporation power of the atmosphere is expressed by the
reference crop evapotranspiration (ETo). The reference crop evapotranspiration represents the
evapotranspiration from a standardized vegetated surface. The ETo is described in detail later
in this Chapter and in Chapters 2 and 4.

Crop factors

The crop type, variety and development stage should be considered when assessing the
evapotranspiration from crops grown in large, well-managed fields. Differences in resistance
to transpiration, crop height, crop roughness, reflection, ground cover and crop rooting
characteristics result in different ET levels in different types of crops under identical
environmental conditions. Crop evapotranspiration under standard conditions (ETc) refers to
the evaporating demand from crops that are grown in large fields under optimum soil water,
excellent management and environmental conditions, and achieve full production under the
given climatic conditions.

Management and environmental conditions

Factors such as soil salinity, poor land fertility, limited application of fertilizers, the presence
of hard or impenetrable soil horizons, the absence of control of diseases and pests and poor
soil management may limit the crop development and reduce the evapotranspiration. Other
factors to be considered when assessing ET are ground cover, plant density and the soil water
content. The effect of soil water content on ET is conditioned primarily by the magnitude of
the water deficit and the type of soil. On the other hand, too much water will result in
waterlogging which might damage the root and limit root water uptake by inhibiting
respiration.

When assessing the ET rate, additional consideration should be given to the range of
management practices that act on the climatic and crop factors affecting the ET process.
Cultivation practices and the type of irrigation method can alter the microclimate, affect the
crop characteristics or affect the wetting of the soil and crop surface. A windbreak reduces
wind velocities and decreases the ET rate of the field directly beyond the barrier. The effect
can be significant especially in windy, warm and dry conditions although evapotranspiration
from the trees themselves may offset any reduction in the field. Soil evaporation in a young
orchard, where trees are widely spaced, can be reduced by using a well-designed drip or
trickle irrigation system. The drippers apply water directly to the soil near trees, thereby
leaving the major part of the soil surface dry, and limiting the evaporation losses. The use of
mulches, especially when the crop is small, is another way of substantially reducing soil
evaporation. Anti-transpirants, such as stomata-closing, film-forming or reflecting material,
reduce the water losses from the crop and hence the transpiration rate.

Introduction to evapotranspiration





8
TABLE 2
Average ETo for different agroclimatic regions in mm/day
Regions Mean daily temperature (°C)
Cool
~ 10°C
Moderate
20°C
Warm
> 30°C
Tropics and subtropics
- humid and sub-humid
- arid and semi-arid

2 - 3
2 - 4

3 - 5
4 - 6

5 - 7
6 - 8
Temperate region
- humid and sub-humid
- arid and semi-arid

1 - 2
1 - 3

2 - 4
4 - 7

4 - 7
6 - 9


BOX 1
Chapters concerning the calculation of the reference crop evapotranspiration (ETo)
PART A -----
Chapter 2 - FAO Penman-Monteith equation:
This chapter introduces the user to the need to standardize one method to compute ETo from
meteorological data. The FAO Penman-Monteith method is recommended as the method for
determining reference ETo. The method and the corresponding definition of the reference surface are
described.

Chapter 3 - Meteorological data:
The FAO Penman-Monteith method requires radiation, air temperature, air humidity and wind speed
data. Calculation procedures to derive climatic parameters from the meteorological data are presented.
Procedures to estimate missing meteorological variables required for calculating ETo are outlined. This
allows for estimation of ETo with the FAO Penman-Monteith method under all circumstances, even in
the case of missing climatic data.

Chapter 4 - Determination of ETo: The calculation of ETo by means of the FAO Penman-Monteith equation, with different time steps, from
the principal weather parameters and with missing data is described. The determination of ETo from
pan evaporation is also presented.

BOX 2
Chapters concerning the calculation of crop evapotranspiration under standard conditions (ETc)
PART B -----
Chapter 5 - Introduction to crop evapotranspiration:
This chapter introduces the user to the 'Kc ETo' approach for calculating crop evapotranspiration. The
effects of characteristics that distinguish field crops from the reference grass crop are integrated into
the crop coefficient Kc. Depending on the purpose of the calculation, the required accuracy, the
available climatic data and the time step with which the calculations have to be executed, a distinction
is made between two calculation methods.

Chapter 6 - ETc - Single crop coefficient (Kc): This chapter presents the first calculation method for crop evapotranspiration whereby the difference in
evapotranspiration between the cropped and reference grass surface is combined into a single crop
coefficient (Kc).

Chapter 7 - ETc - Dual crop coefficient (Kc = Kcb + Ke): This chapter presents the other calculation method for crop evapotranspiration. Kc is split into two
separate coefficients, one for crop transpiration (i.e., the basal crop coefficient Kcb) and one for soil
evaporation (Ke).

Crop evapotranspiration





9
The amount of water required to compensate the evapotranspiration loss from the
cropped field is defined as crop water requirement. Although the values for crop
evapotranspiration and crop water requirement are identical, crop water requirement refers to
the amount of water that needs to be supplied, while crop evapotranspiration refers to the
amount of water that is lost through evapotranspiration. The irrigation water requirement
basically represents the difference between the crop water requirement and effective
precipitation. The irrigation water requirement also includes additional water for leaching of
salts and to compensate for non-uniformity of water application. Calculation of the irrigation
water requirement is not covered in this publication, but will be the topic of a future Irrigation
and Drainage Paper.

Crop evapotranspiration can be calculated from climatic data and by integrating directly
the crop resistance, albedo and air resistance factors in the Penman-Monteith approach. As
there is still a considerable lack of information for different crops, the Penman-Monteith
method is used for the estimation of the standard reference crop to determine its
evapotranspiration rate, i.e., ETo. Experimentally determined ratios of ETc/ETo, called crop
coefficients (Kc), are used to relate ETc to ETo or ETc = Kc ETo.

Differences in leaf anatomy, stomatal characteristics, aerodynamic properties and even
albedo cause the crop evapotranspiration to differ from the reference crop evapotranspiration
under the same climatic conditions. Due to variations in the crop characteristics throughout its
growing season, Kc for a given crop changes from sowing till harvest. The calculation of crop
evapotranspiration under standard conditions (ETc) is discussed in Part B of this handbook
(Box 2).

Crop evapotranspiration under non-standard conditions (ETc adj)
The crop evapotranspiration under non-standard conditions (ETc adj) is the evapotranspiration
from crops grown under management and environmental conditions that differ from the
standard conditions. When cultivating crops in fields, the real crop evapotranspiration may
deviate from ETc due to non-optimal conditions such as the presence of pests and diseases,
soil salinity, low soil fertility, water shortage or waterlogging. This may result in scanty plant
growth, low plant density and may reduce the evapotranspiration rate below ETc.

The crop evapotranspiration under non-standard conditions is calculated by using a
water stress coefficient Ks and/or by adjusting Kc for all kinds of other stresses and
environmental constraints on crop evapotranspiration. The adjustment to ETc for water stress,
management and environmental constraints is discussed in Part C of this handbook (Box 3).


DETERMINING EVAPOTRANSPIRATION
ET measurement
Evapotranspiration is not easy to measure. Specific devices and accurate measurements of
various physical parameters or the soil water balance in lysimeters are required to determine
evapotranspiration. The methods are often expensive, demanding in terms of accuracy of
measurement and can only be fully exploited by well-trained research personnel. Although the
methods are inappropriate for routine measurements, they remain important for the evaluation
of ET estimates obtained by more indirect methods.

Introduction to evapotranspiration





10

BOX 3
Chapters concerning the calculation of crop evapotranspiration under non-standard conditions
(ETc adj)
PART C -----
Chapter 8 - ETc under soil water stress conditions: This chapter discusses the reduction in transpiration induced by soil moisture stress or soil water
salinity. The resulting evapotranspiration will deviate from the crop evapotranspiration under standard
conditions. The evapotranspiration is computed by using a water stress coefficient, Ks, describing the
effect of water stress on crop transpiration.

Chapter 9 - ETc for natural, non-typical and non-pristine vegetation: Procedures that can be used to make adjustments to the Kc to account for less than perfect growing
conditions or stand characteristics are discussed. The procedures can also be used to determine Kc for
agricultural crops not listed in the tables of Part B.

Chapter 10 - ETc under various management practices: This chapter discusses various types of management practices that may cause the values for Kc and
ETc to deviate from the standard conditions described in Part B. Adjustment procedures for Kc to
account for surface mulches, intercropping, small areas of vegetation and management induced stress
are presented.

Chapter 11 - ETc during non-growing periods: This chapter describes procedures for predicting ETc during non-growing periods under various types
of surface conditions.


FIGURE 5
Schematic presentation of the diurnal variation of the components of the energy balance above a
well-watered transpiring surface on a cloudless day

0 4 8 12 16 20 24
time (hour)
en
erg
y a
rriv
ing
/le
av
ing
th
e s
urf
ac
e
Rn
ET
H
G

Crop evapotranspiration





11
Energy balance and microclimatological methods
Evaporation of water requires relatively large amounts of energy, either in the form of
sensible heat or radiant energy. Therefore the evapotranspiration process is governed by
energy exchange at the vegetation surface and is limited by the amount of energy available.
Because of this limitation, it is possible to predict the evapotranspiration rate by applying the
principle of energy conservation. The energy arriving at the surface must equal the energy
leaving the surface for the same time period.

All fluxes of energy should be considered when deriving an energy balance equation.
The equation for an evaporating surface can be written as:

Rn − G - λET − H = 0 (1)

where Rn is the net radiation, H the sensible heat, G the soil heat flux and λET the latent heat
flux. The various terms can be either positive or negative. Positive Rn supplies energy to the
surface and positive G, λET and H remove energy from the surface (Figure 5).

In Equation 1 only vertical fluxes are considered and the net rate at which energy is
being transferred horizontally, by advection, is ignored. Therefore the equation is to be
applied to large, extensive surfaces of homogeneous vegetation only. The equation is
restricted to the four components: Rn, λET, H and G. Other energy terms, such as heat stored
or released in the plant, or the energy used in metabolic activities, are not considered These
terms account for only a small fraction of the daily net radiation and can be considered
negligible when compared with the other four components.

The latent heat flux (λET) representing the evapotranspiration fraction can be derived
from the energy balance equation if all other components are known. Net radiation (Rn) and
soil heat fluxes (G) can be measured or estimated from climatic parameters. Measurements of
the sensible heat (H) are however complex and cannot be easily obtained. H requires accurate
measurement of temperature gradients above the surface.

Another method of estimating evapotranspiration is the mass transfer method. This
approach considers the vertical movement of small parcels of air (eddies) above a large
homogeneous surface. The eddies transport material (water vapour) and energy (heat,
momentum) from and towards the evaporating surface. By assuming steady state conditions
and that the eddy transfer coefficients for water vapour are proportional to those for heat and
momentum, the evapotranspiration rate can be computed from the vertical gradients of air
temperature and water vapour via the Bowen ratio. Other direct measurement methods use
gradients of wind speed and water vapour. These methods and other methods such as eddy
covariance, require accurate measurement of vapour pressure, and air temperature or wind
speed at different levels above the surface. Therefore, their application is restricted to
primarily research situations.

Soil water balance
Evapotranspiration can also be determined by measuring the various components of the soil
water balance. The method consists of assessing the incoming and outgoing water flux into
the crop root zone over some time period (Figure 6). Irrigation (I) and rainfall (P) add water
to the root zone. Part of I and P might be lost by surface runoff (RO) and by deep percolation

Crop evapotranspiration






15
Part A

Reference evapotranspiration (ETo)


Part A deals with the evapotranspiration from the reference surface, the so-called reference crop
evapotranspiration or reference evapotranspiration, denoted as ETo. The reference surface is a
hypothetical grass reference crop with an assumed crop height of 0.12 m, a fixed surface resistance of
70 s m-1 and an albedo of 0.23. The reference surface closely resembles an extensive surface of green,
well-watered grass of uniform height, actively growing and completely shading the ground. The fixed
surface resistance of 70 s m-1 implies a moderately dry soil surface resulting from about a weekly
irrigation frequency.

ETo can be computed from meteorological data. As a result of an Expert Consultation held in
May 1990, the FAO Penman-Monteith method is now recommended as the sole standard method for
the definition and computation of the reference evapotranspiration. The FAO Penman-Monteith
method requires radiation, air temperature, air humidity and wind speed data. Calculation procedures
to derive climatic parameters from meteorological data and to estimate missing meteorological
variables required for calculating ETo are presented in this Part (Chapter 3). The calculation
procedures in this Publication allow for estimation of ETo with the FAO Penman-Monteith method
under all circumstances, even in the case of missing climatic data.

ETo can also be estimated from pan evaporation. Pans have proved their practical value and
have been used successfully to estimate ETo by observing the water loss from the pan and using
empirical coefficients to relate pan evaporation to ETo. However, special precautions and
management must be applied.

Crop evapotranspiration





17
Chapter 2

FAO Penman-Monteith equation




This chapter introduces the user to the need to standardize one method to compute reference
evapotranspiration (ETo) from meteorological data. The FAO Penman-Monteith method is
recommended as the sole ETo method for determining reference evapotranspiration. The
method, its derivation, the required meteorological data and the corresponding definition of
the reference surface are described in this chapter.


NEED FOR A STANDARD ETO METHOD

A large number of more or less empirical methods have been developed over the last 50 years
by numerous scientists and specialists worldwide to estimate evapotranspiration from different
climatic variables. Relationships were often subject to rigorous local calibrations and proved
to have limited global validity. Testing the accuracy of the methods under a new set of
conditions is laborious, time-consuming and costly, and yet evapotranspiration data are
frequently needed at short notice for project planning or irrigation scheduling design. To meet
this need, guidelines were developed and published in the FAO Irrigation and Drainage Paper
No. 24 'Crop water requirements'. To accommodate users with different data availability,
four methods were presented to calculate the reference crop evapotranspiration (ETo): the
Blaney-Criddle, radiation, modified Penman and pan evaporation methods. The modified
Penman method was considered to offer the best results with minimum possible error in
relation to a living grass reference crop. It was expected that the pan method would give
acceptable estimates, depending on the location of the pan. The radiation method was
suggested for areas where available climatic data include measured air temperature and
sunshine, cloudiness or radiation, but not measured wind speed and air humidity. Finally, the
publication proposed the use of the Blaney-Criddle method for areas where available climatic
data cover air temperature data only.

These climatic methods to calculate ETo were all calibrated for ten-day or monthly
calculations, not for daily or hourly calculations. The Blaney-Criddle method was
recommended for periods of one month or longer. For the pan method it was suggested that
calculations should be done for periods of ten days or longer. Users have not always
respected these conditions and calculations have often been done on daily time steps.

Advances in research and the more accurate assessment of crop water use have revealed
weaknesses in the methodologies. Numerous researchers analysed the performance of the
four methods for different locations. Although the results of such analyses could have been
influenced by site or measurement conditions or by bias in weather data collection, it became
evident that the proposed methods do not behave the same way in different locations around
the world. Deviations from computed to observed values were often found to exceed ranges
indicated by FAO. The modified Penman was frequently found to overestimate ETo, even by

FAO Penman-Monteith equation





18
up to 20% for low evaporative conditions. The other FAO recommended equations showed
variable adherence to the reference crop evapotranspiration standard of grass.

To evaluate the performance of these and other estimation procedures under different
climatological conditions, a major study was undertaken under the auspices of the Committee
on Irrigation Water Requirements of the American Society of Civil Engineers (ASCE). The
ASCE study analysed the performance of 20 different methods, using detailed procedures to
assess the validity of the methods compared to a set of carefully screened lysimeter data from
11 locations with variable climatic conditions. The study proved very revealing and showed
the widely varying performance of the methods under different climatic conditions. In a
parallel study commissioned by the European Community, a consortium of European research
institutes evaluated the performance of various evapotranspiration methods using data from
different lysimeter studies in Europe.

The studies confirm the overestimation of the modified Penman introduced in FAO
Irrigation and Drainage Paper No. 24, and the variable performance of the different methods
depending on their adaptation to local conditions. The comparative studies may be
summarized as follows:

• The Penman methods may require local calibration of the wind function to achieve
satisfactory results.
• The radiation methods show good results in humid climates where the aerodynamic term
is relatively small, but performance in arid conditions is erratic and tends to
underestimate evapotranspiration.
• Temperature methods remain empirical and require local calibration in order to achieve
satisfactory results. A possible exception is the 1985 Hargreaves’ method which has
shown reasonable ETo results with a global validity.
• Pan evapotranspiration methods clearly reflect the shortcomings of predicting crop
evapotranspiration from open water evaporation. The methods are susceptible to the
microclimatic conditions under which the pans are operating and the rigour of station
maintenance. Their performance proves erratic.
• The relatively accurate and consistent performance of the Penman-Monteith approach in
both arid and humid climates has been indicated in both the ASCE and European studies.

The analysis of the performance of the various calculation methods reveals the need for
formulating a standard method for the computation of ETo. The FAO Penman-Monteith
method is recommended as the sole standard method. It is a method with strong likelihood of
correctly predicting ETo in a wide range of locations and climates and has provision for
application in data-short situations. The use of older FAO or other reference ET methods is
no longer encouraged.


FORMULATION OF THE PENMAN-MONTEITH EQUATION

Penman-Monteith equation

In 1948, Penman combined the energy balance with the mass transfer method and derived an
equation to compute the evaporation from an open water surface from standard climatological
records of sunshine, temperature, humidity and wind speed. This so-called combination

Crop evapotranspiration





19
method was further developed by many researchers and extended to cropped surfaces by
introducing resistance factors.

The resistance nomenclature distinguishes between aerodynamic resistance and surface
resistance factors (Figure 7). The surface resistance parameters are often combined into one
parameter, the ‘bulk’ surface resistance parameter which operates in series with the
aerodynamic resistance. The surface resistance, rs, describes the resistance of vapour flow
through stomata openings, total leaf area and soil surface. The aerodynamic resistance, ra,
describes the resistance from the vegetation upward and involves friction from air flowing
over vegetative surfaces. Although the exchange process in a vegetation layer is too complex
to be fully described by the two resistance factors, good correlations can be obtained between
measured and calculated evapotranspiration rates, especially for a uniform grass reference
surface.


The Penman-Monteith form of the combination equation is:












+γ+∆
−ρ+−∆
=λ
a
s
a
aspan
r
r1
r
)e(ecG)(R
ET (3)

where Rn is the net radiation, G is the soil heat flux, (es - ea) represents the vapour pressure
deficit of the air, ρa is the mean air density at constant pressure, cp is the specific heat of the
air, ∆ represents the slope of the saturation vapour pressure temperature relationship, γ is the
FIGURE 7
Simplified representation of the (bulk) surface and aerodynamic resistances for water vapour flow

soil
stomatal
air
flow
r
r
s
a
(bulk) surface
resistance
aerodynamic
resistance
reference
level
evaporating
surface
cuticular

Crop evapotranspiration





21
BOX 4
The aerodynamic resistance for a grass reference surface
For a wide range of crops the zero plane displacement height, d [m], and the roughness length
governing momentum transfer, zom [m], can be estimated from the crop height h [m] by the
following equations:
d = 2/3 h
zom = 0.123 h

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

zoh = 0.1 zom

Assuming a constant crop height of 0.12 m and a standardized height for wind speed,
temperature and humidity at 2 m (zm = zh = 2 m), the aerodynamic resistance ra [s m-1] for the
grass reference surface becomes (Eq. 4):

222
a u
208
u(0.41)
(0.12)0.123(0.1)
(0.12)2/3-2ln(0.12)0.123
(0.12)2/3-2ln
r =














=

where u2 is the wind speed [m s-1] at 2 m.



active
1s LAI
rr = (5)

where rs (bulk) surface resistance [s m-1],
rl bulk stomatal resistance of the well-illuminated leaf [s m-1],
LAIactive active (sunlit) leaf area index [m2 (leaf area) m-2 (soil
surface)].

The Leaf Area Index (LAI), a dimensionless quantity, is the leaf area (upper side only)
per unit area of soil below it. It is expressed as m2 leaf area per m2 ground area. The active
LAI is the index of the leaf area that actively contributes to the surface heat and vapour
transfer. It is generally the upper, sunlit portion of a dense canopy. The LAI values for
various crops differ widely but values of 3-5 are common for many mature crops. For a given
crop, green LAI changes throughout the season and normally reaches its maximum before or
at flowering (Figure 8). LAI further depends on the plant density and the crop variety.

The bulk stomatal resistance, rl, is the average resistance of an individual leaf. This
resistance is crop specific and differs among crop varieties and crop management. It usually
increases as the crop ages and begins to ripen. There is, however, a lack of consolidated
information on changes in rl over time for the different crops. The information available in
the literature on stomatal conductance or resistance is often oriented toward physiological or
ecophysiological studies.

The stomatal resistance, rl, is influenced by climate and by water availability. However,
influences vary from one crop to another and different varieties can be affected differently.
The resistance increases when the crop is water stressed and the soil water availability limits

Crop evapotranspiration





23
REFERENCE SURFACE
To obviate the need to define unique evaporation parameters for each crop and stage of
growth, the concept of a reference surface was introduced. Evapotranspiration rates of the
various crops are related to the evapotranspiration rate from the reference surface (ETo) by
means of crop coefficients.

In the past, an open water surface has been proposed as a reference surface. However,
the differences in aerodynamic, vegetation control and radiation characteristics present a
strong challenge in relating ET to measurements of free water evaporation. Relating ETo to a
specific crop has the advantage of incorporating the biological and physical processes
involved in ET from cropped surfaces.

Grass, together with alfalfa, is a well-studied crop regarding its aerodynamic and surface
characteristics and is accepted worldwide as a reference surface. Because the resistance to
diffusion of vapour strongly depends on crop height, ground cover, LAI and soil moisture
conditions, the characteristics of the reference crop should be well defined and fixed.
Changes in crop height result in variations in the roughness and LAI. Consequently, the
associated canopy and aerodynamic resistances will vary appreciably with time. Moreover,
water stress and the degree of ground cover have an effect on the resistances and also on the
albedo.

To avoid problems of local calibration which would require demanding and expensive
studies, a hypothetical grass reference has been selected. Difficulties with a living grass
reference result from the fact that the grass variety and morphology can significantly affect
the evapotranspiration rate, especially during peak water use. Large differences may exist
between warm-season and cool-season grass types. Cool-season grasses have a lower degree
of stomatal control and hence higher rates of evapotranspiration. It may be difficult to grow
cool-season grasses in some arid, tropical climates.
The FAO Expert Consultation on Revision of FAO Methodologies for Crop Water
Requirements accepted the following unambiguous definition for the reference surface:
"A hypothetical reference crop with an assumed crop height of 0.12 m, a fixed
surface resistance of 70 s m-1 and an albedo of 0.23."
The reference surface closely resembles an extensive surface of green grass of uniform
height, actively growing, completely shading the ground and with adequate water. The
requirements that the grass surface should be extensive and uniform result from the
assumption that all fluxes are one-dimensional upwards.
The FAO Penman-Monteith method is selected as the method by which the
evapotranspiration of this reference surface (ETo) can be unambiguously determined, and as
the method which provides consistent ETo values in all regions and climates.

FAO PENMAN-MONTEITH EQUATION
Equation
A consultation of experts and researchers was organized by FAO in May 1990, in
collaboration with the International Commission for Irrigation and Drainage and with the
World Meteorological Organization, to review the FAO methodologies on crop water
requirements and to advise on the revision and update of procedures.

Crop evapotranspiration





25
The reference evapotranspiration, ETo, provides a standard to which:
• evapotranspiration at different periods of the year or in other regions can be compared;
• evapotranspiration of other crops can be related.

The equation uses standard climatological records of solar radiation (sunshine), air
temperature, humidity and wind speed. To ensure the integrity of computations, the weather
measurements should be made at 2 m (or converted to that height) above an extensive surface
of green grass, shading the ground and not short of water.

No weather-based evapotranspiration equation can be expected to predict
evapotranspiration perfectly under every climatic situation due to simplification in formulation
and errors in data measurement. It is probable that precision instruments under excellent
environmental and biological management conditions will show the FAO Penman-Monteith
equation to deviate at times from true measurements of grass ETo. However, the Expert
Consultation agreed to use the hypothetical reference definition of the FAO Penman-Monteith
equation as the definition for grass ETo when deriving and expressing crop coefficients.

It is important, when comparing the FAO Penman-Monteith equation to ETo
measurements, that the full Penman-Monteith equation (Equation 3) and associated equations
for ra and rs (Equations 4 and 5) be used to enable accounting for variation in ET due to
variation in height of the grass measured. Variations in measurement height can significantly
change LAI, d and zom and the corresponding ETo measurement and predicted value. When
evaluating results, it should be noted that local environmental and management factors, such
as watering frequency, also affect ETo observations.

The FAO Penman-Monteith equation is a close, simple representation of the physical and
physiological factors governing the evapotranspiration process. By using the FAO Penman-
Monteith definition for ETo, one may calculate crop coefficients at research sites by relating
the measured crop evapotranspiration (ETc) with the calculated ETo, i.e., Kc = ETc/ETo. In
the crop coefficient approach, differences in the crop canopy and aerodynamic resistance
relative to the hypothetical reference crop are accounted for within the crop coefficient. The
Kc factor serves as an aggregation of the physical and physiological differences between crops
and the reference definition.

Data
Apart from the site location, the FAO Penman-Monteith equation requires air temperature,
humidity, radiation and wind speed data for daily, weekly, ten-day or monthly calculations.
The computation of all data required for the calculation of the reference evapotranspiration is
given in Chapter 3. It is important to verify the units in which the weather data are reported.
Factors to convert common units to the standard unit are presented in Annex I.

Location
Altitude above sea level (m) and latitude (degrees north or south) of the location should be
specified. These data are needed to adjust some weather parameters for the local average
value of atmospheric pressure (a function of the site elevation above mean sea level) and to
compute extraterrestrial radiation (Ra) and, in some cases, daylight hours (N). In the
calculation procedures for Ra and N, the latitude is expressed in radian (i.e., decimal degrees
times π/180).

Crop evapotranspiration





31
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 PARAMETERS
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.

Atmospheric pressure (P)

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:


26.5
293
z0065.02933.101P







−
= (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).

Latent heat of vaporization (λ)

The latent heat of vaporization, λ, 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 λ varies only slightly over normal temperature ranges a
single value of 2.45 MJ kg-1 is taken in the simplification of the FAO Penman-Monteith
equation. This is the latent heat for an air temperature of about 20°C.

Psychrometric constant (γ)

The psychrometric constant, γ, is given by:

Meteorological data





32
P10x665.0Pc 3p −=
λε
=γ (8)

where γ psychrometric constant [kPa °C-1],
P atmospheric pressure [kPa],
λ latent heat of vaporization, 2.45 [MJ kg-1],
cp specific heat at constant pressure, 1.013 10-3 [MJ kg-1 °C-1],
ε 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 cp =
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 1 800 m.

With:
From Eq. 7:
From Eq. 8:
z =
P = 101.3 [(293 - (0.0065) 1800)/293]5.26 =
γ = 0.665 10-3 (81.8) =
1 800
81.8
0.054
m
kPa
kPa °C-1
The average value of the atmospheric pressure is 81.8 kPa.
The psychrometric constant, γ, is 0.054 kPa/°C.



AIR TEMPERATURE

Agrometeorology is concerned with the air temperature near the level of the crop canopy. In
traditional and modern 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 24-hour
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 24-hour maximum and minimum values.

Due to the non-linearity of humidity data required in the FAO Penman-Monteith
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 (Tmax) and daily minimum air temperature (Tmin) are, respectively,
the maximum and minimum air temperature observed during the 24-hour period, beginning at
midnight. Tmax and Tmin for longer periods such as weeks, 10-day's or months are obtained by
dividing the sum of the respective daily values by the number of days in the period. The mean

Crop evapotranspiration





35
The actual vapour pressure (ea ) 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
(ea) to the saturation (eo(T)) vapour pressure at the same temperature (T):


(T)oe
ae100RH = (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.

Measurement

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 so-called 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.

Meteorological data





36
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 modern 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.

Calculation procedures
Mean saturation vapour pressure (es )
As saturation vapour pressure is related to air temperature, it can be calculated from the air
temperature. The relationship is expressed by:









+
= 3.237T
T27.17exp6108.0)T(eo (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 non-linearity 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:

( ) ( )2
TeTee min
omaxos
+
= (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.
Determine the saturation vapour pressure for that day.
From Eq. 11 e°(Tmax) = 0.6108 exp[17.27(24.5)/(24.5+237.3)] 3.075 kPa From Eq. 11 e°(Tmin) = 0.6108 exp[17.27(15)/(15+237.3)] 1.705 kPa From Eq. 12 es = (3.075 + 1.705)/2
Note that for temperature 19.75°C (which is Tmean), e°(T) =
2.39

2.30
kPa

kPa
The mean saturation vapour pressure is 2.39 kPa.

Meteorological data





38
where apsy is a coefficient depending on the type of ventilation of the wet bulb [°C-1], and P is
the atmospheric pressure [kPa]. The coefficient apsy depends mainly on the design of the
psychrometer and rate of ventilation around the wet bulb. The following values are used:

apsy = 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 non-ventilated 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 1 200 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:
Then:
z=
P =
1 200
87.9
m
kPa
From Eq. 11 (Table 2.3), for
Then:
Twet = e°(Twet) =
19.5
2.267
°C
kPa
Ventilated psychrometer apsy = 0.000662 °C-1
From Eq. 15:

ea = 2.267 - 0.000662 (87.9) (25.6 - 19.5)=

1.91

kPa

The actual vapour pressure is 1.91 kPa.


Actual vapour pressure (ea ) 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 RHmax and RHmin:

( ) ( )
2
100
RHTe100
RHTe
e
minmaxomaxmino
a
+
= (17)

where ea actual vapour pressure [kPa],
e°(Tmin) saturation vapour pressure at daily minimum temperature [kPa],
e°(Tmax) saturation vapour pressure at daily maximum temperature [kPa],
RHmax maximum relative humidity [%],
RHmin minimum relative humidity [%].

For periods of a week, ten days or a month, RHmax and RHmin are obtained by dividing the
sum of the daily values by the number of days in that period.

• For RHmax:
When using equipment where errors in estimating RHmin can be large, or when RH data
integrity are in doubt, then one should use only RHmax:

Meteorological data





40

BOX 7
Calculation sheet for vapour pressure deficit (es - ea)
Saturation vapour pressure: es (Eq. 11 or Table 2.3)

Tmax



°C








+
= 3.237T
T27.17exp6108.0)T(e
max
maxmaxo




kPa

Tmin



°C








+
= 3.237T
T27.17exp6108.0)T(e
min
minmino




kPa

saturation vapour pressure es = [e°(Tmax)+e°(Tmin)]/2 Eq.12



kPa

Actual vapour pressure: ea

1. ea derived from dewpoint temperature (Eq. 14 or Table 2.3)

Tdew



°C








+
= 3.237T
T27.17exp6108.0e
dew
dewa




kPa

OR 2. ea derived from maximum and minimum relative humidity

RHmax



% ( ) 100
RHTe maxmino



kPa

RHmin



% ( ) 100
RHTe minmaxo



kPa

ea = [e°(Tmin) RHmax/100 + e°(Tmax) RHmin/100] / 2 Eq. 17



kPa

OR 3. ea derived from maximum relative humidity (errors in RHmin)

RHmax



%

ea = e°(Tmin) RHmax/100 Eq. 18



kPa

OR 4. ea derived from mean relative humidity (less recommended)

RHmean



%

ea = es (RHmean)/100 Eq. 19



kPa

Vapour pressure deficit: (es - ea)



kPa

Crop evapotranspiration





41
RADIATION
Concepts
Extraterrestrial radiation (Ra )
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, Ra.

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, Ra change as well. Extraterrestrial radiation is thus a function of latitude, date
and time of day. Daily values of Ra throughout the year for different latitudes are plotted in
Figure 13.

Solar or shortwave radiation (Rs )
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, Rs. Because the sun emits energy by means of
electromagnetic waves characterized by short wavelengths, solar radiation is also referred to as
shortwave radiation.

FIGURE 13
Annual variation in extraterrestrial radiation (Ra) at the equator, 20 and 40° north and south
0
10
20
30
40
50
R a
(
MJ
m
-2
d
ay
-1 )
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
40°S
20°S
20°N
40°N
Equator

Meteorological data





42
For a cloudless day, Rs 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 (Rs /Rso )
The relative shortwave radiation is the ratio of the solar radiation (Rs) to the clear-sky solar
radiation (Rso). Rs is the solar radiation that actually reaches the earth's surface in a given
period, while Rso 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 Rn, 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 Rs, the relative
sunshine duration, n/N, is often used to derive solar radiation from extraterrestrial radiation.
As with extraterrestrial radiation, the daylength 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.
FIGURE 14
Annual variation of the daylight hours (N) at the equator, 20 and 40° north and south
40 S
20 S
20 N
40 N
o
o
o
o
8
9
10
11
12
13
14
15
16
N
da
yli
gh
t h
ou
rs

(ho
urs
/da
y)
Equator
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Crop evapotranspiration





43
Albedo (α) and net solar radiation (Rns)
A considerable amount of solar radiation reaching the earth's surface is reflected. The fraction,
α, 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 0.20-0.25. For the green grass reference crop, α is
assumed to have a value of 0.23.

The net solar radiation, Rns, is the fraction of the solar radiation Rs that is not reflected
from the surface. Its value is (1-α)Rs.

Net longwave radiation (Rnl)
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 (Rl,up) is absorbed by the atmosphere or is lost into space. The longwave
radiation received by the atmosphere (Rl,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, Rnl. As the outgoing longwave radiation is
almost always greater than the incoming longwave radiation, Rnl represents an energy loss.

Net radiation (Rn)
The net radiation, Rn, 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 (Rns) and the net
outgoing longwave (Rnl) radiation (Figure 15). Rn is normally positive during the daytime and
negative during the nighttime. The total daily value for Rn 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 Rn 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 Rn when estimating
evapotranspiration.

Units
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.

Crop evapotranspiration





49
EXAMPLE 9
Determination of daylight hours

Determine the daylight hours (N) for 3 September at 20°S.
From Example 8: ωs = arccos[-tan(-0.35)tan(0.120)] = 1.527 rad
From Eq. 34: N = 24/π (1.527) = 11.7 hour
The number of daylight hours is 11.7 hours.

BOX 9
Calculation sheet for extraterrestrial radiation (Ra) and daylight hours (N)
Latitude Degrees and minutes are
+ positive for northern hemisphere
- negative for southern hemisphere

Degrees



°

---------------------------------------->



°

Minutes



'

-−-------------- / 60 -------------->



°

Decimal degrees = Sum(degrees+minutes/60)



°

ϕ = π/180 * [decimal degrees] Eq. 22



rad
Day of the year


Day









Month





J Table 2.5 (Annex 2)





dr = 1+0.033 cos(2πJ/365) Eq. 23





δ = 0.409 sin(2πJ/365 - 1.39) Eq. 24



rad

sin(ϕ)sin(δ)





cos(ϕ)cos(δ)





ωs = arccos[-tan(ϕ)tan(δ)] Eq. 25



rad

(24 (60)/π) Gsc

37.59

MJ m-2day-1
Extraterrestrial radiation: Ra

[ ])(sin)(cos)(cos)(sin)(sindG)60(24R ssrsca ωδϕ+δϕω
π
=
Eq.21



MJ m-2day-1
Daylight hours: N


s
24N ω
π
= Eq. 34



hour/day

Meteorological data





54

Soil heat flux (G)
Complex models are available to describe soil heat flux. Because soil heat flux is small
compared to Rn, 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:

where G soil heat flux [MJ m-2 day-1],
cs soil heat capacity [MJ m-3 °C-1],
Ti air temperature at time i [°C],
Ti-1 air temperature at time i-1 [°C],
∆t length of time interval [day],
∆z effective soil depth [m].

As the soil temperature lags air temperature, the average temperature for a period should
be considered when assessing the daily soil heat flux, i.e., ∆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, ∆z, is only 0.10-0.20 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 ten-day periods:
As the magnitude of the day or ten-day soil heat flux beneath the grass reference surface is
relatively small, it may be ignored and thus:

• 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:

or, if Tmonth,i+1 is unknown:

where Tmonth,i mean air temperature of month i [°C],
Tmonth,i-1 mean air temperature of previous month [°C],
Tmonth,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:
zt
TTcG 1iis ∆∆
+
=
− (41)
0Gday ≈ (42)
)TT(07.0G 1i,month1i,monthi,month −+ −= (43)
)TT(14.0G 1i,monthi,monthi,month −−= (44)

Crop evapotranspiration





55

and during nighttime periods as:

Where the soil is warming, the soil heat flux G is positive. The amount of energy
required for this process is subtracted from Rn 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:
Gmonth = 0.07 (18.8 - 14.1) =

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 SPEED

Measurement

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 profile relationship

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 2 m, a logarithmic wind speed
profile may be used for measurements above a short grassed surface:

nhr R1.0G = (45)
nhr R5.0G = (46)

Meteorological data





56

where u2 wind speed at 2 m above ground surface [m s-1],
uz 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.



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:
And:
Then:
uz = z =
Conversion factor = 4.87 / ln(67.8 (10) - 5.42) =
3.2
10
0.75
m/s
m
-
From Eq. 47: u2 = 3.2 (0.75) = 2.4 m/s
The wind speed at 2 m above the soil surface is 2.4 m/s.

)42.5z8.67(ln
87.4uu z2
−
= (47)
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)
0
1
2
3
4
5
6
7
8
9
10
0.7 0.8 0.9 1 1.1 1.2 1.3
conversion factor
me
as
ure
d h
eig
ht
ab
ov
e g
rou
nd
su
rfa
ce
(m
)

Crop evapotranspiration





57
CLIMATIC DATA ACQUISITION

Weather stations

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 non-reference sites for computing ETo.

Agroclimatic monthly databases

Starting in 1984, FAO has published mean monthly agroclimatic data from 2 300 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 (A-J) (1987),
Volume 2: Agroclimatic data for Asia (K-Z) (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 ETo calculated with the FAO Penman-Monteith method are listed on the diskettes
for mean long-term conditions.

Crop evapotranspiration





63
The method is only appropriate for monthly calculations. The constant relation between Rs
and Ra does not yield accurate daily estimates.

Missing wind speed data
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
several-day 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
light to moderate wind
moderate to strong wind
strong wind

... ≤ 1.0 m/s
1 – 3 m/s
3 – 5 m/s
... ≥ 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 2 000 weather stations around the globe.

In general, wind speed at 2 m, u2, should be limited to about u2 ≥ 0.5 m/s when used in
the ETo equation (Equation 6). This is necessary to account for the effects of boundary layer
instability and bouyancy 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 u2 ≥ 0.5 m/s in the ETo equation improves the estimation
accuracy under the conditions of very low wind speed.

Meteorological data





64
Minimum data requirements

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 Penman-
Monteith method.

An alternative equation for ETo when weather data are missing
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, ETo can
be estimated using the Hargreaves ETo equation where:

a5.0minmaxmeano R)TT()8.17T(0023.0ET −+= (52)

where all parameters have been previously defined. Units for both ETo and Ra in Equation 52
are mm day-1. Equation 52 should be verified in each new region by comparing with estimates
by the FAO Penman-Monteith 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 ETo = a +
b ETo Eq.52, where the “Eq. 52” subscript refers to ETo 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 Penman-Monteith
equation) will provide somewhat more accurate estimates as compared to estimating ETo
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 ETo estimate made using Equation 6.

Equation 52 has a tendency to underpredict under high wind conditions (u2 > 3 m/s) and
to overpredict under conditions of high relative humidity.

Crop evapotranspiration





67

BOX 11
Calculation sheet for ETo (FAO Penman-Monteith) using meteorological tables of Annex 2
Parameters


Tmax



°C



Tmin



°C

Tmean = (Tmax+Tmin)/2



°C

Tmean



°C

∆ (Table 2.4 of Annex 2)



kPa/°C

Altitude



m

γ (Table 2.2 of Annex 2)



kPa/°C

u2



m/s

(1 + 0.34 u2)




∆ / [ ∆ + γ (1 + 0.34 u2)]
γ / [ ∆ + γ (1 + 0.34 u2)]
[ 900 / (Tmean + 273) ] u2

Vapour pressure deficit


Tmax



°C

e°(Tmax) (Table 2.3)



kPa

Tmin



°C

e°(Tmin) (Table 2.3)



kPa

Saturation vapour pressure es = [(e°(Tmax) + e°(Tmin)]/2



kPa

ea derived from dewpoint temperature:

Tdew



°C

ea = e°(Tdew) (Table 2.3)



kPa
OR ea derived from maximum and minimum relative humidity:

RHmax



%

e°(Tmin) RHmax/100



kPa

RHmin



%

e°(Tmax) RHmin/100



kPa



ea: (average)



kPa

OR ea derived from maximum relative humidity: (recommended if there are errors in RHmin)

RHmax



%

ea = e°(Tmin) RHmax/100



kPa
OR ea derived from mean relative humidity: (less recommended due to non-linearities)

RHmean



%

ea = es RHmean/100



kPa
Vapour pressure deficit (es - ea) kPa

Determination of ETo





68


Radiation

Latitude °
Day Ra (Table 2.6) MJ m-2 d-1
Month N (Table 2.7) hours
n hours n/N
If no Rs data available: Rs = (0.25 + 0.50 n/N) Ra MJ m-2 d-1
Rso = [0.75 + 2 (Altitude)/ 100 000] Ra MJ m-2 d-1
Rs / Rso
Rns = 0.77 Rs MJ m-2 d-1
Tmax σ Tmax,K4 (Table 2.8) MJ m-2 d-1
Tmin σ Tmin,K4 (Table 2.8) MJ m-2 d-1
(σTmax,K4 + σTmin,K4)/2 MJ m-2 d-1
ea kPa (0.34 - 0.14 √ea)
Rs/Rso (1.35 Rs/Rso - 0.35)
Rnl = (σTmax,K4 + σTmin,K4)/2 (0.34 - 0.14 √ea) (1.35 Rs/Rso - 0.35) MJ m-2 d-1
Rn = Rns - Rnl MJ m-2 d-1
Tmonth °C Gday (assume) 0 MJ m-2 d-1
Tmonth-1 °C Gmonth = 0.14 (Tmonth - Tmonth-1) MJ m-2 d-1
Rn – G MJ m-2 d-1
0.408 (Rn - G) mm/day

Grass reference evapotranspiration

[ ])GR(408.0)u34.01( n2 −






+γ+∆
∆
mm/day
[ ])ee(u273T
900
)u34.01( as22
−







+







+γ+∆
γ
mm/day
)u34.01(
)ee(u273T
900)GR(408.0
ET
2
as2n
o
+γ+∆
−
+
γ+−∆
=



mm/day

Determination of ETo





74
Hourly time step

In areas where substantial changes in wind speed, dewpoint or cloudiness occur during the day,
calculation of the ETo equation using hourly time steps is generally better than using 24-hour
calculation time steps. Such weather changes can cause 24-hour means to misrepresent
evaporative power of the environment during parts of the day and may introduce error into the
calculations. However, under most conditions, application of the FAO Penman-Monteith
equation with 24-hour data produces accurate results.

With the advent of electronic, automated weather stations, weather data are increasingly
reported for hourly or shorter periods. Therefore, in situations where calculations are
computerized, the FAO Penman-Monteith equation can be applied on an hourly basis with good
results. When applying the FAO Penman-Monteith equation on an hourly or shorter time scale,
the equation and some of the procedures for calculating meteorological data should be adjusted
for the smaller time step. The FAO Penman-Monteith equation for hourly time steps is:

)u34.01(
)e)T(e(u273T
37)GR(408.0
2
ahro2
hr
n
oET +γ+∆
−
+
γ+−∆
= (53)

where ETo reference evapotranspiration [mm hour-1],
Rn net radiation at the grass surface [MJ m-2 hour-1] (Equation 40),
G soil heat flux density [MJ m-2 hour-1] (Equations 45 and 46),
Thr mean hourly air temperature [°C],
∆ saturation slope vapour pressure curve at Thr [kPa °C-1] (Equation 13),
γ psychrometric constant [kPa °C-1] (Equation 8),
e°(Thr) saturation vapour pressure at air temperature Thr [kPa] (Equation 11),
ea average hourly actual vapour pressure [kPa] (Equation 54),
u2 average hourly wind speed [m s-1].

Given relative humidity measurements, the actual vapour pressure is determined as:

where ea average hourly actual vapour pressure [kPa],
e°(Thr) saturation vapour pressure at air temperature Thr [kPa] (Equation 11),
RHhr average hourly relative humidity [%].

The net radiation is the difference between the net shortwave radiation (Rns) and the net
longwave radiation (Rnl) at the hourly time steps. Consequently:

• If Rns and Rnl need to be calculated, the extraterrestrial radiation value (Ra) for the
hourly period (Equation 28) should be used.
• In the computation of Rnl by means of Equation 39, (σTmax,K4 + σTmin,K4)/2 is
replaced by σThr,K4 and the Stefan-Boltzman constant becomes:
σ = (4.903/24) 10-9 = 2.043 10-10 MJ m-2 hour-1.

100
RH)T(ee hrhroa = (54)