Calculation approach

Mid-season
stage - Adjustment for sparse vegetation

Mid-season
stage - Adjustment for stomatal control

Late season stage

Estimating ET_{c
adj} using crop yields

Non-typical refers to types or arrangements of agricultural crops that are not listed or described in Tables 12 and 17. Non-pristine vegetation is defined, in the usage here, as vegetation having less than perfect growing conditions or stand characteristics (i.e., relatively poorer conditions of density, height, leaf area, fertility, or vitality) as compared to 'pristine' conditions.

The approach whereby a crop is characterized by a crop coefficient, K_{c}, and the crop evapotranspiration is given by the product of K_{c} and the reference evapotranspiration ET_{o}, provides a simple and convenient way of also characterizing the evapotranspiration from natural vegetation and for non-typical cultivation practices. This chapter presents procedures for estimating K_{c} values for natural vegetation and for agricultural vegetation for which K_{c} values are not available.

Initial growth stage

Mid and late season stages

Water stress conditions

As described in Figure 27, the first step in the K_{c}ET_{o} approach is the estimation of lengths of growth stages. This also applies to natural and other vegetation. The next step is the development of crop coefficient curves that represent the ratios of ET_{c} to ET_{o} during the various growth stages of the vegetation.

The procedure to estimate crop coefficients for the initial growth stage for natural, non-typical and non-pristine vegetation is identical to that described in Chapter 6 (single crop coefficient K_{c ini}) or Chapter 7 (dual crop coefficient, K_{cb ini} + K_{e}). The crop coefficient in this stage is primarily determined by the frequency with which the soil is wetted.

The K_{c} during the mid-season period (K_{c mid} and K_{cb mid}) and to a lesser extent the K_{c} during the late season period differ from that described in previous chapters. As the ground cover for natural and non-pristine vegetation is often reduced, the K_{c} is affected to a large extent by the frequency of precipitation and/or irrigation and by the amount of leaf area and ground cover.

*Dual crop coefficient approach*

The determination of K_{c} for natural, non-typical or non-pristine vegetation should ordinarily follow the approach described in Chapter 7 whereby separate transpiration (K_{cb}) and evaporation (K_{e}) coefficients are used. The effects of evaporation from the soil surface can be directly estimated as such.

Two procedures that can be used to adjust the basal crop coefficient (K_{cb mid adj}) for sparse vegetation are presented in this section. In these approaches, K_{cb mid adj} is estimated either from LAI (Equation 97) or from effective ground cover (Equation 98). After the determination of K_{cb mid adj}, the soil evaporation coefficient, K_{e}, should be determined to obtain the crop coefficient for the mid-season stage: K_{c mid adj} = K_{cb mid adj} + K_{e}. Procedures for calculating K_{e} are presented in Chapter 7.

Even where the estimated K_{cb mid adj} is small, the total K_{c adj} (= K_{cb adj} + K_{e}) following precipitation may sometimes be as high or higher than the K_{c} for pristine vegetation due to surface evaporation from among sparse vegetation.

*Single crop coefficient approach*

When the single crop coefficient K_{c} of Chapter 6 is used, the average effects of soil wetting are incorporated into a general mean K_{c}. Some guidelines for the estimation of K_{c adj} are given in the following sections. The single crop coefficient can also be derived from the adjusted K_{cb} by considering the frequency of soil wetting, i.e., during the midseason period, K_{c adj} = K_{cb adj} + 0.05 for infrequent wetting and K_{cb adj} + 0.10 for wettings of up to once a week. For more frequent wettings, the dual crop coefficient approach should be used.

Alternatively, Equations 97 and 98 can be used to determine K_{c} instead of K_{cb}. Then, K_{c min} in Equations 97 and 98 can be set equal to K_{c ini}, where K_{c ini} is estimated from Figure 29 or 30. The use of K_{c ini} incorporates effects of soil evaporation and therefore serves as a lower limit on the estimate for K_{c mid}.

Where rainfall or irrigation is low, water stress might be induced and the evapotranspiration will drop below the standard crop evapotranspiration, ET_{c}. The reduction in the value for K_{c} under conditions of low soil water availability is determined using the stress coefficient K_{s} as described in Chapter 8.

Adjustment from simple field observations

Estimation of K_{cb mid}from Leaf Area Index (LAI)

Estimation of K_{cb mid}from effective ground cover (f_{c eff})

Estimation of K_{cb full}

Conclusion

As a rough approximation for K_{c} during the mid-season stage for crops that usually nearly completely shade the soil under pristine conditions, but where cover is reduced due to disease, stress, pests, or planting density, the values for K_{c mid} and K_{cb mid} can be reduced by a factor depending on the actual vegetation development:

K_{c adj}= K_{c}- A_{cm}(94)

where

K_{c}the K_{c}from Table 12 (K_{c mid}) or 17 (K_{cb mid}) after adjusting it for climate (Equation 62 or 70),

K_{c adj}the adjusted K_{c}(K_{c mid adj}or K_{cb mid adj}).

The K_{c} adjustment using Equation 94 does not apply when crops are frequently wetted and increased soil evaporation compensates for the reduced ground cover. Under these conditions Equation 94 should be applied only to K_{cb}.

The adjustment coefficient, A_{cm}, is estimated from:

_{}(95)

where LAI is the actual leaf area index (Box 17) and LAI_{dense} is the leaf area index expected for the same crop under normal, standard crop management practices. The values for LAI in the above equation can be replaced by values for the ground cover fraction (f_{c}):

_{}(96)

**EXAMPLE 40. First approximation of the crop coefficient for the mid-season stage for sparse vegetation**

A tomato crop was grown at Davis, California, United States in 1980 and only developed 50% ground cover during the midseason period (Pruitt et al., 1984). The height of the tomato crop was 0.75 m. The typical percentage of ground cover for tomatoes at effective full cover at Davis is 85 to 90% and corresponds to the K |

From Tables 12 and 17, K Following adjustments for climate (Eq. 62 and Eq. 70) where u K The ground cover fraction implied in the tabulated values for tomatoes grown under pristine conditions is about 85% (f From Eq. 96 A The K K Compare the results with Example 42 where a more precise derivation of K |

As a first estimate, the crop coefficient is expected to be 20% lower than the value under pristine conditions. |

Natural vegetation typically has less leaf area or fraction of ground cover than does agricultural vegetation that has been developed for full ground cover and for soil water conditions favouring vigorous growth. This is especially true in semi-arid and arid climates. The value for K_{cb mid} for natural or non-pristine vegetation should be reduced when plant density and/or leaf area are lower than for full cover conditions (generally defined as when LAI ³ 3). Where LAI can be measured or approximated, a peak K_{cb mid} for natural, non-typical or non-pristine agricultural vegetation can be approximated similar to a procedure used by Ritchie as:

K_{cb mid}= K_{c min}+ (K_{cb full}- K_{c min})(1 - exp[-0.7 LAI]) (97)

where

K_{cb mid}estimated basal K_{cb}during the mid-season when plant density and/or leaf area are lower than for full cover conditions,K

_{cb full}estimated basal K_{cb}during the mid-season (at peak plant size or height) for vegetation having full ground cover or LAI > 3 (Equations 99 and 100),K

_{c min}the minimum K_{c}for bare soil (K_{c min}» 0.15 - 0.20),LAI actual leaf area index, defined as the area of leaves per area of underlying ground surface averaged over a large area. Only one side of leaves is counted [m

^{2}m^{-2}].

Equation 97 is recommended for annual types of vegetation that are either natural or are in a non-pristine state due to sparse density or effects of some type of environmental stress on growth.

The relationship expressed in Equation 97 produces results similar to those suggested by Ritchie (1974). For LAI > 3, K_{cb mid} » K_{cb full}. The LAI used in Equation 97 should be the 'green' LAI representing only healthy leaves that are active in vapour transfer.

LAI can be measured directly by harvesting all green healthy leaves from vegetation over a measured or prescribed area, for example, 1 m In the absence of measurements for LAI, LAI can be estimated for sparse, annual vegetation as:
where LAI Population number of plants per unit area of soil surface under the actual growing conditions [No. m Population a a = 0.5 when population is formed from vigorous growing plants; a = 1 when plants are less vigorous. The 0.5 exponent in the equation simulates the tendency for vegetation to compensate for reduced stand density by increasing the size and total leaf areas for individual plants. Therefore, LAI does not fall in direct proportion to plant population. Under conditions where the plant size does not increase with reduced stand density, the 'a' exponent in the equation should be set to 1 (a = 1). These latter conditions may occur where soil fertility is poor or where soil salinity, soil water stress, or waterlogging inhibit both growth and stand density, so that the growth of individual plants is retarded. |

Where only estimates of the fraction of soil surface effectively covered by vegetation are available, the following approximation for K_{cb mid adj} can be used:

_{}(98)

where

K_{cb mid}estimated basal K_{cb}during the mid-season when plant density and/or leaf area are lower than for full cover conditions,K

_{cb full}estimated basal K_{cb}during the mid-season (at peak plant size or height) for vegetation having full ground cover or LAI > 3 (see Equations 99 and 100),K

_{c min}the minimum K_{c}for bare soil (in the presence of vegetation) (K_{c min}» 0.15-0.20),f

_{c}observed fraction of soil surface that is covered by vegetation as observed from nadir (overhead) [0.01 - 1],f

_{c eff}the effective fraction of soil surface covered or shaded by vegetation [0.01-1] (see Box 18),h the plant height [m].

Stomatal conductance and water transport within plants may limit ET under conditions of sparse, tall vegetation. Under these conditions, K_{cb mid} is limited by the "2f_{c}" term in Equation 98. Equation 98 applies well to trees and shrubs.

f f
where f HWR height to width ratio of individual plants or groups of plants when viewed from the east or from the west [], tan(h) tangent of the mean angle of the sun, h, above the horizon during the period of maximum evapotranspiration (generally between 11.00 and 15.00 hours) []. For most applications, h can be computed at solar noon (12.00 hours). HWR is computed as:
where h Width mean width of the canopy of a plant or group of plants (e.g., row) [m] G angle of plant row from east-west direction [rad] (for east-west rows, G = 0; for north-south rows, G = p /2)
For north-south rows, the HWR would be zero, as cos(p /2) = 0. This implies that rows of plants that run from north to south would have f For trees or vegetation that do not have canopies that extend to the ground, h |

For
where sin(h) is the sine of the mean angle of the sun, h, above the horizon during the period of maximum evapotranspiration (generally between 11.00 and 15.00) [] |

The sine of h can be calculated for any specific time of day as: sin(h) = sin(j)sin(d) + cos(j)cos(d)cos(w) where j latitude [rad] (negative for southern latitudes) Generally, f sin(h) = sin(j)sin(d) + cos(j)cos(d) The value for h can be obtained by taking the arcsine of the above equation. |

*Agricultural crops:*

Non-pristine agricultural crops represent crops that have not developed to their potential due to environmental stresses caused by soil water shortage, fertility, disease, grazing or insect damage or due to low plant density. The value for K_{cb full} in Equations 97 and 98 can be taken as the K_{cb mid} value listed for any "full-cover" crop (f_{c eff} ~ 1) in Table 17, after adjusting it for climate (Equation 70):

_{}(99)

where

u_{2}mean value for wind speed at 2 m height during the mid-season [m s^{-1}],

RH_{min}mean value for minimum daily relative humidity during the mid-season [%].

h mean maximum plant height [m].

*Natural vegetation and crops not listed in Table 17:*

For natural vegetation, nonfull-cover crops, or for crops not listed in Table 17, K_{cb full} can be approximated as a function of climate and mean plant height for areas of vegetation that are greater than a few hectares:

_{}(100)

where

K_{cb, h}K_{cb mid}for full cover vegetation (LAI > 3) under sub-humid and calm wind conditions (RH_{min}= 45% and u_{2}= 2 m s^{-1}), (Equation 101),u

_{2}mean value for wind speed at 2 m height during the mid-season [m s^{-1}],RH

_{min}mean value for minimum daily relative humidity during the mid-season [%]h mean maximum plant height [m].

The value for K_{cb, h} is estimated as:

K_{cb, h}= 1.0+0.1 h for h £ 2 m (101)

where K_{cb, h} is limited to £ 1.20 when h > 2 m. The value of 1.2 represents a general upper limit on K_{cb mid} for tall vegetation having full ground cover and LAI > 3 under the sub-humid and calm wind conditions. This limit of 1.2 is adjusted for other climatic conditions in Equation 100. Equations 100 and 101 produce a general approximation for the increase in K_{cb full} with plant height and climate. The form of these equations adheres to trends represented in Equation 70.

For small, isolated stand sizes, K_{cb full} may need to be increased beyond the value given by Equation 99 or 100, as discussed in Chapter 10.

Equations 97 and 98 can be used to estimate or to reduce K_{cb} for non-pristine agricultural vegetation. The exponents in Equations 97 and 98 reflect the effects of microscale advection (transfer) of sensible heat from dry soil surfaces between plants toward plant leaves, thereby increasing ET per unit leaf area, and the effects of increased aerodynamic roughness as the value for LAI decreases. Equation 98 suggests that as h increases, total leaf area and effective roughness of vegetation increase, thereby increasing the crop coefficient. In addition, as h increases, more opportunity for microadvection of heat from soil to canopy occurs and turbulent exchange within the canopy increases for the same amount of ground coverage. All of these factors affect the relative magnitude of K_{cb mid}.

Equations 97 and 98 should be used with caution as they provide only an estimate of the maximum K_{cb} expected during peak plant growth for vegetation with healthy transpiring leaves and a dry soil surface. Where stomatal control is greater than for typical agricultural vegetation, then the K_{cb} should be further reduced using the recommendations set out in the next section (Equation 102).

**EXAMPLE 41. Estimation of mid-season crop coefficient**

Estimate K | |

On day J = 200 at latitude (j = 40 (p /180) = 0.70 radians (40°N), from Eq. 24, the solar declination d = 0.36 radians. At solar noon (w = 0): sin(h) = sin(j)sin(d) + cos(j)cos(d) = 0.94 The value for h by taking the arcsine of above value is 1.24 radians and tan(h) = 2.8. If f | |

From Eq. 101 |
K |

From Eq. 100 |
K |

Therefore, K K This value does not need any further adjustment for climate. | |

K K depending on the frequency of soil wetting. | |

The estimated crop coefficients for the mid-season stage are K |

**EXAMPLE 42. Estimation of mid-season crop coefficient for reduced ground cover**

A more precise estimate of K What is the adjusted K |

On day J = 201 (20 July) at latitude j = 38.5 (p /180) = 0.67 radians (38.5°N), from Eq. 24 the solar declination d = 0.36 radians. At solar noon (w = 0): sin(h) = sin(j)sin(d) + cos(j)cos(d) = 0.95 The value for h by taking the arcsine of the above value is 1.26 radians. Therefore, for the observed HWR = 1 and f f The K K From Eq. 98 and using K K |

The results K |

The value for K_{cb full} in Equations 97 and 98 may need to be reduced for vegetation that has a high degree of stomatal control. For vegetation such as some types of desert vegetation or trees with leaf resistance significantly greater than that of most agricultural crops where r_{l} is commonly about 100 s m^{-1}, the K_{cb mid} estimated using Equations 97 and 98 can be modified by multiplying by a resistance correction factor, F_{r}. The resistance correction factor is developed based on the FAO Penman-Monteith equation:

_{}(102)

where

r_{l}mean leaf resistance for the vegetation in question [s m^{-1}].

The mean leaf resistance r_{l} is 100 s m^{-1} for the grass ET_{o} reference and for many agricultural crops. Values for r_{l} for many agricultural and non-agricultural plants can be found in Körner *et al.* (1978) and Allen *et al.* (1996). Equation 102 reflects the fixed aerodynamic roughness of grass rather than the roughness of the specific vegetation, since the adjusted K_{c} is multiplied by the grass ET_{o} and the K_{c} already reflects the effects of the aerodynamic roughness for the specific vegetation.

**EXAMPLE 43. Estimation of K**_{cb mid}** from ground cover with reduction for stomatal control**

A grove of olive trees has a tree spacing of 10 m. The horizontal diameter of the trees as viewed from overhead is 5 m. The tree height is 5 m. The lower 1.5 m of the trees have no foliage. The ground cover between the trees is bare. The mean u Estimate K |

On day J = 180 (29 June) at latitude (j = 30 (p /180) = 0.52 radians (30°N) and from Eq. 24 the solar declination d = 0.405 radians. At solar noon (w = 0): sin(h) = sin(j)sin(d) + cos(j)cos(d) = 0.99 As olive trees have somewhat round shapes, the effective fraction of ground cover (Box 18) can be estimated as f f From Eq. 101: K From Eq. 100: K From Eq. 98 and using K Körner
The K K |

The value K |

If the olives had been planted on a 5x10 m spacing, as is common in California, the United States, and which is reflected in the K |

The basal crop coefficient, K |

The equation would underestimate F_{r} (overestimate the reduction in K_{cb}) if used with the actual roughness of the vegetation when r_{l} > 100 s m^{-1} because of the lack in Equation 102 of feedback effects that reduced ET_{c} has on temperature and vapour pressure deficit profiles over the crop. These parameters generally increase with decreasing ET_{c} and therefore dampen the reduction in ET_{c}.

During the late season stage, the K_{cb} begins to decrease until it reaches K_{cb end} at the end of the growing period. Values for K_{cb end} can be scaled from K_{cb mid} in proportion to the health and leaf condition of the vegetation at termination of the growing season and according to the length of the late season period (i.e., whether leaves senesce slowly or are killed by frost). Values for K_{c end} can be similarly scaled from K_{c mid}; however, the reduction in K_{c end} will be affected by the frequency of wetting by irrigation or precipitation and K_{c end} may be proportionally less.

If estimated from Equations 97 and 98, K_{cb end} should be reduced if it is to represent K_{c} values for plants with stomatal control that is greater than that for agricultural vegetation (where r_{l} » 100 s m^{-1}) or to reflect effects of ageing and senescence on stomatal control. In these situations, the estimated K_{cb end} values should be multiplied by the F_{r} from Equation 102. Alternatively, they can be reduced by about 10% for each doubling of r_{l} above 100 s m^{-1} when mean daily air temperature (T_{mean}) is about 30° C and by about 20% for each doubling of r_{l} above 100 s m^{-1} when T_{mean} is about 15° C.

Alternatively, the value for K_{cb end} can be reduced relative to the calculated value for K_{cb mid} in proportion to the fraction of green healthy leaves remaining at the end of the late season stage relative to that during the mid-season. This can often be based on a visual survey of me field and may therefore be a subjective observation.

The f_{c} parameter and h are probably the simplest indices to estimate in the field. Again, Equations 97 and 98 should be used only as general or preliminary estimates of K_{cb end}.

A simple, linear crop-water production function was introduced in the FAO Irrigation and Drainage Paper No. 33 to predict the reduction in crop yield when crop stress is caused by a shortage of soil water. This function was presented earlier as Equation 90:

_{}(90)

where

Y_{a}= actual yield of the crop [kg ha^{-1}]Y

_{m}= maximum (expected) yield in absence of environmental or water stressesK

_{y}= yield response factor []ET

_{c}=potential (expected) crop evapotranspiration in the absence of environmental or water stresses (K_{c}ET_{o})ET

_{c adj}= actual (adjusted) crop evapotranspiration as a result of environmental or water stresses

Values for K_{y} have been reported in Paper No. 33 for a wide range of crops for predicting the effect of water stress and associated reduction in ET_{c adj} on crop yield. Factors are presented there for predicting yield reductions for when stress occurs in only one crop growth stage, or when stress is distributed throughout the growing period. Seasonal yield response functions are summarized in Table 24.

Many environmental stresses such as water shortage, salinity, low fertility and disease impact yield by reducing the amount of ET_{c adj} relative to the potential amount ET_{c}. The same can be true for when yields are reduced due to the use of low densities for plant populations. Therefore, for very general estimates of ET_{c adj}, one can invert Equation 90 and solve for the stress factor, K_{s}:

_{}(103)

where K_{s} is multiplied by K_{cb} or by K_{c} in equations 80 or 81 to predict the ET_{c adj} in the presense of the water or other environmental stresses or for low plant populations or virility. The ET_{c adj} predicted using K_{s} from equation 103 provides only a very general and approximate estimate of monthly or even seasonal evapotranspiration. Equation 103 works best for forage or other indiscriminate crops where the value for K_{y} is relatively constant during the season.

Equation 103 is generally only valid for use in predicting actual crop evapotranspiration for use in regional water balance studies, for studies of ground-water depletion and recharge, or for estimating historical water use. The procedure is not valid for predicting ET_{c} for daily or weekly time periods due to the very general nature of the K_{y} coefficient and the seasonal time scale of the crop yield. The procedures presented previously for adjusting ET_{c} using a daily soil water balance, salinity functions, or reductions in K_{c} based on leaf area or fraction of ground cover are recommended over the use of equation 103.

**EXAMPLE 44. Approximate estimation of K**_{s}** from crop yield data**

An irrigation scheme (project) cultivates dry, edible beans. There is known to be a shortage of irrigation water and a corresponding reduction in crop yield. The reported yield for the scheme averages 1100 kg/ha. The potential yield for the region and variety of beans, in the absence of water or environmental stresses and with good soil fertility is 1800 kg/ha. From FAO Irrigation and Drainage Paper No. 33 or Table 24, the K
Therefore, the ET |

The estimated seasonal ET |