# Chapter 9 - ETc for natural, non-typical and non-pristine vegetation

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, Kc, and the crop evapotranspiration is given by the product of Kc and the reference evapotranspiration ETo, 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 Kc values for natural vegetation and for agricultural vegetation for which Kc values are not available.

## Calculation approach

As described in Figure 27, the first step in the KcETo 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 ETc to ETo during the various growth stages of the vegetation.

### Initial growth stage

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 Kc ini) or Chapter 7 (dual crop coefficient, Kcb ini + Ke). The crop coefficient in this stage is primarily determined by the frequency with which the soil is wetted.

### Mid and late season stages

The Kc during the mid-season period (Kc mid and Kcb mid) and to a lesser extent the Kc 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 Kc 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 Kc for natural, non-typical or non-pristine vegetation should ordinarily follow the approach described in Chapter 7 whereby separate transpiration (Kcb) and evaporation (Ke) 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 (Kcb mid adj) for sparse vegetation are presented in this section. In these approaches, Kcb mid adj is estimated either from LAI (Equation 97) or from effective ground cover (Equation 98). After the determination of Kcb mid adj, the soil evaporation coefficient, Ke, should be determined to obtain the crop coefficient for the mid-season stage: Kc mid adj = Kcb mid adj + Ke. Procedures for calculating Ke are presented in Chapter 7.

Even where the estimated Kcb mid adj is small, the total Kc adj (= Kcb adj + Ke) following precipitation may sometimes be as high or higher than the Kc for pristine vegetation due to surface evaporation from among sparse vegetation.

Single crop coefficient approach

When the single crop coefficient Kc of Chapter 6 is used, the average effects of soil wetting are incorporated into a general mean Kc. Some guidelines for the estimation of Kc adj are given in the following sections. The single crop coefficient can also be derived from the adjusted Kcb by considering the frequency of soil wetting, i.e., during the midseason period, Kc adj = Kcb adj + 0.05 for infrequent wetting and Kcb 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 Kc instead of Kcb. Then, Kc min in Equations 97 and 98 can be set equal to Kc ini, where Kc ini is estimated from Figure 29 or 30. The use of Kc ini incorporates effects of soil evaporation and therefore serves as a lower limit on the estimate for Kc mid.

### Water stress conditions

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

## Mid-season stage - Adjustment for sparse vegetation

### Adjustment from simple field observations

As a rough approximation for Kc 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 Kc mid and Kcb mid can be reduced by a factor depending on the actual vegetation development:

Kc adj = Kc - Acm (94)

where

Kc the Kc from Table 12 (Kc mid) or 17 (Kcb mid) after adjusting it for climate (Equation 62 or 70),

The Kc 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 Kcb.

The adjustment coefficient, Acm, is estimated from: (95)

where LAI is the actual leaf area index (Box 17) and LAIdense 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 (fc): (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 Kcb mid listed in Table 17 for tomatoes. The mean values for wind speed and minimum relative humidity during the midseason period were u2 = 1.1 m/s and RHmin = 30%. The latitude at Davis is 38.5° N and the midpoint of the midseason occurs on July 20. What is an adjusted Kcb mid for tomatoes that reflects the 50% ground cover condition? From Tables 12 and 17, Kc mid = 1.2 and Kcb mid = 1.15. Following adjustments for climate (Eq. 62 and Eq. 70) where u2 = 1.1 m/s, RHmin = 30% and mean crop height = 0.75 m, Kc = Kc, Table + [0.04(1.1 - 2) - 0.004(30 - 45)](0.75/3)0.30 = Kc, Table + 0.02 yields,Kc mid = 1.22 and Kcb mid = 1.17. The ground cover fraction implied in the tabulated values for tomatoes grown under pristine conditions is about 85% (fc dense = 0.85). For a sparse tomato crop where fc = 0.50, From Eq. 96 Acm = 1 - (0.50/0.85)0.5 = 0.23 The Kcb mid adj and Kc mid adj for 50% ground cover is (from Eq. 94) Kcb mid adj = 1.17 - 0.23 = 0.94Kc mid adj = 1.22 - 0.23 = 0.99 Compare the results with Example 42 where a more precise derivation of Kcb mid adj is made. As a first estimate, the crop coefficient is expected to be 20% lower than the value under pristine conditions.

### Estimation of Kcb mid from Leaf Area Index (LAI)

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 Kcb 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 Kcb mid for natural, non-typical or non-pristine agricultural vegetation can be approximated similar to a procedure used by Ritchie as:

Kcb mid = Kc min + (Kcb full - Kc min)(1 - exp[-0.7 LAI]) (97)

where

Kcb mid estimated basal Kcb during the mid-season when plant density and/or leaf area are lower than for full cover conditions,

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

Kc min the minimum Kc for bare soil (Kc 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 [m2 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, Kcb mid » Kcb full. The LAI used in Equation 97 should be the 'green' LAI representing only healthy leaves that are active in vapour transfer.

 BOX 17. Measuring and estimating LAI LAI can be measured directly by harvesting all green healthy leaves from vegetation over a measured or prescribed area, for example, 1 m2 or 10 m2, and then measuring and summing the areas of individual leaves using photometric methods or by measuring areas of several representative leaves, averaging, and then multiplying by the total number of leaves counted. In the absence of measurements for LAI, LAI can be estimated for sparse, annual vegetation as: where LAIdense LAI for the particular plant species under normal, 'dense' or pristine growing conditions. LAIdense can be obtained from various physiological sources and textbooks on crops and vegetation. Population number of plants per unit area of soil surface under the actual growing conditions [No. m-2]. Populationdense number of plants per unit area of soil surface under the 'dense' or pristine growing conditions [No. m-2]. 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.

### Estimation of Kcb mid from effective ground cover (fc eff)

Where only estimates of the fraction of soil surface effectively covered by vegetation are available, the following approximation for Kcb mid adj can be used: (98)

where

Kcb mid estimated basal Kcb during the mid-season when plant density and/or leaf area are lower than for full cover conditions,

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

Kc min the minimum Kc for bare soil (in the presence of vegetation) (Kc min » 0.15-0.20),

fc observed fraction of soil surface that is covered by vegetation as observed from nadir (overhead) [0.01 - 1],

fc 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, Kcb mid is limited by the "2fc" term in Equation 98. Equation 98 applies well to trees and shrubs.

 BOX 18. Measuring and estimating fc eff fc eff should normally represent the fraction of the soil surface that is shaded by vegetation. This value is generally larger than fc, the actual fraction of the soil surface that is covered by vegetation as observed from directly overhead. The amount of shading represents the amount of solar radiation intercepted by plants for potential conversion into evapotranspiration. The total fraction of shading is a function of the sun angle and the horizontal size and shape of individual plants (or rows) relative to their height. fc eff for 'rectangular' shaped canopies such as most agricultural plant rows can be approximated as: where fc observed fraction of soil surface that is covered by vegetation as observed from nadir (overhead) [0.01-1], 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 hcanopy mean vertical height of the canopy area of the plant [m], 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 fc eff » fc as all soil is exposed to the sun at various times of the day and as the shaded area is the same as the fraction of vegetation cover at midday. For trees or vegetation that do not have canopies that extend to the ground, hcanopy does not include the lower trunk length, but only the active canopy. Therefore, in these situations, hcanopy < h where h is mean plant height. For round or spherical shaped canopies such as trees, fc eff can be estimated as: 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) [] Mean angle of the sun above the horizon during the period of maximum evapotranspiration (h) 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)d solar declination [rad] (Eq. 24)w solar time angle [rad] (Eq. 31) Generally, fc eff can be calculated at solar noon (12.00), so that w = 0 and the above equation reduces to: 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.

### Estimation of Kcb full

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 Kcb full in Equations 97 and 98 can be taken as the Kcb mid value listed for any "full-cover" crop (fc eff ~ 1) in Table 17, after adjusting it for climate (Equation 70): (99)

where

u2 mean value for wind speed at 2 m height during the mid-season [m s-1],
RHmin 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, Kcb 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

Kcb, h Kcb mid for full cover vegetation (LAI > 3) under sub-humid and calm wind conditions (RHmin = 45% and u2 = 2 m s-1), (Equation 101),

u2 mean value for wind speed at 2 m height during the mid-season [m s-1],

RHmin mean value for minimum daily relative humidity during the mid-season [%]

h mean maximum plant height [m].

The value for Kcb, h is estimated as:

Kcb, h = 1.0+0.1 h for h £ 2 m (101)

where Kcb, h is limited to £ 1.20 when h > 2 m. The value of 1.2 represents a general upper limit on Kcb 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 Kcb full with plant height and climate. The form of these equations adheres to trends represented in Equation 70.

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

### Conclusion

Equations 97 and 98 can be used to estimate or to reduce Kcb 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 Kcb mid.

Equations 97 and 98 should be used with caution as they provide only an estimate of the maximum Kcb 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 Kcb should be further reduced using the recommendations set out in the next section (Equation 102).

EXAMPLE 41. Estimation of mid-season crop coefficient

 Estimate Kcb mid and Kc mid for rectangular shaped 2 m tall vegetation that is as tall as it is wide, where 30% of the ground is covered by vegetation (fc = 0.3) on 19 July (day 200 of the year) and at latitude 40°N. RHmin = 55% and u2 = 1.5 m/s during the mid-season period. 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 fc = 0.3 and the HWR for the vegetation is 1, then fc eff from Box 18 is: 0.3(1 + 1/2.8) = 0.41. From Eq. 101 Kcb, h = 1.0 + 0.1(2) = 1.20 (£ 1.20, so OK) From Eq. 100 Kcb full = 1.20 + (0.04(1.5 - 2) - 0.004(55 - 45)) (2/3)0.3 =1.15 Therefore, Kcb mid estimated by Eq. 98 for Kc min = 0.15 is Kcb mid = Kc min + (Kcb full - Kc min) min[1, 2fc, (fc eff) (1/(1+h))] = 0.15 + (1.15 - 0.15) min[1,2(0.3), (0.4) (1/(1+2))] = 0.75 This value does not need any further adjustment for climate. Kc mid (where Kc mid includes average wetting effects) can be derived from Kcb mid using the guidelines presented in the calculation procedures at the beginning of this chapter Kc mid = Kcb mid + (0.05... 0.10) = 0.80 .. 0.85 depending on the frequency of soil wetting. The estimated crop coefficients for the mid-season stage are Kcb mid = 0.75 and Kc mid = 0.80 to 0.85.

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

 A more precise estimate of Kcb mid for the tomato crop of Example 40 that only developed 50% ground cover at Davis, California, the United States can be calculated if one knows that the tomato crop was planted in 1.52 m rows running east-west, that the crop reached a plant height of 0.75 m and that the height to width ratio of the tomato crop can be estimated as about 1.0 for the east-west rows. The latitude is 38.5°N and the midpoint of the mid-season occurs on July 20. What is the adjusted Kcb mid for tomatoes that reflects the 50% ground cover condition 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 fc = 0.5, the effective soil cover for the east-west rows can be calculated as (Box 18): fc eff = 0.5 [1 + 1/tan(1.26)] = 0.66 The Kcb mid in Table 17 representing a full cover tomato crop is 1.15 and the average h for fully developed tomatoes (this variety) is about 0.75 also. Following adjustment for climate (using Eq. 99) Kcb full = 1.15 + [0.04(1.1 - 2) - 0.004(30 - 45)] (0.75/3)0.30 = 1.17 From Eq. 98 and using Kc min = 0.15, the adjusted Kcb mid for 50% ground cover would be: Kcb mid adj = 0.15 + (1.17 - 0.15) min(1, 2(0.5), 0.661/(1+0.76)) = 0.95 The results Kcb mid adj = 0.95 for 50% ground cover are similar to the first estimate calculated in Example 40 and compare with the measured Kcb mid ~ 0.90 to 1.00 as determined by precision lysimeter by Pruitt et al. (1984).

## Mid-season stage - Adjustment for stomatal control

The value for Kcb 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 rl is commonly about 100 s m-1, the Kcb mid estimated using Equations 97 and 98 can be modified by multiplying by a resistance correction factor, Fr. The resistance correction factor is developed based on the FAO Penman-Monteith equation: (102)

where

rl mean leaf resistance for the vegetation in question [s m-1].

The mean leaf resistance rl is 100 s m-1 for the grass ETo reference and for many agricultural crops. Values for rl 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 Kc is multiplied by the grass ETo and the Kc already reflects the effects of the aerodynamic roughness for the specific vegetation.

EXAMPLE 43. Estimation of Kcb 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 u2 during the mid-season growth stage is 2 m/s and the mean RHmin = 25%. The midpoint of the mid-season growth stage is on 29 June (i.e., day 180 of the year). The latitude of the location is 30°N. Estimate Kcb mid using Eq. 98 for the 10x10 m and for a 5x10 m spacing. 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 fc eff = fc/(sin(h)). fc = area of canopy/area tree spacing = (p (5)2/4)/(10)(10) = 0.196fc eff = 0.196/0.99 = 0.20 From Eq. 101: Kcb, h = 1.0 + 0.1(5) > 1.2 or Kcb, h = 1.2 From Eq. 100: Kcb full = 1.2 + [0 - 0.004(25 - 45)](5/3)0.3 = 1.29 From Eq. 98 and using Kc min = 0.15: Kcb mid = 0.15 + (1.29 - 0.15) min(1,2(0.196), (0.20)1/(1+5)) = 0.60 Körner et. al. (1979) indicate that olives (Olea europaea) have rl of about 420 s/m. Therefore, assuming that average Tmean = 25°C and that the elevation of the grove is 0 m (sea level), so that D = 0.189 kPa (Eq. 13) and g = 0.0676 kPa (Eq. 8), Fr is estimated from Eq. 102 as: The Kcb mid adjusted for increased stomatal control using Fr is then Kcb mid adj = Fr Kcb mid = 0.67 (0.60) = 0.40 The value Kcb mid adj = 0.40 estimated for fc = 0.20 is less than the value for Kcb mid in Table 17 for olives for fc = 0.40 to 0.67, due to the differences in fc. The value from Table 17 is 0.70, which after adjustment for climate using Eq. 70 equals 0.79. 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 Kcb values for olives in Table 17, then fc = 0.39, fc eff = 0.40, and Kcb mid from Eq. 98 is Kcb mid = 1.04, so that the estimated Kcb mid adjusted for stomatal control using Fr = 0.67 is Kcb mid adj = 0.67(1.04) = 0.70. This value compares with the value of 0.79 obtained from Table 17 for mature trees, after adjustment for climate. The basal crop coefficient, Kcb mid, taking the low density, climatic condition and stomatal control into account is 0.40. It increases to 0.70 for the 5x10 m spacing.

The equation would underestimate Fr (overestimate the reduction in Kcb) if used with the actual roughness of the vegetation when rl > 100 s m-1 because of the lack in Equation 102 of feedback effects that reduced ETc has on temperature and vapour pressure deficit profiles over the crop. These parameters generally increase with decreasing ETc and therefore dampen the reduction in ETc.

## Late season stage

During the late season stage, the Kcb begins to decrease until it reaches Kcb end at the end of the growing period. Values for Kcb end can be scaled from Kcb 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 Kc end can be similarly scaled from Kc mid; however, the reduction in Kc end will be affected by the frequency of wetting by irrigation or precipitation and Kc end may be proportionally less.

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

Alternatively, the value for Kcb end can be reduced relative to the calculated value for Kcb 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 fc 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 Kcb end.

## Estimating ETc adj using crop yields

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

Ya = actual yield of the crop [kg ha-1]

Ym = maximum (expected) yield in absence of environmental or water stresses

Ky = yield response factor []

ETc = potential (expected) crop evapotranspiration in the absence of environmental or water stresses (Kc ETo)

ETc adj = actual (adjusted) crop evapotranspiration as a result of environmental or water stresses

Values for Ky have been reported in Paper No. 33 for a wide range of crops for predicting the effect of water stress and associated reduction in ETc 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 ETc adj relative to the potential amount ETc. 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 ETc adj, one can invert Equation 90 and solve for the stress factor, Ks: (103)

where Ks is multiplied by Kcb or by Kc in equations 80 or 81 to predict the ETc adj in the presense of the water or other environmental stresses or for low plant populations or virility. The ETc adj predicted using Ks 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 Ky 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 ETc for daily or weekly time periods due to the very general nature of the Ky coefficient and the seasonal time scale of the crop yield. The procedures presented previously for adjusting ETc using a daily soil water balance, salinity functions, or reductions in Kc based on leaf area or fraction of ground cover are recommended over the use of equation 103.

EXAMPLE 44. Approximate estimation of Ks 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 Ky for dry beans, assuming that stresses are distributed uniformly through the growing season, is 1.15. Therefore, from Equation 103, the estimated Ks to apply with Equation 80 for the growing season is: Therefore, the ETc adj for the season is predicted to be only 0.66 of maximum ETc under pristine growing conditions. The estimated seasonal ETc adj is predicted to be ETc adj = 0.66 ETc where ETc is predicted as ETc = Kc ETo.