This annex illustrates in more detail the application of the various equations for calculating K_{cb}, K_{e} and ET_{c} using the dual K_{c} approach of Chapter 7. The example is in the form of a computer spreadsheet and is applied to the dry, edible bean crop that was used in example boxes 15 and 16 of Chapters 6 and 7. The spreadsheet is shown in Figure 8.1, where the irrigation schedule is determined using the daily soilwater balance procedure described in Chapter 8. The timing of irrigations is based on the management allowed depletion (MAD) of the available water that can be stored in the root zone. The irrigation schedule and the corresponding estimated wet soil evaporation are different from the actual values shown in Box 16 of Chapter 7, since Box 16 represents the actual irrigation schedule used at Kimberly during 1974. The actual schedule deviated somewhat from the theoretical schedule of Figure 8.1.
The spreadsheet formulas used for calculations and the references to equations in the text are indicated in Box 8.1. The variable names used for parameters follow the same convention used in Chapters 1 to 9. The variable names are defined in the List of principal symbols and acronymns in the introduction to this paper. A few exceptions are defined in Table 8.1.
The spreadsheet in Figure 8.1 includes columns for variables T_{max}, u_{2} and T_{dew}. The T_{max} and T_{dew} columns are used to calculate daily RH_{min}. The u_{2} and RH_{min} columns are used to adjust K_{cb mid} and K_{cb end} using Equation 70 of Chapter 7 and to calculate K_{c max} using Equation 72 on a daily basis. The data in the first 7 rows of Figure 8.1 that appear within boxes represent the specific crop and soils information that is entered by the user for a particular crop and soil combination. All other information (outside of boxes) is calculated automatically by the spreadsheet program. The columns having double underlined headings represent the data that are input by the user into the spreadsheet.
The calculations in Figure 8.1 can be used to verify other computer programs or spreadsheet calculations for K_{e}, K_{c} and ET_{c}. Small differences may result, depending on the assumptions of timing of irrigations. The spreadsheet of Figure 8.1 presumes that all irrigation and precipitation events occur early in the morning. The scheduling and magnitudes of irrigations are based on the soil water depletion at the end of the previous day. The spreadsheet also presumes that all drainage from the root zone due to excess precipitation occurs on the day of the precipitation event. It is assumed that runoff from precipitation is zero. If necessary, procedures for predicting precipitation runoff can be entered into the spreadsheet using procedures described in most standard hydrology textbooks. It is assumed that the net depth of irrigation that is retained in the crop root zone is exactly equal to the depletion depth of the previous day. This assumption presumes perfect knowledge of soil water depletion by the irrigator or that all irrigations are adequate or excessive. This assumption may not hold for some irrigation conditions and can be changed by the user as needed.
Spreadsheet formulas used to create the spreadsheet of Figure 8.1 are listed in Box 8.1 for the Microsoft Excel language (versions 5 and higher). Formulae for other types of spreadsheets would be similar. Formulae for the Corel QuattroPro language (versions 5 and higher) can be downloaded from the FAO internet site.
BOX 8.1. Spreadsheet formulas and corresponding equations for Excel spreadsheet programs. Formulas for Rows 1 to 15 of Figure 8.1 (for Microsoft Excel, versions 5/95 and later) Underlined numeric values are input by the user 

Equation in text or footnote 
Cell 
Text, value, or formula 

A1: 
Example Spreadsheet for Calculating ET_{c} = (K_{cb} + K_{e}) ET_{o} and an Irrigation Schedule 
P2: 
Computed Dates for Stages: 

A3: 
Crop: 

B3: 
Dry, Edible Beans 

F3: 
Table 11: 

I3: 
Table 12: 

J3: 
Following Adjustment: 

P3: 
J_{plant} 

Table 2.5 
Q3: 
= TRUNC(275*C5/9  30 + C6) + IF(C5 > 2,2,0) + IF(MOD(C14, 4) = 0, + 1,0) 

V3: 
f_{w} (irrig.): 
X3: 
0.5 

AE3: 
Root_{min} 

AF3: 
0.2 

AG3: 
m 

AH3: 
MAD during Initial Stage 

AK3: 
70 

AL3: 
% 

E4: 
L_{ini} 

F4: 
25 

H4: 
K_{cb ini} 

I4: 
0.15 

J4: 
= I4 

L4: 
K_{c min} 

M4: 
= J4 

P4: 
J_{Dev} 

Q4: 
= Q3 + F4 

V4: 
REW: 

X4: 
8 

Y4: 
mm 

AE4: 
Root_{max} 

AF4: 
0.8 

AG4: 
m 

AH4: 
MAD after Initial Stage 

AK4: 
45 

AL4: 
% 

A5: 
Planting: 

B5: 
Month 

C5: 
5 

E5: 
L_{dev} 

F5: 
25 

H5: 
K_{cb mid} 

I5: 
1.1 

Eq. 70 
J5: 
= I5+(0.04*($K$8  2)0.004*($K$9  45))*($M$5/3)^0.3 

L5: 
Max. Ht.: 
M5: 
0.4 

N5: 
m 

P5: 
J_{Mid} 

Q5: 
= Q4 + F5 

V5: 
TEW: 

X5: 
22 

Y5: 
mm 

AE5: 
Avail. Water 

AF5: 
160 

AG5: 
mm/m 

B6: 
Day 

C6: 
22 

E6: 
L_{mid} 

F6: 
30 

H6: 
K_{cb end} 

I6: 
0.25 

Eq.70 
J6: 
= IF(16 < 0.45, 16, 16 + (0.04*($K$8  2)  0.004*($K$9  45))*($M$5/3)^0.3) 

P6: 
J_{Late} 
Q6: 
= Q5 + F6 

V6: 
initial D_{e}: 

X6: 
18 

Y6: 
mm 

E7: 
L_{late} 

F7: 
20 

P7: 
J_{Harv} 

Q7: 
= Q6 + F7 

V7: 
Initial f_{w}: 

X7: 
1 

H8: 
Midseas. Av. Wind Speed: 

^{ (1)} 
K8: 
= (VLOOKUP(Q6, D14: AP183, 38)  VLOOKUP(Q5, D14: AP183, 38))/(Q6  Q5) 

L8: 
m/s 
M8: 
<Computed automatically from Lookup on column AO 

AH8: 
(Irrigation that is needed is presumed applied at beginning of next day) 

H9: 
Midseas. Av. RH_{min}: 

^{ (1)} 
K9: 
= (VLOOKUP(Q6, D14: AP183, 39)  VLOOKUP(Q5, D14: AP183, 39))/(Q6  Q5) 

L9: 
% 
M9: 
<Computed automatically from Lookup on column AP 

First row of formulas (row 14) Note: some formulas in row 14 (first day) vary from those in rows 15 onward. See row 15 for example calculations for all subsequent days. 


A14: 
5 
B14: 
15 

C14: 
74 

Table 2.5 
D14: 
= TRUNC(275*A14/9  30 + B14) + IF(A14 > 2,2,0) + IF(MOD(C14, 4)=0, +1,0) 

E14: 
10 
F14: 
5.7655 

G14: 
0 

H14: 
3.4 

Eq. 14 
I14: 
= 0.6108*EXP((17.27*G14)/(G14 + 237.3)) 
Eq. 11 
J14: 
= 0.6108*EXP((17.27*E14)/(E14 + 237.3)) 
Eq.63 
K14: 
= I14/J14*100 

L14: 
0 
Eq. 66^{ (2)} 
O14: 
= IF(D14 < $Q$4,$J$4, IF(D14 < $Q$5,$J$4 + (D14  $Q$4)/$F$5*($J$5 $J$4), IF(D14 < $Q$6, $J$5, IF(D14 < $Q$7, $J$5 + (D14  $Q$6)/$F$7*($J$6  $J$5), $J$4)))) 
^{ (3)} 
P14: 
= MAX(014/$J$5*$M$5, P13) 
Eq. 72 
Q14: 
= MAX(1.2 + (0.04*(F14*0.9  2)  0.004*(K14  45))*(P14/3)^0.3, O14 + 0.05) 
^{ (4)} 
R14: 
0 
Eq. 76 
S14: 
= MAX(((O14  M$4)/(Q14  M$4))^(1 + 0.5*P14), 0.01) 
^{ (5)} 
T14: 
=IF(R14 > 0, X$3, IF(L14 > 0, 1, X7)) 
Eq. 75 
U14: 
= MIN(1  S14, T14) 
^{ (6)} 
V14: 
= X6 
Eq. 74 
W14: 
= MAX(IF(V14 < X$4, 1, (X$5  V14)/(X$5  X$4)),0) 
Eq. 71 
X14: 
= MIN(+ W14*(Q14  014), U14*Q14) 

Y14: 
= X14*H14 
Eq. 79 
Z14: 
= MAX(L14 + R14, 0) 
Eq. 77^{ (6)} 
AA14: 
= V14  L14  R14 + Y14/U 14 + Z14 
^{ (7)} 
AB14: 
= 014 + X14. 
Eq. 69^{ (7)} 
AC14: 
= AB14*H14 
Eq. 8.1^{ (8)} 
AE14: 
= MAX((O14  $J$4)/($J$5  $J$4)*($AF$4  $AF$3) + $AF$3, AE13) 
Eq. 82 
AF14: 
= MAX(IF(D14 < Q$4, AK$3, AK$4)/100*AE14*$AF$5, AF13) 
Eq. 85^{ (9)} 
AG14: 
= $X$6  L14 + AC14 
^{ (10)} 
AH14: 
= IF(D14 >= Q$3, IF(D14 < (Q$6 + Q$7)/2, IF(AG14 > AF14, AG14, 0), 0), 0) 
Eq. 88 
AI14: 
= MAX(+ L14  AC14  $X$6, 0) 
Eq. 84^{ (11)} 
AJ14: 
= IF(AG14 > AF14, (AE14*AF$5  AG14)/(AE14*AF$5  AF14), 1) 
Eq. 80 
AK14: 
= X14 + 014*AJ14 
Eq. 85^{ (9)} 
AU 4: 
= +$X$6  L14 + AK14*H 14 + AI14 
^{ (12)} 
AO14: 
= F14 
^{ (12)} 
AP14: 
= K14 
Second row of formulas All rows below row 15 are similar. 


A15: 
5 
B15: 
16 

C15: 
74 

Table 2.5 
D15: 
= TRUNC(275*A15/9  30 + B15) + IF(A15 > 2,2,0) + IF(MOD(C15, 4)= 0, +1,0) 

E15: 
13.3 
F15: 
2.2175 

G15: 
5 

H15: 
4.1 

Eq. 14 
I15: 
= 0.6108*EXP((17.27*G15)/(G15 + 237.3)) 
Eq. 11 
J15: 
= 0.6108*EXP((17.27*E15)/(E15 + 237.3)) 
Eq. 63 
K15: 
= I15/J15*100 

L14: 
0 
Eq. 66^{ (2)} 
015: 
= IF(D15 < $Q$4, $J$4, IF(D15 < $Q$5, $J$4 + (D15  $Q$4)/$F$5*($J$5  $J$4), IF(D15 < $Q$6, $J$5, IF(D15 < $Q$7, $J$5 + (D15  $Q$6)/$F$7*($J$6  $J$5), $J$4)))) 
^{ (3)} 
P15: 
= MAX(015/$J$5*$M$5, P14) 
Eq. 72 
Q15: 
= MAX(1.2 + (0.04*(F15*0.9  2)  0.004*(K15  45))*(P15/3)^0.3, 015 + 0.05) 
^{ (4)} 
R15: 
= IF(AH14 > 0, AH14/$X$3, 0) 
Eq. 76 
S15: 
= MAX(((015  M$4)/(Q15  M$4))^(1 + 0.5*P15), 0.01) 
^{ (5)} 
T15: 
= 1F(R15 > 0, X$3, IF(L15 > 0, 1, T14)) 
Eq. 75 
U15: 
= MIN(1  S15, T15) 
^{ (6)} 
V15: 
= MAX(AA14  L15  R15,0) 
Eq. 74 
W15: 
= MAX(IF(V15 < X$4, 1, (X$5  V15)/(X$5  X$4)), 0) 
Eq. 71 
X15: 
= MIN(+W15*(Q15  015),U15*Q15) 

Y15: 
= X15*H15 
Eq. 79 
Z15: 
= MAX(L15 + R15  AA14, 0) 
Eq. 77^{ (6)} 
AA15: 
= AA14  L15  R15 + Y15/U15 + Z15 
^{ (7)} 
AB15: 
= 15 + X15 
Eq. 69^{ (7)} 
AC15: 
= AB15*H15 
Eq. 81^{ (8)} 
AE15: 
= MAX((015  $J$4)/($J$5  $J$4)*($AF$4  $AF$3) + $AF$3, AE14) 
Eq. 82 
AF15: 
= MAX(IF(D15 < Q$4, AK$3, AK$4)/100*AE15*$AF$5, AF14) 
Eq. 85^{ (9)} 
AG15: 
= AK14  L15  AH14 + AC15 
^{ (10)} 
AH15: 
= IF(D15 >= Q$3, IF(D15 < (Q$6 + Q$7)/2, IF(AG15 > AF15, AG15, 0), 0), 0) 
Eq. 88 
AI15: 
= MAX(+ L15 + AH 14  AC 15  AK14, 0) 
Eq. 84^{ (11)} 
AJ14: 
= IF(AG15 > AF15, (AE15*AF$5  AG15)/(AE15*AF$5  AF15), 1) 
Eq. 80 
AK14: 
= X15 + 015*AJ15 
Eq. 85^{ (9)} 
AU 5: 
= +AL14  L15  AH14 + AK15*H15 + AI15 
^{ (12)} 
AO15: 
= AO14 + F15 
^{ (12)} 
AP15: 
= AP14 + K15 
Footnotes: 

^{ (1)} 
Cells K8 and K9 use the vertical lookup function to automatically calculate the average wind speed and average daily minimum relative humidity during the midseason period. The lookup function uses cumulative totals of wind speed and RH_{min} that are calculated in columns AO and AP. 

^{ (2)} 
The formula to calculate K_{cb} for each day uses a series of imbedded IF statements to determine which growing period the day is in. Linear interpolation is applied when the day is within the development and late season growing periods. 

^{ (3)} 
The crop height on any day is calculated as proportional to the value of K_{cb} on that day to the K_{cb mid} value, multiplied by the maximum crop height that has been entered by the user in cell M5. The value for crop height is not allowed to decrease with time. Hence, the MAX() function is employed, comparing with the value of the previous day. 

^{ (4)} 
The value for irrigation depth (divided by f_{w} to express the depth over the wetted fraction of the soil, only) is presumed to occur early in the day. This value is based on a decision made at the end of the previous day (column AH), based on whether or not the ending soil water depletion on the previous day has exceeded the readily available water (RAW). The irrigation depth on the first day is presumed to be zero. 

^{ (5)} 
The value for f_{w} is determined according to the last occurrence of precipitation or irrigation, as described in Chapter 7. 

^{ (6)} 
The depletion of the evaporation layer (top soil layer) at the beginning
of the day is presumed to equal the depletion at the end of the previous
day less any precipitation or irrigation, which is assumed to occur very
early in the day. The value for D_{e, i} is limited to ³
0. 

^{ (7)} 
The value for K_{c} is calculated as K_{c} = K_{cb} + K_{e} and the value for ET_{c} is calculated as K_{c} x ET_{o}. 

^{ (8)} 
The depth of the effective root zone on any day is calculated as being proportional to the ratio of the value of the K_{cb} on that day (above the value of K_{c min}) to the K_{cb mid}  K_{c min}, as described in Eq. 1 of this annex. The rooting depth is not allowed to decrease with time. Therefore, the MAX() function is utilised, where the value for the previous day is compared. 

^{ (9)} 
The "first" estimate for ending depletion of the root zone (D_{r, i}) is estimated using Eq. 85, with drainage assumed to be zero and with ET_{c} for nonstressed conditions. The value for D_{r, i}, is then recalculated in Column AK, after any drainage loss is estimated and after any reduction in ET_{c}, to account for low soil water content. The value for D_{r, i} in column AK represents depletion of the root zone at the end of the day. 

^{ (10)} 
The net depth of irrigation needed is based on the value of soil water depletion at the end of the day. It is assumed that irrigation will be applied at the beginning of the following day. The formula in column AH checks to insure that the specific day is within the growing period. The formula assumes that no irrigation will be desired during the last onehalf of the lateseason period. This assumption may need to be modified for some other crops. The value for management allowed depletion is allowed to have a different (normally higher) value during the initial period as compared to during the rest of the growing season. 

^{ (11)} 
The stress coefficient K_{s} represents the K_{s} under the current conditions of soil water. The value for K_{s} is reduced below 1.0 using Eq. 84 if the depletion of the root zone (following any irrigation or precipitation earlier in the day) is greater than the readily available water (RAW). It is presumed that the stress point, p, is the same as the value entered for MAD. This presumption can be modified as needed. 

^{ (12)} 
Columns AO and AP contain cumulative sums of daily wind speed and daily minimum relative humidity. These columns are used to calculate mean values for u_{2} and RH_{min} during the midseason period (footnote 1). 
TABLE 8.1. List of variable names in the spreadsheet that are not included in the List of principal symbols and acronyms in the introduction section of this paper.
Avail. Water 
water available to plant (field capacity  wilting point) [mm/m] 
J_{Plant} 
number of day of the year at time of planting [] 
J_{Dev} 
number of day of the year at beginning of development period [] 
J_{Mid} 
number of day of the year at beginning of midseason period [] 
J_{Late} 
number of day of the year at beginning of late season period [] 
J_{Harv} 
number of day of the year at time of harvest or death [] 
Max. Ht. 
mean height of vegetation during the midseason period [m] 
MAD during initial stage 
management allowed depletion fraction during the initial growing period [] 
MAD after initial stage 
management allowed depletion fraction following the initial growing period (during all other periods) [] 
Root_{min} 
average depth of "effective" root zone during the initial period (also described as Z_{r min}) [m] 
Root_{max} 
maximum depth of "effective" root zone (also described as Z_{r max}) [m] 
FIGURE 8.2. Daily values for K_{cb} from the calculation example of Figure 8.1
The daily values calculated for K_{cb} and K_{c} are illustrated in Figure 8.2. The daily soil water depletion at the end of each day calculated in the spreadsheet example is graphed in Figure 8.3. Figure 8.3 illustrates the effect of an increasing root zone on the allowable depletion. The allowable depletion is the same as the readily available water (RAW) when it is assumed that MAD = p, the evapotranspiration depletion factor. The depth of the effective root zone is calculated on each day as:
_{} (81)
and
Z_{r i} = Z_{r max} for J ³ J_{mid} (82)
where
Z_{r i} effective depth of the root zone on day i [m]Z_{r min} initial effective depth of the root zone (at the beginning of the initial period (planting))
Z_{r max} maximum effective depth of the root zone during the midseason period (from Table 22 of Chapter 8)
J Day of year [1 to 366]
Z_{r min} is the same as variable Root_{min} that is used in Figure 8.1 and Z_{r max} is the same as Root_{max}. Equations 81 and 82 presume that the development of the root zone increases in proportion to the increase in K_{cb}. This implies that the maximum effective root depth is reached by the beginning of the midseason. Other approaches to estimate Z_{r} can be used, including interpolations based on time of season, for example:
_{} (83)
and
Z_{r i} = Z_{r min} when J < J_{start}, and Z_{r i} = Z_{r max} when J > J_{max}
where:
J_{start} Day of year at beginning of the increase in Z_{r i} beyond Z_{r min}
J_{max} Day of year at the attainment of maximum rooting depth
Z_{r min} for annual crops should represent the depth of seed placement plus an additional depth of soil that may contribute water to the seed as it extends its initial roots downward following germination. For many annual crops, Z_{r min} can be estimated as 0.15 to 0.20 m.
The value used for MAD is given a separate and larger value during the initial period to account for the ability of roots for some crops to extract water at relatively dry water contents during germination and during the initial period with little impact by stress. In this example, it is assumed that p = MAD.
The irrigation period for the bean crop is presumed to begin at planting and to terminate halfway through the late season period. Therefore, the last irrigation date is on day 225. The bean crop exhibited only a small amount of stress following day 225, since the K_{c} was declining. The stress coefficient (K_{s}) is calculated in column AJ of the spreadsheet.
The fact that irrigations are not applied in the spreadsheet until the the soil water depletion at the end of the previous day is greater than or equal to RAW occasionally causes a small amount of stress on the day prior to irrigation (see K_{s} in column AJ). The impact of K_{s} on K_{c adj} was small before planting and near the end of the growing season because K_{cb} is small relative to the potential value for K_{e} during these periods.
This particular example is intended only to demonstrate how to apply the soil evaporation equations during scheduling of irrigations. The procedure used to determine the irrigation schedule and the assumptions used may not always be appropriate. The reader should modify the irrigation scheduling procedure to fit the conditions of the local area.
FIGURE 8.3. Soil water depletion at the end of each day calculated in Figure 8.1