Annex 8. Calculation example for applying the dual K_c procedure in irrigation scheduling

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 soil-water 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, u₂ and T_dew. The T_max and T_dew columns are used to calculate daily RH_min. The u₂ 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 Quattro-Pro language (versions 5 and higher) can be downloaded from the FAO internet site.

FIGURE 8.1. Example Spreadsheet for Calculating ET_c = (K_cb + K_e) ET_o and an Irrigation Schedule⁽¹⁾

FIGURE 8.1, continued.

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.
The depletion of the evaporation layer at the end of the day is calculated according to Eq. 77, with root extraction of plant transpiration from the evaporation layer assumed to equal zero.

⁽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 one-half of the late-season 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 u₂ 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:

(8-1)

and

Z_{r i} = Z_{r max} for J ³ J_mid (8-2)

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 8-1 and 8-2 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:

(8-3)

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 half-way 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

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:	= 0.6108EXP((17.27G14)/(G14 + 237.3))
Eq. 11	J14:	= 0.6108EXP((17.27E14)/(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(F140.9 - 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), U14Q14)
	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$4)/100AE14$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, (AE14AF$5 - AG14)/(AE14AF$5 - AF14), 1)
Eq. 80	AK14:	= X14 + 014*AJ14
Eq. 85⁽⁹⁾	AU 4:	= +$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:	= 0.6108EXP((17.27G15)/(G15 + 237.3))
Eq. 11	J15:	= 0.6108EXP((17.27E15)/(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$5, P14)
Eq. 72	Q15:	= MAX(1.2 + (0.04(F150.9 - 2) - 0.004(K15 - 45))(P15/3)^0.3, 015 + 0.05)
⁽⁴⁾	R15:	= IF(AH14 > 0, AH14/$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),U15Q15)
	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$4)/100AE15$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, (AE15AF$5 - AG15)/(AE15AF$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. The depletion of the evaporation layer at the end of the day is calculated according to Eq. 77, with root extraction of plant transpiration from the evaporation layer assumed to equal zero.
⁽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 one-half of the late-season 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 u₂ and RH_min during the midseason period (footnote 1).

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]

Annex 8. Calculation example for applying the dual Kc procedure in irrigation scheduling

Annex 8. Calculation example for applying the dual K_c procedure in irrigation scheduling