N o : AOfb Cote

. 8. Statistical Models of the Climatic Growing Period .Crop Potentials Pierre Franquin

PRESENTATIOh (The E d i t o r s )

This paper p r e s e n t s a s t a t i s t i ca l model f o r c a l c u l a t i n g t h e o v e r a l l water ba lance f o r a crop growth p e r i o d . It a l s o d e f i n e s the growing season i n terms of t h e p r o b a b i l i t y of o b t a i n i n g a f a v o r a b l e per iod f o r non- i r r iga ted c rops between two g iven d a t e s of t h e y e a r .

When cons ider ing t h e s u c c e s s i v e 10-day p e r i o d s which make up a growing season, i t i s p o s s i b l e t o c h a r a c t e r i z e each elementary per iod i n terms of t h e p r o b a b i l i t y of o b t a i n i n g dur ing t h a t p e r i o d p r e c i p i t a t i o n l e v e l s s u p e r i o r o r e q u a l t o a q u a n t i t y of water con- s i d e r e d s u f f i c i e n t f o r good crop growth. A s e r i o u s problem arises, however, when a t tempt ing t o c a l c u l a t e t h e p r o b a b i l i t y of o b t a i n i n g p r e c i p i t a t i o n l e v e l s throughout t h e accumulated consecut ive 10-day p e r i o d s which c o n s t i t u t e a growing season ( e . g . , 1 4 consecut ive 10- day p e r i o d s ) . That i s , t h e elementary p r o b a b i l i t i e s c h a r a c t e r i s t i c of each 10-day per iod are not independent of, each b t h e r , so t h a t t he composite p r o b a b i l i t y c a l c u l a t e d on t h e base of t h e s e elementary p r o b a b i l i t i e s has no meaning.

D r . Franquin proposes a method which a l lows u s t o determine an index t h a t expresses t h e p r c b a b i l i t y of ob ' ta ining s u f f i c i e n t pre- c i p i t a t i o n throughout t h e l e n g t h of t h e growing season, d e f i n e d by a s t a r t i n g d a t e and a n ending d a t e . It i s p o s s i b l e t o vary t h e s e dates--f ive days on e i t h e r s i d e , f o r example--and t o c a l c u l a t e t h e c h a r a c t e r i s t i c i n l e x f o r t h e p e r i o d under c o n s i d e r a t i o n each t i m e . The i n t e g r a t i o n of t h i s index i n t o a g l o b a l water b a l a n c e model a l lows f o r r a t i o n a l management of a g r i c u l t u r a l water resources .

BACKGROUND (The Author)

In 1967, fo l lowing p u b l i c a t i o n of "A Study of Agroclimatology of t h e Semiarid Area South o f t h e Sahara i n West A f r i c a , " t h e French Minis t ry of Cooperat ion and ORSTOM ( O f f i c e of Overseas S c i e n t i i i c and Technica l Research) concluded t h a t some major gaps i n t h e

The a u t h o r would l i k e t o thank Mira Shah, t r a n s l a t o r of ICRISAT I n d i a , f o r a s s i s t i n g with t h e t r a n s l a t i o n of t h i s paper i n t o English:

93

meteoro logica l and agronomic d a t a e x i s t e d . They decided t o s tan- d a r d i z e and develop c l imato logy proper i n t h e 13 French-speaking Afr ican c o u n t r i e s i n o r d e r t o g i v e more r e l i a b l e foundat ions t o agrocl imatology. The o p e r a t i o n fol lowed t h e s e p r i n c i p l e s : s tan- ,

d a r d i z a t i o n of equipment and methods i n a l l s t a t i o n networks, d a t a c e n t r a l i z a t i o n by one o r g a n i z a t i o n , p u b l i c a t i o n of a l l d a t a i n t h e same monthly r e p o r t , a v a i l a b i l i t y of any pr imary o b s e r v a t i o n s , and d a t a process ing through h igh-ef f ic iency means. For i n s t a n c e , a l l t h e d a i l y ,measurements of r a i n f a l l made from t h e o u t s e t , i n about 1,500 s t a t i o n s , were eva lua ted and then recorded on card and mag- n r t i c t a p e s . T h i s f i r s t s t e p was necessary b e f o r e under tak ing a v a r i e t y of d e t a i l e d a g r o c l i m a t i c s t u d i e s , whatever t h e t i m e and space s c a l e s might be . Various programs were c a r r i e d out i n o r d e r t o meet d i f f e r e n t needs--above a l l i n t h e f i e l d of f requency analy- sis of r a i n f a l l and water balance--for a g r i c u l t u r a l o f f i c e s i n most of t h e c o u n t r i e s , as w e l l as f o r i n t e r n a t i o n a l o r g a n i z a t i o n s (FhO, CIEH-USAID, OMVS, CHAD BASIN, e t c . ) . Moreover, ORSTOPl cteveloped a methodology--which tias recextrly been a p p l i e d t o t h e whole Ivory Coast--in o r d e r t o under take g e n e r a l a g r o c l i m a t i c s t u d i e s .

INTRODUCTION

I n agrocl imatology , d a t a process ing h a s allowed the e s t a b l i s h - nient of i n c r e á s i n g l y more d i v e r s i f i e d models t h a t can answer spe- c i f i c q u e s t i o n s . l a i l e t h i s e f f o r t should be encouraged, i t should not d e t r a c t ' a t t e n t i o n from t h e g e n e r a l q u e s t i o n of c h a r a c t e r i z i u g t h e " c l i m a t i c growing per iod ," o r season.

Considerable improvements have been made s i n c e t h e t i m e when t h e growing p e r i o d tiad t o be def ined through niean r a i n f a l l , tempera- t u r e , and humidity curves , o r even curves o i sunshine d u r a t i o n o r g l o b a l r a d i a t i o n . Curves o€ potent , ia l c v a p o t r a n s p i r a t i o n (PET). and c l i m a t i c water b a l a n c e (PET-R) were used l a t e r . The p r e s e n t d i scus- s i o n about s t a t i s t i c a l models of t h e growing per iod r e s u l t s from t h e use of d a t a process ing i n - s t a t i s t i c a l o p e r a t i o n s . Now i t i s pos- s i b l e t o s ta te a t t h e o u t s e t t h e requirements that niust b e m e t by a n i n t e r a n n u a l "frequency expression" of t h e p e r i o d under coiisidera- t i o n : (1) t o g i v e a d e t a i l e d ( i f not d a i l y ) accouI?tl of t h e i n t r a - annual v a r i a b i l i t y of c l i m a t i c c o n d i t i o n s ; (2) t o be cont inuous i n time, l i k e t h e growing per iod i t s e l f , which shows a cont inuous i n t e - g r a t i o n of c l i m a t i c e lements; ( 3 ) t o g i v e a s imple b u t e f f i c i e n t account of t h e p o s s i b i l i t i e s of crop development and growth, i n terms of " p r o t i b i l i t y " ; ( 4 ) n o t t o be a mere v i s u a l r e p r e s e n t a t i o n , b u t to- become an o p e r a t i o n a l ins t rument i n a g r i c u l t u r a l p lanning , r e s e a r c h , and ex tens ion; (5) t o be a p p l i e d t o a s y n t h e s i s of t h e c o n s t i t u e n t s of the- growing-per iod , inc luding some s o i l water c h a r a c t e r i s t i c s , because of t h e d i f f i c u l t i e s r a i s e d by compound p r o b a b i l i t i e s .

Fram t h e p o i n t of view , o f p r o b a b i l i t f e s , any a g r i c u l t u r a l s i t u a t i o n i s complex, re la t ive b o t h t o c l i m a t e and mois ture condi- t i o n s , even i f t h e l a t t e r are l i m i t e d t p a s i n g l e r a i n f a l l expressed i n s t a t i s t i ca l terms.

E

94

STATISTICAL ANALYSIS OF RAINFALL

The s i m p l e s t method f o r c h a r a c t e r i z i n g mois ture v a r i a b i l i t y i s a s t a t i s t i c a l a n a l y s i s of r a i n f a l l . S i n c e a frequency model of t h e growing per iod m u s t account bo th f o r i n t r a - and i n t e r a n n u a l v a r i - a b i l i t i e s , time-step i n t e r v a l s t h a t are chosen must s a t i s f a c t o r i l y r e f l e c t in t ra -annual change. of &mual r i i i n f a l l totals is not adequate . It i s more u s e f u l t o s tudy t h e d i s t r i b u t i o n s of monthly t o t a l s ; d i s t r i b u t i o n s of t o t a l s

E s t a b l i s h i n g S t a t i s t i c a l d i s t r i b u t i o n s

' over 10-day, weekly, o r 5-day i n t e r v a l s are even more u s e f u l . Time d i s t r i b u t i o n i s a c o n d i t i o n of " d i s c o n t i n u i t y .I' Agronomic

e v e n t s must b e f i x e d w i t h i n , a t most, a 10-day i n t e r v a l i n r e l a t i o n t o cropping schedule: l and p r e p a r a t i o n , sowing, t e c h n i c a l opera- t i o n s , p e s t c o n t r o l t r e a t m e n t s , i r r i g a t i o n , h a r v e s t , e t c . The grow- i n g p e r i o d and t h e phases of crop development a r e a l s o cont inuous, and t h e i r p r o b a b i l i t i e s of s u c c e s s should be es t imated . T h i s i s a c o n d i t i o n of " c o n t i n u i t y . ' I

I n o r d e r t o d e a l w i t h t h e s e c h a r a c t e r i s t i c s of c o n t i n u i t y and d i s c o n t i n u i t y , ORSTON h a s formulated a progral.: f o r t h e s t a t i s t i c a l a n a l y s i s of r a i n f a l l . Its advantage i s t h a t i t ana lyzes r a i n f a l l (accord ing t o t h e incomplete t r u n c a t e d gamma f u n c t i o n ) w i t h i n any i n t e r v a l of n days (n from 1 t o 365) , which can move from m t o m days (m from 1 t o n ) . T h i s method i s employed a t Ouagadougou sta- t i o n (Upper Vol ta ) f o r 10-day i n t e r v a l s t h a t may v a r y 5 days on e i t h e r s i d e accord ing CO c o n d i t i o n s ( f i g u r e 8 . 1 ) .

of t h e growing p e r i o d are d iscussed i n t h i s paper . For i n s t a n c e , f i g u r e 8.2 shows t h e Ouagadougou s t a t i o n (12'21'N), which h a s a t r o p i c a l c l i m a t e w i t h a s i n g l e r a i n f a l l (R) peak. The growing per iod i s expressed i n s t a t i s t i c a l r a i n f a l l and e v a p o t r a n s p i r a t i o n (PET) terms. The s t a t i s t i ca l r a i n f a l l i s based on two t i m e scales: t h e IO-day t ime s t e p moving from 10 t o 10 days , and t h e 30-day t ime s t e p , a l s o moving from 10 t o 10 days. Also shown a r e : (1) t h e median r a i n f a l l curve, i . e . , r a i n f a l l t h a t i s exceeded by PET f o r one out of two years ( p r o b a b i l i t y of PET exceeding 0 .50) ; (2) t h e curve of r a i n f a l l t h a t i s exceeded f o r f o u r out of f i v e y e a r s (prob- a b i l i t y of PET exceeding 0 .80) ; and (3) t h e curves f o r mean poten- t i a l e v a p o t r a n s p i r a t i o n v a l u e s (PET and PETIZ) c a l c u l a t e d accord jng t o Penman. S ince PET i s less v a r i a b l e t h a n R, i t i s n o t analyzed i n C ~ S U S of frequency . d e s c r i b e t h e growing per icd mainly a s a r e s u l t of t h e i r i n t e r s e c - t i o n s , which d i v i d e t h i s per iod i n t o subperiods. However, t h i s d i v i s i o n v a r i e s f o r time i n t e r v a l s of 10 and 30 days. There may be, i n f a c t , a 20-day d i f f e r e n c e i n d u r a t i o n and p o s i t i o n of two cor- responding subper iods . S ince t h e IO-day t i m e i n t e r v a l i s s h o r t e r , i t g i v e s a more adequate d e s c r i p t i o n of t h e in t ra -annual change than a n i n t e r v a l of 30 days, b u t i t i s l e s s adequate t h a n ari i n t e r v a l of 7 o r 5 dzys.

The p r o b a b i l i t i e s of exceeding PET l e v e l s g i v e a more advanced s t a t i s t i c a l r e p r e s e n t a t i o n of t h e growing per iod and t h u s f a c i l i t a t e t h e d i s c u s s i o n of t h i s q u e s t i o n .

Only t h e program a p p l i c a t i o n s r e l a t e d t o t h e frequency modeling

R a i n f a l l f requency curves , combined wi th PhT and PETIZ curves ,

s t a t i o n Number 200238 Upper Volta Ouagadougou S t a t i o n

Dirplaceinent Days

Date o f the First Day 10s I O * I I I I I & I P L

Number of Consecutive Days

Median Level 7.69 A.51 8.99 12.66 15.b2

Range o f Observations 83.5 .2.5 .I.2 13.6 10.0 IR.. ,6.* ,..L I l . 9

1.1 9.2 a.*

7.5 7 . 2 6 .7 5.0 4.6 * . 6 1.0 3.b 1.5 ,.LI ,.a I . , 1.2 l . 2 0 .8 0 . 1 o.* 0 .0 0 . 0 0 . 0 0 . 0 0.0 0.0 0 . 0 0 . 0 ".O ".O 0 . 0 0 .0

a."

8J.S .n.o 16.6 3 0 . 0 26.3 20.1 ,P." 1e.9 I..* , , . O 10.7

*.e Q.4 0.3 R.9 8.5 7 .5 6.2 6. o 3.2 5.6 5.6 ..6 4.. ..o 3.6 2 . . 2.3

I . , L.2 I . , 0 .6 ".. o.. 0.0 0 . 0 0 . 0 0 . 0 0." O . " O."

t.n

53.2 *C.> 3'.. 10.1 2I.O 21.0 *o.- ?O." 17.1 ,a.. ,e., 15.5 12.' 10.5

9.8 A.* 8 . 7 e.* 6.5

.4.L e. I 6 . l 5 . 0 ..c ..> ..o 1.8 2.7 *.6 2.. 2.1 t.7 0.P 0.U d .2 u.0 0 .0 0.0 0.0 0 .0 0 .0 o. *

.,.o L5.9 15.1 ..,? l7.S 1C.Z I..* 27." >a.* d0.J I5.J a..> >..o 23.. I I . 0 I . . , 13 .2 IZ.6 12.1

9.6 ".? 4.2 7 . 6 7 .0 I .5 5 .6 5 . I 5 . 0 4 . a L.5 1.2 2 .7 2.b 7 . . >.a I . ? I .+ 0.- 0 .6 0 . 0 q . 0 0 .0

h6.5 . 51.7 50.9 -6.7 ... 6 3 3 . t 10.9 .o., 11.1 ..:., 2e.e >... >I.* L..5 >..O E l . . 1 2 . 2 21.4 a o . , * * . O 15.5 i . . , I ? . . ,? .S I 2 . 5 12 .7 L 2 . l ,o.- 10 .1

5.* 4.2 J.2 3.0 2.4 I . ? I . = I . . , 0.9 n.7 0 .7 " . I ".O

Date o f the F i r s t Day

Gama Parameter

l o b

0 . B I O

,..575

O . J O 0

0.0

0.0

0.0

*.o 0.0

0.0

0.0

0.0

1.11

3..7

6 - 1 6

9.3.

12.22

11.17

21.53

I I . 0 8

L6.15

56 .60

FIGURE 8.1 Example o f s t a t i s t i c a l analysis o f r a i n f a l l according to the incomplete truncated gamma function. for nonexceedence of r a i n f a l l values.

The lO-dav time-step interval may move from 5 to 5 days on e i ther side. Probabil i t ies are

-,

FIGURE 8.2 Growing period a t Ouagadouyou (Upper Volta. 12'21'N), which has a t ropica l c l i - mate with a single r a i n f a l l peak. (probabil i ty 0.50) and four out of f i v e years (probabil i ty 0.80). intersected by curves o f PET and PET/2 values. The subperiods defined by these intersections have d i f f e r e n t posi- tions and durations according t o whether the time-step interval i s 10 o r 30 days.

Curves show r a i n f a l l which i s exceeded one out o f two

, , i.

. ' . G 96

PROBABILITIES OF EXCEEDING PET LEVELS

Figure 8.2 shows exceeded r a i n f a l l w i t h p r o b a b i l i t i e s of O .50 and 0 .80 i n r e l a t i o n t o PET and PETIZ. Conversely, i t i s p o s s i b l e t o r e p r e s e n t t h e p r o b a b i l i t i e s of exceeding r a i n f a l l , which i n t h i s c a s e a r e e q u a l t o PET and PETI2 ( f i g u r e 8 . 3 ) . The curves f o r t h e s e p r c b z b i l i t i e s of exceeding r a i n f a l l ( i f t h e s e can be t r a c e d ) f o r 10-day and 30-day time-step i n t e r v a l s , r e s p e c t i v e l y , do not coin-.

~ However, t h e t r u e p r o b a b i l i t i e s yere c,âlcula&>ed as. a c c p r a t e l y . c i d e .

*as poss ib le , . i r r e s p e c t i v e " o f the: time i n t e x v a l (10 o r 30' da$s). curves a r e d i f f e r e n t because t h e p r o b a b i l i t y c a l c u l a t e d f o r a 30-day i n t e r v a l does n o t account f o r t h e r a i n f a l l d i s t r i b u t i o n i n eûch of t h e t h r e e 10-day i n t e r v a l s . There i s - n o r u l e a l lowing t h e combina- t i o n of t h e t h r e e elemectary p r o b a t i l i t i e s ( r e l a t e d t o t h e t h r e e 10-day i n t e r v a l s ) i n t o a compound p r o b a b i l i t y f o r 30 days t h a t would account f o r t h e r a i n f a l l d i s t r i b u t i o n i n each of t h e t h r e e 10-day i n t e r v a l s . Moreover, t h i s p o s s i b i l i t y i s r e j e c t e d because, even though t h e r e i s no s t a t i s t i c a l dependence (au tocorre la . t ion) betweqn t h e r a i n f a l l of s u c c e s s i v e 10-day i n t e r v a l s , t h e p r o b a b i l i t i e s of exceedicg PET o r PETI2 are l i n k e d .

F igure 8.4 shows t h a t t h e sequence observed i n t h e s u c c e s s i v e 10-day i n t e r v a l s of + and - s i g n s , i n d i c a t i n g excess o r d e f i c i t r a i n f a l l , i s n o t random. Therefore , i f t h e time-step i n t e r v a l i s reduced towards d i s c o n t i n u i t y , t o g i v e a more r e a l i s t i c d e s c r i p t i o n of t h e in t ra -a imual change, t h e elementary p r o b a b i l i . t i e s of t h e time-step i n t e r v a l cannot be e x t r a p o l a t e d f o r longer i n t e r v a l s . But whi le t h e time i n t e r v a l i s i n c r e a s e d towards cont inui ty- - to cover t h e cont inuous phases of crop development f o r t h e purpose of eva l - u a t i n g t h e p r o b a b i l i c i e s of success- - i t does n o t t a k e i n t o account t h e r a i n f a l l d i s t r i b u t i o n dur ing t h e s e c rop phases . .*

One' of the s o l u t i o n s , t o t h i s c o n t i n u i t y - d i s c o n t i n u i t y problem i s t h e u s e of t h e "frequency growing per iod ," based p r e c i s e l y on t h e observa t ion t h a t t h e sequence of exceeding PET l e v e l s (and b e t t e r s t i l l AET/PET l e v e l s ) i s n o t random.

T h k a ,

..

THE FRE.QUEVCY GROWlNG PERIOD .<

3'. y ,f . . P r i n c i p l e

T h i s s t a t i s t i c a l d e s c r i p t i o n of t h e growing p e r i o d meets the requirements mentioned i n t h e i n t r o d u c t i o n . The frequency growing per iod ' i s n o t a discont int tous e x p r e s s i o n i n t i m e l i k e , t h e prev ious models, bu t a cont inuous express ion t h a t e n a b l e s t h e e v a l u a t i o n of t h e p r o b a b i l i t y of a water a v a i l a b i l i t y l e v e l f o r any t i m e i n t e r v a l .

T h i s system c o n s i d e r s sequence and n o t frequency of occurrence o f an expected event w i t h i n t h e s u c c e s s i v e time i n t e r v a l s ( f i g u r e 8 . 4 ) . I n f a c t , f requency i s cons idered i n t h e f i r s t and l a s t i n t e r - v a l s of a uniform sequence, which mark t h e c r o s s i n g s of t h r e s h o l d s r e p r e s e n t i n g 710uts tanding" events . These e v e n t s , which c h a r a c t e r i z e t h e growing per iod by d e f i n i n g i t s subper iods , are taken indepen- d e n t l y and expressed i n terms of f requency b e f o r e be ing cons idered t o g e t h e r . Any c l i m a t i c o r phenologica l event--fact of crop devel- opmenL r e l a t e d t o c l i m a t i c event--that i s of t e c h n i c a l , economic,

I

. .

I

97

I

FIGURE 8.3 Growing period at Ouagadougou. Curves show probabilities of rainfall exceeding PET and PET/'L values, according to whether the time-step interval is 10 or 30 days. because the probability calculated for a 30-day interval does not ac- count for the rainfall distribution in each of the three 10-day intervals.

These curves do not coincide

FIGURE 8.4 the successive ]&day intervals. deficit, is not random.

In each year of the rainfall sample, rainfall is compared with PET and PET/P values in The sequence of t and - signs, indicating rainfall excess or

L

98

exper imenta l , o r t h e o r e t i c a l i n t e r e s t is considered as a n outs tand- i n g event f o r d i v i d i n g t h e grcjwing p e r i o d i n t h e annual cyc le . There a r e a s many p a r t i c u l a r growing pericjäs f o r t h e came l o c a t i o n a s t h e r e a r e s p e c i f i c p r o j e c t s .

The growing per iod can be g e n e r a l i z e ä , however, i f t h e e v e n t s considered a r e v a l i d f o r a l l c a s e s . T h i s i s p o s s i b l e s i n c e they are r e l a t e d n o t co phenology, wliich i s c r o p - s p e c i f i c , bu t t o t h e growth o r .product.&ion of dry ,mat ter ( i n r e l t i t i o n t o AET/PET), which i s con- t r o l l e d , by u.ore g e n e r a l laws. An example i s t h e ,.two e v e n t s nark ing t h e i n t e r s e c t i o n s of a n i n c r e a s i n g and d e c r e a s i n g r a i n f a l l curve and t h e PET curve ( f i g u r e 8.5). I n a f i r s t approximation, t h e s e two e v e n t s have t h e same s i g n i f i c a n c e f o r c rops gr0wir.g under c losed cover: f o r e , i n theory , optinium d r j niiitter product ion . t h e r e l a t i v e c h a r a c t e r of an event should be s t r e s s c d : the .accurhcy of t h e phenomena (space and t i m e s c a l e s , p r e c i s i o n of

. i' measurements, r e p r e s e n t a t i v e n e s s of formula, 1 e t c . ) ,of the^ .cropL . c l i a r a c t e r i s t i c s , maximum e v a p o t r a n s p i r a t i o n , s o i l type , e t c . '

-. L

t h t ae te r_ i i ia t ion of t h e subperiod when AET = PET and t h e r e - " I n theory" because

' in f luence of

Cons t ruc t i o n

The frequency exprcss ion of t h e growing per iod f i t s i n t o a system of c o o r d i n a t e s whose x-axis r e f e r s t o t i m e and whose y-axis i$ 2 s c a l e of r e l a t i v e f r e q u e n c i e s o r p r o b a b i l i t i e s . I n t h i s sys- tem, t h e v ú r i a b i l i t y of each o u t s t a n d i n g event can b e r e p r e s e n t e d by :

1. A h is togram of d e n s i t y f r e q u e n c i e s . Over s u c c e s s i v e y e a r s , t h e d i s t r i b u t i o n of che p o s i t i o n i n t ime of an event def in3ng e i t h e r t h e beginning, t h e end, o r an i n t e r m e d i a t e s ta te i s r e p r e s e n t e d by a frequency d i s t r i b u t i o n his togram based on an i n t e r v a l of 15, 10, 7 ,

FIGURE 8.5 Curves of r a i n f a l l and PET. The i r i n t e r s e c t i o n s (ou t - standing events) d e f i n e che humid subperiod 8182, when AET equals ?ET, f o r crops growing under c losed cover ( i n general) .

~~ ~ ~~

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o r even 5 days (a l though f o r s t a t i s t i c a l purposes , t h e t i m e v a r i a b l e i s cont inuous) . The e m p i r i c a l d i s t r i b u t i o n s of e v e n t s may v a r y wide ly , b u t f o r p r a c t i c a l purposes , they need n o t s t r i c t l y cor res - pond t o t h e o r e t i c a l d i s t r i b u t i o n laws .' (While t h i s r a r e l y a p p l i e s t o r a i n f a l l , i t i s more common f o r temperature events . ) per iod t h e r e f o r e comprises f requency d i s t r i b u t i o n s of e v e n t s which, taken i n twos n o t n e c e s s a r i l y i n success ion , s t a t i s t i c a l l y ckarac-

( f i g u r e 8-.6). -The number of- thebe e v e n t s i s l i m i t e d - b y t h e i r dcgree of dependence, s i n c e two e v e n t s a r e more l i k e l y t o be l i n k e d when c l o s e r i n time.

2. An i n t e g r a l polygon of r e l a t i v e f r e q u e n c i e s , ob ta ined by cumulating t h e r e l a t i v e f r e q u e n c i e s represented i n t h e d e n s i t y h i s - togram. For example, i f t h e t ime í n t r v a l i n c l i n e s towards zero , and i f t h e s a a p l e i s s u f f i c i e n t l y l a r g e , i t i s p o s s i b l e t o t r a c e a smooth curve of cumulat ive r e l a t i v e krequencies . This i s a sigmoid

, curve p l o t t e d by e l i m i n a t i n g minor i r r e g u l a r i t i e s and p o s s i b l y by f i t t i n g t o i t a t h e o r e t i c a l d i s t r i b u t i o n l a w .

The growing

. t e r i z e t h e beginning and end of t h e per iod arte each subperiod

f-

R e l a t i v e polygon i n t e g r a l s g ive a more e f f i c i e n t frequency express ion of t h e growing per iod than d e n s i t y his tograms. Given t h e r e p r e s e n t a t i o n of t h e s e i n t e g r a i s (polygons o r sigmoid curves) f o r each e v e n t , t h e y-axis g i v e s t h e p r o b a b i l i t y t h a t t h i s event lias a l r e a d y been achieved a t a d a t e g iven i n t h e x-axis ( f i g u r e 8 . 6 ) . The sigmoid curve f o r t h e "beginning" (B) event of a per iod g i v e s . t h e p r o b a b i l i t i e s t h a t t h e p e r i o d lias a l r e a d y begun, and t h e sigmoid curve € o r t h e "end" (E) event of a p e r i o a g i v e s t h e p r o b a b i l i t i e s t h a t i t h a s a l r e a d y ended. I n t h i s l a s t case , t h e focus I s on t h e complementary p r o b a b i l i t y t h a t t h e per iod i s s t i l l open; t h i s i c

FIGURE 8.6 Frequency growing per iod. Top: Frequency dens i ty histograms of outstanding events ( B = beginning event, E = end event, I = in te rmed ia te event) and t h e i r i n t e g r a l s igmoid curves; Sottom: The area and the shape o f the surface enclosed between two opening and c l o s i n g s i q r o i d curves i n t e g r a t e the p o s i t i o n s and dura t ions o f the period and subperiods.

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r e p r e s e n t e d by t h e sigmoid curve t h a t i s s y m i e t r i c a i t o t h e hor i - z o n t a l l i n e .

I n t e r p r e t a t i o n

T h i s geometr ic f requency model g i v e s a s t a t i s t i c a l representa- t i o n of t h e growing per iod t h a t i s b o t h a n a l y t i c a l and s y n t h e t i c . It shows a l l t h e p o s s i b l e durac ions and p o s i t i o n s of t h e per iod , + r i n g which t h e f a c t s of t h e cropping schedule , as w e l l a s t h e

, phenological. phases o i crop development, occur . On t h e o t h e r hand, t h e area and shape of t h e s u r f a c e enclosed by two "opening" á c d "c los ing" s i g n o i d curves in tegra t : t h e v a r i a b i l i t y of t h e p o s i t i o n and d u r a t i o n of t h e per iod under c o n s i d e r a t i o n .

The p r o b a b i l i t y t h a t che per iod h a s a l r e a d y begun a t a given d a t e , i r r e s p e c t i v e of i t s d u r a t i o n ( i . e . , i t s end) . i s g iven by t h e beginning sigmoid curve; tlic p r o b a b i l i t y t h a t i t i s s t i l l open, i r r e s p e c t i v e of i t s d u r a t i o n ( i . e . , i t s begir ining) , i s g iven by t h e c1osir.p. sigmoid curve. T h i s i s v a l i d whether o r no t t h e beginning and end a r e independent events . Eut t h e p r o b a b i l i t y t h a t t h e per iod i s open between two g iven d a t e s ( i . e . , a l r e a d y open b e f o r e t h e f i r s t d a t e and s t i l l open a f t e r t h e secoud d a t e ) w i l l be t h e prcduct of t h e p r o b a b i l i t i e s r e l a t e d t o t h e s e t w G à a t e s , i f t h e beginning and t h e ecd are independent o r can be cons idered a s such. In case of dependence, t h e c o n d i t i o n a l sigmoid curves of t h e opening sigmoid

' curve may be drawn. This model can be designed i n terms of water c o n d i t i o n s (gener-

a l l y t h e c a s e f o r t r c p i c a l r e g i o r s j ; energy c o n d i t i o n s ( f o r tem- p e r a t e r e g i o n s ) ; o r both water and encr4y c o n d i t i o n s ( f o r subt ropi - c a l , )Lediterracean, and h i g h - a l t i t u d e t r o p i c a l r e g i o n s ) . k'hcn t h e model is Lased on water c o n d i t i o n s , i t cali Le e a s i l y c o n s t r u c m d from very elementáry informat ion such a s r a i n f a l l d c t a , i f t h e r a i n - f a l l i a high enough t o e l i n i i n s t e random sequences. For t h i s , r a i n - f a l l t h r e s h o l d s t h a t c h a r a c t e r i z e che begirining and t h e end of each per iod and subperiod should be determined.

t h e enà i n terms of PET-level t h r e s h o l d s t o b e exceeded by r a i n f a l l , t h e model w i l l prove t o b e t h e most e i f i c i e n t i f i t i s based on AET/ PET l e v e l s ob ta ined from a s i m u l a t i o n of water ba lance . S ince t h e r e l a t i v e e v a p o t r a n s p i r a t i o n r a t i o (AET/PET) i s i n d i c a t i v e of dry nlcltter product ion , t h e s u r f a c e a r e a cf t h e geometr ic model repre- s e n t s t h e dry matte1 product ion c a p a c i t y . l t i s t h e r e f o r e con- s i d e r e d a r e l a t i v e p r o d u c c i v i t y - r e l a t e d c l i m a t i c index ( a l l t h i n g s be ing e q u a l ) . t e r i s t i c s ( tempera ture , g l o b a l o r p h o t o s y n t h e t i c rzc i ia t ion , e t c . ) .

same a r e a . The a d a p t a t i o n of a c u l t i v a r t o t h e c o n d i t i o n s repre- s e n t e d depends on t h e s e dimensions. I n p r o b a b i l i s t i c terms, t h e h o r i z o n t a l dimensions ( t i n e , sum of temperatures o r r a d i a t i o n s ) account f o r che p o s s i b i l i t i e s of crGp development, w h i l e tile v e r t i - c a l dimensions ( p r o b a b i l i t i e s of exceeding AET/PET l e v e l s ) account f o r t h e p o s s i b i l i t i e s of dry m a t t e r product ion .

requirements mentioned i n t h e i n t r o d u c t i o n : i n g from 5 t o 10 days, used t o e s t a b l i s h t h e d e n s i t y h is tograms,

Although i t i s more u s e f u l t o c h a r a c t e r i z e t h e beginning and

This a r e a could be f u r t h e r weighted by energy charac-

However, che dimensions, and t h e r e f o r e shapes, may vary f o r t h e

T h i s f requency express ion a l r e a d y meets he f j rst t h r e e t h e t i m e i n t e r v a l rang-

101

e x p l a i n s in t ra -annual v a r i a b i l i t y of t h e c l i m a t i c c o n d i t i o n s under c o n s i d e r a t i o n . However, f o r t h e i n t e r v a l between t h e sigmoid curves , t h i s f requency express ion i s cont inuous i n t ime and enables t h e e v a l u a t i o n of compound p r o b a b i l i t i e s f o r any t i m e i n t e r v a l ; i n p r o b a b i l i s t i c terms, i t a l s o g i v e s t h e p o s s i b i l i t i e s cjf development as w e l l as crop growth.

Before d e a l i n g w i t h a p p l i c a t i o n s ( f o u r t h requirement) der ived from t h e o p e r a t i o n a l a s p e c t of t h i s f requency e x p r e s s i o n , w e s h a l l s tudy examples which show t h a t i t can be e f f e c t i v c l y a p p l i e d t o t h e most synthes ized informat ion based on t h e water ba lance ( f i f t h requirement) .

EXAMPLES OF THE CONSTRUCTION OF TIIE NODEL

Relative e v a p o t r a n s p i r a t i o n (AET/PET) is eva lua ted by es tab- l i s h i n g t h e water b a l a n c e , where

R = r a i n f i l l I = p o s s i b l e i r r i g a t i o n

RF = runoff DR = dra inage A\$ = v a r i a t i o n of a v a i l a b l e s o i l water

AET = a c t u a l e v a p o t r a n s p i r a t i o n

The water b a l a n c e i s expressed as:

R + (I) 1 RF 2 DR ? AW = AET

T h i s equat ion , which i s v a l i d f o r t h e c u l t i v a t e d p l o t as w e l l as t h e watershed, i s v.ore o r less approximated by the numerous cur- r e n t l y used models proposed i n l i t e r a t u r e on t h e s u b j e c t . Whatever these models may be, they o p e r a t e accord ing t o a t ime-step i n t e r v a l ranging from a day ( i d e a l l y ) t o a month. For p r a c t i c a l reasons , t h e i n t e r v a l i s u s u a l l y 10, 7 , o r 5 days. F igure 8.7 shows such a model whose terms a r e as fo l lows:

P: r a i n f a l l dur ing i n t e r v a l s of 10 , 7 , o r 5 days ( n a t u r a l o r ca lendar )

smaller of t h e two v a l u e s (RS + P ) and AWC HD: a v a i l a b l e s o i l water above t h e w i l t i i i g p o i n t = t h e

HR: r e l a t i v e s o i l mois ture = HD/Ak;C ETP: c l i m a t i c p o t e n t i a l e v a p o t r a n s p i r a t i o n ( o r PET), l i m i t

ETM: maximum e v a p o t r a n s p i r a t i o n of t h e crop cover ( o r MET, 1 I of ETM

varies w i t h crop development), upper l i m i t of ETR K: crop c o e f f i c i e n t s = ETM/ETP

ETR: a c t u a l e v a p o t r a n s p i r a t i o n (o r BET) = f(HR, ETP)

RDR: runoff + dra inage = (RS + P) - AWC

D(RS): s o i l water d e f i c i t = AWC - RS

product ion index

RS: r e s i d u a l s o i l water = HD - ETR

RCDR: cumulat ive runoff + dra inage

ETR/ETP: r e l a t i v e e v a p o t r a n s p i r a t i o n ( o r AET/PET) = dry m a t t e r

FIGURE 8.7 grrowina n e t i o d i s d i v i d e d according t o t h e f o l l o w i n g events: (1) sowing ( th resho ld 0.50); ( 2 ) AETIPET r i s e s and remains h igher than 0.90; (3) AET/PET re tu rns t o less than 0.90; ($) AET/PET re tu rns t o less than 0.50.

Examole o f a water balance w i t h ALK = 100 mn, f o r any year a t Ouagadougou. The

E'iTI-ETI?: crop e v a p o t r a n s p i r a t i o n d e f i c i t , supplemented by i r r i g a t i o n

(ETPI-FTR)/ETPI: r e l a t i v e e v a p o t r a n s p i r ü t i o n d e f i c i t AWC: a v a i l a b l e water c a p a c i t y , v a r i a b l e o r c o n s t a n t

depending on whether t h e r o o t system i s annual o r p e r ennia 1.

Using t h i s model, t h e water ba lance can be s imula ted f o r each y e a r of t h e r a i n f a l l sample i f t h e i n t e r a n n u a l v a l u e s remain con- s t a n t f o r PET, which i s much less v a r i a b l e than r a i n f a l l . l a c i o n of t h e biilancr over s e v e r a l y e a r s i s u s e f u l f o r o b t a i n i n g p r o b a b i l i s t i c in format ion . For t h i s purpose, it i s not . t h e i n p u t v a r i a h l e s of t h e b a l a n c e - - r a i n f a l l , p o s s i b l e i r r i g a t i o n - - t h a t âre anzlyzed s t a t l s t i c a l l y , b u t t h e output variables--RS, D(RS), RDR, ETR, ETPI - ETR, and, as f a r as we a r e concerned h e r e , AET/PET. A program f r e q u e n t i a l l y c l a s s i f i e s (by f r a c t i l e s ) t h e i n t e r a n n u a l v a l u e s of each v a r i a b l e f o r each time i n t e r v a l ; i t i s t h u s p o s s i b l e t o p l o t curves similar t o those shown i n f i g u r e 8 . 3 ( p a r t i c u l a r l y kor AETIPET) and t h e r e f o r e having t h e same inadequac ies .

I t i s a l s o p o s s i b l e t o apply t h e p r i n c i p l e of the- f requency growing per iod t o t h e AETjPET v a r i a b l e . In t h e examples t h a t fo l low, we cons ider a n annual crop a t Ouagadougou s t a t i o n and a t Bouak; s t a t i o n a p e r e n n i a l crop (meadow o r p e r e n n i a l v e g e t a t i o n ) .

Example of a n Annual Crop a t Ouagadougou

The simu-

'

Tn t h e casc o: an annual c r o p , t h e major probleril I s t h e evalua- t i o n of MET v a l u e s (maxiniuu, AET) and K v a l u e s (K = kÍET/PET) f o r û

developing crop. h c e t h i s i 5 done, t h e sowir;g can be s i n d a t e d i n r e l a t i o n t o t h e expecced requi rements of t h e crop. I n our example, t h e crop was sowiì from 1 May ir; t h e f i r s t 10-day i n t e r v a l w i t h a t l eas t : 31: nim sf r a i n f d . 1 , which i s e q u a l t o PET/2. (Other combina- t i o n s can b e d e v i s e d . )

103

The frequency express ion of t h i s sowink event (1) i s repre- sen ted by t h e opening,s igmoid curve of t h e growing p e r i o d under c o n s i d e r a t i o n . T h i s per iod i s c h a r a c t e r i z e d by t h r e e o t h e r out- s t a n d i n g e v e n t s whose t i m e of occurrence a r e t h e 10-day p e r i o d s (unoer l ined i n f i g u r e 8 .7) where ( 2 ) P.ET/PET rises and remains h igher than 0.90, ( 3 ) &l'/PET r e t u r n s t o less than 0.90, and ( 4 ) AET/PET r e t u r n s t o less than 0.50.

The i n t e r a n n u a l v s r i a b i l i t y of t h e p o s i t i o n s of each of t h e s e e v e n t s i s represented by a f requency d e n s i t y his togram (not shoxn) and by a sigmoid curve. Corresponding t o an ANC of 50, 100, and 200 mm, t h r e e c a s e s w i l l appear ( f i g u r e 8 . 8 ) . The subpcr iods g r a d u a l l y become loiiger and t h e r e f o r e more s u i t e d f o r t h e f i t t i n g of l o n g e r crop c y c l e s ( o r t h e f i t t i n g of t h e sanie crop d u r a t i o n w i t h a h i g h e r p r o b a b i l i t y of s u c c e s s ) .

In o r d e r t o show t h e important r o l e of AWC i r 1 determining t h e s i z e of t h e growing p e r i o d , two s e p a r a t e graphs are g iven f o r Ouagadougou (average r a i n f a l l , 875 nun) showing (1) top: t h e sub- per iods c a l l e d "subhumid," c h a r a c t e r i z e d by t h e BETIPET p r o b a b i l i - t i es t o b e h igher than 0.50; and (2 ) bottom: t h e subperiods c a l l e d "humid," c h a r a c t e r i z e d by t h e AET/P'ET p r o b a b i l i t i e s t o be h i g h e r than 0.90 ( s e e f i g u r e 8 .8) . H o r i z o n t a l and v e r t i c a l dimensions a r e s i g n i f i c a n t l y improved when t h e a v a i l a b l e s o i l water c a p a c i t y i s increased .

Example of a Permanent Crop a t Bouaké

Since t h e crop cover i s considered t o be permanent, MET i s s a i d t o be e q u a l t o PET; t h e r e f o r e , K = 1. ,An example of t h e water ba l - ance i s g iven wi th an AWC of 60 nun f o r any year taken from t h e 50 which c o n s t i t u t e t h e r a i n í á 1 1 sample ( f i g u r e 8 .9) . T h i s same sample i s used a g a i n w i t h AWCs of 120 and 200 mm (not shown).

The Bouaké s t a t i o n (average r a i n f a l l , 1,200 nun) g e n e r a l l y h a s two r a i n y seasons s e p a r a t e d by a s h o r t d ry season. There are e i g h t ou ts tanding e v e n t s ( i n s t e a d of f o u r a t Ouagadougou), with one s i n g l e r a i n f a l l peak. Each of them i s under l ined , ill t h e columns f o r 10- clay p e r i o d s and f o r AET/PET (ETR/ETF), t o mark t h e c r o s s i n g of a t h r e s h o l d . These e i g h t e v e n t s a r e :

F i r s t r a i n y season 1. AET/PET rises ind remains more than b.50 2. AET/PET rises and remains more than 0.90 3 . AET/PET r e t u r n s t o less than 0.90 4 . AET/PET r e t u r n s t o l ess than 0.50

Second r a i n y season 5. AET/PET rises and remains niore than 0.50 6 . AET/PET r i s e s and remains more t h a n 0.50 7 . AET/PET r e t u r n s t o less than 0.90 8 . AET/PET r e t u r n s t o less t h a n 0.50 *

Between t h e s e two r a i n y seascns , t h e r e i s g e n e r a l l y á s h o r t dry season ( a decrease of r a i n f a l l ) c h a r a c t e r i z e d by t h e f r e q u e n c i e s of Occurrence of events 4 and 5. I n very dry o r very w e t y e a r s , some of t h e s e e i g h t e v e n t s do not occur ; t h e r e f o r e , some sigmoid curves

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FIGURE 3.9 7"41'N), which has a subequatorial climate with two rainfall peaks. standing events (2 x 1).

Example of a water balance with AUC 60 nun, for any year at BOUak6 (Ivory Coast, There are einht Out-

FIGURE 8.3 Frequency growing period at Ouagadougou, with three dif- ferent AWCs (50, 100, 200 mn). Too: subhumid subperiods (probability for AET/PET to be higher than 0.50); Bottom: humid subperiods (prob- ability for AET/PET to be higher than 0.90). Curves were calculated and plotted manually. ~~

105

may n o t reach 100 percent and/or do not s t a r t from O p e r c e n t (events 3 , 4 ; 5; a i i d . 5 . f o r t h e l a t i t e r ) .

The combinatiori df t h e s e e i g h t sigmoid curves shows the growing per iod t o - b e O .50 and 0.90. -of AET/PET. ' (Otlier valÜes can be devised . ) The t h r e s h o l d s of AET/PET h igher t h a n Oz90 can seldom be considered because of t h e d i s c o n t i n u i t y of t h e i r sequences, b u t any t h r e s h o l d e q u a l t o o r less than t h i s v a l u e c o u l d s e r v e as a u s e f u l c r i t e r i o n f o r a s p e c i f i c problem.

While t h e -frequency models f o r OuaFadougou were c a l c u l a t e d and cons t ruc ted manually, those t o r Bouaké were e s t a b l i s h e d e n t i r e l y by , microcomputer (HP 9845) , from t h e poinr: of d a t a e n t r y f o r d i v i d i n g t h e growing per iod f o r each of t h e 50 years of t h e r a i n f a l l sample-- 8 numbers from 1 t o 35 of t h e 10-day i n t e r v a l s f o r each AWC-year- s t a t i o n , t o t a l i n g 400 v a l u e s over 50 years--through p l o t t i n g of t h e curves. These curves a r e represented as fo l lows f o r an AWC of 6G mm:

1. F igure 8.10: (Top) I n t e r s e c t i n g rough curves f o r t h e subhumid subperiod ( t h r e s h o l d 0.50) . (Bottom) I n t e r s e c t i n g rough curves f o r t h e humid per iod ( t h r e s h o l d O . 90) .

2 . F igure 8.11: (Top) Adjusted curves f o r bo th t h e subhumid and humid subper iods , which t o g e t h e r c o n s t i t u t e t h e growing per iod . Here opening and c l o s i n g sigmoid curves i n t e r s e c t and d e f i n e a lower area on t h e one hand and an upper a r e a on t h e o t h e r hand ( f i g u r e 8 .10) . from t h e lower a r e a , which i s p o s i t i v e , r e s u l t i n g i n a "usefu l" f i n a l a r e a . (Bottom) Curves of p r o b e b i l i t i e s f o r d u r a t i o n s of t h e subllumid and humid subper iods and of t h e s h o r t d ry season, whatever t h e t ime p o s i t i o n of t h e d u r a t i o n under c o n s i d e r a t i o n .

The u p p e r . a r e a i s n e g a t i v e and must be s u b t r a c t e d

LPIITATIONS TO THE SYSTDi

The random nacure of t h e sequences of t h e e v e n t ' s occurrence r e p r e s e n t s a major problem f o r t h e system shown h e r e . O i ~ l y when nonrandom, uniform, m o r e , o r l e s s long sequences a r e found--for most y e a r s , w i t h i n c e r t a i n t ine regions-- is i t p o s s i b l e t o determine t h e frequency d i s t r i b u t i o n of t h e begilining and end of t h e evenc, i . e . , t o g i v e a s t a t i s t i c a l amount of t h e d u r a t i o n s and time p o s i t i o n s of the,se sequences ( f i g u r e 8 . 4 ) .

R a i n f a l l i s o f t e n t o o e r r a t i c t o c o n s t i t u t e uniform sequences, mainly i n t h e low r a i n f a l l c o u n t r i e s , such as t h e n o r t h e r n and southern l i m i t s of t h e Sahara ana probably t h e B r a z i l i a n Nor theas t . Sequences where r a i n f a l l exceeds PET l e v e l s are longer and more uniform s i n c e t h e p r o b a b i l i t i e s of exceeding PET are more c l o s e l y l inked . Because of mois ture c o n t e n t due t o t h e water s t o r e d i n s o i l ( e s p e c i a l l y s i n c e AWC i s h i g h q r ) , t h e use of t h e water ba lance i s t h e most e f f i c i e n t method f o r c o n s t r u c t i n g the model.

Under t h e c o n d i t i o n s i n t h e Sahara and t h e B r a z i l i a n E o r t h e a s t , however, even t h e water ba lance nethod cou1.d ricit be a p p l i e d . Here t h e p r o b a b i l i t i e s of exceeding PET levels are independent and i t i s p o s s i b l e to ' work w i t h t h e i r r e s p e c t i v e curves .

Another problem l i e s i n t h e s t a t i s t i c a l . dependence o f c e r t a i n sigmoid curves . Genera l ly , they a r e not r e c i p r o c a l curves (events 1 and 4 , 2 and 3 , 4 arid 5 , 5 and 8 , aiid 5 and 7 ) wliich sometimes

.’ 106

FIGURE 8.10 curves were c a l c u l a t e d and p l o t t e d by microcomputer. subhumid subperiod ( th resho ld 0.50) ; Bottom: I n t e r s e c t i n g rough curves f o r the humid sub- per iod ( th resho ld 0.90). The upper area determined by two i n t e r s e c t i n g rec ip roca l s i g m i d curves i s neaat ive and must be subt rac ted from the lower area which i s p o s i t i r e .

Frequency growing per iod a t Bouaké, w i t h AWC = 60 mm. Un l ike Ouagadcugou, Top: I n t e r s e c t i n g rough curves fo r the

107

FIGURE 8.11 Top: Adjusted curves for subhumid and humid subperiods which c o n s t i t u t e :he frequency growing per iod. f u l ” curves. and o f the shor t d ry season, whatever the t ime p o s i t i o n o f the dura t ion being considered.

The subt rac t ion o f the areas (see f i g u r e 8.10) r e s u l t s i n use- Bottom: P r o b a b i l i t y curves f o r dura t ions o f the subhumid and humid subperiods

t

I . 108

i n t e r s e c t . In t h e l a s t c a s e , t h e r e i s a d e f i n i t e , though n o t h igh , dependence between t h e sigmoid curves . For t h e curves t o b e inde- pendent , i t should be p o s s i b l e t o connect any p o i n t of t h e c l o s i n g sigmoid curve fsom any p o i n t of t h e opening sigmoid curve wi thout t u r n i n g backwards. .However, f o r t h e s e r e c i p r o c a l i n t e r s e c t i n g curves , i t appears (a l though t h e r e i s s t i l l no mathematical p roof ) t h a t t h e a r e a c o r r e c t i o n i s r e l a t e d a t t h e same time t o t h e cor rec- t i o n of dependence t h a t i s i n h e r e n t t o t h e i n t e r s e c t i o n . This ques- t ior i of dependence comes up more o f t e n f o r s u c c e s s i v e n o n r e c i p r o c a l

- curves t h a t are more o r less p a r a l l e l and move i n t h e same d i r e c t i o n (e.g., e v e n t s 1 and 2 , and 2 and 7 ) .

APPLICATIONS

' Thè frequency growing p e r i o d i s p a r t i c u l a y l y su i ted . , f ,o$ 'appl i - J . . c a t i o n s i n p lanning a t any l e v e l . e v a l u a t i n g n a t u r a l r e s o u r c e p o t e n t i a l i t i e s should provide both t h e p o t e n t i a l c l i m a t i c l e v e l of p r o d u c t i v i t y i n space and t h e l o c a l v a r i a b i l i t y i n time. The frequency growing per iod f u l l y meets t h i s double requirement . It a l s o e f f i c i e n t l y i n t e g r a t e s d i t h e repre- s e n t a t i v e elements through water ba lance , which c o n s i d e r s t h e s o i l c h a r a c t e r i s t i c s t h a t determine AWC. For i n s t a n c e , us ing d a t a from 50 r a i n f a l l s t a t i o n s , t h e Ivory Coast w a s descr ibed by t a k i n g as a b a s i s t h r e e AWCs (60, 120, and 200 mm) and e s t a b l i s h i n g an i n t e r - p o l a t i o n a t l a s among t h e SO s t a t i o n s . Moreover, t h e areas regre- sen ted by t h e models can be weighted by energy c h a r a c t e r i s t i c s ' s u c h a s tempera ture and g l o b a l and p h o t o s y n t h e t i c r a d i a t i o n .

l i s h e d , no t on ly f o r d i f f e r e n t s o i l s covered by t h e same s t a t i o n , Lut a l s o f o r v a r i o u s c r o p s having s p e c i f i c requirements . With d a t a process ing , i t i s p o s s i b l e t o r a p i d l y determine ( f o r the same wcather s t a c i o n ) a s many models as t h e r e are s o i l s and crops , mul t i - p l i e d by the number of s i m u l a t i o n s of p lan t i i igs faced w i t h i n c r e a s - i n g r i s k s .

The problem of planning t h e cropping schedule i n c l u d e s land p r e p a r a t i o n and i t s maintenance, p e s t c o c t r o l t r e a t m e n t s , h a r v e s t c o n d i t i o n s , and s a t i s f a c t i o n of water requirements . ( I t i s n o t pos- s i b l e t o show i n t h i s paper how models can d e s c r i b e t h e p l a n t and p a r a s i t e phenology, o r how s i m u l a t i o n s of i r r i g a t i o n l e a d t o modif i - c a t i o n s i n t h e models' c h a r a c t e r i s t i c s , i n r e l a t i o n t o t h e growth and development of c u l t i v a r s . )

P. s p e c i a l a p p l i c a t i o n i s t h e f i t t i n g of c u l t i v a r vegetacior , c y c l e s t o t h e p o s s i h i l i t i e s of crop development and growth, i n termo of p r o b a b i l i t y . The process of f i t t i n g " t o t h e b e s t p r o b a b i l i t y " w i l l d i f f e r depending on t h e photoper iodic c h a r a c t e r of t h e c u l t i - var . A s t r i c t l y photoper iodic v a r i e t y (having a c r i t i c a l photo- per iod) cün be made t o always f lower a t t h e same d a t e (wi th a d i f - f e r e n c e of on ly a few d a y s ) , provided i t - i s not sown too l a t e . On t h e o t h e r hand, a non- o r h a r d l y photoper iodic v a r i e t y of constai i t d u r a t i o n f lowers a t a d i f f e r e n t d a t e accord ing t o t h e d a t e of p l a n t - i n g ,

On a n s t i o n a s s c a l e , ' a sy$tem'*for

O n a regional . s c a l e , more s p e c i a l i z e d models c2n b e es tab-

109

E i t t i n g of a Photoper iodic C u l t i v a r

In t h e Ouagadougou r e g i o n , t h e t r a d i t i o n a l v a r i e t i e s of sor- ghum, which are s t r i c t l y photoper iodic viith t h e b e s t p r o d u c t i v l t y p o t e n t i a l , head around 15 September. P r o d u c t i v i t y depends i n c r e a s - i n g l y on t h e d u r a t i o n of t h e growth c y c l e (mainly, t h e p u r e l y vege- t a t i v e p h a s e ) , which i s determined a t p l a n t i n g . T h i s da te of head- ing , which r e s u l t s from an a d a p t a t i o n , corresponds t o t h e end of t h e heavy r a i n f a l l ( r a i n f a l l h i g h e r than PET when expressed over a 10- day i n t e r v a l ) . Flowering and seed s e t t i n g most o f t e n do not occur dur ing t h e r a i n y per iod when p ü n i c l e d i s e a s e s are l i a b l e t o occur . Seed s e t t i n g i s ensured, however, by edaphic mois ture and t h e r a i n s ( l a s t r a i n f a l l lower t h a n PET over a 10-day i n t e r v a l ) . The ATJC f o r t h e s e sorghum crops ranges from 50 t o 160 mm o r more, depending on t h e s o i l . .

These c u l t i v a r s are c l ìa rac te r ieed by a 4G-dayr shobt idglheadingj f lower ing phase, fol lowed by a ?O-day f r u i t i n g per iod ( g r a i n f i l l i n g b e f o r e matur i ty ; . F igure 8.12 shows t h t f i t t i n g of t h e c y c l e of such a c u l t i v a r , as determined by t h e d a t e of headingl f lower ing on 15 September. The p r o b a b i l i t i e s of succ(?ss of t h e 40-day and 20-day p e r i o d s a r e c a l c u l a t e d over t h e s e phases , which a r e represented by l i n e segments. ( P r o b a b i l i t i e s of t h e 20-day phase a r e only s i g n i f i - c a n t because of t h e s t a t i s t i m l dependence betweer t h e l a s t two s i g - moid c c r v e s . )

For a g iven d a t e of heading/€iowering (here 15 September), these p r o b a b i l i t i e s depend only on AWC and a r e t h u s independect of t h e p l a n t i n g d a t e ( u n l e s s i t i s t c o l a t e ) . The durat ior i of t h e p u r e l y v e g e t a t i v e phase, o c c u r r i n g b e f o r e shoot ing and f i x e d by t h e d a t e of p l a n t i n g , determines t h e product ion c a p a c i t y , which depends on t h e t o t a l number of nodes o r l e a v e s . F igure 8.12 shows t h i s number, expressed accord ing t o t h e n y c t i p e r i o d Ñ ( i n r e l a t i o n t o t h e c r i t i c a l n y c t i p e r i o d No) and t h e sum of tempera tures from germina- t i o n t o shoot ing . T h i s phase l a s t s 65 days when t h e crop i s sown on 1 June, b u t i t s p r o b a b i l i t y of s u c c e s s i s only about 20 p e r c e n t . In t h e c a s e of a 1 J u l y p l a n t i n g d a t e , t h e phase l a s t s only 35 days, wi th a p r o b a b i l i t y of about 75 percent : buî i t s product ion c a p a c i t y i s reduced.

F i t t i n g of- a Nonphotoperiodic C u l t i v a r

A nonphotoperiodic sorghum w i t h a t o s a l d u r a t i o n of 1GG days i n c l u d e s a 30-day p u r e l y v e g e t a t i v e phase, a 40rday shoot ing/ headingl f lower ing phase, and a 20-day f r u i t i n g phase ( f i g u r e 8.13). Unlike a photoper iodic c u l t i v a r (heading a t a f i x e d d a t e ) , t h e p r o b a b i l i t i e s of success of t h e 40- and 20-day phases depend on t h e d a t e of p l a n t i n g and AWC. T h i s makes i t p o s s i b l e t o sow on such a d a t e t h a t t h e shooting/heading/flower?ng phase, by f a r che mobt c r i t i c a l p e r i o d , i s f i t t e d a t t h e b e s t p r o b a b i l i t y between t h e twc c e n t r a l sigmoid curves e n c l o s i n g t h e humid subperiod (AEï/PET e q u a l t o o r h igher than 0.90).

c r i t i c a l phase ( t h e two r e c i p r o c a l sigmoid curves be ing independent) 2re:

Depending on kW, t h e b e s t p r o b a b i l i t i e s f o r f i t t i n g t h i s

110 s ' 111

AWC 50 mm: (1.60 x 0.60 = 0.36 AWC 100 mni: 0.85 x 0.75 = 0.55 AWC 200 mm: 0.85 x 0.95 = 0.80

Thus, t h i s f i t t i n g de t e rmines (1) t h e optimum sowing d a t e (So) 30 days e a r l i e r : ANC 50 , 2516, p r o b a b i l i t y 0 . 6 0 ; ANC 100, 117, p r o b a b i l i t y 0.75; AWC 200, 117, p r o b z b i l i t y 0.75 (The sowing d a t e remains p r a c t i c a l l y t h e same i n t h e t h r e e c a s e s ) ; and (2) t h e d a t e of t h e f r u i c i n g phase 20 days later: AGlC 5 0 , 2519, p r o b a b i l i t y 0 . 3 0 ; AWC 100, 1 /10 , p r o b a b i l i t y 0.55; AWC 200, 1/10, p r o b a b i l i t y

*, 0.95 (Once more t h e d a t e s , u n l i k e t h e p r o b á b i l i t i e s , v a r y s l i g h t l y ) . I n f a c t , i f t h e optir;.um sowing d a t e s (So) 2nd t h e f r u i t h g

d a t e s a r e c o r r e c t , qthe p r c b a b i l i t i e s are only s ig i i i f i can r : because of t h e dependence between t h e f i r s t 2rLd t h e l a s t two sigmoid curves. In any case , they cannot b e combined w i t h t h e compound p r o b a b i l i t y f o r t h e 40-day c r i t i c h l phase, except as a rough guide.

S ince t h e c u l t i v a r i s gonphotoperiodic , t h e r e i s no c r i t i c a l photoperLod !No = O ) . I f N o = O i n t h e formula f o r f i g u r e 8.12, which g i v e s t h e number of nodes, number of nodes = no. T h i s para- reter corresponds t o IC,, which meascres t h e sum o f t empera tu res of

between two successLve l e a v e s I s k = k , /n . c a l r e g i o n s , I:owever, temperature sums need n o t be cons ide red f o r eepressir ig t h e d u r a t i o n of phases i n terms of days.

- - the e a r l y phase, where T(Ti-T,) = k, . The s l i m of t empera tu res For l o c a t i o n s i n t r o p i -

Applicät ioi i t o Energ) F i e l d >

Figure 8.14 shows a frequency model of growing p e r i o d s (more p r e c i s e l y , p e r i o d s of n o n l e t h a l temperatures) f o r Pocahontas, Iowa, [ISA (temperdite c l i m a t e ) , s i t u a t e d between t h e thermal t h r e s h o l d s of 16", 24", and 32"F, based on d a t a from Thom and S h m (1958). S ince t h e frequency d i s t r i b u t i o n f o r pass ing threshold-value even t s i s nore ia l , t h e a u t h o r s p l o t t e d s t r a i g h t I . ines corresponding t o t h e t i t t e d sigmoid cu rves . The s t anda rd d e v i a t i o n s p r o p o r t i o n a l t o t h e s l o p e s a c e indicaLed a long t h e s e s t r a i g h t l i n e s . Takeli i n twos a t random, t h e s e "opening" and "c los ing" s t r a i g h t l i n e s d e f i n e n i n e mailì i requency p e r i o d s , which can b e d i v i d e d i n t o subperiods.

o l d s of germinatior, a t t h e beginning of t h e growth p e r i o d , and t h r e s h o l d s of ma tu r i ty h t t h e end. These are expressed i n d a i l y , 5-day, weekly, and 16-day terms. Thresholds can be b a s e tenipera- t u r e s (To ) of c u l t i v a r s , t e n p e r a t u r e s determining phenologica l e v e n t s l'or annual 01' perenn ia l c rops , ar,d even t h e phenology of p a r ü s i t e s , i n s e c t s , o r cryptogams. Sigmoid cu rves f o r temperacure t h r e s h o l d s san be coniLined w i t h sigraoid cu rves f o r t h r e s h o l d s of r a d i a t i o n a d even water e v a i l a b i l i t y .

The p r e s e n t system can be a p p l i e d t o f r e e z i n g p o i n t s , t h re sh -

SUPWARY

The i n t r a - nnd i n t e r a n n u a l v a r i a b i l i t i e s of c l i m a t e are obscac le t o a g r i c u l t u r a l development even when c h e i r zmpli tude i s n o t l i k e l y t o brii ,g about c a t a s t r o p h e s . The re fo re , i t i s a d v i s s b l e t o model t h e s e v a r i a b i l i t i e s i n o r d e r t o mahc t h e b e s t u s e of them i n a g r i c u l t u r n l r e s e a r c h , p racE ice , acd planning. I n t r o p i c a l

I I

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8 Agroclimate Information ' for Development

Reviving the Green Revolution

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edited' by David F. Cusack

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16-BR Westview Press / Boulder, Colorado - I A & * 2 4 - i