Agricultural Water Management, 15 (1988) 165-175 165 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

Irr igat ion Schedul ing under a Limited Water Supply

N.H. RAO l, P.B.S. SARMA 1 and SUBHASH CHANDER 2

1 Water Technology Centre, Indian Agricultural Research Institute, New Delhi 110-012 (India) "~Department of Civil Engineering, Indian Institute of Technology, New Delhi (India)

(Accepted 19 November 1987)

ABSTRACT

Rao, N.H., Sarma, P.B.S. and Chander, S., 1988. Irrigation scheduling under a limited water supply. Agric. Water Manage., 15: 165-175.

The problem of scheduling irrigation at weekly intervals for a single crop when water supply is limited is considered. The mathematical formulation is based on a dated water-production func- tion, weekly soil-water balance, and a heuristic assumption that water stress in the early weeks of a crop-growth stage leads to suboptimal yields. The allocation problem is solved at two levels, growth stages, and weeks. At the first level, the dated water-production function is maximized by dynamic programming to obtain optimal allocations for growth stages. At the second, the water allocated to each growth stage is re-allocated to satisfy weekly water deficits within the stage in a sequential order, beginning with the 1st week of the stage. Water delivery and soil-water storage constraints are included at both levels. The model is applied to a field problem to derive weekly irrigation programmes for cotton under various levels of seasonal water supply and initial soil moisture.

INTRODUCTION

T h e i r r iga t ion- schedu l ing p r o b l e m m a y be def ined as d e t e r m i n i n g the t im ing a n d a m o u n t of i r r iga t ion wa te r for a given crop. T h e m a i n fac tors which de- t e r m i n e its so lu t ion are c l imate , type of c rop a n d soil, a n d avai lab le wa t e r sup- ply. T h e wa te r supp ly is adequa t e if i r r iga t ions can be scheduled such t h a t the so i l -wate r c o n t e n t in the c rop ' s roo t zone can be m a i n t a i n e d t h r o u g h o u t the season a t levels which do no t l imi t c rop growth. W h e n the avai lab le supply is l imited, wa t e r defici ts are unavo idab l e in some per iods of the growing season. T h e schedul ing p r o b l e m t h e n becom es complex because i r r iga t ion decis ions need to be b a s e d on the c rop ' s sens i t iv i ty to wa t e r defici ts in d i f fe ren t per iods of its growth. T h i s requi res an eva lua t ion of a l t e rna t ive i r r iga t ion schedules a n d choos ing the schedule which m a x i m i s e s yields for the given level of wa t e r

0378-3774/88/$03.50 © 1988 Elsevier Science Publishers B.V.

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supply. This paper deals with irrigation scheduling in weekly intervals with the objective of maximizing crop yield.

Considerable effort has been devoted, in the recent past, to the mathematical formulation of the problem of irrigation scheduling from a limited seasonal supply of water as a multi-stage sequential decision process and its solution by system simulation and optimization (Jones, 1983). The formulation consists of three steps: (1) Determining the appropriate water-deficit index to quantify crop water

stress in specified periods of the growing season based on crop water use (actual evapotranspiration).

(2) Deriving a functional relationship, called the dated water-production func- tion, to evaluate the effects of alternative combinations of crop water def- icits in the various periods on crop yield.

(3) Determining the optimal water allocations by maximizing the above func- tion by dynamic programming.

Although the above procedure has been used by several workers to determine optimal water allocations for specified periods in the crop growing season, there are difficulties in extending it to irrigation scheduling for short and regular intervals such as weeks, fortnights etc. This paper deals with the development of a procedure to solve this problem.

CONCEPTUAL BASIS

The dated water-production functions commonly used in irrigation optimi- zation models are derived empirically from irrigation experiments designed to determine crop sensitivity factors to water deficits in specified periods of growth. The periods usually coincide with the physiological stages of crop growth. Doorenbos and Kassam (1979) derived the stress sensitivity factors for differ- ent physiological growth stages for 26 crops from a large number of experi- mental data. The dated water-production function is formulated by postulating that, individually, the effects of water deficits in each period (growth stage) on crop yield are independent and, sequentially, they are either multiplicative or additive. Several dated production functions derived in this way have been reviewed by Vaux and Pruit t (1983). A difficulty with such dated water-pro- duction functions is that irrigation water allocations based on them hold only for the physiological crop growth stages. This limits practical applicability when irrigation decisions are required at regular intervals of weeks or fortnights (Tsakiris, 1982 ).

There are only a few reports in the literature which deal with the problem of irrigation water allocation in short and regular intervals, when the seasonal water supply is limited. They estimate stress sensitivity factors for short in- tervals from experimentally derived growth-stage sensitivity factors for the

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crop by interpolation (Rhenals and Bras, 1981; Tsakiris and Kiountouzis, 1984) or assume that these are equal to the corresponding growth-stage factor for all intervals of the stage (Bras and Cordova, 1981; Allen 1986). Such approaches, though mathematically attractive, are not realistic because the growth-stage yield-response factors are dependent on the durations of growth stages them- selves. Further, interstage stress effects may be significant at short intervals and simple additive or multiplicative dated production functions may not be appropriate.

The approach adopted in this study is based on an important characteristic of many non-forage crops, namely developmental plasticity (Turner and Burch, 1983). This characteristic indicates that plants may shed tillers, lose leaves, or abort flowers and seeds when water stress develops. If water stress is allowed to develop early in any growth stage, irreparable damage to the yield-contrib- uting characters would have already occurred (fewer tillers, leaves, flowers, seeds) leading to sub-optimal yields for a given level of water use (Stegman, 1983). Thus, within any growth stage, the earlier weeks have a priority for irrigation over the later weeks. This has been experimentally observed for many crops (Marani and Horowitz, 1963; Chaturvedi et al., 1981; Sneed and Patter- son, 1983 ). Such priorities are also usually enforced by farmers and in routine agronomic recommendations (Sinha et al., 1985).

Thus, two major characteristics of crop yield response to water of relevance to irrigation optimization models are utilized in the mathematical formulation of this study: ( 1 ) For longer periods coinciding with physiological growth stages, the simple

additive or multiplicative dated water-production functions are approxi- mately representative of crop yield response to water.

(2) For shorter periods (weeks), such simple relationships cannot be derived. However, experimental evidence suggests that the earlier weeks in each growth have a priority for irrigation if optimal yields are to be realized.

Keeping these points in mind, the irrigation scheduling problem is consid- ered to occur at two levels, growth stages, and weekly intervals within each growth stage.

To solve the first-level problem, a simple multiplicative dated water-produc- tion function model (Rao et al., 1988 ) is maximized for a given level of seasonal supply to obtain optimal irrigation-water allocations during each growth stage by dynamic programming. The constraints of the optimization model are de- rived from a weekly soil-water balance model (Rao, 1987).

At the second level, the water allocated to each growth stage is reallocated to meet water deficits in the standard weekly intervals of the stage. This is done in the sequential order of the intervals, till the supply allocated to the stage at the first level is exhausted. These allocations also are subject to the soil-water balance, irrigation system, and other physical constraints.

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MATHEMATICAL FORMULATION

The crop's growing season is divided into N periods which coincide with its physiological growth stages. Each growth period (i) is further partitioned into irrigation intervals (Mi, i= 1, N) which coincide with the standard weeks of the year. Let Qo (mm) be the total depth of water available for irrigation at the beginning of the growing season and Wo (mm/cm) the initial available soil water.

To solve the first-level problem, the dated water-production function model used is (Rao et al., 1987):

N Y / Y M =- ~ [ l - K / (1--AET/PET)i] (1)

i=1

Mi AETi= ~ eai j (2)

j=l

Mi PET/= ~ epii (3)

j=l

where Y is actual and YM is maximum (when AET=PET) crop yield, Ki is the yield sensitivity factor taken from Doorenbos and Kassam (1979), eaij and epij are, respectively, the actual and potential evapotranspiration in the j th weekly interval of the ith growth stage.

The value of e,i~ is estimated from daily pan evaporation data by the proce- dure described by Doorenbos and Pruit t (1974). The value of eaij is estimated from a soil-water balance model using the procedure given by Doorenbos and Kassam (1979). According to this procedure, evapotranspiration is considered to occur at its potential rate until the soil moisture is depleted to a threshold level, below which it falls off in a linear fashion. The threshold moisture con- tent is determined by an empirically derived soil-water depletion factor, which varies with the crop and the prevailing potential evapotranspiration rate. Val- ues of this factor for several crops, as tabulated by Doorenbos and Kassam (1979), are used in this study. The soil-water balance model assumes a uniform soil reservoir of depth (Z~i) equal to the effective root depth of the crop in the period under consideration. The expected rainfall in this period (rij) and irri- gation applied (U~j) are assumed to be added to the soil reservoir at the begin- ning of the interval. For these conditions, the model calculates the actual evapotranspiration each day, based on the soil-water content and the soil-water depletion factor, before cumulating them to the value for the period. The depth of the soil reservoir (Zo) considered for each period is determined by a linear root-growth model. Details of these calculations ae given by Rao (1987).

If X~ is the water allocated to the ith stage:

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Mi X,= 2 U~i, i=l,...,N (4)

J

To obtain optimal allocations of water to growth stages, equation (1) is max- imized by dynamic programming using the backward recursion procedure and the recursive equation:

PETJiA O<~Xi<~Q (5) O<Q<~Qo

and i = N - 1 , N - 2 , ..., 1

fN(Q, WN)=Max[1--KN(1--AET~ 1 ~_ \ P E T , I N

O ~ X N ~ Q O<~Q<~Qo (6)

In equations (5) and (6), AETi is a function ofXi. Q and W are the two state variables, namely available water supply over the remaining season and the available soil water, respectively, at the beginning of each stage. Thus:

Wi+l =Wi.M, (7)

where Wi,Mi is the available soil water at the end of the M~th (last) interval of the ith growth stage and is obtained from the water balance model for the successive weekly intervals {j = 1, .., M~), using:

=Min ~ (Wi,j--1)Zi,j_l q-rij q- Uij -I- W o ~Zij -eaij) /Zij wij [ We

(8)

mZij = Zij - Zi, j- 1,

AZi,1 = Z i l - - Z i - l , M i 1,

j ¢ l (9)

j = l

where W~ is the field capacity relative to the permanent wilting point, and eaij is obtained from the soil-water balance model as described earlier; AZ is ob- tained from the linear root growth model (Rao, 1987).

The irrigation depth (Uii) is zero if the available soil water at the beginning of the weekly interval is adequate to maintain evapotranspiration at its poten- tial rate throughout the interval. Otherwise it is restricted by the available soil- moisture storage capacity, the supply allocated to the stage (Xi), and the car- rying capacity of the physical farm water distribution system (AWC), i.e.

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Uo=O if Zowo>~(1-p)WcZo+epo (10)

[ WcZij~_ lWi,j_ l Zi,j_ l - ro -- Wo AZi~

= M i n 1Xi - - t~ lUi t [ A w c i J o t h e r w i s e

Equations (5) and (6) are optimized subject to (7)- (10). A detailed state- ment of the optimization procedure is given in the Appendix. The optimal allocations to each growth stage so obtained are designated as (X*, i= 1, ...., N).

T A B L E 1

Growth stage (i), weekly interval (j), corresponding s tandard week of the year, probable rainfall ( r o), potent ia l evapotranspi ra t ion (epo) and root depth (Z~ i) for cot ton

Growth stage Interval Week r o %o Zii No. (i) No. (]) No. (mm) (mm) (cm)

1 1 29 8 7.8 7.5 2 30 10 6.4 15.0 3 31 8 14.7 22.5 4 32 5 16.1 30.0 5 33 10 24.3 37.5 6 34 7 23.3 45.0 7 35 8 22.2 52.5 8 36 1 3.7 60.0

2 1 36 5 18.3 64.3 2 37 4 25.2 68.6 3 38 9 24.5 72.9 4 39 8 20.3 77.1 5 40 9 27.0 81.4 6 41 7 22.2 85.7 7 42 2 21.9 90.0 8 43 0 22.3 94.3 9 44 1 24.4 98.6

10 45 0 22.2 102.9 11 46 0 19.6 107.1 12 47 0 16.5 111.4 13 48 0 15.6 115.7 14 49 0 12.5 120.0

3 1 49 0 5.0 120.0 2 50 0 15.1 120.0 3 51 0 11.5 120.0 4 52 0 6.1 120.0

Total 102 448.7

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TABLE 2

Optimal irrigation water allocation (X*) to growth stage (i) of cotton for different levels of sea- sonal supply (Qo) and initial soil moisture content (Wo)

Qo Wo ( m m / X* (mm) Y/YM (mm) cm)

Xi X2 X3 Xi

200

300

400

0.4 0 160 40 200 0.70 1.1 0 160 40 200 0.73 1.8 0 160 40 200 0.75 2.5 0 140 60 200 0.79

0.4 0 280 20 300 0.87 1.1 0 280 20 300 0.90 1.8 0 280 20 300 0.92 2.5 0 280 20 300 0.94

0.4 96 279 4 379 1.00 1.1 95 279 2 376 1.00 1.8 87 276 0 373 1.00 2.5 80 279 11 370 1.00

The allocations to the standard weekly intervals comprising each stage ( U~, i= 1, ..., N; j= 1, ..., Mi) which represent the solution of the second-level problem are obtained sequentially from X* and the water-balance equations (7 ) - (10 ) .

APPLICATION

The procedures described in the previous section are applied to cotton for the soil and rainfall data of an irrigation project in India. The area is charac- terized by semi-arid to semi-humid climatic conditions. The crop, sown on 16 July (29th week of the year), has a duration of 165 days. The crop has three physiological growth stages: vegetative (50 days, K1 = 0.2 ); flowering (95 days, K2=0.5); and ripening (20 days, K3=0.25). The values of K/ ( i=1 , ..., 3), taken directly from Doorenbos and Kassam (1979), are assumed to hold for the situation of this study. The soils of the area are deep black clay with 2.5 mm/cm of available water. The weekly irrigation programmes for the season were developed for different levels of seasonal water supply for pre-specified conditions of weekly rainfall (rij) and potential evapotranspiration (epij). The effective root depth (Zii) determines the dimension of the soil reservoir (Table 1).

Values of rij used in this study are estimates of weekly rainfall at 75% ex- ceedence probability obtained after fitting the gamma distribution to weekly

1 7 2

" F A B L E 3

W e e k l y i rr igat ion schedules for d i f ferent levels o f seasonal supply (Qo) and t w o c o n d i t i o n s of ini t ia l soi l mo i s ture ( W o ) (AWC = 5 0 m m for all w e e k s )

W e e k N o .

Qo ( m m ) =

Wo = 0 . 4 m m / c m Wu = 2 . 5 m m / c m

2 0 0 3 0 0 4 0 0 2 0 0 3 0 0 4 0 0

2 9 . . . .

3O . . . .

31 - 39 - - - 32 . . . .

3 3 - 5 0 - - 5 0

3 4 . . . .

3 5 . . . . 3 0

3 6 4 3 4 3 5 0 4 3 4 3 4 3

3 7 5 0 5 0 5 0 5 0 5 0 -

3 8 5 0 5 0 - 4 7 5 0 5 0

3 9 . . . .

4 0 5 0 - - 5 0

4 1 . . . .

4 2 17 5 0 - - 5 0 -

4 3 5 0 - - -

4 4 5 0 - - 5 0 5 0

4 5 5 0 - - -

4 6 3 7 - - 3 7 5 0

4 7 . . . . .

4 8 . . . .

4 9 14 1 4 3 6 1 4 1 4 4 7

5 0 2 6 6 - 4 6 6 -

5 1 . . . . .

5 2 5 - - -

rainfall. Values of epij are based on average daily pan evaporation data over 12 years. Values of Zij are obtained from a linear model of root growth. It is pre- sumed that the irrigation system permits the application of a uniform depth of irrigation of up to 50 mm at the beginning of each weekly interval, i.e. AWC = 50 mm/week for all weeks (equation 10).

For the above conditions, the optimal irrigation-water allocations (X*) to the three growth stages (i) of cotton, for different levels of seasonal supply (Qo) and initial soil moisture (Wo), are derived using equations (4) - (8) (Ta- ble 2 ). The weekly irrigation schedules are derived from X* and equations (6) to (8). The results for two extreme initial moisture conditions, Wo (near-dry, and field capacity), are presented in Table 3.

D I S C U S S I O N

The two-level approach to irrigation scheduling in weekly intervals devel- oped in this study facilitated solving the above allocation problem for cotton

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by a three-stage dynamic programming model. The cotton crop has a duration of 24 weeks. In the absence of the two-level approach, the dynamic-program- ming formulation using weekly sensitivity factors would have included 24 de- cision stages. Thus, besides using the available experimental information on crop response to water in a realistic mathematical formulation, the procedure is also computationally more efficient in comparison to the conventional pro- cedures of weekly irrigation programming.

From Tables 2 and 3, it appears that Wo, though it affects crop yield, does not significantly influence weekly irrigation sequences, except when the sea- sonal supply (Qo) is nearly adequate. This is because the incremental soil water ( W0 AZ) is small compared to the total soil-water content of the effective root zone. A second factor is the fixed capacity of the irrigation water delivery sys- tem (AWCii) compared to the available soil-storage capacity. This is more so for deep-rooted crops like cotton. The result is that, under such conditions, when available supplies are highly deficient, irrigation decisions are governed more by crop and irrigation system-related factors than by the soil-water avail- ability. However, when supplies approach adequate levels, irrigation frequen- cies rise and Wo influences the timing of irrigations significantly.

The mathematical formulation developed in this study is essentially deter- ministic. In applying the model to practical situations, uncertainty of weather data may affect the model results. The two input variables affected by this uncertainty are evapotranspiration (epij) and rainfall rij). Rhenals and Bras (1981) have shown that inclusion of evapotranspiration uncertainty may not significantly influence weekly irrigation decisions if P E T estimates are based on long-term averages. In the case described above, epi2 values were derived from 12 years of pan evaporation data. Thus, with this data, ET uncertainty may not significantly affect the model results presented above.

Uncertainty of weekly rainfall (ri~), on the other hand, is critical in irrigation scheduling in semi-arid to semi-humid climates. Prediction of weekly rainfall by stochastic or other methods has not advanced to a stage that it can be used at operational level. Although it is feasible to include rainfall uncertainty ex- plicitly by expanding the model considerably, its usefulness is doubtful. In- stead, a probabilistic approach was adopted and weekly rainfall values at specified probabilities were used. An alternate approach to include rainfall un- certainty would be to use the probabilistic model in real-time with on-line in- teractive management of irrigation by mid-season adjustments. This would require updating the model with real-time data of rainfall in successive weeks, while assuming the probable value of weekly rainfall for subsequent weeks of the growing season. The mathematical formulation presented above is partic- ularly suited for such adaptive management.

REFERENCES

Allen, R.G., 1986. Sprinkler irrigation project design with production functions. J. Irrig. Drain. Div. ASCE, 112(IR4): 305-321.

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Bras, R.L. and Cordova, J.L., 1981. Intraseasonal water allocation in deficit irrigation. Water Resour. Res., 17: 866-874.

Chaturvedi, G.S., Aggarwal, P.K., Singh, A.K., Joshi, M.G. and Sinha, S.K., 1981. Effect of irri- gation on tillering in wheat, triticale and barley in a water limited environment. Irrig. Sci., 2: 225-235.

Doorenbos, J. and Kassam, A.H., 1979. Yield response to water. FAO Irrig. Drain. Pap. 33, 193 pp.

Doorenbos, J. and Pruitt, W.O., 1977. Guidelines for crop water requirements. FAO Irrig. Drain. Pap. 24, 193 pp.

Jones, J.W., 1983. Irrigation options to avoid critical stress: optimisation of on-farm allocation to crops. In: H.M. Taylor et al. (Editors), Limitations to Efficient Water Use in Crop Production. pp. 507-516.

Marani, A. and Horowitz, M., 1963. Growth and yield of cotton as affected by the time of a single irrigation. Agron. J., 55: 219-222.

Rao, N.H., 1987. Field test of a simple soil water balance model for irrigated areas. J. Hydrol., 91: 179-186.

Rao, N.H., Sarma, P.B.S. and Chander, S., 1988. A simple dated water-production function for use in irrigated agriculture. Agric. Water Manage., 13: 25-32.

Rhenals, A.E. and Bras, R.L., 1981. The irrigation scheduling problem and evapotranspiration uncertainty. Water Resour. Res., 17: 1328-1333.

Sinha, S.K., Aggarwal, P.K. and Khanna-Chopra, R., 1985. Physiological and phenological basis of irrigation in India. In: D. Hillel (Editor), Advances in Irrigation, 3. Academic Press, New York, pp. 129-212.

Sneed, R.E. and Patterson, R.P., 1983. The future role of irrigation in a humid climate. In: C.D. Raper Jr. and P.J. Kramer (Editors), Crop Reactions to Water and Temperature Stress in Humid Temperate Climates. West View Press, Boulder, CO.

Stegman, E.C., 1983. Irrigation scheduling applied timing criteria. In: D. Hillel (Editor), Ad- vances in Irrigation, 2. Academic Press, New York, pp. 1-30.

Tsakiris, G.P., 1982. A method for applying crop sensitivity factors in irrigation scheduling. Agric. Water Manage., 5: 335-343.

Tsakiris, G.P. and Kiountouzis, E., 1984. Optimal intraseasonal water distribution. Adv. Water Resour., 7 (2): 89-92.

Turner, N.C. and Burch, G.J., 1983. The role of water in plants. In: I.D. Teare and M.M. Peet (Editors), Crop-Water Relations. Wiley, pp. 73-126.

Vaux, H.J. and Pruitt, W.O., 1983. Crop water production functions. In: D. Hillel (Editor), Ad- vances in Irrigation, 2. Academic Press, New York, pp. 257-272.

APPENDIX

Equation (6) is first maximized by direct search. To accomplish this, the value of X N which maximizes this equation is determined for several values of the two state variables Q (0 ~< Q ~< Qo) and W (0~< W~< We) subject to equations (7)-(10) and (2)-(4) . A two-dimensional table is ob- tained which provides the values of optimal irrigation water allocations X N to this stage and the corresponding value of the yield function fN, for each pair of values of Q and W.

After the first stage optimization is complete, the principle of optimality is applied and, to provide the values of recursive function with two stages remaining, (5) is maximized by setting i = ( N - 1 ). In this equation, Wi+ 1 is the soil moisture content at the beginning of the Nth stage. This is also the moisture content at the end of the ( N - 1 )th stage as a result of allocating XN 1 units of water to this stage. Wi+l can, therefore, be determined from water balance in the ( N - 1 )th stage. The numerical value of fi+ ~ ( Q - Xi, Wi+ 1 ) represents the maximum value of this function

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for the pair of state variables (Q-Xi, Wi +1). This is obtained by two-dimensional linear inter- polation from the recursive function table for the Nth stage for available irrigation water supply (Q-Xi) and an initial soil moisture of Wi+l. Equation (5) is maximized by this procedure of direct search and interpolation for the range of values of Q and W and another two-dimensional recursive function table is obtained for this stage. When this maximization is accomplished, the computations are progressed with i successively set equal to N - 2, N - 3,..., 1 etc. and the process repeated to obtain the recursive function tables for all the growth stages of the crop. For each value of i, the optimizing values of irrigation allocations X~ and the corresponding yield function fi are stored, for the range of values of Q and W.

The actual irrigation water allocation (X*) for each stage for a given level of supply (Q0) over the crop season and an initial uniform soil moisture ( Wo ) are obtained from the recursive function tables by two-dimensional interpolation and water balance as follows: ( 1 ) Beginning at stage 1, for initial soil moisture Wo and seasonal supply level Qo, the irrigation

allocation to this stage (X`;) is determined by two-dimensional interpolation from the re- cursive function table for the first stage (i= 1 ). The soil moisture (W2) at the end of the first stage is also determined from the soil water balance model and equations (7)- ( 10 ) and (2)- (4).

(2) The optimal irrigation water allocation in stage 2 is determined by two-dimensional inter- polation from recursive function table of this stage for the values of the two state variables given by Q - X`; ) and W2. The soil moisture content W:~ at the end of the second stage is determined as in (1) above.)

(3) The optimal irrigation water allocations for the successive stages are similarly determined. The state variables take on successively the pairs of values (Q-X`;-X'~, W:~), (Q-X`; - X ~ - X L W4) ..... (Q-X'f ..... -X'N_1, WN).