Abstract Estimates of the effects of alternative discreteirrigation water scheduling options on consumptive use orevapotranspiration and on crop yield are developed for anortheastern Colorado case study. The analysis proceedsfrom the premise that farmers, rather than considering ir-rigation water as a continuously variable input, tend to treatirrigations as discrete events, and make scheduling deci-sions as choices among numbers of irrigations of approx-imately equal volume. The van Genuchten-Hanks modelis employed to develop a transient-state water-crop pro-duction function model. Results for two crops – corn grainand edible dry beans – are presented here. Findings are thatthe effect of the number of irrigations on evapotranspira-tion and yield per hectare varies widely, depending uponthe timing of applications. When farmers can choose theoptimal timing of irrigations, a reduced number of irriga-tions has a relatively limited adverse effect on crop pro-duction until irrigations are reduced to less than four perseason. However, there are many situations in which an in-ability to apply water can result in a very large reductionfrom potential maximum yield, particularly if water iswithheld early in the season and/or during the rapid growthperiod of the crops. In many contexts of irrigation watermanagement, water policy analysts will wish to considerthe more realistic discrete-input simulation model for pol-icy evaluation.

Introduction

For water resource planning and management, water-cropproduction functions can play an important role in bothproduction decisions and policy analysis. A productionfunction, which mathematically or graphically representsthe relation between inputs and output(s) in a productionprocess, serves as a basis for describing, explaining, andpredicting the output expected from a specified level of in-puts. Production managers need to have a rough under-standing of the production function in order to design op-timum input allocation strategies. Economists and policyanalysts employ production functions as building blocksfor models of farmer response to alternative irrigation wa-ter management policies.

Production functions for irrigated agricultural crops canbe derived directly from experiments, from statistical anal-ysis of secondary data, or indirectly by mathematical sim-ulation models (whose parameters are normally obtainedfrom direct observations). Reviews of early approaches todeveloping water-crop production functions are found inCarruthers and Clark (1981) and Vaux and Pruitt (1983),while Boggess et al. (1993). Many analysts prefer to relyon experimental production functions because they aredeemed more realistic and reliable. However, experimentsare expensive and time consuming, and may not be read-ily generalizable beyond the local experimental conditions.Hexem and Heady (1978) describe studies of experimen-tally based water-crop production functions.

There are few water-crop production function analysesbased on secondary data because of the scarcity of data(see Moore et al. 1994 for an example). While valuable forbroad-scale testing of hypotheses of interest to policy mak-ers, water-crop production functions based on secondarydata are usually aggregated over crops and production re-gions and are seldom suitable for disaggregation into otherplanning models.

As more precise scientific knowledge of the factors af-fecting crop growth becomes available and the costs ofcomputing fall, simulated water-crop production functions

Irrig Sci (1997) 18: 23–31 © Springer-Verlag 1997

Received: 1 November 1996

Susanne M. Scheierling · Grant E. CardonRobert A. Young

Impact of irrigation timing on simulated water-crop production functions

ORIGINAL PAPER

S. M. ScheierlingInternational Irrigation Management Institute, c/o CIMMYT, Apdo. Postal 6-641, CP 066000, Mexico DF, Mexico

G. E. Cardon (½)Department of Soil and Crop Sciences, Colorado State University, Fort Collins, CO 80523, USAFax: +1-9 70/4 91-05 64

R. A. YoungDepartment of Agricultural and Resource Economics, Colorado State University, Fort Collins, CO 80523, USA

become a more feasible approach in many situations. Simulation models can be readily adapted for specific soiland climatic conditions, and so provide a flexible and rel-atively inexpensive method of producing production func-tions for varying local conditions (Letey 1993). Econo-mists, agronomists, and agricultural engineers haveworked separately and jointly on developing simulationmodels of crop response to irrigation. For discussions ofthe particular issues arising in estimating water-crop pro-duction functions see Just (1991), Letey and Dinar (1986),and Dinar and Letey (1994).

Simulation models used by economists to compute water-crop production functions have typically treatedthe irrigation water input as a single value of water ap-plied during the season, but have given less considera-tion to the timing of water applications. For a review ofboth seasonal and transient-state models which allow fordiscrete water applications see Letey (1991) and Leteyet al. (1990). However, the approach of computing yieldsfrom seasonal water application is often not suitable foranalyzing problems in irrigation water management be-cause it does not accord with typical field conditions. Thenumber and timing of irrigation water applications areunderstood by farmers to be important decision variables.The amount of water applied is varied mainly by adjust-ing the number of irrigations. Therefore, production func-tions which are constructed from transient-state modelsimulations, which can capture the effects of irrigationtiming as discrete-input events, will be more representa-tive of farmers’ actual decision practices. Although pre-vious examples of the use of transient-state models forsuch applications are very few, these models can be read-ily adapted to modeling discrete-input water-crop pro-duction functions.

In addition to the conventional relationship between wa-ter application and crop yield, water-crop production func-tion simulations can provide estimates of another policy-relevant variable: the consumptive use or evapotranspira-tion (ET) associated with a given irrigation schedule. ETrepresents the difference between water application (eitherprecipitation and/or irrigation) and change in soil water con-tent, surface runoff, and deep percolation. Because of its influence on downstream water availability for further in-stream and offstream uses, ET is an important variable forbasin-scale planning exercises (Booker and Young 1994).

The purpose of the research reported here is to demon-strate, by means of a set of case study simulations, the im-portance of accounting for timing effects on the relation-ship between irrigation water application and crop yieldnecessary for determining the economically optimum al-location of irrigation water. The first section briefly de-scribes the simulation model used for computing the pro-duction functions. It is followed by an outline of the appli-cation of the model to the production of corn grain (Zeamays) and edible dry beans (Phaseolus vulgaris) in north-eastern Colorado. The final sections present model resultsfor corn and bean yields as a function of alternative irriga-tion water schedules and discuss their implications for ir-rigation water management.

Model description

In this study, a transient-state simulation model originallyformulated by Cardon (1990) is employed to estimate dis-crete-input water-crop production functions. Because themodel is based on a combination and modification of pre-vious models of van Genuchten and those by Hanks andcolleagues, it is referred to as the van Genuchten-Hanksmodel. Cardon and Letey (1992 a, b) and Minhas and Gupta(1993) compared model predictions to experimentallymeasured data and found that predicted and measured val-ues correlated well.

The main features of the van Genuchten-Hanks modelinclude the modeling of water and solute movementthrough the soil and the modeling of simultaneous wateruptake (root extraction) by plants. Solute movement is handled by the convection/dispersion equation. The modelcalculates water flow by the Darcy-Richards equation andaccounts for water removal by transpiration by adding awater uptake term:

(1)

where C is soil water capacity (mm–1), h is matric poten-tial (m), t is time (days), z is soil depth (mm, positive downwards), K is hydraulic conductivity (mm day–1), andS is water uptake (day–1).

The water uptake term, which links soil water status andcrop yield, is formulated as:

S = α (h, π) · Smax (2)

where π is osmotic potential (m), Smax is maximum wateruptake per unit depth of the root zone for no-stress condi-tions (day–1), α (h, π) is a dimensionless stress responsefunction and

(3)

where PET is potential (or non-stressed) crop ET (mmday–1), L is rooting depth (mm), and

PET = ET0 · CC (4)

where ET0 is reference ET (mm day–1), and CC is an as-sociated mean crop coefficient.

The relation α (h, π) is based on an empirical relation-ship between water uptake and salinity stress. In a studyof the salt tolerance response function of various crops,van Genuchten and Hoffman (1984) suggested a smoothS-shaped expression for relative water uptake (i.e., uptakerelative to the maximum uptake under non-saline condi-tions) as a function of the average salt concentration of theroot zone. Converting salt concentration to osmotic poten-tial, their equation is:

(5)SS

p=+

max

150

ππ

SLmax = PET

C ht z

K hz

K Sδδ

δδ

δδ

= −

−

24

where π50 is the osmotic potential at which S is reduced by50% (m), and p is an empirical constant which is approx-imately 3 for many crops.

Assuming that water uptake is affected similarly by bothmatric and osmotic stresses, then for dynamic rootingdepths and plant water stress susceptibility, the water up-take term can be written as:

(6)

where λ (z, t) is the depth- and time-dependent fractionalroot distribution, and a (t) is a time-variable weighting co-efficient that accounts for any differences in crop responseto h or π.

Since the van Genuchten-Hanks model is not formu-lated to calculate crop yield directly, values of water up-take, S (z, t) need to be summed for the season and thenconverted to yield. The authors follow Doorenbos and Kas-sam (1979) assuming S/Smax = ET/PET and the followinglinear relationship between yield and ET:

(7)

where Y/Ymax is relative yield, and c is a crop-specific yieldresponse coefficient. Based on this relationship and withan assumption for Ymax, yield values can be calculated bysubstituting cumulative S divided by cumulative Smax forET/PET in Eq. (7), using the c value suggested by Dooren-bos and Kassam (1979) for the respective crops.

Values of Ymax for a given area are generally taken fromreported yields from nearby fields that have received “highlevels of crop and water management” (Doorenbos andKassam 1979). These values could be taken from the recordof highest yields from neighboring farms, local researchplot yields, or can be based on an average yield goal fromnearby growers. Growers in most states are now encour-aged to base yield goals on an average of their last 5 years(excluding anomalously low yields due to crop failurefrom, e.g., pest damage, hail, frost) increased by 5% to 10%to account for improved methods, crop varieties, or opti-mal seasonal conditions. Chosing a Ymax value based onthe yield goal for a given area may be the most desirablemethod, since it will include the inherent cultural, soil, andclimatic idiosyncracies of the area.

Application to study area

The New Cache-La Poudre Irrigation District in the SouthPlatte River basin near Greeley in Weld County, Colorado,was chosen as the study area. Flows from the South PlatteRiver and its tributaries serve the most important agricul-tural region and major urban-industrial centers in Colo-rado. Agriculture is the largest water user of South Platteflows and also accounts for the bulk of consumptive wa-

YY

c cmax

= − +

1 ET

PET

S z tS t

a t ht

z t( , )( )

( )( )

( , )max=

+ +

⋅1

50

3ππ

λ

ter use (Dennehy et al. 1993). Over 80% of the water di-verted is used for irrigating the semiarid plains of north-eastern Colorado. Farmers in the study area almost exclu-sively use surface irrigation methods, particularly openditches with siphons and flexible and gated pipes, and ap-ply several irrigations per crop and season, each with amore or less fixed amount of water. Irrigation water isdrawn from several sources, including river flow, reservoirwater, and groundwater from the unconfined alluvial aquifer along the South Platte River. The major crops pro-duced in the study area are corn grain and edible dry beanswhich together cover about 50% of the irrigated land. Othercrops grown include alfalfa, corn silage, and sugarbeet. Forthis study, the van Genuchten-Hanks model is applied tothe two major crops. A discussion of the model applica-tion to the other crops is given by Scheierling (1995).

Input parameters

The time-dependent tabular input data used for corn andbeans in the model are presented in Tables 1 and 2, respec-tively. In the region around Greeley, the typical growingseason dates for corn are from 30 April to 16 Septemberand for beans from 20 May to 2 September (US Depart-ment of Agriculture, Soil Conservation Service 1988). Thenumber of irrigations applied to a particular field duringthe growing season varies considerably in the area. (Forexamples see Crookston and Hoffner 1992). For the sim-ulation of the effect of the potential range of numbers ofirrigations up to nine irrigations during the season are al-

25

Table 1 Time-dependent input data for simulating corn production,with the time interval for which each set is operative (ET0 is refer-ence evapotranspiration, CC is crop coefficient (unadjusted), L is themaximum (non-stressed) rooting depth, π50 is the osmotic potentialat which the yield is reduced by 50%, h50 is matric potential at whichthe yield is reduced by 50%)

Interval Effective ET0 CC L π50 h50(days) rainfall (mm (mm) (m) (m)

(mm) day–1)

32 – 3.5 0.27 300 –42 –421 60 3.5 0.27 350 –42 –42

17 – 6.9 0.62 600 –42 –421 52 6.9 0.62 650 –42 –428 – 7.8 0.82 900 –42 –421 14 7.8 0.82 950 –42 –428 – 7.6 0.91 1200 –32 –321 11 7.6 0.91 1250 –32 –328 – 7.4 0.93 1550 –32 –321 18 7.4 0.93 1600 –32 –328 – 7.2 0.93 2000 –42 –421 23 7.2 0.93 2000 –42 –428 – 6.5 0.92 2000 –42 –421 15 6.5 0.92 2000 –42 –42

13 – 4.5 0.85 2000 –42 –421 20 4.5 0.85 2000 –42 –42

13 – 4.4 0.69 2000 –42 –421 17 4.4 0.69 2000 –42 –427 – 2.6 0.55 2000 –42 –42

lowed for. These values for the number of possible irriga-tions, though they represent the upper limit of the range ofgrower practice, are not uncommon. For the purposes ofthis study, the nine irrigations were programmed to takeplace on the following dates: 1, 20, 30 June, 10, 20, 30 July,10, 25 August, and 8 September. Thus corn may be irri-gated from zero up to nine times. Beans, due to their shortergrowing period, may be irrigated as many as eight times.Each irrigation event is assumed to consist of the sameamount of net water infiltration into the soil, becomingavailable for plant water uptake or deep percolation. Ac-cording to Colorado State University extension agents inWeld County, typical net infiltration is about 3 in (76 mm).The amount of water which actually needs to be applied toachieve this net infiltration is higher, and depends on theirrigation method used. For example, open ditch with si-phons with an average application efficiency of 40% in thestudy area would require 190 mm of water per irrigation,whereas surge with gated pipe with an average applicationefficiency of 60% would require 127 mm. The authors rec-ognize that there are also considerable differences betweenirrigation systems in terms of distribution uniformity. Forthe purposes of the case study simulations performed aspart of this research, irrigation distribution was considereduniform for surface irrigation systems and the net infiltra-tion really only represents an average value on a givenfield. The variation in the distribution from top to bottomon a field can be simulated with the van Genuchten-Hanksmodel if desired. It is assumed that the other inputs to thecrop production process besides water are managed at alevel so that water is the only limiting factor.

With loam soil being the predominant type in the studyarea (Dennehy et al. 1993), a further model assumption isthat all soils are loam soil. The initial soil moisture condi-

tions at the beginning of the irrigation season in mid-Mayare set at 70% of field capacity (volumetric water con-tent = 0.17), reflecting typical conditions for the study areaat the planting times for corn and beans. The sensitivity ofthese crops to salinity requires the modeling of salinity ef-fects on water uptake. Initial soil salinity is set at 1 dS/m,and irrigation water salinity in the area is typically0.3 dS/m.

Precipitation supplies additional water for the crops.According to unpublished data from the Colorado ClimateCenter for the period from 1967 to 1992, Greeley receivesa mean annual precipitation of 350 mm with about 44%occurring during the irrigation season from mid-May tomid-September. The mean daily precipitation values forthe 123 days of the irrigation season are used to estimatemean daily effective precipitation. Effective precipitation,which is the amount of rainfall not lost by surface runoff,unnecessary deep percolation, or instantaneous surfaceevaporation, is estimated with the ET/precipitation methoddeveloped by the US Department of Agriculture, Soil Con-servation Service (1970) and recommended by Doorenbosand Pruitt (1977). It relates effective monthly precipitationto total monthly precipitation, crop consumptive use, andnet depth of application. Based on an earlier study whichemployed this method to estimate effective rainfall in thestudy area (Michelson 1988), an average effectiveness of75% is used for calculating mean daily effective precipi-tation. Because effective precipitation is relatively limitedin the region around Greeley, the model does not take intoaccount its stochastic nature. Instead, mean daily effectiverainfall values are summed over the preceding irrigationinterval and applied at the end of the interval as shown inTables 1 and 2.

Time-dependent values for reference ET are another in-put requirement. Reference ET is defined as the ET for anyspecified day from a well-watered reference crop whichfully covers the soil surface. In Colorado, the common re-ference crop is alfalfa, and a frequently used technique tocalculate daily reference ET is the Jensen-Haise equation(Duke 1987) given by

ET0 = Ct · (Tavg − Tx) · SR/HV (8)

where ET0 is daily reference ET (mm day–1), Tavg is meandaily temperature (°C) calculated as the average of dailymaximum and daily minimum temperatures, SR is dailysolar radiation (MJ m–2 day–1), HV is latent heat of vapor-ization (2.45 MJ kg–1), and Ct and Tx are empirical constants taken as 0.0251 °C–1 and –9.4 °C, respectively,based on Duke (1987). Values for Tavg and SR are acquiredfor Greeley during the period 1978 to 1989 from the COAGMET Historical Weather Data Files of the US Department of Agriculture Agricultural Research Servicelocated in Fort Collins, Colo. Based on these data, dailyET0 values are calculated for the 123 days of the irrigationperiod of every year available, and then averaged. The dailyaverages are used to calculate the model inputs for the dif-ferent time intervals given in Tables 1 and 2.

To calculate daily crop coefficients, CC, Duke (1987)recommends two equations which are based on measure-

26

Table 2 Time-dependent input data for simulating bean production,with the time interval for which each set is operative (ET0 is refer-ence evapotranspiration, CC is crop coefficient (unadjusted), L is themaximum (non-stressed) rooting depth, π50 is the osmotic potentialat which the yield is reduced by 50%, h50 is matric potential at whichthe yield is reduced by 50%)

Interval Effective ET0 CC L π50 h50(days) rainfall (mm (mm) (m) (m)

(mm) day–1)

12 – 3.5 0.19 300 –26 –261 60 3.5 0.19 350 –26 –26

17 – 6.9 0.40 600 –26 –261 52 6.9 0.40 650 –26 –268 – 7.8 0.72 900 –19 –191 14 7.8 0.72 950 –19 –198 – 7.6 0.89 1200 –19 –191 11 7.6 0.89 1250 –19 –198 – 7.4 0.92 1500 –26 –261 18 7.4 0.92 1500 –26 –268 – 7.2 0.82 1500 –26 –261 23 7.2 0.82 1500 –26 –268 – 6.5 0.66 1500 –26 –261 15 6.5 0.66 1500 –26 –26

13 – 4.5 0.45 1500 –26 –261 20 4.5 0.45 1500 –26 –267 – 4.4 0.15 1500 –26 –26

ments with lysimeters in eastern Colorado. One equationdescribes CC from planting to full cover, the other fromfull cover to harvest. They can be expressed as:

CC = aR 3 + bR 2 + cR + d . (9)

For the time before full cover, R is the fraction of time from planting to full cover (0.0 – 1.0); for the time after full cover, R is days after full cover. In Colorado, corn onaverage achieves full cover 10 days after tasseling andbeans 50 days after planting. Parameters for the constantsa, b, c, and d are taken from Duke (1987), and daily val-ues are calculated for the growing period of each crop. Theinput data for CC presented in Tables 1 and 2 are based onthese daily values.

Tables 1 and 2 also give the assumed maximum root-ing depth, L, of corn and beans at the input-specifiedtimes. Furthermore, it shows the osmotic potential andmatric potential which result in a 50% yield reduction(π50 and h50). Based on Maas (1986), the π50 chosen forcorn is –42 m and for beans –26 m. Matric and osmoticpotentials are assumed to have equal effects on water up-take, with a (t) in Eq. (6) set at 1.0. Growth-stage-specificstress tolerances are taken into account by adjusting thevalues of π50 and h50 . Corn has been shown to be ex-tremely sensitive to water deficits during the tasseling pe-riod (Rhoads and Bennett 1990). Based on Claassen andShaw (1970) and Stewart et al. (1975), the π50 and h50values for corn during tasseling are reduced by 25% fromthose given by Maas (1986). The critical period for beansis from flowering to pod filling (US Department of Ag-riculture, Soil Conservation Service 1988). Water stressduring this period has been shown to reduce yields byabout 25% (Halterlein 1983). Therefore, π50 and h50 val-ues for beans are reduced by 25% between days 52 and71 after planting.

The use of the Darcy-Richards equation requires datafor the hydraulic conductivity- and matric potential-watercontent relationship for the soil modeled. The modified vanGenuchten-Hanks model uses the equations proposed byCampbell (1974) as modified by Hutson and Cass (1987).The specific soil properties for the loam soil in the studyarea are reflected by choosing appropriate values for theparameters in these equations (for details see Scheierling1995).

Yield values for corn and beans are obtained using Eq.(7) to first transform ET estimates into values for relativeyield, Y/Ymax. Relative yield values are then multiplied bymaximum yield, Ymax, to obtain absolute yield values. Theapplication of Eq. (7) requires values for the stress-ad-justed potential ET, Smax, the crop-specific yield responsecoefficient, c, and maximum yield, Ymax. Estimates forSmax are obtained from model calculations. Values for care chosen based on Doorenbos and Kassam (1979); forcorn, c is 1.25 and for beans 1.15. Values for Ymax are basedon suggestions of Colorado State University extensionagents in Weld County; the typical maximum yield perhectare is assumed to be 10 670 kg for corn and 2690 kgfor beans.

Results and discussion

On each of the nine specified irrigation days during thegrowing season, an irrigation event may or may not occur.This results in 29 = 512 alternative irrigation schedules forcorn, and 28 = 256 alternative irrigation schedules forbeans, ranging from no irrigation to nine or eight irriga-tions, respectively. An irrigation schedule of (0 1 1 1 1 11 0 0) for corn, for example, would imply irrigation eventsof 76 mm net infiltration on 20 June, 30 June, 10 July, 20July, 30 July, and 10 August, but no irrigation events on 1 June, 25 August, and 8 September. Each of the possiblecombinations is used as an input to the simulation modelto estimate the respective values for ET and, subsequently,yield for corn and beans. By using the method proposedby Doorenbos and Kassam (1979) (Eq. (7)), yield valuesare simply a linear transformation of the ET values.

Model estimates must approximate field conditionswith an adequate degree of accuracy to be useful. Table 3therefore presents predicted yields for the “extreme” caseswith no irrigation and full irrigation (that is, nine irriga-tions for corn and eight irrigations for beans during thegrowing season) and measured yields for non-irrigated(dryland) and irrigated crops in the study area. Data onmeasured yields are reported in the Colorado AgriculturalStatistics issued yearly by the US Department of Agricul-ture and the Colorado Department of Agriculture. Usingrecent data, the mean is calculated for corn and beans underconditions of no irrigation and full irrigation. A compari-son between measured and predicted yields shows that themodel predicts reasonably well actual crop yields for theextreme cases. Based on this result, the model estimatesfor the cases with one to eight (or seven) irrigation eventsare also believed to adequately represent actual conditions.

Yield estimates for all possible combinations of irriga-tion events are displayed in Fig. 1 for corn and in Fig. 2for beans. Each point represents a water-crop productionfunction providing a relationship between crop yield andthe quantity of infiltrated irrigation water. The special char-acteristic of these water-crop production functions is thatthey show yield not only as a function of seasonal net ir-rigation water infiltration, but also of the number and tim-

27

Table 3 Comparison between measured and predicted yields ofcorn and beans. Measured yield data are from the Colorado Agricul-tural Statistics Service. Means for no irrigation are based on data fornortheastern Colorado. Means for full irrigation are based on datafor Weld County, Colorado

Measured yield Predicted yield(mean for 1989 – 1992)(kg/ha) (kg/ha)

CornNo irrigation 3439 3653Full irrigation 9603 9465

BeansNo irrigation 1009 1223Full irrigation 2477 2495

ing of irrigations during the growing season. Thus, theysuggest the dynamic aspects of irrigation timing and theyield effect of alternative irrigation schedules.

Selected model results for the irrigation combinationswhich achieve the lowest and highest yields with a givennumber of irrigations are shown in Table 4 for corn and inTable 5 for beans, including associated ET and yield val-ues, irrigation schedules and soil moisture in the profile atthe end of the irrigation season. Space limitations precludecomplete enumeration. However, the results presented pro-vide some useful insights.

First, depending on the timing of irrigation, the effectof a given number of irrigations on yield varies widely, par-ticularly in the range between two and five irrigations. Forexample, corn yield resulting from four irrigations may

range from 4795 kg to 9326 kg/ha. The lowest yield forany number of irrigation events is generally attained whenthe irrigations take place in the latter part of the season;the highest yield is achieved when the irrigation water isapplied towards the beginning of the season when crop wa-ter needs are highest and/or crops are highly sensitive towater stress. Irrigations after 20 July have little effect iffour irrigations have already occurred, because the soil pro-file has accumulated enough moisture to carry the cropthrough the critical periods and maturation.

Second, when examining the optimally timed irrigationschedules (those which achieve the highest yield for a givennumber of irrigations), it becomes obvious that the incre-ments in yield from additional irrigation events diminishas the number of irrigations increases. For example, corn

28

Fig. 1 Computed yield of cornas a function of the number andtiming of irrigations

Fig. 2 Computed yield of beansas a function of the number andtiming of irrigations

yield increases by 364 kg (from 8962 kg/ha to 9326 kg/ha)if the number of optimally timed irrigations is increasedfrom three to four, but it increases by only 107 kg (from9326 kg/ha to 9433 kg/ha) if the number of optimally timed irrigations is further increased to five. Alternativelystated, when farmers can choose the optimal timing of ir-rigations, reducing the number of irrigations from nine tofive has a relatively limited adverse effect on yield; how-ever, the decrement in yield increases rapidly as the irri-gations are reduced to less than four per season. This im-plies a diminishing marginal productivity of irrigation wa-ter in the case of optimally scheduled irrigation events.

Third, values for ET (and yield) and soil moisture dif-fer significantly depending on the irrigation schedule cho-sen. As the number of irrigations increases, ET (and yield)as well as the moisture remaining in the soil profile at theend of the season increase but at different rates. In the caseof irrigation schedules resulting in highest yields, the in-crease in ET values diminishes as more and more irriga-tions are applied, whereas the increase in soil moisture values continues to rise. In the case of irrigation schedulesresulting in lowest yields, this relationship is reversed.

Estimates of ET are significant for several irrigation wa-ter allocation issues. The difference between total irriga-tion water applied and ET goes to either soil storage, sur-face runoff, or deep percolation, of which the latter twomay be available to downstream users. The amount of ir-rigation water contributing to runoff and deep percolationrises with higher numbers of irrigations (particularly ifmore than four optimally scheduled irrigations are ap-plied). Even though the exact amount of water running offor percolating out of the modeled soil profile cannot be es-timated with a simulation model based on net infiltrationof irrigation water, it is clear from Tables 4 and 5 that notonly the amount of seasonal irrigation water applied, butalso the timing of irrigation events during a season stronglyinfluences the amount of runoff and deep percolation,which in many circumstances is an important influence onthe quantity and quality of water available downstream.

The model results indicate that there is only a weak cor-relation between the amount of irrigation water infiltratedon the one hand and ET and yield on the other. In particu-lar, when framers have the option to optimally time irriga-tion events for each crop, an additional irrigation towards

29

Table 4 Selected model results of corn production for the cases with lowest and highest yield from a given number of irrigations. Each irrigation results in 76 mm of net infiltration

Number Lowest yield Highest yieldofirrigations Evapo- Yield Associated Ending soil Evapo- Yield Associated Ending soil

transpiration (kg/ha) irrigation moisture in transpiration (kg/ha) irrigation moisture in(mm) schedule 2.5-m profile (mm) schedule 2.5-m Profile

(mm) (mm)

0 243 3653 000000000 351 243 3653 000000000 3511 244 3716 000000001 417 330 5969 010000000 3422 252 3910 000000011 482 400 7789 101000000 3533 266 4293 000000111 543 445 8962 101100000 3874 285 4795 000001111 602 459 9326 111010000 4465 311 5454 000011111 652 463 9433 111110000 5096 352 6477 000111111 696 464 9458 111111000 5707 406 7939 001111111 717 464 9465 111111100 6228 437 8755 011111111 741 464 9465 111111110 6789 464 9465 111111111 750 464 9465 111111111 750

Table 5 Selected model results of bean production for the cases with the lowest and highest yield from a given number of irrigations. Eachirrigation results in 76 mm of net infiltration

Number Lowest yield Highest yieldofirrigations Evapo- Yield Associated Ending soil Evapo- Yield Associated Ending soil

transpiration (kg/ha) irrigation moisture in transpiration (kg/ha) irrigation moisture in(mm) schedule 2.5-m profile (mm) schedule 2.5-m Profile

(mm) (mm)

0 187 1223 00000000 397 187 1223 00000000 3971 188 1230 00000001 469 257 1834 01000000 4042 193 1278 00000011 539 302 2224 10100000 4343 204 1366 00000111 605 322 2399 11100000 4834 223 1536 00001111 665 331 2475 11110000 5445 251 1777 00011111 715 333 2492 11111000 6046 291 2126 00111111 742 333 2495 11111100 6587 319 2372 01111111 762 333 2495 11111110 7038 333 2495 11111111 770 333 2495 11111111 770

the end of the season requires 76 mm of water infiltrationbut contributes negligibly to an increase in ET and yield;it results in increases in soil moisture and deep percola-tion. When considering the amount of irrigation waterwhich – depending on the application efficiency of the ir-rigation method used – actually needs to be applied toachieve a net infiltration of 76 mm, the correlation is evenless strong. This implies that when irrigation water isscarce, farmers can to some extent substitute optimal tim-ing of irrigation events and switching to irrigation meth-ods with higher application efficiencies for additional ir-rigation water without significantly impacting ET and cropyield.

However, there are circumstances in which it is not pos-sible for farmers to optimally time irrigation events. Anexample would be when irrigation water cannot be drawnfrom several sources and made available at any time dur-ing the growing period, as in the study area. In such cases,farmers would have to make trade-offs between, for exam-ple, applying irrigation water to either corn or beans on acertain irrigation day. To determine the most profitableintraseasonal allocation of irrigations between two or morecrops, the application of an economic model would be required which uses the water-crop production functionsestimated with the discrete-input simulation model as aninput.

Summary and conclusions

Crop yield and ET are important considerations in select-ing water management policies, both for farm managersand policy analysts. The purpose of this research is to dem-onstrate the importance of accounting for irrigation appli-cation timing effects on water-crop production functionsto estimate ET and yield. Simulations of a range of dis-crete irrigation scheduling options for a northeastern Col-orado case study suggest the relationships between irriga-tion water application and yield necessary for determiningoptimal intraseasonal allocation of irrigation water. The approach is based on the premise that, rather than consid-ering irrigation water as a continuously variable input,farmers, under certain circumstances, tend to treat irriga-tions as discrete events and make irrigation scheduling decisions as choices among numbers of irrigations of ap-proximately equal size.

To reflect the perspective of irrigations as discrete in-puts, a transient-state water-crop simulation is formulatedby employing the van Genuchten-Hanks model. Resultsfor yields and, implicitly, ET of two important crops, corngrain and edible dry beans, have been presented. They in-dicate that, given a certain number of irrigations, cropyields vary enormously depending on the timing of irriga-tions, and, given the optimal timing of irrigations, the in-crease in crop yield from an additional irrigation event di-minishes as the number of irrigations increases. This im-plies that the number of irrigations, and thus the amountof water applied, can be reduced over a certain range with-

out impacting ET and crop yield significantly. In the caseof corn, the yield from four optimally timed irrigations isnot appreciably different from that resulting from nine ir-rigations. However, there are many circumstances in whichan inability to apply water can cause a very large reduc-tion in yield. An application of four irrigations scheduledin the least optimal way can yield as little as half that ofthe optimal four-irrigation schedule.

The limitations of the crop-water production functionanalysis presented here result to a large degree from its de-terministic assumptions. Rainfall is assumed to reflect thedaily average pattern. Irrigations can take place only onspecified dates. The net irrigation water infiltration is alsoassumed to be uniform for all locations in the study area.The authors have not yet considered the effects of alterna-tive irrigation schedules on the relative amounts of deeppercolation and evaporation. In further research, it will beuseful to test the sensitivity of the results to these assump-tions. In addition, it would be worthwhile to apply themodel to multiyear scenarios which could be used to ex-amine the longer-term effect of alternative irrigation wa-ter schedules on ET and yield as well as soil water content.

Although a transient-state simulation model requiresmore analytic and modeling effort than do seasonal mod-els, it reflects important realities of irrigated crop produc-tion. It is clear that water policy analysts will often wishto consider the more realistic model for evaluating issuesin irrigation water management.

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