- Email: [email protected]
Crop production functions and the allocation and use of irrigation water
Agricultural Water Management, 3 (1980) 53---64
53
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
CROP PRODUCTION FUNCTIONS AND THE ALLOCATION AND USE OF IRRIGATION WATER
J.W. HUGH BARRETT and GAYLORD V. SKOGERBOE*
Sinclair Knight and Partners Pry. Ltd. Cimanuk River Basin Development Project, P.O. Box 9, Bandung (Indonesia) *Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, Colo. 80523 (U.S.A.) (Accepted 18 October 1979)
ABSTRACT Barrett, J.W.H. and :Skogerboe, G.V., 1980. Crop production functions and the allocation and use of irrigation water. Agric. Water Manage., 3: 53--64. The economically optimal depth of irrigation water to apply depends on the relationship between crop yield and water use. Past research efforts to formulate and to explain the factors influencing irrigated crop production functions have therefore been briefly reviewed. Although it is not possible to obtain a unique relationship, by considering a possible range of functions, and by understanding the factors causing variations in the form of these functions, valuable conclusions can be drawn relating to the optimal depth of water application and the relative magnitude of benefits derived from efficient water management.
INTRODUCTION
Irrigation planners who have considered the problem of the optimal depth of irrigation application are divided on the answer. Many would appear to agree with Stewart et al. (1973) who state in the case where water is limited relative to the available land area, that "the optimal seasonal [irrigation] depth for profit maximization will be one unit (or as few units as practicable) greater than [irrigation] at the economic break-even point."
On the other hand, Hillel (1972) feels that "Traditionally, the great fallacy in water management has been the tendency to save water per unit of land area, in order to "green u p " more land. Some of the agricultural planners in Israel, as well as in other arid countries, have fallen prey to this fallacy. We must remember that our basic aim is not to save water but to increase production efficiency by optimizing the water supply (and other environmental variables) so as to maximize plant response. The best chance of increasing production efficiency by water management is to obviate water stress and prevent water from becoming a limiting factor in plant growth." 0378--3774/80/0000---0000/$02.25 © 1980 Elsevier Scientific Publishing Company
54 A qualitative assessment of the economically optimal allocation of irrigation can be based on the functional relationship between crop yield and water use. In practice, however, the function for any crop is far from unique, varying from year to year, area to area and farmer to farmer and, consequently, applying numerical methods to a specific function becomes meaningless. For a given field in a given year, the management practices alone can cause a wide variation in the form of the function, depending on the efficiency of application. Therefore, by considering a range of functions representing different levels of efficiency, a more general indication of the optimal depth of application can be obtained and the value of good management can be demonstrated. This first requires an understanding of the factors influencing crop production functions. CROP YIELD-WATER USE FUNCTIONS
Linear functions Some of the earliest production functions were obtained b y researchers investigating the effects of precipitation and soil moisture on dryland crop yields. Cole (1938) studied the relationship between annual precipitation and the yields of spring wheat on the American Great Plains and obtained a linear relationship. Similarly, Leggett (1959), when relating wheat yields in the eastern Washington dryland areas to available soil moisture in the spring, plus rainfall in the growing season, obtained a linear correlation. Attempts to develop production functions for irrigated crops have been numerous. One of the most comprehensive studies of the dry matter yield-transpiration relationship was undertaken b y De Wit (1958), who studied a wide range of data collected for c o m m o n field crops grown in containers from which evaporation was controlled or measured. In the very few cases where the relationship varied from linear, De Wit considered this to be associated with p o o r aeration of the r o o t system. Arkley (1963) extended the work of De Wit b y plotting the yield of dry matter versus transpiration corrected for mean relative atmospheric humidity during the period of most active growth. Graphs were plotted from sites around the world and for a wide range of crops, including barley, oats, wheat, maize, millet, alfalfa, carrots, snapdragons and weeds. All the graphs showed linear relationships with very high correlation coefficients. In most cases, the line of best fit also passed through the origin, showing that yield of dry matter was directly proportional to the transpiration. Arkley concluded that deviation from a straight line with increasing moisture revealed the effect of overirrigation, while reversal in the slope of the line suggested waterlogging or poor aeration in the soil. A similar relationship between dry matter yield and transpiration was obtained by Hanks et al. (1969) for grain sorghum at Akron, Colo. Linear relationships also existed for wheat, oats, millet and grain sorghum when cumulative dry matter yield was plotted against evapotranspiration (ET) as shown in Fig.l, although the line no longer passed through the origin.
55 • z~ 0 V a
8000
7000
Lysimeters Lysimeters Dry Lysimeters Irr, Sampled Dry Sampled ~[rr.
6000
T
=o
1966 1967 1967 1967 1967 &. /,
5000
4000
200C ~_
I000
j~/ 0
Yield = 208 ET- 1640 r z = 0.977
0 0
I00
200
300
400
500
ET (mm)
Fig.1. Relation of cumulative dry matter yield of grain sorghum to evapotranspiration (from Hanks et al., 1969). A linear relationship between both dry matter and grain yield and evapotranspiration was obtained by Hillel and Guron (1973) experimenting with maize in Israel. Experiments with maize by the Consortium for International Development (1976) showed a similar result. Shalhavet et al. (1976) also obtained linear relationships between relative yield (actual yield as a percentage of maximum yield) and net water application for wheat, grain sorghum, grain and forage maize, cotton and tomatoes.
Non-linear functions Numerous experiments have indicated a non-linear relationship between crop yield and water use. Although, for example, Musick and Dusek (1971) found that lower yielding treatments could be represented by a linear relationship, the higher yielding treatments of their experiments indicated that the seasonal yield--water use curve is a curvflinear diminishing return relationship under conditions of good water management. The analysis was expanded by Musick et al. (1976) to include more extensive data. Relative grain yields of grain sorghum, wheat and soybeans were plotted against seasonal soil water depletion from the 0 to 1.2 m soil depth and a quadratic equation fitted to all scatter diagrams with a high degree of success. Similarly, Shipley and Regier (1975) and Shipley (1977) obtained diminishing curvilinear relationships between yield and applied water for maize,
56
grain sorghum and wheat. Different yields were obtained for a given water quantity, depending on the stage of growth during which the water was applied. By taking only the peak yield associated with a given depth of application, a quadratic expression fitted the results.
Timing of deficits The scatter in the results obtained by the m a n y researchers is undoubtedly largely due to the timing of water deficits. Numerous investigators have shown that a water deficit occurring at one stage of growth will have a different effect on yield as compared with a deficit occurring at another stage. For the same a m o u n t o f evapotranspiration, the yield may be expected to differ. For example, Robins and Domingo (1962) found that soil moisture depletion to the wilting point for 1 or 2 days during the tasseling or pollination period of maize results in as much as a 22% reduction in grain yield, while periods of 6 to 8 days gave a yield reduction of about 50%. Following maturity, the depletion of the available soil moisture had no effect on yield. Denmead and Shaw (1960) found that maize subjected to moisture stress at silking was the most severely affected as far as grain was concerned. D o w n e y (1972) substantiated these results for maize, although he found that water stress during male meiosis (or while the crop was young) increased water use efficiency. In an experiment in which grain sorghum was grown in lysimeters, Howell and Hiler (1975) found only a weak correlation between yield and seasonal evapotranspiration. T h e y concluded that the timing of the occurrence of a water deficit had more influence on the yield than the magnitude o f the deficit and t h e y made water management recommendations accordingly. Similarly, Stewart et al. (1974) made a n u m b e r of recommendations for managing water in maize production, to ensure that unavoidable deficits are timed to occur at the least damaging times. These recommendations for grain sorghum and maize are in general agreement with the conclusions of Salter and Goode (1967), who summarized research on the yield response to water at different growth stages for a wide variety of crops, ranging from cereals to orchards and vegetables to flowers.
The functional relationship To obtain a functional relationship between crop yield and water use, two problems evident in the literature reviewed above must be overcome, viz. (i) to explain h o w some researchers obtained a curvilinear function between yield and water use while m a n y others obtained a linear relationship, and (ii) to show h o w a unique relationship between yield and evopotranspiration can be developed when it is known that different yields can be obtained for the same a m o u n t of evaportranspiration, depending on the timing of deficits.
57 The first problem is due to a combination of two causes. Firstly, when water is applied in excess of the a m o u n t required for m a x i m u m yield, "water use" (and to some extent evapotranspiration) increases while yield remains constant or decreases. A decrease in yield is particularly apparent if waterlogging reduces soil aeration {Downey, 1972). This, coupled with the scatter of data due to the timing of deficits, allows a curvilinear function to be fitted to the plotted data. Secondly, in most cases the abscissa has not been evapotranspiration, but rather applied water or some similar parameter. In this case, the curvilinear nature o f the function m a y be due to a portion of the applied water being unavailable for crop evapotranspiration (e.g., deep percolation losses). II000
(Ymax, ETmox) (Ymex, Irr'rnax)
~ lOOO0
ET
n
9000 8000 7000 6000
5000 4000
o~
3.s
7.61 ,,.e
,5.6 ,e,~.6
~-(ET from ASWP + R ) - ~ Season Irr. Depth (Inches]
3000
2000 I000 0{~
i
I
i
i
i I0
i
1
IT i
i
i 20
i
14 .~4 2 3___/7 Field Water Supply (Inches)
i
1 30
i
i 34
3 2 .~
Legend: ASWP
=
Depth of Available Soil Water at Plantin9
R
=
Rainfall Depth
Irr.
ffi Water Supplied by Irrigation
FWS
ffi Field Wafer Supply
Fig. 2. Relations between yield, evapotranspiration and irrigation taking into account stored water and rainfall during season (from Stewart and Hagan, 1973).
The difference between the two functions is well illustrated in Fig.2, in which the curved line represents the relationship between yield and the seasonal depth of irrigation water applied, or yield and field water supply (which includes available soil water at planting plus rainfall) and the straight line represents the relationship between yield and ET. The difference between the straight line and the curved line, shown as "non-ET" on the figure, represents the water applied (or supplied) that is not consumed (i.e water losses). The convexity of the applied water function illustrates that these losses increase percentagewise as the point of fully satisfying ET demands is approached. The losses are likely due to deep percolation below the root zone,
58
but could also consist of unmeasured surface runoff. When Hillel and Guron (1973) t o o k particular care to calculate the drainage c o m p o n e n t of the field water balance, the relationship between grain yield of maize and seasonal evapotranspiration was strongly linear. The problem of developing a unique relationship between yield and evapotranspiration has received considerable attention from Stewart and associates at Davis, Calif.. They recognized that irrigation programming registers a dual effect on yield (Stewart and Hagan, 1973). The first effect is inevitable, the second manageable. The primary effect is that of water shortage, per se. Thus, any seasonal ET deficit is inevitably associated with some minimum fractional reduction in yield below maximum. A secondary reduction in yield may result from the timing of the ET deficits, with those occurring during more sensitive or "critical" growth stages of the crop in question causing a relatively larger decrease in yield. Such losses are avoidable through improved water management practices, which has the effect of orchestrating the sequence of ET deficits so that yield loss is minimized (Stewart et al., 1976). The Davis experiments have shown that when ET deficit sequencing is optimal (i.e. any deficits are timed so that they cause the least possible reduction in crop yield) the relationship between yield and seasonal ET is quite well represented by a straight line function for maize, grain sorghum and pinto beans. If the upper bound of yield is related to the depth of water applied, rather than ET, a curvilinear relationship will result. The analysis and discussion to follow are based on functions representing this upper bound of the yield--water use relationship (i.e., those representing irrigation regimes in which deficits are sequenced to occur at the least damaging times). This premise is reasonable in that it represents the ultimate goal of the irrigation farmer. However, should irrigation timing be non-optimal, the function will fall lower and these effects may also be readily assessed. The linear y i e l d - w a t e r use (ET) function may be thought of as the ideal, with curvilinear relations representing deviations from the ideal. The relative proximity of the field production function to the ideal allows an evaluation of irrigation managem e n t practices and irrigation system performance. OPTIMAL DEPTH OF WATER APPLICATION
Astute water management practices are of particular significance in the c o m m o n case where the available water is limited relative to the irrigable land area. The a m o u n t of water allocated to each unit of land (i.e. the seasonal depth of application) determines the area of land irrigated (Ax)
Q A I =-x
where Q = total quantity of water available (fixed) and x = depth of water applied (variable).
(1)
59 The optimal area is that which maximizes net revenue (P) in the objective function, P = Returns-Costs.
(2)
The returns per unit area are the price received for the crop (Vc) times the yield per unit area (y). Over the irrigated area, Returns = Vc y AI.
(3)
The costs per unit area in the objective function consist of some yield dependent costs (cost of seed, fertilizing, harvesting, marketing), some constant costs (seedbed preparation, pesticides, interrow cultivation) and some area dependent costs (the costs of water, energy and labour are relatively constant for a fixed volume of water and hence are area dependent). The capital cost of the irrigation system will increase as the size of the system increases with expanding area. However, as the quantity of available water is fixed, the range in which the optimal land area can be expected to lie is limited, and hence, the cost can be assumed to increase linearly with area over this increment. (This is only for mathematical simplicity, as in fact any function could be incorporated in the subsequent graphical procedure.) Therefore, Costs =
vi y + k l + -~i
(4)
A1
where vi = yield dependent costs per unit area, k ~ = constant costs per unit area, k2 = area dependent costs per unit area. Substituting eqs. (3) and (4) into eq. (2) gives (5)
P = (Vc--V1) y A i - - k l A i - k 2
setting a P / ~ A I = 0, i.e.
(vc-vj) y + (vc-vj) 0y - k , = 0
(6)
which may be readily manipulated to show that the optimal land area is obtained where marginal revenue equals marginal costs, or by collecting terms, may be expressed as (0Y
AI + y
)
(Vc-V1)-k1=
0.
(7)
That is, as the return is increased b y expanding irrigation to an additional unit of area, because the total quantity of water is fixed, yield on the remaining area must decline. The optimal area to irrigate is where the t w o factors (area and yield) combine for the greatest return. To obtain the "optimizing equation" in terms o f x and y, substitute eq.(1) into eq.(5) and set a P / a A i = O. Therefore,
g~
x-y
( v c - v i) + kl = O.
(8)
60 A given production function in terms of x and y and given cost coefficients may be substituted in this equation and solved for x, the optimal depth of irrigation water to apply. Unfortunately, this is impossible for the general case. The production function relevant to the farmers' field relates yield to applied water. As mentioned at the outset, this function is so subjective that it is of little value for quantitative purposes, depending on the amount of rainfall during the growing season, the a m o u n t of available soft moisture at planting and, not insignificantly, on the water application techniques of the individual irrigator. The contribution of all of these factors to the water supply will be influenced by the soil type. Furthermore, transferring the results obtained from research plots to irrigated fields will be unrealistic due to the significant difference in efficiency and uniformity to be expected. However, a quantitative assessment m a y be made of the effect of production functions on agricultural income by considering the relative difference between functions representing different irrigation efficiencies. For convenience, eq.(5) will be divided into the two components,
0
Returns = (Vc-Vj) y -x
(9)
and
(10)
Costs = k, Q - k s , x after substituting eq.(1) into eq.(5). EFFECT OF PRODUCTION FUNCTIONS ON AGRICULTURAL INCOME
To incorporate the general case, a range of production functions is shown in Fig. 3. Curves 1 and 2 have been obtained from experimental data from IO00C
8000 Io
6000 ._~ :>=
4000
/I/i// /
/ /
/ I /I I 411
2000
ii i i / i
%
I/
.41
200
I
1
40o
I
6;0 ' 8;o ' , 0 ~ 0 ' , 2 0 o
Depth of Irrigotio. W=ler Applied
(x}
(ram)
Fig.3. Production functions for maize at different irrigation application efficiencies.
61 Grand Junction, Colo. (Barrett and Skogerboe, 1978) while curves 3 and 4 are similar to the results obtained by other researchers (e.g. reported by Hexem and Heady, 1978), representing production functions more typical of field conditions. A constant fertility level is assumed. All curves have been plotted on a c o m m o n abscissa so that, contrary to Fig.2, curves 1 and 2 are no longer tangential at lower values of x. Considering x as the depth of water applied, Curve 1, orginally being the plot of yield versus ET, represents a production function at 100% irrigation application efficiency. The seasonal irrigation application efficiency for the regime represented by Curve 2 is 82% (574 m m / 7 0 0 mm), with 71% for Curve 3 and 48% for Curve 4. Values of x and y from each of the curves in Fig.3. have been substitued into eq.(8) and plotted as the return functions shown in Fig.4. The cost 120
,
~
,
I00 0 0
/
o eo
.0,urn
Fon0,ion,
_= 60 o == 40 _=
~/
/
Net Revenue
,r~...~or /
",, i 4e% ~ - ~ o s t
Function
20
200
400
600
800
I000
1200
Depth of Irrigation Wafer Applied (x)(mm)
Fig.4. Variation in returns and costs with depth of irrigation water applied.
function, as derived in eq.(9), is independent of y. The net revenue is the vertical difference between the respective return function and the cost function. Typical costs and prices have been used in plotting the curves, but this is not particularly significant as these will only affect the relative position of the return functions to the cost function and will n o t change the form of either. Similarly, fixed overheads not connected with irrigation (such as building, fences, etc) have been ignored, as these will have just the same effect. Also, for simplicity, the price of water has been taken as constant per unit volume. Breaking the total cost of watering into c o m p o n e n t parts (water, labour, energy and fixed costs of the system) only has the same effect o f changing the relative position of the cost function. The point of m a x i m u m net revenue is where the return and cost functions are separated b y the greatest distance, i.e. where the curves have the same
62 slope. This point is circled on the four return functions. The corresponding value read on the abscissa is the optimal depth of water to apply. Referring back to Fig.3, for Curve 1 this can be seen to equal the amount corresponding to Ymax. The other regimes have optimal depths of water application (Fig.4) lower than the depths giving maximum yield (Fig.3), although for the regimes corresponding to Curves 2 and 3, it is not greatly less (91 and 86% of Xmax, respectively, where Xmax is the depth of application corresponding to Ymax). For Curve 4, it is considerably less, with X o p t = 0.58 Xmax ( X o p t = 700 mm and Xmax = 1200 mm). In general then, in a water-short area, the optimal irrigation policy is to apply close to the amount of water giving maximum yield only if irrigation application efficiencies are high. If efficiencies are low, less water should be applied. The optimal depth of irrigation water to apply depends solely on the form of the production function. The effect of the production function may be readily demonstrated by referring to Fig.4. If the irrigator follows a practice resulting in a production function the same as Curve 4 in Fig.3, a maximum net revenue of $30 000 may be scaled from Fig.4. A different practice resulting in a production function the same as Curve 2 would result in a maximum net revenue of $52 000, or a 73% increase. As pointed out earlier, fixed overhead costs have been deleted for simplicity; as these would have the effect of raising the cost function, the actual effect of increasing water application efficiency would be to increase the net revenue an even greater percentage than calculated. With costs and production functions available, and return functions derived from these functions as shown in Fig.4, the detrimental effect of low irrigation efficiencies may be readily evaluated. The losses in revenue due to poor irrigation practices are startlingly apparent. In most cases where water is in short supply, the irrigator will endeavor to improve practices so that little water is wasted. With relatively high efficiency so attained, the correct policy is then to apply sufficient water so that close to maximum yield is obtained. If the irrigation system is such that high efficiencies are not possible and capital is not available for improvement, the optimal depth of water to apply is significantly less than the depth corresponding to Ymax and could be obtained from a plot such as Fig.4, or derived mathematically if a reliable expression for the production functions were available. Unfortunately, however, such information is generally not available and can easily be shown to give misleading results (Barrett, 1977). Notwithstanding, a significant general conclusion may be drawn. That is, contrary to those who advocate spreading the available water over as large an area as possible (subject to applying the minim u m depth which is practical), the optimal depth to apply in a system with inherent high losses is a little greater than that which should be applied under more efficient regimes and greater than the crop ET. Regardless of irrigation efficiency, the optimal depth of application actually lies in a fairly narrow range. Furthermore, it is safer to err on the side of applying a greater depth of water than on the side of applying less, as the return and cost functions in Fig.4 converge far more slowly on the right hand side of the optimal depth than on the left hand side.
63
If the timing of irrigations is non-optimal, the production functions will fall below the curves shown in Fig.3. For a given efficiency, the economic losses will be far greater. Fortunately, this cause of reduced income may easily be reduced or eliminated b y acquiring the necessary knowledge of the effect of improper irrigation timing on yields. The extent of the effect of inefficiencies of irrigation application on farm income will determine the extent to which remedial measures are justified. In many cases, improved management is the only additional input required. The economic losses illustrated in Fig.4 show the extent to which the increased costs of better management are justified. Apart from optimal sequencing of any deficits, such practices as beginning the growing season with adequate moisture in the crop root zone, irrigating to bring the root zone to less than field capacity to take advantage of rainfall, sprinkling at night to avoid evaporation and wind drift, avoiding over-watering, and terminating irrigation as soon as crop needs are met at the end of the season, all contribute to improved efficiency of application. CONCLUSION
A number of management practices are available which require no more capital than knowledge and foresight to allow more efficient use of irrigation water. Although efficient irrigation is generally accepted as a desirable goal, the benefits are often expressed in somewhat nebulous or qualitative terms. A consideration of the nature of production functions reflecting efficient and inefficient irrigation regimes allows the relative magnitude of the benefits due to efficient water management to be expressed in terms of net revenue. The optimal depth of water to apply will always be in excess of the potential ET of the crop, increasing slightly with decreasing efficiencies but always less than that giving maximum yield. With methods having low application efficiency, this will result in deficits occurring during the season. Particular care will be needed to ensure that these deficits occur at the least damaging times. By applying an amount of water less than that necessary to achieve maximum yield, it is quite likely, in fact, that the seasonal efficiency of application will rise, as a higher proportion of the applied water may actually be used by the crop. This, in combination with good management practices, will have the effect of transforming a strongly curvilinear crop production function to one more nearly linear. The lower depth of water application will allow a larger land area to be irrigated and increase profits substantially.
REFERENCES Arkley, R.J., 1963. Relationships between plant growth and transpiration. Hilgardia, 34(13): 559--584. Barrett, J.W.H., 1977. Crop yield functions and the allocation and use of irrigation water. Unpublished Ph.D. Dissertation, Colorado State University, F o r t Collins, Colo., July, 210 pp.
64 Barrett, J.W.H. and Skogerboe, G.V., 1978. Effect of irrigation regime on maize yields. J. Irrigation Drainage Div., ASCE 104 (IR2): 179--194. Cole, J.S., 1938. Correlations between annual precipitation and the yield of spring wheat in the Great Plains. U.S. Dept. Agric. Tech. Bull. 636, 40 pp. Consortium for International Development, 1976. Water and salinity production functions. Technical completion report prepared for U.S. Dept. of Interior, Office of Water Research and Technology, Washington, D.C. 20240, November, 191 pp. Denmead, O.T. and Shaw, R.H., 1960. The effects of soil moisture stress at different stages of growth on the development and yield of corn. Agron. J., 52: 272--274. De Wit, C.T., 1958. Transpiration and crop yields. Institute of Biological and Chemical Research on Field Crops and Herbage, Wageningen, The Netherlands. Versl. Landbouwk. Onderz. No. 646, 's Gravenhage, 88 pp. Downey, L.A., 1972. Water--yield relations for nonforage crops. J. Irrigation Drainage Div., ASCE 98 (IRI): 107--115. Hanks, R.J., Gardner, H.R. and Florian, R.L., 1969. Plant growth--evapotranspiration relations for several crops in the central Great Plains. Agron. J., 61: 30--34. Hexem, R.W. and Heady, E.O., 1978. Water Production Functions for Irrigated Agriculture. Iowa State University Press, Ames, Iowa, 215 pp. Hillel, D., 1972. The field water balance and water use efficiency. In: D. Hillel (Ed.), Optimizing the Soil Physical Environment Toward Greater Crop Yields. Academic Press, New York and London, pp. 79---100. Hillel, D. and Guron, Y., 1973. Relation between evapotranspiration rate and maize yield. Water Resour. Res., 9 (3): 743--748. Howell, T.A. and Hiler, E.A., 1975. Optimization of water use efficiency under high frequency i r r i g a t i o n - I. Evapotranspiration and yield relationship. Trans. Am. Soc. Agric Eng., 18(5): 873--878. Leggett, G.E., 1959. Relationship between wheat yield, available moisture and available nitrogen in eastern Washington dryland areas. Washington Agric. Exp. Stn Bull. 609, 16 pp. Musick, J.T. and Dusek, D.A., 1971. Grain sorghum response to number, timing and size of irrigations in the southern high plains. Trans. Am. Soc. Agric. Eng., 14(3): 401--404, 410. Musick, J.T., New, L.L. and Dusek, D.A., 1976. Soil water depletion-yield relationships of irrigated sorghum, wheat and soybeans. Trans. Am. Soc. Agric. Eng., 19(3): 489--493. Robins, J.S. and Domingo, C.E., 1962. Moisture and nitrogen effects on irrigated spring wheat. Agron. J., 54: 135--138. Salter, P.J. and Goode, J.E., 1967. Crop response to water at different stages of growth. Commonwealth Agricultural Bureau, Farnham Royal, Bucks, Great Britain, 245 pp. Shalhevet, J., Mantell, A., Bielorai, H. and Shimshi, D., 1976. Irrigation of field and orchard crops under semi-arid conditions. International Irrigation Information Center Publ. No. 1, 110 pp. Shipley, J.L., 1977. Scheduling irrigations with limited water. Draft prepared for the Proceedings of the High Plains Grain Conference, Guymon, Okla., February. Shipley, J. and Regier, C., 1975. Water response in the production of irrigated grain sorghum, High Plains of Texas. Texas Agricultural Experiment Station, MP-1202, June, 8 pp. Stewart, J.I. and Hagan, R.H., 1973. Functions to predict effects of crop water defictis. J. Irrigation Drainage Div., ASCE, 99 (IR4): 421--439. Stewart, J.I., Hagan, R.H., Pruitt, H.O. and Hall, W.A., 1973. Water production functions and irrigation system design and for increased efficiency in water use. U.S. Dept. of Interior, Bureau of Reclamation, Engineering and Research Center, Denver, Report 14-06-D-7329, 164 pp. Stewart, J.I., Hagan, R.M. and Pruitt, W.O., 1974. Functions to predict optimal irrigation programs. J. Irrigation Drainage Div., ASCE 100 (IR2): 197--199. Stewart, J.I., Hagan, R.M. and Pruitt, W.O., 1976. Water production functions and predicted irrigation programs for principal crops as required for water resources planning and increased water use efficiency. U.S. Dept. of Interior, Bureau of Reclamation, Engineering and Research Center, Denver, Report 14-06-D-7329, 80 pp.
53
Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
CROP PRODUCTION FUNCTIONS AND THE ALLOCATION AND USE OF IRRIGATION WATER
J.W. HUGH BARRETT and GAYLORD V. SKOGERBOE*
Sinclair Knight and Partners Pry. Ltd. Cimanuk River Basin Development Project, P.O. Box 9, Bandung (Indonesia) *Department of Agricultural and Chemical Engineering, Colorado State University, Fort Collins, Colo. 80523 (U.S.A.) (Accepted 18 October 1979)
ABSTRACT Barrett, J.W.H. and :Skogerboe, G.V., 1980. Crop production functions and the allocation and use of irrigation water. Agric. Water Manage., 3: 53--64. The economically optimal depth of irrigation water to apply depends on the relationship between crop yield and water use. Past research efforts to formulate and to explain the factors influencing irrigated crop production functions have therefore been briefly reviewed. Although it is not possible to obtain a unique relationship, by considering a possible range of functions, and by understanding the factors causing variations in the form of these functions, valuable conclusions can be drawn relating to the optimal depth of water application and the relative magnitude of benefits derived from efficient water management.
INTRODUCTION
Irrigation planners who have considered the problem of the optimal depth of irrigation application are divided on the answer. Many would appear to agree with Stewart et al. (1973) who state in the case where water is limited relative to the available land area, that "the optimal seasonal [irrigation] depth for profit maximization will be one unit (or as few units as practicable) greater than [irrigation] at the economic break-even point."
On the other hand, Hillel (1972) feels that "Traditionally, the great fallacy in water management has been the tendency to save water per unit of land area, in order to "green u p " more land. Some of the agricultural planners in Israel, as well as in other arid countries, have fallen prey to this fallacy. We must remember that our basic aim is not to save water but to increase production efficiency by optimizing the water supply (and other environmental variables) so as to maximize plant response. The best chance of increasing production efficiency by water management is to obviate water stress and prevent water from becoming a limiting factor in plant growth." 0378--3774/80/0000---0000/$02.25 © 1980 Elsevier Scientific Publishing Company
54 A qualitative assessment of the economically optimal allocation of irrigation can be based on the functional relationship between crop yield and water use. In practice, however, the function for any crop is far from unique, varying from year to year, area to area and farmer to farmer and, consequently, applying numerical methods to a specific function becomes meaningless. For a given field in a given year, the management practices alone can cause a wide variation in the form of the function, depending on the efficiency of application. Therefore, by considering a range of functions representing different levels of efficiency, a more general indication of the optimal depth of application can be obtained and the value of good management can be demonstrated. This first requires an understanding of the factors influencing crop production functions. CROP YIELD-WATER USE FUNCTIONS
Linear functions Some of the earliest production functions were obtained b y researchers investigating the effects of precipitation and soil moisture on dryland crop yields. Cole (1938) studied the relationship between annual precipitation and the yields of spring wheat on the American Great Plains and obtained a linear relationship. Similarly, Leggett (1959), when relating wheat yields in the eastern Washington dryland areas to available soil moisture in the spring, plus rainfall in the growing season, obtained a linear correlation. Attempts to develop production functions for irrigated crops have been numerous. One of the most comprehensive studies of the dry matter yield-transpiration relationship was undertaken b y De Wit (1958), who studied a wide range of data collected for c o m m o n field crops grown in containers from which evaporation was controlled or measured. In the very few cases where the relationship varied from linear, De Wit considered this to be associated with p o o r aeration of the r o o t system. Arkley (1963) extended the work of De Wit b y plotting the yield of dry matter versus transpiration corrected for mean relative atmospheric humidity during the period of most active growth. Graphs were plotted from sites around the world and for a wide range of crops, including barley, oats, wheat, maize, millet, alfalfa, carrots, snapdragons and weeds. All the graphs showed linear relationships with very high correlation coefficients. In most cases, the line of best fit also passed through the origin, showing that yield of dry matter was directly proportional to the transpiration. Arkley concluded that deviation from a straight line with increasing moisture revealed the effect of overirrigation, while reversal in the slope of the line suggested waterlogging or poor aeration in the soil. A similar relationship between dry matter yield and transpiration was obtained by Hanks et al. (1969) for grain sorghum at Akron, Colo. Linear relationships also existed for wheat, oats, millet and grain sorghum when cumulative dry matter yield was plotted against evapotranspiration (ET) as shown in Fig.l, although the line no longer passed through the origin.
55 • z~ 0 V a
8000
7000
Lysimeters Lysimeters Dry Lysimeters Irr, Sampled Dry Sampled ~[rr.
6000
T
=o
1966 1967 1967 1967 1967 &. /,
5000
4000
200C ~_
I000
j~/ 0
Yield = 208 ET- 1640 r z = 0.977
0 0
I00
200
300
400
500
ET (mm)
Fig.1. Relation of cumulative dry matter yield of grain sorghum to evapotranspiration (from Hanks et al., 1969). A linear relationship between both dry matter and grain yield and evapotranspiration was obtained by Hillel and Guron (1973) experimenting with maize in Israel. Experiments with maize by the Consortium for International Development (1976) showed a similar result. Shalhavet et al. (1976) also obtained linear relationships between relative yield (actual yield as a percentage of maximum yield) and net water application for wheat, grain sorghum, grain and forage maize, cotton and tomatoes.
Non-linear functions Numerous experiments have indicated a non-linear relationship between crop yield and water use. Although, for example, Musick and Dusek (1971) found that lower yielding treatments could be represented by a linear relationship, the higher yielding treatments of their experiments indicated that the seasonal yield--water use curve is a curvflinear diminishing return relationship under conditions of good water management. The analysis was expanded by Musick et al. (1976) to include more extensive data. Relative grain yields of grain sorghum, wheat and soybeans were plotted against seasonal soil water depletion from the 0 to 1.2 m soil depth and a quadratic equation fitted to all scatter diagrams with a high degree of success. Similarly, Shipley and Regier (1975) and Shipley (1977) obtained diminishing curvilinear relationships between yield and applied water for maize,
56
grain sorghum and wheat. Different yields were obtained for a given water quantity, depending on the stage of growth during which the water was applied. By taking only the peak yield associated with a given depth of application, a quadratic expression fitted the results.
Timing of deficits The scatter in the results obtained by the m a n y researchers is undoubtedly largely due to the timing of water deficits. Numerous investigators have shown that a water deficit occurring at one stage of growth will have a different effect on yield as compared with a deficit occurring at another stage. For the same a m o u n t o f evapotranspiration, the yield may be expected to differ. For example, Robins and Domingo (1962) found that soil moisture depletion to the wilting point for 1 or 2 days during the tasseling or pollination period of maize results in as much as a 22% reduction in grain yield, while periods of 6 to 8 days gave a yield reduction of about 50%. Following maturity, the depletion of the available soil moisture had no effect on yield. Denmead and Shaw (1960) found that maize subjected to moisture stress at silking was the most severely affected as far as grain was concerned. D o w n e y (1972) substantiated these results for maize, although he found that water stress during male meiosis (or while the crop was young) increased water use efficiency. In an experiment in which grain sorghum was grown in lysimeters, Howell and Hiler (1975) found only a weak correlation between yield and seasonal evapotranspiration. T h e y concluded that the timing of the occurrence of a water deficit had more influence on the yield than the magnitude o f the deficit and t h e y made water management recommendations accordingly. Similarly, Stewart et al. (1974) made a n u m b e r of recommendations for managing water in maize production, to ensure that unavoidable deficits are timed to occur at the least damaging times. These recommendations for grain sorghum and maize are in general agreement with the conclusions of Salter and Goode (1967), who summarized research on the yield response to water at different growth stages for a wide variety of crops, ranging from cereals to orchards and vegetables to flowers.
The functional relationship To obtain a functional relationship between crop yield and water use, two problems evident in the literature reviewed above must be overcome, viz. (i) to explain h o w some researchers obtained a curvilinear function between yield and water use while m a n y others obtained a linear relationship, and (ii) to show h o w a unique relationship between yield and evopotranspiration can be developed when it is known that different yields can be obtained for the same a m o u n t of evaportranspiration, depending on the timing of deficits.
57 The first problem is due to a combination of two causes. Firstly, when water is applied in excess of the a m o u n t required for m a x i m u m yield, "water use" (and to some extent evapotranspiration) increases while yield remains constant or decreases. A decrease in yield is particularly apparent if waterlogging reduces soil aeration {Downey, 1972). This, coupled with the scatter of data due to the timing of deficits, allows a curvilinear function to be fitted to the plotted data. Secondly, in most cases the abscissa has not been evapotranspiration, but rather applied water or some similar parameter. In this case, the curvilinear nature o f the function m a y be due to a portion of the applied water being unavailable for crop evapotranspiration (e.g., deep percolation losses). II000
(Ymax, ETmox) (Ymex, Irr'rnax)
~ lOOO0
ET
n
9000 8000 7000 6000
5000 4000
o~
3.s
7.61 ,,.e
,5.6 ,e,~.6
~-(ET from ASWP + R ) - ~ Season Irr. Depth (Inches]
3000
2000 I000 0{~
i
I
i
i
i I0
i
1
IT i
i
i 20
i
14 .~4 2 3___/7 Field Water Supply (Inches)
i
1 30
i
i 34
3 2 .~
Legend: ASWP
=
Depth of Available Soil Water at Plantin9
R
=
Rainfall Depth
Irr.
ffi Water Supplied by Irrigation
FWS
ffi Field Wafer Supply
Fig. 2. Relations between yield, evapotranspiration and irrigation taking into account stored water and rainfall during season (from Stewart and Hagan, 1973).
The difference between the two functions is well illustrated in Fig.2, in which the curved line represents the relationship between yield and the seasonal depth of irrigation water applied, or yield and field water supply (which includes available soil water at planting plus rainfall) and the straight line represents the relationship between yield and ET. The difference between the straight line and the curved line, shown as "non-ET" on the figure, represents the water applied (or supplied) that is not consumed (i.e water losses). The convexity of the applied water function illustrates that these losses increase percentagewise as the point of fully satisfying ET demands is approached. The losses are likely due to deep percolation below the root zone,
58
but could also consist of unmeasured surface runoff. When Hillel and Guron (1973) t o o k particular care to calculate the drainage c o m p o n e n t of the field water balance, the relationship between grain yield of maize and seasonal evapotranspiration was strongly linear. The problem of developing a unique relationship between yield and evapotranspiration has received considerable attention from Stewart and associates at Davis, Calif.. They recognized that irrigation programming registers a dual effect on yield (Stewart and Hagan, 1973). The first effect is inevitable, the second manageable. The primary effect is that of water shortage, per se. Thus, any seasonal ET deficit is inevitably associated with some minimum fractional reduction in yield below maximum. A secondary reduction in yield may result from the timing of the ET deficits, with those occurring during more sensitive or "critical" growth stages of the crop in question causing a relatively larger decrease in yield. Such losses are avoidable through improved water management practices, which has the effect of orchestrating the sequence of ET deficits so that yield loss is minimized (Stewart et al., 1976). The Davis experiments have shown that when ET deficit sequencing is optimal (i.e. any deficits are timed so that they cause the least possible reduction in crop yield) the relationship between yield and seasonal ET is quite well represented by a straight line function for maize, grain sorghum and pinto beans. If the upper bound of yield is related to the depth of water applied, rather than ET, a curvilinear relationship will result. The analysis and discussion to follow are based on functions representing this upper bound of the yield--water use relationship (i.e., those representing irrigation regimes in which deficits are sequenced to occur at the least damaging times). This premise is reasonable in that it represents the ultimate goal of the irrigation farmer. However, should irrigation timing be non-optimal, the function will fall lower and these effects may also be readily assessed. The linear y i e l d - w a t e r use (ET) function may be thought of as the ideal, with curvilinear relations representing deviations from the ideal. The relative proximity of the field production function to the ideal allows an evaluation of irrigation managem e n t practices and irrigation system performance. OPTIMAL DEPTH OF WATER APPLICATION
Astute water management practices are of particular significance in the c o m m o n case where the available water is limited relative to the irrigable land area. The a m o u n t of water allocated to each unit of land (i.e. the seasonal depth of application) determines the area of land irrigated (Ax)
Q A I =-x
where Q = total quantity of water available (fixed) and x = depth of water applied (variable).
(1)
59 The optimal area is that which maximizes net revenue (P) in the objective function, P = Returns-Costs.
(2)
The returns per unit area are the price received for the crop (Vc) times the yield per unit area (y). Over the irrigated area, Returns = Vc y AI.
(3)
The costs per unit area in the objective function consist of some yield dependent costs (cost of seed, fertilizing, harvesting, marketing), some constant costs (seedbed preparation, pesticides, interrow cultivation) and some area dependent costs (the costs of water, energy and labour are relatively constant for a fixed volume of water and hence are area dependent). The capital cost of the irrigation system will increase as the size of the system increases with expanding area. However, as the quantity of available water is fixed, the range in which the optimal land area can be expected to lie is limited, and hence, the cost can be assumed to increase linearly with area over this increment. (This is only for mathematical simplicity, as in fact any function could be incorporated in the subsequent graphical procedure.) Therefore, Costs =
vi y + k l + -~i
(4)
A1
where vi = yield dependent costs per unit area, k ~ = constant costs per unit area, k2 = area dependent costs per unit area. Substituting eqs. (3) and (4) into eq. (2) gives (5)
P = (Vc--V1) y A i - - k l A i - k 2
setting a P / ~ A I = 0, i.e.
(vc-vj) y + (vc-vj) 0y - k , = 0
(6)
which may be readily manipulated to show that the optimal land area is obtained where marginal revenue equals marginal costs, or by collecting terms, may be expressed as (0Y
AI + y
)
(Vc-V1)-k1=
0.
(7)
That is, as the return is increased b y expanding irrigation to an additional unit of area, because the total quantity of water is fixed, yield on the remaining area must decline. The optimal area to irrigate is where the t w o factors (area and yield) combine for the greatest return. To obtain the "optimizing equation" in terms o f x and y, substitute eq.(1) into eq.(5) and set a P / a A i = O. Therefore,
g~
x-y
( v c - v i) + kl = O.
(8)
60 A given production function in terms of x and y and given cost coefficients may be substituted in this equation and solved for x, the optimal depth of irrigation water to apply. Unfortunately, this is impossible for the general case. The production function relevant to the farmers' field relates yield to applied water. As mentioned at the outset, this function is so subjective that it is of little value for quantitative purposes, depending on the amount of rainfall during the growing season, the a m o u n t of available soft moisture at planting and, not insignificantly, on the water application techniques of the individual irrigator. The contribution of all of these factors to the water supply will be influenced by the soil type. Furthermore, transferring the results obtained from research plots to irrigated fields will be unrealistic due to the significant difference in efficiency and uniformity to be expected. However, a quantitative assessment m a y be made of the effect of production functions on agricultural income by considering the relative difference between functions representing different irrigation efficiencies. For convenience, eq.(5) will be divided into the two components,
0
Returns = (Vc-Vj) y -x
(9)
and
(10)
Costs = k, Q - k s , x after substituting eq.(1) into eq.(5). EFFECT OF PRODUCTION FUNCTIONS ON AGRICULTURAL INCOME
To incorporate the general case, a range of production functions is shown in Fig. 3. Curves 1 and 2 have been obtained from experimental data from IO00C
8000 Io
6000 ._~ :>=
4000
/I/i// /
/ /
/ I /I I 411
2000
ii i i / i
%
I/
.41
200
I
1
40o
I
6;0 ' 8;o ' , 0 ~ 0 ' , 2 0 o
Depth of Irrigotio. W=ler Applied
(x}
(ram)
Fig.3. Production functions for maize at different irrigation application efficiencies.
61 Grand Junction, Colo. (Barrett and Skogerboe, 1978) while curves 3 and 4 are similar to the results obtained by other researchers (e.g. reported by Hexem and Heady, 1978), representing production functions more typical of field conditions. A constant fertility level is assumed. All curves have been plotted on a c o m m o n abscissa so that, contrary to Fig.2, curves 1 and 2 are no longer tangential at lower values of x. Considering x as the depth of water applied, Curve 1, orginally being the plot of yield versus ET, represents a production function at 100% irrigation application efficiency. The seasonal irrigation application efficiency for the regime represented by Curve 2 is 82% (574 m m / 7 0 0 mm), with 71% for Curve 3 and 48% for Curve 4. Values of x and y from each of the curves in Fig.3. have been substitued into eq.(8) and plotted as the return functions shown in Fig.4. The cost 120
,
~
,
I00 0 0
/
o eo
.0,urn
Fon0,ion,
_= 60 o == 40 _=
~/
/
Net Revenue
,r~...~or /
",, i 4e% ~ - ~ o s t
Function
20
200
400
600
800
I000
1200
Depth of Irrigation Wafer Applied (x)(mm)
Fig.4. Variation in returns and costs with depth of irrigation water applied.
function, as derived in eq.(9), is independent of y. The net revenue is the vertical difference between the respective return function and the cost function. Typical costs and prices have been used in plotting the curves, but this is not particularly significant as these will only affect the relative position of the return functions to the cost function and will n o t change the form of either. Similarly, fixed overheads not connected with irrigation (such as building, fences, etc) have been ignored, as these will have just the same effect. Also, for simplicity, the price of water has been taken as constant per unit volume. Breaking the total cost of watering into c o m p o n e n t parts (water, labour, energy and fixed costs of the system) only has the same effect o f changing the relative position of the cost function. The point of m a x i m u m net revenue is where the return and cost functions are separated b y the greatest distance, i.e. where the curves have the same
62 slope. This point is circled on the four return functions. The corresponding value read on the abscissa is the optimal depth of water to apply. Referring back to Fig.3, for Curve 1 this can be seen to equal the amount corresponding to Ymax. The other regimes have optimal depths of water application (Fig.4) lower than the depths giving maximum yield (Fig.3), although for the regimes corresponding to Curves 2 and 3, it is not greatly less (91 and 86% of Xmax, respectively, where Xmax is the depth of application corresponding to Ymax). For Curve 4, it is considerably less, with X o p t = 0.58 Xmax ( X o p t = 700 mm and Xmax = 1200 mm). In general then, in a water-short area, the optimal irrigation policy is to apply close to the amount of water giving maximum yield only if irrigation application efficiencies are high. If efficiencies are low, less water should be applied. The optimal depth of irrigation water to apply depends solely on the form of the production function. The effect of the production function may be readily demonstrated by referring to Fig.4. If the irrigator follows a practice resulting in a production function the same as Curve 4 in Fig.3, a maximum net revenue of $30 000 may be scaled from Fig.4. A different practice resulting in a production function the same as Curve 2 would result in a maximum net revenue of $52 000, or a 73% increase. As pointed out earlier, fixed overhead costs have been deleted for simplicity; as these would have the effect of raising the cost function, the actual effect of increasing water application efficiency would be to increase the net revenue an even greater percentage than calculated. With costs and production functions available, and return functions derived from these functions as shown in Fig.4, the detrimental effect of low irrigation efficiencies may be readily evaluated. The losses in revenue due to poor irrigation practices are startlingly apparent. In most cases where water is in short supply, the irrigator will endeavor to improve practices so that little water is wasted. With relatively high efficiency so attained, the correct policy is then to apply sufficient water so that close to maximum yield is obtained. If the irrigation system is such that high efficiencies are not possible and capital is not available for improvement, the optimal depth of water to apply is significantly less than the depth corresponding to Ymax and could be obtained from a plot such as Fig.4, or derived mathematically if a reliable expression for the production functions were available. Unfortunately, however, such information is generally not available and can easily be shown to give misleading results (Barrett, 1977). Notwithstanding, a significant general conclusion may be drawn. That is, contrary to those who advocate spreading the available water over as large an area as possible (subject to applying the minim u m depth which is practical), the optimal depth to apply in a system with inherent high losses is a little greater than that which should be applied under more efficient regimes and greater than the crop ET. Regardless of irrigation efficiency, the optimal depth of application actually lies in a fairly narrow range. Furthermore, it is safer to err on the side of applying a greater depth of water than on the side of applying less, as the return and cost functions in Fig.4 converge far more slowly on the right hand side of the optimal depth than on the left hand side.
63
If the timing of irrigations is non-optimal, the production functions will fall below the curves shown in Fig.3. For a given efficiency, the economic losses will be far greater. Fortunately, this cause of reduced income may easily be reduced or eliminated b y acquiring the necessary knowledge of the effect of improper irrigation timing on yields. The extent of the effect of inefficiencies of irrigation application on farm income will determine the extent to which remedial measures are justified. In many cases, improved management is the only additional input required. The economic losses illustrated in Fig.4 show the extent to which the increased costs of better management are justified. Apart from optimal sequencing of any deficits, such practices as beginning the growing season with adequate moisture in the crop root zone, irrigating to bring the root zone to less than field capacity to take advantage of rainfall, sprinkling at night to avoid evaporation and wind drift, avoiding over-watering, and terminating irrigation as soon as crop needs are met at the end of the season, all contribute to improved efficiency of application. CONCLUSION
A number of management practices are available which require no more capital than knowledge and foresight to allow more efficient use of irrigation water. Although efficient irrigation is generally accepted as a desirable goal, the benefits are often expressed in somewhat nebulous or qualitative terms. A consideration of the nature of production functions reflecting efficient and inefficient irrigation regimes allows the relative magnitude of the benefits due to efficient water management to be expressed in terms of net revenue. The optimal depth of water to apply will always be in excess of the potential ET of the crop, increasing slightly with decreasing efficiencies but always less than that giving maximum yield. With methods having low application efficiency, this will result in deficits occurring during the season. Particular care will be needed to ensure that these deficits occur at the least damaging times. By applying an amount of water less than that necessary to achieve maximum yield, it is quite likely, in fact, that the seasonal efficiency of application will rise, as a higher proportion of the applied water may actually be used by the crop. This, in combination with good management practices, will have the effect of transforming a strongly curvilinear crop production function to one more nearly linear. The lower depth of water application will allow a larger land area to be irrigated and increase profits substantially.
REFERENCES Arkley, R.J., 1963. Relationships between plant growth and transpiration. Hilgardia, 34(13): 559--584. Barrett, J.W.H., 1977. Crop yield functions and the allocation and use of irrigation water. Unpublished Ph.D. Dissertation, Colorado State University, F o r t Collins, Colo., July, 210 pp.
64 Barrett, J.W.H. and Skogerboe, G.V., 1978. Effect of irrigation regime on maize yields. J. Irrigation Drainage Div., ASCE 104 (IR2): 179--194. Cole, J.S., 1938. Correlations between annual precipitation and the yield of spring wheat in the Great Plains. U.S. Dept. Agric. Tech. Bull. 636, 40 pp. Consortium for International Development, 1976. Water and salinity production functions. Technical completion report prepared for U.S. Dept. of Interior, Office of Water Research and Technology, Washington, D.C. 20240, November, 191 pp. Denmead, O.T. and Shaw, R.H., 1960. The effects of soil moisture stress at different stages of growth on the development and yield of corn. Agron. J., 52: 272--274. De Wit, C.T., 1958. Transpiration and crop yields. Institute of Biological and Chemical Research on Field Crops and Herbage, Wageningen, The Netherlands. Versl. Landbouwk. Onderz. No. 646, 's Gravenhage, 88 pp. Downey, L.A., 1972. Water--yield relations for nonforage crops. J. Irrigation Drainage Div., ASCE 98 (IRI): 107--115. Hanks, R.J., Gardner, H.R. and Florian, R.L., 1969. Plant growth--evapotranspiration relations for several crops in the central Great Plains. Agron. J., 61: 30--34. Hexem, R.W. and Heady, E.O., 1978. Water Production Functions for Irrigated Agriculture. Iowa State University Press, Ames, Iowa, 215 pp. Hillel, D., 1972. The field water balance and water use efficiency. In: D. Hillel (Ed.), Optimizing the Soil Physical Environment Toward Greater Crop Yields. Academic Press, New York and London, pp. 79---100. Hillel, D. and Guron, Y., 1973. Relation between evapotranspiration rate and maize yield. Water Resour. Res., 9 (3): 743--748. Howell, T.A. and Hiler, E.A., 1975. Optimization of water use efficiency under high frequency i r r i g a t i o n - I. Evapotranspiration and yield relationship. Trans. Am. Soc. Agric Eng., 18(5): 873--878. Leggett, G.E., 1959. Relationship between wheat yield, available moisture and available nitrogen in eastern Washington dryland areas. Washington Agric. Exp. Stn Bull. 609, 16 pp. Musick, J.T. and Dusek, D.A., 1971. Grain sorghum response to number, timing and size of irrigations in the southern high plains. Trans. Am. Soc. Agric. Eng., 14(3): 401--404, 410. Musick, J.T., New, L.L. and Dusek, D.A., 1976. Soil water depletion-yield relationships of irrigated sorghum, wheat and soybeans. Trans. Am. Soc. Agric. Eng., 19(3): 489--493. Robins, J.S. and Domingo, C.E., 1962. Moisture and nitrogen effects on irrigated spring wheat. Agron. J., 54: 135--138. Salter, P.J. and Goode, J.E., 1967. Crop response to water at different stages of growth. Commonwealth Agricultural Bureau, Farnham Royal, Bucks, Great Britain, 245 pp. Shalhevet, J., Mantell, A., Bielorai, H. and Shimshi, D., 1976. Irrigation of field and orchard crops under semi-arid conditions. International Irrigation Information Center Publ. No. 1, 110 pp. Shipley, J.L., 1977. Scheduling irrigations with limited water. Draft prepared for the Proceedings of the High Plains Grain Conference, Guymon, Okla., February. Shipley, J. and Regier, C., 1975. Water response in the production of irrigated grain sorghum, High Plains of Texas. Texas Agricultural Experiment Station, MP-1202, June, 8 pp. Stewart, J.I. and Hagan, R.H., 1973. Functions to predict effects of crop water defictis. J. Irrigation Drainage Div., ASCE, 99 (IR4): 421--439. Stewart, J.I., Hagan, R.H., Pruitt, H.O. and Hall, W.A., 1973. Water production functions and irrigation system design and for increased efficiency in water use. U.S. Dept. of Interior, Bureau of Reclamation, Engineering and Research Center, Denver, Report 14-06-D-7329, 164 pp. Stewart, J.I., Hagan, R.M. and Pruitt, W.O., 1974. Functions to predict optimal irrigation programs. J. Irrigation Drainage Div., ASCE 100 (IR2): 197--199. Stewart, J.I., Hagan, R.M. and Pruitt, W.O., 1976. Water production functions and predicted irrigation programs for principal crops as required for water resources planning and increased water use efficiency. U.S. Dept. of Interior, Bureau of Reclamation, Engineering and Research Center, Denver, Report 14-06-D-7329, 80 pp.