Abstract
ABSTRACT. Several factors, including multi-year drought
Abstract
77). A linear relationship between the cumulative ET w and fraction of season (function of days after emergence) was found. This relationship developed for a given location could be used to extrapolate seasonal ET w for in-season irrigation management. Poor correlation was found between CWP and other variables such as total irrigation, rain + irrigation, and total water. The results of this study can provide useful information for soybean irrigators to make better management decisions under deficit irrigation conditions.
Abstract
everal factors, including multi-year drought, declining aquifer levels, and new water regulations, are reducing availability of irrigation water in the semi-arid area of west-central Nebraska. Since many farmers in this area now do not have enough water to meet the seasonal water requirements of agronomic crops such as corn and soybean, maximizing the yield produced per unit of water under deficit irrigation conditions is becoming crucial to the long-term viability of the local and state economy. Deficit irrigation has been suggested as an alternative strategy for making better use of irrigation water. For instance, Zwart and Bastiaanssen (2004) reviewed measured crop water productivity (CWP) (yield per unit of seasonal evapotranspiration) for several crops around the world and concluded that the CWP could be significantly increased if irrigation was reduced and crop water deficit was intentionally induced. Similarly, Schneekloth et al. (1991) compared the yields of soybean under dryland, deficit irrigation (limited to 150 mm) and full irrigation at North Platte, Nebraska, using sprinkler irrigation. They found the same yields for the deficit and full irrigation treatments during a three-year study. Deficit irrigation increased dryland yields from 0.4 to 2.8 Mg ha −1 , while full irrigation only increased yields by 0 to 0.4 Mg ha −1 over the deficit irrigation treatment. Deficit irrigation represented an average seasonal water savings of 119 mm compared with full irrigation. For the same location, Hergert et al. (1993) found that during a field experiment conducted from 1983 to 1991, an allocation of 150 mm of irrigation, which represented 53% of the water applied to a full irrigation treatment, produced an average soybean grain yield of 88% of that obtained using full irrigation.
Abstract
When water is limited, it is important to maximize storage of rainfall in the soil profile and to use techniques to conserve soil water, such as conservation tillage, terraces (where needed), improved residue management, and effective weed control programs. It is also important to select crops and crop rotations that conserve water, and use irrigation water efficiently by minimizing losses that do not contribute to crop S

strategy for making better use of irrigation water. For instance, Zwart and Bastiaanssen (2004) reviewed measured crop water productivity (CWP) (yield per unit of seasonal evapotranspiration) for several crops around the world and concluded that the CWP could be significantly increased if irrigation was reduced and crop water deficit was intentionally induced. Similarly, Schneekloth et al. (1991) compared the yields of soybean under dryland, deficit irrigation (limited to 150 mm) and full irrigation at North Platte, Nebraska, using sprinkler irrigation. They found the same yields for the deficit and full irrigation treatments during a three-year study. Deficit irrigation increased dryland yields from 0.4 to 2.8 Mg ha −1 , while full irrigation only increased yields by 0 to 0.4 Mg ha −1 over the deficit irrigation treatment. Deficit irrigation represented an average seasonal water savings of 119 mm compared with full irrigation. For the same location, Hergert et al. (1993) found that during a field experiment conducted from 1983 to 1991, an allocation of 150 mm of irrigation, which represented 53% of the water applied to a full irrigation treatment, produced an average soybean grain yield of 88% of that obtained using full irrigation.

When water is limited, it is important to maximize storage of rainfall in the soil profile and to use techniques to conserve soil water, such as conservation tillage, terraces (where needed), improved residue management, and effective weed control programs. It is also important to select crops and crop rotations that conserve water, and use irrigation water efficiently by minimizing losses that do not contribute to crop S yield, such as runoff, deep percolation, wind drift, and soil evaporation. These losses can be minimized by using efficient irrigation systems, and by improving irrigation system management and irrigation scheduling. It is also important to know if there is a time during the growing season when application of a limited amount of water would make the largest contribution to the final marketable yield, so that irrigations can be timed accordingly. The question of whether water stress timing has an impact on crop yield, and the magnitude of the impact, has been the subject of research for decades. For instance, Jensen (1968) proposed that limiting soil moisture during a growth stage would reduce water use during that stage, which would have an effect on marketable crop yield. For a determinate flowering crop, Jensen proposed that the effect of water stress at different stages on crop yield could be explained as: 

where Y = actual yield (kg ha −1 ), Y o = yield when soil water is not limiting (kg ha −1 ), W et = actual water use (mm), W oc = water use when soil water is not limiting (mm), (W et /W oc ) i = relative total evapotranspiration during a given stage of physiological development, l i = relative sensitivity of the crop to water stress during the stage of growth i (unitless), n = number of growth stages, and P means that the right side of the equation is a product instead of a sum. Jensen, however, only provided l i values for grain sorghum. Based on this procedure, Nairizi and Rydzewski (1977) derived l i values, which they called the "sensitivity index," and created l i curves from planting to harvest for different crops. They found that the magnitude and time of occurrence of peak sensitivity to water stress varied considerably for different crops. For soybean, they found that the sensitivity index curve could be defined by a fifth-degree polynomial function of percent duration of growing season (DGS). The index was low until approximately 50% of DGS and then rapidly increased, peaking at about 87% of DGS, and then dropped rapidly as the crop matured. Several later studies in Nebraska used equation 1 to derive l i values, which were called "crop-specific drought index," for several crops including corn (Meyer et al. 1993) , sorghum (Paes de Camargo and Hubbard, 1999) , and wheat (Xu, 1996) . Similarly, Doorenbos and Kassam (1979) proposed that the effect of water stress on yield could be quantified as:

(1 − Y a /Y m ) = ky(1 − ET a /ET m )

where Y a = actual yield (kg ha −1 ), Y m = maximum yield (kg ha −1 ), ET a = actual evapotranspiration (mm), ET m = maximum evapotranspiration (mm), and ky = empirical yield response factor that varies depending on the growth stage when water stress occurs (unitless). For soybean, they reported ky values of 0.2, 0.8, 1.0 for the vegetative, flowering, and yield formation stages, respectively, indicating that yield was more affected by water stress during the yield formation stage than at any other time. This model suggests that if water is limited, the irrigator should time irrigation to minimize stress during the most sensitive stages.

Other studies supporting the philosophy that stress timing has an impact on soybean yield include that of Sionit and Kramer (1977) , who water-stressed soybean at different stages of growth in a controlled environment. They found that soybean grain yield was reduced most by stress during early grain formation and pod filling. However, they did not evaluate the relationship between yield and total seasonal water use. Ashley and Ethridge (1978) also studied the effect of starting irrigation at different stages of growth for soybean and found that the effect of stressing the crop at a given stage varied from season to season. They did not relate yield to total seasonal evapotranspiration either. Korte et al. (1983) and Kadhem et al. (1985) studied the effects of irrigation timing on soybean development and yield in Nebraska. They found significant increases in plant height, nodes per plant, and lodging for indeterminate cultivars when irrigation was applied during the vegetative stage. A single irrigation (gravity irrigation) to refill the crop root zone depth to field capacity was most effective when applied during the period from mid-pod elongation to just before seed enlargement. Korte et al. (1983) found that irrigation timing affected the number of seeds per plant and the 100-seed weight. These studies, however, were conducted in eastern Nebraska, which has a much more humid climate than west-central Nebraska, and there is usually a full moisture profile in the crop root zone at the beginning of the growing season. Therefore, delaying irrigation until the reproductive stage in this environment does not necessarily mean that the crop was under water stress early in the season. These studies did not relate yield to actual water availability and water use. They also applied irrigation at a given stage, regardless of crop needs. Evans et al. (1990) determined the susceptibility of soybean to wet stress at different growth stages using the following model:

where SDI = stress-day-index (unitless), SD i = stress days factor for period i, and CS i = crop susceptibility factor for period i (unitless). During a five-year study, Evans et al. (1990) found that soybean was more susceptible to wet stress during pod development and pod fill. They also found that these results were similar to those previously reported for drought stress by Sudar et al. (1979) , indicating that this model could also be adapted to quantify crop susceptibility to drought stress conditions. Other studies, however, suggest that the total amount of seasonal crop evapotranspiration or transpiration is what determines crop yield, and these researchers have paid less attention to stress timing. For instance, Hanks (1974) proposed that crop dry matter yield could be related to water by a function that did not depend on the stage of growth at which stress occurred:

where Y = actual dry matter yield (kg ha −1 ), Y p = potential dry matter yield (kg ha −1 ), T = actual seasonal transpiration (mm), and T p = potential seasonal transpiration (mm). It is important to notice that Hanks referred to dry matter yield, while others have referred to grain or marketable yield. Klocke et al. (1989) evaluated the feasibility of starting irrigation at different growth stages at four locations in Nebraska, from sub-humid eastern locations to the semi-arid west-central areas of the state. They found that for the semi-arid region, the best irrigation strategy was to use full-season irrigation to meet crop evapotranspiration. For the sub-humid areas, they found that irrigation could be delayed until flowering in deep, medium to fine-textured soils if there was a full soil water lewisjuan@example.com. They found that soybean relative yields for all four locations were related to seasonal crop evapotranspiration by a quadratic function (R 2 = 0.75) independent of stress timing:

where Y = relative yield (%), ET = seasonal evapotranspiration (mm), and a, b, and c are empirical coefficients. Schneekloth et al. (1991) related soybean grain yield to seasonal evapotranspiration by a linear function independent of stress timing:

where Y = yield (kg ha −1 ), ET = seasonal evapotranspiration (mm), a = slope, and b = intercept. The R 2 values for this relationship were 0.53, 0.14, and 0.60 during the three years of the study. A similar linear equation was used by Stone (2003) to relate seasonal evapotranspiration to crop yield for soybean. Irmak et al. (2002) also found a linear relationship (R 2 > 0.48) between plant-available soil water at different times during the reproductive period and yield of rainfed soybean. The objectives of this study were: (1) to quantify the grain yield response of soybean to deficit irrigation, and (2) to determine which of several water variables correlated best to soybean grain yield under deficit irrigation conditions.
METHODS

SITE DESCRIPTION
Field data for this study were collected in 2002 at Curtis (40.6° N, 100.5° W, elevation = 784 m), and in 2003 and 2004 at North Platte, Nebraska (41.1° N, 100 .8° W, elevation = 861 m). At each site, the experiment was conducted using a randomized complete block design with four replications. Nine irrigation treatments were studied in 2002 and 2003, and eight in 2004 . Each year, treatments included a dryland treatment, which received no irrigation. Irrigation treatments consisted of different irrigation timing and target root zone depletions under deficit irrigation conditions (table 1). None of the treatments received irrigation during the vegetative stage, which is a common practice in the area. Soybean growth stages and the number of days needed to reach each stage in Nebraska have been described by Martin (1997) . Treatments were designed to create stress at different stages of crop development, and to create a range of seasonal available soil water that was wide enough to develop meaningful quantitative relationships between crop yield and seasonal water variables such as irrigation, rain + irrigation, total water, evapotranspiration, and evaporation, among others.
SITE DESCRIPTION
In 2002, the field experiment at Curtis was conducted at the Nebraska College of Technical Agriculture, University of Nebraska-Lincoln, in a field previously planted to hybrid sorghum-sudan that was harvested for hay. The soil at Curtis is a Hall silt loam, with an average water content at field capacity of 0.32 m 3 m −3 and permanent wilting point of 0.11 m 3 m −3 . The soybean variety Pioneer 9294 RR was planted in lewisjuan@example.com m row spacing and a depth of approximately 2.5 cm. In 2002, soybean was planted on May 20 and harvested on September 27. The center two rows of each plot were harvested using a plot combine. Plot size at Curtis were 3 m wide by 9 m long, which accommodated four rows per plot.
SITE DESCRIPTION
Plots at Curtis were irrigated using a surface drip irrigation system, which was designed and managed to wet most of the soil surface to simulate a sprinkler-irrigated field. One drip lateral was installed next to each crop row. Drip laterals were 17 mil Typhoon 630 thinwall dripperlines (Netafim USA, Fresno, Cal.) with emitters spaced every 457 mm. The nominal flow rate of the emitters was 3.8 L h −1 at a nominal pressure of 69 kPa. Water for this system was pumped from the Ogallala aquifer and was filtered using a 51 mm diameter (120 mesh) disc filter (Netafim USA, Fresno, Cal.). After the filter, the mainline was divided into four sub-mains. Each sub-main had a flowmeter and a pressure regulator. Each sub-main was further divided into two branches with a manual valve on each. Each branch irrigated one treatment (four plots). This arrangement allowed two treatments to share the same flow meter and pressure regulator. PVC pipe was used for the sub-main, and laterals were connected to 25 mm diameter polyethylene tubing. The laterals for all four replications of the same treatment were connected to the same branch. Pressure regulators were adjusted to apply water just fast enough to wet the soil surface without generating runoff.
SITE DESCRIPTION
In (Martin, 1997) .
SITE DESCRIPTION
Platte is a Cozad silt loam (Fluventic Haplustolls) with water content at field capacity of 0.29 m 3 m −3 and a permanent wilting point of 0.11 m 3 m −3 (Klocke et al. 1999) . The soybean variety Renze 2600 RR was lewisjuan@example.com m row spacing and a depth of approximately 2.5 cm. In 2003, soybean was planted on May 27 and harvested on October 8. In 2004, soybean was planted on May 21 and harvested on October 5. The center four rows of each plot were harvested using a plot combine. The soybean was irrigated using a solid-set sprinkler system, which was arranged in a 12.2 × 12.2 m grid. At each of the four sites, each experimental plot was surrounded by a "border" plot of the same size. The inclusion of "border" plots precluded water from different contiguous irrigation treatments from overlapping within a given experimental plot. Sprinkler heads were installed at the four corners of each plot on 3.35 m risers.
SITE DESCRIPTION
Water for the system was pumped from the Ogallala aquifer using an electric turbine pump with a capacity of 38 L s −1 at 480 kPa of pressure. The mainline at the pumping station was instrumented with two pressure gauges, two flowmeters, a pressure relief valve, a chemigation check valve, and "high" and "low" pressure switches. The flowmeters measured both the instantaneous flow rate and the cumulative volume of water pumped. The irrigation system was turned on and off using an automatic control panel hard-wired to electric valves installed at each plot. Each electric valve controlled the four sprinklers of each plot, which were connected to the same water supply line. This design allowed independent irrigation of individual plots and re-randomization of the plots from year to year to accommodate a particular statistical design. The control panel was also connected to a manual relay panel. This combination allowed both manual and automatic operation of the irrigation system.
IRRIGATION SCHEDULING
Irrigation scheduling at both sites was performed with a scheduling spreadsheet that used the calculated daily evapotranspiration (ET) as the input to estimate daily average soil water content in the crop root zone. The spreadsheet estimated daily ET using the single crop coefficient (K c ) approach presented in FAO-56 (Allen et al. 1998) . When actual soil water measurements were available, it was possible to adjust the spreadsheet calculations to reflect the actual measurements. Soil water measurements were made approximately every two weeks during the growing season using the neutron scattering method. Soil water readings were taken from 50 mm diameter aluminum access tubes installed at the center of two of the four replications of each treatment. Readings were lewisjuan@example.com m depth increments to a depth of 1.8 m. Weather data were obtained from an automatic weather station located at each research site, within a distance of 1.5 km. These two weather stations were part of the High Plains Regional Climate Center (HPRCC) weather network. Daily weather data were downloaded from the HPRCC web site (www.hprcc.unl.edu/home.html) and included daily maximum and minimum air temperature, relative humidity, wind speed, rainfall, and solar radiation. Rainfall was also measured at each research site using manual rain gauges.
DATA ANALYSES
The statistical analyses of yield data, which included analysis of variance (ANOVA) and separation of means by the Duncan's new multiple range method, were conducted using the SAS System for Windows statistical software (SAS Institute, Inc., Cary, N.C.). Regression analyses were conducted using Microsoft Excel. A computer program was written in Microsoft Visual Basic to model the daily soil water status and to calculate all of the water variables that were then related to crop yield. Inputs to the program included daily weather data, rainfall, irrigation, soil water profile at crop emergence, and crop-specific and site-specific information such as planting date, maturity date, soil parameters, maximum rooting depth, etc. Based on these inputs, the rooting depth and the water balance in the crop root zone were calculated on a daily basis.
DATA ANALYSES
Grass-reference evapotranspiration (ET o ) was calculated using the procedure presented in FAO-56 (Allen et al. 1998; Wright, 1982) . Since this is a long procedure, the details of all computations will not be repeated here and the reader is referred to the original sources for details. According to this procedure, crop evapotranspiration is obtained as the product of the evapotranspiration of a reference crop (ET o ) (grass) and a crop coefficient (K c ). ET o is calculated using the weather data as the input to the Penman-Monteith equation, and the K c is used to adjust the estimated ET o for the reference crop to that of other crops at different growth stages and growing environments. In this study, the dual crop coefficient approach was used to separate the two components of evapotranspiration, namely evaporation (E) and transpiration (T). This procedure also linearly reduces crop evapotranspiration when the available soil water in the crop root zone is less than 50%, which is used to quantify the effect of water stress on crop water use. The dual crop coefficient procedure also accounts for the sharp increase of the evaporation component due to a wet soil surface following a rain or irrigation event.
DATA ANALYSES
This procedure permitted calculation of the daily crop evapotranspiration and transpiration when soil water is not limiting (ET w and T w ), and the daily actual crop evapotranspiration, transpiration, and evaporation (ET d 
DATA ANALYSES
where p table = p value taken from table 22 in FAO-56, ET w is in mm day −1 , and 0.1 < p calc < 0.8. This equation suggests that if ET w is less than 5 mm on a given day, it is easier for the crop to sustain ET w rates, and therefore higher soil water depletion levels could be allowed without yield loss, as previously proposed by Doorenbos and Kassam (1979) . This differs from the common procedure of using a fixed p value for a given crop for the entire growing season. In this study, the daily p calc values were calculated, which were taken as the theoretical optimum depletion value for a given day. In addition, a daily actual p (p actual ) was calculated as:
DATA ANALYSES
where D = soil water depletion in the crop root zone (mm), θ FC = soil water content at field capacity (m 3 m −3 ), θ PWP = soil water content at permanent wilting point (m 3 m −3 ), Z r = crop rooting depth (m), and TAW = total available soil water in the crop root zone (mm). In this study, it was expected that p diff (= p actual − p calc ) accumulated for the entire season (seasonal p diff ) only for days when p diff > 0 would be a good indicator of the level of crop stress and would, therefore, relate to crop yield. The greater the seasonal p diff for a given treatment, the more stress. Seasonal p diff was therefore calculated for each treatment.
DATA ANALYSES
The effect of the different irrigation treatments on crop yield was also evaluated by calculating the crop water productivity (CWP) (Zwart and Bastiaanssen, 2004) as:
DATA ANALYSES
where Y a = crop yield (kg ha −1 ), seasonal ET d is in mm, and CWP is in kg ha −1 mm −1 . The effects of several other water variables on soybean grain yields were also evaluated. These variables included seasonal ET d , T d , E, irrigation applied, irrigation + rainfall, and total water available to the crop during the growing season. 
RESULTS AND DISCUSSION

WEATHER CONDITIONS

IRRIGATION
During the first two years of the experiment (2002 and 2003) , the irrigation strategies described in table 1 were closely followed. In 2004, however, a broken water supply mainline when the first irrigation started at beginning bloom (in mid-July) caused a delay of irrigation of about a week, and rainfalls in mid-to late July further delayed irrigation until early August. By that time, soybean was already in the beginning seed (R5) stage, and irrigations were applied to create different levels of stress among treatments late in the growing season. During the three years of the study, the beginning bloom (R1) stage started in mid-July ( supplied by irrigation, for each treatment during the three seasons, are given in Site  Date  T1  T2  T3  T4  T5  T6  T7  T8 T9 [a] = water available to the crop from all sources, including irrigation, rainfall and water already stored in the soil at crop emergence (mm). W_R_I = rainfall plus irrigation water (mm). W soil = water stored in the soil profile at crop emergence, to a depth equal to the maximum root depth (mm). E = actual seasonal evaporation (mm). T w = seasonal transpiration when soil water is not limiting. T d = seasonal actual transpiration (mm). D p = seasonal deep percolation (mm). p diff = daily positive difference between the actual and the theoretical fraction of total available soil water in the root zone that can be depleted before moisture stress occurs, accumulated for the entire season (unitless). CWP = crop water productivity (kg ha −1 mm −1 ). 
cont). Seasonal irrigation (mm) applied to soybean at Curtis (2002) and North Platte (2003 and 2004) for each irrigation treatment (T1 to T9).
ETw ETd_1 ETd_2 ETd_3 ETd_4 ETd_5 ETd_6 ETd_7 ETd_8 ETd_9
Curtis, 2002
Cumulative ET (mm) fig. 2 . The seasonal ET w varied from season to season and among treatments within the same season. Small variations in seasonal ET w among treatments for the same site and season were mainly due to differences in irrigation frequency, which affected the evaporation component of ET w . Variations among seasons were mainly the result of changes in weather conditions. The highest seasonal ET w value during the study occurred at Curtis during 2002, which was 801 mm. The lowest seasonal ET w value occurred at North Platte during 2004, with a maximum of 585 mm. This is a significant difference in ET w of 216 mm for the same crop between the two site years. The seasonal ET w values for North Platte were higher than the 500-520 mm previously reported by Klocke et al. (1989) for soybean for the same location. Differences are possibly due to season-toseason variations and to the use of different methodologies to determine ET w .
Curtis, 2002
Seasonal ET d under the deficit irrigation conditions of this study varied among treatments and seasons depending on the total water available to the crop and prevailing weather conditions (table 4 and fig. 2 ). For all seasons, treatment T1 always had the highest ET d , and the dryland treatment (T9 in 2002 and 2003, and T8 in 2004) For a given year, yields with the same letter are not significantly different. [b] No significant yield differences due to irrigation were found in 2004.
Curtis, 2002
T7, resulting from rainfall events occurring during the crop emergence period. In 2004, deep percolation occurred in all treatments except for the dryland treatment (T8)
Curtis, 2002
Total water available to the crop included irrigation, in-season rainfall, and water stored in the crop root zone at planting time, minus in-season deep percolation. All of these water sources for all treatments and seasons are detailed in (fig. 2) . Water stored in the soil profile at planting time (W soil ) can be a significant source of water to meet seasonal ET w . W soil represented approximately 10%, 23%, and 19% of ET w for 2002, 2003, and 2004, respectively . At Curtis, all treatments had the same W soil since no irrigation treatments were applied at this location in the previous year and the research area was planted to the same crop. However, at North Platte, irrigation treatments were applied to the research plots in the previous year, and therefore treatments had different W soil . At North Platte, there was more W (table 5) . Since all treatments were either dryland or deficit-irrigated, all yields were lower than the yield that would be expected for a fully-irrigated crop, suggesting that water was the main factor limiting yields. Statistically significant yield differences among treatments were only observed in the two driest seasons (2002 and 2003) . There were no significant yield differences in 2004, which could be due to the narrow range of seasonal ET d values among treatments obtained during that season (fig. 2) .
GRAIN YIELD AND CROP WATER PRODUCTIVITY
Crop water productivity ( ( fig. 4b) . Similarly, CWP correlated well to T d /T w (R 2 = 0.72) and correlated best to p diff (R 2 = 0.77). The CWP, however, was poorly correlated to seasonal irrigation (R 2 = 0.04), W all (R 2 = 0.36), and rain + irrigation (R 2 = 0.22). The poor correlation of CWP with the latter variables indicates that the crop was not able to utilize some of those water sources in yield production. This point is illustrated in figure 5 Figure 5 shows that for both treatments T1 and T7, differences between the cumulative ET d and ET w curves started in July. For treatment T1, the two lines are parallel to each other after that time, indicating that enough water was supplied to that treatment to keep up with ET w after the initial stress. However, for that treatment, the cumulative ET d was never able to equal the cumulative ET w after the initial stress, and therefore the seasonal ET d was less than the seasonal ET w , even though the seasonal W all was almost equal to the seasonal ET w . For treatment T7, starting in July, the difference between cumulative ET d and cumulative ET w became larger as the season advanced, indicating that water inputs were not enough to keep up with ET w . For treatment T7, the limited amount of water was applied early enough for the crop to be able to use it all in yield production before maturing. These results suggest that it is important to apply water early enough in the growing season for the crop to be able to use it.
GRAIN YIELD AND CROP WATER PRODUCTIVITY
After crop emergence, the cumulative ET w increased almost linearly with time ( fig. 5 ). Time after crop emergence can be expressed in several ways, such as days after emergence or cumulative growing degree days (GDD), or it can be normalized by calculating the fraction of season (F s ), which can be calculated as the ratio of the cumulative GDD since emergence and the GDD the crop needs to reach maturity (Stegman, 1988) . Figure 6 shows a linear relationship between the cumulative ET w and F s for each of the three seasons, although the slope of the line was greater for Curtis than for North Platte. The linear function developed for a given location could potentially be used to estimate seasonal ET w for in-season irrigation management, especially when F s > 0.5. None of these variables, however, were consistent across seasons and sites, which is due to differences in seasonal ET w between seasons and sites. It may be possible to obtain a unique function between yield and some of these Cumulative ET w (mm) Figure 6 . Relationship between cumulative crop evapotranspiration when soil water is not limiting (ET w ) and fraction of season (F s ) for soybean at Curtis and North Platte, Nebraska. The F s value for a particular day was calculated as the ratio of the cumulative growing degree days (GDD) since emergence and the cumulative GDD at crop maturity. GGD values were calculated using a base temperature of 50°F (10°C) with no upper temperature limit (Stegman, 1988) . The dashed line is the regression line.
RELATIONSHIP BETWEEN GRAIN YIELD
variables by normalizing yields, as suggested by Klocke et al. (1989) , who showed that the relative soybean yields (%) for four different locations in Nebraska were related to crop water use (same as ET d ) by a quadratic function (R 2 = 0.75). In this study, we could not normalize yields because we did not have a fully-irrigated treatment and, therefore, did not know the potential yield when soil water was not limited for each season. 
CONCLUSION
Soybean grain yields across years and sites were best related to the seasonal ratio of the actual crop evapotranspiration and the crop evapotranspiration when water was not limiting (ET d /ET w ), and to the seasonal ratio of actual crop transpiration and crop transpiration when water was not limiting (T d /T w ). Both of these seasonal ratios were linearly correlated to grain yield with R 2 = 0.91 when combining data for all seasons. Other variables such as seasonal ET d , total water, seasonal irrigation, seasonal irrigation + rain, seasonal T d , and seasonal E were linearly related to crop yield for each season, except for 2004, but were not consistent across seasons and sites. Normalizing yields could probably improve consistency for some of those variables, but this could not be tested in this study since we did not know the potential yield when soil water was not limited for each season. It is recommended to include a fully-irrigated treatment in future work so that the potential yield can be determined. The crop water productivity (CWP) (yield per unit of seasonal ET d ) linearly increased with both seasonal ET d /ET w (R 2 = 0.72) and T d /T w (R 2 = 0.72), but was best correlated to the cumulative seasonal difference between the actual and the theoretical fraction of total available soil water in the root zone that can be depleted before moisture stress occurs (seasonal p diff ) (R 2 = 0.77). Poor correlation was found between CWP and variables such as total irrigation, rain + irrigation, and total water. In this study, delaying irrigation until beginning bloom or later growth stages resulted in early stress, which decreased seasonal ET d . The fact that seasonal ET d was linearly related to crop yield for a given season then suggests that stress at any growth stage would reduce yield. Therefore, it is important not to delay irrigation at any stage to the point where seasonal ET d is reduced, since once seasonal ET d is reduced, there is no way to bring it back to match seasonal ET w , and there is no way to recuperate yield that has already been lost by stress occurring earlier in the season.
CONCLUSION
The results obtained in this study suggest that the greater the value of ET w when stress occurs, the greater the impact of water stress on soybean yield. This is because seasonal water variables such as ET d , T d , and ET d /ET w would be reduced most, which could explain some of the effects of stress timing on soybean yield that other researchers have reported. The results also indicate that the larger the amount of water that is available early enough in the growing season to contribute to increase seasonal ET d , the higher the soybean yield. Therefore, in this study, the irrigation treatments that received more water during a given season and received that water early enough in the growing season for soybean to have time to use it, and before yield components were established, resulted in higher yield.

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