Published online 31 October 2006
Published in Agron J 98:1620-1626 (2006)
DOI: 10.2134/agronj2005.0358
© 2006 American Society of Agronomy
677 S. Segoe Rd., Madison, WI 53711 USA
Production Papers
Performance of Double-Cropped Winter WheatSummer Maize under Minimum Irrigation in the North China Plain
Xiying Zhang*,
Dong Pei,
Suying Chen,
Hongyong Sun and
Yonghui Yang
The Center for Agricultural Resources Research, Institute of Genetic and Developmental Biology, The Chinese Academy of Sciences, 286 Huaizhong Rd., Shijiazhuang 050021, China
* Corresponding author (xyzhang{at}ms.sjziam.ac.cn)
Received for publication December 28, 2005.
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ABSTRACT
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The double cropping of winter wheat (Triticum aestivum L.) and summer maize (Zea mays L.) in the North China Plain (NCP) requires intensive irrigation that results in rapidly depleting aquifers and threatens the sustainable agricultural development in the region. This study investigated the possibility of growing winter wheat and maize with minimum irrigation (MI) by bringing soil moisture in the top root zone to field capacity at sowing with no further irrigation afterwards. Results over 8 yr (19972005) showed that grain yield of winter wheat was over 5000 kg ha1 and maize was over 6000 kg ha1 in most of the seasons under MI. The average yield was decreased by 14% for winter wheat and 13% for maize compared with the fully irrigated treatment (FI). Water use efficiency (WUE) under MI was increased by 15% for winter wheat and 10% for maize compared with that under FI. Average seasonal evapotranspiration (ET) was 335 mm under MI and 447 mm under FI for winter wheat, and 319 mm and 403 mm for maize, respectively. The annual irrigation requirement of MI was only half that of FI. Yield reduction in MI was negatively related to seasonal rainfall (significant at P < 0.05). The success of MI depended on the deeper root system of winter wheat using soil moisture that accumulated below the shallower rooted maize over the summer rainfall season. The results showed that an MI strategy that would be simple to implement for farmers would contribute significantly to the sustainability of the groundwater resource.
Abbreviations: ET, evapotranspiration FI, full irrigation MI, minimum irrigation NCP, the North China Plain SWD, soil water depletion WUE, water use efficiency
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INTRODUCTION
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THE NORTH CHINA PLAIN (NCP) is among the most densely inhabited and most developed regions in China. Both population and economic activity have grown markedly in the past 25 yr. Much of this development has depended heavily on the groundwater resources of the NCP. In consequence the groundwater table has fallen rapidly and has become the principal concern regarding the available water resource in this region (Foster et al., 2004). In agriculture, groundwater is mainly used for growing winter wheat and maize in a double cropping system. The annual water requirement of the two crops is about 870 mm (Liu et al., 2002). The annual rainfall is approximately 480 to 550 mm. Consequently, supplementary irrigation is required, especially for winter wheat, which grows during the dry season. Generally six to eight irrigations (about 400450 mm) are applied annually in a maizewheat double crop system. Groundwater use is not controlled and irrigation has contributed greatly to the continued lowering of the ground water table.
The need to reduce agricultural water use has resulted in many studies evaluating ways and means to improve crop WUE (Wang et al., 2001, 2002; McVicar et al., 2002; Mao et al., 2003; Zhang et al., 2003a, 2003b, 2005; Pei et al., 2004; Hu et al., 2005). However, even with the use of regulated irrigation deficit strategies for winter wheat and maize, the average annual irrigation requirement of 350 to 400 mm considerably exceeds the annual recharge of about 195 mm (Hebei Water Resource Bureau, 2004). Further increases in WUE would help to improve the sustainability of groundwater resources. A number of studies have shown that water supply limitations may affect WUE. When WUE under water limitation was compared with well-watered checks, the results showed that WUE increased (Fischer, 1980; Barraclough et al., 1989; Yuan et al., 1992; Zhang et al., 1999; Mao et al., 2003; Pei et al., 2004). Abbate et al. (2004) found WUE of wheat was greater for experiments with water limitation and relative dry matter was not linearly related to relative water use. Other studies showed the possibility of combining residual soil moisture with limited rainfall in obtaining high yields of crops. The availability of residual soil moisture depended on earlier rainfall, soil texture and the effective root depth (Sharma and Ghildyal, 1977; Chaudhary and Bhatnagar, 1980; Mishra et al., 1995; Mugabe, 1998; Mugabe and Nyakatawa, 2000). The aim of the current study is to quantify the grain yield and WUE of the winter wheat and maize double cropping system using only a presowing or postsowing irrigation for each crop. This simple strategy was in part adopted because farmers could more readily apply these results than a more complicated scheduling of deficit irrigation. In this study we assess the potential of MI to reduce groundwater use in winter wheatmaize double crop system.
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MATERIALS AND METHODS
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Experimental Site
The study was conducted from 1997 to 2005 at Luancheng Agro-Eco-Experimental Station of the Chinese Academy of Sciences. The station is located in the northern part of NCP at the base of Mt. Taihang (37°53'N, 114°40'E; 50 m above sea level). The annual number of sunshine hours is 2608 h and the annual average temperature is 12.3°C with a monthly maximum of 26.2°C in July, and a minimum of 3.9°C in January. The cumulative temperature over 0°C is 4710°C d. These conditions meet the needs of most cereal crops. The average annual rainfall is 482 mm with 352 mm falling in the growing season of maize between June and September. Rainfall during the growing season of winter wheat is about 130 mm between October and May in the following year.
Soil at the station is classified as Gleysol. It is moderately well-drained loamy soil with a deep profile that is considered highly suitable for crop production. The average organic matter content in the tillage layer (020 cm) was about 1.7% and the available N, P and K were about 75 mg kg1, 25 mg kg1 and 140 mg kg1, respectively. The soil has a pH around 8 and has a field capacity at 35% (v/v) and wilting point at 13% (v/v) for the upper 2-m soil profile.
Winter wheat was generally sown during early October using a seeder with a row spacing of 0.15 m and a density of 300 seeds m2. Chemical fertilizers were applied before cultivation at a base rate of 145 kg ha1 for N [using urea plus DAP (diammonium phosphate)], 108 kg ha1 for P2O5 (using DAP) and 66 kg ha1 for K (using K2SO4). Nitrogen was applied again at the jointing stage with a rate of 107 kg ha1. Winter wheat was harvested during early June. Maize was planted immediately after wheat harvesting. The final stand of maize was 6.5 to 7 plants m2, with a 0.6-m row spacing. In the middle of July, maize was fertilized with N at a rate of 180 kg ha1. After harvesting at the end of September the remaining maize residue was cut into small pieces by a stalk chopper and was plowed into the soil to a depth of about 20 to 30 cm. At the beginning of October, winter wheat was sown again to begin the double crop rotation for the next year. The wheat cultivars used in the study were 4185 (19972002) and Shixin733 (20032005). The maize cultivars were Yedan4 (19971998), Zhengdan958 (19982001, 20042005), Laiyu2 (2002) and Nongda108 (2003). All the cultivars were commonly used in the region.
Experimental Design and Implementation
The control treatment was considered to be fully irrigated (FI). The FI plots received an irrigation at sowing and, depending on rainfall, a further three to five irrigations for wheat and two to four irrigations for maize. Generally 60 to 70 mm was applied in one of the typical irrigations to FI. Timing of the irrigation was based on soil moisture. Irrigation was applied when the main root zone soil moisture was about 23 to 25% (v/v). The MI treatment was irrigated before sowing for winter wheat and immediately after sowing for maize. The irrigation amount was calculated to bring the top 0.8- to 1.0-m soil profile to field capacity. Generally 90 to 120 mm was applied in one of the irrigations at sowing. The MI treatment then received no more irrigation until harvest. At harvest wheat had generally extracted most of the available soil moisture to a depth of 1.5 m and irrigation was necessary for maize seedling establishment. Before sowing winter wheat, the MI treatment also required irrigation to create more favorable soil moisture conditions in the deep soil profile.
The two treatments (MI and FI) were part of an irrigation experiment (total of 6 treatments) conducted at the station from 1997 to 2005. Plots were arranged in a completely randomized design. Each treatment was replicated four times. Each plot measured 5 by 8 m. Plots were separated by a 2-m wide zone managed as MI to minimize the effects of two adjacent plots.
Soil moisture was monitored using a neutron probe (IH-II, Cambridge, UK) in a 2- to 2.3-m deep access tube installed in the center of each plot. Soil moisture content was measured once every 7 to 10 d at 0.2-m intervals down to 1.8 to 2.0 m. Surface soil moisture (020 cm) was measured gravimetrically by taking soil samples using a soil auger. Irrigation water was applied via a 10-cm plastic hose. A flow meter recorded the amount of irrigation water used.
Individual plots were harvested manually and grain was separated using a thresher. Subsequently the grain was air-dried and the yield recorded. A random sample of grain was taken to determine average seed weight. Before harvest, spike (or ear) numbers per unit area were counted and 40 spikes for wheat and 10 ears for maize were collected at random from each plot to determine kernel numbers per spike (ear).
Total evapotranspiration was determined from the initial and final soil water content, precipitation, irrigation, runoff, drainage and capillary rise using the following equation (Zhang et al., 1999):
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where ET = total evapotranspiration in mm for the growing period of each crop; P = precipitation, mm; I = irrigation, mm; SWD = soil water content at sowing minus the value at harvest to a depth of 2 m, mm; R = runoff, mm; D = drainage below the root zone (mm) and CR = capillary rise to the root zone, mm. Plots were banked and the soil has a high infiltration rate (1.1 m d1), runoff was not observed in the plots. Capillary rise was considered negligible because the groundwater table was at 32 m and soil water extraction did not occur below 2 m. Drainage from the root zone was calculated based on the relation of unsaturated water conductivity using volumetric soil moisture at 2 m in the soil profile (Kendy et al., 2003). Thus ET was taken to equal P + I + SWD D under these experimental condition. Rainfall was recorded at a meteorological station about 100 m away from trial plots. Water-use efficiency was defined as crop yield divided by total water use:
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where Y is grain yield (kg ha1) and WUE was calculated as kg mm1 ha1or kg m3 (1 kg mm1 ha1 = 10 kg m3).
Root distribution of wheat and maize was sampled by taking soil cores according to the method described by B
hm (1979). The diameter of the corer was 10 cm. For each treatment four cores were taken, one from each plot at least 1 m away from the border. Two cores were taken from the plant row and two between rows. After the cores had been taken the holes were firmly packed with similar soil to minimize the effect on water infiltration during subsequent irrigations. The depth of sampling was based on the average maximum rooting depth at different growing stages and samples were divided into intervals of 10 cm (Zhang, 1996).
The soil cores were taken to the laboratory and the soil was removed in washing cans. The resulting mixture of roots and organic debris (partially decomposed roots, stalks, stems, leaves and husks etc., from previous crops) were then placed in a polyethylene bag and preserved in a refrigerator (4°C) until it could be sorted, generally within 7 d. Roots colored from white to midbrown were deemed to be active and any black, dark brown or gray roots were discarded. After separating the live roots the length was measured based on the line-intersect method using a 1.27-cm grid (Tennant, 1975). Roots were then oven-dried at 80°C to determine dry weight.
All the data collected were statistically analyzed as a completely randomized design with four replications using analysis of variance to evaluate the effects on grain yield, ET, WUE, SWD and yield components between the two treatments (Clewer and Scarisbrick, 2001). Means were compared using LSD at P < 0.05 probability. Correlation was used to determine the relationship between yield reduction under MI with seasonal rainfall.
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RESULTS
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Rainfall and Irrigation
The NCP is located in the monsoon climatic zone, which is characterized by highly variable seasonal rainfall. Seasonal rainfall during the eight seasons for winter wheat varied from 50 to 210 mm, and for maize from 190 to 427 mm (Table 1). The 19981999 and 19992000 seasons were very dry with less than half of the long-term average rainfall in the winter wheat season. The other six seasons were normal or wetter than normal. For maize, 1998 and 2001 were dry seasons with rainfall less than 60% of the average. The 2004 season was wet and the other five seasons were near normal with 80 to 90% of the average rainfall. Because of the variation in seasonal rainfall presowing or postsowing irrigation was quite different between seasons and irrigation for the FI treatment also varied greatly (Table 1). In the FI treatment the soil was always wetter at the time of sowing the next crop and thus the depth of the irrigation at sowing was less than that required in the MI treatment. The mean annual irrigation application over 8 yr was about 223 mm for MI and 472 mm for FI.
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Table 1. Rainfall, irrigation at sowing for minimum irrigation (MI) and total irrigation for full irrigation (FI) during 1997 to 2005.
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Root Distribution and Soil Water Utilization
The soil at the study site is alluvial with a deep profile that favors root growth. The maximum rooting depth for winter wheat was 2 m and for maize was 1.2 m (Fig. 1
). The combination of shallow-rooted maize in the wet season with deep-rooted wheat in the dry season made the double-cropping system in the MI treatment more efficient in utilizing soil moisture. Winter wheat extracted soil moisture that was stored in the deep soil profile during rainy season. The shallower root system of maize utilized soil moisture mostly in the top 0.6 to 0.8 m of the soil profile. Rain in excess of maize water consumption was stored lower in the profile and utilized by the following wheat crop. Figure 2
shows the soil moisture distribution at sowing and harvest for winter wheat in the dry season of 19992000 and the wet season of 20032004. Either in dry or wet seasons, winter wheat extracted soil moisture to 1.6 to 1.8 m. The average total soil moisture used was over 200 mm and the average soil moisture content over the root depth of 2 m was reduced to about 40 to 50% of field capacity.

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Fig.1. The distribution of root length density for winter wheat and maize at mature under minimum irrigation (MI) in 20022003.
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Fig. 2. Soil moisture distribution at sowing and harvesting for winter wheat under minimum irrigation (MI) in the dry season of 19992000 (seasonal rainfall was 50.1 mm) and the wet season of 20032004 (seasonal rainfall was 210.6 mm).
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After winter wheat was harvested, the following maize crop was planted into this dry soil. Usually, moisture in the top 50 cm of the profile was near wilting point. The irrigation immediately after sowing brought the profile to field capacity to about 0.6 to 0.8 m. Below that point, soil would be wetted gradually during the rainy season. Generally at winter wheat harvest soil moisture reached its lowest level for the year. With the irrigation at sowing for maize and the addition of summer rain soil moisture was recharged to a relative higher level and this situation continued over winter. From early April onward water use increased rapidly with crop development and increasing evaporative demand. Soil moisture decreased rapidly, especially at the top soil layer, thus completing the annual cycle to the irrigation for maize seedling establishment.
Yield and Water Use Efficiency
Comparing MI and FI treatments the yield of wheat ranged from 5000 to 6800 kg ha1 (WUE, 1.51.8 kg m3) and 5200 to 7100 kg ha1 (WUE, 1.411.56 kg m3), respectively. Similarly, the yield of maize ranged from 5200 to 8200 kg ha1 (WUE, 1.52.5 kg m3) and 5600 to 9200 kg ha1 (WUE, 1.32.2 kg m3), respectively (Fig. 3
, Table 2). On average over the eight seasons the yield of wheat and maize from the MI treatment was lower than FI by 14 and 13%, respectively. During some dry seasons, yield reduction in MI was as high as 20 to 25% while in two out of the eight seasons the yield was not significantly reduced (P < 0.05).

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Fig. 3. Yield of winter wheat and maize under minimum irrigation (MI) and full irrigation (FI) from 1997 to 2005 (yield difference of winter wheat between MI and FI was significant at P < 0.05 for six seasons, except in 20022003 and 20032004; for maize, except in 2002 and 2003, yield difference between MI and FI was significant at P < 0.05; bars represent standard deviation).
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Table 2. Soil water depletion (SWD), total evapotranspiration (ET), and water use efficiency (WUE) for winter wheat and maize under minimum irrigation (MI) and full irrigation (FI).
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The average ET under MI was 25% lower for winter wheat and 21% lower for maize than that under FI. The reduction in ET was greater than that of grain yield. Thus, for winter wheat the WUE was generally higher under MI than under FI by an average of 15% (Table 2). The WUE for maize in the very dry 1998 season was lower for MI than FI. In all other seasons, WUE was higher under MI than that under FI by an average 10% (Table 2).
Yield of winter wheat under MI was reduced as a result of less spikes per area and a lower grain number per spike. During the eight seasons, the spike density and the kernel numbers per spike were on average 12 and 7.8% lower in MI than in FI treatments. However, the average 1000-seed weight under MI was 6% greater than that under FI. The increased weight per grain was considered to result from a tendency for water stressed wheat to flower earlier and thus gain several days during the grain filling stage, before high temperature in June limited further grain fill and accelerated crop maturity. Previous studies at the same site also showed that the maximum grain yield of winter wheat was not obtained with well-watered treatments, but with treatments in which water stress was imposed at development stages that were not sensitive to water deficit (Yuan et al., 1992; Zhang et al., 2003a, 2003b, 2005). Generally, the prewintering irrigation increased soil evaporation during the long dormant period, and good soil moisture conditions at the recovery stage increased tillers in spring that were usually not effective. It was concluded that reducing the irrigations at prewintering (November) and recovery stage (March) in this region could favor grain production of winter wheat (Zhang et al., 2003b).
Yield reduction in MI maize was associated with an increase in the number of plants without ears and with less kernels per ear compared to FI. There was no significant difference (P < 0.05) in 100-seed weight between FI and MI treatments. Significant reduction (P < 0.05) in ear numbers per area was observed under MI during 1998 to 2001. In those years average ear numbers per area and seeds per ear were about 8 and 5% less under MI than that under FI, respectively.
Seasonal rainfall was negatively correlated with the yield reduction (P < 0.05) for both winter wheat and maize (Fig. 4
). The 20022003 and 20032004 seasons were two wet seasons for winter wheat with above average rainfall, and yields of FI and MI were similar. In the driest season, 19992000, yield of MI over FI was reduced by 25%, the highest reduction in the eight seasons. In 19971998, 20002001, 20012002, and 20042005, the seasonal rainfall was similar, but the yield reduction in MI was about 9% greater in 20002001 and 20012002 than that in 19971998 and 20042005. Yield potential of winter wheat has been shown to be most sensitive to water stress from jointing to anthesis (English and Raja, 1996; Ghahraman and Sepaskhah, 1997; Kirda, 2002; Wang et al., 2001). The rainfall in 19971998 and 20042005 was distributed more during this critical period and increased soil moisture at this time compared to the other two seasons when rainfall occurred more during earlier growing stages, such as in 20002001 season, 64% rainfall fell before winter. Both the total seasonal rainfall and its distribution affected the grain yield of winter wheat under MI.

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Fig. 4. The correlation of yield reduction under minimum irrigation (MI) compared with full irrigation (FI) with seasonal rainfall for winter wheat and maize from 1997 to 2005.
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Maize yield was also affected by seasonal rainfall and its distribution under MI. In some seasons such as 2004 (Table 1), the total rainfall was quite high but yield was reduced relative to FI. Maize responded differently to water stress at different growing stages (Pei et al., 2004) and it is most sensitive to water availability during tasseling to silking (Otegui et al., 1995; Calviño et al., 2003). In 2004, the rainfall during this critical time was only 14% of the total seasonal rainfall. The yield of maize in 2002 and 2003 under FI was lower than that in other seasons that was possibly caused by reasons other than water supply. The new cultivar of maize used in 2002 (Laiyu 2) had not been successful in this region and both MI and FI yields were low. In 2003 the maize cultivar (Nongda108) used required about 10 d longer than other cultivars to reach maturity and it may not have been able to reach its yield potential under the region's limited growing period.
Evapotranspiration
The annual ET of double-cropped of winter wheat and maize was about 550 to 820 mm under MI and 760 to 920 mm under FI (Table 3). The average ET of the eight seasons was 654 mm under MI and 850 mm under FI. The difference between ET and the rainfall under MI and FI was 237 and 433 mm, respectively. Generally 230 mm of supplemental irrigation was required annually under MI and 430 mm under FI. In very dry seasons under MI about 270 mm irrigation was required to replenish the depleted soil moisture. Under FI the total irrigation amount was around 600 mm, a considerable sum compared with the annual average groundwater recharge estimated at 195 mm (Hebei Water Resource Bureau, 2004). The average annual rainfall of the 8 yr was 417 mm that was less than the long-term average of 482 mm (19702005). If the mean annual precipitation during the 8 yr had been similar to the long-term average, then the supplementary irrigation could have been reduced in both MI and FI. Kendy et al. (2003) suggested that depending on inputs, a proportion of precipitation plus irrigation drained through the soil profile to the groundwater table in this region and would contribute to aquifer recharge. Under MI the volume of such recharge would be reduced. However, the demonstrated reduction in ET under MI combined with lower groundwater extraction represents a clear benefit to aquifer stabilization.
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Table 3. The annual total evapotranspiration (ET) of double-cropping winter wheat and maize under minimum irrigation (MI) and full irrigation (FI) and the difference between ET and annual rainfall (ETrain).
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The Feasibility of Growing Winter Wheat and Maize under Minimum Irrigation
Due to the deep soil profile and high water holding capacity within the root zone of winter wheat the double cropping of winter wheat and maize can produce high grain yields with only the irrigation at sowing even in a dry year. The yield reduction of MI over FI was negatively correlated with seasonal rainfall (Fig. 4). This relationship indicates that progressive yield reductions of 5, 10, 15, and 20% would be obtained in rainfall years of 166, 136, 107, and 77 mm for the winter wheat season and 418, 355, 293, and 230 mm for the maize season, respectively. Figure 5
illustrated the probability of receiving a cumulative seasonal precipitation for each crop using the rainfall records from 1960 to 2005. Combining Fig. 4 and 5 indicates that under MI a normal yield would be obtained in 1 out of 5 to 6 yr. Yield would be reduced by less than 10% for 1 in 3 yr and by less than 15% for 1 in 2 yr. Other results at the same site have shown that, compared to MI, one additional irrigation at the most sensitive stage to water stress improved the grain yield of winter wheat by 15 to 20% in very dry seasons (Zhang et al., 2003b). Similar results were obtained for maize (Pei et al., 2004). Thus, if a subregional irrigation advisory system based on rainfall and reference evapotranspiration was used to inform farmers whether an extra irrigation was needed in the stress sensitive period then an income loss of less than 10% would be incurred for a maximum of 1 yr in 6 only. In addition, farmers would also use less electricity for water pumping with a saving estimated at 6% of the cash value of both crops.

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Fig. 5. The probability of receiving a cumulative seasonal precipitation for winter wheat and maize based on rainfall recording from 1960 to 2005.
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CONCLUSIONS
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The results show that under MI the double cropping of winter wheat and maize used about 200 mm less water than that of under FI annually. On average there was a 13% yield reduction under MI, but WUE was improved by 12% under MI. The overall supplemental irrigation amount was reduced by 40%. The deep soil profile and high water holding capacity of the soil are essential for this MI practice. Similar and reasonably uniform conditions occur over a large proportion of the NCP.
Especially during the growing period of winter wheat, rainfall fluctuated greatly and in most of the seasons rainfall was less than 150 mm. The deep root system of winter wheat could extract over 200 mm of the stored soil moisture at sowing over the season and more than 5000 kg ha1 of gain was produced without irrigation during the growing season. When profile recharge is incomplete due to a shortfall of rain during the maize growing season the pre-irrigation for wheat could be increased to compensate. The following crop maize grows during the rainy season, with irrigation for the seedling establishment, without further irrigation maize grain production could exceed 6000 kg ha1 for most of the years. The use of appropriately sized presowing irrigations for both winter wheat and maize effectively stabilized profile moisture content.
With the increase in population, China faces the pressure to supply more food to the people. But the rapidly declining groundwater table is threatening the sustainable agricultural development on the piedmont plain in the NCP, which has experienced a water table decline of more than 20 m over the past 30 yr (Evans and Han, 1999). It has been suggested that if groundwater availability for irrigation was restricted to current recharge, yields could be reduced by as much as 50% in a growing season of average rainfall (and more in dry years) (Foster et al., 2004). But results from this study showed that reducing total supplemental irrigation by 40% resulted in less than a 15% yield reduction in normal years. The irrigation water use under MI was similar with the annual recharge to groundwater in the region. Besides, the economic return of each unit of water used in grain production was low in comparison with that in horticultural products and industry. Studies have also shown that in this region the volume of irrigation used to obtain maximum yield was higher than that required to optimize economic return because of the relative higher cost in irrigation (Zhang et al., 2002). Thus, irrigation applications for grain production should be based on both irrigation water availability and economic returns in NCP. An irrigation advisory system should be implemented to augment MI by informing farmers when to apply a single additional irrigation in particularly dry seasons to minimize yield loss.
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ACKNOWLEDGMENTS
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This research was funded by the National Basic Research Program of China (project no. 2006CB403406). The authors acknowledge the anonymous referees for their valuable comments. Special appreciation is extended to the associate editor (Dr. Neal Eash) for his critical review and constructive suggestions. The authors are very grateful to Dr. Peter Jerie for his help in improving and correcting the paper.
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[Abstract]
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