Application of the grey theory to dynamic analyses of the Baiquan Spring flow rate in Xinxiang
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摘要:
新乡百泉具有供水、农灌、人文、旅游及生态等多种功能，研究其泉水流量动态，建立泉水流量动态预测模型，对泉域水资源评价和泉水资源保护具有重要意义。为了进一步研究新乡百泉天然状态下泉水流量动态特征、评价泉域岩溶水资源，基于1964—1978年泉水年均实测流量和泉域年均降水量资料，通过逐步回归分析，确定泉水流量的主要影响因素为前1年降水量，并建立了逐步回归模型，其回归效果显著；在逐步回归分析的基础上建立了泉水流量动态预测的GM(1, 2)模型、NSGM(1, 2)模型和GM(0, 2)模型。结果表明：1964—1978年百泉泉水流量动态主要受泉域降水控制，且泉水流量滞后降水1年，反映了天然状态下泉水动态特征。3种灰色模型的精度等级均为最高级（优）。1964—1978年百泉泉水实测流量为2.347~6.448 m³/s，平均为3.904 m3/s；逐步回归模型预测值为1.882~6.383 m3/s，平均为3.904 m3/s；GM(1，2)模型预测值为2.327~6.448 m3/s，平均为3.939 m3/s；NSGM(1，2)模型预测值为2.133~6.448 m3/s，平均为3.927 m3/s；GM(0，2)模型预测值为1.787~6.448 m3/s，平均为3.907 m3/s。逐步回归模型和前述3种灰色模型的平均相对误差分别为7.794%、7.292%、7.122%、7.797%，均<10%，可用于泉水流量动态预测；其中NSGM(1，2)模型精度更高、对曲线“拐点”的拟合更好。根据4种模型预测的1964—2030年泉水流量，从保泉角度评价百泉泉域岩溶水的开采资源量不得超过1.69 m3/s。研究成果可为泉水流量动态预测和泉域水资源评价提供科学依据，也可为类似地区地下水动态研究提供参考。
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关键词:
- 新乡百泉 /
- 逐步回归分析 /
- GM(1, N)模型 /
- NSGM(1, N)模型 /
- GM(0, N)模型 /
- P-Ⅲ型曲线
Abstract:The Baiquan Spring in Xinxiang has many functions, such as water supply, agricultural irrigation, humanities, tourism and ecology. It is of great significance to study the dynamics of the spring flow rate and establish a dynamic prediction model for the spring water resources evaluation and protection. In order to further study the dynamic characteristics of the Baiquan Spring flow rate in Xinxiang and evaluate karst water resources in the spring area, based on the data of annually measured spring flow rate and annual average precipitation in the spring area from 1964 to 1978, the main influencing factors of the spring flow rate are determined by using the stepwise regression analysis, and a stepwise regression model is established, with remarkable regression effect. On the basis of the stepwise regression analysis, this paper establishes the GM(1, 2) model, NSGM(1, 2) model and GM(0, 2) model for the dynamic prediction of the spring flow rate. The results show that (1) from 1964 to 1978, the Baiquan spring flow rate was mainly controlled by the precipitation in the spring area, and the spring flow rate lagged behind the precipitation for one year, reflecting the dynamic characteristics of the spring water in the natural state. (2) The accuracy levels of the three grey models are the highest (excellent). (3) From 1964 to 1978, the measured discharge of the Baiquan spring ranged from 2.347 to 6.448 m3/s, with an average of 3.904 m3/s. The predicted values of the stepwise regression model range from 1.882 to 6.383 m3/s, with an average of 3.904 m3/s. The predicted value of the GM(1, 2) model varies between 2.327 and 6.448 m3/s, with an average of 3.939 m3/s. The predicted values of the NSGM(1, 2) model range from 2.133 to 6.448 m3/s, with an average of 3.927 m3/s. The predicted values of the GM(0, 2) model range from 1.787 to 6.448 m3/s, with an average of 3.907 m3/s. (4) The average relative errors of the stepwise regression model and the three grey models mentioned above are 7.794%, 7.292%, 7.122% and 7.797% respectively, all of which are less than 10%, indicating that they can be used for dynamic prediction of the spring water. Among them, the NSGM(1, 2) model has a higher accuracy and better fitting to the inflection point of the curve. (5) According to the spring flow rate from 1964 to 2030 predicted by the four models, the exploitation resources of the karst water in the Baiquan spring area should not exceed 1.69 m3/s from the angle of spring protection. The research results can not only provide scientific basis for spring flow dynamic prediction and spring area water resources evaluation, but also provide reference for the study of groundwater dynamics in similar areas.
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表 1 1979—2021年百泉泉水复流情况
Table 1. Statistics of re-flow of the Baiquan Spring from 1979 to 2021
复流年份 复流情况 1982 8月15日复流至次年5月断流，最大出水量为6.65 m3/s 1988 8月中旬复流，次年断流（断流日期不详），最大出水量约为5 m3/s 1996 8月3日复流，次年4月17日断流，最大出水量为3.10 m3/s 1998 9月2日复流，次年1月8日断流，最大出水量为1.91 m3/s 2000 9月21复流，10月26日断流，最高湖水位仅为40 cm 2003 10月26复流，次年3月28日断流，最大出水量为2.35 m3/s 2004 8月28日复流，次年1月31断流，最大出水量为1.40 m3/s 2021 百泉泉域遭遇有气象记录以来的特大降雨，7月23日复流，最大出水量为10 m3/s。2022年7月，仍有近4 m3/s 
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表 2 灰色模型精度等级评价
Table 2. Evaluation of the accuracy grade of the grey model
精度等级 P C 
优 0.95<P≤1.00 0.00<C≤0.35 0.90< 
≤1.00良 0.80<P≤0.95 0.35<C≤0.50 0.80< 
≤0.90中 0.70<P≤0.80 0.50<C≤ 0.65 0.70< 
≤0.80差 0.00<P≤0.70 0.65<C≤1.00 0.00< 
≤0.70
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表 3 1964—1978年泉水流量预测值与实测值对比结果
Table 3. Comparison results between the predicted and measured values of the spring flow rate from 1964 to 1978
年份 年均降水量
/mm实测值
/（m³·s−1）逐步回归模型 GM（1, 2）模型 NSGM（1, 2）模型 GM（0. 2）模型 预测值/ （m³·s−1） 相对误差/% 预测值/ （m³·s−1） 相对误差/% 预测值/（m³·s−1） 相对误差/% 预测值/（m³·s−1） 相对误差/% 1963 1240.0 — — — — — — — — — 1964 886.0 6.448 6.383 1.016 6.448 0.000 6.448 0.000 6.448 0.000 1965 340.0 4.575 4.612 0.817 4.156 9.158 4.678 2.251 4.656 1.770 1966 470.0 2.370 1.882 20.583 2.327 1.814 2.133 10.000 1.787 24.599 1967 540.0 2.347 2.532 7.892 2.585 10.141 2.520 7.371 2.470 5.241 1968 590.0 2.504 2.882 15.016 2.876 14.856 2.870 14.617 2.838 13.339 1969 755.0 3.537 3.132 11.443 3.126 11.620 3.126 11.620 3.101 12.327 1970 807.0 3.976 3.957 0.469 3.996 0.503 3.899 1.937 3.968 0.201 1971 935.0 4.349 4.217 3.027 4.271 1.794 4.198 3.472 4.241 2.483 1972 660.0 4.711 4.857 3.108 4.948 5.031 4.806 2.017 4.914 4.309 1973 560.0 3.733 3.482 6.716 3.493 6.429 3.602 3.509 3.468 7.099 1974 666.0 2.833 2.982 5.269 2.964 4.624 3.049 7.624 2.943 3.883 1975 1030.0 3.215 3.512 9.247 3.525 9.642 3.487 8.460 3.500 8.865 1976 950.0 5.232 5.332 1.920 5.451 4.186 5.181 0.975 5.413 3.459 1977 735.0 5.509 4.932 10.466 5.028 8.731 4.954 10.074 4.993 9.366 1978 548.0 3.219 3.857 19.830 3.890 20.845 3.956 22.895 3.863 20.006 最大值 1240.0 6.448 6.383 19.830 6.448 20.845 6.448 22.895 6.448 20.006 最小值 340.0 2.347 1.882 0.469 2.327 0.000 2.133 0.000 1.787 0.000 平均值 732.0 3.904 3.904 7.794 3.939 7.292 3.927 7.122 3.907 7.797
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表 4 灰色模型检验指标计算结果
Table 4. Calculation results of the grey model test index
模型 P C 
GM（1, 2） 1.0000 0.2653 0.9974 NSGM（1, 2） 1.0000 0.2004 0.9997 GM（0, 2） 1.0000 0.2747 0.9963
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表 5 不同保证率泉水流量
Table 5. Spring flow rate with different guarantee rates
保证率/% 泉水流量/ （m³·s−1） 逐步回归 GM（1, 2） NSGM（1, 2） GM（0, 2） 平均 10 4.91 4.94 4.94 4.94 4.93 20 4.35 4.38 4.37 4.37 4.37 50 3.42 3.44 3.43 3.41 3.43 75 2.80 2.81 2.81 2.77 2.80 95 2.08 2.09 2.11 2.02 2.08 99 1.69 1.69 1.74 1.62 1.69
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