工程科学与技术   2019, Vol. 51 Issue (1): 75-82

1. 四川大学 水利水电学院，四川 成都 610065;
2. 四川大学 水力学与山区河流开发保护国家重点室，四川 成都 610065

Direct Method of Calculating the Manage-flood Based on Division of Dlood Period
WANG Wensheng1,2, YAO Ruihu1, QIN Guanghua1,2, LIANG Shuqi1, DING Jing1
1. College of Water Resource & Hydropower, Sichuan Univ., Chengdu 610065, China;
2. State Key Lab. of Hydraulics and Mountain River Eng., Sichuan Univ., Chengdu 610065, China
Abstract: The manage-flood is very important for flood control safety and economic effectiveness of reservoir engineering. There are two approaches which can be used to calculate the manage-flood at present, i.e., one is based on Copula functions, and the other is based on the principle of total probability. The latter can be divided into direct method and indirect method. The direct method based on the principle of total probability is explored in this article. The flood data for calculation of manage-flood have been divided into three condition:Sufficient data condition, relatively sufficient data condition and insufficient data condition. For sufficient data conditions, the length of flood data series in main flood season is greater than 30 years and the length of flood data series in pre-main flood season and post-main flood season is no less than 20 years, respectively; For relatively sufficient data condition, the length of flood data series in main flood season is greater than 30 years and the length of flood data series in non-main flood season is no less than 20 years; For insufficient data condition, the length of flood data series in main flood season is greater than 30 years and the length of flood data series in non-main flood season is less than 20 years. The calculating methods of manage-flood for three flood data conditions have been presented based on the principle of total probability in this paper. For sufficient data condition, the mathematical expect formula of total probability has been adopted with frequency analysis; for relatively sufficient data condition, the mathematical expect formula of total probability with combined data in non-main flood season was applied; for insufficient data condition, the back calculation approach was presented, and the formula of manage-flood based on the principle of total probability has been suggested. The case studies for calculating of manage-flood in different flood data conditions have been given for different basins. The comparison of the suggested method to the method of Copula function has been discussed. The results have showed that the suggested method is better than the method of Copula function, and the designed value of manage-flood is reasonable.
Key words: division of flood period    manage-flood    total probability    copula function    flood frequency

1 推求管运洪水的直接法

 图1 洪水分期示意图 Fig. 1 Schematic diagram on the flood stage

 ${P_4}({x_4}) \!=\! {P_1}({{{x_1}} / {{A_1}}})P({A_1}) \!+\! {P_2}({{{x_2}} / {{A_2}}})P({A_2}) \!+\! {P_3}({{{x_3}} / {{A_3}}})P({A_3})$ (1)

2 推求方法

 $S = \frac{{Cv}}{{k\sqrt n }}B \times 100$ (2)

1）资料充足情况。主汛期A2的年最大洪水样本容量为30年以上，而前汛期A1和后汛期A3的样本容量分别为20年以上。

2）资料相对充足情况。A2的样本容量为30年以上，而A1A3的样本容量均未达到20年的要求。

3）资料不足情况。A2的样本容量为30年以上，而A1A3的样本容量之和尚未达到20年。

2.1 资料充足情况

A1A2A3期分别有n1n2n3年年最大洪水值资料，总资料数n=n1+n2+n3。对每一分期进行洪水频率计算，将适线法结果绘于图2

 图2 全汛期和各分期洪水频率曲线示意图 Fig. 2 Diagram of the flood frequency curves in different seasons

1）由适线法[14]获得频率曲线时所用的经验频率公式分别为：

A1期：

 $P = \frac{{{m_1}}}{{{n_1} + 1}}$ (3)

A2期：

 $P = \frac{{{m_2}}}{{{n_2} + 1}}$ (4)

A3期：

 $P = \frac{{{m_3}}}{{{n_3} + 1}}$ (5)

2）主汛期通常有历史洪水资料可以利用，其处理办法和通常情况类似。资料充足情况下，对所获得的成果P1P2P3可以在借助式（1）做合理性分析的基础上进行调整，使成果更为可靠。图2中的A1A2A3期频率曲线分别相当于式（1）中的P1(x1/A1)、P2(x2/A2)和P3(x3/A3)；式（1）中的P4(x4)实际上为整个汛期（A1+A2+A3）年最大洪水频率曲线（由容量n的样本系列通过频率计算获得），为了与分期频率曲线相比较，该线亦绘于图2上并标记为“Y”。由于“Y”频率曲线是依据n年资料推得，一般较可靠。

 ${P_4}({x_0}) \!=\! {P_1}({{{x_0}} / {{A_1}}})P({A_1}) \!+\! {P_2}({{{x_0}} / {{A_2}}})P({A_2})\! +\! {P_3}({{{x_0}} / {{A_3}}})P({A_3})$ (6)

2.2 资料相对充足情况

A1A3期的资料长度达不到分别直接进行频率计算要求时，应设法延长资料。若无法延长资料，但当A1A3期洪水特性类似时，可将A1A3期的洪水资料合并，代表整个次汛期（A1+A3）的洪水系列，再进行频率分析计算。设（A1+A3）合并为A5，且P5(x5/A5)为A5期的频率曲线，那么有：

 ${P_4}({x_4}) = {P_5}({{{x_5}} / {{A_5}}})P({A_5}) + {P_2}({{{x_2}} / {{A_2}}})P({A_2})$ (7)

A5期和A2期的资料条件属于上述资料充足情况，因此基于式（7）可做类似于式（6）的分析计算。但需说明2点：1）基于式（7）对P1P3频率曲线进行合理性分析较基于式（6）要方便得多，其可靠性更高；2）获得的P5(x5/A5)既表示P1(x1/A1)，也表示P3(x3/A3)，即A1期的频率曲线和A3期的频率曲线合并为一条。具体将在实例分析中进一步论述。

2.3 资料不足情况

 ${P_5}({{{x_5}} / {{A_5}}}) = \frac{1}{{P({A_5})}}\left[ {{P_4}({x_4}) - {P_2}({{{x_2}} / {{A_2}}}){P_2}({A_2})} \right]$ (8)

A2期和全年的洪水资料长度均达到要求，对应的频率曲线是相对可靠的，加之P(A4)和P(A5)相对可靠，这样基于式（8）可直接推求得到P5，即推得P1P3。这是直接反推而获得的成果，故称此为反算手段。下面将以实例具体说明。

3 实例分析

3.1 资料相对充足情况下分析计算

1）年最大洪水分别发生在主汛期A2和次汛期A5的频率推算

 $P({A_2}) = \frac{{{n_2}}}{{{n_1} + {n_2} + {n_3}}} = 0.647,$
 $\;\;\;P({A_5}) = \frac{{{n_5}}}{{{n_1} + {n_2} + {n_3}}} = 0.353{\text{。}}$

2）全年洪水频率曲线Y[P4(x4)]的推算

 图3 沱江流域某站1941—2008年全年最大洪峰流量频率曲线 Fig. 3 Frequency curves of annual maximum discharge in the flood season at one station of Toujiang River during 1941—2008

3）主汛期洪水频率曲线P2(x2)的推算

 图4 沱江流域某站主汛期、次汛期及全汛期年最大洪峰流量频率曲线 Fig. 4 Flood frequency curves of each flood period at one station of Toujiang River

4）次汛期年最大洪峰流量频率曲线P5(x5)的推算

5）成果合理性分析

3.2 资料不足情况下分析计算

1）年最大洪水分别发生在主汛期和次汛期的频率推算

 $P\left( {{A_2}} \right) = \frac{{{n_2}}}{{{n_1} + {n_2} + {n_3}}} = 0.807,$
 $P({A_5}) = \frac{{{n_5}}}{{{n_1} + {n_2} + {n_3}}} = \frac{{{n_1} + {n_3}}}{{{n_1} + {n_2} + n_3^{}}} = 0.193{\text{。}}$

2）全年洪水频率曲线P4(x4)的推算

 图5 某水库控制站1957—2013年全年最大洪峰流量频率曲线 Fig. 5 Frequency curves of annual maximum discharge in the whole flood season at a control station of one reservoir during 1957—2013

3）主汛期年最大洪峰流量频率曲线P2(x2)的推算

 图6 某水库控制站主汛期年最大洪峰流量频率曲线 Fig. 6 Flood frequency curves of annual maximum discharge in main flood period at a control station of one reservoir

4）次汛期年最大洪峰流量频率曲线P5(x5)的反算

 图7 某水库控制站次汛期年最大洪峰流量频率曲线 Fig. 7 Flood frequency curves of annual maximum discharge in non-main flood period at control station of some reservoir

 ${P_5}({{{x_0}} / {{A_5}}}) = \frac{1}{{P({A_5})}}\left[ {{P_4}({x_0}) - {P_2}({{{x_0}} / {{A_2}}}){P_2}({A_2})} \right]$ (9)

5）合理性分析

 图8 某水库控制站主汛期、次汛期及全汛期年最大洪峰流量频率曲线 Fig. 8 Flood frequency curves of main、non-main and whole flood period at control station of some reservoir

6）两种结果的比较

 ${P_{{\text{年}}}} = 1 - C\left( {u,v} \right)$ (10)

4 结　论

1）第1类途径同第2类途径中的直接法相比，各有优缺点。前者不满足防洪标准要求为先天性的缺陷，后者能满足要求为独特的优势。但前者对资料条件要求较宽，而后者对资料条件要求较严，当前影响其实际应用。

2）应用直接法时，要求资料的长度随情况而变。若汛期分为3期，初步建议主汛期的洪水资料的长度超过30年，而次汛期的洪水资料长度超过20年。

3）在前汛期和后汛期洪水特性类似的条件下，合并手段和反算手段放宽了直接法对资料的要求，扩大了该法的适用范围。

4）以直接法获得的结果必须进一步做定量检查，其依据在于全汛期和各分期的频率之间的定量关系受到全概率公式的制约。这在一定程度上保证结果的合理性，是直接法的另一种不可忽视的优势。

5）本研究所得的结果是初步的，不少问题尚需进一步研究。特别是对资料长度的要求，资料不足时如何应用等问题有待深入探讨。

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