﻿ 部分线性矩在黔南部地区洪水频率分析中的应用

# 部分线性矩在黔南部地区洪水频率分析中的应用The Application of Partial L-Moments for Flood Frequency Analysis in the South of Guizhou

Abstract: The south of Guizhou province suffers serious and frequent flood disasters. In order to provide an effi-cient and reliable theoretical basis for design of the flood control project in this area, the Partial L-moments are applied at the Bamao, Gaoche, Huishui, Guiyang, Libo, Caopingtou and Baben stations for flood frequency analysis. By estimating the parameters of Generalized Extreme Value (GEV) distribution and matching Partial L-Moments to annual maximum flow series of 7 hydrological stations, the design flood is calculated and the flood frequency curve is fitted. The cumulative squares error is used to evaluate the fitting ability of Partial L-moments and L-moments. The results show that as censored level F0 value increases, the relative deviation of the design value is smaller, except the hydrological stations of Gaoche and Caopingtou. Partial L-Moments can describe the data better in flood analysis and improve the estimation precision of design flood. Partial L-moments is a reasonable and effective method of flood frequency analysis in the south of Guizhou province.

1. 引言

2. 部分线性矩

${{\lambda }^{\prime }}_{1}=\frac{1}{{}^{n}C{}_{1}}\underset{i=1}{\overset{n}{\sum }}{x}_{\left(i\right)}^{*}$ (1)

${{\lambda }^{\prime }}_{2}=\frac{1}{2}\frac{1}{{}^{n}C{}_{2}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{1}-{}^{n-i}C{}_{1}\right){x}_{\left(i\right)}^{*}$ (2)

${{\lambda }^{\prime }}_{3}\text{​}\text{​}\text{​}\text{​}\text{​}\text{​}=\frac{1}{3}\frac{1}{{}^{n}C{}_{3}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{2}-2{}^{i-1}C{}_{1}{}^{n-i}C{}_{1}+{}^{n-i}C{}_{2}\right){x}_{\left(i\right)}^{*}$ (3)

${{\lambda }^{\prime }}_{4}=\frac{1}{4}\frac{1}{{}^{n}C{}_{4}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{3}-3{}^{i-1}C{}_{2}{}^{n-i}C{}_{1}+3{}^{i-1}C{}_{1}{}^{n-i}C{}_{2}-{}^{n-i}C{}_{3}\right){x}_{\left(i\right)}^{*}$ (4)

${x}_{\left(i\right)}^{*}=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{\left(i\right)}\le {x}_{0}\\ {x}_{\left(i\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }{x}_{\left(i\right)}>{x}_{0}\end{array}$ (5)

${}^{n}C{}_{i}=\frac{n!}{i!\left(n-i\right)!}$ (6)

3. 广义极值分布及其部分线性矩

$F\left(x\right)=\left\{\begin{array}{l}\mathrm{exp}\left\{-{\left[1-\frac{k}{\alpha }\left(x-\xi \right)\right]}^{\frac{1}{k}}\right\};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }k\ne 0\\ \mathrm{exp}\left\{-\mathrm{exp}\left[-\frac{1}{\alpha }\left(x-\xi \right)\right]\right\};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=0\end{array}$ (7)

$x\left(F\right)=\left\{\begin{array}{l}\xi +\frac{\alpha }{k}\left[1-{\left(-\mathrm{ln}F\right)}^{k}\right];\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\ne 0\\ \xi -\alpha \mathrm{ln}\left(-\mathrm{ln}F\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }k=0\end{array}$ (8)

Wang (1990) [11] 提出了GEV分布部分概率权重矩，当 $k\ne 0$ 时，其表达式为

${{\beta }^{\prime }}_{r}=\left(\xi +\frac{\alpha }{k}\right)\frac{1}{r+1}\left(1-{F}_{0}^{r+1}\right)-\frac{\alpha }{k}\frac{\Gamma \left(1+k\right)}{{\left(1+r\right)}^{1+k}}P\left(1+k,-\left(1+r\right)\mathrm{log}{F}_{0}\right)$ (9)

$P\left(1+k,-\left(1+r\right)\mathrm{log}{F}_{0}\right)={\int }_{0}^{-\left(1+r\right)\mathrm{log}{F}_{0}}\frac{{x}^{k}{\text{e}}^{-x}}{\Gamma \left(1+k\right)}\text{d}x$ (10)

$r=0,\text{1},\text{2}$ 代入式(9)，可得

${{\beta }^{\prime }}_{0}=\left(\xi +\frac{\alpha }{k}\right)\left(1-{F}_{0}\right)-\frac{\alpha }{k}\Gamma \left(1+k\right)P\left(1+k,-\mathrm{log}{F}_{0}\right)$ (11)

$\frac{2{{\beta }^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\beta }^{\prime }}_{0}}{1-{F}_{0}}=-\frac{\alpha }{k}\left[\frac{P\left(1+k,-2\mathrm{log}{F}_{0}\right)}{{2}^{k}\left(1-{F}_{0}^{2}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}\right]$ (12)

$\frac{\frac{2{{\beta }^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\beta }^{\prime }}_{0}}{1-{F}_{0}}}{\frac{3{{\beta }^{\prime }}_{2}}{1-{F}_{0}^{3}}-\frac{{{\beta }^{\prime }}_{0}}{1-{F}_{0}}}=\frac{\frac{P\left(1+k,-2\mathrm{log}{F}_{0}\right)}{{2}^{k}\left(1-{F}_{0}^{2}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}}{\frac{P\left(1+k,-3\mathrm{log}{F}_{0}\right)}{{3}^{k}\left(1-{F}_{0}^{3}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}}$ (13)

$z=\frac{\frac{2{{\beta }^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\beta }^{\prime }}_{0}}{1-{F}_{0}}}{\frac{3{{\beta }^{\prime }}_{2}}{1-{F}_{0}^{3}}-\frac{{{\beta }^{\prime }}_{0}}{1-{F}_{0}}}$ (14)

$z=\frac{\frac{P\left(1+k,-2\mathrm{log}{F}_{0}\right)}{{2}^{k}\left(1-{F}_{0}^{2}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}}{\frac{P\left(1+k,-3\mathrm{log}{F}_{0}\right)}{{3}^{k}\left(1-{F}_{0}^{3}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}}$ (15)

${F}_{0}\ne 0$ 时，给定k的取值范围 $-0.5\le k\le 0.5$ ，分别令 ${F}_{0}=0.1~0.5$ ，根据式(15)分别计算各组取值对应的z值，按式(16)拟合曲线，求得曲线拟合系数如表1所示。

$k={a}_{0}+{a}_{1}z+{a}_{2}{z}^{2}+{a}_{3}{z}^{3}+{a}_{4}{z}^{4}$ (16)

Hosking [12] (1990)给出线性矩与概率权重矩的前4阶关系为

${\lambda }_{1}={\beta }_{0}$ (17)

${\lambda }_{2}=2{\beta }_{1}-{\beta }_{0}$ (18)

${\lambda }_{3}=6{\beta }_{2}-6{\beta }_{1}+{\beta }_{0}$ (19)

${\lambda }_{4}=20{\beta }_{3}-30{\beta }_{2}+12{\beta }_{1}-{\beta }_{0}$ (20)

$z=\frac{\frac{{{\lambda }^{\prime }}_{2}+{{\lambda }^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\lambda }^{\prime }}_{1}}{1-{F}_{0}}}{\frac{\frac{1}{2}\left({{\lambda }^{\prime }}_{3}+3{{\lambda }^{\prime }}_{2}+2{{\lambda }^{\prime }}_{1}\right)}{1-{F}_{0}^{3}}-\frac{{{\lambda }^{\prime }}_{1}}{1-{F}_{0}}}$ (21)

$\stackrel{^}{z}=\frac{\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{2}+{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}}}{\frac{\frac{1}{2}\left({{\stackrel{^}{\lambda }}^{\prime }}_{3}+3{{\stackrel{^}{\lambda }}^{\prime }}_{2}+2{{\stackrel{^}{\lambda }}^{\prime }}_{1}\right)}{1-{F}_{0}^{3}}-\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}}}$ (22)

$\stackrel{^}{\alpha }=-\frac{\stackrel{^}{k}}{\Gamma \left(1+\stackrel{^}{k}\right)}\frac{\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{2}+{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}^{2}}-\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}}}{\frac{P\left(1+k,-2\mathrm{log}{F}_{0}\right)}{{2}^{k}\left(1-{F}_{0}^{2}\right)}-\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}}$ (23)

$\stackrel{^}{\beta }=\frac{{{\stackrel{^}{\lambda }}^{\prime }}_{1}}{1-{F}_{0}}+\frac{\stackrel{^}{\alpha }}{\stackrel{^}{k}}\left[\Gamma \left(1+\stackrel{^}{k}\right)\frac{P\left(1+k,-\mathrm{log}{F}_{0}\right)}{1-{F}_{0}}-1\right]$ (24)

${F}_{0}=0.0$ 时，部分线性矩转化为普通线性矩。Hosking [12] (1990)推出了GEV分布下普通线性矩为

${\lambda }_{1}=\xi +\alpha \left[1-\Gamma \left(1+k\right)\right]/k$ (25)

${\lambda }_{2}=\alpha \left(1-{2}^{-k}\right)\Gamma \left(1+k\right)/k$ (26)

${\tau }_{3}=2\left(1-{3}^{-k}\right)/\left(1-{2}^{-k}\right)-3$ (27)

${\tau }_{4}=\left[5\left(1-{4}^{-k}\right)-10\left(1-{3}^{-k}\right)+6\left(1-{2}^{-k}\right)\right]/\left(1-{2}^{-k}\right)$ (28)

$-0.5<{\tau }_{3}<0.5$ 时，三个参数的估计量计算公式分别为

$\stackrel{^}{k}=7.8590C+2.9554{C}^{2}$ (29)

$\stackrel{^}{\alpha }={\lambda }_{2}\stackrel{^}{k}/\left[\left(1-{2}^{-\stackrel{^}{k}}\right)\Gamma \left(1+\stackrel{^}{k}\right)\right]$ (30)

$\stackrel{^}{\xi }={\lambda }_{1}-\stackrel{^}{\alpha }\left[1-\Gamma \left(1+\stackrel{^}{k}\right)\right]/\stackrel{^}{k}$ (31)

4. 实例应用

4.1. 绘制频率曲线

4.2. 拟合效果分析

$\delta =\underset{i=i|P=50%}{\overset{i|P=98%}{\sum }}{\left(\frac{{x}_{i}-{\stackrel{^}{x}}_{i}}{{x}_{i}}\right)}^{2}$ (32)

Table 2. Lengths of annual maximum flows and comparison of quantile errors using different F0

(a) 八茂 (b) 高车 (c) 惠水 (d) 贵阳 (e) 荔波 (f) 草坪头 (g) 把本

Figure 1. Flood frequency plot of annual maximum flows in northern Shaanxi

5. 结论

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