﻿ 高阶线性矩法在陕北地区洪水频率分析中的应用

# 高阶线性矩法在陕北地区洪水频率分析中的应用Using Higher-Order L-Moments for Flood Frequency Analysis in Northern Shaanxi

Abstract: In order to provide an efficient and reliable theoretical basis for design floods in the northern Shaanxi province, the higher-order L-Moments are applied in flood frequency analysis based on the principles of higher-order L-Moments. The annual maximum flood series of 8 hydrological stations at Jiaokou, Zhangjiashan, Zhaoshiyao, Sudie, Liujia, Zhangcunyi, Linjiacun and Shenmu Rivers are selected for case study. The parameters of Generalized Extreme Value (GEV) distribution and the design floods are estimated. The flood frequency curves are fitted, and the cumulative of squares error is regard as an indicator to evaluate the effect, and compared with the traditional Method of moments. The results show that higher-order L-Moments possess good statistical performance, which can describe the data series much better than lower-order L-Moments in flood analysis. Consequently, this method is reasonable and feasible, and would be provided the basis for the flood quantile calculation.

1. 引言

2. 高阶线性矩

$E\left[{X}_{i,n}\right]=\frac{n!}{\left(i-1\right)!\left(n-i\right)!}{\int }_{0}^{1}x\left(F\right){F}^{i-1}{\left(1-F\right)}^{n-i}\text{d}F$ (1)

${\lambda }_{1}^{\eta }=E\left[{X}_{\left(\eta +1\right)\left(\eta +1\right)}\right]$ (2)

(3)

${\lambda }_{3}^{\eta }=\frac{1}{3}E\left[{X}_{\left(\eta +3\right)\left(\eta +3\right)}-2{X}_{\left(\eta +2\right)\left(\eta +3\right)}+{X}_{\left(\eta +1\right)\left(\eta +3\right)}\right]$ (4)

${\lambda }_{4}^{\eta }=\frac{1}{4}E\left[{X}_{\left(\eta +4\right)\left(\eta +4\right)}-3{X}_{\left(\eta +3\right)\left(\eta +4\right)}+3{X}_{\left(\eta +2\right)\left(\eta +4\right)}-{X}_{\left(\eta +1\right)\left(\eta +4\right)}\right]$ (5)

$\eta =0$ 时，高阶线性矩转化为普通线性矩(Hosking, 1990)。随着 $\eta$ 增高，高阶线性矩对随机变量的较大值更为依赖。高阶线性矩的变差系数 ${\tau }_{2}^{\eta }$ ，偏态系数 ${\tau }_{3}^{\eta }$ 和峰态系数 ${\tau }_{4}^{\eta }$ 分别为

${\tau }_{2}^{\eta }=\frac{{\lambda }_{2}^{\eta }}{{\lambda }_{1}^{\eta }}$ (6)

(7)

${\tau }_{4}^{\eta }=\frac{{\lambda }_{4}^{\eta }}{{\lambda }_{2}^{\eta }}$ (8)

${\stackrel{^}{\lambda }}_{1}^{\eta }=\frac{1}{{}^{n}C{}_{\eta +1}}\underset{i=1}{\overset{n}{\sum }}{}^{i-1}C{}_{\eta }{x}_{\left(i\right)}$ (9)

${\stackrel{^}{\lambda }}_{2}^{\eta }=\frac{1}{2}\frac{1}{{}^{n}C{}_{\eta +2}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{\eta +1}-{}^{i-1}C{}_{\eta }{}^{n-i}C{}_{1}\right){x}_{\left(i\right)}$ (10)

${\stackrel{^}{\lambda }}_{3}^{\eta }=\frac{1}{3}\frac{1}{{}^{n}C{}_{\eta +3}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{\eta +2}-2{}^{i-1}C{}_{\eta +1}{}^{n-i}C{}_{1}+{}^{i-1}C{}_{\eta }{}^{n-i}C{}_{2}\right){x}_{\left(i\right)}$ (11)

${\stackrel{^}{\lambda }}_{4}^{\eta }=\frac{1}{4}\frac{1}{{}^{n}C{}_{\eta +4}}\underset{i=1}{\overset{n}{\sum }}\left({}^{i-1}C{}_{\eta +3}-3{}^{i-1}C{}_{\eta +2}{}^{n-i}C{}_{1}+3{}^{i-1}C{}_{\eta +1}{}^{n-i}C{}_{2}-{}^{i-1}C{}_{\eta }{}^{n-i}C{}_{3}\right){x}_{\left(i\right)}$ (12)

${}^{n}C{}_{i}=\frac{n!}{i!\left(n-i\right)!}$ (13)

3. 广义极值分布及其高阶线性矩

$F\left(x\right)=\left\{\begin{array}{l}\mathrm{exp}\left\{-{\left[1-\frac{k}{\alpha }\left(x-\xi \right)\right]}^{\frac{1}{k}}\right\};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\ne 0\\ \mathrm{exp}\left\{-\mathrm{exp}\left[-\frac{1}{\alpha }\left(x-\xi \right)\right]\right\};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }k=0\end{array}$ (14)

$x\left(F\right)=\left\{\begin{array}{l}\xi +\frac{\alpha }{k}\left[1-{\left(-\mathrm{ln}F\right)}^{k}\right];\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\ne 0\\ \xi -\alpha \mathrm{ln}\left(-\mathrm{ln}F\right);\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }k=0\end{array}$ (15)

$k\ne 0$ 时，即为Hosking (1985) [17] 给出的GEV分布概率权重矩(PWM)

${\beta }_{r}={\int }_{0}^{1}x\left(F\right){F}^{r}\text{d}F$ (16)

(17)

$\left(r+1\right){\beta }_{r}=\xi +\alpha \left[\epsilon +\mathrm{ln}\left(r+1\right)\right]$ (18)

${\lambda }_{1}^{\eta }=\xi +\frac{\alpha }{k}\left[1-\Gamma \left(1+k\right){\left(\eta +1\right)}^{-k}\right]$ (19)

${\lambda }_{2}^{\eta }=\frac{\left(\eta +2\right)\alpha \Gamma \left(1+k\right)}{2!k}\left[-{\left(\eta +2\right)}^{-k}+{\left(\eta +1\right)}^{-k}\right]$ (20)

${\lambda }_{3}^{\eta }=\frac{\left(\eta +3\right)\alpha \Gamma \left(1+k\right)}{3!k}\left[-\left(\eta +4\right){\left(\eta +3\right)}^{-k}+2\left(\eta +3\right){\left(\eta +2\right)}^{-k}-\left(\eta +2\right){\left(\eta +1\right)}^{-k}\right]$ (21)

$\begin{array}{c}{\lambda }_{4}^{\eta }=\frac{\left(\eta +4\right)\alpha \Gamma \left(1+k\right)}{4!k}\left[-\left(\eta +6\right)\left(\eta +5\right){\left(\eta +4\right)}^{-k}+3\left(\eta +5\right)\left(\eta +4\right){\left(\eta +3\right)}^{-k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-3\left(\eta +4\right)\left(\eta +3\right){\left(\eta +2\right)}^{-k}+\left(\eta +3\right)\left(\eta +2\right){\left(\eta +1\right)}^{-k}\right]\end{array}$ (22)

${\lambda }_{1}^{\eta }=\xi +\alpha \left[\epsilon +\mathrm{ln}\left(\eta +1\right)\right]$ (23)

${\lambda }_{2}^{\eta }=\frac{\left(\eta +2\right)\alpha }{2!}\left[\mathrm{ln}\left(\eta +2\right)-\mathrm{ln}\left(\eta +1\right)\right]$ (24)

${\lambda }_{3}^{\eta }=\frac{\left(\eta +3\right)\alpha }{3!}\left[\left(\eta +4\right)\mathrm{ln}\left(\eta +3\right)-2\left(\eta +3\right)\mathrm{ln}\left(\eta +2\right)+\left(\eta +2\right)\mathrm{ln}\left(\eta +1\right)\right]$ (25)

$\begin{array}{c}{\lambda }_{4}^{\eta }=\frac{\left(\eta +4\right)\alpha }{4!}\left[\left(\eta +6\right)\left(\eta +5\right)\mathrm{ln}\left(\eta +4\right)-3\left(\eta +5\right)\left(\eta +4\right)\mathrm{ln}\left(\eta +3\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3\left(\eta +4\right)\left(\eta +3\right)\mathrm{ln}\left(\eta +2\right)-\left(\eta +3\right)\left(\eta +2\right)\mathrm{ln}\left(\eta +1\right)\right]\end{array}$ (26)

$k={a}_{0}+{a}_{1}\left[{\tau }_{3}^{\eta }\right]+{a}_{2}{\left[{\tau }_{3}^{\eta }\right]}^{2}+{a}_{3}{\left[{\tau }_{3}^{\eta }\right]}^{3}{a}_{4}{\left[{\tau }_{3}^{\eta }\right]}^{4}$ (27)

$\stackrel{^}{\alpha }=\frac{{\lambda }_{2}^{\eta }×\stackrel{^}{k}×2!}{\Gamma \left(1+\stackrel{^}{k}\right)\left(\eta +2\right)\left(-{\left(\eta +2\right)}^{-\stackrel{^}{k}}+{\left(1+\eta \right)}^{-\stackrel{^}{k}}\right)}$ (28)

$\stackrel{^}{\xi }={\stackrel{^}{\lambda }}_{1}^{\eta }-\frac{\stackrel{^}{\alpha }}{\stackrel{^}{k}}\left(1-\Gamma \left(1+\stackrel{^}{k}\right){\left(\eta +1\right)}^{-\stackrel{^}{k}}\right)$ (29)

4. 实例应用

4.1. 绘制频率曲线

$P=\frac{m}{n+1}×100$ (30)

Table 1. Coefficients , a 1 , a 2 , a 3 and a 4 for Eq. (27)

Table 2. Comparison of quantile errors using different orders L-Moments

(a) 交口河(b) 张家山 (c) 赵石窖(d) 绥德 (e) 刘家河(f) 张村译 (f) 林家村(g) 神木

Figure 1. Fitting of the GEV distribution to annual maximum flows for different orders L-Moments

4.2. 拟合效果分析

$\delta =\underset{i=i|P=50%}{\overset{i|P=98%}{\sum }}{\left(\frac{{x}_{i}-{\stackrel{^}{x}}_{i}}{{x}_{i}}\right)}^{2}$ (31)

5. 结论

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