﻿ 暴雨极值分析 Extreme Value Analysis of Rainstorm

Climate Change Research Letters
Vol. 08  No. 02 ( 2019 ), Article ID: 29068 , 8 pages
10.12677/CCRL.2019.82018

Extreme Value Analysis of Rainstorm

Wenpeng Zhao, Yonglai Zheng*, Yubao Zhou

College of Civil Engineering, Tongji University, Shanghai

Received: Feb. 8th, 2019; accepted: Feb. 21st, 2019; published: Feb. 28th, 2019

ABSTRACT

In the context of environmental processes, non-stationarity of precipitation is often apparent because of seasonal effects like monsoon, typhoon, etc. Because non-stationarity violates the assumption of extreme models, this paper adopts the seasonal models that divide the rainfall data into four seasons and are examined by likelihood test. GEV is employed to analyze the monthly maximum rainfall. In addition, degree of experience is used to estimate the maximum return period, that is, how large the extrapolation limit is. The result shows that based on the 90-month rainfall data, the summer rainfall is subject to the Gumbel distribution with an inaccurate extrapolation, while the maximum extrapolation periods in spring, autumn and winter are 54.2, 68.6 and 84.6 months respectively. Therefore, it is recommended that degree of experience could be applied to engineering because of its reliability and practicability.

Keywords:Extreme Analysis, Seasonal Models, Degree of Experience, Return Period, Confidence Interval

1. 引言

2. 研究方法

2.1. 泊松分布与极值分布

${F}_{n}\left(y\right)={p}_{n}\left(0\right)=\frac{{\left\{{\lambda }_{n}\left(y\right)\right\}}^{0}}{0!}{\text{e}}^{-{\lambda }_{n}\left(y\right)}$ (1)

${\lambda }_{n}\left(y\right)=\left\{\begin{array}{cc}-{\left(1+\xi \frac{y-{\mu }_{n}}{{\sigma }_{n}}\right)}^{-\frac{1}{\xi }}& \xi \ne 0\\ \mathrm{exp}\left(-\frac{y-{\mu }_{n}}{{\sigma }_{n}}\right)& \xi =0\end{array}$ (2)

${H}_{u}\left(y\right)=\frac{\lambda \left(y,{\theta }_{N}\right)}{\lambda \left(u,{\theta }_{N}\right)}=\begin{array}{cc}\frac{\lambda \left(y,{\theta }_{1}\right)}{\lambda \left(u,{\theta }_{1}\right)}& y>u\end{array}$ (3)

2.2. Delta估计

$\stackrel{^}{\varphi }\sim N\left(\varphi ,{V}_{\varphi }\right)$ (4)

${V}_{\varphi }=\nabla {\varphi }^{T}{V}_{\left(\mu ,\sigma ,\xi \right)}\nabla \varphi$ (5)

$\nabla \varphi ={\left[\frac{\partial \varphi }{\partial \mu },\frac{\partial \varphi }{\partial \sigma },\frac{\partial \varphi }{\partial \xi }\right]}^{T}$ (6)

${V}_{\left(\mu ,\sigma ,\xi \right)}$$\left(\mu ,\sigma ,\xi \right)$ 的方差协方差矩阵，计算时均以估计值代入，此方法即为delta法。

2.3. 季度模型

${Z}_{t}\sim \text{GEV}\left(\mu \left(t\right),\sigma \left(t\right),\xi \left(t\right)\right)$ (7)

${Z}_{t}\sim \text{GEV}\left(\mu \left(t\right),\sigma ,\xi \right)$ (8)

2.4. 似然比检验

2.5. 模型检测

$\stackrel{˜}{{Z}_{t}}=\frac{1}{\stackrel{^}{\xi }\left(t\right)}\mathrm{log}\left\{1+\stackrel{^}{\xi }\left(t\right)\left(\frac{{Z}_{t}-\stackrel{^}{\mu }\left(t\right)}{\stackrel{^}{\sigma }\left(t\right)}\right)\right\}$ (9)

$\left\{i/\left(m+1\right),\mathrm{exp}\left(-\mathrm{exp}\left(-{\stackrel{˜}{z}}_{\left(i\right)}\right)\right);i=1,\cdots ,m\right\}$ (10)

$\left\{{\stackrel{˜}{z}}_{\left(i\right)},-\mathrm{log}\left(-\mathrm{log}\left(i/\left(m+1\right)\right)\right);i=1,\cdots ,m\right\}$ (11)

2.6. 经验度试算

$K=\frac{{E}^{2}\left(\lambda \right)}{V\left(\lambda \right)}$ (12)

3. 实例应用

3.1. 模型拟合

Figure 1. Maximum daily rainfall of every month in different seasons

${\mu }_{春}={\mu }_{冬}$ (9)

${\mu }_{夏}={\mu }_{秋}$ (10)

Table 1. Estimation of seasonal model parameters

$2\left(-1227.37-\left(-1273.54\right)\right)>qchisq\left(0.95,2\right)$ (11)

(a) 概率图 (b) 分位数图

Figure 2. Model check plot

3.2. 经验度计算

(a) (b) (c) (d)

Figure 3. Degree of experience curve

(a) 春 (b) 夏 (c) 秋 (d) 冬

Figure 4. Return level curve

4. 结论

Extreme Value Analysis of Rainstorm[J]. 气候变化研究快报, 2019, 08(02): 160-167. https://doi.org/10.12677/CCRL.2019.82018

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10. NOTES

*通讯作者。