湖泊科学   2015, Vol. 27 Issue (2): 352-360. 0

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HUANG Qiang, CHEN Zishen. Multivariate flood risk assessment based on the secondary return period. Journal of Lake Sciences, 2015, 27(2): 352-360.
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2014-02-27 收稿
2014-08-11 收修改稿

### 码上扫一扫

(中山大学水资源与环境系，广州 510275)

Multivariate flood risk assessment based on the secondary return period
HUANG Qiang , CHEN Zishen
(Department of Water Resources and Environment, Sun Yat-sen University, Guangzhou 510275, P.R.China)
Abstract: Frequency analysis is a useful tool for flood risk assessment, but the definition and calculation of return periods and design values in a multivariate framework are difficult tasks. In this paper, by constructing the joint distribution of flood duration, peak discharge and volume, we introduced the definitions of multivariate "OR", "AND" and secondary return periods and discuss their differences in identifying the safe and dangerous regions in a critical level. The rationality and reliability of information provided from them in flood risk management and engineering design were then analyzed, respectively. The traditional "OR-AND" approach in multivariate return periods definition is limited in the identification of safe and dangerous regions, while the secondary return period based on the Kendall measure would be more rational. Consequently, mistakes in the recognition of safe and dangerous event can be avoided by the use of second return period, which, obviously, is better for the flood risk management. For a given second return period, the combination of flood duration, peak discharge and volume design values calculated from the most-likely approach, may have an advantage to meet the requirement that with low cost to bear greater risks in engineering design. Furthermore, the information provided from multivariate design values would be more considerate and reliable than the univariate ones.
Keywords: Multivariate flood characteristics    extreme value distribution    safe and dangerous regions    Kendall function    secondary return period    multivariate design values

1 洪水特征属性

 $D={{t}_{\text{e}}}-{{t}_{\text{s}}}$ (1)
 $V=\left[\sum\limits_{i={{t}_{\text{s}}}}^{{{t}_{\text{e}}}}{{{q}_{i}}-\frac{1}{2}}\left( {{q}_{{{t}_{\text{s}}}}}+{{q}_{{{t}_{\text{e}}}}} \right) \right]-\frac{1}{2}D\left( {{q}_{{{t}_{\text{s}}}}}+{{q}_{{{t}_{\text{e}}}}} \right)$ (2)
 图 1 洪水过程及相应的特征属性 Fig.1 The flood hydrograph and characteristics

2 边缘分布与联合分布

 F(x)=\left\{ \begin{align} &\text{exp}\left\{-{{\left[1-k\left( \frac{x-\mu }{\sigma } \right) \right]}^{\frac{1}{k}}} \right\}, k\ne 0 \\ &\text{exp}\left[-\text{exp}\left( \frac{x-\mu }{\sigma } \right) \right], k=0 \\ \end{align} \right. (3)

 $C({{\mathit{u}}_{1}}, \cdots, {{\mathit{u}}_{j}})=\varphi _{\theta }^{[-1]}\left[{{\varphi }_{\theta }}({{\mathit{u}}_{\text{1}}})+\cdots +{{\varphi }_{\theta }}({{\mathit{u}}_{j}}) \right]$ (4)

 $C({u}_1^t, \cdots ,{u}_j^t) = {\left[ {C({{u}_1}, \cdots ,{{u}_j})} \right]^t}$ (5)

 $C({u_1}, \cdots ,{u_j}) = {{e}^{ - {{[{{({\rm{ln}}{{u}_1})}^\theta } + \cdots + {{( - {\rm{ln}}{{u}_j})}^\theta }]}^{\frac{1}{\theta }}}}}$ (6)

3 多变量重现期与分位值 3.1 传统多变量重现期

 ${{T}^{\vee }}=\frac{1}{P(\mathit{D>}{{\mathit{d}}^{*}}\vee \mathit{Q>}{{\mathit{q}}^{*}}\vee \mathit{V>}{{\mathit{v}}^{*}})}=\frac{1}{1-C({{\mathit{u}}_{d}}, {{\mathit{u}}_{q}}, {{\mathit{u}}_{v}})}$ (7)
 ${{T}^{\wedge }}=\frac{1}{P(\mathit{D>}{{\mathit{d}}^{*}}\wedge Q>{{q}^{*}}\wedge V>{{v}^{*}})}=\frac{1}{\hat{C}(1-{{u}_{d}}, 1-{{\mathit{u}}_{q}}, 1-{{\mathit{u}}_{v}})}$ (8)

 图 2 “或”重现期(a)和“且”重现期(b)的安全与危险域识别(灰色部分为危险域) Fig.2 Identification of safe and dangerous regions for "OR" return period(a) and "AND" return period(b) (the grey part represents the dangerous one)

 图 3 “或”重现期(a)和“且”重现期(b)对安全危险域识别的局限性 Fig.3 Limitations on the identification of safe and dangerous regions for "OR" return period(a) and "AND" return period(b)
3.2 分位值曲面

3.3 二次重现期

 ${{T}_{\text{K}}}=\frac{1}{P[\mathit{C}\text{(}{{\mathit{u}}_{d}}\text{, }{{\mathit{u}}_{q}}\text{, }{{\mathit{u}}_{v}}\text{)}>\mathit{p}]}=\frac{1}{1-{{K}_{\text{C}}}(\mathit{p})}$ (9)
 图 4 二次重现期的安全与危险域识别(灰色部分为危险域) Fig.4 Identification of safe and dangerous regions for secondary return period (the grey part represents the dangerous one)

TK称为二次重现期(Secondary return period)或Kendall重现期，KC为Kendall分布函数[16]

 ${{K}_{\text{C}}}(\mathit{p})=p+\sum\limits_{i=1}^{k-1}{\frac{{{(-1)}^{k}}}{i!}{{[\varphi (\mathit{p})]}^{k}}{{h}_{i}}(p)}$ (10)
 ${h_i}({p}) = \left\{ \begin{array}{l} \frac{{\partial p}}{{\partial \varphi (p)}},i = 1\\ {h_1}(p)\frac{{\partial {h_{i - 1}}({p})}}{{\partial p}},i \ge 2 \end{array} \right.$ (11)

 ${{K}_{\text{C}}}(p)=p-\frac{p(3\theta-\text{ln}\mathit{p}\text{-1})\text{ln}\mathit{p}}{2{{\theta }^{2}}}$ (12)

3.4 多变量设计值

 $({{d}_{ml}}, {{q}_{ml}}, {{v}_{ml}})=\underset{(d, q, v)\in \mathit{S}_{p}^{\vee }}{\mathop{\text{arg}\ \text{max}\mathit{f}\text{(}\mathit{d, q, v}\text{)}}}\,$ (13)
 $f(d, q, v)=\mathit{c}\text{(}{{\mathit{u}}_{d}}, {{u}_{q}}, {{u}_{v}}\text{)}{{\mathit{f}}_{d}}\text{(}\mathit{d}\text{)}{{\mathit{f}}_{q}}\text{(}\mathit{q}\text{)}{{\mathit{f}}_{v}}\text{(}\mathit{v}\text{)}$ (14)

(1) 设定一个二次重现期Tp，通过公式(9)和(12)推求出pKC[－1](1－1/Tp)；

(2) 利用式(4)，计算所有使C(uduquv)＝p成立的概率组合(uduquv)；

(3) 由式(15)计算出使f(dqv)达到最大值的一组(uduquv)；

(4) 最后分别再根据边缘分布的反函数推求出历时、洪峰和洪量的分位值(dmlqmlvml)，dmlF－1(ud)、qmlF－1(uq)、vmlF－1(uv).

4 案例研究

 图 5 GEV对洪水历时、洪峰和洪量样本的拟合优度 Fig.5 The goodness of fit of GEV for flood duration, peak and volume
 图 6 Gumbel-Hougaard copula对洪水历时、洪峰和洪量样本的拟合优度 Fig.6 The goodness of fit of Gumbel-Hougaard copula for flood duration, peak and volume

5 结论

1) 传统的“或”和“且”多变量重现期对安全与危险域的识别具有局限性，重现期与危险域范围大小的矛盾会造成对安全事件与危险事件的错误识别.

2) 根据C(uduquv)＝p临界条件并利用Kendall函数定义的二次重现期，具有同一临界水平下任意历时、洪峰与洪量组合对安全与危险域识别的一致性，重现期越大，对应的危险域则越小，避免了对安全事件与危险事件的错误识别，更有利于指导洪水风险的管理.

3) 给定二次重现期条件下的洪水事件可以具有不同的历时、洪峰与洪量组合，在工程设计中，与每个特征属性同时取较大值相比，依据出现概率最大的原则推算的设计值组合可以满足以较低成本承受较大风险的追求.相比于单变量设计值，考虑了洪水多个属性联合特征的多变量设计值为防洪工程设计提供了更加全面和可靠的参考信息.

6 参考文献

 [1] 郭生练, 闫宝伟, 肖义等. Copula函数在多变量水文分析计算中的应用及研究进展. 水文, 2008, 28(3): 1-6. [2] Zhang L, Singh VP. Bivariate flood frequency analysis using the copula method. Journal of Hydrologic Engineering, 2006, 11(2): 150-164. DOI:10.1061/(ASCE)1084-0699(2006)11:2(150) [3] Zhang L, Singh VP. Trivariate flood frequency analysis using the Gumbel-Hougaard copula. Journal of Hydrologic Engineering, 2007, 12(4): 431-439. DOI:10.1061/(ASCE)1084-0699(2007)12:4(431) [4] 侯芸芸, 宋松柏, 赵丽娜等. 基于Copula函数的3变量洪水频率研究. 西北农林科技大学学报:自然科学版, 2010, 38(2): 219-228. [5] Shiau JT. Fitting drought duration and severity with two-dimensional copulas. Water Resources Management, 2006, 20: 795-815. DOI:10.1007/s11269-005-9008-9 [6] Salvadori G, De Michele C, Durante F. On the return period and design in a multivariate framework. Hydrology and Earth System Sciences, 2011, 15: 3293-3305. DOI:10.5194/hess-15-3293-2011 [7] Corbella S, Stretch DD. Multivariate return periods of sea storms for coastal erosion risk assessment. Natural Hazards and Earth System Sciences, 2012, 12: 2699-2708. DOI:10.5194/nhess-12-2699-2012 [8] Salvadori G, Tomasicchio GR, Alessandro FD. Multivariate approach to design coastal and off-shore structures. Journal of Coastal Research, 2013, 65: 386-391. DOI:10.2112/SI65-066.1 [9] Yue S, Ouarda TBMJ, Bobée B et al. The Gumbel mixed model for flood frequency analysis. Journal of Hydrology, 1999, 226(1): 88-100. [10] Salvadori G, de Michele C, Kottegoda N et al. Extremes in Nature: An approach using copulas. Dordrecht: Springer, 2007, 15-30. [11] Favre AC, Adlouni SE, Perreault L et al. Multivariate hydrological frequency analysis using copulas. Water Resources Research, 2004, 40: W01101. DOI:10.1029/2003WR002456 [12] Nelson RB. An introduction to copulas: Second edition. New York: Springer, 2006, 115-143. [13] Salvadori G, de Michele C. Multivariate multiparameter extreme value models and return periods: A copula approach. Water Resource Research, 2010, 46: W10501. DOI:10.1029/2009WR009040 [14] Salvadori G, de Michele C. Frequency analysis via copulas: Theoretical aspects and applications to hydrological events. Water Resources Research, 2004, 40: W12511. DOI:10.1029/2004WR003133 [15] Chebana F, Ouarda TBMJ. Multivariate quantiles in hydrological frequency analysis. Environmetrics, 2011, 22: 63-78. DOI:10.1002/env.v22.1 [16] Genest C, Quessy JF, Rémillard B. Goodness-of-fit procedures for copula model based on the probability integral transformation. Scandinavian Journal of Statistics, 2006, 33(2): 337-366. DOI:10.1111/sjos.2006.33.issue-2 [17] Graler B, van den Berg MJ, Vandenberghe S et al. Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation. Hydrology and Earth System Sciences, 2013, 17: 1281-1296. DOI:10.5194/hess-17-1281-2013