# Analysis of Flood Risk of Urban Agglomeration Polders Using Multivariate Copula

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}, accounting for more than 60% of the entire drainage area. Due to increasing polders, river network density and network complexity have showed downward trends, and water levels are continuously rising. According to historical data, floods in the main stem of the Qinhuai River kept reaching new records, and flood stages in the Dongshan Station frequently exceeded the flood warning level. Water resources managers are faced with the challenge of increasing floods.

#### 2.2. Methodology

#### 2.2.1. Archimedean Copula

_{d}: (0,1)

^{d}→(0,1) can be defined as:

#### 2.2.2. Dependence and Ranks

- Chi-plot

- K-plot

#### 2.2.3. Goodness of Fit

_{0}are the probability density functions of F and ${F}_{0}$, respectively.

_{0}are the probability density functions of F and ${F}_{0}$, respectively. The distribution function with better goodness of fit is selected by the minimum values of RMSE and AIC.

## 3. Results

#### 3.1. Dependence of Flood Characteristics

#### 3.2. Marginal Distribution

#### 3.3. Joint Distribution

## 4. Discussion

#### 4.1. Impacts on Flood Risks of Polders

^{3}, are calculated and the conditional probability diagrams are shown in Figure 5. Figure 6 presents the corresponding contour based on the diagram of Figure 5. For both probabilities, the scenarios with UAPs showed higher risks. The contour of ‘OR’ conditional probability is denser than that of ‘AND’ conditional probability. The UAP will intensify the flood risk of the whole basin, with ‘OR’ exceedance scenario particularly easy to be influenced.

^{3}/s for a return period of 20 years. Conditional probabilities of peak flow ‘AND/OR’ water level for these design values for different return periods are shown in Table 5. With UAPs, the risk of either peak flow or water level exceeding the flood warning is higher than that of the scenarios without UAPs. As the return period increases, the relative difference between scenarios with UAPs and without UAPs reduces, showing that the impact of UAPs on integrated risk attenuates with increasing flood magnitudes. This may be due to the fact that draining water from the polder area to external rivers is prohibitive when water levels are extremely high in the entire watershed.

#### 4.2. Impacts on Flood Risks of Polder Area

- (a)
- Jurong City Circle only;
- (b)
- Jurong, Qianhancun City Circle combined;
- (c)
- Jurong, Qianhancun, Dongshan City Circle combined; and
- (d)
- Jurong, Qianshancun, Dongshan, Lishui City Circle combined.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Variable | With UAP | Without UAP | ||
---|---|---|---|---|

Spearman | Kendall | Spearman | Kendall | |

V&P | 0.921(4.5 × 10^{−6}) | 0.762(5.8 × 10^{−8}) | 0.877(9.5 × 10^{−7}) | 0.686(2.4 × 10^{−6}) |

V&Z | 0.893(5.1 × 10^{−8}) | 0.73(4.4 × 10^{−6}) | 0.821(3.4 × 10^{−6}) | 0.619(3.2 × 10^{−5}) |

P&Z | 0.974(1.2 × 10^{−13}) | 0.921(6.6 × 10^{−9}) | 0.926(4.7 × 10^{−6}) | 0.8(6.0 × 10^{−9}) |

Statistics | Without UAP | With UAP | ||||
---|---|---|---|---|---|---|

V | P | Z | V | P | Z | |

Mean | 126.03 | 591.21 | 6.99 | 143.99 | 729.01 | 7.84 |

Std. | 112.81 | 350.50 | 1.45 | 118.17 | 419.10 | 1.41 |

Skewness | 1.90 | 0.59 | 0.30 | 1.73 | 0.74 | 0.18 |

Kurtosis | 6.20 | 2.32 | 2.20 | 5.64 | 2.72 | 2.32 |

## References

- Luo, P.P.; Zhou, M.M.; Deng, H.Z.; Lyu, J.; Cao, W.Q.; Takara, K.; Nover, D.; Schladow, S.G. Impact of forest maintenance on water shortages: Hydrologic modeling and effects of climate change. Sci. Total Environ.
**2018**, 615, 1355–1363. [Google Scholar] [CrossRef] [PubMed] - Luo, P.P.; He, B.; Duan, W.; Takara, K.; Nover, D. Impact assessment of rainfall scenarios and land-use change on hydrologic response using synthetic Area IDF curves. J. Flood Risk Manag.
**2018**, 11, S84–S97. [Google Scholar] [CrossRef] - Luo, P.P.; Mu, D.R.; Xue, H.; Ngo-Duc, T.; Dang-Dinh, K.; Takara, K.; Nover, D.; Schladow, G. Flood inundation assessment for the Hanoi Central Area, Vietnam under historical and extreme rainfall conditions. Sci. Rep. Nat.
**2018**, 8, 12623. [Google Scholar] [CrossRef] [PubMed] - Jiao, T. Influence of construction in low-lying region on aquatic environment of city and countermeasures. Jiangsu Environ. Sci. Technol.
**2006**, S2, 121–123. [Google Scholar] - Luo, P.P.; He, B.; Takara, K.; Xiong, Y.E.; Nover, D.; Duan, W.L.; Fukushi, K. Historical assessment of Chinese and Janpanese flood management policies and implications for managing future floods. Environ. Sci. Policy
**2015**, 48, 265–277. [Google Scholar] [CrossRef] - Van Manen, S.E.; Brinkhuis, M. Quantitative flood risk assessment for Polders. Reliab. Eng. Syst. Saf.
**2005**, 90, 229–237. [Google Scholar] [CrossRef] - Gao, Y.Q.; Yuan, Y.; Wang, H.Z.; Schmidt, A.R.; Wang, K.X.; Ye, L. Examining the effects of urban agglomeration polders on flood events in Qinhuai River basin, China with HEC-HMS model. Water Sci. Technol.
**2017**, 75, 2130–2138. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gao, Y.Q.; Yuan, Y.; Wang, H.Z.; Zhang, Z.X.; Ye, L. Analysis of impacts of polders on flood processes in Qinhuai River Basin, China, using the HEC-RAS model. Water Sci. Technol. Water Supply
**2018**, 18, 1852–1860. [Google Scholar] [CrossRef] - Xing, W.B.; Xu, W.Y.; Wang, K.; Yan, X. Risk analysis of hydrological failures of levees in external Qinhuai River. J. Hohai Univ. Nat. Sci.
**2006**, 3, 262–266. [Google Scholar] - Xu, J.X.; Yao, S.P. A method to determine the mode of wipe off waterlogging on dyke. J. Agric. Mech. Res.
**2008**, 6, 61–63. [Google Scholar] - Zhao, G.F.; Jiang, Z.R.; Ding, Y.; Liang, G.Q. Assessment on risk analysis and risk evaluation of levees. China Water Transp. (Second Semimonthly)
**2010**, 11, 182–184. [Google Scholar] - Zhang, G.F. Polder construction and its ecological and social effects—An investigation of joint river embankment in Taihu Drainage Area (1950s–1970s). J. Minzu Univ. China (Philos. Soc. Sci. Edit.)
**2012**, 4, 36–41. [Google Scholar] - Xu, H.; Yang, S.J. Exploring the evolution of river networks in plain polders of Taihu Lake basin. Adv. Water Sci.
**2013**, 3, 366–371. [Google Scholar] - Yuan, Y.; Gao, Y.Q.; Wu, X. Flood simulation of flood control model for polder type based on HEC-HMS Hydrological Model in Qinhuai River Basin. J. China Three Gorges Univ. (Nat. Sci.)
**2015**, 5, 34–39. [Google Scholar] - Chen, H.R.; Wang, S.L.; Han, S.J. Assessment of field waterlogging risk in Lixiahe Plain Lake Region, Jiangsu Province: Case study from Yundong Plain in Gaoyou. J. Drain. Irrig. Mach. Eng.
**2017**, 10, 887–896. [Google Scholar] - Reddy, M.J.; Ganguli, P. Bivariate Flood Frequency Analysis of Upper Godavari River Flows Using Archimedean Copulas. Water Resour. Manag.
**2012**, 26, 3995–4018. [Google Scholar] [CrossRef] - Ganguli, P.; Reddy, M.J. Probabilistic assessment of flood risks using trivariate copulas. Theor. Appl. Climatol.
**2013**, 111, 341–360. [Google Scholar] [CrossRef] - Clayton, D.G. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika
**1978**, 65, 141–151. [Google Scholar] [CrossRef] - Frank, M.J. On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequationes Math
**1979**, 19, 194–226. [Google Scholar] [CrossRef] - Genest, C. Frank’s family of bivariate distributions. Biometrika
**1987**, 74, 549–555. [Google Scholar] [CrossRef] - Chowdhary, H.; Escobar, L.A.; Singh, V.P. Identification of suitable copulas for bivariate frequency analysis of flood peak and flood volume data. Hydrol. Res.
**2011**, 42, 193. [Google Scholar] [CrossRef] - Genest, C.; Favre, A. Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng.
**2007**, 12, 347–368. [Google Scholar] [CrossRef] - Klein, B.; Pahlow, M.; Hundecha, Y.; Schumann, A. Probability analysis of hydrological loads for the design of flood control systems using copulas. J. Hydrol. Eng.
**2010**, 15, 360–369. [Google Scholar] [CrossRef] - Fisher, N.I.; Switzer, P. Graphical assessment of dependence: Is a picture worth 100 tests? Am. Stat.
**2001**, 55, 233–239. [Google Scholar] [CrossRef] - Genest, C.; Boies, J.C. Detecting dependence with Kendall plots. Am. Stat.
**2003**, 57, 275–284. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Multimodel inference—Understanding AIC and BIC in model selection. Sociol. Methods Res.
**2004**, 22, 261–304. [Google Scholar] [CrossRef] - Suriya, S.; Mudgal, B.V. Impact of urbanization on flooding: The Thirusoolam sub watershed—A case study. J. Hydrol.
**2012**, 412, 210–219. [Google Scholar] [CrossRef] - Nanjing Urban Flood Control Planning Report. Available online: http://max.book118.com/html/2017/1105/139066558.shtm (accessed on 10 1 2018).

**Figure 3.**Chi-plots and K-plots of pair-wise flood characteristics for scenarios with and without polders: (

**a**) Graphical representation of strength of dependence using chi-plot (upper row) and K-plot (lower row), the pair-wise variables are V&P, V&Z and P&Z from left to right in order without UAPs; (

**b**) graphical representation of strength of dependence using chi-plot (upper row) and K-plot (lower row), the pair-wise variables are V&P, V&Z and P&Z from left to right in order with UAPs.

**Figure 4.**The correlation of empirical and theoretical frequency of marginal distribution: (

**a**) The marginal fitting plots of the flood variables without UAPs; (

**b**) the marginal fitting plots of the flood variables with UAPs.

**Figure 5.**Two conditional probabilities of flood volume <300 m

^{3}: (

**a**) ‘OR’ conditional probability of flood volume <300 m

^{3}; (

**b**) ‘AND’ conditional probability of flood volume <300 m

^{3}.

**Figure 6.**Two conditional probability contours of flood volume <300 m

^{3}: (

**a**) ‘OR’ conditional probability contour of flood volume <300 m

^{3}; (

**b**) ‘AND’ conditional probability contour of flood volume <300 m

^{3}.

**Figure 9.**The map of polder distribution: (

**a**) Jurong City Circle only; (

**b**) Jurong City Circle in the upper reach and Qianhancun City Circle in the middle reach; (

**c**) Jurong City Circle in the upper reach, Qianhancun City Circle in the middle reach, and Dongshan City Circle in the lower reach; (

**d**) Jurong and Lishui City Circle in the upper reach, Qianhancun City Circle in the middle reach, and Dongshan City Circle in the lower reach.

Copula | Trivariate Coupla Function |
---|---|

GH | $C\left({u}_{1},{u}_{2},{u}_{3}\right)=\mathrm{exp}\left(-{\left[{\left(-\mathrm{ln}\left({u}_{1}\right)\right)}^{\mathsf{\theta}}+{\left(-\mathrm{ln}\left({u}_{2}\right)\right)}^{\mathsf{\theta}}+{\left(-\mathrm{ln}\left({u}_{3}\right)\right)}^{\mathsf{\theta}}\right]}^{\frac{1}{\mathsf{\theta}}}\right)\mathsf{\theta}\in \left(0,\infty \right)$ |

Clayton | $C\left({u}_{1},{u}_{2},{u}_{3}\right)={\left({u}_{1}{}^{-\mathsf{\theta}}+{u}_{2}{}^{-\mathsf{\theta}}+{u}_{3}{}^{-\mathsf{\theta}}-2\right)}^{-1/\mathsf{\theta}}\mathsf{\theta}\in \left(1,\infty \right)$ |

Frank | $C\left({u}_{1},{u}_{2},{u}_{3}\right)=-\frac{1}{\mathsf{\theta}}\mathrm{ln}\left(1+\frac{(\mathrm{exp}\left(-\mathsf{\theta}{u}_{1}\right)-1)(\mathrm{exp}\left(-\mathsf{\theta}{u}_{2}\right)-1)(\mathrm{exp}\left(-\mathsf{\theta}{u}_{3}\right)-1)}{(\mathrm{exp}\left(-\mathsf{\theta}\right)-1)}\right)$ |

Without UAPs (Urban Agglomeration Polders) | With UAPs (Urban Agglomeration Polders) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Location | Scale | Shape | KS | P | Location | Scale | Shape | KS | P | ||

V | P-Ш | 28.38 | 0.004 | 0.44 | 0.23 | 0.20 | 14.40 | 0.006 | 0.83 | 0.20 | 0.34 |

GEV | 72.60 | 53.86 | 0.31 | 0.12 | 0.88 | 87.17 | 61.07 | 0.28 | 0.12 | 0.86 | |

LN | 4.51 | 0.87 | - | 0.14 | 0.79 | 4.68 | 0.80 | - | 0.14 | 0.79 | |

P | P-Ш | −247.93 | 0.006 | 4.63 | 0.15 | 0.71 | 32.16 | 0.003 | 2.16 | 0.16 | 0.56 |

GEV | 424.66 | 263.84 | 0.05 | 0.16 | 0.61 | 526.58 | 301.26 | 0.09 | 0.19 | 0.39 | |

LN | 6.18 | 0.70 | - | 0.14 | 0.73 | 6.42 | 0.63 | - | 0.15 | 0.66 | |

Z | P-Ш | 0.56 | 2.49 | 16.00 | 0.14 | 0.79 | −0.21 | 3.63 | 29.22 | 0.16 | 0.63 |

GEV | 6.42 | 1.29 | −0.17 | 0.16 | 0.63 | 7.33 | 1.33 | −0.24 | 0.17 | 0.49 | |

LN | 1.92 | 0.21 | - | 0.16 | 0.63 | 2.04 | 0.18 | - | 0.17 | 0.54 |

Copula | Without UAPs | With UAPs | ||||
---|---|---|---|---|---|---|

Theta | RMSE | AIC | Theta | RMSE | AIC | |

GH | 3.680 | 0.015 | −111.45 | 3.149 | 0.015 | −111.42 |

Clayton | 2.773 | 0.022 | −94.58 | 2.463 | 0.018 | −102.31 |

Frank | 15.905 | 0.013 | −117.43 | 14.040 | 0.014 | −113.36 |

JRP (/a) | Without UAPs | With UAPs | ||||
---|---|---|---|---|---|---|

V (m^{3}) | P (m^{3}/s) | Z (m) | V (m^{3}) | P (m^{3}/s) | Z (m) | |

10 | 331.03 | 1514.57 | 9.80 | 370.81 | 1738.81 | 10.40 |

20 | 473.94 | 2010.97 | 10.63 | 514.92 | 2232.61 | 11.13 |

50 | 702.69 | 2683.57 | 11.51 | 738.23 | 2884.44 | 11.89 |

100 | 914.87 | 3216.57 | 12.05 | 940.25 | 3392.11 | 12.35 |

200 | 1172.74 | 3783.76 | 12.55 | 1180.64 | 3925.00 | 12.74 |

**Table 5.**The conditional probability of peak flow and water level at different return periods of flood volume.

Flood Volume Return Period | ‘OR’ Exceedance Probability | ‘AND’ Exceedance Probability | ||||
---|---|---|---|---|---|---|

No UAPs | UAPs | Δ (%) | No UAPs | UAPs | Δ (%) | |

10 | 0.0224 | 0.0425 | 89.91 | 0.0113 | 0.0234 | 28.67 |

20 | 0.0401 | 0.0667 | 66.27 | 0.0107 | 0.0222 | 17.31 |

50 | 0.0545 | 0.0842 | 54.36 | 0.0103 | 0.0215 | 13.29 |

100 | 0.0600 | 0.0904 | 50.83 | 0.0102 | 0.0213 | 12.25 |

200 | 0.0628 | 0.0936 | 49.14 | 0.0102 | 0.0212 | 11.77 |

Average | - | - | 62.10 | - | - | 16.66 |

Flood No. | Scenario (a) | Scenario (b) | ||||||

Flood Volume (m^{3}) | Peak Flow (m^{3}/s) | Water Level (m) | Integrated Risk | Flood Volume (m^{3}) | Peak Flow (m^{3}/s) | Water Level (m) | Integrated Risk | |

1989 | 0.6636 | 0.8256 | 0.8077 | 0.6510 | 0.6796 | 0.8492 | 0.8140 | 0.6682 |

1987 | 0.8582 | 0.8372 | 0.8405 | 0.7934 | 0.8642 | 0.8449 | 0.8664 | 0.8102 |

1991 | 0.9603 | 0.9773 | 0.9731 | 0.9565 | 0.9644 | 0.9807 | 0.9828 | 0.9626 |

Flood No. | Scenario (c) | Scenario (d) | ||||||

Flood Volume (m^{3}) | Peak Flow (m^{3}/s) | Water Level (m) | Integrated Risk | Flood Volume (m^{3}) | Peak Flow (m^{3}/s) | Water Level (m) | Integrated Risk | |

1989 | 06990 | 0.8662 | 0.8652 | 0.6933 | 0.7140 | 0.8910 | 0.9515 | 0.7123 |

1987 | 0.8715 | 0.8579 | 0.8937 | 0.8290 | 0.8773 | 0.8743 | 0.9081 | 0.8446 |

1991 | 0.9632 | 0.9847 | 0.9838 | 0.9622 | 0.9645 | 0.9877 | 0.9937 | 0.9642 |

Scenario | (1) Ratio of Area Protected by Polders | (2) Integrated Risk | (1) × (2) | ||||
---|---|---|---|---|---|---|---|

1989 | 1987 | 1991 | 1989 | 1987 | 1991 | ||

a | 0.13 | 0.651 | 0.7934 | 0.9565 | 0.56 | 0.69 | 0.83 |

b | 0.23 | 0.6682 | 0.8102 | 0.9626 | 0.52 | 0.63 | 0.74 |

c | 0.34 | 0.6933 | 0.829 | 0.9622 | 0.46 | 0.55 | 0.64 |

d | 0.45 | 0.7123 | 0.8446 | 0.9642 | 0.39 | 0.47 | 0.53 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gao, Y.; Wang, D.; Zhang, Z.; Ma, Z.; Guo, Z.; Ye, L.
Analysis of Flood Risk of Urban Agglomeration Polders Using Multivariate Copula. *Water* **2018**, *10*, 1470.
https://doi.org/10.3390/w10101470

**AMA Style**

Gao Y, Wang D, Zhang Z, Ma Z, Guo Z, Ye L.
Analysis of Flood Risk of Urban Agglomeration Polders Using Multivariate Copula. *Water*. 2018; 10(10):1470.
https://doi.org/10.3390/w10101470

**Chicago/Turabian Style**

Gao, Yuqin, Dongdong Wang, Zhenxing Zhang, Zhenzhen Ma, Zichen Guo, and Liu Ye.
2018. "Analysis of Flood Risk of Urban Agglomeration Polders Using Multivariate Copula" *Water* 10, no. 10: 1470.
https://doi.org/10.3390/w10101470