# 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 |

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**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 |

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## 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