# Multivariate Flood Risk Analysis at a Watershed Scale Considering Climatic Factors

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}. The average annual precipitation of the basin is 1031 mm, and the precipitation is unevenly distributed over a year. The rainfall from April to September accounts for 70.6% of the total annual precipitation. The Qinhuai River has two headwaters, that is, the Qinshui River and the Jurong River. The Qinhuai River, with a length of 34 km, starts in a northwestern village of Jiangning District where the Qinshui River meets the Jurong River. The main channel of the Qinhuai River is divided into two branches in Dongshan Town, Jiangning District. The north branch passes through the Wudingmen Gate and the west branch flows into the Yangtze River via the Qinhuaixinhe Gate.

#### 2.2. Methodology

#### 2.2.1. Definition of flood risk

#### 2.2.2. Archimedean Copula

_{U}, the hazardous event E

_{U}indicates that one of the variables exceeds the set threshold (Q

_{0}or V

_{0}), and the joint recurrence period can be represented by the Copula function as:

_{∩}, the danger event E

_{∩}denotes that both of the variables exceed the set threshold. The Copula function can be used to represent the co-occurrence recurrence period as:

#### 2.2.3. Parameter Estimation and Goodness-of-fit Test

_{emp}and C are the empirical and theoretical cumulative probability, respectively; and n is the sample size. The smaller the RMSE value, the better the fitting.

#### 2.2.4. Scenario Hypothesis

#### 2.2.5. Climate Scenarios

## 3. Results

#### 3.1. Flood Response Analysis Under Different Climate Scenarios

#### 3.2. Hydrological Response of Different Climate Scenarios

#### 3.2.1. Analysis of the Impact of Temperature Changes on Floods

#### 3.2.2. Analysis of the Impact of Rainfall Changes on Floods

#### 3.2.3. Analysis of the Combined Effects of Temperature and Rainfall

#### 3.3. Copula Function Fitting

## 4. Discussion

#### 4.1. Multivariate Flood Risk Analysis

#### 4.1.1. Flood Risk Analysis Under Current Situation

^{3}) is 0.1121, 0.1579, and 0.1837, respectively. Given a certain flood threshold, the smaller the magnitude of the flood peak, the smaller the flood exceedance probability. When the flood peak is small, the flood volume is less likely to exceed the threshold.

#### 4.1.2. Flood Risk Analysis Considering Climate Change Impact

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Flood | Relative Flood Change (%) | ||||||
---|---|---|---|---|---|---|---|

Rank | Number | T-2P0 | T2P0 | T5P0 | |||

Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | ||

light | 20040618 | 15.43 | 16.75 | −17.15 | −19.11 | −20.69 | −24.06 |

medium | 19870701 | 2.40 | 5.65 | −0.16 | −2.48 | −1.91 | −7.45 |

heavy | 19910630 | 0.25 | 1.41 | −0.89 | −1.55 | −1.59 | −2.98 |

average | 6.03 | 7.94 | −6.07 | −7.71 | −8.06 | −11.50 |

Floods | Relative Flood Change (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Rank | Number | T0P2 | T0P4 | T0P8 | T0P14 | ||||

Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | ||

light | 20040618 | 4.33 | 4.62 | 8.89 | 9.41 | 18.43 | 19.47 | 33.11 | 35.16 |

medium | 19870701 | 4.23 | 3.82 | 8.66 | 7.64 | 17.50 | 15.27 | 30.02 | 26.80 |

heavy | 19910630 | 2.74 | 2.72 | 5.41 | 5.48 | 11.01 | 11.20 | 19.34 | 24.62 |

average | 3.76 | 3.72 | 7.65 | 7.51 | 15.64 | 15.31 | 27.49 | 28.86 |

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**Figure 1.**The hydrological response of temperatures changes for different-scaled floods: (

**a**) Relative change of flood peak under temperature changes; (

**b**) Relative change of flood volume under temperature changes.

**Figure 2.**The hydrological response of precipitation changes for different-scaled floods: (

**a**) Relative change of flood peak under precipitation changes; (

**b**) Relative change of flood volume under precipitation changes.

**Figure 3.**The hydrological response of increased temperature and precipitation for different-scaled floods: (

**a**) Relative change of flood peak under increased temperature and precipitation scenario (temperature rises 2 °C); (

**b**) Relative change of flood volume under increased temperature and precipitation scenario (temperature rises 2 °C); (

**c**) Relative change of flood peak under increased temperature and precipitation scenario (temperature rises 5 °C); (

**d**) Relative change of flood volume under increased temperature and precipitation scenario (temperature rises 5 °C).

**Figure 6.**(

**a**) Peak-volume co-occurrence distribution; (

**b**) The contour of Peak-volume co-occurrence period.

Copula | Expression |
---|---|

GH [26,27] | $C\left({u}_{1},{u}_{2}\right)=\mathrm{exp}\left(-{\left[{\left(-\mathrm{ln}\left({u}_{1}\right)\right)}^{\theta}+{\left(-\mathrm{ln}\left({u}_{2}\right)\right)}^{\theta}\right]}^{\frac{1}{\theta}}\right),\theta \in \left(0,\infty \right)$ |

Clayton [28] | $C\left({u}_{1},{u}_{2}\right)={\left({{u}_{1}}^{-\theta}+{{u}_{2}}^{-\theta}-1\right)}^{-1/\mathsf{\theta}},\theta \in \left(1,\infty \right)$ |

Frank [29] | $C\left({u}_{1},{u}_{2}\right)=-\frac{1}{\theta}\mathrm{ln}\left(1+\frac{(\mathrm{exp}\left(-\theta {u}_{1}\right)-1)(\mathrm{exp}\left(-\theta {u}_{2}\right)-1)}{(\mathrm{exp}\left(-\theta \right)-1)}\right)$ |

Distribution | Probability Density Function | Denotes |
---|---|---|

P-III | $f(x)=\frac{{\beta}^{\alpha}}{\Gamma (\alpha )}{(x-{a}_{0})}^{\alpha -1}{e}^{-\beta (x-{a}_{0})}$ where, $\Gamma (y+1)={\displaystyle {\int}_{0}^{\infty}{t}^{y}}{e}^{-t}dt,y+1>0$ | $\alpha $, $\beta $, ${a}_{0}$ are shape scale and location parameter respectively. $\begin{array}{l}{a}_{0}<x<\infty \\ \alpha >0,\beta >0\end{array}$ |

GEV | $f(x)=\frac{1}{a}{[1+k(\frac{x-u}{a})]}^{-1/k-1}\mathrm{exp}\{-{[1+k(\frac{x-u}{a})]}^{-1/k}\}$ | k, a and u are shape scale and location parameter respectively. |

LN | $f(x)=\frac{1}{x{\delta}_{y}\sqrt{2\pi}}\mathrm{exp}\{\frac{-{[\mathrm{ln}x-{\mu}_{y}]}^{2}}{2{\delta}_{y}^{2}}\},x>0$ | ${\mu}_{y}$ and ${\sigma}_{y}$ are the expectation and variance of the logarithm of the original sample |

Copula | Expression | ${\mathit{\tau}}_{\mathit{n}}$ |
---|---|---|

GH | ${\tau}_{n}$ = $1-\frac{1}{\theta}$ | [0, 1] |

Clayton | ${\tau}_{n}$ = $\frac{\theta}{\theta +2}$ | [−1, 1]\{0} |

Frank | ${\tau}_{n}$ = $1+\frac{4}{\theta}\left[\frac{1}{\theta}{\displaystyle {\int}_{0}^{\theta}\frac{t}{{e}^{t}-1}}dt-1\right]$ | [−1, 1]\{0} |

ΔT (°C) | ΔP (%) | ||||
---|---|---|---|---|---|

0 | 2% | 4% | 8% | 14% | |

−2 | T-2P0 | T-2P2 | T-2P4 | T-2P8 | T-2P14 |

0 | T0P0 | T0P2 | T0P4 | T0P8 | T0P14 |

2 | T2P0 | T2P2 | T2P4 | T2P8 | T2P14 |

5 | T5P0 | T5P2 | T5P4 | T5P8 | T5P14 |

Scenario | Heavy (19910630) | Median (19870701) | Light (20040618) | |||
---|---|---|---|---|---|---|

Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | |

T-2P0 | 1240.3 | 12.21 | 831.5 | 6.00 | 843.1 | 2.80 |

T-2P2 | 1274.2 | 12.55 | 866.0 | 6.22 | 871.7 | 2.91 |

T-2P4 | 1308.8 | 12.90 | 901.6 | 6.44 | 900.4 | 3.02 |

T-2P8 | 1377.3 | 13.69 | 974.3 | 6.88 | 958.5 | 3.24 |

T-2P14 | 1578.7 | 16.23 | 1081.2 | 7.56 | 1079.0 | 3.64 |

T0P0 | 1237.2 | 12.04 | 812.0 | 5.68 | 730.4 | 2.40 |

T0P2 | 1271.0 | 12.37 | 846.3 | 5.90 | 762.0 | 2.51 |

T0P4 | 1304.1 | 12.70 | 882.3 | 6.11 | 795.3 | 2.63 |

T0P8 | 1373.3 | 13.39 | 954.1 | 6.55 | 865.0 | 2.87 |

T0P14 | 1476.4 | 15.00 | 1055.7 | 7.20 | 972.2 | 3.24 |

T2P0 | 1226.1 | 11.85 | 810.7 | 5.54 | 605.1 | 1.94 |

T2P2 | 1260.0 | 12.18 | 843.2 | 5.75 | 630.2 | 2.03 |

T2P4 | 1293.0 | 12.50 | 876.0 | 5.96 | 655.4 | 2.13 |

T2P8 | 1360.7 | 13.16 | 942.0 | 6.40 | 711.7 | 2.34 |

T2P14 | 1463.8 | 14.22 | 1041.4 | 7.06 | 795.3 | 2.65 |

T5P0 | 1217.5 | 11.68 | 796.5 | 5.26 | 579.3 | 1.82 |

T5P2 | 1250.5 | 12.01 | 828.0 | 5.46 | 602.9 | 1.90 |

T5P4 | 1284.4 | 12.36 | 860.2 | 5.68 | 626.6 | 1.98 |

T5P8 | 1352.1 | 13.03 | 925.1 | 6.14 | 677.4 | 2.15 |

T5P14 | 1453.6 | 14.01 | 1023.1 | 6.81 | 757.1 | 2.44 |

Scenario | Relative Peak Flow Change (%) | Average | Relative Flood Volume Change (%) | Average | ||||
---|---|---|---|---|---|---|---|---|

Light 20040618 | Medium 19870701 | Heavy 19910630 | Light 20040618 | Medium 19870701 | Heavy 19910630 | |||

T2P2 | −13.7 | 3.8 | 1.80 | −2.70 | −15.30 | 1.20 | 1.20 | −4.30 |

T2P4 | −10.3 | 7.9 | 4.50 | 0.70 | −11.20 | 5.00 | 3.80 | −0.80 |

T2P8 | −2.6 | 16 | 10.00 | 7.80 | −2.60 | 12.70 | 9.30 | 6.47 |

T2P14 | 8.9 | 28.3 | 18.30 | 18.50 | 10.40 | 24.40 | 18.10 | 17.63 |

T5P2 | −17.5 | 2 | 1.10 | −4.80 | −20.80 | −3.80 | −0.20 | −8.27 |

T5P4 | −14.2 | 5.9 | 3.80 | −1.50 | −17.50 | 0.00 | 2.60 | −4.97 |

T5P8 | −7.3 | 13.9 | 9.30 | 5.30 | −10.40 | 8.20 | 8.20 | 2.00 |

T5P14 | 3.7 | 26 | 17.50 | 15.73 | 1.70 | 19.80 | 16.40 | 12.63 |

Copula | θ | RMSE | AIC |
---|---|---|---|

G-H | 4.9273 | 0.009 | −134.272 |

Clayton | 3.1097 | 0.011 | −123.354 |

Frank | 17.7637 | 0.009 | −132.292 |

Return Period | Univariate Design Value | Bivariate Design Value | Joint Return Period | Co-occurrence Return Period | ||
---|---|---|---|---|---|---|

Q (m^{3}/s) | V (10^{8} m^{3}) | Q (m^{3}/s) | V (10^{8} m^{3}) | |||

5 | 825 | 3.93 | 868 | 4.38 | 4.41 | 5.76 |

10 | 1049 | 6.95 | 1089 | 7.72 | 8.76 | 11.66 |

20 | 1253 | 11.87 | 1292 | 13.17 | 17.44 | 23.44 |

30 | 1367 | 16.11 | 1404 | 17.83 | 26.13 | 35.22 |

50 | 1504 | 23.54 | 1541 | 26.09 | 43.50 | 58.78 |

100 | 1683 | 39.20 | 1718 | 43.41 | 86.94 | 117.67 |

Scenario | Flood Risk | ||||||||
---|---|---|---|---|---|---|---|---|---|

Light (20040618) | Medium (19870701) | Heavy (19910630) | |||||||

Q (m^{3}/s) | V (10^{8} m^{3}) | Joint | Q (m^{3}/s) | V (10^{8} m^{3}) | Joint | Q (m^{3}/s) | V (10^{8} m^{3}) | Joint | |

T-2P0 | 0.8251 | 0.6740 | 0.6725 | 0.6342 | 0.8891 | 0.6341 | 0.9649 | 0.9665 | 0.9606 |

T-2P2 | 0.8356 | 0.6873 | 0.6859 | 0.6499 | 0.8943 | 0.6498 | 0.9681 | 0.9679 | 0.9633 |

T-2P4 | 0.8456 | 0.6997 | 0.6985 | 0.6654 | 0.8992 | 0.6654 | 0.9711 | 0.9693 | 0.9657 |

T-2P8 | 0.8639 | 0.7233 | 0.7224 | 0.6951 | 0.9081 | 0.6950 | 0.9761 | 0.9720 | 0.9698 |

T-2P14 | 0.8954 | 0.7584 | 0.7579 | 0.7342 | 0.9194 | 0.7342 | 0.9865 | 0.9788 | 0.9783 |

T0P0 | 0.7769 | 0.6170 | 0.6146 | 0.6250 | 0.8782 | 0.6249 | 0.9646 | 0.9657 | 0.9600 |

T0P2 | 0.7916 | 0.6340 | 0.6319 | 0.6410 | 0.8843 | 0.6409 | 0.9678 | 0.9671 | 0.9627 |

T0P4 | 0.8061 | 0.6507 | 0.6489 | 0.6571 | 0.8900 | 0.6570 | 0.9707 | 0.9685 | 0.9650 |

T0P8 | 0.8332 | 0.6819 | 0.6806 | 0.6871 | 0.9002 | 0.6870 | 0.9759 | 0.9711 | 0.9690 |

T0P14 | 0.8679 | 0.7232 | 0.7224 | 0.7254 | 0.9129 | 0.7253 | 0.9819 | 0.9758 | 0.9748 |

T2P0 | 0.7080 | 0.5349 | 0.5313 | 0.6243 | 0.8703 | 0.6242 | 0.9635 | 0.9648 | 0.9588 |

T2P2 | 0.7233 | 0.5528 | 0.5495 | 0.6395 | 0.8771 | 0.6394 | 0.9668 | 0.9663 | 0.9616 |

T2P4 | 0.7379 | 0.5710 | 0.5679 | 0.6543 | 0.8833 | 0.6542 | 0.9698 | 0.9677 | 0.9640 |

T2P8 | 0.7677 | 0.6069 | 0.6042 | 0.6822 | 0.8946 | 0.6821 | 0.9750 | 0.9703 | 0.9681 |

T2P14 | 0.8061 | 0.6534 | 0.6515 | 0.7203 | 0.9089 | 0.7202 | 0.9813 | 0.9737 | 0.9728 |

T5P0 | 0.6915 | 0.5108 | 0.5073 | 0.6175 | 0.8593 | 0.6173 | 0.9626 | 0.9640 | 0.9578 |

T5P2 | 0.7067 | 0.5270 | 0.5238 | 0.6325 | 0.8674 | 0.6323 | 0.9659 | 0.9656 | 0.9607 |

T5P4 | 0.7212 | 0.5433 | 0.5402 | 0.6473 | 0.8749 | 0.6471 | 0.9690 | 0.9671 | 0.9632 |

T5P8 | 0.7499 | 0.5758 | 0.5732 | 0.6753 | 0.8882 | 0.6752 | 0.9744 | 0.9697 | 0.9675 |

T5P14 | 0.7894 | 0.6239 | 0.6219 | 0.7136 | 0.9038 | 0.7135 | 0.9807 | 0.9731 | 0.9722 |

Scenario | Flood Characteristics | Light (20040618) | Medium (19870701) | Heavy (19910630) | |||
---|---|---|---|---|---|---|---|

Risk | Relative Change (%) | Risk | Relative Change (%) | Risk | Relative Change (%) | ||

T0P0 | Q (m^{3}/s) | 0.7769 | - | 0.6250 | - | 0.9646 | - |

V (10^{8} m^{3}) | 0.6170 | - | 0.8782 | - | 0.9657 | - | |

Joint | 0.6146 | - | 0.6249 | - | 0.9600 | - | |

T-2P0 | Q (m^{3}/s) | 0.8251 | 6.21 | 0.6342 | 1.47 | 0.9649 | 0.03 |

V (10^{8} m^{3}) | 0.6740 | 9.24 | 0.8891 | 1.24 | 0.9665 | 0.08 | |

Joint | 0.6725 | 9.42 | 0.6341 | 1.47 | 0.9606 | 0.06 | |

T0P14 | Q (m^{3}/s) | 0.8679 | 11.71 | 0.7254 | 16.06 | 0.9819 | 1.80 |

V (10^{8} m^{3}) | 0.7232 | 17.21 | 0.9129 | 3.95 | 0.9758 | 1.05 | |

Joint | 0.7224 | 17.55 | 0.7253 | 16.07 | 0.9748 | 1.54 | |

T-2P14 | Q (m^{3}/s) | 0.8954 | 15.25 | 0.7342 | 17.48 | 0.9865 | 2.27 |

V (10^{8} m^{3}) | 0.7584 | 22.91 | 0.9194 | 4.70 | 0.9788 | 1.35 | |

Joint | 0.7579 | 23.32 | 0.7342 | 17.49 | 0.9783 | 1.91 |

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**MDPI and ACS Style**

Gao, Y.; Guo, Z.; Wang, D.; Zhang, Z.; Liu, Y.
Multivariate Flood Risk Analysis at a Watershed Scale Considering Climatic Factors. *Water* **2018**, *10*, 1821.
https://doi.org/10.3390/w10121821

**AMA Style**

Gao Y, Guo Z, Wang D, Zhang Z, Liu Y.
Multivariate Flood Risk Analysis at a Watershed Scale Considering Climatic Factors. *Water*. 2018; 10(12):1821.
https://doi.org/10.3390/w10121821

**Chicago/Turabian Style**

Gao, Yuqin, Zichen Guo, Dongdong Wang, Zhenxing Zhang, and Yunping Liu.
2018. "Multivariate Flood Risk Analysis at a Watershed Scale Considering Climatic Factors" *Water* 10, no. 12: 1821.
https://doi.org/10.3390/w10121821