# Joint Flood Risks in the Grand River Watershed

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Grand River Watershed

^{2}. The area has five major urban settlements: Guelph, Kitchener, Waterloo, Cambridge, and Brantford. Agricultural land makes up 70% of the watershed. This study selects the Speed River and Grand River as case studies. These rivers meet in the City of Cambridge, and simultaneous flooding of these rivers can cause disastrous impacts in the highly populated urban settlements (Figure 1). The annual peak flows of the rivers at the meeting station (Galt) are analyzed for their interdependence structure and to estimate the bivariate return period. The annual extreme flow data has been obtained from the website of the water survey of Canada [35]. Additionally, the annual extreme flows at the two rivers, Speed and Grand at Galt, Cambridge, are analyzed to explore the probability of joint occurrence of extreme events at the two rivers.

#### 2.2. Copula in Bivariate Frequency Analysis

_{1}and Y

_{2}, with distribution functions F

_{1}(Y

_{1}) and F

_{2}(Y

_{2}), respectively. As per Sklar’s theorem, there always exists a copula function (C) such that,

_{1}= y

_{1}, Y

_{2}= y

_{2}) = C(F

_{1}(y

_{1}), F

_{2}(y

_{2}))

_{l},u

_{2}) is itself a distribution function, where u

_{1}and u

_{2}are F

_{1}(y

_{1}) and F

_{2}(y

_{2}) respectively.

#### 2.3. Joint and Conditional Return Period Using Copula

#### 2.3.1. Joint Return Period Using Copula

_{1}and Y

_{2}). The copula describing the interdependence of the variables is given as C

_{12}. The joint return period for “AND” and “OR” cases and the conditional return period for different conditions can be calculated using the relations given below:

_{1}and Y

_{2}are exceeded) case can be expressed as follows:

#### 2.3.2. Conditional Return Period Using Copula

## 3. Results and Discussion

#### Bivariate Copula in Estimating Joint Flood Risks

^{3}/s. Beyond this point, the magnitude of the Speed River flow events remained constant for all return periods. This implies that the Grand River flow has a limiting effect on the magnitude of extreme events in the Speed River flow. The inference may suggest that the interdependence and interaction between the two rivers are complex and depend on various factors, such as the magnitude of extreme flow events and the temporal and spatial scales of the analysis. Further research may be required to understand these relationships in more detail. Similarly, the conditional return period analysis of Grand River extremes given the Speed River flow (Figure 9) showed that as the Speed River flow extremes increase, the magnitude of a t-year return period event for the Grand River flow increases. This indicates that the flows of the Speed River can influence the flood outcome of the Grand River, similar to the effect of tributary flows on the main river channel.

^{3}/s), respectively, in the area, it is ten years, whereas the univariate return period is 15 years and 12 years for Speed and Grand flow, respectively. Furthermore, it is worth noting that the 6-year return period flow of 77 m

^{3}/s and 607 m

^{3}/s in Speed and Grand rivers (while considering univariate analysis) can translate to a 9-year return period of simultaneous flow, which also needs to be considered. Therefore, the joint return period is essential in accurately assessing flood risk and planning appropriate management strategies, particularly in areas with high flow conditions.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Annual peak flow events at the Speed River and Grand River in the Grand River watershed in Canada.

**Figure 4.**Probability distribution function (PDF) of the developed bivariate distribution (copula) of the Speed and Grand Rivers flow extremes.

**Figure 5.**Cumulative distribution function (CDF) of the developed bivariate distribution of the Speed and Grand Rivers flow extremes.

Goodness of Fit Tests | p-Value | Test Statistic |
---|---|---|

Cramer-von Mises | 0.96 | 0.024 |

Kolmogorov-Smirnov | 0.96 | 0.437 |

Speed Flow (m^{3}/s) | Grand Flow (m^{3}/s) | T_{AND}(Year) | T_{OR}(Year) | T_{S}(Year) | T_{G}(Year) |
---|---|---|---|---|---|

52 | 420 | 2.8 | 1.7 | 2 | 2.2 |

65 | 518 | 5 | 2.7 | 3.2 | 3.8 |

77 | 607 | 9.5 | 4.7 | 6.1 | 6.5 |

90 | 700 | 20.7 | 10 | 15 | 12.2 |

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**MDPI and ACS Style**

Unnikrishnan, P.; Ponnambalam, K.; Agrawal, N.; Karray, F.
Joint Flood Risks in the Grand River Watershed. *Sustainability* **2023**, *15*, 9203.
https://doi.org/10.3390/su15129203

**AMA Style**

Unnikrishnan P, Ponnambalam K, Agrawal N, Karray F.
Joint Flood Risks in the Grand River Watershed. *Sustainability*. 2023; 15(12):9203.
https://doi.org/10.3390/su15129203

**Chicago/Turabian Style**

Unnikrishnan, Poornima, Kumaraswamy Ponnambalam, Nirupama Agrawal, and Fakhri Karray.
2023. "Joint Flood Risks in the Grand River Watershed" *Sustainability* 15, no. 12: 9203.
https://doi.org/10.3390/su15129203