# Evaluation of Return Period and Risk in Bivariate Non-Stationary Flood Frequency Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

## 3. Methodology

#### 3.1. Trend Analysis

#### 3.1.1. Univariate Trend Test

#### 3.1.2. Multivariate Trend Test

^{(u)}(u = 1, 2,…, d) for each component of the variables is calculated first. S = (S

^{(1)}, S

^{(2)},…, S

^{(d)}) is a d-dimensional vector with a covariance matrix C

_{M}= (c

_{u,v})

_{u,v=}

_{1,}

_{……d}, in which c

_{u,v}= cov(M

^{(u)},M

^{(v)}). The covariate terms are calculated with the following equations [23]:

_{m}, given as:

_{m}is asymptotically normal with zero mean and its variance is given as:

#### 3.2. Time Varying Copula

_{1}( ) and F

_{2}( ) are marginal distributions for the variables Y

_{1}and Y

_{2}, C( ) is the copula function; ${\theta}_{1}^{t}$ and ${\theta}_{2}^{t}$ are parameters for the marginal distributions, and ${\theta}_{c}^{t}$ is the parameter for the copula function, and the superscript t means the parameters are time varying. The procedure for how to construct a time varying copula model is presented in Figure 2.

#### 3.2.1. Marginal Distribution

^{T}is described as a function of explanatory variables and random effects. In case there are no additive terms, the distribution parameters can be denoted by a monotonic link function g

_{k}( ) as:

_{k}denote the explanatory variables, β

_{k}are polynomial coefficients, p is the number of parameters, q is the degree of the polynomial, and q = 1 in this study to avoid over-parameterization. The location and scale parameters were related to the time covariate, including either one of them is time varying or both of them as time varying, and the shape parameter was treated as a constant. The stationary model was obtained by assuming that the parameters are independent of the explanatory variables. The optimal distribution was determined by the Akaike Information Criterion (AIC) [28] and the goodness-of-fit test was performed by the worm plot [29]. PDFs of the marginal distributions and their link functions are listed in Table 1.

#### 3.2.2. Copula Function

_{0}and β

_{1}are polynomial coefficients and t is the time. A brief description of these copulas and their link functions are presented in Table 2.

#### 3.3. Return Period, Risk of Failure and Reliability

#### 3.3.1. Non-Stationary Return Period

_{i}is the exceeding probability of a design event for each year. The EWT return period T of the design event can thus be expressed as:

_{1}≥ y

_{1}and Y

_{2}≥ y

_{2}) and case OR (Y

_{1}≥ y

_{1}or Y

_{2}≥ y

_{2}), which can be defined by two exceeding probabilities using copula theory as follows:

_{i}for each year are exactly the same so that both the two interpretations lead to the same result, T = 1/P

_{i}, which can be divided into two cases, specifically, using the following equations [34]:

#### 3.3.2. Risk of Failure and Reliability

_{i}in the ith year is given as:

_{i}can be either P

_{AND}or P

_{OR}given above in bivariate cases. Correspondingly, the reliability, r, is defined as the probability that no design flood event will happen in the next n years, given as:

## 4. Results

#### 4.1. The Trend Test

#### 4.2. Marginal Distribution

#### 4.3. Copula Modeling

#### 4.4. Flood Risk Assessment

#### 4.4.1. The Non-Stationary Return Period

^{3}/s) are 62,202 and 63,127, respectively, and the designed flood volumes (10

^{6}m

^{3}/s) are 33,054 and 31,165, respectively. Model 2 is characterized by a higher flood peak and lower flood volume compared with model 1, which may be attributed to the time varying marginal distributions.

#### 4.4.2. Risk of Failure and Reliability

^{3}/s) and volume (33,738 × 10

^{6}m

^{3}/s) with 20 years of univariate return period under the stationary assumption, the risk of failure of the design flood events for model 1 and model 2 in the next 50 years are calculated and plotted in Figure 6. Figure 6a shows the risk of failure for case AND. Under the stationary assumption, the risk of failure R increased rapidly with the design life n, as shown by the red line. However, due to non-stationarities, the flood risk is smaller than the stationary model, and the rate of growth decreased as presented by the blue line. On the other hand, Figure 6b shows the risk of failure for case OR, the conclusions of which are completely different from that of case AND. In total, the R values of case OR are nearly the same between model 1 and model 2, indicating that the non-stationary properties have a very small influence on the risk of failure for the flood event case OR. Figure 7 presents the results of reliability for the two models, which are similar to those of Figure 6. The results suggested that the hydrologic facilities controlled by design flood event case OR are more stable under the changing environment in the study region, but the risks are much bigger than that of case AND.

## 5. Discussion

^{AND}and T

^{OR}and any other kind of return period are only different descriptions of the underlying mechanisms of design structures and must be used accordingly. It is important to figure out what type of event is used in the practical application and how to evaluate the effect of non-stationarity on flood risk under this specific condition.

## 6. Conclusions

- (1)
- Both univariate and multivariate M-K trend tests were used to examine the temporal variations between flood characteristics, and the results revealed severe non-stationary properties, indicating that the stationary assumption is invalid, and a non-stationary copula model is needed for hydrological modeling in this region.
- (2)
- The flood peak and volume series were fitted by 6 marginal distributions under stationary and non-stationary situations, and the appropriate distributions vary correspondingly. The Weibull and Logistic distributions were selected for flood peak and volume under the stationary assumption, while the Gamma and Logistic distributions were selected under the non-stationary assumption. The flood peak and volume series presented decreasing trends in their location parameters. Four copula functions were applied to investigate the dependence structure between flood variables, and the Frank copula was selected as the most appropriate. The non-stationary model performs better than the copula with constant parameters according to AIC values. The copula parameters of the non-stationary model detected a decreasing trend, which means the dependence structure between flood variables also weakened over time.
- (3)
- The joint non-stationary return period was calculated for comparison, and the results varied with different design flood events. Compared with the stationary model, in case AND, the flood risk of non-stationary models decreased as a result of the decreased parameters of both marginal distributions and copula functions. While in case OR, the effect of the non-stationary properties is almost negligible. As for the design values, the non-stationary model is characterized by a higher flood peak and lower flood volume. The bivariate risk of failure of 50 years of design life was also estimated, and the conclusions were similar to those obtained by the return period. The effect of non-stationarity is more inclined to case AND rather than case OR, and the non-stationary model is safer than the stationary model in case AND.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Worm plots for the time varying marginal distribution of (

**a**) the flood peak series and (

**b**) the flood volume series.

**Figure 4.**A comparison of empirical and theoretical probabilities for the Frank copula (

**a**) under the stationary condition and (

**b**) under non-stationary condition.

**Figure 5.**The univariate exceeding probability against the non-stationary return period for (

**a**) case AND and (

**b**) case OR.

Distribution | Probability Density Function, f(x) | Link Function | |
---|---|---|---|

g(μ) | g(σ) | ||

Gamma | $f(x)=\frac{1}{{({\sigma}^{2}\mu )}^{1/{\sigma}^{2}}}\frac{{x}^{\frac{1}{{\sigma}^{2}}-1}{e}^{-x/({\sigma}^{2}\mu )}}{\mathsf{\Gamma}(1/{\sigma}^{2})}$ | ln(μ) | ln(σ) |

Weibull | $f(x)=\frac{\sigma {x}^{\sigma -1}}{{\mu}^{\sigma}}\mathrm{exp}{(-\frac{x}{\mu})}^{\sigma}$ | ln(μ) | ln(σ) |

Gumbel | $f(x)=\frac{1}{\sigma}\mathrm{exp}[\left(\frac{x-\mu}{\sigma}\right)-\mathrm{exp}(\frac{x-\mu}{\sigma})]$ | μ | ln(σ) |

Logistic | $f(x)=\frac{1}{\sigma}\left\{\mathrm{exp}\left[-\frac{x-\mu}{\sigma}\right]\right\}{\left\{1+\mathrm{exp}\left[-\left(\frac{x-\mu}{\sigma}\right)\right]\right\}}^{-2}$ | μ | ln(σ) |

Lognormal | $f(x)=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}\frac{1}{x}\mathrm{exp}\left\{-\frac{{\left[\mathrm{log}(x)-\mu \right]}^{2}}{2{\sigma}^{2}}\right\}$ | μ | ln(σ) |

Pearson III | $f(x)=\frac{2}{\mu \sigma \epsilon \mathsf{\Gamma}(4/{\epsilon}^{2})}{\left[\frac{2\left(x-\mu \right)}{\mu \sigma \epsilon}+\frac{4}{{\epsilon}^{2}}\right]}^{\frac{4}{{\epsilon}^{2}}-1}\mathrm{exp}\left[-\left(\frac{2\left(x-\mu \right)}{\mu \sigma \epsilon}+\frac{4}{{\epsilon}^{2}}\right)\right]$ | ln(μ) | ln(σ) |

Copula | C(u,v) | Range of θ | g(θ) |
---|---|---|---|

Clayton | $max({[{u}^{-\theta}+{v}^{-\theta}-1]}^{-\frac{1}{\theta}},0)$ | θ > 0 | ln(θ) |

GH | $\mathrm{exp}\{-{[{(-\mathrm{ln}u)}^{\theta}+{(-\mathrm{ln}v)}^{\theta}]}^{1/\theta}\}$ | θ ≥ 1 | ln(θ) |

Frank | $-\frac{1}{\theta}\mathrm{ln}[1+\frac{({e}^{-\theta u}-1)({e}^{-\theta v}-1)}{{e}^{-\theta}-1}]$ | θ ≠ 0 | ln(θ) |

Joe | $1-{[{(1-u)}^{\theta}+{(1-v)}^{\theta}-{(1-u)}^{\theta}{(1-v)}^{\theta}]}^{1/\theta}$ | θ ≥ 1 | ln(θ) |

Series | Type | Distribution | μ | σ | AIC |
---|---|---|---|---|---|

P (m^{3}/s) | Stationary | Weibull | 53,585.83 | 6.74 | 1341.9 |

Non-stationary | Gamma | exp(10.90 − 0.0026t) | exp(−2.20 + 0.0116t) | 1339.9 | |

V (10^{6} m^{3}) | Stationary | Logistic | 26,542.00 | 2444.02 | 1258.4 |

Non-stationary | Logistic | 28,239.69 − 53.43t | 2387.50 | 1256.6 |

Model | Copula Function | θ | AIC |
---|---|---|---|

1 | Clayton | 3.70 | −94.31 |

GH | 3.06 | −87.33 | |

Frank | 12.88 | −101.80 | |

Joe | 3.43 | −63.52 | |

2 | Clayton | exp(−0.0064t + 1.24) | −76.98 |

GH | exp(−0.0048t + 1.34) | −98.05 | |

Frank | exp(−0.0039t + 2.69) | −102.36 | |

Joe | exp(−0.0052t + 1.55) | −77.45 |

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**MDPI and ACS Style**

Kang, L.; Jiang, S.; Hu, X.; Li, C.
Evaluation of Return Period and Risk in Bivariate Non-Stationary Flood Frequency Analysis. *Water* **2019**, *11*, 79.
https://doi.org/10.3390/w11010079

**AMA Style**

Kang L, Jiang S, Hu X, Li C.
Evaluation of Return Period and Risk in Bivariate Non-Stationary Flood Frequency Analysis. *Water*. 2019; 11(1):79.
https://doi.org/10.3390/w11010079

**Chicago/Turabian Style**

Kang, Ling, Shangwen Jiang, Xiaoyong Hu, and Changwen Li.
2019. "Evaluation of Return Period and Risk in Bivariate Non-Stationary Flood Frequency Analysis" *Water* 11, no. 1: 79.
https://doi.org/10.3390/w11010079