Bivariate Nonstationary Extreme Flood Risk Estimation Using Mixture Distribution and Copula Function for the Longmen Reservoir, North China
Abstract
:1. Introduction
2. Study Area and Flood Data
2.1. Extreme Flood
2.2. Statistical-Physical Heterogeneity Analysis
3. Methodology
3.1. Mixture Distribution as Marginal Distribution
3.2. Bivariate Copula Functions
3.3. Goodness of Fit for Models
3.4. Bivariate Nonstatioarny Return Period
4. Results and Discussion
4.1. Univarite Mixture Distribution Flood Frequency Analysis
4.2. Fitting Bivariate Joint Distribution
4.3. Estimating Bivariate Nonstationary Return Period and Design Flood
4.4. Estimating Joint and Condtional Probabilities
5. Conclusions
- (1)
- The 1-day AMFV exhibits the highest significant correlation with AMFP, which demonstrates the desirability and indispensability of bivariate flood frequency analysis. In addition, the underlying surface changes in the Longmen Reservoir contribute to the heterogeneity of flood generation identified by the statistical methods and physical basis analysis. A significant change point is detected in the year 1979 for 1-day AMFV, but the AMFP is shown to be homogenous.
- (2)
- From univariate nonstationary flood frequency analysis of 1-day AMFV, the fitting performance of mixture distribution is superior to the traditional stationary P-III distribution. Due to the increase of forest land area and some hydraulic engineering construction, the design floods of 1-day AMFV with different return periods estimated by MD are generally smaller than the ones estimated by P-III distribution.
- (3)
- In the case of bivariate analysis, copula-based joint distribution was developed and performed using the stationary P-III distribution for AMFP and nonstationary MD for 1-day AMFV as marginal distributions. There is a relatively large increase for the design floods estimated by bivariate nonstationary joint distribution compared with the ones estimated in a univariate nonstationary context, which can be concluded and proved by rigorous mathematical formula derivation. Furthermore, the results of joint and conditional probabilities demonstrate that, assuming the flood peak and volume share the same return period, the conditional probability of 1-day AMFV exceeding the threshold is likely to be high when the AMFP exceeds the design flood associated with the return period.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Flood Series | Kendall Correlation Test | Spearman Correlation Test |
---|---|---|
1-day AMFV | 0.84 | 0.96 |
3-day AMFV | 0.79 | 0.93 |
6-day AMFV | 0.77 | 0.92 |
Methods | AMFP | 1-Day AMFV |
---|---|---|
Hurst exponent value | 0.67 (no variation) | 0.73 (medium variation) |
MWP | — | 1959–1971, 1974, 1977–1983 |
Brown–Forsythe | — | 1996, 1964 |
Moving rank test | — | 1964, 1979, 1998 |
Change points | — | 1964, 1979 |
Flood | α | EX1 | |||||
---|---|---|---|---|---|---|---|
AMFP (m3/s) | 265.77 | 2.88 | 6.04 | ||||
1-day AMFV (P-III) (108 m3) | 0.12 | 2.3 | 5.2 | ||||
1-day AMFV (MD) (108 m3) | 0.34 | 0.18 | 1.7 | 5.1 | 0.09 | 1.95 | 4.00 |
Flood | Distribution | Return Periods (Year) | |||||
---|---|---|---|---|---|---|---|
2000 | 1000 | 100 | 50 | 20 | 10 | ||
1-day AMFV (108 m3) | P-III | 3.02 | 2.61 | 1.34 | 0.99 | 0.58 | 0.32 |
MD | 2.79 | 2.36 | 1.14 | 0.84 | 0.51 | 0.31 | |
Difference (%) | −7.6 | -9.6 | −14.9 | −15.2 | −12.1 | −3.1 |
Cases | Parameter (θ) | K-S Test (D) | OLS | AIC |
---|---|---|---|---|
Stationary | 6.26 | 0.3214 | 0.1371 | −220.55 |
Nonstationary | 6.26 | 0.1419 | 0.0604 | −312.3 |
Return Period (yr) | Design Flood of Univariate Marginal Distribution | Design Flood of Bivariate Joint Distribution | Difference (%) | |||
---|---|---|---|---|---|---|
Q (m3/s) | W1 (108 m3) | Q (m3/s) | W1 (108 m3) | Q (m3/s) | W1 (108 m3) | |
2000 | 9332 | 2.79 | 9546 | 2.86 | 2.29 | 2.51 |
1000 | 7996 | 2.36 | 8208 | 2.43 | 2.65 | 2.97 |
100 | 3855 | 1.14 | 4038 | 1.19 | 4.75 | 4.39 |
50 | 2752 | 0.84 | 2920 | 0.89 | 6.10 | 5.95 |
20 | 1473 | 0.51 | 1610 | 0.55 | 9.30 | 7.84 |
Return Period (Year) | 2000 | 1000 | 100 | 50 | 20 | |
---|---|---|---|---|---|---|
Design flood Wp (108 m3) | 2.79 | 2.36 | 1.14 | 0.84 | 0.51 | |
Conditional probability (%) | 100-year AMFP | 5.02 | 10.05 | 88.00 | 99.63 | 99.99 |
2000-year AMFP | 88.51 | 99.60 | 99.99 | 99.99 | 99.99 |
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Li, Q.; Zeng, H.; Liu, P.; Li, Z.; Yu, W.; Zhou, H. Bivariate Nonstationary Extreme Flood Risk Estimation Using Mixture Distribution and Copula Function for the Longmen Reservoir, North China. Water 2022, 14, 604. https://doi.org/10.3390/w14040604
Li Q, Zeng H, Liu P, Li Z, Yu W, Zhou H. Bivariate Nonstationary Extreme Flood Risk Estimation Using Mixture Distribution and Copula Function for the Longmen Reservoir, North China. Water. 2022; 14(4):604. https://doi.org/10.3390/w14040604
Chicago/Turabian StyleLi, Quan, Hang Zeng, Pei Liu, Zhengzui Li, Weihou Yu, and Hui Zhou. 2022. "Bivariate Nonstationary Extreme Flood Risk Estimation Using Mixture Distribution and Copula Function for the Longmen Reservoir, North China" Water 14, no. 4: 604. https://doi.org/10.3390/w14040604