Deriving Design Flood Hydrographs Based on Copula Function: A Case Study in Pakistan
Abstract
:1. Introduction
2. Study Basin and Hydrological Data
3. Methods
3.1. Selection of Marginal Distribution
3.2. Traditional Methods of Deriving DFH
3.3. Peak and Volume Amplitude Method
3.4. Deriving Joint Design Floods Based on Copulas
3.4.1. EFC Method
3.4.2. MLC Method
3.5. Joint Return Period
- (1)
- “OR” case, either Q > q or W > w, i.e.,
- (2)
- “AND” case, both Q > q or W > w, i.e.,
3.6. Procedure for DFH Generation
- (1)
- The ratio between the hydrological pairs of flood peak and volume is determined.
- (2)
- The normalized ratio RT varying within (0, 1) is estimated by:
- (3)
- The normalized ratio between M hydrological pairs of the observed hydrograph is also determined using steps 1 and 2 and is represented as , k = 1, …, M.
- (4)
- The square sum of deviations between and for M pairs are calculated as follows for each simulated hydrological pair (, ) covered by single JRP .
- (5)
- The observed TFH offering narrowest ratio (min) is chosen.
- (6)
- The hydrological pair (, ) is transformed into corresponding DFH by amplifying the selected observed TFH values.
4. Application and Discussion
4.1. Selection of Marginal Distribution
4.2. Comparison of Copulas
4.3. Derivation of DFH
4.4. Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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River | Variable | Data Length (Years) | Sample Statistics Values | ||
---|---|---|---|---|---|
Mean | L-Cv | L-Cs | |||
Indus | Q (m3/s) | 29 | 10,474.62 | 0.1384 | 0.6383 |
W3d (108 m3) | 29 | 2.651 | 0.1371 | 0.6817 | |
W7d (108 m3) | 29 | 5.867 | 0.135 | 0.536 | |
W15d (108 m3) | 29 | 11.67 | 0.147 | 0.369 | |
Jhelum | Q (m3/s) | 34 | 970.69 | 0.378 | 0.670 |
W3d (108 m3) | 34 | 0.229 | 0.349 | 0.345 | |
W7d (108 m3) | 34 | 0.493 | 0.335 | 0.320 | |
W15d (108 m3) | 34 | 0.976 | 0.334 | 0.382 |
Copula Function | |||
---|---|---|---|
Gumbel-Hougaard | |||
Clayton | |||
Frank |
Location | Criteria | Probability Distributions | |||||
---|---|---|---|---|---|---|---|
GPA | GNO | GLO | WEI | P3 | GEV | ||
Indus River | AIC | −184.90 | −199.85 | −191.44 | −197.11 | −199.90 | −200.18 |
RMSE | 0.0372 | 0.02874 | 0.0332 | 0.03013 | 0.0287 | 0.02858 | |
Jhelum River | AIC | −223.05 | −229.96 | −216.66 | −224.79 | −225.68 | −224.70 |
RMSE | 0.0344 | 0.0311 | 0.0378 | 0.0335 | 0.0331 | 0.0336 |
River | Variable | G–H | Clayton | Frank | ||||||
---|---|---|---|---|---|---|---|---|---|---|
θ | Dn | Pv | θ | Dn | Pv | θ | Dn | Pv | ||
Indus | Qmax-W3d | 22.56 | 0.031 | 0.012 | 43.11 | 0.037 | 0.024 | 88.55 | 0.039 | 0.022 |
Qmax-W7d | 5.21 | 0.032 | 0.023 | 8.41 | 0.039 | 0.029 | 19.02 | 0.035 | 0.031 | |
Qmax-W15d | 2.57 | 0.028 | 0.031 | 3.14 | 0.030 | 0.035 | 8.23 | 0.037 | 0.029 | |
Jhelum | Qmax-W3d | 14.38 | 0.033 | 0.024 | 26.77 | 0.039 | 0.029 | 55.84 | 0.037 | 0.033 |
Qmax-W7d | 6.45 | 0.029 | 0.039 | 10.90 | 0.040 | 0.033 | 24.03 | 0.035 | 0.041 | |
Qmax-W15d | 4.49 | 0.031 | 0.033 | 6.98 | 0.033 | 0.035 | 16.12 | 0.038 | 0.037 |
River | Indus | Jhelum | |||||||
---|---|---|---|---|---|---|---|---|---|
Return Period | 100 | 50 | 30 | 10 | 100 | 50 | 30 | 10 | |
Q (m3/s) | P3/LM | 14,975 | 14,267 | 13,729 | 12,491 | 2117 | 1935 | 1798 | 1481 |
MLC | 15,229 | 14,842 | 14,084 | 12,606 | 2080 | 1918 | 1796 | 1512 | |
EFC | 15,942 | 15,143 | 14,711 | 12,993 | 2157 | 1977 | 1839 | 1527 | |
W3D (108 m3) | P3/LM | 3.68 | 3.52 | 3.34 | 3.14 | 0.46 | 0.42 | 0.40 | 0.34 |
MLC | 3.75 | 3.62 | 3.46 | 3.18 | 0.47 | 0.43 | 0.41 | 0.34 | |
EFC | 3.71 | 3.57 | 3.42 | 3.16 | 0.47 | 0.43 | 0.40 | 0.34 | |
W7D (108 m3) | P3/LM | 8.02 | 7.78 | 7.55 | 6.75 | 0.96 | 0.89 | 0.84 | 0.72 |
MLC | 8.25 | 7.98 | 7.68 | 7.02 | 0.99 | 0.92 | 0.86 | 0.74 | |
EFC | 8.21 | 7.90 | 7.59 | 6.95 | 0.97 | 0.90 | 0.85 | 0.73 | |
W15D (108 m3) | P3/LM | 16.3 | 15.6 | 15.1 | 14.0 | 1.87 | 1.75 | 1.65 | 1.42 |
MLC | 16.6 | 15.9 | 15.4 | 14.3 | 1.95 | 1.82 | 1.71 | 1.46 | |
EFC | 16.3 | 15.7 | 15.3 | 14.2 | 1.90 | 1.78 | 1.68 | 1.45 |
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Rizwan, M.; Guo, S.; Yin, J.; Xiong, F. Deriving Design Flood Hydrographs Based on Copula Function: A Case Study in Pakistan. Water 2019, 11, 1531. https://doi.org/10.3390/w11081531
Rizwan M, Guo S, Yin J, Xiong F. Deriving Design Flood Hydrographs Based on Copula Function: A Case Study in Pakistan. Water. 2019; 11(8):1531. https://doi.org/10.3390/w11081531
Chicago/Turabian StyleRizwan, Muhammad, Shenglian Guo, Jiabo Yin, and Feng Xiong. 2019. "Deriving Design Flood Hydrographs Based on Copula Function: A Case Study in Pakistan" Water 11, no. 8: 1531. https://doi.org/10.3390/w11081531
APA StyleRizwan, M., Guo, S., Yin, J., & Xiong, F. (2019). Deriving Design Flood Hydrographs Based on Copula Function: A Case Study in Pakistan. Water, 11(8), 1531. https://doi.org/10.3390/w11081531