# Using Copulas to Evaluate Rationality of Rainfall Spatial Distribution in a Design Storm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Distributing Schemes

_{D}represents the design rainfall we try to derive, X

_{B}and X

_{N}are the design rainfall of the whole basin and northern part of the basin, respectively, F and F

_{N}are the area of the whole basin and northern part of the basin respectively, n is the number of non-prioritized sub-basins, i is the ordinal of the non-prioritized sub-basins, x

_{i}and f

_{i}are the typical rainfall and area of each non-prioritized sub-basins, X

_{M}represents the typical rainfall of the non-prioritized sub-basins.

_{M}, that is, if the formal scheme is chosen, X

_{M}is derived using the rainfall information of a typical outstanding year, if the latter, an average information of rainfall over the past decades is applied to X

_{M}.

#### 2.2. Maginal and Joint Distribution

_{X}(x) = u and F

_{Y}(y) = v represent cumulative distribution functions of variable X and Y, respectively, their Copula function can be described as follows:

_{θ}represent the parameter of the Copula function and the Copula function itself.

#### 2.3. Goodness of Fit

_{emp}(x

_{i}

_{1}, x

_{i}

_{2}, …, x

_{i}

_{m}), C( u

_{i}

_{1}, u

_{i}

_{2}, …, u

_{i}

_{m}) are empirical and theoretical cumulative probabilities respectively; m is the dimension of the function; n is the number of observations; k is the number of parameters; MSE is the root mean square error. The goodness of fit is believed to be better when a smaller AIC or OLS value is obtained.

_{i}, q

_{j}, z

_{k}), the empirical frequency can be calculated using the equation below:

_{emp}(p

_{i}, q

_{j}, z

_{k}) is the empirical frequency, N is the number of observations, n

_{ijk}represents the times when measured data is less than (p

_{i}, q

_{j}, z

_{k}) simultaneously.

#### 2.4. Conditional Probability

_{1}, X

_{2}, X

_{3}), the probability of X

_{1}≤ x

_{1}under condition of X

_{2}= x

_{2}, X

_{3}= x

_{3}can be derived using Equation (7).

_{3}≤ u

_{3}under the condition of U

_{1}= u

_{1}, U

_{2}= u

_{2}. The following derivations are based on Clayton Copula because it was proven to outperform the other two types, Frank and Gumbel Copula in this study (see Section 4.2 for discussion).

_{3}over u

_{1}, u

_{2}is derived:

_{2}over u

_{1}, u

_{2}is derived:

## 3. Study Domain and Data

#### 3.1. Study Domain

^{2}, which is definitely a rather large one. Figure 2 depicts the relative geographical location of TLB and its seven sub-basins which are divided according to terrain and flood control needs. The area of North, YCDM, SH, HJH, HQ and ZX is 11,510, 4314, 4466, 7480, 3192 and 5930 km

^{2}, respectively. As is revealed from the elevation, the TLB shows a high altitude in the West mountainous area while the rest is basically plain areas.

#### 3.2. Data Collection

## 4. Results and Discussion

#### 4.1. Marginal Distribution

#### 4.2. Joint Distribution

#### 4.3. Conditional Probability

_{2}≤ u

_{2}under the condition of U

_{1}> u

_{1}, while in this study, we derived the conditional probability of U

_{3}≤ u

_{3}under the condition of (U

_{1}= u

_{1}, U

_{2}= u

_{2}) instead. The equation has to be derived in this way because it needs to keep consistent with the rainfall we are studying.

## 5. Conclusions

- For the “typical year scheme”, the probability analysis of extreme rainfall shows huge gaps among non-prioritized sub-basins, especially for the max-30-day rainfall of 50 years return period. If this scheme is applied to hydrologic modelling for TLB, sub-basin YCDM and SH are considered to take a higher flood risk than they should, which can lead to unnecessary public spending on hydraulic structures in these sub-basins. Meanwhile, sub-basin ZX is taking a lower flood risk than it should, and it can be dangerous since this sub-basin is suffering from torrential flood disasters in reality. Generally, the scheme shows a different degree of rationality when it comes to different durations or different return periods, thus it is in urgent need to be redistributed spatially over the five non-prioritized sub-basins.
- Conditional probabilities of the non-prioritized sub-basins can be more uniform after rainfall is redistributed based on long-term information of data. This new scheme shows a more rational spatial distribution, in which flood risks taken by different non-prioritized sub-basins are much more uniform and are better for the flood prevention of the TLB as a whole.
- The proposed 3-d Copula-based method is proven to be very useful to evaluate spatial distribution of design storm in a large-scale drainage basin where sub-basins need to be considered separately for design storm. Further studies are needed to propose a more reasonable scheme for design storm of sub-basins where rainfall events are not independent.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Moradkhani, H.; Sorooshian, S. General Review of Rainfall-Runoff Modeling: Model Calibration, Data Assimilation, and Uncertainty Analysis. In Hydrological Modelling and the Water Cycle; Springer: Berlin/Heidelberg, Germany, 2009; pp. 1–24. [Google Scholar]
- Pumo, D.; Viola, F.; Noto, L.V. Generation of Natural Runoff Monthly Series at Ungauged Sites Using a Regional Regressive Model. Water
**2016**, 8, 209. [Google Scholar] [CrossRef] [Green Version] - Bureau of the Tai Lake Basin, Water Resources Ministry of China. The Flood Control Planning of the Tai Lake Basin; Bureau of the Tai Lake Basin, Water Resources Ministry of China: Shanghai, China, 2008. [Google Scholar]
- Department of Irrigation and Drainage, Ministry of Natural Environment Malaysia. Hydrological Procedure NO. 1-Estimation of Design Storm in Peninsular Malaysia; Department of Irrigation and Drainage, Ministry of Natural Environment Malaysia: Kuala Lumpur, Malaysia, 2010.
- Bureau Veritas North America, Inc. Analysis of Results for the County of San Diego Rainfall Distribution Study Project; Bureau Veritas North America, Inc.: San Diego, CA, USA, 2013. [Google Scholar]
- Urbonas, B. Reliability of design storms in modeling. In Proceedings of the International Symposium on Urban Storm Runoff, Lexington, KY, USA, 23–26 July 1979. [Google Scholar]
- Cheng, Q.W. Analysis of the design storm time-intensity pattern for medium and small watersheds. J. Hydrol.
**1987**, 96, 305–317. [Google Scholar] - Hromadka, T.V.; Whitley, R.J. The design storm concept in flood control design and planning. Stoch. Hydrol. Hydraul.
**1988**, 2, 213–239. [Google Scholar] [CrossRef] - Kang, M.S.; Goo, J.H.; Song, I.; Chun, J.A.; Her, Y.G.; Hwang, S.W.; Park, S.W. Estimating design floods based on the critical storm duration for small watersheds. J. Hydrol. Environ. Res.
**2013**, 7, 209–218. [Google Scholar] [CrossRef] - Rogger, M.; Kohl, B.; Pirkl, H.; Viglione, A.; Komma, J.; Kirnbauer, R.; Merz, R.; Blöschl, G. Runoff models and flood frequency statistics for design flood estimation in Austria—Do they tell a consistent story? J. Hydrol.
**2012**, 456, 30–43. [Google Scholar] [CrossRef] - Carbone, M.; Turco, M.; Brunetti, G.; Piro, P. A cumulative rainfall function for subhourly design storm in Mediterranean urban areas. Adv. Meteorol.
**2015**, 2015. [Google Scholar] [CrossRef] - Arnaud, P.; Bouvier, C.; Cisneros, L.; Dominguez, R. Influence of rainfall spatial variability on flood prediction. J. Hydrol.
**2002**, 260, 216–230. [Google Scholar] [CrossRef] - Sangati, M.; Borga, M.; Rabuffetti, D.; Bechini, R. Influence of rainfall and soil properties spatial aggregation on extreme flash flood response modelling: An evaluation based on the Sesia River Basin, North Western Italy. Adv. Water Resour.
**2009**, 32, 1090–1106. [Google Scholar] [CrossRef] - Cristiano, E.; Veldhuis, M.-C.T.; van de Giesen, N. Spatial and temporal variability of rainfall and their effects on hydrological response in urban areas—A review. Hydrol. Earth Syst. Sci.
**2017**, 21, 3859–3878. [Google Scholar] [CrossRef] - Zhang, J.; Han, D. Assessment of rainfall spatial variability and its influence on runoff modelling—A case study in the Brue catchment, UK. Hydrol. Process.
**2017**, 31, 2972–2981. [Google Scholar] [CrossRef] - Favre, A.-C.; El Adlouni, S.; Perreault, L.; Thiémonge, N.; Bobée, B. Multivariate hydrological frequency analysis using copulas. Water Resour. Res.
**2004**, 40. [Google Scholar] [CrossRef] [Green Version] - Genest, C.; Favre, A.C. Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask. J. Hydrol. Eng.
**2007**, 12, 347–368. [Google Scholar] [CrossRef] [Green Version] - Dupuis, D.J. Using Copulas in Hydrology: Benefits, Cautions, and Issues. J. Hydrol. Eng.
**2014**, 12, 381–393. [Google Scholar] [CrossRef] - Jeong, D.I.; Sushama, L.; Khaliq, M.N.; Roy, R. A copula-based multivariate analysis of Canadian RCM projected changes to flood characteristics for Northeastern Canada. Clim. Dyn.
**2014**, 42, 2045. [Google Scholar] [CrossRef] - Kao, S.-C.; Govindaraju, R.S. Trivariate statistical analysis of extreme rainfall events via the Plackett family of copulas. Water Resour. Res.
**2008**, 44. [Google Scholar] [CrossRef] [Green Version] - Wahl, T.; Jain, S.; Bender, J.; Meyers, S.D.; Luther, M.E. Increasing risk of compound flooding from storm surge and rainfall for major US cities. Nat. Clim. Chang.
**2015**, 5, 1093–1097. [Google Scholar] [CrossRef] - Salvadori, G.; De Michele, C. Frequency analysis via copulas: Theoretical aspects and applications to hydrological events. Water Resour. Res.
**2004**, 40. [Google Scholar] [CrossRef] [Green Version] - Salvadori, G.; De Michele, C. Multivariate multiparameter extreme value models and return periods: A copula approach. Water Resour. Res.
**2010**, 46, 46. [Google Scholar] [CrossRef] - Xu, K.; Yang, D.; Xu, X.; Lei, H. Copula based drought frequency analysis considering the spatio-temporal variability in Southwest China. J. Hydrol.
**2015**, 527, 630–640. [Google Scholar] [CrossRef] - Kao, S.-C.; Govindaraju, R.S. A copula-based joint deficit index for droughts. J. Hydrol.
**2010**, 380, 121–134. [Google Scholar] [CrossRef] - Xu, K.; Ma, C.; Lian, J.; Bin, L. Joint probability analysis of extreme precipitation and storm tide in a coastal city under changing environment. PLoS ONE
**2014**, 9, e109341. [Google Scholar] [CrossRef] [PubMed] - Tu, X.; Du, Y.; Singh, V.P.; Chen, X. Joint distribution of design precipitation and tide and impact of sampling in a coastal area. Int. J. Climatol.
**2018**, 38, e290–e302. [Google Scholar] [CrossRef] - Papalexiou, S.M.; Koutsoyiannis, D. Entropy based derivation of probability distributions: A case study to daily rainfall. Adv. Water Resour.
**2012**, 45, 51–57. [Google Scholar] [CrossRef] - Papalexiou, S.M. Unified theory for stochastic modelling of hydroclimatic processes: Preserving marginal distributions, correlation structures, and intermittency. Adv. Water Resour.
**2018**, 115, 234–252. [Google Scholar] [CrossRef] - Cai, W.; Liu, K. Studies on planning and layout of water sources for urban agglomeration in the Taihu Basin. China Water Resour.
**2017**, 19, 53–56. [Google Scholar] - Xu, X.; Yang, G.; Tan, Y.; Tang, X.; Jiang, H.; Sun, X.; Zhuang, Q.; Li, H. Impacts of land use changes on net ecosystem production in the Taihu Lake Basin of China from 1985 to 2010. J. Geophys. Res. Biogeosci.
**2017**, 122, 690–707. [Google Scholar] [CrossRef] - Harvey, G.L.; Thorne, C.R.; Cheng, X.; Evans, E.P.; Simm, J.D.; Han, S.; Wang, Y. Qualitative analysis of future flood risk in the Taihu Basin, China. J. Flood Risk Manag.
**2009**, 2, 85–100. [Google Scholar] [CrossRef] - Sun, S.; Mao, R. An Introduction to Lake Taihu. In Lake Taihu, China; Qin, B., Ed.; Springer: Dordrecht, The Netherlands, 2008; Volume 87, pp. 12–15. [Google Scholar]
- Vandenberghe, S.; Verhoest, N.E.C.; De Baets, B. Fitting bivariate copulas to the dependence structure between storm characteristics: A detailed analysis based on 105 year 10 min rainfall. Water Resour. Res.
**2010**, 46. [Google Scholar] [CrossRef] [Green Version] - Ghosh, S. Modelling bivariate rainfall distribution and generating bivariate correlated rainfall data in neighbouring meteorological subdivisions using copula. Hydrol. Process.
**2010**, 24, 3558–3567. [Google Scholar] [CrossRef]

**Figure 1.**Flow chart of how spatial distribution rationality is evaluated using conditional probability analysis.

**Figure 4.**AIC values of fitting for extreme rainfall by 3 types of 3-dimensional Copulas. (

**a**) Maximum 30 day; (

**b**) Maximum 60 day; (

**c**) Maximum 90 day.

**Figure 5.**OLS values of fitting for extreme rainfall by 3 types of 3-dimensional Copulas. (

**a**) Maximum 30 day; (

**b**) Maximum 60 day; (

**c**) Maximum 90 day.

**Figure 6.**Scatterplots of theoretical-empirical frequencies fitted by 3-dimensional Clayton Copulas. (

**a**) Sub-basin YCDM; (

**b**) Sub-basin HQ; (

**c**) Sub-basin SH; (

**d**) Sub-basin HJH; (

**e**) Sub-basin ZX.

**Figure 7.**Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 1 ”typical year scheme”). (

**a**) 50 years return period; (

**b**) 100 years return period.

**Figure 8.**Radar plots of conditional probabilities of the five non-prioritized sub-basins encountering a corresponding rainfall (scheme 2 ”average scheme”). (

**a**) 50 years return period; (

**b**) 100 years return period.

Copula Function | Dimension | Cumulative Distribution Expressions |
---|---|---|

Clayton Copula | 2-dimensional | $F(p,z)=C({u}_{1},{u}_{2})={({u}_{1}{}^{-\theta}+{u}_{2}{}^{-\theta}-1)}^{-1/\theta}$ |

3-dimensional | $F(p,z,r)=C({u}_{1},{u}_{2},{u}_{3})={({u}_{1}{}^{-\theta}+{u}_{2}{}^{-\theta}+{u}_{3}{}^{-\theta}-2)}^{-1/\theta}$ | |

Gumbel Copula | 2-dimensional | $F(p,z)=C({u}_{1},{u}_{2})=\mathrm{exp}\{-{[{(-\mathrm{ln}{u}_{1})}^{\theta}+{(-\mathrm{ln}{u}_{2})}^{\theta}]}^{1/\theta}\}$ |

3-dimensional | $F(p,z,r)=C({u}_{1},{u}_{2},{u}_{3})=\mathrm{exp}\{-{[{(-\mathrm{ln}{u}_{1})}^{\theta}+{(-\mathrm{ln}{u}_{2})}^{\theta}+{(-\mathrm{ln}{u}_{3})}^{\theta}]}^{1/\theta}\}$ | |

Frank Copula | 2-dimensional | $F(p,z)=C({u}_{1},{u}_{2})=-\frac{1}{\theta}\mathrm{ln}[1+\frac{({e}^{-\theta {u}_{1}}-1)({e}^{-\theta {u}_{2}}-1)}{({e}^{-\theta}-1)}]$ |

3-dimensional | $F(p,z,r)=C({u}_{1},{u}_{2},{u}_{3})=-\frac{1}{\theta}\mathrm{ln}[1+\frac{({e}^{-\theta {u}_{1}}-1)({e}^{-\theta {u}_{2}}-1)({e}^{-\theta {u}_{3}}-1)}{{({e}^{-\theta}-1)}^{2}}]$ |

Indexes | Max-30 (1951~2014) | Max-60 (1951~2014) | Max-90 (1951~2014) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

P-III | LOG | LOGN | WEI | P-III | LOG | LOGN | WEI | P-III | LOG | LOGN | WEI | ||

BASIN | K-S | 0.092 | 0.089 | 0.073 | 0.115 | 0.054 | 0.063 | 0.052 | 0.095 | 0.083 | 0.081 | 0.067 | 0.112 |

AIC | 702.1 | 704.6 | 700.2 | 716.6 | 739.0 | 741.9 | 738.9 | 747.3 | 770.1 | 773.3 | 769.0 | 782.1 | |

NORTH | K-S | 0.099 | 0.079 | 0.119 | 0.122 | 0.094 | 0.079 | 0.088 | 0.136 | 0.100 | 0.082 | 0.094 | 0.145 |

AIC | 722.1 | 723.6 | 725.3 | 732.3 | 755.9 | 756.7 | 758.1 | 764.3 | 781.5 | 782.7 | 781.3 | 791.8 | |

YCDM | K-S | 0.093 | 0.098 | 0.073 | 0.129 | 0.067 | 0.070 | 0.064 | 0.095 | 0.065 | 0.060 | 0.083 | 0.106 |

AIC | 723.4 | 728.5 | 723.3 | 730.6 | 756.4 | 760.5 | 757.1 | 760.9 | 779.7 | 783.1 | 780.3 | 785.4 | |

HQ | K-S | 0.074 | 0.073 | 0.067 | 0.121 | 0.064 | 0.083 | 0.081 | 0.096 | 0.067 | 0.077 | 0.054 | 0.124 |

AIC | 719.5 | 723.7 | 716.6 | 734.3 | 751.1 | 754.3 | 750.9 | 760.2 | 776.9 | 780.3 | 775.4 | 790.4 | |

SH | K-S | 0.066 | 0.044 | 0.078 | 0.105 | 0.055 | 0.062 | 0.060 | 0.098 | 0.077 | 0.090 | 0.070 | 0.107 |

AIC | 715.4 | 716.8 | 714.3 | 728.3 | 751.8 | 756.1 | 751.1 | 760.8 | 771.3 | 774.3 | 769.4 | 786.1 | |

HJH | K-S | 0.098 | 0.111 | 0.084 | 0.105 | 0.056 | 0.069 | 0.048 | 0.087 | 0.086 | 0.096 | 0.071 | 0.090 |

AIC | 715.3 | 720.6 | 714.0 | 725.5 | 749.1 | 753.7 | 748.6 | 757.1 | 779.4 | 783.7 | 777.7 | 791.2 | |

ZX | K-S | 0.066 | 0.077 | 0.065 | 0.119 | 0.070 | 0.078 | 0.069 | 0.095 | 0.091 | 0.095 | 0.082 | 0.105 |

AIC | 728.5 | 733.0 | 726.3 | 741.9 | 764.9 | 769.6 | 763.8 | 774.6 | 793.2 | 797.5 | 791.0 | 807.0 |

**Table 3.**Conditional probabilities of the five non-prioritized sub-basins encountering corresponding rainfall in scheme 1 (“typical year scheme”).

Return Period | Duration of Extreme Rainfall | Conditional Rainy Zone (mm) | Rainfall (mm) of Non-Prioritized Sub-Basins and Corresponding Conditional Probabilities | |||||
---|---|---|---|---|---|---|---|---|

Basin | North | YCDM | HQ | SH | HJH | ZX | ||

50 years | Max 30-day rainfall | 514.8 | 540.8 | 601.0 | 546.1 | 532.5 | 447.6 | 456.3 |

Conditional probability | - | - | 1.4% | 4.9% | 2.7% | 18.7% | 44.3% | |

Max 60-day rainfall | 727.7 | 740.4 | 768.2 | 708.3 | 723.8 | 667.4 | 763.0 | |

Conditional probability | - | - | 4.1% | 12.2% | 4.2% | 12.8% | 16.6% | |

Max 90-day rainfall | 908.1 | 936.3 | 946.3 | 881.1 | 889.8 | 820.1 | 964.9 | |

Conditional probability | - | - | 5.3% | 14.9% | 4.9% | 22.6% | 24.9% | |

100 years | Max 90-day rainfall | 975.1 | 1000.0 | 1013.1 | 943.3 | 952.5 | 877.9 | 1033.0 |

Conditional probability | - | - | 2.2% | 7.1% | 2.0% | 12.2% | 12.9% |

**Table 4.**Conditional probabilities of the five non-prioritized sub-basins encountering corresponding rainfall in scheme 2 (“average scheme”).

Return Period | Duration of Extreme Rainfall | Conditional Rainy Zone (mm) | Rainfall (mm) of Non-Prioritized Sub-Basins and Corresponding Conditional Probabilities | |||||
---|---|---|---|---|---|---|---|---|

Basin | North | YCDM | HQ | SH | HJH | ZX | ||

50 years | Max 30-day rainfall | 514.8 | 540.8 | 480.7 | 482.7 | 465.1 | 484.7 | 569.0 |

Conditional probability | - | - | 13.7% | 15.5% | 10.5% | 10.2% | 9.0% | |

Max 60-day rainfall | 727.7 | 740.4 | 676.7 | 688.2 | 754.9 | 689.4 | 819.6 | |

Conditional probability | - | - | 15.6% | 16.3% | 12.0% | 10.2% | 9.0% | |

Max 90-day rainfall | 908.1 | 936.3 | 860.7 | 860.7 | 815.5 | 871.3 | 964.9 | |

Conditional probability | - | - | 17.7% | 17.2% | 13.2% | 12.9% | 10.5% | |

100 years | Max 90-day rainfall | 975.1 | 1000.0 | 904.6 | 924.1 | 875.6 | 935.5 | 1125.7 |

Conditional probability | - | - | 9.5% | 9.2% | 7.2% | 7.1% | 5.4% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, L.; Hu, Q.; Wang, Y.; Zhu, Z.; Li, L.; Liu, Y.; Cui, T.
Using Copulas to Evaluate Rationality of Rainfall Spatial Distribution in a Design Storm. *Water* **2018**, *10*, 758.
https://doi.org/10.3390/w10060758

**AMA Style**

Wang L, Hu Q, Wang Y, Zhu Z, Li L, Liu Y, Cui T.
Using Copulas to Evaluate Rationality of Rainfall Spatial Distribution in a Design Storm. *Water*. 2018; 10(6):758.
https://doi.org/10.3390/w10060758

**Chicago/Turabian Style**

Wang, Leizhi, Qingfang Hu, Yintang Wang, Zhenduo Zhu, Lingjie Li, Yong Liu, and Tingting Cui.
2018. "Using Copulas to Evaluate Rationality of Rainfall Spatial Distribution in a Design Storm" *Water* 10, no. 6: 758.
https://doi.org/10.3390/w10060758