# The Copula Application for Analysis of the Flood Threat at the River Confluences in the Danube River Basin in Slovakia

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

_{maxup}–Q

_{maxtr}, where Q

_{maxup}is the maximum annual discharge of the mainstream above the confluence, Q

_{maxtr}is the maximum annual discharge of the tributary above the confluence, and Q

_{maxdwn}represents the maximum annual discharge of the mainstream below the confluence. The Váh River and its tributary, the Belá River, showed the highest differences in the dates when the annual maximum discharges occur (Figure 3a). In addition, the Nitra River and its tributary, the Bebrava River, showed the lowest difference in the dates when the annual maximum discharges occurred (Figure 3b).

#### 2.2. Method

_{i}and y

_{i}; and n

_{ml}is the number of occurrences of the combinations of x

_{i}and y

_{j}.

_{(x)}, F

_{(y)}) is the bivariate joint distribution function expressed as a copula function and F

_{(x)}and F

_{(y)}are the marginal distribution functions of the variable X and Y. The operator of the penetration (X $\ge $ x and Y $\ge $ y) is used in Equation (3), and therefore, this formula was used to calculate the joint return period if both investigated quantities exceeded a certain threshold value. Equation (4), on the other hand, works with the operator of the unification (X $\ge $ x or Y $\ge $ y). For this reason, it was used to calculate the joint return period if only one of the monitored values exceeded the given threshold value. These relationships indicate that different combinations of the numbers x and y can cover the same return period.

_{(x)}, F

_{(y)}) is the copula function of the random variables X and Y. An equivalent formula for the conditional return period of Y $\le $ y, given X $\le $ x, can thus be obtained.

## 3. Results

#### 3.1. Univariate Statistical Analysis of Flood Hazards

#### 3.2. Bivariate Statistical Analysis of Flood Hazards at River Confluences Using Gumbel–Hougaard Copula

_{maxup}−Q

_{maxtr}between the Váh River and the Belá River reached the lowest value of the Kendall rank correlation (Figure 5). The created combinations of the hydrological variables were used in the bivariate frequency analysis to investigate how the relationship of the hydrological characteristics may affect the size of extreme hydrological situations.

_{maxup}−Q

_{maxtr}are illustrated in Figure 6a–d. According the above-mentioned criteria, the Gumbel–Hougaard copula was deemed to be a suitable statistical tool to calculate the joint probability distribution of the discharges in our study.

_{maxup}−Q

_{maxtr}were estimated. Figure 7 illustrates the scatter plot of the monitored annual maximum discharges and values generated by using the Gumbel–Hougard copula. Figure 7 also illustrates the isolines of the joint return periods “or” and “and”, which are the level curves of the Gumbel–Hougaard copula of interest if the variables exceeded the outward bounds and inward bounds, respectively.

_{50}, Q

_{100}, Q

_{200}, Q

_{500}, and Q

_{1000}) based on the univariate approach and the copula approach for gauging stations below the selected mainstream river confluences. For the traditional univariate method, the resulting discharge for the selected return period was calculated as a reciprocal of the probability of exceedance.

## 4. Discussion

## 5. Conclusions

- -
- The copula-based joint probability approach for the confluence flood estimation performed well for the selected river basins;
- -
- The copula-based joint probability approach provides a way to estimate the confluence flood without the discharge records needed for the mainstream below the confluence and without difficult computations such as flow routing;
- -
- The copula functions for the multivariate analyses enable the use of various types of marginal distributions and thus release the limitation of the others in the case of multivariate approaches where the margins follow the same type of distributions. In our study, based on the selected criterions and the tests, the same type of probability distribution fit the analyzed data, except for Nitrianska Streda Station, situated below the Nitra–Bebrava confluence;
- -
- The joint return periods calculated using copulas could be used to determine the severity of floods based on the desired relations between the mainstreams and their tributaries, looking for the exceedance of both variables.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Maximum annual discharges for the mainstreams and their tributaries during the analyzed periods: Morava–Myjava, Váh–Belá, Nitra–Bebrava, Hron–Slatina.

**Figure 3.**Occurrence of the annual maximum discharges in Julian days for (

**a**) Váh: Liptovský Hádok (1950–2011) and Belá: Podbanské (1950–2011); (

**b**) Nitra: Chynorany (1940–2011) and Bebrava: Nadlice (1940–2011).

**Figure 4.**Comparison of the empirical exceedance probabilities (points) and the theoretical exceedance probability curves (line) of the maximum annual discharges for the selected mainstreams and tributaries during the analyzed periods.

**Figure 5.**(

**a**) Pearson correlation R of the maximum annual discharges and (

**b**) Kendall’s rank correlation τ of the maximum annual discharges.

**Figure 6.**The joint empirical probability (points) and Gumbel–Hougaard copula at selected rivers and their tributaries (Q

_{maxup}−Q

_{maxtr}).

**Figure 7.**Scatter plots of 9000 data pairs generated from Gumbel–Hougaard copula and monitored data of the selected pairs Q

_{maxup}−Q

_{maxtr}. The contours in the return periods (

**a**) in the case of “or” (X ≥ x or Y ≥ y, only one investigated variables exceeded a certain threshold value) and (

**b**) in the case of “and” (X ≥ x and Y ≥ y, both investigated variables exceeded a certain threshold value).

**Table 1.**Selected mainstreams and their tributaries, stations, analyzed periods, annual maximum discharge, river kilometer, and basin area.

River | Gauging Station | Period [Year] | Q_{max} [m ^{3} s^{−1}] | River Kilometer [rkm] | Area [km ^{2}] |
---|---|---|---|---|---|

Morava | Strážnica (up) | 1968–2019 | 901 | 134.3 | 9146.92 |

Moravský Svätý Ján (dwn) | 1968–2019 | 1400 | 67.15 | 24,129.30 | |

Myjava | Šaštín Stráže (tr) | 1968–2019 | 82 | 15.15 | 644.89 |

Váh | Liptovský Hrádok (up) | 1950–2019 | 240 | 359.3 | 638.38 |

Liptovský Mikuláš (dwn) | 1950–2019 | 540 | 343.6 | 1107.21 | |

Belá | Podbanské (tr) | 1950–2019 | 170 | 21.35 | 93.49 |

Nitra | Chynorany (up) | 1951–2019 | 279 | 106 | 1134.28 |

Nitrianska Streda (dwn) | 1951–2019 | 324 | 91.1 | 2093.71 | |

Bebrava | Nadlice (tr) | 1951–2019 | 128 | 6.2 | 598.8 |

Hron | Banská Bystrica (up) | 1972–2019 | 260 | 175.2 | 1766.48 |

Žiar nad Hronom (dwn) | 1972–2019 | 636 | 131.5 | 3310.69 | |

Slatina | Zvolen (tr) | 1972–2019 | 220 | 12.1 | 790.16 |

Copula Function | C (u, v, θ) | Parameter θ | Kendall’s τ | Generator φ(t) |
---|---|---|---|---|

Gumbel–Hougaard | $\mathrm{exp}\left[-{({(-\mathrm{ln}u)}^{\theta}+{(-\mathrm{ln}v)}^{\theta})}^{1/\theta}\right]$ | [1, $\infty $) | $\frac{\theta -1}{\theta}$ | ${(-\mathrm{ln}t)}^{\theta}$ |

**Table 3.**Estimated univariate designed discharge Q

_{max}for various return periods T and p-values of the Kolmogorov–Smirnov test.

Confluence | Q [m ^{3} s^{−1}] | Distr. | p Value | Estimated Q_{T}[m ^{3} s^{−1}] | Monitored Q_{max} [m ^{3} s^{−1}] | |||||
---|---|---|---|---|---|---|---|---|---|---|

Q50 | Q100 | Q200 | Q500 | Q1000 | Q_{max} | T [year] | ||||

Morava–Myjava | Q_{maxup} | JSB | 0.932 | 815 | 892 | 966 | 1059 | 1127 | 901 | 145 |

Q_{maxtr} | JSB | 0.812 | 80 | 85 | 90 | 95 | 98 | 82 | 70 | |

Q_{maxdwn} | JSB | 0.911 | 1221 | 1351 | 1473 | 1621 | 1723 | 1400 | 160 | |

Váh–Belá | Q_{maxup} | JSB | 0.24 | 200 | 232 | 259 | 290 | 313 | 240 | 160 |

Q_{maxtr} | JSB | 0.95 | 136 | 160 | 185 | 213 | 234 | 170 | 160 | |

Q_{maxdwn} | JSB | 0.87 | 372 | 435 | 499 | 587 | 652 | 540 | 310 | |

Nitra–Bebrava | Q_{maxup} | JSB | 0.92 | 225 | 247 | 268 | 295 | 314 | 279 | 220 |

Q_{maxtr} | JSB | 0.91 | 119 | 125 | 130 | 134 | 137 | 128 | 140 | |

Q_{maxdwn} | Weib. | 0.76 | 322 | 346 | 368 | 386 | 400 | 324 | 60 | |

Hron–Slatina | Q_{maxup} | JSB | 0.97 | 266 | 277 | 285 | 294 | 299 | 268 | 50 |

Q_{maxtr} | JSB | 0.83 | 206 | 221 | 234 | 247 | 256 | 220 | 100 | |

Q_{maxdwn} | JSB | 0.86 | 590 | 643 | 692 | 756 | 806 | 636 | 90 |

**Table 4.**The Gumbel–Hougaard copula parameters for the selected variable combinations, mean absolute errors (MAE) values, and K–S test.

Confluence | Pair | Kendall’s τ | Parameter Copula | MAE [%] | p-Value |
---|---|---|---|---|---|

Morava–Myjava | Q_{maxup}–Q_{maxtr} | 0.366 | 1.577 | 4.02 | 0.73 |

Váh–Belá | Q_{maxup}−Q_{maxtr} | 0.225 | 1.290 | 2.27 | 0.96 |

Nitra–Bebrava | Q_{maxup}−Q_{maxtr} | 0.476 | 1.908 | 5.94 | 0.052 |

Hron–Slatina | Q_{maxup}−Q_{maxtr} | 0.366 | 1.567 | 4.94 | 0.78 |

**Table 5.**Comparison of design discharges (Q

_{50}, Q

_{100}, Q

_{500}, Q

_{1000}) based on univariate (Uni.) and based on the copula method at the mainstream stations below the confluences Q

_{maxdwn}.

Confluence (Station on Mainstream below the Confluence) | Method/Differences | Estimated Q_{T}[m ^{3} s^{−1}] | ||||
---|---|---|---|---|---|---|

Q_{50} | Q_{100} | Q_{200} | Q_{500} | Q_{1000} | ||

Morava–Myjava (Morava: Moravský Sv. Ján) | Uni−SB distr. | 1221 | 1351 | 1473 | 1621 | 1723 |

copula G–H | 1369 | 1500 | 1623 | 1722 | 1878 | |

Difference [%] | 12 | 11 | 10 | 6 | 9 | |

Váh–Belá (Váh: Liptovský Mikuláš) | Uni−JSB distr. | 372 | 435 | 499 | 587 | 652 |

copula G–H | 446 | 508 | 570 | 651 | 712 | |

Difference [%] | 20 | 17 | 14 | 11 | 9 | |

Nitra–Bebrava (Nitra: Nitrianska Streda) | Uni−weib. distr. | 322 | 346 | 368 | 386 | 400 |

copula G–H | 336 | 354 | 369 | 388 | 401 | |

Difference [%] | 4 | 2 | 0 | 1 | 0 | |

Hron–Slatina (Hron: Žiar nad Hronom) | Uni−JSB distr. | 590 | 643 | 692 | 756 | 806 |

copula G–H | 649 | 699 | 747 | 808 | 853 | |

Difference [%] | 10 | 9 | 8 | 7 | 6 |

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**MDPI and ACS Style**

Bačová Mitková, V.; Halmová, D.; Pekárová, P.; Miklánek, P.
The Copula Application for Analysis of the Flood Threat at the River Confluences in the Danube River Basin in Slovakia. *Water* **2023**, *15*, 984.
https://doi.org/10.3390/w15050984

**AMA Style**

Bačová Mitková V, Halmová D, Pekárová P, Miklánek P.
The Copula Application for Analysis of the Flood Threat at the River Confluences in the Danube River Basin in Slovakia. *Water*. 2023; 15(5):984.
https://doi.org/10.3390/w15050984

**Chicago/Turabian Style**

Bačová Mitková, Veronika, Dana Halmová, Pavla Pekárová, and Pavol Miklánek.
2023. "The Copula Application for Analysis of the Flood Threat at the River Confluences in the Danube River Basin in Slovakia" *Water* 15, no. 5: 984.
https://doi.org/10.3390/w15050984