# Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Study Area and Data

^{2}and its most important tributary—the Mura River—it connects the Alpine region with the Pannonian region. The Drava River can be divided into three sections by river regime: the upper part which exhibits the Alpine nival-pluvial regime characterized by the construction of numerous dams and reservoirs, which are mainly used for hydroelectric production; the middle part affected by the tributaries (from 255,050 rkm to 53,800 rkm in Croatia); the lower part that stretches to the confluence with the Danube River and is under the influence of the Danube River backwater effect [41]. Prior to the hydro-regulation of the Drava River, the frequency of flood events was quite high. Nowadays, changes in the hydrological regime of the Drava River in Croatia can be described as predominantly influenced by anthropogenic factors and more recently attributed to climate change [42]. Moreover, Bonacci and Oskorus [43] reported the reduction of minimum and average discharges and all characteristic water levels of the lower Drava River, which agrees with similar results of other authors who conclude that the discharge analysis of the Drava River shows a negative trend [44,45]. Additionally, the analysis of the highest flood events in the period before and after the construction of large reservoirs for hydropower plants shows similar results [46].

^{2}), and together Sava and Danube form a large part of the Black Sea drainage basin area. The upstream part of the Sava River exhibits the Alpine nival-pluvial regime, which changes along the downstream Sava River into the Peripannonian pluvial-nival regime, while in the middle course of the Sava River near the town of Jasenovac it gradually changes into the Pannonian pluvial-nival regime [38,47]. Hydrological research on the part of the Sava River that flows through Croatia mainly focused on the changes in floods at the Zagreb gauging station, where it was concluded that the discharge regime is under the influence of engineering works carried out as part of the flood protection system of the Zagreb city area [48,49,50,51,52]. Moreover, Segota and Filpcic [53] reported that the discharge regime at the Zagreb gauging station is linked to scientifically observed climate changes, while Oresic et al. [54] extended that conclusion to the middle section of the Sava River in the 1931–2010 period. Additionally, the analysis of flood waves along the middle course showed strong variability on multiple scales [55,56] and a preliminary approach to clustering of flood hydrograph shapes was conducted at gauging station Zagreb [57].

^{2}and the Sava River has a wider range, from 12,316 to 62,891 km

^{2}, due to the large tributaries from Bosnia and Herzegovina. The mean AM discharge values for the Drava River stations range from 1430 m

^{3}/s at GS Botovo to 1242 m

^{3}/s at GS Belisce. However, the mean AM discharge series on the Sava River stations show a wider range of values, from 1648 m

^{3}/s at GS Podsused to 2679 m

^{3}/s at GS Zupanja. In addition, there is a considerable range in observed mean AM duration and volume data, influenced by tributaries (especially right tributaries with large water yield e.g., Una, Vrbas, Bosna, Drina) and the Middle Sava flood protection system retention areas located upstream of GS Jasenovac. It can be seen that the operation of the hydropower plants has an influence on the formation of the hydrograph, since the peak discharge values of the Drava River do not increase with the catchment area (Table 1). This is in agreement with a previous study conducted by Potočki et al. [46].

#### 2.2. Flood Characteristics

- Divide the mean daily discharge data into non-overlapping blocks of N days and calculate the minima for each of these blocks, and let them be called Q
_{1}, Q_{2}, Q_{3}, … Q_{i}. - Consider in turn (Q
_{1}, Q_{2}, Q_{3}), (Q_{2}, Q_{3}, Q_{4}), … (Q_{i − 1}, Q_{i}, Q_{i + 1}), etc. - In each case, if 0.9·Q
_{i}< Q_{i − 1}and 0.9·Q_{i}< Q_{i + 1}, then the central value is an ordinate for the baseflow line. Continue the procedure until a derived set of baseflow ordinates QB_{1}, QB_{2}, QB_{3}, … QB_{n}is provided with different time periods between them. - Apply linear interpolation between each QB
_{i}value and estimate each daily value of QB_{1}… Q_{1}. - If QBi > Qi, then set QBi = Qi.

Baseflow Separation Method | Acronym | Reference |
---|---|---|

Baseflow index method | BFI | Gustard et al. [61]; Koffler and Laaha [63] |

Recursive digital filter method | RDF1 | Lyne and Hollick [64] |

RDF2 | ||

RDF3 | ||

Sliding interval method | HYSEP1 | Sloto and Crouse [70]; Source code available at: https://github.com/USGS-R/DVstats/blob/main/R/hysep.R, accessed on 1 June 2022 |

Fixed interval method | HYSEP2 | |

Local minimum method | HYSEP3 |

- The sliding interval method (HYSEP1) finds the lowest discharge in one-half of the interval minus 1 day before and after the day being considered and assigns it to that day.
- The fixed interval method (HYSEP2) assigns the lowest discharge in each interval to all days in that interval starting with the first day of the period of record.
- The local minimum method (HYSEP3) checks each day to determine if it is the lowest discharge in one-half of the interval minus 1 day before and after the day is considered. If it is, then it is a local minimum and is connected by straight lines to adjacent local minimums.

- Erase all data points of daily streamflow with $\frac{{d}_{y}}{{d}_{t}}\ge 0$, where $\frac{{d}_{{y}_{i}}}{{d}_{t}}=\frac{{y}_{i+1}-{y}_{i-1}}{2}$ that represents the slope of the curve between two consecutive points.
- Eliminate the previous 2 points before points with $\frac{{d}_{y}}{{d}_{t}}\ge 0$, as well as the next three points.
- Eliminate 5 points after major events that were identified by flood peaks greater than the 90th quantile of all streamflow observations [61].
- Exclude data points followed by a data point with smaller $\frac{{d}_{y}}{{d}_{t}}$, namely $\frac{{d}^{2}y}{d{x}^{2}}\ge 0$.

_{1}, t

_{2}, …, t

_{n}represent the date of the nth strict baseflow point, ${Q}_{m}^{t}$ is the baseflow estimated from a baseflow separation method in date t, ${Q}_{0}^{t}$ is the value of the strict baseflow point in date t, $\overline{{Q}_{0}}$ is the mean value of all strict baseflow points. The NSE ranges from −∞ to 1, where an NSE value of 1 indicates a perfect match of the baseflow estimated from the baseflow separation method and the strict baseflow points.

_{s}and q

_{e}are the daily discharge values on the starting and ending dates of the flood event, q

_{j}denotes the jth day daily discharge value, and D represents the duration of the flood event.

_{0}assuming there is no trend in the sample, and the null hypothesis for the Ljung–Box test is that the time series are not autocorrelated. The significance level of 0.05 was used in this study.

#### 2.3. Marginal Probability Distributions

#### 2.4. Copulas

_{n}was used to check the adequacy of the selected copula functions, and the parameters of all copulas were estimated using the method of moments with the use of the Kendall correlation coefficient [14,16,88]. The best-fitted copula function was then selected using the model selection criterion (i.e., function xvCopula), which is based on the k-fold cross-validation and is also implemented in the R package “Copula” [89,90,91].

_{and}and T

_{or}[92,93], where T

_{and}represents the return period where both u and v are exceeded, and T

_{or}where only u or v is exceeded [16,87,94,95]. T

_{and}and T

_{or}return periods are defined by equations where u and v are marginal distributions:

## 3. Results and Discussion

#### 3.1. Selection of Baseflow Separation Method for Computing Flood Hydrograph Characteristics

#### 3.2. Selection of Marginal Probability Distributions for Q, D and V Series

#### 3.3. Copula Model Estimation

_{n}) applied with the R package “Copula” (Table 9) at a 0.05 significance level. The stations on the Drava River have the same best copula function for the Q–D (Huesler–Reiss) and V–D (Normal) pairs. Moreover, different copulas show the best fit for the Q–V pair. While for the Sava River, the results are more diverse, and no dominant copula function could be identified in this case. Moreover, it can be seen that none of the tested copulas could be rejected by the selected statistical test and taking into consideration the significance level of 0.05.

#### 3.4. Joint Return Periods

_{and}and T

_{or}were calculated using best-fitting copulas for each pair of variables (i.e., Q–D, Q–V, and V–D). Different T

_{and}(T

_{and}(Q

_{10}, V

_{10}), T

_{and}(Q

_{100}, V

_{100}), T

_{and}(Q

_{10}, V

_{100}), T

_{and}(Q

_{100}, V

_{10}) and T

_{or}(T

_{or}(Q

_{10}, V

_{10}), Tor (Q

_{100}, V

_{100}), T

_{or}(Q

_{10}, V

_{100}) and T

_{or}(Q

_{100}, V

_{10}) were calculated for all stations on the Drava River and the Sava River (Table 10). According to Salvadori et al. [92], which is in accordance with the results of the analysis, the relationship between univariate and primary (bivariate) return periods can be written as: ${T}_{u,v}^{OR}<{T}^{UNI}<{T}_{u,v}^{AND}$. The results show that the corresponding peak discharge, hydrograph duration and hydrograph volume values for a return period of 10 years (${T}^{UNI}$) are in the range 1778.1–2054.7 m

^{3}/s, 38.7–39.4 days and 920.9–970.2 10

^{6}m

^{3}for stations on the Drava River, and in the range 2278.8–3450.0 m

^{3}/s, 83.2–98.9 days and 3185.8–4821.4 10

^{6}m

^{3}for stations on the Sava River, respectively. To get the values of the observed flood variables for the future scour analysis around bridges for a fixed value of the OR and AND return periods (${T}_{u,v}^{OR},{T}_{u,v}^{AND})$, different combinations of variables can be selected. It can be seen that the specific design discharge values (e.g., Q

_{10}) decrease with increasing catchment area while different results can be obtained for higher return periods (Table 10).

#### 3.5. Preliminary Methodology for the Bridge Scour Analysis Using Copulas

_{100}and V

_{100}, respectively. At the same time, this hydrograph corresponds to the bivariate return periods T

_{AND}and T

_{OR}256 years and 62 years, respectively (Table 10). Therefore, this design hydrograph can be used further as input to the hydraulic model for the bridge scour analysis.

## 4. Conclusions

_{and}and T

_{or}were calculated using the best-fitting copulas (and marginal distributions) for each pair of variables (i.e., Q–D, Q–V and V–D).

- The HYSEP1 baseflow separation method can be regarded as an appropriate choice for baseflow separation for stations on the Drava River. In order to apply the baseflow evaluation criterion proposed by Xie et al. [59] at the stations in the middle part of the Sava River, additional analyses should be performed, or the proposed rules should be modified to correspond to the complex flood regime that prevails there. This indicates the importance of the visual inspection of the results, especially in the case of rivers where there are significant effects of dam operation and/or flood protection systems on flood hydrograph characteristics. Additionally, some of the tested baseflow separation methods did not yield useful results. Hence, it is advised that further studies that deal with flood hydrograph characteristics test multiple baseflow separation methods since extracted V and D variables can be highly sensitive to the selection of the baseflow separation methods. The differences among tested methods can yield V and D values that differ by an order of magnitude. Hence, this can lead to over- or under-estimation of the design variables.
- The Huesler–Reiss copula from the extreme-value family of copulas was selected as the most suitable copula for modelling peak discharges and hydrograph durations at all stations of the Drava River, while the most appropriate copula for modelling hydrograph volumes and hydrograph durations seems to be the Normal copula from the elliptical family of copulas. On the other hand, for the Sava River, more diverse results were obtained indicating non-uniform flow characteristics along the Sava River in Croatia.
- Different combinations of variables Q, D and V derived from the bivariate copula results for each station can eventually be computed if there is a need in practical applications (e.g., design, scour analysis, etc.). Hence, a preliminary methodology for the implication of the bivariate flood frequency analysis using copulas for the bridge scour analysis is proposed. As an example, the design hydrograph for one station on the Sava River is derived.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of the observed gauging stations on the Drava River and the Sava River in Croatia.

**Figure 2.**Example of the applied baseflow separation methods and strict baseflow points at gauging station Terezino polje on the Drava River for the year 2014.

**Figure 3.**Example of QQ plots for the selected marginal distributions for the Botovo GS on the Drava River.

**Figure 5.**Example of design hydrograph for the Zupanja GS on the Sava River that was constructed based on the measured AM flood hydrograph from the year 2014. The surface runoff volume corresponds to the V

_{100}(i.e., 7454 × 10

^{6}m

^{3}) while peak discharge corresponds to Q

_{100}(i.e., 5060 m

^{3}/s). At the same time, the design hydrograph corresponds to the bivariate return periods T

_{AND}and T

_{OR}256 years and 62 years, respectively.

**Table 1.**Basic properties of the observed annual maximum series of hydrological variables: peak discharge (maximum—Q

_{max}; mean—Q

_{mean}and standard deviation—Q

_{sd}), hydrograph duration (maximum—D

_{max}; mean—D

_{mean}and standard deviation—D

_{sd}) and hydrograph volume (maximum—V

_{max}; mean—V

_{mean}and standard deviation—V

_{sd}) for gauging stations on the Drava and Sava River.

River | Station | Watershed Area [km^{2}] | Period; Years of Data | Q_{max}; Q_{mean}; Q_{sd} (m^{3}/s) | D_{max}; D_{mean}; D_{sd} (day) | V_{max}; V_{mean}; V_{sd} (10 ^{6} m^{3}) |
---|---|---|---|---|---|---|

Drava | Botovo | 31,038.0 | 1962–2019; 47 | 2551; 1430; 464.6 | 58; 25; 10.3 | 1408; 558; 299.4 |

Terezino polje | 33,916.0 | 1962–2019; 47 | 2778; 1379; 493.5 | 62; 26; 11.1 | 1511; 563; 315.8 | |

Donji Miholjac | 37,142.0 | 1962–2019; 46 | 2140; 1269; 371.5 | 56; 27; 10.2 | 1367; 530; 282.3 | |

Belisce | 38,500.0 | 1962–2019; 46 | 2035; 1242; 326.9 | 57; 25; 10.2 | 1355; 508; 273.0 | |

Sava | Podsused | 12,316.0 | 1951–2019; 55 | 3038; 1648; 488.1 | 59; 25; 12.2 | 1195; 560; 221.7 |

Jasenovac | 38,953.0 | 1951–2019; 53 | 2759; 1884; 371.1 | 113; 55; 20.8 | 4059; 1975; 850.5 | |

Mackovac | 40,838.0 | 1951–2019; 53 | 3100; 1803; 393.7 | 129; 54; 23.4 | 5600; 2020; 1015.5 | |

Zupanja | 62,891.0 | 1951–2019; 54 | 5317; 2679; 657.4 | 174; 63; 29.0 | 6274; 2956; 1326.5 |

Copula Family | Copula | C_{θ} (u, v) |
---|---|---|

Archimedean | Gumbel–Hougaard | $\mathrm{exp}(-{({\left(-\mathrm{ln}u\right)}^{\theta}+{\left(-\mathrm{ln}v\right)}^{\theta})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.})$, θ ∈ [ 1, ∞) |

Clayton | ${\left[{u}^{-\theta}+{v}^{-\theta}-1\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.}$, θ ∈ [ −1, ∞) \ {0} | |

Extreme value | Galambos | $u\xb7\mathrm{exp}(-{({\left(-\mathrm{ln}u\right)}^{\theta}+{\left(-\mathrm{ln}v\right)}^{\theta})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.})$, θ ∈ [ 0, ∞) |

Huesler–Reiss | $\mathrm{exp}\left[-\tilde{u}\Phi \left\{\frac{1}{\theta}+\frac{\theta}{2}\mathrm{ln}\left(\frac{\tilde{u}}{\tilde{v}}\right)\right\}-\tilde{v}\Phi \left\{\frac{1}{\theta}+\frac{\theta}{2}\mathrm{ln}\left(\frac{\tilde{u}}{\tilde{v}}\right)\right\}\right]$, θ ∈ [ 0, ∞) | |

Tawn | $uv\xb7\mathrm{exp}(-{({\left(-\mathrm{ln}u\right)}^{\theta}+{\left(-\mathrm{ln}v\right)}^{\theta})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\theta $}\right.})$, θ ∈ [ 0;1] | |

Elliptical | Normal | ${{\displaystyle \int}}_{-\infty}^{{\Phi}^{-1}\left(u\right)}{{\displaystyle \int}}_{-\infty}^{{\Phi}^{-1}\left(v\right)}\frac{1}{2\pi \ast \sqrt{\left(1-{\theta}^{2}\right)}}\mathrm{exp}\left\{-\frac{{s}^{2}-2\theta st+{t}^{2}}{2\left(1-{\theta}^{2}\right)}\right\}\mathrm{d}s\mathrm{d}t$, θ ∈ [ −1;1] |

River | Station | Evaluation Results | BFI | RDF1 | RDF2 | RDF3 | HYSEP1 | HYSEP2 | HYSEP3 |
---|---|---|---|---|---|---|---|---|---|

Drava | Botovo | NSE | −0.216 | 0.325 | 0.119 | −0.224 | 0.350 | 0.324 | 0.216 |

Rank | 6 | 2 | 5 | 7 | 1 | 3 | 4 | ||

Terezino polje | NSE | −0.189 | 0.337 | 0.118 | −0.248 | 0.407 | 0.367 | 0.238 | |

Rank | 6 | 3 | 5 | 7 | 1 | 2 | 4 | ||

Donji Miholjac | NSE | −0.109 | 0.361 | 0.149 | −0.207 | 0.440 | 0.413 | 0.237 | |

Rank | 6 | 3 | 5 | 7 | 1 | 2 | 4 | ||

Belisce | NSE | −0.277 | 0.266 | 0.023 | −0.378 | 0.387 | 0.313 | 0.164 | |

Rank | 6 | 3 | 5 | 7 | 1 | 2 | 4 | ||

Sava | Podsused | NSE | −0.003 | 0.444 | 0.290 | 0.039 | 0.530 | 0.525 | 0.264 |

Rank | 7 | 3 | 4 | 6 | 1 | 2 | 5 | ||

Jasenovac | NSE | 0.220 | 0.615 | 0.499 | 0.313 | 0.654 | 0.624 | 0.488 | |

Rank | 7 | 3 | 4 | 6 | 1 | 2 | 5 | ||

Mackovac | NSE | 0.188 | 0.627 | 0.507 | 0.315 | 0.678 | 0.650 | 0.480 | |

Rank | 7 | 3 | 4 | 6 | 1 | 2 | 5 | ||

Zupanja | NSE | 0.319 | 0.687 | 0.581 | 0.402 | 0.693 | 0.662 | 0.531 | |

Rank | 7 | 2 | 4 | 6 | 1 | 3 | 5 |

**Table 5.**Example of derived AM samples for flood hydrograph duration and flood hydrograph volume for seven different baseflow separation methods on the Terezino Polje GS on the Drava River.

Year | Peak Date | Peak Value | Variable | BFI | RDF1 | RDF2 | RDF3 | HYSEP1 | HYSEP2 | HYSEP3 |
---|---|---|---|---|---|---|---|---|---|---|

1962 | 6 June 1962 | 1367 | D (day) | 39 | 39 | 39 | 245 | 26 | 21 | 26 |

V (10^{6} m^{3}) | 1327 | 1041 | 1192 | 4700 | 414 | 328 | 505 | |||

1963 | 15 March 1963 | 1683 | D (day) | 30 | 31 | 31 | 151 | 21 | 21 | 21 |

V (10^{6} m^{3}) | 970 | 846 | 923 | 2533 | 684 | 789 | 807 | |||

1964 | 29 October 1964 | 1932 | D (day) | 40 | 69 | 69 | 69 | 25 | 23 | 25 |

V (10^{6} m^{3}) | 1683 | 1618 | 1831 | 2132 | 788 | 842 | 875 | |||

1965 | 6 August 1965 | 2471 | D (day) | 29 | 29 | 29 | 37 | 29 | 22 | 29 |

V (10^{6} m^{3}) | 1119 | 980 | 1020 | 1200 | 998 | 1001 | 1121 | |||

1966 | 23 August 1966 | 2525 | D (day) | 53 | 62 | 62 | 104 | 62 | 25 | 52 |

V (10^{6} m^{3}) | 2047 | 1608 | 1763 | 2986 | 1404 | 1173 | 2086 | |||

1967 | 4 June 1967 | 1398 | D (day) | 24 | 50 | 50 | 161 | 9 | 9 | 9 |

V (10^{6} m^{3}) | 486 | 933 | 1027 | 2678 | 267 | 205 | 248 | |||

1968 | 18 June 1968 | 811 | D (day) | 45 | 50 | 68 | 137 | 12 | 20 | 12 |

V (10^{6} m^{3}) | 815 | 641 | 884 | 1958 | 161 | 221 | 160 | |||

1969 | 22 May 1969 | 979 | D (day) | 41 | 41 | 41 | 41 | 18 | 12 | 18 |

V (10^{6} m^{3}) | 860 | 656 | 741 | 847 | 213 | 176 | 182 | |||

1970 | 14 August 1970 | 1390 | D (day) | 52 | 52 | 100 | 100 | 14 | 14 | 14 |

V (10^{6} m^{3}) | 1228 | 855 | 1173 | 1367 | 353 | 324 | 354 | |||

1971 | 25 March 1971 | 717 | D (day) | 36 | 99 | 99 | 239 | 18 | 18 | 18 |

V (10^{6} m^{3}) | 444 | 922 | 1058 | 2181 | 203 | 165 | 214 | |||

1972 | 19 July 1972 | 2882 | D (day) | 57 | 58 | 58 | 119 | 36 | 16 | 36 |

V (10^{6} m^{3}) | 2415 | 1925 | 2093 | 2748 | 1511 | 978 | 2040 | |||

1973 | 30 September 1973 | 1749 | D (day) | 49 | 49 | 78 | 78 | 27 | 24 | 27 |

V (10^{6} m^{3}) | 1729 | 1272 | 1635 | 1871 | 906 | 827 | 1085 | |||

1974 | 23 October 1974 | 1192 | D (day) | 47 | 29 | 61 | 61 | 15 | 15 | 15 |

V (10^{6} m^{3}) | 1214 | 347 | 1030 | 1160 | 290 | 336 | 298 | |||

1975 | 5 July 1975 | 2578 | D (day) | 50 | 70 | 70 | 121 | 50 | 25 | 50 |

V (10^{6} m^{3}) | 1932 | 1607 | 1776 | 2613 | 1174 | 879 | 1964 | |||

1976 | 29 April 1976 | 1108 | D (day) | 35 | 73 | 125 | 125 | 18 | 13 | 14 |

V (10^{6} m^{3}) | 630 | 867 | 1380 | 1684 | 290 | 316 | 259 | |||

1977 | 11 April 1977 | 1137 | D (day) | 26 | 26 | 86 | 107 | 14 | 21 | 14 |

V (10^{6} m^{3}) | 384 | 307 | 1193 | 1556 | 224 | 335 | 229 | |||

1978 | 14 June 1978 | 1226 | D (day) | 36 | 36 | 88 | 251 | 26 | 25 | 26 |

V (10^{6} m^{3}) | 932 | 731 | 1535 | 3533 | 319 | 365 | 394 | |||

1979 | 22 November 1979 | 1428 | D (day) | 29 | 45 | 77 | 77 | 45 | 24 | 45 |

V (10^{6} m^{3}) | 822 | 803 | 1121 | 1285 | 710 | 447 | 1028 | |||

1980 | 16 October 1980 | 1593 | D (day) | 27 | 30 | 70 | 112 | 30 | 21 | 30 |

V (10^{6} m^{3}) | 1367 | 1248 | 1863 | 2562 | 1002 | 930 | 1439 | |||

1981 | 22 July 1981 | 1259 | D (day) | 49 | 50 | 67 | 67 | 29 | 22 | 29 |

V (10^{6} m^{3}) | 822 | 645 | 762 | 840 | 502 | 517 | 632 | |||

1982 | 10 October 1982 | 1190 | D (day) | 40 | 48 | 48 | 48 | 24 | 21 | 24 |

V (10^{6} m^{3}) | 1026 | 835 | 922 | 1025 | 520 | 589 | 632 | |||

1983 | 27 May 1983 | 862 | D (day) | 29 | 72 | 72 | 185 | 12 | 21 | 12 |

V (10^{6} m^{3}) | 404 | 646 | 713 | 2046 | 142 | 164 | 140 | |||

1984 | 24 May 1984 | 1223 | D (day) | 41 | 41 | 94 | 94 | 21 | 22 | 21 |

V (10^{6} m^{3}) | 995 | 777 | 1431 | 1713 | 296 | 421 | 290 | |||

1985 | 11 May 1985 | 1414 | D (day) | 45 | 45 | 45 | 83 | 16 | 22 | 12 |

V (10^{6} m^{3}) | 1141 | 969 | 1119 | 1891 | 396 | 474 | 272 | |||

1986 | 19 June 1986 | 1370 | D (day) | 20 | 53 | 53 | 154 | 46 | 25 | 29 |

V (10^{6} m^{3}) | 496 | 718 | 787 | 3564 | 661 | 492 | 672 | |||

1987 | 8 August 1987 | 1331 | D (day) | 31 | 31 | 31 | 66 | 21 | 12 | 21 |

V (10^{6} m^{3}) | 659 | 572 | 616 | 1031 | 410 | 263 | 450 | |||

1988 | 9 June 1988 | 1058 | D (day) | 27 | 27 | 70 | 70 | 27 | 18 | 27 |

V (10^{6} m^{3}) | 396 | 352 | 637 | 723 | 357 | 338 | 398 | |||

1989 | 8 July 1989 | 1772 | D (day) | 35 | 39 | 39 | 71 | 22 | 25 | 22 |

V (10^{6} m^{3}) | 1174 | 1059 | 1147 | 1855 | 800 | 649 | 925 | |||

1990 | 4 November 1990 | 1321 | D (day) | 32 | 47 | 47 | 99 | 25 | 23 | 25 |

V [(0^{6} m^{3}) | 712 | 747 | 800 | 1832 | 519 | 479 | 625 | |||

2003 | 4 November 2003 | 947 | D (day) | 28 | 28 | 36 | 91 | 16 | 16 | 16 |

V (10^{6} m^{3}) | 401 | 389 | 450 | 1124 | 309 | 323 | 322 | |||

2004 | 28 June 2004 | 1155 | D (day) | 92 | 92 | 92 | 200 | 76 | 23 | 76 |

V (10^{6} m^{3}) | 2551 | 1353 | 1553 | 3473 | 842 | 400 | 1998 | |||

2005 | 27 August 2005 | 1585 | D (day) | 36 | 36 | 36 | 36 | 36 | 24 | 36 |

V (10^{6} m^{3}) | 1109 | 916 | 985 | 1062 | 772 | 574 | 1113 | |||

2006 | 2 June 2006 | 1185 | D (day) | 31 | 124 | 124 | 169 | 17 | 25 | 17 |

V (10^{6} m^{3}) | 699 | 1802 | 2031 | 2868 | 287 | 385 | 231 | |||

2007 | 21 September 2007 | 749 | D (day) | 32 | 63 | 70 | 70 | 8 | 10 | 8 |

V (10^{6} m^{3}) | 428 | 718 | 830 | 947 | 102 | 107 | 100 | |||

2008 | 9 June 2008 | 780 | D (day) | 78 | 85 | 139 | 139 | 26 | 20 | 26 |

V (10^{6} m^{3}) | 1367 | 964 | 1437 | 1706 | 228 | 227 | 279 | |||

2009 | 29 June 2009 | 1129 | D (day) | 43 | 32 | 43 | 43 | 32 | 24 | 31 |

V (10^{6} m^{3}) | 1240 | 706 | 982 | 1080 | 570 | 496 | 879 | |||

2010 | 22 September 2010 | 1634 | D (day) | 38 | 49 | 49 | 82 | 31 | 18 | 31 |

V (10^{6} m^{3}) | 1095 | 937 | 1006 | 1585 | 762 | 639 | 994 | |||

2011 | 22 June 2011 | 789 | D (day) | 68 | 68 | 103 | 103 | 54 | 20 | 54 |

V (10^{6} m^{3}) | 1182 | 773 | 1127 | 1323 | 590 | 337 | 990 | |||

2012 | 9 November 2012 | 1637 | D (day) | 82 | 82 | 82 | 116 | 31 | 25 | 30 |

V (10^{6} m^{3}) | 2310 | 1393 | 1559 | 2380 | 770 | 736 | 1063 | |||

2013 | 11 May 2013 | 1313 | D (day) | 65 | 65 | 140 | 221 | 33 | 24 | 29 |

V (10^{6} m^{3}) | 2021 | 1359 | 2222 | 3825 | 358 | 337 | 436 | |||

2014 | 18 September 2014 | 2322 | D (day) | 41 | 43 | 76 | 76 | 30 | 18 | 30 |

V (10^{6} m^{3}) | 2209 | 1696 | 2459 | 2738 | 968 | 879 | 1114 | |||

2015 | 18 October 2015 | 1357 | D (day) | 35 | 35 | 35 | 105 | 35 | 25 | 35 |

V (10^{6} m^{3}) | 1087 | 867 | 931 | 1415 | 755 | 579 | 1087 | |||

2016 | 4 May 2016 | 1045 | D (day) | 22 | 22 | 55 | 104 | 8 | 8 | 8 |

V (10^{6} m^{3}) | 385 | 332 | 596 | 1464 | 197 | 176 | 160 | |||

2017 | 22 September 2017 | 1424 | D (day) | 39 | 44 | 44 | 44 | 44 | 25 | 44 |

V (10^{6} m^{3}) | 874 | 761 | 818 | 879 | 671 | 655 | 984 | |||

2018 | 2 November 2018 | 1335 | D (day) | 46 | 63 | 63 | 86 | 30 | 24 | 30 |

V (10^{6} m^{3}) | 1061 | 877 | 966 | 1169 | 631 | 640 | 834 | |||

2019 | 21 November 2019 | 1513 | D (day) | 45 | 51 | 51 | 51 | 43 | 25 | 37 |

V (10^{6} m^{3}) | 1659 | 1284 | 1431 | 1624 | 896 | 572 | 1485 |

River | Station | Mann–Kendall Test | Ljung–Box Test | ||||
---|---|---|---|---|---|---|---|

Variable | Test Statistic (S) | Z Value | p-Value | Test Statistic (Q) | p-Value | ||

Drava | Botovo | Q | −44 | −0.394 | 0.693 | 0.417 | 0.518 |

D | 143 | 1.304 | 0.192 | 0.297 | 0.586 | ||

V | 65 | 0.587 | 0.557 | 0.893 | 0.345 | ||

Terezino polje | Q | −121 | −1.101 | 0.271 | 1.292 | 0.256 | |

D | −1 | 0.000 | 1.000 | 0.064 | 0.801 | ||

V | −86 | −0.780 | 0.436 | 0.394 | 0.530 | ||

Donji Miholjac | Q | −135 | −1.269 | 0.205 | 1.125 | 0.289 | |

D | −137 | −1.290 | 0.197 | 0.508 | 0.476 | ||

V | −117 | −1.098 | 0.272 | 0.322 | 0.571 | ||

Belisce | Q | −221 | −2.083 | 0.037 | 2.053 | 0.152 | |

D | −30 | −0.275 | 0.783 | 2.017 | 0.156 | ||

V | −94 | −0.881 | 0.379 | 1.297 | 0.255 | ||

Sava | Podsused | Q | 187 | 1.350 | 0.177 | 0.409 | 0.522 |

D | 448 | 3.248 | 0.001 | 0.241 | 0.623 | ||

V | 301 | 2.178 | 0.029 | 0.025 | 0.874 | ||

Jasenovac | Q | 151 | 1.151 | 0.250 | 0.058 | 0.810 | |

D | 144 | 1.098 | 0.272 | 0.861 | 0.354 | ||

V | 122 | 0.928 | 0.353 | 1.157 | 0.282 | ||

Mackovac | Q | 76 | 0.576 | 0.565 | 0.012 | 0.911 | |

D | 85 | 0.645 | 0.519 | 0.041 | 0.840 | ||

V | 34 | 0.253 | 0.800 | 0.004 | 0.949 | ||

Zupanja | Q | −162 | −1.201 | 0.230 | 1.531 | 0.216 | |

D | −28 | −0.202 | 0.840 | 1.168 | 0.280 | ||

V | −115 | −0.851 | 0.395 | 0.043 | 0.837 |

**Table 7.**Selected distribution functions as marginal distributions of flood peak discharge, hydrograph duration, and hydrograph volume.

River | Station | Q | D | V |
---|---|---|---|---|

Drava | Botovo | GEV | GEV | Pearson 3 |

Terezino polje | GLO | GEV | Pearson 3 | |

Donji Miholjac | Pearson 3 | GLO | Pearson 3 | |

Sava | Jasenovac | log-Pearson 3 | Pearson 3 | log-Pearson 3 |

Mackovac | GLO | log-Pearson 3 | log-Pearson 3 | |

Zupanja | GLO | GLO | log-Pearson 3 |

River | Gauging Station | Sample Size | Q–D | Q–V | V–D |
---|---|---|---|---|---|

Drava | Botovo | 47 | 0.23 | 0.61 | 0.56 |

Terezino polje | 47 | 0.36 | 0.72 | 0.56 | |

Donji Miholjac | 46 | 0.27 | 0.70 | 0.44 | |

Sava | Jasenovac | 53 | 0.21 | 0.49 | 0.47 |

Mackovac | 54 | 0.15 | 0.36 | 0.56 | |

Zupanja | 54 | 0.09 | 0.31 | 0.55 |

River | Station | Q–D | Q–V | V–D | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Copula | S_{n} | p-Value | Copula | S_{n} | p-Value | Copula | S_{n} | p-Value | ||

Drava | Botovo | Huesler–Reiss | 0.038 | 0.937 | Huesler–Reiss | 0.033 | 0.078 | Normal | 0.023 | 0.471 |

Terezino polje | Huesler–Reiss | 0.020 | 0.809 | Gumbel | 0.021 | 0.381 | Normal | 0.017 | 0.873 | |

Donji Miholjac | Huesler–Reiss | 0.020 | 0.895 | Normal | 0.022 | 0.369 | Normal | 0.022 | 0.674 | |

Sava | Jasenovac | Gumbel | 0.031 | 0.251 | Normal | 0.022 | 0.538 | Tawn | 0.033 | 0.413 |

Mackovac | Tawn | 0.035 | 0.166 | Tawn | 0.020 | 0.873 | Gumbel | 0.035 | 0.052 | |

Zupanja | Huesler–Reiss | 0.040 | 0.067 | Tawn | 0.043 | 0.055 | Normal | 0.018 | 0.793 |

Drava | Sava | |||||
---|---|---|---|---|---|---|

Return Period | Botovo | Terezino Polje | Donji Miholjac | Jasenovac | Mackovac | Zupanja |

Q_{10} (m^{3}/s) | 2054.7 | 1967.4 | 1778.1 | 2382.8 | 2278.8 | 3450.0 |

Q_{10} (m^{3}/s/km^{2}) | 0.0662 | 0.0580 | 0.0479 | 0.0612 | 0.0558 | 0.0549 |

D_{10} (day) | 38.7 | 40.8 | 39.4 | 83.2 | 86.4 | 98.9 |

V_{10} (10^{6} m^{3}) | 970.2 | 999.3 | 920.9 | 3185.8 | 3366.2 | 4821.4 |

V_{10} (m^{3}/km^{2}) | 31,258.3 | 29,464.3 | 24,793.0 | 81,786.2 | 82,429.3 | 76,662.3 |

Q_{100} (m^{3}/s) | 2910.9 | 3296.2 | 2411.4 | 2858.8 | 3199.2 | 5060.176 |

Q_{100} (m^{3}/s/km^{2}) | 0.0938 | 0.0972 | 0.0649 | 0.0734 | 0.0783 | 0.0805 |

D_{100} (day) | 51.5 | 59.9 | 58.5 | 111.6 | 127.5 | 157.0 |

V_{100} (10^{6} m^{3}) | 1554.0 | 1674.2 | 1483.7 | 4554.1 | 5617.7 | 7451.7 |

V_{100} (m^{3}/km^{2}) | 50,068.7 | 49,362.8 | 39,945.4 | 116,912.3 | 137,560.2 | 118,485.4 |

Huesler–Reiss | Huesler–Reiss | Huesler–Reiss | Gumbel | Tawn | Huesler–Reiss | |

T_{AND} (Q_{10}D_{10}) | 30 | 22 | 26 | 28 | 36 | 51 |

T_{OR} (Q_{10}D_{10}) | 6 | 7 | 6 | 6 | 6 | 6 |

T_{AND} (Q_{100}D_{100}) | 364 | 238 | 303 | 329 | 462 | 835 |

T_{OR} (Q_{100}D_{100}) | 58 | 63 | 60 | 59 | 56 | 53 |

T_{AND} (Q_{10}D_{100}) | 148 | 114 | 131 | 151 | 226 | 271 |

T_{OR} (Q_{10}D_{100}) | 10 | 10 | 10 | 10 | 9 | 9 |

T_{AND} (Q_{100}D_{10}) | 148 | 114 | 131 | 151 | 226 | 271 |

T_{OR} (Q_{100}D_{10}) | 10 | 10 | 10 | 10 | 9 | 9 |

Huesler–Reiss | Gumbel | Normal | Normal | Tawn | Tawn | |

T_{AND} (Q_{10}V_{10}) | 14 | 13 | 15 | 21 | 21 | 23 |

T_{OR} (Q_{10}V_{10}) | 8 | 8 | 8 | 7 | 7 | 6 |

T_{AND} (Q_{100}V_{100}) | 147 | 127 | 193 | 375 | 224 | 256 |

T_{OR} (Q_{100}V_{100}) | 76 | 82 | 68 | 58 | 64 | 62 |

T_{AND} (Q_{10}V_{100}) | 100 | 100 | 102 | 128 | 121 | 136 |

T_{OR} (Q_{10}V_{100}) | 10 | 10 | 10 | 10 | 10 | 10 |

T_{AND} (Q_{100}V_{10}) | 100 | 100 | 102 | 128 | 121 | 136 |

T_{OR} (Q_{100}V_{10}) | 10 | 10 | 10 | 10 | 10 | 10 |

Normal | Normal | Normal | Tawn | Gumbel | Normal | |

T_{AND} (V_{10}D_{10}) | 20 | 19 | 24 | 19 | 15 | 19 |

T_{OR} (V_{10}D_{10}) | 7 | 7 | 6 | 7 | 7 | 7 |

T_{AND} (V_{100}D_{100}) | 316 | 312 | 479 | 199 | 156 | 306 |

T_{OR} (V_{100}D_{100}) | 59 | 60 | 56 | 67 | 74 | 60 |

T_{AND} (V_{10}D_{100}) | 118 | 117 | 145 | 109 | 102 | 116 |

T_{OR} (V_{10}D_{100}) | 10 | 10 | 10 | 10 | 10 | 10 |

T_{AND} (V_{100}D_{10}) | 118 | 117 | 145 | 109 | 102 | 116 |

T_{OR} (V_{100}D_{10}) | 10 | 10 | 10 | 10 | 10 | 10 |

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## Share and Cite

**MDPI and ACS Style**

Lacko, M.; Potočki, K.; Škreb, K.A.; Bezak, N.
Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe. *Water* **2022**, *14*, 2481.
https://doi.org/10.3390/w14162481

**AMA Style**

Lacko M, Potočki K, Škreb KA, Bezak N.
Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe. *Water*. 2022; 14(16):2481.
https://doi.org/10.3390/w14162481

**Chicago/Turabian Style**

Lacko, Martina, Kristina Potočki, Kristina Ana Škreb, and Nejc Bezak.
2022. "Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe" *Water* 14, no. 16: 2481.
https://doi.org/10.3390/w14162481