Joint Modelling of Flood Hydrograph Peak, Volume and Duration Using Copulas—Case Study of Sava and Drava River in Croatia, Europe
Abstract
:1. Introduction
2. Data and Methods
2.1. Study Area and Data
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 Q1, Q2, Q3, … Qi.
- Consider in turn (Q1, Q2, Q3), (Q2, Q3, Q4), … (Qi − 1, Qi, Qi + 1), etc.
- In each case, if 0.9·Qi < Qi − 1 and 0.9·Qi < Qi + 1, then the central value is an ordinate for the baseflow line. Continue the procedure until a derived set of baseflow ordinates QB1, QB2, QB3, … QBn is provided with different time periods between them.
- Apply linear interpolation between each QBi value and estimate each daily value of QB1 … Q1.
- 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 , where that represents the slope of the curve between two consecutive points.
- Eliminate the previous 2 points before points with , 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 , namely .
2.3. Marginal Probability Distributions
2.4. Copulas
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
3.4. Joint Return Periods
3.5. Preliminary Methodology for the Bridge Scour Analysis Using Copulas
4. Conclusions
- 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
References
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River | Station | Watershed Area [km2] | Period; Years of Data | Qmax; Qmean; Qsd (m3/s) | Dmax; Dmean; Dsd (day) | Vmax; Vmean; Vsd (106 m3) |
---|---|---|---|---|---|---|
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 | Cθ (u, v) |
---|---|---|
Archimedean | Gumbel–Hougaard | , θ ∈ [ 1, ∞) |
Clayton | , θ ∈ [ −1, ∞) \ {0} | |
Extreme value | Galambos | , θ ∈ [ 0, ∞) |
Huesler–Reiss | , θ ∈ [ 0, ∞) | |
Tawn | , θ ∈ [ 0;1] | |
Elliptical | Normal | , 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 |
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 (106 m3) | 1327 | 1041 | 1192 | 4700 | 414 | 328 | 505 | |||
1963 | 15 March 1963 | 1683 | D (day) | 30 | 31 | 31 | 151 | 21 | 21 | 21 |
V (106 m3) | 970 | 846 | 923 | 2533 | 684 | 789 | 807 | |||
1964 | 29 October 1964 | 1932 | D (day) | 40 | 69 | 69 | 69 | 25 | 23 | 25 |
V (106 m3) | 1683 | 1618 | 1831 | 2132 | 788 | 842 | 875 | |||
1965 | 6 August 1965 | 2471 | D (day) | 29 | 29 | 29 | 37 | 29 | 22 | 29 |
V (106 m3) | 1119 | 980 | 1020 | 1200 | 998 | 1001 | 1121 | |||
1966 | 23 August 1966 | 2525 | D (day) | 53 | 62 | 62 | 104 | 62 | 25 | 52 |
V (106 m3) | 2047 | 1608 | 1763 | 2986 | 1404 | 1173 | 2086 | |||
1967 | 4 June 1967 | 1398 | D (day) | 24 | 50 | 50 | 161 | 9 | 9 | 9 |
V (106 m3) | 486 | 933 | 1027 | 2678 | 267 | 205 | 248 | |||
1968 | 18 June 1968 | 811 | D (day) | 45 | 50 | 68 | 137 | 12 | 20 | 12 |
V (106 m3) | 815 | 641 | 884 | 1958 | 161 | 221 | 160 | |||
1969 | 22 May 1969 | 979 | D (day) | 41 | 41 | 41 | 41 | 18 | 12 | 18 |
V (106 m3) | 860 | 656 | 741 | 847 | 213 | 176 | 182 | |||
1970 | 14 August 1970 | 1390 | D (day) | 52 | 52 | 100 | 100 | 14 | 14 | 14 |
V (106 m3) | 1228 | 855 | 1173 | 1367 | 353 | 324 | 354 | |||
1971 | 25 March 1971 | 717 | D (day) | 36 | 99 | 99 | 239 | 18 | 18 | 18 |
V (106 m3) | 444 | 922 | 1058 | 2181 | 203 | 165 | 214 | |||
1972 | 19 July 1972 | 2882 | D (day) | 57 | 58 | 58 | 119 | 36 | 16 | 36 |
V (106 m3) | 2415 | 1925 | 2093 | 2748 | 1511 | 978 | 2040 | |||
1973 | 30 September 1973 | 1749 | D (day) | 49 | 49 | 78 | 78 | 27 | 24 | 27 |
V (106 m3) | 1729 | 1272 | 1635 | 1871 | 906 | 827 | 1085 | |||
1974 | 23 October 1974 | 1192 | D (day) | 47 | 29 | 61 | 61 | 15 | 15 | 15 |
V (106 m3) | 1214 | 347 | 1030 | 1160 | 290 | 336 | 298 | |||
1975 | 5 July 1975 | 2578 | D (day) | 50 | 70 | 70 | 121 | 50 | 25 | 50 |
V (106 m3) | 1932 | 1607 | 1776 | 2613 | 1174 | 879 | 1964 | |||
1976 | 29 April 1976 | 1108 | D (day) | 35 | 73 | 125 | 125 | 18 | 13 | 14 |
V (106 m3) | 630 | 867 | 1380 | 1684 | 290 | 316 | 259 | |||
1977 | 11 April 1977 | 1137 | D (day) | 26 | 26 | 86 | 107 | 14 | 21 | 14 |
V (106 m3) | 384 | 307 | 1193 | 1556 | 224 | 335 | 229 | |||
1978 | 14 June 1978 | 1226 | D (day) | 36 | 36 | 88 | 251 | 26 | 25 | 26 |
V (106 m3) | 932 | 731 | 1535 | 3533 | 319 | 365 | 394 | |||
1979 | 22 November 1979 | 1428 | D (day) | 29 | 45 | 77 | 77 | 45 | 24 | 45 |
V (106 m3) | 822 | 803 | 1121 | 1285 | 710 | 447 | 1028 | |||
1980 | 16 October 1980 | 1593 | D (day) | 27 | 30 | 70 | 112 | 30 | 21 | 30 |
V (106 m3) | 1367 | 1248 | 1863 | 2562 | 1002 | 930 | 1439 | |||
1981 | 22 July 1981 | 1259 | D (day) | 49 | 50 | 67 | 67 | 29 | 22 | 29 |
V (106 m3) | 822 | 645 | 762 | 840 | 502 | 517 | 632 | |||
1982 | 10 October 1982 | 1190 | D (day) | 40 | 48 | 48 | 48 | 24 | 21 | 24 |
V (106 m3) | 1026 | 835 | 922 | 1025 | 520 | 589 | 632 | |||
1983 | 27 May 1983 | 862 | D (day) | 29 | 72 | 72 | 185 | 12 | 21 | 12 |
V (106 m3) | 404 | 646 | 713 | 2046 | 142 | 164 | 140 | |||
1984 | 24 May 1984 | 1223 | D (day) | 41 | 41 | 94 | 94 | 21 | 22 | 21 |
V (106 m3) | 995 | 777 | 1431 | 1713 | 296 | 421 | 290 | |||
1985 | 11 May 1985 | 1414 | D (day) | 45 | 45 | 45 | 83 | 16 | 22 | 12 |
V (106 m3) | 1141 | 969 | 1119 | 1891 | 396 | 474 | 272 | |||
1986 | 19 June 1986 | 1370 | D (day) | 20 | 53 | 53 | 154 | 46 | 25 | 29 |
V (106 m3) | 496 | 718 | 787 | 3564 | 661 | 492 | 672 | |||
1987 | 8 August 1987 | 1331 | D (day) | 31 | 31 | 31 | 66 | 21 | 12 | 21 |
V (106 m3) | 659 | 572 | 616 | 1031 | 410 | 263 | 450 | |||
1988 | 9 June 1988 | 1058 | D (day) | 27 | 27 | 70 | 70 | 27 | 18 | 27 |
V (106 m3) | 396 | 352 | 637 | 723 | 357 | 338 | 398 | |||
1989 | 8 July 1989 | 1772 | D (day) | 35 | 39 | 39 | 71 | 22 | 25 | 22 |
V (106 m3) | 1174 | 1059 | 1147 | 1855 | 800 | 649 | 925 | |||
1990 | 4 November 1990 | 1321 | D (day) | 32 | 47 | 47 | 99 | 25 | 23 | 25 |
V [(06 m3) | 712 | 747 | 800 | 1832 | 519 | 479 | 625 | |||
2003 | 4 November 2003 | 947 | D (day) | 28 | 28 | 36 | 91 | 16 | 16 | 16 |
V (106 m3) | 401 | 389 | 450 | 1124 | 309 | 323 | 322 | |||
2004 | 28 June 2004 | 1155 | D (day) | 92 | 92 | 92 | 200 | 76 | 23 | 76 |
V (106 m3) | 2551 | 1353 | 1553 | 3473 | 842 | 400 | 1998 | |||
2005 | 27 August 2005 | 1585 | D (day) | 36 | 36 | 36 | 36 | 36 | 24 | 36 |
V (106 m3) | 1109 | 916 | 985 | 1062 | 772 | 574 | 1113 | |||
2006 | 2 June 2006 | 1185 | D (day) | 31 | 124 | 124 | 169 | 17 | 25 | 17 |
V (106 m3) | 699 | 1802 | 2031 | 2868 | 287 | 385 | 231 | |||
2007 | 21 September 2007 | 749 | D (day) | 32 | 63 | 70 | 70 | 8 | 10 | 8 |
V (106 m3) | 428 | 718 | 830 | 947 | 102 | 107 | 100 | |||
2008 | 9 June 2008 | 780 | D (day) | 78 | 85 | 139 | 139 | 26 | 20 | 26 |
V (106 m3) | 1367 | 964 | 1437 | 1706 | 228 | 227 | 279 | |||
2009 | 29 June 2009 | 1129 | D (day) | 43 | 32 | 43 | 43 | 32 | 24 | 31 |
V (106 m3) | 1240 | 706 | 982 | 1080 | 570 | 496 | 879 | |||
2010 | 22 September 2010 | 1634 | D (day) | 38 | 49 | 49 | 82 | 31 | 18 | 31 |
V (106 m3) | 1095 | 937 | 1006 | 1585 | 762 | 639 | 994 | |||
2011 | 22 June 2011 | 789 | D (day) | 68 | 68 | 103 | 103 | 54 | 20 | 54 |
V (106 m3) | 1182 | 773 | 1127 | 1323 | 590 | 337 | 990 | |||
2012 | 9 November 2012 | 1637 | D (day) | 82 | 82 | 82 | 116 | 31 | 25 | 30 |
V (106 m3) | 2310 | 1393 | 1559 | 2380 | 770 | 736 | 1063 | |||
2013 | 11 May 2013 | 1313 | D (day) | 65 | 65 | 140 | 221 | 33 | 24 | 29 |
V (106 m3) | 2021 | 1359 | 2222 | 3825 | 358 | 337 | 436 | |||
2014 | 18 September 2014 | 2322 | D (day) | 41 | 43 | 76 | 76 | 30 | 18 | 30 |
V (106 m3) | 2209 | 1696 | 2459 | 2738 | 968 | 879 | 1114 | |||
2015 | 18 October 2015 | 1357 | D (day) | 35 | 35 | 35 | 105 | 35 | 25 | 35 |
V (106 m3) | 1087 | 867 | 931 | 1415 | 755 | 579 | 1087 | |||
2016 | 4 May 2016 | 1045 | D (day) | 22 | 22 | 55 | 104 | 8 | 8 | 8 |
V (106 m3) | 385 | 332 | 596 | 1464 | 197 | 176 | 160 | |||
2017 | 22 September 2017 | 1424 | D (day) | 39 | 44 | 44 | 44 | 44 | 25 | 44 |
V (106 m3) | 874 | 761 | 818 | 879 | 671 | 655 | 984 | |||
2018 | 2 November 2018 | 1335 | D (day) | 46 | 63 | 63 | 86 | 30 | 24 | 30 |
V (106 m3) | 1061 | 877 | 966 | 1169 | 631 | 640 | 834 | |||
2019 | 21 November 2019 | 1513 | D (day) | 45 | 51 | 51 | 51 | 43 | 25 | 37 |
V (106 m3) | 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 |
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 | Sn | p-Value | Copula | Sn | p-Value | Copula | Sn | 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 |
Q10 (m3/s) | 2054.7 | 1967.4 | 1778.1 | 2382.8 | 2278.8 | 3450.0 |
Q10 (m3/s/km2) | 0.0662 | 0.0580 | 0.0479 | 0.0612 | 0.0558 | 0.0549 |
D10 (day) | 38.7 | 40.8 | 39.4 | 83.2 | 86.4 | 98.9 |
V10 (106 m3) | 970.2 | 999.3 | 920.9 | 3185.8 | 3366.2 | 4821.4 |
V10 (m3/km2) | 31,258.3 | 29,464.3 | 24,793.0 | 81,786.2 | 82,429.3 | 76,662.3 |
Q100 (m3/s) | 2910.9 | 3296.2 | 2411.4 | 2858.8 | 3199.2 | 5060.176 |
Q100 (m3/s/km2) | 0.0938 | 0.0972 | 0.0649 | 0.0734 | 0.0783 | 0.0805 |
D100 (day) | 51.5 | 59.9 | 58.5 | 111.6 | 127.5 | 157.0 |
V100 (106 m3) | 1554.0 | 1674.2 | 1483.7 | 4554.1 | 5617.7 | 7451.7 |
V100 (m3/km2) | 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 | |
TAND (Q10D10) | 30 | 22 | 26 | 28 | 36 | 51 |
TOR (Q10D10) | 6 | 7 | 6 | 6 | 6 | 6 |
TAND (Q100D100) | 364 | 238 | 303 | 329 | 462 | 835 |
TOR (Q100D100) | 58 | 63 | 60 | 59 | 56 | 53 |
TAND (Q10D100) | 148 | 114 | 131 | 151 | 226 | 271 |
TOR (Q10D100) | 10 | 10 | 10 | 10 | 9 | 9 |
TAND (Q100D10) | 148 | 114 | 131 | 151 | 226 | 271 |
TOR (Q100D10) | 10 | 10 | 10 | 10 | 9 | 9 |
Huesler–Reiss | Gumbel | Normal | Normal | Tawn | Tawn | |
TAND (Q10V10) | 14 | 13 | 15 | 21 | 21 | 23 |
TOR (Q10V10) | 8 | 8 | 8 | 7 | 7 | 6 |
TAND (Q100V100) | 147 | 127 | 193 | 375 | 224 | 256 |
TOR (Q100V100) | 76 | 82 | 68 | 58 | 64 | 62 |
TAND (Q10V100) | 100 | 100 | 102 | 128 | 121 | 136 |
TOR (Q10V100) | 10 | 10 | 10 | 10 | 10 | 10 |
TAND (Q100V10) | 100 | 100 | 102 | 128 | 121 | 136 |
TOR (Q100V10) | 10 | 10 | 10 | 10 | 10 | 10 |
Normal | Normal | Normal | Tawn | Gumbel | Normal | |
TAND (V10D10) | 20 | 19 | 24 | 19 | 15 | 19 |
TOR (V10D10) | 7 | 7 | 6 | 7 | 7 | 7 |
TAND (V100D100) | 316 | 312 | 479 | 199 | 156 | 306 |
TOR (V100D100) | 59 | 60 | 56 | 67 | 74 | 60 |
TAND (V10D100) | 118 | 117 | 145 | 109 | 102 | 116 |
TOR (V10D100) | 10 | 10 | 10 | 10 | 10 | 10 |
TAND (V100D10) | 118 | 117 | 145 | 109 | 102 | 116 |
TOR (V100D10) | 10 | 10 | 10 | 10 | 10 | 10 |
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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
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 StyleLacko, 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