# The Use of a Uniform Technique for Harmonization and Generalization in Assessing the Flood Discharge Frequencies of Long Return Period Floods in the Danube River Basin

^{*}

## Abstract

**:**

## 1. Introduction

- -
- Maximum annual discharges must be independent and stochastic;
- -
- Processes influencing the runoff process are stationary with respect to time (homogeneity of the series);
- -
- Statistical characteristics of the measured data series (series of maximum annual discharge) represent the past, presence, and future.

## 2. Materials and Methods

#### 2.1. Materials

^{2}(Figure 1). The river originates in the Black Forest in Germany at the confluence of the Briga and the Breg streams. The Danube then discharges southeast for 2872 km (1785 mi), passing through four Central European capitals before emptying into the Black Sea via the Danube Delta in Romania and Ukraine. The Danube River Basin landscape geomorphology is characterized by a diversity of morphological patterns, and the river channel itself can be divided into six sections (Figure 1). The territory of the Danube River Basin is also one of the most flood-endangered regions in Europe. Therefore, it is vital to have complete data of the flood regime to be able to generalize such information on the basis of long-term observations from the whole Danube territory. The occurrence of large floods on the Danube River is described in detail in many publications [8,11,36,37,38,39,40,41,42].

^{3}s

^{−1}(the year 1501). During the period of 1900–2013, the largest peak discharge was measured on the upper Danube at Kienstock (11,450 m

^{3}s

^{−1}in 2013) and on the lower Danube at Ceatal Izmail (15,900 m

^{3}s

^{−1}in 2006).

#### 2.2. Methods

#### 2.2.1. Log-Pearson III Probability Distribution

_{max}discharge series distribution function, we used Log-Pearson Type III distribution. The LPIII distribution is used to estimate the extremes in many natural processes and is the most commonly used frequency distribution, especially in hydrology. The Log-likelihood function of LPIII with estimation of its parameters was developed in [27]. In [43], a frequency factor-based method for hydrological frequency analysis for the random generation of five distributions (normal, lognormal, extreme value type 1, Pearson Type III and Log-Pearson Type III) is presented. The LPIII distribution was also used in flood frequency analysis in [28,33,34]. Use of one type of distribution also allows the value of the T-year maximum discharges to be estimated for parts of the river without observations on the basis of the long-term average of maximum annual discharge and distribution parameters from the neighboring gauging stations.

#### 2.2.2. Conditions of Q_{max} Series

_{i}is the exceedance probability of variable observations X

_{i}ranked from largest (i = 1) to smallest (i = n), and α is a plotting position parameter (0≤ α ≥ 0.5).

#### 2.2.3. Parameter Estimation: Simple Case

#### 2.2.4. Historical Floods

#### 2.2.5. Skew Coefficients in Log-Pearson III Distribution—Regionalization

## 3. Results

#### 3.1. Estimation of the T-Year Design Discharges along the Danube River

_{T}design values. The design values of selected T-year annual maximum discharges along the Danube River with station skew coefficients Gs are listed in Table 2.

^{3}s

^{−1}and for a return period of T = 1000 years was 1730 m

^{3}s

^{−1}. Our investigation showed that the inclusion of historical floods changed the curvature of the LPIII distribution curves and changed the design discharge.

#### 3.2. Regionalization of the Skew Coefficients of the LPIII Probability Curves for the Danube River

_{T}design values change along the Danube River. The ratio k of Q

_{T}/Q

_{a}(Qa: long-term mean discharges) for selected stations are presented in Figure 6a. The 1000-year discharge is 15 times higher than the mean annual discharge at the Berg station, seven times higher at the Bratislava station, and only three times at the Reni station. Subsequently, we individually plotted the course of the skew parameter Gs for each station (Figure 6b).

## 4. Discussion

## 5. Conclusions

_{5%}–Q

_{95%}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diagram of the Danube River Basin, water gauging stations along the Danube River, and the six sections of the Danube Basin based on the landscape geomorphology.

**Figure 2.**The maximum annual discharge series for selected gauging stations located in the upper, middle and lower Danube (with long data series).

**Figure 3.**Course of various extreme floods along the Danube River: (

**a**) floods in 1954, August 2002 on the upper Danube and floods in 1942, 1981 on the lower Danube; (

**b**) floods in 1965, 2006 and 2013 along the entire length of the Danube River.

**Figure 4.**Examples of the theoretical LPIII exceedance probability curves of the Danube maximum annual discharges without historical data (

**left**) and with historical data (red points,

**right**).

**Figure 5.**Differences in the estimated maximum discharges with return periods of (

**a**) 100 years and (

**b**) 1000 years along the Danube River, estimated according to LPIII distribution with historical data (11 red points) and without historical data (20 blue points).

**Figure 6.**The course of the (

**a**) ratio k of Q

_{T}/Q

_{a}at gauge stations along the Danube River, and (

**b**) station skew coefficient G with and without historical floods along the Danube River.

**Figure 8.**Dependence of the skew coefficient G on the runoff depth at Danube River gauges: (

**a**) Gs: without historical floods; (

**b**) Gh: with historical floods in some gauging stations.

**Figure 9.**Examples of the theoretical LPIII exceedance probability curves of the Danube maximum annual discharges with (

**a**) station skew parameter Gs and (

**b**) regionalized skew parameter Ghr for the Danube at Hofkirchen.

**Figure 10.**(

**a**) Travel times of selected floods, and (

**b**) comparison of rating curves of the Danube River at Bratislava (red point: maximum hydrological values of the flood in June 2013).

**Table 1.**List of the gauging stations along the Danube River. LAT: latitude; LONG: longitude;

**Q**: long-term average of the maximum annual discharge.

_{amax}Gauge | River km | Period | Country | Area [km^{2}] | LAT | LONG | Elevation [m a.s.l] | Q_{amax} [m^{3}s^{−1}] |
---|---|---|---|---|---|---|---|---|

Berg | 2613 | 1930–2007 | GE | 4047 | 48.27 | 9.73 | 489.48 | 204 |

Ingolstadt | 2458.3 | 1940–2007 | GE | 20,001 | 48.75 | 11.42 | 359.97 | 1110 |

Regensburg-Schwabelweis | 2376.1 | 1924–2007 | GE | 35,399 | 49.02 | 12.14 | 324.06 | 1532 |

Pfelling | 2300 | 1926–2007 | GE | 37,757 | 48.88 | 12.75 | 307.73 | 1516 |

Hofkirchen | 2256.9 | 1826–2013 | GE | 47,496 | 48.68 | 13.12 | 299.17 | 1896 |

Achleiten | 2150 | 1901–2007 | GE | 76,653 | 48.58 | 13.5 | 287.27 | 4146 |

Linz | 2135.2 | 1821–2013 | AT | 79,490 | 48.31 | 14.3 | 247.06 | 3670 |

Stein-Krems (Kienstock) | 2002.7 | 1828–2006 | AT | 96,045 | 48.38 | 15.46 | 193.32 | 5372 |

Wien-Nussdorf | 1934.1 | 1828–2006 | AT | 101,731 | 48.25 | 16.3 | 157 | 5301 |

Devin/Bratislava | 1868.8 | 1876–2013 | SK | 131,338 | 48.14 | 17.1 | 132.86 | 5884 |

Nagymaros | 1694.6 | 1893–2007 | HU | 183,534 | 47.78 | 18.95 | 99.37 | 5598 |

Mohács | 1446.8 | 1930–2007 | HU | 209,064 | 46 | 18.67 | 79.19 | 5063 |

Bezdan | 1425.5 | 1940–2006 | SR | 210,250 | 45.85 | 18.87 | 79.29 | 4974 |

Bogojevo | 1367.4 | 1940–2006 | SR | 251,593 | 45.53 | 19.08 | 76.11 | 5675 |

Pancevo | 1153.3 | 1940–2006 | SR | 525,009 | 44.87 | 20.46 | 65.98 | 10,147 |

Veliko Gradiste | 1060 | 1931–2007 | SR | 570,375 | 44.8 | 21.4 | 60.83 | 10,529 |

Orsova-Turnu Severin | 955 | 1840–2006 | RO | 576,232 | 44.7 | 22.42 | 44.76 | 10,295 |

Zimnicea | 554 | 1931–2010 | RO | 658,400 | 43.63 | 25.36 | 16.06 | 11,087 |

Reni | 132 | 1921–2010 | UKR | 805,700 | 45.45 | 28.27 | 4 | 11,217 |

Ceatal Izmail | 72 | 1931–2010 | RO | 807,000 | 45.22 | 28.73 | 0.2 | 11,173 |

**Table 2.**Design values of selected T-year annual maximum discharges in [m

^{3}s

^{−1}] along the Danube River (R: runoff depth, Gs: station skew coefficient).

Without Estimated Historical Maxima | R [mm] | Gs | 10 [Year] | 50 [Year] | 100 [Year] | 200 [Year] | 500 [Year] | 1000 [Year] |
---|---|---|---|---|---|---|---|---|

Berg | 296 | −0.30 | 324 | 432 | 476 | 518 | 573 | 613 |

Ingolstadt | 494 | 0.15 | 1514 | 1891 | 2050 | 2209 | 2421 | 2583 |

Regensburg-Schwabelweis | 396 | −0.46 | 2125 | 2530 | 2675 | 2809 | 2969 | 3081 |

Pfelling | 392 | −0.23 | 2144 | 2649 | 2846 | 3034 | 3273 | 3447 |

Hofkirchen | 425 | 0.09 | 2356 | 3547 | 3897 | 4250 | 4724 | 5091 |

Achleiten | 587 | 0.39 | 5486 | 6835 | 7422 | 8020 | 8835 | 9473 |

Linz | 581 | 0.26 | 4641 | 7352 | 8205 | 9092 | 10,323 | 11,304 |

Stein-Krems (Kienstock) | 621 | 0.39 | 7397 | 9605 | 10,592 | 11,613 | 13,028 | 14,154 |

Wien-Nussdorf | 596 | 0.27 | 7187 | 9046 | 9847 | 10,658 | 11,756 | 12,610 |

Devin/Bratislava | 492 | 0.18 | 8116 | 10,273 | 11,192 | 12,119 | 13,365 | 14,328 |

Nagymaros | 401 | −0.05 | 7325 | 8712 | 9257 | 9783 | 10,457 | 10,955 |

Mohács | 355 | −0.08 | 6548 | 7708 | 8157 | 8589 | 9138 | 9541 |

Bezdan | 354 | 0.30 | 6452 | 7847 | 8437 | 9029 | 9823 | 10,435 |

Bogojevo | 363 | 0.19 | 7334 | 8810 | 9418 | 10,020 | 10,815 | 11,418 |

Pancevo | 320 | 0.15 | 12,611 | 14,661 | 15,483 | 16,285 | 17,326 | 18,105 |

Veliko Gradiste | 307 | 0.02 | 13,128 | 15,167 | 15,962 | 16,728 | 17,708 | 18,430 |

Orsova-Turnu Severin | 307 | −0.19 | 12,901 | 14,754 | 15,445 | 16,094 | 16,901 | 17,481 |

Zimnicea | 288 | −0.09 | 13,776 | 15,769 | 16,528 | 17,248 | 18,155 | 18,815 |

Reni | 262 | −0.40 | 13,918 | 15,596 | 16,183 | 16,715 | 17,352 | 17,793 |

Ceatal Izmail | 251 | −0.21 | 13,677 | 15,492 | 16,161 | 16,785 | 17,557 | 18,108 |

**Table 3.**Design values of selected T-year annual maximum discharges in [m

^{3}s

^{−1}] along the Danube River (R: runoff depth, Gh: historical skew coefficient).

With Estimated Historical Maxima | Gh | 10 [Year] | 50 [Year] | 100 [Year] | 200 [Year] | 500 [Year] | 1000 [Year] | Year of Historical Max. |
---|---|---|---|---|---|---|---|---|

Regensburg-Schwabelweis | 0.40 | 2306 | 3140 | 3525 | 3931 | 4505 | 4970 | 1845, 1850, 1862, 1882, 2013 |

Pfelling | 0.45 | 2320 | 3224 | 3651 | 4105 | 4756 | 5290 | 1845, 1862, 1850, 1882, 2013 |

Achleiten | 1.17 | 5778 | 7754 | 8710 | 9743 | 11,244 | 12,495 | 1862, 1899, 2013 |

Linz | 0.76 | 4621 | 7880 | 9027 | 10,276 | 12,104 | 13,639 | 1501 |

Stein-Krems (Kienstock) | 0.71 | 7545 | 10,255 | 11,554 | 12,952 | 14,974 | 16,650 | 1501, 1787, 2013 |

Wien-Nussdorf | 0.64 | 7328 | 9620 | 10,678 | 11,792 | 13,366 | 14,642 | 1501, 1787, 2013 |

Devin/Bratislava | 0.24 | 8194 | 10,485 | 11,477 | 12,487 | 13,860 | 14,931 | 1501, 1787 |

Nagymaros | 0.06 | 7421 | 8955 | 9574 | 10,182 | 10,975 | 11,570 | 2013 |

Mohács | 0.04 | 6658 | 7966 | 8491 | 9004 | 9669 | 10,167 | 2013 |

Reni | 0.05 | 14,156 | 16,485 | 17,403 | 18,291 | 19,434 | 20,281 | 1897 |

Ceatal Izmail | −0.03 | 13,816 | 15,905 | 16,712 | 17,484 | 18,466 | 19,186 | 1876 |

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**MDPI and ACS Style**

Bačová Mitková, V.; Pekárová, P.; Halmová, D.; Miklánek, P. The Use of a Uniform Technique for Harmonization and Generalization in Assessing the Flood Discharge Frequencies of Long Return Period Floods in the Danube River Basin. *Water* **2021**, *13*, 1337.
https://doi.org/10.3390/w13101337

**AMA Style**

Bačová Mitková V, Pekárová P, Halmová D, Miklánek P. The Use of a Uniform Technique for Harmonization and Generalization in Assessing the Flood Discharge Frequencies of Long Return Period Floods in the Danube River Basin. *Water*. 2021; 13(10):1337.
https://doi.org/10.3390/w13101337

**Chicago/Turabian Style**

Bačová Mitková, Veronika, Pavla Pekárová, Dana Halmová, and Pavol Miklánek. 2021. "The Use of a Uniform Technique for Harmonization and Generalization in Assessing the Flood Discharge Frequencies of Long Return Period Floods in the Danube River Basin" *Water* 13, no. 10: 1337.
https://doi.org/10.3390/w13101337