# An Extended Flood Characteristic Simulation Considering Natural Dependency Structures

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## Abstract

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## 1. Introduction

## 2. Extended Flood Characteristic Simulation according to Bender and Jensen

_{r}; the shape parameter of the rising branch, s

_{r}; the peak discharge, Q

_{P}; and the shape parameter of the descending branch, s

_{d}. Synthetic hydrographs are then generated based on these parameters [14].

_{P}. The shape parameters s

_{r}and s

_{d}are dimensionless input parameters used in the hydrograph function and describe the characteristics of the rising and falling branches of the flood wave, respectively. The interaction between all these parameters (rising duration, t

_{r}; the shape parameter of the rising branch, s

_{r}; peak discharge, Q

_{P}; peak duration, t

_{p}; and the shape parameter of the descending branch, s

_{d}) and the resulting shape of the hydrograph is illustrated in Figure 1 [5].

#### 2.1. Hydrograph Functions

_{P}) and shape parameters (s

_{r}and s

_{d}) as the primary parameters. The other input variables remain nearly constant and are fixed values. The maximum discharge, which needs to be determined at the beginning of the parameter estimation process, is purely a calculated quantity. A standard value of Q

_{0}= $2\times {\mathrm{Q}}_{\mathrm{P}}\left(\mathrm{H}\mathrm{H}\mathrm{Q}\right)$ is often used as an example [15]. The term “HHQ” refers to the highest known value, which signifies the maximum recorded value for the particular flow time series used.

_{P}) generally forms a slightly decreasing or increasing plateau. Consequently, the applied method records all values lower than 0.5% below the flood peak (Q

_{P}). In cases where a stagnant peak discharge is not observed, a peak duration of t

_{P}= 0 is assigned [5].

_{tot}) can be calculated using integrals. In this context, it is necessary to distinguish between the rising branch (V

_{r}, as modeled using the Kozeny function) and the descending branch (V

_{d}, as represented using a hyperbolic function). The volume calculation at the peak can be calculated relatively easily with ${\mathrm{V}}_{\mathrm{P}}={\mathrm{Q}}_{\mathrm{P}}\times {\mathrm{t}}_{\mathrm{P}}$.

_{end}), subdivided into its components, for the respective hydrograph.

#### 2.2. Parameter Identification and Creation of a Synthetic Hydrograph

_{P}) exceeding a certain threshold. According to Leichtfuss and Lohr, a reasonable threshold for parameterization is ${\mathrm{Q}}_{\mathrm{P}}\ge 2\times {\mathrm{Q}}_{\mathrm{M}}$ [14]. As the current study aims to generate extremely rare synthetic hydrographs with a low probability of occurrence, the principle of the annual maximum series (AMS or AMAX) is used to parameterize the hydrographs [18]. Additionally, it is possible to use critical water levels for river levee dimensioning as a threshold for parameterization.

_{r}, s

_{r}, Q

_{P}, t

_{p}, and s

_{d}). Upon successful fitting, a large number of synthetic hydrographs can be generated by using the principle of Monte Carlo simulation [20]. Including a sufficient number of synthetic events ensures a desired design event (e.g., HQ

_{10000}) among the generated hydrographs. The probability of occurrence of the synthetic hydrographs is directly related to the probability of occurrence of the flood peak (Q

_{P}).

#### 2.3. Constraints and Enhancement Strategies for the Methodology

#### 2.3.1. Limitations in Modifying the Descending Branch of the Hydrograph

_{r}, s

_{d}, and c) are now determined using an iterative, generation-based optimization method.

- Genetic algorithm

_{r}) and descending branches (s

_{d}and c). Each member is assigned a route with scores in any order. Elite selection identifies members with the highest fitness and forms the elite pool. From this pool, parents are chosen for the new generation, and each child inherits part of its solution from one parent while filling in the missing part from the other. The population is replenished, and the cycle of fitness calculation and reproduction is repeated over multiple generations. With each generation, the deviation between observed and simulated values diminishes.

#### 2.3.2. Seasonal Statistics

#### 2.3.3. Assumption of Statistical Independence for Individual Parameters

_{r}, s

_{r}, Q

_{P}, t

_{p}, and s

_{d}) in the collected data have been assumed to be statistically independent of each other [14]. Further investigations should determine the extent to which adjusting the parameters is beneficial for the Monte Carlo simulation. Consequently, the correlation among all existing parameters, including the newly adapted compression factor, c, was subsequently assessed.

- Correlation

- Copula functions

_{r}, s

_{r}, Q

_{S}, t

_{p}, s

_{d}, and c) are first fitted independently to each sample using univariate statistical methods [31]. The observed pairs of variables (X and Y) are subjected to a transformation process that maps them into a range between zero and one. This transformation is achieved by directly applying the obtained distribution functions to the observed values. The corresponding copula model is fitted to these transformed values. Parameter estimation in common copula models often employs Kendall’s τ [32].

_{1}, X

_{2}, and X

_{3}, where their pairwise dependencies are captured through the bivariate copulas ${\mathrm{C}}_{{\mathrm{X}}_{\mathrm{1,2}}}$ and ${\mathrm{C}}_{{\mathrm{X}}_{\mathrm{2,3}}}$. Mathematical mapping is illustrated in the figure [45].

#### 2.3.4. Multi-Peak Flood Events and Uncertainties

## 3. Results

_{d}and c. Furthermore, the findings derived from the examination of hydrological dependencies within the data and their impact on the results of the extended flood characteristic simulation are discussed in detail. Both adjustments are based on a direct comparison of the simulation accuracy using the detected flood events from the Emskirchen gauge (Middle Aurach).

#### 3.1. Consideration of Combined Genetic Algorithms in Adapting the Descending Branch

_{d}and c) genetic algorithms, a greater level of fitting accuracy can be attained in the descending branch, as compared to the previously suggested method by Bender and Jensen, wherein solely the shape parameter is fitted (Figure 5). This improvement is demonstrated through a correlation analysis using Kendall’s (τ) method and the Kling–Gupta efficiency (KGE) metric [29,48]. Here, the observed and simulated flood volumes are considered. In Figure 5, both the observed and simulated flood hydrographs are displayed, depicting the peak discharge (Q

_{P}) and volume (V).

_{fix}= 1. In contrast, the lower section shows a variable adjustment of the shape parameter and compression factor, c

_{var}, for the hyperbolic function of the descending branch. For both winter and summer events, a clear improvement in the simulated flood volumes becomes evident due to the variable compression factor. For example, KGE for summer events improves from 0.666 with compression factor c

_{fix}= 1 to 0.814 with a variable compression factor. The correlation also improves from 0.89 to 0.951.

_{d}and c

_{fix}(green), as well as s

_{d}and c

_{var}(orange). By utilizing a variable compression factor c

_{var}$(0\le \mathrm{c}\le 1)$ and s

_{d}, there is an improvement in the adaptation of the descending branch to the observed flood hydrograph. This can be justified by acknowledging that both variables have a direct impact on the outcome of the hyperbolic function. With c

_{fix}, it is not always possible to achieve a complete correction solely through adjustments of the shape parameter (s

_{d}) for certain hydrographs. Thus, the fixed adjustment (c

_{fix}) in the example below tends to slightly overestimate the hydrograph in the descending branch.

_{var}) within the range of 0.1 to 1 with two constant shape parameters for the descending branch (s

_{d}= 0.05 and 0.40).

_{d}can also encompass a significantly broader spectrum of hydrographs. In contrast, this improvement can not only be corrected in the same way via the variable adjustment of the compression factor but can also be amplified. Consequently, it is possible to not only ascertain the extent of the descending branch within a specified period but also facilitate significant stretching of the originally compressed shape, exerting a substantial influence on the rate of the descent of the descending branch. The simultaneous application of both variables leads to an improvement, as exemplified in Figure 5.

#### 3.2. Consideration of the Dependency Structure between Correlated Random Variables in Terms of Copulas

#### 3.2.1. Utilizing D Vines for the Present Application

_{r}, s

_{r}, Q

_{P}, t

_{p}, s

_{d}, and c) for the winter period [49]. The detailed structure, including the appropriate copulas, is further described by using the subsequent visualization (Figure 8).

#### 3.2.2. Plausibility Check

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Influencing parameters of the hydrograph [5].

**Figure 4.**Hierarchical nesting of bivariate copulas in the construction of a 3D vine copula including mathematical mapping.

**Figure 5.**Comparison of s

_{d}and c

_{fix}with s

_{d}and c

_{var}using Kendall’s τ and KGE (Middle Aurach, Emskirchen).

**Figure 7.**Adjustment of c

_{var}to the fitting of the descending branch at a constant s

_{d}= 0.05 (

**left**) and 0.40 (

**right**).

**Figure 8.**D Vine structure of the parameters (t

_{r}, s

_{r}, Q

_{P}, t

_{p}, s

_{d}, c) for the summer period of the river Middle Aurach (Emskirchen).

**Figure 9.**Correlation of observed and simulated (EFCS and EFCSC) volume and peak discharges at the Emskirchen (Middle Aurach) gauge.

Author | Initial Parameters |
---|---|

Fouroud and Broughton (1981) [3] | Precipitation data |

Bender and Jensen (2012) [5] | Catchment area conditions |

Wyncoll and Couldby (2013) [6] | Geomorphological characteristics |

Candela et al. (2014) [7] | Land cover data, digital elevation model |

Chatzichristaki et al. (2015) [8] | Numerical simulations |

Bhuyan et al. (2015) [4] | Statistical analysis based on observation data |

Aranda and Garcia-Barthual (2018) [9] Shatnawi and Ibrahim (2022) [10] Fischer and Schumann (2023) [11] |

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**MDPI and ACS Style**

Öttl, M.A.; Simon, F.; Bender, J.; Mudersbach, C.; Stamm, J.
An Extended Flood Characteristic Simulation Considering Natural Dependency Structures. *Hydrology* **2023**, *10*, 233.
https://doi.org/10.3390/hydrology10120233

**AMA Style**

Öttl MA, Simon F, Bender J, Mudersbach C, Stamm J.
An Extended Flood Characteristic Simulation Considering Natural Dependency Structures. *Hydrology*. 2023; 10(12):233.
https://doi.org/10.3390/hydrology10120233

**Chicago/Turabian Style**

Öttl, Marco Albert, Felix Simon, Jens Bender, Christoph Mudersbach, and Jürgen Stamm.
2023. "An Extended Flood Characteristic Simulation Considering Natural Dependency Structures" *Hydrology* 10, no. 12: 233.
https://doi.org/10.3390/hydrology10120233