# Incorporating Uncertainty of the System Behavior in Flood Risk Assessment—Sava River Case Study

^{*}

## Abstract

**:**

^{2}of otherwise safe area, resulting in the increased flood risk.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study Area

^{3}of water flooded the hinterland. Estimated monetary damage from the 2014 flood on the Sava River was, only in Croatia, more than 300 million Euro, with more than 500 million Euro in total considering all affected countries [33]. Although many lessons have been learned from the 2014 flood event, there still exists a significant threat of flooding, despite building additional structural measures.

#### 2.2. Framework for Uncertainty Analysis

#### 2.2.1. Identification of Uncertainty

#### 2.2.2. Integration of the Identified Sources of Uncertainty

_{1}, …, V

_{n}), each representing the particular vulnerability (dark grey) of endangered assets, property, or people, which are under influence of uncertainty as well.

_{1}, …, U

_{i}represent identified sources of uncertainty in the observed system. This set of variables is selected and structured by mapping on the uncertain parameters of the flood risk model factors. Furthermore, the generic model may contain additional set of subvariables U

_{1}

^{1}, …, U

_{i}

^{n}in the next hierarchical level (Figure 6), indicating that each system uncertainty source may contain multiple causes leading to its occurrence. For example, if we consider U

_{1}to be a weir failure, as one of the system uncertainties, the U

_{1}

^{1}and U

_{1}

^{2}may represent the mechanical and human failure, respectively.

#### 2.2.3. Quantification of Uncertainty

_{1}… X

_{n}) is the joint probability distribution over a set of variables, X

_{i}are the variables, and parents(X

_{i}) is the set of variables directed to X

_{i}.

_{i}belonging to a certain fuzzy set A. However, in case of one membership function describing a range of values belonging to a certain fuzzy set, the transformation to probability can be performed by the process of defuzzification, which is used to estimate crisp values of element x

_{i}for a certain degree of membership. In the case of multiple membership functions used to define some fuzzy set A, the stated belief can be described as the chance that a value of x

_{i}belongs to the particular membership function within the fuzzy set. In our case, the transformation from fuzzy variable to a probabilistic one is performed based on general normalization procedure proposed by Maskey [62]:

_{A}(x

_{i}) is a probability function, µ

_{A}(x

_{i}) is a membership value of variable x

_{i}for a particular membership function considered, which is divided by the sum of n membership values considered for variable x

_{i}. This methodology is characterized with a practicability for efficient transformation of fuzzy values to probabilistic ones, suitable for rapid integration in a Bayesian network model.

- (a)
- In the first step the prior values of parameters that are related to flood risk model factors (dark grey, fixed variables in Figure 6) are estimated for a chosen flood event. This is performed by using the results of flood models, available data, and/or expert judgement;
- (b)
- In the second step, the set of variables marked as U
_{1}… U_{i}in Figure 5 is included, representing sources of uncertainty in the managed system. This considers assigning the appropriate mathematical function to each variable, based on the experts’ judgements, modelling results, and/or historical data and events; - (c)
- The following step considers estimation of conditional probabilities between variables U
_{i}and parameters related to flood risk model factors (marked dark grey in Figure 5); - (d)
- Once the conditional probabilities are estimated, the further step is estimation of the posterior values of parameters related to flood risk model factors, quantifying the impact of variables U
_{i}on these particular parameters; - (e)
- In the final step, the posterior results from the previous step are propagated to flood risk factors (dark grey variables in Figure 5), obtaining the flood risk value and completing the analysis.

## 3. Framework Validation and Results

#### 3.1. Identification of Sources of Uncertainty for the Slavonski Brod Area

#### 3.2. Integration of the Identified Sources of Uncertainty

_{1}, which is related to the population sensitivity. The remaining risk factors (hazard and exposure) were defined by their fixed parameters; water depth and number of the exposed people (Figure 8). Flood risk model factors (dark grey) contain their uncertainty as previously defined.

#### 3.3. Quantification of Uncertainty in the Flood Risk Assessment for the Slavonski Brod Area

^{2}, 46% from 250 to 500 people per km

^{2}, and 37% of the area contains more than 500 people per km

^{2}.

^{2}), 30% moderate (250–500 per km

^{2}), and 44% high population density (>500 per km

^{2}). The vulnerability component remained the same as from Figure 13. The final flood risk, due to the weir failure, is equal to 24% low, 24% moderate, and 52% high states.

_{MOD}= 0.50, and µ

_{HIGH}= 0.25 (Figure 14). Following (6) the probability that evacuation rate of 82% belongs to the “moderate evacuation” fuzzy set state is equal to:

^{2}was reduced to approximately 10%. Furthermore, disabled people are considered to be evacuated first, so the overall population sensitivity was significantly reduced. In other words, by including some evacuation activity in the potentially flooded area we can move the flood risk value more towards the low and moderate flood risk state as compared to the no evacuation measures. This could be a very important assessment for emergency decision makers in selecting the risk management measures and their relation to the actual cost of the evacuation process.

## 4. Discussion

## 5. Concluding Remarks

- The proposed approach for the Sava River case study in Croatia shows that additional sources of uncertainty representing the system behavior could appear in the managed area for future flood events.
- Considering the scenario of infrastructure failure of the present weir alone, the Sava river would flood approximately 1 km
^{2}of the safe area, resulting in 52% of high risk, 24% of moderate risk, and 24% of low risk on the flooded area. - By combining the evacuation efficiency, as a nonstructural measure, with the weir failure event, the results indicate that for an estimated evacuation rate of 82%, the flood risk is shifted towards the low risk stage, decreasing the high flood risk value from 52% to 24%.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Estimated flooded area of city of Slavonski Brod during the flood event in the Sava River in 2014.

Funtowicz and Ravetz [34] | Ling [35], Simonovic [36,37] | Walker et al. [38] | Van Asselt [39], Brugnach et al. [40] | Baecher and Christian [41] |
---|---|---|---|---|

Inexactness, unreliability, and border with ignorance | Variability and ambiguity | Location, level, and nature of uncertainty | Incomplete knowledge, unpredictability, ambiguity | Inherent randomness, incomplete knowledge |

Flood Model Uncertainty | Parametric Uncertainty | Unexpected or Random Events | Decision Uncertainty | Ambiguity and Vagueness |
---|---|---|---|---|

Sensitivity analysis | Probability distribution | Boolean variable, probability distribution | Boolean variable, probability distribution | Fuzzy set theory |

Sources of Uncertainty | Context | Type | |
---|---|---|---|

Flood risk model | Water depth distributionEstimation of water depth distribution over an inundated area. | Hazard assessment | Flood model |

Number of people exposed to floodingEstimation of flood extent proportions in relation to population density on an inundated area. | Exposure assessment | Flood model/parametric | |

Population sensitivity to floodsAssessment of exposed population sensitivity in respect to population age and number of disabled people. | Vulnerability assessment | Parametric | |

System | Ice jamAppearance of ice on the river surface jamming on the bridge pillars. | Natural | Unexpected or random event |

Weir failurePossibility of weir closure failure due to mechanical error on the operating mechanism, or due to human operating error. | Technological/social | Unexpected or random event/decision uncertainty | |

Evacuation efficiencyRate of evacuation of people from endangered area before the incoming flood. | Social | Decision uncertainty/ambiguity |

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**MDPI and ACS Style**

Kekez, T.; Knezić, S.; Andričević, R.
Incorporating Uncertainty of the System Behavior in Flood Risk Assessment—Sava River Case Study. *Water* **2020**, *12*, 2676.
https://doi.org/10.3390/w12102676

**AMA Style**

Kekez T, Knezić S, Andričević R.
Incorporating Uncertainty of the System Behavior in Flood Risk Assessment—Sava River Case Study. *Water*. 2020; 12(10):2676.
https://doi.org/10.3390/w12102676

**Chicago/Turabian Style**

Kekez, Toni, Snježana Knezić, and Roko Andričević.
2020. "Incorporating Uncertainty of the System Behavior in Flood Risk Assessment—Sava River Case Study" *Water* 12, no. 10: 2676.
https://doi.org/10.3390/w12102676