# Spatial Representation of Coastal Risk: A Fuzzy Approach to Deal with Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Spatial Representation of Coastal Erosion Risk

**Figure 1.**An example of coastal erosion risk representation: (

**a**) tessellate the region into well-defined polygons, (

**b**) spatial representation of risk zones by aggregating a series of these polygons with the same level of risk [29].

#### 2.2. Uncertainty Characterization

**Figure 2.**A comprehensive UML class diagram of spatial uncertainty in spatial data modeling and the methods to handle it.

## 3. Spatial Fuzzy Object

_{A}(x) which associates with each x in X a real number in the interval [0,1], with the membership value at x representing the “degree of membership” of x in A [47]:

_{A}(x)|x ∈ X ∧ µ

_{A}: X → [0,1])}

## 4. Fuzzy Representation of Coastal Erosion Risk: A Conceptual Framework

**Figure 3.**UML activity diagram of conceptual framework for spatial fuzzy representation of coastal risk zones.

1. Draw the grid on the region with respect to the identified hazard and elaborated vulnerability index, |

2. For I = 1:number of vulnerable indicators, |

a. For j = 1:length of the grid, |

i. Determine the fuzzy membership value for each cell of the grid using defined fuzzy membership function of each indicator, |

ii. Assign membership value to center of each cell for each indicator, |

iii. Represent the risk value for each indicator, |

b. End |

3. End |

4. Aggregate the risk value of different indicators based on assign operator, |

i. Elaborate risk formula, |

ii. Apply IF-THEN rules, |

iii. Calculate the risk value, |

5. Represent the aggregated result, |

6. End. |

#### 4.1. Tessellation

**Figure 4.**UML class diagram of a generic schema of coastal erosion risk assessment adapted from [5].

#### 4.2. Fuzzy Representation

#### 4.2.1. Fuzzification

**Figure 5.**A graphical example of membership functions of some indicators and their crisp classifications: (

**a**) Elevation and (

**b**) Erosion Rate.

_{c},Y

_{c}). Indeed, the Gaussian function is used to feed the cell neighbors with respect to the inverse of weighted distance from the cell’s center (see Figure 6a). MAX (union) operator is applied to choose the membership value of neighboring cells. A risk zone in this case is generated by aggregating a set of cells with the same values (see Figure 6b). In Figure 6b, the color hue represents the risk value. Dark red represents higher risk with a membership value close to 1 and light blue represents lower risk with membership values closer to 0.

#### 4.2.2. Fuzzy Aggregation

IF (HydroNetwork is VH) and (ProtectStructure is VH) and (DistObjVul is VH) and (ErosionRate is H) |

THEN (Use “MAX” operation for “Erosion Risk” calculation) |

VH: very high, H: high, A: Average, L: low, VL: very Low |

**Figure 7.**(

**a**) Representation of five different indicators. (

**b**) Fuzzy aggregation of these indicators: an overlay operation (union, intersection, mean, and weighted mean).

Linguistic Expression | Crisp Value | Fuzzy Risk Value |
---|---|---|

Very Low Risk of Erosion | Risk(Erosion) = 1 | 0 ≤ Risk(Erosion) ≤ 0.175 |

Low Risk of Erosion | Risk(Erosion) = 2 | 0.175 < Risk(Erosion) ≤ 0.375 |

Average Risk of Erosion | Risk(Erosion) = 3 | 0.375 < Risk(Erosion) ≤ 0.575 |

High Risk of Erosion | Risk(Erosion) = 4 | 0.575 < Risk(Erosion) ≤ 0.775 |

Very High Risk of Erosion | Risk(Erosion) = 5 | 0.775 < Risk(Erosion) ≤ 1 |

## 5. Results: A Case Study

#### 5.1. Study Site

^{2}surface, a population density equal to 7.7 people per km

^{2}, and a total population of 3312 [62].

#### 5.2. Implementing Proposed Framework on Study Site

Source | Extracted Information |
---|---|

LiDAR Data | Slop, DEM, erosion rate |

Technical and Research Reports | Protection structure, Infrastructure situation, type of coastline, state of coastline, land use information, distance coast and 5 m depth, distance coast element at risk |

Geobase | Hydrology network and drainage, land use |

Quebec Prov. Transport Dept. | Road network |

_{i}is the vulnerability indicator and w

_{i}is the weight value. Table 5 presents the list of risk parameters, associated weights, and defined membership functions derived from technical reports from the Quebec Provincial Transport Department that are used in our case study [66]. In this case study, we do not apply any Fuzzy IF-THEN rules, because there were no stakeholder concerns to take into account. The calculated fuzzy risk values are then aggregated using the “Mean Weighted” operator as it is the best fit for the risk formula (Equation (3)). A fuzzy representation of the risk zones is shown in Figure 9. It is worth stating that the list of vulnerability indicators provided in Figure 5 is a complete list of coastal vulnerability index. Nevertheless, not all of them are available in the case studies.

**Table 5.**Risk parameters, their weight and membership functions used in the case study (adapted from [66]).

Risk Parameters | Weight (w_{t}) | Membership Function |

Protection Structure | 34% | |

Distance coast and element at risk | 17% | |

Mean Slop | 13% | |

DEM | 10% | |

Geology Type | 8% | |

Land Cover | 7% | |

Hydrology Network | 6% | |

Distance coast to 5 m depth | 5% | |

Erosion Rate |

#### 5.3. Results Interpretation

## 6. Discussion and Remarks

- Spatial uncertainty associated with object definition is explicitly dealt with through the fuzzy approach. It is also possible to attach a probability density to the values of position and measurement uncertainty. If this is the case, before using this data in CERA, cleaning the data using probability approaches with an accepted confidence level is recommended.
- Membership function definition issues are resolved by converting the crisp classification of vulnerability index to a fuzzy classification. Accordingly, the integration of multiple criteria is performed by aggregating their respective membership values using fuzzy aggregation operators. If the vulnerability index classification is not available, methods such as Fuzzy C-Mean and Fuzzy K-Mean are recommended to define the required membership functions based on available data.
- Elaborating risk formula and then constructing IF-THEN rules of associated indicators allows direct control over the entire CERA process. In addition, this provides more flexibility if one or more indicators or their classifications are changed. In this case, updating the desired information by re-running the fuzzification step or modifying the IF-THEN rules by re-executing the fuzzy aggregation step will be sufficient.
- The proposed approach allows performing multiple fuzzy aggregation operators (union, intersection, mean, mean weighted) that is required in any CERA process. The result in Figure 7b shows how significantly the choice of fuzzy operators can affect the end result. Therefore, with regards to the needs of decision makers and the emergence of protection actions, the choice of these operators is also varied.
- The flexibility of fuzzy set theory to characterize and handle inherent spatial uncertainty through the entire assessment process widely increases the confidence levels of adapted strategies for protection regions under study. It also accelerates the implementation of response plans in case of a disaster through the interpretation of the results and the prioritization of planning actions based on expert perception. From another point of view, traditional risk assessment methods lead to crisp decisions, i.e., “Yes” or ”No” while the fuzzy approach leads to smooth transitions between these two extremes.
- Fuzzy risk representation is a relatively new concept for decision makers. In this new context, decision-making processes need to be adapted and meaningful criteria need to be established to accept and manipulate fuzzy risk values. Changing the decision making culture to use fuzzy results requires finding evidence to convince the decision makers of the benefits of this new approach. The defuzzification step explained briefly in this paper is an alternative in this regard to translate fuzzy values to measurable values, making them understandable for decision makers. Kentel and Aral [2] propose a risk-tolerance measure method based on a crisp compliance guideline, which is already available in some domains, such as the health system.

- The proposed fuzzy representation is tested only on regular tessellation. The neighborhood relation is implicit, based on the ID of a cell. If an irregular tessellation is needed, more effort in neighborhood concepts and topological predicates are required.
- The temporal aspect of the fuzzy object is not taken into account in this approach. This paper only discusses the spatial extent of fuzzy objects and the situations in which the fuzzy classification is due to the multi-criteria nature of CERA and spatial uncertainty associated with object definition. This means that the risk zones are represented spatially as a snapshot of a given time period. How to handle fuzzy objects that change in different time periods needs more investigation.
- The proposed approach is employed only on a small region with a given level of detail (scale). When the analysis of extremely large amounts of data within a hierarchical system is required, the proposed approach needs to be adjusted with respect to selected technology. In this regard, efforts are mainly needed on fuzzy aggregation operators such as “Fusion” where the multi-scale representation is required.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Jadidi, A.; Mostafavi, M.A.; Bédard, Y.; Shahriari, K.
Spatial Representation of Coastal Risk: A Fuzzy Approach to Deal with Uncertainty. *ISPRS Int. J. Geo-Inf.* **2014**, *3*, 1077-1100.
https://doi.org/10.3390/ijgi3031077

**AMA Style**

Jadidi A, Mostafavi MA, Bédard Y, Shahriari K.
Spatial Representation of Coastal Risk: A Fuzzy Approach to Deal with Uncertainty. *ISPRS International Journal of Geo-Information*. 2014; 3(3):1077-1100.
https://doi.org/10.3390/ijgi3031077

**Chicago/Turabian Style**

Jadidi, Amaneh, Mir Abolfazl Mostafavi, Yvan Bédard, and Kyarash Shahriari.
2014. "Spatial Representation of Coastal Risk: A Fuzzy Approach to Deal with Uncertainty" *ISPRS International Journal of Geo-Information* 3, no. 3: 1077-1100.
https://doi.org/10.3390/ijgi3031077