# A Spatial Structure Variable Approach to Characterize Storm Events for Coastal Flood Hazard Assessment

^{*}

## Abstract

**:**

## 1. Introduction

_{s}− water levels ζ) statistical analysis, in the framework of coastal flooding hazard assessment.

_{s}, ζ) characterizing each stormy event.

_{s}, ζ) that are instead frequently observed in the same time-series from which the pairs are derived.

## 2. Critical Issues Related to Sea Storm Identification and Sampling

#### 2.1. Declustering Scheme for Sea Storm Identification

_{s}and sea-level ζ.

_{s}. A wave storm is defined as a sequence of wave states during which H

_{s}is above a given threshold, named h

_{CRIT}. Usually, the threshold h

_{CRIT}is related to the average significant wave height < H

_{s}> calculated from its time-series, so it depends on the characteristics of the recorded sea states (e.g., h

_{CRIT}= 1.5 < H

_{s}>, Boccotti, [16]). In order to preserve independency of the data, some technicalities are also considered. It is still the same storm if H

_{s}falls below the threshold but only for a short time interval Δt

_{CRIT}(Boccotti, [16], Mendoza et al. [17]), as shown in the central panel of Figure 1. For the choice of Δt

_{CRIT}, it is suggested to analyze the autocorrelation function of H

_{s}and find the microscale λ (Taylor [18]) (λ is defined as the intercept of the parabola that matches the curvature of the autocorrelation at the origin with the time axis). For simplicity, it was assumed Δt

_{CRIT}= λ.

_{CRIT}, that clearly point out the occurrence of sea storms in the wind time series. The sea level ζ is broken down into two sub-processes: the astronomical tide (ζ

_{A}) and the residual (ζ

_{R}). The latter (ζ

_{R}) represents the sum of different contributions (Pasquali et al., [19]), such as storm surge induced by wind and pressure, basin seiching (Bajo et al. [20]), etc. For instance, in the illustrative figure, seiches are evident in the oscillations that appear in the ζ

_{R}series after day 9 and their amplitude fades after 4 days.

#### 2.2. Sampling Procedure within Sea Storms

_{s}can cause flooding.

- Maximum H
_{s}and simultaneous value of ζ; - Maximum ζ and simultaneous value of H
_{s}; - Maxima of H
_{s}and ζ during the storm regardless of their concomitance.

_{s}, consists of defining a third variable function of these drivers. Similar to Mazas and Hamm [2], instead of considering an output variable (i.e., the overtopping discharge), a function that weights the effect of the variables on the flooding can be used for the sampling:

_{s}− < H

_{s}>)

_{s}and ζ that cause flooding.

_{FLOOD}exceeds a certain value h

_{LIM}, which is chosen in this study as 0.2 or 0.5 m, identifying two limit states.

_{s}, ζ) = 0 defined as:

**X**) = g(H

_{s}, ζ) = h

_{LIM}− h

_{FLOOD}(H

_{s}, ζ) = 0

_{FLOOD}reached in each grid cell of the investigated area can be calculated with a numerical model for a set of pairs of H

_{s}and ζ, representative of the typical range for the site. For the present study, the h

_{FLOOD}values are computed with the Simplified Shallow Water (SSW) model developed by Favaretto et al. [21]. This raster-based inundation model solves a simplified form of the Shallow-Water Equations (suited for GPU acceleration) for each cell (pixel) of the domain. Several validations show that this model is capable of simulate wet/dry transitions (Favaretto et al. [22]) and real flood events (Favaretto et al. [21]).

_{FLOOD}(H

_{s}, ζ) for an illustrative pixel, located on the beach. In the right panel of the figure, dots are the h

_{FLOOD}(H

_{s}, ζ) reached for 90 SSW simulations carried out with different H

_{s}and ζ, the black line is the “limit state” considering a critical value of 0.2 m and the grey area is the unsafe region.

_{s}relatively to ζ, and it is found based on the slope of the “limit state” function for each point of the vulnerability map. Actually, the slope should be evaluated at the point that is most likely to induce flooding. This point can be defined exactly once the joint statistics of H

_{s}and ζ is completely known, and it can be assessed with subsequent iterations. In the first iteration, all data points are used without declustering, and the pairs [H

_{s}, ζ] are considered independent. The proposed procedure follows the First Order Reliability Method (FORM [23]). The variables [H

_{s}, ζ] are transformed to equivalent independent standard normal random variables U = (U

_{H}, U

_{ζ}), and the “limit state” function g(H

_{s}, ζ) is linearized (in similitude with the approach used in Martinelli et al. [24]). The left of Figure 3 shows the standard space of the “limit state” computed with the SSW model and linearized. The point that has the highest failure probability is the one with the shortest distance from the limit state to the origin in the standard space. At this point in the physical space, the slope of the “limit state” (a) is computed (Figure 3 Right). The value r

_{LIM}indicates the limit of the structure variable r over which failure occurs.

_{s}and ζ that determines the maximum flooding in that particular pixel. During each sea storm, the declustering is carried out identifying the maximum r, as shown in Figure 4 and Figure 5.

_{s}and ζ for each location, i.e., for each value of the coefficient a. In theory, these pairs could be used iteratively to define more approximate joint statistics and assuming the actual correlation structure. However, we consider this step unnecessary.

## 3. Case Study

#### Available Data

_{s}time series is available from October 1987 to December 2007 with a recording time step Δt of 3 h and from January 2008 to March 2017 with a Δt = 30 min. The sea levels are available from January 1980 to December 2018 with a Δt = 1 h. For the 2018 event, the H

_{s}is available during October ([31]) every 1 h. For the 2019 event (from 1 November to 1 December 2019), the H

_{s}and ζ are available every 10 min and measured by the “Centro Previsioni e Segnalazioni Maree—CPSM, Venice” institution. To get a homogeneous bivariate series, the Hs dataset is firstly interpolated every 3 h and then the time step is re-set equal to 1 h as for the ζ dataset. Only the simultaneous recording period is taken into account, i.e., from October 1987 to March 2017. The two extreme events (2018 and 2019) are included in the statistical analysis and their representative values of H

_{s}and ζ are taken every 1 h in accordance with the 1987–2017 dataset.

_{MSL}for the Venetian littoral [36] are available from 1870 to 2017 measured at “Punta della Salute” station, situated approximately in front of the St. Mark square in the city of Venice. The ζ

_{MSL}is the long-term trend, described as a function of time, whereas ζ* (defined as ζ* = ζ − ζ

_{MSL}) is a stationary stochastic variable.

## 4. Results

_{s}ranging from 0 m to 7 m and ζ* ranging from 0 m to 2.5 m.

^{6}cells); the Gauckler–Strickler coefficient Ks used is 50 m

^{1/3}/s. Two different boundary conditions were considered: one at the shoreline, i.e., sea levels and significant wave heights, and the other at the two mouths (Livenza and Falconera), i.e., only the sea levels. Each simulation covers only 6 h and the flooding maps include the maximum flooded depth reached during the simulations.

_{LIM}value of the structure variable, considering h

_{LIM}= 0.2 m, i.e., a “nuisance flooding” (Moftakhari et al. [37]). This type of flooding refers to low levels of inundation that do not cause notable threats to people or extensive damages, but it can disturb daily activities, add strain on infrastructures (roads, sewers, drainage systems, etc.) and cause minor damages to private properties.

_{LIM}for the areas with a = 0.12 to 0.17, corresponding to the yellow area in Figure 7.

_{CRIT}equal to 1 m and selecting Δt

_{CRIT}equal to 12 h from the analysis of the autocorrelation function. The total number of independent storms identified is 1057, i.e., ~34 events per year. Four samples r can be derived, corresponding to the different zones highlighted in Figure 7, in order to establish coastal hazard maps.

_{LIM}}, i.e., in this case the probability that a flooding level equal to 0.2 m is exceeded. Figure 9 shows the hazard map for the Caorle area, expressed as a function of the return period T

_{R}.

#### Discussion on the October 2018 and November 2019 Events

_{LIM}= 1.1 m. The two events caused the failure since both exceeded the limit state.

_{0}), exceedance probability P (r > r

_{0}), and return period T

_{R}(where λ is the number of events per year). Table 1 shows also, in the last three rows, the results carried out with three more traditional sampling procedures. In detail, the following samples were taken into account: (S1) Maximum H

_{s}and simultaneous value of ζ*; (S2) Maximum ζ* and simultaneous value of H

_{s}; (S3) Maxima of H

_{s}and ζ* during the storm regardless of their concomitance. For these samples, copulas are used to describe the dependence structure associated with the joint distribution of the two variables. A thorough introduction to Copula modelling and a large selection of the most common families are provided in [38,39]. Similar to the analysis of De Michele et al. [40], the Gumbel–Hougaard distribution, i.e., an Archimedean extreme value copula, is selected as the best candidate to model the data. To estimate the model parameters θ, the maximum likelihood estimation method (MLE) guarantees that the dataset is the most probable under the assumed statistical model. The 2018 event was characterized by extreme waves and high sea levels; conversely, the 2019 event was characterized by extreme sea levels and high waves. For the 2018 event, the return period (T

_{R}) based on sample S1 is very high. Analyzing the same event considering sample S2, the associated T

_{R}is low and the exceptionality is not confirmed. For the 2019 event, the T

_{R}based on sample S2 is very high and the T

_{R}based on sample S1 low. Both events appear to be very rare with the analysis on sample S3, assuming that the maxima occur at the same time in the event.

_{LIM}= 1.4 m. From Table 1, the r

_{0}of November 2019 is larger and therefore flooding is expected. The associated return period in this area is T

_{R}= 47 years. In fact, marine ingression occurred along the Caorle coastline during November 2019: both the beaches were completely flooded and the waves reached the church at the cusp (pictures in Figure 11).

## 5. Conclusions

_{s}and the water level ζ. In turn, the bivariate statistics is significantly affected by the methodology used for the selection of the independent sample pair H

_{s}, ζ during a sea storm.

_{s}and ζ that is the most critical for flooding. This study proposes a procedure to find a spatial structure variable, that is a linear combination of water level and wave height, specific for each location, based on the slope of the failure function at the design point.

_{s}, and hence the relative flooding occurrence is mainly affected by the maximum water levels statistics. Conversely, for an area located just behind a seawall where overtopping is relevant, the role played by H

_{s}is dominant, and the relative flooding occurrence is strongly affected by the maximum significant wave height statistics.

_{s}and ζ significantly affects the return period of the critical storm and the sampling method should therefore be considered as an important aspect for the evaluation of the flooding hazard. The comparison between traditional sampling methods and the proposed one highlights that the latter gives more consistent results in terms of return periods.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of sea storm and declustering: the upper panel shows the wind velocity time series (the meteorological forcing), the central panel shows the significant wave height time series and the lower panel shows the sea level time series and its contributions. The grey boxes highlight two independent wave storms, defined by the exceedance of h

_{CRIT.}

**Figure 2.**

**Left**: Example of schematic topographic map formed by pixels.

**Right**: Simulated flooding levels (h

_{FLOOD}) for a set of H

_{s}and ζ pairs, at the selected beach pixel. The black line is the “limit state” and the grey area is the unsafe region, considering a critical value h

_{LIM}= 0.2 m.

**Figure 3.**Standard space (

**left**): limit state function calculated with the SSW model (black dots) and linearized. The red segment highlights the point with the shortest distance from the origin. Physical space (

**right**): limit state function derived from back-transformation in the standard space. The dashed line highlights the slope of the “limit space” and the red line is the structure variable limit over which failure occurs. In both spaces also the assumed joint probability density function of U

_{H}and U

_{ζ}and of H

_{s}and ζ is drawn.

**Figure 4.**Choice of the samples for each sea storm. The red line is the structure variable determined for an illustrative pixel and used for the sampling. The red stars are the chosen samples.

**Figure 5.**Example of how storms 1 and 2, identified in Figure 4, appear for a specific pixel: (

**left**) in the standard space and (

**right**) back-transformed in the physical space.

**Figure 9.**Hazard map for the Caorle coastline, i.e., the probability that a flooding level 0.2 m is exceeded, expressed as a function of the return period.

**Figure 10.**Measurements of the 28 October 2018 and 12 November 2019 stormy events (

**left**and

**right**) and their evolution in the Hs- ζ plane.

**Table 1.**Return periods for the two extreme events occurred along the Venetian littoral in 2018 and 2019 evaluated considering different types of samples.

Samples | October 2018 | November 2019 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

H_{s} (m) | ζ* (m) | r_{0}(m) | P (r > r_{0}) | T_{R} (y)=1/[λ P (r >r _{0})] | H_{s} (m) | ζ* (m) | r_{0}(m) | P (r > r_{0}) | T_{R} (y)=1/[λ P (r >r _{0})] | |

a = 0 | 2.80 | 1.10 | 0.83 | 0.0887 | 6 | 2.34 | 1.42 | 1.15 | 0.0029 | 181 |

a = 0.05 | 2.80 | 1.10 | 0.95 | 0.0770 | 7 | 2.34 | 1.42 | 1.25 | 0.0031 | 166 |

a = 0.1 | 5.05 | 0.93 | 1.13 | 0.0455 | 11 | 2.34 | 1.42 | 1.35 | 0.0053 | 97 |

a = 0.15 | 5.05 | 0.93 | 1.36 | 0.0217 | 24 | 2.34 | 1.42 | 1.45 | 0.0109 | 47 |

a = 0.2 | 5.05 | 0.93 | 1.60 | 0.0125 | 41 | 2.34 | 1.42 | 1.55 | 0.0189 | 27 |

a = 0.25 | 5.05 | 0.93 | 1.84 | 0.0079 | 66 | 2.34 | 1.42 | 1.65 | 0.0317 | 16 |

a = 0.3 | 5.05 | 0.93 | 2.07 | 0.0078 | 66 | 2.34 | 1.42 | 1.75 | 0.0493 | 10 |

a = 0.35 | 5.05 | 0.93 | 2.31 | 0.0084 | 62 | 2.34 | 1.42 | 1.85 | 0.0681 | 8 |

a = 0.4 | 5.05 | 0.93 | 2.54 | 0.0108 | 48 | 2.34 | 1.42 | 1.95 | 0.0865 | 6 |

a = 0.45 | 5.05 | 0.93 | 2.78 | 0.0096 | 54 | 2.34 | 1.42 | 2.05 | 0.1093 | 5 |

a = 0.5 | 5.05 | 0.93 | 3.01 | 0.0110 | 47 | 2.34 | 1.42 | 2.15 | 0.1304 | 4 |

S1: Max Hs and simultaneous ζ | 5.29 | 0.81 | - | 0.0026 | 258 | 3.16 | 0.72 | - | 0.1020 | 7 |

S2: Max ζ and simultaneous Hs | 0.72 | 1.10 | - | 0.0377 | 13 | 2.34 | 1.42 | - | 0.0050 | 102 |

S3: Maxima of Hs and ζ | 5.29 | 1.10 | - | 0.0024 | 91 | 3.16 | 1.42 | - | 0.0011 | 197 |

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**MDPI and ACS Style**

Favaretto, C.; Martinelli, L.; Ruol, P.
A Spatial Structure Variable Approach to Characterize Storm Events for Coastal Flood Hazard Assessment. *Water* **2021**, *13*, 2556.
https://doi.org/10.3390/w13182556

**AMA Style**

Favaretto C, Martinelli L, Ruol P.
A Spatial Structure Variable Approach to Characterize Storm Events for Coastal Flood Hazard Assessment. *Water*. 2021; 13(18):2556.
https://doi.org/10.3390/w13182556

**Chicago/Turabian Style**

Favaretto, Chiara, Luca Martinelli, and Piero Ruol.
2021. "A Spatial Structure Variable Approach to Characterize Storm Events for Coastal Flood Hazard Assessment" *Water* 13, no. 18: 2556.
https://doi.org/10.3390/w13182556