1. Introduction

2. Materials and Methods

2.4. Tilted Bathtub Approach

To evaluate inundation risk and severity, the bathtub approach is a common method used to define the flood-prone area. This technique posits that all coastal areas connected to the sea and lying below the total water level are susceptible to flooding. In the context of wave-induced coastal flooding, the total water level is often referred to as the wave run-up. This term signifies the highest vertical rise of the water level oscillating in the nearshore due to ocean waves. Nonetheless, this method tends to overestimate flooding predictions [8,24]. The primary cause of this overestimation stems from its disregard for wave dynamics, including temporal and spatial variations, due to its inherent static nature. The model presumes that the overtopping water volume uniformly fills the landscape below the peak run-up level, thereby neglecting the dynamic nature of the phenomenon.

An effective strategy to improve the representation of flooding involves modifying the bathtub approach by incorporating historical records of flooding. By considering the known maximum overwash extension and its correlation with corresponding runup values at a particular event, a more accurate depiction of the affected area can be achieved for following events. This approach allows for potential expansion of the affected area based on higher wave-runup values, introducing an offset to the flooding extension. Therefore, the run-up estimation is required.

2.4.1. Wave Run-Up Estimation

Wave run-up is evaluated by testing several of the most well-established empirical formulas available in the literature by comparing with reports of wave overtopping in the last three decades [25,26,27,28,29,30,31,32].

Early laboratory experiment and studies [25,33] have provided evidence that the runup phenomenon primarily relies on specific factors such as the significant wave height (H₀), wave period (T), wavelength (L₀), and the local beach slope or steepness (β). These variables are interrelated through a linear dispersion relationship, with the wave period playing a crucial role. Together, they contribute to the calculation of the Iribarren number (ξ). The Iribarren number serves as a critical dimensionless indicator of the wave transformation processes occurring on sloped beaches, particularly the nature of wave breaking, which can manifest as spilling, plunging, collapsing, or surging. Additionally, this relationship sheds light on the dynamic interaction between the beach profile and incoming waves, determining whether the beach exhibits a dissipative, intermediate, or reflective response.

$$L_0=\frac{gT²}{2\pi }$$

$$\xi =tan\beta /\sqrt{H_0{/L}_0}$$

Subsequent studies have extensively explored the relationship between these essential variables, incorporating empirical coefficients to account for roughness and permeability factors. These investigations have encompassed various coastal environments, including gravel, sand, and rocky beaches, as well as the presence of hard defence structures [25,26,27,28,34,35,36]. Consequently, multiple empirical formulas have been developed to estimate wave run-up, each tailored to specific environments and objectives. In this study, six of the most widely recognized empirical formulas were evaluated to identify the one that best fits the requirements of the case study. In the context of wave runup, $R_2$ represents the elevation exceeded by the top 2% of wave runup events, serving as a critical metric to assess the potential for wave-induced coastal inundation.

$$(Holman,1986)R_2=0.83tan\beta \sqrt{H_0+L_0}+0.2H_0$$

$$(Nielsen and Hanslow,1991)R_2=\left\{\begin{array}{c}1.0005\beta {\left(H_0{L}_0\right)}^{0.5}for\beta \ge 0.1;\\ 0.0834\beta {\left(H_0{L}_0\right)}^{0.5}for\beta \ge 0.1\end{array}\right.$$

$$(Ruggiero et al.2001)R_2=0.27\sqrt{{\beta H}_0{L}_0}$$

$$(Vousdoukas et al.,2012)R_2=0.53\beta \sqrt{H_0{L}_0}+0.58\xi \sqrt{H_0³{/L}_0}+0.45$$

$$(Stockdon et al .,2006)R_2=\left\{\begin{array}{c}1.1\left(0.35\beta {\left(H_0{L}_0\right)}^{0.5}+\frac{H_0+L_0{\left(0.563{\beta }^2+0.004\right)}^{0.5}}{2}\right)for\xi >0.3\\ 0.043{\left(H_0{L}_0\right)}^{0.5}for\xi <0.3\end{array}\right.$$

$$(Power et al.,2019)x_1=\frac{H_0}{L_0}{x}_2=\beta x_3=\frac{r}{H_0}$$

The study employed a 30-year hindcast of hourly wave climate data as inputs for the five selected empirical formulas. Additionally, the beach slope was derived from the aforementioned Digital Terrain Model (DTM). A preliminary analysis was conducted to evaluate the performance of each formula, taking into account factors such as probability of occurrence and comparison with previous studies, including the work by [2]. Historical sea-level was also incorporated in the calculations, although some lack of data for a few observations were identified. Most of the formulas only consider wave and topography parameters, however, [32] also considers a roughness, or friction, coefficient, which in this case, r = 3e-5m, observed by [37]) for an asphalt bed.

2.4.2. Flood Extension Estimation

Once the most representative formula was validated for the case study, an analysis of extreme values was conducted to determine the expected wave run-up values for different return periods, specifically 5, 10, 25, 50, and 100 years. The analysis utilized the Peak Over Threshold (POT) univariate sampling method, which involved identifying run-up values exceeding a critical threshold defined by the crest of the seawall in this particular scenario. Subsequently, a probabilistic curve was fitted to the data to estimate the expected values corresponding to the different return periods, using a Generalized Paretto Distribution. Furthermore, the expected flood extension was estimated using the rules of triangles resemblance. This involved measuring the relationship between the observed flood extension and its corresponding characteristic run-up and extrapolating it to account for extreme run-up conditions, creating a sort of offset in the flooded area.

Figure 3. Scheme of the Tilted Bathtub Approach, highlighting the relationship between observed flood extension and its wave run-up, as well as the expected flood extension (red) with its run-up value, respectively.

Figure 3. Scheme of the Tilted Bathtub Approach, highlighting the relationship between observed flood extension and its wave run-up, as well as the expected flood extension (red) with its run-up value, respectively.

3. Results and Discussion

3.2. Tilted Bathtub Approach

3.2.1. Run-Up Estimations

The empirical predictors, here named as NS, PW, HM, RG, VK, and ST, respectively to Nielsen’s, Power’s, Holman’s, Ruggiero’s, and Vousdoukas’ formulas, are subsequently examined in relation to the run-up levels. For clarity and ease of visualization, Figure 6 displays the maximum run-up value every three days, considering only values exceeding 3.5m. A critical threshold of 10m was adopted, approximately matching the height of the current coastal defense structure—a seawall aligned parallel to the shoreline. This evaluation is conducted using the range of formulas applied to the representative profile, the same from the SWASH setup.

Among these formulas, VK, RG, HM, and ST formulas did not surpass the critical threshold level, indicating no occurrence of wave overtopping during the analysis period (1990-2021). Consequently, these formulas are deemed unsuitable for this specific case, given their underestimation. However, NS and PW formulas not only surpassed the critical threshold but also exhibited a relatively good agreement with observed overtopping events documented by APA (2014), Pinto (2022), and Tavares et al. (2021b). Furthermore, the event on February 2nd 2014, exhibited one of the highest run-up levels throughout the entire time series, 11.28 m and 11.66 m for PW and NS, respectively. This serves as an indicator of appropriateness given that this day was recognized as the most severe coastal flooding episode according to the Municipality.

Both formulas seem apt for this case study as they yield closely matched results (Figure 6). However, NS tends to project higher estimations of run-up levels in milder sea-states, while the PW formula generally produces more pronounced peaks during rough sea-states, while less run-up in average conditions. NS was developed based on measurements taken on sandy beaches along the New South Wales Coast, in Australia. These beaches are typically flat, aligning with Rayleigh’s distribution for the maximum levels reached by individual waves. It was found out that categorizing beaches according to its steepness, between “steep” (β > 0.1) and “flat” (β < 0.1), can enhance the formula’s accuracy.

It is also worth noting that PW is a more recent study, employing machine learning algorithms, specifically a genetic/programming-based methodology that integrates datasets from various prior authors. Consequently, it is considered to be more robust in nature.

The return periods were then obtained through the extreme values analysis, providing wave run-up levels expected for return periods of 5, 10, 25, 50, and 100 years.

3.2.2. Flood Extension

The flood extensions expected for the different run-up values were determined across four sections (Table 2). These sections were chosen based on the affected perimeter and their distinct lengths, as certain streets exhibited more extensive flooding than others. The flooded area expansion was performed in the streets, preserving the blocks as they were impermeable, apart from the 100-year return period, which covered a whole new block limit. The maximum flood extent from the validation case was 325m away from the crest of the seawall, which may be extended for up to 430m for a storm with 100-year return period. Overall, the offset has not presented a significant increase in the flooded area, which also aligns with the numerical models’ results. The expected wave run-up for a 5-year return period was less than the February 2014 event, which for run-up values was more similar to 10 years return period storm.

Figure 8. Flood map with the TBTA for different return periods.

Figure 8. Flood map with the TBTA for different return periods.

While several authors have advanced methods for assessing overwash-induced flood extensions in natural environments [9,38,39,40], many of these techniques rely on empirical assumptions or complex inputs like wave overtopping discharge, introducing significant uncertainty. The 'Tilted Bathtub Approach' is a process-based indicator rooted in the observation of a previous hazardous event, ensuring a strong alignment with real-life applications. However, it introduces uncertainty when predicting potential future events, such as expected storms with varying return periods.

4. Conclusions

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