the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
The influence of infragravity waves on the safety of coastal defences: a case study of the Dutch Wadden Sea
Christopher H. Lashley
Sebastiaan N. Jonkman
Jentsje van der Meer
Jeremy D. Bricker
Vincent Vuik
Many coastlines around the world are protected by dikes with shallow foreshores (e.g. salt marshes and mudflats) that attenuate storm waves and are expected to reduce the likelihood and volume of waves overtopping the dikes behind them. However, most of the studies to date that assessed their effectiveness have excluded the influence of infragravity (IG) waves, which often dominate in shallow water. Here, we propose a modular and adaptable framework to estimate the probability of coastal dike failure by overtopping waves (P_{f}). The influence of IG waves on overtopping is included using an empirical approach, which is first validated against observations made during two recent storms (2015 and 2017). The framework is then applied to compare the P_{f} values of the dikes along the Dutch Wadden Sea coast with and without the influence of IG waves. Findings show that including IG waves results in 1.1 to 1.6 times higher P_{f} values, suggesting that safety is overestimated when they are neglected. This increase is attributed to the influence of the IG waves on the design wave period and, to a lesser extent, the wave height at the dike toe. The spatial variation in this effect, observed for the case considered, highlights its dependence on local conditions – with IG waves showing greater influence at locations with larger offshore waves, such as those behind tidal inlets, and shallower water depths. Finally, the change in P_{f} due to the IG waves varied significantly depending on the empirical wave overtopping model selected, emphasizing the importance of tools developed specifically for shallow foreshore environments.
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1.1 Background
Coastal defences (e.g. dikes or seawalls), fronted by wide, shallow foreshores, protect many coastlines around the world. Examples include the sandy foreshores along the Belgian coast (Altomare et al., 2016); the wide shelves of the Mekong Delta, Vietnam (Nguyen et al., 2020); and the intertidal flats of the Wadden Sea along the northern coast of the Netherlands (Vuik et al., 2016). These bodies of sediment reduce the water depth in front of the structure such that large incident waves are forced to break. The foreshore contributes to a reduced wave load at the structure and is therefore expected to reduce the likelihood of failure. If vegetation is present, the drag forces exerted by stems, branches and leaves further enhance this attenuation effect (Dalrymple et al., 1984; Mendez and Losada, 2004). Several studies sought to quantify the hazard mitigation potential of shallow foreshores, with and without vegetation, including physical model tests (Möller et al., 2014), numerical simulations (Vuik et al., 2016; Willemsen et al., 2020) and field measurements (Garzon et al., 2019). While these studies assessed the ability of the foreshore to attenuate the height of windsea and swell (hereafter, SS) waves – with frequencies typically greater than 0.05 Hz – they neglected the influence of infragravity (hereafter, IG) waves.
Under extreme conditions, with large offshore waves and very shallow local water depths, the shoaling and subsequent breaking of SS waves results in energy transfer to lower frequencies and the growth of IG waves, also referred to in literature as “surf beat” (Bertin et al., 2018; Van Dongeren et al., 2016). IG waves are widely recognized as the driving force behind coastal erosion and flooding along shallow coastlines. Recent reports of their impact include the following: unexpectedly high wave runup at the coast of the island of Bannec, France (Sheremet et al., 2014); extensive damage and casualties along the coralreeflined coast in the Philippines during Typhoon Haiyan (Roeber and Bricker, 2015); the erosion and overwash of several dunes along the western coast of France (Baumann et al., 2017; Lashley et al., 2019); and damage along the Seisho coast of Japan during Typhoon Lan (Matsuba et al., 2020). Despite this knowledge, IG waves are often not considered in the risk assessment of coastal defences. This oversight is linked to the common practice where phaseaveraged wave models are used to estimate wave parameters at the toe of the structure. These parameters are then used as input to empirical formulae (EurOtop, 2018). As phaseaveraged models (e.g. Simulating Waves Nearshore, SWAN; Booij et al., 1999) tend to exclude IGwave dynamics, the impact of IG waves on the safety of coastal defences is naturally not considered. An empirical approach developed for shallow foreshores using offshore forcing parameters would implicitly account for the intermediate processes, including the effects of IG waves (Mase et al., 2013; Goda, 2010). However, the more widely used formulae are based on parameters at the structure toe (EurOtop, 2018).
In the Netherlands, coastal defences are typically designed to resist the volume of water expected to pass over the crest of (or overtop) the structure due to wave action during storms associated with a very high return period (e.g. 2000 to 10 000 years). This phenomenon, referred to as wave overtopping, is typically represented by a mean (timeaveraged) discharge per metre width of structure, q (m^{3} s^{−1} m^{−1} or L s^{−1} m^{−1}). The probability of failure due to wave overtopping is then determined by assessing the likelihood that the actual discharge (q_{a}) exceeds some critical value (q_{c}), which is dependent on the erosion resistance of the grasscovered landward slope. Following a recent policy revision, the safety standard for the coastal defences in the Netherlands is now defined by an (acceptable) probability of failure. For example, typical values for the Dutch Wadden Sea coast – a shallow, intertidal area in the north of the country – are failure probabilities of $\mathrm{1}/\mathrm{1000}$ and $\mathrm{1}/\mathrm{3000}$ per year. This approach usually considers multiple failure mechanisms (e.g. scour, armour unit instability and slope instability); however, here we limit the analysis to dike failure by wave overtopping, which typically governs the design process.
From a design perspective, the presence of IG waves typically results in higher characteristic values of the two main parameters used to estimate q_{a}: namely, the significant wave height and spectral wave period, both assessed at the structure toe (Van Gent, 1999; Hofland et al., 2017). Vuik et al. (2018b) assessed the overtopping failure probability of an idealized the dike–foreshore system, representative of the Dutch Wadden Sea coast, considering the effects of vegetation. This study considered the influence of IG waves on the wave period at the toe using the Hofland et al. (2017) empirical model but neglected their influence on the wave height. Furthermore, Nguyen et al. (2020) later showed that the Hofland et al. (2017) formulae tend to underestimate the development of longer spectral wave periods on foreshore slopes milder than 1:250 (Nguyen et al., 2020) – which is a typical characteristic of the Wadden Sea. As a result, the true influence of IG waves along the Dutch Wadden Sea coast remains unknown.
Oosterlo et al. (2018) carried out a similar probabilistic assessment of a dike with a sandy foreshore, in the south of the Netherlands, but directly included the IG waves using the XBeach Surfbeat numerical model (Roelvink et al., 2009) to estimate the wave parameters at the toe. The authors found, for the considered case, that accounting for the IG waves resulted in 10^{3} times higher overtopping failure probabilities compared to methods that neglected them. This rather striking finding requires further investigation, particularly, to determine if the large IGwave influence reported by Oosterlo et al. (2018) is valid or if it was merely an artefact of the method used.
Lashley et al. (2020b) demonstrated that the influence of IG waves on wave parameters at the toe could be accurately estimated using a combined numerical and empirical approach. In this approach, the phaseaveraged wave model (SWAN) is used to simulate the dissipation of SS waves in shallow water, while the IG component is estimated using empirical formulae. Since this approach allows for the accurate representation of IG waves at the dike toe but maintains the utility and speed of phaseaveraged wave modelling, it can be applied on a large scale with little computational effort. In the present study, this approach is extended and used as a key component to assess the influence of IG waves on the probability of dike failure along the Dutch Wadden Sea coast.
1.2 Objective and approach
Previous studies either neglected the influence of IG waves on the probability of failure by wave overtopping or yielded inconclusive results (Oosterlo et al., 2018). Consequently, the influence of IG waves on the safety of coastal defences remains unknown. To remedy this, it is the primary aim of the current paper to investigate the influence of IG waves on the probability of failure due to wave overtopping for coastal defences (dikes) with shallow foreshores. This is achieved by first augmenting the probabilistic framework developed by Vuik et al. (2018b) by incorporating newly validated empirical formulae that capture the influence of IG waves on design parameters, following the approach of Lashley et al. (2020b). The modified framework is then used to estimate the probability of dike failure by wave overtopping along the Dutch Wadden Sea coast.
1.3 Outline
This paper is organized as follows: Sect. 2.1 describes the geographic area that will be the focus of this study. Section 2.2 provides a detailed description of the dike–foreshore system under consideration and the probabilistic framework applied, including descriptions of the numerical and empirical models therein. Section 2.3 describes the field dataset used to validate the empirical approach for the inclusion of IG waves. In Sect. 3, the results of the validation are presented, followed by the application of the framework to the wider Dutch Wadden Sea coast. Section 4 discusses the results and their implications for practice, and Sect. 5 concludes the paper by addressing the overall research objective, stating limitations and identifying areas for future work.
2.1 Study area
The Dutch Wadden Sea is a shallow, mildly sloping intertidal zone situated between the Netherlands (mainland) and several barrier islands, which shield the area from large North Sea waves (Fig. 1). Along the Wadden Sea coast, a system of dikes fronted by salt marshes and mudflats exists. In the present study, we consider the stretch of dikes with shallow foreshores – that is, with bed levels at the toe either just below or a few metres higher than mean sea level (NAP, a Dutch abbreviation for the Normal Amsterdam Water Level) – that are typically impacted by northwesterly waves during storms (Fig. 1). Further information on storm conditions in the study area is presented in Table 1 and Sect. 2.3.
The analysis includes the dikes from the city of Harlingen (in the province of Friesland) to those west of Eemshaven (in the province of Groningen), but it excludes the flood defences in front of harbours and areas referred to as summer polders (Fig. 1). Summer polders are lowlying, embanked areas situated in front of the dike and are usually dry in the summer months but may flood during winter storms. As these polders extend for several kilometres, the 1 km transect approach taken here would not be representative.
2.2 Model framework and system description
The model framework used to compute the probability of flooding due to wave overtopping is presented in Fig. 2 and is modified after Vuik et al. (2018b) to include the effect of IG waves. Boundary conditions of offshore wave heights, periods and water levels are transformed over the foreshore to the structure toe using SWAN. These SWAN estimates at the toe are then modified using empirical formulae to account for IG waves. These estimates are then used as input to calculate the actual overtopping discharge, q_{a}. The probability of failure by wave overtopping, P(Z<0), is then obtained using the firstorder reliability method (FORM; Hasofer and Lind, 1974).
While the above framework follows that of Vuik et al. (2018b), there are noteworthy differences between the two approaches. Firstly, the influence of IG waves on both the wave height and the period at the toe is considered – using empirical formulae that are valid for a wide range of foreshore slopes ($\mathrm{10}\le \mathrm{cot}\left(m\right)\le \mathrm{1000}$). Secondly, the effect of wind on wave transformation is neglected here due to close proximity to the shoreline – within 1 km. Lastly, wave attenuation by vegetation is not included in the wider probabilistic analysis because of the following: (i) storms generally occur in the winter season, where there is little vegetation present, and (ii) it is very likely that almost all vegetation present will flatten or break under extreme forcing (Möller et al., 2014; Vuik et al., 2018a). That said, the effect of vegetation (should it be present and remain standing) is demonstrated for one location (Uithuizerwad field site, Fig. 1) in Sect. 3.2 and treated as part of the discussion (Sect. 4.1). The individual components of the model framework are described in detail below. A visual representation of the dike–foreshore system, as well as the various framework components, is provided in Fig. 3.
2.2.1 Boundary conditions
Offshore waves and water levels
To obtain the offshore water levels ($\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$), wave heights (${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$) and periods (${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$), HydraNL (Duits, 2019) – a probabilistic application designed specifically for the assessment and design of flood defences in the Netherlands – was applied. It uses statistics of wind speed, wind direction, water level and their respective correlations to find the corresponding wave characteristics in a precalculated database, obtained using the phaseaveraged numerical model SWAN (Booij et al., 1999). The tool provides estimates along the entire Dutch Wadden coast, every 250 m, a few hundred metres offshore. To reduce the overall computational time of the probabilistic calculations (around 10 min per dike section), the output locations were reduced to one every 1.5 km (Fig. 1).
The wave heights estimated by HydraNL were reverseshoaled, using linear wave theory, to an offshore point approximately 1 km from the dike toe (Fig. 3). This was done to obtain estimates of the offshore wave height (${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$) for use with Eq. (12). Compared to the HydraNL estimates (a few hundred metres offshore), the reverseshoaled significant wave heights (1 km offshore) were on average 2 % higher, with a maximum increase of 4 % and maximum reduction of 9 %. In this approach, the effects of friction and refraction are neglected. Likewise it is assumed that no local generation or wave breaking occurred. The assumption of nonbreaking waves is considered reasonable, since the average ratio of the wave height to water depth at the HydraNL output location was 0.37, while the average ratio for breaking waves – referred to as the breaker index – was estimated as 0.79 (using Eq. 4). In addition, due to the close proximity to the shoreline (within 1 km), it is unlikely that local generation by wind would be significant.
For each location, five exceedance probabilities were considered in HydraNL: $\mathrm{1}/\mathrm{100}$, $\mathrm{1}/\mathrm{300}$, $\mathrm{1}/\mathrm{1000}$, $\mathrm{1}/\mathrm{3000}$ and $\mathrm{1}/\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{000}$ per year (Table 1). Using the $\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$, ${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ estimates for each probability of exceedance, Weibull distribution parameters – namely, scale and shape parameters – were derived to accurately describe the extremes. The range of the scale and shape parameter values is provided in Table A1.
Dike–foreshore characteristics
The bathymetry of the Dutch coast, from dry land up to the 20 m isobath in the North Sea, is continuously measured (at least once every 7 years) by the Dutch government (Rijkswaterstaat). This dataset, referred to as “Vaklodingen” (Wiegmann et al., 2005), covers the Wadden Sea with a 20 m grid resolution (Fig. 4). Crossshore transects of approximately 1 km, at alongshore intervals corresponding to the HydraNL output locations, were extracted considering a NW–SE orientation – inline with the dominant wind–wave direction during storms (NW) (Vuik et al., 2018b). By aligning the transect with the dominant wind–wave direction, the influence of wave obliqueness on wave overtopping, which is typically taken into account using a correction factor, may be neglected (Willemsen et al., 2020). It should be noted that assuming a NW–SE orientation results in an artificially milder dike slope for dikes that are not perpendicular to the assumed transect. This is taken into account in the average dike slope calculation.
For each transect, the mean elevation (z_{b}) and average foreshore slopes (tan (m)) were obtained. To account for variations in bathymetry and measurement inaccuracies, z_{b} was treated as a normally distributed parameter with a standard deviation of 0.2 m (Vuik et al., 2018b). The average slope was determined as the bestfit line (leastsquares method) considering the foreshore elevation data points between the dike toe and 1 km offshore. While the actual bathymetry is used for the numerical modelling of the SS waves, the estimated tan (m) is necessary for use with the empirical formulae for the influence of the IG waves (Sect. 2.2.2). Given the range of validity of the empirical formulae (Eqs. 12 and 18), a minimum foreshore slope of $\mathrm{1}/\mathrm{1000}$ (or 0.1 %) is considered here. This is in line with the common approach where slopes milder than or equal to 0.1 % are treated equally as (nearly) flat (Keimer et al., 2021; Steendam et al., 2004). Note that the calculated foreshore slopes ranged from −0.04 % to 4 % with an average of 0.14 %, where a negative slope indicates a slight downward slope towards the dike toe.
Given the significant influence of the dike geometry on the calculated probability of failure (Sect. 3.2.2), the crest level (z_{c}) and average dike slope (tan (α)) were treated here as deterministic parameters with the same values applied to each location. This was done to remove the influence of variations in dike geometry on the calculated failure probabilities and allow the analysis to focus on what occurs over the foreshore. The crest levels were set to 6 m + NAP, corresponding to the required safety level (probability of failure less than $\mathrm{1}/\mathrm{1000}$ per year). Similarly, the dike slopes were set to $\mathrm{1}/\mathrm{7}$ to represent the average slope of a typical Wadden Sea dike, which is often characterized by 1:4 upper and lower slopes separated by a mildly sloping berm. This value also accounts for the NW–SE orientation, which results in an artificially milder average dike profile. Note that an analysis of the sensitivity of the estimated probability of failure to variations in z_{c}, including its treatment as deterministic versus stochastic, is provided in Sect. 3.2.2. It should be noted that the actual crest levels of the Dutch Wadden Sea dikes typically exceed 8 m + NAP; however, a crest level this high would result in extremely small failure probabilities, which would distract from the findings herein.
2.2.2 Wave transformation
Numerical model for SS waves: SWAN
SWAN is a thirdgeneration, phaseaveraged wave model used to estimate the generation (by wind), propagation and dissipation (by depthinduced breaking and bottom friction) of waves from offshore to the structure toe (Booij et al., 1999). This includes wave–wave interactions, in both deep and shallow water, and waveinduced setup but neglects waveinduced currents and the generation or propagation of IG waves. SWAN computes the spectral evolution of wave action density (A) in space and time. For stationary 1D simulations, the governing equations follow
where c_{x} is the propagation velocity of wave energy in the x direction; ω is the frequency; and S_{tot} may include dissipation terms due to depthlimited wave breaking (${\stackrel{\mathrm{\u203e}}{D}}_{\mathrm{w}}$), vegetation (${\stackrel{\mathrm{\u203e}}{D}}_{\mathrm{v}}$) and bottom friction, and energy transfer terms.
To simulate depthlimited wave breaking, SWAN uses the following parametric dissipation model, by default (Battjes and Janssen, 1978)
and Q_{b} is estimated as
where f_{mean} is the mean wave frequency, H_{max}=γ_{bj}h and E_{tot} is the total wave energy variance. Here, the breaker parameter (γ_{bj}) is based on the offshore wave steepness, ${s}_{\mathrm{0}}={H}_{\mathrm{rms}}/{L}_{\mathrm{0}}$ (Battjes and Stive, 1985), as
where H_{rms} is the root mean square wave height, with ${H}_{\mathrm{rms}}=\sqrt{\mathrm{8}\cdot {E}_{\mathrm{tot}}}$. Following Vuik et al. (2018b), γ_{bj} is treated as a normally distributed parameter with a standard deviation of 0.05 and mean value determined using Eq. (4).
As recommended by BaronHyppolite et al. (2018), the explicit vegetation representation in SWAN – which was implemented by Suzuki et al. (2012) – is applied. This method represents vegetation as rigid cylinders, following the approach of Dalrymple et al. (1984) modified for irregular waves by Mendez and Losada (2004). In this approach, the mean rate of energy dissipation per unit horizontal area due to wave damping by vegetation (ϵ_{v}) is given by the following:
where ρ is the density of water, g is the gravitational acceleration, k is the mean wave number, h is the water depth, C_{D} (0.4) is the drag coefficient, b_{v} (3 mm) is the stem diameter, N_{v} (1200 stems m^{−2}) is the vegetation density and h_{v} (0.3 m) is the vegetation height; the values in parentheses are representative of salt marshes in the Netherlands (Vuik et al., 2016).
Deepwater processes such as white capping, wind and quadruplet wave–wave interactions were disabled, while triad wave–wave interactions, a shallow water process, were activated. All other model parameters were kept at their default values. For all simulations, a constant grid spacing of 2.5 m was applied. This corresponded to approximately 15–20 grid cells per deepwater wavelength.
Empirical formulae for the influence of IG waves
The influence of the IG waves may be represented by an increase in the design parameters, namely the total significant wave height (${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$) and spectral wave period (${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$) at the toe based on incident waves, where
and
where
where C_{ηη}(f) is the wave energy density and 0.05 Hz is the frequency separating SS and IG motions. It should be noted that for conditions with a single, clearly defined peak frequency (f_{p}), the frequency separating IG and SS motions is typically taken as ${f}_{\mathrm{p}}/\mathrm{2}$. However, as wave spectra in the Dutch Wadden Sea typically show multiple peaks, a separation frequency of 0.05 Hz is typically used to avoid contaminating the IG signal with that of swell (with periods around 10 s or 0.1 Hz). This choice of split frequency is consistent with previous studies in the area (Engelstad et al., 2017; De Bakker et al., 2014) and coincides with the minimum in spectral density observed in the field data (Fig. B1).
Influence on significant wave height at the toe
The relative magnitude or significance of IG waves at the toe of the dike (${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$) may be expressed as the following:
Using a numerical dataset of 672 XBeach nonhydrostatic (Smit et al., 2010) simulations with varied ${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$, ${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$, h_{toe}, m, α, wave directional spreading (σ_{dir}) and width of vegetated cover (W_{veg}) parameter values, Lashley et al. (2020a) derived the following formulae to estimate ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$:
where the coefficient (C) is 0.36 m^{−0.5} and ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathit{\sigma}}$, ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{h}}$, ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{f}}$, ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{v}}$ and ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{d}}$ are influence factors for wave directional spreading, water depth at the toe, foreshore slope, vegetated cover and structure slope, respectively. For the probabilistic analysis, Eq. (12) is multiplied by a normally distributed factor (f_{IG}) with a mean (0.99) and standard deviation (0.18) based on the bias and scatter observed during its derivation, respectively (Lashley et al., 2020a).
The individual factors are detailed below:
where the wave directional spreading, σ_{dir}=24^{∘} to represent the windsea conditions of the Wadden Sea.
where W_{veg} is measured as the horizontal, crossshore width of the vegetated cover. It should be noted that the influence of vegetation on IG waves was assessed for very shallow conditions (${h}_{\mathrm{toe}}/{h}_{\mathrm{v}}=\mathrm{3.3}$); therefore, the performance of Eq. (16) for vegetation with larger ratios of water depth to stem height is yet to be verified. Lastly, to account for the influence of waves reflected at the dike slope,
For an analysis of incident waves only, ${\stackrel{\mathrm{\u203e}}{\mathit{\gamma}}}_{\mathrm{d}}=\mathrm{1}$.
The above approach (Eq. 12) accounts for IGwave generation by the following: (i) boundwave shoaling over mildly sloping bathymetry and (ii) the temporal variation in the location of breaking waves, known as the breakpoint forcing mechanism (Battjes, 2004). However, it does not account for IG waves that may be refractively trapped, known as edge waves, or those reflected from a distant coast, known as leaky waves (Elgar et al., 1992; Bertin et al., 2018; Reniers et al., 2021). The relevance of these free IG waves to the Dutch Wadden Sea coast is still to be confirmed.
Influence on the spectral wave period at the toe
The existing method to estimate the increase in the spectral wave period at the toe due to IG waves in shallow water, developed by Hofland et al. (2017), was based on laboratory tests with foreshore slopes, $\mathrm{35}<\mathrm{cot}\left(m\right)<\mathrm{250}$. While the method proved accurate within this range, it tended to underestimate ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ for slopes gentler than 1:250 (Nguyen et al., 2020). As foreshores in the Wadden Sea are typically 1:500 to (nearly) flat, a new formulation for ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ is derived here – using the abovementioned numerical dataset (Lashley et al., 2020a).
Since ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ and ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ both describe the amount of energy in the IG band compared to the SS band, it stands to reason that a simple relation should exist between the two parameters. From the Lashley et al. (2020a) numerical dataset with $\mathrm{10}<\mathrm{cot}\left(m\right)<\mathrm{1000}$, the following relationship between ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$, ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ and cot α_{fore} was found (R^{2}=0.92):
Further details on the derivation of Eq. (18) and its performance in comparison to the Hofland et al. (2017) model are provided in Appendix C. For the probabilistic analysis, Eq. (18) is multiplied by a normally distributed factor (${f}_{{T}_{m}}$) with a mean (0.99) and standard deviation (0.17) based on the bias and scatter observed during its derivation, respectively.
2.2.3 Wave overtopping
Empirical formulae for actual wave overtopping
In the present study, the overtopping formula proposed by Van Gent (1999) (Eq. 19) is applied. This formula was chosen because it (i) was developed specifically for shallow foreshores, (ii) considers the influence of both SS and IG waves, and (iii) is considered valid for a wide range of breaker parameter (${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$) values.
where
where g is the gravitational constant of acceleration, α is the dike slope, ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$ is the Iribarren number (also referred to as the breaker parameter) and ${L}_{m\mathrm{1},\mathrm{0}}$ is a fictitious wavelength based on the spectral wave period at the toe. It is important to note that ${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ in the above equations are based on the incident waves (i.e. without the influence of wave reflection at the structure). The empirical coefficient (c) is a normally distributed parameter with a mean of −0.92 and a standard deviation of 0.24. Here, Eq. (19) is applied to all locations, regardless of the ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$ value. However, it should be noted that this approach does not coincide with the current standard (EurOtop, 2018). EurOtop (2018) recommends that different formulae be applied depending on the ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$ value (Van der Meer and Bruce, 2014; Altomare et al., 2016). However, due to the gentle dike (1:7) and foreshore slopes (1:600, on average) considered here, applying the EurOtop (2018) approach proved challenging. This is discussed in detail in Sect. 4.2.
The dikes of the Dutch Wadden Sea are typically grasscovered and therefore treated as smooth (without roughness elements). To simplify the calculation, the dikes are assumed to be uniformly sloping (without a berm). However, it should be noted that the presence of berms and roughness elements could significantly reduce the overtopping discharge (Bruce et al., 2009; Chen et al., 2020; Van der Meer, 2002).
Critical wave overtopping
The erosion resistance of the grasscovered landward slope of the dike is described by a critical or tolerable overtopping discharge (q_{c}). Given the significant influence of this parameter on the probability of dike failure by wave overtopping (Sect. 3.2.2), it is treated here as a deterministic parameter with a value of 50 L s^{−1} m^{−1} for each location. In this way, the influence of other parameters, such as those linked to the IG waves, can be better assessed. An analysis of the sensitivity of the estimated probability of failure to changes in q_{c} is provided in Sect. 3.2.2. This analysis also demonstrates how the probability of failure would change if q_{c} were instead treated as a stochastic parameter.
2.2.4 Probabilistic methods
FORM
The opensource implementation of FORM, part of OpenEarthTools (Van Koningsveld et al., 2010), is used to evaluate the limit state function (LSF) for any possible combination of input variables, which are each described by probability distributions. The following LSF is considered here (Oosterlo et al., 2018):
where Z is the limit state considering the critical (q_{c}) and actual (q_{a}) overtopping discharges, which represent the resistance and load, respectively, and the probability of failure by wave overtopping, ${P}_{\mathrm{f}}=P(Z<\mathrm{0})$ or P(q_{a}>q_{c}).
FORM simplifies the mathematical problem by linearizing the LSF and transforming all probability distributions to equivalent normal distributions. P_{f} is then expressed in terms of a reliability index (β), which represents the minimum distance from the most probable failure point on the limit state surface (Z=0), referred to as the design point, to the origin of the transformed coordinate system (Hasofer and Lind, 1974).
where μ_{z} and σ_{z} are the mean and standard deviation of the limitstate function (Z), respectively, and
where Φ is the cumulative distribution function for a standard normal variable.
FORM starts in a userdefined position in the probability density functions of all variables (e.g. the mean value). It then uses an iterative procedure to update the design point until convergence is achieved (Vuik et al., 2018b). In each iteration, FORM tests how strong the LSF responds to a perturbation of each individual variable, X_{i}. The response is expressed in terms of the partial derivative $\partial Z/\partial {X}_{\mathrm{i}}$, which is then used to calculate sensitivity factors (α_{sf,i}):
where α_{sf,i} represents the relative importance of the uncertainty in each stochastic parameter such that $\sqrt{\sum _{i=\mathrm{1}}^{n}{\mathit{\alpha}}_{\mathrm{sf},\mathrm{i}}^{\mathrm{2}}}=\mathrm{1}$. Uncertainties in parameters with large α_{sf} values – that is, values closer to 1 – are considered to be significant such that a small change in the uncertainty of that parameter would result in a relatively large change in the reliability index (β). However, the uncertainty in parameters with α_{sf} values close to 0 have minor relative importance, and those parameters may be treated as deterministic (Kjerengtroen and Comer, 1996). An analysis of the sensitivity factors determined for the filed site location at Uithuizerwad is presented in Appendix D.
Dependencies
The following (Gaussian) dependencies between variables are imposed (Table 2); all other variables are considered independent.
2.2.5 Foreshore scenarios
In order to investigate the effect of IG waves on the P_{f} value, we consider the following two scenarios (Fig. 2):

SSwave breaking, where the influence of the foreshore bathymetry on incident SS waves is considered but IG waves are neglected, and

SSwave breaking and IG waves, where the influence of the foreshore bathymetry on both SS and IG waves are considered.
Note that the influence of bottom roughness on wave propagation is included in both scenarios and represented by default parameter values in the numerical model. In the Netherlands, Scenario 1 represents standard practice, as the influence of IG waves are typically not considered during safety assessments. By assessing the difference in P_{f} between Scenario 1 and Scenario 2 – hereafter, referred to as ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ and ${P}_{{\mathrm{f}}_{\mathrm{2}}}$, respectively – the influence of the IG waves may be quantified. Note that in both scenarios vegetation is assumed to be flattened, broken or not present (mudflats) in the analysis of the wider Wadden Sea coast. However, the influence of standing vegetation on the P_{f} value is demonstrated for a single case at the Uithuizerwad location in Sect. 3.2.
2.3 Field data for model validation
The performance of the combined numerical and empirical wave modelling approach is assessed by comparing estimates to storm data measured at Uithuizerwad, the Dutch Wadden Sea (Fig. 1) – where the dike is fronted by vegetated foreshore with an average foreshore slope of 1:600. In this way, the ability of the approach to accurately represent the processes occurring over the foreshore is verified as namely (i) the decrease in SS waves due to depthinduced breaking over the foreshore and (ii) the increase in wave height and period at the toe due to IG waves. This dataset is described below.
Two field campaigns were carried out in winter 2014/15 and 2016/17 at Uithuizerwad (Fig. 1), capturing severe storms on 11 January 2015 and 13 January 2017, both with exceedance probabilities of approximately $\mathrm{1}/\mathrm{5}$ per year (Zhu et al., 2020). Here, we consider two transects of wave gauges that captured the change in wave conditions from the marsh edge to dike toe (Fig. 5). In January 2015, a transect of five pressure gauges (Ocean Sensor Systems, Inc., USA) was deployed nearshore, each sampling at 5 Hz over a period of 7 min, every 15 min (Fig. 5a). In January 2017, the setup of the experiment was slightly altered with four gauges deployed, each sampling continuously at 5 Hz (Fig. 5b).
The pressure signal from each gauge was translated into time series of surface elevations, using linear wave theory to adjust for attenuation of the pressure signal with depth. After that, a Fourier transform was performed to transform the data from the time domain to the frequency domain (Hann window, 50 % overlap). To improve the frequency resolution of the resulting wave spectra, measurements from two successive bursts were combined into a single time record. For the 2015 dataset, this yielded spectra with 19 degrees of freedom and a frequency resolution of 0.011 Hz, while the analysis of the 2017 dataset yielded spectra with 31 degrees of freedom and a frequency resolution of 0.0089 Hz (Appendix B). The measured wave and water level conditions at the marsh edge for the 2015 and 2017 winter storms are summarized in Table 3.
3.1 Validation of wave modelling
In this section, the comparison between the combined numerical and empirical modelling approach for wave transformation (Sect. 2.2.2) and the field measurements (Sect. 2.3) is presented. For both the 2015 and 2017 storms, SWAN is set up as a transect (1D) in line with wave sensors (Fig. 5). In each simulation, the numerical model is forced at its boundary with the measured wave spectra at the most offshore wave sensor. SWAN is able to capture the dissipation of SS waves due to the combined effects of the shallow bathymetry and vegetation (Fig. 6). In 2015, the modelled SSwave attenuation from the most offshore gauge to the dike toe was 56 %, half of which (28 %) was due to depthinduced wave breaking over the shallow bathymetry alone. Similarly, modelled SSwave attenuation in 2017 was 63 % with 39 % due to depthinduced wave breaking alone.
At the toe of the dike, Eqs. (12) and (18) are used to estimate the increase in the relative magnitude of the IG waves, ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ (Fig. 7), and the associated increase in the spectral wave period relative to its deepwater value, ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (Fig. 8), respectively. Compared to the measurements at the toe (Eq. 11), Eq. (12) produced an average error of −5 %; that is, predictions were, on average, 5 % lower than the measurements. In Fig. 8, estimates of ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ made by SWAN and the Hofland et al. (2017) formula are also presented for comparison. For the two storms, SWAN produced an average error of −48 % compared to Eq. (18) with 11 % error, thus indicating the relevance of IG waves. Similarly, the Hofland et al. (2017) formula produced an average error of −55 %. As the Hofland et al. (2017) formula is based on tests with $\mathrm{35}\le \mathrm{cot}\left(m\right)\le \mathrm{250}$, these results further indicate that it should not be applied outside of this range and highlights the added value of Eq. (18) – which considers slopes as gentle as 1:1000.
3.2 Probability of failure: Uithuizerwad case
3.2.1 Influence of IG waves on the probability of failure
Using the full probabilistic framework (Sect. 2.2), the annual probabilities of failure due to wave overtopping (P_{f}) at Uithuizerwad are presented in Fig. 9a for the two scenarios considered (Sect. 2.2.5) and an additional scenario to assess the influence of standing vegetation. The calculated P_{f} value is presented alongside estimates of the wave height and period at the toe for a proxy storm with an exceedance probability of $\mathrm{1}/\mathrm{3000}$ per year (Fig. 9b), which is in line with the safety standard.
Scenario 2 (SS + IG) results in a P_{f} value 1.3 times larger than that of Scenario 1 (SS) (Fig. 9a). This increase corresponds well with the increase in the spectral wave period at the toe (${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$) and, to a lesser extent, the increase in wave height at the toe (${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$) due to IG waves (Fig. 9b). If the effects of standing vegetation are considered – with (SS + IG + Veg) or without IG waves (SS + Veg) – the P_{f} value is reduced by 1 order of magnitude (Fig. 9a). This is due to the wave attenuation effect of the vegetation, which reduces both ${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ compared to Scenario 1 (SS) or Scenario 2 (SS + IG) alone (Fig. 9b).
3.2.2 Influence of parameter values and uncertainty on the probability of failure
The influence of the dike crest level (z_{c}) on the calculated P_{f} value is presented in Fig. 10a. It can be seen that the influence of the IG waves increases with an increasing z_{c} value. This is because the large load (q_{a}) needed for failure of a higher dike is reached earlier when IG waves are included. On the other hand, the difference between Scenario 1 and Scenario 2 remains rather constant with varying values for the critical wave overtopping discharge, q_{crit}, while the magnitude of the calculated P_{f} value decreases by a factor of O(10) when q_{crit} is increased by the same magnitude (Fig. 10b).
Overall, the results of the validation suggest that Eqs. (12) and (18) may be applied to the area with reasonable accuracy. Likewise, the results of the application of the model framework to the case at Uithuizerwad are in line with expectations. The calculated failure probability for Scenario 1 (SS) is similar to the assumed safety standard (less than $\mathrm{1}/\mathrm{1000}$ per year), and the differences observed between the scenarios show clear relationships with physical wave parameters at the dike toe, namely the significant wave height and spectral wave period that determine the magnitude of wave overtopping (Fig. 9b). With confidence in the model framework, it is applied to the wider Dutch Wadden Sea area (Fig. 1) for a spatial analysis of the P_{f} value.
3.3 Probability of failure for the wider Wadden Sea area
As a next step, the dikes of the wider Dutch Wadden Sea area are considered from the city of Harlingen to those west of Eemshaven in the city of Groningen. Again, we apply the assumption of a constant dike height of 6 m (above NAP) and a slope of 1:7 for all the dikes in the area. For Scenario 1 (only SS waves), the probability of failure due to wave overtopping (${P}_{{\mathrm{f}}_{\mathrm{1}}}$) ranges from $\mathrm{1.1}\times {\mathrm{10}}^{\mathrm{4}}$ to $\mathrm{3.1}\times {\mathrm{10}}^{\mathrm{2}}$ per year with an average value of $\mathrm{4.6}\times {\mathrm{10}}^{\mathrm{3}}$ per year (Fig. 11a). These variations in ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ are due to the following: (i) the level of exposure, where areas behind inlets are exposed to higher values of ${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ compared to those behind the barrier islands; (ii) variations in the mean water level ($\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$), where values in the west can be approximately 0.5 m lower than those in the east for the same return period event; and (iii) the amount of wave dissipation that occurs due to depthinduced wave breaking over the foreshore, where attenuation is greater at locations with higher foreshore elevations.
To explain this further, we examine the variations in ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ against physical parameters for a proxy storm with an exceedance probability of $\mathrm{1}/\mathrm{3000}$ per year. In Fig. 12a, an offshore forcing parameter (${H}_{{m}_{\mathrm{0}},\mathrm{deep}}^{\mathrm{2}}{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$), which is proportional to the offshore energy flux, is used to represent the combined influence of ${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$. In Fig. 12b, the influence of variations in $\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$ and the bed level at the toe (z_{b,toe}) are represented by ${h}_{\mathrm{toe}}=\stackrel{\mathrm{\u203e}}{\mathit{\eta}}{z}_{\mathrm{b},\mathrm{toe}}$. The calculated ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ value shows a strong positive relationship with ${H}_{{m}_{\mathrm{0}},\mathrm{deep}}^{\mathrm{2}}{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (R^{2}=0.65), meaning that higher forcing results in higher failure probabilities. Though the correlation with h_{toe} is lower (R^{2}=0.43), there is a trend of increasing ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ with increasing h_{toe}. This is because larger h_{toe} (lower z_{b,toe}) values lead to higher wave heights at the toe due to less wave breaking. Likewise, higher water levels ($\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$) associated with larger h_{toe} values also lead to lower freeboards, which results in higher overtopping volumes. Figure 12 also highlights that dikes fronted by mudflats typically have higher ${P}_{{\mathrm{f}}_{\mathrm{1}}}$ values than those with salt marshes, as salt marshes accrete higher bed levels, which in turn promote more SSwave attenuation by breaking.
In order to identify the influence of the IG waves, the probability of failure by wave overtopping in Scenario 2 relative to that of Scenario 1 (${P}_{{\mathrm{f}}_{\mathrm{2}}}{P}_{{\mathrm{f}}_{\mathrm{1}}}$) is assessed. Figure 11b shows that ${P}_{{\mathrm{f}}_{\mathrm{2}}}/{P}_{{\mathrm{f}}_{\mathrm{1}}}$ ranges from 1.1 to 1.6, with an average value of 1.2. This increase in P_{f} is due predominantly to the increase in ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (at the design point) between Scenario 2 and Scenario 1 – represented by ${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$ (Fig. 13), where $\stackrel{\mathrm{\u0303}}{T}={T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ and subscripts 1 and 2 represent Scenario 1 and Scenario 2, respectively. Figure 13a shows a strong positive relationship between ${P}_{{\mathrm{f}}_{\mathrm{2}}}/{P}_{{\mathrm{f}}_{\mathrm{1}}}$ and ${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$ (R^{2}=0.76), where an increase of a factor of 2 in the spectral wave period (${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$) corresponds to an increase of 1.4 times in the failure probability (${P}_{{\mathrm{f}}_{\mathrm{2}}}/{P}_{{\mathrm{f}}_{\mathrm{1}}}$). On the other hand, the increase in wave height at the toe due to the IG waves between the two scenarios (${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{1}}$) was negligible (0.5 % to 4.5 %, Fig. 13c) compared to the increase in wave period.
As ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ depends largely on the offshore wave height, water depth at the toe and foreshore slope (Eq. 18), the spatial variations in ${P}_{{\mathrm{f}}_{\mathrm{2}}}/{P}_{{\mathrm{f}}_{\mathrm{1}}}$ (Fig. 11b) are due to variations in local bathymetric and forcing conditions. This is further demonstrated in Fig. 13b by examining the relationship between the increase in the spectral wave period (${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$) and the relative water depth under proxy storm conditions ($\mathrm{1}/\mathrm{3000}$ per year). The relative water depth parameter, which takes into account the variations in both local water depth (h_{toe}) and offshore wave height (${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$), shows a clear negative relationship with ${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$ (R^{2}=0.61). Therefore, areas with low water depths at the toe relative to large offshore waves are expected to have a greater IGwave influence on ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$. This is also seen by the areas with a higher IGwave influence to the west in Fig. 11b, which correspond to points with higher offshore waves (>2.4 m, at the design point) and shallower water depths (<1.4 m), compared to the other locations (with offshore wave heights typically <2 m and water depths at the toe >3 m).
It should be noted that the increase in the spectral wave period due to IG waves is also sensitive to the estimated foreshore slope (Eq. 18). However, ${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$ showed little correlation with the foreshore slope here (R^{2}<0.1), since the foreshores of the Dutch Wadden Sea can all be considered very gentle (1:600, on average).
4.1 Modelling approach
The combined numerical and empirical approach to wave transformation proved accurate when compared to the 2015 and 2017 storm data at Uithuizerwad, also highlighting the growth of ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ (Fig. 7) and the associated increase in ${T}_{m\mathrm{1},\mathrm{0}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (Fig. 8) as the water depth becomes shallower. Of particular note is the difference in ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ calculated by the phaseaveraged wave model SWAN compared to measurements. While the measurement data are likely contaminated by IG waves reflected from the dike, leading to longer wave periods, there is still a gross underestimation of ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ by SWAN due to its exclusion of IGwave dynamics (Lashley et al., 2020b). Recent works also indicate that the mismatch between the ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ predictions made by SWAN and measurements may be partially explained by its misrepresentation of the frequency dependence of wave dissipation by vegetation (Ascencio, 2020; Jacobsen and McFall, 2019) – where the presence of vegetation significantly influences the shape of the wave spectrum. However, this topic is still under investigation. Despite this underestimation of ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$, SWAN was able to accurately model SSwave transformation over the foreshore (Fig. 6). Likewise, the growth of ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ (Fig. 7) and ${T}_{m\mathrm{1},\mathrm{0}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (Fig. 8) at the dike toe are accurately captured using Eqs. (12) and (18), respectively.
The probabilistic method FORM was able to compute the P_{f} value within 20 to 30 iterations with a computation time of under 10 min per dike section. Other methods, such as crude Monte Carlo or numerical integration are known to be much more computationally demanding. However, other approaches such as Adaptive Directional Importance Sampling (Den Bieman et al., 2014) may also prove to be equally suitable for this application. This short computation time is also attributed to the use of a phaseaveraged wave model (SWAN), which is roughly 100 times faster than its phaseresolving counterparts (e.g. SWASH – Simulating WAves till SHore – or XBeach nonhydrostatic) (Lashley et al., 2020b).
As the dike characteristics (crest level, slope and critical overtopping discharge) typically dominate the probabilistic analysis, their treatment as deterministic variables here allowed for the analysis to focus on the influence of foreshore parameters. Furthermore, by treating the influence of the IG waves as a separate module (Sect. 2.2.2), calculations with and without IG waves could be easily performed. Such a modular approach allows for the framework to be easily modified or adapted to varying conditions. For example, the module to calculate the actual overtopping discharge could be extended with the formulae of Lashley et al. (2021) for environments where the conditions at the structure toe are extremely shallow or in the case of vertical seawalls rather than sloping structures. Likewise, another numerical or empirical model more suited to the specific area of application could replace the model used here for SSwave transformation (SWAN). This makes the overall approach easily adaptable and applicable to other coastlines where IG waves may play a critical role, such as the Belgian coast (Altomare et al., 2016), Japanese coast (Mase et al., 2013), and northern and southern coasts of Vietnam (Nguyen et al., 2020).
To account for the error in the empirical models, the estimates were multiplied by normally distributed factors with mean values and standard deviations to represent the bias and scatter (errors) associated with each model (Section 2.2). This uncertainty was also shown in Figs. 6 to 8 as error bars. As the overall approach is a succession of different numerical and empirical models, it is important to note the combined error. The combined error (or uncertainty) may be expressed using a coefficient of variation, which is equal to the combined standard deviation normalized by the combined mean. If we consider the means and standard deviations of Eqs. (4), (12), (18) and (19), the combined coefficient of variation or uncertainty is 0.15 or 15 %.
4.2 Applicability of formulae for the actual overtopping discharge
In the present study, the original overtopping formula of Van Gent (1999) (Eq. 19) was applied to all locations. This formula was selected because it was developed specifically for shallow foreshores considering the influence of both SS and IG waves and is considered valid for a wide range of breaker parameter (${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$) values. However, applying Eq. (19) here – to locations with ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}<\mathrm{5}$ at the design point – does not coincide with the current European standard (EurOtop, 2018). In EurOtop (2018), different formulae are applied depending on the ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$ value (Van der Meer and Bruce, 2014; Altomare et al., 2016). An analysis of the different approaches revealed the following points:

For ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}<\mathrm{1.8}$, which is typical for cot (α)=7, the spectral wave period showed a considerable influence on the overtopping discharge (q_{a}) calculated using the EurOtop (2018) approach. Figure 14a shows that an increase of 1.5 times in wave period (due to IG waves) (${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$) resulted in an order of magnitude increase of P_{f} using the EurOtop (2018) approach. Since the EurOtop (2018) formula for ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}<\mathrm{1.8}$ (Van der Meer and Bruce, 2014) was not derived for shallow foreshore conditions (with IG waves), this significant increase in the P_{f} value is likely incorrect and requires further research.

For $\mathrm{1.8}<{\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}<\mathrm{5}$, which is typical for cot (α)=3, the wave period no longer influences the EurOtop (2018) calculation, as a maximum q_{a} value is reached. This is evident in Fig. 14b, as no clear trend between ${\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{2}}/{\stackrel{\mathrm{\u0303}}{T}}_{\mathrm{1}}$ and ${P}_{{\mathrm{f}}_{\mathrm{2}}}/{P}_{{\mathrm{f}}_{\mathrm{1}}}$ is visible for the EurOtop (2018) calculations. In these cases, the differences between EurOtop (2018) and the original Van Gent (1999) calculations are much smaller (Fig. 14).

For ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}>\mathrm{7}$ or wave steepness at the toe less than 0.01, the modified version of the Van Gent (1999) formula based on an equivalent slope concept (Altomare et al., 2016) and described in EurOtop (2018) is only applicable to foreshore slopes steeper than or equal to 1:250. As the foreshore slopes of the Wadden Sea are typically gentler than 1:500, the modified formulae could not be used here. Therefore, locations meeting these criteria were excluded for the EurOtop (2018) calculations.

The results using the original Van Gent (1999) formula were of the same order of magnitude for both dike slopes considered (Fig. 14), suggesting that the formula was not very sensitive to changes in ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}$. However, it should be noted that this formula was derived using a limited dataset with cot (α) values of 2.5 and 4 and cot (m) values of 100 and 250. Therefore, future studies should verify its performance for conditions with values of cot (α)>4 and cot (m)>250.
The above findings suggest that the EurOtop (2018) approach may be incorrect for shallow foreshore conditions with gentle dike slopes (e.g. 1:7), which often have ${\mathit{\xi}}_{m\mathrm{1},\mathrm{0}}<\mathrm{1.8}$. The source of this uncertainty lies in the sensitivity of the formulae to ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$, a parameter whose magnitude increases proportionally with the magnitude of the IG waves (Eq. 18, Fig. C2). As Oosterlo et al. (2018) applied the EurOtop (2007) formulae to a dike with an average slope of 1:8, it can be concluded that the large influence of the IG waves – where including IG waves increased the failure probability by 10^{3} – reported by the authors was indeed due to the method used.
4.3 Influence of IG waves on design parameters
The influence of IG waves may be represented as an increase in the magnitude of both design parameters (${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$), compared to a situation where the IG waves are neglected. This was demonstrated by Lashley et al. (2020b), where the relative magnitude of the IG waves $({\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}={H}_{{m}_{\mathrm{0}},\mathrm{IG},\mathrm{toe}}/{H}_{{m}_{\mathrm{0}},\mathrm{SS},\mathrm{toe}})>\mathrm{0.5}$ had a notable influence on both parameters. In the present study, ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ was much lower, ranging from 0.14 to 0.35 with a mean value of 0.19 (considering proxy storm conditions with $\mathrm{1}/\mathrm{3000}$ per year exceedance probability). As a result, the impact of the IG waves on the total wave height at the toe was negligible (0.5 % to 4.5 %), since ${H}_{{m}_{\mathrm{0}},\mathrm{toe}}=\sqrt{{H}_{{m}_{\mathrm{0}},\mathrm{IG},\mathrm{toe}}^{\mathrm{2}}+{H}_{{m}_{\mathrm{0}},\mathrm{SS},\mathrm{toe}}^{\mathrm{2}}}$. That said, there was still a notable increase in ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$. This is attributed to the following: (i) the sensitivity of ${T}_{m\mathrm{1},\mathrm{0}}$ to wave energy density at low frequencies, by definition (Eq. 9), and (ii) the influence of the foreshore slope on the shape of the wave spectrum at the toe – where gentler foreshore slopes lead to wider surf zones and more energy transfer to lower frequencies. This makes ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ more sensitive to very gentle foreshore slopes compared to the ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ parameter alone (Appendix C). This is further demonstrated in Fig. 15 using the results of two numerical simulations (XBeach nonhydrostatic). The increase in ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ due to IG waves is larger for the 1:500 foreshore slope than the one of 1:50, despite having similar ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ values (Fig. 15, Table 4). Table 4 also highlights that while the influence of the IG waves on ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ is noteworthy, their influence on the total wave height at the toe (${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$) is negligible.
The main takeaway here is that while IG waves may have a negligible influence on the design wave height at the structure, their influence on the design wave period can be considerable and should therefore not be neglected, particularly on gentle foreshore slopes.
4.4 Influence of salt marsh vegetation
Another discussion point is the influence of salt marsh vegetation and whether its effects should be considered for very high return period events. Figure 9 suggests that safety could be significantly improved by standing salt marsh vegetation; however, these findings must be interpreted with caution. In their analysis based on dikes with foreshores in the Wadden Sea, Vuik et al. (2018b) included a stembreakage model and concluded that it was very likely that almost all vegetation would break at this location under extreme forcing – resulting in P_{f} values similar to that of nonvegetated foreshores. This flattening and breaking of salt marsh vegetation under storm conditions was also reported by Möller et al. (2014), who conducted largescale flume experiments with transplanted Wadden Sea vegetation. Moreover, though the vegetation component of Eq. (12) was able to capture the influence of vegetation on IG waves for the two storms considered here (Figs. 6 to 8), its performance for more extreme events requires further validation. This is due to the low ratio of stem height to water depth (${h}_{\mathrm{toe}}/{h}_{\mathrm{v}}=\mathrm{3.3}$) considered in its derivation. As a result, Eq. (12) may overestimate the influence of vegetation for high return period events with ${h}_{\mathrm{toe}}/{h}_{\mathrm{v}}\approx \mathrm{13}$. Thus, the influence of salt marsh vegetation on coastal safety under extreme forcing remains an important issue for future research.
4.5 Implications for practice
Including the effects of IG waves (Scenario 2) increases of P_{f} by up to 1.6 times, compared to Scenario 1 (Figs. 9 and 11b). This effect is considerably smaller than that reported by Oosterlo et al. (2018), where including the IG waves increased the P_{f} value by 3 orders of magnitude. Results here suggest that Oosterlo et al. (2018) likely overestimated the influence of the IG waves due to an inappropriate use of empirical overtopping formulae that were not formulated specifically for situations with IG waves (Van der Meer, 2002; EurOtop, 2007), as demonstrated in Sect. 4.2. But what does the presence of IG waves mean for practice? In general, the reliability of the existing defences may be overestimated, since IG waves are largely neglected in their assessment. By interpolating the results of Fig. 10 logarithmically, the required crest level at Uithuizerwad for a fixed target probability of failure can be determined. For a target annual failure probability of $\mathrm{1}/\mathrm{1000}$ per year (which corresponds to the safety standard), a crest level of 6.3 m (+ NAP) is needed for Scenario 1 (SS). For Scenario 2 (SS + IG), the required crest level is 6.5 m. Therefore, the influence of the IG waves may be alternatively seen as an increase in the required crest level of around 0.2 m with a cost of the order of magnitude of EUR 1 million per kilometre (Jonkman et al., 2013). If the influence of the IG waves on the P_{f} value were 1 order of magnitude larger, as suggested by the EurOtop (2018) formula (Fig. 14a), then the increase in the required crest level would be around 0.8 m with an order of magnitude increase in cost (EUR 10 million per kilometre).
This increase in P_{f} is attributed to the growth of ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ due to the IG waves and the wellknown relationship between wave overtopping and wave period, where longer waves (larger ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ values) result in more overtopping (Sect. 2.2.3) and, by extension, higher P_{f} values. These findings suggest that attention should also be given to changes in wave period – and not to wave height attenuation alone – when considering the influence of shallow foreshores on safety. However, it is important to stress that this effect is highly dependent on local conditions, as ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ (Eq. 18) is dependent on the offshore wave height, the water depth at the toe and the estimated foreshore slope. Therefore, it should be assessed on a casebycase basis rather than assumed constant over a large area. This spatial variability is demonstrated in Fig. 11b. Additionally, the calculated ${P}_{{\mathrm{f}}_{\mathrm{2}}}$ value was found to be somewhat sensitive to the uncertainty in Eqs. (12) and (18) (Appendix D), which are based primarily on numerical simulations, since field and physical model data are lacking. Future studies should carry out experiments to further validate and improve the empirical formulations presented here and, if possible, reduce the uncertainty (scatter) in their estimates.
Even though vegetation itself was neglected in the probabilistic analysis of the wider Wadden Sea area, findings here still advocate for the importance of maintaining salt marshes. As (higher) salt marshes trap sediment, their raised platforms attenuate more SS waves than lower mudflats, which results in lower P_{f} values – even when IG waves are taken into account (Fig. 16). These findings support the arguments of Zhu et al. (2020) for the net positive impact of shallow foreshores on coastal safety. However, the estimated increase in safety due to the foreshore may be reduced when IG waves are included in the analysis – in particular where wave overtopping is concerned. In planning and implementing foreshore systems, it is therefore important to consider the effects of IG waves on safety as well.
By combining several numerical and empirical models, the influence of infragravity (IG) waves on the probability of dike failure by wave overtopping (P_{f}) was quantified for a shallow intertidal area: the Dutch Wadden Sea. The model framework was first validated for a single location using data collected during two storms, each with exceedance probabilities of approximately $\mathrm{1}/\mathrm{5}$ per year. The approach accurately estimated the effects of shallow foreshores on wave propagation, namely (i) the dissipation of incident SS waves by depthinduced breaking, (ii) the increase in the relative magnitude of the IG waves (${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$) and (iii) the increase in the relative spectral wave periods (${T}_{m\mathrm{1},\mathrm{0}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$). The framework was then applied to the wider Dutch Wadden Sea for a spatial analysis. Including the IG waves increased the P_{f} value by a factor of 1.1 to 1.6, suggesting that safety is indeed overestimated when they are neglected. This increase is attributed to the influence of the IG waves on the design wave period and, to a lesser extent, the wave height at the dike toe. The spatial variation in this effect, observed for the case considered, highlights its dependence on local conditions – with IG waves showing greater influence at locations with larger offshore waves, such as those behind tidal inlets, and shallower water depths.
For the mild dike slope considered (1:7), the change in P_{f} due to the IG waves showed high sensitivity to the empirical wave overtopping model applied – where the use of a model that was not developed for shallow foreshores and IG waves (EurOtop, 2018) resulted in failure probabilities up to 55 times higher compared to a method from Van Gent (1999). It is thus important that practitioners consider both the impact of IG waves and the appropriateness of the models used when assessing flood risk along coastlines with shallow foreshores. The methods proposed in this paper can aid in this by allowing practitioners to quickly identify areas where IG waves – and therefore tools which account for them – should be included in the analysis. Furthermore, given the modular characteristic of the approach, it could be easily fitted with different tools or adapted to other coastlines where IG waves may play a significant role. Examples of such sites include the sandy foreshores along the Belgian coast (Altomare et al., 2016); the wide shelves of the Mekong Delta, Vietnam (Nguyen et al., 2020); and the steep foreshores found in Japan (Mase et al., 2013).
Despite the utility of the proposed approach and importance of the findings herein, the results are subject to certain limitations. For instance, the analysis was conducted assuming a fixed dike height with a uniform slope at each location. Therefore, it is recommended that the analysis be repeated using the actual dike geometries where the crest level is notably higher, as findings here suggest that the influence of the IG waves would likely be higher (Fig. 10a). Furthermore, we have validated and applied the framework to a shallow intertidal area, but it is recommended that the framework be applied to sites with different hydrodynamic and geomorphological features, such as open coasts or those fronted by coral reefs. It must also be noted that the current study did not consider edge or leaky (free) IG waves (Reniers et al., 2021). Therefore, additional field campaigns focused on measuring IG waves are recommended to determine the contribution, if any, of free IG waves to wave conditions in the Dutch Wadden Sea. Finally, while vegetation had a notable influence on wave attenuation for storms with a relatively high probability of exceedance ($\mathrm{1}/\mathrm{5}$ per year, Fig. 6), it was assumed to be flattened or broken under more extreme conditions (Vuik et al., 2018a). Further research is required to assess the attenuation effects of salt marsh vegetation under extreme water level and wave forcing.
Hofland et al. (2017) showed that the ratio of the spectral wave period at the structure toe to its deepwater equivalent (${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$) may be empirically modelled as a function of relative water depth and foreshore slope (Eqs. C1 to C3). For longcrested waves (no directional spreading), this is
and for cases with shortcrested wave, it is
where
However, as Eqs. (C1) to (C3) were based on tests with $\mathrm{35}\le \mathrm{cot}{\mathit{\alpha}}_{\mathrm{fore}}\le \mathrm{250}$, they tend to over and underestimate ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ for steep (cot α_{fore}<35) and very gentle slopes (cot α_{fore}>250), respectively, with R^{2}=0.30 when applied to the numerical dataset (Fig. C1).
This inaccuracy, particularly for very gentle slopes (cot α_{fore}>250), has also been reported by Nguyen et al. (2020) and suggests that a new formulation is required for application to the Dutch Wadden Sea – where foreshore slopes are typically 1:500 or gentler.
Since both ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ and ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ describe the amount of energy in the IG band compared to the SS band, it stands to reason that a simple relation should exist between the two parameters. From the Lashley et al. (2020a) numerical dataset, it can be seen that ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$ increases with increasing ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ (R^{2}=0.76) but with scatter related to the foreshore slope (Fig. C2a). Based on these trends, the following relation is proposed:
where the exponents were determined empirically, by minimizing scatter. Including the foreshore slope term significantly reduces the scatter in the data (R^{2}=0.92, Fig. C2b) and gives a better representation for mild slopes. This is due to the influence of the foreshore slope, not only on the relative magnitude of the IG waves – as already accounted for in Eq. (12) – but also on the spectral shape. As the area over which shoaling occurs increases with gentler foreshore slopes, energy transfer by nonlinear (difference) triad interactions occurs over a longer duration than on steeper slopes. This causes the spectral peak to migrate to lower frequencies and results in larger values of ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ for gentler foreshore slopes (Battjes, 2004), despite having similar ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ values.
It should also be noted that for deepwater cases, where ${h}_{\mathrm{toe}}/{H}_{{m}_{\mathrm{0}},\mathrm{deep}}>\mathrm{1}$, ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}/{T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}\approx \mathrm{1}$ and is independent of the foreshore slope and ${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$ parameters (Fig. C2b). This is consistent with the findings of Lashley et al. (2021) which suggest that the foreshore's influence only becomes significant for cases with ${h}_{\mathrm{toe}}/{H}_{{m}_{\mathrm{0}},\mathrm{deep}}\le \mathrm{1}$.
With respect to the stochastic parameters, the sensitivity of the calculated P_{f} value to uncertainties in each parameter is assessed using the FORM α_{sf} values (Sect. 2.2.4, Fig. D1). Negative α_{sf} values represent variables that contribute to the load by increasing the actual overtopping discharge (q_{a}). In both scenarios, the uncertainty in the offshore water level ($\stackrel{\mathrm{\u203e}}{\mathit{\eta}}$) dominates the probability of failure with $\mathit{\alpha}\le \mathrm{0.96}$. This is expected, since the dike is unlikely to fail without extreme water levels (i.e. a severe storm). In Scenario 1 (Fig. D1a), the variables that also contribute to the load are the following: the empirical wave overtopping coefficient (c), since larger c values increase q_{a} (Sect. 2.2.3); the SWAN breaker parameters (γ_{BJ}), which controls the magnitude of breaking waves such that higher γ_{BJ} leads to larger wave heights at the structure toe and thus larger q_{a}; and the offshore wave forcing parameters (${H}_{{m}_{\mathrm{0}},\mathrm{deep}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{deep}}$).
As expected, when the influence of the IG waves is included in the analysis (Scenario 2, Fig. D1b) the uncertainty in factors for the relative IGwave height, ${f}_{\mathrm{IG}}(\mathit{\alpha}=\mathrm{0.03}$), and spectral wave period, ${f}_{\mathrm{TM}}(\mathit{\alpha}=\mathrm{0.02}$), also contribute to the load, as larger ${H}_{{m}_{\mathrm{0}},\mathrm{toe}}$ and ${T}_{m\mathrm{1},\mathrm{0},\mathrm{toe}}$ values increase q_{a}. This suggests that the calculated P_{f} value is indeed sensitive to the accuracy of Eqs. (12) and (18). The uncertainty in the bed level (z_{b}) also contributes to the load due to its influence on the water depth at the toe, which directly influences the relative magnitude of the IG waves at the dike toe (${\stackrel{\mathrm{\u0303}}{H}}_{\mathrm{IG}}$).
All data, models or code that support the findings of this study are available from the corresponding author upon reasonable request.
CHL prepared the manuscript with contributions from all coauthors. CHL designed and carried out the numerical simulations. CHL performed the formal analysis. CHL, JDB, SNJ and JvdM conceptualized the research. SNJ, JvdM and JDB provided supervision. VV and CHL developed the model code (MATLAB). VV designed and carried out the field experiments.
The contact author has declared that neither they nor their coauthors have any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the special issue “Coastal hazards and hydrometeorological extremes”. It is not associated with a conference.
This work is part of the Perspectief research programme AllRisk (project no. B2), which is (partly) financed by NWO Domain Applied and Engineering Sciences, in collaboration with the following private and public partners: the Dutch Ministry of Infrastructure and Water Management (RWS), Deltares, STOWA, the regional water authorities Noorderzijlvest and Vechtstromen, It Fryske Gea, the consultancy HKV, Natuurmonumenten, and the water board HHNK.
This paper was edited by Joanna Staneva and reviewed by two anonymous referees.
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 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 Conclusion
 Appendix A: Overview of stochastic variables
 Appendix B: Wave spectra at Uithuizerwad (Wadden Sea) during 2015 and 2017 storms
 Appendix C: Derivation of formulae for the spectral wave period at the toe
 Appendix D: Analysis of sensitivity factors for the Uithuizerwad case
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Special issue statement
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Materials and methods
 Results
 Discussion
 Conclusion
 Appendix A: Overview of stochastic variables
 Appendix B: Wave spectra at Uithuizerwad (Wadden Sea) during 2015 and 2017 storms
 Appendix C: Derivation of formulae for the spectral wave period at the toe
 Appendix D: Analysis of sensitivity factors for the Uithuizerwad case
 Data availability
 Author contributions
 Competing interests
 Disclaimer
 Special issue statement
 Financial support
 Review statement
 References