# Probabilistic Assessment of Overtopping of Sea Dikes with Foreshores including Infragravity Waves and Morphological Changes: Westkapelle Case Study

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{m−}

_{1,0}wave period was recommended as a measure in these situations to calculate the wave run-up or wave overtopping [8,9,10]. This period is defined as T

_{m}

_{−1,0}= m

_{−}

_{1}/m

_{0}, with m

_{k}being the kth moment of the wave spectrum. The period is commonly used in coastal structure design [11], and the use of the T

_{m}

_{−1,0}wave period is incorporated in manuals like the EurOtop manual [12]. Furthermore, it was found, that on high foreshores where intense breaking occurs, the wave set-up becomes important for the wave run-up and overtopping as well [13].

## 2. Materials and Methods

#### 2.1. Case Study Description

^{−1}m

^{−1}(return period of 1:4000 years). As the landward slope of the dike consists of bare sand, a larger discharge than 0.1 ls

^{−1}m

^{−1}could not be accepted [12]. In 2008, it was decided to strengthen the dike with a foreshore with a volume of 2.5 million cubic meter of sand in front of the dike. The initial height and width of the foreshore were approximately 4 m+NAP and 75 m respectively, with an average foreshore slope of approximately 1:17. During normal conditions, this gives a dry beach in front of the dike.

#### 2.2. Set-Up and Validation of Model Framework

#### 2.2.1. Hydro- and Morphodynamics: XBeach and h/2-Assumption

_{m}

_{0}is half the water depth h. The assumption does not include the complex hydrodynamics such as infragravity waves or wave set-up, nor does it include morphological changes. Also, changes in wave period are not calculated by the model. The h/2-assumption was compared with XBeach 1D without the infragravity wave forcing for several situations. The models showed good agreement. The h/2-assumption calculation rule is used in this paper, to compare to the results of XBeach surfbeat, which does include both the complex hydrodynamics like infragravity waves, and morphological changes of the foreshore.

_{m}

_{−1,0}> 7 was based. XBeach showed good skill in predicting the H

_{m}

_{0}and T

_{m}

_{−1,0}. For a more detailed description of the validation, refer to Oosterlo [32].

_{m}

_{0}) and wave period (T

_{m}

_{−1,0}) at the dike toe are determined, which are used as input for the overtopping calculations.

#### 2.2.2. Wave Overtopping: EurOtop

_{m}

_{−1,0}wave period. In the recent 2016 second edition [12], some equations have been updated. For sloping structures, where the freeboard is at least half the wave height, the differences between the first edition EurOtop [35] formulae and the new ones is quite small. For that reason one may also continue to use the first edition EurOtop [35] formulae in such a situation [12]. Because the previous is always the case for the situations considered in this paper, the equations of the first edition of EurOtop [35] are still used here. A brief description of the used EurOtop [35] equations is given in Appendix A.

_{f}= 1.0 [12]), with which the wave overtopping discharge is calculated for each hour of the storm. The wave overtopping discharges during the storm are then compared and the largest one is used further in the ADIS calculation of the limit state function (see next subsubsection).

#### 2.2.3. Probabilistic Calculation Method: Adaptive Directional Importance Sampling

^{−1}(1 − P

_{f}), with Φ indicating a standard normal distribution) is a commonly used probabilistic measure of safety, in addition to the probability of failure P

_{f}. The value of β indicates the shortest distance from the origin to the failure space. The β-sphere limits the area of the sampling domain for which XBeach calculations are performed, based on a certain β-value. The β-sphere is used when an adaptive response surface with a good fit has been found. In that case, the adaptive response surface is used instead of the limit state function, but still a small number of XBeach (limit state function) evaluations will be made to maintain precision. These XBeach evaluations are performed for the most important points, i.e., the points inside the β-sphere, the points that have a small β-value (and a large contribution to the probability of failure). Hence, the β-sphere defines the area that is excluded from the sampling domain, which reduces the number of exact XBeach limit state simulations needed. Thus, the response surface is applied in the areas outside of the β-sphere, and inside the sphere, the real limit state function using XBeach calculations is used. For a more extensive description of ADIS, refer to Grooteman [36].

_{crit}− q, with q

_{crit}a critical wave overtopping discharge depending on the type of layer and quality of the layer on the landward slope of the dike. For Westkapelle, this limit is 0.1 ls

^{−1}m

^{−1}, which was taken as a constant in the calculations. The probabilistic methods showed to have difficulties with fitting to this function. Therefore, the limit state function that is used for this paper is changed into Z = log(q

_{crit}) − log(q), which has a much smoother behavior. This is consistent with the logarithmic dependence of q on the hydraulic parameters as indicated in the EurOtop [12] equations. Choosing this different limit state function showed to have no influence on the resulting probabilities of failure. The complete set-up of the model framework is schematically presented in Figure 4.

#### 2.3. Input of Probabilistic Calculations

#### 2.3.1. Boundary Conditions

^{−1}is presented in Figure 6.

_{m}

_{0}is correlated with the water level, as for a given surge level, different wave heights can occur, due to the effects of wind speed, wind direction and wind duration. It was found that this correlation could be approximated by a normal distribution with a mean of 0 and a standard deviation of 0.6 m [46]. The relation between surge level and wave height was taken from Steetzel et al.; Weerts & Diermanse [42,46] and uses several statistical parameters as given in Table 2. The wave period T

_{p}is correlated with the wave height. A normal distribution with a mean of 0 and a standard deviation of 1.0 s was used for the relation between wave peak period and significant wave height, according to Stijnen et al. [47]. The statistical parameters of the conditional Weibull distribution for the water level, and the equations describing the relation between water level and wave height, and wave height and wave period for Westkapelle are given in Table 2. For an overview of the equations and the statistical parameters at all the measurement stations, refer to Steetzel et al. [42].

_{storm}is given as a lognormal distribution to prevent negative values from occurring. The surge profile is composed of a tidal effect and a surge effect. For this paper, an adjusted version of the formula from Steetzel [48] was made, with a random phase of the tide at the moment of the maximum surge, see also Figure 6:

_{0}is the mean water level [m] (here set to zero), h

_{a}is the tidal amplitude [m], t the time [s], t

_{tide}the duration of the tide [s], ϕ the time shift in the occurrence of the maximum tidal water level opposed to the maximum surge level [s], h

_{s}the surge amplitude [m] and t

_{storm}the storm duration [s].

_{peak}was set to half the storm duration t

_{storm}. Thus, the maximum surge level coincides with a random phase of the tide level, because of the uniform distribution of ϕ. The variation of the significant wave height and wave peak period during the storm was modelled by the equations of Steetzel [48], see also Figure 6 for the profile.

_{50}and Manning bed friction factor n, modelled with a lognormal and uniform distribution, respectively (according to Vrouwenvelder et al.; Roelvink et al. [43,44]).

- Set 1, Offshore conditions, 3 stochastic variables: z
_{s}, H_{m}_{0}, T_{p}. - Set 2, Model and wave parameters, 7 stochastic variables: C1, C2, C3, C4, γ, n, s.
- Set 3, Storm parameters, 2 stochastic variables: t
_{storm}, ϕ. - Set 4, Morphological parameters, 3 stochastic variables: D
_{50}, FacAs, FacSk.

## 3. Results

_{f}and the reliability index β are given (columns 9,10), as well as the number of simulations necessary and the calculation duration in days (columns 11,12).

#### 3.1. Influence of the Complex Hydrodynamics

^{3}, see Table 3). The dominant parameter in the calculation with the h/2-assumption was the water level, which was very large (7.33 m+NAP), see Figure 7. As can be seen from Figure 7, approximately the same amount of wave overtopping occurs for much less extreme hydraulic conditions (a water level that almost 2 m lower), when including the complex hydrodynamics. The difference in offshore wave height was not as large as for the water level (6.25 m versus 6.09 m). However, the wave set-up in XBeach leads to a somewhat larger wave height at the toe. The main difference lies in the wave period. Offshore, the difference is almost negligible, but due to the inclusion of the wave period transformation and generation of infragravity wave energy, the wave period at the toe is much larger with XBeach. The severe wave breaking on the foreshore leads to a transfer of wave energy to the lower frequencies, which increases the T

_{m}

_{−1,0}wave period strongly (10.50 s versus 42.82 s, see Figure 7). Hence, compared to a calculation rule, using a process-based model that includes complex hydrodynamics like wave set-up, infragravity waves and wave period transformation leads to larger wave heights and much larger wave periods at the toe of the dike. The ratio between the high frequency and low frequency wave heights at the toe (H

_{m}

_{0,HF}/H

_{m}

_{0,LF}) as calculated by XBeach 1D is only 0.5 in this case. The infragravity waves have a significant contribution to the total wave energy in this 1D case with foreshore. Thus, it can be concluded that it is of importance to include these hydrodynamic processes for these kinds of situations with severe wave breaking on a shallow foreshore. This means that a complex hydrodynamic model is necessary, which further necessitates an efficient probabilistic model.

#### 3.2. Influence of the Model and Wave Parameters

_{max}at incipient breaking) and thus larger wave heights at the toe of the dike and more wave overtopping. The difference in probability of failure shows that it can be important to include more stochastic variables than just the offshore conditions. Again, a large wave set-up and strong generation of infragravity wave energy is visible. As a result of this, the T

_{m}

_{−1,0}wave periods at the toe are again quite large. The ratio between the high frequency and low frequency wave heights at the toe (H

_{m}

_{0,HF}/H

_{m}

_{0,LF}) as calculated by XBeach 1D is only 0.4 in this case. This again shows that it is of importance to include hydrodynamic processes like infragravity waves and wave set-up in the calculations.

#### 3.3. Influence of the Storm Duration and Time of the Peak of the Storm

_{storm}and time of the peak of the storm ϕ as stochastic variables, leading to a calculation with 12 stochastic variables (calc. 4). Including these parameters as stochastic led to a probability of failure that was approximately 2 times smaller than the calculation with 10 stochastic parameters (calc. 3), see Table 3. Hence, the influence of including these parameters is rather small. The reduction in probability of failure can be explained by the time of the peak of the storm relative to the tide. When this parameter is considered as deterministic, it is assumed that the maximum surge occurs at the same moment as tidal high water. When this parameter is considered stochastic, the moment of maximum surge does not necessarily correspond to the moment of the maximum tidal water level, which reduces the probability of failure. The rest of the stochastic parameter values in the design point of the calculations with 10 and 12 stochastic parameters corresponded almost exactly, which explains the small difference in probability of failure. The small difference in probability of failure shows that for this case, the storm duration and time of the peak of the storm can be neglected as stochastic variables.

#### 3.4. Influence of Morphological Changes

_{m}

_{−1,0}wave period (38.85 s versus 51.58 s). In this case, the effects of the difference in wave height and wave period almost cancel one another out. This leads to the small difference in probability of failure, despite the quite large differences in the bathymetry due to erosion of the foreshore. Concluding, for this case, the inclusion of the morphological development of the foreshore and related stochastic variables only led to small differences in the probability of failure.

#### 3.5. Influence of Wave Directional Spreading

_{m}

_{−1,0}wave periods are found at the toe of the dike when using a 1D simulation. The 1D XBeach calculations can be considered as a wave flume, with long-crested waves without directional spreading. This causes a stronger forcing of the infragravity waves compared to a situation with directional spreading. In reality, wind waves in extreme storms are short-crested. Therefore, an extra calculation was performed with XBeach 2DH with the same three stochastic parameters. With a 2D simulation, the wave directional spreading is included. Thus, the main differences between a 1D and 2D simulation lie within the fact that the 1D simulation does not include the wave directional spreading (hence, long-crested waves), which the 2D simulation does (hence, short-crested waves). For the 2D model, in the alongshore direction, a section of 1 km was modelled with grid cell sizes of 50 m. The wave directional information was solved in 10 degree bins, spanning 110 degrees (55 degrees to both sides of shore normal). Note that the mean wave angle in all simulations was shore-normal and that we use the methodology of Roelvink et al. [34] to resolve the propagation of wave groups in the model domain. The results are shown in Figure 10.

^{−5}for the 1D case and 1.6 × 10

^{−6}for the 2D case, a difference of a factor 10. The lower probability of failure as found for the 2D case can be explained by the forcing of the infragravity waves, which is less than in the 1D case, because the 2D XBeach calculations include the wave directional spreading. This leads to less low-frequency energy and therefore a smaller wave period at the toe (15.83 s versus 42.82 s), which results in less wave overtopping. For instance Hofland et al. [11] show as well that the generation of low-frequency wave energy is decreased by the directional spreading. With the inclusion of the wave directional spreading, the results of a 2D simulation are more realistic compared to a 1D simulation, since wind waves in extreme storms are short-crested (see also e.g., Van Dongeren et al. [50]). The ratio between the high frequency and low frequency wave heights at the toe (H

_{m}

_{0,HF}/H

_{m}

_{0,LF}) is 1.6 for the 2D case (compared to 0.5 for the 1D case). Thus, even in 2D the infragravity waves have a significant contribution to the total wave energy.

#### 3.6. Influence of the Probabilistic Method

## 4. Discussion

_{m}

_{−1,0}on mildly sloping shallow foreshores was determined [11]. This equation can be used for simple cross-shore profiles and could be implemented in the model framework to save calculation time for such cases. For more complex cross-shore profiles, XBeach will remain necessary.

_{m}

_{−1,0}wave period at the toe and uncertainties in the sensitivity of the EurOtop equations with respect to this wave period. Since the T

_{m}

_{−1,0}wave period is very sensitive to the low frequencies, it is unclear if this wave period is still applicable for situations with large amounts of low frequency energy, such as the ones considered in this paper. Treating the complex combination of infragravity wave energy and wind wave energy with a single parameter as the T

_{m}

_{−1,0}wave period might be an oversimplification, however the T

_{m}

_{−1,0}wave period is the wave period that is currently used in all the most commonly used wave run-up and wave overtopping equations. A new equivalent slope concept was determined for wave overtopping at shallow foreshores in Altomare et al. [51] and could be considered a first step. However, still more research is required to be able to validate the influence of the infragravity waves on wave overtopping, for this case, as well as for other cases.

## 5. Conclusions

^{3}). This difference was mainly caused by the difference in wave period at the toe of the dike, which is not accounted for by the simple calculation. Hence, this indicates that it can be important to include the complex hydrodynamics such as infragravity waves and wave set-up for these kinds of situations.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. EurOtop (2007) Equations

_{m}

_{0}is the significant wave height [m] and L

_{m}

_{−1,0}is the deep-water wave length [m]. The equations that are used for this paper are given by:

_{m}

_{−1,0}< 5, with q the average overtopping discharge [m

^{3}s

^{−1}m

^{−1}], g the gravitational acceleration [ms

^{−2}], R

_{c}the crest freeboard [m], γ

_{f}a correction factor for roughness and permeability [-], γ

_{b}a correction factor for a berm [-], γ

_{β}a correction factor for oblique wave attack [-], γ

_{v}a correction factor for a vertical wall on top of the crest [-]. The reliability is described by assuming the coefficients C1 = 4.75 and C2 = 2.6 to be normal distributed with a standard deviation of 0.5 and 0.35 respectively. The different correction factors are described in more detail in [12]. Furthermore, for shallow and very shallow foreshores (ξ

_{m}

_{−1,0}> 7):

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**Figure 1.**The Netherlands, with the coast of Walcheren indicated (

**left**). The Westkapelle sea defence, with the bathymetrical transects indicated (

**right**). Transect 1832 is shown in red.

**Figure 2.**Measured 2009 bathymetric transect 1832 of the Westkapelle dike-foreshore system (black) and the profile used for the calculations (blue), with the models that were used in the model framework and the processes which they model. Note that only the section from −500 m to 150 m is shown.

**Figure 3.**A schematized and arbitrary example of the ADIS method, with two stochastic variables (U1 and U2). The standard normal space is given, together with the limit state (red line), the design point (red asterisk), β-sphere (blue circle), random directions (dashed arrows) and XBeach and ARS evaluations (blue circles and black triangles). The areas where Z < 0 and Z > 0 and the limit state Z = 0 are indicated as well (after Den Bieman et al.; Grooteman [21,36]).

**Figure 4.**Model framework, using ADIS, XBeach and EurOtop to calculate the probability of dike failure due to wave overtopping.

**Figure 5.**The cross-shore profile, with the locations where the stochastic variables of Table 1 act on, indicated by their tags. Note that only the section between −500 m and 100 m is shown.

**Figure 6.**Hydraulic boundary conditions during a storm. The top plot shows the total water level (solid black line), composed of the tide (dashed blue line) and surge (dashed red line). The middle plot shows the significant wave height (solid black line) and this wave height divided into one-hour sections (dashed blue line). The bottom plot shows the same for the wave peak period. Here, the tidal peak coincides with the surge peak. Storm shape based on Steetzel et al. [42].

**Figure 7.**h/2-model (solid lines) versus XBeach 1D (dashed lines) with the offshore conditions as stochastic variables and a morphostatic foreshore at the design point. The dike toe is indicated by the dotted line.

**Figure 8.**XBeach 1D calculations with 3 (solid lines) and 10 (dashed lines) stochastic variables at the design point. The dike toe is indicated by the dotted line. The parameters offshore and at the toe of the dike are given in the title of the figure.

**Figure 9.**XBeach 1D with 12 stochastic variables and a morphostatic foreshore (solid lines) versus 15 stochastic variables and a morphodynamic foreshore (dashed lines), in the design point. The dike toe is again indicated by the dotted line. The parameters offshore and at the toe of the dike are given in the title of the figure.

**Figure 10.**XBeach 1D (long-crested waves) with the three offshore conditions as stochastic variables (calc. 2) versus XBeach 2D (short-crested waves) with the three offshore conditions as stochastic (calc. 6) at the design point. The dike toe is indicated by the dotted line. The parameters offshore and at the toe of the dike are given in the title of the figure.

**Table 1.**Stochastic variables as used in the probabilistic calculations and their accompanying probability distributions.

Description | Parameter | Distribution | Reference | Location |
---|---|---|---|---|

Model uncertainty | ||||

XBeach wave breaker index [-] | γ | N(0.54;0.054) | [28] | C |

XBeach calibration factor wave skewness in sediment transport [-] | facSk | L(−1.024;0.29) | [40] | C |

XBeach calibration factor wave asymmetry in sediment transport [-] | facAs | L(−2.14;0.29) | [40] | C |

EurOtop coefficient ξ_{m}_{−1,0} < 5 [-] | C1 | N(4.75;0.5) | [35] | D |

EurOtop coefficient ξ_{m}_{−1,0} < 5 [-] | C2 | N(2.6;0.35) | [35] | D |

EurOtop coefficient ξ_{m}_{−1,0} > 7 [-] | C3 | N(−0.92;0.24) | [35] | D |

EurOtop coefficient zero freeboard [-] | C4 | N(1;0.14) | [35] | D |

Inherent uncertainty: Time | ||||

Storm duration [s] | t_{storm} | L(3.94;0.34) | [41] | A |

Maximum water level [m] | z_{s}_{,max} | W(ω,ρ,α,σ) | [42] | A |

Wave height [m] | H_{m}_{0} | N(0;0.6), related to z_{s} (see main text) | [42] | A |

Wave period [s] | T_{p} | N(0;1), related to H_{m}_{0} (see main text) | [42] | A |

Wave directional spreading [-] | s | U(1.5;10) | A | |

Location tidal peak relative to surge peak [s] | ϕ | U(−t_{tide}/2;t_{tide}/2) | A | |

Inherent uncertainty: Space | ||||

Sediment size [m] | D_{50} | L(−8.12;0.15) | [43] | B |

Manning friction factor [sm^{−1/3}] | n | U(0.01;0.03) | [44] | B |

**Table 2.**Statistical parameters for water level, wave height and wave period during a storm [42]. α is a shape parameter [-], ω is a threshold above which the conditional Weibull function is valid [m], σ is a scale parameter [m] and ρ is the frequency of exceedance of the threshold level ω [year

^{−1}]. a [m], b [-], c [-], d [m] and e [-] are statistical parameters. α [s] and β [sm

^{−1}] are statistical parameters as well. All parameters taken from Steetzel et al. [42].

Water Level | |||||

Station | ω [m] | ρ [year^{−1}] | α [-] | σ [m] | |

Vlissingen | 2.97 | 3.91 | 1.04 | 0.280 | |

Hoek van Holland | 1.95 | 7.24 | 0.570 | 0.0158 | |

Wave Height | |||||

Station | a [m] | b [-] | c [-] | d [m] | e [-] |

Vlissingen | 2.40 | 0.35 | 0.0008 | 7 | 4.67 |

Hoek van Holland | 4.35 | 0.60 | 0.0008 | 7 | 4.67 |

Wave Period | |||||

Station | α [s] | β [sm^{−1}] | |||

Vlissingen | 3.86 | 1.09 | |||

Hoek van Holland | 3.86 | 1.09 |

**Table 3.**Overview of the different probabilistic (ADIS) calculations that were performed with the framework and their results.

Calc. no. [-] | Type of Model | No. of Stoch. Vars. | Complex Hydrodyn. | Set 1, Offsh. Cond. | Set 2, Model Params | Set 3, Tide, Storm | Set 4, Morph. Changes | P_{f} [year^{−1}] | β [-] | No. of Simulations [-] | Calc. Duration [days] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | h/2 | 3 | X | 2.3 × 10^{−8} | 5.47 | 23 | 0.004 | ||||

2 | XB 1D | 3 | X | X | 1.6 × 10^{−5} | 4.15 | 112 | 1 | |||

3 | XB 1D | 10 | X | X | X | 2.9 × 10^{−4} | 3.45 | 341 | 1 | ||

4 | XB 1D | 12 | X | X | X | X | 1.2 × 10^{−4} | 3.67 | 391 | 5 | |

5 | XB 1D | 15 | X | X | X | X | X | 5.3 × 10^{−4} | 3.27 | 484 | 7 |

6 | XB 2D | 3 | X | X | 1.6 × 10^{−6} | 4.66 | 43 | 1 |

**Table 4.**Comparison of the probabilities of failure as calculated with ADIS and as calculated with an approach that uses the same probability exceedance value for each of the stochastic parameters (fully correlated). Results for XBeach 1D with the three offshore parameters as stochastic variables.

Calculation | Offshore z_{s} [m+NAP] | Offshore H_{m}_{0} [m] | Offshore T_{p} [s] | Toe H_{m}_{0} [m] | Toe T_{m}_{−1,0} [s] | P_{f} [year^{−1}] | β [-] |
---|---|---|---|---|---|---|---|

XBeach 1D fully correlated | 5.16 | 6.32 | 11.74 | 1.84 | 46.62 | 2 × 10^{−4} | 3.54 |

XBeach 1D ADIS | 5.46 | 6.09 | 11.11 | 1.81 | 42.82 | 1.6 × 10^{−5} | 4.15 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Oosterlo, P.; McCall, R.T.; Vuik, V.; Hofland, B.; Van der Meer, J.W.; Jonkman, S.N. Probabilistic Assessment of Overtopping of Sea Dikes with Foreshores including Infragravity Waves and Morphological Changes: Westkapelle Case Study. *J. Mar. Sci. Eng.* **2018**, *6*, 48.
https://doi.org/10.3390/jmse6020048

**AMA Style**

Oosterlo P, McCall RT, Vuik V, Hofland B, Van der Meer JW, Jonkman SN. Probabilistic Assessment of Overtopping of Sea Dikes with Foreshores including Infragravity Waves and Morphological Changes: Westkapelle Case Study. *Journal of Marine Science and Engineering*. 2018; 6(2):48.
https://doi.org/10.3390/jmse6020048

**Chicago/Turabian Style**

Oosterlo, Patrick, Robert Timothy McCall, Vincent Vuik, Bas Hofland, Jentsje Wouter Van der Meer, and Sebastiaan Nicolaas Jonkman. 2018. "Probabilistic Assessment of Overtopping of Sea Dikes with Foreshores including Infragravity Waves and Morphological Changes: Westkapelle Case Study" *Journal of Marine Science and Engineering* 6, no. 2: 48.
https://doi.org/10.3390/jmse6020048