# Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm

^{1}

^{2}

^{3}

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^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}/s/m or in l/s/m. The mean overtopping discharge depends on local wave conditions (wave height, wave period, water depth) but also on the layout and main features of the coastal defense, such as slope of the seaward face ([12]), slope of the foreshore before the defense ([13]), structural crest freeboard ([12,14]), presence of storm walls ([15]), presence of a berm ([16]), blocks or any artificial element on the surface of the structure that might increase its roughness to the wave run-up ([17]), etc. A sketch of the overtopping over a sea dike is represented in Figure 1. Both deep-water wave height and period (H

_{m}

_{0,o}and T

_{m}

_{−1,0,o}) and local wave characteristics (H

_{m}

_{0,t}and T

_{m}

_{−1,0,t}) are indicated. The latter ones correspond to wave height and period at the toe of the structure; h

_{t}corresponds to the local water depth or depth at the toe of the structure. Meanwhile, R

_{c}defines the crest freeboard, i.e., vertical distance between the still water level (SWL) and the crest of the defense. The spectral wave period is preferred to the peak period, since it was proved to show better performance for wave overtopping predictions for a different kind of wave spectra (e.g., single or double peak spectrum), as detailed in [18].

_{c}/H

_{m}

_{0,t}, and the surf similarity parameter, which is defined as the ratio between the structural slope and the square root of the wave steepness, H

_{m}

_{0,t}/L

_{m−}

_{1,0,t}, in which L

_{m}

_{−1,0,t}is the wavelength at the toe of the structure calculated using the deep-water formula gT

_{m}

_{−1,0,t}

^{2}/2π.

## 2. Average Wave Overtopping Assessment in Existing Literature

#### 2.1. Goda (2009)

_{c}is the freeboard, H

_{m}

_{0,t}is the incident spectral wave height at the dike toe, and g is the gravity acceleration. The coefficients A and B are a function of the dike slope, the foreshore slope, and the dimensionless toe depth:

_{t}is the still water depth at the dike toe, and α is the dike slope angle. The Equations (2)–(5) are valid in the range 0 ≤ cot α ≤ 7. Both A and B coefficients increase up to a constant value if the relative toe depth increase. The datasets employed to fit Equations (2)–(5) are characterized by a dimensionless toe depth h

_{t}/H

_{m}

_{0,t}bigger than 1.0, lacking data for very and extremely shallow water conditions.

#### 2.2. Mase et al. (2013)

_{t}< 0). An imaginary slope is defined between the sea bed at the breaking location and the point reached by the maximum run-up height; see Figure 2. The imaginary slope is employed to overcome the difficulties in schematizing complex dike and foreshore layouts. The formulae proposed by [22] are based on deep-water wave characteristics. Overtopping discharge is a function of the freeboard, the run-up, and the deep-water wave height.

_{m}

_{0,o}and L

_{m}

_{−1,0,o}are the deep-water wave height and wavelength, respectively, and θ

_{im}refers to the imaginary slope as schematized in [22]. The use of deep-water conditions and calculation of wave run-up to assess the average discharge are key aspects of [22] that will be considered later on in this work.

#### 2.3. EurOtop (2018)

#### 2.3.1. Breaking and Non-breaking Wave Conditions

_{m}

_{−1,0,t}is the deep-water wavelength calculated based on the spectral period T

_{m}

_{−1,0,t}. The influence factors presented in [19] are here omitted to take into account the effects of wave obliqueness, berm presence, and roughness, among others.

#### 2.3.2. Very (shallow) Foreshores

_{t}/H

_{m}

_{0,t}is smaller than 1.5; otherwise, the dike slope only should be considered for further calculations. The new equivalent slope concept can be applied also to cases of dry toe (= negative water depth at the toe). The relative toe depth ranges between −0.25 m and 3.65 m. Mean wave overtopping can be assessed using the following equation:

_{m}

_{0,t}and the run-up level R

_{u}

_{2%}can be expressed as follows:

_{m−1,0}is assessed replacing the dike slope with the value of cotθ

_{eq}.

#### 2.4. Gallach (2018)

- a
_{Gallach}= 0.0109 − 0.035(1.05 − cot α) and a_{Gallach}= 0.109 for cot α > 1.5; - b
_{Gallach}= 2 + 0.56(1.5 − cot cot α)1.3 and b_{Gallach}= 2 for cot α > 1.5; and - c
_{Gallach}= 1.1

_{c}/H

_{m}

_{0}< 2.5, and 0 < H

_{m}

_{0,t}/h

_{t}< 0.55. Influence factors are omitted here.

## 3. Scaling Laws in Wave Overtopping

#### 3.1. Volume Flux and Deficit in Freeboard

_{0}= H

_{m}

_{0,o}L

_{p}/2π, of the incident waves. H

_{m}

_{0,o}is the offshore wave height and L

_{p}is the deep-water wavelength calculated starting from the peak period T

_{p}. The authors start from the assumption that scaling parameters for overtopping typically include the local wave height and period (or wavelength), the beach slope, and the crest freeboard. Usually, overtopping discharge is scaled on √(gH

_{m0,t}

^{3}), which was employed based on the weir discharge equation and assuming critical flow conditions at the crest. However, the authors observed that except for cases of submerged structures, overtopping is usually characterized by subcritical flows for non-breaking wave conditions and supercritical flows in case of broken waves. Besides, the authors demonstrate that to use the deficit in freeboard, 1-R

_{c}/Ru

_{2%}, instead of the R

_{c}/H

_{m}

_{0}reduces substantially the scatter of data around the best fit. The deficit in freeboard has been already used in [22]; see Equation (6).

_{u}

_{2%}.

_{m}

_{−1,0}, in which T

_{m}

_{−1,0}is the mean spectral wave period. By replacing L

_{p}with L

_{m}

_{−1,0}(=gT

_{m}

_{−1,0}

^{2}/2π), the scaling for the mean discharge can be finally expressed as follows:

#### 3.2. Dimensional Analysis

- (1)
- Q* = 4π
^{2}q/gH_{m}_{0,o}T_{m}_{−1,0,o}, based on [26], see Equation (14): deep-water wave characteristics are employed, after having verified that there is almost no difference (R^{2}= 99%) between the volume flux calculated employing the deep-water wave height and period with the one calculated based on local wave conditions at the toe. - (2)
- Dimensionless freeboard, R
_{c}/H_{m}_{0,t}. - (3)
- In [22] and [26], the overtopping is proved to be a function of the deficit of the freeboard, namely 1-R
_{c}/Ru_{max}, rather than the dimensionless freeboard. Notice that we use here the maximum run-up according to [22]. Considering the maximum run-up, the effects of wave periods, foreshore, and dike slope on overtopping discharge are included, as also in [13]. - (4)
- The slope parameter proposed in [27], β = θT
_{m}_{−1,0,o}√(g/H_{m}_{0,o}), is employed to consider the influence of long waves generated by the presence of mild slopes on wave overtopping. - (5)
- Relative toe depth employed as shallowness criteria, h
_{t}/H_{m}_{0,o}, is employed as proposed by [27]. - (6)
- The ratio between the local water depth, the dike toe, and the local wave height, h
_{t}/H_{m}_{0}_{,t}, is considered as it was proven to have an influence on overtopping discharge [1], see Equations (2)–(3). - (7)
- The combined effect of the foreshore slope and dike slope is taken into account by considering the equivalent slope defined in [13] and expressed as cotθ
_{eq}. As an alternative, the imaginary slope described in [22] and expressed as cotθ_{im}will be considered. By means of Evolutionary Polynomial Regression (EPR) application, the influence of the two conceptual slopes on the assessment of mean overtopping discharge will be evaluated and discussed. - (8)

_{eq}; meanwhile, Equation (7) is employed for the imaginary slope calculation, cotθ

_{im}. The differences between the two methods are presented in Section 6. The most probable maximum run-up Ru

_{max}= (Ru

_{max})

_{99%}, assuming a Rayleigh distribution, is finally equal to 1.54Ru

_{2%}.

## 4. Methodology

#### 4.1. The Evolutionary Polynomial Regression Paradigm

_{j}are the numerical coefficients to be estimated, X

_{i}are the candidate explanatory variables, ES(j,z) is the exponent of the z

^{th}input within the j

_{th}term in Equation (19), and f is a function selected by the user among a set of possible alternatives (including no function selection). The exponents ES(j,z) are selected from a user-defined set of candidate values (which should include 0).

_{i}is basically deselected from the resulting equation (i.e., the exponent ES(j,k) = 0 and the relevant input X

_{kj}= 1). This is crucial when investigating a physical phenomenon and the most important inputs to characterize it. In fact, during its search in the space of solutions, EPR can deselect a less important input by assigning a null exponent.

#### 4.2. Model Performance

_{predicted,i}and q

_{estimated,i}are respectively the i-th predicted and measured mean overtopping discharge. The higher the value of the CoD (with a maximum of 1), the better the model performance. The average error is defined as:

## 5. Experimental Datasets for Fitting Overtopping Formula

- EurOtop [19] database: the new extended database comprises about 13,500 tests on wave overtopping and extends the original CLASH database [40]. The tests include different dike geometries and kinds of structures, 14% of which represent smooth dikes. By excluding cases with zero overtopping, oblique waves, berms, or storm walls and considering only core data, finally 1128 data have been extracted.
- Data described in [13] and corresponding to tests on sea dikes with shallow foreshores. The tests correspond to five different experimental campaigns carried out at Flanders Hydraulics Research, in Antwerp, and at the Department of Civil Engineering of Ghent University, in Ghent (both in Belgium). These experimental campaigns were conducted to study the influence of mild foreshore slopes and very or extremely shallow water conditions on wave overtopping and loading on smooth sea dikes. The geometry of the dikes resembled typical layouts from the Belgian coast.
- Data from [41] corresponding to coastal defenses in Japan built in extremely shallow foreshores or on land (emergent toe), namely seawalls with slopes of 1:3, 1:5, and 1:7 with uniform foreshore slope of either 1:10 or 1:30. A Bretschneider–Mitsuyasu spectrum was used to generate random wave trains, differently from other datasets here employed, where a JONSWAP spectrum was employed.
- Data from [42] corresponding to the experimental campaign carried out in the multi-directional wave basin, at Flanders Hydraulics Research. A 1:2 sloping dike after a 1:35 foreshore slope was employed. All data correspond to cases with mild foreshore slope (β < 0.62) in very and extremely shallow water conditions (h
_{t}/H_{m}_{0,o}< 1). The experimental campaign was conceived to characterize the overtopping discharge of long- and short-crested waves proceeding very obliquely with respect to the dike crest. For the present study, only perpendicular long-crested wave cases are considered. - Data from Altomare et al. (under review) [43], where steep foreshore (β > 0.62) is used in combination with a rather steep dike (1:1) and very or extremely shallow waters (h
_{t}/H_{m}_{0,o}< 1). The datasets include cases with crests larger than 0, which have been excluded from the present analysis. The experiments were carried out at the Maritime Engineering Laboratory of Universitat Politècnica de Catalunya – BarcelonaTech (LIM/UPC), in Barcelona, Spain. The scope of the experimental campaign was to measure the overtopping flow properties and analyze the related safety of pedestrian standing on the promenade of a sea dike.

_{t}/H

_{m}

_{0,o}(shallowness criteria), foreshore and dike slopes, dimensionless freeboard, R

_{c}/H

_{m}

_{0,t}, and slope parameter, β = θT

_{m}

_{−1,0,o}√(g/H

_{m}

_{0,o}).

_{t}≤ 0), for which it is important to provide some details on the methodology followed to measure local wave characteristics. For emergent toes and more in general for extremely shallow water cases, the waves break heavily before reaching the dike. These broken waves present an unusual shape that is similar to a bore rather than to an oscillatory phenomenon around the mean water level. Their oscillatory nature can be no longer distinguished, yet the spectral momentum and, therefore, the wave height and period can be defined. The accuracy of this choice is demonstrated after showing the overtopping results in [13]. For some of the data employed for the present work, the information at the dike toe was lacking. In some cases, specifically in those from [13], the SWASH model (http://swash.sourceforge.net/) was employed to get the incident wave height and period at the toe. For other cases from [41], the method developed by [25] was applied. This method allows calculating the wave height at the toe even when the water depth is 0. Citing [25]: "A finite wave height is considered at the shoreline that originates from the presence of stationary wave set-up and time-varying surf-beat there." In the case of a dry or emergent toe, the same method from [25] has been applied, replacing the dry toe (h

_{t}< 0) with h

_{t}= 0. In other words, the solution is forced at h

_{t}= 0. A similar procedure has been found in [44].

_{c}= 1 m using Froude’s similarity law in order to compare different datasets. The selected variables are mean overtopping discharge (expressed in logarithmic scale), deep-water wave height and period, local wave height and period, foreshore and dike slope, local water depth, and maximum wave run-up. The last one has been calculated using Equation (7) and is plotted in logarithmic scale. It can be noticed that despite the large variety of data and range of each dataset, a strong relationship can be still identified between the discharge values and the local wave height. Even better correlation is noticeable in the wave run-up, where the calculation of wave run-up depends not only on the wave height but also on the wave period, foreshore, and dike slope.

^{2}q/gH

_{m}

_{0,o}T

_{m}

_{−1,0,o}against the dimensionless freeboard; meanwhile, in the right plot, the dependence is shown in terms of deficit in freeboard. For the former one, all data but the ones from dataset #3 are quite clustered. Dataset #3 corresponds to the data from [22], mainly with sea dikes and walls with extremely shallow waters and even an emergent toe. In the right plot of Figure 4, the maximum run-up has been calculated with Equation (7) (Ru

_{max,im}). In this case, the data belonging to each dataset seem to follow a different trend but converge for value of the deficit in freeboard equal to 1 (i.e., freeboard equal to 0). The relationship between discharge, dimensionless freeboard, and deficit in the freeboard will be investigated further and discussed in Section 5 and Section 6. It is worthwhile to notice that employing the dimensionless discharge Q* scaled by the volume flux leads to less data scatter than employing the most used Q = q/√(gH

_{m}

_{0,t}

^{3}), especially for low freeboards and large overtopping rates (where measurement uncertainties are less and the test repeatability is usually high): see the comparison between Figure 4 (left plot) and Figure 5.

_{predicted}/q

_{measured}< 10, except for data from datasets #1 and #3, which correspond to the two datasets having extremely shallow water conditions with emergent toes. The inaccuracy to predict the mean overtopping discharge of these two datasets is due to the range of applicability of Equation (1) and the range of hydraulic and geometrical parameters used to derive the coefficients expressed in Equations (2)–(5), where a clear lack of data in very or extremely shallow water conditions has been noticed in the dataset employed by [23].

## 6. A Set of Unified Formulas for the Preliminary Assessment of Wave Overtopping on Smooth Sea Dikes and Vertical Walls

#### EPR Setting and Results

_{max,im}, or Equations (16)–(18), Ru

_{max,eq}. A set of model expressions returned by EPR is reported in Table 2. Looking for model parsimony, here, only expressions limited to a maximum of 3 terms (excluding the constant) and 4 inputs are reported. The values of CoD and AVG for both training and test data are reported, and the results overall are very similar to each other.

_{m}

_{0,o}/H

_{m}

_{0,t}has been selected, as derived from the ratio between the h

_{t}/H

_{m0,t}and h

_{t}/H

_{m0,o}. Notwithstanding, models VIII, IX, and X do not add further accuracy, and the dependence of the dimensionless discharge on H

_{m}

_{0,o}/H

_{m}

_{0,t}seems rather the result of model overfitting than something physically based.

_{predicted}/q

_{estimated}calculated based on GEO and GSD for each model expression is reported. The model expression VII results are the most accurate, having a GEO almost equal to 1 (=0.99) and the lowest standard deviation. For model VII, assuming the overtopping rate as normally distributed, 90% of the predicted overtopping rate is to be located in the range between 0.20 and 4.84 times the measured overtopping discharge. Excluding models VIII, IX, and X, the second most accurate model expression is model III, which is a function of the deficit in freeboard and the equivalent slope. The accuracy of Equation (1) and Equation (6) has been also assessed, showing a larger confidence interval for both equations, with a tendency to overestimate overtopping discharge when Equation (1) is employed. The differences between equivalent and imaginary slopes will be discussed later. Yet, it is evident that a slope—a combination of foreshore slope and dike slope, and the definition of which depends also on the local water depth—is a key variable to express the mean overtopping discharge. The dependence of discharge on θ, α, and h

_{t}was already remarked in [13,23]. The results of predicted mean discharge rates employing model III and VII are plotted against the measured ones in Figure 7 and Figure 8, respectively.

## 7. Discussion

_{m}

_{0,t}

^{3}), here, the scaling factor includes the wave period as a key variable to explain overtopping discharge. Assuming the extreme case of zero freeboard for different wave periods but keeping all other hydraulic and geometrical characteristics fixed, the mean discharge will be different; namely, it will be larger for longer wave periods. This result is actually corroborated by the evidence that if we assume sea storms of the same duration, which are expressed as the number of waves in one wave train, longer waves will lead to larger volumes than shorter ones. However, only breaking wave formulas, such as Equation (8), express similar relationships. For non-breaking waves, the wave period does not play a role at all. For very and shallow water conditions, the wave period is considered only within the exponent; hence, there is no influence in case of zero freeboard.

_{c}and local wave height H

_{m}

_{0,t}. Employing instead the deficit in freeboard includes the wave run-up calculation and hence the influence of wave period, local water depth, foreshore slope, and dike slope. This means proper consideration of the wave transformation and eventual wave breaking to which the waves are subjected before reaching the dike toe and dike crest ([13]).

_{b}. For the equivalent slope, if the local water depth is bigger than 1.5H

_{m}

_{0,t}, only the dike slope is assumed. In case of vertical walls, this will correspond therefore to a value equal to 0. Besides, the calculation of the equivalent slope and the imaginary slope requires an iterative process, since the wave run-up, which is necessary for the slope definition, is not known a priori and depends on the same slope. Uncertainties related to applicability range of the same run-up formulas raise questions about the accuracy of one method over another. A comparison between the calculated imaginary slope and equivalent slope is reported in Figure 10. It can be noticed that for most of the data, the two calculations lead to similar results; however, for some data, the imaginary slope is far larger than the equivalent one. This is due to the calculation of the breaking depth and its distance from the dike toe, especially for very mild foreshores: the area underneath the foreshore will actually increase significantly, giving more weight to the foreshore slope than the dike slope (see Figure 2). In general, however, EPR models show similar dependence of the overtopping rates on the imaginary and equivalent slopes: namely, the larger the slope is, the smaller the overtopping. Larger equivalent or imaginary slopes correspond to cases where the foreshore slope is rather influential; hence, heavier wave breaking and bigger wave energy dissipation are expected. This leads to reduced overtopping discharge. As an example, in countries such as Belgium or The Netherlands, beach nourishment is employed in such cases as a soft countermeasure to reduce overtopping and flooding. In fact, the resulting very long and mild beaches before sea dikes induce heavy wave breaking that decrease overtopping rates.

_{max,im}and Ru

_{max}

_{,eq}, respectively. Further research aiming to derive a unified formula for wave run-up is advised, covering the whole range of sloping dikes and vertical walls, for all water conditions and even an emergent toe. For it, ad hoc experimental tests and new measurements will be required. Despite the clear difference and independently of the way it is calculated, the deficit in freeboard remains the most important explanatory variable for the process at stake. From a practical point of view, model expression III might be preferred to model expression VII for two main reasons: (1) uncertainty in the calculation of the breaking depth required to define the imaginary slope in model expression VII; and (2) model expression VII requires using both deep-water and local wave conditions. The former ones are needed to calculate the run-up based on Equation (7), and therefore, the deficit in freeboard; the latter ones define the dimensionless freeboard. Instead, the terms of the right-hand side in model expression III are expressed as a function of local wave conditions only, and due to the almost identity of volume flex expressed either with deep-water or local wave conditions (Figure 9), the latter ones can be used to scale the mean discharge (left-hand side in model III).

## 8. Conclusions

- Mean overtopping discharge is scaled by the volume flux, including both wave height and period ([26]): dimensionless discharge can be defined as Q* = 4π
^{2}q/gH_{m}_{0,o}T_{m}_{−1,0,o}. In this way, the wave period is always considered as an explanatory variable for wave overtopping assessment also for those cases where it is usually not taken into account; see Equation (9) for non-breaking waves. - Employing deep-water wave characteristics rather than local wave characteristics at the dike toe to define the volume flux and, consequently, the scaling law for overtopping, leads to differences that are negligible. The reader can employ either the former ones or the latter ones, depending on the availability and accuracy of the employed information.
- EPR analysis confirms that an exponential structure, the most employed in literature (see [19]), leads to the best model representation.
- Both the dimensionless freeboard, R
_{c}/H_{m}_{0,t}, and the deficit in freeboard, 1-R_{c}/Ru_{max}, are key explanatory variables for wave overtopping assessment. The main difference is that the definition of the deficit in freeboard requires the assessment of wave run-up, which is not only depending on the wave height but also on the wave period, local water depth, foreshore slope, and dike slope. - Therefore, the study confirms previous findings from [13,22,23], and partly from [12], where other variables such as dike and foreshore slope and local water depth play an important role for the calculation of mean discharge rates. By combination these variables in one equivalent or imaginary slope concept, the model accuracy increases.
- EPR results confirm that the imaginary or equivalent slopes provided improve the estimate of mean discharge values, confirming somehow the physical meaning of such concepts. Overtopping reduces when the slope (defined as a cotangent of the angle with the horizontal) increases (heavier wave braking and dissipation over the foreshore slope).

## Author Contributions

## Funding

## Conflicts of Interest

## Disclaimer

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**Figure 2.**Scheme for definition and calculation of equivalent slope (Altomare et al., 2016) and imaginary slope (Mase et al., 2013). For the latter, the area A shadowed in red is employed.

**Figure 4.**Variation of the dimensionless overtopping Q* discharge versus with the dimensionless freeboard (left) and deficit in freeboard (right) for each of the five employed datasets.

**Figure 5.**Variation of the dimensionless overtopping discharge q/√(gH

_{m}

_{0}

^{3}) versus with the dimensionless freeboard.

**Figure 6.**Comparison of Goda (2009) and measured overtopping discharge (solid line: prediction using Equation (1) equal to measured rate; dashed lines: prediction using Equation (1) equal to 10 times and 0.1 times the measured rate).

**Figure 7.**Comparison of overtopping prediction employing EPR model expression III and measured overtopping discharge (solid line: prediction equal to measured rate; dashed lines: prediction equal to 10 times and 0.1 times the measured rate).

**Figure 8.**Comparison of overtopping prediction employing EPR model expression VII and measured overtopping discharge (solid line: prediction equal to measured rate; dashed lines: prediction equal to 10 times and 0.1 times the measured rate).

**Figure 9.**Comparison between the volume flux calculated employing deep-water wave characteristics and local ones.

**Figure 11.**Comparison of dimensionless run-up values calculated with formulations proposed in EurOtop (2018) and Mase et al. (2013) and employing the equivalent (Ru

_{max,eq}) and imaginary slope (Ru

_{max,eq}), respectively.

**Figure 12.**Dimensionless freeboard versus deficit in freeboard, where maximum run-up is calculated employing the formulas from EurOtop (2018) and the equivalent slope from Altomare et al. (2016).

Dataset ID | Dataset Source | # of Data | h_{t}/H_{m0,o} | cot α | cotθ | R_{c}/H_{m0,t} | β |
---|---|---|---|---|---|---|---|

1 | Altomare et al. (2016) | 170 | (−0.14)–0.86 | 2–6 | 35–50 | 0.41–3.50 | 0.29–0.62 |

2 | EurOtop (2018) | 1128 | 0.27–22.73 | 0–7 | 10–1000 | 0–6.24 | 0.01–3.5 |

3 | Tamada et al. (2009) | 230 | (−0.27)–0.50 | 3, 5, 7 | 10–30 | 0.93–8.04 | 0.44–1.92 |

4 | Altomare et al. (under review) | 26 | 0.10–0.36 | 1 | 15 | 2.95–6.19 | 1.13–1.42 |

5 | Altomare et al. (2020) | 125 | 0.12–0.58 | 1.97 | 35 | 0.53–3.24 | 0.37–0.58 |

**Table 2.**Model expressions returned by Evolutionary Polynomial Regression (EPR). AVG: average error, CoD: coefficient of determination.

Model # | Model Expression | Training | Test | ||
---|---|---|---|---|---|

CoD | AVG | AVG | CoD | ||

I | ${Q}^{*}=\mathrm{exp}\left(8.42{\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},eq}}\right)}^{1.5}-9.80\right)$ | 0.69 | 0.18 | 0.70 | 0.18 |

II | ${Q}^{*}=\mathrm{exp}\left(-4.19\sqrt{\frac{{R}_{c}}{{H}_{m0,t}}}-0.37\right)$ | 0.65 | 0.20 | 0.66 | 0.20 |

III | ${Q}^{*}=\mathrm{exp}\left(8.10{\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},eq}}\right)}^{1.5}-0.14\mathrm{cot}{\theta}_{eq}-9.00\right)$ | 0.75 | 0.17 | 0.74 | 0.17 |

IV | ${Q}^{*}=\mathrm{exp}\left(-4.6\sqrt{\frac{{R}_{c}}{{H}_{m0,t}}}+0.87\beta -0.29\right)$ | 0.70 | 0.20 | 0.70 | 0.20 |

V | ${Q}^{*}=\mathrm{exp}\left(-\left(4.95-0.69\beta \right)\sqrt{\frac{{R}_{c}}{{H}_{m0,t}}}+0.14\right)$ | 0.71 | 0.20 | 0.71 | 0.19 |

VI | ${Q}^{*}=\mathrm{exp}\left(3.68\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},im}}\right)-2.94\sqrt{\frac{{R}_{c}}{{H}_{m0,t}}}-4.11\right)$ | 0.73 | 0.18 | 0.74 | 0.18 |

VII | ${Q}^{*}=\mathrm{exp}\left(-\frac{{R}_{c}}{{H}_{m0,t}}+3.94\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},im}}\right)-0.08\mathrm{cot}{\theta}_{im}-5.80\right)$ | 0.76 | 0.16 | 0.74 | 0.16 |

VIII | ${Q}^{*}=\left(-2.49\sqrt{\frac{{H}_{m0,0}}{{H}_{m0,t}}}+7.23{\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},im}}\right)}^{1.1}-6.65\right)$ | 0.75 | 0.17 | 0.74 | 0.17 |

IX | ${Q}^{*}=\mathrm{exp}\left(1.56{\frac{{H}_{m0,0}}{{H}_{m0,t}}}^{-2}+7.89{\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},eq}}\right)}^{1.5}-10.67\right)$ | 0.74 | 0.16 | 0.74 | 0.16 |

X | ${Q}^{*}=\mathrm{exp}\left(-1.80\sqrt{\frac{{H}_{m0,0}}{{H}_{m0,t}}}+5.37\left(1-\frac{{R}_{c}}{R{u}_{\mathrm{max},im}}\right)-0.68\frac{{R}_{c}}{{H}_{m0,t}}-5.38\right)$ | 0.76 | 0.15 | 0.76 | 0.16 |

**Table 3.**Geometric mean and standard deviation for each EPR model expression. The 90% interval of q

_{predicted}/q

_{estimated}is reported. GEO: geometric mean, GSD: geometric standard deviation.

Model # | GEO | GSD | q_{pred}/q_{EXP} | |
---|---|---|---|---|

μ − 1.64σ | μ + 1.64σ | |||

I | 1.00 | 3.40 | 0.18 | 5.58 |

II | 1.15 | 3.60 | 0.19 | 6.77 |

III | 0.99 | 3.08 | 0.20 | 5.01 |

IV | 1.00 | 3.38 | 0.18 | 5.53 |

V | 1.00 | 3.29 | 0.19 | 5.39 |

VI | 0.99 | 3.14 | 0.19 | 5.10 |

VII | 0.99 | 2.98 | 0.20 | 4.84 |

VIII | 0.85 | 3.00 | 0.17 | 4.18 |

IX | 0.93 | 3.28 | 0.17 | 5.00 |

X | 0.89 | 3.11 | 0.17 | 4.54 |

Equation (1) | 1.59 | 8.03 | 0.12 | 20.94 |

Equation (6) | 0.96 | 24.55 | 0.02 | 38.65 |

© 2020 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/).

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**MDPI and ACS Style**

Altomare, C.; Laucelli, D.B.; Mase, H.; Gironella, X. Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm. *J. Mar. Sci. Eng.* **2020**, *8*, 570.
https://doi.org/10.3390/jmse8080570

**AMA Style**

Altomare C, Laucelli DB, Mase H, Gironella X. Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm. *Journal of Marine Science and Engineering*. 2020; 8(8):570.
https://doi.org/10.3390/jmse8080570

**Chicago/Turabian Style**

Altomare, Corrado, Daniele B. Laucelli, Hajime Mase, and Xavi Gironella. 2020. "Determination of Semi-Empirical Models for Mean Wave Overtopping Using an Evolutionary Polynomial Paradigm" *Journal of Marine Science and Engineering* 8, no. 8: 570.
https://doi.org/10.3390/jmse8080570