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THE UNCERTAINTY IN THE PREDICTION OF THE DISTRIBUTION
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OF INDIVIDUAL WAVE OVERTOPPING VOLUMES USING A NON-
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LINEAR SHALLOW WATER EQUATION SOLVER
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Hannah E. Williams*, Riccardo Briganti*, Tim Pullen† and Nicholas Dodd*
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*
Infrastructure, Geomatics and Architecture Division,
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Faculty of Engineering, University of Nottingham,
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Nottingham, NG7 2RD, U.K.
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[email protected]
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†
HR Wallingford, Howbery Park, Wallingford, OX10 8BA, U.K.
LRH: Williams, Briganti, Pullen and Dodd RRH: Uncertainty in the prediction of overtopping volumes using NLSWE ABSTRACT This work analyses the uncertainty of the prediction of individual overtopping volumes using
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the Non-Linear Shallow Water Equations. A numerical model is used to analyse the
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variability due to seeding. The effect of the incident wave height distribution on the
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individual overtopping volumes distribution is also considered. The numerical results are then
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compared with both the laboratory tests and available empirical methods. A large variability
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across the distributions is found, which produces some results showing a significant diversion
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from the empirical prediction methods. The magnitude of this departure is seen to directly
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relate to the accuracy of the numerical model in reproducing the incident wave height
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distribution at the toe of the structure from the physical model.
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ADDITIONAL INDEX WORDS Random waves, Smooth slope, Wave height distribution, Time series reconstruction.
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INTRODUCTION
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The main overtopping parameter considered in the design of coastal defence structures is the
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discharge, i.e. the volume of water that flows over the structure crest over a certain period of
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time (Pullen et al., 2007). When the safety of people or property to the direct impact of an
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overtopping wave is to be assessed the distribution of the individual overtopping volumes
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and, in particular, the maximum expected volume, are used. As also pointed out in Victor,
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van der Meer, and Troch (2012), this distribution is very important in the optimisation of
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wave energy conversion devices based on wave overtopping (OWEC).
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A significant amount of work based on physical modelling has been carried out by a number
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of researchers in order to describe the probability distribution of the overtopping volumes.
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One of the first studies on the subject was carried out by van der Meer and Janssen (1994).
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They found that the overtopping volume per wave can be expressed by a Weibull
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distribution, based on two parameters, the scale factor, a, which describes the magnitude of
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the individual volumes and depends on the wave attack parameters and percentage of
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overtopping waves, and the shape factor, b, which describes the shape of the distribution of
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the volumes. A more detailed analysis by Besley (1999) considering a range of existing data
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based on commonly occurring seawall types, such as different sloping and vertical structures,
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found fairly consistent behaviour for all those tested, confirming the findings of van der Meer
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and Janssen (1994). The van der Meer and Janssen (1994) formulation for the scale and shape
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factors has been included in Pullen et al. (2007). More recently, research has been carried out
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on specific conditions: Victor, van der Meer, and Troch (2012) considered steep structures
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with relatively small crest freeboards and Nørgaard, Lykke Andersen, and Burcharth (2014)
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considered the effect of depth limited conditions on the distributions. These two works have
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proposed modifications to the parameters used for the Weibull distribution.
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Wave resolving numerical models, such as those based on non-linear shallow water equations
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(NLSWE), are intrinsically capable of predicting individual overtopping volumes. However,
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there is very limited research on their accuracy in describing the volume distribution.
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Numerical accuracy in overtopping prediction is usually assessed by comparing the total
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predicted discharge, the number of overtopping events and, sometimes, the maximum
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overtopping volume, with experimental data. Although the probability of overtopping and the
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time history of the overtopping volumes have been studied (see McCabe, Stansby, and
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Apsley, 2013; Williams, Briganti, and Pullen, 2014), there are no specific studies on the
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accuracy and the uncertainty of the modelled distribution. One of the reasons for this is that,
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often, only the total overtopping volume during a physical model test is considered due to
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individual volumes being difficult to measure accurately.
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When wave resolving models are used, the results vary with the sequence of waves at the
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seaward boundary. Williams, Briganti, and Pullen (2014) found that, when wave energy
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density spectra are used to obtain an input free surface time series, overtopping parameters
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vary due to the seeding of the phases of the spectral components. This variability was
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quantified by carrying out a Monte-Carlo analysis. The previous work focused on the
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following parameters; overtopping discharge, probability of overtopping and the maximum
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individual overtopping volume. The variability in these parameters was found to vary by up
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to a factor of more than 50, depending on the level of overtopping present. The individual
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overtopping volume distribution was not analysed. This work will present such analysis.
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To the knowledge of the authors, the uncertainty in the distribution of the overtopping
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volumes has never been investigated in numerical models. This work will show how the
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reconstruction of the free surface time series at the offshore boundary and the resulting
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incident wave height distribution affects the variability in the distribution of the overtopping
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volumes. Guidelines on how to address uncertainty in the modelling will be given.
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This work is organised as follows; the next section provides an overview of the various
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prediction methods. The third section presents the results of the laboratory experiments and
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the numerical experiments. The fourth section discusses the findings; and the final section
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will present the conclusions of this research. In addition, explanations of the abbreviations
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and nomenclature used throughout this work can be found at the end of article.
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METHODS
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This work is mostly based on a numerical investigation; however the wave conditions
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modelled were compared to laboratory tests. Therefore, these are briefly described before the
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numerical approach is defined. Also, the predictions have been compared to the existing
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empirical formulas to assess how numerical and empirical predictions agree at different levels
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of overtopping.
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Laboratory Tests
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The analysis of the individual overtopping volumes is carried out using reference laboratory
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tests in which the hydraulic input and output conditions are known. These tests were carried
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out in a two dimensional (2D) wave flume at HR Wallingford, UK.
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The experiments were first presented in Williams, Briganti, and Pullen (2014) where further
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details of the set-up can be found; here only an overview is provided. The physical model 3
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layout is shown in Figure 1. The structure tested was a simple concrete 1:2.55 impermeable
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slope, and the wave conditions were chosen so as to be those of shallow water and therefore
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suitably modelled with a NLSWE solver. Each test ran for a time length equivalent to 1000
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mean wave periods. The free surface was measured at various points along the flume using
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eight wave gauges (WG). The wave gauge at the toe of the structure (WG7) is used to
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provide the offshore boundary conditions to the numerical model.
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During the experiments, overtopping events were directed into two separate tanks (see Figure
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1). As explained in Williams, Briganti, and Pullen (2014), due to the higher accuracy
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measurement obtained using tank 2, it was decided to only use this tank in the analysis of
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volumes. A wave gauge placed in the tank allowed the estimation of overtopping events,
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following the procedure used in Briganti et al. (2005).
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Table 1 summarises the characteristics of the incident waves used, along with the measured
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parameters in the physical model. Tp is the peak period, Hm0/dt is the local wave height to
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local water depth ratio, Rc/Hm0 is relative freeboard, s0 is the wave steepness defined as
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and g is the gravitational acceleration. ξm-1,0 is the surf similarity parameter
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defined as
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overtopping discharge, Q* is the dimensionless overtopping defined as
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is the maximum individual volume.
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This analysis will focus on a subset of tests that are representative of different levels of
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overtopping. In tests with few overtopping events, e.g., Tests 002 and 003, it is not possible
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to obtain an individual overtopping volume distribution, therefore only medium and high
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levels of overtopping are considered for this study.
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Numerical Tests
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The numerical simulations are carried out using the solver of the NLSWE based on a finite
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volume scheme using a Weighted Averaged Flux (WAF) proposed in Briganti et al. (2011).
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The hydrodynamics equations are:
, where Tm-1,0 is the mean spectral period. q is the mean
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and Vmax
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x is the horizontal abscissa, t is time, U denotes the depth-averaged horizontal velocity and zB
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is the bed level. h=d+η where d is the still water depth and η the free surface. τb the bottom
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shear stress and ρ is the water density. The numerical model is discretised into a grid with
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cells of equal width, ∆x=0.01m, and using the Courant number Cr=0.8.
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Two sets of tests were performed. In both, the toe of the structure, which is also the position
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of WG7, was the seaward boundary of the model. The choice of the location of the offshore
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boundary has been done following the NLSWE application of Dodd (1998) and Shiach et al.
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(2004). Both works indicate that best results are obtained when the offshore boundary is close
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to the structure to be simulated in order to obtain shallow water conditions for the incoming
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wave attack.
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In the first set of tests, η measured at WG7 during the laboratory tests provides the free
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surface offshore boundary conditions to the model. The corresponding U was obtained using
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the shallow water approximation, i.e.
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seaward boundary condition that prescribes the total water depth and velocity is applied (see
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Dodd, 1998). These results will be referred to as Measured Offshore Boundary Conditions
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(MOBC).
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In the second set of tests, each energy density spectrum obtained from the incident time series
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of η at WG7 measured during the calibration tests was used to generate a population time
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series of η. These time series of random waves were generated by computing directly from
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the incident wave spectrum the amplitudes of the spectral components. An initial seed value
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was required for a random number generator to generate a uniform distribution of starting
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phases between 0 and 2π across the domain. The components were then combined to produce
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the different times series of η.
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Since the incident spectrum was used, an absorbing generating boundary condition was used
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following the approach in Dodd (1998); Kobayashi and Wurjanto (1989). This means that
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waves enter the domain at this point, but at the same time any reflected wave from the
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structure can leave the domain, without any additional effect on the processes. In addition to
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this, a transmissive boundary is present on the landward side at a distance behind the crest,
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which means that waves that have overtopped the structure can leave the domain.
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These tests will be referred to as the Reconstructed Offshore Boundary Conditions (ROBC).
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To allow a statistically relevant comparison of the variability in the results due to the
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reconstructed time series, 500 ROBC tests were carried out for each wave spectra measured
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in the laboratory.
. To allow this time series to be used, a
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In order to measure the wave overtopping in the numerical model, a virtual wave gauge was
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located at the crest of the structure. Here, h and U were used to measure each overtopping
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event. The individual overtopping volume was computed by integrating in time the discharge
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of each event Q=hU, for the duration of the event itself.
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Empirical Methods
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As previously mentioned, the probability distribution of individual overtopping volumes
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(Vov) has been investigated by a number of researchers. All found that the probability
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distribution of Vov can be described by a two parameter Weibull distribution:
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where Pv is the probability of the overtopping volume per wave of
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similar to Vov. The scale factor, a, and the shape factor, b, vary depending on the empirical
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method chosen which is selected based on the exact conditions present. Details of the values
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of a and b for the different empirical methods can be found in Table 2. q is the average
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overtopping discharge for each test, Pov is the probability of overtopping and Tm is the mean
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wave period, α is the angle of the sloped structure and H1/10 is the mean of the largest 10% of
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the waves.
being greater than or
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RESULTS
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Before the analysis of individual volumes distributions is presented, the wave height
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distribution at the toe of the structure is analysed as this has an important role in the Vov
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distribution (Nørgaard, Lykke Andersen, and Burcharth, 2014). The results presented here are
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focussed on Test 006 as representative of a medium level of overtopping, where occasionally
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waves overtop the structure and produces a dimensionless discharge in the order of 10-5. Test
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007 is used as representative of the higher overtopping level tests, with consistent
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overtopping waves resulting in a dimensionless discharge in the order of 10-3.
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Wave Distribution at the Toe of the Structure
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In deep water, wave heights generally follow a Rayleigh distribution. In shallow water this
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approximation can no longer be assumed to be valid. Battjes and Groenendijk (2000) looked
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at numerous wave height distributions on shallow foreshores and found a composite Rayleigh
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and Weibull distribution is suitable to model these conditions.
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In the present tests the conditions at the toe of the structure are depth limited as the values of
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Hm0/dt in Table 2 indicate, this means that the Battjes and Groenendijk (2000) distribution is 6
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expected to provide a better match to the wave height distribution than the Rayleigh
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distribution.
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Incident MOBC
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The wave height distributions measured at WG7 for Tests 006 and 007 during the calibration
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tests, referred to as Incident MOBC are shown in Figure 2. Note that the individual wave
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heights, H, have been normalised with the mean wave height Hm. The measured wave height
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distributions are compared with both the Battjes and Groenendijk (2000) distribution and the
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Rayleigh distribution.
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Panel (a) shows the wave height distribution for Test 006. Here the waves in the tests follow
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well the Battjes and Groenendijk (2000) distribution. Panel (b) looks at the measured wave
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height distribution for Test 007. The data follows reasonably well the Battjes and
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Groenendijk (2000) distribution as well confirming that the waves at the toe of the structure
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in the physical model are subject to shallow water conditions.
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ROBC Numerical Tests
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Figure 2 also considers the wave height distributions modelled in the ROBC tests. Panel (a)
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shows the distributions of 10 randomly selected ROBC runs of Test 006. It is clear from this
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graph that there is a large variability in the wave height distribution across these test runs.
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Some of the time series appear to better follow the Battjes and Groenendijk (2000)
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distribution, while others appear to better follow the Rayleigh distribution. However, the
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empirical distributions themselves generally fall between the two distributions. In all cases,
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similar to the incident MOBC tests, the smaller wave heights are better described by the
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distributions than the larger wave heights.
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In Panel (b), 10 randomly selected ROBC runs from Test 007 are considered. These runs
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show a similar level of variability to Test 006. This time none of the time series follow a
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similar shape to the Battjes and Groenendijk (2000) distribution, with most of them more
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closely described by the Rayleigh distribution. Again, most of the results lay between the two
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distributions in the higher wave heights, with none matching that measured in the physical
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experiments.
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Individual Overtopping Volume Distribution
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For irregular waves the quantity of water that overtops a structure will vary from wave to
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wave, this can be described by probability distributions as seen earlier in this paper. The
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various empirical prediction methods mentioned earlier will each produce a distribution of
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Vov which will be compared with the results from the numerical tests.
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To quantify this similarity between prediction methods, a two sample Kolmogorov-Smirnov
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(K-S) test has been carried out. The null hypothesis is deemed to be true if the two
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distributions being compared match. The test provides a statistic, Dn, defined as the
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maximum absolute difference between the two considered distributions. Γ is the test decision
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for the null hypothesis and it is equal to 0 if the null hypothesis is accepted and 1 otherwise.
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MOBC Numerical Tests
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To assess the accuracy of the NLSWE solver, the distribution of the individual volumes from
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the physical model is compared with the results produced from the MOBC tests. These are
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plotted in the form of cumulative distribution functions in Figure 3, which show that the
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numerical model provides similar results to those observed in the physical model in both of
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the tests considered. Both tests indicate that the MOBC shows slightly larger individual
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volumes present than in the physical model. The higher overtopping test shows a greater
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number of overtopping events in the MOBC than the physical model, this is most likely
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caused by the increased accuracy in the measurement of overtopping in the numerical model,
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which cannot be achieved with the physical model.
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Table 3 shows the results of the K-S test for the two test cases considered. This test confirms
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that the distribution of the volumes in the MOBC tests matches that of the physical model for
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both test cases, by the achievement of the null hypothesis and therefore confirms that the
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individual volumes can be well modelled by the NLSWE solver when MOBC are used.
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The results from the MOBC tests are also compared with the empirical formulae. In Panel (a)
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of Figure 3, the Vov results from the MOBC run of Test 006 are compared. As the wave
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conditions at the toe are the same as those in the physical model, the Nørgaard, Lykke
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Andersen, and Burcharth (2014) does not provide a modification in this case.
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Both the Pullen et al. (2007) and Victor, van der Meer, and Troch (2012) prediction methods
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provide a reasonable approximation to the results with the latter giving the best results. The
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accuracy is quantified using the K-S test shown in Table 3, which shows that although both
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formulae produce the null hypothesis, the Dn value for Victor, van der Meer, and Troch
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(2012) is smaller.
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In Panel (b), the Vov results from the MOBC run of Test 007 are compared with the empirical
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formulae. Both of the formulae appear to give a reasonable approximation of the distribution
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of the MOBC tests with the Victor, van der Meer, and Troch (2012) method providing the
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closest match. This time the null hypothesis is not obtained for the Pullen et al. (2007).
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ROBC Numerical Tests
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Now we consider the results of the ROBC tests to examine the variability of the Vov
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distributions. We already saw that the incident wave height distribution showed large
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variability among tests. In Figure 4, the ROBC results are plotted as cumulative distribution
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functions. It can be seen that the medium overtopping tests (004, 005 and 006) show a large
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variability in the distribution of the overtopping volumes. The variability in the higher level
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of overtopping (001, 007 and 008) is lower, with a narrower band of results shown on the
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graphs. This agrees with the observation in Williams, Briganti, and Pullen (2014) that the
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variability is related to the level of overtopping.
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The three empirical prediction methods have also been compared to these results, for this the
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values used have been taken from the mean values obtained from all of the ROBC tests. For
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the moderate levels of overtopping it can be seen that all three methods provide results
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approximately in the centre of those observed in the ROBC tests. To test how well the ROBC
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results produce a Weibull distribution, the three different empirical methods have been
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compared against all of the ROBC distributions using the K-S test. The percentage of the 500
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ROBC tests for each condition that achieved the null hypothesis for each of the empirical
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methods can be seen in Table 4. It was found that the Victor, van der Meer, and Troch (2012)
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formula provided the best match in all three of the moderate test conditions with the highest
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percentages of the ROBC runs achieving the null hypothesis.
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In addition the ROBC distributions have been compared to see if the shape factor for each
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test (bt) is smaller or larger than the value of b for the empirical prediction. The shape factor
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is a function of both kurtosis (peakedness) and skewness, meaning that a higher value
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represents a narrower distribution with positive skewness (i.e. a larger number of smaller
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volumes are predicted). It can be seen in Table 4 that compared with the Pullen et al. (2007)
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the shape factor for the ROBC distributions are generally larger, whilst with regard to the
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Victor, van der Meer, and Troch (2012) formula they are fairly evenly distributed either side,
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and with the Nørgaard, Lykke Andersen, and Burcharth (2014) formula the ROBC
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distributions are generally smaller.
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To examine this distribution of Vov in the ROBC in more detail to establish why some of the
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tests follow the empirical distributions, Figure 5, (a) looks at the results from the 10 randomly
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selected runs of the ROBC considered earlier for test 006 (moderate overtopping). It can be
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seen here that a large variability is still present, although as expected some of the empirical
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distributions closely match the empirical formulae. This is quantified in Table 5 which shows
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the results of a K-S test comparing these runs with the empirical formulae. It can be seen that 9
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seven of these tests obtain the null hypothesis. If we consider the specific tests that achieved
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the null hypothesis, we can see that it is those that produced a wave height distribution at the
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toe of the structure closest to the Battjes and Groenendijk (2000) distribution (Figure 2, Panel
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(a)).
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In the higher overtopping tests (See Figure 4), there is less variability in the individual
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overtopping volumes of each of the test runs. The three prediction methods have been
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compared to the results using test 007 as a representative test (Figure 5, Panel (b)). This time
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none of the formulae provide a good match for the distribution regardless of the wave height
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distribution of the selected tests. In Table 4 it can also be seen that the shape factors for the
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ROBC distributions are larger than the Pullen et al. (2007) and Victor, van der Meer, and
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Troch (2012) formula, but smaller than that given by the Nørgaard, Lykke Andersen, and
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Burcharth (2014) formula. The K-S tests also confirm that none of the empirical methods
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match the ROBC results (See Table 5).
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DISCUSSION
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The comparison between the physical model and the MOBC tests shows good agreement. In
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both cases the distribution of the individual overtopping volumes can be best modelled by the
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Victor, van der Meer, and Troch (2012) methods for both of the levels of overtopping. This is
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as expected due to the steep geometry of the slope in these experiments which is similar to
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those used in Victor, van der Meer, and Troch (2012).
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The variability of the overtopping volume distribution in numerical models was studied; it is
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found that when reconstructed offshore η time series from energy density spectra are used,
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the seeding has a significant effect on the distribution of Vov. It is possible for different
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distributions to be produced from the same incident spectra, some of which are shown to
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significantly diverge from the Weibull distributions usually used for overtopping analysis.
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The lower level of overtopping produced more variation between the numerical model runs
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as expected.
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It is noted that in the moderate overtopping conditions, the numerical model can over or
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underestimate the probability of higher individual volumes with respect to the empirical
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formulae. This should be taken into account when using a NLSWE solver at this level of
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overtopping, as a single run could produce an over or underestimate of the overtopping.
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For the moderate level of overtopping the most suitable empirical method for predicting the
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numerical results was that of Victor, van der Meer, and Troch (2012), which showed
10
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agreement up to 98%. This was expected due to similarities in the conditions present with
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those used in Victor, van der Meer, and Troch (2012) and the based on the physical model
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results; however it should also be noted that in certain conditions up to 25% of the results do
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not achieve the null hypothesis. The accuracy compared with the other empirical methods is
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significantly lower, with far fewer tests obtaining the null hypothesis.
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The accuracy of the prediction of the distribution of the Vov in the moderate level of
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overtopping (
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reconstructed time series matches the shallow water wave height distribution, i.e. those
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ROBC tests showing a better match to the Battjes and Groenendijk (2000) wave height
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distribution found in the experiments, also provides a better match to the Vov distribution.
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For higher levels of overtopping none of the suggested distributions have been found to
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produce a good match to the ROBC tests. This is due to the difficulty in accurately
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reconstructing the wave height distribution at the toe of the structure due to the larger waves
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that are present rather than the resulting higher overtopping of these conditions.
) ROBC tests was found to be dependent on how closely the
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CONCLUSIONS
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This work has highlighted two issues in the prediction of Vov using a NLSWE model,
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resulting in the requirement to carry out more than a single numerical test to justify a design.
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Firstly, it has been shown that the variability of the distribution of Vov decreases with the
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increasing level of overtopping considered. This is consistent with the findings of Williams,
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Briganti, and Pullen (2014) about q, which lead to the recommendation to carry out a
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sensitivity analysis for conditions with a value of Pov
