Uncertainties of wave overtopping of coastal structures

Ines Corne

Uncertainties of wave overtopping of coastal structures

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Peter TrochDepartment of Civil Engineering

Master of Science in Civil EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: David Gallach SanchezSupervisor: Prof. dr. ir. Andreas Kortenhaus

I

Permission for use on loan

“The author gives permission to make this master dissertation available for consultation and to copy

parts of this master dissertation for personal use.

In the case of any other uses, the copyright terms have to be respected, in particular with regard to

the obligation to state expressly the source when quoting results from this master dissertation.”

Ines Corne, June 2015

II

Uncertainties of Wave Overtopping of Coastal

Structures Ines Corne

Master’s dissertation submitted in order to obtain the academic degree of

Master of Science in Civil Engineering

Academic year 2014-2015

Supervisor: Prof. dr. ir. Andreas Kortenhaus

Department of Civil Engineering

Chairman: Prof. dr. ir. Peter Troch

Abstract

Coastal structures are designed to protect coastal regions against wave attack, storm surges,

flooding and erosion. Due to the climate changes, the sea level is rising and more severe

storms occur (see Carter et al., 1988). This emphasizes the importance of the design of these

protective structures. The amount of sea water transported over the crest of a coastal

structure, referred to as ‘wave overtopping’, is a critical design factor in that context. The

European Manual for the Assessment of Wave Overtopping (“EurOtop”) gives guidance on

analysis and/or prediction of wave overtopping for flood defences attacked by wave action.

The prediction models for overtopping are empirical based on physical model data. Hence

inherent scatter has to be taken into account, this scatter can be seen as the reliability of the

equations. Reliable overtopping prediction methods are indispensable to provide safety of

densely populated coastal regions. Increased attention to flood risk reduction and to wave

overtopping in particular, have increased interest and research in this area. As a result,

sufficient new research results on the subject are available today to justify a revision of the

current manual. The main goal of this master’s thesis is to update the uncertainties of the

prediction models in the EurOtop (2007) manual. The influence of new data collected at the

University of Ghent on the uncertainties of the EurOtop (2007) models is examined first.

Following, it is investigated whether the revised formulae by Van der Meer and Bruce (2014)

improve the uncertainties in the prediction.

Key words wave overtopping; sloping structures; vertical structures; EurOtop manual;

uncertainties.

III

PREFACE

Due to the climate changes, the sea level is rising and more severe storms occur (see Carter

et al., 1988). This emphasises the importance of protective coastal structures. The amount of

sea water transported over the crest of a coastal structure, referred to as ‘wave overtopping’,

is a critical design factor. Reliable prediction methods are indispensable to provide safety of

densely populated regions.

The overtopping manual, EurOtop (2007), gives guidance on analysis and/or prediction of

wave overtopping for flood defences attacked by wave attack. The manual is now used

worldwide. However, given the timeliness of the subject and the lack of reliable overtopping

models, a lot of research is still going on. As a result, sufficient new research results on the

subject are available to justify a revision of the current manual.

In this report, the influence of more recent data collected at the University of Ghent on the

uncertainties in the prediction is examined. Then it also checked whether the revised formulae

by Van der Meer and Bruce (2014) improve the reliability of the prediction.

Knowing the practical purpose of my work was a great motivator and made it also very

interesting. I am glad to have had the opportunity to be a part of it and I hope I have been able

to contribute to the revision of the EurOtop (2007) manual through my work. Furthermore, I

am convinced that I developed skills that will help me in my further professional life.

I would like to thank my promotor Prof. dr. ir. Andreas Kortenhaus who supervised and guided

me with great knowledge through the past months.

I would also like to thank my parents for supporting me through my studies.

IV

Uncertainties of Wave Overtopping of Coastal

Structures

Ines Corne

Supervisor: Prof. dr. ir. Andreas Kortenhaus

Abstract: This article synthesizes a master thesis in which the uncertainties on wave overtopping are

studied. The prediction models of the EurOtop (2007) manual as well as the more recent formulae by

Van der Meer and Bruce (2014) are examined considering data of the CLASH database and the UG

datasets.

Author keywords: wave overtopping; sloping structures; vertical structures; EurOtop manual;

uncertainties.

I. Introduction

Coastal structures are designed to protect coastal regions against wave attack, storm surges, flooding and erosion. Due to climate changes, the sea level is rising and more severe storms occur (see Carter et al., 1988). This emphasizes the importance of the design of these protective structures. The amount of sea water transported over the crest of a coastal structure, referred to as ‘wave overtopping’, is a critical design factor in this context. [1]

The European Manual for the Assessment of Wave Overtopping (“EurOtop”) gives guidance on analysis and/or prediction of wave overtopping for flood defences attacked by wave action [1, 2].

The prediction methods in the manual are empirical equations derived from physical model data. Hence inherent scatter has to be taken into account, this scatter can be seen as the reliability of the equations. This scatter has been described by statistical distributions for the parameters occurring in the models.

In the EurOtop (2007) manual there is each time one parameter that is assumed to be stochastic and normally distributed, this parameter is then described by a mean and a standard deviation. Its standard deviation is an indicator of the reliability of the considered formula.

These empirical formulae typically describe a relation between a relative overtopping discharge Q∗ and a relative freeboard Rc

∗ .

Figure 1: Mean value approach and confidence band

Two different approaches can be considered: 1) Deterministic approach, where output corresponds to mean values plus one standard deviation and 2) Mean value approach, where output is exceeded by 50% of all results. In this report, the considered prediction formulae give the average overtopping in accordance to the mean value approach. The uncertainties are then typically presented by a confidence band in the corresponding plots. The mean value approach and its associated confidence band is illustrated in Figure 1.

The data used to derive the EurOtop (2007) formulae and their uncertainties are included in the CLASH database. The CLASH database exists out of data for more than 10,000 test results of wave overtopping tests with vast ranges of geometries and wave characteristics.

V

II. Objectives

It is clear that increased attention to flood risk reduction, and to wave overtopping in particular, have increased interest and research in this area. As a result, there is a lot of new research results available today. This justifies a revision of the current manual. [2]

The main goal of this master dissertation is to update the uncertainties in the EurOtop (2007) manual in the context of its revision.

The following research questions are used as guidance:

1) Is the EurOtop (2007) approach used so far still valid today?

2) Is there an update needed for ‘probabilistic’

and ‘deterministic’ parameters due to more

and new data and due to modified methods?

3) Is the assumption of normally distributed

parameters valid or do we need adjustments

here?

III. Methodology

First a literature survey on wave overtopping is set up. As well the prediction models as the overtopping data used to derive these models are discussed. The focus for the prediction models is on the overtopping formulae given in the EurOtop (2007) manual and the revised formulae by Van der Meer and Bruce (2014). Further, the CLASH database as well as the UG datasets are examined.

The structures considered here with governing overtopping equations are sloping structures and vertical structures. The overtopping equations exist out of pairs of formulae, depending on the predicted regime, breaking waves and non-breaking waves.

For each type of structure, the uncertainties of the EurOtop (2007) formulae are derived first considering data from the CLASH database. For sloping structures, three different datasets are used: 1) Simple, smooth sloping structures; 2) Smooth, sloping structures and 3) Sloping structures.

For vertical structures, two different datasets are considered: 1) Plain vertical walls and 2) Plain and composite vertical walls.

Then, the database is widened with the UG datasets and the uncertainties. The UG data have no mounds or roughness elements and can thus be added to the first datasets considered for both sloping structures and vertical structures, i.e. respectively simple, smooth sloping structures and plain vertical walls. The influence of including the UG data to these first datasets on the uncertainties is examined.

Finally, the uncertainties for the more recent formulae by Van der Meer and Bruce (2014) are derived using the same datasets considered before from CLASH together with the UG data. It is investigated whether they are more reliable to predict overtopping.

As mentioned before, the uncertainties are described by a standard deviation on a stochastic parameter. This standard deviation is obtained by determining the value of the other parameter occurring in the formulae by a trend line first. This parameter is further assumed constant. The values for the stochastic parameter are then calculated for each data point based on the value for the other parameter from the trend line. The mean value and the standard deviation of the stochastic parameter can then be calculated as well. The relative standard deviation is also calculated, this allows for better comparison in between the results of different formulae.

Next to these values, other means are used in this master thesis to analyse the reliability of the formulae. First, histograms are used to check for the assumption of a normal distribution. Then also measured against predicted overtopping plots are used as an additional check for the reliability of the considered formula.

V. Prediction models

Sloping structures

The EurOtop (2007) formulae are of the exponential type for sloping structures:

Q∗ = a ∙ exp[−(b ∙ Rc∗ )] 1

VI

with Q∗ the relative overtopping discharge and Rc

∗ the relative freeboard made nondimensional according to EurOtop (2007) and depending on breaking or non-breaking conditions, the parameters a and b fitted coefficients. The reliability of the equations is described by a standard deviation on the parameter b. Exponential equations give a straight line in a log-linear graph.

The more recent formulae by Van der Meer and Bruce (2014) are of the Weibull type which give a curved line in a log-linear graph:

Q∗ = a ∙ exp(−(b ∙ Rc∗)c) 2

with the coefficient c a constant. The relative overtopping discharge Q∗ and the relative freeboard Rc

∗ are nondimensionalized in the same way as in EurOtop (2007).

The formulae by Van der Meer and Bruce should fit better for small freeboards whereas the EurOtop (2007) formulae over predict.

Vertical structures

The EurOtop (2007) formula in the non-impulsive regime for vertical structures is again of the exponential type. The formulae in the impulsive regime, however, is of the power law type:

Q∗ = a ∙ Rc∗ (−b) 3

The relative overtopping discharge Q∗ and the relative freeboard Rc

∗ are nondimensionalized differently here.

The scatter in the logarithm of the data is described by a standard deviation on the parameter a. Power law equations give a curved line in a log-linear graph.

A different approach needs to be followed for composite vertical walls than for plain vertical walls. A vertical wall is treated as composite only if the mound is significant.

The more recent formulae by Van der Meer and Bruce (2014) are again either of the exponential type or of the power law type.

For the new formulae an additional distinction is made between structures with or without a foreshore. Also, a different formulae is

suggested each time for small relative freeboards compared to larger relative freeboards.

VI. Main results

Sloping structures

The relative standard deviations obtained for each dataset are on average 5% larger than the relative standard deviations given in the EurOtop (2007) manual. Furthermore, the relative standard deviations are consistently larger for non-breaking waves. The latter is also observed in the EurOtop (2007) manual.

For breaking waves, including structures with berms, increases the relative standard deviations the most. The difference is, however, not significant.

For non-breaking waves, including rough slopes, leads to a significant increase of the amount of data and a lot of scatter in the plots (Figure 2). Correspondingly the relative standard deviation has increased considerably.

Figure 2: Scatter due to rough slopes, non-breaking waves

The UG data increases the relative standard deviation for both regimes. The effect is the largest for non-breaking waves are the UG data generally have steeper slopes. The scatter in the associated plots has not increased considerably due to the UG data (Figure 3).

Note that these points are located below the EurOtop (2007) curve, indicating an over prediction.

VII

Figure 3: CLASH and UG data for non-breaking waves

The reason for the increased relative standard deviation when UG data are included, is the increased amount of data with small relative freeboards.

In our approach, the parameter a in the exponential equation is determined by a trend line and is further assumed constant. The parameter a in an exponential equations represents the intersection point with the relative discharge axis. The required slopes or the values of parameter b to go through each data point are then calculated with a fixed parameter a. As a consequence, the deviations for the parameter b are the largest for small relative freeboards. This effect is illustrated in Figure 4.

Figure 4: Effect of fixing parameter a on parameter b

The formulae by Van der Meer and Bruce (2014) fit better for small relative freeboards. The derived relative standard deviations are the same order of magnitude as the ones obtained for the different datasets from CLASH while the UG data is included too this time in the analysis.

All the histograms show more or less a bell-shaped curve. Therefore, the assumption of a normal distribution seems acceptable at first

sight. The histogram for the dataset considering only simple, smooth sloping structures for breaking waves is given as an example in Figure 5.

Figure 5: Histogram simple, smooth sloping structures for breaking waves

Vertical structures

For vertical structures, there is less data available in the CLASH database resulting in smaller datasets.

The relative standard deviation in the non-impulsive regime considering only plain vertical walls (no berm or toe) is only two third of the relative standard deviation indicated in the EurOtop (2007) manual. When, however, the composite vertical walls are added to the analysis, the resulting standard deviation is the same order of magnitude as the one of EurOtop (2007).

In the non-impulsive regime, the relative standard deviation is significantly larger than all previously obtained relative standard deviation. This gives the impression that the power law equation is not very reliable.

The UG data increase the relative standard deviation in the non-impulsive regime for the same reason as for sloping structures: the UG data increase the amount of data with small relative freeboards.

The formulae by Van der Meer and Bruce (2014) do not give clear improvements. The considered datasets are also too small to be able to see clear changes.

Most of the histograms, again, show more or less a bell-shaped curve. There are, however, some cases with a distribution more to the right.

VIII

VII. Conclusions

Sloping structures

As the uncertainties, even if we are only considering simple, smooth sloping structures, are larger than the uncertainties in the EurOtop (2007) formulae, a revision of these uncertainties is recommended for these formulae.

Besides, adding rough slopes, leads to significant scatter for non-breaking waves. This makes us conclude that either these data are less reliable or that the roughness factor is not determined right.

The EurOtop (2007) formulae fit less good for small relative freeboards. The formulae by Van der Meer and Bruce (2014) fit better over a larger range of relative freeboards and are therefore recommended.

Vertical structures

For vertical structures, a lot of more data is needed in the first place to be able to make conclusions.

Further, it is recommended to reconsider the power law formulae and the way their reliability is expressed as the relative standard deviations are very large now compared to all others results.

Finally, the formulae by Van der Meer and Bruce (2014) do not give a clear improvement.

References

[1] Verhaeghe, H. (2005): Neural Network

Prediction of Wave Overtopping at Coastal

Structures, PhD, University of Ghent,

Promotor Prof. dr. ir. Julien De Rouck

[2] European Overtopping Manual. Die Küste.

Archiv für Forschung und Technik an der

Nord- und Ostsee, vol. 73, Pullen, T.; Allsop,

N.W.H.; Bruce, T.; Kortenhaus, A.;

Schüttrumpf, H.; Van der Meer, J.W.,

www.overtopping-manual.com.

http://www.overtopping-manual.com/

IX

Table of contents

Chapter 1: Introduction ........................................................................................................................... 1

1.1 Background .................................................................................................................................... 1

1.2 Definition of wave overtopping..................................................................................................... 1

1.3 EurOtop manual ............................................................................................................................ 2

1.4 CLASH database ............................................................................................................................. 4

1.5 Objectives ...................................................................................................................................... 5

1.6 Methodology ................................................................................................................................. 5

Chapter 2: Literature ............................................................................................................................... 7

2.1 Introduction ................................................................................................................................... 7

2.2 Prediction of overtopping ............................................................................................................. 7

2.2.1 Introduction ............................................................................................................................ 7

2.2.2 EurOtop (2007) ....................................................................................................................... 7

2.2.3 Van der Meer and Bruce (2014) ........................................................................................... 15

2.2.4 Uncertainty of the prediction ............................................................................................... 20

2.3 Overtopping data ........................................................................................................................ 21

2.3.1 CLASH database .................................................................................................................... 21

2.3.2 UG data ................................................................................................................................. 28

2.4 Conclusions .................................................................................................................................. 29

Chapter 3: Sloping structures ................................................................................................................ 31

3.1 EurOtop (2007) formulae applied to CLASH data........................................................................ 31

3.1.1 Introduction .......................................................................................................................... 31

3.1.2 Calculation procedure .......................................................................................................... 32

3.1.3 Filtering of data .................................................................................................................... 37

3.1.4 Uncertainty analysis ............................................................................................................. 44

3.2 EurOtop (2007) formulae applied to CLASH and UG data ........................................................... 49

3.2.1 Introduction .......................................................................................................................... 49




3.3 Van der Meer and Bruce (2014) formulae applied to CLASH and UG data ................................. 54

3.3.1 Introduction .......................................................................................................................... 54




3.4 Summary results .......................................................................................................................... 58

3.5 Discussion .................................................................................................................................... 59

X

3.5.1 Local wave length ................................................................................................................. 59

3.5.2 Distinction regime ................................................................................................................ 61

3.5.3 Uncertainty analysis approach ............................................................................................. 61

3.5.4 Influence factor for roughness ............................................................................................. 62

3.5.5 Normality tests ..................................................................................................................... 62

Chapter 4: Vertical structures ............................................................................................................... 65

4.1 EurOtop (2007) formulae applied to CLASH data........................................................................ 65

4.1.1 Introduction .......................................................................................................................... 65




4.2 EurOtop (2007) formulae applied to CLASH and UG data ........................................................... 76

4.2.1 Introduction .......................................................................................................................... 76




4.3 Van der Meer and Bruce (2014) formulae applied to CLASH and UG data ................................. 78

4.3.1 Introduction .......................................................................................................................... 78




4.4 Summary results .......................................................................................................................... 87

4.5 Discussion .................................................................................................................................... 87

4.5.1 Uncertainty analysis approach ............................................................................................. 87

4.5.2 Normality tests ..................................................................................................................... 88

Chapter 5: General conclusions ............................................................................................................. 91

5.1 Summary...................................................................................................................................... 91

5.2 General conclusions .................................................................................................................... 92

5.2.1 Sloping structures ................................................................................................................. 92

5.2.2 Vertical structures ................................................................................................................ 93

5.2.3 General remarks ................................................................................................................... 93

References ............................................................................................................................................. 95

Appendix A - Sloping structures Appendix B - Vertical structures

XI

List of figures

Figure 1. 1 Definition of wave overtopping at coastal structures [2]...................................................... 2

Figure 1. 2 EurOtop manual (2007) [2] .................................................................................................... 3

Figure 1. 3: EurOtop (2007) approach [2] ............................................................................................... 4

Figure 2. 1 Breaking versus non-breaking waves on a slope [1] ............................................................. 8

Figure 2. 2 Definition of angle of wave attack β [2] .............................................................................. 10

Figure 2. 3 New formulae scheme [3] ................................................................................................... 20

Figure 2. 4 Determination of B [m], Bh [m], tanαb [-], hb [m] [1]........................................................ 27

Figure 2. 5 Determination of Rc [m], Ac [m] and Gc [m] [1] ................................................................. 28

Figure 2. 6 Determination of the structure slope parameters [1] ........................................................ 28

Figure 3. 1 Wave overtopping data and mean value approach with its confidence band [10] ............ 31

Figure 3. 2 Interdependencies calculation procedure .......................................................................... 33

Figure 3. 3 One-way calculation procedure .......................................................................................... 33

Figure 3. 4 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with its

5% under and upper exceedance limits, effect of CF ............................................................................ 38

Figure 3. 5 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with its

5% under and upper exceedance limits, effect of RF ............................................................................ 39

Figure 3. 6 Wave overtopping data for sloping structures, non-breaking waves and equation 3.4

with its 5% under and upper exceedance limits ................................................................................... 40

Figure 3. 7 Wave overtopping data for sloping structures, non-breaking waves and equation 3.4

with its 5% under and upper exceedance limits, effect of rough slopes .............................................. 41

Figure 3. 8 Wave overtopping data for smooth sloping structures, breaking waves and equation 3.3

with its 5% under and upper exceedance limits, effect of berms ......................................................... 42

Figure 3. 9 Wave overtopping data for simple, smooth sloping structures, breaking waves and

EurOtop (2007) data, equation 3.3 with its 5% under and upper exceedance limits ........................... 43

Figure 3. 10 Wave overtopping data for simple, smooth sloping structures, non-breaking waves

and EurOtop (2007) data, equation 3.4 with its 5% under and upper exceedance limits .................... 43

Figure 3. 11 Filtering scheme sloping structures ................................................................................... 44

Figure 3. 12 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with

its 5% under and upper exceedance limits, effect of rough slopes ...................................................... 47

Figure 3. 13 Histogram ∆b’ for sloping structures, breaking waves with its mean value and 90%

confidence interval ................................................................................................................................ 47

Figure 3. 14 Measured against predicted relative overtopping for simple, smooth sloping

structures, breaking waves ................................................................................................................... 48

Figure 3. 15 Wave overtopping data for simple, smooth sloping structures, CLASH and UG,

breaking waves and equation 3.3 with its 5% under and upper exceedance limits ............................. 51

Figure 3. 16 Wave overtopping data for simple, smooth sloping structures, CLASH and UG, non-

breaking waves and equation 3.4 with its 5% under and upper exceedance limits ............................. 51

Figure 3. 17 Measured vs calculated relative overtopping for simple, smooth sloping structures,

CLASH and UG, breaking waves ............................................................................................................ 53

Figure 3. 18 Wave overtopping data for simple, smooth sloping structures, CLASH and UG data,

breaking waves, comparison equation 3.3 and equation 3.15 ............................................................. 55

Figure 3. 19 Measured vs predicted relative overtopping for simple, smooth sloping structures,

CLASH and UG, breaking waves ............................................................................................................ 58

XII

Figure 3. 20 Wave overtopping data for simple, smooth sloping structures, breaking waves and

equation 3.3 with its 5% under and upper exceedance limits, comparison deep water and local wave

length………….......................................................................................................................................... 61

Figure 4. 1 Wave overtopping data for plain vertical walls in the impulsive regime and the EurOtop

(2007) equation with its 5% under and upper exceedance limits [2] ................................................... 65

Figure 4. 2 Final filtering scheme .......................................................................................................... 69

Figure 4. 3 Wave overtopping data for composite vertical walls in the impulsive regime and

equation 4.7 with its 5% under and upper exceedance limits, dataset 505-__ highlighted ................. 70

Figure 4. 4 Wave overtopping data for vertical walls in the non-impulsive regime and equation 4.5

with its 5% under and upper exceedance limits, scatter highlighted ................................................... 71

Figure 4. 5 Histogram ∆b’ for the dataset containing plain and composite vertical walls in the non-

impulsive regime with its mean value and 90% confidence interval .................................................... 74

Figure 4. 6 Measured vs predicted relative overtopping for the dataset containing plain and

composite vertical walls in the non-impulsive regime, under prediction highlighted .......................... 75

Figure 4. 7 Wave overtopping data for plain vertical walls, CLASH and UG data, in the non-impulsive

regime with equation 4.1 with its 5% under and upper exceedance limits .......................................... 77

Figure 4. 8 Decision chart new formulae .............................................................................................. 80

Figure 4. 9 Wave overtopping data for vertical structures with a foreshore in the impulsive regime

with the corresponding equations 4.13 and 4.14 ................................................................................. 81

Figure 4. 10 Wave overtopping data for vertical structures with a foreshore in the non-impulsive

regime and equation 4.10 ..................................................................................................................... 82

Figure 4. 11 Wave overtopping data for structures without a foreshore and composite vertical

structures with a foreshore in the non-impulsive regime and the corresponding equations 4.10 and

4.11…………………..................................................................................................................................... 83

Figure 4. 12 Wave overtopping data for composite vertical structures with a foreshore in the

impulsive regime and the corresponding equations 4.15 and 4.16 ...................................................... 84

Figure 4. 13 Effect of the fixed value of parameter a on the uncertainty of parameter b ................... 88

1

Chapter 1: Introduction

1.1 Background

Coastal structures are designed to protect coastal regions against wave attack, storm surges, flooding

and erosion. Due to climate changes, the sea level is rising and more severe storms occur (see Carter

et al., 1988). This emphasizes the importance of the design of these protective structures. The amount

of sea water transported over the crest of a coastal structure, referred to as ‘wave overtopping’, is a

critical design factor in this context. [1]

The design of coastal structures should lead to an ’acceptable’ overtopping amount. Which amount is

assessed as acceptable is revealed by socio-economic reasons. High crested coastal structures

preventing any overtopping are preferably avoided as these structures are extremely expensive.

Moreover, such structures impose visual obstructions where the broad view on the sea is an important

tourist attraction with an economic impact. However, the design of (lower crested) coastal structures

should provide safety for people and vehicles on the structure, and avoid structural damage as well as

damage to properties behind the structure. The preservation of the economical function of the

structure under bad weather conditions is also an important factor and has an additional influence on

the design. [1]

Hence a detailed knowledge of these volumes of water that may pass the coastal defence structures

under different wave conditions is required. To that end, different models were developed to predict

the amount of overtopping. Most frequently applied for structure design are empirical models, set up

based on laboratory overtopping measurements [1]. Empirical models are theoretical curves that

match the data as closely as possible. There will, however, always be some scatter or uncertainty in

the prediction.

1.2 Definition of wave overtopping

Wave overtopping occurs because of waves running up the face of a coastal structure. Hence

overtopping is related to the wave run-up as overtopping occurs when wave run-up levels are high

enough to reach and pass over the crest of the structure. [1, 2] This defines the ‘green water’

overtopping case where a continuous sheet of water passes over the crest [2].

2

Figure 1. 1 Definition of wave overtopping at coastal structures [2]

A second form of overtopping occurs when waves break on the seaward face of the structure and

produce significant volumes of splash. These droplets of water may then be carried over the structure

crest either under their own momentum or as a consequence of an onshore wind. [2]

Another less important method by which water may be carried over the crest is in the form of spray

generated by the action of wind on the wave crests immediately offshore of the wall. However, even

with strong wind the volume is not large and this spray will not contribute to any significant

overtopping volume. [2]

Overtopping rates predicted by the various empirical formulae described within this report will include

green water discharges and splash, since both were recorded during the model tests on which the

prediction methods are based. [2]

Two approaches to measuring and assessing wave overtopping at coastal structures can be

distinguished. The first approach considers the overtopping volume per overtopping wave. The second

and most applied approach considers mean overtopping discharges over certain time intervals and per

meter structure width, i.e. q in m³/s/m or l/s/m. The uneven distribution of overtopping in time and in

space caused by irregular wave action is the basic reason for the assessment of overtopping by means

of mean overtopping discharges. [1]

Within this report the mean overtopping discharge per meter run, q, expressed in m³/s/m, is used. This

corresponds to the most common approach to design coastal structures, i.e. based on mean

overtopping discharges. [1]

1.3 EurOtop manual

The European Manual for the Assessment of Wave Overtopping (“EurOtop”) gives guidance on analysis

and/or prediction of wave overtopping for flood defences attacked by wave action [1, 2]. The manual

was issued free on the internet in 2007 and is now used worldwide. It was the result of synthesis of

existing Dutch, UK and German guidance and new research findings arising out of projects such as the

EC FP7 “CLASH” project [2].

3

Figure 1. 2 EurOtop manual (2007) [2]

The manual has been intended to assist coastal engineers analyse overtopping performance of most

types of sea defence found around Europe. The manual defines types of structure, provides definitions

for parameters, and gives guidance on how results should be interpreted. Users may be concerned

with existing defences, or considering possible rehabilitation or new-build. [2]

The prediction models in the manual are empirical based on physical model data. Hence inherent

scatter has to be taken into account, this scatter can be seen as the reliability of the equations. This

scatter, or reliability of the equations, has been described by statistical distributions for the parameters

occurring in the models. These parameters then have a mean and a standard deviation. The reliability

of the formulae is described by a standard deviation on the presumed stochastic parameter. Two

approaches are classified with regard to exceedance probabilities:

Deterministic, where output corresponds to mean values plus one standard deviations and

Mean value or probabilistic, where output is exceeded by 50% of all results.

In Figure 1. 3 both approaches are illustrated. The mean value approach or probabilistic approach is

given together with its 5% under and upper exceedance limits or thus the 90% confidence interval. The

deterministic approach is given as well in which the the uncertainty of the prediction is included by

adding one standard deviation resulting in a more conservative prediction. In this report, only formulae

that give the average overtopping, in accordance with the mean value approach, are considered.

4

Figure 1. 3: EurOtop (2007) approach [2]

The CLASH database contains the physical model data used in the preparation of the manual.

The empirical formulae are typically only applicable for a certain type of structure. This is also the case

in the EurOtop (2007) manual where the last three chapters each deal with a different type of

structure: smooth sloping structures; rubble mound structures and vertical structures.

1.4 CLASH database

The project “CLASH” (Crest Level Assessment of coastal Structures by full scale monitoring, neural

network prediction and Hazard analysis on permissible wave overtopping), supported by the European

Commission, was intended to improve the knowledge on the phenomenon of overtopping [1].

The CLASH project, under contract no. EVK3-CT-2001-00058, ran from January 2002 until December

2004 (www.clash-eu.org) [1].

More than 10,000 test results of wave overtopping tests with vast ranges of geometries and wave

characteristics have been collected during the CLASH project (De Rouck and Geeraerts, 2005) and

gathered in the CLASH database (Van der Meer et al. 2005) [3].

One of the objectives within the CLASH project was to develop a generally applicable overtopping

prediction method based on many existing datasets gathered in a database on wave overtopping. This

objective required in a first phase the set-up of a database on existing overtopping information.

Besides gathering overtopping information, thorough screening of the data was carried out. [1]

http://www.clash-eu.org/

5

1.5 Objectives

It is clear that increased attention to flood risk reduction, and to wave overtopping in particular, have

increased interest and research in this area. As a result, sufficient new research results on the subject

are available today to justify a revision of the current manual. [2]

The University of Ghent is one of the partner universities working on this updated version of the

manual. The main goal of this master’s thesis is to update the uncertainties in the EurOtop (2007)

manual.

The following research questions are used as a guidance throughout this work:

1) Is the EurOtop (2007) approach used so far still valid today?

2) Is there an update needed for ‘probabilistic’ and ‘deterministic’ parameters due to more and

new data and due to modified methods?

3) Is the assumption of normally distributed parameters valid or do we need adjustments here?

1.6 Methodology

To meet the objectives mentioned in previous section, several steps are taken. An overall view of the

methodology and the contents of this thesis is presented here.

In a first phase, a literature survey on wave overtopping is set up. Therefore the EurOtop (2007) manual

has been read thoroughly, as well as other related materials. Different prediction models as well as

overtopping data are discussed. This summary of research performed on wave overtopping is

described in Chapter 2.

Chapter 3 treats sloping structures, this includes smooth sloping structures as well as rubble mound

structures (rough slopes). First, the uncertainties of the EurOtop (2007) formulae are derived for

different datasets from the CLASH database. Then, the influence of new data on the EurOtop (2007)

formulae and their uncertainties is examined. This concerns three datasets UG10, UG13 and UG14

collected at the University of Ghent. Finally, it is investigated whether the revised formulae by Van der

Meer and Bruce (2014) give more reliable predictions.

Chapter 4 deals with vertical structures in the same way as sloping structures are dealt with in Chapter

3.

Finally, in Chapter 5 a summary is given and the general conclusions of this thesis are compiled.

6

7

Chapter 2: Literature

2.1 Introduction

A literature survey on wave overtopping is set up in this chapter. Therefore the EurOtop (2007) manual

has been read thoroughly, as well as other related materials.

In the first part, different prediction models are explained. More specifically the EurOtop (2007)

overtopping formulae and the revised formulae by Van der Meer and Bruce (2014) are discussed. The

uncertainties in the prediction models are also briefly addressed at the end.

Then the experimental overtopping data used to derive these models are described. The CLASH

database and more in particular the various parameters in the database are discussed. Consecutively,

some more recent data collected at the University of Ghent is discussed (the UG datasets).

2.2 Prediction of overtopping

2.2.1 Introduction

An exact mathematical description of the wave overtopping process for coastal dikes or embankment

seawalls is not possible due to the stochastic nature of wave breaking and wave run-up and the various

factors influencing the wave overtopping process. Therefore, wave overtopping for coastal dikes and

embankment seawalls are mainly determined by empirical formulae. [2]

Within this report, wave overtopping is described by an average wave overtopping discharge q, which

is given in m³/s per m width [2] and is easy to measure in a laboratory wave flume or basin. The actual

process of wave overtopping is much more dynamic. Only large waves will reach the crest of the

structure and will overtop with a lot of water in a few seconds. [1]

First, the EurOtop (2007) prediction formulae are discussed in detail. Then the more recent formulae

by Van der Meer and Bruce (2014) are discussed. A short discussion of the uncertainties in the

prediction follows at the end.

A distinction is made each time between prediction models for sloping structures and prediction

models for vertical structures.

2.2.2 EurOtop (2007)

The EurOtop (2007) manual focuses on the research performed on simple regression models for

overtopping.

8

Two frequently appearing types of regression models are distinguished, i.e. [1]:

- exponential Q∗ = a ∙ exp[−(b ∙ Rc∗)]

- power law Q∗ = a ∙ Rc∗ (−b)

where Q∗ refers to a dimensionless mean overtopping discharge per meter structure width and Rc∗ to

a dimensionless crest freeboard. The parameters a and b are fitted coefficients.

The last three chapters of the EurOtop (2007) manual each deal with a different type of structure:

sloping structures; rubble mound structures and vertical structures. Since the prediction formulae for

rubble mound structures are practically the same as for sloping structures, they are further treated

together as one type of structure, sloping structures.

2.2.2.1 Sloping structures

Van der Meer (1993) developed an overtopping model of the exponential type for overtopping at

impermeable smooth slopes. Van der Meer (1993) finds that ‘plunging conditions’, corresponding to

waves breaking on a slope, show a different overtopping behaviour from ‘surging conditions’ or non-

breaking waves on a slope. [1]

Figure 2. 1 Breaking versus non-breaking waves on a slope [1]

Van der Meer (1993) proposes a set of two equations, both of the exponential type, which are

extended for application for rough and bermed slopes as well. The research performed on run-up and

overtopping by De Waal and Van der Meer (1992) is on the basis of the proposed overtopping model.

The data from Owen (1980) and Führboter et al. (1989) are also used. The original model proposed by

Van der Meer (1993) has been improved subsequently (see Van der Meer and Janssen, 1995, and TAW,

1997) resulting in the most recent form in TAW (2002). [1]

9

The TAW (2002) formulae are given in the EurOtop (2007) manual. These equations are only valid for

breaker parameters smaller than 5, i.e. ξm−1,0 < 5 [1, 2]:

q

√g∙Hm0³=

0.067

√tan α∙ γb ∙ ξm−1,0 ∙ exp (−4.75 ∙

Rc

ξm−1,0∙Hm0∙γb∙γf∙γβ∙γv) 2.1

with a maximum of q

√g∙Hm0³= 0.2 ∙ exp (−2.6 ∙

Rc

Hm0∙γf∙γβ) 2.2

with q the mean overtopping discharge [m³/m∙s];

Hm0 an estimate of the significant wave height from spectral analysis [m];

tan α the slope steepness, α the slope of the front face of the structure;

ξm−1,0 = tan α /(sm−1,0) 1/2 the breaker parameter;

sm−1,0 = Hm0 L0⁄ the wave steepness (ratio of wave height to wave length);

L0 =g∙Tm−1,0²

2π the deep water wave length;

Tm−1,0 the mean period from spectral analysis;

γb the influence factor for a berm;

γf the influence factor for roughness elements on a slope;

γβ the influence factor for oblique wave attack in case of run-up;

γv the influence factor for a vertical wall and

Rc the crest freeboard [m].

The TAW (2002) equations give the average overtopping in accordance with the mean value approach,

hence they are supposed to give the average of the measured data. The reliability of the equations is

described by taking the coefficients 4.75 and 2.6 as normally distributed stochastic parameters with

means of 4.75 and 2.6 and standard deviations σ = 0.5 and σ = 0.35 respectively. [1, 2]

The model proposed by TAW (2002) advises to use spectral wave parameters. For the significant wave

height the spectral value Hm0 and for the wave period the mean period Tm−1,0 is advised. These

parameters account for the shape of the wave spectrum in the best way. [1] Furthermore, it is

indicated that the parameters determined at the toe of the structure have to be used.

The influence factors account for additional influences on overtopping which were not considered in

the original model [1]. Their value is always smaller than or equal to 1, that is why they are also referred

to as reduction factors. The maximum occurring reduction is 60% which corresponds to a minimum

value of 0.4 for each possible combination of influence factors. This it not only applicable to the above

equations, but to any equations in which they occur.

10

Effect of oblique waves

The angle of wave attack β is defined at the toe of the structure after any transformation on the

foreshore by refraction or diffraction as the angle between the direction of the waves and the

perpendicular to the long axis of the dike or revetment. [2]

Figure 2. 2 Definition of angle of wave attack β [2]

Hence the direction of wave crests approaching parallel to the dike axis is defined as β = 0°

(perpendicular wave attack). The influence of the wave direction on the wave run-up or wave

overtopping is defined by an influence factor γβ. [2]

The influence factor for oblique wave attack γβ in the case of wave overtopping is determined by [2]:

γβ = 1 − 0.0033 ∙ |β| for 0° ≤ β ≤ 80° 2.3

γβ = 0.736 for β > 80° 2.4

In the case of wave run-up, it is determined by [2]:

γβ = 1 − 0.0022 ∙ |β| for 0° ≤ β ≤ 80° 2.5

γβ = 0.824 for β > 80° 2.6

Until now the effect of oblique waves on run-up and overtopping on smooth slopes (including some

roughness) has been discussed. For rough slopes, the influence factor for oblique wave attack γβ in

the case of wave overtopping is determined differently. The reduction is much faster with increasing

angle for rubble mound structures:

γβ = 1 − 0.0063 ∙ |β| for 0° ≤ β ≤ 80° 2.7

γβ = 0.496 for β > 80° 2.8

11

Effect of wave walls

In some cases a vertical or very steep wall is placed on the top of a slope to reduce wave overtopping.

Vertical walls on top of the slope are often adopted if the available place for an extension of the basis

of the structure is restricted. The knowledge about the influence of vertical or steep walls on wave

overtopping is quite limited and only a few model studies are available. The influence factor for a

vertical wall γv can be determined as follows [2]:

γv = 1.35 − 0.0078 ∙ αwall 2.9

However, this is only valid for very specific conditions.

Effect of roughness

The influence factor for roughness elements on a slope γf depends only on the surface roughness. It is

supposed to give an indication of the roughness and the permeability of the structure. The idea is that

the rougher and more permeable a structure is, the lower the overtopping will be, as more energy is

dissipated. This is incorporated in a lower value of the parameter γf [2] and thus a larger reduction.

Effect of composite slopes and berms [2]

Many dikes do not have a straight slope from the toe to the crest, but consist of a composite profile

with different slopes, a berm or multiple berms. A characteristic slope is required to be used in the

equations for composite profiles or profiles with berms to calculate wave run-up or wave overtopping.

This characteristic slope does not take into account berms. [2]

Theoretically, the run-up process is influenced by a change of slope from the breaking point to the

wave run-up height. Therefore, often is has been recommended to calculate the characteristic slope

from the point of wave breaking to the wave run-up height Ru2%. The breaking point can be chosen

1.5 times the wave height Hm0 below the still water line. The run-up has yet to be determined. This

approach needs some calculation effort, because of the iterative solution since the wave run-up height

Ru2% itself depends on the characteristic slope tan αexcl through the breaker parameter ξm−1,0 [2]:

Ru2% = 1.65 ∙ Hm0 ∙ γb ∙ γf ∙ γβ ∙ ξm−1,0 2.10

with a maximum of Ru2% = Hm0 ∙ γf ∙ γβ ∙ (4 −1.5

√ξm−1,0) 2.11

The characteristic slope tan αexcl is determined in two steps. In the first one, the run-up Ru2% is

estimated with 1.5 times the wave height Hm0 above the still water line [2]:

1st estimate tan αexcl =3∙Hm0

Lslope−B 2.12

12

In the second step, the characteristic slope tan αexcl is estimated using the run-up Ru2% resulting from

the first estimate of the slope [2]:

2nd estimate tan αexcl =1.5∙Hm0+Ru2%(from 1st estimate)

Lslope−B 2.13

If the run-up height Ru2% or 1.5 times the wave height Hm0 comes above the crest level, then the crest

level must be taken as the characteristic point above SWL.

Berms reduce wave run-up and wave overtopping. Until now, they have not been considered. The

effect of a berm is introduced through an influence factor γb which is determined as follows [2]:

γb = 1 − rB ∙ (1 − rdb) for 0.6 < γb < 1.0 2.14

with rB =Bh

Lberm ;

Bh width of the horizontally schematized berm;

Lberm the horizontal length between two points on slope, Hm0 above and Hm0 below

the middle of berm, Lberm = Bh + Hm0 toe ∙ cot αd + Hm0 toe ∙ cot αu;

cot αd the cotangent of the slope of the structure downward of the berm;

cot αu the cotangent of the slope of the structure upward of the berm;

A berm below SWL (hb ≥ 0): if |hb| > 2 ∙ Hm0 toe, rdb = 1;

else rdb = 0.5 − 0.5 ∙ cos (π|hb|

2∙Hm0 toe) ;

A berm above SWL (hb < 0): rdb = 0.5 − 0.5 ∙ cos (π|hb|

Ru2%) .

2.2.2.2 Vertical structures

In this section the prediction models for vertical and steep-fronted coastal structures such as caisson

and blockwork breakwaters and vertical seawalls, as well as composite vertical wall structures (where

the emergent part of the structure is vertical, fronted by a modest berm) are discussed. [2]

In assessing overtopping on sloping structures, a distinction is made between breaking waves and non-

breaking waves. Similarly, for assessment of overtopping at steep-fronted and vertical structures the

regime of the wave/structure interaction must be identified first, with quite distinct overtopping

response expected for each regime. [2]

On steep walls (vertical, battered or composite), “non-impulsive” conditions occur when waves are

relatively small in relation to the local water depth, and of lower wave steepnesses. In contrast,

“impulsive” conditions occur on vertical or steep walls when waves are larger in relation to local water

depth. Under these conditions, some waves will break violently against the wall. Lying in a narrow band

13

between non-impulsive and impulsive conditions are “near-breaking” conditions. These conditions

usually give overtopping in line with impulsive (breaking) conditions. [2]

In order to proceed with assessment of overtopping, it is therefore necessary first to determine which

is the dominant overtopping regime (impulsive or non-impulsive) for a given structure and design sea

state. No single method gives a discriminator which is 100% reliable. The suggested procedure includes

a transition zone in which there is significantly uncertainty in the prediction of dominant overtopping

regime and thus a “worst-case” is taken. [2]

Further, a distinction is also made between plain vertical walls and composite verticals walls. Since it

is well-established that a relatively small toe or berm can change wave breaking characteristics, thus

substantially altering the type and magnitude of wave loadings. [2]

Plain vertical walls

The following discriminator is used for plain vertical walls [2]:

h∗ = 1.35 ∙h

Hm0 toe∙

2π ∙ h

g ∙ Tm−1,0² 2.15

with h the water depth at the toe.

Non-impulsive conditions dominate at the walls when h∗ is larger than 0.3, and impulsive conditions

occur when h∗ is smaller or equal to 0.2. In the transition zone we take the “worst-case”, hence the

one where the relative discharge Q∗ is maximum. [2]

For simple vertical breakwaters in the non-impulsive regime, the following equation is given [2]:

q

√g∙Hm0³= 0.04 ∙ exp (−2.6 ∙

Rc

Hm0) valid for 0.1 <

Rc

Hm0< 3.5 2.16

The coefficient of 2.6 for the mean prediction has an associated standard deviation of σ = 0.8 [2].

In the impulsive regime, the following equation is given:

q

h∗²∙√g∙h³= 1.5 ∙ 10−4 ∙ (h∗ ∙

Rc

Hm0)

−3.1 valid for 0.03 < h∗ ∙

Rc

Hm0< 1.0 2.17

The scatter in the logarithm of the data about the mean prediction is characterized by a standard

deviation of 0.37 [2].

14

Composite vertical walls

For vertical composite walls where a berm or significant toe is present in front of the wall, an adjusted

version of the method for plain vertical walls should be used. A modified “impulsiveness” parameter

d∗ is defined in a similar manner to the parameter h∗ used for plain vertical walls. [2]

d∗ = 1.35 ∙hc

Hm0 toe∙

2π ∙ h

g ∙ Tm−1,0² 2.18

with h the water depth at the toe and

hc the water depth on the berm or on the toe.

Non-impulsive conditions dominate when d∗ is larger than 0.3, and impulsive conditions occur when

d∗ is smaller or equal to 0.2. In the transition zone overtopping is again predicted for both conditions

and the larger value assumed. [2]

When non-impulsive conditions prevail, overtopping can be predicted by the standard method given

previously for non-impulsive conditions at plain vertical structures. [2]

For conditions determined to be impulsive, a modified version of the impulsive prediction method for

plain vertical walls is recommended to account for the presence of the mound by use of d and d∗ [2]:

q

d∗²∙√g∙h³= 4.1 ∙ 10−4 ∙ (d∗ ∙

Rc

Hm0)

−2.9 valid for 0.05 < d∗ ∙

Rc

Hm0< 1.0 2.19



Effect of oblique waves

For non-impulsive conditions, an adjusted version of equation 2.16 should be used to take into account

the influence of oblique wave attack [2]:

q

√g∙Hm0³= 0.04 ∙ exp (−2.6 ∙

1

γβ∙

Rc

Hm0)

where γβ is the reduction factor for angle of wave attack and is given by [2]:

γβ = 1 − 0.0062 ∙ |β| for 0° ≤ β ≤ 45° 2.20

γβ = 0.72 for β > 45° 2.21

and β is the angle of attack relative to the normal, in degrees.

For impulsive conditions, a more complex picture emerges (Napp et al., 2004). Within this report, the

effect of oblique waves is not considered in the impulsive regime for vertical structures.

15

2.2.3 Van der Meer and Bruce (2014)

2.2.3.1 Sloping structures

Research in CLASH resulted in a lot of new data and in prediction formulae for slopes, for breaking

waves as well as non-breaking waves [4]. For sloping structures, the EurOtop (2007) formulae are of

the exponential type. These exponential type equations fit the data nicely, except for the data points

at very low and zero freeboard, where the formulae would significantly over predict [5].

The theoretical analysis of Battjes (1974) demonstrated that wave overtopping at gentle, smooth

slopes should be a curved line on a log-linear graph [4]. An exponential fit for larger relative freeboards

(a straight line) would be close the curve proposed by Battjes, but such a fit would deviate for low

freeboards [5]. However, a polynomial fit as in Battjes/TAW describes the data, but is not easy to use

for comparison between different formulae [4]. A new fit for low freeboards only, with an extra set of

formulae would solve the problem [4]. It is more elegant and more physically rational, however, to

propose a curved line in an easy way [5]. As the exponential function is a special case of the Weibull

distribution [4], Weibull-type formulae were proposed by Van der Meer and Bruce (2014), describing

wave overtopping at slopes for the whole range Rc∗ ≥ 0 [4]. Such a function looks still very much like

the current EurOtop (2007) prediction formulae and is described by [4]:

Q∗ = a ∙ exp(−(b ∙ Rc∗ )c) 2.22

This type of equation needs fitting of the correct shape factor, c, and then a re-fit of coefficient a and

exponent b. Analysis gave a shape factor of c = 1.3 for a good fit for both breaking and non-breaking

waves. It should be noted that this is not necessarily the best fit, but there is advantage in having the

same value for both equations. [4] Overtopping on sloping structures with zero and positive freeboard

is then described by the revised formulae of Van der Meer and Bruce (2014) [5]:

q

√g∙Hm0³=

0.023

√tan α∙ γb ∙ ξm−1,0 ∙ exp (− (2.7 ∙

Rc

ξm−1,0∙Hm0∙γb∙γf∙γβ∙γv)

1.3

) 2.23

with a maximum of

q

√g∙Hm0³= 0.09 ∙ exp (− (1.5 ∙

Rc

Hm0∙γf∙γβ)

1.3

) 2.24

The reliability of equation 2.23 is given by σ(0.023) = 0.003 and σ(2.7) = 0.20, and of equation 2.24 by

σ(0.09) = 0.013 and σ(1.5) = 0.15 [4].

These formulae give almost the same wave overtopping as the original TAW (2002) formulae given in

the EurOtop (2007) manual, except that they give a better prediction for Rc∗ < 0.8 [4].

16

In general, there is no need to replace the original equations by the new ones, as they give similar

predictions. Only for low and zero freeboards the new formulae will be better. But the new equations

give better insight in wave overtopping over the full range of zero and positive freeboards. [4]

2.2.3.2 Vertical structures

A reformulation of the procedures for overtopping analysis of vertical walls is presented offering a

simpler and more physically rational procedure. A distinction is made between overtopping at

structures without influence of foreshore and those where the foreshore plays a role in the

hydrodynamics at the structure. This leads to an improved method for higher freeboard situations.

Furthermore, composite vertical structures are revisited, and a procedure proposed that integrates

their analysis with that of the updated procedure for plain vertical walls. [6]

In the EurOtop (2007) manual non-impulsive conditions are described by a familiar exponential

formula, whereas impulsive (breaking or impacting) wave overtopping is better described by a power

law formula (Pullen et al. 2007). An analogous approach described overtopping at composite vertical

breakwaters. The downside of this two-formula approach is that it is not at all easy to compare on a

single plot, in an visual/intuitive way, the overtopping behavior of a single structure because conditions

move between impulsive and non-impulsive conditions (different nondimensionalization schemes are

used for both discharge and freeboard axes). [5]

Plain vertical walls

For overtopping at verticals walls in the non-impulsive regime, the easy exponential formulation with

simple values for a and b has become a trusted design formula [6]. EurOtop (2007) gives a = 0.04 and

b = 2.6. The early works of Franco et al. (1994) and Allsop et al. (1995) also remain reliable references.

Franco et al. (1994) gave a = 0.2 and b = 4.3 for relatively deep water, whereas Allsop et al. (1995) gave

a = 0.05 and b = 2.78 in conditions of shallower water. [5]

In the impulsive regime, a power law formula is used. The coefficient a and the exponent b change

depending on structure and wave conditions considered. The exponent b in the EurOtop (2007)

formulae takes values of 3.1 for impulsive overtopping at plain verticals walls; 2.7 for broken waves at

plain vertical walls; and 2.9 for composite vertical structures [2, 5]. These exponents are simply the

result of fitting to the data; the differences have no basis in any analytical framework or in physical

reasoning. The fact that the exponents are all different makes it difficult to carry out a direct, e.g.,

graphical, comparison between the different but closely related structures and their associated

overtopping responses. [5]

17

Whether an exponential or a power law should be used is determined by some discriminating

parameter h∗. This parameter is used as a measure of impulsiveness [4]:

h∗ ≈ 1.3 ∙h

Hm0∙

h

Lm−1,0

The fact that discharge and freeboard are nondimensionalized in different ways for impulsive and non-

impulsive conditions has until now prevented simple comparison for the formulae.

As mentioned before, the exponents b in the power law equations are all very close to 3; fixing b = 3

enables the equations to be manipulated algebraically [5]:

q

h∗2∙√g∙h3= a ∙ (h∗ Rc

Hm0)

−3

↔q

√g∙h3= a ∙ (h∗)−1 ∙ (

Rc

Hm0)

−3

↔q

√g∙h3= a ∙

Hm0

h∙

Lm−1,0

h∙ (

Rc

Hm0)

−3

↔q

√g∙Hm03

= a ∙ √Hm0

h∙

1

sm−1,0∙ (

Rc

Hm0)

−3

This equation is much clearer than the formula with h∗ on both sides. The coefficient a and the

equation itself will be reexamined using existing data of the CLASH database. A first reanalysis of

verticals walls with CLASH data found that the influence of steepness was better represented by

√sm−1,0 in the power law equation [5]:

q

√g∙Hm03

= a ∙ √Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3

A part of the CLASH database relates to tests with a vertical or battered wall, which can be found by

filtering on cot αd = 0 or on very small values of cot αd (i.e. battered walls). Individual analysis of all

filtered datasets led to one clear conclusion: there is a distinct difference between vertical structures

with and without a sloping foreshore. The results with a sloping foreshore always gave larger

overtopping. Within the group of datasets without a foreshore slope, there was no notable difference

between composite type and plain vertical walls. On the basis of this conclusion, the datasets were

split into two groups, and each group was then analyzed separately. [5]

Vertical structures without foreshore

In the case of vertical structures without a foreshore, it was established that Franco et al. (1994) will

over predict overtopping for lower freeboards. Allsop et al. (1995), however, covers this area well.

18

Hence both formulae are valid for vertical structures without a sloping foreshore but each has their

own range of application. [5]

The description of wave overtopping is then given by [5]:

q

√g∙Hm0³= 0.05 ∙ exp (−2.78 ∙

Rc

Hm0)

Rc

Hm0< 0.91 (Allsop et al. 1995) 2.25

q

√g∙Hm0³= 0.2 ∙ exp (−4.3 ∙

Rc

Hm0)

Rc

Hm0> 0.91 (Franco et. 1994) 2.26

with reliability of equation 2.25 and equation 2.26 respectively σ(2.78) = 0.17 and σ(4.3) = 0.6 [5].

Vertical structure on sloping foreshore

First, h² (Hm0 ∙ Lm−1,0) = 0.23⁄ is proposed as a discriminator between non-impulsive and impulsive

wave conditions, this is approximately equivalent to h∗ = 0.3 [5].

For values larger than 0.23, non-impulsive waves, research showed that Allsop et al. (1995) describes

the wave overtopping for these kind of structures and for given wave conditions very well [5].

For values larger than 0.23, impulsive waves, a distinction is made between low and larger freeboards.

It is clear that a power function cannot give the trend line for small or zero freeboards because it will

not cross the vertical axis, but rather, it uses the vertical axis as an asymptote. It is for this reason it

was decided to keep the power function for larger freeboards and to introduce the common

exponential function for zero and low freeboards. The formulae are described by [5]:

q

√g∙Hm0³= 0.011 ∙ √

Hm0

h∙sm−1,0∙ exp (−2.2 ∙

Rc

Hm0) for

Rc

Hm0< 1.35 2.27

q

√g∙Hm0³= 0.0014 ∙ √

Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3 For

Rc

Hm0≥ 1.35 2.28

with reliability of equation 2.27 σ(0.011) = 0.0045 and of equation 2.28 σ(0.011) = 0.0006 [5].

Composite vertical structures

Whether an exponential or a power law should be used is determined by some discriminating

parameter d∗ in the EurOtop (2007) manual. This parameter is used as a measure of impulsiveness.

d∗ ≈ 1.3 ∙d

Hm0∙

h

Lm−1,0

19

Similarly as for plain vertical walls, the power law formula in the impulsive regime can be rewritten:

q

d∗2∙√g∙h3= a ∙ (d∗ Rc

Hm0)

−3

↔q

√g∙h3= a ∙ (d∗)−1 ∙ (

Rc

Hm0)

−3

↔q

√g∙h3= a ∙

Hm0

d∙

Lm−1,0

h∙ (

Rc

Hm0)

−3

↔q

√g∙Hm03

= a ∙ √d

h∙ √

Hm0

h∙

1

sm−1,0∙ (

Rc

Hm0)

−3

The vertical wall reanalysis of the preceding section found that the influence of steepness was better

represented by √sm−1,0. The similarity of the physical situation suggests that this adjustment should

also be included for the composite structures, giving a tentative prediction equation [5]:

q

√g∙Hm03

= a ∙ √d

h∙ √

Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3

The only difference with above is the constant multiplier and simple factor of √d h⁄ , which becomes

unity for plain vertical walls with zero berm height (h = d). [5]

The two predictors coincide at a value of d h⁄ ≈ 0.6, this suggests that the mound’s influence should

cease for conditions where d > 0.6 ∙ h, which seems physically sensible. [5]

EurOtop (Pullen et al. 2007) gives the discriminator d∗ < 0.3 for impulsive conditions. Plotting data

according to this discriminator showed a group of data at higher freeboards that is significantly under

predicted. Research showed that resetting the switch upward to a value of 0.85 improved the success

of the predictor in identifying apparently impulsive conditions. [6]

q

√g∙Hm0³= 1.3 ∙ √

d

h∙ 0.011 ∙ √

Hm0

h∙sm−1,0∙ exp (−2.2 ∙

Rc

Hm0) for

Rc

Hm0< 1.35 2.29

q

√g∙Hm0³= 1.3 ∙ √

d

h0.0014 ∙ √

Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3 for

Rc

Hm0≥ 1.35 2.30

The scheme for composite structures is thus now aligned with the improved vertical scheme. In

summary, in cases where the mound is small, the structure is treated as vertical. For d h⁄ > 0.6, in the

absence of a foreshore and possible breaking, the structure is again treated as plain vertical. In the

case of possible breaking, however, the overtopping is arrived at according to the method for plain

walls but with a factor of 1.3 ∙ √d h⁄ included. The multiplier of 1.3 allows composite and vertical

formulas to coincide at d h⁄ = 0.6. [5]

20

A decision chart summary of the proposed unified scheme for plain vertical and composite structures

is illustrated in Figure 2. 3.

Figure 2. 3 New formulae scheme [5]

2.2.4 Uncertainty of the prediction

The recommended formulae for wave run-up height and wave overtopping calculations are empirical

formulae are based on a large (international) dataset. Due to the large dataset of all kind of structures,

a significant scatter is present, which cannot be neglected for applications. [2] Uncertainties or scatter

in data is the result from a lot of things such as measurement accuracy, inaccuracies in model setup

and water levels, determination of wave parameters, etc.

The model uncertainty is considered as the accuracy, with which a model or method can describe a

physical process. Therefore the model uncertainty describes the deviation of the prediction from the

measured data due to this method. These differences between predictions and data observations may

results from either uncertainties of the input parameters or model uncertainty [2].

This reliability of the equations has been described by statistical distributions for the parameters

occurring in the prediction models. Hereby it is assumed that the parameters are normally distributed.

Two implications for design can then be considered: Probabilistic design values for all empirical models

used in this manual describe the mean approach for all underlying data points. This means that, for

21

normally distributed variables, about 50% of the data points exceed the prediction by the model, and

50% are below the predicted values. The deterministic design value for all models will be given as the

mean value plus one standard deviation, which in general gives a safer approach, and takes into

account that model uncertainty for wave overtopping is always significant. [2]

These approaches are necessary because of the uncertainties inherent in all the data and the scatter

in overtopping results which can be considerably large in wave overtopping processes. [2]

Within this report, only formulae that give the average overtopping are considered in accordance to

the mean value approach. As a consequence, the value of the stochastic parameters in the model is

their mean value.

2.3 Overtopping data

2.3.1 CLASH database

2.3.1.1 Background

Many physical model tests have been performed all over the world, both for scientific (idealized)

structures and real applications or designs. The European CLASH project resulted in a large database

of more than 10,000 wave overtopping tests on all kind of structures. Some series of tests have been

used to develop the empirical methods for prediction of overtopping. [4] Each test was described by

31 parameters as hydraulic and structural parameters, but also parameters describing the reliability

and complexity of the test and structure. The database includes more than 10,000 tests and was set-

up as an Excel database. The database, therefore, is nothing more than a matrix with 31 columns and

more than 10,000 rows. [2] This chapter discusses the set-up of the extensive CLASH database on wave

overtopping at coastal structures.

During the last 30 years, overtopping at coastal structures has been the subject of extensive research,

resulting in a lot of overtopping information available at different universities and research institutes

all over the world. The first phase of composing a database consisted therefore in collecting as much

of these present data as possible. Overtopping data were gathered from partners within the CLASH

project as well as from other authorities in and outside Europe. [1]

In the first phase of the set-up of the database, about 6500 tests were gathered. During the second

and last phase, not only 4000 new overtopping tests were added, but also some parameters were

improved, resulting in an extended and improved final database. [1]

To obtain a complete and reliable overtopping database as much information as possible was gathered

for all test series. Not only information about wave characteristics, test structure and corresponding

overtopping discharges, but also information concerning the test facility used to perform the tests, the

22

processing of the measurements and the precision of the work performed was gathered. Depending

on this information, each test could be assessed on reliability and complexity. This was taken into

account in the database by defining a Reliability Factor, RF and a Complexity Factor, CF for each test,

which are respectively a measure of the reliability of the performed test and the complexity of the

overtopping structure. [1]

How the data was gathered, screened and put together has been described in the background report

“Database on wave overtopping at coastal structures” by Verhaeghe, 2003. One is referred to that

report to become familiar with the database. [7]

The final database consists of 10532 overtopping tests which are represented by an equal number of

rows in a spreadsheet.

It could be important for researchers using the overtopping database to know which parameter values

concern real measured values, which ones concern calculated values and which ones concern

estimated values. This cannot be checked by the value of the reliability factor RF as this factor only

gives an overall indication of the reliability of the test. To distinguish such cases from each other, colors

were used to mark the calculated and estimated values. [1]

Beside the 31 columns already mentioned (resulting from 11 hydraulic parameters, 17 structural

parameters and 3 general parameters), 2 more columns were added to the spreadsheet. [1]

The first added column, column 32, is called ‘Remark’ and contains a remark additional to the test,

mainly bearing in mind a neural network application of the database. As model and scale effects may

affect small scale overtopping measurements in specific cases, prototype measurements should be left

out from a neural network development. Further also laboratory measurements performed with

artificial wind generation should not be considered for a neural network development. The fact that

wind is no parameter of the database can be mentioned as reason for this. Finally, a part of the

laboratory tests concerns test sections not appearing in reality (i.e. a synthetic test set-up in the

laboratory), and should consequently also be left out from a neural network application. [1]

A second added column, column 33, is called ‘Reference’. For public tests, column 33 contains a

reference to a report or paper describing the tests. This allows interested researchers to find more

information on specific tests or test series. [1]

An overview of all the information summarized in the CLASH database is given in Table 2. 1.

23

Table 2. 1 Information summarized in the CLASH database [1]

2.3.1.2 Parameters

Each test has been described by 31 parameters. Three groups of parameters were defined: hydraulic

parameters, structural parameters and general parameters. The hydraulic parameters describe the

wave characteristics and the measured overtopping, whereas the structural parameters describe the

test structure. The general parameters are related to general information about the overtopping test.

[1] In the following, all parameters are described.

General parameters [1, 7]

There are 3 general parameters:

1 Name The first parameter, Name, assigns a unique name to each test.

It consists of a basic test series number, which is the same for all the tests within the same test series,

followed by a unique number for each test. The parameter Name is always composed of 6 characters.

24

E.g. test 36 from test series 178 has the unique code: 178-036.

This parameter is only meant to recognize each test but has no further meaning.

2 CF [-] The complexity factor CF gives an indication of the complexity of the structure.

It can adopt the values 1, 2, 3 or 4. A value of 4 means unreliable, not to be used for the Neural

Network.

The factor refers to the degree of approximation which is obtained by describing a test structure by

means of structural parameters in the database. It should be mentioned that only the structure section

itself is considered, i.e. an approximation of the foreshore is not accounted for in the value of CF.

Table 2. 2 gives an overall view of the values the reliability factor CF can adopt. For each value a short

explanation is given.

Table 2. 2 Values of the complexity factor CF and their corresponding meaning [1]

3 RF [-] The reliability factor RF gives an indication of the reliability of the test.

It can adopt the values 1, 2, 3 or 4. A value of 4 means a too complex structure or situation, not to be

used in the Neural Network.

Table 2. 3 gives an overall view of the values the reliability factor RF can adopt. For each value a short

explanation is given.

25

Table 2. 3 Values of the reliability factor RF and their corresponding meaning [1]

The reliability factor RF is determined by several factors:

- the precision of the measurements/analysis of the researcher who performed the overtopping

test;

- the restrictions/possibilities of the test facility used to perform the test;

- the estimations/calculations which had to be made because of missing parameter values.

Hydraulic parameters [1, 7]

There are 11 hydraulic parameters:

1 Hm0 deep Significant wave height from spectral analysis, determined at deep water.

2 Tp deep [s] Peak period from spectral analysis at deep water.

3 Tm deep [s] Mean period from spectral analysis at deep water.

4 Tm −1,0 deep [s] Mean period from spectral analysis at deep water.

5 β [°] Angle of wave attack relative to the normal on the structure.

6 Hm0 toe [m] Significant wave height from spectral analysis at the toe of the structure.

7 Tp toe [s] Peak period from spectral analysis at the toe of the structure.

8 Tm toe [s] Mean period from spectral analysis at the toe of the structure.

9 Tm −1,0 toe [s] Mean period from spectral analysis at the toe of the structure.

26

10 q [m³/s∙m] Overtopping volume per second per meter width.

11 q [%] Percentage of the waves resulting in overtopping.

Often several of these parameters were not available in the corresponding report of the test, simply

because they were not measured or not written down during performing the test. With the aim of

obtaining a database as complete as possible, if possible an acceptable value was searched for these

missing parameters. Well-founded assumptions based on previous research and extra calculations

were used to achieve this. [1]

These estimations, however, clearly had an influence on the reliability of the values, this fact was

incorporated in the database by adapting the value of the reliability factor RF. If any calculations or

estimations were needed, a minimum value of 2 was assigned to the factor RF. [1]

Nevertheless, in some cases it was simply not possible to estimate missing hydraulic parameters

accurately. In such cases, preference was given to leave the value of the missing parameter blank in

the database. [1]

Note that to distinguish calculated and estimated parameters from measured parameters in the

database, the former values are marked with specific colors in the Excel file, depending on the type of

the calculation and estimation. [1]

Structural parameters [1, 7]

There are 17 structural parameters:

1 hdeep [m] Water depth at deep water.

2 m [-] Slope of the foreshore.

3 h [m] Water depth at the toe of the structure.

4 ht [m] Water depth on the toe of the structure.

5 Bt [m] Width of the toe of the structure.

6 γf [-] Roughness/permeability factor of the structure.

7 cot αd [-] Cotangent of the slope of the structure downward of the berm.

8 cot αu [-] Cotangent of the slope of the structure upward of the berm.

9 cot αexcl [-] Mean cotangent of the slope of the structure, excluding the berm.

10 cot αincl [-] Mean cotangent of the slope of the structure, with contribution of the berm.

27

11 Rc [m] Crest freeboard of the structure.

12 B [m] Width of the berm.

13 hb [m] Water depth on the berm.

14 tan αb [-] Tangent of the slope of the berm.

15 Bh [m] Width of the horizontally schematized berm.

16 Ac [m] Armour crest freeboard of the structure.

17 Gc [m] Width of the structure crest.

Note that the water depth on the berm hb is measured in the middle of the berm. If the berm is

situated above the still water level (SWL), the water depth is negative. This is illustrated in Figure 2. 4.

Figure 2. 4 Determination of B [m], Bh [m], tan αb [-], hb [m] [1]

An illustration of the crest parameters and of the structure slope parameters is also given in

respectively Figure 2. 5 and Figure 2. 6.

28

Figure 2. 5 Determination of Rc [m], Ac [m] and Gc [m] [1]

Figure 2. 6 Determination of the structure slope parameters [1]

2.3.2 UG data

More data has been collected since the CLASH database was created. At the University of Ghent some

research was carried out that resulted in extra overtopping data. Three different datasets UG10, UG13

and UG14 will be used in this work and are briefly discussed here.

2.3.2.1 UG10

Research carried out at Ghent University resulted in a first dataset UG10 (Victor & Troch, 2012a,

2012b).

Extensive knowledge is available on the overtopping behaviour of traditional smooth impermeable sea

defence structures, such as mildly sloping dikes and vertical walls, both typically featuring a high crest

freeboard to reduce wave overtopping. The design of Ocean Wave Energy Concerts (OWECs) requires

an inverse thinking compared to traditional sea defence structures, since wave overtopping in this case

needs to be maximized. Correspondingly, OWECs typically feature a low crest freeboard and a smooth

impermeable uniform slope. Furthermore, their slopes are rather steep to avoid energy loss due to

29

wave breaking on the slopes. To date, only relatively limited knowledge is available on the overtopping

behaviour of those structures. [3]

The new experiments, referred to as the UG10 test series, have been carried out in a wave flume at

the Department of Civil Engineering at Ghent University (Belgium).

2.3.2.2 UG13

More research carried out at Ghent University resulted in another dataset UG13 (Troch et al., 2014).

The main goal of this research was to extend the existing overtopping datasets to steep slopes and

vertical walls by performing additional overtopping experiments in the wave flume of Ghent

University. The average wave overtopping performance was studied for sloping coastal structures

characterised by very steep slopes and very small freeboards. [8]

This research extends earlier research of Victor & Troch (2012a, 2012b) resulting in dataset UG10, who

investigated the cases of steep slopes and small freeboards for smooth sloping coastal structures. [8]

2.3.2.3 UG14

The dataset UG14 has been generated at Ghent University in order to improve the knowledge of wave

overtopping on different structures for various wave conditions. The UG14 dataset was obtained in

tests performed in the wave flume of Ghent University within a PhD thesis (Victor, 2012). The UG14

dataset completes previous research carried out at Ghent University that resulted in the UG10 (Victor

& Troch, 2012a, 2012b) and the UG13 (Troch et al., 2014) datasets, that were focused on rather deep

water conditions. [9]

The existing overtopping datasets were extended to steep slopes and vertical walls for the case of

shallow water wave conditions by performing additional overtopping experiments. Note that shallow

water wave conditions have been very limitedly studied in past literature. [9]

A small overlap in the UG14 experiments with UG10 and UG13 datasets is present to additionally verify

these previous test results. [9]

2.4 Conclusions

Depending on the type of structure, different prediction models need to be applied. The two main

types of structures are sloping and vertical structures. These type of structures will therefore be

treated separately in what follows.

Further, the calculation procedures are not always well-defined and clear. Therefore, it will be

necessary each time to start with defining a straightforward calculation procedure.

30

Besides, the CLASH database contains data for all types of structures. The considered formulae,

however, are only applicably for a certain type of structure only. Hence, a filtering scheme needs to be

defined such that the data valid for each formula can be filtered out.

When the calculation procedure and the filtering scheme are defined, the uncertainties can be derived

for each formula. The scatter or the uncertainty is described by a standard deviation on the stochastic

parameters occurring in the prediction models.

The reliability of the equations in the EurOtop (2007) manual are described by a standard deviation for

one of the two parameters occurring in the equations. In some of the new formulae by Van der Meer

and Bruce (2014), however, the reliability is described by a standard deviation for both parameters.

Both parameters are then considered stochastic instead of just one. The goal of this master’s thesis is

to update the uncertainties in the EurOtop (2007) manual. Therefore, the reliability of the formulae by

Van der Meer and Bruce (2014) will be expressed similarly as the reliability of the EurOtop (2007)

formulae, assuming one of the two parameters is stochastic and the other constant.

Both the EurOtop (2007) formulae as the more recent formulae by Van der Meer and Bruce (2014) use

the CLASH database. Hence, the influence of more recent collected data on the uncertainties has not

been studied.

The next steps are now to: (1) derive the uncertainties of the EurOtop (2007) formulae considering

data from the CLASH database; (2) investigate the influence of the UG datasets on these uncertainties

and finally (3) see whether the more recent formulae by Van der Meer and Bruce (2014) reduce the

uncertainties.

This is done consecutively for sloping structures in Chapter 3 and for vertical structures in Chapter 4.

31

Chapter 3: Sloping structures

3.1 EurOtop (2007) formulae applied to CLASH data

3.1.1 Introduction

The EurOtop (2007) formulae for wave overtopping calculations are based on a large dataset. Due to

the large dataset of all kind of sloping structures, a significant scatter is present, which cannot be

neglected for applications.

The formulae are of the exponential type for sloping structures:

Q∗ = a ∙ exp[−(b ∙ Rc∗ )] 3.1

where Q∗ and Rc∗ were made non-dimensional according to EurOtop (2007) and depending on breaking

or non-breaking conditions. The formulae discussed here describe the average overtopping discharge,

in accordance to the mea value approach.

The uncertainties or scatter are presented by a confidence band in the EurOtop (2007) plots. These

upper and under exceedance limits are constructed by considering parameter a which determines the

intersection point, constant and parameter b which represents the slope stochastic and normally

distributed. Hence the 90% confidence interval can be constructed as follows:

Q∗ = a ∙ exp[−(b ± 1.64 ∙ σ) ∙ Rc∗ )] 3.2

Exponential equations give a straight line in a log-linear graph. Figure 3. 1 gives an overall view of the

resulting plot with the mean value approach and its 90% confidence band.

Figure 3. 1 Wave overtopping data and mean value approach with its confidence band [10]

The CLASH database contains the wave overtopping data used to derive the EurOtop (2007) formulae.

32

3.1.2 Calculation procedure

EurOtop (2007) gives the following overtopping formulae for breaking and non-breaking waves [1, 2]:

q

√g∙Hm0³=

0.067

√tan α∙ γb ∙ ξm−1,0 ∙ exp (−4.75 ∙

Rc

ξm−1,0∙Hm0∙γb∙γf∙γβ∙γv) 3.3

with a maximum of q

√g∙Hm0³= 0.2 ∙ exp (−2.6 ∙

Rc

Hm0∙γf∙γβ) 3.4

These formulae are only valid for breaker parameters ξm−1,0 smaller than 5. The reliability of the

equations is described by taking the coefficients 4.75 and 2.6 as normally distributed stochastic

parameters with means of 4.75 and 2.6 and standard deviations σ = 0.5 and 0.35 respectively [1, 2].

The maximum occurring reduction due to the influence factors is 60%. This it not only applicable to

the above equations, but to any equations in which they occur.

Equations 3.3 and 3.4 can be rewritten in the general form of equation 3.1. The nondimensional axes

in the case of breaking waves can then be described by:

the relative overtopping rate Q∗ =q

√g∙Hm0 toe³∙ √

Hm0 toe Lm−1,0 toe⁄

tan α∙

1

γb

the relative freeboard Rc∗ =

Rc

Hm0 toe∙

√Hm0 toe Lm−1,0 toe⁄

tan α∙

1

γb∙γf∙γβ∙γv

In the case of non-breaking waves, the axes are given by:

the relative overtopping rate Q∗ =q

√g∙Hm0 toe³

the relative freeboard Rc∗ =

Rc

Hm0 toe∙

1

γf∙γβ

In order to reconstruct the EurOtop (2007) plots, all variables occurring in the above expressions need

to be known. Some of them are simply given in the CLASH database, others still need to be calculated.

Some of these still to be calculated variables can be derived easily, others require iterative calculations.

The calculation procedures indicated in the EurOtop (2007) manual are not always well defined and

clear, a lot of ambiguities are detected. Therefore, we start with defining a consistent calculation

procedure.

More in particular, the problem is the interdependencies between the tangent of the slope tan α, the

breaker parameter ξ, the influence factor for a berm γb and the run-up Ru. Both the tangent of the

slope tan α as the influence factor accounting for a berm γb, depend upon the run-up Ru. While the

run-up height Ru in its turn depends on the influence factor for a berm γb and on the breaker

parameter ξ and hereby also on the tangent of the slope tan α. These interdependencies are illustrated

in Figure 3. 2.

33

Figure 3. 2 Interdependencies calculation procedure

A clear one-way procedure existing out of two steps is proposed which is summarized in Figure 3. 3. In

a first iteration, the tangent of the slope tan α and influence factor for a berm γb are estimated based

on the wave height only. Thus, the wave height Hm0 was used to estimate the unknown run-up Ru. In

the second iteration, the tangent of the slope tan α and influence factor for a berm γb are estimated

again, however, this time based on the first estimate of the run-up Ru too, instead of only the wave

height Hm0. Below, the calculation procedure is described in detail.

Figure 3. 3 One-way calculation procedure

In the first iteration, the wave height Hm0 is used to estimate the unknown run-up Ru.

The influence of a berm is taken into account by defining an equivalent slope which yields an equivalent

breaker parameter [11]. The breaker parameter is calculated for the slope not considering the berm

(characteristic slope), the influence factor for a berm than takes into account the berm (equivalent

slope) [11]:

ξeq = yb ∙ ξav 3.5

34

The influence factor for a berm γb gives the combined influence of the berm width and the berm depth

[1, 2, 11]:

γb = 1 − rB ∙ (1 − rdb) for 0.6 < γb < 1.0 3.6

I. rB = Bh Lberm⁄ the reduction of the average slope caused by the berm width (zero if no berm)

with Lberm = Bh + Hm0 toe ∙ cot αd + Hm0 toe ∙ cot αu,

i.e. the horizontal length between two points on the slope, 1.0 Hm0 toe

above and 1.0 Hm0 toe below middle of berm;


cot αd the cotangent of the slope of the structure downward of the berm and

cot αu the cotangent of the slope of the structure upward of the berm.

II. rdb the reduction of the influence of the berm caused by its depth (zero if berm at SWL)

- Berm above SWL (hb < 0)

If |hb| > 2 ∙ Hm0 toe, rdb = 1 (no influence on the run-up and overtopping)

Else rdb = 0.5 − 0.5 ∙ cos (π|hb|

2∙Hm0 toe) ≤ 1

- Berm below SWL (hb ≥ 0)


Else rdb = 0.5 − 0.5 ∙ cos (π|hb|

2∙Hm0 toe) ≤ 1

The tangent of the slope tan α can be found as follows [2]:

- Berm above SWL (hb < 0):

If |hb| > 1.5 ∙ Hm0 toe then only the lower slope is relevant, thus tan α1 =1

cot αd

Else tan α1 =3∙Hm0 toe

Lslope−Bh

with Lslope = (1.5 ∙ Hm0 toe + |hb|) ∙ cot αd + Bh + (1.5 ∙ Hm0 toe − |hb|) ∙ cot αu

- Berm below SWL (hb ≥ 0):

If |hb| > 1.5 ∙ Hm0 toe then only the upper slope is relevant, thus tan α1 =1

cot αu

Else tan α1 =3∙Hm0 toe

Lslope−Bh

with Lslope = (1.5 ∙ Hm0 toe − |hb|) ∙ cot αd + Bh + (1.5 ∙ Hm0 toe + |hb|) ∙ cot αu

A first estimate of the breaker parameter is now made:

ξ1 =

tan α1

(Hm0 toe Lm −1,0⁄ )½

3.7

35

Using all these estimates, a first estimate of the run-up Ru is determined:

Ru2% = 1.65 ∙ Hm0 toe ∙ γb ∙ γβ ∙ γf ∙ ξ1 3.8

with a maximum of Ru max = 1.00 ∙ Hm0 toe ∙ γf ∙ γβ ∙ (4.0 −1.5

√ξ1) 3.9

Ru1 = min (Ru2%; Ru max)

The influence factor for oblique wave attack in the case of run-up is determined as follows [2]:

γβ = 1 − 0.0022 ∙ |β| for 0° ≤ β ≤ 80° 3.10a

γβ = 0.824 for β > 80° 3.10b

The influence factor for roughness γf is given in the CLASH database.

When we have a first estimate of the run-up, the second iteration starts with the influence factor for

a berm γb [2]:

γb = 1 − rB ∙ (1 − rdb) for 0.6 < γb < 1.0 3.6

I. rB = Bh Lberm⁄ the reduction of the average slope caused by the berm width (zero if no berm)

with Lberm = Bh + Hm0 toe ∙ cot αd + Hm0 toe ∙ cot αu,

i.e. the horizontal length between two points on the slope, 1.0 Hm0 toe

above and 1.0 Hm0 toe below middle of berm;


cot αd the cotangent of the slope of the structure downward of the berm and

cot αu the cotangent of the slope of the structure upward of the berm.

II. rdb the reduction of the influence of the berm caused by its depth (zero if berm at SWL)

- Berm above SWL (hb < 0)

If |hb| > Ru1, rdb = 1 (no influence on the run-up and overtopping)

Else rdb = 0.5 − 0.5 ∙ cos (π|hb|

Ru1) ≤ 1

- Berm below SWL (hb ≥ 0)


Else rdb = 0.5 − 0.5 ∙ cos (π|hb|

2∙Hm0 toe) ≤ 1

36

Next the tangent of the slope tan α is now determined as follows [2]:

- Berm above SWL (hb < 0):

If |hb| > Ru1 then only the lower slope is relevant tan α2 =1

cot αd

Else tan α2 =1.5∙Hm0 toe+Ru1

Lslope−Bh

with Lslope = (1.5 ∙ Hm0 toe + |hb|) ∙ cot αd + Bh + (Ru1 − |hb|) ∙ cot αu

- Berm below SWL (hb ≥ 0):

If |hb| > 1.5 ∙ Hm0 toe then only the upper slope is relevant tan α2 =1

cot αu

Else tan α2 =1.5∙Hm0 toe+Ru1

Lslope−Bh

with Lslope = (1.5 ∙ Hm0 toe − |hb|) ∙ cot αd + Bh + (Ru1 + |hb|) ∙ cot αu

Then the breaker parameter can be calculated again using the second estimate of the tangent. The

resulting value is the one that will be used in any further calculations.

So far the proposed two-step procedure, all other variables can be calculated rather straight forward.

The wave length L can be readily calculated as follows [2]:

L0 =g∙Tm−1,0 toe²

2π 3.11

This is the deep water wave length which is prescribed in the EurOtop (2007) manual.

The influence factor for oblique wave attack γβ in the case of overtopping is determined as follows for

smooth slopes (limited roughness) [2]:

γβ = 1 − 0.0033 ∙ |β| for 0° ≤ β ≤ 80° 3.12a

γβ = 0.736 for β > 80° 3.12b

When the influence factor for roughness γf is smaller than 0.9, the influence factor for oblique wave

attack γβ is determined differently [2]:

γβ = 1 − 0.0063 ∙ |β| for 0° ≤ β ≤ 80° 3.13a

γβ = 0.496 for β > 80° 3.13b

The influence factor of a wave wall γv is not considered in this report. As it is only applicable for certain

conditions and only very few data, if not none, will satisfy these conditions.

Now that all the variables can be calculated, the distinction between breaking and non-breaking waves

can be made. To that end, the EurOtop (2007) formulae are used. When the right hand side of equation

37

3.3 for breaking waves is smaller than the right hand side of equation 3.4 for non-breaking waves, the

waves are assumed to be breaking. Otherwise, the waves are supposed to be non-breaking.

3.1.3 Filtering of data

The CLASH database exist out of 10,532 data, whereas the EurOtop (2007) plots together exist out of

a little over 1000 data (considering both breaking and non-breaking waves).

Since the EurOtop (2007) formulae are based on the CLASH database, we try to get more or less the

same plots both in terms of magnitude and scatter.

The following basic filters were applied first, datasets matching one of these criteria were not further

considered:

1. If the angle of wave attack relative to normal on structure β was not provided;

2. If the significant wave height from spectral analysis determined at the toe of the structure

Hm0 toe was not provided – since this parameter is needed for the calculations – or zero –

since in this case there is no wave, stand-alone wave overtopping;

3. If the average wave period from spectral analysis at the toe of the structure Tm −1,0 toe was

not provided, since this is another parameter needed for the calculations and

4. If a remark was given that is was not to be used in the Neural Network, this let us believe

that these data are rather unreliable.

5. If there is no measured overtopping discharge.

Besides, since this chapter treats sloping structures, vertical walls and steep slopes are excluded as

well. Also, the EurOtop (2007) formulae are only valid for breaker parameters ξm −1,0 smaller than 5.

Hence, the following additional filters are proposed such that only data where the following conditions

are valid remain:

- Cotangent slope not considering the berm cot αexcl < 1;

- Cotangent slope downward the berm cot αd < 0.2;

- Cotangent slope upward the berm cot αu < 0.2 and

- Breaker parameter ξm −1,0 < 5

Two remarks here: 1) A typical cotangent for rubble mound structures is 1.5 and 2) If only one of the

two slopes is vertical or steep, they do not fall under the category of ‘steep slopes’ or ‘vertical walls’

as well. They might be considered ‘seawalls’, but this needs a closer look. In this section, however,

these kind of structures are excluded.

38

In a next step, the effect of the complexity factor and reliability factor on the scatter is analyzed, in the

knowledge that:

RF or CF = 1 very good; reliable set-up and measurements; parameters cover cross-section;

RF or CF = 2 good; but some parameters are calculated or assumed or not completely

schematized;

RF or CF = 3 data can be used; but less reliable or very complex structures and

RF of CF = 4 not to be used for Neural Network.

It is decided to exclude data with either a reliability factor or a complexity factor equal to 4. But first,

the effect of both factors on the scatter are analysed, starting with the complexity factor. High

complexity factors lead to some scatter. This is illustrated in Figure 3. 4 for breaking waves. The

EurOtop (2007) formulae are based upon model tests conducted on generic structural types [2]. Hence,

the methods presented in the manual will not predict overtopping performance with the same degree

of accuracy in the case of more complex structures with as a consequence more scatter.

Figure 3. 4 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with its 5% under and upper exceedance limits, effect of CF

Large reliability factors also lead to scatter. This is illustrated in Figure 3. 5 for breaking waves.

39

Figure 3. 5 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with its 5% under and upper exceedance limits, effect of RF

Because of these observations, two additional basic filters are proposed:

6. CF > 1 and

7. RF > 2.

The above two filters, together with the first five filters, will be the seven so-called basic filters that

will be used throughout this work. These 7 filters already halve the CLASH database from 10,532 data

to 4883 data. Applying the 4 other filters such that only sloping structures remain, reduces the dataset

with another 10% to 3620 data. 2655 of these data are supposed to have non-breaking waves, hence

955 data breaking waves. These plots still exists out of a lot more data than the EurOtop (2007) plots.

Besides, there is still a wide band of scatter in Figure 3. 6 for non-breaking waves. Hence further

filtering is suggested.

40

Figure 3. 6 Wave overtopping data for sloping structures, non-breaking waves and equation 3.4 with its 5% under and upper exceedance limits

Analysing the data, leads to the observation that a lot of the remaining data have small roughness

factors γf. Therefore, it was proposed to remove rough structures by excluding data where the

roughness factor γf is smaller than or equal to 0.9. Only smooth, sloping structures remain. The dataset

is further reduced to 1244 data (an additional reduction of 66%).

Non-breaking waves occur when slopes are steeper and wave steepness’s small, at the same time

steeper slopes are mostly rubble mound structures or in other words rough slopes. As a consequence,

excluding rough slopes has the largest impact in the case of non-breaking waves. The size of the dataset

is reduced with 80% (from 2665 to 520) for non-breaking waves. A distinction is made between rough

and smooth slopes in the case of non-breaking waves in Figure 3. 7. It is clear excluding rough slopes

for non-breaking waves reduces the scatter significantly.

41

Figure 3. 7 Wave overtopping data for sloping structures, non-breaking waves and equation 3.4 with its 5% under and upper exceedance limits, effect of rough slopes

Finally, structures with a berm are filtered out. Hence only simple, smooth sloping structures remain.

The resulting datasets for breaking and non-breaking waves are called the minimum datasets. The

filtered data is further reduced to 1003 data (an additional reduction of 19%).

Breaking waves occur when the slope is flat or the wave steepness large (short waves or large wave

heights) and berms are usually built for this type of slopes. Excluding structures with a berm has

therefore the largest effect on the size of the dataset for breaking waves (a reduction of 32%). At the

same time excluding them has as good as no impact in the case of non-breaking waves. In Figure 3. 8

a distinction is made between structures with a berm and simple sloping structures for breaking waves.

42

Figure 3. 8 Wave overtopping data for smooth sloping structures, breaking waves and equation 3.3 with its 5% under and upper exceedance limits, effect of berms

A final thing to do, is plot the minimum datasets together with the EurOtop (2007) data. This is done

respectively in Figure 3. 9 for breaking waves and in Figure 3. 10 for non-breaking waves. For breaking

waves, EurOtop (2007) has approximately 650 data, while our minimum dataset has 492 data. The

minimum dataset has more scatter for small freeboards and less scatter for large freeboards in

comparison to the EurOtop (2007) data.

43

Figure 3. 9 Wave overtopping data for simple, smooth sloping structures, breaking waves and EurOtop (2007) data, equation 3.3 with its 5% under and upper exceedance limits

For non-breaking waves, EurOtop (2007) has approximately 430 data, while our minimum dataset has

511 data. The scatter of the minimum data is again larger for smaller freeboards.

Figure 3. 10 Wave overtopping data for simple, smooth sloping structures, non-breaking waves and EurOtop (2007) data, equation 3.4 with its 5% under and upper exceedance limits

It is also noticed that in both plots some data points overlap.

44

The final filtering scheme is summarized in Figure 3. 11.

Figure 3. 11 Filtering scheme sloping structures

3.1.4 Uncertainty analysis

3.1.4.1 Approach

In the previous section, the filtering scheme used to derive the different datasets is explained. The

three different datasets that will be considered are:

1. Simple, smooth sloping structures (no rough slopes nor berms), the so-called minimum

dataset;

2. Smooth sloping structures (taking into account berms, no rough slopes) and

3. Sloping structures (taking into account berms and rough slopes).

For each of these datasets, the uncertainties are derived in this section.

As mentioned before, the formulae are of the exponential type:

and the reliability of the equation is given by a standard deviation on the parameter b.

Q∗ = a ∙ exp(−b ∙ Rc

∗ ) 3.1

45

Hence the following approach was used to derive the uncertainties:

- First the value for parameter a which represents the intersection point, is estimated by an

exponential trend line through the relevant data set.

It is established that adding a maximum value for the relative freeboard here, decreases the

uncertainties by improving the fit for large relative freeboards. In this way, more weight is given to the

data points with large overtopping which justifies this action. A threshold value of 2.2 is used for

breaking waves and 3 for non-breaking waves. Note that this threshold value is only applied when

determining the trend line.

- Next, the values for the parameter b which represents the slope of the curve, are calculated

and the corresponding mean value and standard deviation is determined using the estimated

value of a:

bi =ln a−ln Qi

∗

Rc,i∗ ;

μb =∑ bi

ni=0

n;

σb = √∑ (bi−μb)2n

i=0

n and σ′b =

σb

μb.

Calculating the value of parameter b, requires dividing by the relative freeboard. Therefore, data with

zero freeboard are excluded in this exercise.

The relative standard deviation is calculated next to the classic standard deviation. Since the relative

standard deviation allows for better comparison of results.

The uncertainties have been presented by confidence bands in EurOtop (2007). There are, however,

other means to present uncertainties, such as a histogram and plotting measured against calculated

values. In this report, histogram will be used to check the normal distribution assumption. If the

parameter b is normally distributed, the frequency histogram should show a bell-shaped curve. The

accuracy of the formula will be checked when plotting the measured against the predicted values of

the relative overtopping discharge Q∗.

Q∗measured vs Q∗ calculated is plotted on a double logarithmic scale where the axes are given by:

Qmeas∗ =

q



tan α∙

1

γb for breaking waves;

Qmeas∗ =

q

√g∙Hm0 toe³ for non-breaking waves;

Qcalc∗ = a ∙ exp(−μb ∙ Rc

∗ ).

The 1:1 slope is indicated in the plots. The data points should be located along this slope.

46

A frequency histogram is constructed for ∆b′i:

∆b′i =μb−bi

μb

The vertical axis gives the ratio between the amount of data within a certain interval of ∆b′ and the

total amount of data, p. The mean value of ∆b’ and its 90% confidence interval assuming a normal

distribution are also indicated on the plots. The standard deviation of ∆b’ is equal to the relative

standard deviation of the parameter b, σ′b . Hence, the 90% confidence interval is obtained as follows:

μ∆b′ ± 1.64 ∙ σ∆b′ = μ∆b′ ± 1.64 ∙ σ′b

All of the above calculations are done in Excel.

3.1.4.2 Results

The results can be summarised as provided below. All the corresponding plots can further be found in

Appendix A. The following list provides the formula for breaking waves with first the EurOtop (2007)

formula in bold and then the results for each of the three data sets discussed before. At the end the

size is of the dataset is provided as well as the size when data with zero freeboard are excluded.

Breaking 𝐐∗ = 𝟎. 𝟎𝟔𝟕 ∙ 𝐞𝐱𝐩(−𝟒. 𝟕𝟓 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟓 and 𝛔′𝐛 = 𝟏𝟎. 𝟓% 3.3

1. Q∗ = 0.0769 ∙ exp(−4.8759 ∙ Rc∗) with σb = 0.6763 and σ′b = 13.9% (492/483 data)



The same can now be done for non-breaking waves.

Non-breaking 𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟑𝟓 and 𝛔′𝐛 = 𝟏𝟑. 𝟓% 3.4

1. Q∗ = 0.2249 ∙ exp(−2.6015 ∙ Rc∗) with σb = 0.4909 and σ′

b = 18.9% (511/506 data)



3.1.4.3 Conclusion

In the case of breaking-waves, the scatter visually increases the most when adding slopes with a berm

(Figure 3. 8, Figure 3. 12). The standard deviation increases correspondingly when structures with a

berm are included in the analysis. Including rough slopes does not increase the scatter in the plot

(Figure 3. 12), but does increase the size of the dataset with the same magnitude as the berms. Hence,

more data is included, less scatter is observed resulting in a decrease in the standard deviation. The

differences in uncertainties are, however, not significant for both. All of the standard deviations are

the same order of magnitude. This is reflected in the histograms where the width of the 90%

47

confidence band is approximately the same for all three considered datasets. The corresponding

histograms can be found in Appendix A.

Figure 3. 12 Wave overtopping data for sloping structures, breaking waves and equation 3.3 with its 5% under and upper exceedance limits, effect of rough slopes

All of the histograms show more or less a bell-shaped curve indicating a normal distribution. The

hypothesis of a normal assumption seems therefore valid. The histogram of the third dataset, sloping

structures (taking into account berms and rough slopes), for breaking waves is given as an example in

Figure 3. 13.

Figure 3. 13 Histogram ∆b’ for sloping structures, breaking waves with its mean value and 90% confidence interval

48

In the measured against predicted relative overtopping plots (Appendix A), the data is located around

the 1:1 slope. A better fit is desired in the zones with large discharges as they represent more

dangerous situations. Furthermore, data points are preferably located above the 1:1 slope, especially

for large discharge, since in this case the measured overtopping is smaller than the predicted one. The

measured vs calculated relative discharge plot for the minimum dataset (simple, smooth sloping

structures), for breaking waves is given in Figure 3. 14.

Figure 3. 14 Measured against predicted relative overtopping for simple, smooth sloping structures, breaking waves

For non-breaking waves, there are only a few structures with a berm. Including slopes with a berm,

increases the size of the dataset only with 1.7%. As a consequence, the differences in the results are

negligible. Adding rough slopes, on the other hand, increases the scatter significantly, see Figure 3. 7,

with as a result a considerably larger standard deviation. This is reflected by a wider 90% confidence

interval in the histograms (Appendix A).

Again, all of the histograms show more or less a bell-shaped curve. In the measured against predicted

relative discharge plots no problems are identified either, the data points are located along the 1:1

slope with a slightly better fit for large discharges.

For both breaking and non-breaking waves, the derived uncertainties are larger than the ones given in

the EurOtop (2007) manual. Also, the relative standard deviation is generally larger for non-breaking

waves. Note that the standard deviation, on the hand, is smaller for non-breaking waves. The smaller

standard deviation simply results from the smaller mean value of the parameter b. Therefore, relative

49

standard deviations are considered too as they allow for better comparison in between different

formulae.

3.2 EurOtop (2007) formulae applied to CLASH and UG data

3.2.1 Introduction

In this section, the influence of new data on the previously derived uncertainties is investigated. To

that end, more recent data collected by the University of Ghent is added to the existing CLASH

database and the uncertainty analysis is repeated. More specifically, this concerns the datasets UG10,

UG13 and UG14 which were each discussed in Chapter 2. For the most part these datasets describe

steeper slopes.


The EurOtop (2007) formulae are considered again, therefore the same calculation procedure as in

Section 3.1.2 can be used.


The same filtering scheme is used as in Section 3.1.3.

Applying the seven basic filters on the UG datasets has no effect:

1. If the angle of wave attack relative to normal on structure β was not provided;

Even if the angle of wave attack β is not always provided, it is known that this parameter was always

zero during these tests.

2. If the significant wave height from spectral analysis determined at the toe of the structure

Hm0 toe was not provided – since this parameter is needed for the calculations – or zero –

since in this case there is no wave, stand-alone wave overtopping;

The significant wave height is always provided and has a value different from zero.

3. If the average wave period from spectral analysis at the toe of the structure Tm −1,0 toe was

not provided, since this is another parameter needed for the calculations;

The average wave period is always provided.

4. If a remark was given that is was not to be used in the Neural Network, this let us believe

that these data are rather unreliable;

No such remarks occur in the UG data.

5. q = 0 m³/m∙s.

There is no data included where there is no overtopping measured.

50

6. CF > 1 and

7. RF > 2.

Both the reliability factor and complexity factor are always equal to one.

Hence, the 7 basic filters do not reduce the UG datasets.

Next, vertical and steep structures are excluded by applying the following filters, such that only data

satisfying the following conditions remain:

1. Cotangent slope not considering the berm cot αexcl < 1;

2. Cotangent slope downward berm cot αd < 0.2 and

3. Cotangent slope upward berm cot αu < 0.2.

As a final filter, because the EurOtop (2007) formulae are only applicable for a certain range of breaker

parameters ξm −1,0, we have:

4. Breaker parameter ξm −1,0 < 5.

These 4 filters reduce the dataset from the total of 1039 data to 280 (26.9% of the total dataset).

All the UG data can be added to the minimum datasets, since no cases with berms occur nor any cases

with roughness. In Figure 3. 15 the minimum dataset from the CLASH database is plotted together with

the UG data for breaking waves. Only 9 data in the UG data fall under the breaking waves regime (the

UG data have steeper slopes). Also, it is noticed that the UG data have smaller relative freeboards.

51

Figure 3. 15 Wave overtopping data for simple, smooth sloping structures, CLASH and UG, breaking waves and equation 3.3 with its 5% under and upper exceedance limits

For non-breaking waves, the same plot is made (Figure 3. 16). The UG datasets especially treat steeper

slopes, therefore few cases occur for breaking waves and at the same time many cases for non-

breaking waves. Also here, the UG data increase the amount of data with smaller relative freeboards.

Figure 3. 16 Wave overtopping data for simple, smooth sloping structures, CLASH and UG, non-breaking waves and equation 3.4 with its 5% under and upper exceedance limits

52


3.2.4.1 Approach

Because it are still the EurOtop (2007) formulae that are being considered, the same approach as in

Section 3.1.4.1 is used.

The discussed datasets, however, differ. Since all the UG data can be added to the minimum datasets,

it is decided to compare the results for these minimum datasets from the CLASH database with the

results from the minimum datasets together with the UG data.

All the corresponding plots are given in Appendix A.

3.2.4.2 Results

In the following, the results are summarized both for breaking as for non-breaking waves. The first

result given each time corresponds to the previously obtained result for the minimum dataset in

Section 3.1.4.2. The second result given is the one for the minimum dataset from CLASH together with

the UG data.

Breaking 𝐐∗ = 𝟎. 𝟎𝟔𝟕 ∙ 𝐞𝐱𝐩(−𝟒. 𝟕𝟓 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟓 and 𝛔′𝐛 = 𝟏𝟎. 𝟓% 3.3



Non-breaking 𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟑𝟓 and 𝛔′𝐛 = 𝟏𝟑. 𝟓% 3.4



3.2.4.3 Discussion

For breaking waves, the relative standard deviation increases due to the UG data. An even higher value

is obtained than the previously obtained maximum. While there is only little additional data and

moreover the UG data do not increase the scatter in the plot (Figure 3. 15). The UG data do, however,

increase the amount of data with small relative freeboards. Due to the small relative freeboards, the

calculated b values increase:

bi =ln a−ln Qi

∗

Rc,i∗

Values up to 11.45 are observed. Note with small freeboards correspond usually large discharges. The

larger the relative discharge, the smaller the nominator. However, this effect is less pronounced. Large

values for b, clearly, have an impact on the standard deviation. As a consequence, it is concluded that

the formula fits less good for data with small relative freeboards. Therefore, it is proposed to limit the

53

range of validity of the formula. If a minimum value of 0.2 for the relative freeboard is introduced, the

standard deviation already decreases to 0.6809:

Q∗ = 0.0753 ∙ exp(−4.8641 ∙ Rc∗) with σb = 0.6809 and σ′b = 14.0%

In the corresponding histogram a wider confidence band is detected indicating the larger uncertainty.

With regard to the normal distribution, the assumption still seems valid (Appendix A).

When the calculated values are plotted against the predicted values, Figure 3. 17, the data with large

discharges originating from the UG data, are located above the slope. Hence the calculated

overtopping is larger than the measured one, which is conservative. The over prediction for data with

small relative freeboards may also be observed in Figure 3.15 where the data with small relative

freeboards is located below the EurOtop (2007) curve.

Figure 3. 17 Measured vs calculated relative overtopping for simple, smooth sloping structures, CLASH and UG, breaking waves

For non-breaking waves, the same phenomena can be observed, only even stronger. The relative

standard deviation increases considerably due to the UG data. Again, the small relative freeboards are

the reason for this, since the scatter in the plot does not increase significantly. Limiting the validity

range is a solution. When a minimum value of 0.5 is assigned for the relative freeboard, the following

results are found:


54

Hence, the standard deviation decreases back to the same order of magnitude as before the UG data

were added. Note that the standard deviation even gets slightly smaller than the before occurring

minimum value. The reason for this is that there is more data, but not more scatter.

The larger uncertainty is reflected in the histogram by a wider confidence band. The histogram also

shows a bell-shaped curve again indicating a normal distribution (Appendix A).

Finally, when the calculated values are plotted against the predicted values of the relative overtopping,

there is more data for small discharges than before. These data points are, however, located close to

the 1:1 slope but more below it, indicating an over prediction (Appendix A).

3.3 Van der Meer and Bruce (2014) formulae applied to CLASH and UG data

3.3.1 Introduction

In this section, the uncertainties of the more recent formulae by Van der Meer and Bruce (2014) are

derived for the CLASH and UG data together. Whereas the EurOtop (2007) formulae are of the

exponential type, the new formulae are of the Weibull type:

Q∗ = a ∙ exp(−(b ∙ Rc∗ )c) 3.14

The only difference here with the exponential equations is the fitted coefficient c, which is a constant

in the Van der Meer and Bruce (2014) equations equal to 1.3.

The revised formulae are supposed to fit the data better for data points at very low and zero freeboard,

The EurOtop (2007) formulae over predict for these data. The better fit is illustrated in Figure 3. 18

where the Van der Meer and Bruce (2014) formula is plotted together with the EurOtop (2007) formula

for breaking waves. The data points for larger relative freeboards are still located along the curve too.

55

Figure 3. 18 Wave overtopping data for simple, smooth sloping structures, CLASH and UG data, breaking waves, comparison equation 3.3 and equation 3.15


The overtopping formulae by Van der Meer and Bruce (2014) for sloping structures are given by [5]:

q

√g∙Hm0³=

0.023

√tan α∙ γb ∙ ξm−1,0 ∙ exp (− (2.7 ∙

Rc

ξm−1,0∙Hm0∙γb∙γf∙γβ∙γv)

1.3

) 3.15

with a maximum of q

√g∙Hm0³= 0.09 ∙ exp (− (1.5 ∙

Rc

Hm0∙γf∙γβ)

1.3

) 3.16

The reliability of the first equation is given by σ(0.023) = 0.003 and σ(2.7) = 0.20, and of the second one

by σ(0.09) = 0.013 and σ(1.5) = 0.15 [5]. Both parameters a and b are considered as stochastic

parameters instead of just the parameter b as seen in EurOtop (2007).

The nondimensionalization of the axes give the same results as for the EurOtop (2007) formulae.

Therefore the same calculation procedure can again be followed for the most part as in Section 3.1.2

The only difference is the distinction between breaking and non-breaking waves. Equations 3.15 and

3.16 are this time used to make the distinction instead of equations 3.3 and 3.4.


The same filtering scheme is followed as in Section 3.1.3. However, due to the different distinction

between breaking and non-breaking waves, the ratio between breaking and non-breaking waves will

differ. Resulting each time in more cases for breaking waves.

56

There is no maximum value for the breaker parameter indicated for the Van der Meer and Bruce (2014)

formulae, but since we want to compare the results with the ones previously obtained, this filter is

applied again such that the same data is considered as in the previous sections.


3.3.4.1 Approach

The different datasets from CLASH considered here are again the following:

1. Simple, smooth sloping structures (no rough slopes nor berms), the so-called minimum

dataset;

2. Smooth sloping structures (taking into account berms) and

3. Sloping structures (taking into account berms and rough slopes).

This time, the UG data is included to all datasets. The uncertainties are then derived for each dataset.

The formulae by Van der Meer and Bruce (2014) are of the Weibull type:

Q∗ = a ∙ exp(−(b ∙ Rc∗)1.3) 3.17

The reliability of the equation is given by a standard deviation on both parameters a and b. In this

report, we will stick to a standard deviation on parameter b, hence assuming parameter a is a constant.

This will allow us better to compare the results obtained in the different sections.

Because of the different formulae, a slightly different approach is used to derive the uncertainties:

- First the value for a is estimated by an exponential trend line through the relevant data set.

Again, a maximum value will be used for the threshold when estimating the value of the parameter a.

A threshold value of 1.8 is used for breaking waves and 2.5 for non-breaking waves. This time also a

minimum freeboard is proposed, the reason for this is explained later on, for breaking waves a value

of 0.1 is proposed, while for non-breaking waves 0.2 is used. The maximum value is only applied when

determining the trend line, the minimum value, however, throughout the whole exercise.

- With this value for a, the values of b can be calculated as well as its mean value and its standard

deviations:

bi =(ln a−ln Qi

∗)1/1.3

Rc,i∗ ;

μb =∑ bi

ni=0

n;

σb = √∑ (bi−μb)2n

i=0

n and σ′b =

σb

μb.

57

Again, histograms and measured against calculated values plots are constructed too. Where in the

latter, the axes are determined as follows:

Qmeas∗ =

q



tan α∙

1

γb for breaking waves;

Qmeas∗ =

q

√g∙Hm0 toe³ for non-breaking waves;

Qcalc∗ = a ∙ exp(−(μb ∙ Rc

∗ )1.3).

The applicability of the formulae is limited in this exercise by a minimum value for the relative

freeboard. Because for small relative freeboards, the relative discharge is larger than the estimated

value for the parameter a, as a consequence the parameter b cannot be calculated:

(ln a − ln Qi∗)1/1.3

since the value between brackets would become negative and thus the solution a complex value. This

can more easily be understood in the following comparison:

√−|𝑥| = (−|𝑥|)1/2

By trial-and-error, threshold values were proposed for the relative freeboard. In the case of breaking

waves there is chosen for a value of 0.1, in the case of non-breaking waves 0.2.

3.3.4.2 Results

The results are summarized below, the amount of data used for the analysis is indicated too:

Breaking 𝐐∗ = 𝟎. 𝟎𝟐𝟑 ∙ 𝐞𝐱𝐩[−(𝟐. 𝟕 ∙ 𝐑𝐜∗)𝟏.𝟑] 3.15

1. Q∗ = 0.0250 ∙ exp[−(2.7235 ∙ Rc∗ )1.3] with σb = 0.4000 and σ′b = 14.7% (547 data)



Non-breaking 𝐐∗ = 𝟎. 𝟎𝟗 ∙ 𝐞𝐱𝐩[−(𝟏. 𝟓 ∙ 𝐑𝐜∗)𝟏.𝟑] 3.16




Note that for the first two datasets for non-breaking waves, one data point 031 from the UG datasets

is excluded. This point has a relatively large discharge in relation to its relative freeboard which is also

rather small. As a result, this point gives problems in the calculations (Qi∗ > a) and requires a large

minimum value for the relative freeboard. Since for both datasets it is the only point that gives a

problem, it is decided to exclude it instead of increasing the minimum value and limit the applicability

of the formula even further.

58

3.3.4.3 Discussion

The relative standard deviations are the same order of magnitude as the ones obtained in Section 3.1

for the EurOtop (2007) formulae considering CLASH data, with that difference that the UG data is also

considered here. In Section 3.2 it is determined that the UG data increase the uncertainties because

of their small relative freeboards. The applicability of the formulae was limited by adding a minimum

value for the relative freeboard in order to restore the uncertainties back to the original level. For the

new formulae, the applicability of the formulae is limited again in order to improve the uncertainties.

However, the minimum values required to restore the uncertainties back to their original level, are

smaller for the Van der Meer and Bruce (2014) formulae. Hence, they give a better fit for small relative

freeboards, but the applicability of the formulae still needs to be limited.

This better fit for small relative freeboards and thus large discharge is also reflected in the measured

against predicted relative discharges plot (Appendix A). This is illustrated in Figure 3. 19 for the

minimum dataset of CLASH together with the UG data for breaking waves.

Figure 3. 19 Measured vs predicted relative overtopping for simple, smooth sloping structures, CLASH and UG, breaking waves

Otherwise, the same trends are observed in between the different datasets as in Section 3.1.

3.4 Summary results

The uncertainty is larger for non-breaking waves than for breaking waves. For both breaking and non-

breaking waves, the derived uncertainties are larger than the ones indicated in the EurOtop (2007)

manual.

59

Breaking waves occur when slopes are steeper, these type of structures typically do not have a berm

and are often rubble mound structures (rough slopes). Therefore adding slopes with a berm has no

impact on breaking waves, but adding rough slopes does. The scatter is very large for the latter case.

There is a lot of data with small roughness factors. For breaking waves, the differences in uncertainty

are less significant.

Due to the UG data, reliability of the formulae decreases both for breaking as for non-breaking waves.

It has, however, the strongest effect on non-breaking waves, since the UG data concerns steeper

slopes. The reason for the increased uncertainties is that the UG data has generally smaller relative

freeboards. Deviations in calculated values for b are the largest for smaller relative freeboards, hence

the standard deviation increases. It is concluded that the formulae fit less good for data points with

small relative freeboards. Limiting the applicability of the formulae by adding a minimum value for the

relative discharge, is an option to improve the reliability of the considered formula.

The derived uncertainties for the new formulae are the same order of magnitude as the ones derived

in the first phase for the EurOtop (2007) formulae considering only CLASH data. While the UG data is

included in the analysis for the new formulae. The UG data has smaller relative freeboards and

therefore increases the uncertainty. The applicability of the EurOtop (2007) formulae is reduced in

order to restore the reliability of the formula back to its original level before the UG data was added.

For the new formulae, the applicability needs to be limited to get a reliability of the same order of

magnitude. However, the minimum relative freeboard required in order to reach this, is smaller.

Hence, the new formulae give a better fit for small relative freeboards meanwhile maintaining the

good fit for larger freeboards. But the applicability of the formulae still needs to be limited.

With regard to the assumed normal distribution of the parameter b, this hypothesis seems acceptable

in all the histograms as they all show more or less a bell-shaped curve.

The considered relative freeboard is for some formulae limited when determining the trend line.

Because of that, there is a better fit for data with large freeboards and the corresponding uncertainty

will decrease. By doing this, more weight is added to data with large overtopping which is conservative.

3.5 Discussion

3.5.1 Local wave length

In EurOtop the deep water wave length is proposed. However, theoretical more correct, the local wave

length should be calculated based on linear wave theory. This depends both on the mean wave period

from spectral analysis at the toe of the structure Tm−1,0 toe and on the local water depth at the toe h.

60

The local wave length could be calculated as follows. First, the wave length is calculated according to

the approximation by Fenton and McKee, 1990:

LFM = L0 ∙ [tanh (2π∙h

L0)

3

4]

2

3

3.18

with L0 =g∙Tm−1,0 toe²

2π the wavelength in deep water.

Next, depending on the ratio of the local water depth at the toe h and the approximated wave length

LFM, the wavelength is given by:

Shallow water if h

LFM<

1

25 then Lm −1,0 toe = Tm −1,0 toe ∙ √g ∙ h

Deep water if h

LFM>

1

2 then Lm −1,0 toe = L0

In the transition zone, iterative computations are required:

xi = xi−1 − [(c1 ∙ tanh (c2

xi−1) − xi−1) (

−c1∙c2

xi−12 ∙cosh(

c2xi−1

)2

−1

)⁄ ]

with c1 = L0 and c2 = 2 ∙ π ∙ h

x0 = LFM, iterations until |xi−1 − xi| < 0.001, then Lm −1,0 toe = xi.

This wave length L has in the first place an immediate influence on the breaker parameter ξ that will

reduce as a consequence of using the local wave length. Hence, more data are considered to have

breaking waves due to the local wave length.

It does also have an influence on the plot for breaking waves, as it occurs in both axes. A different wave

length does not have an influence on the non-breaking waves plots. When plotting the results from

before based on the deep water wave length together with the resulting values when using the local

wave length for breaking waves, the points are shifted upwards due to usage of the local wave length.

Thus the points are shifted to larger overtopping, hence indicating that the existing formulae are not

safe. Figure 3. 20 shows this effect for the minimum dataset of CLASH for breaking waves.

61

Figure 3. 20 Wave overtopping data for simple, smooth sloping structures, breaking waves and equation 3.3 with its 5% under and upper exceedance limits, comparison deep water and local wave length

3.5.2 Distinction regime

We have done the distinction between breaking and non-breaking based on the EurOtop (2007)

formulae or the new formulae by Van der Meer and Bruce (2014) without updating the parameter

occurring in them. Even though, we updated these parameters in the uncertainty analysis. We have

chosen to the distinction like this since it should not depend on parameters.

Others use the breaker parameter to make the distinction between breaking and non-breaking.

In practice, there are a lot of factors that have an influence on this distinction, so the real distinction is

less straightforward. The approach that is used in this work is hence not ideal either, however, we

believe it is more consistent than other approaches.

3.5.3 Uncertainty analysis approach

In our approach, we get the value for the parameter a out of the trend line and further consider it as

a constant. Parameters a and b are interdependent and treating one as a constant, does have an impact

on the standard deviation.

The goal of this work is to derive the uncertainties only, therefore the approach we follow, aligns with

the one EurOtop (2007) used, where also only parameter b is considered stochastic.

Note that another option could have been keeping the value of coefficient a out of the formula.

62

3.5.4 Influence factor for roughness

A lot of data in the CLASH database have small roughness factors. Rough slopes lead to a lot of scatter

for non-breaking waves. This let us believe that either these data are less reliable or that the roughness

factor is determined wrongly.

It is known that the roughness factor only takes into account the types of stones on the surface. The

way the stones are set and the underlying layers clearly also have an influence on the roughness and

permeability of the slope.

3.5.5 Normality tests

Regarding the assumption of a normal distribution, a histogram is just one of the means to check this.

The histograms do not contradict the hypothesis of a normal distribution.

Other ways to check for the normality of the distribution is a Shapiro-Wilk test or normal probability

plots.

The null hypothesis in the Shapiro-Wilk test is that the considered sample, here the calculated b values,

(x1, x2, … , xn) comes from a normally distributed population. The test statistic is

W =(∑ ai ∙ x(i))²n

i=1

∑ (xi − x)²ni=1

with x(i) the i’th smallest number in the sample

x the sample mean

ai the weight based on the value of n, these can be found in Shapiro-Wilk table

The smaller the resulting the p-value (i.e. the probabilty the null hypothesis will be accepted), the more

it appears that the considered sample does not come from a normally distributed sample.

Carrying out Shapiro-Wilk tests in R, resulted in very small p-values for each of the considered datasets.

The results of these tests can be found in Appendix A. The Shapiro-Wilk is, however, just one of the

tests available. Moreover, a well-known issue with these tests is that for large amounts of data, very

small deviations from normality can be detected. With as a consequence the larger your sample, the

larger the chances of rejecting the null hypothesis becomes. Hence, it is no coincidence that the p value

reaches it maximum value for the minimum dataset in CLASH for breaking waves, since this is the

smallest dataset.

Finally, also normal probability plots are constructed in R for all the above obtained results, they can

be found in Appendix A as well. The points should be located along the straight line, which is clearly

not the case.

63

Hence the assumption of a normal distribution is taken into question. For practical purposes, however,

the assumption of a normal distribution seems acceptable when only the histograms are considered.

Furthermore, the goal here it not to look into the correct distribution of the stochastic parameter, but

to find the right formula such that the distribution of the data around the curve is normally distributed.

64

65

Chapter 4: Vertical structures

4.1 EurOtop (2007) formulae applied to CLASH data

4.1.1 Introduction

For vertical structures a distinction is made between two regimes: the impulsive regime and the non-

impulsive regime. Whereas for sloping structures, all formulae are exponential, the formula in the

impulsive regime for vertical structures is of the power law-type. The non-impulsive formula is of the

exponential type and can thus be treated similar as the formulae discussed in Chapter 3 for sloping

structures.

The power law formula in the impulsive regime can be described as follows:

Q∗ = a ∙ Rc∗ −b 4.1

where Q∗ and Rc∗ were made non-dimensional according to EurOtop (2007). The scatter here is

described in the logarithm of the data, assuming parameter a is stochastic and b a constant.

The reliability of the formula is presented by a confidence band in the EurOtop (2007) plots. The scatter

about the mean prediction is described by a standard deviation in the logarithm of the data σlog a. The

90% confidence interval can then be constructed as follows:

Q∗ = (a ×/÷ 101.64∙σlog a) ∙ Rc∗ −b 4.2

The power law equation gives a curved line in a log-linear graph. The EurOtop (2007) plot for wave

overtopping data in the impulsive regime at plain vertical walls is given as an example in Figure 4. 1.

Figure 4. 1 Wave overtopping data for plain vertical walls in the impulsive regime and the EurOtop (2007) equation with its 5% under and upper exceedance limits [2]

66


Depending on the type of vertical wall, plain vertical wall or composite vertical wall, a different

calculation procedure is given in the EurOtop (2007) manual. Both calculation procedures are

explained, starting with plain vertical walls.

In the case of plain vertical walls, the distinction between the two regimes (impulsive/non-impulsive)

is based on the following “impulsiveness” parameter h∗ [2]:

h∗ = 1.35 ∙h

Hm0 toe∙

2π ∙ h

g ∙ Tm−1,0 toe² 4.3

with h the water depth at the toe.

Non-impulsive conditions dominate at the walls when h∗ is larger than 0.3, impulsive conditions occur

when h∗ is smaller or equal to 0.2 [2]. In the transition zone, the “worst-case” is assumed, in other

words the regime where the calculated relative overtopping discharge Q∗ is maximum.

EurOtop (2007) then gives the following equation and its corresponding validity range in the non-

impulsive regime [2]:

q

√g∙Hm0³= 0.04 ∙ exp (−2.6 ∙

Rc

Hm0) valid for 0.1 <

Rc

Hm0< 3.5 4.4

The coefficient of 2.6 for the mean prediction has an associated standard deviation of σ = 0.8 [2].

In the impulsive regime, the following equation and its corresponding validity range is given [2]:

q

h∗²∙√g∙h³= 1.5 ∙ 10−4 ∙ (h∗ ∙

Rc

Hm0)

−3.1 valid for 0.03 < h∗ ∙

Rc

Hm0< 1.0 4.5


deviation of σlog a = 0.37 [2].

When there is a significant mound, the structure is treated as a composite vertical wall. A different

calculation procedure is then specified. For composite vertical walls, the distinction between the two

regimes is based on the “impulsiveness” parameter d∗ [2]:

d∗ = 1.35 ∙

hc

Hm0 toe∙

2π ∙ h

g ∙ Tm−1,0 toe² 4.6


hc the water depth on the berm/toe.

67

Non-impulsive conditions dominate when d∗ is larger than 0.3, impulsive conditions occur when d∗ is

smaller or equal to 0.2 [2]. In the transition zone, overtopping is predicted in both regimes and the

larger value assumed.

When non-impulsive conditions prevail, overtopping can be predicted by the standard method given

previously for non-impulsive conditions at plain vertical structures [2].

For conditions determined to be impulsive, a modified version of the impulsive prediction method for

plain vertical walls is recommended to account for the presence of the mound by use of d and d∗ [2]:

q

d∗²∙√g∙h³= 4.1 ∙ 10−4 ∙ (d∗ ∙

Rc

Hm0)

−2.9 valid for 0.05 < d∗ ∙

Rc

Hm0< 1.0 4.7



Finally, the effect of oblique waves is only considered in the non-impulsive regime, by using an adjusted

version of the previously given equation 4.4:

q

√g∙Hm0³= 0.04 ∙ exp (−2.6 ∙

1

γβ∙

Rc

Hm0)

where γβ is the influence factor for the angle of wave attack is given by [2]:

γβ = 1 − 0.0062 ∙ |β| for 0° ≤ β ≤ 45° 4.8a

γβ = 0.72 for β > 45° 4.8b

with β is the angle of wave attack relative to the normal, in degrees.


First the seven basic filters defined in Section 3.1.3 are applied again, this reduces the CLASH database

from 10,532 data to 4883 data. Since vertical structures are treated in this chapter, sloping structures

as well as overhanging structures are filtered out. Because of the different calculation procedures

depending on the type of structure, a distinction is made between plain vertical walls and composite

vertical walls. Both should have a vertical slope upward the berm, therefore the following filter is

applied first:

Cotangent slope upward the berm cot αu ∈ [−0.2,0.2].

764 data remain after applying the above filter, which is only 7.3% of the total CLASH database and

considerably smaller than the amount of data that is considered for sloping structures.

68

Plain vertical walls should also have a vertical slope downward the berm. In this context, plain vertical

walls are considered to have no berm or toe at all, even not an insignificant one. Therefore, additional

filters are applied in order to distinguish plain vertical structures, such that only data satisfying the

following conditions remain:

Cotangent slope downward the berm cot αd ∈ [−0.2,0.2];

The width of the toe Bt = 0.0 m and

The width of the berm B = 0.0 m.

Only plain vertical structures remain, we have 236 data from which 152 are assumed to be in the

impulsive regime and 84 in the non-impulsive regime.

Every structure with a berm or toe, is considered a composite one. Composite vertical walls with a

berm can be filtered out according to the following conditions:

The width of the berm B ≠ 0.0 m;

The width of the toe Bt = 0.0 m and

The water height on the berm hb > 0 m (berm below SWL).

Composite vertical walls with a toe are distinguished using these conditions:

Cotangent slope downward the berm cot αd ∈ [−0.2,0.2];

The width of the toe Bt ≠ 0.0 m

The width of the berm B = 0.0 m

As a result, 324 data are filtered out as composite vertical walls, i.e. vertical structures with either a

berm or a toe.

The resulting filtering scheme is summarized in Figure 4. 2.

69

Figure 4. 2 Final filtering scheme

In the above filtering scheme all vertical structures that have a mound are considered composite

vertical walls. In the EurOtop (2007) manual structures with a mound are only treated as considered

composite vertical structures if the mound is significant.

In all the EurOtop (2007) plots for vertical structures, the overtopping data used is given. For plain

vertical walls datasets from CLASH are indicated. For composite vertical walls VOWS (Violent

Overtopping of Waves at Seawalls) data is used. This gives the impression that the data used, VOWS

data, is not included in the CLASH database.

Furthermore, when all the data filtered out as composite vertical structures according to the filtering

scheme in Figure 4. 2 is assumed to have a significant mound, a large part of the data do not fit the

EurOtop (2007) equations for composite vertical structures. The resulting plot in the impulsive regime

is given in Figure 4. 3. Only one dataset 505-__ fits more or less the EurOtop (2007) equation.

When the same data is treated as plain vertical structures, hence assuming the mound is insignificant,

the resulting plot in the impulsive regime looks much better. The data points fit the corresponding

EurOtop (2007) equation, including these from dataset 505-__.

Because of the above mentioned reasons, all the filtered data are treated as plain vertical structures

from now on.

70

Figure 4. 3 Wave overtopping data for composite vertical walls in the impulsive regime and equation 4.7 with its 5% under and upper exceedance limits, dataset 505-__ highlighted

Further analysis of the filtered data indicates a couple of datasets that lead to inexplicable scatter.

In the plot of wave overtopping data in the non-impulsive regime, Figure 4. 4, there is scatter below

the 5% under exceedance limit from EurOtop (2007). Datasets 355-__ and 356-__ are the reason of

this scatter. In the impulsive regime, the same conclusion can be made concerning dataset 354-__. No

reason could be assigned for this scatter, therefore these datasets are excluded for further analysis.

71

Figure 4. 4 Wave overtopping data for vertical walls in the non-impulsive regime and equation 4.4 with its 5% under and upper exceedance limits, scatter highlighted

As mentioned before, the datasets used to construct the EurOtop (2007) plots are indicated. This

allows for comparison between the EurOtop (2007) data and our filtered data.

The datasets used in the EurOtop (2007) plots for plain vertical walls are:

Non-impulsive 028-__, 106-__, 224-__, 225-__, 351-__, 402-__, 502-__ and

Impulsive 028-__, 224-__, 225-__, 351-__, 502-__, 802-__.

The datasets we filtered out as plain vertical walls (and as composite vertical walls) are:

Non-impulsive 224-__, 351-__, 402-__, 502-__, 503-__, 504-__, 507-__

(+ 043-__, 105-__, 223-__, 229-__, 505-__, 509-__, 914-__) and

Impulsive 224-__, 351-__, 502-__, 503-__, 504-__, 507-__, 802-__

(+ 043-__, 505-__, 509-__).

Three datasets from the datasets considered in the EurOtop (2007) plots are not included in our

filtered data. Each of these datasets have a high reliability factor and are therefore filtered out due to

our filtering scheme.

72


4.1.4.1 Approach

In the previous section, the filtering scheme used to derive the different datasets is explained. For the

uncertainty analysis, two different datasets in each regime (impulsive/non-impulsive) are treated:

1. Plain vertical walls and

2. Plain vertical walls and composite vertical walls.

In the non-impulsive regime the EurOtop (2007) formula is of the exponential type. How to derive the

uncertainties, is already explained in the previous chapter (cfr. Section 3.1.4.1). The only difference is

the maximum relative freeboard considered when determining the trend line, which is 2.5 here.

In the impulsive regime, the formula is of the power law-type:

Q∗ = a ∙ Rc∗ −b 4.1

with the reliability of the equation described with a standard deviation on the parameter a for the

logarithm of the data. Hence a different approach is defined to derive the uncertainties:

- First the value of b is estimated by a power law trend line through the relevant data set.

- Next, the value of parameter a are calculated for each data points together with the

corresponding mean value and standard deviation using the estimated value of b:

ai = Q∗ (Rc∗ )−b⁄ ;

μa =∑ ai

ni=0

n;

σa = √∑ (ai−μa)2n

i=0

n and σ′a =

σa

μa.

The uncertainties are presented by confidence bands in EurOtop (2007). Two additional means are

used in this work to analyse the uncertainties: a histogram and a measured against calculated values

plot.

The relative discharge Q∗measured and the relative discharge Q∗ calculated are plotted on a double

logarithmic scale where in the impulsive regime the axes are determined as follows:

Qmeas∗ =

q

h∗²∙√g∙h³

Qcalc∗ = μa ∙ (Rc

∗ )−b

73

For clarity, the axes to be used in the non-impulsive regime are also given since they differ from the

ones given in Section 3.1.4.1:

Qmeas∗ =

q

√g∙Hm0 toe³

Qcalc∗ = a ∙ exp(−μb ∙ Rc

∗ )

The histogram is constructed analogously as the previously obtained histograms in Chapter 3, the only

difference the parameter being considered.

4.1.4.2 Results

The results are summarized below. The following list provides the EurOtop (2007) formula for the non-

impulsive regime in bold followed by the results for the two datasets discussed, being the dataset with

only plain vertical walls and the dataset with both plain and composite vertical walls.

Filtering out plain vertical walls, resulted in 84 data in the non-impulsive regime. However, the

considered formula is only valid for a certain range of relative freeboards, as a consequence only 80 of

the 84 data remain. The same applies to the others datasets. The amount of data used for the analysis

is indicated in between brackets at the end.

Non-impulsive 𝐐∗ = 𝟎. 𝟎𝟒 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟖 and 𝛔′𝐛 = 𝟑𝟎. 𝟖% 4.4

1. Q∗ = 0.049 ∙ exp(−2.5646 ∙ Rc∗ ) with σb = 0.3357 and σ′b = 13.1% (80 data)

2. Q∗ = 0.0498 ∙ exp(−2.5494 ∙ Rc∗) with σb = 0.7773 and σ′b = 30.5% (284 data)

Likewise the results are summarized in the impulsive regime.

Impulsive 𝐐∗ = 𝟏. 𝟓 ∙ 𝟏𝟎−𝟒 ∙ (𝐑𝐜∗)−𝟑.𝟏 with 𝛔𝐥𝐨𝐠 𝐚 = 𝟎. 𝟑𝟕 4.5

1. Q∗ = 3.036 ∙ 10−4 ∙ (Rc∗ )−3.144 with σa = 2.101 ∙ 10−4, σ′a = 69.2%

and σlog a = 0.2958 (148 data)

2. Q∗ = 2.648 ∙ 10−4 ∙ (Rc∗ )−3.234 with σa = 1.872 ∙ 10−4, σ′a = 70.71%


The corresponding plots can be found in Appendix B.

4.1.4.3 Discussion

In the non-impulsive regime, the uncertainty increases considerably due to the data for composite

vertical walls. In the corresponding plots, the scatter does not increase significantly. But there is more

data with small relative freeboards for composite vertical structures. The calculated values for the

parameter b for these data, are rather small, values up to -0.8 are observed.

74

With small relative freeboards correspond usually large overtopping discharges, the larger the relative

discharge, the smaller the values for the parameter b get:

bi =ln a−ln Qi

∗

Rc,i∗

On the other hand, small relative freeboards increase the values for b. The large discharges, seem to

be, however, decisive for the values of b whereas in Section 3.2 the small freeboards are. The

difference is that overtopping for small relative freeboards is under predicted here as you can see in

Figure 4. 4 whereas it was over predicted in Section 3.2.

The resulting relative standard deviation for only plain vertical walls was, however, is much smaller

than the one given in the EurOtop (2007) manual. When considering both plain and composite vertical

walls, the resulting relative standard deviation is the same order of magnitude as the one of EurOtop

(2007).

The larger relative standard deviation is reflected by a wider confidence band in the histogram. With

regards to the normal assumption, the histograms shows a bell-shaped curve. The histogram for the

second dataset considering both plain and composite vertical walls is given in Figure 4. 5.

Figure 4. 5 Histogram ∆b’ for the dataset containing plain and composite vertical walls in the non-impulsive regime with its mean value and 90% confidence interval

In the measured against predicted values plot for the dataset containing both plain and composite

vertical walls, data with large discharges are located below the 1:1 slope indicating an under prediction

for these data (Figure 4. 6).

75

Figure 4. 6 Measured vs predicted relative overtopping for the dataset containing plain and composite vertical walls in the non-impulsive regime, under prediction highlighted

Excluding small freeboards by limiting the validity range of the formula is an option to improve the

reliability of the formula. For example, when a minimum value of 0.5 for the relative freeboard is

introduced, the uncertainty improves but is still relatively large compared to the one for the dataset

only containing plain vertical walls:


In the impulsive regime, the relative standard deviation for plain vertical walls alone is already

relatively large. This gives the impression that the power law formula is not very reliable. Including

composite vertical walls does not have a significant influence on this uncertainty or whatsoever.

Due to these large uncertainties, the histograms do not longer contain the 90% confidence intervals

when plotted on the same scale as previously obtained histograms. A zoom-out is required to interpret

the distribution. Therefore the histogram are given in a different scale than the others (Appendix B).

In the measured against predicted relative discharges plots, the largest discharges are located above

the line corresponding to an over prediction (Appendix B). For slightly smaller, but still relatively large

discharges, data points are located below the line, indicating an under prediction for these data. The

same trends are observed in the traditional overtopping plot.

76

4.2 EurOtop (2007) formulae applied to CLASH and UG data

4.2.1 Introduction

Here the influence of more recent gathered data at the uncertainties is investigated. To that end, data

collected at the University of Ghent is added to the existing CLASH database and the uncertainty

analysis is repeated. More specifically, this concerns the datasets UG10, UG13 and UG14 which were

discussed in Chapter 2. These datasets generally describe steeper slopes and therefore are expected

to increase the amount of data more for vertical structures than for sloping structures.


There are no changes to the calculation procedure in Section 4.1.2, since formulae considered are

again the ones from the EurOtop (2007) manual.


In principal, the same filtering scheme is followed as before. However, the seven basic filters do not

have an impact on the UG data as previously remarked (cfr. Section 3.2.3). Furthermore, there are no

cases with a berm or toe. Hence, all the filtered data will fall under the category of plain vertical walls.

The following filters are applied such that only data for vertical structures remain:

Cotangent slope upward the berm cot αu ∈ [−0.2,0.2]

Cotangent slope downward the berm cot αd ∈ [−0.2,0.2]

The combined dataset of 1039 is reduced to 319 data due to these filters, 288 from these are in the

non-impulsive regime and 31 in the impulsive regime.

When the UG data and the CLASH data for plain vertical walls are plotted together in the non-impulsive

regime, the amount of data considered increases considerably (Figure 4. 7). Furthermore, a large part

of the UG data has small relative freeboards. In the impulsive regime, on the other hand, the UG data

do not really affect the image of the plot as there is only little extra data.

77

Figure 4. 7 Wave overtopping data for plain vertical walls, CLASH and UG data, in the non-impulsive regime with equation 4.4 with its 5% under and upper exceedance limits


4.2.4.1 Approach

With regard to the uncertainty analysis, no changes are made compared to the approach described in

Section 4.1.4.1 as the same formulae (EurOtop (2007) are considered.

4.2.4.2 Results

The results for the dataset with only plain vertical walls from CLASH, derived before, is compared to

the results when the UG data is added. Again, only the data that fall under the validity range of the

corresponding formulae is considered. The results are summarized below with first the results for the

plain vertical walls from CLASH only and following the results when the UG data is added.

Non-impulsive 𝐐∗ = 𝟎. 𝟎𝟒 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟖 and 𝛔′𝐛 = 𝟑𝟎. 𝟖% 4.4

1. Q∗ = 0.049 ∙ exp(−2.5646 ∙ Rc∗ ) with σb = 0.3357 and σ′b = 13.1% (80 data)


78

1. Q∗ = 3.036 ∙ 10−4 ∙ (Rc∗ )−3.144 with σa = 2.101 ∙ 10−4, σ′a = 69.2%


2. Q∗ = 3.338 ∙ 10−4 ∙ (Rc∗ )−3.038 with σa = 2.483 ∙ 10−4, σ′a = 74.4%


4.2.4.3 Discussion

The UG data have the largest impact in the non-impulsive regime where the uncertainty increases

considerably. In the corresponding plots, however, the scatter is not significantly increased. The UG

data increase the amount of data with small relative freeboards resulting in larger uncertainties. The

reason for this is already discussed in Section 4.1.4.3. Besides, the overtopping is under predicted for

these data (Appendix B).

In the impulsive regime, the UG data have less impact. The uncertainty increases somewhat, but the

difference is not significant.

4.3 Van der Meer and Bruce (2014) formulae applied to CLASH and UG data

4.3.1 Introduction

In this third section, the CLASH and UG data are analysed against the more recent formulae by Van der

Meer and Bruce for vertical structures described in Chapter 2.

For the new formulae, not only a distinction needs to be made between plain vertical structures and

composite vertical structures, but also one between structures with and without a foreshore.

Furthermore, the formulae are now each time given in pairs, with one formula for large discharges and

the other for small discharges.


The first distinction is the one between plain vertical walls and composite vertical walls (significant

berm or toe):

hc

h> 0.6 4.9


hc the water depth on the berm/toe.

If the above expression is true, the structure will be treated as a plain vertical wall, otherwise it will be

treated as a composite vertical wall. A different calculation procedure needs to be followed depending

on the type of structure. The calculation procedure for plain vertical walls will be explained first.

Impulsive 𝐐∗ = 𝟏. 𝟓 ∙ 𝟏𝟎−𝟒 ∙ (𝐑𝐜

∗)−𝟑.𝟏 with 𝛔𝐥𝐨𝐠 𝐚 = 𝟎. 𝟑𝟕 4.5

79

Within the plain vertical walls, structures with and without a foreshore are distinguished.

In the case of vertical structures without a foreshore, there are two given formulae with each their

range of application [5]:

q

√g∙Hm0³= 0.05 ∙ exp (−2.78 ∙

Rc

Hm0)

Rc

Hm0< 0.91 (Allsop et al. 1995) 4.10

q

√g∙Hm0³= 0.2 ∙ exp (−4.3 ∙

Rc

Hm0)

Rc

Hm0> 0.91 (Franco et. 1994) 4.11

with reliability of equation 4.10 and equation 4.11 respectively σ(2.78) = 0.17 and σ(4.3) = 0.6 [5].

In the case of vertical structures with a foreshore, the following expression is used as a discriminator

between the impulsive and the non-impulsive regime:

h² (Hm0 ∙ Lm−1,0) = 0.23⁄ 4.12

For values larger than 0.23, non-impulsive waves, the formula of Allsop is given, equation 4.10.

For values larger than 0.23, impulsive waves, a distinction has to be made between low and larger

freeboards. Depending on the value of the freeboard, the following formulae need to be applied [5]:

q

√g∙Hm0³= 0.011 ∙ √

Hm0

h∙sm−1,0∙ exp (−2.2 ∙

Rc

Hm0)

Rc

Hm0< 1.35 4.13

q

√g∙Hm0³= 0.0014 ∙ √

Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3

Rc

Hm0≥ 1.35 4.14

with reliability of equation 4.13 and equation 4.14 respectively σ(0.011) = 0.0045 and σ(0.0014) = 0.006

[5].

For composite vertical structures, structures with and without a foreshore are distinguished again.

Structures without a foreshore are then treated the same as plain vertical structures without a

foreshore.

In the case of composite structures with a foreshore, the same “impulsiveness” parameter as

mentioned in EurOtop (2007) is used to make a distinction between the impulsive and the non-

impulsive regime:

d∗ = 1.35 ∙

hc

Hm0 toe∙

2π ∙ h

g ∙ Tm−1,0 toe² 4.6

with the difference being the value considered. When this parameter is larger than 0.85, the impulsive

regime is assumed.

80

In the impulsive regime, the following formulae are given for composite vertical structures with a

foreshore [5]:

q

√g∙Hm0³= 1.3 ∙ √

hc

h∙ 0.011 ∙ √

Hm0

h∙sm−1,0∙ exp (−2.2 ∙

Rc

Hm0)

Rc

Hm0< 1.35 4.15

q

√g∙Hm0³= 1.3 ∙ √

hc

h∙ 0.0014 ∙ √

Hm0

h∙sm−1,0∙ (

Rc

Hm0)

−3

Rc

Hm0≥ 1.35 4.16

In the non-impulsive regime, the structures are treated like structures without a foreshore.

A decision chart summary of the calculation procedure is given in Figure 4. 8.

Figure 4. 8 Decision chart new formulae


It is decided to work with the same filtered data as in Section 4.2.3, since the objective is to compare

all the results obtained. 560 data results from the CLASH database and 319 data from the UG data,

together 808 data that is considered here.

Further filtering of the data happens according to Figure 4. 8.

For the first pair of equations, equation 4.13 and 4.14, vertical structures with a foreshore in the

impulsive regime, we have 21 data with small relative freeboards and 134 data with large relative

freeboards. The plot of this is given in Figure 4. 9.

81

Figure 4. 9 Wave overtopping data for vertical structures with a foreshore in the impulsive regime with the corresponding equations 4.13 and 4.14

Vertical structures with a foreshore in the non-impulsive regime should follow Allsop, equation 4.10,

we have 64 data here and this is plotted in Figure 4. 10.

82

Figure 4. 10 Wave overtopping data for vertical structures with a foreshore in the non-impulsive regime and equation 4.10

Structures without a foreshore and composite structures with a foreshore but in the non-impulsive

regime are represented by the combination of Allsop and Franco (equation 4.10 and equation 4.11

respectively), we have 314 data with small relative freeboards here and 174 data with large relative

freeboards. The resulting plot is given in Figure 4. 11.

83

Figure 4. 11 Wave overtopping data for structures without a foreshore and composite vertical structures with a foreshore in the non-impulsive regime and the corresponding equations 4.10 and 4.11

Finally for the last pair of equations, equation 4.15 and 4.16, valid for composite structures with a

foreshore in the impulsive regime, we have only 3 data for small relative freeboards and 98 data for

large relative freeboards. The resulting plot is given in Figure 4. 12.

84

Figure 4. 12 Wave overtopping data for composite vertical structures with a foreshore in the impulsive regime and the corresponding equations 4.15 and 4.16

The datasets from CLASH used to derive the new formulae are given. A comparison of the considered

datasets is therefore possible. 24 different datasets from CLASH were considered to derive the new

formulae [5], while our filtered exists out of 15 different datasets. 12 of these datasets coincide. The

datasets we considered plain vertical structures and composite vertical structures also align, except

for dataset 351-__ which is considered as a plain vertical structure in our filtered data but as a

composite vertical structure in the analysis of the new formulae. Our classification seems the correct

one since this dataset is used in the EurOtop (2007) plots for plain vertical walls (cfr. Section 4.1.3).

The other datasets used to derive the new formulae that are not included in our filtered data, all have

either a high complexity factor or a high reliability factor. Note that one dataset, 102-__ they

considered, is even not a vertical structure (cotangent of the slope equal to 4).

Our filtered data 043-__(C), 105-__ (C), 223-__ (C), 224-__, 229-__ (C), 351-__, 402-__, 502-__,

503-__, 504-__, 505-__ (C), 507-__, 509-__ (C), 802-__, 914-__ (C)

New formulae [5] 006-__, 028-__, 043-__ (C), 044-__ (C), 102-__, 106-__, 107-__, 108-__,

113-__ (C), 224-__, 225-__, 228-__ (C), 229-__ (C), 315-__ (C), 351-__ (C),

380-__ (C), 402-__, 502-__, 503-__, 504-__, 505-__ (C), 507-__, 509-__ (C),

802-__, 914-__ (C)

85


4.3.3.1 Approach

All the formulae are either of the exponential-type or of the power-law type, for both types calculation

procedures have already been explained (cfr. Section 4.1.3.1).

4.3.3.2 Results

As a result of the filtering, we have data for each formula. The uncertainty for every formula is then

derived based on these data. The only exception is for the Allsop formula, since this formula has two

different instreams (see Figure 4. 8). One comes straight from the non-impulsive vertical structures

with a foreshore, no distinction is made in the relative freeboards for this data, hence this data includes

data with large freeboards. The other comes from the structures without a foreshore and the

composite structures with a foreshore in the non-impulsive regime, here we have only small relative

freeboards since a distinction is made in the step before.

Not every formula will be discussed, since some of the datasets are too small in order to be statistically

relevant. This concerns equations eq. 4.13 and 4.15.

The results for the remaining formulae are summarized below.

𝐐∗ = 𝟏. 𝟒 ∙ 𝟏𝟎−𝟑 ∙ (𝐑𝐜∗)−𝟑 with 𝛔𝐚 = 𝟎. 𝟔 ∙ 𝟏𝟎−𝟑 and 𝛔′𝐚 = 𝟒𝟐. 𝟗% 4.14

Q∗ = 1.551 ∙ 10−3 ∙ (Rc∗ )−2.689 with σa = 0.944 ∙ 10−3, σ′a = 60.9% and σlog a = 0.2799 (134 data)

𝐐∗ = 𝟎. 𝟎𝟓 ∙ 𝐞𝐱𝐩(−𝟐. 𝟕𝟖 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟏𝟕 and 𝛔′𝐛 = 𝟔. 𝟏% 4.10

First instream of vertical structures with a foreshore in the non-impulsive regime (all relative

freeboards):

Q∗ = 0.0854 ∙ exp(−3.9581 ∙ Rc∗) with σb = 2.2578 and σ′b = 57.0% (263 data)

Second instream of structures without a foreshore and composite structures with a foreshore in the

non-impulsive regime and the relative freeboard limited to values smaller than 0.91:


𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟒. 𝟑 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟔 and 𝛔′𝐛 = 𝟏𝟒. 𝟎% 4.11


When the relative freeboard considered is limited to 1.5 when determining the trend line, the following

result is obtained:


86

𝐐∗ = 𝟏. 𝟒 ∙ 𝟏𝟎−𝟑 ∙ (𝐑𝐜

∗)−𝟑 4.16


When the relative freeboard considered is limited to 3.5 when determining the trend line, the following

results is obtained:


4.3.3.3 Discussion

Power law equations give larger uncertainties in general. The derived uncertainties with the new

formulae are slightly smaller but the differences are not significant.

Until now, formulae of the power law type have only been considered for small relative freeboards

(impulsive regime). In the new formulae, power law formulae are used as well only in the impulsive

regime but now in a combination with an exponential formula for the smaller freeboards. The relative

freeboards considered here as larger are, however, still relatively small.

When the Allsop formula (equation 4.10) is considered the instream with limited relative freeboards,

the derived relative standard deviation is large, while the scatter it not in the corresponding plots

(Appendix B). The Allsop formula is of the exponential type. The reason for this large uncertainty is

most probably the fact that data with small relative freeboards leads to larger deviations in the

calculated values for parameter b. The assumption of a normal distributed parameter b seems valid in

the histogram (Appendix B).

When the Allsop formulae is considered for its second instream (unlimited relative freeboard), the

relative standard deviation is very low. The scatter, however, is of the same order of magnitude as in

the previous case (Appendix B). But the considered relative freeboards are larger now.

In the case of the Franco formula (equation 4.11), the relative standard deviation is relative large when

the relative freeboard is not limited when determining the trend line. In addition, data points with

large discharges are under predicted. Limiting the maximum value for the relative freeboard for the

trend line, results in a better fit for data points with large discharges and corresponding a smaller

uncertainty. There is still some under prediction for data with large overtopping, but it has already

decreased.

Finally, equation 4.16 is again of the power law type and the relative standard deviation again large,

even though only larger relative discharges are considered. Limiting the relative freeboard for

87

determining the trend line does give a certain improvement. The difference is, however, not

significant.

4.4 Summary results

The uncertainty in the non-impulsive regime considering only plain vertical walls from CLASH is much

smaller than the uncertainty given in EurOtop (2007). Adding composite vertical walls to this dataset

increases the standard deviation significantly because of data with small freeboards. However, the

final uncertainty is the same order of magnitude as the uncertainty given in EurOtop (2007). Limiting

the applicability of the formula is an option to improve the reliability of the formula.

In the impulsive regime, where we have the power law formula, the reliability given in the EurOtop

(2007) manual is already very low compared to the reliability of other formulae. The uncertainties

obtained in this work are the same order of magnitude. This gives the impression that the power law

formula is not ideal.

Adding the additional UG datasets, has the strongest impact in the non-impulsive regime. The

uncertainty increases significantly because of data with small freeboards. In the impulsive regime, the

impact of the UG data have is negligible.

With the new formulae, the obtained uncertainties are again considerably larger for power law

formulae. The uncertainties are smaller for the new formulae, but the differences are not significant.

Until now, formulae of the power law type have only been considered for small relative freeboards

(impulsive regime). In the new formulae, power law formulae are used as well only in the impulsive

regime but now in a combination with an exponential formula for the smaller freeboards. This could

be the reason for the small improvement in the reliabilities.

It is concluded that the new formulae do not give a clear improvement.

The datasets are smaller than the ones considered for sloping structures because there is less data

available for vertical structures. More data would give a better view.

4.5 Discussion

4.5.1 Uncertainty analysis approach

In our approach for exponential type formula, the value of parameter a is derived first using a trend

line and assumed constant for the further analysis. Parameter a determines the intersection point on

the vertical axis of the exponential equation.

88

For small relative freeboards, the value of parameter b gets either very large or either very small as a

consequence even though the scatter is not larger for these points. In our approach, the equations

with the preassigned parameter a, are forced through these data points by adjusting the parameter b

which represents the slope of the curve in the log-linear graph. Hence the smaller the relative

freeboards, the larger the difference in the calculated b value even though the scatter is not larger. So

the standard deviation increases artificially in this approach. This effect is illustrated in Figure 4. 13.

Figure 4. 13 Effect of the fixed value of parameter a on the uncertainty of parameter b

Hence, when the amount of data for small relative freeboards increases, even though the scatter does

not, the standard deviation increases. Treating parameter a as a constant has thus an impact on the

standard deviation.

In that regard, it would make more sense to consider both parameters stochastic. For example this can

be done by determining the mean and standard deviation for the data with zero freeboards and like

this determine parameter a. Then both parameters would be considered as the stochastic parameters

they actually are.

In order to improve the reliability of the equations, the validity range of the formula can be limited by

excluding small relative freeboards. These values are, however, chosen rather arbitrary.

However, we do not want to change the formula, we only want to derive the uncertainties. EurOtop

(2007) also only considers parameter b as a stochastic one. Hence the followed approach aligns in that

sense with the one the manual used.

4.5.2 Normality tests

Additional Shapiro-Wilk normality tests are executed and normal probability plots are made in R

(Appendix B). Both give the impression that the normal assumption is not acceptable.

89

However, as remarked before for sloping structures in Chapter 3, the goal is not to look closer into this

distribution but to find a formula such that the scatter is normally distributed along the curve.

90

91

Chapter 5: General conclusions

5.1 Summary

In this master dissertation the uncertainties on wave overtopping on coastal structures are analysed

in the context of a revision of the EurOtop (2007) manual.

In Chapter 2 the prediction models in the EurOtop (2007) manual as well as the more recent models

by Van der Meer and Bruce (2014) are discussed. These models are often applicable for a certain type

of structure only. In that context, a distinction is made between the two main types of structures:

sloping structures and vertical structures. Within these models for a certain type of structure, another

distinction is made each time between two regimes. For sloping structures, this concerns breaking

waves and non-breaking waves. Likewise for vertical structures, we have the impulsive and the non-

impulsive regime. Besides the different prediction models, the available overtopping data are

addressed too in Chapter 2, more specifically this concerns the CLASH database and the UG datasets.

The different overtopping models considered are all empirical based on physical model data. Hence,

inherent scatter has to be taken into account for applications. The scatter or uncertainties are

described by statistical distributions on the parameters occurring in the models. The parameters are

thus assumed to be stochastic, further it is also assumed that they are normally distributed. The

stochastic parameters then have a mean and a standard deviation. In the formulae considered in this

report, the mean values of the parameters are used in the formulae, hence these formulae give the

average overtopping in accordance to the mean value approach. In this report, the relative standard

deviations are considered too, since this allows for better comparison of the results.

The uncertainties are usually presented by a confidence band in the overtopping plots. In this report

additional means are used to present and analyse the uncertainties. More in particular, histograms are

as well as measured vs predicted relative discharge plots are considered. The histograms are used to

check for the hypothesis of a normal distribution, a bell-shaped curve indicates a normal distribution.

The measured vs predicted plots are used as an additional check for the reliability of the prediction.

The prediction models and their corresponding uncertainties are then examined consecutively for

sloping structures in Chapter 3 and for vertical structures in Chapter 4.

For each type of structure, the uncertainties of the EurOtop (2007) formulae are derived first

considering data only from the CLASH database. Three different datasets from CLASH are considered

for sloping structures which can be described as follows: 1) Simple, smooth sloping structures (no

berms or rough slopes); 2) Smooth sloping structures (taking into account berms, no rough slopes) and

92

3) Sloping structures (taking into account berms and rough slopes). For vertical structures, two

different datasets are considered for the uncertainty analysis: 1) Plain vertical walls (no berm or toe)

and 2) Plain and composite vertical walls (vertical structures with a mound included). It is assumed that

the composite vertical walls considered here have no significant mound and are therefore treated as

plain vertical walls. The results for the different datasets are compared with each other as well as with

the EurOtop (2007) formula and its reliability.

In a second step, the database is widened and the influence of these extra data on the uncertainties is

investigated. More in particular, the datasets UG10, UG13 and UG14 collected at the University of

Ghent are added. There are no structures with a mound or a rough slope included in the UG data.

Therefore the extra data are added with the first considered dataset for sloping structures (simple,

smooth sloping structures) and with the first considered dataset for vertical structures (plain vertical

structures). The results obtained in the first phase for the first considered datasets from CLASH are

then compared with the results for these same datasets together with the UG data.

Finally, the uncertainties for the more recent formulae by Van der Meer and Bruce (2014) are derived

and investigated whether they give improvements. The same different datasets from CLASH are

considered here together with the UG data. The results are compared to all previously obtained results

as well as with the EurOtop (2007) formulae and its reliability.

5.2 General conclusions

5.2.1 Sloping structures

The uncertainties for the EurOtop (2007) formulae are already larger when we are only considering

simple, smooth sloping structures than the uncertainties indicated in the manual. The relative standard

deviations are on average 5% larger. A revision of the uncertainties is recommend for these formulae.

For non-breaking waves, the relative standard deviation increases considerably when rough slopes are

included, an increase of approximately 10% is observed. There is a lot of data with rough slopes and

thus small roughness factors, as a consequence the size of the dataset has increased fivefold due to

the including of rough slopes. The scatter is also substantially larger in the corresponding plot. This let

us believe that either these data with small roughness factors should be assigned a higher reliability

factor or the determination of the factor itself must be revised.

The UG data increase the amount of data points with small relative freeboards. The data with small

relative freeboards are located below the EurOtop (2007) formula. Hence the EurOtop (2007) formulae

over predict for data with small relative freeboards. Although, the UG data does not increase the

scatter in the plots, the resulting standard deviations are larger. The reason for this is the data with

93

small relative freeboards. When the applicability of the formula is limited by a minimum relative

freeboard, the uncertainty improves. It is concluded that the EurOtop (2007) formulae fit less good for

data with small relative freeboards.

The more recent formulae by Van der Meer and Bruce (2014) fit the data points for smaller relative

freeboards better in the corresponding plots. The corresponding uncertainties are also better. Hence

the more recent formulae by Van der Meer and Bruce (2014) are more reliable over a greater range of

relative freeboards and are therefore recommended.

5.2.2 Vertical structures

In general, the sizes of the datasets considered for vertical structures are much smaller than for sloping

structures indicating a lack of data for these type of structures. The size of the datasets is insufficient

in order to make good decisions.

In the impulsive regime, the relative standard deviations are large compared to all other results. The

difference is the considered formula is now of the power law type. This gives the impression that the

power law formula is not very reliable. It is recommended to reconsider the equations of the power

law type and the way their reliability is expressed.

The UG data do not increase the scatter in the corresponding plots, but they do increase the relative

standard deviations. The UG data increase the number of data points with small relative freeboards.

Hence, the EurOtop (2007) formulae fit less good for small relative freeboards. In the non-impulsive

regime, data points with small relative discharges are located above the EurOtop (2007) curve in the

corresponding plots indicated an under prediction.

For the revised formulae by Van der Meer and Bruce (2014) the uncertainties are again large when

power law formulae are considered. They are however slightly smaller than the ones obtained with

the EurOtop (2007) formulae. Further, no clear improvement is observed.

5.2.3 General remarks

In the approach used for the uncertainty analysis, one parameter is assumed each time to be stochastic

and the other constant. As both parameters are interdependent, fixing the one parameter has an

impact on the standard deviations for the other. In that context, other approaches can be suggested.

The goal of this master dissertation, however, is to update the uncertainties given in the EurOtop

(2007) manual. In the manual, one parameter is assumed to be stochastic each time. Hence, the

followed approach aligns with the approach in the EurOtop (2007) manual.

94

The hypothesis of a normally distributed stochastic parameter seems acceptable in most histograms.

However, additional tests such as the Shapiro-Wilk normality test and normal probability plots

contradict this assumption. The goal, however, is not to look further into this distribution, but to find

a formula such that the data points are normally distributed along its curve.

95

References

[1] Verhaeghe, H. (2005): Neural Network Prediction of Wave Overtopping at Coastal Structures,

PhD, University of Ghent, Promotor Prof. dr. ir. Julien De Rouck

[2] European Overtopping Manual. Die Küste. Archiv für Forschung und Technik an der Nord- und

Ostsee, vol. 73, Pullen, T.; Allsop, N.W.H.; Bruce, T.; Kortenhaus, A.; Schüttrumpf, H.; Van der

Meer, J.W., www.overtopping-manual.com.

[3] Victor, L.; Troch, P. (2012): Experimental study on the overtopping behaviour of steep slopes -

transition between mild slopes and vertical walls. Proceedings 33rd International Conference

on Coastal Engineering (ICCE), ASCE, Santander, Spain.

[4] van der Meer, J.W.; Bruce, T.; Allsop, N.W.H.; Franco, L.; Kortenhaus, A. (2013): EurOtop

revisited. Part 1: Sloping structures. Proc. ICE, Coasts, Marine Structures and Breakwaters

2013, Edinburgh, UK.

[5] J. and Bruce, T. (2014): New Physical Insights and Design Formulas on Wave Overtopping at

Sloping and Vertical Structures. Journal of Waterway, Port, Coastal, Ocean Eng., Volume 140,

Issue 6

[6] Bruce, T.; van der Meer, J.W.; Allsop, N.W.H.; Franco, L.; Kortenhaus, A.; Pullen, T.;

Schuttrumpf, H. (2013): EurOtop revisited. Part 2: Vertical structures. Proc. ICE, Coasts, Marine

Structures and Breakwaters 2013, Edinburgh, UK.

[7] van der Meer, J.W. (2001): CLASH Overtopping Database report, 1st draft

[8] Troch, P.; Mollaert, J.; Peelman, S.; Victor, L.; van der Meer, J.W.; Gallach-Sánchez, D.;

Kortenhaus, A. (2014): Experimental study of overtopping performance for the cases of very

steep slopes and vertical walls with very small freeboards. Proceedings International

Conference on Coastal Engineering (ICCE), ASCE, Seoul, Republic of Korea, 8 p.

[9] Gallach-Sánchez, D.; Troch, P.; Vroman, T.; Pintelon, L.; Kortenhaus, A. (2014): Experimental

study of overtopping performance of steep smooth slopes for shallow water wave conditions.

Proceedings International Conference on the Application of Physical Modelling to Port and

Coastal Protection (Coastlab14), Varna, Bulgaria, 10 p.

[10] Kortenhaus, A. (2014): Uncertainties, Ghent University, Department of Civil Engineering

[11] de Waal, J.P.; van der Meer, J.W. (1992): Wave runup and overtopping at coastal structures.

Coastal Engineering 1992, p. 1758-1771

http://www.overtopping-manual.com/

Appendix A – Sloping structures

First an overview of all the results obtained in Chapter 3 is given:

EurOtop (2007) formulae applied to CLASH data

Breaking 𝐐∗ = 𝟎. 𝟎𝟔𝟕 ∙ 𝐞𝐱𝐩(−𝟒. 𝟕𝟓 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟓 and 𝛔′𝐛 = 𝟏𝟎. 𝟓%

1. Q∗ = 0.0769 ∙ exp(−4.8759 ∙ Rc∗) with σb = 0.6763 and σ′b = 13.9%



Non-breaking 𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟑𝟓 and 𝛔′𝐛 = 𝟏𝟑. 𝟓%




EurOtop (2007) formulae applied to CLASH and UG data

Breaking 𝐐∗ = 𝟎. 𝟎𝟔𝟕 ∙ 𝐞𝐱𝐩(−𝟒. 𝟕𝟓 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟓 and 𝛔′𝐛 = 𝟏𝟎. 𝟓%


Non-breaking 𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟑𝟓 and 𝛔′𝐛 = 𝟏𝟑. 𝟓%


Van der Meer and Bruce (2014) formulae applied to CLASH and UG data

Breaking 𝐐∗ = 𝟎. 𝟎𝟐𝟑 ∙ 𝐞𝐱𝐩[−(𝟐. 𝟕 ∙ 𝐑𝐜∗)𝟏.𝟑]

9. Q∗ = 0.0250 ∙ exp[−(2.7235 ∙ Rc∗)1.3] with σb = 0.4000 and σ′b = 14.7%



Non-breaking 𝐐∗ = 𝟎. 𝟎𝟗 ∙ 𝐞𝐱𝐩[−(𝟏. 𝟓 ∙ 𝐑𝐜∗)𝟏.𝟑]




In the following, for each of the above results the corresponding plots and results can be found:

i. The numerical results described above;

ii. Wave overtopping data and the corresponding formula with its 90% confidence interval;

iii. Wave overtopping data used to determine the trend line;

iv. Histogram ∆b’;

v. Measured vs predicted relative discharge;

vi. P-value Shapiro-Wilk normality test and

vii. Normal probability plot


Shapiro-Wilk test: p = 5.949E-07







-3 -2 -1 0 1 2 3

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Normal Q-Q Plot

Theoretical Quantiles

Sa

mp

le Q

ua

ntile

s




Shapiro-Wilk test: p < 2.2E-16





9. Q∗ = 0.0250 ∙ exp[−(2.7235 ∙ Rc∗)1.3] with σb = 0.4000 and σ′b = 14.7% (581/547 data)





Shapiro-Wilk test: 1.924E-08







Appendix B – Vertical structures

First an overview of all the results obtained in Chapter 4 is given:

EurOtop (2007) formulae applied to CLASH data

Non-impulsive 𝐐∗ = 𝟎. 𝟎𝟒 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟖 and 𝛔′𝐛 = 𝟑𝟎. 𝟖%



Impulsive 𝐐∗ = 𝟏. 𝟓 ∙ 𝟏𝟎−𝟒 ∙ (𝐑𝐜∗)−𝟑.𝟏 with 𝛔𝐥𝐨𝐠𝐚 = 𝟎. 𝟑𝟕

3. Q∗ = 3.036 ∙ 10−4 ∙ (Rc∗)−3.144 with σa = 2.101 ∙ 10−4, σ′a = 69.2%

and σloga = 0.2958 (148 data)

4. Q∗ = 2.648 ∙ 10−4 ∙ (Rc∗)−3.234 with σa = 1.872 ∙ 10−4, σ′a = 70.71%


EurOtop (2007) formulae applied to CLASH and UG data

Non-impulsive 𝐐∗ = 𝟎. 𝟎𝟒 ∙ 𝐞𝐱𝐩(−𝟐. 𝟔 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟖 and 𝛔′𝐛 = 𝟑𝟎. 𝟖%


Impulsive 𝐐∗ = 𝟏. 𝟓 ∙ 𝟏𝟎−𝟒 ∙ (𝐑𝐜∗)−𝟑.𝟏 with 𝛔𝐥𝐨𝐠𝐚 = 𝟎. 𝟑𝟕

6. Q∗ = 3.338 ∙ 10−4 ∙ (Rc∗)−3.038 with σa = 2.483 ∙ 10−4, σ′a = 74.4%


Van der Meer and Bruce (2014) formulae applied to CLASH and UG data

Eq. 4.14 𝐐∗ = 𝟏. 𝟒 ∙ 𝟏𝟎−𝟑 ∙ (𝐑𝐜∗)−𝟑 with 𝛔𝐚 = 𝟎. 𝟔 ∙ 𝟏𝟎−𝟑

7. Q∗ = 1.551 ∙ 10−3 ∙ (Rc∗)−2.689 with σa = 0.944 ∙ 10−3, σ′a = 60.9%


Eq. 4.10 𝐐∗ = 𝟎. 𝟎𝟓 ∙ 𝐞𝐱𝐩(−𝟐. 𝟕𝟖 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟏𝟕 and 𝛔′𝐛 = 𝟔. 𝟏%



Eq. 4.11 𝐐∗ = 𝟎. 𝟐 ∙ 𝐞𝐱𝐩(−𝟒. 𝟑 ∙ 𝐑𝐜∗) with 𝛔𝐛 = 𝟎. 𝟔 and 𝛔′𝐛 = 𝟏𝟒. 𝟎%


Eq. 4.16 𝐐∗ = 𝟏. 𝟒 ∙ 𝟏𝟎−𝟑 ∙ (𝐑𝐜∗)−𝟑

11. Q∗ = 5.203 ∙ 10−3 ∙ (Rc∗)−3.963 with σa = 3.037 ∙ 10−3, σ′a = 58.4%


In the following, for each of the above results the corresponding plots and results can be found:

i. The numerical results described above;

ii. Wave overtopping data and the corresponding formula with its 90% confidence interval;

iii. Wave overtopping data used to determine the trend line;

iv. Histogram ∆b’;

v. Measured vs predicted relative discharge;

vi. P-value Shapiro-Wilk normality test and

vii. Normal probability plot.


Shapiro-Wilk test: p = 0.0002252



3. Q∗ = 3.036 ∙ 10−4 ∙ (Rc∗)−3.144 with σa = 2.101 ∙ 10−4, σ′a = 69.2% and σloga = 0.2958

(148 data)


4. Q∗ = 2.648 ∙ 10−4 ∙ (Rc∗)−3.234 with σa = 1.872 ∙ 10−4, σ′a = 70.71% and σlog a = 0.3290

(185 data)





(175 data)



(134 data)









(98 data)


Documents

Uncertainties of wave overtopping of coastal structures