2015-06-08_MSc_Thesis_Olij_Final.pdf

Wave climate reduction for medium
term process based morphodynamic
simulations
- With application to the Durban coast -
MSc Thesis
Derrick Jan Carl Olij

Wave climate reduction for medium term process
based morphodynamic simulations
- With application to the Durban coast -
DERRICK JAN CARL OLIJ
MASTER THESIS
For the degree Master of Science (MSc) in Hydraulic Engineering, Faculty Civil Engineering,
Delft University of Technology
Delft June 8, 2015
In cooperation with Deltares (Delft, the Netherlands) and the University of Kwazulu-Natal
(Durban, South Africa)
Graduation Committee:
Prof. Dr. Ir. A. Reniers (TU Delft and Deltares)
Ir. D. Walstra (TU Delft and Deltares)
Dr. Ir. M. Tissier (TU Delft)
Prof. Dr. Ir. D. Stretch (University of Kwazulu-Natal)
Dr. Ir. S. Corbella (University of Kwazulu-Natal)
Figure on the frontpage is made in Durban, South Africa by Justin Pringle. c all rights reserved.

Acknowledgments
With this report my student life has come to an end. It has been a great time where I have learned a lot.
In this chapter I would like to thank several persons that have made it possible for me to write this thesis.
First of all I would like to thank Henk-Jan Verhagen. Due to his contacts I came in touch with the
university of Kwazulu-Natal. I also want to thank him for his help with defining my graduation topic.
Secondly I would like to thank prof. Derek Stretch, Stef Corbella, Ooma Chetty, Justin Pringle and
Kruschen Govender for their warm welcome and unlimited support in Durban. We had some great discus-
sions about several topics in hydraulic engineering and life in general that were very useful for my report
and my personal development. Besides that we had a lot of fun. I feel very fortunate to have been part of
such a high quality and nice research group and community.
Thirdly I would like to thank my supporting committee from the TU Delft: prof. Ad Reniers, Dirk-Jan
Walstra and Marion Tissier. During our meetings we always had interesting discussions that brought up
new ideas. Besides that you were always ready to support me by answering my questions. I am also grate-
ful for your flexibility that made it possible for me to do most of my work in South Africa, without loosing
a lot of time.
As last I would like to thank my family. Without them I wouldn’t have had the opportunity to go to
South Africa. I truly appreciate that.
Derrick Olij
Amsterdam, June 8, 2015
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Executive Summary
The dynamics of the morphology of a beach is important for many reasons. For example the hinterland
safety and tourism is dependent on it at certain locations. To predict the evolution of the morphology
of a beach many models are available. Process based models are often used, because they take into ac-
count physical phenomena that influence the dynamics of the morphology. One large disadvantage of
these models is that they have a large computation time. A method that is commonly used to decrease
the computation time is reducing the size of the input data set. In this report is investigated how one can
reduce a input dataset of waves, while minimizing the prediction error of the process-based model that is
introduced by this reduction.
For this investigation the coast of Durban is used as a case study. This is a highly energetic coast with a
large seasonal variation in the wave climate. The morphology is wave dominated and shows bar forming.
Because the seasonal variation is strong and bar forming is occurring, it is assumed that the influence of
wave conditions on the morphology is a cyclic process. The process based model is therefore executed in
a morphodynamic setup. This means that a reduced wave climate is used as input for one simulation, in
which all wave conditions are present.
The investigation has as a goal to reduce a wave climate to 10 representative wave conditions, while
minimizing the prediction error of a process based model. That is approached by answering three sub-
questions:
1. What algorithm reduces the wave climate in such a way that the prediction error is minimized?
2. What is the most realistic way to order the wave conditions in a reduced wave climate, so that the
prediction error is minimized?
3. What influence does the number of reduced wave conditions have on the prediction error?
The prediction error is determined in two ways. First a bulk transport model is used. It compares the
output of a bulk transport formula (e.g. CERC) of the reduced wave climate with the full wave climate.
Secondly the process-based model Delft3D is used. Here model runs that have a reduced wave climate
as input are compared with a model that uses the original wave climate as input (a brute force simu-
lation). The two models are compared with 8 evaluation variables: The along and cross shore transport
rate, the total erosion and accretion and the location and size of a trough near the beach and a bar offshore.
In the report eight reduction algorithms are evaluated. Algorithm 1 bins the observed wave conditions
based on their wave height and direction. It then uses a bulk transport formula to determine how much bulk
alongshore transport is generated by the observations in the bins. The algorithm then selects the 10 bins
that generate the most bulk alongshore transport. The representative wave conditions are constructed by
taking the average of the observed wave conditions in the selected bins. Algorithm 2 determines 10 reduced
wave conditions by clustering the observed wave conditions based on their equality. The clustering is
restricted in such a way that the total amount of bulk alongshore sediment transport that the observations
in a cluster generate is the same for every cluster. Algorithm 3 splits the time series of observed wave
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conditions into blocks of one season. The blocks are represented by a wave conditions that is the average
of the observations in a block. Algorithm 4 bins the observed wave conditions like algorithm 1. The bin
sizes are chosen in such a way that only 10 bins exist. The algorithm then finds the representative wave
condition for every bin taking the average wave conditions in a bin for the wave period and direction, and
by using a non-equal nonlinear weighting for the wave height. Algorithm 5 is the crisp k-means algorithm.
This iterative method finds its representative wave conditions by averaging over the wave conditions in
a cluster. An observation is assigned to a cluster by using the Euclidean distance. Since the location of
representative wave conditions is changing every iteration, observations are constantly allocated to other
clusters during different iterations. Algorithm 6 is the fuzzy k-means algorithm. That algorithm is similar
to algorithm 5, only now observations can be part of multiple clusters during the same iteration. Algorithm
7 is the k-harmonic means algorithm. This algorithm is similar to algorithm 6. The difference is that it
uses boosting. Boosting means that observations that are far from a representative wave condition in the
observation space get more weight during the determination of the next location of the representative wave
condition. Algorithm 8 is the Maximum Dissimilarity Algorithm (also known as MDA). This algorithm finds
the 10 wave conditions that are most dissimilar to each other in the dataset, and uses them as representative
wave conditions.
In the report also three methods are used to determine the order of the wave conditions in a reduced
wave climate. These are so called ’sequencing methods’. Method 1 is called Low to High sequencing (LtH).
It orders the wave conditions from the condition with the smallest wave height to the condition with the
largest wave height. Method 2 is called Markov Chain (MC) sequencing. It orders the wave condition
based on their Markov transition probabilities. Method 3 is called Fourth Variable (FV) sequencing. This
method is only applied to the crisp k-means and the MDA. The method adds a fourth variable to a wave
condition. This variable shows at which day in a year an observation has occurred. This leads to clusters
that are filled with observation that are not only similar to each other due to their wave height, period and
direction, but also because they have occurred during the same time period.
As last it is investigated what the influence of the chosen number of wave conditions is on the perfor-
mance of the reduction algorithms and the sequencing methods.
In the report six important conclusions are drawn.
1. It is concluded that reducing a wave climate introduces a large prediction error. The size and vari-
ability of the prediction error depends on the evaluation variable.
2. The alongshore transport is reproducible with several algorithms and sequencing methods. Most
algorithms do not reproduce this variable accurately though. When a LtH sequence is used, algorithm
5 performs best. When a MC sequencing method is used, algorithm 7 performs best.
3. The cross shore transport is mostly under estimated by the algorithms. When a LtH sequence is used
algorithm 7 reproduces the cross shore transport the best. When a MC sequence is used, algorithm 5
and 7 perform best. That is only the case when an unequal weighting scheme is used in the distance
function of the clustering algorithm.
4. The trough is in general better reproduced than the bar. Both phenomena are under estimated by
most algorithms. This under estimation is expressed with a trough depth that is too low, a bar height
that is too small, and the locations of the two patterns that are positioned too close to the western
side of the grid. The trough is as best reproduced when a Markov chain sequencing is used. When
a LtH sequencing is used, algorithm 7 performs best, while when a MC sequencing is used, model 5
performs best.
5. The fourth variable sequencing method does not work well, since the error that it introduces is large.
6. Doubling the number of representative wave conditions from 10 to 20 does mostly not lead to better
results.
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Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research questions and approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Structure report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Case study beach description 4
2.1 Location of case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 The wave climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Sediment transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Cross shore transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Alongshore transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Rivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Methodology 17
3.1 Reduction algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1 Notation, Normalization and Distance measures . . . . . . . . . . . . . . . . . . . . . . 18
3.1.2 Bulk transport formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.3 Algorithm 1: Conditions with the largest transport contribution . . . . . . . . . . . . . 21
3.1.4 Algorithm 2: Grouping with equal sediment influence . . . . . . . . . . . . . . . . . . . 22
3.1.5 Algorithm 3: The representative wave approach . . . . . . . . . . . . . . . . . . . . . . 23
3.1.6 Algorithm 4: Non-linear wave bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.7 Algorithm 5: Crisp k-means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.8 Algorithm 6: Fuzzy k-means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.9 Algorithm 7: K-harmonic means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.10 Algorithm 8: Maximum Dissimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Sequencing reduced wave climates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Markov Chain sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 Time as fourth variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Bulk alongshore transport model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Delft3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Results 38
4.1 Evaluation reduction algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.1 Overview evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.2 Algorithm 1: Conditions with the largest transport contribution . . . . . . . . . . . . . 40
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4.1.8 Algorithm 7: K-Harmonic means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.9 Algorithm 8: Maximum dissimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.10 Intermediate conclusions reduction algorithms . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Sequencing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1 Influence of sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.2 Markov Chain Sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Sequencing with a fourth variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.4 Intermediate conclusions sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Influence number of wave conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Low to High with 20 wave conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.2 Markov chain sequencing with 20 wave conditions . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Sequencing with a fourth variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.4 Intermediate conclusions k=20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Discussion 79
5.1 Reduction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Algorithm 1: Conditions with largest transport transport contribution . . . . . . . . . 79
5.1.7 Algorithm 7: K-Harmonic means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.8 Algorithm 8: Maximum dissimilarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.2 Sequence based on wave height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.3 Markov chain sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.4 4th variable sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Evaluation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3.1 Bulk sediment transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3.2 Process based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Conclusion 86
6.1 Reduction algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Number of wave conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Best performing setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Recommendation 89
7.1 Application of methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Improvement of the methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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A Discussion Delft3D model setup 94
A.1 Tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A.2 Transport formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.3 Roughness formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.4 Wave breaker parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.5 Eddy viscosity and diffusity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B Unused setups sequencing with four variables 107
B.1 Comparison different measures for time of the year Crisp k-means . . . . . . . . . . . . . . . . 107
B.2 soft membership cluster methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B.3 Comparison model 5 and 8 using 4th variable sequencing . . . . . . . . . . . . . . . . . . . . . 112
C Comparison Delft3D and bulk transport 115
D Morphostatic approach 118
E Symbol list 120
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1 | Introduction
1.1 Motivation
The dynamics of the morphology and the sediment transport of a beach have been studied by many re-
searchers, since they are important for the environment, the safety of people and financial reasons. The
morphology has for example influence on the hinterland safety, since it protects certain coastal areas against
storms from sea. The beach width is an important factor for tourism, because this gives tourists the space
to recreate on the beach. Also swimmer safety is influenced by the morphology. For example every year
numerous people drown due to rip currents. To study the morphology dynamics many tools are available.
The tools are used to predict and analyze the behavior of the sediment transport and the morphology.
Commonly used tools to make predictions about the sediment transport rate and the morphology are
process-based models. These are numerical models that simulate physical processes of transport rates,
hydro and morpho dynamics. These models use spacial and forcing data as input. Spacial data contains
information about the bathymetry, sediment layers and other variables that describe the spacial properties
of the model environment. Force data contains information about the forces that act on the model space.
Examples of these forces are waves, (tidal) currents and wind.
A large disadvantage of the process based model is their extensive computation time. The computation
time is usually so extensive that most users employ methods to reduce this computation time. These
methods unfortunately commonly decrease the accuracy of the predictions of these process based models.
One of the most common used methods to decrease the computation time is a reduction of the size of the
input data. This can be done by reducing the spacial data (e.g. enlarging the grid size), or by reducing
the size the forcing input data set (e.g. waves, currents, tides and wind). Although the reduction of input
data is so common used in practice, not much knowledge is available about this topic in the literature.
Therefore in this report the implications of input reduction on the prediction accuracy of a process based
model is further investigated. The report focuses on the reduction of the wave input of a model.
This wave input is called a wave climate. This is a time series of snap shots of the sea state during a
certain period. These sea states are described by a wave condition. That is a set of variables that describe
a sea state and commonly exists of the significant wave height, peak period and mean direction. In the
report eight algorithms for wave climate reduction are evaluated. Four of these algorithms are already
used in practice by coastal engineers. In the report a better understanding is given about their reduction
behavior and their implication on the prediction error. The other four algorithms are gained from the
literature of other research fields such as machine learning, biology and chemistry. These algorithms are
more sophisticated than the first four algorithms. In the report these algorithms are modified so that they
are usable in the field of coastal engineering. The goal of the introduction of these algorithms is to decrease
the prediction error of process based models that is introduced by the reduction of its input data.
In the report is also the influence of the chronology of the wave conditions in a reduced wave climate
examined. There are four methods used in the report that determine the sequence of the wave conditions
in the input time series. Two of these sequencing methods are new and designed by the author of this
report.
Thirdly it is investigated what the influence of the number of conditions in a reduced wave climate is
1

on the prediction errors of a process based model. In the core of the report reduced wave climates that
exist of 10 wave conditions are used. During this sensitivity analysis it is determined how prediction errors
change when 20 wave conditions are used in a reduced wave climate.
To evaluate methods to reduce a wave climate, the coast of Durban in South Africa is used as a case
study. The wave climate at this coast is highly energetic and the morphology is wave dominated. That is the
reason why for the input reduction the focus lies on the reduction of the wave input data. Besides its highly
energetic character, three other properties of this coast have possibly an influence on the dynamics of the
sediment transport and morphology of the coast: 1. Bar forming during large wave events. 2. Significant
seasonal variation in the wave climate. 3. A strong bimodal wave spectrum. A bimodal wave climate is a
wave climate that exists of both swell waves and locally generated wind waves. A goal of this thesis is to
find and construct methods that take these properties into account during the wave climate reduction.
1.2 Research questions and approach
The main objective of this thesis is to better understand how input reduction effects the behavior of a
process based model. The main question that is answered is the following:
"How can a wave climate be reduced in a way that the effects on the morphology and the
transport rates are minimized, while the reduction of the computation time of a process based
model is maximized?"
This question cannot be answered directly, since multiple effects play a role. There are three sub
questions that are answered. These are the following:
1. Which wave climate reduction algorithm introduces the smallest prediction error in the sediment
transport and the morphology?
2. How should the wave conditions of a reduced wave climate be ordered so that a realistic time series
is created and the prediction errors of the sediment transport and morphology in a process based
model are minimized?
3. What is the influence of the number of used wave conditions in a reduced wave climate on the
performance of a process based model?
Subquestion 1: Reduction algorithms The reduction algorithm reduces a wave climate to a number, k,
of conditions. In this report a value of k = 10 is used and eight different algorithms are evaluated. Some of
them are commonly used in literature, others are only used in other research fields and are adapted so that
they also become useful for the reduction of wave climates. The evaluation of the reduction algorithms
is done with two models. First a bulk transport formula is used. The estimates of the bulk alongshore
sediment transport based on the full and reduced wave climate are compared. Secondly the process based
model Delft3D is used to evaluate the reduction algorithms. As a benchmark, a model run with the complete
wave climate is executed. The estimated transport rates and morphology are compared with results from
models that use reduced wave climates that are generated with different reduction algorithms.
Subquestion 2: Sequencing methods In chapter 2 it is argued that morphological response due to
a wave condition depends on the morphological responses to previous wave conditions. Because of the
interdependence of the morphological responses on the wave conditions, the Delft3D model is executed in a
morphodynamic set up. This means that one simulation is executed with a constantly evolving bathymetry.
The prediction error that is introduced by reducing a wave climate therefore depends on the order
of the wave conditions in the input time series. In this report four sequencing methods are evaluated.
The sequencing methods are only evaluated with a process based model, and not with the bulk transport
2

formulas. This because the bulk transport formulas assume that wave conditions are independent of each
other and thus no history effect is present.
Subquestion 3: Number of wave conditions In practice users construct process based models that
usually have between 10 and 20 wave conditions (with a bias to 10 conditions). In the report all reduction
and sequencing methods are evaluated with 10 wave conditions. In theory double the number of wave
conditions should improve the accuracy of the made predictions. In reality this does not have to be the
case. Therefore in this report it is investigated how much the performance of the reduction algorithms
changes when not 10 but 20 wave conditions are used.
1.3 Structure report
First in chapter 2 an analysis of the case study beach is made. Here is analyzed what the main properties
of the wave climate and morphological developments are. Secondly in chapter 3 the methodology is de-
scribed. In that chapter is explained how the reduction and sequencing methods work, and how they relate
to the literature. In the methodology chapter it is also explained how the model results are evaluated. In
chapter 4 the results of the evaluation procedures are reported and analyzed. Then in chapter 5 the setup
of report and the investigation are discussed and the strong and weak points of the research are pointed
out. The reports main findings are listed in the chapter 6 ’Conclusion’. As last, recommendations are
made about the implementation of the reports findings and methods that could improve the wave climate
reduction.
3

2 | Case study beach description
2.1 Location of case study
In this thesis the coast of the city Durban in South Africa will be used as a case-study. Durban is located at
the east coast of South Africa in the Kwazulu-Natal province (figure 2.1). The wave climate at this coast
is highly energetic. Both large swell waves and locally generated wind waves are commonly observed there.
For the investigation a representative beach of the Durban coast is used. The beach that is selected as
the ’representative beach’ is located North of the Umgeni and South of the Ohlanga river (figure 2.1). The
beach is located between the Beachwood and Virginia beaches. At this location the average grain size is
D50 = 200µm and the direction of the coastline is θc = 30o
. In this report it all directions are presented in
degrees and are relative to the Northern direction, which has direction of 0o
.
This beach is chosen as the representative beach because of three main reasons: The first reason is
that the beach is located at the Durban coast and is not far from the Waverider buoy that provides the
wave data for the investigation. The second reason is that a recently measured bathymetry of this location
is available. The third reason is that the beach does not have many unique properties: It has a relative
homogeneous alongshore profile (the depth contours are relatively parallel), the man-made structures of
the Durban coast (the groynes and the harbour) are too far from the beach to have a significant influ-
ence on the beach morphology and no nourishments take place near the case study beach (nourishments
are being executed between the harbour and the Umgeni river). Due to the more general properties of the
case study beach, it becomes possible to relate the results of this investigation to other beaches in the world.
4

Figure 2.1: Overview maps of the case study location. Top left corner: Map of Africa. Left side: Map of
South Africa. Right side: Overview map of the Durban Coastline.
2.2 The wave climate
There are three storm sources for the coast of KwaZulu-Natal. 1. Cold fronts and coastal lows cause storms
from the southern direction with relatively low wave heights and shorter periods, because the cold fronts
and coastal lows mainly occur near the coast 2. Cut-off low pressure systems cause swell from south-eastern
direction. These cut-offs are occurring further from the coast and therefore the generated swell has a rel-
ative large wave height and relative long period. 3. Tropical cyclones generate large and long waves from
the north east and east-north-east direction. Most literature indicates that the tropical cyclones generate
the largest storm events. The occurrence of these cyclone generated waves are rare, therefore the cut-off
low pressure systems have the largest influence on morphology the coast.
In table 2.1 the general statistics of the Durban wave climate are shown. The mean wave condition has
a significant wave height Hs = 1.68m, a peak period Tp = 10.0s and direction θ = 132o
. The difference
between the direction of the mean wave direction and the angle of the normal of the coastline(30o
+90o
=
120o
) is 12o
. This angle has influence on the alongshore sediment transport the incoming waves generate.
The larger this angle, the more transport the waves cause.
The mean wave heights during winter(1.76m) and autumn (1.76m) are larger than during summer
(1.61m)and spring (1.59m). The mean periods are largest during spring (10.3s) and winter (10.8s) and
lower during summer (9.2s) and autumn (9.6s). The mean wave direction comes from the south-east
direction. Winter has the largest average angle with the normal of the coastline, followed by the spring
season.
The median wave heights and periods show the same pattern as the mean wave values. The longer
periods during spring and winter suggest that during this period more swell waves (with longer periods)
arrive at the Durban coast.
The 99% percentile values show that the wave heights during extreme events are highest during winter
and autumn. It is also notable that most 99% percentile periods are 16.6s. This is not a realistic value, and
can be explained by the fact that the Waverider buoys have trouble with observing long wave periods.
In figure 2.2 the average two-dimensional wave spectrum of the Durban wave rider is plotted for the
different seasons. The cosine-2s spreading function is used to create these plots. It would be more informa-
5

Table 2.1: General statistics of the Durban Waverider
Mean Median 99% perc
Hs Tp θ Hs Tp θ Hs Tp
Full 1.68 10.0 132 1.61 10.0 140 3.17 16.6
Summer 1.61 9.2 126 1.56 9.0 125 2.76 15.3
Spring 1.59 10.3 135 1.53 10.5 143 3.02 16.6
Winter 1.76 10.8 138 1.66 11.1 151 3.53 16.6
Autumn 1.76 9.6 130 1.69 9.0 137 3.19 16.6
99% perc is the 99% percentile. Full stands for the complete sample.
tive to use observational data for the spreading of energy over the directions, but this is not possible since
the wave rider does not distribute the measured data over different directional bins, but rather only gives
the mean direction of the observed wave energy in a certain frequency bin. The directional coefficients can
also not be derived from the Fourier coefficients (Kuik, 1988), since these are also not provided. Corbella &
Stretch (2014) show in an indirect way that the cosine 2s spreading function is the most suitable spreading
function for the eastern coast of South Africa.
The average of all the seasons (a) indicates that the spread of wave energy reaches from 45o
till 190o
.
Most energy comes from the southern directions (between 130o
and 180o
), which suggests that wave
conditions with the larger wave heights are coming from these directions. The largest energy peak has
a direction between 150o
and 160o
. Also notable is that there is not much wave energy coming from
directions smaller than 90o
, while the largest wave events are usually from that direction (storms caused
by the tropical cyclones). This indicates that these large wave events do not occur often.
A seasonal cycle can be observed in figure 2.2. During summer the wave energy spreading over the
directions is the largest of all seasons, indicating that waves come from a wide variation of directions. The
energy is also widely spread over the frequencies. This indicates that during this season both swell waves
(low frequencies) as wind waves (high frequencies) occur. During autumn both the directional energy
spreading as the spreading over the frequencies declines relative to the summer climate. More wave energy
are swell waves and most wave energy has a incoming wave angle between 125o
and 180o
. During winter
the wave energy spreading is the smallest of all the seasons. During this season most wave energy comes
from swell waves and the directional spreading is the lowest of all seasons. There is a energy peak with
wave angles between 135o
and 170o
. During spring both the directional spreading as the spreading over
the frequencies increases again. Autumn and spring can be seen as transitional seasons from summer,
which has largest spread of wave energy over the frequencies and directions, and winter, which has the
most intense energy peaks.
6

1.20
0°
90°
315°
225°
45°
135°
0
1.05
0.15
0.90
0.75
0.60
0.30
0.45
0.05
0.10
0.15
0.20
0.25
270°
180°
0.8
0°
90°
315°
225°
45°
135°
0
0.7
0.1
0.6
0.5
0.4
0.2
0.3
0.05
0.10
0.15
0.20
0.25
270°
180°
1.20
0°
90°
315°
225°
45°
135°
0
1.05
0.15
0.90
0.75
0.60
0.30
0.45
0.05
0.10
0.15
0.20
0.25
270°
180°
1.75
0°
90°
315°
225°
45°
135°
0
1.50
1.25
1.00
0.75
0.25
0.50
0.05
0.10
0.15
0.20
0.25
270°
180°
0.90
0°
90°
315°
225°
45°
135°
0
0.75
0.60
0.45
0.30
0.15
0.05
0.10
0.15
0.20
0.25
270°
180°
(a) Full dataset (b) Summer (c) Autumn
(d) Winter (e) Spring
Figure 2.2: The three dimensional mean wave spectra of the the Durban Waverider, based on the
cosine-2s spreading function
The seasonal cycle suggests that the occurrence of large storm events is not uniformly distributed over
the year. When large wave events are not uniformly distributed, the occurrence of these events are possibly
clustered. This clustering could mean that wave events with large wave heights and periods are mostly
happening closely after each other (most likely during winter and autumn). This could mean the time
between storms is under estimated.
When two storms occur relatively close after each other then the beach has less time to restore, and
therefore the clustering of wave events can have an effect on the morphological response of the beach.
Figure 2.4 shows the distribution of wave events (wave heights larger than 3.5m) over the years. The
plot shows that most storm event are clustered in groups, and are thus not occurring uniformly spread
over the year. This is confirmed by figure (b) and (c). These figures show the occurrence of large wave
events during different months and different seasons. It is difficult to determine what exactly a large wave
event is. In literature it is in general assumed that a large wave event for Durban coast have a significant
wave height larger than 3.5 m (Corbella & Stretch, 2012c). In figure 2.4 the number of wave events with
a wave height larger than 3.5 and 2.5 m per season and per month are plotted. This plot thus shows
events occurring using two different threshold wave heights. The storm events with wave heights larger
than 3.5m occur mostly in winter, followed by spring. The storm events with wave heights of at least 2.5m
are occurring mostly in spring, followed by winter. The differences between the number of storm events
happening during winter and spring, compared with summer and autumn is obvious. During winter the
occurrence of storm events with waves is five times as high than during summer. The occurrence of storm
events with a wave height larger than 2.5 m occurs about twice as much as during summer. From this can
be concluded that the occurrence of large wave events are clearly not equally distributed over the seasons.
7

Date
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
Peak
storm
wave
height
[m]
0
2
4
6
8
10
Figure 2.3: Peak wave height of large wave events. A wave event is defined as a set of observed wave
conditions with a wave height larger than 3.5 m. The data includes data from Richards bay.
J F M A M J J A S O N D
Average
number
of
storm
events
0
2
4
6
8
10
Per month
Sum Aut Win Spr
Average
number
of
storm
events
0
5
10
15
20
25
Per season
Hthres
= 3.5 m
Hthres
= 2.5 m
Figure 2.4: The average number of large storm events for different months (left) and seasons (right)
2.3 Sediment transport
The morphology of a coast is determined by the transport of sediment. In this report transport in two
directions is considered: Transport in the alongshore direction and transport in the cross shore direction.
2.3.1 Cross shore transport
The cross shore transport can have a large influence on the morphology of a coast. Cross shore transport
act in two directions: onshore and offshore.
The offshore directed sediment transport is driven by large flow velocities near the bottom. These
velocities are caused by the undertow. The undertow is directly related to the wave energy (Ener g y ∼ H2
).
Offshore directed transport is dominant during swell storm events, with wave conditions with large wave
heights and long periods. In Durban this effect causes bar-forming.
The onshore directed transport is mainly driven by the skewness of short waves (Roelvink & Stive,
1989). The onshore transport thus mainly occurs for wave conditions short periods. When the undertow
8

is not strong these short waves move sand onshore and are therefore restoring the beach profile after large
swell events.
The bar forming is a short term phenomenon that happens during the few days that a storm hits in.
The recovery of a beach is a long term process. Until a beach is recovered the bar in front of the beach
will influence the water and sediment dynamics in the surf zone, because large waves will break above
this bar. The lower the bar is (and thus the more recovery has taken place) the less influential the bar will
be. For both the cross and the alongshore transport and the morphology it is thus important to take this
recovery into account. This can only be done by selecting not only highly energetic wave conditions, but
also smaller waves so that recovery can take place.
Corbella & Stretch (2012a) have investigated the time it takes for the Durban beaches to recover.
They concluded that this depends on several properties of the beach. On average the recovery time is
about 2 years. Beaches with rocky outcrops tend to recover slower than the other beaches. Urbanized
beaches recover slower than vegetated beaches. They also think that the recovery depends on the amount
of alongshore transport and the intensity of cross-shore effects (which depends on the sediment size and
the steepness of the beach).
2.3.2 Alongshore transport
Changes in the morphology are, next to the cross shore transport, also dependent on the alongshore trans-
port. Changes in the morphology occur when there is a gradient in the alongshore transport direction. This
change occurs when the alongshore flow velocities are different at certain locations along a beach. For the
Durban coast the alongshore transport is forced by a current along the coast. This current is generated by
breaking waves that are coming in from with an angle relative to the normal of the coastline. A difference
in the generated alongshore current at two locations at the beach can therefore be caused by a difference
in the incoming wave condition (different wave height, period or direction) or a difference in the mor-
phology which leads to a difference in the location and or intensity of the breaking process between the
two locations. In this section it is investigated what kind of wave conditions mainly cause the alongshore
transport so that these factors can be taken into account during the wave climate reduction.
Some wave climate reduction techniques use bulk alongshore reduction techniques to construct re-
duced wave time series. Besides that the wave climate reduction techniques that are presented in this
report are also evaluated with alongshore transport formulas. To give a better understanding of the bulk
alongshore transport, first these complete wave data set is used to determine the alongshore transport. It
is also investigated how the seasonality of the waves influences the alongshore transport.
There are several bulk transport formulas known in literature. In this report three formulas are used:
the CERC formula, the Kamphuis formula and the Bijker formula.
In table 2.2 the estimated annual transport rates for the different formulas are reported. Based on
measurements from the sand trap near the harbour (see paragraph 2.5) the net annual longhore transport
is 500,000m3
per year. It must be taken into account that alongshore transport measurements are difficult
to obtain, and therefore the standard error of this value is likely to be large. This uncertainty does also
apply to alongshore bulk transport formulas. It is for example mentioned in lots of cases in literature
that the CERC formula gives about twice the amount of transport than the Kamphuis formula. The CERC
formula results in a net northern transport of 1.27 × 106
m3
. This is more than two times the measured
transport. The Northern transport is about twice the size of the southern transport, suggesting that most
sediment transport is caused by waves coming from a southern direction.
The Kamphuis formula gives a net northern transport of 4.15 × 105
m3
. the northern transport is three
times as large as the southern transport. This net transport is similar to the measured values at the sand
trap.
The Bijker formula gives about the same net transport as the Kamphuis formula (7.86 × 105
m3
). The
northern transport rate is now only a bit larger than the southern transport. This suggests that the waves
9

with angles smaller than 120o
have relatively more influence when the Bijker formula is used, than other
formulas.
Table 2.2: Annual longshore transport
rates
CERC Kamphuis Bijker
Net 13.1 4.1 7.9
Gross 24.4 10.2 30.3
Northern 18.7 7.2 19.1
Southern -5.6 -3.0 -11.2
All values are in [105
m3
/year]. North-
ern (Southern) is the annual average sed-
iment transport in northern (southern)
direction SNo (SSo). Net is the net an-
nual average sediment transport: SNet =
SNo − SSo. Gross is the gross annual aver-
age sediment transport SGro = SNo − SSo.
In figure 2.5 the cumulative sediment transport of the different transport formulas is shown. The
transport formula react with different strengths to the wave conditions. The shape of the cumulative
transport formulas is very similar. It can be seen that certain periods in time the cumulative sediment
transport quickly increases. This is especially true for the CERC transport. These periods seem to determine
most of the net sediment transport.
2008 2009 2010 2011 2012 2013 2014 2015
Cum.
Sediment
transport
×106
0
2
4
6
8
CERC
Kamphuis
Bijker
Figure 2.5: Cumulative sediment transport for three different bulk sediment transport formulas over
the sample.
To determine what happens during these periods of fast increasing cumulative sediment transport,
the influence of different wave parameters is analyzed. In figure 2.6 (a) the relative influence of wave
conditions with a certain wave height to the total transport is shown. In the figure can be seen that waves
with height of 1.5 m and 2.0 m have the largest influence on the sediment transport. This is surprising
since in all the transport formulas the wave height scales with a power, and therefore one could suspect
that the wave conditions with large wave heights would have a larger influence on the total transport.
Wave conditions with small wave heights have a larger influence on the sediment transport when the
Bijker formula is used, while the large waves have the largest influence when the Kamphuis formula is
used.
In 2.6 (b) the breakdown of the influence per period is visualized. The transport by the Bijker formula
is relatively the most influenced by short period waves of all the transport formulas. The difference with
the other bulk transport formulas is relatively large.
Figure 2.6 (c) shows that the waves from the southern directions have most influence on the transport
for all three transport formulas. The Bijker formula gives more weight to the waves from a southern
direction. This could be caused by the fact that most waves with directions lower than 120o
are waves
10

with lower periods, which in the Bijker formula have less influence on the sediment transport.
The influence distribution for the seasons is shown in figure 2.6 (d). The relative influence on the
sediment transport is the highest during winter and autumn. This suggests that not only large wave events
are clustered, but also the alongshore transport. In section 2.2 was concluded that most large wave events
were occurring during winter and spring. Most of the sediment transport is occurring during winter and
autumn. This suggests that wave events with large wave heights could drive the alongshore sediment
transport, but are not always doing that, when a bulk transport formula is used.
The difference between the bulk transport formula are observable. This is especially the case for the
period. Here the Bijker shows to be much more influenced by waves with short period than the other
transport formulas. This is an important fact to take into account when wave climate reduction decisions
are made. When choosing for the Bijker formula, one has to expect that non-swell waves will be more
represented in the reduced wave climate.
(a) Wave height [m]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Influence
on
transport
[%]
0
10
20
30
40
CERC
Kamphuis
Bijker
(b) Period [s]
2 4 6 8 10 12 14 16 18
Influence
on
transport
[%]
0
5
10
15
20
25
30
(c) Direction [o
]
4
0
6
0
8
0
1
0
0
1
2
0
1
4
0
1
6
0
1
8
0
2
0
0
2
2
0
Influence
on
transport
[%]
0
10
20
30
40
50
60
(d) Season
Sum Spr Win Aut
Influence
on
transport
[%]
0
5
10
15
20
25
30
35
Figure 2.6: Breakdown of the influence of different input variables on the sediment transport. The
influence is defined as the relative amount of the the total transport caused by a wave condition with
the property reported on the x-axis.
In figure 2.7 the breakdown is shown of the influence the wave conditions on the gross alongshore
11

transport during a specific season. The figure confirms most conclusions that have been drawn from pre-
vious analyses.
During winter and autumn the conditions have more influence on the sediment transport, than during
the other seasons. Still the wave conditions with wave heights between 0.75 and 2.25 m have the largest
influence on the sediment transport of that season. This suggests that, to produce a representative wave
climate, one can not simply select the conditions with large wave heights, since this not reproduces the
natural behavior of the morphodynamics of the coast.
The influence of the different periods on the sediment transport also varies over the seasons. In section
2.2 is concluded that during winter and autumn the wave energy was less spread over the frequencies
and was mostly located around the lower (swell) frequencies, while during summer and spring the energy
was distributed over both the small and the large frequencies. The influences of the sediment transport
show slightly different results. During summer indeed the short periods dominated the sediment transport
(especially when the Bijker formula is used). For sediment during winter and spring the wave conditions
with high periods dominate the sediment transport, while during autumn the wave conditions with low
periods dominate again. The dominance of the lower periods during autumn is notable, since most waves
that are observed during that season have mostly long periods relative to the other seasons. The high
variation of influence during the seasons suggests that it is important to take into account the period of the
waves. This is sometimes not done when wave climates are reduced.
The variation of the influence of wave conditions are shown in the most right column of figure 2.7.
The figure shows that, as expected, the waves from the southern directions have most influence on the
sediment transport. This effect is the strongest during winter, where more than 50% of the transport is
caused by waves that come from a direction between 150o
and 170o
. The spreading of influence over the
directions is the strongest during summer.
12

Wave height [m]
0 0.5 1 1.5 2 2.5 3 3.5 4
Influence
[%]
0
20
40
Summer
CERC
Kamphuis
Bijker
Period [s]
2 4 6 8 1012141618
Influence
[%]
0
20
40
Summer
Direction [o
]
40
60
80
100
120
140
160
180
200
220
Influence
[%]
0
20
40
60
Summer
Wave height [m]
0 0.5 1 1.5 2 2.5 3 3.5 4
Influence
[%]
0
20
40
Spring
Period [s]
2 4 6 8 1012141618
Influence
[%]
0
20
40
Spring
Direction [o
]
40
60
80
100
120
140
160
180
200
220
Influence
[%]
0
20
40
60
Spring
Wave height [m]
0 0.5 1 1.5 2 2.5 3 3.5 4
Influence
[%]
0
20
40
Winter
Period [s]
2 4 6 8 1012141618
Influence
[%]
0
20
40
Winter
Direction [o
]
40
60
80
100
120
140
160
180
200
220
Influence
[%]
0
20
40
60
Winter
Wave height [m]
0 0.5 1 1.5 2 2.5 3 3.5 4
Influence
[%]
0
20
40
Autumn
Period [s]
2 4 6 8 1012141618
Influence
[%]
0
20
40
Autumn
Direction [o
]
40
60
80
100
120
140
160
180
200
220
Influence
[%]
0
20
40
60
Autumn
Figure 2.7: Breakdown of the influence different variables have on the total transport during a certain
season. The influence is defined as the relative amount of the the total transport caused by a wave
condition with the property reported on the x-axis.
13

2.4 Rivers
The case study beach is located in between two rivers: The Ohlanga and the Umgeni river. In (Theron
et al., 2008) is estimated that the sand discharge of the the Umgeni river is about 25000 m3
/y and that
of the Ohlanga about 0 m3
/y. The Umgeni is dammed at several locations. These dams trap most of the
sand. The Ohlanga sediment discharge is significantly influenced by the large sand mining activities. From
this can be concluded that both rivers will not add an extensive amount of sediment to the Durban coastal
area, relative to the wave driven alongshore sediment transport that is occurring there. Their influence
can therefore be neglected.
2.5 Data
Two sorts of data is used in this report: Wave data and morphological data.
The wave data are measured with a directional wave rider buoy at 35 m depth in front of the Durban
harbour. The location of the buoy is indicated in figure 2.1). The wave data are provided by the operator
of the Durban harbour Transnet. Wave data are measured with a frequency of half an hour and is available
from 23 August 2007 until 30 November 2014. The length of the dataset is thus 7.3 years. In (Corbella &
Stretch, 2012c) it is argued that the wave data from the Durban waverider has a large correlation with the
wave data from a wave rider buoy in Richards bay. This dataset could thus increase the length of the wave
data set. In (Corbella & Stretch, 2014) it is shown though, that the Richards bay wave dataset contains
less wave energy from the eastern direction in Durban. The wave energy from the east is the driver for the
sediment transport in the southern direction. It is therefore concluded that the Richards bay data can only
be used for analyses that do not have the wave direction as an input variable.
The morphology data consist of a recent bathymetry of the case study beach and the dredged mea-
surements of the sand trap. The bathymetry is provided by the eTwingeni Metro and is measured by the
Envionmental mapping & Surveying company during the autumn of 2014.
Alongshore transport rates can be determined from the dredged data of the sand trap, which is located
south of the Durban harbour (figures 2.1 and 2.8 (a) and (b)). Schoonees (2000) argues that these rates
can be interpreted as net transport rates. This data will be used for an approximation of the alongshore
transport rate of the case study beach, because the coastal orientation and beach properties are similar to
the case study beach. In figure 2.9 the measured transport rates are shown. The mean measured transport
rate is 5.3·105
m3
/y and the standard deviation is 1.8·105
m3
/y. The volatility of the sediment transport
is not large, which suggests the mean value of the transport is a realistic representation. It has to be taken
into account however, that the measurement error is possibly large (possibly, because it is unknown). The
measurement error is potentially large due to three main reasons. First the amount of sand captured by
the sand trap depends on the amount of sediment that is already in the sand trap. The sand trap is a
dredged hole. The more filled up it becomes, the less sand will settle in it, and therefore it is likely that not
all transported sediment is captured by the sand trap. The second reason is that the measured amounts
of sand are based on estimations by dredging companies. These companies could have an (unproven)
bias that overestimates the amount of sand dredged, because this could lead to more profits. Thirdly it is
not clear whether the measured rates are northern, gross or net transport rates. The measured data are
northern transport rates when only the transport in the northern direction is captured by the sand trap,
and no sediment leaves the sand trap. The measured data are gross transport rates when both the northern
and the southern transport are captured by the sand trap, while no sediment escapes from the sand trap.
The measured data are net transport rates when the northern transport rates are captured by the sand trap,
and none of the southern transport are captured, while only the sediment in southern direction can escape
from the sand trap.
In reality most of the northern transport rates will be captured by the sand trap, because the northern
sediment flow is blocked by the harbours southern breakwater. There will be not much transport in the
southern direction captured, because this form of transport is generated by waves coming from the (north)
14

east. At the north eastern side of the sand trap depths are relatively large, which in practice means that
not a lot of transport will occur, since waves will not break there, causing no bed velocities. Sand could
escape the from the sand trap, especially when it is nearly full. In that case waves from the north could
generate transport that will let some sediment escape from the sand trap in the southern direction. The
escaping sediment is likely to be in the southern part of the sand trap, since that is the location where most
sediment is settled. Waves from the southern direction could also cause this effect, but the sediment that
is then moving to the south first has to pass the complete sand trap, and could settle more northern in the
sand trap. In practice the measured transport rate is likely a mix between northern and net transport.
(a) Bathymetry around the sand (b) Overview sandtrap
Figure 2.8: Overview of the location of the sand trap. Source: (Schoonees, 2000)
1980 1985 1990 1995 2000 2005 2010
Dredged
volumes
[m
3
/y]
×105
0
2
4
6
8
10
Observed
Mean
Figure 2.9: Sediment dredged from the sand trap
2.6 Conclusions
Three conclusions can be drawn from this chapter.
The first conclusion is that not only alongshore, but also cross shore dynamics have influence on the
morphology of this coast. Most large waves come in from an angle close to the normal of the coast. This
leads to extensive cross shore hydraulic velocities and is probably one of the reasons that bar forming is
observed. Wave conditions that generate mostly cross shore sediment transport must therefore not be left
15

out of the reduced wave climates.
The second conclusion is that the wave conditions in a reduced wave climate can probably not be seen
as independent events, since they probably influence each others morphological response. This conclusion
is supported by three observations.
The first observation is that bar forming is observed. When a bar is present the morphology reacts
different on wave conditions, then when it is not present. The morphological response on a wave condition
is then thus dependent on the changes in the morphology due to previous occurring wave conditions.
Secondly is observed that both the observed wave climate and the bulk transport rates show cyclic
behavior. This will have influence on the size and location of the bar. This means that the morphology does
not have one equilibrium state. This suggests that wave conditions are dependent on the morphological
response on previous wave conditions, since that determines the height and the location of the bar.
Thirdly is observed that large wave conditions are clustered and mainly occur close to each other in
time. When the period between two storm events is short, the effects on the cross shore profile by a storm is
not restored when another storm hits. The beach is then more exposed to this second incoming storm and
will respond differently. This conclusion is taken into account by running the Delft3D model in a morpho-
dynamic setup instead of a morphostatic setup. A morphodynamic setup means that the wave conditions in
a reduced wave climate are forced on the model in one simulation. The bathymetry that follows from the
force of the first wave condition is the initial bathymetry for the simulation of the second wave condition
and so on.
The third conclusion is that not only swell waves, but also locally generated wind waves have influence
on the morphology. In the bulk transport analysis is observed that wave conditions with low period waves
are responsible for a significant amount of the total alongshore sediment transport.
16

3 | Methodology
In the chapter three parts can be distinguished. First eight reduction algorithms are described. The meth-
ods will be implemented with different parameter setups. A description of that can be found in section
3.1. Secondly several methods will be evaluated that optimize the sequence of the reduced wave climate.
The description of these models can be found in section 3.2. As last is in section 3.3 explained how the
performance of the reduction algorithms and the sequencing methods will be evaluated.
3.1 Reduction algorithms
In practice several input reduction methods are available. Methodologies can be divided into two groups:
Methods with a sediment target and methods without a sediment target. A sediment target is a target
that is used as a benchmark for the measurement of the performance of the climate reduction. When the
method does not use a sediment target, it bases its reduction on the retaining of the statistical properties
of the data set. All methods try to find user defined number of k wave conditions that represent the full
observed data set the best.
A wave climate is a collection of observations that describe the sea state over time. In a multivariate
approach these sea states are described with multiple variables. The collection of these variables at a
certain time is called a wave condition. Examples of these variables are the significant wave height, peak
period, dominant direction, direction and strength of wind waves from another direction and the duration
of the storm event. In the report three basis variables are used: The significant wave height (Hs), the peak
period (T) and the wave direction (θ).
There can be two approaches distinguished in the multivariate data reduction literature: 1. The clus-
tering approach and 2. The dissimilarity approach. In the clustering approach similar observed wave
conditions are bundled in one cluster. Every cluster of wave conditions will be represented by one repre-
sentative wave condition. The dissimilarity approach selects k observed wave conditions that, by scaling
up their probability of occurrence, represent the observed data set of conditions the best. In the approach
observations are selected that are the most dissimilar of each other.
In the clustering literature five main methods can be distinguished: 1. Partional methods, 2. Hierar-
chical methods, 3. Density-based methods, 4. Grid based methods and 5. Model based methods.
The partional clustering algorithms distribute a set of observations iterative over k clusters, based on
their similarity. The method has a hard membership function when every observation belongs to maximal
one cluster, the method has a soft membership function when a observation can belong to more than one
cluster. Examples of these methods are the (fuzzy) k-means and (fuzzy) k-centroids approaches.
In hierarchical clustering all N observations are divided into N individual clusters. In an iterative
process the clusters are merged until k clusters remain. Examples of common used algorithms are BIRCH
(Zhang et al., 1996), CURE (Guha et al., 1998) and CHAMELEON (Karypis et al., 1999).
The density based clustering methods select k observations as initial clusters. The methods then let the
clusters grow by adding neighbor observations to the cluster. Neighbors are added until the clusters reach
17

each others boundaries. In this way, clusters with equal sizes can be constructed. Examples of density
based models are DBSCAN (Ester et al., 1996) and OPTICS (Ankerst et al., 1999).
Grid based models divide the observation space in grids. It then clusters these grids. An example of
grid based clustering method is STING (Wang et al., 1997).
For the model-based clustering methods a model is assumed that describes every cluster. Observations
are added to the clusters when they fit best to the assumed model. An example is the Self Organizing Map
approach (Kohonen, 2001).
The described methods can all be used to cluster observations in a static approach. A static approach
assumes that the observed wave conditions is a data set are independent and that the sequence of the ob-
servations is negligible. When sequence is important the data set is time dependent. Time dependent data
sets can be clustered with special variants of the clustering approaches. There are two disadvantages of
using time dependent clustering methods relative to static clustering methods: 1. It is likely more reduced
wave conditions k are needed to describe the original data set to find accurate results. 2. The internal error,
which measures the sum of the differences between the observed wave conditions and the representative
wave condition of their cluster, is likely to be larger than that of the static approach.
In the next subsection general notation and algorithm features that are part of multiple reduction meth-
ods are explained, so that a better understanding of the working of the algorithms can be obtained. Then
the algorithms themselves are described. The algorithms are implemented in with different parameters,
these are called different algorithm setups.
3.1.1 Notation, Normalization and Distance measures
3.1.1.1 Notation
In the description of the methods that are used in this report a standard notation will be used. Define
xi as a vector that describes a measured wave condition with multiple variables xi = {xi,1, xi,2,..., xi,z}.
Here z is the number of variables that is used to describe a wave condition. A wave condition can for
example be described with the wave height (Hs), peak period (Tp) and average wave direction (θ). In
that case z = 3 and xi = {Hs,i, Tp,i,θi}. All the wave conditions that are available are collected in database
X = {x1, x2,..., xN }, where N is the total number of observed wave conditions. All reduction methods
describe the original data set X with k representative wave conditions, which in the data mining liter-
ature are called centroids. Define a centroid as a vector that describes a representative wave condition
νi = {νi,1,νi,2,...,νi,z}. The centroids are collected in the database V = {νi,ν2,...,νk}. When the same
example with z = 3 is used: νj = {H
rep
s,j
, T
rep
p,j
,θ
rep
j
}, where the superscript rep means that the variable is
representing a group of observations.
3.1.1.2 Distance measures
Some reduction methods use distance measures. A distance measure is a technique to determine the
distance between two points in a multivariate space. The algorithms use this distance to determine how
similar observations are to each other: The smaller the distance between two observations the more similar
they are. In literature multiple distance measures are available. Because the observations used in the
algorithms are time independent and can be freely located in the complete observation space the Euclidean
distance is used.
A distance measure between an observation and for example a centroid xi and νj will be indicated with
two vertical bars: ||xi,νj|| . The Euclidean distance can be generalized as:
||xi,νj|| =
q
(w1R1(xi,1,νj,1))2 + (w2R2(xi,2,νj,2))2 + ... + (wzRz(xi,z,νj,z))2 (3.1)
Here R stands for the distance measure of a variable that is used to describe a wave condition and wi for a
18

user defined weight a certain variable has in the distance function. The distance measures for the different
variables used in this report are specified as RH (distance measure for the wave height), RT (distance
measure for the periods), Rθ (distance measure for the direction) and RQ (distance measure for the day of
the year indicator):
RH(xi,νj) = H0
i − H
rep 0
j
(3.2)
RT (xi,νj) = T0
i − T
rep 0
j
(3.3)
Rθ (xi,νj) = min
€
|θ0
i − θ
rep 0
j
|,2 − |θ0
i − θ
rep 0
j
|
Š
(3.4)
RQ(xi,νj) = min
€
|Q0
i − Q
rep 0
j
|,2 − |Q0
i − Q
rep 0
j
|
Š
(3.5)
Here the accent indicates that the input variables are normalized. The variables in these formula are
defined in the next section.
3.1.1.3 (De) Normalization
The distance between observations is determined in a space that is determined by multiple variables. There-
fore all reduction methods that use a distance function also use normalized input data. Non circular vari-
ables (the wave height and period) are scaled to values between [0,1]. This is conducted with a simple
linear transformation. First define minimum and maximum values of a set of wave heights H and periods
T:
Hmin
= min(H), Hmax
= max(H) (3.6)
Tmin
= min(T), Tmax
= max(T) (3.7)
The normalization is conducted with feature scaling. Resulting in a normalized wave height H0
and
period T0
:
H0
=
H − Hmin
Hmax − Hmin
(3.8)
T0
=
T − Tmin
Tmax − Tmin
(3.9)
Variables that are circular are transformed differently. It has to be taken into account that these variables
describe a circle and thus that the distance between two values can be measured clockwise and counter
clock wise. This for example means that the maximum difference between two radial directions is π.
Therefore the distances between two angles are normalized between [0,1] and the absolute values of the
angle will be normalized between [0,2]. It is assumed that the angles are measured in radians or that
sexagesimal degrees are converted to radians by multiplying them with a factor π/180. The normalized
angle θ0
is found by converting the observed angle θ.
θ0
=
θ
π
(3.10)
The day-of-the-year indicator (Q) is a variable which describes at which day in a certain year a certain
wave condition is observed. This variable thus reaches from 1 until 365. The variable will be modeled as a
circular variable (like the angle). It is then for example taken into account that day 1 (1 January) is more
close to day 365 (31 December) than day 30 (30 January). The distance between two variables will be
19

normalized between [0,1] and the absolute values will be scaled between [0,2].
Q0
=
2(Q)
max(Q)
(3.11)
The denormalization procedure is the opposite of the normalization procedure:
H = H0
· Hmax
− Hmin

+ Hmin
(3.12)
T = T0
· Tmax
− Tmin

+ Tmin
(3.13)
θ = θ0
· π (3.14)
Q =
Q0
max(Q)
2
(3.15)
3.1.2 Bulk transport formulas
In this report bulk transport formulas are used by the wave climate reduction methods with a sediment
budget and for the evaluation of multiple reduction techniques. Three formulas are used. The first is the
CERC formula (Komar Inman, 1970), second the Kamphuis formula (Kamphuis, 1991), the third is the
Bijker formula (Bijker, 1971). Recently Mil-Homens et al. (2013) have re-evaluated the most used bulk
transport formulas. Their findings have been implemented in the formulas described here.
1. The CERC formula The CERC formula is defined as:
SB =
J
16(s − 1)(1 − p)
v
t g
γ
sin(2(θ − θC ))H2.5
(3.16)
Here SB is the bulk sediment transport, s the relative sediment density, p the porosity, γ the breaker
parameter and θC the angle of normal of the coastline. J is based on a polynomial fit done by (Mil-Homens
et al., 2013):
J =

2232.7

H
L
‹1.45
+ 4.505
−1
(3.17)
2. The Kamphuis formula The Kamphuis formula is defined as:
SB =
17.5tan(α)0.86
D−0.69
50
ρ(s − 1)(1 − p)
H2.75
T0.89
sin(2(θ − θC ))0.5
(3.18)
Here tan(α) is the beach slope at the breaker point.
3. The Bijker formula The Bijker formula is originally not a bulk transport formula. But by averaging
it over the cross-shore, it can be seen as a bulk transport formula. First of all three parameters have to
be determined: the ripple factor (µ), the Bijker wave parameter (e) and the roughness height for currents
(Rc):
µ =

C
C90
1.5
, e =

C fw
2g
‹0.5
, rc =
g
C2
(3.19)
20

Where C and C90 are the Chezy value based on the D50 and D90 respectively. g is the acceleration of gravity
and fw is Johnsons friction factor. The orbital velocity at the bottom (u0) can then be determined:
u0 =
H gTcosh
€
2π(d+H)
L
Š
2Lcosh 2πd
L
cos(θ − θC ) (3.20)
Here d is the water depth, H the wave height, and L the wavelength. The wave forced longshore current(Vc)
is then:
Vc = 20.7tan(α)gH0.5
sin(2(θ − θC )) (3.21)
The bulk transport can be determined with:
SB = S1 + S2 (3.22)
S1 and S2 are defined as:
S1 =
BD · D50V
p
g
C
exp
−0.27
ρs−ρ
ρ D50ρg
µτcw
!
(3.23)
S2 = 1.83S1

I1ln

33d
rc
‹
+ I2
‹
(3.24)
BD a user specified variable. The other variables are specified as:
I1 = 0.216
rc
d
z∗
−1
1 −
rc
d
z∗
Z 1
rk/d

1 − y
y
‹z∗
d y (3.25)
I2 = 0.216
rc
d
z∗
−1
1 −
rc
d
z∗
Z 1
rk/d
ln(y)

1 − y
y
‹z∗
d y (3.26)
µ∗
=
τcw
ρ
0.5
, z∗
=
ws
κµ∗
(3.27)
τcw = ρrcV2

1 + 0.5e
u0
V

(3.28)
Here κ is the von Karman constant.
3.1.3 Algorithm 1: Conditions with the largest transport contribution
Algorithm 1 is a grid based model that works with a sediment budget. It first distributes the observed wave
conditions over a set of bins with predefined boundaries. The observations in a bin are described with
one descriptive wave condition. Per wave bin can be determined how much it is contributing to the total
sediment transport. This is executed by estimating both the sediment transport caused by the waves in a
certain bin and the total transport that is caused by the full wave climate with a simple sediment transport
formula, so that the relative influence becomes clear. The relative influence on the total transport by waves
of a certain bin i, Wi can then be calculated as follows:
Wi =
P(H = Hs,i,θ = θi)S(Hs,i,θi)
Stot
(3.29)
21

Stot =
N
X
i=1
P(H = Hs,i,θ = θi) S(Hs,i,θi) (3.30)
Here P(H = Hs,i,θ = θi) is the probability of occurrence of wave bin, which depends on the represen-
tative wave height (Hs,i) and wave direction (θi) of the bin. S(Hs,i,θi) is the sediment transport caused by
the representative wave condition of wave bin i. This can be determined with a simple sediment transport
formula as the CERC formula. Stot is the total sediment transport, caused by all the representative wave
conditions. N is the number of original wave conditions.
Based on these formulas the k bins with the highest influence on the total sediment transport are
selected. The reduced wave climate will consist of the representative wave conditions of the selected bins.
To create a well-balanced wave climate the probability of occurrences of the reduced wave conditions have
to to be scaled up in order to ensure the total sediment transport caused by the reduced wave climate is
the same as the sediment transport caused by the full wave climate.
Pup,i = P(Hs,i,θi)
Stot
Pk
i=1 P(Hs,i,θi)S(Hs,i,θi)
(3.31)
Pup,i is the new probability of occurrence of the waves in bin i.
This algorithms will we applied in three different setups. Model 1C uses the CERC formula, Model 1K
the Kamphuis formula and Model 1B the Bijker formula to determine the alongshore sediment transport.
3.1.4 Algorithm 2: Grouping with equal sediment influence
Algorithm 2 is a distribution based algorithm. In the literature mostly the bivariate variant is used (with
wave height and direction as variables). The algorithm will form k clusters. It starts by selecting k initial
wave conditions as individual clusters. Every iteration of the algorithm every cluster picks one observation
that will become part of that cluster. The algorithm will results in k clusters that all will cause the same
amount of sediment transport. The following algorithm is used for this method:
1. Determine the sediment transport that every observation is causing (S(xi)). This is done with one
the three bulk sediment transport formulas (CERC, Kamphuis or Bijker). The total transport will
be Sabs,tot =
PN
i=1 |S(xi)|. The algorithm will form k clusters that all represent a certain number of
observations. The sum of the absolute sediment transport of a certain cluster Sj will be Sclu,j ≈ Stot/k
since the clusters represent an equal amount of sediment transport.
2. Randomly select k observations as the initial clusters centroids V and the first observations that
belong to these clusters.
3. An iterative process starts. During every iteration every cluster selects an observation that is not
in a cluster yet and has the smallest distance to the clusters centroid. When due to the selection
of an observation for cluster j applies: Sclu,j ≥ Sabs,tot/k the selected observation will not join the
cluster. Instead an observation with the second smallest distance to centroid νj is selected. This
observation can also join the cluster as long as Sclu,j ≥ Sabs,tot/k. When this is not the case the third
closest observation is selected. After every iteration the cluster centroids V are updated by taking
the average values of the wave conditions that are appointed to the cluster.
4. When no observations can join a cluster anymore, because for every cluster Sclu,j ≥ Sabs,tot/k , j =
1...k the remaining observations join the cluster to which they have the smallest distance to.
This algorithms will be applied in three different setups. Model 2C uses the CERC formula, Model 2K
the Kamphuis formula and Model 2B the Bijker formula to determine the alongshore sediment transport.
22

3.1.5 Algorithm 3: The representative wave approach
The representative wave approach is introduced by Brown Davies (2009). They divide the data set in
sections over time. In Brown Davies (2009) every section represents a season. Per section one repre-
sentative wave height, wave direction and wave period is determined, by taking the (weighted) average
value. In this approach the chronology of the original data set is therefore retained, it is therefore a time
series clustering method. The wave period and wave direction are determined by taking the unweighted
average value of one section. The representative wave height of a section is determined as follows:
νj,1 = H
rep
s,j
=



P
i:xi∈Cj
(fiH
p
s,i
)
P
g:xg ∈Cp
fi



1
p
(3.32)
Here fi is the frequency of occurrence of the significant wave height Hs,i of observed wave condition xi.
Cj is a database of the observations that occurred during section j. Here p is the power of the sediment
transport formula that is used in the morphology model.
The representative wave approach will be implemented with two setups. For algorithm 3L p = 1, while
for algorithm 3NL p = 2.5.
3.1.6 Algorithm 4: Non-linear wave bins
The non-linear wave approach is introduced by Walstra et al. (2013) and is a grid based cluster approach.
This method uses bins with predefined boundaries. All observations are divided over the bins to determine
the occurrence of the bins. The representative wave conditions of the bins are determined with a non-linear
weighting formula:
νj,1 = H
rep
s,j
=



P
i:xi∈Cj
fiH
p
s,i
P
i:xi∈Cj
fi



1
p
(3.33)
Model 4 will be implemented with only one setup. In that setup p = 2.
3.1.7 Algorithm 5: Crisp k-means
The k-means algorithm partional crisp cluster method. The algorithm starts with k initial randomly selected
centroids. Then all observations X are assigned to the cluster they have the smallest distance to. Next the
values of the centroids will be updated by taking the average of the wave conditions of the observations
that are assigned to the clusters.
The crisp k-means algorithm has a ’hard’ membership function, which means that observations can only
be a member of one cluster. The hard membership function can lead to poor clustering and makes that
method very dependent on the initial centroids of the algorithm. There are several measures to bypass these
problems. First of all an algorithm with a soft membership function can be chosen, in that case observations
can be part of multiple clusters. In this report two k-means algorithms with a soft membership function
are used: the Fuzzy k-means and the K-harmonic means algorithm. A second measure to increase accuracy
and independence of initialization is by smarty choosing the initial centroids, which pushes the algorithm
in the right direction. A third measure is to repeat the algorithm several times, so that different outcomes
can be compared. The run that results with the lowest error will then be selected as the final reduced wave
climate.
23

In (Kearns et al., 1998) is the difference between a hard and a soft membership function evaluated. It
is concluded that a hard membership results in less accurate clusters, but that the clusters do show less
overlap when compared with the soft membership k-means algorithms.
Several authors have proposed initialization methods for k-means. In Lozano Larranaga (1999)
and Meila Heckerman (2013) multiple initialization methods are discussed. Higgs Bemis (1997)
and Snarey et al. (1997) propose to use the MaxMin maximum dissimilarity algorithm as initialization
algorithm. This algorithm is a reduction method on its own (algorithm 8). Because this algorithm is not
simple, in literature usually other less complex algorithms are used. Currently the k-means++ algorithm
is commonly used (Arthur Vassilvitskii, 2007). Therefore this method is used in this report. Unpublished
tests have shown that the two initialization methods do not give large differences for the reduced wave
climates.
The algorithm can be described as follows:
1. Select the initial centroids using the k-means++ algorithm (Arthur Vassilvitskii, 2007)
(a) Draw uniformly an observation from X. This will be the first centroid ν1.
(b) Compute the distance between ν1 and all observations in X : D = ||xi,ν1||, i = 1,..., N. Here
||.|| stands for a multivariate distance between two vectors.
(c) Draw the second centroid ν2 from X with probability:
P(xi = ν2) =
||xi,ν1||2
PN
j=1 ||xj,ν1||2
, i = 1,..., N
(d) Assign the observations to the cluster for which the distance between the observation and the
clusters centroid is the smallest. Draw the remaining centroids iteratively with the probability:
P(xi = νg) =
||xi,νp||2
P
g:xg ∈Cp
||xg,νp||2
, g = 1,..., k − 2, i = 1,..., N, p = 1,..., k − g.
Here Cp is the database of observations that are in the cluster p.
2. Compute the distance between all observations and all centroids. Assign all observations to the
cluster of the centroid they are most close by.
3. Determine the new representative wave conditions V by averaging the the wave conditions of the
observations in a cluster:
νg =
P
g:xg ∈Cp
xg
Np
, p = 1,..., k
Where Np is the number of observations in cluster p.
4. Determine the internal error:
it =
Pk
p=1
P
g:xg ∈Cp
||xg,νp||
N
Here it stands for the it iteration of the algorithm. Now determine the improvement of V over the
iteration: ∆ = it−1 − it.
5. Repeat step 2, 3 and 4 until ∆ ≤ min. Here min is predefined by the user.
The crisp k-means is implemented with two setups. Both setups use three variables as input the wave
height, period and direction. Setup Model 5A uses an equal weighting for all variables in the distance
measure. This means that w1 = w2 = w3 = 1 in equation 3.1. Model 5B tries to take into account that the
wave height has more influence in sediment transport than the period and the wave direction. It therefore
uses an unequal weighting in the distance function. In this case w1 = 4, w2 = w3 = 1 in equation 3.1.
24

3.1.8 Algorithm 6: Fuzzy k-means
The fuzzy k-means is a variant on the crisp k-means algorithm.
The fuzzy k-means algorithm allows observations to be assigned to more than one cluster. With this
method transition wave conditions are possibly better selected.
(b) Compute the distance between ν1 and all observations in X : D = ||xi,ν1||, i = 1,..., N.
P(xi = ν2) =
||xi,ν1||2
PN
j=1 ||xj,ν1||2
, i = 1,..., N
(d) Draw the remaining centroids iteratively with the probability:
P(xi = νg) =
||xi,νp||2
P
g:xg ∈Cp
||xg,νp||2
, g = 3,..., k, i = 1,..., N, p = 1,..., k − j.
2. Determine how much each observation belongs to every cluster, this is called the membership func-
tion. o is a (N×k) matrix that quantifies these memberships. The weights in this matrix are restricted:
0 ≤ Mi,g ≤ 1. And the total contribution of one observation sums up to one: Mi =
Pk
j=1 Mi,g = 1.
Mi,g =
(1/||xi,νj||2
)1/(o−1)
Pk
j=1(1/||xi,νj||2)1/(o−1)
, i = 1,..., N, g = 1,..., k
3. Determine the new representative wave conditions V:
νg =
PN
i=1(Mi,g)o
xi
PN
i=1(Mi,g)o
, g = 1,..., k
it =
PN
i=1
Pk
g=1(Mi,g)o
||xi,νg||2
PN
i=1
Pk
g=1(Mi,g)o
5. Repeat step 2, 3 and 4 until ∆ ≤ min
Like the Crisp k-means algorithm the fuzzy k-means algorithm will be implemented with two setups.
Again it is tried to take into account the unequal influence of the wave describing parameters on the
morphology. The algorithm uses again the standard three input variables wave height, period and direction.
Model 6A uses an equal weighting in the distance measure. This means that w1 = w2 = w3 = 1 in equation
3.1. Model 6B uses an unequal weighting in the distance function. In this case w1 = 4, w2 = w3 = 1 in
equation 3.1. Visual inspection has lead to the fuzzy parameter o = 2.1.
3.1.9 Algorithm 7: K-harmonic means
The k-harmonic means algorithm is another way to decrease the dependency of a partional cluster method
on its initialization. The algorithm is a extension of the fuzzy k-means algorithm. It is using a dynamic
weight function. This means that in total some observations have a larger influence on the determination of
the centroid values during an iteration than other do. Dynamic weighting for partional clustering methods
25

is introduced by Zhang (2000). The weighting of the observations is done with a method called boosting
(Freund et al., 1999). This process gives more weight to data points that are not well-clustered yet. It
has been shown in several papers that the harmonic k-means gives accurate results of the partial methods
described in this report, while it is almost independent of its initialization (Zhang et al., 1999), (Zhang,
2000), (Hamerly Elkan, 2002) and (Ünler Güngör, 2009).
(b) Compute the distance between ν1 and all observations in X : D = ||xi,ν1||, i = 1,..., N.
P(xi = ν2) =
||xi,ν1||2
PN
j=1 ||xj,ν1||2
, i = 1,..., N
(d) Draw the remaining centroids iteratively with the probability:
P(xi = νg) =
||xi,νp||2
P
g:xg ∈Cp
||xg,νp||2
, g = 3,..., k, i = 1,..., N, p = 1,..., k − j.
2. Determine how much each observation belongs to every cluster, this is called the membership func-
tion. M is a (N × k) matrix that quantifies these memberships. The weights in this matrix are
restricted: 0 ≤ Mi,g ≤ 1. And the total contribution of one observation sums up to one: Mi =
Pk
j=1 Mi,g = 1.
Mi,g =
(||xi,νj||)−o−2
Pk
j=1(||xi,νj||)−o−2
, i = 1,..., N, g = 1,..., k
3. Determine the weights of the observations. Some observations will have more influence on the value
determination of the centroids (boosting). The matrix K is a (N × 1) matrix that contains these
weights. Ki =
Pk
j=1 ||xi − νj||−o−2
€Pk
j=1 ||xi − νj||o
Š2
4. Determine the new representative wave conditions V:
νg =
PN
i=1 Mi,g Ki xi
PN
i=1 Mi,g Ki
, g = 1,..., k
it =
N
X
i=1
k
Pk
j=1
1
||xi−νj||o
6. Repeat step 2, 3 and 4 until ∆ ≤ min
Model 7 uses the wave height, period and direction as input variables. Model 6A uses an equal weight-
ing in the distance measure. This means that w1 = w2 = w3 = 1 in equation 3.1. Model 7B uses an unequal
weighting in the distance function. In this case w1 = 4, w2 = w3 = 1 in equation 3.1.
26

3.1.10 Algorithm 8: Maximum Dissimilarity
The goal of MDA is to create a sub-set V of k centroids that represents the full diversity of the full data
sample X. This is done by maximizing the dissimilarity between the the vectors in the subset. This method
is introduced by Kennard Stone (1969). There are many methods available to measure the dissimilarity
between vectors. A lot of these variants of the MDA are discussed in Willett (1996). One of the most
accurate and efficient variants is the MaxMin algorithm. This algorithm is used in this report:
1. Determine the distances D of all observations in X with each other:
Di,m = ||xi, xm||2
, i = 1,..., N, m = 1,..., N
This distance can be interpreted as the dissimilarity between the observations. The initial centroid
ν1 is the observation with the largest total distance with the other observations.
2. Determine the dissimilarity of the remaining observations and the subset of centroids Vj:
Di,ν1
= min

||xi,ν1||2

, i = 1,..., N
At this step Vj consists of one centroid (ν1), so j = 1. The observation with the largest dissimilarity
is chosen as ν2.
3. To determine values of the next j = 3,..., k centroids the efficient algorithm of Polinsky et al. (1996)
is used. The observation with the largest Dmin
i,Vj
is chosen a the next centroid:
Dmin
i,Vj
= min
”
Di,νj
, Dmin
i,Vj−1
—
i = 1,..., N − j, j = 3,..., k
To start this iterative procedure Dmin
i,V2
= Di,ν1
.
The MDA algorithm uses the wave height, period and direction as input variables. It uses an equal
weighting in the distance function.
3.2 Sequencing reduced wave climates
3.2.1 Markov Chain sequencing
Markov chain sequencing is an attempt to order the wave conditions in the way they most likely would
appear in the full sample. It creates a sequence where a wave condition is followed by another wave
condition that is most likely to follow him. The procedure can be described as follows:
1. Number the reduced wave conditions that are stored in database V from 1:k.
2. Determine for every observation in the observation database X which of the reduced wave climates
in V is most similar to it. A new one-variate time series F with size (N × 1) is created where per
observation the number of the most similar wave condition is stored:
Fi =
k
X
j=1
j I[||xi,νj|| = min(||xi, V||)]
Here I[.] is an indicator that is 1 when the equation between the bracket is true and 0 otherwise.
3. Now determine the Markov transitions for the wave conditions in time series F. The Markov transi-
tions are stored in a Markov transition matrix G of size (k × k):
G(m, n) = P(Ft+1 = n|Ft = m) =
N−1
X
t=1
I[Ft = m] I[Ft+1 = n]
N − 1
, m = 1,..., k and n = 1,..., k
4. The algorithm will now step-by-step create a chronology in the time series. It does that by starting
with one wave condition. It then selects the wave condition that has the largest probability of oc-
curring after the wave condition that was selected first. It then selects the wave condition that most
likely occurs after the wave second selected wave condition and so on.
27

To do this first define two time series matrices: The first matrix AS will contain the numbers that
are assigned to the wave conditions in step 1 in the sequence determined by the algorithm. Since
no sequencing is determined yet this matrix starts empty. The second matrix ANS contains the wave
numbers assigned to wave wave conditions in step 1 at the start of the algorithm. When a wave
condition is selected by the algorithm its number will be deleted from matrix ANS and added to
matrix AS.
5. Let the first wave condition in AS,1 be the wave condition with the lowest wave height. The number
assigned to that wave condition will now be deleted from matrix ANS that thus reduces to the size
(k − 1 × 1).
6. The next wave condition that is selected for the time series has the largest Markov transition proba-
bility, conditional on the previous selected wave condition. Also this wave condition must not have
been selected yet:
AS,t =
k−t+1
X
q=1
ANS,q I[G(AS,t−1,ANS,q) = min(G(ANS,ANS)]
7. Reorder the wave conditions in database V in such a way that their assigned numbers match matrix
AS.
3.2.2 Time as fourth variable
The time as fourth variable sequencing method is only applicable for clustering methods that use use three
variables to describe the wave climate. The algorithm adds a fourth variable to describe a wave condition.
This variable describes where the observation has occurred in a year relative to the other observations.
Wave conditions can now not only be classified as similar when they have similar wave heights, periods
and directions, but also when they have occurred during the same time of the year. So observations in
winter will more likely be clustered with other observations in winter.
Since this concept has never been used in literature, three different measures for time of the year have
been tried out: The quarter of the year, the month of the year and the day of the year indicator. The
difference between these variables is the precision of describing the moment the observation has occurred.
The quarter of the year indicator has four different values: 1, 2 3 or 4, while the day of the year indicator
can have 365 different values. In section 3.1.1.3 the implementation of the day of the year indicator is
described. In appendix B.1 is shown that the accuracy of the description of the time is important for the
quality of the clustering. The more accurate the time description, the better the clustering. Therefore in
the report the day of the year variable is used.
The day of the year indicator is applied for the k-means, fuzzy k-means, k-harmonic means and MDA
algorithm. In appendix B.2 is shown that the algorithms with soft membership functions (fuzzy k-means
and k-harmonic means) do not generate realistic clusters. This is possibly caused by two reasons: First
of all a property of the algorithms with the soft functions is that they show more overlap in their final
results than the algorithms with hard membership functions. This property is discussed in Kearns et al.
(1998). The second reason is that cluster algorithms always have problems with clustering in more than
three dimensions. This is especially true for soft membership cluster algorithms. Due to these problems,
only the results of the crisp k-means and MDA algorithms are reported.
3.3 Evaluation
In this section the evaluation of the algorithm results are described. The performance of the wave climates
resulting from the reduction methods will be evaluated with two measures: First the generated transport
28

rates are evaluated, then the implications for the morphology are evaluated. This is done with two models:
First in a alongshore bulk transport model, and second in a Delft3D model.
The alongshore bulk transport is the amount of transport that is generated with a bulk transport for-
mula when the reduced wave climate is used as input. The generated transport will be compared with a
benchmark transport rate. This benchmark rate is the amount of transport that is generated by the bulk
transport formulas with the full sample of observed wave conditions as input.
The transport rates and morphology changes of a Delft3D model that uses a reduced wave climate are
compared with the transport rates and morphological changes of a Delft3D model with brute force input.
This brute force model can be seen as a benchmark. A brute force model is a model for which the wave
input has not been reduced and for which no morfac is used. It is thus a real time simulation.
Because a brute force simulation is computation intensive, it is not possible to run the complete avail-
able dataset of 7 years. Instead one representative year has been selected for the brute force simulation,
which is the year 2010. The input wave conditions for the wave climate reduction methods are only the
observations from the year 2010. Also the bulk transport results are only based on the observation sample
of the year 2010.
The transport rates and morphological changes are, as explained above, all evaluated by comparing
the results of a brute force and reduced wave climate in different model setups. This is done by comparing
different evaluation variables. A evaluation variable can be a transport rate(e.g. net alongshore transport,
gross cross shore transport) or a variable that describes the morphology (e.g. maximum height berm, the
average location of the berm).
This comparison of the different evaluation variables is quantified with the error rate (ε):
ε = 100% ·
Yred − Yf ull
Yf ull
(3.34)
Here Yred is the evaluation variable that is resulting form a model that uses a reduced wave climate and
Yf ull the same evaluation variable, only now resulting from a model that uses the complete wave climate
as input.
It will be evaluated how well the reduction methods can reduce the wave climate to k = 10 and k = 20
wave conditions. These numbers are determined based on practical experience: most users use between
10 and 20 wave conditions, so that the simulation time of their process based models can be increased.
For the evaluation of the reduction algorithms first the wave conditions are sequenced from low wave
height to high wave height. This because the large wave heights will cause most of the sediment transport,
which can influence the location of the wave breaking and thus the morphology. To reduce the complexity
of the comparison of the wave reduction methods influence of bar forming is now minimized, since only
the last wave conditions are effected by it.
In the next sections the bulk transport formulas and the Delft3D model will be described.
3.3.1 Bulk alongshore transport model
The bulk alongshore transport model setup is evaluated with four variables: The net (SB,net), gross (SB,gro),
northern (SB,no) and southern (SB,so) transport. Here the northern (southern) transport is all along trans-
port that is moving in the northern (southern) transport. All the variables have the dimension [m3
/year].
The net and gross transport are defined as follows:
SB,net = SB,no − SB,so and SB,gro = SB,no + SB,so (3.35)
The transport rates are determined with the three bulk transport formulas that are described in section
29

3.1.2.
3.3.2 Delft3D model
The Delft3D model is calibrated based on the alongshore transport and its morphology. The calibration
is done based on a reduced wave climate (Algorithm 1B), since calibrating with a brute force model is to
computational intensive. Besides that it is visually inspected whether the evolution of the morphology is
realistic. This means that bar forming takes place during large storm events. This phenomenon can not be
quantified since there are no usable measurements available.
After the calibration of the model with a reduced wave climate several brute force simulations have
been executed to optimize the model parameters for that setup.
3.3.2.1 Grid
The Delft3D model exists of three nested grids. The first grid is a large scale wave grid with a resolution
(200x200 m), a length (alongshore) of 20 km and a width (cross-shore) at the northern boundary of 8 km.
The water depth near the eastern edge of this grid is about 35 m for most of the grid, which is the same
water depth as the wave rider buoy that observed the wave input data.
The second grid is medium sized and has a resolution of (50x50 m), is 5 km long and 2 km wide. The
resolution of the grid is finer and is chosen in such a way that refraction effects on the waves are better
captured by the wave module, while the computation time is not influence much by this grid resolution.
The third grid has a resolution of (15x15 m) at the lateral side. The cross shore grid length decreases
linearly towards 7.5 m at the beach side of the grid, so that the resolution increases to (7.5x15 m) at the
beach. The grid length and width are both 1000 m. It is a wave and a flow grid, which means that not
only the wave dynamics are resolved in this grid, but that also the tide and morphology are resolved. The
wave grid is nested in the medium size grid. The resolution of the grid is chosen in such a way that the
model is stable, while the large flow structures are still resolved.
Figure 3.1 shows the bathymetry of the three grids. It can be seen that the bathymetry of the small and
the medium sized grids are fairly uniform in the alongshore direction. The large grid shows a steep ridge
in the north eastern part. It is assumed that this ridge does not have a large influence on the transport
rates and morphology, since only a few waves come from the (south) east direction.
30

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