analysis of ship responses by means of wavelet transforms

Analysis of ship responses bymeans of wavelet transforms

Jorinna Gunkel

Kongens Lyngby 2016

Technical University of Denmark

Department of Applied Mathematics and Computer Science

Richard Petersens Plads, building 324,

2800 Kongens Lyngby, Denmark

Phone +45 4525 3031

[email protected]

www.compute.dtu.dk

Summary (English)

Until today, the FFT is the unchallenged method to analyse ship responsemeasurements but wavelet transforms have been designed to deal with non-stationary signals and provide a signal representation as a function of time andfrequency simulatenously. Therefore, they have the potential to improve vesselresponse analysis by extracting new information and the objective of this thesisis to assess the capability of wavelet transforms to improve the estimation of seastates and future ship responses.

This thesis evaluates the potential of wavelet transforms in three steps. At �rst,the theoretical background and conceptual design of wavelet transforms arestudied compared to the conventional FFT. Afterwards, it is assessed whetherthe three relevant methods and their output �t to the expectations of shipresponse analysis. The last study attempts to determine the potential of wavelettransforms to provide simple estimates of the sea state or future ship responsesbased on full-scale ship response measurments and corresponding wave radardata.

The results demonstrate that the concept of wavelet transforms is useful becausethe energy of response measurements can be represented in its temporal develop-ment. Only one out of three transforms is, however, found to provide su�cientaccuracy in this context. The last study implies that wavelet transforms arenot suitable to signi�cantly simplify response prediction or sea state estimation.Nonetheless, it is recommended to conduct further studies on a more complexlevel to obtain a better impression of the potential of wavelet transforms.

Keywords: ship responses, predictions, sea state estimation, wavelet transforms

Preface

This thesis was prepared at the Section of Fluid Mechanics, Coastal and MarineEngineering, Department of Mechanical Engineering at DTU in Kgs. Lyngby,Denmark, in ful�lment of the requirements for acquiring an M.Sc. in MaritimeEngineering within the Nordic Master program. The work load is equivalent to30 points in the European Credit Transfer System (ECTS) and the project wascarried out between January and June 2016 under the supervision of AssociateProfessor Ulrik Dam Nielsen.

The thesis introduces wavelet transforms to the �eld of maritime engineering andprovides insight into the calculation method and its potential for the applicationin ship response ananlysis. Starting without any previous knowledge about thetopic of wavelet transforms, the author aimed to provide an illustrative reportfor other maritime engineers who would like to gain insight into this particulartechnique.

The thesis consists of 6 chapters including the introduction and conclusions. Thework has mainly been conducted by the author though the MATLAB script foravering WaMoS sea state estimates has been provided by Associate ProfessorUlrik Dam Nielsen. As this project is based on an intensive literature survey,the results of research of others is integrated into the text and the author hasdone her best to provide accurate references to these sources.

iv

Lyngby, 25-June-2016

Jorinna Gunkel

Acknowledgements

At this point, the author would like to thank a number of people who have beenhelpful and supportive in the working and writing process.

• First of all, I would like to express my gratitude to my supervisor AssociateProfessor Ulrik Dam Nielsen for his continuous support, his willingness tohelp whenever problems turned up and his valuable input given in ourweekly meeting. Furthermore, I would like to say thanks for the quickreplies to my emails when guidance was needed spontaneously.

• Secondly, I would like to thank Professor Sverre Steen from the Depart-ment of Marine Technology at NTNU in Trondheim who followed theprogress of this thesis on behalf of NTNU and was willing to provide in-put and help at all time.

• Special thanks goes to my brother, Philipp Gunkel, who spent his valu-able time proofreading this report and give important input to ensure thecomprehensibility of the documentation.

• Besides my brother, I would like thank my good friend Maria Berndt whohad a critical eye on the layout and language of this document.

• At last, I am sincerely grateful for the encouragement and support pro-vided by my parents and for the motivation inspired by Lea Beglerovic, afellow Nordic Master student.

Nomenclature

FFT Fast Fourier transform

WT Wavelet transform

CWT Continuous wavelet transform

DWT Discrete wavelet transform

DWPT Discrete wavelet packet transform

STFT Short time Fourier transform

DFT Discrete Fourier transform

f, ω Wave frequency

t Time

n Discretised time

N Number of samples

k Discretised frequency

Es Signal energy

ψ0(t) Mother wavelet

Ψ(ω) Fourier transform of wavelet function

ψa,b(t) Daughter wavelet

a Wavelet scaling factor

b Wavelet shifting factor

fc Wavelet center frequency

fa Pseudo frequency

viii

Wψx (a, b) Wavelet coe�cients per scale and shift

W̄ 2(a) Wavelet energy per scale

GWS Global wavelet spectrum

j Discrete wavelet transform scale parameter; also level of DWT

ψa,b(t) Discretised daughter wavelet

Dj Detail of discrete wavelet transform per level j

Aj Approximation of discrete wavelet transform per level j

J Maximum number of decomposition levels (DWT)

dj Detail coe�cients of discrete wavelet transform per level

aj Approximation coe�cients of discrete wavelet transform per level

φa,b(t) Discretised scaling function

Vs Ship speed

µs, µw, µr Ship, wave and relative heading

Hs Signi�cant wave height

Tz, Tp Zero-crossing period, peak period

ζ Wave amplitude

S(f, θ) Directional wave energy spectrum

θ Wave direction

S(f), S(ω) 1-D wave energy spectrum as function of f or ω

σ2 Variance

σ Standard deviation

mn Spectral moment of order n

WaMoS Wave Monitoring System

D Ship draught

DB Daubechies wavelet

Sym Symlet

SR(ω) Response Spectrum

RAO,φR(ω) Response Amplitude Operator

x Contents

Contents

Summary (English) i

Preface iii

Acknowledgements v

Nomenclature vii

1 Introduction 11.1 Thesis Background and Problem Formulation . . . . . . . . . . . 11.2 Methodology and Report Outline . . . . . . . . . . . . . . . . . . 31.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Wavelet Transforms - Theoretical Considerations 72.1 The Potential of Wavelets . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Brief Review of the FFT . . . . . . . . . . . . . . . . . . . 82.1.2 Resolution Problems of Fourier Transforms . . . . . . . . 11

2.2 Introduction to Wavelet Transforms . . . . . . . . . . . . . . . . 122.2.1 What is a Wavelet? - The Concept of Scales . . . . . . . 122.2.2 The Continuous Wavelet Transform . . . . . . . . . . . . 152.2.3 The Discrete Wavelet Transform . . . . . . . . . . . . . . 202.2.4 The Discrete Wavelet Packet Transform . . . . . . . . . . 26

2.3 Discussion of Intermediate Results . . . . . . . . . . . . . . . . . 28

3 Full-Scale Measurements and Related Notation 313.1 Background of the Measurements . . . . . . . . . . . . . . . . . . 313.2 Ship Operational Conditions . . . . . . . . . . . . . . . . . . . . 323.3 Full-Scale Ship Response Data . . . . . . . . . . . . . . . . . . . 333.4 Sea State Measurements . . . . . . . . . . . . . . . . . . . . . . . 34

xii CONTENTS

3.4.1 Wave Parameters . . . . . . . . . . . . . . . . . . . . . . . 343.4.2 Onboard Installation of the WaMoS System . . . . . . . 363.4.3 Stationary Time Periods . . . . . . . . . . . . . . . . . . . 37

4 Determination of the most appropriate wavelet transform method 394.1 Approach of the study . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 Input data and settings . . . . . . . . . . . . . . . . . . . 404.1.2 Assessment Criteria . . . . . . . . . . . . . . . . . . . . . 42

4.2 Results of the CWT Test . . . . . . . . . . . . . . . . . . . . . . 434.3 Results of the DWT Test . . . . . . . . . . . . . . . . . . . . . . 464.4 Results of the DWPT Test . . . . . . . . . . . . . . . . . . . . . . 484.5 Comparison and selection of a Wavelet Transform . . . . . . . . . 50

5 The potential of the CWT for ship motion analysis 535.1 Procedures to analyse the relevance of the CWT for decision sup-

port models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.1 Objectives, approach and limitations . . . . . . . . . . . . 545.1.2 Input data and further assumptions . . . . . . . . . . . . 55

5.2 Evaluation of the CWT's relevance for decision support systems . 565.2.1 Evaluation related to simple response predictions . . . . . 575.2.2 Evaluation related to simple sea state estimates . . . . . . 605.2.3 Summary and discussion of the �ndings . . . . . . . . . . 62

6 Conclusions 656.1 Findings and discussions . . . . . . . . . . . . . . . . . . . . . . . 666.2 Recommendations for further research . . . . . . . . . . . . . . . 67

A Appendix - Chapter 2 69A.1 Center Frequency of the Morlet Wavelet . . . . . . . . . . . . . . 69A.2 Schematic representation of th wavelet translation in the CWT . 70A.3 Schematic representation of wavelet scaling in the CWT . . . . . 71A.4 Schematic representation of wavelet scaling in the CWT . . . . . 72

B Appendix - Chapter 3 73B.1 Main particulars of the CMA CGM Rigoletto . . . . . . . . . . . 74B.2 Center Frequency of the Morlet Wavelet . . . . . . . . . . . . . . 75B.3 Stationary sea state time periods in WaMoS data . . . . . . . . . 76

C Appendix Chapter 4 79C.1 Wavelet and scaling function of the DB8 . . . . . . . . . . . . . . 79C.2 Wavelet and scaling function of the DB12 . . . . . . . . . . . . . 80C.3 Wavelet and scaling function of the DB20 . . . . . . . . . . . . . 80C.4 Wavelet and scaling function of the Sym12 . . . . . . . . . . . . . 81C.5 Wavelet and scaling function of the Sym20 . . . . . . . . . . . . . 81C.6 Wavelet and scaling function of the Coif5 . . . . . . . . . . . . . 82

CONTENTS xiii

C.7 CWT assessment: 20.Aug, stationary, heave . . . . . . . . . . . . 83C.8 CWT assessment: 20.Aug, not stationary, heave . . . . . . . . . . 84C.9 CWT assessment: 02.Oct, stationary, heave . . . . . . . . . . . . 85C.10 CWT assessment: 02.Oct, not stationary, heave . . . . . . . . . . 86C.11 CWT assessment: 20.Aug, stationary, pitch . . . . . . . . . . . . 87C.12 CWT assessment: 20.Aug, not stationary, pitch . . . . . . . . . . 88C.13 CWT assessment: 02.Oct, stationary, pitch . . . . . . . . . . . . 89C.14 CWT assessment: 02.Oct, not stationary, pitch . . . . . . . . . . 90C.15 Reconstructed heave sequence using the Morlet wavelet . . . . . 91C.16 Reconstructed pitch sequence using the Morlet wavelet . . . . . . 92C.17 Reconstructed heave sequence using the Sym12 wavelet . . . . . 93C.18 Reconstructed pitch sequence using the Sym12 wavelet . . . . . . 94C.19 Reconstructed heave sequence using the DB20 wavelet . . . . . . 95C.20 Reconstructed pitch sequence using the DB20 wavelet . . . . . . 96C.21 DWT assessment: 02.Oct, not stationary, heave . . . . . . . . . . 97

D Appendix - Chapter 5 99D.1 Data collection for response prediction; 20.August, heave . . . . . 100D.2 Data collection for response prediction; 20.August pitch . . . . . 103D.3 Data collection for response prediction; 20.August, roll . . . . . . 106D.4 Data collection for response prediction; 02.October, heave . . . . 109D.5 Data collection for response prediction; 02.October, pitch . . . . 112D.6 Data collection for response prediction; 02.October, roll . . . . . 115

E Appendix MATLAB 119E.1 Code to read and prepare the vessel response measurements . . . 120

Bibliography 141

xiv CONTENTS

Chapter 1

Introduction

This thesis studies di�erent wavelet transform methods as a complement oralternative for the fast Fourier transform and explores their potential to improvespectral analysis models in di�erent �elds of application in a marine context.The background of this project, its main objectives and the thesis' methodologyincluding a brief literature survey are presented in the following.

1.1 Thesis Background and Problem Formulation

Operating a ship requires careful monitoring of the related costs while ensuringa high level of safety. Shipboard decision support systems may enable the ship'screw to reduce costs and minimise risks while sailing. Within this �eld, it isof particular importance to estimate the surrounding sea state and to predictfuture responses of the ship in terms of motions. As decision support systemsformulate their guidance based on mathematical models and measured data,it has to be ensured that the estimation methods and measurements are asaccurate as possible.

Many approaches for decision support systems use time domain ship responsemeasurements and apply the fast Fourier transform to convert the data to the

2 Introduction

frequency domain. The frequency domain is convenient when it comes to ob-taining statistical parameters based on which guidance is provided to the crew.However, it is known that the fast Fourier transform does not handle non-stationary data, such as ship motions in heavy seas, properly. Moreover, theFFT is a very exact representation in the frequency domain but it is not able todepict the temporal development of the measured quantity. In signal process-ing, di�erent forms of wavelet transforms serve as an alternative to conventionalFourier transforms as they address some of the drawbacks of the fast Fouriertransform and are able to highlight the temporal development of di�erent fre-quency components by a joint time-frequency representation.

Recently, the two studies [Iseki, 2015] and [Xu und Iseki, 2015] have exploredthe application of the discrete wavelet transform with respect to main enginepower �uctuations and non-stationary ship response analysis. They hypothesisethat the discrete wavelet transform might reveal information in measured datawhich has been hidden so far.

As [Iseki, 2015] and [Xu und Iseki, 2015] drew interest towards wavelet trans-form methods as complements or alternatives to the fast Fourier transform, thequestion quickly arises whether wavelet transform may be used to improve theprediction of future responses or the estimation of sea states. Scanning basicmaterial about wavelets furthermore shows that there are, in fact, three rele-vant methods: the continuous wavelet transform, the discrete wavelet transformand the discrete wavelet package transform. Until now, wavelet transforms havehardly ever been used in maritime engineering and knowledge about the di�er-ent wavelet transform methods and their capabilities is scarce. Thus, there is agap of knowledge and based on this chain of thoughts, the following two mainquestions of this thesis are formulated:

• Are wavelet transform methods useful for ship response analysis from ageneral point of view and which method is the most suitable one?

• How could wavelet transforms be used to improve the analysis of shipresponse measurements with respect to spectral analysis?

The �rst question refers to the conceptual design of wavelet transforms. Wavelettransforms would have to provide new pieces of information presented in a usefulmanner in order to be applicable to the �eld of response analysis. Thus, answer-ing question one ensures that these transforms are theoretically and practicallyable to produce new input from response measurements. The second questionconnects the general �ndings about wavelet transforms to ship response analysis,in particular to ship response prediction and sea state estimation.

1.2 Methodology and Report Outline 3

If the above questions are answered, it is possible to give a detailed commenton the potential of wavelet transforms in a decision support context and, if theresult is positive, a new technique for the analysis of ship responsible is madeavailable.

1.2 Methodology and Report Outline

The following section presents the scope, limitations and methodology of thisthesis and outlines the structure of this report.

The purpose of this study is to extend the current knowledge about wavelettransforms to eventually comment on their potential for spectal and responseanalysis in a decision support context. By using a thorough literature reviewand small, speci�c examples, the answers to the main thesis questions are foundstep by step. Due to a limited time frame, the thesis focuses on qualitativestatements and observations, as all applications of wavelet transforms can onlybe conducted in small scale.

Due to the fact that wavelet transforms have hardly ever been applied in mar-itime engineering, a thorough literature survey is the key to answering the �rstof the two questions. Understanding what wavelet transforms are, how theyfunction and how they di�er from the common fast Fourier transform regardingspectral analysis, is essential to ensure their correct and e�cient application.Afterwards, the application of wavelet transforms to relevant data is supposedto tell which wavelet transform is most useful for this thesis. A second studyis then performed that actually uses the selected wavelet transform to analyseship response data, in this case, with respect to ship response prediction andsea state estimation. These three steps are supposed to allow for a performancerating of the wavelet transforms' potential within the given �elds of application.

It can be seen that the thesis is formulated in a quite open manner. The outcomeof the individual steps is unknown in the beginning and not �nding a positiveanswer to any of the questions is always an option.

This thesis works with full-scale ship responses measurements, wave radar seastate estimates and operational conditions that have been collected on a shipfor a di�erent project. All computations are performed in MATLAB version 8.5released in 2015 which includes a wavelet toolbox and related documentation.The main MATLAB code divided in six sub�les is given in Appendix E.

Resulting from the above approach, the thesis is structured as described in the

4 Introduction

following.

Chapter 2 provides a brief and basic theoretical background on the currentlyused fast Fourier transform in contrast to the three wavelet transforms men-tioned previously. The chapter focuses on aspects related to spectral analysisand highlights the advantages, disadvantages and di�erences of the di�erentmethods to ensure they are all theoretically applicable to the analysis of shipmotions.

Chapter 3 introduces the full-scale measurements and related notations usedthroughout this thesis.

Chapter 4 focuses on the practical application of the three wavelet transformmethods to identify which wavelet transform method is most suitable for fur-ther analysis. The transforms are evaluated based on the usefulness of theiroutput regarding spectral analysis and an evaluation of their implementation inMATLAB.

Chapter 5 presents the application of wavelet transformation to ship responsedata with the aim to decide whether wavelet transforms have the potential toimprove simple sea state estimation or response prediction.

Chapter 6 summarises the �ndings regarding the main question of this thesisand indicates further possibilities for research.

1.3 Literature Survey

The literature survey constitutes an essential element of this project, as thereis no previous knowledge regarding wavelets to rely on. In the course of theproject, a large number of publications are consulted but only the most relevantones are mentioned in this chapter. The literature can further more be dividedin two categories: literature which is directly referenced in the main part ofthis thesis and literature which signi�cantly in�uenced the development of thisthesis.

The latter includes [Iseki, 2015] and [Xu und Iseki, 2015]. [Iseki, 2015] is a basicstudy applying the discrete wavelet transform methods to ship response dataconcluding that they are useful tools to analyse non-stationary data. The secondstudy applies the discrete wavelet transform to main engine power �uctuationsand tries to estimate the fuel consumption without direct measurements. Thesestudies initiated this very thesis, because they gave rise to assumption that

1.3 Literature Survey 5

wavelet transform may be an important technique to understand and integratein response analysis methods.

The studies [Nielsen u. a., 2013] and [Nielsen und Iseki, 2015] present currentwork on the topics of sea states estimates and vessel response predictions andthereby help to develop new ideas on how wavelets could be applied in these�elds. Moreover, both studies utilise the same measurements as this thesis andhelp to understand and use the data appropriately.

The theoretical background on wavelet transforms in relation to Fourier trans-forms is mainly based on the following publications.

The lecture notes [Nielsen, 2010] from the "Ship operations" course at DTUelaborate on navigational guidance and decision support system including theunderlying models. Thereby, this document provides the practical frame inwhich this project is situated. In addition, it is the main source regarding thefast Fourier transform, as it explains its main features by using the example ofship response analysis.

The theoretical background of wavelet transforms is a well-studied �eld thoughmost of publications follow a quite mathematical approach. [Mallat, 2009] isprobably one of the most comprehensive textbooks available which focuses onmulti-resolution analysis and the discrete wavelet transform methods. Thoughhighly mathematical, it provides useful information on the mathematic de�ni-tions and advantages of the multi-resolution approach which is closely linked todiscrete wavelet transforms. [Bergh u. a., 1999] also mainly deals with the dis-crete wavelet transform but attempts to explain the mathematical backgroundwith slightly less detail to enable a larger �eld of interested readers to un-derstand wavelet theory. The textbook [Alessio, 2016], in contrast, covers allrelevant wavelet transforms including a comparison to Fourier transforms andfocuses speci�cally on spectral analysis. It furthermore provides illustrative ex-planations, useful for an engineer with a particular interest in the applicationof wavelet transforms.

The handbook [Misiti u. a., 2009] explains basic theoretical aspects using agraphic approach with little mathematical detail but a lot of practical infor-mation on how to link the theory to the actual utilisation of the MATLABcommands available in the wavelet toolbox.

6 Introduction

Chapter 2

Wavelet Transforms -Theoretical Considerations

Navigational guidance is usually provided through statistical measures obtainedfrom measurements converted to the frequency domain. The common trans-formation method, the fast Fourier transform (FFT) , struggles to extract allrelevant information from non-stationary data which is why wavelet transformsare considered as a valuable complement or alternative.

The mathematical background of wavelet transforms(WT) is excessively dealtwith in literature but this thesis focuses on the application in an engineeringcontext. It is therefore necessary to limit the mathematical de�nitions and fo-cus on the functionalities, advantages and disadvantages of the relevant wavelettransforms, the continuous wavelet transform(CWT), the discrete wavelet trans-form(DWT) and the discrete wavelet packet transform (DWPT).

This chapter �rst summarises the FFT and its problems with respect to timeresolution before wavelets and wavelet transforms are introduced.

8 Wavelet Transforms - Theoretical Considerations

2.1 The Potential of Wavelets

The FFT is the currently common method to convert ship response measure-ments to the frequency domain. The FFT is not ideal when it comes to analysingnon-stationary data. This chapter brie�y summarises the functionality of theFFT to derive the source and extent of this problem. The short-time Fouriertransform (STFT) will be mentioned as an attempt to solve the problem beforewavelets may enter the stage.

2.1.1 Brief Review of the FFT

The general idea of all Fourier transformation methods, continuous or discrete,is to decompose a time domain function or signal x(t) into sinusoidal compo-nents which are ordered by their frequencies. These Fourier transforms can beused to obtain an energy spectrum in the frequency domain showing how mucha sine wave of a speci�c frequency contributes to the original signal. Conse-quently, the time and the frequency domain are two di�erent representations ofthe same signal and converting the signal from one domain to the other mayreveal di�erent information and characteristics of the underlying physical pro-cesses[Wavelets]. The frequency domain has furthermore proven to be e�cientfor statistical analysis of signals.

In real life, signals are not described by a continuous function but by discretemeasurements which have �nite energy. Mathematically, they are thus partof the space L2 of square-integrable functions. The energy of the discrete sig-nal x(n) containing N elements is de�ned below. The discretised time is nowdenoted n.

Es = 〈x(n), x(n)〉 =

∞∑n=−∞

|x(n)|2 <∞ (2.1.1.1)

Signals for which (2.1.1.1) is valid are called energy signals.

If such a data sequence is �nite, periodic and the samples are equally-spaced,the Discrete Fourier Transform (DFT) may be used to analyse the signal andobtain a discrete, periodic spectrum in the frequency domain. Using the so-called frequency synthesis, it is possible to retrieve the original time history fromthe spectrum. The analysis of a discrete sequence uses complex exponentialswhich leads to complex Fourier Transforms X(k) with k denoting the discretisedfrequencies. For a discrete sequence, the synthesis and analysis are de�ned as

2.1 The Potential of Wavelets 9

[Alessio, 2016].

x[n] =1

N

N−1∑k=0

X[k]ei2πkN n (2.1.1.2)

X[k] =

N−1∑n=0

x[n]e−i2πnN k (2.1.1.3)

n = 0, 1, 2, ..., N − 1

k = 0, 1, 2, ..., N − 1

The above equations (2.1.1.2) and (2.1.1.3) show that calculating the DFT re-quires O(N2) arithmetic operations which is impractical for most applications.Therefore, the fast Fourier transformation was established which provides ef-�cient algorithms for the calculation of the DFT by reducing the number ofoperations to O(N log N) [Bingham, 2015]. Consequently, the FFT is not anindependet transform but an optimised version of the DFT and both expressionsare often used interchargeably.

The coe�cients X[k] and thus the importance of a speci�c sine wave componentare comprehensively illustrated in the energy density spectrum which uses therelation between energy and amplitude and relates each frequency componentto a spectral ordinate according to (2.1.1.4) [Nielsen, 2010].

S[k] =1

N2|X[k]|2 (2.1.1.4)

The spectrum is commonly visualised by plotting the spectral ordinates over thefrequencies. The total area underneath this spectrum is then proportional tothe total energy contained in the wave system. This relation resembles (2.1.1.1)and it implies that the total energy of the system can either be calculated fromthe sum of the squared time-domain coe�cients or from the sum of the squaredFourier transforms.

N−1∑n=0

|x[n]|2 =1

N

N−1∑k=0

|X[k]|2 (2.1.1.5)

Equation 2.1.1.5 is known as Parseval's theorem.

To illustrate the use of the FFT and the information it extracts from data, arather simple example of superimposed sine waves is presented in the following.This example is going to be adopted during the presentation of the wavelettransforms to visualise how they treat this simple set of data. The focus isspeci�cally on how the signal is represented in the frequency domain in termsof a spectrum, though other outputs might be discussed, if necessary.


Example - Superimposed sine waves analysed by the FFT:

A signal is constructed of two sine waves with f1 = 20Hz and f2 = 60Hzaccording to (2.1.1.3).

x(t) = sin(2πf1t) + 2 · sin(2πf2t) (2.1.1.6)

The considered period is 1.5s with a sampling time of 0.001s. The FFT isperformed in MATLAB using in-built functions and appropriate scaling. Thescaling factor is de�ned as the number of positive frequencies times the samplingfrequency [Nielsen, 2010]. In fact, this example is rather simple but appropriateto show the full power of the FFT. The signal is built up of sine waves i.e. thereis a perfect match between the signal's constituents and the FFT's analysingfunctions. Figure 2.1 presents a 0.2s long extract of the time history and theenergy spectrum of the signal following eq.(2.1.3).

(a) Extract of the time history (b) Energy density spectrum

Figure 2.1: Heave response measurements in common time and frequency do-main representations

The FFT exactly identi�es the two frequencies f1 and f2 and the average signalenergy related to them. By ensuring proper scaling in MATLAB, the energy ofthe signal and the FFT spectrum is identical which is why the above spectrumis a trustworthy representation of the signal in the frequency domain.

In fact, Figure 2.1 illustrates the great strength of the FFT, namely the perfectfrequency resolution. As the FFT uses sinusoids which are all de�ned by singlefrequencies, it is able to detect exactly these frequency components perfectly ina signal. In consequence, the FFT averages signal power over time and providea spectral estimate per frequency.

2.1 The Potential of Wavelets 11

2.1.2 Resolution Problems of Fourier Transforms

Especially with regards to non-stationary signals, the perfect frequency resolu-tion of the FFT constitutes a major disadvantage. This issue is further discussedin the following including a description of why the problem cannot be solvedwithin the range of Fourier transforms.

To understand the time and frequency resolution provided by a FFT, it is neces-sary to look at the analysing functions. Sinusoids are in�nite and if a particularsinusoid is found to be part of the signal, it is naturally assumed that the signalis a�ected by this component at all points of time. It is therefore not necessaryto specify the time at which a particular wave is present in the signal. In mea-sured data, however, transcient oscillations at varying amplitudes might occurwhich contribute to the signal energy only in a very short interval. It is thennot accurate to approximate them by in�nite sinusoids. In the cases of non-stationary data sequences, it is important to also locate the wave and its energyin time, not only in the frequency domain. Yet, the FFT does not provide anytime resolution and thus wrongly assumes that even transcient waves are presentforever and may be represented by an average power.

The lack of a time resolution of the FFT is di�cult to solve, as, in fact, it istechnically not possible to provide perfect frequency and perfect time resolutionat the same time. This phenomenon is referred to as Heisenberg's uncertaintyprinciple which is usually associated with quantum mechanics. In terms ofsignal processing, Heisenberg's uncertainty principle implies that it is impossibleto accurately determine the frequency and the point of time at which a waveoccurs[FriendlyGuide]. Therefore, perfect frequency resolution, as provided bythe FFT must always result in a complete lack of time resolution. Every attemptto improve the time resolution will always lead to a certain loss in frequencyresolution.

Real signals or measurements are usually assumed to be quasi-stationary in or-der to apply the FFT. The less accurate the assumption of quasi-stationarityis, the more relevant it becomes to represent the constituents of the signalin both domains to account for the instationarity. In order to overcome thenon-existence of time resolution in Fourier transforms, the so-called Short-TimeFourier Transform (STFT) was invented. It uses a window function to split asignal into smaller quasi-stationary segments and analyses the segments individ-ually [Alessio, 2016] by a FFT. The discrete STFT is mathematically describedin (2.1.2.1) where m represents the shifting factor of the window.

X[k] =

N−1∑n=0

x[n]w[n−m]e−j(2πnN )k (2.1.2.1)


The window function w[n] is non-zero in only a short interval which is called thewindow width. By multiplying the original signal x[n] by the window function,only the signal components inside the window are non-zero and Fourier trans-formed, resulting in a regular FFT spectrum. Afterwards, the window is shiftedalong the timeline in order to transform the next set of elements. Thus, theoriginal signal is subdivided in smaller intervals corresponding to the windowwidth and information about the temporal development of the signal's spectralfeatures is provided through separate spectra for each interval.

Nevertheless, the window function only improves the time resolution but doesnot solve the problem for non-stationary signals. As the window width is con-stant, it is still di�cult to identify high and low frequency components of thesignal equally well. If a small window is selected, high frequency oscillationsare captured adequately but it is not possible to cover su�ciently many low fre-quency oscillations to determine their frequency properly. The other way round,large windows are bene�cial when low frequency components are to be identi-�ed but with regards to high frequency wave components, the same problemsas known from the FFT are faced.

This constant window width problem may be encountered by using an adjustable"window" which is translated along the time line and �tted to the wave com-ponents at a each time segment. Wavelets are in fact "adjustable windows"[Alessio, 2016].

2.2 Introduction to Wavelet Transforms

Wavelets are indeed functions that provide the desired adaptive type of time-frequency representation of measured data. Therefore, wavelet functions are �rstintroduced separately, before the actual wavelet transforms, the CWT, DWTand DWPT, are introduced.

2.2.1 What is a Wavelet? - The Concept of Scales

This chapter introduces wavelets which are the analysing functions of all wavelettransforms and thus represent the counterpart to sinusoids in the FFT.

The name wavelet can be translated to "little wave" which implies that the waveis not in�nite but represents a short, transcient oscillation of �nite energy, i.e.wavelets belong to the space L2 mathematically. Unlike sinusoids, the wavelet is

2.2 Introduction to Wavelet Transforms 13

concentrated in time and covers a certain frequency band [Alessio, 2016] insteadof representing one particular frequency. This is important to bear in mind forthe interpretation of wavelet transform results. Wavelets, usually denoted ψ(t),have to ful�l the zero-mean condition given in (2.2.1.1) in order to be called awavelet [FriendlyGuide]. ∫ ∞

−∞ψ(t)dt = 0 (2.2.1.1)∫ ∞

−∞|ψ(t)|2dt = 1 (2.2.1.2)

The zero mean value indicates that the function oscillates around the time axisand thus may be called a wave. In fact, the zero mean also implies that thewavelet's Fourier transform Ψ(ω) vanishes at Ψ(0) which indicates that waveletshave a bandpass spectrum [Alessio, 2016]. This property will be taken up againin Chapter 2.2.3. Moreover, normalising wavelets according to 2.2.1.2 ensuresthat the wavelet transform is only a�ected by the amplitudes of the signal andnot weighted by the wavelet itself.

Knowing that wavelets are short and transcient instead of in�nitely long wavesdoes not yet explain how they provide an adjustable window to decompose asignal. In the FFT, the analysing sinusoids change their frequency to reveal thecorresponding components in the signal. Wavelets cannot change their frequencyin the same manner but they can be scaled i.e. stretched or compressed. Figure2.2 sketches the e�ect by means of the popular Morlet wavelet. Compressingthe wavelet as shown in the top left plot means reducing its extent in time andthus improving its time resolution. At the same time, the corresponding spec-trum in the frequency domain covers slightly higher frequencies and increases inbandwith compared to the unscaled wavelet. The latter statement indicates adecrease in frequency resolution. Again, Heisenberg's uncertainty principle re-veals itself and it should be noted that scale and frequency are in fact inverselyrelated. Thus, stretching the wavelet as shown in the lowest plot has exactly theopposite e�ect on the wavelet than compression. The adjustable window is aninherent property of the wavelets and is also called the multiresolution property.

To analyse a complete signal, the wavelet is shifted along the time domain justlike the window in the STFT. The unscaled wavelet is usually called motherwavelet ψ0(t), while the scaled and shifted versions of it each represent a newdaughter wavelet ψa,b(t). In Figure 2.2 , the top and the bottom wavelet arethus the daughters of the wavelet in the middle. The parameter a representsthe scale factor and b the translation factor. Thus, b de�nes at which positionon the time line the signal is analysed. In conclusion, a wavelet is �rst scaledby a and afterwards translated along the signal by b. To change the frequencyband that the wavelet analyses, it is scaled and shifted again. This procedure


(a) The Morlet wavelet in the time domain (b) Calculated Pressure due to Thickness

Figure 2.2: The corresponding representations of the Morlet wavelet in thefrequency domain

is visualised in 2.3.

The basic idea of wavelet transform thus is a comparison between the originalsignal and the daughter wavelets. Mathematically, this process is described in(2.2.1.3).

ψa,b(t) =1√aψ0

(t− ba

)(2.2.1.3)

The factor 1√aensures that the normalisation given in 2.2.1.2 is ful�lled.

From the above, it can be seen that wavelets analyse a signal by changingtheir scale. Consequently, wavelet transforms convert a signal from the timeto the scale domain rather than the frequency domain. However, representinga signal as a function of frequency is more intuitive. Even though it is notpossible to associate the wavelet with a particular frequency, it is often feasibleto determine a center frequency fc of the mother wavelet which captures thedominant oscillation of ψ0(t). Figure A.1 in appendix A illustrates the centre


Figure 2.3: Comparison of a signal to a shifted and scaled wavelet [Alessio,2016]

frequency of the Morlet wavelet by comparison to the sinusoid that �ts best toits major oscillation. For the daughter wavelets, the centre frequency is scaledto fa, usally called pseudo-frequency, according to 2.2.1.4.

fa =fcadt

(2.2.1.4)

It is thus possible to represent wavelets and the results of wavelet transforms interms of frequency instead of scale.

Wavelets thus di�er considerably from sinusoids, especially due to the represen-tation of a signal in the scale instead of the frequency domain. The followingchapters explain how wavelets are used in practise.

2.2.2 The Continuous Wavelet Transform

In the following, the functionality of the continuous wavelet transform is pre-sented with a particular focus on its use in spectral analysis.

The generic CWT is de�ned for a continuous signal x(t) which is multiplied byshifted and dilated complex conjugates of the mother wavelet ψ(t).

Wψx (a, b) =

∫ ∞−∞

x(t)1√aψ∗0

(t− ba

)dt (2.2.2.1)


However, real life signals x[n] are of �nite length and are sampled at discretepoints of time n′ = [0, N − 1]. It follows that b and a must be discretised inorder to enable numerical calculation [Alessio, 2016]. This is re�ected in 2.2.2.2:

Wψx (a, b) =

1√a

N−1∑n′=0

x[n′]ψ∗0

[n− n′

a

](2.2.2.2)

Now, n denotes the discretised translation factor and states at which temporalpositions the signal is analysed by the wavelet. Usually, a dense scheme is as-sumed for the shifts implying that the signal is analysed at all points in n′. Thisscheme causes a signi�cant amount of redundancy in the wavelet coe�cients.Figure A.2 in appendix A shows that translating the wavelet in small stepschanges the analysed signal sequence only slightly for two subsequent wavelets.

The values of the scaling factor a may, in general, be discretised arbitrarilybut again a dense scheme is preferred. Consequently, the wavelet is stretchedand compressed in small steps as shown in A.3 in appendix A. The frequencybands have similar bandwidths and cover similar intervals in the frequency do-main. Thus, the individual wavelets in A.2 reveal only little new frequencyinformation, as their frequencies bands overlap signi�cantly. The high degree ofredundancy caused by dense scaling and dense shifting makes the CWT a verytime-consuming business.

However, the dense discretisation schemes are required for the inverse CWT.The theoretical, non-discreticed inverse CWT indeed allows for the perfect re-construction of the original signal by using special analytic wavelets. But assoon as scales and translations are discretised, information of the signal getslost and the reconstruction is no longer perfect implying that energy is lost atthe same time. Consequently, the inverse discretised CWT only approximatesthe signal and requires as many coe�cients as possible to achieve acceptable ac-curacy. Usually, the the loss of information is small but computing the inverseCWT and comparing it to the original data set is a good way of checking thequality of the selected translation and scale values.

At this stage, it seems odd to still label the CWT "continuous" because itoperates on a discrete signal using discrete sets of scale and translation fac-tors. However, the possibility to arbitrarily select the number of translationand especically scale values distinguishes the CWT from the discrete wavelettransforms. This is going to be addressed in further detail in Chapter 2.2.3.

Until now, all plots used the Morlet wavelet as stated in Chapter 2.2.1. TheMorlet wavelet is by far the most common choice in the CWT, as it is analyti-cally described and basically is a product of a Gaussian window and a complex


exponential, as shown in (2.2.2.3). It is therefore linked to Fourier transforms.

ψ(t) = eiωte−t22 (2.2.2.3)

However, a strength of wavelet transforms in general is the possibility to selecta wavelet out of a large variety of functions and it is even possible to selectdi�erent wavelets for the decomposition and reconstruction of a signal due tothe large degree of redundancy in a CWT.

For this project, the most important feature of wavelet transforms is the abilityto provide a frequency-based representation of the signal energy. [Alessio, 2016]provides a detailed description of how spectral analysis is conducted by meansof the CWT. By squaring the coe�cients Wψ

x (a, n) a wavelet energy spectrumis obtained.

Eω(a, n) = |Wψx (a, n)|2 (2.2.2.4)

This spectrum speci�es signal's energy content at a particular scale and positionin time. The most common method to visualise the wavelet energy spectrum isthe scalogram which represents the temporal development of the signal energyper scale. The scalogram is easy to interpret but not the most useful plotto be used in this project. For comparison to FFT results, a global waveletspectrum(GWS) is more suitable. Both the scalogram as well as the GWSare described in further detail at the of this chapter by means of an example.However, to create a GWS, the instantaneous wavelet energy spectrum valuesare summed up for each scale and doing this for all scales leads to a spectrumcomparable to the FFT' frequency spectrum. The GWS for a discretised CWTis calculated according to 2.2.2.5

W̄ 2(a) =1

N

N−1∑n=0

X(k)|Wψx (a, n)|2 (2.2.2.5)

As scales are inversely related to (pseudo-)frequency, it is then possible to obtaina spectrum in the frequency domain.

Example - Superimposed sine waves analysed by the CWT:

The CWT is performed on the same signal introduced in Chapter 2.1.1. Itshould be noted beforehand that wavelet transforms are not able to analyseperiodic signals perfectly, just as the sinusoids in the FFT struggle to representa short transcient oscillation. The example is supposed to illustrate this andshow how the CWT output is usually visualised.

This CWT uses the Morlet wavelet for decomposition and reconstruction andthe scales range from 0.002 to 100 with an interval of 0.004 between the scale


values. The signal is analysed at all points of time in the signal vector, thus thetranslation scheme is maximally dense.

First of all, the inverse CWT, as an idicator of the transform quality, showsthat the accuracy of the computation is promising. The reconstructed signal,shown in red, in Figure 2.4 (a) varies so little from the original signal, given inblue, that the deviation is only visible in a zoomed plot 2.4 (b). The averagedi�erence between the signal and its CWT reconstruction is 0.375% for thissimple example. In consequence, the selected scaling scheme is acceptable.

(a) Reconstructed singal (b) Close-up of the amplitude peaks show-ing the degree of deviation

Figure 2.4: Quality check of the of CWT using the reconstructed signal

As stated previously, the scalogram is the most common CWT plot but the GWSis expected to be more useful in this project. The following results explain thisassumption.

Figure 2.5 shows the scalogram, which is automatically produced by MATLAB.The scalogram is a plot in the scale-shift plane which indicates the importanceof each |Wψ

x (a, n)| by colours. If the plot is dark, the coe�cient is close to zero.The lighter the plot is at a particular combination of scale and position, thelarger the value of |Wψ

x (a, n)|. The scalogram does not show the signal energybut how active the component of a particular scale or frequency is [Alessio,2016]. In this case, two bands of scales are active over the whole time period.The CWT thus detects the two sine waves but it does not associate them withspeci�c frequencies but with bands. The scalogram also shows the periodicityof the signal, as the active bands are regularly interrupted by darf "stripes".

To get information about the average energy in the signal, the GWS is certainlymore useful. The GWS in Figure 2.6 is clearly similar to Figure 2.1 (b), eventhough it is signi�cantly smoother. The energy concentrates in two frequency


Figure 2.5: Scalogram of the analysed superimposed sine waves signal

Figure 2.6: GWS of the superimposed sine waves signal

bands which is in line with the scalogram. It is evident that the peak at ap-prox. 60Hz is lower than it is expected from the FFT spectrum. As compressedwavelets are linked to bad frequency resolution they tend to spread the infor-mation over larger frequency bands which results in a greater smoothing in thecorresponding part of the spectrum. The CWT is thus not recommended whenenergy peaks have to be identi�ed [Alessio, 2016].

The main advantages and disadvantages of the CWT are summarised in Table2.1. In general, the CWT provides all required functionalities for the analysisof ship response data. However, the loss of energy and the computation timeneed to be analysed in order to decide whether the CWT is of practical use.


Table 2.1: Summary of the CWT's major advantages and disadvantages

CWT

Advantages

CWT

Disadvantages

- results are intuitive and

easy to interpret- high computation time

- suitable for spectral analysis

(GWS)- energy not completely preserved

- provides a smooth

time-frequency

representation of the signal

- energy of high frequency

components not recognised

properly

2.2.3 The Discrete Wavelet Transform

The considerable amount of redundant information in CWT coe�cients eventu-ally led to the development of the DWT which is computationally signi�cantlymore e�cient. As the DWT is a complex mathematical topic which further-more makes use of di�erent signal processing concepts, this chapter can onlygive a brief overview. The focus therefore is on highlighting the computationaland conceptual di�erences between the DWT and CWT to enable the reader tounderstand the advantages and disadvantages of the method.

The DWT uses a signi�cantly di�erent approach towards signal decompositionthan the CWT. The mathematician Stéphane Mallat recognised the bandpasscharacter of wavelets and combined it with common signal processing techniquesto eliminate the redundancy from the wavelet transform. His idea was to con-sider the scaled wavelets as bandpass �lters. By scaling wavelets so that theirfrequency bands only overlap minimally, it is possible to extract all required in-formation with the smallest possible e�ort. Figure A.4 visualises how waveletsfollowing this scheme separate the signal into frequency bands. By this, thenumber of computations and wavelet coe�cients is signi�cantly reduced. Toobtain minimally overlapping frequency bands, the scale factor should be apower of two, so a = 2j with scale parameter j ∈ Z. Equation (2.2.3.1) re�ectsthis adjustment.

ψj,k(t) =1√2jψ

(t− k2j

2j

)(2.2.3.1)

It is seen that the translation factor is now depending on the scales as well,


so b = k2j with k representing the integer shift parameter. Now, the scale andtranslation parameters assume discrete values which is why this method is called"discrete" [FriendlyGuide], even though the CWT also works with discretisedfactors.

Stretching a wavelet by a factor of two means the corresponding frequency bandis halved and shifted down to lower frequencies by the same factor as sketched inA.4. It should be noticed that fn denotes the Nyquist frequency. Thus, in orderto split the signal into frequency bands all the way down to zero, an in�nitenumber of wavelets would be required. By introducing a scaling function whichacts like a low pass �lter, this problem is solved. The wavelets decompose thesignal until su�ciently low freuquency bands have been produced. The scalingfunction then captures the remaining information in the low frequency range[FriendlyGuide]. The red box in A.4 indicates how the scaling function coversthe frequency band that is not reached by the wavelets.

The wavelet and scaling functions thus form a �lter bank i.e. a set of �ltersthat breaks up the signal into di�erent, disjoint frequency bands. The iterativeprocess of splitting up the signal using �lter banks is called subband coding[FriendlyGuide]. In consequence, the nature of the DWT wavelets must becompletely di�erent from CWT wavlets. The wavelet and scaling functionshave to represent �lters that are able to extract all information relevant forsubsequent reconstruction which also means that the signal energy is preservedin the process. Usually, the wavelet functions are designed to be high pass�lter, as this simpli�es the process. Figure 2.7 demonstrates the actual DWT�ltering process with h[n] and g[n] representing the low and the high pass �lter.The signal of length N is at �rst passed through both �lters. The coe�cientsobtained from the high pass �lter are saved as details D1 while the outputfrom the low pass �lter is �ltered again. Thus, a DWT decomposition is aniterative process because �ltering the signal only once would result in a toocoarse representation. Figure 2.7 shows three �lter iterations implying thatthis DWT has three levels. At level three, the wavelet coe�cients are storedas D3 while the remaining low frequency components of the signal are savedas approximation coe�cients A3. Thus, no information is lost and the signalcontents is broken up in four frequency bands. Furthermore, the DWT makesuse of downsampling. If a signal of length N is just passed through two �lters,both outputs would be of length N again. The number of wavelet and scalingcoe�cients would then double with every iteration. Downsampling by two, i.e.storing only every second coe�cient after �ltering, ensures a minimum number ofcoe�cients without loss of information required for a subsequent reconstructionof the signal.

The mathematical description of the inverse DWT is shown in 2.2.3.3. Themaximum number of level J de�nes how many scales are used, as j must increase


Figure 2.7: Block diagram showing the generic DWT �ltering process

from 1 to J. Consequently, aJ represents the coe�cients at the �nal level ofdecomposition, while dj stands for the details at each level. The original signalis thus reconstructed by using the relationship between the approximations andthe scale function φJk(t) and the detail coe�cients and the wavelet functionψjk(t). The details and the approximation are often summarised as the vectors


Dj and AJ .

x(t) =∑k

aJ [k]φJk(t) +

J∑j=1

dj [k]ψjk(t) (2.2.3.2)

Dj =

J∑j=1

dj [k]ψjk(t) (2.2.3.3)

AJ =∑k

aJ [k]φJk(t) (2.2.3.4)

The scaling and wavelets functions are not de�ned explixitely and there arenumerous restrictions which ensure that they reconstruct the signal perfectlywhile ensuring a critical sampling scheme.

At this point, the most important pieces of information on the DWT are known.The DWT is quick because it selects wavelets in such a way that there is prac-tically no overlap in the frequency information they take out of the signal. Thisand the applied downsampling reduces the number of coe�cients used and in-creases the e�ciency of the method. However, this e�ciency is linked to splittingthe frequency spectrum into bands with �xed bandwidth because the bandwidthof the spectrum at level j is always half the bandwidth of level j − 1. It shouldbe checked whether this scheme is su�cient for a project that requires a certainaccuracy in the frequency domain. Furthermore, DWT is hardly ever linked tospectral analysis in literature and seems to be most relevant for signal de-noisingand compression.

Again, the example presented in Chapter 2.1.1 is used to illustrate the outputof a DWT decompostion i.e. the details which are the main result. It will thenbe seen that the computation method of the DWT trades o� a lot of frequencyresolution for computational e�ciency.

Example - Superimposed sine waves analysed by the DWT:

MATLAB provides good tools to conduct a DWT including a large number ofwavelets that could be used to analyse the signal, from which one may selectfreely. In this case the symlet "Sym8" is selected. More information about thisand other wavelets is given in Chapter 4. For now, it is enough to know thatthis wavelet is symmetric,comparatively regular in shape and in general, it is avalid selection for this sort of investigation. Knowing the wavelet and the lengthof the signal, MATLAB is able to determine the maximum number of levels upto which the wavelet is able to extract interpretable information. In the givencase, this is six.

The major output from a DWT is not a spectral representation but a plot of wave


components that the DWT expects the signal to be built up by. Figures 2.8 and2.9 therefore show that the DWT assumes most of the signal to be made up outof the waves D3 to D5 which have the largest amplitudes. The details are linkedto certain frequency bands. For the given signal of sampling rate 1000Hz, the�rst detail D1 covers high frequencies between 0.5fn and 1fn. Consequently,the details D3 to D5 cover the frequency bands 62.5-125Hz,31.25-62.5Hz and15.625-31.25Hz. It follows that the DWT expects the signal to be made up ofcomponents with frequencies between 15.625 and 125 Hz. This is certainly thecorrect range but it is also very coarse. Knowing that the original signal is built

Figure 2.8: Details D1 to D3 from a 6-level DWT for the superimposed sinewave example

up of just two sine waves, the shape of the identi�ed wave components mightbe surprising. However, it should be noted that it is not possible to exactlyreproduce an in�nite sinusoids by a �nite wavelet just a sinusoids struggle toidentify transcient oscillations. This shows the di�erence between the FFTand all wavelet transforms, as the CWT provides a huge number of similarcomponents. However, especially D3 and D4 show the periodicity of the signal.The DWT therefore highlights that WT are not suitable for the analysis ofclearly stationary signals.

Apart from this, it is important to state that a frequency representation ofthe DWT results is neither part of the automatic MATLAB output nor arethere instructions or descriptions to be found. Considering the coarse frequencyresolution of just seven frequency bands(six details and one approximation), onemay conclude that the DWT is not ideal for spectral analysis.


Figure 2.9: Details D4 to D6 from a 6-level DWT for the superimposed sinewave example

The main advantages and disadvantages of the DWT are therefore summarisedin Table 2.2.

Table 2.2: Summary of the DWT's major advantages and disadvantages

DWT

Advantages

DWT

Disadvantages

- e�cient computation method- splits the signal in very few

and thus large frequency bands

- perfect signal reconstruction

including energy preservation

- no information about its use in

spectral analysis found


2.2.4 The Discrete Wavelet Packet Transform

This chapter brie�y introduces the DWPT which was developed as a compromisebetween the CWT and the DWT to provide more details than the DWT withoutlosing the e�ciency of Mallat's algorithm.

As described previously, the DWT seems to be de�cient in the time-frequencyanalysis due to the coarse frequency intervals it creates. A richer analysis maybe created, if the bandwidths of the frequency bands were smaller. As a con-sequence, the DWPT performs the iterative �lter process not only on the lowpass �lter approximations but also on the high pass �lter output [Bergh u. a.,1999]. This results in a decomposition tree structure shown in Figure 2.10. The

Figure 2.10: Generic 3-level Wavelet packet decomposition

comparison to Figure 2.7 highlights the fundamental di�erence between the twomethods because the DWPT continues to break up the former details Dj untilthe maximum level of decomposition is reached. Thereby, the frequency bandsbecome narrower. In the end, the DWPT will provide 2j sets of coe�cients butas the downsampling process is still included, the �nal number of coe�cientsis identical in the DWT and DWPT. Thus, the DWPT avoids redundancy butprovides a �ner discretisation in the frequency domain. In Figure 2.7, the co-e�cients are no longer stored as pure details(D) or approximations(A) but thenaming scheme indicates the �ltering order.

It is thus expected that the DWPT is superior to the DWT when it comes to a


frequency representation of the signal. This is further investigated by applyingthe DWPT to the superimposed sine waves.

Example - Superimposed sine waves analysed by the DWPT:

The DWPT can be performed using the same wavelet and levels of decomposi-tion as the DWT. MATLAB provides an in-built function for a wavelet packetspectrum. The corresponding plot is shown in Figure 2.11 and it shows a highdegree of similarity to the CWT scalogram. Again, the spectrum is shown in atop view of the frequency-time plane and the relative importance of a coe�cientis indicated by colours. While cyan areas imply that there is no signal energyrelated to the corresponding frequency and time, violet in di�erent shades ofbrightness presents where the DWT expects the signal energy to be locatedtemporally and in which frequency range. The highest coe�cients are found

Figure 2.11: Wavelet packet spectrum for the superimposed sine waves

right above the 50Hz mark but also the 20Hz component is identi�ed by theDWPT. From the above, it is evident that the DWPT is less coarse than theDWT and is meant to be used for time-frequency representation of a signal.


2.3 Discussion of Intermediate Results

The previous sections gave an overview of wavelets and described the threetransform methods individually. To e�ectually apply wavelets to ship responsedata, the most suitable method should be determined. The theoretical aspectsregarding this decision are discussed in the following.

Both the FFT and wavelet transforms have individual strengths and weaknesses.The FFT is able to decompose a signal into exact spectral components and re-late an average portion of the signal energy to them. This averaging processof the FFT is a disadvantage of the analysed data sequence is non-stationarybecause mean energy per frequency might not su�ciently explain what is hap-pening in reality. The WT are able to localise energy in time and are representsignal energy as an evolutionary process. Due to this it might be possible todiscover developments inside the signal which are valuable for the analysis ofship response data. If, for example, a WT reveals that the energy inside thefull-scale response or performance measurements increases or decreases by trend,estimations on the future development of the analysed of these quantities mightbe improved. As wavelet transforms trade o� frequency resolutions to providethis time resolution, it is suggested that they complement but not fully replacea FFT analysis.

All wavelet transforms have been applied to a simple example and comparedto FFT results. While the CWT and the DWPT naturally provide a time-frequency representation of the wavelet coe�cients, the DWT is only used tovisualise approximations of the signal's constituents. The time-frequency plotsobtained from the CWT and the DWPT are furthermore not directly applicableto spectral analysis and it should be aimed to create an easily usable represen-tation for all WT. In general, none of the WT was able to identify the two sinewaves accurately but this is not to be expected from them. Apart from this, thecalculation of the example showed that there is a large variety of wavelets andit has to be determined how the most suitable one is selected for an analysis.

Therefore, the following questions need to be answered before wavelet transformscan actually be used for any kind of prediction or estimation:

• Is the CWT too slow when applied to full-scale measurements? Or if theCWT can be used, is it an accurate representation of the signal consideringthe loss of energy?

• Do the DWT or DWPT provide su�cient accuracy or are they too coarse?

• How are WT successfully set up e.g. with respect to the selection of a

2.3 Discussion of Intermediate Results 29

suitable wavelet?

• Is the MATLAB code easy to use or is a lot of knowledge about wavelettheory required to apply it?

The above questions address the accuracy of the results as well as practical as-pects regardings the MATLAB implementation. In case that wavelet transformsare useful for the analysis of full-scale data and could be applied in a decisionsupport context, it is important that the implementation of wavelet transformsis simple and that the computation is quick.

Chapter 3

Full-Scale Measurementsand Related Notation

To test the wavelet transforms e�ciently, relevant full-scale data is required. Inall further studies conducted within this project, ship response and wave radarmeasurements taken from intensive studies on the vessel CMA CGM Rigolettoare used. This chapter gives an overview of the data and presents the requirednotation along the way.

3.1 Background of the Measurements

The input data used in this thesis originates from European research programsfor which the CMA CGM Rigoletto, a container vessel, was equipped with ex-tensive instrumentation systems. First, strain sensors and accelerometers wereinstalled onboard the Rigoletto in 2006 for the Lashing@Sea project, a joinedindustry project focusing on lashing physics. In 2010, the EU FP7 projectTULCS (Tools for Ultra Large Container Ships) started for which additionalstrain sensors and sea state assessment devices were integrated into the measur-ing system. The full-scale data therefore comprises operational, ship response,and wave data.

32 Full-Scale Measurements and Related Notation

During the TULCS project, the Rigoletto with main dimensions as per B.1sailed on a route between the North Sea and the Sea of China. Out of thelarge amount of collected data �ve sets of 24h continuous measurements wereselected based on the criteria that the operational conditions (e.g. speed anddraught) were approximately constant, that all sensors were on-line and that thesea conditions were di�erent from each other. An overview of the operationalparameters on the �ve days including visual observations of the surrounding seastate is given in Table 3.1.

Table 3.1: Operational parameters on the �ve selected days

Date

(in 2011)

Geograpical

Position

Ship Speed

[knots]

Draught

[m]

Sea state

(visual obs.)

12thAug. Gulf of Aden 21.0 - 23.5 14.2 Moderate

20thAug. Mediterranean Sea 24.0 - 25.0 14.2 Mild

16thSep. Gulf of Aden 17.0 - 18.0 14.0 Mild/moderate

20thSep. South of India 11.5 - 13.5 14.0 Mild

02ndOct. O� Hong Kong 9.5 - 14.0 15.0 Severe

In later chapters, the speci�c data sets will be referred to by mentioning the date.The 20th August and the 2nd October are particularly focused on because thesurrounding sea states are noticeably di�erent.

3.2 Ship Operational Conditions

For this thesis, the speed and heading conditions are of interest, as they mightgive information about operational changes a�ecting the response measure-ments. The WaMoS system explained in 3.4.2 stores averaged values of themean ship speed Vs, the mean ship's heading µs and the mean wave headingµw. Both ship and wave heading are de�ned with respect to cardinal directions.However, it is common to relate heading to the angle between the wave head-ing and the ship heading which is called relative mean wave heading µr. Therelative mean wave heading is classi�ed for portside as:

• 150-180 deg, head seas; wave and ship heading are exactly oppsite

• 120-150 deg, bow seas

• 60-120 deg, beam seas

3.3 Full-Scale Ship Response Data 33

• 30-60 deg, quartering seas

• 0-30 deg following seas

Because of the ship's symmetry around its center line, these de�nitions can bemirrored to cover 180-360 degrees on starboard.

The assumption that the important operational conditions are su�ciently con-stant has been made prior to this thesis and has not been validated any further.

3.3 Full-Scale Ship Response Data

In the following, the notion regarding ship motions is introduced before theresponse measurements particularly used in this thesis are described.

The motions of a ship and thus its responses to wave excitation are de�ned bythree translations of and three rotations around the ship's centre of gravity. Thetranslational motions are called surge x1 (x-direction), sway x2(y-direction) andheave x3(z-direction), whereas the rotations are named pitch x5 (around y-axis),roll x4 (around x-axis) and yaw x6 (around z-axis). Figure 3.1 represents thesede�nitions schematically for the right-handed, ship-�xed coordinate system. For

Figure 3.1: Ship motions and coordinate system

most ships, port-starboard symmetry can be readily assumed because of whichthe motions in the symmetry plane - heave, surge and pitch - are symmetricmotions.

The �ve data sets comprise the ship motions heave, pitch, sway and roll andvertical bending moment information. This project only considers ship motions,


which were measured by accelerometers and saved in continuous time histories,one for each day. The heave and sway responses are given in meters, the rolland pitch motion in radians and all of them were recorded in intervals of ∆t =0.2s. It should be noted that the heave measurements are not fully independentof pitch and roll because the motions were recorded only on the starboard side.

As these response measurements have been used in several projects prior to thisthesis, they are expected to be reliable.

3.4 Sea State Measurements

As a part of the TULCS project, two sea state measurement systems - onerecording the instantaneous wave heights at the ship's bow and one wave radarsystem - were installed on the Rigoletto. This project uses the wave radar dataobtained from a Wave Monitoring System (WaMoS). The following subchaptersintroduce the sea state parameters used in this thesis and explain how they areobtained from the WaMoS installation.

3.4.1 Wave Parameters

This section provides a brief overview of important quantities characterising asea state and how they are calculated by statistical analysis of ocean wave datarecords.

Figure 3.2 shows an extract of measured ocean wave elevations and the infor-mation derived from it. The record is clearly irregular with signi�cantly varyingwave heights H, wave amplitudes ζ and wave periods T . While there is di�erentways to de�ne T , H is measured as the vertical distance between a wave peakand the subsequent trough. The amplitude ζ is de�ned as the vertical distancebetween the mean water level and a peak or trough. In this thesis, the charac-teristics of such records are expressed by three average quantities obtained fromstatistical analysis. The signi�cant wave height Hs represents the average valueof the highest third of all measured H. Tz constitutes the mean value of thetime measured between to successive up-crossings. In contrast to that, the peakperiod Tp quanti�es the average time between two successive peaks. For all ofthese quantities, a large number of measurements has to be considered.

The above quantities are usually derived from a spectral representation of thewave elevation record. As further explained in the next subchapter, WaMoS pro-

3.4 Sea State Measurements 35

Figure 3.2: Ocean waves time history and peak-trough analysis

vides the wave data in the form of a directional wave energy spectrum S(f, θ)which represents the wave energy as a function of the wave frequency f andthe wave direction θ. Integrating this spectrum with respect to θ results ina frequency-dependent wave energy spectrum S(f) which indicates the impor-tance of the sine components making up the record.

S(f) =

∫ 360

0

S(f, θ)dθ (3.4.1.1)

Figure 3.3 expresses the interpretation of such a spectrum. For a given rangeof frequencies δf , the area under the curve is proportional to the energy de-livered by the wave components of frequencies within δf . The area below thecomplete curve is thus proportional to the total energy of the recorded wavemeasurements. Most commonly, this 1-D spectrum is represented through the

Figure 3.3: Ocean waves time history and peak-trough analysis

angular frequency ω instead of the "true" frequency f . During the conversion,the energy contained within a frequency band δf must be the same as the energy


inside the corresponding segment δω. The conversion is performed as stated in(3.4.1.2) and (3.4.1.3).

ω = 2πf (3.4.1.2)

S(ω) = S(f)δf

δω(3.4.1.3)∫ ∞

0

S(f)df =

∫ ∞0

S(ω)dω = m0 = σ2 (3.4.1.4)

Equation (3.4.1.4) expresses the important relation between the wave spectrumand the statistical measure of variance which is also called 0-th order spectralmoment m0. This relationship may be applied to all cases in which time domainmeasurements are expressed in the frequency domain and must also hold whenresponse measurements are analysed. The general formula for spectral momentsis

mn =

∫ ∞0

ωnS(ω)dω (3.4.1.5)

Using the spectral moments, the ocean wave characteristics can be calculated.

Hs = 4.0√m0 (3.4.1.6)

Tz = 2π

√m0

m2(3.4.1.7)

Tp = 2π

√m2

m4(3.4.1.8)

The formula forHs is valid under the assumption that Tz and Tp are about equalwhich has been shown by [Jensen, 2001] to give a reasonably good estimation.

3.4.2 Onboard Installation of the WaMoS System

The wave radar system WaMoS analyses radar images to provide an analysisof the spatial and temporal development of the sea surface out of which a di-rectional wave spectrum is created. From this, the above described sea statecharacteristics are obtained. Therefore, a X-band radar system was installed onthe compass deck of the Rigoletto as shown in Figure B.2 in appendix B. The�nal position was chosen in order to avoid of the disturbance of other devicessuch as communication antennas or the compass.

WaMoS operates at wind speeds of more than 3 m/s and is able to detect wavelengths from 15m to 600m and wave periods from 4s to 200s. Rough weatherconditions, harsh sea states or darkness do not a�ect the functionality of theWaMoS system.

3.4 Sea State Measurements 37

X-band radars usually provide �ne temporal and spatial resolution so that se-quences of collected radar images containing the spatial sea state developmentof the ocean surface can be converted to the frequency domain by Fourier trans-form methods. From this, the directional wave energy spectrum given in theWaMoS data �les are derived. These WaMoS �les have been created every �veminutes based on 32 radar images, so there is more than 270 WaMoS �les perday. Furthermore, WaMoS provides Hs and TP for each of these �les based onthe approach outlined in the 3.4.1. The data contained in the data �les can fur-thermore be used to calculate Hs,Tz and Tp manually and provide a 20-minutesaverage of these sea state parameters. Usually, 20 minutes are more commontime intervals for the evaluation of e.g. response measurements by the FFT.

In general, wave radar measurements are believed to deliver su�ciently accurateestimates. However, the above explanation shows that the FFT is used togenerate spectral representations of the radar images and as it is discussed inChapter 2, the FFT has its drawbacks. So, one should keep in mind, thatWaMoS sea state parameters are just approximations, even though they aregood and reliable.

3.4.3 Stationary Time Periods

As WaMoS covers 24h continuously, most of the data must represent non-stationary processes due to variations in the sea state or slight changes in shipspeed or headings. However, it is natural to assume that there is time periodswithin the data record for which neither the sea state parameters nor relevantoperational conditions change. These time periods may be considered stationaryand they are supposed to be used in the wavelet transform functionality checkexplained in Chapter 4. This subchapter brie�y introduces how stationary timeperiods have been identi�ed within the 5 data sets.

Stationary time periods were identi�ed within the WaMos data and are expectedto last 30 minutes to three hours [Nielsen, 2010]. This work has already beenperformed master thesis "Transfer functions of a large containership" by KasperFønss Bach submitted in March 2015 to DTU. The following passages thereforesummarise his approach towards identifying the stationary time periods whichare listed in B.3.

In the present case, stationarity requires that the wave parameters Hs and Tzand the operational parameters D, Vs and µ do not change with time. Thesequantities are determined for all WaMoS data �les and compared to the condi-tion in the previous and the subsequent time period [Bach, 2015]. Bach assumesthat the directional wave spectra in the WaMoS �les contain information about


the previous 20 minutes of measurements. Thus, time period i is stationary if allabove parameters are identical to those in time period i−1 and i+1. Figure 3.4illustrates this procedures with black circles representing the di�erent WaMoS�les for one day of continuous measurements [Bach, 2015]. While D, Vs and

Figure 3.4: Time line of the WaMoS data �les created every �ve minutes

µ are directly stated as average values in each WaMoS �le, Hs and Tz are cal-culated from the spectra as outlined in 3.4.1. The comparison is performed bymeans of the statistical measures of mean value, standard deviation σ and thecoe�cient of variation CoV which denotes the ratio of σ to the mean value andthus constitutes a relative standard deviation. Those time periods for whichthe lowest standard deviation or CoV is obtained are considered stationary.This is done for all �ve days. As seen in B.3, the largest numbers of stationarytime periods are identi�ed for the 20th August and the 20th September which,according to Table 3.1, are associated with calm sea states.

Chapter 4

Determination of the mostappropriate wavelettransform method

The literature survey and theoretical assessment of wavelet transforms did notmake it possible to decide which wavelet transform method is most suitable forthe analysis of full-scale response measurements. Therefore, a study is performedthat uses response data, di�erent wavelets and all three WT methods to assessthe di�erent methods. The focus of this study is on the time-frequency repre-sentation of the signal energy. Global wavelet spectra as shown for the CWTare good indicators of the transforms capabilities but they do not present hid-den information. The strength of wavelet transforms is the property to localisesignal constituent in time and frequency to a certain degree. The previouslydescribed ways to illustrate this, e.g. the scalogram, do not particularly workwith signal energy, so it is desirable to create a di�erent way to represent thedata.

The following study decides on the WT with which this thesis is continuedby comparing the developed 3D representations as well as computation-relatedaspects such as calculation time.

40 Determination of the most appropriate wavelet transform method

4.1 Approach of the study

As the outcome of this chapter signi�cantly in�uences the further work of thisproject, the analysis has to be in depth and reliable. Therefore, a great varietyof parameters need to be de�ned and checked. This chapter explains the selecteddata and settings and the criteria based on which the perforamce of the wavelettransforms is evaluated.

4.1.1 Input data and settings

The reliability and accuracy of the CWT, DWT and DWPT can only be assessedproperly, if the study is based on relevant data i.e. full-scale ship responsemeasurements as presented in Chapter 3. To keep the study within reasonablelimits, only two data sets out of �ve are considered and split into intervals of20 minutes. Out of these intervals, six are selected per day and only heave andpitch motions are considered. A summary of the selected time periods takenfrom the 20th August and the 02nd October is given in Tables 4.1 and 4.2.

Table 4.1: Selected responses data from 20th August

20. August

WaMoS FilesTime Periods in

Response DataStationary

11-15 3300 - 4500 yes

12-16 3600-4800 yes

72-76 21600-22800 yes

31-35 9300-10500 no

117-121 35400-36600 no

197-201 59100-60300 no

The tables indicate which time periods (in seconds) are accessed in the responsemeasurements. The interval is assumed to be stationary, if at least one of thestationary time periods described in Chapter 3 lies within the considered 20minutes. As the stationarity check was performed for 5-minutes of sea statemeasurements and not for 20 minutes of response measurements, there is noguarantee that the responses are actually stationary. However, the probably thatthey are is increased compared to randomly selected time periods. As wavelettransforms perform better on non-stationary signals, it would be interesting to

4.1 Approach of the study 41

Table 4.2: Selected responses data from 2nd October

02. October

WaMoS FilesTime Periods in

Response DataStationary

16-20 4800 - 6000 yes

62-66 18600-19800 yes

63-67 18900-20100 yes

37-41 11100-12300 no

188-192 56400-57600 no

189-193 56400-57600 no

see whether the results are a�ected by an assumed degree of stationarity. Thetwo days, 20th August and 02nd October, have been selected because the seastates were reported to be signi�cantly di�erent, as seen in 3.1. This mighta�ect the CWT as well, as the probability that the sea state and ship motionsare non-stationary is larger on the 2nd October.

The calculations are completely performed in MATLAB (APPENDIX). It hasalready been mentioned that it is possible to select di�erent kinds of waveletsbut hints on how to select them are hardly ever found. The response analysis istherefore conducted based on up to six mother wavelets per wavelet transformto gain detailed insight into how the wavelet selection in�uences the results.The following mother wavelets are selected:

• CWT: Morlet, DB12, DB20, Sym12, Sym20

• DWT: DB8, DB12, DB20, Sym12, Sym20, Coif5

• DWPT: DB8, DB12, DB20, Sym12, Sym20, Coif5

The Morlet wavelet is already known. Because it is an analytic wavelet anddoes not represent a �lter bank, the discrete wavelet transform cannot use theMorlet wavelet. The abbreviation DB stands for Daubechies wavelets and thenumber indicates the number of vanishing moments. Vanishing moments arean important concept for discrete wavelets. They indicate the degree of dif-ferentiability of the wavelet and therefore its smoothness [Alessio, 2016]. TheDaubechies wavelets are named after Ingrid Daubechies, one of the main con-tributors to discrete wavelet theory. The wavelets denoted "Sym" are calledsymlets and are modi�ed versions of the Daubechies wavelets as they are more


symmetric than these [Misiti u. a., 2009]. Both, DB-wavelets and symlets maybe used to decompose a signal with the CWT but they are not permissible forthe reconstruction. Therefore, the inverse CWT always uses the Morlet wavelet.In addition to this, Daubechies developed coi�ets which show a high degree ofsymmetry [Misiti u. a., 2009] and do not support the CWT. The discrete waveletand scaling functions are illustrated in C.1 to C.6 in Appendix C. The functionshave normalised amplitude and are plotted over their support length in e.g.time.

4.1.2 Assessment Criteria

The wavelet transforms are evaluated based on two types of decision criteria:measures to determine the accuracy of the results and measures to determinecomputational aspects.

One of the most important criteria is the availability of new pieces of informationin the results. The underlying question of this study is basically whether wavelettransforms are able to reveal and present new information of measured data thatcould be of use for response prediction or sea state estimates. Therefore, onepart of this study is to enable MATLAB to create a 3D representation of thesignal energy as a function of frequency and time.

All in all, the following the following aspects have to be assessed:

• Visual evaluation of 3D plot: accuracy, reliability

• Signal reconstruction and energy preservation issues

• E�ect of the mother wavelet selection on results

Energy preservation is checked by comparing the signal variance to the areaunder the spectrum i.e. the GWS. This relationship is already known fromEq. (eq:equalenergy). Equation 4.1.2.1 expresses the variance of a measuredsequence in terms of the heave motion x3 and its mean value x̄3. This equationcan be equivalently used for all other types of motions.

σ2 =1

N

N∑n=1

(x3 − x̄3)2 (4.1.2.1)

Furthermore, the applicability of the MATLAB code is assessed. This includesthe following items:

4.2 Results of the CWT Test 43

• Computation time

• Required manual work an knowledge

The latter point assess how practical the wavelet transforms and the relatedMATLAB scripts are. If the a lot of manual ajustments are required or thewavelet selection a�ects the results noticeably so that well-founded choices arenecessary, the transform might not be applicable in practise. In the case thatWT can be used for decision support, they need to be ready for integration inonboard systems and should work in the background.

4.2 Results of the CWT Test

The theoretical considerations in Chapter 2 suggested that the CWT is able toprovide a �ne evoulotionary spectrum of the signal but it also assumed highcomputation time and imperfect energy preservation which create doubt thatthis transform is practically applicable. Therefore the performance of the CWThas to be evaluated in detail and a method to preserve the energy needs to bedetermined.

Therefore, the signal reconstruction and energy preservation have to be assessedbefore the time-frequency representation can be developed. Recalling that thematch between the signal and its CWT representation is assessed for all timeperiods listed in 4.1 and 4.2, for heave and pitch measurements and for �vedi�erent mother wavelets, it is important to keep track of the �ndings. Hence,data sheets are used to record the main �ndings. One data sheet shows theanalysis for one day, either stationary or non-stationary time periods, one typeof motion and all applied wavelets. The sheets are provided in C.7 to C.14 andFigure 4.1 exempli�es the contents of these data sheets for one analysis usingthe Morlet wavelet. In the following, it is explained how the information notedin these tables is used to assess the quality of the CWT computations. Thecolumns three to wave relate to the energy preservation issue. Just like theFFT spectrum, the GWS needs sclaing to represent the signal energy properly.While the FFT is conveniently scaled by the sampling frequency Fs and thenumber of positive frequencies in the spectrum nf , it is not as simple to �nda scaling factor for the CWT. The scaling factor, 2.2/N , works reasonably wellbut does not ensure energy preservation. Therefore, the CWT spectral estimatesare scaled a second time by the ratio between the signal and the CWT variance.In consequence, the CWT spectral estimates match the signal energy. In fact,[Alessio, 2016] states that it is necessary to calculate individual scaling factorsfor the di�erent wavelets to compensate for the energy loss between signal and


Figure 4.1: Exemplary presentation of the tables used to analyse the CWTperformance

CWT representation related to them. Because the determination of these scal-ing factors was not performed successfully in this project, the two-step scalingprocess explained before is applied throughout this thesis.

The signal reconstruction is considered as a general measure of quality for theCWT and may help to decide which mother wavelet is most suitable for thistransform method. Checking for the agreement between the original and recon-structed signal actually requires to compare them by plots. To simplify this,column six as shown in Figure 4.1 contains a value that quanti�es the maximumdeviation between signal and reconstruction. This measure itself should not beused for decision making but it is a reliable indicator of the quality approxi-mation. Extremely high values of e.g. more than 100 % indicate that at leastone peak in the original signal is not recognised and studying plots of such re-constructions shows that there is a general mismatch between the original andsynthesis. The symlets are usually exceeding the 200% and as expected, theirreconstructions do not match the signal well (see C.17 and C.18). It is there-fore concluded that the symlets are not recommended for this kind of data. Incontrast to this, the Morlet wavelet usually creates only small maximum devia-tions and the Figuers C.15 and C.16 show that there is a quite good agreementbetween signal and reconstruction. So, if the maximum deviation is lower than30%, the synthesis is acceptable. The Daubechies wavelets show di�erent ten-dencies. While the DB12 is almost as good as the Morlet wavelet, the DB20fails to capture the signal with the same accuracy. The reconstruction of theDB12 is shown in C.19 and C.20 in Appendix C. All in all, the Morlet waveletprovides the best results and is considered as a reliable choice.

As mentioned in Chapter 4.1.1, it was attempted to test the e�ect of stationarity

4.2 Results of the CWT Test 45

in the data by selecting time periods which may be assumed to contain stationaryintervals. However, the current study does not provide clear information aboutthis. It is assumed that the CWT would be better at handling data that is notstationary, so the quality of the reconstruction is expected to be better in thesecases. Tables C.7 to C.14 indicate that the measure given in column six in 4.1is slightly worse for the time periods called stationary. However, the trend isnot very clear. A better tendency is seen when comparing the qualitiy ofthereconstruction between the two selected days. Especially for the heave motion,it is remarkable that the reconstruction is better on the 02nd October which islinked to a harsher sea state and signi�cant variation in the wave and motionamplitudes.

The last part of the CWT results check refers to the time-frequency represen-tation of the signal energy. It was decided to develop a 3-dimensional diagrambased on Eq.(2.2.2.5) describing the calculation of GWS. To obtain a GWS,the coe�cients Wψ

x (a, n) are averaged above all points of time. In the samemanner, it is possible to take mean values of Wψ

x (a, n) for shorter intervals, e.g.per minute or every �ve minutes. In this way, short sequence wavelet spectraare created and one can represent the signal as a sequence of spectra. This isa rather interesting way of presenting the data because it allows for statisticalanalyses at di�erent points of time within the data set. In general, this approachis similar to what the STFT produces. Figure 4.2 shows two examples for suchdiagrams. .

(a) Local wavelet spectrum created every10s

(b) Local wavelet spectrum created everyminute

Figure 4.2: Time-frequency representation of a heave motion sequence bymeans of the CWT

Of course, the intervals over which the coe�cients are averaged should be chosenreasonably, as too many spectra might be di�cult to handle. Generating a highnumber of short term spectra slows down the calculation to some extent. Ingeneral, the computation time was not found to be a problem and performing


the CWT for a 20-minutes data sequences is completed within very few sec-onds. Generating a plot like 4.2(a) takes up to 20s. It is thus assumed thatanalysing several motions at the same time including the computation of shortsegment spectra may take up to a minute. Considering that the CWT is easyto automate, the method seems rather suitable.

Each of the spectra in the spectra sequence represents a fragment of the signalenergy related to a certain time interval e.g. one minute or ten seconds as shownin 4.2. These spectra can be evaluated individually, just as shown for wavespectra in Chapter 3.4.1. Therefore, this representation is considered useful forfuture applications.

4.3 Results of the DWT Test

In Chapter 2 it is assumed that the DWT's decomposition scheme is not �neenough to create useful time-frequency representations. This chapter thus dis-cusses the obtained 3D plots and evaluates whether they are useful for furtherstudies.

One of the strengths of the DWT is the perfect reconstruction and energy preser-vation that is provided through the �ltering process. It is therefore not necessaryto track the DWT's performance in a Table comparable to C.7. However, onedata sheet is provided in Figure C.21 to ensure that the DWT calculations areactually conducted correctly. As there is no signi�cant deviation, the DWTcalculations are accepted.

The critical issue in this case is the representation of the signal in an evolutionaryspectrum or by a sequence of wavelet spectra as described in 4.2. Chapter 2already points out that literature hardly ever deals with spectral analysis by theDWT and the subsequently described test underlines why the DWT is indeednot expected to be suitable for this sort of application.

As described in 2.2.3, the DWT is based on decomposition levels for whichdetail coe�cients are stored. For the mother wavelets as per Chapter4.1.1, themaximum number of levels J varies between six and eight depending on theselected wavelet. Consequently, the 20-minutes measurements are �ltered anddownsampled six to eight times and the coe�cients per level j are stored in Dj .For signals with a sampling time of 0.2s, the Dj are then linked to the frequencybands as shown in Table 4.3. The displayed values correspond to J = 8. If Jis smaller, e.g six, the last details are stored as D6 and the approximation, nowA6, covers the complete frequency band from zero to 0.0195, as this is where

4.3 Results of the DWT Test 47

the breaking-up of the frequency domain ends.

Table 4.3: DWT frequency bands for a signal with a sampling time of 0.2s

Level j

Details/

Approximations

per j

Frequency

Bands

[Hz]

1 D1 1.25 - 2.5

2 D2 0.625 - 1.25

3 D3 0.3125 - 0.625

4 D4 0.1562 - 0.3125

5 D5 0.0781 - 0.1562

6 D6 0.0391 - 0.0781

7 D7 0.0195 - 0.0391

8 D8 0.0098-0.0195

8 A8 0.00 - 0.0098

Table 4.3 reveals that the frequency spectrum is broken up into rather coarsebands. For example, D5 represents a large range of those frequencies that areimportant for heave motion analysis. The meaning of this is further illustratedby Figure 4.3 which shows the GWS obtained from a DWT in comparison tothe FFT's spectrum of heave motion measured on the 02. October. It is evidentthat the DWT is not able to di�erentiate between frequencies in a frequencyrange where details are required. Instead the DWT averages the energy whichit detects in the individual frequency bands. The corresponding 3D plot furtherillustrates this de�ciency. Figure 4.4 sketches the temporal development of thesignal by showing the energy coe�cients at decompostion levels four to seven.The 3D representation is not only very discontinuous and coarse, it is alsodi�cult to work with. As the DWT uses downsampling, the number of storedcoe�cients changes from level to level. The energy content is therefore spreadover di�erent time intervals depending on the level and it is challenging to "cut"through the plot at a particular point of time and get a local spectrum. It istherefore not easily possible to create short segment spectra as described inChapter 4.2. Because the complete DWT decomposition is not deemed to be�ne enough for further studies in this thesis, the lack of short segment spectrais not of importance.


Figure 4.3: Comparison of a heave motion FFT spectrum to the GWS from aDWT

Figure 4.4: Time-Frequency representation of the DWT coe�cients for heavemotions

4.4 Results of the DWPT Test

Previously, the DWPT has been identi�ed as a compromise between the DWTand the CWT. It is therefore to be determined to what extent it adopts theindividual advantages and shortcomings of the two methods.

The DWPT uses the same computational method as the DWT which meansthat perfect signal reconstruction is no problem for this method. As it appliesthe iterative �ltering to approximations and details, it is noticeably more de-tailed than the DWT. Furthermore, the tree structure of the decomposition

4.4 Results of the DWPT Test 49

method ensures that all frequency bands are represented by the same numberof coe�cients and that their bandwidth is constant. This simpli�es the work inMATLAB signi�cantly.

In general, it is thus possible to generate a GWS and 3D representation con-taining several spectra. An example for the latter is given in Figure 4.5. The

Figure 4.5: Time-frequency representation of a heave motion sequence by theDWPT

similarity to the corresponding CWT plots is striking though the lack of de-tail in the DWPT is observable. Even though the DWPT is much �ner thanthe DWT, it acutally provides average energy per frequency band. To get abetter insight in the shape of DWPT spectra, Figure 4.6 shows two GWS gen-erated from the same data set but using two di�erent wavelets, the DB20 andthe Sym12. These diagrams highlight the e�ect of the wavelet selection for theDWPT which splits the frequency domain into 2j frequency bands. The symletmay go down to level eight, resulting in 28 equi-distant frequency bands. Incontrast to this, the DB20 only works properly until level seven. The di�erencethat this selection makes is seen in Figure 4.6. Figure 4.6 thus shows that thefrequency resolution of the DWPT depends on the selected wavelet. It should benoted that according to Heisenberg's uncertainty principle the time resolutionmust be worse for a Sym12-decomposition than for a DB20-analysis. Providingmore detail in the frequency domain is naturally linked to losing time resolutionand vice versa. This a�ects the number of di�erent spectra seen in the kind ofplots shown in Figure 4.5. Selecting a wavelet for the DWPT therefore requiresa weighing up process that compares the need for good frequency resolution tothe desire to understand the temporal development. This can be challenging


(a) DWPT with DB20 (b) DWPT with Sym12

Figure 4.6: Comparison of FFT and DWPT spectra for heave motion anddi�erent wavelets

and is a potential drawback of the method, even though the representation asa sequence of spectra is an indicator that the DWPT is potentially interesting.

4.5 Comparison and selection of aWavelet Trans-

form

This section compares the �ndings regarding the individual wavelet transformsand consequently decides on which method is most suitable for further studieswhich require a time-frequency representation of the signal energy.

First of all, the main �ndings of this study summarised in Table 4.4. TheCWT and the DWPT both represent the signal energy as a sequence of waveletspectra. Both therefore allow for statistical analysis of the the measurement'senergy at di�erent points of time. Because the DWPT is less detailed, theCWT is preferred. Another advantage of the CWT is the possibility to performall calculations with the same wavelet, the Morlet wavelet. In general, themost critical aspect related to the CWT is the computation time. The hereinpresented study does not reveal signi�cant problems. The CWT is slower thanthe discrete methods but the amount of data utilised in the present project doesnot create problems.

In general, the CWT using the Morlet wavelet is thus the preferred method forfurther studies. However, it should be noted that the CWT should be seen asa complement to the FFT rather than as an alternative. The wavelet spectra

4.5 Comparison and selection of a Wavelet Transform 51

Table 4.4: Comparison of the performance of the CWT, DWT and DWPT

Assessment

CriterionCWT DWT DWPT

Time-frequency

representation

(quality and type)

useful and

detailed;

sequence of

wavelet spectra

coarse in time and

frequency;

limited in

degree of

detail but ok;

sequence of

wavelet spectra

Signal

reconstructionapproximation perfect perfect

Energy

preservation

perfect after

particular

scaling method

perfect perfect

Wavelet

selection

Morlet works

best

no clear

best selection;

de�nes level

of detail in

time & frequency

de�nes level

of detail in

time & frequency

Computation

timeacceptable very good very good

Manual work

(MATLAB)none

if wavelet

is changednone

created by the CWT are rather smooth an cannot identify energy peaks in thefrequency domain as accurately as the FFT. Therefore, it seems recommendableto use both methods and bene�t from their individual advantages.

Chapter 5

The potential of the CWTfor ship motion analysis

The previous chapters have dealt with wavelet transforms on a very general leveland highlighted the impression that especially the CWTmight complement anal-yses which are currently solely based on the FFT. As the FFT always calculates atime average of the signal energy per frequency, it is de�cient when the analysedmeasurements display a non-stationary character. The CWT trades o� some ofthe accuracy in the frequency domain to also locate the signal energy in timeand therefore it presents instationarity in a more realistic manner. In Chapter4, a means to represent the signal energy as sequence of spectra is establishedthat allows for the assessment of the CWT output by statistical measures. Thequestion arising at this point is therefore: What sort of information is actuallyuncovered by the CWT and how can this be of practical use?

This chapter tries to provide an answer to this question by presenting a studywhich assesses ways to integrate the information produced by the CWT in re-sponse prediction and sea state estimation. In the following, the approach andideas of the study are described and the �ndings are discussed and evaluated.

54 The potential of the CWT for ship motion analysis

5.1 Procedures to analyse the relevance of the

CWT for decision support models

The following sections describe the objectives and the approach of the studyto illustrate its general framework. Furthermore, an overview is given of theconsidered data.

5.1.1 Objectives, approach and limitations

As this thesis works in the context of decision support systems, it is importantto �nd a link between their needs and requirements and the output of the CWT.This section provides an overview about the context, approach, objective andlimitations of the study that is supposed to enlighten the applicability of theCWT in the given �eld.

At �rst before the actual study approach is presented, a side note is given on why,of all potential �elds of application, sea state estimates and response predictionsare selected as a context for this examination. In general, it is imaginable toapply the CWT whenever time domain measurements are statistically analysedand the non-stationary character of the data is supposed to be accounted for. In[Xu und Iseki, 2015], for example, the DWT is used to investigate possibilities toestimate the main engine power without requiring direct measurements. How-ever, this study considers the CWT with respect to ship response prediction andsea state estimation and thus mainly in the context of safety. Sea state estimatesform the input to various aspects of decision support, also to more sophisticatedvessel response predictions (see [Nielsen und Iseki, 2015]). The ship's responseto the surrounding sea has a large e�ect on e.g. its structural integrity andstability. Studying the CWT with a focus on the selected to �elds of applica-tion allows to deal with the CWT from two di�erent viewpoints. Thereby, itis expected that the insight into the output of the CWT is improved. In thisproject, the available data base encourages the consideration of both aspects inone study. As described in Chapter 3, the available measurements are ratherdetailed and cover di�erent conditions. Furthermore, the data has previouslybeen used di�erent studies and may safely be deemed reliable.

Due to the tight time frame of this thesis, the study assessing the applicabilityof the CWT has to be limited. It is therefore examined to what degree the CWTcould be used to provide simple estimates of sea states and future responses witha particular focus on getting deeper insight into the information presented bythe CWT. In consequence, the purpose of the study is to give recommendations

5.1 Procedures to analyse the relevance of the CWT for decision support

models 55

on how the CWT could be applied instead of developing a complete calculationmethod. The study is therefore mainly of a qualitative nature.

The presented work is split in two parts: the CWT's potential for simple re-sponse prediction methods and its potential for providing simple sea state esti-mates. Both parts work on the same set of data, described further in Chapter5.1.2. This data is prepared to give di�erent kinds of input to the study, as canbe seen in Appendix D as Table D.1 to D.6. The contents of these data sheetsis going to be discussed in the following sections.

The �rst part focusing on response prediction concentrates on using the sequenceof spectra provided by the CWT. It is investigated if it is possible to deducethe future development of the response from this plot. The most importantparameter is the standard deviation of the measured response sequence and it istried to deduce the development of this measure for the next sequence. So, fortime period i+ 1 an estimate is provided bast on the measurements recorded intime period i.

To estimate the sea state present in time period i, the CWT spectra sequence isused. To conclude on the surrounding sea state, as much of the data recordedin Tables D.1 to D.6 as possible is tried to integrate in the estimation.

The result of the study is a comment on the potential of the CWT to enablesimple, empirical estimation of sea states and future responses.

5.1.2 Input data and further assumptions

As mentioned previously, the study requires di�erent kinds of data. While re-sponse prediction mainly works on the full-scale response measurements, devel-oping sea state estimation methods require information of Hs and Tz to verifythe approach. In addition, knowledge of Vs and the relative heading µr com-pletes the data basis. Therefore, all kinds data presented in Chapter 3 areintegrated in this study.

Due to the tight time frame, the data analysed in this thesis has to be limited toensure a su�cient analysis. The following list provides an overview of importantselections.

• 25 time periods of the 20th August and 30 of the 2nd October

• heave, roll and pitch motions


The time periods represent 500-600 minutes of measurements per motion andday and are not extracted completely randomly from the full 24h measurements.Usually, 3-5 consecutive time periods are selected to illustrate the general de-velopment of the sea state and responses.

Even though the study has to be limited, three out of four available motion mea-surements are selected. Restricting the number of included responses requirescareful consideration. Analysing heave motions is a rather common choice, asthey are related to safety issues for various types of marine operations. Further-more, large damping forces make it possible to consider them as quasi-staticwhich is convenient for many kinds of analysis. However, this study does notonly consider responses but also tries to establish a link to the sea state. Be-cause of the in�uence of the heading on the heave motion, it is important toalso include roll and pitch. Together with the heading as recorded by WaMoS,the data basis is expected to be as good as possible.

Regarding the WaMos data �les, Chapter 3 points out that new estimates arecreated every �ve minutes. However, a MATLAB code provided by AssociateProfessor Ulrik Dam Nielsen calculates the mean sea state parameters based onfour subsequent WaMos �les. This code has been utilised in di�erent studiesprior to this project and therefore provides reliable estimtates of the sea stateparameters, mainly Hs and Tz. Furthermore, µr and Vs are recorded withineach WaMoS �le. The selected time periods in the ship motion measurementsthus have to match the data given by WaMoS.

As mentioned in the previous section, D.1 to D.6 in Appendix D give an overviewof the analysed data. Ship speed and relative heading are recorded for every�ve minutes to increase the level of detail when their in�uence on the motionsis required.

5.2 Evaluation of the CWT's relevance for deci-

sion support systems

The following sections present the di�erent steps undertaken to check the CWT'scapabilities in response and sea state estimation. The presentation includes thereasoning behind each step and eventually explains whether the CWT is deemeduseful for further work.

5.2 Evaluation of the CWT's relevance for decision support systems 57

5.2.1 Evaluation related to simple response predictions

This section presents an invesitgation which assess what kind of information theCWT may provide for simple, emiricial motion response prediction methods.The �ndings are discussed in order to evaluate the applicability of the CWT.

A simpli�ed approach for motion prediction is scaling the standard deviation ofthe analysed time period i, σi , to obtain an estimate of the motion's standarddeviation in the subsequent 20-minutes interval denoted σi+1.

σi+1 = σi(1± factor) (5.2.1.1)

This study attempts to �nd a way to produce the scaling factor inlcuding its signbased on the CWT sequece of spectra. Following from the previous chapters,the CWT presents the energy of the motion as a function of time. So, it mightbe able to deduce a general tendency from the temporal development of themeasured sequence.

At �rst, the accuracy with which the CWT represents the signal's developmentis evaluated to ensure that the representation is, at all, useful. Figure 5.1 illus-trates a comparison between a 20-minutes roll motion sequence taken from the02nd October and the corresponding CWT spectra sequence. The CWT is setup to split the signal in twenty segments, so one spectrum shows the average en-ergy of one minute of measurements. It is clearly visible that the CWT re�ectsthe development of the signal. Around t = 300s and t = 600s, the roll motionis particularly low but after 700s the it increases noticeably. The same develop-ment is visible in the CWT representation 5.1 (b) which shows extremely smallspectra for approximately time segements 5 and 10 and large spectra from aboutsegment 13 on. In consequence, the CWT might be able to give important inputto the motion prediction, if the trend of the standard deviation is deducible fromthe temporal evoluation of the signal energy. In particular, it should be testedwhether a general statement can be made on whether the standard deviation oftime interval i+ 1 is increasing or decreasing compared to σi.

The data collection in Tables D.1 to D.6 in Appendix D provides an overviewof the standard deviation of each time segment and indicates the developmentof σ beteen consecutive time periods. The �rst attempt to derive whether theCWT provides information about the future energy development is a simplevisual check. Creating a simple estimation method, requires little and easilyinterpretable input to the generation of a scaling factor. Pre-checking the spectrasequence of several time periods, gave rise to the hope that the CWT plots couldbe divided into three categories based on which the development of the standarddeviation can be deduced.


(a) Time history of roll motion (b) Time-frequency representation by theCWT

Figure 5.1: Comparison of a CWT sequence of spectra to the original mea-sured roll motion (2nd October, time period 26)

a) If the spectra in the last third of the time line are signi�cantly larger thanthe previous spectra, the respective motion is often has a larger standarddeviation in the following time interval.

b) If the largest spectra are located in the �rst third of the sequence, the mo-tion's standard deviation is likely to be lower in the following 20 minutes.

c) If the spectrum is more balanced and the peaks are not as pronounced,the standard deviation usually does not vary much between the respectivetime period and the following interval.

These categorised are exempli�ed in Figures 5.2 to 5.3

Of course, this categorisation is very coarse but this step allows to assess howeasy or di�cult it actually is to deduce the future motion development based onthe CWT. Consequently, the CWT is applied to all time periods and motionsmentioned in Chapter 5.1.2 and the tables in Appendix D show whether theplotted spectra sequences indicate increasing (+), decreasing (-) or stable ( )standard deviation in the subsequent time interval. As this is a post-voyageanalysis rather than a prediction, the match between the expected developmentand the actual development is evaluated by comparing to the given σ in columneight. If the expectation derived from the spectra sequence is not true, thecorresponding �eld is marked orange. By following this simple method, it ispossible to decide whether this direction should be headed for further.

As there is a lot of good matches, the next step is to remove as many mismatchesas possible by using more data contained in the CWT results. Therefore, the


(a) Sequence illustrating �nding a) (b) Sequence illustrating �nding b)

Figure 5.2: CWT sequence of spectra for roll motion, 02nd October, timeperiod 26 and 27

Figure 5.3: CWT sequence of spectra for roll motion, 02nd October, timeperiod 28,refers to �nding c)

second step of the investigation tries to eliminate some of the oragne mismatchesby recording the maximum, minimum and mean standard deviations found inthe respective time interval. These parameters are hoped to give further insightinto the signal development. In some of the orange cases, the mismatch occuredbecause extremely high motions occured in a small interval which changed theoverall impression of the spectrum. Analysing mean, maximum and minimumspectra is exptected to give further clues. The following is assumed:

If the mean and minimum standard deviation of time interval i are aboutequal to those of the previous ones in i−1 but the maximum is signi�cantlyhigher, this might be considered as an "outlier", a sudden temporary mo-tion and not a part of the general development.

If the mean and/or minimum standard deviation increase, this could be


an indicator of a general increase in the motion energy level.

If all values are not changing signi�cantly, this is an indicator that theoverall signal energy does not change much in the near future.

As can be seen from above, this approach requires knowledge of what happened20 minutes previous to the analysed time interval i because it is di�cult to judgewhether increases or decreases in the standard deviation are large or small ifthere is nothin to compare to.

The results of this are given in columns 16 and 17 of the data sheets in Appendix.In general, only few of the mismatches were cleared by this method. Mostly, themean, maximum and minimum standard deviations supported the impressionalready provided by the plots of the spectra sequence.

All in all, it is di�cult to judge whether it is possible to conclude the futuredevelopment of the ship motion energy's standard deviation. The general im-pression is that the presented simple approach is not a bad indicator. However,the categorisation of the spectra sequences in one of the categories was oftennot as easy as the Figures 5.2 to 5.3 imply. Many plots were noticeably lessdistinct so that it is expected that an implementation in Matlab would requiremore if-statements as given in a,b and c. In general, the CWT is able to illus-trate the previous development of the motion energy and to some extent thiscontains implications of the future motion. However, further studies would berequired to ensure that the identi�ed trends are usable and it is expected thatthe method would not be as simple as outlined in the beginning of this section.

Furthermore, it should be noted that any such prediction method cannot be ex-pected to estimate future responses perfectly. Sudden, unexpected motions mayalways occur and especially such simple methods will not be able to account forthese. Therefore, an uncertainty measure accounting for abrupt changes in themotion amplitude and the general possibility that the previous measurementsare misinterpreted has to be integrated.

5.2.2 Evaluation related to simple sea state estimates

Apart from predicting future ship motions, this study assesses whether the seastate can be identi�ed from the response measurements through the CWT rep-resentation and thus, whether the CWT can be used to estimate a sea statewithout direct measurements.

Usually, a linear relation ship between the amplitude of the encountered waves


and the ship's motion response is assumed. This allows for the establishment ofa Response Amplitude Operator (RAO) approximates the frequency responseof a particular ship to regular waves. The response spectrum for a speci�cmotion can then be expressed by the wave spectrum as shown below whereφR(ω) denotes the RAO.

SR(ω) = Sζ(ω)|φR(ω)|2 (5.2.2.1)

The response amplitude operator φR(ω) is de�ned for a particular ship, loadingcondition, heading and it furthermore depends on the ship speed. The RAOillustrates the complexity of the relation between ship response and sea stateand it is evident why simple approaches are desirable.

However, the herein described study with all its limitations could not de�ne asuitable method to estimate the sea state in a simple and quick manner by useof the CWT.

In general, the study was limited to estimating the signi�cant wave height tosimplify the study even further. In Tables D.1 to D.6, a ratio between thesigni�cant wave height and the standard deviation of the particular motion isgiven. The basic idea was to approximate these ratios by means of the CWTto eventually scale the standard deviation of a time period to �nd Hs. Lookingclosely at these ratios shows that their values vary noticeably even for subsequenttime periods i.e. Hs might increase continuously but the response does not. Thisis not unexpected but it complicates the procedure, as the scaling factor has tobe calculated uniquely for each time period and a standardised or constant factoris not an option. In contrast, the scaling factor must account for the particulardevelopment of the response inside the measured interval.

Chapter 3 already describes that the response data is known to be linked toconstant loading conditions. The datasheets in D.1 to D.6 illustrate furthermorethat the speed changes only slightly within 20 minutes, just like the relativeheading. The basic conditions for the analysis are consequently good, as suddenchanges in the operational conditions do not a�ect the responses, at least in themajority of the considered time intervals. Still, it was di�cult to separate thesea state e�ect on the response from the total response and to utilise the headingand speed data e�ectively, as no direct e�ect from both could be identi�ed inthe data.

The general approach to estimating the sea state again incorporated using thesequence of spectra and the statistical parameters derived from them. As thedevelopment of the sea state does not necessarily correlate with the calculatedstandard deviation of the 20-minutes interval, the �ndings described in Chapter5.2.1 are of no particular use in this case. However, all attempts to manipulate


the standard deviation of the spectra remain unsuccessful and no particularroute for estimating the sea state based on the CWT is identi�ed.

5.2.3 Summary and discussion of the �ndings

To conclude the study, the main �ndings are brie�y discussed and the overallapplicability of the CWT is evaluated.

In general, the CWT is able to generate a reliable representation of the signalenergy in its temporal development. Nevertheless, a few additional considerationneed to be stated. First of all, the study has been limited with respect to theanalysed data. Even though the described tendencies are recognisable in a greatnumber of time periods, the �ndings should be veri�ed through other data.Secondly, the length of the considered time interval is exptected to be limited.Predicting the motions for much more than 20 minutes, seems optimistic, if themethod is based on 20-minutes of measurements. However, it might be worthto try and use more than 20 minutes of measurements and see whether usingmore information about the previous development helps to predict the motionsover a longer time period.

The outcome of the attempt to identify the CWT's usefulness for simple responseprediction methods is not easy to de�ne but shows that there is some potential.However, this potential is not quanti�ed and further work would be required tocon�rm it.

The application to sea states is less successful because it has not been possibleto suitably link the given data on operational conditions to the CWT spectra.It is, in general, believed that CWT may play a role in sea state estimationbut the general models for assessing operational conditions and other externale�ects would have to be more complex. The sea state is not easily identi�ablein the response measurements, even when the time axis is included. The lack ofsuccess is deemed to be related to the overall too simple approach of the study,which was, however, required, as the main objective was the evaluation of theCWT's capabilities. It can therefore be saved that a strikingly simpli�ed seastate estimation method cannot be developed solely based on the CWT.

All in all, the CWT is not able to provide a simple sea state estimation methodand its ability to simplify response prediction is not proved. However, it isquestionable whether the presented studies are at all able to give a proof of thereal potential of wavelet transforms. Of course it is desirable to get simple andreliable sea state estimates but the chosen approach was probably too simple.In the same way, it is at least questionable whether the potential of the CWT


for response prediction can be assessed based on the simple approach describedbefore. After all, the CWT is similar to the FFT in many ways and mostprobably applications in which the FFT is mormally used are more suitable toassess the potential of the CWT when it comes to providing new information.

Chapter 6

Conclusions

This thesis was set out to explore wavelet transforms as a complement or alter-native to the fast Fourier transform and to identify their potential for the �eldof ship response analysis.

In particular, the following two main questions had to be answered by a thoroughliterature review and studies on the performance of the three wavelet transforms:

• Are wavelet transform methods useful for ship response analysis from ageneral point of view and which method is the most suitable one?

• How could wavelet transforms be used to improve the analysis of shipresponse measurements with respect to spectral analysis?

These questions are addressed in detail in chapters three to �ve which dealwith the theoretical background, a study on the actual results given by thethree relevant wavelet transforms and a study on the application of one wavelettransform to analyse full-scale ship response measurements.

The results and additional recommendations for future work are presented inthe following sections.

66 Conclusions

6.1 Findings and discussions

This section synthesises the main �ndings and implications of this thesis byparticularly addressing the two main research questions. Detailed summariesof the �ndings regarding the theoretical background, the comparison of thethree wavelet transforms and their potential for ship response analysis are foundat the end of the respective chapters. The studies identi�ed advantages anddisadvantages of wavelet transforms, their di�erences to the FFT and theirability to produce accurate time-frequency representations of the signal energyincluding their usefulness in a ship response analysis context.

The �rst question addresses the conceptual design of wavelet transforms andmainly concerns their ability to reveal new pieces of information from measureddata compared to the FFT. The presentation of the results in a useful manner isan additional criterion for the assessment of the transforms. Wavelet transformsare indeed found to uncover signal information that is not captured by theconventional FFT. As opposed to Fourier transforms, wavelet transforms aredesigned to handle non-stationary data by representing the signal and its energyas a function of time and frequency simultaneously. The signal energy can thusbe analysed in its temporal development. As ship response measurements arelikely to be non-stationary to some extent, it is expected that knowledge of thedevelopment of the response energy is useful for di�erent applications.

However, it was found that wavelet transforms should not be seen as a replace-ment of the FFT but as a complement. The great strength of the FFT is itsability to link the signal energy to particular frequencies. All wavelet transformslack this accuracy in the frequency domain in return for the localisation of theenergy in time. The three considered wavelet transform methods are found todi�er noticeably in their accuracy in both domains. As the representation of thesignal energy should be as precise as possible, the continuous wavelet transformis de�ned as the most suitable transform from a spectral analysis point of view.The CWT is able to produce spectra similar to the FFT but for several pointsof time within a given time interval. Therefore, the development of the signalenergy can be represented as a sequence of spectra which is deemed to be arather useful manner because it allows for statistical analysis.

It should be mentioned that the CWT is known to su�er from computationaline�ciency, even though this was not found to be a problem within this project.It is, however, recommended to subject the CWT to larger amounts of data totest its limits of applicability.

The second question is an attempt to further explore the output of the CWT andto assess its potential to provide input to simple estimates of the surrounding sea

6.2 Recommendations for further research 67

state and ship motions. The related studies are performed based on the samesets of data. Considering two di�erent �elds of application allows for insightinto the CWT's results from di�erent perspectives. Nevertheless, it is di�cultto use the CWT to develop simpli�ed estimation methods for both, sea statesand response predictions. While all attempts to use the CWT output to get areliable method were unsuccessful, the CWT showed some potential for responseprediction. Illustrating the temporal development of motion energy, the CWTis found to be able to give trends for the future development of the analysedmotion. Still, the results need further veri�cation and the study conductedwithin this thesis may only be seen as a �rst indicator of the CWT's potential.

In conclusion, it is not possible to give a clear answer to main question two. Theconducted studies do not imply that sea state estimation or response predictioncan be signi�cantly simpli�ed by use of the CWT, at least this is not provedfor the response prediction. Still, it is believed by the author that the CWTis an interesting topic of further studies. As the time frame of this thesis islimited and the CWT applicability studies had to be simple enough to provideat least a tendency, the studies themselves are most probably unsuitable to give aclear statement. It is therefore recommended to see the CWT as a complementto the FFT and analyse whether it provides new information if it is used inapplications which are currently solely based on the FFT. The CWT is, ingeneral, an interesting technique which at least theoretically has the potential toimprove current methods by treating non-stationarity with increased accuracy.

6.2 Recommendations for further research

The previous discussion of the results regarding the assessment of the CWT'spotential to improve sea state estimation and response prediction showed thatapplying the method in a more sophisticated context might be more suitableto identify the CWT's capabilities. Therefore, two possible topics for furtherstudies are described in the following.

The �rst recommendation for further studies addresses the estimation of seastates by means of the wave buoy analogy. In this method, the sea state isestimated based on the transfer functions and the response spectrum of themotions of a particular given ship. A common method to provide responsespectra is converting motion measurements to the frequency domain by the FFT.Therefore, a certain length of the time history is required. As the CWT useswavelets to decompose the signal, it is possible to produce sea state estimatesfor noticeably smaller time intervals which could be of interest if the sea stateis harsh and the responses cannot be readily assumed to be quasi-stationary. In

68 Conclusions

such cases, the CWTmight be an advantage and the sea state might be identi�edin greater detail. As the CWT is not as accurate in the frequency domain asthe FFT, it would be important to see whether this impairs the applicability ofthe CWT in this context and would therefore provide a relevant implication onthe real potential of the CWT.

A second potential application is derived from [Nielsen und Iseki, 2015] whichdeals with response prediction including uncertainty factors re�ecting a level ofcon�dence. Again, the FFT is used to derive the response spectrum of mea-sured motion and the CWT could provide updates on this spectrum at shorterintervals. The presented uncertainty factors are based on the 0th order spectralmoment of the spectrum and therefore on the response spectrum. If the analysedresponses contain a certain degree of non-stationarity, it might be interesting toanalyse the measurement with the CWT and assess whether the prediction aremore accurate due to better recognition of the temporal motion development.

Appendix A

Appendix - Chapter 2

A.1 Center Frequency of the Morlet Wavelet

70 Appendix - Chapter 2

A.2 Schematic representation of th wavelet trans-

lation in the CWT

A.3 Schematic representation of wavelet scaling in the CWT 71

A.3 Schematic representation of wavelet scaling

in the CWT


A.4 Schematic representation of wavelet scaling

in the CWT

Appendix B



B.1 Main particulars of the CMA CGM Rigo-

letto

Name CMA CGM Rigoletto

Owner CMA CGM

Operator CMA CGM

Flag France

IMO Number 9299654

Delivery Date 29/9/2006

Builder Hyundai Heavy Industries Ltd. Co, South Korea

Max. TEU Capacity 9415

TEU Capacity at 14t 6919

Summer Deadweight 113,890tons

Gross Tonnage 99,500tons

Length Overall 350.00m

Beam Overall 42.80m

Max. Draught 14.50m

Max. Speed 25.4kn

Main Engine Type 12K98MC6-TI

Main Engine Power 68,666kW

B.2 Center Frequency of the Morlet Wavelet 75

B.2 Center Frequency of the Morlet Wavelet


B.3 Stationary sea state time periods in WaMoS

data

WaMoS

File No.

#

Sign.

wave height

H_s

[m]

Zero crossing

period

T_z

[m]

Mean

ship speed

V_s

[kn]

Heading µ

[deg]

Draught D

[m]

12th August

246 2.1 6.6 22.6 105 14.2

265 1.4 6.6 23.0 101 14.2

20th August

13 1.1 6.5 24.8 143 14.2

14 1.2 6.5 24.8 143 14.2

15 1.2 6.5 24.8 143 14.2

16 1.1 6.5 24.8 143 14.2

26 1.1 6.5 24.7 142 14.2

41 1.1 6.5 24.3 143 14.2

42 1.1 6.5 24.2 143 14.2

54 1.1 6.5 24.2 141 14.2

55 1.1 6.5 24.2 141 14.2

65 1.2 6.5 24.2 151 14.2

66 1.2 6.5 24.2 152 14.2

69 1.1 6.5 24.2 153 14.2

70 1.2 6.5 24.3 153 14.2

74 1.2 6.5 24.3 152 14.2

75 1.2 6.5 24.3 152 14.2

110 1.2 6.5 24.3 149 14.2

16th September

159 2.8 6.7 17.7 94 14.0

214 1.7 6.8 17.7 103 14.0

216 1.7 6.8 17.7 105 14.0

251 2.8 6.9 17.6 105 14.0

255 2.9 6.9 17.8 104 14.0

265 3.3 6.9 17.9 99 14.0

B.3 Stationary sea state time periods in WaMoS data 77

Table B.1: My caption

WaMoS

File No.

#

Sign.

wave height

H_s

[m]

Zero crossing

period

T_z

[m]

Mean

ship speed

V_s

[kn]

Heading µ

[deg]

Draught D

[m]

20th September

66 1.3 7.3 11.6 91 14.0

67 1.3 7.3 11.6 91 14.0

123 1.3 7.2 12.0 93 14.0

124 1.4 7.2 11.9 94 14.0

130 1.4 7.2 12.1 97 14.0

131 1.4 7.2 12.1 98 14.0

136 1.4 7.1 12.3 99 14.0

137 1.4 7.1 12.3 100 14.0

152 1.4 7.1 12.0 94 14.0

179 1.6 7.0 12.8 104 14.0

180 1.6 7.1 12.8 104 14.0

181 1.6 7.1 12.8 104 14.0

182 1.6 7.1 12.8 105 14.0

232 1.6 7.0 12.3 95 14.0

233 1.6 7.0 12.3 94 14.0

264 1.6 6.8 12.5 100 14.0

2nd October

5 1.6 6.5 15.9 164 15.0

20 2.2 6.5 16.7 161 15.0

66 4.6 8.2 14.4 142 15.0

67 4.6 8.3 14.2 142 15.0

68 4.7 8.2 14.0 142 15.0

101 6.6 8.7 12.6 138 15.0

Appendix C

Appendix Chapter 4

C.1 Wavelet and scaling function of the DB8

80 Appendix Chapter 4



C.4 Wavelet and scaling function of the Sym12 81

C.4 Wavelet and scaling function of the Sym12

C.5 Wavelet and scaling function of the Sym20


C.6 Wavelet and scaling function of the Coif5

C.7 CWT assessment: 20.Aug, stationary, heave 83

C.7 CWT assessment: 20.Aug, stationary, heave

nf = number of positive frequency (FFT); Fs = sampling frequency


C.8 CWT assessment: 20.Aug, not stationary,

heave


C.9 CWT assessment: 02.Oct, stationary, heave 85

C.9 CWT assessment: 02.Oct, stationary, heave



C.10 CWT assessment: 02.Oct, not stationary,

heave


C.11 CWT assessment: 20.Aug, stationary, pitch 87

C.11 CWT assessment: 20.Aug, stationary, pitch


C.12 CWT assessment: 20.Aug, not stationary,

pitch

C.13 CWT assessment: 02.Oct, stationary, pitch 89

C.13 CWT assessment: 02.Oct, stationary, pitch


C.14 CWT assessment: 02.Oct, not stationary,

pitch

C.15 Reconstructed heave sequence using the Morlet wavelet 91

C.15 Reconstructed heave sequence using the Mor-

let wavelet


C.16 Reconstructed pitch sequence using the Mor-

let wavelet

C.17 Reconstructed heave sequence using the Sym12 wavelet 93

C.17 Reconstructed heave sequence using the Sym12

wavelet


C.18 Reconstructed pitch sequence using the Sym12

wavelet

C.19 Reconstructed heave sequence using the DB20 wavelet 95

C.19 Reconstructed heave sequence using the DB20

wavelet


C.20 Reconstructed pitch sequence using the DB20

wavelet

C.21 DWT assessment: 02.Oct, not stationary, heave 97

C.21 DWT assessment: 02.Oct, not stationary,

heave

Appendix D



D.1 Data collection for response prediction; 20.Au-

gust, heave

D.1 Data collection for response prediction; 20.August, heave 101

D.2 Data collection for response prediction; 20.August pitch 103


gust pitch

D.2 Data collection for response prediction; 20.August pitch 105



gust, roll

D.3 Data collection for response prediction; 20.August, roll 107

D.4 Data collection for response prediction; 02.October, heave 109

D.4 Data collection for response prediction; 02.Oc-

tober, heave

D.4 Data collection for response prediction; 02.October, heave 111



tober, pitch

D.5 Data collection for response prediction; 02.October, pitch 113

D.6 Data collection for response prediction; 02.October, roll 115


tober, roll

D.6 Data collection for response prediction; 02.October, roll 117

Appendix E

Appendix MATLAB

120 Appendix MATLAB

E.1 Code to read and prepare the vessel response

measurements

E.1 Code to read and prepare the vessel response measurements 121

122 Appendix MATLAB

124 Appendix MATLAB

126 Appendix MATLAB

128 Appendix MATLAB

130 Appendix MATLAB

132 Appendix MATLAB

134 Appendix MATLAB

136 Appendix MATLAB

138 Appendix MATLAB

140 Appendix MATLAB

Bibliography

[FriendlyGuide ] : A Really Friendly Wavelet Guide. � URL http://www.

polyvalens.com/blog/wavelets/theory/#note3

[Alessio 2016] Alessio, Silvia M.: Digital Signal Processing and SpectralAnalysis for Scientists. 2016. � Forschungsbericht

[Bach 2015] Bach, Kasper F.: Transfer functions of a large containership.2015. � Forschungsbericht

[Bergh u. a. 1999] Bergh, Jöran ; Ekstedt, Fredrik ; Lindberg, Martin:Wavelets. 1999. � Forschungsbericht

[Bingham 2015] Bingham, Harry B.: Supplemental Lecture Note for: "WaveLoads on Ships and O�shore Structures. 2015. � Forschungsbericht

[Iseki 2015] Iseki, Toshio: Non-stationary Ship Motion Analysis Using Dis-crete Wavelet Transform. In: Proceedings of the 12th International Conferenceon the Stability of Ships and Ocean Vehicles, 2015, S. 673�680

[Mallat 2009] Mallat, Stéphane: A wavelet tour of signal processing. 2009.� Forschungsbericht

[Misiti u. a. 2009] Misiti, Michel ; Misiti, Yves ; Oppenheim, Georges ;Poggi, Jean-Michel: Wavelet Toolbox 4 User's Guise. 2009. � Forschungs-bericht

[Nielsen 2010] Nielsen, Ulrik D.: Ship Operations - Engineering Analysisand Guidance. 2010. � Forschungsbericht

[Nielsen u. a. 2013] Nielsen, Ulrik D. ; Andersen, Ingrid Marie V. ; Koning,Jos: Comparisons of Means for Estimating Sea States from an Advancing

http://www.polyvalens.com/blog/wavelets/theory/#note3

http://www.polyvalens.com/blog/wavelets/theory/#note3

142 BIBLIOGRAPHY

Large Container Ship / Technical University of Denmark and MARIN. 2013.� Forschungsbericht

[Nielsen und Iseki 2015] Nielsen, Ulrik D. ; Iseki, Toshio: Prediction ofFirst-Order Vessel Responses with Applications to Decision Support Systems/ Technical University of Denmark and Tokyo University of Marine Scienceand Technology. 2015. � Forschungsbericht

[Xu und Iseki 2015] Xu, Peng ; Iseki, Toshio: Multi-Resolution Analysisof Main Engine Power Fluctuations in Waves / Tokyo University of MarineScience and Technology. 2015. � Forschungsbericht

analysis of ship responses by means of wavelet transforms

Documents