spectral analysis in engineering, concepts and case studies

Spectral Analysis in Engineering

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Spectral Analysis in Engineering Concepts and Cases

Grant E Hearn Andrew V Metcalfe University of Newcastle upon Tyne

ELSEVIER

Elsevier Ltd. Linacre House, Jordan Hill, Oxford OX2 8DP 200 Wheeler Road, Burlington, MA 01803

Transferred to digital printing 2004 �9 1995 Grant E Hearn and Andrew V Metcalfe

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Whilst the advice and information in this book is believed to be true and accurate at the date of going to press, neither the authors nor the publisher can accept any legal responsibility or liability for any errors or omissions that may be made.

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0 340 63171 6

1 2 3 4 5 95 96 97 98 99

Typeset in 10/12pt Times by Wearset, Boldon, Tyne and Wear.

Contents

About the authors

Preface

Notation and nomenclature

Why understand spectral analysis?

1.1 Introduction 1.2 Overview

Relationships between variables

2.1 Introduction 2.2 Discrete bivariate distributions

2.2.1 Modelling discrete bivariate populations 2.2.2 Expectation for discrete distributions

2.3 Continuous bivariate distributions 2.3.1 2.3.2

2.3.3 2.3.4

2.4 2.5 2.6

2.7 2.8

Modelling continuous bivariate distributions Justification and geometric interpretation for continuous conditional densities Sample correlation for continuous variables Expectation for continuous distributions

Linear functions of random variables Bivariate normal distribution Confidence intervals for population correlation coefficient Multivariate normal distribution Exercises

3 Time varying signals

3.1 Introduction 3.2 Why study time series? 3.3 Estimation of seasonal effects and trends

3.3.1 Moving average method 3.3.2 Standardizing method 3.3.3 Multiple regression method 3.3.4 Estimation of trend

3.4 Moments of a discrete random process 3.4.1 The ensemble

xi

. o

Xll

8 9

10 12 15 18

20 20 24 27 31

35 36 36

39

39 40 41 42 42 43 43 47 47

vi Contents

3.4.2 Moments of a discrete random process 3.5 Stationarity and ergodicity

3.5.1 Stationarity 3.5.2 Ergodicity

3.6 ARIMA models for discrete random processes 3.6.1 Discrete white noise (DWN) 3.6.2 Random walk 3.6.3 Moving average processes 3.6.4 Autoregressive processes 3.6.5 ARIMA (p,d,q) processes 3.6.6 Gaussian and non-Gaussian processes 3.6.7 Relationship between MA and AR processes

3.7 Estimation of parameters of models for random processes 3.7.1 Estimation of the autocovariance function 3.7.2 Bias of the autocovariance function 3.7.3 Estimation of the autocorrelation function 3.7.4 Estimation of parameters in ARMA models 3.7.5 Determining the order of ARMA processes

3.8 Simulations 3.9 Further practical examples 3.10 Models for continuous time random processes

11

3.10.1 3.10.2 3.10.3 3.10.4 3.10.5 3.10.6 11

The Dirac delta Autocovariance function Estimation of the mean and autocovariance function Wiener process White noise in continuous time Linear processes

Exercises

47 49 49 49 51 51 51 51 53 57 60 61 62 62 62 64 64 70 73 74 83 83 83 84 84 85 86 87

Describing signals in terms of frequency

4.1 Introduction 4.2 Finite Fourier series

4.2.1 Fourier series for a finite discrete signal 4.2.2 Parseval's theorem 4.2.3 Leakage

4.3 Fourier series 4.4 The Fourier transform

4.4.1 Fourier transform 4.4.2 Generalized functions 4.4.3 Convolution integrals

4.5 Discrete Fourier transform 4.5.1 Discrete Fourier transform for an infinite sequence 4.5.2 Nyquist frequency 4.5.3 Convolution integral results for infinite sequences 4.5.4 The discrete Fourier transform

4.6 Exercises

91

91 91 91 94 95 97

100 100 103 104 105 105 106 106 107 108

Contents vii

5 Frequency representation of random signals

5.1 Introduction 5.2 Definition of the spectrum of a random process

5.2.1 The periodogram 5.2.2 The relationship between the sample spectrum

and autocovariance function 5.2.3 The spectrum of a random process

5.3 Estimation of the spectrum from the sample autocovariance function 5.3.1 The need for smoothing 5.3.2 Smoothed autocovariance based spectral estimators 5.3.3 Alternative lag windows 5.3.4 Comparing windows 5.3.5 Confidence intervals for the spectrum

5.4 Estimation of the spectrum from the periodogram 5.4.1 Smoothing the periodogram 5.4.2 Segment averaging 5.4.3 Use of the fast Fourier transform to estimate the periodogram

5.5 High resolution spectral estimators 5.5.1 The maximum entropy method 5.5.2 The maximum likelihood method

5.6 Exercises

109

109 110 110

111 114 115 115 119 121 123 127 128 128 129 130 136 136 138 141

6 Identifying system relationships from measurements 6.1 Introduction 6.2 Discrete processes

6.2.1 Generalization of the covariance concept 6.2.2 Cross-spectrum

6.3 Linear dynamic systems 6.4 Application of cross-spectral concepts 6.5 Estimation of cross-spectral functions

6.5.1 Estimating cross-correlograms and spectra 6.5.2 Estimation of linear system transfer functions

6.6 Exercises

143

143 145 145 147 148 150 152 152 153 157

7 Some typical applications 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Introduction Calculating the sample autocovariance function Calculating the spectrum Calculating the response spectrum The spectrum and moving observers Calculation of significant responses Exercises

161

161 161 162 163 170 173 179

Wave directionality monitoring 8.1 Introduction 8.2 Background

184

184 184

viii Contents

8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11

The technical problem Reduction of the monitoring problem to a mathematical problem Application of the mathematical model The probe arrangements deployed Analysis of Loch Ness data MLM-based spectral analysis formulations Cross-spectral density simulation Simulation results and alternative probe management Final comments

9 Motions of moored structures in a seaway

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

9.11 9.12

Introduction Background Modelling moored structures Equations of motion Determination of time dependent wave force Evaluation of the quadratic transfer function (QTF) Simulation of a random sea Why the probabilistic method of simulation? Statistical analyses of the generated time series Sensitivity analysis of a moored tanker and a moored barge to integration time step Effects of wave damping on the surge motion Final comments

I0 Experimental measurement and time series acquisition

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Introduction Background Experimental facilities set-up Data collection and principles of analysis Six degrees-of-freedom SELSPOT motion analysis Data acquisition and SELSPOT calibration Practical aspects Some typical results Final comments

I I Experimental evaluation of wide band active vibration controllers

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Introduction Background Techniques for active vibration control Why use a spectral analyser? Experimental rig Some typical results Final comments

187 189 192 193 194 197 198 199 206

208

208 209 211 212 213 215 216 218 219

221 221 232

233

233 233 235 236 239 240 242 245 248

253

253 253 254 255 255 257 257

Contents ix

12 Hull roughness and ship resistance

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Introduction Background An introduction to surface metrology Process bandwidth Surface topography and fluid drag Measures of texture Filtering and filter assessment Final comments

Appendices

Appendix I:

Appendix II: Appendix III:

Appendix IV: Appendix V: Appendix VI:

Mathematics revision Arithmetic series Geometric series Harmonic series Taylor series Even and odd functions Complex numbers Generating pseudo-random numbers Impulse responses

Inflows to the Font reservoir Chi-square and F-distributions

Chi-square distributions F-distribution Relationship with the t-distribution Comparison of variances

The sampling theorem Wave tank data Sampling distribution of spectral estimators

Discrete white noise A linear stochastic process Smoothed spectral estimators

References

Further reading

Index

Answers to exercises are available on the Internet at" h ttp://www, ncl. ac. uk/~- ne ngm/publish/

261

261 261 262 263 264 267 272 273

274

274 274 274 275 275 276 277 279 280 283 285 285 286 287 287 289 291 292 292 292 293

294

299

303

About the authors

Professor Grant E. Hearn BSc MSc CMath FIMA CEng FRINA trained as an industrial mathematician, graduating from Bath University of Technology. He also studied at Sheffield University and was a Research Fellow in Applied Mathematics before joining Strathclyde University as the DTI Research Fellow in Naval Architecture. At BSRA he was promoted to Head of Mathematics Group and joined Newcastle University from BSRA. He has worked in the aircraft industry, telecommunications industry and glass industry before committing himself to ship and offshore related hydrodynamics and design. He is currently Professor of Hydrodynamics and Head of the Department of Marine Technology.

Dr Andrew Metcalfe BSc PhD CStat AFIMA is a Senior Lecturer in the Department of Engineering Mathematics which he joined in 1978. His current research involves the applications of statistical methods to engineering problems, but past work has included theoretical and practical investigation of active vibration controllers. He has considerable experience of acting as a statistical consultant to industry, through personal consultancies and Teaching Company schemes.

Preface

Twelve years ago, we suggested a course called 'Probability and Spectral Techniques' as part of a Science and Engineering Research Council programme of postgraduate training for engineers. We thought the ability to model wind and wave forces dynamically, rather than as a static wave, was an essential skill for modern engineering. Other important applications ranged from the design of vibration controllers to the measurement of surface finish. We thoroughly enjoyed giving the course, at the University of Newcastle upon Tyne, on several occasions, and hope the participants learnt as much as we did.

This book is based on the course notes, which have evolved over the years. Peter Gedling, who then worked for the British Ship Research Association, was a guest lecturer. His clear exposition, and physical insight, are the basis of the sections on the maximum entropy and maximum likelihood methods for estimating spectra. One of the engineers on the first course, John Medhurst, has contributed the case study on measuring ship hull roughness, and Bob Townsin gave us helpful comments on our presentation of this work. We thank them for permission to use their work. We also thank Ann Satow for organizing the courses so efficiently, Richard Carter for valuable library support, and Diane Sevenoaks for all her secretarial work.

The Newcastle short courses were followed by a special 'in-house' version for BP International in London. We thank the Company for taking such an interest, and Colin McFarlane, now Professor of Subsea Engineering at Strathclyde University, for arrang- ing this. Our own undergraduates were, and still are, taught about spectral analysis during their second year, and Professors Caldwell and Jeffrey, then our respective Heads of Departments, suggested we write a book on the subject. This we followed up, and we thank them for their support. We also thank John Roberts of Edward Arnold for his enthusiasm for the project.

In the first half of the book we mix the physical and mathematical development with practical applications. The reader needs no more than a knowledge of elementary calculus, and basic statistics and probability, to follow this material, which is accessible to undergraduates at any stage of their courses. We thank many people for the data and permission to base examples on their work. The later chapters describe case studies. These are valuable, because they demonstrate the type of thinking, and the sort of compromises that have to be made, to solve real engineering problems. We do not expect readers to be faced with identical challenges. What we do hope is that this book will help them solve new problems in their careers as engineers.

Preparing the text took longer than we expected, so we also have to thank two subsequent editors, David Mackin and Russell Parry, for their forbearance. We are also grateful to all our undergraduate students for their perceptive comments, to Brenda Robson for typing the manuscript, and to the production staff at Arnold.

Grant E. Hearn Newcastle upon Tyne

June 1994

Andrew V. Metcalfe

xi

Notation and nomenclature

A summary of the main symbols used in the text is given below. i

Sample Population (number of data N) (usually considered infinite)

.~ mean 6 ~ variance a 2 or Var 6r standard deviation tr scov covariance Cov r correlation p c(k) autocovariance ~,(k) r(k) autocorrelation p(k) C(to) spectrum F(00)

�9 Random variables and their values in a particular case, data, are usually distinguished by upper and lower case letters respectively, e.g. X and x. Upper case letters are also used for transforms of time signals, e.g. X(to) for the transform of x(t), but this should be clear from the context.

�9 The limits of summations are not explicitly given if the summation is clearly from l t o N .

�9 The equals sign is used for definitions, assignments, identities and equations. Which of these is meant should be clear from the context.

�9 g is usually gravitational acceleration. �9 The case studies often involve many parameters, and some of the notation used is

specific to particular case studies. �9 In the context of sampled signals, frequency to is dimensionless and expressed as

radians per sampling interval. If the sampling interval is 1 second, then 00 is in radians per second, whereas a sampling interval of 0.01 second implies that to is in units of 100 radians per second. In the physical context of many of the case studies to is in radians per second.

�9 For those not familiar with the motions of rigid bodies (aeroplanes or ships), the terms surge, sway and heave, and the terms roll, pitch and yaw, refer to oscillatory translations and rotations of the body, respectively.

�9 Water waves in the engineering context are designated: incident, scattered or diffracted, and radiated waves. An incident wave is the wave approaching the structure. The scattered, or diffracted, wave is the wave resulting from the interaction of the incident wave with a floating or fixed structure. The radiation waves are generated by the motions of the structure, and there is one radiation wave system for each degree of freedom. The incident and scattered waves provide the wave excitation forces and moments. The radiation waves provide the reactive forces and moments. Second order forces are designated drift forces or added resistance forces, according to the absence or presence of forward speed, respectively.

xii

Why understand spectral analysis?

I. I I n t r o d u c t i o n

If you can hear the difference between Chopin and Beethoven you are responding to the different frequency compositions of changes in air pressure. As well as our hearing, our sight is also very sensitive to the frequency, and amplitude, of signals. Visible light is electromagnetic radiation between frequencies of approximately 6 x 10 ~4 and 10 ~5 cycles per second. Within this frequency range our eyes can detect all the colours of the rainbow and their mixtures. A rainbow is a common and beautiful sight, and is readily explained as refraction of sunlight in water droplets. Since the amount of refraction is greater for shorter wavelengths (higher frequencies) the sunlight is split into its different components--a spectrum. Most other creatures are responsive to light and sound signals~bats are a good example of the latter. We are also sensitive, as far as our health is concerned, to frequencies of electromagnetic radiation that we cannot sense directly. For example, the higher frequencies, including X-rays and gamma rays, are known to be dangerous in anything but very small doses.

Physical systems are also highly sensitive to signal frequency, and we refer to such systems throughout this book. The techniques described here can be used to distinguish a repeating pattern in a signal from background noise. However, we concentrate on using these techniques to describe an average frequency composition of non-deterministic disturbances, such as wind gusts and wave motion. Anyone who doubts the need to understand such phenomena should think about the collapse of the Tacoma Narrows Bridge. In 1940, four months after the bridge opened, a mild gale set up resonant vibrations along the half-mile centre span of the bridge, which caused it to collapse within a few hours. However, this was not the first case of resonance causing a bridge to collapse. In 1831, a column of soldiers marched across the Broughton Bridge near Manchester, UK and set up a forced vibration whose frequency was close to a natural frequency of the bridge. The Broughton Bridge collapsed, and the advice to break step when marching across bridges is now traditional.

An oscillatory motion, under a retarding force proportional to the amount of displacement from an equilibrium position, is known as simple harmonic motion. Very many natural oscillations are well modelled in this way. Some examples are: a mass on a spring which is given an initial displacement and then released, a pendulum making small oscillations, and a spindly tree branch after a bird lands on it. The displacement of a body undergoing simple harmonic motion, plotted against time, is a sine curve (also known as a sinusoid or harmonic). The bird on the branch will also be subject to considerable natural damping, which reduces the amplitude of the oscillation quite rapidly. If this is modelled by assuming damping is proportional to velocity, then the resulting motion is a

2 Why understandspectralanalysis?

harmonic whose amplitude decays exponentially. The essential harmonic nature of the disturbance is retained. Another example is radio transmitters, which emit signals that are the sum of harmonic waves.

In spectroscopy, elements are excited at their natural frequency and emit harmonic radiation of a 'pure' colour. One reason for taking a harmonic as the standard shape for a wave in spectral analysis is its frequent natural occurrence. A harmonic signal is completely described by its amplitude, its frequency and, when it is being considered relative to some time origin or other signals, its phase. The most common (SI) scientific unit for frequency is now cycles per second, hertz, but old wireless sets are usually calibrated in terms of wavelength. (If f is the frequency in hertz the wavelength in metres is c/ f where c is the speed of light in m s-~; that is, approximately 3 x 108ms-~.) However, from a mathematical point of view, it is more convenient to use radians---one cycle is 27r radians~because the use of radians eliminates factors of 2zr from arguments of exponential functions in the formulae. A further remark about units is that not all waves occur in time. Measurement of surface roughness, from aeroplane runways to painted surfaces on ship hulls, is an important application of spectral analysis and the appropriate units are cycles (or radians) per unit length.

An alternative approach to describing a harmonic signal, and any other signal, is to give its value over time. In principle, this could be a continuous trace~as recorded by an oscilloscope~but it is now more usual to give sampled values. The sampling interval must be small compared with the time taken for a single cycle at the highest frequency present in the signal, so that we avoid ambiguity in our analyses. The discrete sequence resulting from the sampling process is known as a 'time series'. The time series can be analysed on a digital computer and is the starting point for the techniques described in this book. Electronic devices that sample a continuous signal are known as analogue-to- digital (A/D) converters, and speeds up to 250000 samples per second are usual on standard equipment. The physical systems described in this book have their natural frequencies well below 100 Hz and can be analysed with such equipment. The theoretical concepts are also relevant for specialist applications such as radio and radar. Although the sampling interval is of crucial importance, it is convenient to work in radians per sampling time unit and thereby avoid introducing the sampling interval explicitly into the formulae. The final results can always be rescaled into radians per second or hertz.

For some purposes, notably the design of digital controllers for systems ranging from inter-connected reservoirs to industrial robots, the time history is usually used. For other purposes a 'frequency domain' description, which excludes the phase information, may be more appropriate. This is provided by calculating a function of frequency, known as the 'spectrum'. A brief explanation of the spectrum is needed if one is to appreciate why it can be a more useful way of looking at data than the data's original time order.

Jean Baptiste Joseph Fourier (1768-1830) first investigated the possibilities of approximating functions by a sum of harmonic components. These ideas can be applied to a time-ordered sequence consisting of an even number (N) of data. If this 'time series' is plotted, datum against sample number, we have N points equally spaced in the horizontal direction but, in general, with no obvious pattern in the vertical direction. However, it is possible to construct a function, which is the sum of a constant and N/2 harmonics of frequencies 2"rr/N, 47r/N, 67r/N . . . . . (N/2)2zr/N radians per sampling interval, whose graph passes through all N points. This function requires a unique specification of the amplitudes and phases of the N/2 harmonics. Throughout this book the time series will be thought of as one of an infinite number that could have been

1.2 Overview 3

generated by some underlying random process. The precise form of the function is therefore irrelevant, particularly as the specific frequencies used depend only on the record length. The sample spectrum is calculated by averaging the squares of amplitudes of the harmonics over sensibly chosen frequency intervals, and it consists of contributions to the variance of the original time series over a continuous frequency range. The sample spectrum is an estimate of the unknown spectrum of the supposed underlying random process.

Thus, the spectrum highlights the frequency composition of a signal. This may be of scientific interest in itself, for example sunspot cycles and periodicities in other data obtained from space exploration. However, the main justification for spectral analysis is its practical importance, emphasized by examples throughout the text and the case studies which form the later part of the book.

Two applications, which demonstrate the possible advantages of calculating the spectrum, rather than relying on the original time series, are mentioned here. The first is 'signature analysis' of rotating machinery. A sudden change in the spectrum of a vibration signal from machinery can provide early warning of a breakdown. A policy of rectifying faults before catastrophic breakdowns contributes to safety and can result in considerable financial savings. If the change in the spectrum is a shift it might not be detected by simply monitoring the amplitude of the time series. However, an additional peak in the spectrum would be linked to an increase in variance of the time series, even though this would be more difficult to detect at an early stage.

The second example concerns the design of offshore structures, such as drilling platforms. These, and many other structures, can reasonably be modelled as l inear~at least within certain limits. A linear structure responds to a harmonic disturbance by vibrating at the same frequency. The amplitude of the induced vibration is proportional to the amplitude of the disturbance, and the constant of proportionality depends on the frequency. The frequencies at which the structure is most responsive are known as its natural frequencies, and these can be calculated theoretically from the design. If the spectra of typical seas are estimated, the structure can be designed so that its natural frequencies are distanced from likely peaks in the sea spectra, and its response can be predicted. It would, admittedly, be feasible to model the response of the structure to typical time series, but this would not give any insight into the design. If the response was excessive it would not be possible to decide which natural frequencies to try and move, unless the spectrum of the response signal was itself calculated. Even if the response appeared satisfactory, the proximity of a natural frequency to possible peaks in the sea spectra would go unnoticed. The design of a vehicle suspension for typical road or runway surfaces is a similar problem.

The objective of this introduction has been to show that, whilst the spectrum is calculated from a time history and cannot contain any additional information, it presents the frequency content very clearly. For many engineering and scientific purposes this is exactly what is required. This is not to deny the value of analyses in the time domain, which are also covered in their own right and as a prelude to spectral analysis.

1.2 Overview

This section provides a brief overview of the contents of the following chapters. Chapter 2, 'Relationships between variables', assumes some background knowledge of

4 Why understand spectralanalysis?

probability and statistics and concentrates on joint probability distributions. The ideas of covariance and correlation (which can be thought of as measures of linear association between random variables) are particularly relevant to what follows. A closely associated method is regression analysis, where we investigate the distribution of the random variable Y for fixed values of x; that is, the conditional distributions of Y. This is a widely used and useful technique.

Chapter 3 is entitled 'Time varying signals'. Throughout this book a time series is considered as a realization of some underlying random (stochastic) process. A full description of a random process is usually very complex and we will concentrate on the first and second moments, known as the mean and the autocovariance function. The latter plays an essential part in spectral analysis. It is particularly important to understand clearly the concepts of stationarity and ergodicity. A random process is second-order stationary if its mean and variance do not change with time, and the covariance depends only on the time lag between variables and not on absolute time. The 'ensemble' is the hypothetical population of all possible time series that might be produced by the underlying random process. A random process is ergodic if time averages tend to averages over the ensemble. Loosely speaking, ergodicity means that a sufficiently long stretch of one record will be representative of the ensemble. It is usual to define the ensemble so that the assumption of ergodicity is reasonalJle. In some situations it may be possible to obtain several realizations, in which case the assumption of ergodicity is open to investigation, although there is always the problem that each realization may not be sufficiently long. An example of this situation might be signals received from mobile pipeline scanners on different occasions. In many other circum- stances, such as river flow records, there may only be the one realization. Before carrying out a spectral analysis it is assumed that the underlying process is stationary. This may require some preprocessing of the data to remove any trends or seasonal variability. Non-stationarity can be checked to some extent by looking through the one time series for obvious trends or changes in variability.

Fourier analysis is covered in Chapter 4, 'Describing signals in terms of frequency'. A finite Fourier series is a finite sum of sine waves with periods and phases chosen so that it coincides exactly with a finite number of datum points. A Fourier series is an infinite sum of such waves which converges to a signal defined for all time on a finite interval. In both the above cases the Fourier series is periodic, with period equal to the time interval of the signal. The Fourier transform is obtained by considering the time interval in the (infinite) Fourier series tending to infinity. Finally, the discrete Fourier transform is defined for an infinite sequence. The complex forms of these results are much easier to handle algebraically. The various statements of Parseval's Theorem are crucial to the arguments that follow and the convolution results are also often used. In spectral analysis, the usual situation is that a sequence of data is available that can be considered a sample from a continuous signal. A potential pitfall is the phenomenon of aliasing. For example, with a sampling interval of 0.2 s a frequency of 4 Hz appears identical to one of frequency 1 Hz, yet a system may respond to a vibration of 1 Hz but be relatively unaffected by the higher frequency vibration. Once a signal is digitized, aliasing cannot be avoided if frequencies higher than the 'Nyquist frequency' are present. To avoid aliasing, the original continuous (analogue) signal must have the higher frequency components filtered out by electrical means, or the sampling interval must be chosen so that the Nyquist frequency is higher than any frequencies present in the signal.

Chapter 5, 'Frequency representation of random signals', begins with a justification of

1.2 Overview 5

the definition of the spectrum of a stochastic process as the Fourier transform of the autocovariance function. The following sections deal with estimation of the spectrum. As the sample size increases, more ordinates are estimated in the unsmoothed sample spectrum, but the variability of the estimates does not decrease. The fidelity increases but the stability does not. Therefore, some smoothing procedure is necessary. One approach is to consider dividing the record into sections, calculating the sample spectrum for each section and averaging the results. The more sections the record is divided into the more stable the estimate will become, but this will be at the expense of smoothing out peaks and troughs. This approach is shown to be a special case of smoothing a sample spectrum estimator by giving decreasing weight to the autocovariances as the lag increases. The weighting function is known as the lag window and leads to a smoothed spectral estimator. Different windows and the effect of bandwidth are discussed. This general approach is the simplest computational method of spectrum estimation. An alternative is to average neighbouring ordinates in the periodogram. This is made computationally easier by the fast Fourier transform algorithm (FFT), which is an efficient computational technique used to evaluate the discrete Fourier transform of the recorded data. A simple derivation of the FFT method is given. The ideas behind the more recent maximum entropy method (MEM) and maximum likelihood method (MLM) of estimating the spectrum are also described. These are valuable with short data sets and for certain spatial problems.

Up to this point all the analyses have been of a single time series. Most practical applications involve the relationships between the inputs and outputs of physical systems. In this book the emphasis is on linear systems. For a linear system, the response to a sum of inputs is the sum of the responses to the individual inputs. Furthermore, the response to a harmonic signal will be at the same frequency, with the amplitude multiplied by some factor, and a phase shift. Both the factor and the phase shift depend on the frequency, but are constant for any given frequency, and are summarized by the 'transfer function'. Linear models provide good approximations to many physical systems. Their theory is particularly important because they are also often used as 'local approximations' to the theoretically more complicated non-linear systems. Chapter 6, 'Identifying system relationships from measurements', includes the theory behind transfer functions and their estimation. This involves looking at two random processes, or time series, and investigating the relationship between them. Concepts such as cross-covariance, cross-spectra and coherency are natural extensions of the previous work. At this stage all the essential theory has been recovered.

Chapter 7, 'Some typical applications', includes all the detailed working required to arrive at a sample spectrum. With an estimate of the spectrum available and some knowledge of a structure's transfer functions the response spectra may be determined. This may then lead to assessment of the probability of the responses exceeding some design or operational threshold. Transformation of the spectrum from one form to another and the ideas of significant responses are introduced and applied to the design of offshore oil rigs.

In the first part of the book, examples have been used to highlight specific points associated with the development of the theory, and to illustrate the application of methods to relatively simple situations. The second part of the book deals with engineering case studies. In addressing engineering problems the convenient partitioning of knowledge into subjects, which form the basis of specific courses, is not always possible. This is because engineering problems do not generally present themselves as

6 Why understandspectra/analysis?

nicely posed questions limited to one particular area of knowledge. The purpose of the case studies presented in Chapters 8 to 12 is to give some insight into the application of spectral analysis to actual engineering problems, mostly tackled by the authors, and simultaneously to provide some appreciation of the variety of roles spectral analysis will play in different situations. In attempting such presentations there is the danger that each problem will demand so much explanation that the relevance of spectral analysis is lost, or its role cannot be appreciated because of the knowledge required to address other integral aspects of the problem. In each case study we shall therefore provide some general background on the engineering problem, identify a number of questions that must be answered via analysis to solve the problem, explain the role of spectral analysis in the solution and then proceed with the applications.

Because the problems solved are real, and the authors human, errors of judgement will have sometimes been made in the initial analysis. In these cases we explain how initial methods of analysis have been modified, or additional analyses undertaken, to quantify the errors or lack of resolution in the engineering quantities of interest. At the time it was carried out, much of this work was innovative. However, subjects develop, and the computing power now available may facilitate more sophisticated analyses. Even so, we think the solutions are still adequate answers to the problems posed. Furthermore, the solutions are included here because they provide insight into how to go about solving problems, not to provide the latest state of forward thinking regarding analysis methods pe r se.

The different case studies presented can be read in any order, depending upon the reader's needs and interest. For those involved in the various disciplines of Marine Technology the ordering of the marine related case studies is both logical and deliberate. However, the book is meant to be of general interest to all engineers. To provide case studies related to other specific disciplines would require us to obtain wider experiences. The lessons learnt in honestly reporting the case studies should be transferable to other situations.

The first case outlines the mathematical modelling required behind the design of a wave monitoring system. The system described was actually built and here we consider the use of the FFT method to estimate the spreading of the wave energy and the problems of resolutions which were overcome using the MLM approach. The second case study considers the simulation of moored offshore structures subject to random excitation and the associated problem of generating realizations representative of a specific spectral form and characteristics. The method was used to show that certain simplified hydrodynamic models for predicting low frequency damping are totally inappropriate for realistic simulations of moored structures. The third case study discusses an investigation of the low frequency damping forces of moored tankers and barges. This case study highlights the difficulties of extracting the required time series from the recorded data and the importance of including the analysis techniques as an integral part of the experimental design. These two studies also, rather more construc- tively, allowed judgements to be made about the appropriateness of the mooring lines' configuration and materials, and their ability to keep the structure on station even if one or two failed.

The fourth case study describes the use of a spectral analyser to investigate the performance of active vibration controllers on a test rig. The final case is concerned with the characterization of the roughness of paint surfaces on the outer hulls of ships. A moderate level of wetted hull roughness may significantly increase a shipowner's fuel bill

1.2 Overview 7

(by 10% or more) as a result of the increased resistance caused by the roughness. A considerable amount of research effort has been devoted to measuring and reducing roughness of ship paint finishes.

2 Relationships between variables

2. I In t roduct ion

We start by imagining that we are responsible for monitoring the quality of drinking water in a region. To do this we will need to fill bottles with samples of water and analyse the contents. Two questions arise immediately and neither is straightforward. The first is: what measurements should we make during our analyses? The second is: where should we fill our bottles? Some of the items we should measure are acidity, chemical content, including traces of lead or other metals, coloration and number of microbes. The point illustrated, and of principal interest in this chapter, is that we are interested in more than one variate for each bottle analysed. The answer to the second question is that the bottles must be filled according to some random sampling scheme. However, complications arise when we start to define the details of an appropriate random sampling scheme. We could begin by identifying all the kitchen mains supply taps in the region. The simplest procedure would be to number all the taps and use computer generated random numbers to select those from which the bottles would be filled on each occasion. Such a procedure is an example of a simple random sampling scheme. Throughout this book we will assume that samples have been obtained from such a scheme unless we state otherwise.

Use of simple random sampling in this case could be criticized because it might lead to all the bottles being filled at taps in a new estate. Although this is unlikely if the sample is large, and most unlikely to occur often in repeated monitoring, unlikely events do occur and such a sample would be unrepresentative in this respect. However, if we generate simple random samples until we obtain a sample we like, we destroy the random nature of the scheme. Randomization is essential so that the sampling can be seen to be 'fair', and as the basis for a measure of the accuracy of estimates. A solution to this dilemma is to divide all the taps into areas and then take simple random samples within each area. This is an example of a stratified random sampling scheme, with the areas forming the strata. It is not necessary for all taps to have the same probabilities of selection provided the results are weighted appropriately. The essential requirement is that they all have a known positive chance of selection.

It is easy to think of other examples where we are interested in more than one variate for each unit in a study. If we are manufacturing bonded razor blades we would take frequent random samples to control quality. For each blade we might measure the geometry~for example, the protrusion from the guard bar at both ends~the sharpness, and the number and depth of any microscopic notches in the profile. Whenever we have more than one variate we are considering multi-variate data. When there are only two variates we can use three-dimensional diagrams to illustrate ideas which generalize algebraically. In all situations we must be mindful of distinctions between populations

2.2 Discrete bivariate distributions 9

and samples, and between discrete and continuous variates. These distinctions will be illustrated by looking at some bivariate data sets in detail.

2.2 Discrete bivariate distributions

Example 2. I Table 2.1 provides an estimate of waterway quality for England and Wales in 1985. It has been taken from the 1987 edition of Social Trends, which is published by the Central Statistical Office. The entries in the table are the number of kilometres (km) of three types of waterway which satisfy certain criteria for quality.

Table 2.1 categories

i

River quality of waterways in England and Wales: estimates of km in each of 12

Classification of waterway

Classification Canals Freshwater Estuaries Total by quality rivers Good 955 26 500 1860 29 315 Fair 1271 8 463 652 10 386 Poor 240 3 315 129 3 684 Bad 31 619 89 739 Total 2497 38 897 2730 44 124

i i | ill

If we identify a 1 km stretch of water as a 'unit', we can associate two discrete variates with each unit. The variates are the 'waterway classification' and the 'pollutant classification'. If we add over the pollutant classification we have the distribution of waterway classification, given in the bottom line of the table. This is known as a marginal distribution. That is, the distribution of kilometres of waterway according to waterway type irrespective of the quality of the waterways. The term 'marginal distribution' thus represents a distribution in which the influence of one or more known variates has been summed out. The marginal distribution of pollutant classification is given by adding over the waterway classification. The result is shown in the right-hand column of the table. The entries for each of the 12 categories give the bivariate distribution. This is illustrated in Fig. 2.1.

The estimates of kilometres of waterway in each category are based on samples. We could use these estimates to postulate a model for the corresponding population. Whilst it is easy to refer to a 'corresponding population' we should give careful thought to its definition. In this case, the population could be all 1 km stretches of waterway in England and Wales. The variates of interest could be the waterway classification and the pollutant classification based on an average throughout 1985. In common with many other engineering examples there is a certain amount of subjectivity in defining the population. The definitions of quality are even more subjective. The populations we define may often be imaginary and infinite, for example, all the items that would be produced if a machine continues indefinitely on its present settings. We must remember that statistical inference is crucially based on the assumption that we have a random sample from the population. Sampling has been used in three respects when constructing Table 2.1. Firstly, not all 1 km stretches of waterway have been monitored. Thus, the results given are based on a

I0 Re/ationships between variables

Fig. 2.1 Bivariate distribution of waterway classification and water quality

sample of all the waterways in the two countries. Within each stretch of waterway chosen for investigation it is only possible to draw samples from spot locations at any one time. Finally, the samples were taken at some unstated points in time and not continuously throughout the year. There is a great deal of detail behind the construction of Table 2.1, and its soundness depends crucially on the sampling schemes used.

2.2.1 Modelling discrete bivariate populations Let X and Y be discrete random variables. The bivariate probability mass function is defined by

Pxv(X, Y) = Pr {X = x and Y = y } (2.1)

If we refer back to Example 2.1 we could define X as taking the values 1, 2 or 3 if a randomly selected 1 km stretch of waterway is a canal, a freshwater river or an estuary respectively. We could define Y as taking the values 1, 2, 3 or 4 according to the classification by pollutant. A probability mass function, which could provide a model for the population from which the data represented by Table 2.1 came, is given in Table 2.2. Each entry has been obtained by dividing the corresponding entry in Table 2.1 by the total number of kilometres of waterway, namely 44124, and rounding to three decimal places.

A line diagram for the probability mass function is illustrated in Fig. 2.2. For any bivariate probability mass function

E Z px~x,y) = 1 y x

and you can check that the numbers in Table 2.2 do satisfy this requirement.

2.2 Discrete bivariate distributions II

Table 2.2 Values of Pxy(X,y) for the water quality model

Type of waterway x Pollutant y 1 2 3 1 0.022 0.600 0.042 2 0.028 0.192 0.015 3 0.005 0.075 0.003 4 0.001 0.014 0.003 ,, , i , , ,

The marginal probability mass function of X is defined by

Px(x) = Z Pxv(x,y) (2.2) y

We can define the marginal probability mass function of Y in a similar fashion. We can also define the conditional probability mass functions of X given y or Y given x. The formal definition of the latter is

Pvl~(y]x) = P r { r = y l X = x}

Pr{ Y = y and X = x} Pxv(x,y) = = (2.3)

Pr{X = x} ex(x)

For our example, the marginal probability mass function for X is

ex(1) = 0.056, ex(2) = 0.881, ex(3) = 0.063

0.8 J l :

0.7 ._r

0.6 ~6 E 0.5

i O.4

E 0.3 o

~ 0.2

I; ~ o.1 ~

/ Waterway quality - Good . . . . : J J ,7 J

/ / / , / / / Fair _~l,,_l___ J~,"

/ /" / / .- poor/_____~c___ ~ c . . . . ~,-

/ / ,,

= a d / . . . . ~,~c___y_____/ . / / " _/" f

/1" , / I I I ( I 6 6 0.0 Canals Rivers Estuaries

Waterway classification Fig. 2.2 Bivariate probability mass function to model waterway classification and water quality


The marginal probability mass function for Y is

Pr(1) = 0.664, Pr(2) = 0.235, ey (3 ) = 0.083, ey (4 ) = 0.018

The conditional probability mass function for Y when x = 1 is

evl~(lll) = 0.393, P r l l ( 2 l 1) = 0.500, Pvl~(3l 1) = 0.089, evl~(411) = 0.018

We see that the conditional probability distribution for Y, when we restrict our attention to canals, is different from the distribution of Y when we consider all waterways. In other words, X and Y are dependent random variables.

In general, the random variables X and Y are independent if and only if

exr(x, y) = ex(x)er(y) (2.4)

This follows immediately from the definition of independence in probability theory. To show this we start from the basic definition introduced earlier, namely

Pxy(x,y) = Pr{X = x and Y = y}

If and only if the events X = x and Y = y are independent does Pr{X = x and Y = y} = P r { X = x} Pr{Y = y}, i.e. P r{X= x and Y = y} = Px(x)Pr(y).

An equivalent definition of independence is, X and Y are independent if and only if PYI~(YlX) = PY(Y) for any x. This follows from the formal definition of conditional probability.

2 . 2 . 2 E x p e c t a t i o n for d i s c r e t e d i s t r i b u t i o n s

Expectation is averaging in a population. An average value of a variate is the sum of the values it takes for each item in a sample, or population, divided by the total number of items. In the case of infinite populations this requires careful interpretation. For the water quality example, a meaningful statistic could be the average quality of the water. The water quality categories can be thought of as being on a scale numbered somewhat arbitrarily, as 1, 2, 3 and 4 for good to bad respectively. Turning to Table 2.1 we have 29315 km of water of category 1 which can be thought of as 29315 km units of water quality 1. The average waterway classification would be:

1 x 29315 + 2 x 10386+ 3 x 3684 + 4 x 739 = 1.45

44124

An equivalent method of expressing the left-hand side of this equation is

29315 10386 3684 739 1 x ~ + 2 x ~ + 3 x ~ + 4 x

44124 44124 44124 44124

and this is the sum of the possible values of the variate multiplied by the relative frequencies of their occurrences. We now consider the model for the population of all waterways, namely Pxr(x,y). The average value of the random variable Y, which represented pollutant, is called the expected value of Y. The expected value of a random variable is the sum of the values it can take, multiplied by the probability of their occurrences. In this case we have

E[Y] = 1 x Pr(1) + 2 x Py(2) + 3 x ey(3) + 4 x ey(4)

= 1 x 0.664 + 2 x 0.235 + 3 x 0.083 + 4 x 0.018 = 1.45

2.2 Discrete bivariate distributions 13

To summarize, when we average in a sample we take the sum of the products of the values of the variate with their relative frequency. When we average in the population, relative frequencies are replaced by probabilities. This ties in with our common sense interpretation of relative frequencies as estimates of probabilities, and probabilities as some sort of limit of the relative frequencies as the sample size becomes very large. In this example, the sample average equals the population average because the population probabilities were inferred from the sample. Good agreement would, in any case, be expected with such a large sample.

We can formally define the expected value of an arbitrary function (g, say) of X and Y for a general bivariate probability mass function as

E[g(X,Y)] = ~, ~, g(x,y) Pxv(x ,y) (2.5) y x

We can easily verify that this definition is consistent when g is a function of only one of the variables, X say. Then

E[g(X) l = X ~, g(x) Pxy(x ,y) y x

= ~ g(x) Px(X), as required x

The expected value of Y is known as the mean of the distribution of Y, written as/~v, thus

~ r = E[Y] (2.6)

Another expected value of considerable interest is the variance of the distribution. The variance is the average value of the squared deviations of Y from its mean. It is written as o'2 and its units are the square of the units of Y. The square root of the variance is called the standard deviation, written as try, and has the units of Y. The reason why the variance is often used as a measure of spread of a distribution is that it is relatively easy to handle mathematically. Formally the variance is defined by

trEy = E [ ( Y - /~ v ) 2] (2.7)

We now define the covariance of X and Y, written as Cov(X, Y), as

Coy(X, Y) = E [ ( X - I ~ x ) ( r - ~v)] (2.8)

This quantity is of crucial importance in later chapters. Here, we should note that if X and Y are independent the covariance is zero. This is readily illustrated by formal application of the expectation operator, E. By definition

Cov(X, Y) = ~, 2 (x - g x ) ( Y - txr) Pxv(x ,y ) y x

and assuming X and Y are independent then

14 Re/adonshipsbetweenvariab/es

~--- E [ ( Y - [Jby )] E [ ( X - ~&x )]

By definition of /xr and/Xx both expected values are zero and the result follows. It is important to note that the converse is not true, i.e. it is not necessarily true that X

and Y are independent if their covariance is zero. It is straightforward to construct a simple discrete distribution to demonstrate this fact. You are asked to do this is in one of the exercises.

The covariance has dimensions of the product of the dimensions of X and Y. For many purposes it is more convenient to scale the covariance to make a dimensionless quantity known as the correlation. The correlation of X and Y is defined by

p(X, Y) = Cov(X, Y)/(trxtrr) (2.9)

and we shall prove later that

- I <~p(X, Y)<~ I

To make a sensible interpretation of Cov(X, Y) we require the values taken by X and Y to be on some meaningful ordered scale.

Example 2.2 At a certain stage of their training, sea cadets of a certain merchant shipping line take two practical and two theoretical examinations. Let X be the number of practical examinations passed and Y be the number of theoretical examinations passed. The probability distribution Pxy(x,y) based on extensive past records is given below.

Theoretical examinations passed, y

Practical examinations passed, x

0 1 2 0 0.0 0.0 0.I 1 0.0 0.I 0.I 2 0.I 0.2 0.4

By processing these probabilities, various concepts and definitions already introduced can be illustrated.

1 The probability that a cadet passes three or four examinations in all is equivalent to determining the probability that x + y I> 3. Formally, we may express this understanding of the situation as

2 2 Pr {passing 3 or 4 examinations} = Y. ~ Pxy(x,y)

x=0 v=3-x

This required probability equals

Pxv(1,2) + Pxv(2,1) + Pxv(2,2) = 0.7

2 The marginal distribution of x is given by summing the probabilities over the possible

Z3 Continuous bivariate distributions 15

y values for each x value in turn; that is, summing each column of the given probability table.

0.1 : x = 0 Thus Px(x) = 0.3 :x = 1

0.6 : x = 2

and the marginal distribution of y is given by

0.1 :y = 1 Pv(Y) = 0.2 :y = 1

0.7 : y = 2

It follows that X and Y are not independent since, for example

Pxv(O,O) = 0

from the original table, and

Px(0) Py(0) = 0.01

using the marginal distribution results. The statistics ~x and O2x are determined from I~x=XxPx(x) and O2x = X(x- /~)2 Px(x) with similar expressions for/~y and o2r. Application leads to

t tx = 1.5 tr2x = 0.45

and

/Xy = 1.6 tr2r = 0.44

By directly using Cov(X,Y) = XxXy(X- tZx)(y- ~Y)Pxv(x,y) the appropriate calculation is

(0--1.5)(0-1.6)(0.0) + . . . + (2-1.5)(1-1.6)(0.1) + (2-1.5)(2-1.6)(0.4) = 0.1

Using the previously calculated variances we have

p(X, Y) = 0.1/[V'(0.45)V'(0.44)] = 0.22

The positive value of p indicates a tendency for higher X to be associated with higher Y. We can also calculate conditional distributions. For example, the conditional distribution of Y, given that x is equal to 2, is

PYIx=2(y l 2) = Pxy(2,y)/ex(2) = Pxr(2, y)/0.6

These are simply the entries in the column corresponding to x = 2 scaled, by division, by the sum of that column.

2.3 Continuous bivariate distributions

In the previous section we introduced the concept of relative frequencies leading to probabilities as the sample size became large. Since only discrete values of variates were used, the sample line chart and population bivariate distribution consisted of vertical lines of varying heights. If a continuous range of values is used we can represent the


Fig. 2.3 (a) Three-dimensional histogram; (b) bivariate probability density function

sample by a set of contiguous blocks, i.e. a three-dimensional histogram. As the sample size becomes larger, we can take more 'blocks' for the histogram and at the same time have more observations in each block. We will assume that, as the sample size tends towards infinity, the histogram will tend towards the volume under a smooth surface. A smooth surface can be represented by an algebraic function of two variables, fxr(x,y). The function fxr in this case is a bivariate probability density function (PDF) and is a model for an infinite population. Figures 2.3(a) and (b) show a three-dimensional histogram and a bivariate PDF.

Example 2.3 The data in Table 2.3 are the depth (in ram) and duration (in minutes) of rainfall events from January 1958 until December 1972, at a site near Finningley in Yorkshire, UK. They have been provided by the Meteorological Office.

From the table you can observe that there is a tendency for the longer rainfall events to be associated with greater depths of rain. This is what we would expect on physical grounds, but we should notice that there is a considerable variation in the intensity of the rainfall. For these data it was not practical to take equal length class intervals and so this must be borne in mind when interpreting the data. We do this by considering relative frequency density rather than frequency as the heights of our histogram 'blocks'. Thus, in constructing a three-dimensional histogram, we erect a block with a height equal to the

2.3 Continuous bivariate distributions 17

Table 2.3 Numbers of rainfall events at Finningley classified by depth and duration , , , , , L ,,, , ,

Depth (0.01 mm)

0 50 100 150 200 300 500 750 1000 1500 2000 Time to to to to to to to to to to to (min) 50 100 150 200 300 500 750 1000 1500 2000 6000 Total

0-15 41 2 3 1 1 15-30 208 24 6 2 5 30-45 206 51 14 7 5 5 1 1 45-60 198 62 20 5 8 5 3 1 1 60-90 266 114 41 21 17 12 5 3 2 90-120 94 93 40 38 19 10 11 3 2

120-240 98 179 168 89 128 99 47 13 7 1 240-360 2 37 46 37 79 93 68 27 14 5 360-720 6 25 14 42 114 87 56 33 18 720-4320 1 4 8 16 22 15 Total 1113 568 363 214 305 342 229 120 81 40

i i i i

1 1 2

11 30 45

48 245 290 303 481 311 830 410 406

96 3420

i

proportion of observations in that cell divided by the area of the cell. In this case, the first cell is 15 minutes by 50 mm and so the height of the first block is 41/((3420)(15)(50)), which is 0.0000160, which can be written as 160 x 10 -7. It follows that the volume of the block is equal to the relative frequency of observations in that cell, and that the total volume of all the blocks equals unity. We can construct two-dimensional histograms for the (marginal) distributions of depth, and of duration, by dividing the relative frequencies by the width of the class interval. For example the height of the first rectangle in the depth distribution is given by 1113/((3420)(50)). The results of these calculations are presented in Table 2.4. The histograms are shown in Figs 2.4(a), (b) and (c).

Example 2.3 described a sample. The corresponding population could be defined as all possible rainfall events that could ever occur at Finningley if underlying climatic mechanisms do not change.

We model such infinite populations with bivariate PDFs and determine marginal

Table 2.4 Relative frequency densities (x 10 7) for the depth and duration of rainfall events at Finningley

Depth (0.01 mm)

0 50 100 150 200 300 500 750 1000 1500 2000 Time to to to to to to to to to to to Marginal (min) 50 100 150 200 300 500 750 1000 1500 2000 6000 densities

O- 15 160 8 12 4 2 9357 15-30 811 94 23 8 10 47758 30-45 803 199 55 27 10 5 1 56530 45-60 772 242 78 19 16 5 2 1 59064 60-90 518 222 80 41 17 6 2 1 46881 90-120 183 181 78 74 18 5 4 1 30312

120-240 48 87 82 43 31 12 5 2 20224 240-360 1 18 22 18 19 11 7 3 1 9990 360-720 1 4 2 3 5 3 2 1 3298 720-4320 78 Marginal dens i t ies 65088 33216 21228 12515 8918 5000 2678 1404 474 234 33


Fig. 2.4 (a) Durations of rainfall events at Finningley; (b) depths of rainfall events at Finningley (c) duration of rainfall events at Finningley

distributions by integration, rather than summing over a finite number of components. Similarly, the concept of conditional distributions can be generalized.

2.3.1 Modelling continuous bivariate distributions The following definitions for bivariate continuous distributions are simply generalizations of the discrete definitions. They differ only in so far as summation of a finite number of terms is replaced by integration. They can be extended to cases of more than two variables. A bivariate cumulative distribution function (CDF) for two continuous random variables, X and Y, is defined by

Fxv(x,y) = Pr{X~<x and Y<-y} (2.10)

The corresponding bivariate probability density function (PDF) is the function fxy(x,y), which satisfies the relationship

Y) belongs to a region A} = f ( f x r ( x , y ) dx dr Pr {(x, J J A

The CDF and PDF are therefore related by


Fig. 2.4 (cont.)

02Fxv(x,y) f xr(x, y) = Ox Oy (2.11)

Any bivariate PDF must also satisfy

and

(i) fxr(x,y)>~O for all x and y,

(ii) / f f xr(x, y) dx dy = l

The probability density function for X alone is

fx(x) = j f(x,y) dy (2.12)

and is called the marginal probability density Junction. The marginal PDF of Y is defined similarly. The conditional probability density junction of Y given that X = x is

20 Re/adonships between variables

Fig. 2.5 Probability that Xwithin 8x of xp is shaded

f vlx(Y Ix) = f xr(x, y) / f x(x) (2.13)

The conditional PDF of X given that Y = y is similarly defined. The random variables X and Y are said to be statistically independent if, and only if

f x~x , y) = f x(x)f e(Y) for all x and y

Whilst all these definitions are just the continuous analogues of those for discrete variables, we will justify the definition of the conditional PDF directly and consider its geometric interpretation.

2.3.2 Justification and geometric interpretation for continuous conditional densities

We start by considering Pr{Y within ~y of YIX within ~x of x)

Pr { Y within 6y of Y and X within ~x of x}

Pr {X within ~x of x }

It can be seen from Fig. 2.5 that the denominator on the right-hand side is approximately fx(x) times (28x), since we consider X is within 8x of x. If we use similar approximations for the other two terms we obtain

fxr(x,y)(46x6y) f vlx(y l X within 6x of x)(Z6y) =

fx(x)(26x)

The approximation becomes exact as 6x and 6y tend to zero. Letting 6x and 6y tend to zero gives us the definition

fxr(x,Y) frl~(YlX) = &(x)

It should be noted that fVlx is a scaled section of the joint PDF of X and Y, cut parallel to the (y,z)-plane through the point x, as shown in Fig. 2.6. The scale factor is [fx(x)] -~.

2,3.3 Sample correlation for continuous variables The data given in Table 2.5 are the compressive strengths (N mm -2) and cement content (per cent by weight) for 24 concrete paving blocks. These blocks were a random sample from a large consignment of nearly 300000 paving blocks. The data pairs are plotted in Fig. 2.7.


Fig. 2.6 Section of joint PDF cut through xp parallel to the yz-plane

Table 2.5 Compressive strength and percentage cement content of paving blocks i. iii ,, i i

Compressive Cement Compressive Cement strength content strength content (N mm -2) (%) (N mm -2) (%)

38.4 14.6 40.5 13.3 75.8 14.6 48.8 13.3 40.0 13.3 35.8 14.8 38.0 15.1 56.8 15.3 60.3 18.7 37.2 14.0 70.6 18.8 48.8 14.3 63.6 15.2 52.0 13.5 59.8 14.6 50.1 13.7 71.0 18.9 74.6 15.1 70.6 18.0 73.6 16.1 48.8 13.3 43.8 16.8 29.3 14.6 60.4 14.6

,. i , | , ,. ,

In Fig. 2.7 the 24 data pairs are represented by

(Xi, Yi) i = 1 . . . . . 24

where xi are the compressive strengths and Yi are the percentage cement contents. The lines x = ,f and y = )~, also shown in Fig. 2.7, divide the area into four quadrants. We can

19.0

18.0 -

~ 1 6 . 0 -

14.0 - �9 O 0

�9 f

g o

~ _ _ _ _ _ _ J . . . . . . . . J _ . . . . .

o~ ID I

i

I I 11 I ! I 30 40 50 ,~ 60 70 80

Compressive strength (N mm -2) Fig, 2,7 Strengths and cement contents of 24 paving blocks

22 Re/ationships between variables

see that, within the range of data available, there is a tendency for larger values of cement content to be associated with larger values of compressive strength. We will explain the sample correlation as a measure of linear association.

We start with the sum

24

E (Xi- .f)(Yi- Y ) i=l

Every point in the upper right-hand quadrant makes a positive contribution to this sum, because xi exceeds ~ and Yi exceeds y and so the product (x i -x) (y i -Y) is positive. Points in the lower left-hand quadrant also make positive contributions to the sum because the products are of two negative quantities. Similar arguments show that points in the upper left-hand and lower right-hand quadrants make negative contributions to the sum. For our data, the sum is 307.23. This reflects the fact that most of the points are in the upper right-hand and lower left-hand quadrants. In general, if there is a tendency for y to increase when x increases, a majority of the points will be in those two quadrants. If there is a tendency for y to decrease as x increases, the majority of the points will be in the upper left-hand and lower right-hand quadrants leading to a 'substantial' negative sum. If there is no association, the points will be scattered about all four quadrants in similar numbers and the summation could be positive or negative, but it will be 'small'. The sum

(Xi - : f ) ( Y i - fi ) i=l

is called the corrected sum of products. The term 'corrected' refers to the fact that the appropriate mean has been subtracted from the values of x and y. It is straightforward to provide an equivalent expression for the corrected sum of products, calculated from a sample of N pairs, which is quicker for hand calculation. By direct algebraic expansion we have

E (Xi- .~) ( Y i - Y) = E(x i - ,f)Yi- Y ~, ( x i - .~)

= ~xiYi--.f~Yi, since Y~ (xi-.~) = 0

: xiyi ( xi)l yiI,N However, great care must be taken when using such formulae, because of rounding errors when x and y have large means and relatively small standard deviations. Formulae of this type should not be programmed on a computer. For hand calculations, the rounding problem can be avoided by scaling x and y by subtracting a convenient constant that is fairly close to their means, e.g. 50 and 10 respectively for the paving block data. The corrected sum of products is of little direct use because it is dependent on the sample size. A more useful quantity is its average value, which is called the sample covariance. We will write this as 'scov' and define it by

scov(x,y) = E(xi - X)(yi-Y)/N (2.14)

The sample covariance is proportional to the choice of units of x and y. A non-


dimensional measure of linear association is the sample correlation, commonly written as r. This is obtained by scaling the deviations from the means by the appropriate sample standard deviations; thus, formally

where

\ e , / \ Cry

= scov (x, y)/(dr, ay)

(2.15)

= X (Yi - y)21N

and we will prove later that - 1 ~< r ~ < 1. We will find it slightly more convenient always to use the divisor N when calculating

sample averages. The unbiased estimator of tr 2, which has N - 1 as the divisor, is usually written as s 2.

For the paving block data, r = 0.53. A value of r that is as far away from 0.00 as 0.53 is unlikely to have occurred by chance in a sample of 24 if there is no association between 'strength' and 'cement content'. We must remember that statistical evidence of a linear association tells us nothing about causation. However, within the limits found for the paving blocks there are physical reasons for expecting an increase in cement to make them stronger. Particular care needs to be taken when interpreting correlation coefficients calculated from data which are taken over a period of time. As an example, consider the findings of the managers of a small chemical company who introduced quality control procedures at all stages of their operations. Over one year there was an improvement in the quality of all their products. There was also an increase in the amount of impurity in the raw material for a particular product. It was known that this impurity has no effect on the quality of the product, yet the correlation coefficient between the 'quality' and the 'amount of impurity', for data over this period, was nevertheless statistically significantly greater than zero!

We should also note that correlation is only a measure of linear association. Points scattered around the vertex of the parabola in Fig. 2.8 have a near zero correlation despite the obvious pattern.

Fig. 2.8 Data clustered around the vertex of a parabola with near zero correlation

24 Relationships between variab/es

2.3.4 Expectation for continuous distributions Let g be an arbitrary function of X and Y. If we had a sample of N pairs (xi, yi) from the bivariate population we could calculate an average value of g from

g = ~ g(xi, Yi)/N

Now, we will suppose that the data have been grouped into cells. Suppose that the x variate has been divided into 'a' categories and the y variate into 'b' categories. Then

b

g ~-- ~ E g(xi, yj) fx,,y/N i=1 j=1

where (xi,yj) are the coordinates of the mid-point of the cell and fx,,y~ is the number of data in that cell, that is, the frequency. The approximation arises because we assume that all the pairs in the cell are at the mid-point. The relative frequency fx,.y~/N is the volume of the three-dimensional histogram above the cell. As the sample size becomes large we assume this volume is equal to the product

fxv(xi,Yj) (cell area)

where fxr(xi,yj) is the value of the bivariate PDF at the mid-point of a cell. The approximate expression for g can be bracketed as

-" ~ ~ g(xi , yy )( f~,. y /N) i=l j=l

and replacing the volume of the histogram by the volume under the bivariate PDF we have

g = ~ ~ g(x,, yj) fxy(Xi)yj)Ax,Ayj i=] j=l

where z~r Ayj are the lengths of the sides of the cells. If we now consider the limit of the right-hand side, as the sample size tends to infinity, we have a natural definition of the expected value of g(X, Y), namely

"i E[g(X,Y)] = f g(x,y)fxr(x,y) dx dy (2.16)

These ideas are illustrated in Fig. 2.9. In particular we have

E[X] = J J xfxv(x,y) dx dy = Jxfx(x) dx = , x (2.17) - o o - o o

and similarly oo

E[(X- /x) 2] = f (x- txx)Zfx(x) dx 02X (2.18)

The mean and variance of Y are defined in the same way. The population covariance of X and Y is defined as

Cov(X, Y) = E[(X- p.,x)(Y- p,y)]


Fig. 2.9 The volume of the block is constructed to be fx,.y,,IN. This is approximately f(xi, y,)&xiAy,

If X and Y are independent then Cov(X, Y) equals zero. The proof follows immediately from the definitions, namely

oo cO

Cov(X, Y) = J J (x- IZx)(y- tXr)fxr(x,y) dx dy - - 0 o ~ o o

and assuming independence O0 OO

= J (x- tZx)fx(X) dx J (y - tzv)fv(Y) dy = (0)(0) = 0 - - o o - - o 0

However, if Cov(X, Y) equals zero, X and Y are not necessarily independent. As an example, we can consider a uniform distribution of probability over a unit circle. From the symmetry of the situation Cov(X, Y) equals zero, whereas it can be seen from the plan view in Fig. 2.10 that

Pr { 1/~/2 ~< X and 1/V'2 ~< Y } = 0, whereas

ar{ 1/V'2 ~< X }. Pr { 1/~/2 ~< r } = 0.008

Example 2.4 The random variables X and Y represent the amounts of cement and plaster sold by a small builders' merchant in a week. Their joint distribution is modelled by

fxv(x,y) = x(1 + 3y2)]4 0~<x~<2, 0<~y~< 1

where the implied units are tens of tonnes. The PDF is drawn in Fig. 2.11. To help decide on stock levels the owner wishes to determine the probability of selling less than 10 tonnes of cement and between 2.5 and 5 tonnes of plaster in the next week. Formally, we require

Pr{X~ < 1 and ~ ~< Y<~ �89


Fig. 2.10 Plane of uniform distribution PDF over unit circle

This is given by the integral 1/2 1

f f lx(l+3y2) dxdy -1 /4 0

which can be evaluated to give 23/512. The marginal PDFs of cement sales and plaster sales are

1

fx(x) = f tx(1 + 3y 2) dy = �89 0~<x~<2 0 2

re(Y) = f tx(1 + 3y 2) dx = �89 + 3y 2) 0 ~< y ~< 1 0

It follows that X and Y are independent. In general, X and Y are independent if and only if the joint PDF can be expressed as a product of a function of x only, with a function of y only, and the domains of the functions do not depend on x or y.

The next example demonstrates the necessity of the condition on the domains.

Fig. 2.11 Cement and plaster PDF for Example 2.4

2.4 Linear functions of random variables 27

v

X

Fig. 2.12 PDF of Example 2.5

Example 2.5

A specialist engineering works produces pistons for outboard motor engines. Each piston is cast and turned and then polished if necessary. The random variables X and Y represent the total labour time to make a piston and the labour time used for the casting and turning respectively. The joint PDF is

fxv(x,y) = 8xy O~x <-1, O<~ y<~x

The region over which the PDF is defined is no longer rectangular. The PDF is sketched in Fig. 2.12.

The marginal distributions are

fx(x) = i 8xy dy = 4x 3 0~<x<~ 1 0

and i

fy(y) = f 8xy dx = 4y(1 _y2) 0~<y~< 1 y

We see that the variables are not independent. For example

frlx(y[x) = 2Y I X 2 0 < ~ y < ~ x

2.4 Linear functions of random variables

Linear combinations of random variables are used in a wide variety of practical problems. We start by defining

W= aX+ bY (2.19)

where a and b are constants, and investigate the mean and variance of W. It turns out that

tXw = a/zx + bIJ, r (2.20)

and

O'{v = a2o'2x + b2crEr + 2ab Cov(X, Y) (2.21)


We will prove this result for the continuous case, the proof for the discrete case being the same with integration replaced by summation. By definition

E[W] = f f (ax + by)fxv(x,y) dx dy

= a f fx fxr (x ,y ) dx dy + b f fy fxy(x ,y ) dx dy

= a~x+ bp.r, and

o~w = Var[W] = f f[(ax + b y ) - (alxx+ b/zr)] 2 fxr(x,y) dx dy

= f f [ a ( x - I~x) + b(y - ~r)]2 fxv(x,y) dx dy

- aEff[(x - i~x)Efxy(x,y) dx dy 4- bEf f (y - i~r)2fxv(x,y) dx dy

+ 2 a b f f ( x - tXx) (y- tXr)fxy(x,y) dx dy

= aEo'Ex + bEtrEy + 2ab Cov(X, Y)

This relationship is the basis for a proof of the statement made earlier that

- 1 ~<p(X,Y)~ 1

Initially, we choose a = O'y a n d b = trx then

O'2w = O'2yO2x + O2xO2r + 2trxtrr, Cov(X, Y)

Now, any variance must be positive, or possibly zero, so tr2w must be non-negative. Setting 0 ~< crEw and dividing by cr2rtr2x implies that

- 1 ~< Cov(X, Y)/(trxtrr)

I f we now choose a = trr and b = - t r x we obtain the other side of the inequality, namely

Coy(X, Y)/(trxtry)<~ 1

The equivalent result for samples, - 1 ~< r ~ < 1, can be proved in a similar manner using sample averages.

If X and Y are independent then Cov(X, Y) equals zero and the result simplifies. The result also extends to more than two variables and this can be formally proved by an inductive argument. An important special case is when variables are independent and the covariance terms are all zero. A consequence is that if { X ~ , . . . , XN } is a random sample from a distribution with mean/Zx and variance tr2x then EXi has mean Nttx, variance No2x, and standard deviation (V'N)trx. Division by n shows that ,(" has a mean ~x and standard deviation trx/V'N. We should remember that random sampling justifies assuming independence.

We notice that these results do not tell us the form of the distributions. However, the normal distribution often comes to our aid. First, under very general conditions, a linear combination of normal random variables is itself normal. Second, the distribution of the sum of a large number of independent random variables, with finite variances, is approximately normal. This is a consequence of the Central Limit Theorem. It is of great practical importance because a large number can usually be interpreted as above 30 and, if the individual distributions are identical and approximately symmetric, above 10. The distribution of the mean is just a scaled distribution of the sum.

Figure 2.13 shows the distribution of the mean of 30 independent exponential random

2.4 Linear functions of random variables 29

l 1 2 r

Fig. 2.13 Exponential distribution, and near normal distribution of means of random samples of 30 from that distribution

variables, each with expected value arbitrarily chosen as 1, together with the distribution of a single exponential random variable.

Example 2.6 An engineering company makes motor bike engines. The nominal diameter of each cylinder is 50 mm. The tolerance limits for the clearance between a piston and cylinder are 1.00-3.00mm and the workshop manager aims for an average of 499 in 500 clearances to be within this range. The target diameter of pistons is 48 mm, and for cylinders it is 50 mm. The standard deviation of the turning process for the pistons can be maintained at 0.1 mm, and the manager has asked us to provide the corresponding standard deviation for the boring process. During assembly, pistons are randomly matched to cylinders.

We assume that the cylinder and piston diameters are normally distributed with means of 50 and 48 and standard deviations of or and 0.1 respectively. We also assume that they are independently distributed because they are matched at random. We let the random variable D represent the clearance. Then

o - - N((50-48) ,(0.1) 2 + o ~)

We require

Pr{1.00<~ D~<3.00} = 0.002

If we standardize D by subtracting its mean of 2 and dividing by its standard deviation of V'((0.1)2 + o -2) we can write,

P r{- 1/[V'((O. 1) 2 + o'2)1 ~< Z ~< 1/[%/((0.1) 2 + tr2)]} = 0.002

where Z is a standard normal random variable; that is, it has a mean and variance of 0 and 1 respectively. From tables of the normal distribution

Pr{-3 .09~ < Z~<3.09} = 0.002

Hence, 3.09= 1/V'((0.1)2+o a) and solving for o-leads to a value of 0.31. The recommended standard deviation for the boring process is therefore 0.31 mm.

We should check that the assumption of normality is reasonable by investigating any available data from the process.


Example 2.7

A large civil engineering company specializes in contract work in Africa and operates a helicopter that can carry ten passengers. The weights of male employees are distributed with a mean a 80 kg and a standard deviation of 13 kg. The weights of female employees are distributed with a mean of 60 kg and a standard deviation of 10 kg. We have been asked to find the probability that the total weight of ten passengers exceeds 800 kg, if they are selected at random from all employees of the company and are equally likely to be male or female. This problem is more awkward than the last because of the mixture of two distributions. We will tackle it in two parts. We first let M, W and Y represent the weights of men, women and a randomly selected passenger respectively, then

I~r = E[Y] = �89 + �89

= 70 kg

The variance of Y can be found from the following identity

o2y = E [ ( Y - /xv) 2] = E[(Y 2 - 21~yY + tzv2)]

= E [ Y 2] - 2 ~ y E [ Y ] + E[~ , ]

= E [ y 2 ] - 2 ~ , + ~2r = E[Y 2] - ~ ,

This is a useful general result, and is often rearranged as

EiY2] = O~y+ ~2y

Half the time Y will equal M and the other half it will equal W. Therefore, the average value of y2 is given by

E[Y 2] = �89 a] + �89 2]

Since

ElM 2] = O'2M +/X2M = 6569

and similarly

E[W2] = 3700

it follows that or, = 234.5. Notice that we cannot say

Y= �89189

even if the helicopter always carried equal numbers of male and female passengers. This is because Y represents a single passenger who must be a woman or a man.

To continue with the problem let T be the total weight, that is

I0

T = Z Y ~ i=1

Because simple random sampling is used it is reasonable to assume the Yi are independent and o'2r is ten times or,, which is 2345. The expected value of T is 700 provided only that men and women are equally likely to be selected.

Z5 Bivariate normal distribution 31

From the Central Limit Theorem the distribution of T is approximately normal and therefore

Pr{800~<T} = P r { 800-700V'2345 ~<Z}

and the required probability is

Pr{2.06~ < Z} - 0.02

Example 2.8 An engineering consultancy has provided a control system for a chemical plant. There are two flow meters in pipes leading into a reactor. On average, both meters provide correct flow readings, but they are subject to errors with mean 0 and standard deviations tr. The correlation between the errors in the two readings is 0.2. The standard deviation of the error of the sum of the two flows is needed to give limits of accuracy for the total liquid flowing into the reactor.

Let X and Y be the two readings. The sum of the two flows is S, where

S = X + Y

Using the main result of this section

tr2s = tr2x + tr2r + 2 Cov(X, Y )

By definition

Cov(X, Y) = PxYtrXtrY

and hence

tr2s = o 2 + o 2 + 2(0.2)02

and or s = 1.55tr

2.5 Bivariate normal distribution

In the previous examples we were able to reduce the problem to the distribution of a single normal random variable. It is not always appropriate to do this, so we introduce the PDF of the bivariate normal distribution

e x p [- 2(1 - 19 2) [ \ t rx / o ' r \ t ry

f xr(x , y ) = 2 r - p 2 ) 1 / 2 (2.22)

The distribution has five parameters/~x, trx, /zv, try and p. As the notation suggests, these are the means and standard deviations of the marginal distributions of X and Y and the correlation of X and Y respectively. We should note that if p in the bivariate normal distribution equals zero then X and Y are independent, and remember that this is not true in general. To study the effect of p on the shape of the distribution it is convenient to choose/Xx,/.tr as zero and trx, try as one. The effect of changing p in this case is shown in Fig. 2.14.

32 Re/adonships between variables

Fig. 2.14 Perspective drawings of standardized bivariate normal density function

Z5 Bivariate normal distribution 33

Fig. 2.14 (cont.)

The conditional distribution of Y for a fixed value of x is normal with mean

E[YIx] = t r+

and variance 0-2r(1 -p2) . You should note that the variance does not depend on the value of x. We will prove this result for the standardized normal distribution. The argument for the general case is identical, but it tends to be obscured by the algebra. The PDF of the standardized bivariate normal distribution is

1 (X 2 -- 2pxy + y2)] exp - 2(1 - p 2 )

fxy(X, y) = ~'(1 - p 2) 1/2 (2.23)

From the definition, and with a little algebra [ 1 ] | | exp - (y -- pX) 2

f 1 _�89 f 2 ( 1 - p 2) fx(x) = -| dy = (2~r)1/2 e _~ [(2~r)(1 _ pE)]U2 dy

The integral in the last expression equals one because the integrand is a normal PDF with mean px and variance (1 - p2) and the limits of integration are from minus infinity to plus infinity. At this stage we have verified that the marginal distribution

1 _~x~ Ix(x) = ( 2 , 0 u 2 e


is normal. We now use the definition of a conditional PDF to obtain the required result, thus

1 f x v ( x , y ) exp [ - 2 ( 1 - p2 ) (y -px )2 ]

fYIAylx) - - fx(x) [(27r)(1 - p2)] u2

This is the PDF of a normal distribution with mean px and variance ( 1 - p2). Returning to the general case, if you substitute (x-/~x)&rx for x and the equivalent

expression for y in the relation y = px, the conditional distribution of Y for a fixed value of x is normal with mean

E[YIx] =/a,v+ p(o'v/O'x)(X- tZx) (2.24)

If we plot the expected value of y against a given x we have the regression line of y on X, that is

Y = ~ v + P(O'Y/OrX)(X-- P'X) (2.25)

The regression line of x on y is

x = ~ x + p( trx/ trv)(y - I~y) (2.26)

A typical pair of regression lines, together with contours of the bivariate normal distribution from which they are obtained, are shown in Fig. 2.15, and the two lines are different unless p equals one. We should also notice that since:

(i) fvlx(Y[X) is a scaled section of the joint PDF cut through x parallel to the ( y - z ) plane (Section 2.3.2);

(ii) fvlx is a normal PDF (proved above); and (iii) the mean and mode of a normal distribution are the same, i.e. f(/z) is the maximum

height off; then

the locus of the mean values of the distributions f r l x will lie along the highest points of the sections cut parallel to the (y - z)-plane. That is, the regression line of y on x joins the

ax x = Px + P ~ (Y - PY)

. . . . . . . . . . . . .

~ (X- gx) = IJv+ PZx x

X

Fig. 2.15 Regression lines for a bivariate normal distribution

Z6 Confidence intervals for population correlation coefficient 35

points at which lines ruled parallel to the y-axis are tangential to the elliptical contours. A similar argument applies to the regression line of x on y. Either this geometric argument, or the fact that p is less than one in absolute value, justifies drawing the regression line of y on x with a smaller slope than the major axis of the ellipse and the regression line of x on y with a steeper slope. This phenomenon is known as regression towards the mean. For example, a distillery sells bottles of whisky with declared contents of 750 ml. For quality control purposes it is much easier to weigh full bottles than to break their seal and measure the volume of their contents. The weights of full bottles and the volumes of their contents have a bivariate normal distribution with a correlation of 0.7. The main reason why the correlation is less than I is that the weights of empty bottles vary independently of the volumes of whisky dispensed. A sub-population of the full bottles are those whose volumes are about two standard deviations below the mean. The average weight of these full bottles will not be as far below the mean weight of all full bottles as two standard deviations. This is because the other major factor influencing it, the average weight of the empty bottles, is independent of the volume. This phenomenon of regression towards the mean is a geometric property of a bivariate normal distribution. It does not imply that the members of the population will tend to become identical as the bottling process continues!

2.6 Confidence intervals for population correlation coefficient

Confidence intervals for p in a bivariate normal distribution can be constructed by using the variable

which is approximately normally distributed with a mean of �89 and variance ( n - 3 ) -l where n is the sample size. This is known as Fisher's transformation of the sample correlation coefficient.

Example 2.9

Measurements of chromium and nickel content were made on a sample of ten rocks taken from the lava fields of a volcano. The correlation coefficient r of the ten data pairs was -0.606. It follows that §

�89 ln\i _ r = -0 .702

and a 95% confidence interval for

�89 1 - 0 is

-0 .7o2 • (1.96)/xJ7 which is

[-1.44, 0.039l


If we are 95% confident that �89 In (l_-~p) is between -1.44 and 0.039 we are 95% confident that p is between

[-0.89, 0.04]

The confidence interval is wide because it is based on a small sample.

2.7 Multivariate normal distribution

The ideas of this chapter can be extended to multivariate distributions. The multivariate distribution is commonly used and will be referred to later. One reason for its importance is that even if data cannot reasonably be assumed to be normally distributed, it may be possible to find transformations of the data that can be normally distributed.

The p-dimensional random variable X has a multivariate normal distribution if its joint PDF is of the form

f(x) = exp[ - �89 ~)T(2)-I(x-- t3,)] (2ar)~2lxl 1'2

(2.27)

where x, ~ are (p x 1) matrices and E is a (p x p) symmetric positive definite matrix. The matrix g is the mean of the distribution, it contains the means of the p marginal distributions. The matrix ~; goes by several names, the dispersion matrix, the variance- covariance matrix or the covariance matrix. The diagonal terms are the variances and the off-diagonal terms the covariances. It is instructive to verify that putting p equal to two does result in the bivariate normal distribution.

2.8 Exercises

Limestone in a certain area can be classified by texture and colour. The proportions of specimens in nine possible categories, based on extensive past records, are shown below.

t , , ,

Coloor

Texture Light Medium Dark

11

Fine 0.03 0.17 0.09 Medium 0.04 0.19 0.10 Coarse 0.18 0.13 0.09

, ,, , , , , , , , , , ,

Let X and Y represent the colour and texture of a randomly selected specimen and code light, medium, dark and fine, medium, coarse as 1, 2, 3.

(i) (ii)

(iii)

Find the marginal distribution of X. Find the conditional distribution of Y given X equals 3. Are X and Y independent?

Consider the plan of bivariate discrete probability mass function in Fig. 2.16. Assume all the spikes have equal height. What must this height be? Demonstrate that X and Y have a zero correlation but are not independent.

2.8 Exercises 37

=

Y

1

1 x

Fig. 2.16 Plan of bivariate discrete uniform probability mass function.

11

1

1

1

0

The random variables X and Y represent the proportions of an aeroplane's passenger and freight payloads used on a flight. They have the joint PDF

f ( x , y ) = cxy 0 < x < l , 0 < y < l , and c is constant

(i) (ii)

(iii)

Find the value of c that makes f (x ,y) a probability density function. Write down the marginal distribution of X. Find the probability that more than half of both the passenger and freight payloads are used.

An electronic system has two different components in joint operation. Let X and Y denote the lifetimes of components of the first and second types respectively. The joint PDF is given by

f ( x , y ) = ~x exp[ - (x+y) /2 ] 0 < x , 0 < y

Find (i) Pr{1 < X a n d 1 < Y} and (ii) Pr {X+ Y<t} for any t>~0. Hence write down the PDF of the random variable T = X + Y. The following are ten pairs of measurements of the carbon content (x%) and the permeability index (y) of ten sinter mixtures

x 4.4 5.5 4.2 3.0 4.5 4.9 4.6 5.0 4.7 511 ~ y 12 14 18 35 23 29 16 12 18 21

Plot the data and calculate the correlation coefficient r. The life of a spacecraft component is normally distributed with a mean of 3000 hours and a standard deviation of 800 hours. A failed component can be replaced, immediately, by a new one during the mission. What is the probability that one spare will suffice for a mission of 3000 hours? The amount of impurity (X) in a diesel fuel oil varies from batch to batch. For each batch the amount of impurity (M) is measured. The measurement technique is subject to random error (E) which has a zero mean and is independent of the actual amount of impurity present. If the standard deviations of M and E are 25 ppm and 15 ppm respectively what is the standard deviation of X? What is the correlation between M and E?


8. An oil company owns a submersible survey vessel with two sets of depth measuring equipment. Let the readings given by these sets of equipment be denoted by X and Y. Now suppose X and Y both give unbiased estimates of the depth, the variance of Y is k 2 times the variance of X and they have a correlation p. Your aim is to find an expression, in terms of k and p, for the coefficients of the linear combination of X and Y which gives a minimum variance unbiased estimator of the depth. You should start by letting the linear combination, W say, equal aX + bY. What must a and b add to if W is to be unbiased? Hence eliminate one of them. Let tr 2 be the variance of X and write down the variance of W in terms of a 2, p and k. Use differentiation to find the minimum.

9. A company manufactures walls for timber-frame houses. A particular design is shown in Fig. 2.17. X~ and X2 are normally distributed with mean 3.00 m and

x

Fig. 2.17

r2

1

10.

standard deviation 0.20 m and Y, and Y2 are normally distributed with mean 2.50 and standard deviation 0.16 m. Find the probability that the difference in heights of two walls exceeds 0.50 m when components are selected at random. Assume the data in Question 5 are a random sample from a population. Calculate a 90% confidence interval for p in this population.



In the previous chapter the order of the data collection was not considered relevant and we were concerned with understanding or modelling the relationships between two or more variates. In this chapter, we shall be concerned with how the data vary with time and whether their order is crucial. Formally, we consider such data as a time series defined as a collection of observations made sequentially in time. We will treat time series as a realization of some underlying random (stochastic) process. It is useful to compare this with other statistical situations. The simplest case is when we are only interested in a single variable defined on each member of a population. If we take a random sample from that population we can model it by independent random variables from a univariate distribution. If we are interested in several variables defined on each member of a population we can model a random sample by independent random variables from a multivariate distribution. We are usually interested in relationships between these variables and we hope to estimate such relationships from our sample.

Our main interest in random processes is in the relationships between the variables at different times. Usually we only have one time series, which can be thought of as a sample of one from the underlying multivariate distribution, which represents the random process. However, a sample of one is not sufficient to estimate all the parameters of a completely general distribution. To progress, it is necessary to make drastic simplifying assumptions, for example, by assuming that changes in the mean of the distribution follow a straight line. It is also necessary to assume that if a time series is sufficiently long it will be representative of the underlying random process, which is the concept of ergodicity. Despite such simplifying assumptions, the models for random processes discussed in this chapter often provide good fits to time series and enable useful predictions to be made.

Models for random processes can be classified into four categories depending on whether time and the variable of interest are modelled as discrete or continuous. Some examples are given in Table 3.1.

In this book we are mainly concerned with continuous variables. Time series can also be classified into the same four categories as models for random processes, although the observed time series need not be in the same category as the model assumed to generate it. For example, a continuous signal from an accelerometer located at a drill head on an oil-rig would probably be passed through an analogue-to-digital converter and stored on a microcomputer as a 'continuous-variable discrete-time sequence'. It would be possible, but not necessary, to use a discrete time dynamic model for the oil rig.

There are many extensions of these ideas. Length is often treated in a similar way to

39


Table 3.1 Classification and examples of models for random processes , ,, ,

Time

Variable Discrete Continuous

Discrete A tropical country has a wet and a dry A model for the number of accidents season each year. The volume of occurring at a new design of road water in a reservoir at the end of each junction any time after it has been dry season is modelled on a scale built. (This is an example of a point from 0 to N in increments of 1. This process.) gives N+ 1 levels, ranging from empty to full. An (N + 1) by (N + 1) matrix of transition probabilities is postulated which takes into account water inputs, water use and evaporation. (This is an example of a Markov chain.)

Continuous A model for monthly rainfall at a selected location in England.

A continuous-time dynamic model for an offshore oil rig operating in heavy sea conditions. One variable of interest might be the resultant vertical displacement at the drill head.

time. Examples include roughness measurements along lines drawn on surfaces, ranging from aircraft runways to paint films. Thinking about length naturally leads us to consider areas and volumes, and it is then usual to talk of spatial series rather than time series. Another common, and extremely useful, extension of elementary ideas is to consider several variables over time.

3.2 Why study time series? To understand any physical phenomenon we must observe it and record our data. As engineers or scientists we will be interested in identifying the random process that produced a given time series, rather than the series as a historical record. Whilst exact knowledge of such a process is unattainable we should be able to choose a suitable form of model for it and then estimate its parameters. The spectrum, which is the main subject of this book, can be considered as one of the parameters of the process. Some typical physical situations where such analyses are used include the following.

(i)

(ii)

The quality of the paint finish on a ship's hull has a significant effect on skin friction and hence fuel consumption. The roughness of the paint film is measured by drawing a stylus across the surface and monitoring its vertical movement. The resulting series is a time series with 'time' replaced by length. Parameters of the underlying random process can be used to describe the surface texture. For many structures it is possible to postulate a linear dynamic model to describe their behaviour, but estimating values of the model parameters needs experimentation. Estimates of model parameters can be made from measurements made on disturbances applied to the structure and the corresponding responses. Such techniques are used to identify the vibration modes of structures such as bridges, ship bulkheads, and loudspeaker diaphragms.

3.3 Estimation of seasonal effects and trends 41

(iii)

(iv)

(v)

(vi)

It is useful to use models of random processes to provide short-term predictions in situations ranging from stock control to flood warning schemes. The parameters of the random process that models the environment, can be used to assess the integrity of the design of structures. For example, if we have knowledge of the wave frequency motion at a particular location, we can design an offshore structure to be compatible with this environment. In particular, we would require the natural frequencies of the structure to be away from expected peak frequencies of wave motion. The model of the random process can be used to generate much longer series of data, which can help in design decisions. Such a procedure might be used when

.

designing storm sewer networks, using hydrological models which need a rainfall input. A plausible range of parameter values must be expected. Parameters of the process may be of scientific interest in their own right, sun-spot cycles being one example. Empirical evidence may lead to useful theories which provide explanations of the phenomena. Tycho Brahe's observations on planetary motion were forerunners of Newton's theory of gravitation.

3.3 Estimation of seasonal effects and trends

It is sometimes useful to split a time series into constituent parts, which may include: a trend, oscillations about the trend such as seasonal effects, and a random component. Even if a series can be represented as a sum of such components it does not follow that they correspond to independently operating causal systems. As an example, we will look at some possible explanations for trends and seasonal effects in hydrological time series. Gradual natural or human-induced changes in the environment can produce trends in time series. A deposit of silt could produce changes in height measurements made at a specific location. Urbanization may produce changes in run off and even, if it is on a large scale, changes in precipitation. Agricultural practices can produce changes in watershed conditions which result in changes in stream-flow. Jumps in a time series may result from any sudden change in the environment such as closure of a new dam, starting to pump groundwater, forest fires and so on. More frivolous examples, which have nevertheless occurred without notification, are a change of position of a measuring pole and a change of units from degrees Fahrenheit to degrees Celsius! Annual cycles are often present in stream-flow, precipitation, evaporation, groundwater level, soil moisture deficit and other hydrological data. Variation within the week and within the day may be present in water use data, such as industrial, domestic or irrigation demands.

Some time series show marked seasonal fluctuations, for example, sales of ice cream, flow of water to reservoirs, and unemployment statistics. It is often useful to make adjustments for these predictable fluctuations and then look for any features of the deseasonalized series. The first step is to model the predictable seasonal variation in some way. For example, we can derive a factor by which data from the high season are increased and factors by which data from the lower seasons are decreased relative to the yearly average. These factors are often called seasonal indices.

Rather than give general formulae, three possible methods for the estimation of monthly 'seasonal' indices from monthly data x, will be demonstrated. You should be able to modify them so that they are applicable to time series with other times between observations.


3.3.1 M o v i n g ave rage m e t h o d

(i) The first step is to calculate a centred 12-month moving average; that is, to evaluate

Sm(xt) = (�89 6 + x t _ 5 -4;- . . . -Jc. xt+5 + �89

Thus, if the data run from January 1970 to December 1987, as they do in Example 3.1, the first 12-month average will be centred on July 1970 and the last will be centred on June 1987. The reasoning behind the formula given is that the average

( X t - 6 q - X t - 5 q- . �9 �9 +xt+5)/12 corresponds to the average value at time t-�89 and the average

(Xt_ 5 + Xt_ 4 " + ' . . . "~- Xt+ 6 ) / 1 2

would correspond to time t + �89 The formula given is the average of these two averages and corresponds to time t.

(ii) It is now necessary to decide whether any seasonal effect is better modelled by additive or multiplicative indices and so calculate either xt-Sm(xt) or xt/Sm(xt) respectively. The authors have found the multiplicative model more useful in applications because it also compensated for seasonal variation in the standard deviation. The choice must depend on the data and the context. If there is a strong trend, for example passenger miles booked with a small but expanding airline, the difference between assuming a fixed increase for summer bookings rather than a proportional increase is important. The proportional increase is the more plausible, but this could be checked by investigating the data.

(iii) If there are 18 years of data there will now be 17 estimates of the seasonal effect for each month. These are averaged to give a single estimate of the effect for each month. It will be necessary to adjust these 12 indices slightly so that they average 0 or 1 for the additive or multiplicative model respectively, as the constructions do not satisfy these common sense requirements identically.

(iv) The data can be 'deseasonalized' by subtracting or dividing by the appropriate index. That is, using a multiplicative model, the deseasonalized January 1970 datum is the original January 1970 datum divided by the January index, and so on. A plot of either the deseasonalized data or the 12-month moving average should indicate any trend. Fitting a trend is dealt with in Section 3.3.4.

This method makes no direct attempt to compensate for a seasonal variation in standard deviation. However, if the standard deviation is approximately proportional to the mean value and the multiplicative model is used, the deseasonalized data could reasonably be assumed to come from a process with a constant standard deviation.

3 .3 .2 S t a n d a r d i z i n g m e t h o d This method estimates a mean and standard deviation for each month, conditional on any trend having been removed. It requires many years of data for the 24 estimates to have acceptable precision, but a compromise is to smooth them. The procedure involves the following steps.

(i) Calculate the means for each calendar year and plot them. If appropriate, fit a trend curve and detrend the data as described in Section 3.3.4.


(ii)

(iii)

For each month, calculate the sample mean and standard deviation. Plot these values and, if it seems appropriate, draw a smooth curve through them and use 'smoothed' estimates read off from the curve. The data are deseasonalized by subtracting the appropriate mean and dividing by the appropriate standard deviation.

3.3.3 Multiple regression method The moving average method involves the estimation of 12 indices. The standardizing method involves the estimation of 24 indices, which is only practical for long series, unless the estimates are smoothed. If it is reasonable to suppose that monthly variation is part of a harmonic cycle we can use the following model

2~rt 27rt X, =/3o +/31 cos - ~ +/32 sin 12 + fl3t + Et

where E, represent random variations with zero mean. There are only two parameters, /3~ and/32, describing the seasonal effects with/33 allowing for a linear trend, which can be fitted simultaneously if required. The parameters can be fitted by least squares. We start by defining a function qJ which depends on four parameters rio . . . . . fi3, by

[ ( c o s (27rt)12 (27rt112 ] + )]2 I/J(~o, ill, ~2, fi3) -" ~ Xt-- ]~()d-/~l -1-/~2 sin\ fist

which is the sum of squared differences between the data and their predictions from the model with the unknown parameters /3,, . . . . . /33 replaced by specific numbers /3o, . . . . fi3. The least squares estimates of/3,, up to/33 are denoted by/3,, . . . . . fi3 and are the values of rio . . . . /33 that minimize qs. That is, rio . . . . . /33 are the values of rio . . . . , fi3 that satisfy the equations

0q, = 0 i = 0 , 1 , 2 , 3

As the model is linear in the parameters 13,, . . . . . /33 these equations are linear for rio . . . . . fi3. Standard statistical tests for multiple regression analysis are not strictly appropriate unless further assumptions about E, are made. The most crucial additional assumption is that the E, are independent of each other. In our applications, the E, represent the random process that generates the deseasonalized data, and we are particularly interested in relationships between the E,. The residuals e,, defined by

) ,12: are the deseasonalized data and can be interpreted as estimates of the values taken by E,.

This method does not allow for any seasonal variation in the standard deviation, it can be extended to allow for a quadratic or other trend curve by adding extra terms.

3.3.4 Estimation of trend Trend curves can be fitted to data by least squares although, for the reasons given in Section 3.3.3, standard statistical tests should not be relied upon. Examples of trend curves include the following.


(i) Polynomials such as

Xt = fl0 + fll t q- f12/2

The simplest case being a straight line.

(ii) Modified exponential

x t = a - b r t 0 <~ r; 0 <~ t

(iii) Logistic

xt = l / ( a + br t) O~r; O<~t

The least squares equations for the parameters a, b and r in (ii) and (iii) have to be solved iteratively. The estimated trend can be removed by subtracting the trend value from the observations. Forecasts can be made by extrapolating trend curves into the future and adjusting for seasonal effects. Unfortunately, there is often no logical basis for choosing a trend curve and several 'plausible' curves may lead to widely different forecasts. Linear extrapolation is often used for short-term forecasts, in the hope that changes in the trend will occur over longer time periods than those periods for which forecasts are required.

Example 3. I Monthly effective inf lows ( m 3 S -1) to the Font reservoir in Northumberland (Grid Reference NZ049938) from January 1909 until December 1980 have been made available by Northumbrian Water and are given in Appendix II. The data are the actual inflows from rivers during each month less an allowance for evaporation from the surface of the reservoir. The purposes of the reservoir are to supply local customers with at least 12000 m 3 day -~, which cannot be provided from alternative sources, and to maintain a minimum of flow of 2270 m 3 day-~ in the river downstream. If the reservoir is full, any excess inflow spills over and is wasted. If the level of water in the reservoir falls below the lowest draw-off pipe, it fails to fulfil the requirements. A typical problem for the water supply industry is to operate all its sources of water supply so as to minimize waste and the time over which minimum requirements are not met.

The 72-year time series for the Font reservoir is probably long enough to make useful comparisons between different operating schemes. However, the overall water supply system involves many inter-linked reservoirs, and other components, for which the available records are much shorter. One possible approach is to investigate different models for the Font series and select the most appropriate to fit to the shorter length records. These fitted models could then be used to generate synthetic series for 72 years. Any correlation between inflows to the different reservoirs at the same time also has to be estimated and used in the generation procedure. It is difficult to produce a set of rules for modelling complex systems and it is better to combine common sense with available statistical techniques.

For the purposes of this example we will restrict our attention to the single series for the Font reservoir, and investigate whether there is any evidence of a trend and compare some of the deseasonalizing procedures. The yearly means are plotted in Fig. 3.1. Fitting the least squares straight line through the data provides rather weak evidence of a negative trend, and there is no evidence of dependence between yearly means after the trend is removed.


.0 -

0.9

0.8

0.7

0.6

0.5

_. 0.4

0.3

0.2

0.1

0.0

�9 � 9 � 9 �9 �9 �9 �9 �9

�9 �9 �9 �9 �9 �9 �9 �9 00 �9 �9 �9 o�9

�9 O r e � 9 �9 �9 �9

�9 0 0 0 0 O 0 0 �9

I I I I I I I I i i t J t I I 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Year

Fig. 3,1 Yearly mean in f lows to Font reservoir

The fitted regression line is

yearly average effective inflow = 0.521-0.00131 (year-1908)

Using the usual results for regression analysis, the probability that the estimated coefficient exceeds 0.00131 in absolute value, if there is no trend, is 0.07. A possible explanation for a negative trend is that the area under pine forest has been increasing. There is no doubt that there is seasonal variation in the inflows. This can be seen clearly in a plot of the first six years of the series shown in Fig. 3.2. The monthly means and

1.8

1.6

1.4 A

7 (n 1.2

3= 1.0

0.6

0.4

0.2

0.0

�9 �9 o o o

O l �9 �9

�9 ~ o �9

~ o o ~ �9 OQo �9 �9 o � 9

i i 1 e l ! 1 o i ,, I I I T e i I Wl 5 10 15 20 25 30 35 40 45" 50 55 60 65 70

Months from January 1909 Fig. 3 .2 Month ly inf lows to Font reservoir 1 9 0 9 - 1 9 1 4


standard deviations are given in Table 3.2 correct to two decimal places, no estimate of trend has been removed. The estimated standard deviations tend to increase with the means but not in proportion to them. The net result is that the coefficients of variation (standard deviation/mean) appear to decrease as the means increase, see Table 3.2.

Table 3.2 Estimated means and standard deviations of monthly inflows to the Font reservoir for each month

, ,, , , , , , L,

Estimated Coefficient Estimated standard of

Month mean deviation variation

January 0.76 0.40 0.53 February 0.70 0.39 0.56 March 0.68 0.50 0.74 April 0.40 0.28 0.70 May 0.28 0.20 0.71 June 0.21 0.21 1.00 July 0.19 0.17 0.89 August 0.31 0.38 1.23 September 0.29 0.25 0.86 October 0.44 0.33 0.75 November 0.69 0.40 0.58 December 0.73 0.41 0.56

i i

The 12-month moving average multiplicative model yields the following estimated indices for the months of January to December 1.73, 1.53, 1.44, 0.88, 0.67, 0.49, 0.46, 0.70, 0.69, 1.01, 1.52, 1.62.

If a harmonic wave is fitted, the equation is

monthly inflow - 0.473 +0.265 cos\ /+0 .129 sin \-7~ ~

where t is the number of the month, starting at 1 for January 1909. Which of the three models is the most appropriate? For this long time series it is

probably best to estimate the means and standard deviations individually for each month. Referring to Table 3.2 the standard deviations are larger for the months with the higher means but not proportionately so. The first six years of the series, deseasonalized by standardization, are shown in Fig. 3.3. The multiplicative moving average method implicitly assumes a constant coefficient of variation and the harmonic wave method requires a constant variance if our deseasonalized series is to have a constant variance. The harmonic wave also assumes the means lie on a sinusoidal curve, which seems a reasonable approximation for these data. For smaller data sets the standardization method could be modified by assuming a relationship between the standard deviation and the mean. This would alleviate the requirement for estimating 12 near-independent variances. In conclusion, even after fitting several methods, it is difficult to select the 'most appropriate', and it would be even harder to do so a priori. Usually, the practical effects of the choice will be fairly small; however, if there are severe non-linearities in the system being simulated, several models might have to be tried, together with variations in their parameters.

5 . 0 0 -

2 . 5 0 - . . . .

lo 0.00

3.4 Moments of a discrete random process 47

Fig. 3 .3

3 4 4 6 9 4 1 678 :5

56 0 2 9 34 2 5 5 - 0 5 7 4 9 78 1

2 789 :5 567 2 8 0 1 1 8 1 6 9 3 O1

6 123

456 8 7 7 9 45

0

. . . . . I , I I I i I 0 . I0 20 30 40 50 60

Months from January 1909

Deseasonalized (standardized)monthly inflows to Font reservoir 1909-1913

3.4 Moments of a discrete random process 3.4.1 The ensemble The ensemble is the imaginary infinite population of all the time series that might be generated by an underlying random process. The observed time series is one particular realization of this random process. A discrete-time random process may be defined as an ordered set of random variables

{Xt} t = . . . - 2 , - 1 , 0, 1, 2, 3 , . . .

A continuous-time random process may be defined as a random function

{X(t)} -oo~<t~<oo

The expected value of the variable, or function, at time t is denoted by

E[Xt] or E[X(t)I and is an average over the ensemble at time t. This is illustrated in Figs 3.4(a) and (b). The usual situation is that there is only one time series available and hence only one observation on the random variable at time t. Direct estimation of E[Xt] would have to be made with the single observation xt and it would not be possible to estimate the variance of Xt, that is E [(Xt - E [Xt ])2].

We will restrict our attention to discrete time random processes until Section 3.10.

3.4.2 Moments of a discrete random process A full description of a stochastic process requires the specification of the joint probability distribution of (X t , , . . . ,Xt ,) for any set of times {tl . . . . ,t,,}. A less ambitious approach is to give the first and second moments. These do not uniquely specify the process, which is hardly surprising when we remember that the mean and variance of a univariate distribution do not specify an exact shape. If, in addition, we specify that the distributions are multivariate normal, the first and second moments do define a unique process, which is usually called Gaussian in the engineering literature. The first moment is the mean function

I~(t) = E[Xt] (3.1)


I S

/ #

/

]I

I

x(t) / . , . ,._ I v - , ~ t

I AJ y' , . , - . .

t /

P / (a)

/

/

to ,I x

x

. . . . . . I 0

I v

/

[3 / t ~ / 0 121 ~

/ 0 / 0

o ~/ o o

/ / / 0 /

/ X / X

/ X /

,/X X /

(b)

/ i

0 0 ~," 0

/ I

V / / V /

I / I V i

/ A i

r

r

V V

[3

t r

r

Fig. 3.4 (a) Ensemble averaging for continuous case; (b) ensemble averaging for discrete case

The second moment is the autocovariance function (acvf)

7(t , , t2) = E[(Xt, - tx(t, ))(Xt2- /x(t2))]

A special case of the acvf is the variance function

o~(t) = 7(t, t)

The autocorrelation function (acf) is

P ( t l , 12 ) = ) ' ( t , , l 2 )/(O'(t I )0"(/2 ))

(3.2)

(3.3)

3.5 Stationariry and ergodicity 49

Equations (3.2) and (3.3) are generalizations of Equations (2.8) and (2.9). Even restricting our enquiries to the first two moments is not a sufficient simplification, as we observed in Section 3.4.1. In practice, we usually assume that a time series is a realization of a random process that can easily be reduced to a stationary process. Stationarity is defined in the next section and the 'easy' reductions include deseasonalizing and removing any trend. If we have only one time series we have to assume ergodicity.

3.5 Stationarity and ergodicity 3.5 .1 S t a t i o n a r i t y

A stochastic process is strictly stationary if the joint distribution of

(x,,,... ,x,.) is the same as the joint distribution of

(x,,+k,..., x,.+k) for any set of times { q , . . . , t,, } and any value of k. As we are limiting our attention to the first two moments we will adopt a rather weaker notion of second-order stationarity. If we are prepared to assume a multivariate normal distribution, then the distributions are equivalent. A stochastic process is second-order stationary if

E[X,] =/x, a constant for all t (3.4)

and y(fi, t2) depends only on the difference between t2 and q , known as the lag and usually denoted by k. That is

Cov(X,, X,+k) = y(k), where k = t2 - tl (3.5)

and the acf becomes

p(k) = y(k)/y(O) (3.6)

It follows from the definition of p(k) that p(0) is 1 and p(-k) is equal to p(k). Also, as p(k) is a special case of a correlation coefficient it is less than or equal to 1 in absolute value. A plot of p(k) against k is called a correlogram. In summary, a random process is second-order stationary if its mean and variance do not change with time and if the covariance depends only on lag and not on absolute time. If a random process includes a trend or seasonal effects, or a change in variance, it is non-stationary. A plot of the data is an important first step in looking for non-stationarity in time series. Figures 3.5(a) and (b) show realizations from non-stationary random processes. The first includes an obvious trend, the second an obvious increase in variance with time.

3 . 5 . 2 E r g o d i c i t y

The mean function of a stochastic process is

Ix(t) = E[X,]

where the expectation refers to an average across the ensemble. If Ix(t) is a constant ~,


I

(a) t

�9 �9 t

(o)

Fig. 3.5 (a) An obvious trend with time" (b) increase in variance with time

the process is stationary in the mean. If a single realization of the process is available, the time average can be calculated. If the probability that the time average

X = Z X t / N

will be close to tz tends to one as N tends to infinity, then the process is ergodic in the mean. To understand this in physical terms, imagine a chemical factory with two discharge pipes for effluent, A and B. On any one day the probability that pollutant discharged from A exceeds a tolerance level is 0.1, and the probability that the pollutant discharged from B exceeds the same tolerance level is 0.3. An inspector only has resources to set up a permanent in-pipe monitoring system in one of the two pipes, which will be chosen with equal probability, 0.5. A random process is defined in the following way. The random variable Xt is defined as 0 if the pollutant is below the tolerance level and 1 if it is above the tolerance level at time t. Then {Xt } is a random process which is stationary in the mean. The mean is

E[X,] = 0.5{(0)(0.9) + (1)(0.1)} +0.5{(0)(0.7)+ (1)(0.3)}

=0 .2

A realization of the process is obtained by choosing one of the pipes at random and then monitoring the effluent from that pipe daily. The time average will tend towards 0.1 if pipe A is chosen and 0.3 if pipe B is chosen. In neither case will the time average tend towards the ensemble mean.

The concept of ergodicity arises naturally when considering situations involving prototypes. A portable drill for water bore-holes has been designed and it is thought that

3.6 ARIHA mode/s for discrete random processes 51

vibration may be a problem. Is a long record on a single prototype going to give reliable information on the general design? If we are restricted to one prototype we would have to assume ergodicity and analyse the data emphasizing the significance of the assumption. If we have more than one prototype, the assumption of ergodicity can be tested by comparing parameter estimates from long records from the different prototypes.

In this book, we shall assume that stationary time series are realizations of random processes that are ergodic in the mean and acf.

3.6 ARIMA models for discrete random processes

At this stage we can, if appropriate, account for trends and seasonal effects with deterministic mathematical models. We now turn to ways of modelling the random nature of the remaining process, which will usually be assumed stationary.

3.6.1 Discrete white noise (DWN) Discrete white noise (DWN) is defined by {Z,} where Zt is a sequence of mutually independent, identically distributed random variables. It has a constant mean, constant variance and the acvf is zero at all lags except 0. The importance of DWN is that it forms a building block for more complicated processes.

3 .6 .2 R a n d o m w a l k

Let (Zt } be DWN with mean t~z and variance O~z. A process {Xt }, defined by

x,= x,_l + Z,

is a random walk. It is usual to start with t equal to zero and to specify X0. The mean and variance change with t, so the process is non-stationary with

E[X,] = t/x + Xo

and

Var [Xt ] = tO~z

It has been found to give a reasonable model for many share prices and for some index numbers. The implication of using this model is that future share prices depend only on today's prices; that is, the historical record of prices before today gives us no further information. A random walk provides a reasonable fit for the oil product prices of Section 3.9.

3 .6 .3 M o v i n g a v e r a g e p r o c e s s e s

Let {Z,) be DWN with mean 0 and variance O'Zz. A process {Xt} defined by

Xt = t-t + floZ, + f l iZ , - i + . . . + flqZ,_q (3.7)

is a moving average process of order q, written as MA(q). It is usually convenient to define/3o = 1. The mean and autocovariance are given by

E[Xt] = /x + floE[Zt] + . . . . + flqE[Zt_q] = tx

and


7(k) = E[(X,- tt)(X,+k- /x)]

= E[(floZ,+... + flqZt-q)([3oZt+k + . . . - I - f lqZt+k_q) ]

Now, since Zt is DWN with mean 0, then

if E[ZtZt+p] = if p 4:0

It follows that if k is between 0 and q, then

q-k 7 ( k ) - ~, ~i~i+kO'2z

i=O

If k exceeds q then ~/(k) = 0. Recalling the definition for p(k), we can write

1 k = 0

p(k) = flifli+k ] ~ [3 2 k = 1 , . . . , q i=0

q<<-k

For this model p(k) = p(-k) when k ~<0. The process is stationary for any finite value of q.

Whilst there is an obvious arithmetic similarity between the centred moving average method of Section 3.3.1 and a MA(q) process there are fundamental differences. The former is a definition of an arithmetic operation but the latter is a model for a random process using DWN as the driving mechanism.

Example 3.2 An aeroplane has a digital altitude meter with displays located in the passenger cabin. Height measurements are taken every minute. When the aeroplane is flying at a constant height /x the measurements are assumed to be subject to error and independently distributed with mean/x and variance tr2z. Every minute an equally weighted moving average of the latest ten measurements is displayed. The display {Xt } is given by

1 1 Xt = ls + - ~ Z t + . . . + i ~ Z t -9

whereas the minute measurements are simply (/z + Zt). Thus

E [ X , ] =

and

1 Var[Xt] = ~ (o'2z+.. . +O'2z) = cr2z/lO

The series Xt is smoother than the minute measurements; that is, it has a smaller variance than the series { tt + Z,). The moving average is centred 4.5 minutes before it is first displayed. It has the advantage of smoothing out measurement errors which might

3.6 ARIMA models for discrete random processes 53

1.0 0.8 O.S

0,. 0.4 0.2 0.0

- - �9

- - �9

_ �9

i I I I I I I I ? ,L ,L ,.._ 1 2 3 4 5 6 7 8 9 10 l ' l r k

Fig. 3.6 Correlogram of MA(10) process of Example 3.2

otherwise give the impression of alarming fluctuations in height. It has the disadvantage of being slow to respond to real changes in height.

The acf of the moving average display is readily evaluated using the above derived theory with/3i = 1/10 for each i. Thus

1 0 - k p(k) -- ~ k = 0, 1 , . . . , 10

10

and p(k) is zero for k greater than 10. The correlogram of the stationary MA(q) process X, is drawn in Fig. 3.6.

3 .6 .4 A u t o r e g r e s s i v e p r o c e s s e s

Let Zt be DWN with mean 0 and variance tr2z. A process {X, } defined by

( S t - / s - " O t l ( X , _ 1 - ]J,) + . . . --I-- Olp(Xt_ p - Is + Z , ( 3 . 8 )

is an autoregressive process of order p, AR(p). The coefficients ot i must satisfy certain conditions if the process is to be stationary. Provided it is stationary, taking expectation of both sides shows that the parameter/z is the mean. The AR(p) process can also be written as

S t --~ [30 + OllX,_ 1 + . . . + o lpXt_p + Z t

where/30 = / z ( 1 - a l - . . . - a v), and this form is used in some statistical software. A useful special case is the first-order autoregressive process. In the following

development it is convenient to take/z as zero, i.e.

X, = + Z,

If we substitute for At_ 1 from

= aX,_2 + Z,_,

and so on, we obtain

S t = Z t + o tZ t -1 + o t2Zt -2 + o t3Zt -3 + . �9 �9

Taking expectation of both sides gives

E[X,] = E[Ztl + ~e [ z , _ , l + . . . = 0

which is consistent with/z equalling zero. A less obvious result is obtained by taking the variance of both sides. Thus

Var [X,] = o2z + a2o2z + a4O2z + . . .


since the Zt are independent. The right-hand side is a geometric progression with common ratio a 2. Provided a 2 is less than 1 the infinite progression converges and

O~x = O2z/(l- a 2)

If a 2 is greater than or equal to I the variance increases with time and the process is not stationary, i.e. the process is stationary if and only if la] < I.

The covariance of Xt and Xt-i is

E[X,X,_, ] = E[(Z, + + . . . ) ( Z , _ , + + + . . . ) ]

= aE[Z2t_, ]+ a3E[Z2t_:]+...

= + +...)

=ao~x

A similar argument shows that the covariancc of Xt and Xt-k iS (Z k O2X.

It follows that the acf is

p(k) = ~ lk l k = . . . - 2 , - 1, 0, 1, 2 , . . .

These results assume the series continues back indefinitely. If it starts at t = 0 when Z0 = 0, Xt would be the finite geometric progression

X t = Z t + oIZt_ 1 + . . . o l t - l z 1

The variance is now given by

Var[X,] = ~ ~2

Strictly speaking, the process would not be stationary, even when lal < 1, because the variance depends on t. We ignore this complication because a 2t will be negligible for all except small values for t, and use the results for the infinite series.

Example 3.3 Daily outputs of sulphur dioxide from a coal-fired power station are unlikely to be independent because underlying conditions such as coal quality will tend to vary slowly over the course of several days. If the sulphur dioxide emission for one day is above average, we would expect the emission for the next day to be above average as well. An AR(1) process with positive a might provide a good model. The correlogram for an AR(1) process with a = 0.7 is shown in Fig. 3.7.

Example 3.4 A process operator in a pharmaceutical company finds that if a good yield is obtained from one batch of chemical reactants there is a tendency for a tarry deposit to remain in the reactor. This adversely affects the yield of the next batch. This situation could be modelled with an AR(1) process with negative a. The correlogram for an AR(1) process with a = -0 .7 is shown in Fig. 3.8.

3.6 ARIt~IA models for discrete random processes 55

1.0 0.8 0.6 0.4 0.2 0.0 I I I 1 I ~ qP 0 6 ~ I

1 2 3 4 5 6 7 8 9 1 0 1 1 ~ k

Fig. 3.7 Correlogram of AR(1) process with a equal to 0.7

Example 3.5 A project is under way in Transport Engineering at the University of Newcastle upon Tyne to provide 'on-line' predictions of waiting time for passengers at bus stops. One aspect of the work involves developing algorithms to make the predictions. It has been observed that, in the peak period, if one bus is relatively slow the next tends to be relatively fast. An explanation is that, as a bus falls behind schedule, there will be more passengers to pick up at each stop, while a following bus will find fewer than usual passengers at each stop. It seems that an AR(1) with negative a may again prove useful. The value of c~ would be allowed to vary throughout the day to incorporate off-peak periods.

Example 3.6 No matter how close the parameter a in the AR(1) model becomes to I the correlogram decays exponentially with the lag. If a is set equal to 1 the process becomes the unstable random walk. Many physical processes exhibit 'persistence', that is, there are still high correlations between variables separated by a large lag. An AR(1) model is not suitable for such processes and fractionally differenced models (Hosking 1981 and Haslett and Raftery 1989) are far more realistic. High order autoregressive models might also be suitable in principle, but the estimation of many parameters is a practical disincentive. Some weather systems tend to persist and consequently many environmental time series, such as river baseflows and wave heights, exhibit persistence.

Suppose we need to estimate parameters of sea states during March at the site of a

1.0'

0.8 0.6 0.4 0.2

Q. 0.0 -0.2 -0.4 - 0 . 6 - 0 . 8

] i i i i 'i' i , I , , ~ d, , ~ 2 3 4 6 �9 8 1011 k

Fig. 3.8 Correlogram of AR(1) process with e equal to -0.7


wave buoy. We would intuitively prefer 10 days from each of three years to 30 days from the same year, and a mathematical explanation for this is persistence in the time series. Unfortunately, practical considerations such as no data being available for a site, and deadlines on projects, often force us to compromise with perhaps the 30 days from one year and previous years' data from a neighbouring site. The limitations of the available data must always be made clear in reports.

The second-order autoregressive process is a suitable model for second-order physical systems provided the driving force is approximately constant over the sampling interval. Also, on some occasions, it is found to give a better empirical fit to a time series than the AR(1) model.

It is again convenient to take a zero mean, in which case

S t = OtlXt-i + a2X,-2 + Zt

After some manipulation and analysis it is possible to show that Xt is stationary if and only if

al+a2~<l, - l < ~ a l - a 2 and -l~<a2

It can also be shown that the acf is given by

p(k) = A O~ kl + BOOk I

where

= 2 01,2 (al + (al + 4a22)1/2)/2

A = [ a , / ( 1 - a 2 ) - 0 2 ] l ( 0 1 - 0 2 ) and B = 1 - A

Example 3.7 A civil engineering contractor has formulated the following economic model for his business

Ik = Ck + ek + Gk

where Ik is the net income in year k, Ck is the profit from work done for private customers in year k, Gk is the profit from work done for the government in year k, and Pk represents private investment raised through rights issues or, in a negative sense, by buying own shares.

I f Ck = 0.51k_

and ek = Ck - Ck- 1

the model simplifies to

Ik = Ik- ~ - 0.51k-2 + Gk

If it is further assumed that the profit from work done for the government is DWN, this is an AR(2) process with a~ = 1.0 and a2 = -0.5. The correlogram is shown in Fig. 3.9.

Example 3.8 A water resource system consists of a river and its tributaries, towns, reservoirs and pumped storage reservoirs. A mathematical model of the system is required to help with

p _

3.6 ARINA models for discrete random processes 57

I I I I I I t ~ 9 & I I ~ I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 k

Fig. 3.9 Autocorrelation of an AR(2) process with el, t~2 equal to 1.0, -0.5

control decisions, such as when to release water from reservoirs and when to abstract water from the river into the pumped storage reservoir. Objectives are to provide the inhabitants and industries of the town with their water requirements, to avoid flooding and to avoid low stream-flows which would cause a build up of pollution created by effluent. The overall model would require models for stream-flow at various points. Autoregressive models have been used in this context. In particular, models of the form

X t = O l l X t _ 1 + o l 2 X t _ 2 -I- o l 3 X t _ 3 + N t + C t

have been used, where Xt represents flow, N, unknown disturbances and Ct a control sequence. If the control sequence is set equal to zero and the N, are assumed DWN this is an AR(3) model. In practice, C, would often be a function of the past values of the flow. In the cases where this is some linear function the 'closed loop' system could itself be modelled by some autoregressive model. Stationarity or, in control terms, 'stability', must be maintained!

3 .6 .5 A R I M A ( p , d, q ) p r o c e s s e s

When modelling random processes we usually try to manage with a small number of parameters, mainly because of the difficulties involved in their estimation. A low-order combination of the AR and MA processes can sometimes provide a better model than higher order AR or MA models on their own. For this reason, the ARMA (p, q) model is defined by

S t = O l l X t - 1 "~" . . . "~" O l p X t - p 4" ZtJt" ~ l Z t - i "~" . . . ~" ~ q Z t _ q (3.9)

Equation (3.9) encompasses the definitions of Equations (3.7) and (3.8). As before Z, is zero mean DWN and, for convenience,/x has been taken as zero. The ARMA model is often written as

= O(B)Z ,

where ~b(B) and O(B) are polynomials in B of order p and q respectively and B is the backward shift operator defined by

BX,= X,_~ (3.10)


The restrictions on the parameters for stationarity are the same as those for an AR(p) model. At this point we will introduce the first difference operator, V, defined by

V Xt = X t - X t_ 1 ( 3 . 1 1 )

First-order differencing is one way of removing a linear trend. Another way is to fit a straight line through the time series and then analyse the differences between the original series and the trend values. Although either method will account for a linear trend, the structures of the detrended processes differ. We can see this by applying both methods to the stochastic process Yt defined by

Yt = a + bt + Zt

where a and b are constants and Z, is DWN with mean 0 and variance a 2. On taking first differences a stochastic process {Xt } is obtained, where

x,= vY,

Substituting for II, gives

Xt = b + Z , - Z,_ 1

It is clear that {Xt} is MA(1). If the second method of removing the trend is used the resulting process would be,

W, = Y t - (a + bt )

and (Wt } is identical to {Zt }, i.e. DWN. In practice, we would be more likely to subtract an estimate of the linear trend, in which case { Wt } would be approximately the same as {Z t}, provided the length of the record from which the trend line is estimated is reasonably long.

If we assume that a continuous time curve x(t) is sampled at t i m e s . . . - t - 1, t, t + 1 and so on to give the discrete time series xt, then Vxt is the slope of x( t )at some time between t - 1 and t. So, first differencing of the time series can be thought of as an approximation to differentiating the continuous signal. There are many more sophisticated methods of numerical differentiation but an inherent problem is that any noise in the signal becomes magnified on differencing. Electronic devices which produce continuous approximations to the derivative of a continuous input signal are available and have been used with success in some control applications, but designers usually try to avoid having to differentiate signals. Signal noise is again a major problem.

The inverse operator to differencing is summation, since

and

t

X (Vx,) = x , - Xo k=l

Higher order differences can be defined by extension but practical applications beyond second-order are rare. Second-order differencing removes a quadratic trend. It is defined by


v2x, = v ( v x , ) = v ( x , -

= ( x , - X , _ l ) - ( x , _ , - x , _ 2 )

= X , - 2)(,_ 1 + St--2

We are now in a position to define the autoregressive integrated moving average process, ARIMA(p,d ,q) . If {Yt } is a stochastic process such that the dth differences are ARMA(p, q) then { Yt } is ARIMA (p, d, q). The simplest example is that of the random walk which is ARIMA (0,1,0). The reason for the term 'integrated' is that the stationary model, which is fitted to the differenced data, has to be summed, or 'integrated', to provide a model for the original time series.

As stated previously, when fitting ARIMA models we aim for ones in which p and q are relatively small, and it is unusual for d to exceed two. A general seasonal ARIMA model denoted by

ARIMA (p, d, q) (P, D, Q)s , is defined by

$(B)$~(/P)(1 - B)d(1 -- IV)~ = O(B)O~(B~)Z,, (3.12)

where 4~(B) and O(B) are polynomials in B of degrees p and q respectively and ff,(/V) and 0~(/V) are polynomials in /V of degrees P and Q respectively, where s is the seasonal period. The operator/V is defined by

B'X, = X,_,

D is almost always 0 or 1 and for monthly variables s is 12. Then

(1 - B 12)X, - Xt - X,_ 12

i.e. the difference between a monthly variable and its value in the same month of the previous year. If d equals 1, ( 1 - B ) ( 1 - B I 2 ) X , are the differences of the sequence {(Xt - Xt-12)}, i.e.

{ (X t - X t _ l) - (Xt_12 - Xt_13) }

These models, which use differencing to represent trend and seasonal effects, are often used by hydrologists, but the simple ARMA models are also widely used~trend and seasonal effects having been accounted for in a preliminary analysis. If the model is to be used for simulation the latter is a preferable approach, because differencing is potentially unstable. To demonstrate this point, imagine a random process X, which is DWN superimposed on a quarterly seasonal effect S~, $2, $3 and $4. In our previous notation

Xl = Sl + Z1

X2 = S2 + Z2

X3 = $3 + Z3

.,,~4 = S 4 + Z 4

X s = S I + Z 5

X6 -- $2 + Z 6


etc. If a seasonal differencing (1 - B 4) is used the resulting process is:

Z s - Z 1

z6-z2

etc. This process is MA(4) with fli equal to 0, 0, 0 and -1 . It is unlikely that these coefficients would be identified exactly from a sample, and there is a strong possibility that the 'integrated' process will be unstable unless suitable constraints are applied during the parameter estimation. Such instabilities are likely to become apparent in long simulations from models whose parameters have been estimated from relatively short time series; see Exercise 12 for an example.

3.6.6 Gaussian and non-Gaussian processes If the sequence {Zt } in an ARMA model is assumed to be normally distributed the joint distribution of

Xt i ~ * , , ~ XtN

will be multivariate normal. The normal distribution is often called the Gaussian distribution after the mathematician Karl Gauss (1777-1855) who investigated the function, although the PDF was first obtained as a limit of the binomial distribution by Abraham De Moivre (1667-1754). If the joint distribution of any set of variables from the random process {Xt } is multivariate normal the process is said to be Gaussian, and is uniquely defined by its mean and acvf. The marginal distribution of the Xt will be normal, but this is not a sufficient condition for {At } to be Gaussian. A MA(30) process with random variation from an exponential distribution could have a near normal distribution.

When we plot a histogram of the data in a time series it can be noticeably asymmetric, in so much as one tail extends far further beyond the peak than the other, indicating that even the marginal distribution cannot reasonably be considered normal. It follows that a Gaussian random process will not be a suitable model. If the longer tail is to the right it is said to be positively skewed, and this is a common shape for environment data. Examples range from airborne pollutants to river flows. Several schemes have been proposed to force skewness on ARMA models. A measure of skewness, Y, is defined by El(X-/x)3]/cr 3. One strategy is to assume the DWN sequence {Zt } is from a skewed distribution. In the case of AR(1), with parameter a, the skewnesses of the distributions of Xt and Zt are related by

Yz(1 - a2) 3/2 - (1 - a3)~x

Using a skewed distribution for the Zt also has the effect of modelling asymmetry or, more formally, time-irreversibility in a random process. Stream-flow data often exhibit rapid increases followed by slower recessions and this can be modelled, to some extent, by taking the Zt as random deviates from an exponential distribution with the mean subtracted. Figure 3.10 is a realization, and histogram of the marginal distribution, from an AR(1) process with ct equal to 0.8 and random variations based on an exponential distribution with a standard deviation of 1.

An alternative procedure is to assume that a stochastic process { Yt } is related to a Gaussian ARMA process {Xt } by some transformation. Common choices involve raising

xt

2 .50 -

0 . 0 0 -

1 2

-2.50 - 3

-5.00 F 0

xt

2.50 -

123

0 .00 - 4

-2.50 -

-5.00 7 6O

6 8 7 9

12 456 0 3

1 23

0

4

56 9

78


89

8 I 7 9 89 7

2 6 OI 7 0 345

2 123 54 456

56 789

0

I I I I I I - ~ 10 20 30 40 50 60 r t

7 8 0 91

3 9 2 2

2 4 0 4 0 45 5 5 3 ! 3 8 6 8 I 5 3 6 4 7 9 7 6 7 3 6 5

e e a e9 6 o 7 9 9 4 7

0 0 I 1

2

I I , I I I 70 80 90 100 110 120~t

Histogram midpoint Count

- 3 4 - 2 20 1 -1 27

0 25 1 23 2 17 } 3 3 L_ 4 1

Fig. 3 .10

N= 120

I

II

Realization of an AR(1) process with exponential noise and a equal to 0.8

the variate to some power or taking logarithms. These are gathered together by the Box--Cox transform

I[(X,- L) ̂ - 11/~ ~ ~ o Yt = [In ( X , - L ) A = 0

for some selected values of h and L. The particular L must be chosen so that X - L is positive, and is often taken as zero. The scaling for non-zero choices of A is not necessary; it is only introduced because it is convenient to have the transform continuous with respect to A when A = 0 for deriving some mathematical results.

3 . 6 . 7 R e l a t i o n s h i p b e t w e e n M A a n d A R p r o c e s s e s

We worked out the acvf of the AR(I ) process by expressing it as a MA process of infinite order. We could easily express any finite order AR process as an infinite order MA process. Conversely, we can express any MA process as an infinite order AR process. At


this point we come across restrictions on the parameters of MA processes to give invertibility. This will be demonstrated for an MA(1) process. Both

Xt = Zt + OZt_ 1

and

1 x ,= z,+-ff z,_,

have the same acf, that is p(1) equals 0/(1 + 02) and for higher lags p(k) equals 0. So we could define such a process {Xt} as MA(1) with/3o = I,/31 = 0 and O~z = ~ or as MA(1) with/30 = 1,/31 = 1/0 in which case O~z = ((02 + 0~)/(02- 1))o ~. If we express the two processes in terms of previous values of Xt by successive substitution we obtain

z, = x , - ox,_~ + 02x,_2-...

1 1 Z t = X t - - - ~ X t _ 1 + - - ~ X t - 2 - . . �9

respectively. If we now arbitrarily assume 101 < 1 we see that it is only the infinite series on the right of the first Zt equation that converges and this corresponds to the first form of the MA(1) process. A requirement that the coefficient of Z,_~ should be smaller than one in a MA(1) process is known as the invertibility condition, and is of relevance for more advanced theoretical work. The estimation procedures which we are about to introduce involve estimating residuals and naturally lead to invertible forms of MA processes. We shall not, therefore, consider the conditions on the parameters for high-order MA processes to be invertible.

3.7 Estimation of parameters of models for random processes 3.7.1 Estimation of the autocovariance function We will assume that the underlying process is stationary and ergodic, and that a record {x, } of length N is available. The sample autocovariance coefficient at lag k, defined by

N-k

c(k) = ~ (X, - X)(X,+k- f()lN, where f( = ~ Xt/N (3.13) t= 1 t= 1

will be used as an estimate of y(k). This definition is not universal and some texts adopt ( N - k ) for the denominator. In the next sub-section we show that the bias is of order 1/N.

3.7.2 Bias of the autocovariance function We now consider the random variable c(k), and write X, for the random variables at time t. The use of upper case emphasizes that we are considering Xt as random variables rather than the actual values in the record, which are represented by x,. Then

N - k

c(k)- E (X,-~-(X-~))(X,+k-~-(X-~))/N t= 1

3.7 Estimation of parameters of models for random processes 63

can be expanded to

I N - k N - k

c(k) = ~, (X, - Ix)(X,+,- tx) - ~, (X,- p,)(f(- tx) - t= 1 t= 1

E (Xt+k- I.O(f(-- I~) + ( N - k)(f(-/z) 2 /N t = l

Now

N

~ - E X,/N t = l

and for large N it is nearly the same as either

N - k N - k

E XtI(N- k) or ~, X,+k/(N- k) t= 1 t= 1

Using these approximations

} c(~) = E (x. - . ) (x ,+, - . ) - ( N - k ) ( 2 - ~) 2 /U t - 1

Taking expectation gives

E[c(k)] - { ( N - k)./(k) - ( N - k) Var(X)}/N

where the expression for

Var(X~ + X2 + . . . + XN) - Var(X1 ) + . . . + Var(XN)

+ 2 Cov (XIX2) + . . . + 2 Cov (XIXN)

+ . . . + 2 Cov(XN_~Xu)

is used to generate the result

Vat(X) = N~0) + 2 ( N - 1)~,(i) /N 2 i=1

Putting these results together gives

_ 1 2 i=! y(i E[c (k ) ] - - (1 -k ) {y (k ) ~ - y ( 0 ) - ~ - ( 1 - 1 ) ~ ' )} (3 14)

Provided YiN=I 1 "),(i) is bounded as n tends to infinity, which is always the case for a stationary process, the bias in the estimator can be seen to be of the order of lln. However, it should be noted that if n is not very large and the y(i) decay rather slowly, as is the case for a persistent process, c(k) will be badly biased downwards. In particular, any estimate of the variance of a persistent process based on a relatively short record will tend to be too small; this is a problem encountered by Abdul Ghani (1988) in the context of flood prediction from river flows recorded at midday over a single season.

A final point is that a feature which an environmental scientist regards as 'persistence' could, in a different context--such as short-term forecasting~be reasonably taken as a 'trend'. To sum up, data from the spring of one year are unlikely to be adequate to make

64 Time var,/ing signals

any useful estimate of flood risks in future years but might provide a useful measure of flood risk for the month ahead.

It can also be shown that

N-1

Cov(c(k), c(l))--- Y~ [3'(j)v(j + l - k) + r(J + l ) y ( j - k)l/N j . - _ ~

3.7.3 Es t imat ion of the au tocorre la t ion function An estimator of the autocorrelation function p(k) is given by r(k) where

r(k) = c(k)/c(O) (3.15)

If the underlying process is discrete white noise then for any non-zero integer k, see Exercise 13

and

E[r(k)l= - I l N (3.16)

Var [r(k)]--- 1/N (3.17)

and r(k) is asymptotically normally distributed provided the distribution for which the DWN is drawn has a finite variance. The sample correlogram is a plot of r(k) against k. Since it is symmetric, it need only be plotted for k~>0. The correlogram for the deseasonalized inputs to the Font reservoir is shown in Fig. 3.11. Dotted lines at -1 /N+2V'N and - 1 / N - 2 ~ / N have been added to the correlogram. If we took realizations of length N from DWN we would find that, on average, 95% of r(k) values would lie within these lines. The approximation for the distribution of r(k) also holds for a specific value of k if p(k) equals 0.

3 .7 .4 Es t imat ion of parameters in A R M A mode l s We again suppose that we have a time series {xt} from a stationary ergodic random process. There are several methods for estimating the parameters of models for random processes but we shall restrict our attention to two of them, the method of moments and the method of least squares. They will be discussed in the context of fitting the AR(1) model.

Estimation of parameters of AR(I) The model can be written

x , - tz = ~(x,_ ~ - ~) + z,

Recall that the parameter/x is the mean of the process and a can be interpreted as the correlation at lag 1. The 'method of moments' uses the sample mean of the time series as an estimate of/z and the sample correlation at lag 1 as an estimate of a. The reason for the name is that moments of the implied multivariate distribution f(xz, x 2 , . . . , xN) are estimated by sample moments. An astute reader may ask why a is estimated by r(1) rather than the square root of r(2). It is usually more 'efficient', a term which will be formally defined later in this section, to use the lowest order moments and the lowest lags possible. The variance of the Zt can be estimated from the sample variance of the xt and the relationship that

= (1 - ~ ) ~

-0.3

1 0.242 2 0.112 3 0.012 4 0.048 5 0.013 6 -0.037 7 0.038 8 0.027 9 -0.000

10 0.011 11 0.029 12 -0.010 13 0.042 14 -0.014 15 -0.006 16 0.026 17 -0.042 18 0.014 19 0.001 20 -0.011 21 0.035 22 0.062 23 0.109 24 0.055 25 0.058 26 0.006 27 -0.009 28 -0,017 29 0.002 30 -0.001 31 -0.037 32 0.003 33 -0.037 34 0.009 35 -0.022 36 0.011 37 0.010 38 0.002 39 -0.002

Fig. 3.11


-0.2 i

r(k) -0.1 0.0 0.1 0.2 0.3

, ,

I ] 1 I 1 , r l l

m

m

m

m

m

i

m

i ,

u

m

m

m

m

Correlogram for the deseasonalized inputs to the Font reservoir

The method-of-moments estimators for the AR(1) process can be summarized as

/2 = 2, a = r(1)

and ~2z = (1 - a 2) c(0)

We now look at the least squares method of estimation. The model can be rearranged as

z , = ( x . - ~) - . ( s , _ ~ - ~)

and the sum of the squared DWN disturbances is

Y~z 2 = Y. ( ( x , - ~ ) - ,~(x,_, - ~)}2

Let/2 and c~ be any guesses for/x and a respectively. The least squares estimates of/~ and a are the values of/2 and c~ which minimize the function ~k defined by

66 Time varying sisnals

N r ~) ---- 2 {(Xt-- ~) - - / I ' (X , - I - t-~)} 2

t=2

if 'errors' defined by

z, = ( x , -

can be thought of as estimates of the values taken by the Z,. The sum of squared errors must be minimized in the sense that if they were defined analogously with any other values substituted for g and a their sum of squares would be greater. Because of the product of c~ and : in 4' it is not possible to solve the equations

= o, [ = 0

to give /2 and & as explicit functions of the data. The equations have to be solved iteratively. An alternative is to apply some minimization procedure directly to the function ~b. The variance of Z,, o2z, should be estimated from ~z2 , / (N - 2).

The method-of-moments estimates have an advantage of simplicity in this context but we need a rather more objective criterion for choosing between the methods. To establish such a criterion you must imagine very many realizations of size n being taken from a random process. For each realization gt and a will be estimated by/J, and ti, but the values of these estimates will vary from realization to realization. That is, ~ and tl considered as random variables have a sampling distribution. Their root mean squared errors are defined by

(E [( /i - /,02]) 1/2 and ( E [ ( t l - a)2]) 1/2

respectively. If they are unbiased, i.e. E[ ~] = ~ and E[ti] = a, their root mean squared errors equal their standard deviations. Root mean squared errors are a useful measure of the performance of estimators, and if one estimator has a smaller root mean squared error than another it is said to be more efficient. The least squares method of estimation is slightly more efficient for AR models and much more efficient for MA, and hence ARMA, models.

Estimation of parameters of AR(2) The model is

( X t - ft) = GI ( X t _ 1 - f t) + ~ 2 ( X t _ 2 - f t) -{- Z t

Method-of-moments estimates are given by

&, = r(1)(1 - r(2))/(1 - (r(1)) 2)

and r = r(2)(1 - r(1)2)/(1 - (r(1)) 2)

The least squares estimates are the values of / i , ~1 and &2 which minimize

N a , , a2) = Z [ ( x , - - a , ( x , _ , -

t=3

In this ease ~ should be estimated from Y.z~ / (N- 3).


Estimation of parameters of MA(I) The model is

x , = ~ + z , + / 3 z , _ ~

Since p(1) =/3/(1 +/32), method-of-moments estimates are given by

and by solving

r(1) =/~/(1 +/~2)

where/3 is chosen to be the root such that It l < 1 and satisfies the invertibility condition. Unfortunately, the method of moments is not efficient for fitting MA models, although

the ensuing, estimates do provide a starting point for the following iterative procedure. Let/2 and/3 be the method-of-moments estimates of ~ and/3.

Set Zo = 0 and define the errors by

Z 1 - " X l - - jt~,

zN " x N - t i - f3zN_ l

Then calculate the error sum of squares

N

z, i=2

Repeat the procedure for different values of/2 and/3. The least squares estimates of 1~ and/3 are the values of/2 and/~ which minimize the error sum of squares. There are several optimization procedures which could be used for the minimization in the NAG subroutine library.

The parameters of more general ARMA models can be estimated in a similar fashion, but convergence to the required 'global minimum' should not be taken for granted. The choice of initial conditions is important and convergence problems generally increase with the number of parameters to be estimated. Anyone who intends doing much time series analysis would find a respectable statistical package such as MINITAB or GENSTAT a sound investment.

A slight refinement to the least squares estimation procedure is to use back-casting. This uses the initial estimates and the reversed time series to predict values before the start of the series. These are known as 'back-casts' and allow ~ to be defined as a sum from 1 to N. The parameters are then re-estimated but the change is usually slight unless the time series is very short. Details can be found in Box and Jenkins (1976).

In all ARMA models an estimate of ~ , corresponding to an unbiased estimator, is

~ = ~ , z 2 1 ( N - l - p - q)

This is the usual statistical result that a 'degree of freedom' is lost for each parameter


375

I , . -

~ 125

- 2

3 3 4 2 5

9 0

I 6 9 " 34567

0

8 3 67

5 4 1 45 9 12

67890 0 I I 10 2O

Week

0 3 5 6 8 8 1

9 2

0 I I I I 0 30 40 50

Fig. 3 . 1 2 Fluor ide con ten t of grass samp les

estimated. It will be shown in the next section that this is an aid to deciding on the appropriate order for a model, but we will first look at some data which are relatively straightforward to analyse.

Example 3.9 (Loosely based on Craggs 1980) A researcher monitored the fluoride content of grass at a site near a chemical plant. Samples of grass were collected once a week over a 50 week period and the measurements of fluoride content (0.1/~gF g-1 dry weight of grass) are given in Table 3.3. The time series is plotted in Fig. 3.12 and there is no obvious seasonality. Histograms of the original data and the square roots of the original data are given in Figs 3.13(a) and (b). The latter looks more like a normal distribution and this is confirmed by the normal probability plots in Figs 3.14(a) and (b). Marginal normality is a necessary, but not a sufficient, requirement for the random variation to be from a normal distribution. There are several reasons why it is convenient to model the random variation by a normal distribution: estimation (by theory or simulation) of the probability that the fluoride content exceeds some critical level being one of them. For this reason, we will consider the transformed data. The correlogram is shown in Fig. 3.15 and its shape suggests that an AR(1) process with a positive value of a might be suitable. This model was fitted and

Midpoint

0 50

100 150 200 250 3O0 350 40O 45O

Fig. 3 . 1 3 samp les

Count

3 g 9 9 6 8 1 - - 4 0 1

I I I

I

I

I I I l l I

Midpoint Count

4 3 ,, 6 6 " 8 5 "

," 10 6 ' 12 10 , ' ' 14 6 ' - 16 9 - 18 3 I 20 2

(a) (b) His tog rams of (a) f luor ide con ten t and (b) t r ans fo rmed f luor ide con ten t of grass

3.7 Estimation of paramecers of models for random processes 6 9

,

0.999 . . . . . . . . . . . . . . . . . . . . . .

i : ! .i..=- : i o . s o . . . . . . . . . . . . . . . . . ~ . . . . . . . i . . . . . . . . . . . , ~ , J " . . . . . . . . . . . . . . . . . . . . . . . . . . ~.

i " " 0.01 " . . . . . . . �9 �9 : . . . . . . . . . . . . . 0.031 �9 . . . . .

, . . . . . f

o so ~6o ~so ~ 250 ~o ~ 400 45o Ruodde Average: 157.72 Anderson-Darling normality test

Std dev: 103.741 A-squared: 0.775 N of data: 50 ,_, p-value: 0.041

0.999

0.99

0.95

i 0.80

0.50

0.20

0.05

0.01

0.001

-4 .... . . . . . : . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . ;. . . . . . . . . . . .~.~.'~":. .........

Average: 11.8173 Std dev: 4.29422 N of data: 50

10 sqrt FI

tb)

15

Andertmn-Darling rmmlality test A-squared: 0.307 p-value: 0.551

Fig. 3.14 Normal probabil i ty plots for (a) f luoride contents of grass and (b) the square root of the original data

Tab le 3.3 Fluoride contents of gross samples

432 256 200 "i68 86 ~2 ' 23 34" 23 3~ 42 184 192 54 48 93 93 131 40 14 62 56 128 80 176 256 336 125 254 224

286 336 248 255 240 240 256 122 102 141 117 93 149 80 144 160 176 144 368 352

70 Time varyin# signals

ACF of C2

1 0.672 2 0.462 3 0.344 4 0.249 5 0.125 6 0.010 7 -0.100 8 -0.064 9 -0.008

10 -0.019 11 -0.008 12 -0.057 13 -0.014 14 0.022 i5 -0.064 16 -0.122 17 -0.138

-0.8 -0.6 -0.4 i " i

-0.2 r(k)

t

0.0 0.2 I

Ill,,, , , , , L J

.

J , L , _

0.4 0.6 0.8 I I ]

Fig. 3.15 Correlogram of transformed fluoride contents

the estimated value of a was 0.79 with a standard deviation of 0.10. A histogram of the errors is given in Fig. 3.16, a plot of the errors against their normal scores is given in Fig. 3.17 and their correlogram is shown in Fig. 3.18. They can plausibly be considered a realization of DWN from a normal distribution.

3.7.5 Determining the order of ARMA processes Any ARMA(p, q) process is a relatively simple model for some complex physical process which exhibits an appreciable degree of 'randomness', and there is no 'true' model. The investigator must be content with choosing a 'suitable' model. This strategy is justified by the useful predictions such models have provided in a wide variety of projects over many years. Several models may all provide a reasonable fit to a data set and it would be inflexible to stipulate definite criteria for selecting the 'best', Fortunately, and in marked contrast to trend curves, different ARMA models, which provide similarly good fits to a time series, usually provide similar predictions, although it is advisable to check this in any particular study. Nevertheless, some general rules applied with common sense should be helpful.

Histogram of C50 N = 50

Midpoint Count - 6 3 -4 6 -2 11

0 13 2 11 4 4 i 6 1 8 1

_

I I 1 |

Fig. 3.16 Histogram of errors after fitting the AR(1) model to transformed fluoride contents


C50

4.0

0.0 W

-4.0

0 0 0 0 0

2 2202oo

o o

2.o2 OOO

0 o

O0 O 0

t i I ! I C51 -1.60 -0.80 0.00 0.80 1.60

Normal score

Fig. 3.17 Normal score plot of errors after fitting AR(1) model to transformed fluoride contents

A plot of the sample acf is a good starting point. For an AR(1) process the theoretical acf decays exponentially and the sample acf should have a similar shape. If a MA(q) model is appropriate the r(k) will fluctuate randomly about a mean of -1/N with a standard deviation of 1/X/N for values of k greater than q. Lines drawn on the acf at -1/N + 2/X/N are essential to decide whether a MA(q) model is appropriate, but it must be remembered that, on average, 5% of r(k) values will lie outside these limits once the k exceeds q. The partial autocorrelation function, pacf, plays a role in deciding the value of p for a purely autoregressive model similar to that played by the acf in deciding the value of q in a purely moving average model.

When fitting an AR(p) model, the estimate of the last coefficient, tip, will be denoted ~p and estimates the excess correlation at lag p which is not accounted for by an A R ( p - 1) model. It is called the estimate of the pth partial autocorrelation coefficient

1 - 0 . 0 8 2 2 -0.002 3 -0.011 4 0.137 5 0.062 6 0.057 7 -0.301 8 0.019 9 0.103

10 -0.044 11 0.039 12 -0.191 13 0.041 14 0.252 15 - 0 . 0 6 8 16 -0.072 17 O. 006

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 i"' , , " , ' , " ,

r(k)

t I

I

R

I I

I

Fig. 3.18 Correlogram of errors after f i t t ing AR(1) model to t ransformed f luoride contents


and when plotted against p gives the sample pacf. It can be seen from the definition that the pacf of an AR(p) process 'cuts off' at lag p, i.e. 7rp+l, . . . are all zero. It can be shown that if 1rp equals zero, ~p has the same distribution as r(k) when p(k) equals zero. An indication of a suitable AR process is the value beyond which the ~p can be reasonably considered a random sample from a normal distribution with mean - 1 / N and standard deviation 1/~/N. Lines should be drawn on the pacf in the same way as for the acf.

If neither the sample acf nor the sample partial autocorrelation appear to 'cut off', an ARMA model may be appropriate.

After fitting a model the errors must be investigated. If the model is appropriate the errors should not provide evidence against a hypothesis that they are a realization of DWN. There are many formal tests, but we should bear in mind that, in large samples, small deviations from the null hypothesis will frequently give rise to statistically significant results which may not be of practical significance. A simple plot of the errors against time is worth drawing and their correlogram should be investigated. As well as our subjective assessment of points within the +2/~/N limits we can also use a 'portmanteau lack-of-fit' test. This combines the autocorrelations of the errors rz(k) into a statistic

L

N E 6(k) k=l

where N is the number of data and L is a lag chosen, subjectively, to include the autocorrelations of interest. The statistic is known as the Box-Pierce statistic. If the residuals are a realization of a Gaussian DWN process a reasonable approximation is that the Box-Pierce statistic is distributed as a chi-square distribution with L - p - q degrees of freedom, denoted by )~L-p-q. This distribution is described in Appendix III.

Now, suppose that several ARMA models all provide plausible fits to a given time series, in so much as their residuals could be a realization of DWN. A reasonable means of choosing the 'best' would be to take the one with the fewest parameters. An alternative criterion would be to select the model with the smallest value of d~z. Notice that whilst Ez 2 can only get smaller if the number of parameters in the model is increased, this is compensated for by the reduction in the degrees of freedom, N-1-p-q . However, it may be that a simpler model is preferred to one that results in a slightly lower value of if the reduction is of little practical significance. An experimenter, who is used to multiple regression analysis, may decide to include an additional parameter only if its estimate is significantly different from zero at some chosen level. As an approximate guide, an estimate of a coefficient is significantly different from zero at the 10% or 5% levels if the ratio of the estimate to the estimated standard deviation of the estimator exceeds 1.7 or 2.0 respectively, but correlations of the estimated parameters must also be taken into consideration, see Exercise 15. As this text is primarily concerned with spectral analysis, some details of ARMA modelling, such as estimation of the standard deviations of estimators, have had to be excluded. Estimated standard deviations and correlations of parameters are provided by most statistical packages and the theory behind this is described by Box and Jenkins (1976).

3.8 Simulations 73

3.8 Simulations We have already noted that models of random processes can be used as part of simulation studies. The basic tool in any simulation is the generation of realizations of DWN. This is achieved through the generation of some random or, more commonly, some pseudo-random number generator. This is discussed briefly in Appendix I. For the moment we assume the availability of DWN realizations. A specific example may demonstrate both the attractions and limitations of simulation procedures.

River water quality and flooding are related to the design of both storm water and foul water sewers. Water quality is particularly dependent on the length of dry spells between rainfall when toxins can build up in the sewer system.

If a long, hot, dry spell is followed by a storm, these toxins can be flushed into the river over a relatively short length of time, killing creatures which live in the river and affecting plant life. A highly regarded hydrological model of sewer systems (WASSP) is available. This allows designers to investigate the effects of improving sewer systems by increasing the diameter of pipes, adding retention tanks and so on. The hydrological model requires a rainfall input at one minute intervals and, if it is to model water quality as well as flow, lengths of dry spells with associated temperatures. Historical records are one possible source of data, and would be adequate for making a choice between competing improvement schemes on the basis of the performance the schemes would have achieved over this period, but such records have limitations. They may not be available at the required locations, they may not be long enough to obtain any indication of the probabilities of events with serious consequences, and if they are long they require a large amount of computer storage.

An alternative procedure is to investigate some of the long series available for several locations in the United Kingdom and identify the form of suitable random process models. The parameters of these models could then be estimated at the required locations by a combination of extrapolation from the parameter estimates at other locations and direct estimates from the limited data available at the site in question. A particular advantage is the possibility of estimating the probabilities of certain unusual events occurring, e.g. extreme floods for use in cost-benefit analysis. Given the non-linearity of sewer systems it would be quite impractical to estimate such events from available historic data. In principle, if the parameters of the random process were known, a very long simulation would allow accurate assessment of such probabilities. Since we only have estimates of parameters of a model of the random process, we could make better use of our computer time by running several shorter simulations and noting the sensitivity of our probability estimates to the parameter values and the assumed model.

The estimated standard deviations of the parameter estimators are extremely useful as a guide to choosing parameter variations. Parameters could be set at their point estimates, plus and minus one and two standard deviations. There is a further refinement that should really be considered here. For example, if two parameter estimates, a~ and a2, have a high negative correlation, then if one is above its expected value the other tends to be below its expected value. In this case, the scenario with both parameters set at two standard deviations above the point estimates is unrealistic. It would be preferable to take points on the boundary of the 95% confidence region; that is, the ellipse within which we are 95% confident the pair (a~, a2) lies. A general construction for a confidence region is now described.


We will denote the p parameters of the AR part of the model by 0 and the q parameters of the MA part of the model by 4). The error sum of squares S depends on the values of/~, 0 and ~b, that is S = S(0, ~b,/z). The least squares estimates of 0, 4) and/x, are denoted by 0, ~ and/2 and they minimize S. We call S(0,~,/J) the residual sum of squares. If the DWN process {Zt} is Gaussian, an approximate ( 1 - e) x 100% confidence region for O and 4) is all the values 0 and ~ such that

I . 1 +

p + q / N - p - q

where Fp+q,N_p_q,l_ei$ the upper e x 100% of the F-distribution with p+q and N - p - q degrees of freedom. The F-distribution is described in Appendix III.

In repeated realizations from a stochastic process, 95% of the vectors of parameters estimates will lie within the 95% confidence region. In practice, however, only the most dedicated researcher will be inclined to do this work for more than two parameters. A compromise is to select the two most highly correlated parameters and to treat the others as independent. Estimated correlations are usually provided by statistical packages.

3.9 Further practical examples At this stage, we will look at some more practical uses of the theory we have been considering.

Example 3. I0 The data presented in Fig. 3.19 are Rotterdam prices for premium gasoline from 1973 to 1985. The monthly prices are given in Table 3.4. These data have been made available by the British Petroleum Company plc. The correiogram for the original series, Fig. 3.20, suggests that we might difference the data. If we do this we obtain the correlogram shown in Fig. 3.21. If the differenced process is DWN, the probability that r(1) is as far from its expected value~approximately 0--as is 0.17, is about 0.04. We could try fitting an MA(1) to the differenced data. However, the overall pattern of the correlogram is not inconsistent with a hypothesis that the differenced series is DWN. This is equivalent to a

Table 3.4 Year

Rotterdam prices for premium gasoline for months from 1973-1985

. . . . . Monthly 'costs '

1973 50 63 76 88 110 100 90 76 76 100 160 150 1974 150 150 180 162 150 140 130 120 110 102 110 120 1975 120 130 140 145 130 124 120 125 133 140 145 140 1976 130 140 150 160 150 145 144 143 142 141 140 139 1977 137 136 138 139 136 135 134 133 132 131 130 130 1978 130 132 136 135 143 150 164 180 200 213 200 230 1979 250 310 280 300 350 390 370 345 360 380 420 390 1980 380 369 343 335 330 321 315 340 360 380 360 370 1981 364 356 350 342 365 380 375 360 358 346 334 322 1982 317 270 300 330 360 345 325 345 342 330 310 285 1983 285 260 265 298 300 310 304 280 286 275 266 253 1984 260 270 275 260 255 252 250 245 240 235 232 228 1985 230 240 250 260 270 275 270 255 270 260 255 250

ii i i i i

0d N

mO

i~.

O4

04

0 m

~

o

04

J I

_N O4 o

N

j ur,)

o,4

eo!Jd eu!loseE)

o ,i=,,,,

iv=.

o 04

iT.= : ,i,,,=

Or)

r,.,. ,l,-

=ti ~E

~J

t-"

O

u') tq tm

E ::3 o~ E q) L_ C

). L~ O

03 q)

.m

L~ (2.

E "O

t,,,. Q

)

O

rr"

I,,,,

im

u._


r(k) -1.0-0.8-0.6-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

, , I I I I I I I I

�9 1 0.975 2 0.946 3 0.919 4 0.898 5 0.876 6 0.849 7 0.824 8 0.800 9 0.776

10 0.752 11 0.733 12 0.711 13 0.688 14 0.664 15 0.647 16 0.627 17 0.606 18 0.587 19 0.568 20 0.546 21 0.521 22 0.495

I I i

I I I I I I I I I

I I I I I

i I

,, , , , ,

m I I

I

I II

Fig. 3.20 Correlogram of Rotterdam oil prices

hypothesis that the original series is a random walk. The mean of the differenced series is 1.29, although there is no statistical evidence against a hypothesis that the mean of the underlying random process is zero (the standard deviation of the mean is 1.31). Overall, our best prediction of the price next month is 1.29 more than this month. However, we are not very confident about this increase.

r(k) -0.2 -0.1 0.0 0.1 0.2

, I 1 ! !

i I i 1 0.170 2 -0.047 3 -0.143 4 0.145 5 0.107 6 -0.108 7 -0.112 8 0.003 9 0.109

10 0.O57 11 0.101 12 0.023 13 0.017 14 -0.103 15 0.022 16 -0.031 17 -0.118 18 -0.045 19 0.025 20 0.105 21 -0.061 22 0.057

I I I I

m

I I

m ==

i ii i

i i

= , , = , , =

i

Fig. 3.21 Correlogram of differenced Rotterdam oil prices

3.9 Futzherpracticalexamples 77

Midpoint -1.5 -1.0 -0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

Count 21

177 236 165 99 64 51 19 16 5 8 2 0 0 0 0 1

, | , , ,

n

m

m

I

I I I

, I |

I

i

Fig. 3.22 Histogram of deseasonalized inflows to Font reservoir

Example 3. I I We continue with Example 3.1 in which we looked at inflows to the Font reservoir. We now look at the data after the estimated linear trend has been removed and the data have been deseasonalized by subtracting the appropriate estimated monthly mean and dividing by the appropriate estimated monthly variance. We think that it is reasonable to consider this series as a realization of a stationary, ergodic random process.

The correlogram shown in Fig. 3.11 suggests that a MA(2) model might be appropriate. The histogram of all the data is shown in Fig. 3.22 and provides an estimate of the marginal distribution. This appears to be highly skewed and there is one striking observation which is considerably greater than the rest. Further enquiries confirmed the datum was correct and so it is kept in the analysis. A square root transformation of the data incremented by 1.51 to make them positive was tried, and the resulting histogram is shown in Fig. 3.23. It is clearly closer to normality. The correlogram of the transformed data is shown in Fig. 3.24; it is very similar to that for the original data. The fitting of a MA(2) model yields

Xt = 1.1674 + Zt+ 0.2333Zt_1 + 0.1037Zt_2

Midpoint Count 0.0 1 0.2 4 0.4 13 0.6 59 0.8 150 1.0 187 1.2 164 1.4 120 1.6 77 1.8 51 2.0 23 2.2 13 2.4 1 2.6 0 2.8 1

. . , . . , .

I I

Fig. 3.23 Histogram of transformed deseasonalized inflows to Font reservoir


-0.3

1 0.240 2 0.106 3 0.020 4 0.066 5 0.022 6 -0.053 7 0.028 8 0.025 9 0.010

10 0.010 11 0.031 12 -0.018 13 0.039 14 -0.005 15 0.005 16 0.013 17 -0.035 18 0.022 19 0.013 20 0.000 21 0.033 22 0.049 23 0.109 24 0.054 25 0.061 26 0.008 27 0.008 28 -0.002 29 -0.005 30 -0.014 31 -0.030 32 0.012 33 -0.024 34 0.005 35 -0.020 36 0.012 37 0.013 38 0.009 39 0.006

r(k) -0.2 -0.1 0.0 0.1 0.2 0.3

m i l I i t , i, i

i

m

= ,

m

m

i l [

m

m

i

m

i

m

m,

Fig. 3.24 Correlogram of transformed deseasonalized inflows to Font reservoir

with

&2 = 0.141

The ~ value is a reduction on the variance of the transformed time series, dr2~ which equals 0.149, demonstrating that the MA(2) model is an improvement on assuming that the transformed time series was itself a realization of DWN, i.e. a 'null' ARMA(0,0) model.

The standard deviations of/31 and/~2 were estimated as 0.04 and are clearly significant on studying parameter-to-parameter standard deviation ratios. The correlogram of the errors is shown in Fig. 3.25 and suggests that the model is adequate. A histogram of the errors is shown in Fig. 3.26, and suggests that the transformation to near normality has been surprisingly successful. Finally, the values of the Box-Pierce statistic, for L = 12, 24, 36, 48 are 14.4, 28.7, 33.9 and 48.3 respectively, should be compared with the upper

3.9 Further practicalexample$ 79

ACFofC12 r(k) -0.2 0.0 0.2

i' I

1 -0.002 2 0.003 3 0.001 4 0.068 5 0,021 6 -0.077 7 0.040 8 0,023 9 -0.004

10 0.004 11 0.035 12 -0.037 13 0.049 14 -0.016 15 0.004 16 0.022 17 -0,049 18 0.030 19 0.011 20 -0.012 21 0.020 22 0.022 23 0.092 24 0.019 25 0.049 26 -0,007 27 0,006 28 -0.002 29 0.000 30 -0.009 31 -0,031 32 0.026 33 -0.029 34 0.014 35 -0.025 36 0.013 37 0.011 38 0.004 39 -0.002

u

i

1 I r !

I

m

n

m

m

Fig. 3.25 Correlograms of errors after fitting MA(2) to transformed deseasonalized inflows to Font reservoir

percentage points of chi-squared distributions with 10, 22, 34 and 46 degrees of freedom respectively. They do not provide any evidence against the adequacy of the MA(2) model.

Example 3.12 This is a simplified account of some work done by Tom Fearn and others at the Flour Milling and Baking Research Association and is reproduced with their permission. The objective was to design a control system for the addition of gluten to flour in a mill.

The background to the problem is that European wheat varieties are low in gluten, the protein necessary to give strength to bread. Gluten is extracted from a proportion of the European wheat crop and added to some of the remainder to provide suitable flour for


Midpoint Count -1.2 1 -1.0 2 -0.8 13 -0.8 30 -0.4 137 -0.2 205

0.0 1091 0.2 134 0.4 84 0.6 48 0.8 24 1.0 12 1.2 4 1.4 1

i Fig. 3.26 Histogram of errors after fitting MA(2) to transformed deseasonalized inflows to Font reservoir

baking. It is important to add the correct amount. Figure 3.27 is a diagram of an automatic control system for adding gluten.

NIROS is a near-infra-red on-line sampler which provides an estimate of the protein content of the flour at 1 minute intervals. The measuring time is 30 s. If a change is made to the gluten feeder at time t = 0, no change is noted by NIROS at time t = 0 but nearly all the change is seen at time t = 2. This system response is shown in Fig. 3.28.

The time series of protein content, less the target value obtained for the NIROS, is denoted by {zt }. An ARIMA(0,1,1) model was found to provide a good fit. (It is of inte, est to note that this is equivalent to the following model for Z,

Zt = Ixt + et

where

/xt =/xt-1 + rh

This model is physically plausible with et representing an independent sequence of

Fig. 3.27 Automatic control system for adding gluten

3.9

, . , , , .

z

J I I 0 1 2 3

Time after step change to gluten feeder

Fig. 3.28 System response to step change in gluten feeder rate

Furtherpracticalexamples 81

sampling and measurement errors and rh corresponding to an independent sequence of random disturbances.)

The ARIMA model is

VZ,= E , - OEt-1

We note that a proportion 0 of E, will disappear in VZ,+~ and this observation led researchers to propose an addition of -AEt, where A = 1 - 0 , as a control to the differenced series. Since E, is not known it is replaced by an estimate, et, obtained from the equation

et = zt - z t - 1 + Oct-1

with e0 taken as 0. The control cannot be physically applied to the differenced series. The equivalent control applied to the flour stream at time t is

t

c, = Z ( -Xe,) 0

which, after some algebra, can be shown to be equivalent to

c, = -A(z , + 0z,_z + z,_2 + . . . + 0'z,,)

Because of the delay in the system, the effect of the control at time t is not noted until time t + 2. Formally, the model for the controlled system is

t+2

Z,+2 = ~', (E,.- OE~_t ) + C, r---0

with

Ek = 0 for k~<O

Analysis of the uncontrolled process suggested that 0 was approximately 0.75. The control law performed well in tests.


Example 3.13 Hipel (1977) investigated the fit of ARIMA (p,d,q) (P,D, Q)s models to monthly flows of the South Saskatchewan River in Canada. An ARIMA (1,0,0) (0,1,1)12 model had been fitted previously, but the errors failed most diagnostic tests. In particular, the correlogram of the residuals contained significant terms up to lag 5. In view of this, an ARIMA (1,0,5) (0,1,1)12 model was tried. This provided a better fit but the errors still exhibited skewness and seasonal changes in variance. A Box-Cox power transformation with )t = -0.16649 and L = 0 was then used, before fitting the ARIMA model, with the result that the residuals gave no evidence against the hypothesis that they were a realization of a Gaussian DWN process. The final model was

where

(1 - 0.95354B)Yt = (1-0.37119B-O.13932B2-O.O8882B 3- 1.0771B 5) (1 - 0.95646B 12)Zt

Yt -" ( 1 - BI2)(Xt 1"6649- 1)/(-1.6649).

A In transform yields similar results to the power transform. The model can be written in the less compact, but more easily understood, manner

where

Yt = 0.95354 Yt-1 + Zt-O.37119Zt_1 -O.13932Zt_2-O.O8882Zt_3-O.lO771Zt_5

- 0.95646Zt_ 12 + 0.35503Zt_ 13 + 0.13325Zt_ 14 + 0.08495Zt_ 15

+ 0.10302Zt_17 (A)

Y,= W,- W,-12 Wt = -6.0064 (Xt 1.6649 _ 1)

and Xt is the river flow in month t. The Zt are zero mean Gaussian DWN.

(B) (c)

Next, we look at how to use this model to generate a synthetic time series {srt }. To use Equations (A), (B) and (C) we equate ~:t, to, and v, with Xt, Wt and Yt respectively. In this notation Xt is the assumed underlying random process, {X t} is the historic time series which is considered a realization from this process and srt is a simulated time series using the process with parameters estimated from {xt }. Similar correspondences hold for { Yt } and {Wt }. {Zt } represents DWN and {zt } are the errors in the least squares fitting procedure and can be thought of as estimates of the values taken by the Zt. A realization of DWN will be denoted by {~t}. First, we must calculate the estimate, dS, of the standard deviation of the DWN. The coefficients of the model for { Yt } are estimated by the least squares procedure and ~ is obtained by squaring the errors, adding them and dividing by the appropriate degrees of freedom. To calculate a value for v~ we need a value for Vo and values to replace Z-~6 up to Z~. We would take Vo equal to the expected value of Yo which is 0. We would then generate 18 random numbers from N(0, ~ ) and call them ~'-16, ~'-15, �9 . . , ~'o, ~'t. We could then use Equation (A) to calculate vl. Then, we would generate a further normal deviate st2 and use Equation (A) to calculate v2 and so on. To obtain t o l , . . . , to12 we could use Equation (B) if we had values for to_~z , . . . , too. We could take to_~ as the average of wt for all the Januaries in the time series, to_ 1o as the average for all the Februaries and so on until too is the average for all

3. I0 Models for continuous time random processes 83

Decembers. There would be no further problems in calculating to~3 onwards. Finally, we would use Equation (C) to transform the tot into ~:,. That is

~t = (1 - 0.16649 tot )-6.0o84

3. I0 Models for continuous time random processes

The theory for random processes in continuous time is, in many ways, more complicated than that for discrete time processes. Their consideration is necessary because deterministic dynamic systems are more easily described by continuous differential equations than by difference equations. We shall restrict ourselves to non-rigorous justification of the results required later.

3.10.1 The Dirac delta The Dirac delta is a generalized function defined by 6(0 such that

I~ x(0 8(0 x(0) dt - - o o

(3.18)

where x(t) is any function which is continuous at O. We see that ~(t) is properly defined in terms of integration, and operates on a function to give a single number. Despite its rather abstract definition we will need to use the Dirac delta on numerous occasions. It is often convenient to think of 6(0 as the limit as n tends to infinity of any sequence of functions such that

I~ (0 6,, = 1 for everyn - - 0 o

and (3.19)

LimSn(t) = { O .--.** t = O t#:O

Two choices of sequences satisfying these conditions appear in the following: a square pulse and a triangular pulse. The former is defined by

~n -1/(2n)~< t<~ l/(2n) 6.(0 (3.20) L0 otherwise

and the latter by

8,(0 = - Itl - l ln<~t <-lln (3.21) otherwise

Their shapes are shown in Figs 3.29(a) and (b).

3.10.2 Autocovariance function For a continuous time stationary random process the acvf is defined by

y(u) = E[(X(t) - tz)(X(t + u) - / x ) l , where tx = E[(X(t))] (3.22)


~(t)

r~ ,,

- 1 0 1 2n 2n

(a)

-1 r/

o ~ "-/

Fig. 3.29 (a) Square pulse; (b) triangular pulse

~(t)

Co)

Equation (3.13) is the discrete form of this equation. The acf is defined as p(u)= y(u)/y(0), and is a generalization of Equation (3.15).

3 .10.3 E s t i m a t i o n of the m e a n and a u t o c o v a r i a n c e func t ion If a continuous signal is available it can be processed by an analogue computer. The sample mean X is defined by

1 [ rx( t ) dt (3.23) "r The sample acvf, which is an estimator of the acvf of the random process, is

c(u) = } (X(t)- X)(X(t + lul)- x ) dt (3.24)

3 .10.4 W i e n e r p rocess The Wiener process is the process in continuous time analogous to the random walk in discrete time. The key to the similarity is that they are both processes with uncorrelated increments. In the notation of Section 3.6.2, typical consecutive increments are Xt- Xt-~ and Xt-1-Xt-2. These are Zt and Zt-~ respectively which are independent and hence uncorrelated. We now consider

I t

X(t) = Z(0) dO (3.25) - - o 0

where X(t) is a continuous process with uncorrelated increments. From this definition

E[(X(tz) - X(tl ) ) ( X ( t 4 ) - X(t3 ))]

will be zero if the intervals (t~, t2) and (t3, t4) do not overlap. If they do overlap, as shown below

t tl

t3

3. I0 Models for continuous time random processes 85

we may write the expectation as

E[(X( tz ) - X(t3 ) + X(t3 ) - X( t l ))(X(t4) - X(t9 ) + X(tz ) - X(t3 ))]

When we use the definition of uncorrelated increments, the above expression reduces to

E[(X(t2 ) - X(t3 )) 2]

We use this as a starting point for an explanation of continuous time white noise.

3.10.5 White noise in continuous time Following on from the previous section we assume

�9 E[(X( t z ) - X(t3))21 = v l t z - t3[ for some constant V. We will show that Z(O) in the definition of the Wiener process is analogous to DWN in the random walk. Equivalently, the 'derivative' of the Wiener process is continuous white noise in an analogous manner to the differences of the random walk being DWN. We therefore start by considering the 'derived process'

Y(t) = X ( t + h) - X(t)

where, for convenience, we take h as positive and therefore h tends to zero from above. Using the assumed form of the expectation and the definition of the acvf we find, by equating q , t2, t3 and t4 with t, t + h, t + u and t + h + u respectively, that

V(h- lu l ) h 2 lul

, /~u) =

0 otherwise

If h is identified with (l/n) we can see that this aevf is the triangular pulse of Fig. 3.29(b) multiplied by a factor of V. If h is now allowed to tend to 0 from above then y(u) tends to V6(u), where 6(u) is the Dirac delta. The process Y(t) was defined to be equivalent to Z(t) in the definition of the Wiener process. A function Z(t) is said to be white noise with intensity V if it has an acvf

7(u) = V6(u) (3.26)

Furthermore, if the increments, X ( t + h ) - X ( t ) have a Gaussian distribution, Z(t) is referred to as Gaussian white noise. Continuous white noise has 'infinite variance' and is physically unrealizable. Nevertheless, it does provide a reasonable approximation to certain physical processes, such as voltage fluctuations in a conductor due to thermal noise and the impulses imparted by fluid molecules colliding with a suspended particle. It plays a similar role to DWN as a driving mechanism for more complicated processes, a topic discussed in the next section. A very important practical application of continuous white noise is in system identification, because of the relationship between the transfer function of a linear system and its response to white noise. Commercial instruments called 'spectral analysers' provide excellent approximations to continuous white noise sources, analyse signals from the transducers on the system and display estimates of the transfer function. Nearly all modern instruments sample the continuous transducer signals and process them digitally. The theory behind this is covered in later chapters.


3.10.6 Linear processes In discrete time, a general linear process is defined by

o o

X t - tz = ~, ~Zt-~ (3.27) i=0

where {Zt } is DWN. You should note that any ARIMA model can be expressed in this form. The analogous process in continuous time is

f- X ( t ) - tx = h(O) Z ( t - 0) dO (3.28) 0

where Z(t) is white noise of intensity V. If E[Z(t)] = 0 then E[X(t)] =/z which justifies the notation. We will now investigate

the acvf. By definition

"y(u) = EI (X( t ) - tx ) (X( t+ u)- /x)]

= h(O)h(~)E[Z(t-O)Z(t+u-4~)]dOd4~ 0

Since Z( ) is white noise, the expectation is the product of its intensity with a delta function whose argument will be zero when ( t - 0 ) equals (t + u - ~b). Hence

~,(u) = h(O)h(4~) V6(u + O- 4)) dO d4~ 0 0

Applying the formal definition of 6 ( ) , Equation (3.18), gives the simplification

y(u) = V h(O) h(u + O) dO 0

In particular

,/(o) = v h2(o) do 0

and

I ~ p(u) - h(O)h(u + o) dO/ h:(O) dO 0 0

The condition for the process X(t) to be stationary is

| I h ( 0 ) l dO <~ M 0

for some finite number M.

Example 3.14 It is reassuring to see that the above results do give 'sensible' results for time averaging white noise of intensity V over an interval T. The time average is given by

3. I I Exercises 117

f !

X ( t ) = (l /T) Z(~k) d~b t - T

which is of the form

I 0 0

X ( t ) -- h(O) Z ( t - O) dO o

0 0 ~ 0

1 where h(O) = -~ 0 ~ O~ T

0 0~T

Applying the equation for 7(0) derived in Section 3.10.6 gives

f r 1 V

Var(X(O) ffi V d# = - r

For comparison, the analogous result in discrete time is

1 Var(Z ) = Var-~ ~ Zk = ----

k=l T

3. I I Exerc ises

Q

0

11

Plot the continuous form of the logistic curve (see Section 3.3.4)

1 Y ~" i 4- (0.8)'

for t ranging from 0 to 20. Sketch the continuous forms of the modified exponential and logistic curves. That is

y "- a - br t and y = l / (a + br t) for 0 < t

The following data are the average flows (m 3 s-~) through the spillway of a dam on the Island of Losden over the past five years. There are two seasons on l .x~en , the dry season which is quite wet, and the wet season which is very wet. During the past five years there have been extensive building programmes in many parts of the catchment area of the dam.

17 148 86 285 97 213 72 375 4 301

(i) Plot the data and comment. (ii) Deseasonalize the data using the centred moving average method, assuming

(a) additive seasonal effects; and (b) multiplicative seasonal effects. Estimate the seasonal effects and plot both sets of deseasonalized data on the same graph as the original data.

(iii) Draw a straight line through whichever deseasonalized set looks the more appropriate.

(iv) Predict the average flows for next year.


~ Sales of paint and varnish in million litres for the quarters of years 1974-1979 are given below.

, , , , , , . , , , . , , , , , , , , , ,.

1st quarter 2nd quarter 3rd quarter 4th quarter

I0.

1974 136.9 143.3 144.1 131.6 1975 134.5 147.4 140.8 128.5 1976 143.4 147.9 152.2 143.8 1977 154.9 161.7 157.0 145.3 1978 155.3 165.2 167.3 148.4 1979 156.1 182.2 176.6 153.6

�9 . , , , , , , , , , , . ,

Calculate the centred moving average of four quarters. Estimate the seasonals assuming a multiplicative model. Produce forecasts for 1980.

5. The following data are a subsequence of wave heights, measured in mm relative to still water level at 0.1 s intervals, in a wave tank.

- 1 2 314 241 375 59 -550 -439 -121 367 478

(i) Plot the data. (ii) Plot xt+l against xt.

(iii) Calculate c(0), c(1), c(2) and hence r(0), r(1) and r(2).

6. Plot a realization of the autoregressive process

X t --" S t _ 1 -- 0 . 8 X t _ 2 + Z ,

where Zt are taken from a table of random standardized normal deviates. 7. If xt = a cos tto and to is a constant in (0, 7r) for this realization, show that

r(k)--> cos kto as N---> oo

Hint: 2 cos A cos B = cos (A + B) + cos (A - B) 8. Find the acf of the MA(2) process given by

2

x, = Z z,_,/3 i=0

9. A company manufactures burners for domestic boilers. A critical dimension is the distance from the front of the burner to the first gas outlet hole (x,). ARMA models have been fitted to data from the past few months and an AR(1) has been found to be a good model for the process {X, }. Find an expression for the mean of a sample of four consecutive readings in terms of O{x and a. In fact, a was estimated as 0.21. Calculate the ratio of the variance of the mean of a sample of four consecutive

readings to the mean of a random sample of four. A random process is composed of an ensemble of sample functions, each of which is a sine wave of constant amplitude a and frequency to. The frequency is the same for all samples but the amplitude varies randomly from sample to sample. Each sample also has a constant phase angle ~b, which also varies randomly from sample to sample. A typical sample is therefore represented by the equation

x(t) = a sin ( tot-4 ')

where a, to and ~ are all constants for the sample. If the joint PDF for a, 4, is

f (a , ~b) = [1 + ( 2 a / L - 1) cosqb]/27rL 0~< ~b~<27r, O<~a<~L

11.

12.

3. I I Exercises 89

determine the ensemble averages E[X2], and E[X2[tbo]. State with reasons whether the process is ergodic and whether it is stationary. The following data are the inflows into a reservoir during the 'wet' and 'dry' seasons of the past three years.

105 65 104 72 123 75

Denote the underlying random process by {Xt }. A hydrologist has fitted the model,

r, = aY,-l + Z,+ ~Z,-i

where

Yt = A t - X t - 2 and {Z t) is DWN with zero mean

to data from a considerably longer record.

(i) What is the model in ARIMA (p, d, q) (P, D, Q)s notation? (ii) Are either of the processes {Xt } or { I:, ) stationary?

(iii) In fact, a and fl were estimated as 0.4 and -0 .3 respectively. Make predictions for the next two years if the most recent residual equals 8. Remember that you were given the recent values of X at the beginning of the question.

Suppose

Yt = mt + Zt

where Zt is zero mean DWN and mt are seasonal effects such that m~ represents the January e f f e c t , . . . , m~2 the December effect, m~3 equals m~ and so on. Then

Vt2 Y, = Z, - Z,_ 12

If this moving average model is correctly identified, we would obtain the following in a simulation

VI2 YI2 -- Z12 - Zo

and taking Zo equal to 0 gives

YI2 -" Yo + ZI2

Similarly

Y24 = Y12 + 224-" ZI2 = Yo + Zt2 + Z24- Zt2 - Yo + Z24

If Yo is identified with m~ for some value of i between I and 12 all is well. However, the model is unlikely to be identified exactly. For example, it is estimated, from a short time series, as

Vi2 Yt = Zt- 'O.9Zt-12

YI2 = Yo + Zl2

as before, but

I:24 = E~+ Z24+0.1Zj2

Y36 = Yo + Z36 + 0.1 Z24 + 0.1 Z i2

etc. What is the variance of Y after n years?

90 Time v~ry'ing signals

13.

14.

15.

16.

Let X1 and X2 be independent random variables with mean 0 and variance o2. Calculate E[c(1)], Var[c(1)], and approximately, E[r(1)] and Var[r(1)]. Research workers at the Laboratorio National de Hidniulica in Venezuela fitted an ARIMA (5,0,0)(0,1,1)73 model to 5-day mean levels of the Orinoco River at Palua.

Let {Y t} represent the corresponding random process and {Zt) represent a DWN process. Express the model in the form Yt equal to a linear combination of past values of Y and the Zt sequence. A random sample of 100 standard normal variates was drawn using the MINITAB statistical package. The sample mean and standard deviation were 0.1959 and 0.9614 respectively. What is the probability of obtaining a sample mean this far from 0? No values in the correlogram up to lag 20 lay outside the -0.01 + 0.2 limits. An ARMA(2,2) model was fitted with the following result

Xt = 0.6651 - 1.4636Xt_1 -- 0 .9601Xt_2 + Zt + 1.4624 Zt-1 + 0.9206Zt_2

The residual mean square was 0.9240. The ratios of the estimated parameters to their estimated standard deviations were 2.05, -33.53, -7.06, -20.01 and -5 .20 respectively. No values in the correlogram of the residuals up to lag 20 lay outside the -0.01 + 0.2 limits. How do you account for these highly statistically significant parameter estimates? What further information would you request? If

T-~t + Y = X

where x and y are functions of time, show that

where

f ao

y(t) "" x(u) h(t- u) du m o o

h(O) ffi {~/t)exp(-#lT) 0 ~ 0

0<0

4 Describing signals in terms of frequency

4.1 Introduction

Since the ideas of Fourier series are so central to spectral analysis, it seems appropriate to say something of the mathematician who first used them. Jean Baptiste Joseph Fourier (1768-1830) was the 19th child of a French family. His father was a tailor but Jean was orphaned at the age of eight and educated in the Benedictine School at Auxerre. He was appointed an Assistant Professor in 1795 at the founding of the Ecole Polytechnique in Paris. In 1807 he published a paper describing how trigonometric series would provide a solution of the heat conduction equation to satisfy given boundary conditions. He later rose to prominence as an administrator. In common with most brilliant mathematical advances, Fourier's work was influenced by predecessors. Daniel Bernoulli had done some work on trigonometric expansions 50 years earlier.

This chapter begins with the idea of a finite Fourier series, which is a finite sum of sine waves with periods and phases chosen so that it coincides exactly with a finite number of data plotted against time. A Fourier series is an infinite sum of sine waves, which converges to a continuous signal defined over a finite time interval. In fact, the signal can have a finite number of discontinuities at which the Fourier series converges to the average of the left and right-hand limits.

Botli these Fourier series are periodic with period equal to the length of time over which the signal is defined. The Fourier transform is obtained by imagining that the duration of the continuous signal tends to infinity. Finally, the discrete Fourier transform is defined for an infinite sequence.

Spectral analysis is primarily concerned with the identification of the different frequency components of a given signal. The phase information cannot be recovered from a spectrum. Various forms of Parseval's theorem are crucial for this. Unfortunate- ly, Marc Antoine Parseval Des Ch6nes' (1755-1836) mathematical career was inter- rupted when he had to flee France after writing poems critical of the Napoleonic government.

4.2 Finite Fourier series 4.2.1 Fourier series for a finite discrete signal The usual physical unit of frequency is Hz, but some of the algebra of Fourier analysis looks less daunting when expressed in radians per sampling interval. The formulae in this book are all in terms of the latter, but results can be scaled and presented with frequencies in Hz.


We start with a finite discrete signal. It is assumed that observations are separated by a constant time interval, which is conveniently taken as one unit. It also simplifies matters to assume that the number of observations (N) is even~if necessary, the first observation can be discarded to achieve this. The duration of the signal, T sampling units, is given by

T = N

The sequence of observations will be written

{xr} r = - n , . . . , - 1 , 0 , 1 , . . . , n - 1

where 2n = N. In many situations Xr will be a sample from a continuous signal x(t)

x( t) - T/2 ~ t <- T/2

in which case

x -x(r)

We hope to find a function of the form

n-1

$(t) = Ao + 2 ~ {Am cosmtolt +Bm sinmtolt} + A, cosntolt (4.1) m = l

with tol = 21r/N such that $(r) equals Xr. In this equation, to1 has a frequency of one cycle per signal length and is known as the fundamental frequency. It is entirely dependent on the record length and therefore only has physical significance if the record length also has significance.

The function i(t) is made up of a constant value, which turns out to be the mean value of {x~ }, and harmonic components at n multiples of the fundamental frequency tol. The coefficients Am and Bm govern the amplitude and phase of the harmonic components, and it seems reasonable to suppose that 2n adjustments are sufficient to obtain a curve that passes through 2n points. Our first objective is to find formulae for the coefficients Am and B,, in terms of the data. The use of complex numbers makes the algebra much easier, and this more than compensates for the increased abstraction. (A review of elementary complex number theory is given in Appendix I.) In addition, nearly all the literature is based on 'complex forms'. We therefore define

Xm = A m - j B m O<~m<~n-1 (4.2)

X _ m = A m + j B m O<~m<~n

so that ~f(t) can be expressed as

n - I

~(t) = ~ Xm exp(j27rmt/N) m ' - - - n

(4.3)

We therefore need to find X,, so that

n - l

Xr = ~, X,, exp(j27rmr/N) (4.4a) m = - - I1

Multiplying both sides by exp(-j21rkr/N), where k is an integer between - n and ( n - 1), and summing over r leads to

4.2 Finite Fourier series 93

n - I n - I n - I

~, x~ exp(-j27rkr/N) = ~, ~, Xm e x p ( j e ~ m - k)r/N) r = - n r= -n m= -n

= s Xm exp (j2"rr(m - k) tiN) m = - - t l r - - - - / 1 . ':

I f m equals k, all the terms in the bracketed sum are one, and the sum is therefore 2n. I f m is not equal to k the fol|owin 8 argument shows that the sum is zero, that is

n-1 2n- I

exp(j2,n'(m- k)r/N) = exp(-j2,n ' (m- k)n/N) ~, exp (j2"tr(m- k)r/N) r = - n r=0

j

and so, using the standard result for the sum of a geometric progression, this can be written as

= exp (-j2"n(m - k)n/N) { 1 - exp (j27r(m - k)2nlN)} I {1 - exp (j2"n(m - k)lN)}

Since 2n equals N and ( m - k ) is an integer, e x p ( j 2 ~ m - k ) 2 n / N ) equals 1, and the numerator of the expression equals zero. The denominator is not zero because ( m - k) is assumed to be non-zero and the range of m and k values restricts it to be at most 2 n - 1, which is less than N. This simplification of the bracket leads to

1 n - l

Xk = ~ ~, XreXp(-j2ctkr/N) r = - - I1

(4.4b)

for - n <~k<~n- 1. The subscript k is just a 'dummy' and can be replaced by m, or any other letter. The introduction of complex numbers has led to m taking both positive and negative values. In expressions like exp(j21rmr/N), where r is reserved to represent time, the absolute value of m is equal to the frequency in terms of cycles per record length, and IXml is the amplitude of the harmonic at this frequency. It is more convenient to allow frequency to take negative values than to refer continually to 'absolute values'. In physical terms, a negative frequency is identical to the positive frequency with the same absolute value.

It is a straightforward exercise to verify that n-1

Am = (l/N) ~, x, cos(21rmr/N) r - - " - - n

and (4.5) n - I

Bm= (1/N) X x, sin (2~rnr/N) /'--'-- - - / 1

for 0 <~ m ~ n, and the particular results that A0 equals s and Bo and B. are zero. As x(t) is a continuous signal which does not oscillate between sampling points it will

provide a reasonable approximation to any underlying continuous signal x(t), provided x(t) does not oscillate between the data points. This requirement can be expressed more formally by the condition that the sampling frequency should be higher than any frequencies in the signal.

If we now define

R,,. = (A2m + B 2 ) 1/2, and ~,,, such that tan ~b,,. = B , . Am

then ,f(t) can be written in terms of Rm and 4~m. That is


n - I

,~(t) = Ro + 2 ~, R,, cos(mtolt+$m) + R,, cosntolt (4.6) m = l

4 . 2 . 2 P a r s e v a l ' s t h e o r e m

Parseval's theorem for a finite Fourier series can be stated as follows

n - I n - I

1IN Z x 2 = R 2+2Z R 2 + R 2 rf -n m = l

Since Ro is equal to the mean value of {Xr } the result can also be written as

n - I n - I

( l /N) ~ (Xr-,~) 2 = 2 E R 2 + R 2 (4.7) r f - - n m = l

When it is written in this form we can see that it is a partitioning of the variance of the sequence {x~ } over the n frequencies. It is the usual mathematical convention to refer to the cosine function at the fundamental frequency, to~, as the first harmonic, the cosine function at the frequency 2to~ as the second harmonic and so on. We will refer to a plot of the contribution to the variance versus the frequency of the harmonic as a Fourier line spectrum.

The proof of Parseval's theorem is much easier if complex numbers are used. It is then expressible in the form

n - 1 n - 1

z Z IX,,,I 2 (4.8) ( l /N) Z Xr = r ' - --/l m "~ --/1

The contribution to the mean square, 2R 2 in the previous statement of the theorem, is divided into two parts. Each is equal to IX~I 2, which itself equals R 2, one is at frequency mto~ and the other at frequency -into1.

We already have the result that

n - I

x~ = ~, Xm exp(j2~rmr/N) m---- --n

The complex conjugate of this result is

n - l

x*= ~, X~ exp(-jE~rpr/N) p=-n

Now, take the products x~x* and sum over r, that is

n - I n - I n - ! n - I

~, x~x*- ~, ~, ~ XmX~ exp( jE~m-p)r /N) r--n rffi-n m=-n p=-n

n-I n-I ( n-I ) = ~ ~ XmX~ ~ e x p ( j 2 ~ ( m - p ) r / N )

/11----71 p - ' - - n r - " - - t l

It was explained earlier in this section that the sum in brackets is zero unless m equals p, in which case it equals N. Hence, since the data Xr are real

n - I n - l

X x 2 = N X IX,,,I 2 r--- - - n m - " - - B

4.2 Finite Fourier series 95

Example 4. I The following data (Carter and Challenor 1979) are averages of the maximum significant wave heights for each month at Seven Stones' Light Vessel, situated a few miles west of Land's End, UK. The averages are taken over seven years up to 1977 and the units are metres.

January . . . . . . 8.39 . . . . . July " '3.92 February 7.44 August 4.33 March 7.54 September 5.58 April 5.94 October 6.05 May 5.20 November 7.02 June 4.23 December 7.94

The 12 values from January to December will be treated as a digital signal with N equal to 12. The formulae of this section lead to the following breakdown.

Source m Contribution

Am Bm to mean square Mean Fundamental 2nd harmonic 3rd harmonic 4th harmonic 5th harmonic 6th harmonic

0 6.131667 0 37.5977 1 -1.022191 0.024063 2.0909 2 -0.031667 0.036084 0.0046 3 -0.057500 0.040833 0.0099 4 -0.028333 -0.093819 0.0192 5 -0.037809 -0.044270 0.0068 6 -0.143333 0 0.0205

39.7496 , i , , | , , , , , , , , , , , , , , , , , , , , , , ,

The data, together with the mean, fundamental and fourth harmonic are shown in Fig. 4.1. The Fourier line spectrum is shown in Fig. 4.2. The mean and the fundamental harmonic give a good description of the seasonal variation.

4.2.3 Leakage Let us suppose that a continuous signal is sampled at equal intervals and the total number of data is N. A finite Fourier series will fit harmonics at frequencies 2r (radians/ sampling interval) for k = 1 , . . . , N/2. If the signal is itself harmonic with frequency equal to 2r for some integer m, then its Fourier line spectrum will consist of a single spike at 2r If the signal is harmonic at some frequency between 2r and 2w(m + 1)/N its line spectrum is not confined to these two frequencies but 'leaks' out to others. The three harmonic signals below illustrate this phenomenon

(a) sin (wt/2) (b) sin (3r (c) sin (5~t/8)

sampled at times t = O, 1, 2, 3, . . . . 7.

96 Describing si#nals in terms of frequency

9Fx(t)

~ 7 |

C

"~ 5

0.2 r-

ill O ll -0.2 L-

F i g . 4 . 1

= 6 .132 + 2{-1.022 cosOzt/6) + 0.024 sin(~t/6)}

Mean level

- �9 Raw data

Mean + first harmonic as defined by x(t)

I I I 1 I i I I, i. ,I I I ! -6 -5 -4 -3 - 2 - 1 0 1 2 3 4 5 6 t

(Jan.) (June) (July) (Dec.) (a)

y(t) = 2 {-0.028 cos(4~t/6) - 0,094 sin(47zt/6)}

.

Fourth harmonic as defined by y(t)

(b)

Average significant wave height at Seven Stones' Eight Vessel and (a) first and (b) fourth harmonics

2.09 . . . . . . F

0.02 ~

o

0

= 0.01 .9 .I0

8

0.00 1 2 3 4 5 6 Harmonic

Fig. 4.2 Fourier line spectrum for significant wave heights

4.3 Fourier series 97

0.5

.9.o 8

0.0

.~ 0.5

.9.o 8

~ 0.0

(a)

1 2 3 4 Harmonic

(b)

1 2 3 4 Harmonic

0.5

C: .Q

8 0.0

Fig. 4.3

(c)

, I 1 2 3 4

Harmonic

Line spectra of harmonic signals; (a) and (b) no leakage; (c)leakage

The corresponding Fourier line spectra are"

(Mean) 2 1st harmonic [z r/4 rad/samp int] 2nd harmonic [~r/2 rad/samp int] 3rd harmonic [3~r/4 rad/samp int] 4th harmonic [Tr rad/samp int]

They are shown in Fig. 4.3.

(a) (b) (c) 0.0000 0.0000 0.0070 0.0000 0 . 0 ( ~ 0.0225 0.5000 0.0(0)0 0.1821 0.0000 0.5000 0.2543 0.0(0)0 0.0000 0.0350

4.3 Fourier series

A continuous signal x(t) defined for - T/2 <<. t <~ T/2 can be represented by an infinite sum of harmonic terms known as its Fourier series representation. The Fourier series is defined for all t and is periodic with period T. The Fourier series converges to x(t) for

- TI2 <~ t <~ TI2, and to the periodic extension of x(t) outside this interval, except at points of discontinuity when it converges to the average of the left and right-hand limits. Convergence at points near a discontinuity is slow, a feature known as the Gibb's phenomenon. In engineering applications, signals are usually in the form of a fluctuating voltage and sudden jumps are common, causing a discontinuity in the record. An


example is the signal from a contactless displacement sensor when various objects on a production line pass beneath it. When the sensor is unaware of any object passing it will register a reference DC level which will jump to a lower value voltage as soon as a passing object is detected. If the voltages at the beginning and end of a record differ, there are artificial discontinuities because the imaginary periodic extensions of the signal do not join at the ends.

We start by assuming that a suitable Fourier series exists and write

x(t) = Ao/2 + ~ (Am cos(2rrmt/T) + Bm sin(2zrmt/T)} m=l

In terms of complex numbers, if we define

Xm = (Am -iBm)~2

and

X-m = (Am + iBm )/2

(4.9)

then co

x(t) = ~, X,. exp(j2zrmt/T) (4.10a)

It remains to find an expression for X,,. Multiplying both sides of the identity for x(t) by exp(-j2~rpt/T), where p is an integer, and integrating from -T/2 up to T/2 gives

f | J T/2 r/2 x(t) exp(-jE1rpt/T) dt = ~, Xm exp(jE~r(m-p)tlT) dt -7"12 m=-= - T12

If m equals p the integral on the right is equal to T. If m is not equal to p the integral is

T exp (j2"rr(m - p)t/T) ] j2,n-(m - p )

7"/2

- 7/2

= T(exp(j2.tr(m-p))-exp(-j~r(m-p)))/(j2"rr(m-p))

and since ( m - p) is a non-zero integer this is zero. Hence

It/2 x(t) exp(-j21rpt/T) dt (4.10b) T X p = - TIE

The whole argument is essentially the same as that given for the finite Fourier series, although the details are, if anything, easier. Again, p is just a 'dummy' integer variable, and taking real and imaginary parts of the identity for Xp gives

f T/2 Am = (2/T) x(t) cos(2~rmt/T) dt -TI2

and

f T/2 B,, = (2/T) x(t) sin (2r dt -TI2

Parseval's theorem may now be written as

4.3 Fourierseries 99

f T/2 | (liT) x2(t) dt ~- ~'. Ix l (4.11) -772 m=-~

The proof of this result is similar to that given for the finite Fourier series case. We have Qo

x(t) = ~, X,. exp(j2mnt/T) mm--oo

and can write

x*(t) = ~, X~, exp(- j2wpt /r )

Then

f ra x2(t) dt ,= x(t)x*(t) dt -T/2 -772

" ~, ~, XmX~, m exp(j2"tr(m-p)tlT) dt m-~- ,o p = - ~ .-TI2

and the result follows.

E x a m p l e 4.2

A signal iS defined by

- 1 - w ~ t ~ < 0 x(t) - 1 O~t<~ w

The jump could represent a change from 'doors open' to 'doors closed' in a metro train. Before calculating the coefficients for a Fourier series it is always worthwhile checking whether the signal has any obvious symmetries. If it is symmetrical about the vertical axis--formally, an even function~its Fourier series will consist of cosine functions only. If it has a rotational symmetry of half a turn about the origin--formally, an odd function--its Fourier series will consist of sine functions only. A rotational symmetry about some other point on the vertical axis would require the addition of a constant term.

The above signal is an odd function and so all the A,,, coefficients are zero. If we apply the formulae we obtain

B2m-I ~" 4/[w(2m- 1)], and B2m = 0

Therefore 0o

x(t) ~" E {4 sin(2m- 1)t}/['rr(2m I)] m--I

The function x(t) and the first few terms of the Fourier series are shown in Fig. 4 .4 .

I 00 Describing signals in terms of frequency

I ,1 - 2 n - ~

I I, I

-1 2~

sin t

v v

-,,...., ~ g sin 3 t }

I ~ A A

\ 4 {sin t + 1 1 ~ sin 3t + ~ sin 5t}

Fig. 4.4 Step function and its Fourier series

4.4 The Fourier transform 4.4.1 Fourier transform We now turn our attention to the case of a signal x(t) defined over the infinite interval - ~ ~<t~< oo. A heuristic approach is adopted to introduce the Fourier transform, rather than the provision of a formal proof of the relationship. We start by writing the Fourier series for x(t) defined for -TI2<~t~ T/2 as

oo

x(t) = ~, (TXm) exp(j27rmt/T) (I/T) m.- .~oo

that is

f TI2 (rXm) = x(t) exp( - j2zrmt / r ) dt -T/2

Now write to for 2rrm/T, dto for (21r/T), define X(to) as TXm and let T tend to infinity. Then

1 [= ej u x(t) = ~ _= x(oo) d,,, (4.12a)

and

I 0o

X(to) = x(t) exp(-jtot) dt - - o o

(4.12b)

X(to) is the Fourier transform of x(t), and so x(t) is considered the inverse Fourier transform of X(to). Together, the two equations form a Fourier transform pair. Although this form arises naturally from our development of the theory, some other authors have

4.4 The Fourier trat~orm I01

the factor 1/2r associated with X(to) and yet others have the factor l / X / ~ associated with both equations. We can see from the definitions that if x(t) has Fourier transform X(to) then 2r is the Fourier transform of X( - t ) .

We now turn to Parsevars identity. For the Fourier series this can be written

i o0 oa rn x2(t)dt= T E IXml2---(1/2 r) E ITXml2(2 /T) - 7 7 2 m ~ - ~ m = - o ~

If we now let T tend to infinity as before we obtain

f| 1 I | 12 x2(t) dt = - - IX(to) dto -| 2r -|

Classical theory requires that

x t dt<~oo

for the Fourier transform pair to exist. However, the use of generalized functions allows this condition to be relaxed. Thus, the Fourier transform of a sine wave can be taken.

The results of this section are crucial to our development of spectral analysis and it is worthwhile spending some time on their interpretation. First we note that

X 2 t d t - o o

is the total variability of the signal rather than the time averaged variability (the variance). In physical terms we are considering energy rather than power. If we revert to Parseval's Theorem, as stated in Section 4.3, we see that a component IXml 2 of the variance of the signal is attributed to the discrete frequency 2mn/T. This component of variance is equivalent to a component TIXml 2 o f the total variability. The limiting process can be thought of as attributing this variability to a frequency band of width 2~/T, i.e. the separation of the spikes. Thus, the average variability is TIX,,, 121(2r which can be thought of as the 'energy' per (radiardsampling interval) over the frequency band centred at 2mn/T. The average variability can be rearranged as lTX,,,12/(2cr). Again, allowing T to tend to infinity we see that the average variability corresponds to (1/2r IX(to)l 2. Thus, the total variability is attributed over a continuous frequency range.

The Fourier transform allows for a continuum of frequencies which will be needed if an infinite length signal is aperiodic. It is not the signal's infinite length itself which leads to a continuum of frequencies, since a periodically extended finite signal has infinite length yet only requires frequencies separated by 2r but its aperiodicity (the lack of a periodically repeating pattern).

Example 4.3 If x(t) = e-Itl

then f0 X(<o) = e t e - j~t dt + e -t e -j '~ dt - - o o 0

2

1 + t o 2


Fig. 4.S- Double exponential and its Fourier transform

The function x(0 and its Fourier transform X(co) are shown in Fig. 4.5.

Example 4.4 The function

{; x(t) = itl>~b has the Fourier transform X(~)= 2asin(~b)loJ. The function and its transform are shown in Fig. 4.6.

~, x(t)

-b b % V : , [ t I ! 1 1 3

2b b 2b Fig. 4.6 Pulse and its Fourier transform

Example 4.5 For x ( t ) = exp(-alt[) c o s ( ~ ) we can show that

X(r = a/{a 2 + [to+ too] 2} + a/{a 2 + [to- ~)]2}

The function and its transform are shown in Fig. 4.7.

~,.V/~jIi/~v.,. (t)

v

f

Fig. 4.7 Damped harmonic and its Fourier transform

4.4 The Fourier transform 103

Example 4.6 The previous three examples were even functions and their Fourier transforms were therefore real even functions. In this example we look at an asymmetrical function defined by

[0 t >i 0 X(t) l e x p ( - 0 0<~t~<oo

Its Fourier transform is X(to) = 1/(1 + jto), which is complex.

4 . 4 . 2 G e n e r a l i z e d f u n c t i o n s

The Fourier transform of 8(0, the Dirac delta function, is

I | e - j ~ dt = 1 6(t) ~co

by definition of 8(0. It follows from the 'symmetry' result of Section 4.4.1, that the Fourier transform of 1 is

2zrS(co). This result can be used to find the Fourier transform of simple harmonic functions.

For instance

Therefore

B u t

and so

x(t) = cos(21rl)t)

= (exp (j2~rl~t) + exp (-j2~f~t))/2

Hence

X(to) = ~ exp(j2zrt~t) exp(-j tot) dt

I oo + exp (-j2"n~t) exp (-jtot) dt

I | 1 exp( - j to t )d t = 21r~(to)

f?| 1 exp [-j(to + 2~rl))t] dt = 2~rS(ta + 2~'fl)

X(to) = It{ 8(to - 2~rfl) + $(to + 27rf/) }

We can check this by using the definition of the inverse transform, i.e.

1 x ( o ) do = ~ - 8( ,o- 2rrt~) + 8(,0 + 2 m ) e-j~ do

2~r -| -| -|


Now substitute the new variable u for w__+ 27r1~ in the respective integrals. Using the definition of 8(u) given in Equation (3.18); that is

I| a(u) f (u) = f (O) du

it follows that

I| 1 ej 2,M)t ,n~t 1 .: X (w) e j'~ dw = ~ { + e -j2 } 277" - |

= cos (j2zr~t)

4 .4 .3 C o n v o l u t i o n i n t eg ra l s

The integral on the right-hand side of

I" y(t) = h( t - ,)x(~') dr (4.13a)

is known as a convolution integral. If the process which produces the output y(t) from an input x0") is linear and stable we can think of h(t - r) as an inpulse response[unction. The condition for a linear system to be stable is

I =

Ih(u) l - - o o

The concept of an inpulse response function is explained further in Appendix I. If we now consider the Fourier transforms of x(t) and y(t) we can provide frequency

dependent relationships between the input and output. In particular, we may show that

Y(w) = H(w)X(w) (4.13b)

where H(w) is the Fourier transform of h(t). H(w) is often referred to as the transfer function.

To prove this, we start from the definition of the Fourier transform of y(t), i.e.

}'(to)= exp(- jwt) h(t - r)x(r) d dt

Next, introduce the variable u defined by

u = t - - , r

That is

} Y(w) = exp(- jwu) exp(-jwz)h(u)x(z) d1" du - o o - o o

since the Jacobian of the transformation is 1. Rearranging we have

s {s ,) Y(w) = h(u) exp(- jwu) x(r) exp( - jw , ) d du

and since the inner integral is a function of w only

4.5 Discrete Fourier transform 105

I = I = Y(a,) = h(u) exp(-j,ou) du x('r) exp(-jcor) dr ~ o o ~ a o

that is

v(,o) = n(~,) X(,,,)

We shall also make frequent use of the similar result relating multiplication in the time domain to convolution in the frequency domain. That is, if

p(t) = w(t)s(t) (4.14)

then

11 - I,(~,) = ~ _. w ( , o - o)s(o) dO (4.15)

The proof is essentially the same as that given above and is left as an exercise.

By symmetry we also have P(tu) = ~ S(to-O)W(O) dO

4.5 Discrete Fourier transform 4.5.1 Discrete Fourier transform for an infinite sequence We now suppose that the signal x(t) defined for - ~ ~ t ~ ~ is sampled at equal time intervals to give the sequence

{x~} r = . . . , - 1 , 0 , 1 , . . .

The discrete Fourier transform of the infinite sequence {x~ } is defined by

XD(to) = Y. x, exp(-jtor) -1r<~ to~< 7r

As usual, the units of to are radians/(sampling interval). The inverse transform is

1 I " X r -" ~ XD(tO ) exp(jtor) dto

- - I r

0o

A sufficient condition for XD(tO) to exist is Y. Ix, I ~ r = - - o o

(4.16a)

(4.16b)

If we compare these definitions with those of the Fourier transform we can see that XD(to) is an approximation to X(to). The approximation arises from the 'errors' in assuming the Xr are constant over the interval [r,r + 1]. If X(to) is zero whenever (to) exceeds It, then Xo(to) and X(to) are identical. This result, which is intuitively reasonable, is proved in Appendix IV.


0V V Fig. 4.11 Aliasing of signals at 1 Hz and 4 Hz when samples are at 0.2 s intervals

4.5 .2 Nyqu i s t f r e q u e n c y The frequency of Ir radian/(sampling interval) is known as the Nyquist frequency. If the length of the sampling interval is A seconds the Nyquist frequency is ~r radians/second or equivalently 1/(2A) Hz. Frequencies which are higher than the Nyquist frequency will, after sampling, be indistinguishable from frequencies below the Nyquist frequency. A single example illustrates all the essential points. We will look at a harmonic wave with a frequency of 1 Hz, which is sampled at intervals of 0.2 seconds. This wave ~oes through 1/5 of a cycle between sampling points. It is therefore indistinguishable from waves that go through ~th + k cycles between sampling points, where k is any integer. The higher frequencies 1 + 5k Hz are known as alias frequencies because they are indistinguishable from 1 Hz. If we take k = 1 and a negative sign we have an alias frequency of - 4 Hz, which is physically equivalent to 4 Hz. Figure 4.8 shows a wave of frequency 1 Hz and a wave of frequency 4 Hz coinciding at every sampling point. In this example the Nyquist frequency is 2.5 Hz.

There are two ways of avoiding the aliasing problem. One is to ensure that our sampling interval is sufficiently small for the Nyquist frequency to exceed any high frequency components in the continuous signal. The other is to filter out components whose frequency exceeds the Nyquist frequency by an analogue filter before it is sampled! It is not possible to achieve a perfect cut-off with physical filters, and digital filters cannot be used to remove the effects of aliasing.

Many modern spectral analysers use analogue pre-filters, which are designed to remove frequencies above 25 kHz. This cannot be achieved exactly so the sampling rate is set at 100000 data per second, which makes an allowance for imperfections in the filter and the leakage which occurs with finite length signals. This is because frequencies near the Nyquist frequency will 'leak' into high frequencies, which will then act as alias frequencies. To compensate for the leakage effect the sampling interval should be somewhat less than that specified by the Nyquist frequency. Practitioners recommend sampling intervals between 1/(4fH) and 1/(2.5fH), where ]'H is the highest frequency that is expected to occur in the signal.

The following example demonstrates why it is essential to avoid aliasing in spectral analysis. A linear model for a lightly damped structure, such as an electricity pylon, has resonances at frequencies fl~, [ ' ~ 2 , �9 �9 �9 , [ ' ~ n ' It is subjected to a harmonic disturbance at some higher frequency fill. If we sample the disturbance so that fill is close to a high frequency alias of one of the lower frequencies, we would predict drastic fluctuations of the structure when, if the linear model is reasonable, it would be relatively unaffected!

4 .5 .3 C o n v o l u t i o n in tegra l resul ts for infinite s equences Let pr be a weighted function of the signal Xr defined by p~ = W,Xr. We will obtain a result for the discrete Fourier transform of pr in terms of those for Wr and Xr. By definition

4.5 Discrete Fourier transform 107

PD(W)= ~ w,xrexp(-jtor) r = - - oo

} = y~ 1 " W(0) exp(j0r) dO xr e•

I" ] 1 W(O) x r e x p ( - j r ( t o - 0)) dO 211" -,, ,=-|

1 I - - 2~r Wo(0) XD(tO - O) dO - - 1 1 ,

This is exactly what we would expect from our work in Section 4.4.3.

4.5.4 The discrete Fourier transform In the course of this chapter we have moved from the finite Fourier series, to the Fourier series, to the Fourier transform and eventually the Fourier transform for an infinite sequence. The obvious next step is to establish a direct relationship between the first and the last. We would hope that as the length of the sequence increases, the finite Fourier series would tend towards the Fourier transform for an infinite sequence. To check that this is so, in Table 4.1 we compare the known results for these two cases.

The infinite sequence form of Parseval's theorem can be deduced from the finite Fourier series result, that is

- ~ E I x ~ I ~ N

2 12 = y. x ( ~ ) ~

and, in the limit, this becomes

1 7 . ,~ = 2,, I -~- X2(to) d~

Tab le 4.1 Comparison of finite Fourier series and discrete Fourier transform for infinite sequence ..

Discrete Fourier transform for Finite Fourier series infinite sequence

Xm = ~ Xr e -j2"ma N r = - - n

n -1

Xr = ~ Xm ej2"trmrfN m=-n

~1 n-1 1 x2 y ix,.,,l~ , N r=-n m=-n

where 2n = N and -~-~ ~o~ ~-. | . . . . . . . i i . . . . . . . . . . . . . . . . . . .

XD(~) = ~ Xr e - j . r r = - ~r

xr = ---2Ir - . XD(to)el'rdt~

x 2 = 2~r IX2(to)dto


If, in the manner of Section 4.4, we write os for 2wm/N and dco for 2w/N as N tends to infinity, we observe that the equivalence of the two sets of relationships requires that the factor 1/N be used in the second equation of the finite Fourier series rather than in the first. Since the equation to which this factor is applied is an arbitrary choice, the equivalence follows. It is common to refer to the first equation of the finite Fourier series pair as the discrete Fourier transform (DFT).

4.6 Exercises

�9

0

0

0

,

6.

A tank holds nine steel rods, in a 3 by 3 formation, for chrome plating. Each rod is surrounded by eight anodes. The depth of chrome was measured (0.01 mm) at 100 intervals around the circumference at the mid point of a plated rod. The data are given below.

150 152 148 146 144 144 143 144 144 145 147 148 145 144 146 148 152 149 146 144 143 142 141 140 136 140 138 142 140 141 136 140 142 144 149 148

(i) Plot the data. (ii) Fit harmonics at 1, 2, 3, 4 and 8 cycles per revolution.

(iii) Draw the Fourier line spectrum. (iv) Can you think of any physical reasons for cycles at these frequencies?

Sketch and find the Fourier series for the function

0 for - , r < x < - , r /2

f(x)= 1 for - , r /2<x<~r /2

0 for w / 2 < x < , r

Hence write down the Fourier series for a similar function defined on [ - L , L] rather than [ -w, ,r]. Sketch and find the Fourier series for the saw'tooth function

Ix~w+1 for - ~ r < x < 0 f (x)=[x/w_l for 0 < x < w

Sketch and find the Fourier series for the triangular function

f(x) = [ : x 0<x<,r-~r<x<0

Sketch and find the Fourier series for e ~ defined on the interval [ -w, ,r]. Prove the results quoted in Examples 4.4, 4.5 and 4.6.


5.1 Introduction

Periodic phenomena pervade the natural world, and the detection of periodicities in data can be traced back to when ancient communities first observed the moon and noted the passage of the seasons. Pythagoras, who is perhaps best known for his famous Theorem, developed laws of musical harmony in 600 BC, but it was not until 1772 that Lagrange proposed a more sophisticated mathematical approach and used it to analyse the orbit of a comet. Fourier's work on continuous functions led to the Fourier analysis of a finite sequence of numbers, the finite Fourier series. In Chapter 4 we chose to present Fourier methods in a different order from their historical development and began with the concept of a finite Fourier series for a discrete signal. Parsevars theorem distributes the variance of such a signal over a finite set of frequencies. If the signal is continuous, the set of frequencies becomes infinite and a Fourier series is appropriate. The next step was to suppose that the duration of the signal tended to infinity. The Fourier series tends to the Fourier transform, but the limiting process involves distributing the total variability, rather than the variance, over a continuous frequency range. As the total variability of the signal is being accounted for, the 'classical' condition for taking the Fourier transform

lim IX(t) l dt < oo U ~ | L

L---,- |

can be loosely interpreted as a requirement that the total variability is finite. The use of generalized functions allows this condition to be relaxed so that the Fourier transforms of harmonic signals are defined. The chapter ended with a discussion of the discrete Fourier transform for a discrete signal as the duration tends to infinity. The essential difference between the Fourier transform and the discrete Fourier transform for an infinite sequence is that the range of frequency for the latter is restricted to [0, 7r].

In this chapter we turn our attention to random processes. A realization from a random process is a finite signal and the results of Chapter 4 would seem directly applicable. However, it should be remembered that the particular frequencies used depend only on the record length and do not usually have any special physical significance. Furthermore, even if the signal is a realization of discrete white noise, quite large amplitudes are likely to be attributed to some frequencies by chance. Some early work by Professor C.G. Knott (1899) was motivated by a study of earthquake vibrations, and Sir Arthur Schuster (1898 and 1906) demonstrated that the data's magnitudes were

109

I I0 Frequency representation of random signals

not statistically significant. He developed his ideas by applying an averaging interpretation to spikes in the line spectrum introduced in Section 4.2.2. This was given the name 'periodogram analysis' and was applied to various sets of data by several research workers. Two long sequences of data, which are well known to time series analysts, consist of annual sunspot numbers from 1700 onwards and indices of wheat prices in Western Europe from 1500 to 1869. A second challenge was to define rigorously a continuous spectrum for the imagined random process which generated the time series. We now explain why this is not a straightforward task. The definition of a random process precludes deterministic periodic components. If we imagine it to be made up from harmonic components, we must allow these to be at random frequencies, of randomly varying amplitude, and to start and stop at random times. The total variability of the infinite process will itself be infinite and the 'classical' condition for taking the Fourier transform will not be satisfied. The assumption of random frequencies means we cannot define the process to consist of harmonics at an infinite set of discrete frequencies, and this rules out defining a Fourier transform in terms of an infinite sequence of delta functions. To summarize, the spectrum of the process cannot be directly defined in terms of the Fourier transform. In 1930 Norbert Wiener published his influential paper, 'Generalised Harmonic Analysis', which included a precise definition of the spectrum of a random process as the Fourier transform of its autocorrelation function.

In the 1940s and 1950s research efforts were again turned towards estimating the spectrum from the time series. Various methods of smoothing the periodogram were proposed, together with approximate sampling distributions of the estimators. One of the first recorded discussions in the UK was by Daniell (1946), whilst, on the other side of the Atlantic, Tukey (1949) made his first of many major contributions to the subject. The increasing use of spectral methods in electrical engineering, and many other fields, stimulated regearch but the extensive computations involved remained a drawback. The publication of a paper by Cooley and Tukey (1965) on the fast Fourier transform led to very significant savings in calculation times and made the routine Fourier analysis of large data sets a feasible proposition. A history of the fast Fourier transform is given by Cooley et al. (1969) and the original ideas can be traced back to the beginning of the century. However, the practical potential of the algorithm does not seem to have been appreciated before the 1965 paper.

Although more powerful computers and the fast Fourier transform have removed the practical problems of carrying out the arithmetic for large sets of data, small data sets still lead to the more usual statistical problem of too few data to make accurate estimates of population parameters. There is no panacea for this, but 'high resolution' strategies such as the Maximum Entropy Method (MEM) introduced by John Burg (1967) and the Maximum Likelihood Method (MLM) published by Capon (1969) have received considerable attention.

The presentation in this chapter follows the historical development: we start with the periodogram and end with the 'high resolution' methods.

5.2 Definition of the spectrum of a random process 5.2.1 The periodogram We start by assuming that we have a time series

{x~} r = - n , . . . , -1 , O, 1 , . . . , n - 1

5.2 Definition of the spectrum of a random process I I I

separated by equal time intervals, from a stationary random process. It is convenient to define the Xr as deviations from the overall mean. We next use the results of Section 4.2.3, namely

1 n--1

X m - - ~ E Xre -j21fnr/N r ~. - - I t

and

n-I 1 Xx2_. X Ix~l ~

N m--" --tl

We could plot Ix l 2 against m to show the contribution to the variance of the sequence at the frequency 2r (radians/sampling interval). However, {x, } is just one realization of the random process and the discrete frequencies to which the variance is attributed are a direct consequence of the record length. Therefore, Ix l = is interpreted as an estimate of the contribution to the variance at frequencies between 2w(m-�89 and 2 ~ m + �89 If a rectangle of height NlXml2/2~r and width 2r centred on 2ran~N, is drawn then the area equals Ix l' and corresponds to the contribution to the variance between these frequencies. We define the periodogram as

t(o) = NIX I /2 2w(m- �89 < O<~ 2 ~ m + �89 and m = - n , . . . n - 1 (5 1) N N ' "

and the unsmoothed sample spectrum as

1 n - 1 12 UC(o)) "- 2,'rrN E Xr e -jt~ --"fl'~ CO~< Ir (5.2)

r----" --n

You should check that

and deduce that the sample spectrum is a smooth curve joining the mid-points of the rectangles of the periodogram, as shown in Fig. 5.1. You should note that we have moved from the line spectrum, a set of lines at n discrete physical frequencies, to the periodogram, which is a non-continuous function defined for all frequencies, O, between - r (r and or- (r The sample spectrum is a continuous function of frequency, to, defined for frequencies from -or to 11" radians/sampling interval. It is the periodogram with the 'steps' smoothed out, and some books refer to it as the periodogram. Unfortunately, the periodogram is not well behaved. As N increases, it gives estimators of the contribution to the variance of the random process over an increasingly fine set of frequencies, but the variance of these estimators does not decrease. There are two obvious ways around this difficulty. One is to average neighbouring strips in the periodogram. The other is to split the time series into several shorter time series, calculate the periodogram for each one and average the results.

5.2.2 The relationship between the sample spectrum and autocovariance function

The point of the following algebra is that it leads us to a definition of the spectrum of a

112 Frequenc X representation of random signals

T l l - - l t

C (e)) is symmetric about (o = 0

ir, l, :/ilil !/lli!l

I ! II

i'l

C(e)) ~ V o l t 2 m d -1

1 / g]

/!!' / l'lill' "'jI I'FI' I I I I I i I I I I I I I I I I I I I I I i I I I I i I I I I I

I I I I I I i I I I I I

I / , ,,T,, : $ !; i

" :'"lil'il '"liI:!l: 'il!l:[rJ' /i/l'l'tlllii ' . . . . . . .

I I I I i I I I I I I I I I I I I I I I I I i I I I i i i

| i i I I I I I i I i I I I I I I I I I I i i I I I I I i I I I I I I i t I i I I I i

I I I I I I I I I I I ! I ! I , 0

Frequency (m rad) (a)

,ji t l l I I

" i1: I !

i l l l 1 / l l ] I / I I r l l

'1'1'!' I I t i I I ~,3 I I I I I I I i i I I i i , ,Ill,1 o I I r n . , l /t

n

Width 1/_~N r -

I 1 I I 1 8 I I I I I I I

~ Enlargement

. r a m ,

. , , . , ,

- T

I C(o) / /

! . . . . " 1 I I I I I I I I I I I I I I I I I 1

r---7 ~ , , I /

,__"_"_~.r._--_C~ I

I I I

I I I I I I

, I . . . .

(m + 1)/N (m + 2)/N m / N m

(b)

Fig. 5.1 (a) Fourier line spectrum (full lines), periodogram (dashed rectangles) and sample spectrum (chain curve). (b) Enlargement of a typical length

random process. Now, for any complex number z, ]zl 2= zz*, so we can write the unsmoothed sample spectrum, defined in the last section, as

If this is written in a different order

I n-I n-I

uc(~ , ) = ~ 2 Y, x~x,, e -j~'~ e j~'' r - ' - - n p----rl

the possibility of making a substitution

r=p+k

to obtain lagged products suggests itself. Then

5.2 Definition of the spectrum of a random process I 13

1 n~ -p n-I -" 2 Xp+kXp e-JagP+k) ejtop UC(co) , - - .

1 n-l-p n-1 = -- E E Xp+kXp e -j''k

N k=-n-p p=-n

If the range of p is modified slightly, the inner summation will become the sample autocovariance at lag k. The trick is to make the range of p depend on k rather than the range of k depend on p, as it does in the last equation. Since p runs from - n up to ( n - 1) and k runs from ( - n - p ) up to ( n - l - p ) , then k can take values between ( n - ( n - 1 ) ) and ( n - 1 - ( - n ) ) . That is, k can take values between - ( N - I ) and ( N - 1), provided we remember that p is now restricted by the relationships

- n - p ~ k < ~ n - l - p

on the first summation sign, in addition to the constraint

- n < ~ p ~ n - 1

on the second summation sign. It is easiest to see the combined effect of these if the range of k is split into - ( N - 1) up to - 1 , and 0 up to ( N - 1). When k is in the former range; that is, when it is negative, the restriction

- n - p < ~ k ~..~ - n - k < ~ p

bounds p below. When k is positive the restriction

k<~n- l - p , -~p< .n - l - k

bounds p above. Therefore

UC(co) = ~ [ ~, ~, Xk+pXp e -j~'k + E k=O p=-n k=-(N-1) p=-n

-, } - - - E c ( k ) e -j k+ E c(k) e -j k

2r k=o k=-(N-~)

1 N-I = ~ ~, c(k) e -j''k - zr<~ to <~ 7r UC(to) 2zr k--r

~lxk+pXp e-J ,~ }

(5.3)

Now, as we discussed in the previous section, UC(to) does not converge in any statistical sense to a limiting value as N---~ 0o. Nevertheless, its expected value does do so, as we now show. We start by taking expectation of both sides of the defining relationship, that is

1 N-I E[UC(to)] = ~ E EIc(k)l e -i''k

kf-tN-n)

We now use the result of Section 3.7.2 to yield

1 N-I E

k=-(N-~) ~ k ) ( 1 - 1 k l / N ) e -j~'k

The next step is to let N--~ oo to obtain the limit of the expected value, namely

114 Frequency representation of random sigr~Is Qo

lim E[UC(to)] = ~ 2 "),(k) e - j ' k N - - * ~

This limit is taken as the definition of the spectrum.

5 .2 .3 T h e s p e c t r u m of a r a n d o m p roces s

For a discrete random process the spectrum is defined as

1 | F(~) = ~ ~ ~k ) e - j ' k - I t< to< 1r (5.4a)

k _-.. _ 0o

that is, the discrete Fourier transform of the autocovariance function. The inverse transformation gives the relationship

I" ~k) = F(to) e i 'k dto (5.4b) - - ' I t

and in particular

I" = o - r ( , o ) d , o

This last result is reassuring, because the area under the periodogram equals the variance of a realization from a random process and the area under the spectrum equals the variance of the random process itself. Remember that whilst the time average of c(k) tends to y(k), as the sample size increases, the Fourier transform UC(~o) of c(k) does not have a limit. This is why the spectrum of the random process has to be defined in such a roundabout way.

An analogous argument with summation replaced by integration leads to the definition of the spectrum for a continuous random process, that is

F(~o) -- ~ _| ~,(u) exp (-j~ou) du - 7r ~ co ~ Ir (5.6a)

(5.5)

which has the inverse transformation

f ~

exp(jtou) dto (5.6b)

If a discrete random process is considered to be embedded in a continuous random process, the spectrum of the discrete process will equal that of the continuous process provided there are no frequencies above the Nyquist frequency in the continuous random process. We finish this section by presenting realizations, the autocorrelation functions (acfs), and the spectra from three simple autoregressive processes. First-order autoregressive processes with positive or negative parameters are intuitively 'low frequency' or 'high frequency' respectively, and their spectra confirm this, see Figs 5.2(a) and (b). Second-order autoregressive processes (Section 3.6.4) can exhibit more variety in their spectra, Fig. 5.2(c) for example.

5.3 Estimation of the spectrum from the sample autocovariance function II 5

5.3 Estimation of the spectrum from the sample autocovariance function

5.3.1 The need for smoothing Since the spectrum is defined as the Fourier transform of the acf, it seems reasonable to try to estimate it directly from the sample acf. The algebraic relationship between the sample spectrum and the sample acf suggests that this will amount to some sort of periodogram averaging, and this turns out to be the case. The methods are computationally simple, and whilst they are inefficient for long sequences of data they often provide a convenient means of analysing historic time series. The value of more efficient methods becomes apparent in 'signal processing' applications such as digitized measurements from transducers. The theory begins with the result for the expected value of the sample spectrum given in Section 5.2.2, namely

1 N - I

= - - ~ 3,(k)(1 -Ikl /N) exp (-jtok) EtVC(t~ 2w k=-(N-l)

This can be written as

1 ~, w(k)y(k)exp(-joJk) e [ u c ( , o ) ] = ~ ~_--.

where the function w(k) defined by

1 -Ikl/N Ikl ~< N, k an integer w(k)=, 0 otherwise

has been introduced. We now use the convolution result of Equation (4.15) to obtain the relationship

I f , , EIUC(,,,)] = ~ w(0) r ( , o - 0) dO --'/t

where W(O) is the Fourier transform of the discrete sequence w(k) and F(0) is the Fourier transform of the discrete sequence ~/(k), that is, the spectrum of the random process.

Using the expression for the Fourier transform of a discrete sequence given in Equation (4.16a) we can write

o o

w(o)= X w(k)e -j~ - , r r< O< r

If we now use the result that

exp(-jOk) = cos(Ok)-j sin(Ok)

together with the facts that w(k) and cos(Ok) are even functions, sin(Ok) is an odd function, and w(k) is zero if k exceeds ( N - 1), we obtain the expression

N - I

W(O) = w(O) + 2 ~, w(k) cos(0k) k=l

This can be plotted for different values of N, but it is no t in a convenient form for making


xt t (a) X Xx x [? x

(b)

F t

A %%.

"%%%%,%, .. , " ~ ~ , ,,

k

Q.

~%'%%*%.,,.,,. �9 .... ~;.:::::::::

k

J 1

,.__,

o _c 0

- 1 I I I ; I 6 0.2~ 0.4n 0.6~ 0.8~ ~ r i I i I ! , . -

- 1 0 ~ 0.2n 0.4n 0.6n 0.8n 7t

Fig. 5.2 (a) Realization, acf and spectrum of AR(1) with e = 0.9. (b) Realization, acf and spectrum of AR(1) with ~x = -0.9. (c) Realization, acf and spectrum of AR(2) al = 1, e2 = -0.5

general observations about its shape. The Fourier transform of the continuous function analogous to w(k), that is

1-[tl/N It[<N w(t)= 0 otherwise

is much easier to deal with. It is close to zero for absolute values of 0 exceeding frequencies of order zrlN, so the Sampling Theorem justifies its use as an approximation to W(0) provided N is reasonably large. This Fourier transform is

f = e-j ot dt W( t)

which can be simplified in the same way as the Fourier transform of the discrete sequence to give

I N

2 (1 - t/N) cos (Ot) dt 0

5.3 Estimation of the spectrum from the sample aur function I I 7

(c)

. I , fl .._

r

k

lOt S t, . . ,

1

0.1 I I I . I ".--- r

0.0 0.2e 0.4e 0.6e 0.8e e

F ig . 5.2 (cont.)

Straightforward integration by parts, and use of the standard trigonometric identity

cos(2A) = 1 - 2(sin A) 2

leads to the approximation

W(O) = N[sin (ON/2)/(ON/2)] 2

This function is drawn in Fig. 5.3, and is known as a spectral window for reasons that will now be explained. We start by making the approximation that W(0) is negligible outside a narrow interval centred on 0, which for illustration we will take as [-4~r/N, 47r/N]. We now assume that F(to) is approximately linear for frequencies over a band of width 87r/N centred on top. Then

E[VC(top)] = ~ W(0)F(tOp- 0) dO --,ff

27r W(O)[r(COp) + bOl dO


w(o)

N

-8~ - 6 n - 4 n -27t

N N N N 2n 4n 6~t 8n ' ~

fi N hi

Fig. 5.3 A natural spectral window

where b is the slope of the linear approximation. This integral can now be written as the sum of two integrals

1 W(0)F(top) d0+~ OW(O) dO 271" -~- - l r

The second term is the integral of an odd function and equals zero, and F(cop) is a constant which can be factored out of the first integral to give

E[UC(top) l = ~ F(tOp) W(O) dO --'n"

Now since W(O) is the Fourier transform of w(k), the inverse transform with 0 substituted for k gives

1I~ w(0) = ~ W(0) dO

Since w(0) is equal to 1 we now have the result

e[ uc(,op )] =

Since top is an arbitrarily selected frequency we may drop the subscript. As stated already, the approximation relies on the spectrum, F(to), remaining approximately linear when 'viewed' through a 'window' of width 8rdN, and this is least acceptable at the peaks and troughs. So far, we have shown that the bias of the sample spectrum is small provided N is sufficiently large, but we have not concerned ourselves with the problem that its variance does not decrease.

The preceding argument suggests the following approach. If the sequence is split into m sub-sequences each of length M and an unsmoothed sample spectrum, UCC~)(to) say, is calculated for each sub-sequence, these sample spectra can be averaged to produce a single estimate with a smaller variance. That is

m

SA C(to) = Y~ VC(')(to)/m i=1

The variance of the estimator decreases as m increases but the attendant reduction in M

5.3 Estimation of the spectrum from the sample autocovariance function II 9

increases the bias and a compromise has to be struck. In practice, the averaging is usually carried out in a slightly different manner.

5.3.2 Smoothed autocovar iance based spectral est imators A smoothed spectral estimator is of the form

1 N-] C(to) = ~ k--~v-~) w(k)c(k) exp (-jtok) - zr< to< Ir (5.7)

where w(k) is a set of weights known as a lag window, defined for integer values of k, and satisfying the following three conditions

(i) w(0)= 1 (ii) w(k)= w(-k)

(iii) w(k) = 0 for Ikl > M, where M~< N

A simple example, which is rarely used in practice, is the Bartlett or triangular window defined by

w(k) = (1 -Ikl/M) Ikl ~<M

This general form of an estimator is a natural development from SAC(to). Both types of estimator only use values of c(k) up to M - 1 ; in SAC(to) the sub-sequences are of length M and it is not possible to calculate c(k) for any larger values, while in C(to) they are excluded by the weighting system. A general feature of lag windows is that they give less weight to c(k) as the modulus of k increases. The estimator SAC(to) also does this, implicitly, because c(k) is estimated in each sub-sequence by the sum of the appropriate M - k products divided by M. The corresponding effect when estimating c(k) up to lag M - 1 from N data is relatively small whenever M is considerably less than N. We now investigate the bias of C(to). The Fourier transform of w(k) occurs as a spectral window in the manner described earlier. The convolution property applied to the definition of C(to) gives

1 I " c(o,) = ~ w (o) u c (~ - o) d o

where UC(to) is the sample spectrum. Taking expectation gives

1 I " elC(,o)] = ~ W(OlelUC( , , , - o)1 dO

and if N is large this is approximately

I" ] W(O)r(oo-O)dO 2"~ -~r

We are now in a position to repeat the argument put forward in Section 5.3.1.. If F(~o) is approximately linear over intervals as short as the width of the spectral window, the estimator will be approximately unbiased. The natural window discussed in Section 5.3.1 occurs when c(k) is used to estimate 7(k) because the denominator in the definition of c(k) is N for all lags k. The Bartlett spectral window has the same shape as that shown in Fig. 5.3 with N replaced by M. It is wider by a factor of N/M, so the estimator will inevitably be more biased. Figure 5.4 shows the Bartlett lag and spectral windows when M equals 20. The advantage of using the Bartlett window is that averaging has taken


"...._....... w(o)

2o-

-20 0 20 k - 8rt - 6rt - 4r~ - 2~ 2rt 4rt 6n 8~ 0 . . . . . . . . . . 20 20 20 20 20 20 20 20

Fig. 5.4 Bartlett lag and spectral windows when M = 20

place, so reducing the sampling variability. To sum up, using a window is an averaging process which reduces sampling error but smooths out peaks and troughs in the spectrum of the underlying random process. The amount of averaging depends on the relationship between M and N and if M is considerably smaller than N, the natural window, which results from using N in the definition of c(k), is negligible compared with the Bartlett window. Sample spectra can be calculated for several values of M, but the usual advice is to start with a value of M approximately equal to 2V'N. The choice of M is analogous to choosing the width of the grouping intervals when drawing a histogram, and the advice given will be interpreted in this context in Section 5.4.1.

The formula for the spectral estimator

1 N- I

C(to) = ~ S'. w(k)c(k) exp(-jwk) k = - ( N - l )

- - 7 r~< t o < ~ ,rr

is not in a suitable form for computational purposes because of the complex exponential. If we use the relationship

exp (-jtok) = cos (tok) - j sin (tok)

and remember that w(k), c(k) and cos(tok) are even functions, sin(tok) is an odd function, w(0) equals unity and w(k) is zero if k exceeds ( M - 1) in absolute value, we obtain the equivalent formula

l 1 1 c(O) + 2 w(k)c(k) cos(tok) Ir~< to~< 7r 2~" k=l

Also, since the spectrum is an even function of frequency it is only necessary to calculate it over the range 0 <~ to ~< 7r. However, to preserve the relationship between the area under the estimated spectrum and the sample variance, it is necessary to double the contribution attributed to each frequency if the range 0~<to~<rr is used. Hence, the computational formula is

C(to) =-1 c(0)+2 w(k)c(k)cos(tok) O<~to<~'tr (5.8) 71" k=l

5.3 Estimation of the spectrum from the sample autocovariance function 121

This formula can be used to calculate an estimate of the spectrum at any value of to within the range 0 to ~r.

Example 5. I The 397 data in Appendix V were sampled by a probe situated at the centre of a wave tank that was equipped with sufficient wave makers to synthesize random sea states. The data are distances from the still water level measured in millimetres (mm) recorded every 0.1s.

The first 50 data are plotted in Fig. 5.5(a). Figure 5.5(b) shows the spectral estimates using a Bartlett window with M equal to 10, 20 and 40 respectively. The smoothing effect of reducing M is clear. A step-by-step account of the calculations is presented in Chapter 7.

5.3.3 Alternative lag windows The three conditions satisfied by the lag window, (i), (ii), (iii) of Section 5.3.2, imply that

'~ 400 -,.z / ,

J !/

i5 t I 0 10

1 6 -

~ ' 1 4 - "0 2 1 2 -

~ 10-

E

r, f, tl [' ''! if lJ

I I 20 3O

Time (t, O. 1 s) (a)

7 I I I

40 50 60 " -

Sampling interval & = 0.1 s I ~ Spectral estimate for Bartlett window

8 - i" "t - - - - - - - - - - M = 10 ' ,'~ /7 ~il I ' ~ , ' l ~ . . . . . . . . M = 2 0

, L Y . . . . . ; , o

0 0.5 1.0 1.5 2.0 2.5 3.0 Frequency (rad/sampling interval)

~o)

Fig. 5.5 (a) First 50 wave data' (b) spect rum est imates for wave data using a Bart lett w i n d o w


T a b l e 5.1 Lag and corresponding spectral windows , ,

Lag window---, Fourier transform ~ Spectral window

(i) v v ( O ) = 1 ~ W(O) dO = 1 2zr -,,

(ii) w(k) = w(-k) VV(to) = W(-to)

(iii) w(k) = 0 for Ik] > M W(to) has a 'width' of order M-1

three conditions must be satisfied by its Fourier transform, the spectral window. These conditions are presented in Table 5.1.

In some ways the most natural lag window might be the rectangular lag window defined by w(k) = 1 for all k between - M and M. This selection corresponds to reducing the formal definition of F(to), namely

1 oo

F(to) = ~ Y~ y(k) exp(-jtok) k--oo

by its approximate finite sum

M

1 ~, c(k) exp(-jtok) =

That is, y(k) has been replaced by its estimate c(k) and only a finite sum considered. This last expression may therefore be written as

1 M C(to) = ~ Y~ w(k) c(k) exp (-jtok)

k = - M

with w(k) = 1. However, the results of Section 4.4.3 lead to

c(,o) = w(o) dO --I1"

with the spectral window W(O) having the undesirable shape

sin (M0) W ( O ) - -

MO

This result follows in a similar manner to the results of Section 5.3.2. If this spectral window is plotted it will be observed that W(O) has oscillatory side lobes which include negative values. Hence w(k) = 1 is not a good choice of lag window for practical use.

Two commonly used windows, which are a considerable improvement over the rectangular window, and in some ways better than the Bartlett window, are named after Tukey and Parzen.

Tukey window The Tukey window is defined by

w(k) = �89 +cos(zrk/M)) Ik} = 0, 1 , . . . , M (5.9a)

It is also known as the Tukey-Hanning window, and is equivalent to the Hanning


window, which is usually offered as one of the choices for data analysis on commercial spectral analysers. The computational procedure is different because most spectral analysers use the fast Fourier transform which is described in Section 5.4. A good approximation to the spectral form of the window can be obtained in a similar manner to that for the Bartlett window. It is

W(O) --" M'tr 2 sin (OM )l[(OM) ('rr 2 - 02M2)] (5.9b)

and using a Taylor expansion for the sine function one may demonstrate that W(0) is properly defined and equals M.

Parzen window The Parzen window is defined by

~ 1 - 6 ( k / M ) 2 +6(k /M) 3 O<<.k<~M/2 w(k) [ 2(1 - k /M )3 M/2 < k < M

and w ( - k ) = w(k). Again, straightforward but rather lengthy integration leads to an approximation for

the spectral form. This is

W(O) = 24(3 + cos ( 0 M ) - 4 cos(OM/2))/(O4M 3) It is easy to verify that W(O) is properly defined and equals 0.75 M by using the Taylor expansion for cosine.

5 .3 .4 C o m p a r i n g w i n d o w s

If we are to make any sensible comparison of spectral windows we must ensure they are of similar 'sizes', and this is not necessarily achieved by setting M to the same value. In order to discuss what we mean by the 'size' of a window we use a result for the distribution of the ratio of the spectrum estimator to the population spectrum. Jenkins and Watts (1968) prove the following approximation which improves as the sample size increases,

vC (to)/F(to) -~ X 2 (5.10a)

This is a chi-squared distribution with nu degrees of freedom. It is a standard result that Var(x,, 2) equals 2v, and so Var(C(to)/F(to)) is equal to 2v/u 2 which simplifies to 2/v. In Appendix VI we show that the appropriate degrees of freedom are defined by

M

=2m/ X w (k) (5.10b) k=-M

The distribution of the ratio of the spectrum estimator to the population spectrum is reminiscent of a result for the ratio of a sample variance to the population variance when sampling from a normal distribution, that is

(n - 1)sZ/tr2~. X2_ 1

This is not altogether surprising when we remember that the spectrum gives the distribution of the variance over frequency. Since the variance of the ratio, C(to)/F(to), equals 2It,, it increases as M increases, but decreases as M decreases. Following substitution of the approximate expressions for w(k) the variance equals 0.75MIN for the


0.6

o . 5

0.4

' .~ i , t ' ' + . . . . . . , . . ,::, .~ . . . . . +,~ . . . . . . - ' ' ' ~

~+ +I~ �9 f+ +s

I~+ +f

f+ +9 § +

If+ +l~

0.7 '+ +' 4" 4"

s t 4" 4-

+1~ t~+ 4. 4.

4.it 1~4. * ' . ~*

4" I 1 4 " , 4 , 4. o o

4. It % +

0.3 4. X 0 Ox 4. 4- 4-

RO 0% 4. 4"

0.2 ,o o, 4. A l t o % 4- 4- 0 X

4. X O 0 X 4. 4" % X 4- 0.1

4. 1% o o XIL 4" 1 v u X ,lkll ~

o . . . . . . . ,~;---?~:++-+ .. . . . . . . . . . , : . . . . . . . . .++::~'*---;+', . . . . . . . -40 -:30 -20 -10 0 10

~ k (a)

25

2O

15

lO

5

0

-51 -1.0

= �9 i i @~ i i �9 | t t e . . t o it

12~:: _,::, . . . . . . ~.,. ....... i ; " ...... '"'"" """' ..... "'""" ":" I "'"'""

-d 8 -d 8 -0.4 -a 2 ~ 0'2 04 016 03 ~0 0 (radians/sampling interval)

Co)

Rg. 5.6 (a )Lag w indows : rectangular, M = 10, ~ ; Bartlett, M = 30, + ; Tukey, M = 27, o; Parzen, M = 38, x . (b) Spectral w indows" rectangular, M = 10, ----; Bartlett, M = 30, - - - ; Tukey, M = 2 7 , - - ; Parzen, M = 38, - . . . . . .

Tukey window and 0.54M/N for the Parzen window. These lag and spectral windows are compared in Figs 5.6(a) and (b) for N = 400, with the values of M chosen as 27 and 38 respectively so that the corresponding estimators have the same variance. Their shapes are similar, but the Parzen spectral window has the advantage of being positive for all


values of 0. The Tukey window could lead to small negative spectral estimates if the unsmoothed spectrum contains an isolated spike, but this is not a serious practical drawback. The lag and spectral windows for the Bartlett window and a rectangular lag window are also shown in the figure with values of M set at 30 and 10, respectively, to give the same variance as the Tukey and Parzen windows. The Bartlett window is noticeably more oscillatory than the Parzen and Tukey windows. The rectangular lag window is clearly unsuitable for practical use.

The bandwidth of a spectral window is defined as the width of the rectangular spectral window, L, which would give an estimator of the same variance. We now explain how this bandwidth, L, is related to the parameter M in the definitions of the windows. A perfect rectangular spectral window would be an 'ideal' band-pass filter, but this is unobtainable in practice. However, averaging neighbouring ordinates in the Fourier line spectrum is a close approximation. It is proved by Jenkins and Watts (1968), and shown informally in Appendix VI, that the ratio of two-times an ordinate in the Fourier line spectrum to the height of the spectrum (in the underlying random process) has a chi-squared distribution with two degrees of freedom. Hence, the variance of the ratio UC(to)/F(to) is 2/v and this is equal to 1. We now assume that the underlying spectrum does not vary much over a frequency band of width L. Neighbouring ordinates in a Fourier line spectrum are approximately independent, and the approximation improves as the length of the time series increases, which partly accounts for the irregular shape of the periodogram. There are NI2 ordinates over a frequency range of 0 to ,r. Therefore, there are NL/(2"tr) ordinates over a band of length L, and the variance of their average will be the variance of a single ordinate divided by this number, that is

I/[NL/(2zr)] = 2zr/(NL)

This is a fundamental equation and we note that the standard error, a term commonly used for the standard deviation of some estimator, of a single smoothed ordinate is inversely proportional to V~-L. It can be reduced by increasing the record length or bandwidth. However, increasing the bandwidth means more averaging and this increases the bias. Thus, peaks will be reduced and troughs increased somewhat.

We now equate this expression for the variance of a spectrum obtained by averaging ordinates in the Fourier line spectrum to the variance of the Tukey window, the relationship is given by

2zrl(NL) = (0.75) M/N

which gives L as 2.677r/M. This is the bandwidth of the Tukey window and it should be shown on the same graph as the spectral estimate. It gives a visual assessment of the width of the frequency interval over which averaging has taken place. The bandwidths of the Parzen, Bartlett and rectangular lag windows are (3.72)~r/M, 37riM and ,riM respectively. All the spectral windows in Fig. 5.6(b) have a bandwidth of about 0.32. The bandwidth determines the amount of smoothing which is applied to the sample spectrum and therefore has a considerable effect on its shape, but once it has been chosen the choice of window often has relatively little effect. Harris (1978) describes his detailed investigation of window shapes.

12

10 . " i

�9 , . , . . .

"o 8

E 6

�9 4

E 2

Frequency representation of random signals

xIO 4

126

-2 o.o o15 11o 210 2'.S 3'.o 3.s

Fig. 5.7

Frequency (rad/sampling interval) Estimates of spectrum for wave data using different windows of bandwidth 0.24

Example 5. I (continued) Figure 5.7 shows spectra for the wave data using Parzcn, Tukey, Bartlett and rectangular lag windows with bandwidths of 0.24. The corresponding values for M are 50, 36, 40 and 13 respectively. There is little to choose between the first three, but the fourth leads to negative estimates which are physically impossible.

The disadvantage of smoothing is that resolution is lost. If there are two peaks in the spectrum of the random process at frequencies to~ and to2, the bandwidth of the estimator must be less than the difference in the frequencies, to~- to2. In practice, it is advisable to set the bandwidth at about half this difference because the spectral window does not have a sharp cut-off, and because of the effects of leakage and sampling error. A criterion about the resolution of the spectral estimator will dictate the bandwidth. The amount of smoothing associated with this bandwidth will depend on the record length but this is not always under the investigator's control. If the record length is fixed, smoothing results in a loss of resolution, and increasing the resolution decreases the smoothing and hence increases the chances of identifying spurious peaks. You must also bear in mind the aliasing problem, discussed in detail in Section 4.5.2, which requires the sampling interval to be less than rr/(highest frequency (rad s -~) present in the signal).

We have now covered sufficient theory, and its practical implications, to calculate sample spectra from univariate time series and to interpret the results of these calculations. We finish this section with some time series, and their associated spectra, which were analysed by Cartwright and Longuet-Higgins (1956). Five time series are shown (below) and their spectra (above) in Figs 5.8(a)-(e).


4

"~ 40

~20 20

0 0 0.5 1.0

(a)

Spectra

40

20

OL 0 0.5 1.0 0

a(s -1) (b)

0.5

(c)

1.0

10

Lu 0

0 0.5 1.0 (d)

15 - lO , z L L 0 1.5 0 0.5 1.0

~(s -1) (e)

I 1.5

5ft water

Time series

. . . . . . . . . . . . . . . . . . . . . . (a)

1 min

20.[

20.[

lo~

�9 , , o

1 rain

1 min

P

1 min

(b)

(c)

(d)

20o I (e) t, , ,

1 min

Fig. 5.8 Typical short sections of the five records chosen for analysis. (a) Pressure on the sea bed off Pendeen, Cornwall, UK, 08:00 to 08:20, 15 March 1945; (b) wave height in the Bay of Biscay, 19:00 to 19:12, 11 November 1954; (c) wave height in the Bay of Biscay, 02:00 to 02:12, 12 November 1954; (d) angle of pitch of RRS Discovery I1o in North Atlantic, 13:21 to 13:33, 25 May 1954; (e) angle of roll of RRS Discovery II, in North Atlantic, 14:05 to 14:17, 21 May 1954


5.3.5 Confidence intervals for the spectrum It follows from the distributional result for the ratio of the sample spectrum to the population spectrum that, to a reasonable approximation

2 1~C(r ~ , 2 ) ' - 1 - a Pr X,,,1-,~/E <~ F(~o) ,,,~/2

It follows that an approximate ( 1 - a) x 100% confidence interval for F ( f ) is given by

2 ,,c(,o)/x~,~/2 ~c(,o)/x 2 , v, 1 -od2 ]

This is a useful result if there are some frequencies of particular interest, such as the natural frequencies of a structure subjected to vibration described by a random process. If there are not, it may be more appropriate to plot the logarithm of the sample spectrum against frequency, in which case the confidence interval is of constant width. This follows because the ( 1 - a ) x 100% confidence interval for In (F(~o)) is

[In (C(to)) + ln(vlx~,,~/2), In (C(~o)) + ln(v/X~,l-,~/2)]

A 90% confidence interval, for example, can be displayed on the graph as a vertical line to complement the bandwidth, which is best shown as a horizontal line. This presentation of a spectrum for the data of Example 5.1 is used in Fig. 5.9, the degrees of freedom for a Parzen window with M and N equal to 50 and 397, respectively, is 29.45.

5.4 Estimation of the spectrum from the periodogram 5.4.1 Smoothing the periodogram It was demonstrated in Section 5.3 that the computational and smoothing methods based on the sample autocovariance function were equivalent to an averaging of periodogram ordinates. A more direct method is to do precisely that. This approach is not only intuitively more direct, but is also more computationally efficient for long time series when the fast Fourier transform (FFT) algorithm is used. An ordinary unweighted average of rn ordinates is known as the Daniell window. That is

C(~o) = 1 ~,I(2zr]/N) (5.11) m

where j varies over m consecutive integers, which are chosen so that the frequencies 2~]/N are symmetric about to. Estimates at the highest frequencies are found by assuming the ordinates are symmetric about zr. These ordinates should be of little practical importance if the sampling interval is small enough to avoid aliasing. The bandwidth of the Daniell window is 2zrm/N. This can be justified by noting that there are N/2 spikes in the Fourier line spectrum over a frequency range of ~r, and therefore m spikes correspond to a frequency range of 2zrm/N. Use of the DanieU window, with a bandwidth of 0.24, is compared with use of the Parzen window on the wave data in Fig. 5.10. The number of ordinates, m, that were averaged was 15. Since the lag window is related to the spectral window by the inverse Fourier transform, it should be possible to find the Daniell lag window. It turns out to be

~1 k = o w ( k ) l s in (m~k /N) / (m sin(Trk/N)) k = 1, 2 , . . . , N - 1

5.4 Estimation of the spectrum from the periodogram 129

Sampling interval A = 0.1 s Parzen window Bandwidth 0.24 I ' ~

121- .= 0.49-T" | ~ 900/0 confidence r / - ~ interval o--~-

- - . 8

c

3.0 Frequency (rad/sampling interval)

Fig. 5.9 In(estimated spectrum) with 90% confidence interval

This is slightly different from the other lag windows we have considered so far, because it does not tend to zero as k tends to infinity. Conversely, given a lag window it should be possible to construct an equivalent weighted average of periodogram ordinates. It was mentioned in Section 5.3.3 that the Hanning window is a weighted average of periodogram ordinates, which is equivalent to the Tukey window.

5.4.2 Segment averaging Another way of smoothing the periodogram is to cut the time series into bits, known as 'segments', calculate the periodogram for each segment and then average these periodograms. The technique can be modified by smoothing the averaged periodogram. Some spectrum analysers are programmed to carry out a FFT on a sequence of 1024 data

16

14

12

lO , p , -

•

8

6 m

0.5

Sampling interval A = 0.1 s DanieU

. . . . . . . Parzen ? I ~ Bandwl~h

I

1. 1.0 1.5 2.0 2.5 3.0

Frequency (rad/sampling interval)

Fig. 5.10 Estimates of spectrum for wave data using Daniell and Parzen windows of bandwidth 0.24


and do so continuously as the data become available. The average of the periodograms is updated by including the latest one with appropriate weighting and the result is displayed with a cathode ray tube. The FFT algorithm is usually programmed with hardware and data are processed very quickly. These machines are far more versatile than this brief .description might suggest. They sample continuous signals at a rate of 25 kHz, or even faster, and can then filter this sampled signal before applying the FFT so that it can zoom in on sections of the spectrum over different frequency ranges. They can apply different windows by smoothing the periodograms before averaging the periodogram with appropriate weights. One of their most useful capabilities is the analysis of correlated signals, and this is discussed in Chapter 6.

5 .4 .3 U s e of the fast F o u r i e r t r a n s f o r m to e s t i m a t e the p e r i o d o g r a m

The fast Fourier transform (FFT) is a very efficient algorithm for calculating the discrete Fourier transform (DFT) of a sequence of data. If you glance back to Section 4.5.4 you will see that the DFT is essentially the first equation of the finite Fourier series pair. However, for present purposes it is more convenient to define the DFT as

X m = - - XrW~l m m = O, 1 , . . . , N - 1 N r=0

In this equation, X m denotes the mth component of the transform, Xr denotes a datum and WN is a convenient shorthand for e -2"j/N. The inverse discrete Fourier transform (IDFT) is given by

N-I X r "- 2 X m W N T M r = 0 , 1 , . . . , N - 1

m=0

The slight difference in definitions between the above equations and those in Section 4.5.4 leads to some subtle arithmetic changes. It is necessary to be well aware of these because most mathematical software is based on the definitions given in this section. Unfortunately, our exposition naturally leads to the definitions in Section 4.5.4. The difference is that the data are now represented by a sequence x0, x l , . . . ,xN-~ rather than x_,,, x_,,+ 1, �9 �9 �9 x,,_l. To investigate the consequences of this we will now refer to the sequence which runs from - n up to n - 1 as {Sr}. That is

X 0 = S _ n

X] = S - n + 1

0 0 4 0 0 0

X n = S 0

X N - 1 = S n - I

Substituting for WN in the DFT and IDFT definitions leads to

~ Xr e x p ( - j 2 1 r m r / N ) O ~ m < ~ N - 1 X ' = N r=O

N-I Xr = ~ X,,, e x p ( j 2 7 r m r / N ) O < ~ r ~ N - 1

m=O


where N = 2n. The equivalent expressions using the definition of the finite Fourier series given in Section 4.5.4 are

1 n-I S,,, = - - ~'. s~ exp (- j2~rmr/N) 0 <~ m <~ n - 1

N r-" --it

n--l

s~ = ~'. S,,, exp ( j2mnr/N) - n <~ r <~ n - 1 t / l ' - --t l

It is fairly straightforward to verify that

- 1)"S,,, 0 ~ m ~ - r N X~ 1)~S,,,_ N 2 " ~ m ~ N

The change of sign for odd values of m is compensated for in the IDFT. It follows that the DFT, defined in this section, is the same as the finite Fourier series. For a real sequence, the equivalence of the result S_,. = S,.* in terms of X is

X N _ m -~- X * m

We are now in a position to explain the FFT algorithm. We will restrict our explanation to the case when N, the number of data, is an exact power of two. This is not a practical restriction because more general algorithms, which can be used with many values of N, are commonly available. If necessary, a few data can be omitted from the beginning or end of a long time series to obtain an acceptable number. The 'trick' that the algorithm relies on is that W~," can only take N distinct values, which can be found very quickly when r m is expressed in binary form, as it always is on a digital computer. Let the discrete time series { X k } be decomposed as in Fig. 5.11 so that the even and odd numbered points are used to form two other discrete series { Y k } and {zk }. That is, if

k = 0, 1 , . . . , ( N - 2 ) / 2 Y k X2k

Zk --- X 2 k + l J

~ ~ o

0

Original time series 0 0 0 0

0 0 0 0

0 0 0

! ! 1 I ~ I I I I ,._ 2 4 6 8 10 12 14 16 r k

! I 0 1 2

01 0

F i g . 5 . 1 1

"Even" sequence o y , = x ,

0 0

! I ,I I 1 ! 3 4 5 6 7 8 "~k

"Odd" sequence Z k = X2k+ l

0 0

0 0 0

0 L .I ! ? 1 I I ! ,.._ 1 2 3 4 5 6 7 8 r k

First decompos i t i on of t ime ser ies


then the DFTs of y and z, in our usual notation, are

(N-2)/2 rr= E ykW~ rk

k=O

(N-2)/2 Zr -- E Zk W 2rk

k=O

r = O, 1 , . . . , ( N - 2 ) / 2

Now the DFT of x is

N - I

Xr-" E XkW~l k k = 0

(N-2)/2 (N-2)/2 = E E

k=O k=O

(N-2 ) /2 (N-2 ) /2

= ~, Yn w2rk + WrN E Zk W 2rk k = 0 k=0

Therefore, Xr = Y~ + g'~NZr.

So we see that the computation of the DFT of N samples can be reduced to computing the DFTs of tWO sequences of N/2 samples. It follows that, if N is equal to 2 to some power n, we can make n such reductions and then use the fact that the DFT of a single datum is the datum itself. This procedure, known as decomposition in time is presented in three stages for the case when N equals 8 in Figs 5.12(a)-(c). In. Fig. 5.12(c) the DFT process has been reduced to simple sets of multiplications and additions and, if computed

Xo = Yo =

8 t- ~D X2 = y l =

g. =~ X4 = Y2 ~" uJ =

x6 = y3~- ' -

X 1 =ZOO-.-----

tO e-- (9 =~- X3= Z 1 ;

"1o Xs = Z:~ e - - - 0 =

X 7 = Z3 e..__..

DFT N=4

DFT N=4

Yo ~_ Xo

1 ~ Xl

Y2 X2 "e-

if) Y3 X3 ,)

z 0 ~ -~- ~ X4 ""

o Z, Xs ,IT

Z2 X6

_ ._ W7 (a)

Fig. 5.12 Decomposition in t ime and the FFT algorithm. (a) First decomposition of time series" (b) second decomposition of t ime series' (c) reduction of DFT to multiplication and addition operations


Xo_- I "i Yo - Xo

N = 2 Xl

x2= / 'i x2

N= 2 •

I DFT " V~, ~VV 4- N= 2 X5

x3= / I Xe

N = 2 X~ x,: i i - ~ , ~ ~w ~

(b)

Xo Yo

W o

Xo

w 1

w 4

x2, - x~

w ~ w 6 X6,

Zo

W 2

~7 x3 W 3

X4

W 4 Xs

Xe

x T , ~ W 4

Fig. 5.12 (cont.)

W ~

(c)

W 7 X7


in this form, requires no additional data space for the intermediate results, since the computations may be performed 'in-place'.

Professionally written FFT programmes are not restricted to N being some power of 2, formally known as radix 2, and a typical NAG routine (C06EAF) will accept any N which can be expressed as a product of up to 19 prime numbers, which themselves have numerical values below 20. The common practice of adding zeros to the mean corrected time series to obtain a suitable value of N will reduce the variance, and the spectrum will need scaling to allow for this. An alternative is to ignore a few data at the beginning or end of the time series. Some programmes also work with complex data sequences, whereas in normal data analysis we deal with real time histories. Suppose the finite record has N data points then defining Yk and Zk as before we may define

X'k = Yk +j Zk

and determine X'r using the FFT. It can be shown by straightforward algebra that

and

, ,. } 2 Yr = X r + X N / 2 - r

t* 2Zr = j ( g N / 2 - r - X'r)

r = 0, 1, 2 , . . . , (N-2) /2

and as we already know how the transform of the original series Xr is related to Yr and Z r

we may determine the DFT of N real points using the FFT on N/2 complex points. We should point out here that the FFT not only reduces the computation time but also substantially reduces round-off errors associated with the computations. Also, using only the first set of relationships in the case when Yk and zk represent distinct quantities, one may process two real signals in parallel by assigning one to the real part and the other to the imaginary part of x'k.

Once the DFT has been calculated, the Fourier line spectrum, defined over positive frequencies only, consists of the spikes

IXol z, 21X~ [ z , . . . , 2lXn- ~ 12

These can be smoothed by one of the methods described in the preceding section.

Example 5. I (concluded) The wave tank time series consists of 397 data. If the first datum is omitted, the requirements of the NAG routine are met because 396 is the product of 2, 2, 9 and 11. The estimate of the spectrum obtained using a Daniell window of bandwidth 0.24 on this, mean corrected, sequence is compared with the spectrum obtained when 396 zeros are added in Fig. 5.13. The only appreciable difference is that the area under the latter spectrum is half that under the former. This is intuitively reasonable because adding zeros gives no information about frequency. It will also reduce the variance. If an equal number of zeros is added to the sequence then the variance will be halved. Nevertheless, we prefer not to add zeros. A differing view is held by Yuen and Fraser (1979), who recommend adding an equal number of zeros so that the Fourier line spectrum contains sufficient information to infer the original time series.

Example 5.2 (Waterfall display) A valuable feature of modern spectral analysers is their ability to provide a waterfall display, obtained from measurements made on rotating machinery. The associated


Fig. 5.13 Spectra for original time series and series with added zeros

calculations rely on the fast Fourier transform. A marker on the shaft enables the analyser to sample an analogue signal at some high power of 2 (usually 1024) observations per revolution. This is because powers of 2 are most convenient for rapid on-line fast Fourier transform calculations. Spectra are repeatedly calculated as rotation speed increases, and plotted with a horizontal scale of 'cycles per revolution' (properly referred to as the kinematic excitation number). An informative waterfall display is shown in Fig. 5.14. The purpose of presenting information in the form of Fig. 5.14 is to demonstrate how attributes of an engineering system can be readily deduced.

Suppose the measurements were noise of a two-bladed wind turbine made with a microphone. Then we might associate the ridge at 2 cycles per revolution with the number of blades. In contrast to this, the curved ridge would indicate the wind turbine has a natural mode of vibration at 2.5 Hz. This ridge follows a hyperbolic curve which starts from 10 cycles per revolution at 15 revolutions per minute, and passes through, for example, 5 cycles per revolution at 30 revolutions per minute.

Fig. 5.14 Waterfall display


5.5 High resolution spectral estimators In the late 1960s, renewed interest in spectral estimators resulted from the development of two new techniques that were found to achieve higher resolution than the established methods. They were found to be invaluable for estimating spectra when the length of the available data is short. The first technique was published by J.P. Burg in 1967 and is termed the M a x i m u m Entropy Method (MEM). The second was published in 1969 by J. Capon and is called the Maximum Likelihood Method (MLM). The mathematics underpinning these techniques are rather more advanced than the other methods explained, and therefore only a brief synopsis is given. However, the final formulae are no more difficult to implement on a computer than the FFT procedure.

The term entropy, long familiar to engineers versed in thermodynamics, made an important reappearance in the 1940s with the work of Shannon and Weaver (1949) on the theory of communication. This was concerned with the capacity of communication systems to transmit information.

In the 1950s the term was brought to the attention of the statistical world by E.T. Jaynes (1957), and it was shown that the different applications of the word were not only mathematically similar but were, in fact, all examples of the same idea.

In statistics, entropy is a measure of the overall uncertainty, or disorder in a probability distribution. If we have a discrete distribution p~: i = 1 , . . . , k its entropy is defined as

k

E = - ~, Pi log Pi i=1

A detailed review of this fascinating topic is given by Kendall and Ord (1990), but we will just note that entropy must be non-negative and demonstrate that maximizing the entropy gives the common-sense solution in one simple example.

Example 5.3 A computer produces random binary digits. Let Po,Pl be the probabilities that the next random digits are 0 and 1, respectively. We only know that the sum of Po and Pl must equal one. The entropy is

E = -Po In P o - P l In P l

If we substitute 1 -Po for Pl , it is easy to show that the entropy takes its maximum value of 0.69 when P0 equals 1/2. This is precisely the value we would attribute to Po by using an 'equally likely' argument.

Maximizing the entropy in this way ensures that the distribution assigned is consistent with the known data but is minimally prejudiced with regard to any other factors.

This is a simple example of Jaynes' principle, that the minimally prejudiced probability distribution is that which maximizes the entropy subject only to the constraints supplied by the given information.

5 .5 .1 T h e m a x i m u m e n t r o p y m e t h o d The traditional approach to spectral analysis is to calculate the autocovariance function, multiply by a lag window and then carry out the Fourier transform. This technique

5.5 High resolution spectral estimators 137

effectively sets the autocovariance function to zero for all lags greater than some maximum value, as well as modifying the known autocovariance.

In the MEM the known autocorrelations are used unchanged and the unknown values are estimated in accordance with Jaynes' principle by maximizing the entropy.

It can be shown that for a stationary process the entropy is given by

f oo

E = log F(to) dto - - o o

where F(to) is the spectral density function. The constraints are

I | F(co) do , t dco = c ( k ) - - o 0

k = l , . . . , N - 1

where c(k) is the sample autocovariance at lag k. The solution is obtained by the calculus of variations using Lagrange multipliers. It

leads to the estimator

C(to) = a0/(21r N - 1

~, ak e -j a,k k=O

2) where a0, a ~ . . . aN-1 are obtained as the solution of

"r aN-1)T = (1 0 . . . O) T

and I7" is the N by N estimated autocovariance matrix of the process. Use of the divisor N for the autocovariances ensures that the matrix I2 is positive definite, and this is one reason for adopting that definition throughout this book. That is, the ( i - j ) t h element of I2 is the sample autocovariance at lag (i- j) , which in our notation is c(i-j). The matrix I7' is an example of a Toeplitz matrix and special methods are available for solving the above system with little computational effort.

Example 5.4 The results of applying the MEM with two sequences of 20 wave data, corresponding to time intervals 10.1 up to 12.0 seconds and 20.1 up to 22.0 seconds, are shown in Fig. 5.15. Despite such short records the resultant spectra are qualitatively similar to those of

< I 4 t,. / Spectrum for data collected 20.t-22.0s

t I p j . \ ~ Spectrum for data collected 10.1-12.0s ._~ I u ~ " ' / i "

0 0.5 1.0 1.5 2.0 2.5 3.0 Frequency (red/sampling interval)

Fig. 5.15 Spectra for subsequences of 20 wave data using MEM


Example 5.1. The difference in areas corresponds to the difference in the two sample variances and would be the same with any other method. If longer sequences are tried the estimates become increasingly spikey. Our intuition that physical spectra should be smooth prompts us to consider subsequencing and averaging.

The results of Example 5.4 are consistent with the generally held view that the MEM is particularly suited to short sequences. Applications to the analysis of spatial data exist in the literature. Another area in which the MEM has been used with success is speech recognition and synthesis for robots.

5 .5 .2 T h e m a x i m u m l i k e l i h o o d m e t h o d

Although the MLM formula was first published by J. Capon in 1969 as a f requency- wave number spectral estimator, it was derived in an earlier paper (1967) by the same author. This paper was concerned with the estimation of a seismic signal among noise, using data from a seismic array. In 1971 R.T. Lacoss showed that the MLM formula is equally applicable to the simpler case of a time series. The essential points of his argument are outlined in the following description, which shows a link with regression analysis.

We start with a model which assumes the average value of a random variable Y is proportional to a non-random variable x; that is, a linear regression through the origin. Then

Yt= flxt+ Wt where the errors, Wt, are assumed to have a mean of zero, to be independent of the xt and each other, to have a constant variance o "2 and to be normally distributed. A standard least squares approach gives the minimum variance unbiased linear estimator of /3 as

ExtY,

Furthermore

o2 E[/3] =/3 and V a r ( # ) = 2 ~

If the errors have a normal distribution, the least squares estimator is also the maximum likelihood estimator; that is, the estimator which is most likely to generate the Y,. It is convenient to write the above results using matrices for the x,, II, and W, values,

X T _ . ( X l , X 2 , . . . , Xn), y T = ( Y I . . . . , Y,,) and W T = ( W l , . . . , W n)

Then/3 and its variance can be written as

"- xTy[(xTx)

and

Var(#) = o2/(xTx)

respectively. The model can be generalized to allow for correlated errors, see Draper and Smith (1981). The covariance matrix V of the errors is now allowed to take the form

5.5 High resolution spectral estimators 139

V ~

E[W~] E[W, W2] . . . E[W, W,,]

E[W2W,] E[WZ~] . . . EIW2W,,]

, , , , , , , , , , , ,

E[W.W,] E[W.Wz] . . . E[W2.]

that is V = E[WW T] instead of being restricted to tr2I. The maximum likelihood estimator of/3 then becomes

fl -- xT V - I y](xT V - I x )

It has a mean value of/3 and a variance

Var (/3) = 1/(x T V - I x )

This is known as a weighted or generalized least squares estimator. It is easy to check that the original results are obtained if V is replaced by o21. We now apply the generalized result to a time series which we suppose to have been generated by a stationary ergodic random process. In this context we will choose some frequency to in the range 0 to r and set

x, = cos(tot) for 1 <~t<~N

Remember that the assumptions of stationarity and ergodicity imply that contributions near frequency to will have randomly varying amplitudes and will stop and start at random times. The consequences of this are that the average value of/3 can be assumed to be zero, the contributions near frequency to will be represented by the relatively large estimated variance of/3 for this choice of frequency, and it is not appropriate to try and estimate a phase because there are no deterministic components. The matrix V is the autocovariance matrix of the process. In fact it is, as usual, rather more convenient to use a complex version of the result. To summarize the argument so far

Var (/3) = 1/(zTV -1 z*)

where z T = [exp(jto), e x p ( j o 2 ) , . . . , exp( j to(N- 1))]. Now, since the average value of/3 is zero

Var (/3) = E [/~21

and we estimate the spectrum at frequency to by

C(to) = K / ( z T ],)'-Iz*) In this last equation, I2 is the sample estimate of V; that is, it contains the estimates of the correlogram up to lag ( N - 1), and the factor K is needed for the area under the spectrum to equal the variance.

The value of K can be found by calculating the area under the spectrum when K is set at 1 and then choosing K to scale this area so that it equals the variance of the time series.

Example 5.5

The results of applying the MLM to the same sequence of 20 data that was used for the MEM in Example 5.4 are shown in Fig. 5.16. The estimated spectra are somewhat smoother than those obtained with the MEM and there is a mathematical relationship


10.0

8.0

6.0

-~1~ (xlo'

/ \

I I I I

t I I I

t I I I

I I

I i I I ! I

I I I I

I I I I

I I

] , . , I~ i I ~l j t I I

\ t I l

. _ ..--

0 , 0 1 ' ..... t " I '" ~ ~ . . . . . ~ . . . .

0.0 0.5 1.0 1.5 2.0 2.5 Frequency (rad/sampling interval)

Fig. 5.16 Spectra for subsequences of 20 wave data using MLM

between the two methods which demonstrates that this will always be the case. If it is applied to longer records the results are more spikey and two spectra calculated from sequences of length 50 are shown in Fig. 5.17. These sequences were the data obtained during the time intervals 7.1 until 12.0 s and 17.1 until 22.0 s.

5.6 Exercises 141

1 6 -

14

12-

10- "<I~E (xlo')

l .g

E

i I I

I I I I

I II

t I I

I I !

| 1 I

I I 1

I I I

I I t

I I I

/ I I I I

I

'l I I I I

I

/ t \

/

o- 1 i 0.0 0.5 1.0

i \ I

t7~ i I

t t I - - - 1.5 2.0 2.5

Frequency (rad/sampling interval)

Fig. 5.17 Spectra for subsequences of 50 data using MLM

5.6 Exercises

0

Find the spectrum of an AR(1) process

X t = a X , _ 1 + Z t

in terms of a and the variance of the D WN. Find the spectrum of the M A (2) process

X, = Z, +/3~Z,_ ~ + fl2Z,_2


in terms of ill, ~2 and the variance of the DWN. Set/3~ equal to a and f12 equal to zero and compare the spectrum with that obtained in Question 1 when a equals 0.5.

3. The bandwidth of a spectrum (e), which must not be confused with the bandwidth of a procedure used to estimate it, is defined by the relationship

e2= l _ m 2 / ( m o m 4 )

where mn are the moments of the spectrum, which are themselves defined by

m, = f to n F(to) dto

The, one-sided, cut-off equilibrium spectrum is defined by

F(to) = Ato -5 for 00o< ~o

Take the limits of integration in the definition of m~ as 0 to infinity and calculate e. 4. A discrete time stochastic process has a one-sided spectrum

F(to) = (~r- to)/'tr 2 for 0 < to< 7r

Find its acvf and hence its acf. 5. The following data are masses in grammes of a set volume of barium titanate taken

from a kiln, which is the final stage of a continuous chemical process, at one hourly intervals over a two-day period. Calculate a sample spectrum.

29.74 31.96 28.20 31.14 34.02 34.46 33.08 31.82 33.16 33.16 32.44 32.22 32.14 32.38 34.18 34.66 34.54 39.10 36.66 36.02 34.42 32.08 32.92 32.14 31.50 31.06 31.41 31.82 32.56 35.54 38.08 37.32 38.60 36.60 37.50 41.40 44.04 43.02 36.84 29.68 30.44 38.54 38.10 38.38 37.98 34.34 34.14 31.64

6 Identifying system relationships from measurements


You should now understand what is meant by a spectrum, and be familiar with a selection of different methods for estimating the spectrum from a time series. We now proceed to the investigation of relationships between measurements of engineering or scientific quantities of interest. In order to identify a relationship it is necessary for at least two quantities to have been measured. It is also usual for certain assumptions to be made about the conditions under which the measurements are made. Thus, Boyle's law, which states that the pressure P experienced by a gas is inversely proportional to the volume V of the gas, assumes the temperature of the gas to be invariant during the measurement of P and V. If this were not the case, then Boyle's law, expressible as

PV = constant

would either have to be replaced by the ideal gas law

PV T = constant

or some more complex model to account for the changes in the temperature T. In any good experimental research programme, measurements are not restricted to the variables of primary interest. Other variables, which the underlying theory may assume to be constant, should also be monitored to check assumptions made.

In this chapter we will consider situations where the measurements are recorded as time series and any two sets of measurements are considered as realizations of two random processes. Often, one of the processes, X(t), will be an input into the system and Y(t) will be a corresponding output. The ideas of autocovariance and its relationship to the spectrum of a time series must therefore be generalized to those of cross-covariance and cross-spectrum, respectively.

Estimation of the cross-spectral density and the spectral density of either the input or the output signal will enable us to estimate the transfer function provided the system is linear in nature. The transfer function is most easily thought of as a plot of the ratio of the amplitude of the response to the amplitude of excitation versus the frequency of the associated excitation. By a linear system we mean that if the system is disturbed at a certain frequency to then the system only responds at that frequency. A doubling of the input amplitude, for example, will lead to a doubling of the response amplitude, and the response to a sum of disturbances is equal to the sum of the individual responses.

143


A familiar linear system is the motion of a mass suspended by a spring, which obeys Hooke's law. This situation corresponds to the free vibration of an undamped system. A more complex situation would be the design of an engine-room raft which supports the main engines of a research vessel. The raft rests on spring mountings and appropriate dampers so that engine-borne vibrations are not transmitted through the hull of the vessel to the surrounding water. For a research vessel, this may be of primary importance in order not to disturb the marine life being observed and to prevent interference with outboard sensing devices. The design of the raft requires identification of appropriate spring and damper characteristics to achieve the required characteristics. In some cases, non-linear systems may have to be investigated by approximating them with a linearized system in order that the ideas of spectral analysis techniques may be used. A well known non-linear system, which is approximated by a linear model, is a pendulum consisting of a mass supported by a string of negligible mass undergoing small angles of oscillation.

However, we are not only interested in relationships between signals to estimate transfer functions. We are also interested in the quality of observations made. The measurement of acceleration at a point on an aeroplane wing provides us with a practical application. If the wing is excited by a shaker at high frequencies an accelerometer would be one possible monitoring instrument, but it has the disadvantages of poor performance at lower frequencies and of adding its own mass to a critical point (perhaps) on the structure. An alternative procedure is to differentiate the observed displacement, which can be measured by non-contacting methods, twice. Whilst modern methods for differentiating signals are quite effective, they are inherently sensitive to noise and resolution problems. The correlations between the signals from the measuring instruments and the control signal to the shaker, over the frequency range to be investigated, help the engineer to decide on the appropriate instrumentation for an experiment. The squared correlation function is known as the coherence. For any 'distributed' structure, such as an aircraft wing, there will be many modes of vibration and transfer functions can be defined and estimated from any point to any other point.

In most experiments, many more than two sets of results will be collected simultaneously. In the situation presented in Fig. 6.1 we must record the x - y coordinate movements of each of four light emitting diodes (LEDs) located on the model, with respect to each camera, plus the strain gauge measurements of four mooring lines and wave data monitored by numerous wave probes, automatically. The experiment was designed to measure surge low-frequency damping, where surge corresponds to the forward-backward harmonic motion of the model. The long-armed spreaders attached to the model were used to secure the mooring lines so that the yaw (angular) motion in the horizontal plane was naturally suppressed. The mooring lines were also kept out of the water because their hydrodynamic loading is non-linear (due to viscous drag) and cannot be treated as a known quantity, nor can it be measured simultaneously with the other quantities. The mooring lines are there to provide a specific natural period and they are of no further interest in this particular experiment. The mooring line springs were made to British Standard design to ensure they obeyed Hooke's law of extension over a wide range of loads. LED motion and mooring line strain measurements ensure redundancy in the monitoring system so that consistency and accuracy of observations can be assessed.

At the start of each experiment the model was slowly pulled back towards the tank wall opposite the wave makers, made secure and allowed to achieve equilibrium. Using a solenoid switch controlled by the data-acquisition, data-storage computer, the model is

6.2 Discrete processes 145

Fig. 6.1 Multi-transducer experimental set-up

released and its damped behaviour in still water and in regular water waves observed. The regular water waves were of selected frequency and amplitude. The experiments were then repeated for different initial model offset positions as well as being repeated for different mooring stiffnesses. Some 1000 experiments were completed with 32 channels of data, each realization consisting of at least 1024 observations. Some 32768000 observations were stored using 12-bit analogue-to-digital (A/D) conversion. The ideas of cross-spectral analysis, now to be explained, can be applied to any pair of such recorded signals.

6.2 Discrete processes 6.2.1 Generalization of the covariance concept We assume that Xt and Yt are stationary random processes with means /Zx and /zy respectively. The cross-covariance function is defined by

y~y(k) = E[(Xt- /Xx)(Yt+k- ply)] (6.1)

The autocovariance function defined in Section 3.4.2 corresponds to Yt being a copy of Xt. As before, k is the lag and our definition of cross-covariance implicitly assumes the processes are second-order stationary (see Section 3.5.1). The cross-covariance function is not symmetric and therefore yxy(k) is not equal to yy~(k). Furthermore

x(t)

(a)

Time

First copy of original subsequence

x(t) t o 0

o o o

o o o t ~ I

l

i, ,~ x t ) o o

o o o o

' 15ositive lag

x'" T '~o, 0 0 0 0

0 0 0

Original subsequence

Negative lag (b)

o

o t

Secondcopy of original subsequence

Copy of second subsequence

x(t) o (~. ., o

y(t) to o o o ' I

o '~o

~, ~ ? t o l I I I I I I i ,,,,t~ o o, o o

o o o I s I Positive lag I I I I o o I

I I , . ~

Negative lag (c)

�9 Original subsequence

r o t

Second subsequence

T ~ i x(t) o

0 0 ,.~

o ~ ~ t ~ I , j ,=

I I I I

I ~ ~ o o o t

Y(t) I o oo I

s o i I o o

I I ', I I

I '~176 x(t) o , o I 0 0 ,

0 9

o t

(d)

6.2 Discrete processes 147

These relationships can be readily understood if we consider the pictorial relationship presented in Fig. 6.2. In Fig. 6.2(a) we have the first ten values of some discrete process. The autocovariance relationship corresponds to taking a copy of the original data and shifting it to the left or right to give a negative or positive lag respectively. We note in Fig. 6.2(b) that the same products are formed in each case except that the order of the data in each pair is reversed. However, since one data set is a copy of the other we have a symmetric relationship for the autocovariance. Figure 6.2(c) presents the corresponding situation for two distinct signals. If we now examine the overlapping parts of the signal representing the negative and positive lags respectively, we note that different products are now formed and so yxy(k) cannot be equal to 3,xy(-k). Figure 6.2(d) demonstrates that reversal of the order in which the signal products are generated, coupled with a change in sign of the lag does bring about equality, that is 3,xy(k) = 3,y~(-k).

Since the magnitude of the cross-covariance is dependent upon the magnitude of the signals Xt and Yt we define the cross-correlation function as

~/xy(k) pxy(k) = (~ /xx (O)~yy (O) ) l , 2 (6.2)

6 .2 .2 C r o s s - s p e c t r u m Having demonstrated why the spectrum of a discrete random process can be defined as the discrete Fourier transform of the autocovariance function (Section 5.2.2), we now define the cross-spectrum as the discrete Fourier transform of the cross-covariance function, that is

oo

Fxy(tO) = ~ Yxy(k) exp(-jtok) for - Ir~< to~< zr (6.3) k - - _ oo

Since 3,~y(k) is a complex function with no symmetry, F/y(to) is expressed as

Fxy(~o) = COxy((o) + j quadxy(tO)

where COxy(tO) is a real function of to called the co-spectrum, and quad~r(to ) is a real function of to, called the quadrature spectrum. The names co-spectrum and quad- spectrum arise from the fact that if a signal at frequency to is input to some linear system, the output will be at the same frequency but may experience changes in amplitude and phase. This output signal may be considered as the sum of a co-signal (a signal in-phase or anti-phase with the input signal) and a quadrature signal (a signal phase shifted by a quarter turn with respect to the input signal). The co-spectrum is then the cross-spectrum between the input and co-output. The quadrature spectrum is the cross-spectrum between the input and a phase-shifted quadrature output. The multiplication of the quadrature spectrum by j restores the phase shift. We show later in this section that the ratio of the modulus of the cross-spectrum to the input spectrum contains relative magnitude information over frequency. Therefore, the cross-spectrum contains both magnitude and phase information.

Since two distinct cross-correlation functions may be evaluated, so too may we

Fig. 6.2 (a) Original discrete signal subsequence; (b) autocovariance for positive and negative lag; (c) positive and negative lag for cross-covariance to demonstrate why ",/xy(k) =/= ~'xv(-k); (d) positive lag for cross-covariance ~'xv(k) and negative lag for cross-covariance "),yx(k) to demonstrate "),xy( k) = "),yx( - k)


evaluate two cross-spectra. However, bearing in mind that Yxy is easily obtained from Yrx we cannot expect the two cross-spectra to display different information. From the definition of F~y(tO) given, it follows that

Fxy(-to) = Fy/(to)

and

= r x , ( O )

Thus, one need only evaluate one cross-covariance function and the corresponding cross-spectrum. From the last relationship and definition of Fxy(tO) it follows that

rxy(to ) -- COxy(to ) + j quadxy(tO)

-

= COxy(-to) - j quadxy(-to)

Therefore CO~y(tO) is an even function of to and quad/y(tO) is an odd function of to and consequently it is usual to consider the frequency range 0 ~< to <~ rr.

By themselves, the co-spectrum and the quadrature spectrum are not convenient to work with. Instead, the coherence and the frequency response function are introduced as they present the information embedded in the cross-spectrum in a more useful way. The coherence function is defined by

IFxy( ~ (6.4) coh/y(tO) = Fx~(to)Fyy(tO)

where Irxy(~O)l is known as the cross-amplitude spectrum. Coherence is a useful concept in terms of providing diagnostic information about the reliability of the transfer function estimation and the quality of the information gathering. The coherence corresponds to the square of the linear correlation between the two signals at frequency to, and therefore lies in the range [0,1]. When the coherence of two signals over a particular frequency range is high then it is relevant to enquire about the relationship between the signals' amplitudes. We shall now do so assuming the system under investigation is linear.

6.3 Linear dynamic systems We assume that we have a stable linear system. A stable system is one which will return to its equilibrium state after being given an initial displacement or 'kick'. The impulse function is a suitable mathematical model for either form of disturbance. Therefore, the impulse response, explained in Appendix I, of a stable system must decay to zero at a sufficiently fast rate. Mathematics leads to a more precise statement; namely, a sufficient condition for stability is

ioo Ih(t)l dt<K m o o

where h(t) is the impulse response and K is a finite constant. Physically we can think of the impulse response as the response of the system to a short sharp shock, such as the single striking of a metal plate by a hammer. If, prior to the hammer striking, the plate

6.3 Linear dynamic systems 149

was in a state of equilibrium, then for t < ~', h ( t - r ) is zero. For t I> ~-the response at a selected point on the plate is represented by h ( t - z ) . If one now considers striking the plate repeatedly, with amplitude of excitation x(z), at time ~', then each of the responses experienced at the selected point will be superimposed, assuming a linear system. Thus, the response at time t is y(t), where Y(0 is the sum of the responses h ( t - ~ ) x ( z ) over all possible times ~. The output y(t) of the system is thus related to the input x(t) by the relationship defined by Equation (4.13a), namely

I y(t) = h ( t - r)x(r) dr = h(O)x( t - O) dO - o o - - o o

If x(r) is a unit impulse represented by 80") then y(t) = h(t). The convolution theorem gives

Y(to) = H(to)X(to)

where H(to) is known as the frequency response function or transfer function. To avoid confusion between a random process and its Fourier transform, in this

section we use lower case x and y to denote both random processes and associated realizations. It is also much easier to prove the results we need for continuous processes. Provided the sampling rate is high enough, analogous discrete time results provide adequate models for linear systems and more easily lend themselves to parameter estimation. It is convenient to assume the input x(t) has zero mean for the derivation of the following results, although they remain true if x(t) has a non-zero mean because all covariance functions are defined in terms of deviations from the mean. The first result shows the cross-spectrum to be the product of the spectrum of the input and the frequency response function. The cross-covariance function is defined by

Yxy(U) = E[(x(t) - tXx)(y(t + u) - ~y)] (6.5a)

and the cross-spectrum is given by

F~,(~o) = -~ y~y(u) e• du (6.5b)

As we have assumed zero means, we start with the cross-covariance function, namely

%y(u) = E[x(t )y( t + u)]

Substitution for y(t + u) leads to

I o o

Yxy(U) = E[x(t) h(O)x(t + u - O) d0] ~ o o

= h(O)EIx(t)x(t + u - 0)1 d0 --0o I oo

= h(O)Yxx(U - O) dO

Hence, the use of the convolution theorem gives

= (6.6)

150 /dentitying system re/ationships from measurements

We now demonstrate a second result relating the input and output spectra. The autocovariance function for the output signal y is

ryy(U) = E[y( t )y( t + u)]

using the definition of Section 3.6.3. Substituting for y(t) and y(t + u) we may write

~ / y y ( U ) - - E h(O)x(t- O) dO h(ck)x(t + u - ok) d ~ o o ~ o o

= h(O)h(4~) E [ x ( t - O)x(t + u - 4~)] dO d4~

That is

~ y y ( U ) - - h(O)h(~) yx~(U + O- dp) dO ddp ~ o o

Taking the Fourier transform of both sides gives

F~y(W) = h(O)h(4~)%x(U + O-40 exp(-jwu) dO d4, du

Use of the identity

exp(-jwu) = exp (-jw(u + 0-~b)) exp(+jw0) exp(-jw~b)

allows F~y(w) to be re-expressed as

Fxy(tO) = {h(0) exp(jw0)} {h(qb) exp(-jwO)) ~ o 0 ~ o o ~ o o

{%~(u + 0 - 6) exp (-jw(u + 0 - 6))} dO d6 du

That is

o r

Fyy(W) = H*(w)H(to) Fxx(W)

Fyy(W) = IH(w)12Vxx(W) (6.7)

The transfer function H(w) of a linear system may therefore be determined using the cross-spectrum and input spectra, or the spectra of the output and input.

The phase spectrum ~bxy(tO) is defined by the equation

tan t~xy(tO ) - - q u a d x y ( t O ) / C O x y ( t O ) (6.8)

and this presents a measure of the extent to which one signal (event) is leading, or lagging behind, the other at various frequencies.

6.4 Application of cross-spectral concepts Having generalized the concepts of spectral analysis to cross-spectral analysis we now consider two quite distinct examples.

6.4 Application of cross-spectral concepts 151

Example 6. I A model for the spatial distribution of rainfall, which has some physical justification, allows for 'rain cells' within storms. The velocities of the rain cells arc not necessarily the same as the velocity of the storm itself. Part of a current research project involves estimates of these velocities. A feature of the estimation can be simplified as follows. At weather station one, the rain depth recorded during minute t, At, is modelled by

Xt = tX + Zl,t

where ZI,, is DWN about a mean/x. At weather station two, synchronous measurements are modelled by

Yt = Xt_d+ Z2,t

where d is the time taken for the rain cell to travel between the two stations. We now assume that {Zl.t} and {Z2.,} are two uncorrelated, purely random, processes with zero means and equal variance of a2z. Then, by definition

")txy(k) = E [ ( X t - #x)(Y,+k-- #r ) ]

with #x = #y = #. Direct substitution for X and Y gives

Yxy(k) = E[Zl , t (Zl , t -d+k + Zz,,)l

= e I Z l , , Z , , , _ ~ + ~ ] + E[Z,,,z~,,I Since Z~ and Z2 are uncorrelated, the second term vanishes identically. By definition of a purely random process (DWN), see Section 3.6.1, Zl,t and Zl,t+d-k have zero covariance for d - k non-zero. Hence

y~y(k) = otherwise

Reverting to the definition of the cross-spectrum we note that since y~y(k) is only non-zero for k = d then

F~y(~O) - o~z exp(-j~od)

and thus

COxy(,O) = ~ cos (,od)

and

quad/y(to) = - a2z sin (rod)

Similarly, by applying the definitions of spectrum and cross-spectrum we can readily show that

=

r.~(~) = 2o~z

and thus

coh~y(co) = �89

and tan~xy(tO) = - tan(rod)


The phase spectrum is thus a straight line with slope -d . We will not always be in the position whereby we either want, or are able, to propose a

probabilistic model for the process under investigation. The next example illustrates the uncertainties and dependencies of the various models used at different stages of an offshore engineering related analysis.

Example 6.2 One may determine the motion response of marine vehicles using free surface dependent hydrodynamic analyses of varying sophistication and complexity, followed by frequency- domain based motion analysis techniques. Thus, one is able to plot the amplitude response per unit wave amplitude of the vessel as a function of the regular sinusoidal excitation frequency co. This plot specifies the so-called transfer function H(co). Conceptualizing a random seaway as the superposition of waves of varying frequency components we can directly apply the earlier established result

Fyy(co) = In (~o)12 F~x(~O)

Often, H(w) is referred to as the response amplitude operator and Fyy and Fxx, which correspond to the input and output spectral densities, as the energy spectrum and response spectrum respectively. Thus, in practice, to predict vessel responses in a seaway, an idealized spectral formulation is required to represent the desired sea severity.

In reality, the shapes of wave spectra measured in the ocean vary considerably and, in fact, are dependent upon environmental conditions such as geographic location, duration and fetch of the prevailing wind, stage of growth and decay of a storm, existence of swell and many other factors. Mathematical descriptions of sea spectra have the form

Fx/(to) = Aw -5 exp[-Bw-4l

with the definitions of the constants A and B specified according to the state of the sea. Numerical examples based on specific forms are presented in Chapter 7. Since the magnitude of the seaway responses is significantly influenced by the shape of the wave spectra for a given severity, various researchers have investigated the implications using different idealized spectral formulations in the context of design; see Ochi and Bales (1977) for example. The area under Fyy(to) represents the variance of the selected responses in a seaway. Provided the spectrum F~x is narrow banded we may use this statistic to determine the probabilities of exceeding selected engineering quantities using the Rayleigh distribution. Thus, spectral analysis is used in some situations as a means of quantifying the integrity of a design or its ability to perform intended tasks within acceptable limits of risk.

6.5 Estimation of cross-spectral functions 6.5.1 Estimating cross-correlograms and spectra We will continue to restrict our discussion of estimation to discrete data. This is far less limiting than it may sound, because even when measurements are continuous it is usual to digitize them and to process the resulting time series with a microcomputer. The classical approach is to calculate an estimate of the cross-covariance function and to transform the cross-covariance with a window, as described for the spectrum in Chapter 5. The estimation formula for the sample cross-covariances are

6.5 Estimation of cross-spectral functions 153

and

N - k

Cxy(k) = 2 (xt-.~) (Yt+k -)~ )/N (6.9a) t= 1

N - k

Cxy(-k) = ~, (Xt+k --.~) (Y , - ~9)/N (6.9b) t=l

The corresponding estimate of the cross-spectrum is

1 N-1

= - - 2 w(k)cxy(k) exp(-joJk) - r to< r Cxy(to) 2~ ,=-(u-,)

upon inclusion of a lag window, w(k). The real and imaginary parts of C~y(to) are handled separately and correspond to estimates of the co-spectrum and quad-spectrum respectively. The other functions can be estimated by replacing their constituent terms with the appropriate sample estimates.

In practice, it is usual to use the FFT to calculate spectral estimates, as we shall now explain. At the beginning of Chapter 5 we justified a sample spectrum

N 12 2ran UC( to ) - - -~ lX m for to = N

In Section 5.2.2 we established the identity

1 N-1 = ~ ~N c(k)e-J~ UC(to) 2"a" k=- -1)

We used this to define the spectrum of the underlying random process. In this chapter, we started by defining an analogous cross-spectrum, and then presented a physical interpretation and demonstrated that it will be invaluable for investigating physical systems. If we now reverse the sequence of identities presented in Section 5.2.2 with one 'x' replaced by 'y' we obtain the identity

1 N-| N 27rm - - E Cxy(k) e -j' 'k = __ Xm Y* for to - 2zr k = - ( N - l) 2zr N

This is a convenient alternative form for computation of the cross-spectrum via the FFT. You should note that X,,, and Y* are as defined in Section 5.2.1 and are therefore related to the output of standard FFT packages in the manner described in Section 5.4.3. These estimates will need smoothing and a typical spectral analyser does this by averaging consecutive estimates based on 1024 data.

6.5.2 Estimation of linear system transfer functions Estimation of a transfer function requires measurements of both input and output. Most FFT-based spectrum analysers estimate the transfer function H(a 0 by Hi(to) where

fl~(a 0 = !Cxy..(~ (6.10) Cxx(~)

Here Cxy(O) and Cxx(~) are estimates of the cross-spectrum F~y(o) and the input spectrum F~x(~). In most laboratory experiments, the investigator can supply the


disturbance, preferably white noise for maximum precision. Most commercially available spectral analysers also have an in-built noise source which has a flat spectrum up to 25 kHz, i.e. band limited white noise. The est imate/~(to) is insensitive to extraneous noise on the response signal, provided such noise is independent of the input. To appreciate this, suppose that y is mixed with a contaminating noise signal, e, which is independent of x and y, that is, the measurement z is the sum of y and e. The spectral ratio Fxz/Fxx therefore satisfies

r = ( , o ) rxy(tO) + rxe(O))

Since x and e are independent, their cross-covariance function and hence their cross-spectrum Fx~(Co) are both zero. It follows that

Irxz( )l Irxr(,o)l = H(w)

which is estimated by/t1(0)). An alternative estimate of H(~o) is/-12(0)) where

1:12((o) = Cyy(~,) (6.11)

Whilst/q~((o) is less sensitive to measurement noise, corrupting the output signal, it has the disadvantage of underestimating peaks in the spectrum. To see this we first observe that the coherence estimate satisfies

That is

c6h~y(o~) = icA,o)l

Cxx(,o)Cyy(,o)

c6hxy(OO) = [Cxy(W)12 Cxx(~O) C2~x(~ Cyy((.o)

Since both the coherence and its estimate always lie between 0 and 1,/~2(~o) is greater than or equal to /~1(~o). To reduce the variability of the spectral estimates some averaging procedure must be applied, with the result that the maxima of the transfer function tend to be underestimated, and the minima to be overestimated. The fact that

suggests that/-t2(~o) is less susceptible to bias than/~1(~,) at frequencies near resonance. The reverse is true around anti-resonances. This also provides an explanation for the fact that estimated coherences at resonances tend to be low when smoothed estimators are used. If estimates are not smoothed, then the estimated coherence will always equal 1, because

IC~y(~O)l z X Y * ( X Y * ) * = = 1

Cxx(,o)c A,o ) x x * r v*

6.5 Estimation of cross-spectral functions 155

Other situations, which give rise to low coherence, include measurement noise, additional inputs to the system and non-linearities in the system causing energy to be generated at additional frequencies.

Example 6.3 The differential equation

y + 0.2~ + lOOy = x

represents a lightly damped single-mode linear structure such as a mass on a spring. A computer simulation was used to generate 100 s of Gaussian white noise input (x) with mean 0 and standard deviation 100 and the response (y). The first 20 seconds are shown in Fig. 6.3. These simulated signals were sampled at 0.05 s intervals to give two time series of length 2001 for x and y. One-sided spectra for x and y, C~(to) and Cyy(O))

- " w ' , . . . . w . . . . . . w . . . . v

-I

-2

-3 0 ..... - 6 8 i"o

v ' v . . . . . . �9 J

i

1'2 .... 1 '41"6 1"8 ?.0

400

300

200

100

-100

-200

-300

-400 0 2 4 6 8 10 12 14 16 18 20

Fig. 6.3 Input (lower) and response (upper) of single-mode structure of Example 6.3

156

" 0

e-

Identifying system relationships from measurements

-10

-20

-30

-40

-50

-60 0 1'0 15 2'0 2~5

rad/s 30

-20

-40

-60

-80 v

-100

-120

-140

-160

-180

. . , , . .

0 5 1'0 1~i 2'0 2'5 30 rad/s

Fig, 6,4 Transfer function of single-mode structure of Example 6.3

respectively, were calculated by the acvf using a Parzen window with M equal to 200 as described in Chapter 5. The one-sided cross spectrum, C~y(tO), was calculated via the cross-covariance function with the same window.

c~(~) = 1 [ -~ %(0) + E w(k)[Cxy(k) + 9~(k)l cos (~ok)

- j~, w(k)[C~y(k) - Cyx(k) ] sin (tok) J

where the summations are from 1 to 200, with

C~y(k) = ~, (x,-.~) (Y,+k -- ~)/N

6.6 Exercises 157

,3

4500-

3750 -

3000

2250

75.0 -

50.0 --

25.0

0.0

567 4 8

3 90 2

!

56 0123456 90 234 78

f 90 89 7 m 890 I ? 89 8

23456 0 567 7 12

456 34

- 1 2 3

I 10

Frequency (rad/s) (a)

3 12 4

5 0

6 9

7 8

- - 6 0

5 I 4 Z

23 3 - 123456789012345678g01 45678901234567890

0 10 Frequency (rad/s)

(b) Fig. 6.5 (a) Spectrum of {x}" (b) spectrum of {y}

20

I

~0

and

c~y(-k) = E (Xt+k -- ~) (y,-- y)/N

and summations are now from 1 to 2001- k. The transfer function is shown in Fig. 6.4. The estimated spectra of {x} and {y} are shown in Figs 6.5(a) and (b). Estimates, H1 and /-/2, of the magnitude of the transfer function (up to 20 rad s-1) are shown in Fig. 6.6(a). The square root of the coherence function is shown in Fig. 6.6(b). The estimated tangent of the phase is shown in Fig. 6.6(c). The exercise was repeated with Gaussian white noise with zero mean and standard deviation equal to 20% of that of the original signals, added to the original signals. The results are shown in Figs 6.7(a)-(c).

6.6 Exercises

l . An offshore structure is subjected to a regular wind gust of amplitude Wo. The tip response of the flare boom has been identified as

y(t) = Yo sin tol(t- to) sin to2 ( t - to)

where the periodicity of the gust pulses, T, equals 21r/try2 (see diagram below). If the duration of the gust GT is extremely small compared with both to and T, show that the associated cross-covariance functions satisfy

-20

-40

10 Frequency (rad/s)

(a)

%%

I 20

0.25 ~r 0

1.00

" I 10


Frequency (rad/s) (b)

I

2O

0.0

A -25.0

t"

7- -50.0 -

- 7 5 . 0 -

0

123456789012345678901234567890 - 12

m

45 6789012345678901234567890

3 I J

10 20

(c)

Fig. 6.6 (a) Estimates H1 (broken curve), H2 (full curve)of the magnitude of the transfer function; (b) coherence for single-mode structure of Example 6.3 using original time series; (c) estimated tangent of phase of transfer function for single-mode structure of Example 6.3 using original time series

and

= WoYoGT sin[to,(z - to ) ] sin [to2(z - t() )] T

= WoYoGT sin[to,(z+ to)] sin [to2(z+ to)] T

Comment. Calculate a sample average over one period by direct integration and recall that c o s O ~ l and sinO~O for very small O. Assume the first gust occurs at t - O .

w0

r

-'F-GT I"~- . . . . . T__ . >] Time

o= ==

-2O

-40 0

v

10 20 Frequency (rad/s)

(a)

0.30 T 0

0.75

J 10 20

Frequency (rad/s) (b)

6.6 Exercises 159

20.0

10.0

-e-

C

F.-- 0.0

-10.0

34567890123456789012345678 9 0

1

2

4 567

89012345678901234567890

, , I . . . . . . . . I

0 10 20

Fig. 6.7 (a) Estimates H1 (broken curve), H2 (full curve) of the magnitude of the transfer functions; (b) coherence for single-mode structure of Example 6.3 using noise-corrupted time series" (c) estimated tangent of phase of transfer function for single-mode structure of Example 6.3 using noise-corrupted time series

D

,

Further investigations of the spatial distribution of the rainfall examined in Example 6.1 suggest that the Xt model is acceptable but

Yt = OllXt-d, + oL2Xt-d2 q- Z2,t

where dl and d 2 are the times taken for the beginning and end of a rain cell to travel between the two stations. Determine the new cross-covariance function. Also, determine the corresponding autocovariance functions for X and Y and hence the cross-correlation function. What ranges of values are valid for a~ and a2? The tanker of Exercise 5, Chapter 7, is heading at 15 knots into a head sea of mixed wave frequencies. The heave and pitch motions of the vessel have been monitored. Five seconds of the recorded signals, sampled at intervals of 0.1 second, are now presented (Table 6.1).

Initially, plot the data. Then, if you have a computer, try to write a computer program to provide the cross-covariance function, the cross-correlation function and the cross-spectrum. Otherwise, select a manageable subset from the record presented to produce the requested quantities using a calculator.

160. Identifying system relationships from measurements

Table 6.1 . . . . . . . .

Time Heave Pitch Time H e a v e Pitch Time Heave Pitch

0.0 -0.1542 0.0091 0.1 0.0608 0.0130 0.3 0 .7161 0.0217 0.4 0.7795 0.0192 0.6 0.1362 -0.0041 0.7 -0.3532 -0.0159 0.9 -1.0577 -0.0130 1.0 -1.2233 0.0006 1.2 -1.3978 0.0218 1.3 -1.4420 0.0199 1.5 -1.2340 -0.0052 1.6 -0.8737 -0.0208 1.8 0.3555 -0.0408 1.9 1.0595 -0.0399 2.1 1.9570 -0.0122 2.2 1.9278 0.0087 2.4 0.9839 0.0380 2.5 0 .3421 0.0405 2.7 -0.5814 0.0261 2.8 -0.7560 0.0139 3.0 -0.6056 -0.0132 3.1 -0.3305 -0.0241 3.3 0.4459 -0.0286 3.4 0.7942 -0.0218 3.6 0.9427 -0.0024 3.7 0.6524 0.0069 3.9 -0.4767 0.0269 4.0 -1.1279 0.0357 4.2 -2.0361 0.0318 4.3 -2.0990 0.0145 4.5 -1.3221 -0.0323 4.6 -0.6299 -0.0480 4.8 0.7566 -0.0448 4.9 1.2612 -0.0284

i , �9 i i

0.2 0.4216 0.0184 0.5 0.5618 0.0096 0.8 -0.7724 -0.0196 1.1 -1.3232 0.0144 1.4 -1.4084 0.0097 1.7 -0.3257 -0.0336 2.0 1.6401 -0.0298 2.3 1.5666 0.0269 2.6 -0.2090 0.0357 2.9 -0.7547 0.0003 3.2 0.0400 -0.0297 3.5 0.9840 -0.0122 3.8 0 .1521 0.0166 4.1 - 1.6848 0.0387 4.4 -1.8464 -0.0092 4.7 0.1010 -0.0522 5.0 1.5777 -0.0077

7 Some typical applications

7. I Introduction

We start this chapter with a step-by-step account of the calculations which lead to the spectral estimates for the wave data shown in Fig. 5.5(b). These were estimated from the sample autocovariance function. A spectrum is then used to investigate the spectral response of a rolling ship, modelled as a one degree-of-freedom system, and the probability of the roll angle exceeding a specified threshold. The relationship between the spectral description of a wave system observed by stationary and moving observers is then established. An explanation of how to compute the integral of an encounter- frequency-based spectrum is discussed. The concept of significant responses is then explained and used in an oil rig design problem.

7.2 Calculating the sample autocovariance function The 397 wave data in Append ix V wi l l be denoted by,

{xt} t = 1 , . . . , 397

The first 50 points were plotted in Fig. 5.5(a) and a plausible wave profile--that is, a smooth curve passing through all the points~is shown in Fig. 7.1. The arithmetic begins

,ooo 2

I I ,, I 1 1 0 10 20 30 40 5o

Time (0.1 s) Fig. 7.1 Plausible wave profile

1 6 1

162 $ome typical applications

Table 7.1 Acvf of wave tank data

lag acvf lag acvf k c(k) k c(k)

lag acvf lag acvf k c(k) k c(k)

0 70790 11 -5687 21 11645 31 4378 1 33360 12 2088 22 8768 32 -635 2 -18466 13 5124 23 3548 33 -3871 3 -35097 14 4615 24 -266 34 -1292 4 -26705 15 -615 25 -4629 35 2608 5 -15319 16 -7279 26 -8332 36 1053 6 -2794 17 -9027 27 -8288 37 -775 7 12540 18 -7849 28 -2641 38 -446 8 19026 19 -3377 29 4648 39 645 9 9142 20 5599 30 6743 40 -293

10 -5307

with the calculation of the autocovariance function up to lag 40, which was the largest value of M used with the Bartlett window. In our usual notation this is

397-k

c(k) = ~, ( x t - ~ ) (X,+k -- ~ )/397 t=l

for k running from 0 to 40. As the water levels were measured from the still water level, ~f was close to zero, -0 .24 in fact. The autocovariance function is given in Table 7.1. The correlogram

r(k) = c(k)/c(O)

is shown in Fig. 7.2. The lines drawn at

- 1/397 + 2/V'(397)

which equal -0.103 and 0.098 respectively, delineate the area within which 95% of the autocorrelations from a discrete white noise process would be expected to lie. The wave data are clearly distinguishable from white noise in this respect. In particular r(1) is positive, as one would expect, and equal to 0.47. The pairs on which this correlation is based a r e (xt, x t+l) and they are plotted in Fig. 7.3.

7.3 Calculating the spectrum Now that we have the autocovariance function we can calculate the spectrum from

= -- c(O)+ 2 ~ w ( k ) c ( k ) cos(tok 7r k=l

This is the one-sided spectrum and co can be set equal to any frequency in the range [0, 17"]. In fact, Fig. 5.5(b) was drawn from values of C(to) calculated for 501 values of to ranging from 0 to 1r in 500 steps of ~r/500. Three values of M (10, 20 and 40) were used, and

w(k) = 1 - k / M

We have not tabulated all 1503 values of C(to), because they can be read off Fig. 5.5(b),

-0 .6 I

1 0.471 2 -0.261 3 -0 .496 4 -0.377 5 -0.216 6 -0.039 7 0.177 8 0.269 9 0.129

10 -0.075 11 -0.080 12 0.029 13 0.072 14 0.065 15 -0.009 16 -0.103 17 -0.128 18 -0.111 19 -0.048 20 0.079 21 0.164 22 0.124 23 0.050 24 -0.004 25 -O.065 26 -0.118 27 -0.117 28 -0.037 29 0.066 30 0.095 31 0.062 32 -0.009 33 -0.055 34 -0.018 35 0.037 36 0.015 37 -0.011 38 -0.006 39 0.009 40 -0.004

-0 .2 i

-0 .4 !

r(k) 0.0

I

I

Fig. 7.2 Correlogram for wave tank data

! L-

I

- - . . - - .

' ' - - ' t

I

I

I

. I I i

I I I

7.4 Calculating the response spectrum 163

0.2 0.4 0.6

but Table 7.2 does contain C(to) for to ranging from 0 to ~r in steps of Ir/25 for M equal to 10 and 40. You may have noticed that, despite a lengthy theoretical development, the actual calculations involved in estimating a spectrum from the autocorrelation function are surprisingly straightforward and simple to program. Furthermore, this method is perfectly respectable for a series of this length, and no advantage would be gained from using a FFT algorithm.

7.4 Calculating the response spectrum We showed in Chapter 6 that the input and response spectrum satisfy the relationship


500 -

E 0 -

+ ~--

- 5 0 0 - �9

F i g . 7 . 3

�9 �9 �9 o 0 0 0 �9 �9 �9

0 0 � 9 1 4 9 1 4 9 0 0 0 0 �9

o o � 9 � 9 � 9 � 9 o o � 9 �9 O � 9

�9 �9 0 � 9 0 0 I H ~ e e o � 9 1 4 9 1 4 9 �9 0 o o �9 �9

�9 o o o o o o o e o � 9 1 4 9 0 � 9 o � 9 � 9

�9 0 � 9 o o 0 0 o o 0 0 o o o o o o o o o o �9 �9

�9 �9 � 9 1 4 9 1 4 9 o o o o o 0 o � 9 0 0 0 e e �9

0 � 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 �9

�9 � 9 o 0 4 ) 0 �9 �9 �9 �9

�9 �9 0 0 0 (De 0 0 �9 �9

I I I I I , 1 - 6 0 0 - 3 0 0 0 300 600 900

xt (ram)

Plot of wave tank data pairs separated by one sampling unit

In order to predict the response spectrum, the transfer function, H(to), and the input spectrum, F~(to), must be assumed known. Obviously, C(to) estimated in Section 7.3, or some theoretical spectral form, might be used for Fx~.

Initially, we suppose the transfer function, H (to), corresponds to the solution of a one degree-of-freedom spring damper system as modelled by the general equation

a 19 + b t9 + c 0 = Mo cos (rod)

Provided b 2 - 4 a c is negative, the general solution of this second-order ordinary differential equation is

0 = e x p ( - n t ) [A sin (todt) + B cos (todt)] + 0, COS (rod-- e2)

where A and B are determined from the assumed initial conditions. The relationships between the parameters of the solution and the governing equation are

2n = b / a , to 2 = to 2 - n 2, to # = c / a and 0~ = 0~t/z

where 0st is the static response of amplitude M o / c and the magnification factor/x satisfies

1 /x = ((1 - A2) 2 + 4K2A2) 1/2

Table 7.2 Spectrum ordinates for wave tank data. Bartlett window with M equal to 10 and 40 i i i i

Frequency M = 10 M = 40 Frequency M - 10 M = 40

0.000 6212.12 2160.25 0.126 7513.76 2406.72 0.251 12532.97 3164.63 0.377 23039.08 6326.19 0.503 38367.43 35714.96 0.628 54095.70 74453.38 0.754 64170.04 67 900.91 0.880 64 886.41 106 759.03 1.005 57 341.20 63 222.70 1.131 46321.73 30 973.74 1.257 36 728.46 29 659.40 1.382 30 565.43 29 734.52 1.508 26 678.92 36 633.45

1 634 1 759 1 885 2011 2 136 2 262 2 388 2513 2 639 2.765 2.890 3.016 3.142

22847.40 23718.69 18018.46 15550.57 12882.00 6130.08

8794.84 5201.60 6439.49 7234.26 5428.41 3564.81 4901.14 2386.77 4299.98 2203.00 3622.29 2751.61 3130.71 2161.38 2964.86 1821.81 3014.32 1742.01 3065.05 1454.59

Z4 Calculating the response spectrum 165

\

t = 0 ~, / ~ . - . ~ t

/

Oa / / / L d / T = 27Z/(0 d

/ /

v'

Fig. 7.4 Free damped oscillation O= Odexp(-nt) sin(~d+8), where Od = ( A2+ B2) 1/2 and tan 8 = B/A

with A = to,/toe and r = n/too. Hence, too is the natural frequency, and A and x represent the tuning factor and the non-dimensionalized damping respectively. By natural frequency we mean the frequency associated with the undamped and unforced motion. This corresponds to the solution for b = 0 and Mo = O, that is

0 = A sin (toot) + B cos (toot)

The dynamic or forced general solution clearly consists of two distinct sets of oscillations. Whilst both are in effect, the motion is referred to as the transient mo t ion . Usually, the first rOd-based oscillation decays quickly, as illustrated in Fig. 7.4, and we arrive at the steady-state motion described by the second to~-based oscillation. By plotting/x against A for different K values (see Fig. 7.5), we note that for K very small/~ is very large. In fact, by differentiating /z with respect to A we find that /x has a maximum value when

P 10 9 - - K = O

/ ~ -K =0.1

- / / _ \ \ = 0 . 3

0.2 0.6 1.0 1.4 1.8 A Fig. 7.5 Magnification factor variation as a function of 'tuning' for different damping levels

166 Some txpical applications

A = (1 - r2) 1/2. That is, for very undamped motions, when b (and hence K) is very small, A tends to 1. This means that the excitation frequency toe and the natural frequency toe are equal. This is considered to be a critical situation because we have resonance. Figure 7.5 indicates that as the damping (and hence r) increases the maximum response is obtained for A less than unity.

For a ship, or any other floating structure which exhibits port and starboard symmetry, the linear governing equation for roll has the form

(/44 + A 4 4 ) 0 + B440+ C440+ D(s2,$6) = Mo cos ( to j + e)

where D($2,$6) indicates that the motions of sway ($2) and yaw (s6) also induce roll motion. Here, for simplicity, we shall assume that D($2,$6)= 0 and take e = 0. The frequency of excitation toe will equal the wave frequency if the ship is stationary and will equal the so-called wave encounter frequency if the ship is advancing in waves. Although not strictly accurate, such models may be used to study the roll response of a ship. In this case, the parameters of our original governing equation satisfy a = 144 + A4a, b = B44 and c = C44. Here, 144 corresponds to the roll moment of inertia of the ship about the centre of gravity.

The hydrodynamic coefficients A44 and B44 are the added moment of inertia and the fluid damping. These quantities are calculated by solving the hydrodynamic problem of force oscillating the vessel in the roll mode in the free surface of otherwise calm water. The hydrostatic stiffness term C44 is associated with the Archimedian restoration moment. That is, the moment generated from buoyancy if the ship is rolled out of its normal state of static equilibrium. The excitation moment Mo represents the excitation of the ship due to a regular train of regular waves hitting the ship and being scattered by the ship. Once again, this hydrodynamic problem can be solved to produce M0 and the true value of e, the phase difference between the incident wave and the generated moment.

Generally, for regular sinusoidal waves the surface profile in the direction of wave propagation, x say, has the form

17 -- r/a exp j[kx- tot]

Clearly, one can select either the cosine or sine wave, or process both simultaneously by maintaining the complex exponential form. Here, k is the wave-number defined by 27r/A, with A denoting the wavelength, and to is the circular frequency of the wave.

Whereas the horizontal harmonic translational sway transfer function might be expressed in terms of sway amplitude per unit wave amplitude, the roll transfer function is generally expressed as roll amplitude per maximum wave slope. The wave slope is simply determined by differentiating r/with respect to x, that is

0r/

Ox = jkr/a exp j [ k x - tot]

Thus, the maximum wave slope is kr/a, and since the system is linear it is quite usual to assume r/a -" 1.

Assuming 0s is the peak ship roll angle and 0w is the maximum wave slope then /x = 0s/0w. If the ship has a natural period of roll of 8.6 seconds and u - 0.14 is assumed, then the ratio 0s/0w can be calculated, using the solution provided, for different excitation wave frequencies toe. In particular, upon noting that Wo = (2zr)/8.6 = 0.7306

7.4 Ca/cu/ating rhe response spectrum 167

T a b l e 7.3 Motion response amplitude operator for rolling of ship with natural roll period 8.6 s

es 1 T toe A 2 (1 -- A2) 2 0.0784A 2 k =

Ow k 7.5 0.838 1.3149 0.0990 0.1031 0.4495 2.22(45) 8.0 0.785 1.1556 0.0240 0.0906 0.3385 2.95(45) 8.5 0.739 1.0237 0.0006 0.0803 0.2844 3.51 (67) 9.0 0.689 0.9131 0.0076 0.0716 0.2814 3.55(33) 9.5 0.661 0.8195 0.0326 0.0643 0.3113 3.21 (25)

10.0 0.628 0.7396 0.0678 0.0580 0.3547 2.81 (96) 10.5 0.598 0.6708 0.1083 0.0526 0.4011 2.49(31) 11.0 0.571 0.6112 0.1511 0.0479 0.4461 2.24(16) 11.5 0.546 0.5592 0.1943 0.0428 0.4880 2.04(92) 12.0 0.524 0.5144 0.2358 0.0403 0.5255 1.90(30) 12.5 0.503 0.4733 0.2744 0.0371 0.5608 1.78(30) 13.0 0.483 0.4376 0.3163 0.0343 0.5920 1.68(90) 13.5 0.465 0.4064 0.3523 0.0319 0.6198 1.61(30) 14.0 0.449 0.3774 0.3877 0.0296 0.6460 1.54(80) 14.5 0.433 0.3518 0.4202 0.0276 0.6692 1.49(44) 15.0 0.419 0.3287 0.4506 0.0258 0.6902 1.44(87) 15.5 0.405 0.3078 0.4791 0.0241 0.7094 1.40(97) ....... , , , i i i

rads -~, and 4r 2= 0.0784, the results in Table 7.3, and hence Fig. 7.6, are readily generated.

With the transfer function H(to) available one can now consider generating the response spectrum using the sample wave slope spectrum defined in Table 7.4. The results are presented in Table 7.5 and Fig. 7.7. Here, transfer function and response amplitude operator (RAO) are used interchangeably.

Just as the area under the input wave spectrum is equal to the variance of the original wave elevation time series, so the area under the response spectrum produces a measure of the variance of the response due to the random excitation represented by the input spectrum. Provided the input spectrum is narrow banded (see Section 7.6), the

4 . 0 ~

3 . 6 -

3 . 2 -

2 . 8 -

2.4-

2.0-

1.6-

1.2 0.3

Fig. 7 . 6

l i I I I 0.4 0.5 0.6 0.7 0.8 1 1 I ~ o~ e

0.9 1.0 1.1

Roll response amplitude operator (RAO)


Table 7.4 Sample wave slope spectrum , ,

Sow x 10 -3 2.55 5.493 10.242 5.588 1.011

to 0.8 0.7 0.6 0.5 0.4 i i , , ,

Table 7.5 Generation of response spectra for ship with RAO defined in Table 7.3 and subject to sea state defined in Table 7.4

i i , ,

to So,,, x 10 -3 RAO (a,&0) 2 Sa, x 10 -3

0.8 2.555 2.73 7.4529 19.042 0.7 5.493 3.54 12.5316 68.836 0.6 10.242 2.42 5.8564 59.981 0.5 5.588 1.78 3.1684 17.705 0.4 1.011 1.395 1.9460 1.967

i

probability density function associated with the peak roll amplitude response can be assumed to be Rayleigh distributed. That is

i0 20~ Pr{0~<0} = ~ exp[-~/o '2] d0~ 0

= 1 - exp[-02/o -2 ]

where o .2 is the variance of the ship roll angle time series (not the variance of the peak roll amplitude responses themselves). Since the input spectrum is more than likely to be an estimate, so the response spectrum is an estimate too. Therefore, in practice, rather than o ~ will be used. An estimate of the area under the response spectrum is 17.40 x 10 -3. Selecting the roll threshold as 0 = 5 ~ and noting that 5 ~ corresponds to 0.0873 radians then

Pr {0~ i> 5 ~ = exp ( - (0.0873)2/17.40 x 10 -3) = 0.6453

That is, the probability of 0s exceeding 5 ~ is 0.65 using a more detailed definition of the spectrum than that presented in Table 7.4. Upon using the response spectrum of Table 7.5 with Simpson integration rules, an estimate of the response variance is 16.64 x 10 -3, and this leads to a probability of exceedence value of 0.6325, that is 0.63.

In undertaking such a simple example the amount of numerical effort required for one

20

80 70 60

x 50 40 30

10

0.2 0.4 0.6 0.8

90-

Fig. 7.7 Wave slope spectra

I I I . . ~ r

1.0 1.2 co

Z4 Ca/cu/aring the response spectrum 169

case, especially if one includes estimation of the hydrodynamic and hydrostatic data, is sufficient to warrant automated computer calculations. This becomes even more necessary when one realizes that it is usual to consider different forward speeds, different wave headings, different sea spectra, and different ship loading conditions in a full assessment of the behaviour of the ship in a seaway. However, once automated there is the danger that the often elegantly presented results will take on a greater respectability due to their inclusion in a published document, such as this book. In good engineering practice a number of questions related to the illustrative calculation need to be put.

First, how well is the physics represented by such a simple model of the roll motion in regular waves. The assumed linear fluid damping implies that viscous roll damping has been omitted. If we include viscous damping then this is proportional to a power of the roll velocity. Often a square law is assumed. Although the non-linear equations can be solved iteratively, the response spectrum-input spectrum relationship used assumes linear dynamic responses. Therefore, the non-linear viscous roll damping is usually linearized by using the first term of its Fourier series. As already indicated, the roll motion of a structure with port-starboard symmetry is not uncoupled from either sway or yaw. Even with fore-aft, as well as port-starboard symmetry, the roll motion remains coupled to sway. The equations of motion, as written, also assume that the mass of the structure and its cargo are arranged symmetrically so that all products of inertia are zero. However, provided the will and the knowledge to calculate all the required additional hydrodynamic and hydrostatic quantities are available, to allow generalization to the dynamic coupled motion equations, the predicted roll transfer function can always be improved.

Second, one must ask how representative of the environment are the selected input sea spectra for the intended operational waters of the ship. The manner in which the original time series data were collected, the season and the duration of the wave observation period(s) will obviously influence the final form of the selected spectra. In looking at any specific operation of a ship in a seaway one needs to know its route and the characteristics of the sea in the different regions it crosses. The spectra and their possible characteristics need to be known for each area. Thus, the probability of exceedence calculations just performed only provide a conditional probability.

That is, the calculated probability only holds for the spectrum selected. We therefore need to know, or be able to estimate, the probability of occurrence of the selected spectrum. Thus, to estimate realistically the probability of exceedence it is necessary to undertake the calculations implied in the following equations, namely

Pr {Os~> O} = E all spectra

Pr { 0~ I> 01 selected spectrum} Pr (spectrum)

Here 'all spectra' means all spectra relevant to the situation under scrutiny for all the environmental conditions of interest, defined to be mutually exclusive.

Next, provided we are confident that the spectrum selected is sufficiently well defined, its moments ought to be calculated to determine the spectral width parameter (see Section 7.6). This would help to decide whether the selected extreme distribution, the Rayleigh distribution, is applicable. The fact that the Rayleigh distribution is traditionally used does not justify our action and we should, in many applications, consider whether a Gaussian or a Cartwright-Longuet-Higgins distribution is more appropriate.


The apparently high conditional probability calculated may be unrealistic for all sorts of different reasons. In general, the volume of work and the apparent sophistication of the methods employed in undertaking such calculations do not by themselves provide validity to the results generated.

7.5 The spectrum and moving observers

Throughout this book it has been implicitly assumed that spectra are determined at some fixed location in space by some 'fixed' device. Consequently, the frequencies associated with the observations are the absolute frequencies of the phenomena under investigation. Suppose that observations are made with respect to a reference point which has a velocity relative to earth equal to Vs. The encounter angle,/3, will be defined by the angle between the direction of wave travel and the heading of the moving observation platform, a ship say. For regular wavefronts this situation is presented in Fig. 7.8. The speed of the ship relative to the waves is Vw-V~ cos/3. The wavelength and the wave velocity are A and Vw respectively. Hence, the wave encounter period, T~, satisfies

Te _.,e- Vw - V~ cos/3

Tw

l_( )cos upon noting that Tw = AIVw. In deriving the expression for the encounter period it is

i - ._o

el . 0

Wave crest ,o~/

.8'/~q,

ks = Ship speed

Vw = Wave s p e e d / f / f J Wave crest

. . . .

J,, / / Wave crest

L = Wave length )~e = Encounter wavelength

Fig. 7.8 Relative advance of ship and regular wave system

Z5 The spectrum and moving observers 171

simply necessary to realize that the time required by the moving observer (or ship) to travel from one crest to another is the wavelength divided by the speed of travel in the direction of wave propagation. This should make the definition of the wave period T~ equally obvious.

From hydromechanics it is known that water waves satisfy the dispersion equation

to E = gk tanh (kd)

This states that wave frequency, to, wave-number, k, and water depth, d, cannot be selected arbitrarily. One may therefore choose any two wave parameters, but the third wave parameter is governed by the dispersion equation. For deep water, water depths greater than half the longest wavelength of interest, this relationship reduces to to 2 = gk for all frequencies of interest. Hence, the deep water wave velocity is given by

Vw =g- tO

The wave encounter frequency toe, equal to 2~r/T~, is thus

= . , 1- cos

Since the wave encounter frequency is a function of ship speed, V~, and the encounter angle, fl, it is worth briefly studying the characteristics of toe before considering the relationship between a wave frequency dependent spectrum and an encounter-frequency based spectrum.

The encounter frequency is only zero when the velocity of the observer in the direction of wave propagation and the wave velocity are equal. On the other hand, when Vw < g~ cos fl, the encounter frequency is negative and the ship overtakes the waves. Although to the ship-borne observer the waves will appear to come off the bow of the ship, this is not the case. Such waves or seas are described as following seas and occur for 0 ~ < fl < 90 ~ and 270 ~ < fl < 360 ~ when cosfl is positive. The encounter frequency is positive for Vw > Vs cos fl. Clearly, Vw and V~ cos fl need not be of the same sign. The encounter frequency is maximal when the derivative dto~/dto is zero, that is when Vw satisfies 0.5 Vw = Vscosfl. Thus, when Vw is greater than V~cosfl the waves will approach the ship from behind and will be travelling so fast that to~ is only slightly less than to. If the waves travel slowly, compared with the ship, then to~ will tend to be small. When Vw and V~cosfl are of opposite direction and sign, such as when the waves approach the bow of the ship, then to~ is everywhere greater than to, that is 90~ fl < 270 ~ and Vw + V~ cos fl is positive and large. Generally, we consider such seas as head seas. A more specific definition of sea types is presented in Fig. 7.9.

Since the waves can be described in absolute or encounter frequency terms, so too may the spectrum. Suppose F~(to~) is the derived encounter frequency spectrum, then the total energy associated with the elemental frequency band dto~ must equal that corresponding to dto. The energy associated with dto is fixed by definition of F(to), and since each to value maps on to to~, the amount of energy associated with dto and dto~ cannot be changed because of a simple change of independent variable. That is

= r do ,

Since we now know that in deep water


Beam seas

Following seas

Quartering s e a s ' ~ I / B~ seas

Ouarter,no / I \ Bow seas seas

Head seas

Beam seas

Fig. 7.9 Definition of seas according to direction of approach relative to vessel

and

(Oe = to 1 - V~ ~ cos g

dol e dtoe = ~ dto

do

then

dtoe [1 2 Vst~ ] = - ~ C O S / 3 do~ g

Hence, the two spectra are related through the equation

r(~) r~(~~ 2~'v~l g cos/3]

Thus, given a particular sea spectrum, F(to), the encounter spectrum as a function of wave encounter frequency, for a given V~ and /3, may be determined using the two derived equations for to~ and Fe(tO~). With some further manipulation, left as an exercise, it is possible to show that

ro(c~ 4 ],/2 1 - - - toe V~ cos/3

g

Having extended the theory to allow for an encounter spectrum, it is worth looking at

7.6 Calculation of significant responses 173

some of its characteristics and apparent difficulties. For a head-on sea, /3 = 180, the energy in the encounter spectrum is spread out over a wider band of higher frequencies than for a stationary point spectrum. For the corresponding following seas,/3 = 0, and the spectral density Fe(toe) is singular at the wave encounter frequency, toes, defined by

1 g ( D e s - - m

4Vs

Here, to~ is positive since V~ is to be treated as a positive quantity irrespective of its direction relative to Vw. However, others may give this same relationship with a negative sign because they prefer to treat V~ as negative under the selected circum- stances. Whatever the notation, tO~s is positive. Following-sea spectra may also be double valued where the same encounter frequency exists for two different incident wavelengths. This must be the case since, if to~ is assumed known, then the encounter frequency equation can be treated as a quadratic equation in wave frequency, to. The encounter frequency spectrum of a one-sided wave spectrum F(to) will also extend into the 'negative frequency' range as Vs increases. The negative spectral density does not pose any problems when integrating the spectrum, to determining the total energy say, provided a path of increasing toe values is followed. Since the product Fe(toe)dtoe will always be positive, we may write ~,

r (,o) dos = ro (toe) dtoe + (toe) dtoe 0 mes

and F(to) is a one-sided spectrum. Here, integration is from left to right in the upper half-plane of F~(to~) versus toe for the first encounter frequency integral, and is from right to left in the corresponding lower half-plane second integral. Numerical application is left as an exercise.

In practical ship response calculations, the ship speeds would be specified and the wave frequencies corresponding to A/Ls between 0 and 3 would be selected, where L~ is the ship length. Having determined the corresponding encounter frequencies, the transfer functions of interest would be determined for the calculated encounter frequencies. After selecting suitable wave spectra, the response spectra would be determined. From examination, or known properties, of the spectra, the appropriate probability density function(s) would be selected. This allows any required 'exceedence' probability calculations to be undertaken. One particularly useful form of statistic, which is determined from the response spectrum, is described as significant response. The underlying ideas of a significant response and a design related application are considered next.

7.6 Calculation of significant responses In certain engineering applications it is convenient to consider the mean of the highest 1/nth of the total number of maxima. If N peak-to-peak height measurements are made, then this quantity, Hi#,, is the average of the N/n highest measurements. Thus, as n--)N so H~/, tends to the maximum measurement in the sample. The peak-to-peak amplitude, Xmax, in a given realization may be defined as half the difference in level between a crest (a local maximum) and the preceding (following) trough (a local minimum). This is not to be confused with the apparent amplitude, which is half the difference between the


x(t)~ Local /'~ +

max [< Tc --~

o0 closs:; t / i Down crossing V v

maxtma

'-" 7"7

Tz "- apparent zero crossing period Tc --- apparent period or crest period hw --- apparent wave height Xpp = peak to peak amplitude

Fig. 7.10 Definitions of apparent quantities

highest crest (absolute maximum) between two consecutive zero crossings and the preceding (following) lowest trough (see Fig. 7.10). Consequently, the 1/nth highest maxima corresponds to those maxima values Xma x greater than hu., where h~/. is such that

Ioo Pr {Xma x ~ hi/,, } = p(Xmax ) dxmax 1

h,,. n

The average value of Xma x for these maxima, Hu. , is therefore

I co H1/. = Xmax p(Xmax) dXmax

hl/n

Evaluation of these quantities requires specification of the probability density function p(Xmax). The general probability density function describing the peak or maxima distribution of the random process x(t) has been shown to satisfy

1 --Xmax p(Xmax) = (2,rrmo)l/2 [e exp( 2 /[2moe2])+

( 1 - e 2) u2 (Xmax/mlo/2) exp (--X2m.x/[2mo ])N]

where

N = -= exp - ~ x dx

and the upper limit is X = (Xmax/emo)(1 - e2) !/2

Z6 Calculation of significant responses 175

The quantity e 2= 1 - m2/(mom4) defines the spectral width parameter of the associated spectrum. In terms of apparent physical oceanographical quantities this parameter is defined as

e2= 1-(Tc/Tz) 2

where Tc is the mean crest period and Tz is the mean zero-upcrossing wave period (see Fig. 7.10). Tc is defined as 2~r(m2/m4) 1/2 and T, = 2~r(mo/m2) 1/2. Here, the spectral moments are defined by

m. = to'F (to) dto 0

assuming F(to) is a two-sided spectrum. Thus, the quantity 2mo is the variance of the random signal x(t). The cited probability density function is a weighted mean of the Rayleigh and the normal probability density distributions. Each extreme distribution corresponds to e--~0 and e--~ 1 respectively (see Fig. 7.11). For the narrow banded case,

0.41-

- 3 - 2 - 1 0

~ e = 0.0 .--, Rayleigh

I l 1 2 3

Fig. 7.11 Probability distribution of the heights of the maxim, 7/= x/(2mo) 1/2, as a function of e = 0 (0.2) 1.0, from Cartwright and Longuet-Higgins

the maximum wave elevation follows a Rayle~gh distribution and therefore numerical integration of the probability density function may be used to show that

HI/3 = Hs = 4(2mo) 1/2

In some texts this result will be quoted as 4m01/2 with mo defined as the variance, that is, the total area under the two-sided spectrum. These factors of 2 (or a half) and hence factors of ~/2 are a recurring problem in those theoretical aspects of spectral analysis associated with providing expressions for H~/,,. This can prove to be quite a problem for student and lecturer alike, especially if one moves from text to text without carefully cross-checking authors' preferred and often implicit, rather than explicit, definitions. Another source of confusion, and hence a factor of 2, is that the authors do not always stipulate whether the spectrum presented uses wave height or wave amplitude as the basic statistic employed in the determination of the spectrum. Assuming the Rayleigh distribution is applicable, the indicated possible variations lead to the following definitions of Hu, for different n (see Table 7.6), with y - 1 or y = 2 according to whether m0 or 2mo defines the total variance of the original signal.


Table 7.6 Definitions of H1/,., for narrow- banded spectra n Amplitude Height

1 1.25 ~/(~/mo) 2.50 V(~,mo) 3 2.00 x/(ym0 ) 4.00 V'(ymo )

10 2.55 %/(ymo) 5.09 V'(yrno) 100 3.34 x/(ymo) 6.67 V'(ymo)

i i , ,

Strictly speaking, significant wave height Hs is defined as four times the root mean square of the wave signal. But H1/3, using the earlier definition of H~/,,, has been shown to satisfy H1/3 = 0.94H~. Therefore, H1/3 = 4V'(ym0) tends to be treated as a literal definition of significant wave height. This convention has been used in the example that follows.

The ideas of significant response are now applied in a small contrived design study. The aim is to investigate four alternative offshore oil rigs with a view to extending their operability. That is, we would like them to continue drilling in larger waves. The sea spectrum chosen is the Pierson-Moskowitz spectrum (see Ochi and Bales 1975) which has the form

with

og2 [ F (~o) = - ~ exp g 4

a = 4,tr 3 gT2 and /3 = 167r 3

For the North Sea, a and /3 may be taken as 0.0081 and 0.74 respectively and by manipulating the definitions of a and/3 we have

2W 2 a W 1 H1/3 = ~ ~ a n d T o = 2 r g [3 g (j~Tr) TM

Here H1/3 is the significant wave height (m), W is the wind speed (m s-l) , conventionally measured at 19.5 m above the still water, and To is the mean wave period. This spectral form represents a fully developed sea as lower frequencies are present.

The four eight-columned rigs have a number of identical features regarding the dimensions of the shape of the submerged pontoons and the lengths of the columns (see Table 7.7). The basic idea behind the design scheme is to try and increase the operability of the rig by increasing the waterplane area.

Table 7.7 Some common rig dimensions Pontoon length (overall) 100 m Pontoon breadth (overall) 10 m Pontoon depth (overall) 10 m Pontoon CB 0.9 Column length 22.5 m Operating draught 20.5 m Deck height 8.0 m

,

7.6 Calculation of significant responses 177

Table 7.8 Computed rig particulars (based on Riiser M.Sc Dissertation)

Calculated particulars . . . . . . . Rig 1 Rig 2 Rig 3 Rig 4

Transverse distance between corner columns 47.4 49.5 49.4 50.1 Column diameters: large 7.0 8.0 9.0 10.0

small 6.0 6.0 6.8 7.2 Waterplane area 267.0 314.0 400.0 477.0 Displacements: columns 2874.0 3381.0 4302.0 5133.0

pontoons 18450.0 18450.0 18450.0 18450.0 total 21 324.0 21 831.0 22 752.0 23583.0

Natural heave period 24.2 22.5 20.3 18.5 Deck area 3643.0 3910.0 4027.0 4205.0 Weight: equipment 2937.0 2937.0 2937.0 2937.0

steel 5296.0 6696.0 7038.0 7396.0 lightship 8283.0 9682.0 10025.10 10383.0

Lightship vertical position of centre of gravity 24.55 25.98 25.95 25.99 Max. transit deck load* -ve -ve 347.0 1738.9 Max. operating deck load* -ve 23.3 1559.0 3167.8 KM transit** 13.27 15.74 18.63 21.75 KM operating (GM = 1.0) 13.27 15.66 17.99 20.34

, , , , , ,

* Assuming no loads in column and deck load centre of gravity = 40.5 m ** GM = O, KM corresponds to pontoon submergence condition.

To maintain initial stability, it is necessary to keep the waterplane inertia constant. Since the longitudinal distance between corner column centrelines is to be kept constant at 60 m, and the waterplane areas are gradually increased from 267 m E for Rig I to 477 m 2 for Rig 4 (see Table 7.8), the transverse distances between column centrelines are adjusted for each rig so that the second moments of area about the longitudinal and transverse platform axes are equal. This is achieved by moving the four smaller inner columns. This procedure represents an approximate method of maintaining the initial stability, by offsetting the increased moments of area resulting from the increased displacements, which arise from the increased waterplane areas. The equipment weights were kept constant for all designs. The steel weights, and hence the lightship weights, were evaluated with the hydrostatics, and the maximum permissible deck load capacities (for both the operating condition and the pontoon submergence condition) together with the heave responses using computer programs developed at Newcastle University. Details of computed rig particulars are presented in Table 7.8.

It follows from earlier comments that H1/a can be generated by simply selecting different wind speeds. Having calculated H1/3, the corresponding response spectrum can be determined as explained in Section 7.4. By determining the area under the response spectrum one may determine the significant heave response h l/a, say. With h~/3 calculated for each 1-11/3 o n e may plot the results and determine the maximum level of H1/3 for which the rigs may be permitted to drill. The criterion used is that drilling may not continue for significant heave responses, h l/a, greater than 2.5 m. Alternatively, we can consider the


1.4

1.0

0.8 g 0.6

0.4

0.2

0.0 0.2

r ;7, ~, - i h i~ "'g I

�9 :

- ir _

- .~~~"! i/':.-.-_-.~ i I" ..," . . . . . . :,~.'?~, V~ / . . . . "~

- ' i ' I I I I I 1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 o~ (rad/s)

Fig. 7.12 Heave response amplitude operators

exercise as seeking to determine how rough the seas may get before drilling must be discontinued!

In Fig. 7.12 we observe a considerable difference in the regular wave heave response characteristics of the four rigs. Also, the natural heave period is decreasing with increasing waterplane area. This results in a shift of the secondary peak of the response and a reduction in response near the secondary peak. However, Fig. 7.13 illustrates rather small differences in the significant heave responses of the rigs. Only the significant heave responses for Rigs 1 and 4 are plotted as these envelop the corresponding responses for Rigs 2 and 3. The large differences in the transfer functions of Fig. 7.12 are clearly smoothed out in Fig. 7.13 because of the integrations undertaken.

In order to keep so many characteristics of the rigs common, and to try and provide some cause and effect understanding, the generated particulars are slightly artificial. Consequently, Rigs 1 and 2 are impractical. The heave responses were also generated

8 - / 7-

6 - Rig

e -

3~ Maximum permissible /4"

2.5 5.0 7.5 10.0 12.5 "- HII3 (m)

Fig. 7.13 Significant rig response hl/3 as a function of significant wave height H1/3

7.7 Exercises 179

using quite basic one degree-of-freedom models. This, and the simple basis for generating the alternative figs, clearly makes the design procedure contrived. However, this type of application, based on more sophisticated ideas of design, with more complex hydrodynamic and motion analyses, and the earlier ideas of predicting probability of 'exceedence', represent fairly typical engineering-based applications of spectral analysis. When designing structures which are known to be subject to random loadings, deterministic methods alone are generally not sufficient, as the example demonstrates. This point will be developed further in some of the case studies presented in Chapter 8.

7.7 Exercises

, Processing a full record of collected wave data (such as that presented in Appendix IV), the observed peak-to-peak wave amplitudes (reading down the columns) are given in Table 7.9.

Table7.9 Exercise 1 peak data

626 3 144 121 87 343 6 545 246 132 523 163

244 119 336 392 175 304 360 458 219 347 170 394 371 261 405 167 190 24 150 303 131 501 79 412 393 163 161 12 473 235

67 21 437 216 60 142 514 335 22 275 206 162 255 23 8 26 75 380 90 124 358 303 559 27

465 400 206 198 814 66 131 123 240 234 495 326 162 255 124 63 223 132 243 293 465 7

�9 , , , , , , , , ,

Group the peak wave amplitude data using the intervals 0-99, 100-199, 200-299, 300-399, 4120-499,500-599 and 600-849.

Draw a histogram ensuring that the total area equals unity. If the original time series is to produce a narrow bandwidth spectrum then the peak wave amplitude, y, given above should exhibit a Rayleigh distribution, which has the probability density function (PDF)

p(y) = Y--- exp (-y2/[2m o ]) mo

Assuming the original wave height data have an estimated variance of 70776 (mm 2) superimpose the Rayleigh PDF on the histogram drawn.

Next, calculate the expected frequencies (to 2 decimal points) for each interval, under the hypothesis that we have a random sample of 88 values from a Rayleigh cumulative distribution function, that is

Pr { y ~< Y } = 1 - exp ( - y2/70776)


Calculate the statistic W given by

P ( O i - E i ) 2 W=E

i=1 Ei

0

where 0 i and E~ are the observed and expected number of occurrences in the i th cell of the histogram, and p is the total number of cells. If the above stated hypothesis is true then the distribution of W is approximated by the X 2 distribution with ( p - 1) degrees of freedom. Those familiar with hypothesis testing may now determine whether the calculated value of W is significant, and hence determine whether there is evidence to support the proposed null hypothesis.

Comment. This exercise indicates how the assumption of narrow-bandedness may be tested directly from the data rather than making a judgement based on the calculated value of e, which involves evaluation of moments of the derived wave spectrum. One may obviously repeat the process using the Cartwright-Longuet- Higgins probability density function defined in the text for different assumed values of e. Within the text the equivalence of the expressions

and

- ~ cos/3 1 - ~ cos/3 g g

1/2

@

11

was assumed to provide the encounter frequency spectrum. By expanding [1 - (2toVs/g) cos/3] 2 and using the specified relationship between toe and tO establish the assumed identities. Next, assume/3 = 0, F(to) is a Pierson-Moskowitz spectrum, and Vs, the ship speed, is equal to g/(2tOa), where tOa is the average of the wave frequency values required to define the stationary point spectrum for your selected wind speed (m s-l). Determine the corresponding encounter frequency spectrum and check that the area under the spectrum is preserved. You may assume that data processed in Exercise 1 were collected at 0.1 s intervals and the associated autocovariance function is given in Table 7.10.

Evaluate the sample spectrum over the frequency range from zero to the Nyquist frequency. In calculating the spectrum, select one of the lag windows discussed in Chapter 5 and repeat the calculation for M equal to 10, 20 and 40. Then plot your results and discuss the influence of M. in theoretical and design orientated studies, internationally accepted analytic forms of spectra are used. Deduce that the peak frequency, tOm, of the general one-sided theoretical sea spectrum

S (tO) = A tO-5 exp [ - n t o - 4 ]

4 ._ 4 B / 5 satisfies tO m

Deduce that the nth moment of the given spectrum satisfies

A ( n ) mn 4B(l_,a4) F 1 ~- n < 4

7.7 Exercises 181

Table7 .10 Autocovariancefunction k dk) k r(k) k dk)

1 0.47 15 -0.010 29 0.07 2 -0.26 16 -0.100 30 0.10 3 -0.50 17 -0.130 31 0.06 4 -0.38 18 -0.110 32 -0.01 5 -0.22 19 -0.050 33 -0.05 6 -0.04 20 -0.080 34 -0.02 7 0.18 21 0.160 35 0.04 8 0.27 22 0.12 36 0.01 9 0.13 23 0.05 37 -0.01

10 -0.07 24 0.00 38 -0.01 11 -0.08 25 -0.07 39 0.01 12 0.03 26 -0.12 40 0.0 13 0.07 27 -0.12 14 0.07 28 -0.04

, , ,

.

1.6

1.4

1.2 d. E ca 1.0

"~ 0.8 d. E �9 0.6

" 1 "

0.4

0.2

by using the gamma function definition

n t -n/4 e - t dt n < 4 r 1 - ] - = o

Assuming r(1) = 1, r(�89 = V'1r and = 1.2254 -- v'(l]) deduce expressions for the mean wave period T~ = 2zrmo/m~ and the zero-upcrossing wave period T, = 2zr(mo/m2) u2. What can you say about m4 and hence the spectral width parameter?

C o m m e n t . These general results can now be used with any particular sea spectrum for which A and B are defined. In this exercise, treat the function F as defined, as the context should not cause any confusion with the population spectrum. A tanker 140 m long with beam and draught equal to 20.0 m and 8.0 m has the heave and pitch transfer functions presented in Figs 7.14 and 7.15. At 15 knots

0.0 0.0

t Heading:Head seas ~ X ~

H e~~et r::sf:iiii~ti~ s l/ ~ .

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Wave length/Ship length

Fig. 7.14 Heave transfer function of ship


1.2 1.1 1.0

~' 0.9- X d. 0.8-

~ 0.7

0.6

~ 0.5- E ~ 0.4 l -

a. 0.3 0.2 0.1

0"00. 0

Heading:Head seas

Pitch transfer functions

�9 Speed = 0 knots �9 Speed = 15 knots

/

! : : --- : 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Wave length/Ship length

116 .... 118 2;.0 212 214

Fig. 7.15 Pitch transfer function of ship

11

(7.7166ms -1) the associated encounter-frequency based response amplitudes calculated for the selected wave frequencies, together with the encounter frequencies (for a head sea) and the stationary point spectrum, C(to), are presented in Table 7.11. Determine the encounter spectrum, the corresponding response spectrum at 15 knots, and hence the significant heave and pitch motions.

Comment. The spectrum defined corresponds to North Atlantic sea number 5 with Hz/3 = 3.25 m, and the most probable modal wave period of 9.7 s has been selected. The annual probability of occurrence of this sea state is 0.2064. In Fig. 7.16 the cross-section of a pneumatic wave energy extraction device is presented. By modelling the internal free surface, IFS, as a massless flat plate, the general responses of the device and the internal free surface may be determined using appropriate hydrodynamic analyses, and suitably generalized coupled equations of motions. The transfer function for the heave and phase amplitude of the device and the free surface are presented in Fig. 7.17.

Generate a sea spectrum for significant wave heights of 3 m, 5 m and 7 m using

Table 7.11 Heave ($3)and pitch ($5) response amplitudes and C(o~) i

Oe e S 3 ( o ) e ) S 5 ( o ) e ' ) c(,,,,) 0.29669 0.41958 0.46911 0.50150 O. 54168 0.56069 0.58185 0.60561 0.63254 0.66342 0.76605 0.93821

0.36595 0.981766 0.008349 0.00379 0.55811 0.958593 0.018640 1.27911 0.64227 0.965995 0.024158 1.81295 0.69940 1.002222 0.028233 1.89436 0.77256 1.138620 0.033577 1.78763 0.80807 1.255210 0.035940 1.69091 0.84826 1.398530 0.038106 1.56645 0.89421 1.456440 0.039524 1.41827 0.94738 1.197950 0.037805 1.25180 1.00974 0.639911 0.028799 1.07347 1.22781 0.092605 0.003763 0.62374 1.63086 0.022953 0.002020 0.25673

, ,

7.7 Exercises 183

Head

waves

-~ Turbine-generator unit [ 7 _ 7 7 , - - 7 ]

_ IFS - - ~

- ' - - - 2a /

/ /

Fig. 7.16 Pneumatic device wave power generator (Hearn and Katory 1978). IFS oscillates vertically due to waves.

the Pierson-Moskowitz spectrum. Determine and plot the significant heave response of both the internal free surface and the device against significant wave height.

The energy extraction clearly depends upon the flow rate of air through the turbine located on top of the device. Given the information supplied, how might one determine the response spectrum for the volumetric air flow to the turbine?

7. Assuming a narrow-banded process, the peak values are distributed according to a Rayleigh distribution. Under these assumptions, indicate how you might determine H1/. for different values of n.

- 9O.0 i s

r a.o i - .

I2o " 1,s I s" >= o.o

i =~-~1~ , , i ~ / I!j ~ , , , , ~ \ X g0.0

I I I 0 0.5 1.0 1.5 2.0 2.5 3.0

Wave frequency (red/s)

Fig. 7.17 Heave amplitude and phase for device(s) and internal free surface (IFs). (Original calculations by Hearn and Katory 1977)

8 Wave directionality monitoring

8. I Introduction

This first case study is concerned with designing a piece of equipment to measure correctly the directions of wave travel, and the amount of energy available to a wave energy extraction device being assessed in Loch Ness, Scotland. Given that the engineers involved in designing the wave power device could measure the amount of work done, or energy extracted, to measure the efficiency of such a device, we needed to know how much energy was actually available.

This case study will clearly demonstrate that the analysis of data should not be considered as trivial or straightforward. In this case, some of the basic ideas of the initial, imposed analysis, method were an integral part of the design of the energy measuring device. Furthermore, it will be shown that constraining one's thinking to acceptable 'established' norms of analysis can lead to failure. In particular, the original device had to be modified to produce more realistic measurements. Also, given the minimal number of transducers in the measuring device we were forced to look at less conventional spectral analysis methods. Since the final design and the final selected analysis were in conflict with other specialist spectral analyst advice, it was necessary to demonstrate, using simulations, how the modified device and alternative analysis would produce a truer picture of the wave climates.

Some of the mathematical detail involved may be considered too advanced by some readers. This should not, however, prevent the reader from appreciating the following points.

1 Spectral analysis cannot always be undertaken as an independent task. 2 Real engineering problems do not respect academic divisions of knowledge. 3 Working in industry often leads to involvement (even responsibility) in projects

which require new knowledge. In this case one works with sufficient minimal understanding and full responsibility.

The reader may take comfort in the fact that the author involved in this case study knew little about spectral analysis at the start of the project~being a hydrodynamics specialist.

8.2 Background Here, we need to explain our understanding of the directional wave spectrum and the need for its monitoring.

Since we shall refer to the spectral density distribution, or simply the spectrum, as a

184

8.2 Background 185

measure of energy per unit area, let us explain the reasoning behind the interchangeabil- ity of these terms. In the case of water waves the total energy in a regular wave is �89 per unit area of sea surface, where ~'~ is the wave amplitude. A one-dimensional (long crested) random sea, ~'(t), may be approximated by a summation of N superposed harmonic wave components with the nth wave component described by

en(l) --'- r cos (tont -I- tonen)

where e,, is the random phase angle. It is obvious (it really is) that E[sr,, ] and hence E[~'] are zero.

The mean square value of a sample, 2T in duration, from this random sea is thus

E[~.2] = ~ T [ r ~.2(t) dt - T

1 f T

_ ~'a,, COS (~o,,t + ~o,,e. dt 2T J - r n-1

That is

1 N

=7 Z & n=|

because of the orthogonal nature of the trigonometric functions. Thus, the total energy in the sample is

1 N 2 pg ~, r = pg E[~.2]

But, we have seen in Chapter 5 that

"=1 = _ . dto = ~ _ ~'2 n

or, for a single-sided sea spectrum

r ( ~ . ) doJ = ~'2 n

for a bandwidth do centred on the component frequency to,,. Since the total energy is directly related to the spectrum it is often referred to as the 'energy' spectrum or 'power' spectrum.

Water waves travel at various directions 0 relative to the predominant wind direction and therefore their spectral density should be related to direction as well as frequency. The total wave energy is thus not only distributed according to frequency, but also direction. If the total energy is to remain the same, one may assume that

r ( , o , o) = r ( c o ) f ( o )

where f (0) is selected to keep the total energy constant. A simple 'spreading function' is

f(O) 2 " ~ COS 2 0 "/T

since

186 Wave directionafity monitoring

2f ,2 COS 2 0 dO = 1

-u l2

and hence

f fr(~o,o) do dO= fr(a , ) da, f f (O) dO= fr(a , ) dto

with appropriate implicit integration limits. In general, one may select other spreading functions such as

f(O) = an cos z" 0

with

1

a,, f ~r cos2n 0 dO

or, more generally, following Mitsuyasu et al. (1975), we may express the directional spreading as

f(O) = G(n) O-Oo 11

where G(n) is again a normalizing function satisfying

2 2" (n!) 2

G ( n ) - 2zr (2n)!

for 00, the mean direction of a spectral component; n is a selected integer value and the 'exclamation mark' denotes factorial. We shall use different parameterizations in this study. The spectrum of a directionally spread sea is often referred to as a short crested spectrum. We may now discuss the need for monitoring such short crested seas.

The assessment of British wave energy extraction devices was based on projected units of electricity costs and a demonstration of their power take-off capability. Theoretical studies based on inviscid free-surface hydrodynamic theory and generalized frequency domain equations of motion may be used to predict the power take-off of a device subject to regular wave excitation. From our understanding of the energy stored and transported in a regular wave one can determine the theoretical efficiency of the device as a function of the frequency and heading of the incident wave. Knowing the regular wave efficiency transfer function, the irregular sea-state efficiency can be determined using the input-output spectral relationships of Chapter 6. Using two-dimensional hydrodynamics to analyse very long wave energy devices, their length being parallel to the incident wavefront, one can formally prove that the maximum efficiency attainable is 50% if the device has fore-aft symmetry and 100% otherwise. At certain wave frequencies the device appears to focus the waves and so efficiencies greater than 100% are attainable if one assumes that the actual device width, or the resolved width of the device in the direction of the wavefront, is the nominal length of the wavefront available for the extraction of wave energy.

In at-sea tests, such as the Salter duck spine studies in Loch Ness, the Cockerell Raft studies in the Solent and the Coventry reservoir tests, it was possible to measure the power produced quite readily. However, rating the efficiency of the device requires a knowledge of the energy available and its directional spread, since most of the then

8.3 The technical problem 187

supported wave energy extraction device responses were wave direction sensitive. The Fluids Structural Loading Technical Advisory Group, one of a number of Technical Advisory Groups (TAGs) set up to advise the Department of Energy's Wave Energy Steering Group, therefore agreed to undertake a wave directionality monitoring programme in Loch Ness in 1977 and later in the Solent. The Loch Ness case study will highlight some of the technical difficulties experienced and the personal lessons learnt from real problem solving under commercial and resource limited pressures. This should not be interpreted as criticism or negative reporting, but a personal record of the enhancements achieved in terms of matching a selected monitoring system to a particular wave climate so as to improve the resolution of the spectral energy spread.

8.3 The technical problem The design of the monitoring system requires consideration of such questions as the following.

(i) (ii)

(iii) (iv) (v)

What quantity is to be monitored? How is the selected quantity to be measured? How is the required information to be extracted from the measured quantities? How are the measurements to be recorded? How is the monitoring equipment to be located and kept on station?

From Airy wave theory we can establish relationships either between wave height and wave-induced dynamic pressure, or between wave height and the resultant forces or moments experienced by small submerged objects, called 'force balls'. Therefore, in deciding what is to be measured, wave height, pressures or forces and moments, we must also consider on-site calibration and handling of the monitoring device from a small boat as well as its maintenance and survival. Some of the wave-height dependent quantities, such as particle velocity and pressure, are dependent upon the actual water depth and the accurate location of the submerged devices. These difficulties and the lack of consensus regarding the different devices available, and the best form of monitoring, led to the decision to measure the wave height directly using capacitance wire techniques since:

(a) (b) (c) (d) (e)

experience of their use existed in the establishment; it will permit recording without pre-amplification; there is no need for digital logging; it will be simpler to assemble and to replace damaged wires; and software already tested under laboratory conditions existed at another research establishment.

The basic wave height sensor was a polythene-insulated conductor wire held under tension. The multi-channel capacitance sensing transducer electronics provided seven channels of wave height information for reasons explained in Section 8.5. The associated electronics had to be tested to demonstrate linearity over a large range of capacitances (0 to 1000 picofarads, pf), and to provide a locally stabilized voltage and locally RC filter probe data before onward transmission ashore through a multicore cable. Resistive transducers were rejected as they were considered more subject to errors owing to the changing conductivity of the water and its temperature. On-shore transmitted signals


Fig. 8.1 Floating wave monitoring system

would also have to be scaled, filtered and DC levelled to suit the selected FM tape recorder.

Having selected the quantity and its means of measurement, and the number of probes, a platform which does not change the wave environment being measured must be devised. We therefore either use a floating device or a bottom mounted device, such as those presented in Figs 8.1 and 8.2. The simpler pole-type device was selected with 'guys' used to stabilize the staff. Both a buckling analysis, to ensure structural integrity, and natural frequency estimates of the mast and the 'plucking' frequencies of the guys, to ensure the wave frequencies of interest and the natural responses of the guys were well separated, were undertaken. The diffraction-borne interference of the wave environment being measured with the more complex floater design was considered problematic. Hence, this design was rejected for being too complicated.

The remaining decisions were the number of transducers to be used, their relative positioning with respect to each other and hence the method of analysis to be used to extract the directional information.

8.4 Reduction of the monitoring problem to a mathematical problem 189

Fig. 8.2 Bottom standing wave monitoring system as installed in Loch Ness

8.4 Reduction of the monitoring problem to a mathematical problem

The monitoring system was initially required to be sensitive to wave periods of 1-4 seconds at least, and was to provide a measure of the spectral density function and its directional spread. In the adopted approach the spatial distribution of wave height was used to obtain the directional information. Having decided on this particular wave parameter, the configuration of the wave-gauge array must be selected. The choice is either a one-dimensional (linear) array or a two-dimensional (polygonal) array. Line arrays can only be used, without ambiguity, when significant amounts of wave energy arrive from one side of the array, that is, such arrangements are sharply tuned for waves generally travelling normal to the line. It was therefore decided that a polygonal array was preferable for more general use. What was now required was a means of deciding on


/)r Probe J

/ I \ Wave direction \ / I j f ' , \ / / I ._~ D~ / / lilY

/ / ' ,

~ / I

. . . . i J , o

'~. Wave front \

, \

X!

q~ -,- wave height at probe I qu --- wave height at probe J

Fig. 8.3

x

Probe and wave orientation relative to fixed reference system

the shape of the polygon and the spacing of the gauges such that the required sensitivity was attained. The constraint that there already existed an available external analysis package to deal with equilateral triangular arrays (at zero cost to the project) decided the shape. One now required to relate the sizing of the triangles to be used to the required sensitivity of the monitoring system. To do this one must appreciate the underlying mathematical model now outlined.

The wave energy is not necessarily concentrated in a finite number of directions and the phasing of the wave components will be random. Consider initially two probes I and J (see Fig. 8.3), and denote the wave elevation at the position (xi, Yl) by qI(xI,yl,t). Assuming that (X, Y) is the position of the Jth probe relative to the Ith probe then

x j = x I + X

and

yj = yl-t- Y

Then the cross-covariance, cIj, corresponds to the expected value of the product of the wave elevations q~ and qj. The cross-covariance is thus expressed as

(- f" 2"rrClj -- 2 CO(tO) COS(tot) dto+ 2 quad (to) sin(tot) dto (8.1) 0 0

where co(to) and quad(to) are the co-spectrum and quadrature-spectrum respectively. That is

I /T

co (to) = r (to, O) cos tx dO 0

and (8.2)

f / t

quad (to) = F (to, 0) sin a dO 0

8.4 Reduction of the monitoring problem to a mathematical problem 191

where a = k(XcosO+ Y sin0), k is the wave-number and F(to,0) is the directional spectral density function. Now, the cross-spectrum, here defined as co(to)- j quad(to), may be obtained as the product of the complex conjugate FFT coefficients of one probe with the FFT coefficient of the other, as argued in Chapter 6. From Equations (8.1) and (8.2) and the identities X = D cos/3 and Y = D sin/3 (see Fig. 8.3), it follows that

co + j quad = F (to, 0) e jkD~~176 ~ dO (8.3) 0

where 0 denotes the wave heading and/3 is the orientation of the line passing through both probes with respect to the x-axis. We now assume that the directional spectrum can be represented as a finite Fourier series of the form

a o N F(~o, 0) - ~ + ~ {a,, cos(n0) + b,, sin(n0)}

n=l

then, substitution into Equation (8.3) and integrating, yields

N

co+j quad= rraoJo(kD)+ Y~ 2N"J,,(kD){a,,cos(nCt)+b,,sin(n~)} (8.4) ~=~

upon noting that

f 2~ cos(n0) dO = 2~j~J~(kD) cos (nr eJkDcos( O-13) 0

with an analogous result for the sine integration. J~(z) is a Bessel function of the first kind of order n.

On expanding the right-hand side of Equation (8.4) it follows, after some manipulation, that

coIj(os) = ~r{Ao, --32, - B 2 , A4, B4, �9 �9 �9 } {ao, a2, b2, a4, b 4 , . . . }T

and (8.5)

quadIj(to) = ~r{Al, B1, - A 3 , - B 3 , A5, B s , . . . } {al , b l , a3, b3, a5, b5 . . . . }T

where the superscript T denotes the row vector is to be transposed. The coefficients A,, and Bn correspond to

A~ = 2 cos nflJ~(kD) t B~ = 2 sin nflJ~(kD) ~ n >I 1

and Ao = Jo(kD). Once D and/3 are specified for each pair of probes then Ao and An and B,, are known coefficients.

From our understanding of the evaluation of the cross-spectrum we may determine numerical values for cou and quadij. The only unknowns are therefore the coefficients a~ and b,, of Equation (8.5). Since there are as many cross-spectra between probes as there are pairs of probes, the greatest number of harmonics which can be represented in F(os,0) cannot exceed the number of probe pairs. In particular, if a triangular arrangement of the probes is to be used then N = 3 and the coefficients to be determined are a0, al , a2, a3 and b 1 and bE. For an equilateral triangle D is constant for all pairs.


8.5 Application of the mathematical model

Equation (8.5) provides a means of generating sets of simultaneous equations whereby the spreading characteristics of the wave energy can be determined at any selected frequency. Now, the problem is easier the greater the number of probes used. The factors which limited the number of probes at the time were the financial resources made available for checking out the design, the provision and installation of the indicated hardware and electronics and the constraint of using existing unseen software. At the time, the theoretical basis of the software was unknown to us and was therefore derived from first principles by Hearn (1977). To achieve credibility with the software owners, the mathematical theory outlined was produced and a typed report despatched to them within 12 hours of the contract being awarded, to demonstrate we were capable of developing our own software if necessary. The availability of this software clearly has advantages and disadvantages. Assuming the software is bug free, time, energy and costs are saved. The disadvantages are that it was limited to three probes, we were not sure it would run on the designated analysis computer and we had to ensure that the monitoring system was capable of resolving the specified wave periods of 1-4 seconds. We must therefore try to understand how the theory is to be applied and how the theory can provide dimensions regarding the appropriate separation of the probes.

The series representation of Equation (8.4) is a truncated form of the cross-spectral density, and to appreciate the order of magnitude of the truncation one must appreciate the behaviour of the Bessel functions associated with the A,, and B, coefficients. From tabulated values of the Bessel functions we recognize the bounding relationships

1.o

10-2~<Jl(z) ~<4.5 • 10 -1

10-5 ~<J3(z) ~< 2.0 • 10 -1

10 -7 ~<J4(z) <~ 7.0 x 10 -2

10 -8 ~<Js(z) ~< 1.7 x 10 -2

for 0~<z~<2.5, see Abramowitz and Stegun (1964). Thus, provided kD is selected such that the higher order neglected functions (n~>4 for the triangular arrangement employed) remain small, the would-be contribution from the neglected terms will remain tolerable. That is, the accuracy of the spreading function may be related to both the separation of the probes D and the wavelengths of the waves to be monitored. That is, in the same way that the sampling interval associated with data collection determines the highest frequency capable of resolution by spectral analysis techniques, so kD controls the resolution of the spreading function. Thus, specifying D effectively determines the tuning of the wave probe configuration. One now has to match the dimension of the triangular geometric configuration to the required wavelengths.

For the (N + 1)st harmonic to be negligible we require kD to be bounded, that is

Ot I ~ kD <~ a2

In general, the wave frequency, to, the wave-number, k, and the fluid depth satisfy the dispersion equation

tO E = gk tanh (kd)

8.6 The probe arrangements deployed 19:3

That is, Airy water wave theory is only applicable when the dispersion equation holds. In turn, the dispersion equation indicates that if we choose any two of the parameters, the third is automatically fixed by the laws of physics describing water waves. Since tanh (kd) tends to unity as d increases indefinitely, then for simplicity, we use here the deep water relationship

to 2 k = ~

g

to assess the approximate tuning influence of D. In this case

gal ga2 ~ ~<D~<----

o) 2 o) 2

or, assuming g / 4 ~ ~ 1/4, then

4D 4D ~ < T 2 ~ < ~

ot 2 ot I

Table 8.1 provides measures of the tuning for a~ = 0.2 and a2 = 3.0 under the assumption that the waves may be described by the linearized deep water theory of progressing waves. In the actual spreading function analysis described, the water depth could be specified and so k was determined from the stated dispersion relationship using an iterative procedure.

Table 8.1 Separation of probes ,,

D (m) Train (s) Tmax (s) 0.0625 0.289 1.118 0.1250 0.408 1.581 0.2500 0.577 2.236 0.5000 0.816 3.162 0.7500 1.000 3.873 1.0000 1.550 4.472 1.2500 1.291 5.000 1.5000 1.414 5.427 1.7500 1.528 5.916 2.0000 1.600 6.320

8.6 The probe arrangements deployed To cover the range of periods 1--4 seconds, with some redundancy for cross checking, required a minimum of seven probes, arranged as indicated in Fig. 8.4. The length of the capacitance probes was 3 m to cover expected maximum wave height and the diameter of the mast was selected to ensure any diffraction caused by the same did not influence measurements undertaken at the probes! The stability of the probes and establishment of a linear relationship between probe capacitance and wave height was also a necessary condition and had to be validated prior to installation in Loch Ness, see Hearn et al. (1977).

The three triangles ABC, ADE and AFG of Fig. 8.4 were to be used with waves in the period ranges presented in Table 8.2.


A

D E

F_ - ~-('-') _ G Mast

Fig. 8.4 Seven-probe arrangement as used in Loch Ness

Table 8.2 Tuning of probe array

Periods Probe arrangements

~ - 2 s AABC of side 187.5 mm 1-4 s AADE of side 750.0 mm 1.5-6 s AAFG of side 1750.0 mm

An inherent weakness of the design is the non-redundant nature of the role of probe A. This was unavoidable given the budgetary constraints. However, at this point we have selected a probe geometry and provided dimensions to the geometry through an ad hoc, somewhat heuristic, application of the spreading function mathematical model. Clearly, the proof of the pudding lies in the eating.

8.7 Analysis of Loch Ness data Once the monitoring system was installed (see Fig. 8.5), records from the system were analysed for each set of three probes and the unidirectional spectrum for each heading plotted. Some debate then arose about the directional resolution of the energy as well as the bandwidth of the energy about the peak frequency. A spectral density estimate based on the mean of the estimates from the seven probes is presented in Fig. 8.6. The MLM-based angular distribution for 0.22 Hz and 0.26 Hz is presented in Figs 8.7 and 8.8 respectively. The directional spread is quite different for small frequency changes. Figure 8.8 demonstrates the unacceptably wide directional spreading of the truncated Fourier series approach. Each set of results was based on 1024 samples per record, with the time interval between scans read by the computer analysis programme equal to 0.32s, compared with a sampling ratio of 0.02 s and several minutes of recording per monitoring session. The computer analysis automatically normalized each probe record with respect to its respective mean RMS value of wave height. The same calibration factor was used for all seven probes, although this was known to vary due to variations in the thickness of the single capacitance wire along its length. The sensitivities of the wires tested lay within

8.7 Analysis of Loch Ness data 195

Fig. 8.5 Installation of wave monitoring system in Loch Ness

. . . . I | �9 �9 " * ~ - -

0Hz

O 3 I - -

N - I -

"E 2

Peak frequency = 0.26 Hz

1Hz = 2~ rad/s

1 Hz

Fig. 8.6 Spectral density function--mean of seven probes as arranged in Fig. 8.4

196

2

T c ) , p . .

1 "

"E

Wave directionafity monitoring

0 30 60

Fig. 8.7

90 120 150 180 210 240 270 300 330 Relative wave heading in degrees

MLM based on angular distribution for a frequency of 0.22 Hz (1.38 rad s-1)

a 5% bandwidth. A difference of 0.127 mm (0.005 inches) in diameter could lead to a 18 to 20 pf capacitance difference for a 3 m long probe. Lack of recalibration in situ at Loch Ness and the application of a single laboratory derived calibration factor was also thought to be a source of the 20% maximum variation between the spectral density estimates based on the individual probes.

Another observation that came to light was the apparent redundancy of the smallest triangular array. The Loch Ness wave-climate observations made suggested that the period range was higher than the 1-4 s range specified when the design was commis- sioned. Although the as-built system exhibited duplication and a longer period range, waves within the interval 0.5-6.2 s could be 'seen' by the system. Was this large enough? If not, then perhaps the triangles were not large enough. Note that this is all hindsight commenting.

The problem now to be resolved was whether dissatisfaction with the results lay in the existing analysis or in the monitoring system. Anyone who has worked on joint industrial ventures will realize that 'corners' are now likely to be defended and thi~refore pure scientific/engineering arguments can become clouded. As far as the authors are

ased analysis

~" 21- j / ~ \ Original Fourier series O 4

E 1 nalysis

0 " - - - - ' ~ ~_~ : . _ . i _ , , �9 �9 J , _ _ _ . ~ = 1 . _ _ = _ - - _ . _ _ _ . . . . - A _ _ _ _ ; -

0 30 60 90 120 150 180 210 240 270 300 330 Relative wave heading in degrees

Fig. 8.8 MLM and Fourier-series based angular distribution for a frequency of 0.26Hz (1.63 rad s- 1)

8.8 MLH-based spectral analysis formulations 197

concerned the real issue is how to decide on a way forward. This was important as the selected site was considered to represent an approximate 1/15th scale of the North Atlantic (an area noted for high average wave energy levels). From the device team point of view, the directional data were required with some confidence bands so that the spectra could be used to assess wave energy extraction performance and be used in computer simulations. Given we had already selected, built and installed the monitoring system, it was decided that we try to improve the analysis of the collected results. This suggestion was not taken to immediately and so it was agreed to demonstrate the performance of the old and new techniques by sampling 'mathematical waves' and analysing the simulated data. The first step of the simulation is the simulation of the covariance matrix so that the corresponding spectrum may be evaluated directly or via the MLM technique and comparisons of the estimates made.

8.8 MLM-based spectral analysis formulations For completeness, the two formulations might be compared. This is not a trivial exercise since the MLM method to be used is to be based upon a frequency wave-number spectral density function. Such a function not only provides information concerning the energy as a function of frequency but also information regarding the propagation of the waves. This is really beyond the scope of this book. Of necessity therefore, the method implemented is only outlined to indicate the steps taken. The reader is not expected to appreciate fully the mathematical detail presented.

The covariance matrix elements cij(to) are identified as

cq(to) = rAi(to; T)A; (to; T)

where Ai is the FFT of the measured wave amplitude on the ith sensor, A~ is the complex conjugate of the FFT of the measured data of the jth sensor, and the length of record T = NA, with A -- 1 for consistency with the mathematics of the earlier chapters. The direct frequency wave-number cross-spectral estimate is provided from

N N C(w;k) = E E cq(w)exp(-jk.rij)

i=1 1=1

where rq is the position vector linking sensor positions i and j and k is the vector wave-number. In order to reduce the effect of calibration errors, the ijth covariance matrix element is normalized using the product tri(to)o).(to). The MLM estimate of the spectrum, using the formulation of Capon (1969), is

=

N N E E/~q(w) exp(- jk . r / j ) ~=l j=l

- l

where f'q(to) are elements of the inverse of the matrix cq(w). If we express the directional spectrum as the sum of N plane waves of power C(to, Oi)

from direction 0i in the frequency band near to, the true spectral density is

N 2 X(to, 0 i) x*T(to, 0 i )C( to , 0 i) i=l

198 Wave directionafitu monitoring

where the ruth component of x is the complex phase lag

Xm(tO, Oi) -- e x p ( - j k i . r m )

where rm is the position vector of the mth probe. The vector wave-number k~ is given by

ki = k (cos 0i, sin 0i)

where k is the usual wave-number. Following the derivation of Lacoss (1971), an MLM estimate of the power in a plane

wave at frequency w from direction 0d is given by

c( ,o , 0d) = Od)Px(o , - '

where P is the matrix consisting of the elements P~y already defined. The MLM estimate of either spectrum is designed to minimize the error of the spectral estimate in a least squares sense, as discussed in Chapter 5.

8.9 Cross-spectral density simulation To simulate the cross-spectral density matrix we consider the height observed on the Ith

kD (0 fl) seconds later, i.e. sensors at time t will be measured by the Jth sensor ~ cos -

qj t+ cos(O-f l ) -= qx(t)

Next suppose that the complex Fourier amplitudes at the Ith and Jth sensors are F~ and Fj respectively. Given 'a' is their amplitude then

FI = a exp(j ~bl)

Fj = a exp(j ~bj)

and hence the elements of the cross-spectral density matrix are given by

FIF~ = a 2 e j ( 6 ' - r

= a 2 e jkD cos(O-/3)

since the phase difference between I and J is - k D cos(0- /3) radians. Hence, for a three-sensor system we have

a 2 a 2 eJkDi2cos(0-/3~2) a 2 eJkDl.,cos(0-/31.~)

C O V I 2 3 = a2eJkD2zcos(O-&~) a 2 a2eJkD23cos(O-/323)

a2 eJkD3~cos( O- /33,) a2 eJkD.~2cos( O- /3.,'.) a 2

whereas the normalized matrix assumes the form

1 eJkDi2cos(0-/312) eJkDi3cos(0-/313)

C O V N 2 3 = eJkO2,cos(0- &,) 1 eJkD2.~cos(0-/32.,)

eJkD.~lcos(0-/331) eJkD32cos( 0-- /3.,2) 1

However, noting that cos ( 0 - flij) = -cos ( 0 - (~r +/3ij)) and ~" +/30 corresponds to interchanging probes i and j, it follows that COV~'23 is a Hermitian matrix.

8. I0 Simulation results and alternative probe management 199

For several waves in a more general simulation we may constrain ~a 2 to equal I giving

1 ~'~aEe jkDc~176 ~aEe jkoc~

covNI23 _~_ ~ a 2 e-Jkmos(0- ~,~) 1 ~ a 2 eJkDcos(0- 823)

'~,a 2 e-ikOco~(0- t~3) ~ a 2e-jkDco~(0- t3~3) 1

where we have now assumed that the three probes form an equilateral triangle of side D. Finally, we suppose there is incoherent noise at 100 R% of the total wave power, then the diagonal terms are replaced by '1 + R'. Given R is small, then (1 + R) -I - 1 - R and so we may use

1 ~,aR ejkDc~176 ~,aR ejkDc~176

c o v N 2 3 = EOtRe-jkDcos(O-,,2) 1 ~aRejkDr ~2,)

~f, age-jkDcos( O- /323) ~,Otne-jkDcos( O- ~,3) 1

where aR = a2(1 -- R). In practice, small perturbations due to quantization error, noise and leakage from

neighbouring frequencies will probably occur. For this reason, R has been included in the final stage of the analysis. Also, one may show that if R is identically zero, and if fewer wave frequencies fall within the FFT window than there are sensors, the cross-spectral density matrix can become singular. The presence or, in this case, the inclusion of the noise term, R, prevents the matrix becoming singular and so its inverse will exist.

As the earlier stages of the presented analysis indicate, we are not restricted to D q - = D. Hence, one may study both the influence of the arrangement of the probes and, by further generalization of the analysis, the influence of the number of probes to be used to undertake the indicated directional spreading assessment.

8. I0 Simulation results and alternative probe management

For comparative purposes the MLM and the original analysis were used to process the same simulated data. In order for the simulation to work, it is necessary to introduce the effect of random noise by slightly depressing the off-diagonal elements of the cross- spectral density matrix. The noise level can clearly be made an input parameter in any computerized analysis. In the study undertaken, the level was reduced from 2.5% to 0.5% in steps of 0.5%, giving a gradual improvement in definition of the spread of the energy at the selected frequency of interest. However, some appreciation of the actual noise levels would be useful.

In the recording of the actual Loch Ness wave data, an FM instrumentation tape recorder was used on-site and an Alpha 16 hybrid computer was used to sample and condition the signal. Owing to the high-frequency attenuation inherent in the overall system, the majority of the system noise was attributed to the analogue-to-digital conversion. The estimated level of noise due to this process was 1%. The remainder of the system was considered to contribute a noise level not exceeding 0.2%, giving a typical overall signal-to-noise ratio of 1.2%.

In the simulation study, the analyses of two waves of equal energy, with wave direction headings separated by 30 ~ , were also investigated to test the sensitivity of the particular


O

"- 1 N -1-.

"E

0 0Hz

Mean spectrum using all seven probes

1 Hz = 2n rad/s

,-~, .. , . ~ , . . . . . . . . . . .'~.,:-..- . . . . . . . . . . . . 1Hz

? Q

, r - -

" 1 0

t , -

N "r" e q

E

Directionality for 0.26 Hz

_ o . , - . _ . . _ !

0 30 60' 90 120 150 180 210 240 270 300 330

Fig. 8.9 Second Loch Ness MLM analysis

seven-probe arrangement selected on the basis of the earlier Fourier series analysis. Given that the two probes, B and C, forming the base of the smallest equilateral triangle ABC were apparently redundant for the Loch Ness wave climate, the base of the largest triangle AFG was extended to form an equally spaced linear array of four probes. This linear array and the intermediate-sized triangle of the three probes, A, D and E, were then used in a seven-probe based MLM analysis for comparison with the original Fourier series analysis based on the two triangular arrangements remaining from the original arrangement, namely AADE and AAFG. In Fig. 8.8, data collected at Loch Ness have been analysed using the original analysis for a frequency of 0.26 Hz. Figure 8.8 also shows the results of analysing the same data with the described MLM procedure. Figure 8.9 presents a second Loch Ness spectrum and the associated MLM spreading for the frequency of 0.26 Hz. The graphs are similar to earlier plots, except in terms of amplitude. In Figs 8.10 and 8.11, a simulated wave of frequency 0.26 Hz is first analysed

8. I0 Simulation results and alternative probe management 2.01

6 T - .5 lo

"r'4

3

0 30 60 90 120 150 180 210 240 270 300 330

Fig. 8.10 Angular distribution of energy for a frequency of 0.26 Hz using MLM analysis and the original probe configuration (noise level 1.25%)

6 T o , p -

-o 5

1- `` 4 cM

E

3

2

i 0 30 60 90 120 150 180 210 240 270 300 330

Fig. 8.11 Angular distribution of energy for a frequency of 0.26 Hz using MLM analysis and the new probe configuration (noise level of 1.25%)


180 degrees 5 0 %

7 150 degrees 50% ~,~

6

5

2

1

0 30 60 90 120 150. 180 210 240 270 300 330

T (3 ,e,-

4

N -1--

"E3

Fig. 8.12 Angular distribution of energy for a frequency of 0.26 Hz using MLM and the new probe configuration (noise level of 1.25%)

T 3

7O

"1"

"E2

150 degrees 50% 180 degrees 50%

0 0 30 60 90 120 150 180 210 240 270 300 330 Fig. 8.13 Angular distribution of energy for a frequency of 0.37 Hz using MLM and the new probe configuration (noise level of 1.25%)


6 T C~

-- 5

~" 4 "E

O 0 30 60 90 120 150 180 210 240 270 300 330

Fig. 8.14 Angular distribution of energy for a frequency of 0.26 Hz using MLM analysis with a noise level of 2.50%

,_ 6 ,i-,,

~ 5 L

N -1- ~" 4 E

0 30 60 90 120 150 180 210 240 270 300 330



6 O

-o5 N -1- ~ 4

"E

0 30 60 90 120 150 180 210 240 270 300 330


,- 6

-0 5

.1- " ~ 4 "E

0[ 0 30 60 90 120 150 180 210 240 270 300 330



using the original probe configuration and then reanalysed with the probes B and C repositioned on an extension of the base line FG.

Figure 8.11 demonstrates that, if a single wave direction is the true situation, the modified probe arrangement provides a much sharper resolution of the directional spread of the spectrum than does the original probe arrangement when using the MLM analysis procedure. On the other hand, Figs 8.8 and 8.9 show that the MLM method provides a significantly less spread directional spectrum than the Fourier series method of analysis, using the same Loch Ness data for the original probe arrangement.

The simulated two-wave system test is quite a stringent one, as Figs 8.12 and 8.13 indicate. Figure 8.12 shows that the two waves could not be separated into distinct spikes at the lower frequency. Figure 8.13 shows that the two waves were partially resolved at the higher frequency. The discrete nature of the two waves makes the test somewhat artificial in the context of the Loch Ness wave climate. A more appropriate test would have been to define the directionality of the energy by a continuous function and modify the simulation programme accordingly.

Figures 8.14 to 8.18 show the effect of noise on the MLM analysis for the original probe arrangement for a single wave of 0.26 Hz for a 180 ~ wave heading. Compared with the first set of Loch Ness data, visual inspection would suggest that a noise level between 1% and 1.5% would match the two analyses. This tends to reinforce the noise level estimate of 1.25% used in the simulation presented in Fig. 8.19.

6 T

-1- 4 ~E

0 30 60 90 120 150 180 210 240 270 300 330



6 T

N "r" ~ 4 E

0 0

D

,,

�9 I | , , . . , , ,L * , , . . . . . . . . . L �9 | s . | | . . . . . ~ * �9

30 60 90 120 150 180 210 240 270 300 330


8.11 Final comments

This particular study highlights the manner in which non-technical (cost-saving) external influences may have a significant impact upon the success or otherwise of a technical research project. At the time the project was undertaken there also existed a general antagonistic attitude towards the MLM approach within key UK Government establishments. It was this attitude that led to the simulation studies. In the USA, the MLM was generally accepted as a useful tool for wave data analysis.

The alternative configuration adopted maximizes the use of the original probes in a manner which improves the analysis capability, when coupled with the MLM, without incurring large modification costs to the installed monitoring system. The reduction of the noise level from a 1.25% level to about 0.6% was later achieved in 1978 by replacing the FM tape recorder with a Gould 6100 logger. A more complete report of the discussed research programme was published by Clarke and Gedling (1981).

Clearly, the availability of a computer analysis package is not necessarily the same as having the ability to analyse a specific situation. The mathematical analysis presented was developed under highly pressurized face-saving conditions. Whilst concentrating the mind, this may not be the best way of progressing. It is also clear that the original specification was a little vague. One requires a better appreciation of the wave environment to be monitored, than simple expected period ranges, to provide a really well-matched monitoring system. However, such wisdom is only born out of hindsight!

As a final comment we quote Borgman (1979) from a paper related to similar work undertaken in the USA, 'The constraints of cost and operational difficulty in oceanic

8. I I Final comments 207

measurement severely limit most data arrays. A problem which would be very simple and straightforward if 100 sensors were available, becomes much more difficult if only five or fewer sensors are provided.' This particular case study echoes such sentiments, but more importantly, indicates the need to understand the principles of spectral analysis in the context of the situation to be investigated.

9

Motions of moored structures in a seaway


An important result for Naval Architects and Offshore Engineers is the response spectrum relationship of Equation (6.7), namely

= IH( (o )12 r~ ,

Given the transfer function, H(aO, is known for a ship, semisubmersible of some other floating structure, then the response in a sea-state with spectrum Fxx can be readily investigated. In particular, knowing the bandwidth of F~x allows one to select an appropriate probability density function to evaluate probabilities of 'exceedence' of peak engineering responses of interest. Hence, ultimately, one can assess the levels of risk associated with a design.

However, one must also appreciate that the transfer function H(to) represents a steady-state envelopment of response amplitudes. Effectively, this assumes that a steady-state solution is obtainable and, furthermore, if the bounds of the envelopment are sufficient to provide a reasonable probability of 'exceedence' values, then the operability of the design is assured. However, this does not allow one to investigate 'acts of God', such as the sudden failure of a mooring line, or provide detailed time histories of the behaviour of the moored structure when subject to the random sea-state represented by F~. To do this requires integration of appropriate time-domain equations of motion or observation of a model of the moored structure in a tank subject to random wave excitation. In either case, one needs to provide explicitly the driving random wave profile as a function of time.

This particular case study is therefore concerned with the time domain behaviour of moored structures in random sea-states, and the effect the inclusion or exclusion of low frequency damping has upon our assessment of the behaviour of moored structures. In particular, are we overdesigning the mooring lines because predicted excursions of the structure are overestimated?

To do this we must consider the inverse spectral analysis problem~that of generating true random wave amplitude signals with prespecified spectrum characteristics. That is, given F~ how does one meaningfully produce the corresponding x(t). Again, we shall observe that there are choices to be made, there is no one universal solution. Developing such a theoretical capability also allows us to appreciate the effect the length of the simulations undertaken has upon the precision of the statistics associated with the behaviour of the moored structure. Once again the engineering problem involves a number of different academic disciplines.

208

9.2 Background 209

9.2 Background To analyse the motions of a moored floating structure, subject to the excitation of a random sea, we need to understand the different forces involved, how to model and solve the relevant equations of motion and how to include the random nature of the sea. Much of the earlier text has been concerned with the processing of realizations of a random process. Thus, one may use 'saved' wave amplitude realizations as part of the input to a time domain analysis, to represent a particular sea-state or, as is more often the case, generate realizations to represent a particular class of spectral forms. A sea spectrum might, for example, be described by the two-parameter ITTC (International Towing Tank Conference) spectrum which has the form

F(~o) = A e x p ( - B / t o 4 ) / t o s O<<. to

for

and

A 172.8 2 4 "- H 1 / 3 / T 1

B = 691.2/T 4

H1/3 is the significant wave height, the average of the highest third of the wave heights, and T~ is the period corresponding to the average frequency of the component waves. It can be shown that

and

where

and

T1 = 2 7rmo/m1

H2/3 = 16mo

mo = ~- Y~ s r2 = r(,o) do, 0

f oo

m l = coF (to) dos 0

Other mathematical spectral forms have been proposed (see Ochi and Bales 1977). The selection of an appropriate form can be related to such parameters as fetch and wind speed. The application of such mathematical spectra requires selection of the parameters Hu3 and T1, or other related parameters. Any particular combination of the parameters is assumed to exist with some probability, and for a given sea area the probability of such a combination is usually available in the form of a scatter diagram (see Fig. 9.1). The numbers in the scatter diagram indicate the expected number of occurrences in a total of 1000 observations, and they therefore sum to 1000.

The particular reason for studying the motions of a moored structure in this case study is the assessment of the influence of low frequency damping upon the horizontal excursions of the moored structure. The reasons for this are twofold. In offshore analyses prior to 1986/87 low frequency damping was accepted as an important phenomenon, but prediction methods were not generally available and the available experimental data


Winters 1969-1976 Famita 57030' N, 03~ E

11.5 ' ' / o+~t'

11.0 1.1o~/ 1.20/ 10.5 10.0 jI o+ o+ l ~ ,~ / o~ Io§

9.0

8.5

8.0

_E 7.5 :E .= 7.0 O)

6.5

6.0

8 5.5 c 5.0 ._m oo

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

Fig. 9.1

/ o, , , . , - , ,~ j o+ o+ / . o~ j~ o+ o~ , , o j

i~ r , o~1, ~, o~ o~1 r

I ~ ~ ,,, ~ '~ o~ ,~ I~~, b ' o+ 2 2 o+ /

| , . I

r ' q o, qr, j , / , , , ,, H ' , ' - , t { , o , # I ~ :1

J, ' , ; ' f / ' i I ~ o+ / / p . f , # , ,~ , / , , ,, o~ r j oir ) t ; ~ ' , ,o ,, ," ~ o,.

l ~ ~ ~ ~ i ~ ~ - ~ ~ i 1 2 3 5 6 7 8 9 10 11 12 13 14 15

Mean zero crossing period (s) 0+ indicates less than 0.5 p.p.t.

Hs versus Tz scatter diagram

were very sparse. The reason for its importance can be explained by the facts that, at low frequency, all radiation fluid damping (attributable to waves generated by the motion of the structure in response to excitation by the incident waves) is very small and, in the horizontal plane, there are no hydrostatic restoration forces such as those attributable to Archimedean laws in the vertical plane. At low frequency, the moored system is therefore likely to resonate. This is especially true as the depth of the water increases, for then the mooring lines become increasingly longer and the natural frequency becomes much lower. When predicted excursions are compared with measured excursions, the difference in RMS values can exceed 100% compared with the measured excursions (see

9.3 Modelling moored structures 211

10

30

20

. f . 1 x

/f x j f / "

Damping of low frequency motions in waves Cx = 13.6 ton. m -1 bo.x = 18.0 .s. m -1

o Computed with still water damping o Computed with still water and wave damping x Measured

A E t~

o 260 ~w21/3 (m 2 )

Fig. 9.2 Influence of wave damping upon computed excursions of moored tanker. Results presented are based upon research undertaken by J. Wichers in The Netherlands

Fig. 9.2). From 1983 onwards, much more hydrodynamic research concentrated on the development of low frequency damping prediction methods. Also, more experimental measurements were undertaken. Having developed a number of different analysis techniques (see Hearn et al. 1988), which provided different estimates, and with some commercial consultants suggesting less rigorous alternative approaches (to exploit existing computer codes), there was an ever increasing need to assess the influence of different estimates of low frequency damping upon the statistics of the excursions of moored structures and the mooring line tensions.

The undertaking of such studies necessitates a generalization of the appropriate time domain analysis with some thought being given to the simulation of realizations to represent the selected sea spectra. The latter could significantly affect the assessment of the importance of the influence of low frequency damping upon the behaviour of moored structures.

9.3 Modelling moored structures

For completeness the equations of motion will be presented, but their solution will not be discussed in detail. To model the moored structure one must have some appreciation of


the physics involved and a method of including the phenomena of importance. Here it is sufficient, from the reader's perspective, to explain the phenomena using descriptive models and to assume the equations presented subsequently are correct. It is not necessary to understand all the hydrodynamic detail as far as this case study is concerned.

A moored structure will be subjected to a number of different waves in a random sea. If we consider the component waves then we would expect the structure to respond at the frequencies of the component waves and, due to non-linear effects, at those frequencies equal to the sums and differences of the component waves. The frequencies associated with the frequency differences will lead to the low frequency responses. The responses at the component frequencies will be those we normally associate with the 'first-order' linear transfer functions. Typically, the low frequency resonances of the moored structure will have periods of the order of 100 s or more. If we restrict our thinking to these two responses we can imagine the structure undergoing relatively high frequency oscillations in each degree of freedom whilst undergoing very low oscillations in the horizontal plane. The mean velocity associated with these very slow horizontal motions is zero over each period, but at each instant over the long period the structure can be thought of as oscillating in each degree of freedom whilst subject to a changing forward speed. This last observation is sufficient to make the connection with the fact that a vessel advancing in waves experiences a resistance that is greater than the resistance resulting from steadily advancing in still water. The difference, termed the added resistance, is directly proportional to the instantaneous wave amplitude squared and is therefore referred to as a second-order force. If we write the second-order force as

F~o2~ (t) = F(2)(t) + b(2)j:(t)

then FC2)(t) refers to the usual second-order 'drift' forces associated with zero speed and b (2) is the low frequency damping. For surge, b C2) can be shown to be proportional to the velocity dependent added resistance gradient

ORw

OU u=o

where Rw is the added resistance.

9.4 Equations of motion In order to analyse the behaviour of a mooring system under first and second-order excitation forces, a time-domain approach has been adopted because of the dependence of the second-order forces on instantaneous values of wave amplitude and/or structural velocity. Cummins (1962) proposed time-domain equations, based on the impulse response functions, which are independent of the forcing function. We showed in Chapter 6 that for any stable linear system, if K(t), the response to a unit impulse, is known, then the response of the system to an arbitrary forcing function f ( t) is given by the convolution integral

f t

x(t) = K ( t - 8) f(8) d~ - - o o

(9.1)

Generalizing Equation (9.1) for the case of six degrees of freedom, the time domain

9.5 Determination of time dependent wave force 213

equations of motion, including the reaction forces of the mooring lines and low frequency damping influences, can be expressed as

[M + m(oo)lJ~(t) + b(2)(t)J~(t) + K ( t - r)Jc(t- T) dr + Cx(t) = F(t) + D(t) (9.2) 0

where each term is defined as follows"

M = mass matrix of the structure; m(oo) = infinite frequency fluid added mass matrix;

C = hydrostatic restoration force coefficient matrix; F(t) = total wave excitation force vector, F(1)(t) + F(2)(t); D(t) = other sources of excitation, e.g. current, wind, non-linear mooring

loads, etc.

Physically F(1)(t) corresponds to the forces associated with the linearized Bernoulli pressure equation, and averages zero over a wave period. F(E)(t) includes the non-linear terms of the Bernoulli equation and other second-order effects. The existence of F(E)(t) necessitates the mooring of floating structures in waves. To solve the second-order differential equation (9.2) the time history of the low frequency damping b(E)(t), the impulse response function K(t) and the excitation forces have to be known. Direct evaluation of some of these quantities in the time domain is extremely difficult and so they are computed in the frequency domain and the corresponding time series are obtained through appropriate Fourier transforms (see Hearn et al. 1987 and 1988). Ogilvie (1964) has shown that the impulse function K(t) is given by

K(t) = _2 b(~o) cos (cot) d~o (9.3) 7r 0

where b(~o) denotes a first-order radiation damping coefficient of the vessel at frequency to, and the constant inertia coefficient is determined using

m ( ~ ) = m(oo') + - - K(t) sin(~o't) dt (9.4) tO r 0

where m(~o') is the frequency dependent added mass coefficient corresponding to the arbitrary selected wave frequency to'.

9.5 Determination of time dependent wave force

The total wave excitation force F(t) can be expressed as a two-term Volterra series (see for example Dalzell 1974), that is, we may express F(t) in the form

f" I;. (;. F(t) = g,(T)71(t--T ) dr+ g2(~',, T2) 7/(t- ~'1)7/(t- ~'2) d~'! dT2 - o o

= F(1)(t) + F(2)(t) (9.5)

when T/(t) is the wave elevation at time t and gl(T) and g2('rl, "r2) are the first and second-order Volterra kernels respectively. These kernels correspond to the impulse response functions of the first and second-order wave excitation forces. The excitation

214 Motions of moored structures in a seawa X

force given by the first integral is therefore linear, whereas that given by the second integral is the non-linear drift force. The computation of the first and second-order wave excitation forces in the time domain is extremely cumbersome and time-consuming, and therefore they are invariably evaluated in the frequency domain. The impulse functions in the time domain are related to the frequency response function through the Fourier transformations defined by

G~(oJ) - g~(r) e• (-j~or) dr (9.6)

G2(~Ol ,o)2) = g2(rl, r2) exp[-j(~olr, + ~o2r2)] dr, dr2 (9.7)

The quantities Gl(~O) and G2(COl ,o)2) are usually known as the linear and the quadratic transfer functions of the wave forces. In terms of these quantities the total wave excitation force in the frequency domain can be written as

F( o) - a (co)A( o) + a 2 ( u , c o - u)A(u)A(oo- u) du (9 .8)

where A(o)) is the Fourier transform of the wave elevation ~(t) defined by

1I~ A(~o) = ~ -~ n(t) exp(-j~ot) dt (9.9)

The quadratic transfer function, G2(~ol,o)2), also satisfies the following symmetry relationships

G2(tol, to2) = G2(t02,091) =

G2(tol,-to1 ) = G2(-tOl, to1)

G2(tOl ,tOE) also has the property of being real for to1 equal to to2, otherwise it is a complex quantity.

From Equation (9.8) it follows that the second-order contribution to the total wave excitation at frequency to originates from all pairs of wave frequencies where the sum of each pair equals to, i.e. all component waves of frequencies to~ and to2 satisfying 0 0 1 + 0 0 2 = tO.

If, momentarily, the input wave system is considered as being biharmonic then

rl(t ) = R e ( A 1 exp (-joJlt) + A2 exp(-j0o2t)) (9.10)

where A1 and A2 are the complex wave amplitudes. The corresponding mean and low frequency drift force due to this biharmonic wave system can be written as

F(2)(t) = �89 l [2G2(OOl, to I ) + [A2]2G2(o92 ,r

+ Re [AIA~ G2(00, ,tOE) exp [j(00,- oJ2)t ] (9.11)

where * denotes the complex conjugate. Consequently, for a general input wave system, it follows that evaluation of the drift force time history from Equation (9.1 1) requires the computation of the quadratic transfer function Gx(oJl, 0oz) for all pairs of waves present in the wave spectrum. This is a formidable task since it will involve the double

9.6 Evaluation of the quadratic transfer function (QTF) 215

summation of all frequencies to~ and to2 from zero to infinity. Furthermore, computation of G2 also requires a knowledge of the second-order velocity potential satisfying the appropriate inhomogeneous free surface boundary condition. This second-order potential is not normally calculated directly. Therefore various computational procedures have been developed to approximate G2(to~, to2) in terms of known quantities, such as the mean quadratic transfer function (QTF).

9.6 Evaluation of the quadratic transfer function (QTF) The best known method of approximating the QTF at low frequency is that due to Newman (1974). If the input wave spectrum is assumed narrow banded, so that the difference frequency contributions, corresponding to the off-diagonal elements G2(to~, to2), are very small compared with the mean, then the low frequency slow varying QTF can be approximated by

G2(tOl, to 2) = G2(tOl, 0) 1 ) + terms in (tOl - to2) and higher powers.

The narrow-banded assumption thus allows the QTF in the irregular wave to be evaluated from the corresponding QTF in the regular wave. Since the QTF in the regular wave is simply given by the mean drift force, which is readily obtainable from the first-order hydrodynamic quantities, this approximation enables the low frequency force to be evaluated in a wide range of problems.

The time history of the total wave force can be computed from Equation (9.8) and the Fourier transform of the wave elevation record using the FFT technique. However, the disadvantage of this method is that both the linear and the quadratic transfer functions need to be evaluated at all frequencies present in the input wave spectrum. The presence of irregular wave frequencies, associated with the boundary integral equation technique at the higher wave frequencies, makes this approach less convenient.

A simpler approach is to compute the time history of the wave force as well as the time history of the wave drift damping b(E)(t) from the wave envelope and the wave frequency of the wave train. This can formally be achieved by using the Hilbert transform of the wave elevation record defined as

~'(t) = _1 f rl(z) d , (9.12a) r j t - ~

The inverse transformation is defined by

r/(t) = ---~rl I ~'(r) d ~ ' t - r (9.12b)

(see Wichers 1982, Marthinsen 1983 and Eatock Taylor and Sincock 1986). Since the wave elevation is normally represented by a Fourier cosine series, namely

N

ag(t) = ~, gmcos(tOmt+ Ore) (9.13) m=l

where 0~ is the random phase angle of the ruth component wave, then the Hilbert transform of r/(t) is simply the corresponding sine series given by


N ~'(t) = ~ Rmsin(tOmt+Om) (9.14)

m=l

From Equations (9.13) and (9.14) it follows that the local wave envelope at any instant in time is evaluated from

a(t) = (r/2(t) + ( ( 0 ) '/2 (9.15)

and the corresponding local wave frequency is given by

w(t) = 0 (~'(t) ) - - tan- 1 Ot \ - ~

~' (t)rl(t) - rl' (t)~(t) ot2(t)

(9.16)

The time derivatives of ~'(t) and ~(t) of Equation (9.16) may be expressed as

N rl' (t) = - ~ Rmcom sin (corot +Om ), and

m=l N

~' (t) = ~ Rm tom COS (tOmt + 0 m ) m=l

The first-order wave force, the low frequency drift force and the slow varying wave drift damping time series are therefore given by

N F(1)(t) = ~ RmF(1)(tOm) COS(tOmt+ Om+ "Ym) ( 9 . 1 7 )

m=l

F(2)(t) = a2(t) F(2)(w(t) ) (9.18)

b(2)(t) = t~2(t)b(2)(w(t)) (9.19)

where ~'m and F(1)(tOm) are respectively the phase lag and amplitude of the first-order wave excitation force. Equation (9.19) is consistent with the procedure outlined previously by Hearn et al. (1987), but here it is the Hilbert transform that is used to determine w(t) rather than the Hsu and Blankern (1970) method.

9.7 Simulation of a random sea

The sea surface is assumed to be Gaussian and the method of simulation is based upon the summation of a finite number of Fourier components to obtain the surface elevation r/(t) given by Equation (9.13). The amplitude of each sinusoidal component wave is given by

R m -" ( 2 F (to m)mtO) 1/2 (9.20)

where F(w,,,) is the input energy spectrum and Aw is an elemental frequency interval centred on the discrete frequency tom. Over the frequency interval Aw the spectrum F(tom) is assumed uniform. In the summation of the series, the value assigned to N~the number of wave components~should be as large as possible in order to achieve a close approximation to a realization of the random Gaussian process.

9.7 Simulation of a random sea 217

In the summation of the component wave series, Equation (9.13), two approaches can be adopted in the selection of the frequency interval Ato; namely, equal spacing or variable spacing. In the equal spacing approach, the number of frequencies and the selection of the frequency values are determined with regard to the time 'sampling' interval between the quantities numerically predicted and the duration of the simulation. Thus, if the total simulation time is T seconds then the equal spacing frequency interval is given by

A to = 2 7r/ T

The simulated wave elevation 7/(t) will therefore repeat itself after a period of T seconds. The total number of frequency components required in the summation is given by toH/Ato , where toll is the highest wave frequency for which the input wave spectrum is defined. Therefore, for long simulation periods the equal frequency approach is computationally very expensive.

The second approach is to use unequally spaced frequency components so that the periods of the sine waves are not harmonically related and the series repeats only after a long time. This approach enables much longer simulation times than the equal frequency spacing method for the same number of frequency components. The unequal frequency interval can be chosen in various ways but the most straightforward method is to let F(to) Ato be constant. In other words the amplitude of each component sinusoidal wave is fixed and equal to the constant value ~ / N , where N is the number of wave components and tr 2 is the total variance of the time series. This variance equals the area under the spectrum.

If the input spectrum is narrow-banded, the unequal spacing approach thus has more frequency components within the frequency band where the spectrum has maximum energy. On the other hand, the equal frequency interval method distributes the frequency components evenly over the entire frequency range.

In this case study the unequal frequency spacing method is adopted. The simulation procedure is therefore as follows.

1 Generate a sequence of paired unit normal distributed random numbers am and tim. 2 Generate a realization of the selected spectrum F(to) using

N/2

7/(t) = ~ {am cos (tomt) + bmsin(tomt)} m = |

with

a . = [ r

bm= tim IF (to,,,)Ato] 1/2

3 Generate R,,, and 0,,, of Equation (9.13) using

g m = (a2,,, + b E)l'2

0m = tan- l (bin ~am)

4 Generate F(E)(t), the second-order drift force from Equation (9.5), m(oo), b(2)(t) and K(t) associated time series ready to solve the equations of motion, Equation (9.2).

Because the amplitudes, Rm, of the component waves are also determined from the values of am and/3,,, both the amplitude and phase are said to be randomized. This


procedure, referred to as the probabilistic method of determining the wave amplitude, is used to produce the results presented later. In contrast, in the deterministic method, the component amplitudes are specified by Equation (9.20) and only the phase is randomized.

9.8 Why the probabilistic method of simulation? So far, we have indicated that the engineering problem of interest can only be realistically approached by an appeal to spectral and probabilistic ideas. The question of how to simulate a realization to represent a selected spectrum is thought by many researchers to be a choice of personal preference. That is, the amplitudes of the component waves can be either deterministic or probabilistic in Equation (9.13). The essential idea behind this equation rests with the development of a statistical signal theory which leads to the development of a Gaussian distribution because of the central-limit theorem.

A single sine wave, x~, has the probability-density distribution function

1 p,(x,) = - x h Ix, I < R,

= 0 otherwise.

So the question raised is whether or not it matters how we select the component wave amplitudes. Surely, if N is large enough the central-limit theorem will ensure that the sum of N sine waves in Gaussian. The problem of determining the probability-density distribution of the sum of a limited number of sinusoidal waves is old and has been studied by Slack (1946), Lyon (1970) and Miller (1963). In the context of simulation requirements, the paper of Flower (1981) looks at the normality of the summed sinusoidal waves of Equation (9.13) using Miller's equivalent mathematical analysis, which provides the result

1 1 ~1[~ jo(kTTRRn) ] (k..~) = - - cos I n l < R p(rl) 2R + R _- ,_-~

where

= 0 otherwise

N

R = ~R, , , n=l

and, as before, J0 is a Bessel function of the first-kind of order zero. Using large- argument based asymptotic formulae for J0, an easier computational equation for the PDF can be generated. It would appear that the deterministic representation of an irregular wave is well understood with sufficient mathematical and numerical skills available to allow investigation of the statistics of finite summed sinusoidal waves.

However, as Tucker et al. (1984) pointed out, the limit N ~ ~ is required for a correct simulation. On this basis, the distribution of the maxima, as studied by Cartwright and Longuet-Higgins (1956) is correct. However, when N is finite it is not clear which wave statistics simulated in this way are seriously in error, but statistics of wave groups are

9. 9 Statistical analyses of the generated time series 219

certainly affected. Consequently, Tucker thinks it is possible that some rather surprising results on wave group lengths found in the literature could be faulty due to the simulation technique used. This problem could be avoided altogether by generating random numbers with a Gaussian PDF and passing the numbers through a linear filter whose frequency response is the required spectrum. There appear to be no fundamental problems with this alternative approach, but we revert to simulations based on summed sinusoidal waves.

Tucker has demonstrated that since the Fourier coefficients a,, and b~ are independent, then R,,, has a Rayleigh distribution with an RMS value of X/2tr~ = 2I'(to,,)Ato~, and the phase e~ is independent with a uniform distribution on (0, 2,0, where F(to) is a two-sided spectrum in his notation. Thus, to fix R,,, = X/2F(to~)Ato,, is incorrect because the randomness of the wave requires more than randomness of the phase alone! The consequence of random phase alone is that the simulated process is not ergodic; that is, an average from one realization over a long time is not equivalent to an average of an ensemble of shorter realizations.

Since we wish to study the influence of low-frequency damping, it is necessary that the statistics associated with such an investigation be sensible. The foregoing overview highlights why the simulation process has been undertaken as described.

9.9 Statistical analyses of the generated time series

The random wave records or realizations used in the simulation are generated for the selected theoretical wave spectrum. The generated time series of the linear and the non-linear hydrodynamic forces, together with the motion responses, must be analysed by statistical means because of their random nature.

The statistical quantities generally considered in the post processing analyses are: mean values, RMS values (standard deviations), maxima and minima, the mean of the variances, the coefficient of variation of the variances (CVV), the autocorrelation functions and associated spectral density functions and their moments. Most of these terms are readily understood from the earlier text or basic statistics, and therefore we only define the mean of the variances and CVV, since we shall discuss these later. Respectively, these two statistics are defined by

n = l

and

CVV = ( 1 E ( ~ - 6~)2) u2

where 0~ is the variance associated with the nth realization of the stochastic process. As already pointed out, one of the important decisions to be made is the selection of

the number of wave components. The number selected must be sufficient to give a close approximation to a realization of the selected wave spectrum. In the present analyses a total of 300 wave components, summed at unequal frequency spacing, are used. The resultant time series of the wave elevation records, for a simulation period of 4.5 hours at every half second, were analysed statistically. In particular, the RMS value and the zeroth and first moments of the spectral density function were calculated. If the


realization of the selected wave spectrum is correct, the computed statistical quantities of significant wave height and the average crossing period should (ideally) be identical to the selected values used with the specified wave spectrum. For the two-parameter ITI'C spectrum defined and a significant wave height of 7.0 m and an average crossing period of 10.2 s, the computed values from the spectral analysis of the generated realization were 7.09m and 10.10s. This simple test indicates that 300 wave components used in the reported analyses are sufficient to provide a realistic representation of the specified wave spectrum.

Because the sea is a random process, the statistical quantities of mean, RMS and the maxima of the responses of the moored structure for different realizations of the input wave spectrum for a given duration of simulation will be different for each simulation. The main reason for the variation of the output statistics is believed to be due to the limited duration time of the simulation. Therefore, if one carries out the simulation for an infinite time duration the computed random surge motion, say, should approach a stationary process. In the low-frequency surge motion of a moored vessel (large natural period) a simulation of several hours may be needed to reduce the variation in RMS values. To illustrate that the differences in the RMS values of the surge motion do reduce as the simulation time is increased, computations were undertaken for ten realizations of the random wave records. Each simulation record has a duration of 10 hours and the exercise was undertaken for both a moored barge and a moored tanker. The first 1/2 hour (tanker) or first 2 hours (barge) of the generated time series of each simulation are considered as the transient portion and neglected in the statistical calculation of the mean and variance. This point will be returned to later when presenting the results of the study.

For each of the 10-hour simulations the time series is divided into intervals of 1 hour, 2 hours, 4 hours and either 8.0 hours (barge) or 9.5 hours (tanker) records. These records were then analysed statistically. The computed root mean variance and CVV for the surge motion of the tanker and the barge are presented in Tables 9.1 and 9.2

Table 9.1 The effects of simulation time on the root mean variance and CW of the surge motion variance of a tanker

' 'R-'otu mean CW . . . . Number~fu ' Simulation duration variance realizations in hours

1.0 13.64 0.325 90 2.0 13.55 0.302 40 4.0 13.63 0.243 20 9.5 13.78 0.164 10 , , i i i i

Table 9.2 The effects of simulation time on the root mean variance and CW values of the surge motion variance of a barge

Simulation Root mean CW Number of duration variance realizations in hours 6-

1 1.451 0.2205 80 2 1.465 O. 1861 40 4 1. 454 O. 1250 20 8 1.495 0.09958 10

i , ,,

9. I I Effects o f wave damping on the surge motion 221

respectively. One can observe from these tables that dr remains rather constant for different realizations of the wave records and CVV tends to decrease with an increasing duration period of the simulation. This indicates an improvement in the reliability of the output statistics for longer simulation periods.

9.10 Sensitivity analysis of a moored tanker and a moored barge to integration time step

The Pierson-Moskowitz spectrum, with a significant wave height of 7.0 m and an average crossing period of 14.5 s, was used in this study. The tanker selected is the tanker model investigated experimentally in the MARIN OceanBasin, The Netherlands, by Wichers (1982). The barge is basically a rectangular box with a truncated wedge form at the fore end. The hydrodynamic quantities of added mass, the radiation damping, the fluid retardation function, the wave drift damping, the wave excitation forces and the drift forces of each dynamic system were computed using a first-order hydrodynamic boundary value problem solver developed by Hearn et al. (1987). In particular, the three-dimensional theoretical low frequency wave drift damping and the mean drift forces were evaluated using the near-field based Added Resistance Gradient Method of Hearn et al. (1987).

Both the barge and the tanker are considered to be moored in a water depth of 100.0 m using six mooring cables spread out symmetrically with respect to the longitudinal plane of symmetry. There are three mooring lines fore and aft. One line has a zero angle of incidence and the other two lines have an incident angle of 0 to port and starboard. For the barge and tanker respectively 0 is 15 ~ and 29 ~ The nominal lengths of the mooring cable are 325 m and 250m for the barge and tanker respectively. Each vessel is positioned so that its heading is coincident with a 180 ~ incident wave. The mooring cables are of identical material and their properties are given in Table 9.3. The geometric particulars of the tanker are well known and those for the barge correspond to Barge A of the Hearn and Tong studies (1987).

Table 9.3 The characteristics of the mooring cables i i i i

Cable weight per unit lengih in air = 1.098 kN m -1 Modulus of elasticity = 5.6 x 10 7 kN m -2 Stress sectional area of the cable = 4.537 x 10 -2 m 2 Cable breaking strength = 2.310 x 104 kN

, , , , , , . , ,

The resultant surge motion time series were analysed statistically for their mean, RMS value and their maxima and minima. The results of the eight simulations for the tanker as a function of time step used in the solution process are presented in Table 9.4. For a large time step, the statistical quantities deviate markedly from the corresponding estimates at the smaller time step. For the selected wave spectrum characteristics a time step of 0.5 s seems appropriate for modelling the equations of motion accurately.

9.11 Effects of wave damping on the surge motion

In the low frequency simulation of the moored vessels it is known that the wave drift damping plays an important role in the motion of the vessels. In order to assess the


Table 9.4 The effects of time stepping on the motion statistics , , , , L , , ,

t Mean RMS Max+ Max-

0.2 - 13.84 11.62 16.43 -33 .04 0.3 - 13.86 11.64 16.69 -32.95 0.4 - 13.83 11.64 16.53 -33.03 0.5 - 13.89 11.61 16.69 -32.82 0.6 -14 .04 11.61 15.62 -32 .95 0.7 - 14.49 10.84 13.47 -32.87 0.8 - 14.27 11.13 14.83 -32.81 1.0 - 15.71 9.755 10.86 -32 .48

degree of sensitivity of the damping on the surge motion, numerical simulations were carried out using different estimates of the wave drift damping coefficients. The five cases considered were:

1 three-dimensional theoretical wave drift damping (ARG method); 2 three-dimensional simplified theoretical wave drift damping (DFG method); 3 constant wave drift damping value assigned; 4 without damping; 5 experimentally measured damping.

The three-dimensional theoretical wave drift damping is based upon the near-field Added Resistance Gradient (ARG) method developed by Hearn and Tong (1986).

The three-dimensional simplified theoretical wave drift damping is calculated from the frequency gradient of the wave drift forces, as described by Standing et al. (1987) and further investigated by Hearn and Tong (1988).

By constant damping we mean the assignment of the peak wave drift damping value, predicted by the three-dimensional ARG method, at all wave frequencies. The experimental damping corresponds to the values published by Wichers and Huijsmans (1984) for the tanker and the values measured by Hearn and Tong (1987) for the barge.

In discussing the simulations, we shall consider one degree-of-freedom simulations initially and then progress to six degrees-of-freedom simulations so that the effects of motion coupling can be appreciated. The wave spectrum used in these simulations is the two-parameter ITTC spectrum. The peak frequency of this spectrum is governed by the average crossing period and not the selected value of the significant wave height. In the simulations, the selected average crossing period corresponds to 10.2 s and this leads to a peak frequency of 0.475 rad s -~.

The statistical RMS values of the surge motion for each simulation are plotted against the input significant wave height as illustrated in Figs 9.3 and 9.4 for the tanker and barge respectively. For the tanker, Fig. 9.3 shows that the RMS values of the surge motion computed using the three-dimensional ARG wave damping and the experimentally measured damping correlate well for significant wave heights less than 9.0 m. At the higher wave height the RMS values of surge motion based on the experimental wave damping is smaller than that of the theoretical damping. The simulations based on the three-dimensional DFG wave drift damping values exhibit large departures from the simulations based on the three-dimensional ARG wave drift damping values. This departure is especially significant at the higher significant wave heights, i.e. in the more severe sea states. For a significant wave height smaller than 3.0 m, the low-frequency

9. I I Effects of wave damping on the surge motion 223

50, . . . . . . . .

40

~'~ 30

r 20

10

3D theoretical wave damping (wd.) �9 3D simplified theoretical w d �9 Constant w.d. = 400kN s/m �9 Without wave damping �9 With experimental w.d.

i t /

/f,,-'-,,, io" t

f , / " ' , . . . . . ;~ ',..

f , j J : "

~ , . . - L ~ . . . .

0 0 5 10 15 20

Significant wave height (m)

Fig. 9.3 The surge motion variance versus the significant wave height for the tanker

surge wave drift damping has negligible effects on the computed surge motion statistics of the tanker.

Of the different wave drift damping values considered, for both the tanker and the barge, the simulations based on the constant assignment of the peak wave drift damping value provide the lowest RMS values. This is to be anticipated, since the wave drift damping values assigned, 40.0 kN m s -~ for the tanker and 97.0 kN m s -1 for the barge, represent the highest damping in the five cases considered. The computed surge motion variance for both the tanker and the barge, when there is no wave drift damping is much larger than the corresponding results with low frequency damping effects incorporated-- the wave drift damping having a much greater influence on the motions of the barge than on the motions of the tanker.

Figures 9.5 and 9.6 present the extreme surge displacement of the tanker and the barge respectively. The extreme displacement, in general, increases with the significant wave height irrespective of the different wave drift damping coefficients used in the simulation. The computed coordinates, based on the three-dimensional ARG damping, agree very

30

�9 3D theoretical wave damping 25 a= Constant w d. = 970kN s/m

�9 With experimental w.d. �9 Without wave damping/10.0

2O r162 E.E " 15 . . . . . . . . . . . . .

j . " .. ,,. e - " " . , h LD-" "" "

oo �9 �9 �9

o ~'4" i r

0 0 ; 1; 20

Significant wave height (m) F i g . 9 . 4 The surge mot ion var iance versus the signif icant w a v e h e i g h t for the barge

224 Mot ions o f m o o r e d structures in a seaway

lOO �84

A 8 0

-r 60 8 E 40 E

=~ 20[

o_

a 3D theoretical wave damping (w.d.) 3D simplified theoretical w.d.

�9 Constant w.d. = 40.0 kN s/m �9 Without wave damping �9 With experimental w.d. , P

�9 ,�9

0 5 10 15 20 Significant wave height (m)

Fig. 9.5 The surge extreme displacement versus the significant wave height for the tanker

well with those using the experimental damping for both the tanker and the barge for the range of significant wave height values considered. The extreme displacement of the tanker computed without the wave drift damping shows a progressive departure from that computed with three-dimensional wave drift damping. These differences in the predicted extreme motions are even more significant for the barge (see Fig. 9.6). The reader should note the additional scaling of the 'without wave drift damping' predictions to permit their presentation with the other results. When the motions are simulated using the three-dimensional DFG wave drift damping, the predictions are largely in disagree- ment with the predictions using the three-dimensional ARG values. In Figs 9.5 and 9.6 the predictions based on a constant wave drift damping value give the lowest extreme displacements.

Figure 9.7 provides the mean surge displacement of the barge for the different wave drift damping values used in the simulations. For significant wave heights less than 3.0 m the predicted mean surge displacements for the different damping values are almost indistinguishable from one another. As the significant wave height increases, the

80,

60 l C

,o E .E_

20

3D theoretical wave damping i Constant w.d. = 97.0 kN s/m �9 With experimental w.d. �9 Without wave damping/10.0

0- 2O

.... , o

.,, ,AI,. o - "

i f . / ' , , s " �9 f s �9 �9

1() 1~5 Significant wave height (m)

Fig. 9.6 The surge extreme displacement versus the significant wave height for the barge

9. I I Effects of w a v e damping on the surge motion 225

40

30

E | 20

10

�9 3D theoretical wave damping s Constant w.d. = 97.0 kN s/m �9 W'dh experimental w.d. �9 Without wave damping

... . . . ,B . ... . , ' ' ' ' ' 4 1

�9 .,, , , . .4P " " �9 41 . . . A L P " " . . . . . - , , 4 P ' " " "

o 20 Significant wave height (m)

Fig. 9.7 The mean surge d i s p l a c e m e n t ve rsus the significant wave height for the barge

constant wave-drift-damping based predictions lead to the highest mean displacement with the three-dimensional ARG damping providing the lowest estimate. The predictions based on the use of the experimental values of wave drift damping fall between the former sets of results. In each case, the mean displacement tends to increase with the significant wave height. The mean displacement, computed without the inclusion of wave drift damping effects, does not follow the same trend as the predictions with the wave drift damping included. In fact, the results are rather erratic over the range of significant wave heights considered.

Figures 9.8, 9.9 and 9.10 provide estimates of the variance, the maxima and the mean value of the mooring line tension for the barge. The results presented correspond to the mooring line subjected to the largest loads for the particular six mooring line configuration adopted. The mooring statistics, for the different wave drift damping values used in the simulations, show the same trends as the corresponding motion statistics for the range of significant wave height values considered. In general, the statistics' values increase with significant wave height and those simulations based on a constant wave

250

200

z 150

8

50

�9 3D theoretical wave damping/10.0 I Constant w.d. = 97.0 kN s/m/10.0 �9 With experimental w.d./10.0 �9 Without wave damping/10..2

sS f O ~ ~*SSD �9 t S ~LD ~ L ~ S ~m~ ~ ~

0 5 10 15 20


Fig. 9.8 The variance of the mooring line (No. 2) tension for the barge


200

150 A z r 0

"~ 100 i Ii) C

3D theoretical wave damping/10..2 �9 Constant w.d. = 97.0 kN stm/10.2 �9 With exped,'nental w.d./lO..2 �9 Without wave damping/lO..2

,,,At ,,,, , ' ~

50 . 7 .......... "-_.--- ' "

s ~ r , . . - - " _ . . . . j ~ . . r

. s . o "~ t .4B,., .,,...' " "

i "t w~rtt t .e . I ..,....,.

0 t=~ .4 r " 0 5 1() ............ 1~i 20


F ig. 9 .9 The m a x i m u m mooring line (No. 2) t ens ion for the barge

drift damping value provide the lowest estimate of maximum line tension, tension variance and mean tension. For significant wave heights smaller than 3.0 m, the line tension statistics are indifferent to the different wave drift damping values used in the simulations.

Figure 9.11 shows the time series of the line tension for the time segment between the third and the sixth hour of a six-hour simulation. Each of the peak line tension values corresponds to a peak displacement of the vessel.

So far, the numerical results presented have been based upon one degree of freedom motion simulations. The coupling effects of heave and pitch on the surge motion have therefore been neglected in the numerical syntheses so far presented. The coupling omitted clearly arises through the hydrodynamic coupling modelled in the added mass and radiation damping terms in the dynamic equation of motions. Because of the large hydrostatic restoring forces (in the vertical plane) the heave and pitch motions persist at the component wave frequencies. On the other hand, the surge motion in the horizontal plane is predominated by the low frequency component as a result of the slow drift surge

200

150

A z ~" 100 r

50

�9 3D theoretical wave damping/10.2 a Constant w.d. = 97.0 kN s/m/10.0 �9 With experimental w.d./10.0 �9 Without wave damping/10..2

o

0 0 lb 20


Fig. 9.10 The mean mooring line (No. 2) tension for the barge

9. I I Effects of wave damping on the surge motion 227

60

50 -~ Une two/lO.O

| i

30

�9 20

,o

_ _ . . . ,

10800 12600 14400 16200 18000 19800 (S)

Fig. 9.11 The times series of the mooring line (No. 2) tension for the barge at the significant wave height 7.0 m

excitation wave force. However, the coupling of the wave frequency vertical motions and the horizontal surge motion at low frequency can be significant in severe sea states.

To illustrate the effects of cross-coupling, the simulations were repeated for the same realization of the wave spectrum, but with the heave and pitch motion effects incorporated into the computations for the tanker. For a head sea and a symmetric mooring configuration, the lateral motions of sway and yaw do not occur. The low-frequency surge wave drift damping values used are the full three-dimensional ARG values. The computed statistics of motion, mean and the motion RMS value, are compared with the earlier one degree-of-freedom motion-based predictions in Table 9.5. It follows from this that the difference in predicted variances is small for significant wave heights less than 11.0 m, whereas the corresponding difference in the predicted mean values is more significant.

Tab le 9.5 The effects of degrees of freedom on the simulated motion

. . . . . . . . . . . . ~3ix degrees of freedom ' Onedegree oi freedom

Significant wave height Mean RMS Mean RMS

3.0 -5.877 5.168 -5.984 5.174 5.0 -8.778 12.30 -8.182 12.93 7.0 -12.48 11.41 -13.11 11.11 9.0 - 13.08 13.67 - 12.15 13.41

11.0 -9.553 16.97 -8.718 16.99 15.0 - 12.28 17.40 - 15.03 16.14

, ,1 , , , , ,,, , ,, , , , , , ,

To appreciate the relative orders of magnitude of first and second-order forces, and the motions, we shall briefly look at samples of the time series generated. Figure 9.12 presents a sample of the time series for the generated wave record of a 4.5 hour period of the simulation. The incident wave elevation is shown as a continuous curve and the broken curve provides the low frequency wave envelope. Figures 9.13 to 9.15 respectively provide the corresponding linear first-order wave excitation forces in surge, heave and pitch for a head sea. Figures 9.16 to 9.18 provide the corresponding second-order


12.5 ,~

10.0

7.5 A E c 5.0 o .,_,

2.5

~ 0.0

Wave elevation ETA ('1") . . . . Wave envelope A (1")

-'"[""'l I' " I"ll'~" r""l" I'"',,' if' 'P'I']' rr',, -5.0

-7.5 I t ,,, ~ ~ i 14400 14700 15000 15300 15600 15900 1

Fig. 9.12

16200 Simulation time (s)

Time series of the generated random wave records

1~176 It . . . . . . . . . . I 80 ]- Surge excitation time series/10..3

60

4O

5. -20 _40[ l''P

-6~ t -80 i ~ I I 1

14400 14700 15000 1 5 3 0 0 1 5 6 0 0 1 5 9 0 0 ' 16'200 Simulation time (s)

Fig. 9.13 The f irst-order surge excitat ion force t ime series for the tanker

100

80

60~

40

E" 20

0

-20

-40

-60" -80

14400

Fig. 9.14

-Heave excitation time series/lO..3

! I " I I, I 14700 15000 15,300 16200 15600 15900

Simulation time (s) The f i rst-order heave excitat ion force t ime series for the tanker

200

150

100

F i g . 9 . 1 5

A 50 z 0

- 5 0

-~oo r -150 I - 2 0 0 ,

14400

100

80

~ ' 6 0

Pitch excitation time series/10..5

I I ! , , I , I 14700 15000 15300 1 5 6 0 0 1 5 9 0 0 16200

Simulation time (s)

The first-order pitch excitation moment time series for the tanker

"~. 40

- - - - Surge excitation time series/10_2

14400 1 4 7 0 0 15000 15300 1 5 6 0 0 1 5 9 0 0 16200

F i g . 9 . 1 6

Simulation time (s) The second-order low-frequency surge drift force time series

100 i

60

40

o,._ r' -40 -60 -80

Heave excitation time series/10..2

- 100 L I I I , 14400 1 4 7 0 0 15000 15300 15600

_L~ lk �9

rio

9. I I Effects of wave damping on the surge morion 229

I 15900 16200

F i g . 9 . 1 7 Simulation time (s)

The second-order heave drift force time series


6 0

50

40

~' 30

20

10

-10 14400

Pitch excitation time sedes/lO.,4

t i , i . . . . . . i, i ,, 14700 15000 15300 15600 15900

Simulation time (s) 16200

F i g . 9.18 The second-order pitch drift moment time series

non-linear drift force in surge, heave and pitch for the same time interval. Figure 9.19 presents the time series of the surge wave drift damping based on the three-dimensional ARG calculations. Figures 9.20 to 9.22 provide samples of the corresponding simulation of the motions of surge, heave and pitch. Figure 9.20 clearly shows that the surge motion is predominated by the low frequency surge motion with the wave frequency component superimposed upon it. On the other hand, the heave and pitch motions only persist at the wave frequencies. The scaling and simulation period shown in Figs 9.21 and 9.22 have been changed to accommodate these differences in order of magnitude.

During a numerical simulation, the wave frequency motions and the low frequency motions can be identified by suppressing the appropriate parts of the wave excitation. In Fig. 9.23 the low frequency excitation was suppressed and the resultant motion represents the wave surge frequency response. It is seen that for the first few hours of the simulation the vessel underwent a slow oscillatory motion due entirely to the transient effect of the simulation. As the transient motion dies down, the vessel responds purely to the wave frequency excitation.

20[ 15

z lO ~..

144OO

' " Surge damping time series/10.2

t 14700 15000 15300 15600 15900 16200

S i m u l a t i o n t ime (s)

F i g . 9.19 The low frequency surge wave drift damping time series for the tanker

9. I I Effects of wave damping on the surge morion 231

50 40 30 20

~

::N VVVVVvVVVIIVVV VVVVVyVI -40[____~ - 5 0 ~L_..____L~L~L__..__LJ__.____L______

0 2000 4000 6000 8000 10000 12000 14000 16000 Simulation time (s)

Fig. 9.20 The low frequency surge response time series for the tanker

A

r-

- 1 "1-

-2

Heave displacement time series

-3L I I t I 14400 14700 15000 15300 15600

Simulation time (s) Fig. 9.21 The heave response time series

I I 15900 16200

0.06['

0 .04 Pitch displacement t ime series

~ 0.02 0.00 -

~ -0.02 t j .e~ O.

-0.04

-0.06 14400

F i g . 9 . 2 2

.1 , I I . . . . I, I , ] 14700 15000 15300 15600 15900 16200

Simulation time (s)

The pitch response time series

_ 4:f 1

Surge displacement time series


.... 0 1 8 0 0 3600 5400 7200 9000 10800 12600 14400 16200 Simulation time (s)

Fig. 9.23 The wave frequency surge response time series

9.12 Final comments

This case study was concerned with the inverse process of generating a time series to represent a random sea state with a particular spectral form. The procedures and the problems identified have nothing to do with the particular two-parameter ITTC spectrum selected. The same rationalization would have had to have been addressed if one used some other possible mathematical form. Once again, we note that the physics of the engineering situation and the techniques of spectral analysis had to be coupled in a meaningful way. The object of the task addressed was the provision of reasonable and realistic simulations to permit assessment of a complex engineering situation which has associated risk factors. Overdesign the moorings and you have financial resource problems, underdesign the moorings and you make international news headlines regarding safety of men, pollution of the sea, etc. Thus, just as one has to argue the case for the selected methods of modelling the moored structure dynamics, the modelling of the hydrodynamic loads, and the modelling of the mooring lines themselves, so too the method of simulation has to be discussed in relation to the number of component waves, the assignment of associated wave amplitudes, the time integration and the ability of the total model to represent the situations of concern. Once the simulations are complete, standard statistical techniques are used to quantify the behaviour.

The arguments related to simulation of spectra could equally apply to other engineering situations, for example, wind loading of buildings or bridges, or earthquake excitation of structures. The specific details will naturally be quite different, but the approach presented might be equally applicable.

l0 Experimental measurement and time series acquisition

10.1 Introduction Throughout this book the availability of the time series for undertaking a spectral analysis has been assumed. Even in the wave monitoring device case study of Chapter 8, the transducers selected measured the required wave amplitude directly at the point of their installation. During that study some care was also taken to avoid contamination of the incident waves being monitored. In particular, the diameter of the principal supporting pole was selected so that it would not buckle, and to avoid diffraction waves arising from the interaction of the incident wave and pole. Similarly, excitation of the mooring lines stabilizing the supporting structure was avoided.

This case study is concerned with recovery of a particular time series and its analysis in the context of observing a moored model structure excited by waves in a tank. A moored structure in waves has six rigid-body degrees of freedom; three translations and three rotations. Mathematically the amplitudes associated with each degree of freedom can be approximately predicted through linearization of the dynamics. However, in an experimental observation of the same event only resultant motions are seen, that is, the vector summations of the individual motions. Here, cameras are used to make records of the motions of light emitting diodes attached to the moored structure. This method of observation was selected as it does not interfere with the interactions being observed. In particular, two very practical aspects are considered.

1 How to separate out the individual motion records from the observed resultant motions.

2 How to allow for 'losses' when estimating the amplitude of the regular incident waves from their spectrum.

The significance of compensating for 'losses' is best appreciated by recognizing that a 5% error in wave amplitude produces a 10% error in quantities with a quadratic dependence upon the wave amplitude, such as the low frequency damping measured in the experiments now discussed.

10.2 Background In the first case study, the time series analysed were directly obtainable from the signals recorded and the associated calibration factors. In the second case study, a time series

233

234 Experimental measurement and time series acquisition

was generated to represent a particular spectrum. This generated time series represented the continuously changing elevation of a random sea at a selected special location. Now, the so-called second-order dynamic loads acting upon a moored structure are proportional to the square of the instantaneous sea elevation. Here, we shall consider the experimental determination of the wave amplitude (squared) dependent surge low- frequency wave drift damping coefficients---designated as b (2) in the second case study. In undertaking such an experiment, it is not possible to prevent the structure having both high and low frequency responses because of the water wave excitation and the mooring line influences. Therefore, direct acquisition of the required low frequency data is not possible. Furthermore, although we may only be interested in one particular degree of freedom, surge, we cannot prevent other degrees of freedom being excited. Conse- quently, the surge motion must be separated out from the data recorded. The sought hydrodynamic data are small and highly sensitive to exciting wave frequency and structural characteristics. The identified characteristics of surge wave drift damping for ship-shaped vessels are:

(a) (b) (c)

(d)

the damping in still water and regular waves is linear; the damping in waves is generally greater than that in still water; the additional wave-borne damping is proportional to the square of the wave amplitude; the wave-borne damping is wave frequency dependent, but independent of the surge motion frequency.

These characteristics were originally identified as a result of Wichers and Van Sluijs undertaking free decaying surge motions of a ship model moored by a linear spring system in still water and in regular waves. The damping coefficients were derived from the logarithmic decrement of the amplitude of the measured surge motion.

The experimental procedure described exhibits in-built redundancy to provide some level of confidence in the values measured. This is necessary because the sought hydrodynamic coefficients for barges and semi-submersibles were known to exist in certain confidential reports, and we fully intended to publish our findings to increase the public domain data significantly. The objectives of the experimental programme were as follows.

(a)

(b)

(c)

(d)

(e)

To establish the nature of surge low-frequency damping for a number of different vessel types. To establish the dependence of the surge low-frequency wave damping coefficients upon wave frequency and wave amplitude for each selected vessel type. To determine the dependence, or otherwise, of low-frequency surge wave damping upon the natural surge frequency of each moored structure. To provide sufficient information for inferences to be drawn on the effect of systematic changes in geometric ratios on the surge low-frequency wave damping of each selected vessel type. To provide sufficient experimental data to allow a reasonable assessment of the validity of any theoretical method of predicting surge low-frequency wave damping.

To fulfil these objectives, free decay tests of the surge motion were carried out in still water and in waves of differing frequency (8 to 12 in number) and amplitude (2 to 4 in number). The complete motion time histories were recorded and saved on the dedicated

10.3 Experimental facilities set-up 235

data acquisition computer for each and every experiment undertaken. These raw data are still available.

10.3 Experimental facilities set-up The main experimental programme was originally planned to be carried out in the Newcastle Towing Tank. Selected runs were then to be repeated in the Edinburgh University Wave Power Tank in order to provide an indication of the effects of tank wall interference. The Newcastle Towing Tank measures 40 m x 3.8 m x 1.25 m. The Edin- burgh University Wide Tank dimensions are 11 m x 27.5 m x 1.2 m, but regarding the useful working area it is 22.0 m by 7.3 m and the water level is I m deep. In the end, the majority of the experiments were carried out in the Edinburgh Wide Tank with the preliminary tests conducted in the Newcastle Towing Tank to rehearse the procedure and to test the adequacy of the planned experimental set-up. Availability and high costs prohibited a repeat testing programme in a significantly bigger ocean engineering testing facility. This is still considered to be very desirable.

Figure 10.1 illustrates the Edinburgh tank and model arrangement. Wave makers (80 in all) are situated on one of the long sides of the tank. The tank is wide so that 'side-wall' reflection effects can be neglected. It is, however, rather short, from wave generators to beach, so that longitudinal reflection of waves will significantly affect our results for those wave frequencies for which poor absorption by the 'beach' occurs.

The model under investigation was moored by four linear springs near the centre of the tank, with the longitudinal centre of the model and the mooring lines perpendicular to the line of wave makers (see Fig. 10.1). One end of each spring was connected to the model via an inextensible string which passed through a grub screw connector attached to the spreader arm screwed to the model. Variation of string length permits variation of pre-tension levels in the mooting line system and prevents the mooring lines dipping into the waves during the experiments, i.e. hydrodynamic loads on the mooring system are eliminated. The other end of the spring was connected directly to a strain beam. The tensions in the springs, measured by strain gauges mounted on the strain beam, were recorded throughout each test. Since the strain gauge readings correspond to the horizontal component of the spring tension they are directly related to the surge motion of the model. The strain gauge readings therefore serve as a set of redundant

Wave probes m m m m m m m m m m m m m m m m m m m m m m m m Wavemakers .... ~...a.---- 1' --Strain gauge beams

�9 �9 �9 I �9 �9 �9

�9 �9 o4." % �9 �9 �9 o ~ o

Scaffold ~ , / ~ Wave probes " Selspo, ~ m e r a ~ i I / ' 4----~ i Mt~ . el&.~ wall - -

m ! \ ! ---- Mooring lines P.C. monitoring--- Release mechanism system

Beach and walkway

Fig. 10.1 Edinburgh Wide Tank and experimental set-up


measurements of the surge motion. The purpose of the spreaders is to prevent significant yaw motions by ensuring the springs provide a sufficiently large restoration moment.

The motion of the model was monitored by measuring the position of four Light Emitting Diodes (LEDs) mounted on the deck of the model. An optical non-contact position tracking system called SELSPOT was used. Using a precalibrated two-camera system, the three-dimensional position of each LED can be measured at a very high frequency (exceeding 100 Hz). The six degrees-of-freedom motion of the model can then be determined from the positions of the four LEDs with some redundancy in the data collected.

The bottom standing scaffolding used to provide a rigid mounting for the SELSPOT opto-electronic cameras was built some 2 m to one side of the centre-line of the model. To reduce interference from light reflected from the free surface of the water, the cameras were situated approximately 2 m above the mean water level. The LEDs were therefore inclined at an angle of 45 ~ upwards to face the cameras. Behind each LED was a dark non-reflecting shield.

A catch-release mechanism mounted on the beach side of the tank was positioned so that the line used to offset the model was aligned with the centre-line of the model. Model centre alignment was checked using overhead video cameras and plumb-bobs, hence model release-induced yaw was negligible. An inextensible string connecting the model and mechanism was used to release the model from some preselected and adjustable initial displacement. The switch used to release the model also triggered the sampling program in the micro-computer used for the data acquisition. This release mechanism was necessary because additional vibrations were picked up if we released the model ourselves by lying on the walkway (see Fig. 10.1).

Wave climate monitoring was provided using two capacitance wave probes placed between the wave makers and the model, marked with a cross in Fig. 10.1, and a permanently fixed set of 13 capacitance wave probes regularly positioned along a line parallel to, and in front of, the wave makers' faces. During the tests, wave elevation at each probe was recorded using our data acquisition system and the Edinburgh University PDP 11/60 based data acquisition system. The wave makers were controlled through the same PDP system with interactive dial-up of wave frequency and wave amplitude. The same runs of wave generation were repeated with wave probes located where the model had been located. This meant that transmission and reflection coefficient data, together with the variation of the wave amplitude over the 'model location' area, could be assessed.

10.4 Data collection and principles of analysis Data sampling was controlled through a dedicated micro-computer, and a specially developed computer program. The program automatically collected the SELSPOT data through a digital-to-analogue (D/A) input/output card taking one 16-bit digital channel, and collected the strain gauge and wave probe data through six analogue channels with 12-bit A/D conversion (see Fig. 10.2). Due to the inflexibility of our 10-year-old SELSPOT system, the computer sampling had to be synchronized with, and be controlled by, the SELSPOT system. Because the basic cycling period of the SELSPOT system is 3.2 ms (312.5 Hz), the dial up sampling rates available to the user are simply integer multiples of this period. Fortunately, the motion periods of interest are so long

10.4 Data collection and principles of analysis 237

Strain gauge

Camera [ Sel'spot Wave - Waveprobe ! Condition!ng signal probe I unit ano input mini LCU driver unit_ J [ amplifier - ~-Power supply and |

, , _ t synchronisation signal Output i Selspot system signal

main unit f

~ Labmaster | c~ ! ! - ! daughter board |

i Labmastercard ----- I motor ~ d

Fig. 10.2 Data acquisition system

(of the order of 10 s) that this restriction does not impose any practical limitations upon sampling choice. However, a great deal of time was required to sort out the synchronization of the computer-based sampling system and the SELSPOT monitoring system. The computer system had to match the SELSPOT system to work!

The x - y coordinates for each of the four LEDs, with respect to each camera, meant a total of 16 numbers were read in from the digital port of the data acquisition system for each sample. Synchronization and data valid flags were checked during the process. The six analogue channels carried the four strain gauge readings and the two wave probe voltages. Thus, 22 pieces of different data per sample were collected.

The aim of each test was to provide the low-frequency surge damping coefficient for the model under the prevailing conditions. With the basic premise that the damping was linear, the damping coefficients were derived using the logarithmic decrement method. The surge motion history of the vessel is obtained from the SELSPOT output of the LED position time series. Auxiliary information can also be derived from the time histories of the mooring spring forces. Identification of the prevailing wave condition in any one experiment was obtained from the energy spectrum of the recorded wave elevations.

Consider a free damped one degree-of-freedom linear vibrating system governed by the equation

m . ~ + b . ~ + k x = 0

The general oscillatory solution for x (see Chapter 7) can be written as

x = x0 exp (-bt/2m) sin(oat + $) (10.1)

where <o 2 = [1 - b2/(4m2o2)]a, 2

and x0 and $ are arbitrary constants, determined from the initial conditions, and to,, = ( k / m ) u2 is the natural undamped frequency of oscillation. The logarithmic decrement, 8, is defined as the natural logarithm of the ratio of any two successive amplitudes, i.e.

8 - log (xn/x,,+ 1 )


with

Xn/Xn+ l = exp (-b'r/2m)

on appealing to Equation (10.1), and z is the period of the damped motion. Clearly

b'/" ~ = 2m

and since the period of the damped oscillation is

2zr

4m262)a'2 it follows that

blr 1/2

For a lightly damped system the damping b will be very small compared with the critical damping given by 2(ink) 1/2, and hence 6 can be approximated by

that is

_ . .

bTr (mk) 1/2

b = (mk)l/2 8

The logarithmic decrements are derived from the experimental data and since

6 = log x0 = log X l = log Xn-1 ~ - - . , , ,

X1 X2 Xn

it follows that n6 = log x 0 - log x, . Hence, if log x~ is plotted against n the slope of the resulting straight line graph is -6 . In the analyses undertaken, the maximum and minimum of the low-pass filtered surge

motion time history were identified by searching for changes in the sign of the surge velocity.

Since the recorded surge motion time history represents a combination of the wave-frequency surge responses and the low-frequency natural decay motion, the wave-frequency response must be removed using a low-pass filter.

A basic non-recursive digital low-pass filter with a Hamming window was selected for the analysis undertaken. The low-pass filter can be written as

to~T N sin (ntOc T) y m = ~ E Wn Xm-n

"n- ,,=-u (ntocT)

where

10.5 Six degrees-of-freedom SELSPOT morion analysis 239

Xm is the input noisy signal to be filtered at the mth time step; Ym is the output filtered signal at the mth step;

T is the sampling period; to~ is the cut-off angular frequency; and (N~r)

147. is defined by the Hamming window, 14,', = 0.54 + 0.46 cos ~ .

The sampling interval used in all the tests was chosen to be 0.1984 s. The natural surge period for all the experiments ranges from 8 s to 24 s and the wave period varies between 0.5 s to 2 s. A filter half span of N - 30 with a cut-off period of 4 s was therefore employed.

10.5 Six degrees-of-freedom SELSPOT motion analysis The six degrees-of-freedom motions of a rigid body can be determined if the position of at least three non-collinear points on the body are known. Let the position of such a point be denoted by Pi. The point is situated at (x~, y~, zi) in a body-fixed frame of reference with origin at the centre of rotation, and is located at (Xi, Y~, Zi) in a space-fixed frame of reference (see Fig. 10.3). Let the rotational angles (g4, ~5, ~:6) be

(X1, Y1, Z1) P, (xl, Yl, zl)

Z Y

X

Fig. 10.3 Fixed frame of reference

small and the six degrees-of-freedom motion be measured with respect to the space-fixed frame of reference. Then, given the resultant measured displacements are ui, vi, wi for camera i (i = 1,2) we can deduce

Ui gi - -Xi ~1 0 --~6 ~5 Xi

Vi = Yi Yi = ~2 + ~6 0 -~4 Yi

Wi Zi zi ~s ~5 ~4 0 z i

where ~:i denotes the ith displacement/rotation of the six degrees of freedom. The matrix equation for each point can be re-written as a set of three simultaneous linear equations in the six unknowns, i.e.

ui = ~i + z i~5- yi~6

v~ = ~ 2 - zi~4 + x~6

Wi = ~3 + Yi~4 + Xi~5


If the position of N points is known, a system of 3N equations is available for the solution of the six unknown responses ~i. A least squares solution to the system can be evaluated. The least squares solution to N unknowns in N + M equations, written as

N ~, ai/xj = bi i = 1, . . . , N + M j=l

can be derived by minimizing the square of residues. In particular, an N by N system of linear equations of the form

N(N+M ) N+M ~, ~ aij. aik Xk = ~ ai/. bi j = I , . . . , N k=l i=1 i=1

i.e. ( A ' r A ) x = A T b , is found. This can be solved using standard Gaussian elimination techniques.

10.6 Data acquisition and SELSPOT calibration

In the experimental programme the position of four LEDs, mounted on the deck of the model, was tracked by the SELSPOT monitoring system and the collected data stored on the hard disc of the dedicated micro-computer. Therefore, position data for four body-fix points are available for finding the six degrees-of-freedom motion of the structure being investigated. Thus, redundancy exists in the data collected. A reference coordinate system (x,y, z) was chosen with its origin at the centre of the lens of Camera 1. The z-axis points vertically upwards and the horizontal x-axis is directed towards Camera 2. The SELSPOT monitoring system has to be calibrated by finding:

(a) the scale factors f x and f y defined by

D D fx = V~--L and fY = VyL

where Vx and Vy are the changes in volts recorded for a known movement of an LED (see Fig. 10.4(b));

(b) the point of intersection of the optical axes of the two cameras; hence, the distances l~ and 12 and the inclinations/3~ and/32 of the optical axis to each camera to the intersection point must be measured accurately;

(c) the distance between the cameras, l~; and (d) the inclination of the camera x-axis to the horizontal, a~ and t~2 respectively.

From Fig. 10.4(a), the position of Camera 2, XcE is (Xc2 , 0.0, Zc2 ) with the x and z coordinates given by

x22 = l 2 - ~ 2 and

Zc2 "- (/2 sin ~2 -- ll sin/31 )

Application of the cosine rule leads to

= 2 _ (/2 cos 132 ) 2] / [2x~2 I I cos/31 ] cos01 [(/1 cos/31 )2 + Xc2

and COS02 = [(/2 COSfl2) 2 + X22 - (/1 COSfll )2]/[2Xc212 COSfl2]

10.6 Data acquisition and SELSPOT calibration 241

/2 . . h

a l . C M 2

�9 ,, ~ 0, 02 / " /~ /2 sin I~ / l s l n p l v , t . . . . . . . . .

(a)

Fig. 10.4 Definition of quantities measured to permit SELSPOT data analysis

I_.. L j

(b)

and hence the direction vectors of the optical axes a l , a2 are given by

al = (c0s/31 cosOl, cos/3] sin01, sin/3]) and

a2 = (-cos/32 cos 02, cos/32 sin 02, sin/32)

Let the direction of the cameras' x and y-axes be denoted by hi, el and b2, e2 respectively. To find b we make use of the orthogonality relationship

a]bl + a2b2 + a3b3 = 0

where the subscripts 1, 2 and 3 denote the scalar components. Since b makes an inclination a with respect to the horizontal (see calibration step (d) above) this implies that

b3 = sin a

and therefore

b E + b 2 = cos 2a

since we have a right-angled orthogonal reference system. Substituting these relationships into the above orthogonality relationship gives

+a] (cos2 a - b22) u2 + a2b2 + a3 sina = 0

or, upon further rearrangement we have

(a 2 + a 2)b 2 + 2b2a2a3 sin a + (a 2 sin 2 a - a 2 cos 2 a) = 0

a quadratic equation in b2. Hence, the solution for b2 can be determined from

2 2 IX) 1/2 -2a2a3 sin a + (4(a 2 + a2)a 2 cos 2 a - 4ala3 sin 2 2(a 2 + a 2)

This leads to two possible solutions for b2 and hence b. However, a further condition that requires to be satisfied is the orthogonality condition

b x a = r

subject to c3 = cos a, but not c3 = - c o s a. Hence we have one solution for b2. The direction vectors b~, el and b2, c2 can all be determined by the above indicated

process. Consequently, the position vector x of a point having SELSPOT voltages V~ and Vy~ with respect to Camera 1 and V~2 and Vy2 with respect to Camera 2 would be given as


X = rl(al +fxlVxlbl q-fylVylC1 ) = r2(a2 +/,2 Gd,2 +/y2G2c2) + x~2

which is effectively a set of three simultaneous linear algebraic equations in each of the unknown scalars rt and r2. This redundant system can again be solved by the least squares method described previously.

Following the indicated procedure, the position of each of the four LEDs can be found from their SELSPOT readings.

10.7 Practical aspects Three different types of vessels operating offshore were tested during the experimental programme. These models included a series of tankers, a series of barges and a semi-submersible. The size of the physical models was selected subject to the limitations of the Newcastle and Edinburgh University tank sizes. A body plan of the basic tanker model is given in Fig. 10.5. This basic model was described as Tanker A. Three more models were obtained by removing different lengths of the parallel middle-body from the basic model. This results in a range of models covering a length-to-beam, L/B, ratio from 7.0 to 5.5 and block coefficient, Ca, values from 0.84 to 0.80. The segmentation of the tanker model is shown in Fig. 10.6. Table 10.1 provides the designation and geometric particulars of the tanker models tested.

A total of nearly 1000 experimental runs were completed. These runs form 22 sets of results for the ten different models tested. One set of results corresponds to one model at a selected draught, examined for one natural frequency of the mooring system in still water and regular waves.

Each set of runs consists of some eight to 12 wave frequencies and three different wave amplitudes for each frequency. Since the theoretical predictions of the low-frequency

t I / I I L / I I k ~ m = / B B B P J ~ r J ~ ' A I ~ E I I 1 1 1 1 1 1 1 1 1 1 1 1 D , z r ib ! !

! l l ~ l k ~ ~ l l b _ ' v " . ~ ~ ~ ~ ~ . . ~ l ~ ~ . B

Ip~4 _ _ .

I - 3

F 0 lh 1 I I 116 2 21/2

Fig. 10.5 Basic Tanker A--ship lines

, , , ~ 205 ~ 3 8 0 , ~ . , . . . . . ,,,1000 L , i , , , ,

h ' , I " , , ' , , ' _,i 2675mm

Fig, 10,6 Segmentation of tanker model

10.7 Practical aspects 2 4 3

damping coefficients were available prior to experimental testing the frequencies selected were those necessary to check the validity of the predictions. Frequencies corresponding to the peaks and troughs of the prediction were deliberately chosen to validate or invalidate the wave damping characteristics predicted.

To derive the measured low frequency wave damping a least squares fit to a straight line through the extreme values on the logarithmic decrement plot was sought. The fitted line was then used in the evaluation of the low frequency damping coefficient using the procedure outlined.

For each run, the wave condition and damping coefficients derived from the SELSPOT motion monitoring system and the mean value of the strain gauge measurements are given. The wave damping coefficient, bw, is given as the difference of total damping coefficient, btot, and the mean value of the still water damping coefficient.

The wave and motion records were passed through a discrete Fourier transform (DFr) algorithm to find their associated frequency and amplitude. Since sampling was performed at a fixed rate of 5.04 Hz and the wave motion frequency ranged from 0.5 Hz to 2.0 Hz, it was not possible to ensure that an integral number of wave cycles had been sampled. Hence, leakage was inevitable in the DFT spectrum. In order to determine the amplitude of the signal accurately, the following procedure was adopted. The centroid of the peak in the DFT spectrum was determined and taken as the frequency of the signal. A sinusoidal signal of unit amplitude and frequency was then generated and its DFT spectrum obtained. The amplitude of the original signal was then taken as the ratio of the area under the DFT spectral peak to that under the DFT spectrum peak for the unit signal. If the DFT had been implemented to produce an energy spectrum then the square of the ratio would have been required. Here, what we refer to as the 'DFT' is really the modulus of the discrete Fourier transform of the original discrete time series.

An important limitation concerning the Edinburgh Wide Tank is its short length and the non-perfect absorption of the beach. This means that the amplitude of the reflected wave onto the model tended to be quite significant. Measurement of the wave condition in the tank, in the absence of the model, showed that the reflected wave amplitude could

T a b l e 10.1 Principal particulars of tanker models

Length Draught 'kG LCG fwd Disp Model Load (m) (m) (m) UB CB midships (kg)

A Full 2.675 0.1595 0.112 7.04 0.839 0.0550 136.1 B Full 2.470 0.1595 0.112 6.50 0.826 0.0546 123.7 C Full 2.295 0.1595 0.112 6.04 0.813 0.0543 113.1 C Half 2.295 0.1276 0.121 6.04 0.795 0.0693 88.5 D Full 2.090 0.1595 0.112 5.50 0.795 0.0533 100.7

, i i i

breadth = 0.380 m; depth = 0.222 m' kG = vertical distance of the centre of gravity measured from the keel line of the tanker.


Surge motion time trace Tanker/A Full load Spring no., 51 Mean long. postn = -1.994m

Run no. 0007 Wave freq = 11.4rad/s Wave amp = 40mm Sampling freq = 5.04Hz

A -1.6 -1.7 4,d

-1.8

- / . ~ "o ID

~~ -,:311 "~-2 .4 ~ 0 20 40 60 80 100 120 140 160

~" -1.6 Time(s) "~ -1.7 = -1.8

~ �9

0 20 40 60 80 100 120 140 160 Time (s)

Fig. 10.7 Tanker A SELSPOT-based surge motion response

Surge motion (from strain gauge) time traces Tanker/A Full load Spring no., 51 Run no. 0007

Wave freq = 11.4 rad/s Wave amp = 40 mm Sampling freq = 5.04 Hz

1.3 .~ 1.2 ~ 1.1

1.0 _~ 0.9 o. 0.8 ._~ "o ~ 0.7 ~ 0.6 r 0.5

0 20 40 60 80 100 120 140 160

~-- 1"3 t E.E.. ~.2 m 1.1 G) E 1.0

0.9 �9 ~ 0.8

r 0.7 o~ 0.6 0.5

0 20 40 60 80 100 120 140 160 Time

Time(s) 11.~ A , . Strainga(uS~e2 "E" !311"Z- A I _ Strain gauge1 E"

1.0

=~ 0:6 ~ 0:6 0 VVVVV " o 0 tvv v " " 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160

Time (s) Time (s)

Fig. 10.8 Tanker A strain-gauge based surge motion response

10.8 Sometypicalresults 245

exceed 10% of the incident amplitude at the low and high frequency ends of the wave maker capability; that is, below 0.65 Hz and above 1.7 Hz. In cases of extreme low frequency, below 0.5 Hz, reflection could exceed 30% of the incident wave amplitude.

10.8 Some typical results A sample of the graphs generated for each set of runs is now presented. The sample consists of the time trace and the logarithmic decrement plot of the SELSPOT-derived results and the corresponding plots for the strain gauge results, see Figs 10.7 to 10.11 for a subset of Tanker results and Figs 10.12 to 10.14 for a subset of Barge results.

Figures 10.7 and 10.8 provide the surge motion responses as measured using the SELSPOT and mooring line strain gauges. In Fig. 10.7 both the unfiltered and filtered signals are presented. Similarly, Figs 10.9 and 10.10 provide the corresponding logarithmic decrement plots. Figure 10.11 shows that only low frequency surge motion exists at a single frequency, the other motions existing only as high frequency (first-order) responses. Figures 10.9 and 10.10 also show that the slopes are consistently predicted to three decimal places accuracy and the damping is linearly dependent upon the velocity.

Figures 10.12 to 10.14 show that the damping of the barge is quite large and, quite unlike the tanker, the motions quickly die away. The damping is also slightly non-linear.

0.40

0.35 "

0.30

0.25-

E 2 ~ 0.20 IID

0.15

Log dec plot for surge decay motion

Tanker/A Full load

Spring no., 51 Wave amp = 40mm Run no. 0007

Wave freq = 11.4rad/s Sampling freq = 5.04 Hz Slope = -0.07680

0.10 L 0 2 4 6 8 10 12

N

Fig. 10.9 Tanker A $ELSPOT-based logarithmic decrement plot


0.40: 0.351

,...,, 0.30~ E 0.25

E

x 0.20

O'J t,,.. 03 0.15

Tanker/A

Run no. 0007 Spring no., 51 Wave amp = 40ram Wave freq = 11.4 rad/s Sampling freq = 5.04 Hz

.. . "�9

P} 9 ,

O9_

Log dec plot for surge decay motion

Full load

)A -%

90 PO

90

0,10 Str~in~aug~ 1

) 2 4 6 8 10 12

'06 9,

00 0,

99

9�9 9, 0

t 90 A

0 , � 9

0.45 0.40 0.35

0.30

E 0.25

~, 0.20

03 0.15 Strain gauge 3

0.10 0 2 4

0.40 0.35

0.30

0.25

0.20

0.15

0.10

Slope, SG 1 = -0.07732 SG 2 = -0.07709 SG 3 = -0.07742 SG 4 = -0.07764

99: ~9

99~

' � 9

Strain gauge 2

0 2 4 6 8 10 12 N

%- FI I

I 900 ! PO o

9, 9 .

0

Strain gauge 4

6 8 10 12 0 2 4 6 N N

O o o

t 8 0 12

Fig. 10.10 Tanker A strain-gauge based logarithmic decrement plot

For the tankers, the theoretical low frequency damping is plotted in Figs 10.15 and 10.16 for frequency non-dimensionalized with respect to ship length, L, and ship beam, B, respectively. The amplitude of the first peak of the wave drift damping coefficient increases as we consider models A to D. This corresponds to an increase in the ship length to ship beam, L/B, ratio. The converse is observed upon studying the amplitude of the second peak of the wave drift damping coefficient. No general trend can therefore be deduced regarding the behaviour of the damping coefficients at higher frequencies. It is also interesting to note that the location of the first peak of the wave drift damping coefficient transfer function occurs at the same value of the non-dimensional frequency to(B/g) 1/2, at a value approximately equal to 1.1. That is, the peak's location is unaffected by any change in ship length. However, the location of the second peak of the wave drift damping coefficient occurs at a value of to(L/g) 1/2 of approximately 3.5 for all four models. Hence, we may conclude that only the second peak is a ship length related phenomenon.

In Figs 10.17 and 10.18, predicted and measured low frequency damping are

10.8 Sometypicalresults 247

Motion response spectra Tanker/A Full load Spring no., 51 Run no. 0007

Wave freq = 11.4 rad/s Wave amp = 40ram Sampling freq = 5.04 Hz

o iL . . . . . . . . . . . . . . . . surge I 0"25 Heave 0.085 Pitch 0.004

=_" 0 2011 0.004 | ]i ! o o ~0"15 0.003

~ ~111i ~ 0"00~0~~ ! + .,,.,,.,.d ,0.002

0" 0 2 4 6 ' 10 12' 1"4 ~" '0 24..=.~.~~dlJl~6 8 I 0"001 �9 " ~ - ~ ~o 12"~4"- o ooor- ' - " . , . , ' 0 2 4 6 8 10 12 14 rad/s rad/s rad/s

I I ! "'~' iMii l / l /

o ~ , ~" . . . . ~ ' ~ ' - o ~ . . . . , , , o = ~ , - , , o , . . 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

rad/s rad/s rad/s

Fig. 10.11 Six degree-of-freedom spectral response for Tanker A

Barge/A Run no. 0713 Wave freq = 9.70rad/s

-1.4

~" -1.5

-1.6

-1.7

-1.8

-1.9 J

0 20 40 60 80 100 120 140

.~ -1.4

-1.5

g, -1.6 e.-

= -1.7 g �9 -1 .8

-1 .9

Surge motion time traces and log dec plot Half load Spring no., 17 Mean long. postn = - 1.627 m

Wave amp = 4ram Samplihg freq = 5.04Hz

0.24 0.20 0.16 0.12

"~ 0.08 ! 0.04

Time (s)

Slope = -0.34558

A

0.00 0 " 2"0 " 40 " 60 " 8-0 "1()0 "120 " 140 0 2 4 6 8 N

Time (s)

Fig. 10.12 SELSPOT-based surge motion responses and logarithmic decrement plot

10

248 Experimental measurement and time series acqubition

0.26' 0.22 0.18 0.14

.EE 0.10 r E

0.06'

o3

0.02

0.26' 0.22 0.18 0.14

E o~o r

E O06

0.02

Log dec plot for surge decay motion* Barge/A

Run no. 0713 Spring no., 17 Wave amp = 4mm Wave freq = 9.70rad/s Sampling freq = 5.04 Hz

1

I.

I' , ~

i i l l l Strain gauge 1 I I I I I

0 2 4 N

.!,!

' i ~ . . . .

I. ,!, T l,

I i ~

,!1 I i gauge

, I I I I 0 2 4

N

Fig. 10.13

e

6 8

1 ! i 1 ....

I i !

"L o

6 8

Half load

Slope, SG 1 = -0.34329 SG 2 = -0.34915 SG 3 = -0.34636 SG 4 = -0.34361

I ! 1 1 Strain gauge 2

! I I 1 2 4

N

0.22] 0.181 0.14]

::] 0.02

~ ..

@,__

ii I "~

J l l _ 1 ! 1 1 [ Strata gauge 4 ,,! I I 1 t

0 2 4 N

luun

l i m a a l i b i

F

T ~

6 8

Barge A strain-gauge based logarithmic decrement plots

compared. The spread of the measured damping values is, in part, due to the non-identification of viscous damping and the problems of generating regular waves in a tank of the Edinburgh dimensions without there being interactions between the model and the wavemakers. A consequence of this programme of work was to develop a new theory which takes into account tank wall and tank bottom effects.

10.9 Final comments

The experiment discussed should demonstrate the important point that the method of analysis, whether the analysis be related to the recovery of the desired signal or the post processing of the sought signal, is an integral part of the design of the experiment. Furthermore the correctness of such analysis must be demonstrated. To do this in advance of undertaking the experiments we predefined the motions in each degree of

10.9 Final comments 249

A 0.20 ~ A g o.15 c 0.10 " AA 0.05 o~ 0.00 . A,., . A , , ~ . A ~. V 4 0 J "~80 \ 1 0 0 V I'20" " ._~ -0.05 60J lo | -o.lo Time Is)

-0.15 o3 -0.20

Surge motion (from strain gauge) time traces Barge/A Haft load Spring no., 17 Run no. 0713

Wave freq = 9.70rad/s Wave amp = 4rnm Sampling freq = 5.04Hz 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05

~176176 f V -0.05 20 40 60 80 100 120 -0.10 Time(s) -0.15

�9 Strain gauge 2 0 40 o1351t"

o~ iI A 0.15 0110

-ii i V--2o . . . . , o -

-o.~o t Time (s) -0.15t

E" 0.20t ~ Strain gauge 1

o.15| Ii = o,o,,,

~' o .oo l | - - o ~ -0.10|| ~ 0 V 4 Time(s)

-~176 V 6 0 J ~ \1 I )0X 'J120 /

-0.15[I

Fig. 10.14 Barge A strain-gauge based surge motion responses

1.6

1.4

1.2

I• 1.0

0.8;

g :D 0.6

~ 0.4 C Q. E

| 0.2

Tanker A - - - Tanker B -- -- Tanker C - - ' - - Tanker D il

ii 'ft

I

Wave damping coefficient Tanker model for experiment

Wave heading 180 Fully loaded

i,,~ ii i/:/ i', ~, il;I ! ii,1 ! i I

o.o ~ -: 2 ,, , 1 "C" " " ~'~~JJ, r~, - - 4 ~ ' 5 , i ~l '6

- 0 . 4 i .v '~

Fig. 10.15 Theoretical predictions of tanker models. Frequency scaled wi th respect to tanker length

4.0

3.5

3.0

I ~ 25 %,

i 2.0 J~

c _~ 1 5 r

~ 1.0 r-

E

"~ 0.5 (ID >

0.0

�9 Tanker A - - - Tanker B -- -- Tanker C - - ' - - Tanker D

Wave damping coefficient Tanker model for experiment

Wave heading 180 !~ Fully loaded / !

�9 ~'J~-- 1 ; ' I / i " 2," ' \ t 1/ ' i~!

-0.5

! % . /

-1.0 ' Fig. 10.16 Theoretical predictions of tanker models. Frequency scaled with respect to tanker beam

.=_ e,s

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0 .4

-~ 30 theory, 330 facet Expt - NCLE 13 s _Expt - EDIN'I 3 s

: _Expt- EDIN' 7.0 s Expt- EDIN', 23 s II Ii

!

I; II �9

0 0 O

i i

4) 0

1 --2== 3 4

cox/~g

$ $ �9 =

e Q �9

Wave damping coefficient Tanker model A for experirnent


5 ~6 7

Fig. 10.17 Compar i son of p red ic ted and measu red w a v e dr i f t damp ing coe f f i c ien ts for Tanker A


1.2 [ -~ 30 theory. 330 facet Expt- EDIN 14 s Expt- EDIN', 20 s

1.01"

Wave damping coefficient Tanker model B for experiment


~ 0.8

ut.P

.~ 0.6

r

0.4 r r

._. Q. E

"O | 0.2

0.0

-0.2

. |

1 "" 2- 311 4 5 / | 6

Fig. 10.18 Comparison of predicted and measured wave drift damping coefficients for Tanker B

freedom, and determined theoretically the resultant motion at the would-be locations of the LEDs on the vessel. Hence, with the LED related time series generated theoretically, the motions could be unscrambled and compared with the known constituent motions.

Another practical point is the fact that it may not always be convenient to obtain a nice suitable number of samples. Also, regarding the logarithmic decrement, one can see that the geometry of the structure determines how many oscillations are actually observed. Since the tanker is of a slender and deeper form it oscillates for quite some time once released. On the other hand, the barge is like a floating football field (at full scale) in area and is relatively shallow, hence its damping is significantly larger and therefore the number of oscillations is quite small. Whilst the experimental investigation of a semi-submersible has not been discussed, it was undertaken. The time traces of the wave makers showed that there was a significant interaction between incident waves diffracted by the semi-submersible and the wave makers generating the incident waves. Compari- son of predicted and measured wave damping in this case was difficult to make and so a new theoretical method was devised to take into account the finiteness of the tank and shallowness of the water depth (see Hearn and Liou 1990).

Within the literature we have also published comparisons of theory and experiment (see Hearn and Tong 1989 or Hearn 1989) and, at this level, the lack of confidence in the ability to produce very pure regular waves in the tank is quite crucial. The 10-15%


variation in wave amplitude over the location of the model implies a 20-30% variation in the second-order forces experienced and hence a similar variation can exist in the measured low frequency damping! Given the self-consistency of the slope of the logarithmic decrement and the less than 0.1% measured variation in the stiffness characteristics of any set of four mooring lines, the accuracy of the measured damping rests entirely upon the assumed wave amplitude. Hence, the in-built redundancy and the careful production and examination of the mooring lines allow confidence levels in the measurement technique. The lack of confidence in the quality of the waves therefore puts comparison of theory and measurement into perspective.

I I Experimental evaluation of wide band active vibration controllers

I I.I Introduction

The first step in evaluating any new ideas in engineering is to try them out on simple systems. If they will not work with a single degree of freedom system, there is little hope for more advanced applications. Computer simulations are a relatively quick means of doing this work, but they are inevitably idealized. No mathematical model can demonstrate all the nuances of a real system, and promising simulation results need to be followed up by laboratory experiments.

In this investigation we needed to measure the closed loop response of a test rig with different vibration controllers. The most basic requirement for any controller is that the system is stable. Only then does a steady-state response, which can be described by a transfer function, have any meaning. The transfer function could, in principle, be estimated by observing the responses to harmonic disturbances~a frequency sweep. This is a laborious way to proceed when a wide bandwidth has to be considered, and would severely curtail the amount of work that can be achieved during the time allowed for a project. The usual approach is to apply a pseudo-random signal and make use of the result of Equation (6.6). Spectral analysers, which provide test signals and analyse the responses, are common equipment in any dynamics laboratory.

11.2 Background There is a long history of passive devices, consisting of springs and dampers, being used to control vibrations in mechanical structures. However, active vibration controllers, which require external energy sources and electronic circuitry, have some advantages which may be crucial for specialist applications. These advantages include the potential for improved performance and the possibility of applying control forces with contactless electromagnets. The latter has been exploited by several research workers. Ellis and Mote (1979) designed an electromagnetic vibration controller for a circular saw, which allowed a thinner blade to be used and resulted in considerable economies. A high speed centrifuge on magnetic bearings was designed and built by Schweitzer and Ulbrich (1980) and has been further developed since then. Electromagnetic dampers for vibration control of transmission shafts have been evaluated on test rigs by Nikolajsen et al. (1979). The advantage of improved performance has led to research and development in the

253

254 Experimental evaluation of wide band active vibration controllers

motor industry. At least one manufacturer now offers an active suspension system on their production cars.

11.3 Techniques for active vibration control

Many techniques have been used to design active vibration controllers. If the spectrum of the disturbance is known, stochastic control theory can be used to design a controller which will minimize some chosen performance criterion (Grimble 1984). The special case of known deterministic frequencies can be treated by using delta functions for the spectrum. Alternative strategies, for example modal control, which make no assumptions about the disturbance and involve modifying the dynamics of the structure, have been widely used. In physical terms, the feedback system adjusts the effective stiffness and damping of the structure. A fundamentally different approach to the problem is that taken by Johnson in a series of papers dating from 1967, which are summarized by Johnson (1976). In these papers, Johnson deals with unknown disturbances which have a 'waveform structure'; that is, they can be approximated by functions where the coefficients occasionally jump in value in a random fashion. The special case of a harmonic disturbance at known frequency with unknown initial conditions was considered by Burdess and Metcalfe (1983). The same authors (1985) presented the design and experimental evaluation of a vibration controller for a single-mode structure which made no assumptions about the disturbing force. The underlying concept is to estimate the instantaneous disturbance with the aid of a device referred to as a 'disturbance observer', and to feed back an equal and opposite force. Figure 11.1 illustrates the idea.

i , 1 ~1 G(s) ' Y

'H'v,,, Fig. 11.1 Conceptual vibration controller

The physical plant is represented by its Laplace transform G(s). The 'disturbance observer' V(s) is an analogue electronic circuit which provides an estimate (g) of the net input to the system (e) from the measured output (y). It follows that

C(s) V(s) = W(s)

1 + kG (s) V (s)

and the naive strategy for disturbance rejection is to increase the gain k. Unfortunately, large gains tend to promote instability so there is a limit to this. They also amplify signal noise, and this is another detrimental effect that has to be considered. A limitation of most mathematical models is that they represent a complex physical structure by a finite number of modes. This may be adequate for many purposes but cannot be relied upon when it comes to stability calculations. The problem is that lightly damped high- frequency modes make a small contribution to the output signal which, when amplified and fed back, creates an unstable system (e.g. Balas 1982). It should also be remembered

I 1.5 Experimental rig 255

that transducers and actuators have their own dynamics which are often ignored when modelling the system. There are continuous mathematical models for simple idealized structures, such as beams, which may provide further insight into these instability problems, but are unlikely to provide accurate models of real systems. A consequence is that laboratory experimentation is an important stage between theory and practical implementation. Burdess and Metcalfe (1990) extended the idea sketched in Fig. 11.1 to two-mode structures. This case study describes the experimental evaluation of such a controller on a test rig consisting of two discs on torsionally flexible rods that were secured at one end (Fig. ll .2(b)). A simpler approach was to model the two-mode structure by a single mode and to apply the vibration controller of Fig. 11.1. This strategy was also evaluated.

11.4 Why use a spectral analyser? The spectral analyser provides an estimate of the transfer function of a system from measurements of the input and output. It also provides a convenient white noise source. It would be possible to estimate the transfer function directly by applying harmonic signals and measuring the input and output. However, this has to be done over the range of frequencies of importance, at a sufficient resolution to pick out peaks in the transfer function, and is a lengthy process. One objective of the research was to compare different designs of disturbance observer and it would not have been possible to complete the work in the time available without using a spectral analyser.

11.5 Experimental rig A test rig was designed to evaluate the performance of vibration controllers. It consisted of aluminium discs and torsionally flexible steel shafts (rods) which could be assembled in one or two-mode configurations. Two DC motors were used to provide the disturbance and control torques. The displacements of the discs were measured with contactless probes and the control signals were generated with analogue electronic circuits. The test rig could be assembled in three configurations: as a one-mode structure (C1), as a two-mode structure with the disturbance and control torques applied at the same point (C2), and as a two-mode structure with the disturbance and control torques applied at different points (C3).

Schematic diagrams of the three configurations are shown in Figs l l.2(a)--(c) and a photograph of the third in Fig. 11.3. Other design guidelines were that the natural frequencies should be around 10 Hz for the single-mode configuration and about 10 Hz and 30 Hz for the two-mode configurations. The central axis was vertical.

Frameless Alnico Magnet DC Direct Drive Torque Motors capable of delivering a peak torque of 10.3 Nm (7.0 lb ft - l ) were driven by servo amplifiers connected in a current amplifier configuration as described in the manufacturer's instructions (Inland Motor-Kollmorgen Corporation). The motors are designed so that the torque delivered is proportional to the current supplied. The amplifiers were factory set to give a current gain of 0.5 A V -~ and all voltages were kept below 15 V, which prevented the peak current of 10.6 A being exceeded. In order to investigate the frequency response of the actuators, accelerometers were attached at the end of a light, but rigid, bar, which was


(a) C1

Disturbance

(b) C2

Disturbance

(c) c3

Disturbance

Fig. 11.2 Configurations of test rig

C = Controller S = Sensor

itself attached to a bush which was fitted on the end of the motor shaft. A spectrum analyser was then used to estimate the transfer function from the voltage supplied at the input of the amplifier to the accelerometer output, and a typical estimate using the Hanning passband shape is shown in Fig. 11.4. The anomalous results at low frequency were attributed to noise in the signals from the accelerometers, which did not work well at low frequencies. There was no distinguishable difference between motor/amplifier pairs, and the estimated transfer functions suggested they could be approximated by

I 1.7 Final comments 257

Fig. 11.3 Test rig assembled in configuration C3

Fig. 11.4 Transfer function for motor

damped one-mode systems with a natural frequency around 800 Hz. The maximum frequency of interest in the experiments was 40 Hz and, within the range 0-40 Hz, the transfer function was approximately constant.

11.6 Some typical results

Figures 11.5, 11.6 and 11.7 show some typical results. DOI and DOII are two designs of disturbance observer and details are available in Metcalfe and Burdess (1990). All transfer functions are from disturbance to angular displacement.

I 1.7 F inalcomments

The performances of the controllers were limited by the gains that could be used without causing instability, or saturation of amplifiers. One possible mechanism for saturation is

258 Experimental evaluation o f wide band active vibration controllers

dB

10 20 Hz 0 I I ,

-10

-20

-30"10~ o .- '" " ' - - . .

,.4.=.: . . . . . . . . . . - S , : . - - - . ..~...:..~.........~...-...-T..;..~..,: ~.b-Tt : . . - - . : . . :..'~.. . . . . . . .

-40

Fig. 11.5 Controller performance on C1 no control--solid curve; DOI (lower gain)--circles DOI (higher gain)--dotted curve DOll (lower gain)--broken curve DOll (higher gain)--dashed curve

dB

20 40 Hz 0 " I ! . . . .

~ . . . . . .. ~. , \ .

-4o- - ;~"\~ _ ~

Upper disc

dB

-20

-40

20 40 Hz I l . . . . . .

" , , ,

-6O Lower disc

Fig. 11.6 Controller performances on C2: no control--solid curve; one-mode controlmbroken curve' two-mode controlwdotted curve

dB

0 20 40 Hz

. I 1 , .

-20

-40

-60 Upper disc


dB

20 40 Hz . . . . . J . . . . I ,

,%.....F, " "

-40

-60 : Lower disc

Fig. 11.7 Controller performances on C3" no controlmsolid curve' DOI--dotted curve; DOll (low frequency, lower gain)--broken curve; DOll (low frequency, higher gain)wdashed curve' DOll (high frequency, lower gain)--full circles" DOll (high frequency, higher gain)reopen circles

the following: the controller would be unstable if there were no limit to the control voltage available. However, the + 15 V limit on the control signal results in a non-linear controller that is stable. Another explanation is that saturation is caused by high amplification of signal noise. In all the stable controllers, a sensor was positioned on a plane, perpendicular to the axis, containing the point about which the control torque was applied. The controller, which was specifically designed for a two-mode structure, required an additional sensor. The ability of controllers to use signals from remote points is one of the advantages of active vibration control.

Controllers based on both DOI and DOll were extremely effective on the one-mode structure (C1) over a frequency range from 0 Hz to 15 Hz. The controllers were detrimental at higher frequencies, but the response of the structure at these frequencies remained negligible in absolute terms. The DOl l controller, based on the highest choice of damping, gave the best performance, which is in agreement with the theory. There was little difference between the performance of this controller and that based on DOI.

The controller designed for a single-mode structure gave a comparable performance to


that designed for the two-mode structure (C2) at low frequencies, provided the signal from the lower sensor, coincident with the point at which the torque was applied, was used. This was to be expected since the first mode predominates at these frequencies and the steady-state vibration is well approximated by a single-mode model. The simpler controller became unstable if the signal from the upper sensor was used. The controller designed for a two-mode structure was based on the ideas of DOll; no stable design based on the multiple input/output version of DOI could be found.

For the third configuration (C3), the controllers based on DOll designed for a single mode at the lower natural frequency gave considerably better performances than those designed for a single mode at the upper natural frequency. However, the controller based on DOI and designed for a single mode at the upper natural frequency did give a reasonable performance. The simple controllers designed for a single-mode structure at the lower frequency are surprisingly effective on this two-mode structure. However, it must be remembered that both the eigenvectors of the structure involve the motion of both discs. Also, the stability of the controllers appeared to depend on the measurement and control being applied at the same point.

Once a controller had been built, its performance did not vary noticeably from day to day. However, different realizations of the same circuit diagrams could lead to variations in performance. This affected the 'tuning' of the controllers, but did not qualitatively affect the results shown.


12. I Introduction The other case studies have been concerned with the spectral analysis of time series data. However, random signals associated with spatial variations are equally important. Here, yet another marine application is considered, in which the smoothness of paint finish was being assessed. This subject is of considerable importance to ship owners, since a poor finish can increase fuel consumption costs by about 10%. It is also of great interest to paint manufacturers who continually strive to improve their product and remain competitive in an international market. The amplitude of ripples is not the only significant feature, the wavelength also plays a part in the resistance of the ship hull.

The ideas of spectral bandwidth and the moments of the spectrum are essential concepts in the associated theory of surface metrology as this case study will illustrate. Similarly, it should be noted that the concept of the correlogram is transformed to a 'structure function', which has some more desirable properties.

12.2 Background

A large proportion of the total resistance of some large, slow, merchant ships is due to viscous effects. This fact may surprise some readers, because in looking out to sea we are generally only aware of the effects of wavemaking resistance. As its name suggests, wavemaking resistance is a consequence of the wave generation which takes place as the ship advances through the surface of the water. Classical hydrodynamics tells us that the flow around the ship hull can be separated into a boundary layer, close to the hull, in which viscous effects dominate, and an outer potential flow in which the viscosity of water has no significant influence. The motions of ships and offshore structures, discussed in the earlier case studies, are predominantly of a potential flow nature with the viscous nature occurring in the form of additional damping. Here viscous effects predominate.

In linking surface metrology and ship resistance there is a need to divide ship hull geometric irregularities into at least two categories. There are those geometric variations that cause changes in the pressure around the hull, but do not otherwise affect the inner viscous boundary layer. Such variations may be designated 'structural roughness', whereas those variations that cause disturbance of the fluid layer near the hull surface, but do not greatly affect the surrounding pressure distribution, may be designated 'surface roughness'. Another way of appreciating these influences is to think of how the

261


wavelength of the geometric hull form changes, and how the wavelength affects the hydrodynamics.

At the long wavelength end of the spectrum we encounter the nominal shape of the hull itself. The boundary layer and the external potential pressure distribution both exist because the hull is present and moving through the water. The next commonly acknowledged wavelength is the often called 'starved horse effect'. That is, the periodic undulation of the hull form between frames, caused by the expansion and contraction of the steel shell plating during welding on the berth during construction. This too is generally treated as an inviscid flow phenomenon.

An order of magnitude smaller is the problem of vertical weld beads, generally referred to as a 'step roughness'. This may be, and usually is, treated as wholly a boundary layer phenomenon. Physicists have given considerable attention to step increases in roughness. Here, we suggest the weld bead step should be treated separately if it cannot be ground back to the surface level prior to painting the hull.

Clearly this brings us to the categorization and nature of micro-topographical surface features. Once again there is a range of associated wavelengths. In this case high frequency will relate to such features as pitting, orange-peel texture and overspraying. Such features may exist on a scale magnitude of several miUimetres down to a few tens of a micrometre. At the low frequency end, defined by wavelengths of say 25 mm and longer, plate waviness may have some influence on roughness. However, such features are generally caused by variations in the film thickness of the paint. Since the nominal film thickness of an antifouling coating may be around 150 ~m there is plenty of scope for such roughness, due to the volume of paint per unit area applied, and the flow of the paint on making contact with the hull before drying. Thus, the scale of roughness which affects frictional ship resistance lies in the range 50/zm to 5/~m. As was the case for the previous case study, the process is wide banded.

12.3 An introduction to surface metrology

In the field of mechanical and production engineering, spectral techniques have found a direct application in the analysis of the surface texture of a wide variety of machined components and also of materials as diverse as marine antifouling paints and articular cartilage. However, random techniques are still 'new' in the sense that they have not yet been embodied in any national or international standards covering surface metrology. None .',he less they are in extensive use in industry and this aspect is well covered in Thomas's (1982) book, which is rapidly becoming adopted as the standard reference work. Spectral analysis related software is now available as a standard option on some of the better laboratory bench stylus surface profilometers.

Engineering surfaces are, of course, three-dimensional artefacts. Whilst it is possible to carry out three-dimensional surface measurement and characterization, this is usually over a small area. The usual technique is to measure a number of adjacent profiles using a stylus machine and to move the specimen laterally a few micrometres in between each traverse. In this way a three-dimensional picture of the surface can be built up on an x - y plotter, and three-dimensional characterization can be carried out using techniques initiated by Cartwright and Longuet-Higgins (1956) for sea waves, and adapted to surface metrology by Nayak (1971).

/2.4 Process bandwidth 263

Until recently, three-dimensional surface profilometers have been the instruments of research establishments, but now they are beginning to become available commercially. However, the technique of surface analysis used almost universally is the measurement and characterization of individual surface profiles. This simplified analysis works well enough if the surface is approximately isotropic and also Gaussian in any direction. If the surface is anisotropic the general theory of Longuet-Higgins (1956) may still be applicable, but five non-parallel profiles will be required to extract the necessary surface parameters. The application of such theory to engineering surfaces is generally limited to contact mechanics and is very well covered in Thomas's (1982) book.

Traditionally, surfaces were characterized by roughness height parameters alone. The British Standard (BS 1134:1972 (revised 1990)), for example, then listed only two parameters, Ra (the mean roughness height) and R~ (the 10 point height). Arguably, these two parameters (or similar ones such as Rq, the root mean square (RMS) height and Rt, the peak-to-valley height) are quite sufficient to characterize the amplitude of the rough surfaces. However, it has long been recognized that two surfaces with similar roughness heights can have very different 'textures' and thus behave very differently in a wide range of engineering applications. It is also now well established that 'texture' is well represented by spatial parameters such as 'average slope' or 'wavelength' and it is this recognition that led to the introduction of spectral techniques into surface metrology.

In production engineering, little use has been made of the periodogram. Production engineers have tended to favour the correlogram, a plot of autocorrelation function (acf), autocovariance function (acvf) divided by variance, against lag. Traditionally, an exponential model was used to describe the acf. The parameter extracted from the acf was/3", the correlation length; that is, the lag over which the acf decays from unity at the origin to some constant value c, where c satisfies 0 < c < 1. Usually, c is selected to equal 0.5, l/e or 0.1. This fitting of a model led to difficulties when transforming the acf to obtain the spectrum. In recent years, digital computation of the acvf has taken over and emphasis has been placed on the 'structure function'. This function presents the same information as the correlogram, but in a slightly different form

S (~-) = 2[o 2 - ),(r) ]

This representation has the advantage that graphical realization of differences in both roughness 'height' and 'texture', as a result of an industrial process, can be achieved on a single plot. Examples of the use of the structure function for industrial surfaces 'before' and 'after' grinding are found in Thomas's (1982) book.

12.4 Process bandwidth

With the exception of artificially produced regular waveforms, surface profiles, whether naturally occurring or produced by some form of industrial process like ship surface painting, are broad-banded and non-stationary (see Chapter 3). That is, all wavelengths are likely to occur and samples taken from one area of the structure cannot necessarily be taken as representative of all other areas. At the long wavelength end of the spectrum, the overall shape of the component itself (be it a bolt or a ship) will influence the spectrum, assuming the profile can be made long enough. At the short wavelength end, surface texture can be detected down through the scanning electron microscope (SEM) scale to the molecular level. Because of this non-stationary characteristic, it is necessary


to define precisely the band of wavelengths that is of interest and to filter the profile carefully before attempting to apply any surface characterization techniques, whether they be characteristics of the profile height distribution or of wave-number distribution. In general, the choice of long wavelength cut-off will have the greatest effect upon height parameters, and short wavelength cut-off upon spatial parameters. The choice of wave-number cut-offs will depend upon the purpose for which the component whose surface is being measured is intended. The term 'functional filtering' has been coined for this process of matching bandwidth to component usage and has become one of the most important concepts in surface metrology and characterization.

Application of spectral analysis to surface metrology has, not surprisingly, centred on the first three even moments of the spectrum and the so-called spectral width parameter, e. The physical significance of the moments is readily visualized in the case of static surfaces. That is, mo represents the square of the RMS roughness height, and m2 and m4 the variance of the distribution of slope and the curvature respectively. Thus, e contains a great deal of information which may be of interest to surface metrologists. However, it has been found that e is extremely sensitive to method divergence and, in particular, to different methods of filtering the surface profile. It will be argued below that, in at least one application, the inclusion of m4 in any texture parameter is superfluous and can lead to parameter instability.

12.5 Surface topography and fluid drag For some years now, the north east of England~through the British Ship Research Association (BSRA), now part of British Maritime Technology (BMT), International Paints, and the Universities of Newcastle upon Tyne and Teesside~has been one of the main centres for the investigation of the effects of surface topography on fluid flow. This topic has profound significance in almost every branch of fluid mechanics, from power station cooling culverts to ship performance~which forms the subject matter of the rest of this section. A very readable account of the state of the art on this topic has been presented by Townsin (1987).

To this day, the effect of irregular surface topography on fluid flow is poorly understood, but in recent years a number of semi-empirical correlations between simple measures of surface topography and the so-caUed 'hydrodynamic roughness function' (also known as the 'velocity loss function') have been proposed. The initiative for much of this work has come from ship research, since some 80--90% of the resistance of large tankers or bulk carriers is of a viscous origin and therefore the fuel bill of such a ship may be increased by 20% as a result of a moderate degree of underwater hull roughness. This roughness may be due to mechanical damage, corrosion, poor initial preparation of the hull surface prior to the painting, and/or careless application of the paint. Organic fouling has an even more disastrous effect on performance. The fuel price rises of the early 1970s led to a revolution in the attitudes of ship operators towards hull condition and the development (on Tyneside) of a revolutionary self-polishing antifouling paint. This substance 'wears away', for want of a better expression (the exact mechanism of 'polishing' is still not fully understood), as its antifouling powers become exhausted, revealing a fresh layer of toxin underneath. As a result, ships are kept reasonably smooth (except for mechanical damage due to contact with anchor chains, docks, tugs, etc) and can stay out of dry-dock for up to 3 years or more without becoming fouled.

12.5 Surface topograph 7 and fluid drag 265

The study of the hydromechanics and surface topography of these self-polishing paints is a particularly rewarding exercise because not only do we learn about the effect of the surface topography on fluid flow, but also about the effect of the flow on surface topography (see Townsin and Dey 1990). The latter study is rewarding for the following reasons. Surface roughness is a broad-banded phenomenon and there is no intuitive method of deciding which range of wavelengths is relevant to the study of fluid flow. An incorrect choice of 'cut-offs' may lead to the inclusion of wavelengths which dominate the calculation of the surface characteristics but have no influence at all on the resistance to fluid flow.

For the last 30 years, the universal measure of ship hull roughness has been the maximum peak-to-valley height in a 50 mm sampling length. This measure was developed by BSRA, in the early 1950s, using smoked glass slides upon which a magnified stylus trace was drawn by a hand-propelled tracked vehicle known as the 'wall gauge'. The peak-to-valley height parameter extracted manually from the slides was called 'mean apparent amplitude' by Lackenby (1962). In the late 1970s the wall gauge was superseded by the 'hull roughness analyser'. The single parameter measurement Rtso, defined above, now enjoys almost unrivalled use worldwide as the only practical means of measuring a sufficiently large proportion of the wetted surface area of a ship in a single day (up to 30000 m) to have any statistical significance. The cut-off of 50 mm has been maintained. The short wavelength resolution of the machine is defined by the stylus diameter of about 1.56 mm. This, in turn, is fixed by the constraints under which the machine has to work. In order to enable it to be used on the flat bottom of a ship in dry-dock, some force has to be applied to the stylus to ensure positive tracking and a large stylus diameter is necessary to reduce tracking pressure and avoid damage to the paint.

The long wavelength cut-off of 50mm was originally chosen by BSRA because it appeared to the 1950s' experimenters that the spectrum derived from the surface measurements using the wall gauge had a peak at about 50 mm. It is now certain that any such peak was a function of the sampling method used and that wavelengths of 50 mm are largely irrelevant to the response of typical ship hull surfaces to fluid flow. However, the primary function of Rtso for the ship operator and paint manufacturer is to maintain a check on the standard of maintenance of the hull and, for this purpose, Rtso appears to be an excellent choice of single roughness parameter. Histograms of between 10 and 12 Rtso measurements from up to 100 locations on a ship's hull were found to be well modelled by a single Gumbel extreme value distribution when the hull is new and well painted. Alex Milne, who carried out this work and is the inventor of the self-polishing paints, remembers quite vividly being advised to use a Gumbel distribution with the roughness data. He was at the bottom of a dry dock in Portugal making measurements of hull roughness in the rain. No one else was working on the ship. Suddenly from nowhere a man appeared and asked him what he thought he was doing. Alex replied 'Who are you?' 'The ship owner', was the equally short reply. Alex therefore gave him his full attention.

Given the very bad weather they both went aboard the ship to address the original question. Alex and the ship owner embarked upon a discourse concerning what exactly the hull roughness gauge was measuring. To Alex's surprise the owner postulated that the measuring device was not measuring average roughness at all, but the maximum deviation from the mean in 50mm transects, and hence he (Alex) should look at Gumbel's publications. This chance discourse, on a rather foul weather day in Portugal


in February 1979, was significant in correcting the then existing naval architects' approach to roughness measurement. Previously, normal distributions had been assumed despite dubious fits of the data. This redirection of the thought processes also led to the later strategy of fitting a mixture of two Gumbel extreme value distributions to data from ships that were not new and had therefore seen some service. The self-polishing paint applied to such ships was designed to become smoother as a result of the flow of water over the hull resulting from the forward speed of the ship. However, certain areas would have been subject to bad painting or mechanical damage and hence would be rougher before the special self-polishing coatings were applied. The interpretation of the mixture of two Gumbel distributions was that one represented the smoother patches and the other the rougher patches of the hull surface. Who was the ship owner? Dr Christopher W.B. Grigson, an engineer and one time academic. For further general technical details and information on the fitting of the two Gumbel distributions see Townsin et al. (1981).

The first serious attempt to find alternative hydrodynamically significant cut-offs was undertaken by Musker at Liverpool University. Musker (1977) proposed the use of 50/zm and 2 mm respectively, based on a correlation between measurements of the surface topography and velocity loss functions of five ship hull replica surfaces in a pipe, using air as the working fluid. The choice of cut-offs was justified by measurement of the micro and macro scales of turbulence, using a hot wire anemometer. The choice of short wavelength cut-off could also have been justified by reference to the region close to the surface known as the 'viscous sublayer', which is related to the surface shear stress, fluid density and viscosity and which, for typical ship hull flow, is of the order of tens of micrometres in thickness. The choice of the long wavelength cut-off requires more complex arguments and one possible approach will be outlined below.

Musker (1977) proposed a hydrodynamically significant four-parameter measure of height and texture, the most important spatial parameter of which was average slope. Subsequently, Byrne (1980) at the University of Newcastle upon Tyne, using Musker's data, proposed an alternative measure of height and texture using Rtso as the amplitude measure and e as the texture parameter. He later extended this approach to propellers, using a shorter long wavelength cut-off (see Byrne et al. 1982).

Byrne also extended Thomas's concept of an inverse square law spectrum into ship hull roughness work. He fitted a number of spectra of hull roughness profiles to a general inverse power law, namely

Fe(y) = 4zrA y - N

Using the fitted spectra the spectral moments could be calculated analytically. Thomas had used this concept to define the concept of 'topothesy', the rate at which roughness increases with long wavelength cut-off for a given surface. This may be expressed mathematically in the form

or = trr + trt = trt + ( k L ) - 1 I N

where o-~ and o't are loosely defined as RMS 'roughness' and 'texture' respectively and are linearly dependent over a limited bandwidth (Sayle and Thomas 1978). The parameters k, L and N are curve fitted. Recent work on ship hull roughness in Norway has defined a combined height and texture measure of hydrodynamic roughness using Rtso and R~ as the 'height' and 'spatial' parameters respectively (see Kauczynski and Walderhaug 1984). This is a direct application of the inverse power spectrum/topothesy concept.

12.6 Measures of texture 267

Recent work at the University of Newcastle upon Tyne has centred on the use of [3,*, which is shown to be strongly correlated with average slope and peak count wavelength, and the ratio of the spectral moments m2 to m0. The most recent work at Newcastle has examined all available experimental data where an adequate defined surface roughness has been published, together with measures of the consequent added drag. The influence of long wavelength cut-off is clearly shown and correlation between added drag and the first three even spectral moments has been attempted by Townsin et al. (1957).

12.6 Measures of texture

The proliferation of different measures of texture is hardly surprising in a subject about which so little is yet known and for which there are, as yet, no agreed standards. The large number of measures of texture in use in rough surface hydromechanics may at first seem difficult to justify until one realizes that they are all, in fact, closely related. The overriding consideration when specifying the use of height and texture parameters is not the choice of parameter itself but careful definition of the choice of long and short wavelength cut-offs and the method of filtering used.

One use for random process analysis is to define the appropriate cut-offs and evaluate the performance of filters. The two activities are closely related. Arguably the most difficult task is to specify the appropriate long wavelength cut-off at which to define hydrodynamically significant roughness parameters. Some years ago, a joint venture was organized between a marine paint manufacturer and a shipping company to send a number of test panels coated with self-polishing antifouling paint to sea, bolted to the keel of a large tanker. By locating the test panel accurately on the bed of a bench surface profilometer before and after a period of 9 months in service, it was possible to digitize nominally identical surface profiles before and after interaction between fluid flow at ship scale and the self-polishing paint. These profiles were subsequently subjected to spectral analysis. An example is shown in Fig. 12.1. A short length of each profile is shown as Fig. 12.1(a) and a comparison of the spectra is provided in Fig. 12. l(b). The total length of the profiles was 20 mm and they were not subjected to any filtering other than limitation of the profile length and setting of the digitizing interval at 50 ktm. It will be observed that most of the 'power' removed from the spectrum has been extracted over a fairly narrow wave-number band, corresponding to wavelengths of between about 250/~m and 2 mm.

Thus, the ablative properties of copolymer antifoulings and the power of spectral analysis combine to give a valuable insight into the hidden workings of fluid mechanics. However, further evidence is needed before restricting the hydrodynamically significant bandwidth to a cut-off as small as, say 2.5 mm. As a new approach to this problem, we began to look at the cumulative integrals of the spectra of painted surfaces. It has been known for some time that surface features which have very small slopes do not contribute to hydrodynamic drag. If gradients are small, the boundary layer is able to follow the surface contours without becoming perturbed. There are a number of studies in the literature by means of which the relationship between the amplitude-to-wavelength ratio of a sinusoidally varying surface and its fluid drag may be quantified.

The cumulative integrals of the spectrum are usually defined as

f YN m. = y"F~(y) dy

o


(a)

250

~" 200

150 . m

t -

| 100 ._>

| 50 n -

O

0.0 2.5 5.0 Length (mm)

7.5 10.0

Total profile length analysed = 25 mm Digitised interval = 50pm

125

E E 100

E

c .g 75

-~ 50

i. ~ 2s

.

o ~o 20 ao 40 5o 6o 7o Wavenumber (rad/mm)

(a) Surface profile before and after 9 months at sea; (b) comparison of spectra Fig. 12.1

Before

After

where here YN is the Nyquist wavelength. We may also introduce the partial integrals

t I y' m . = m . ( y ' ) = y"F~(y) dy o

where y' may assume any value between zero and YN. These functions were given the

12.6 Measures o f t e x t u r e 269

formal symbol m.(y') because they are functions of wave-number, but they are usually abbreviated to m~, m~, m~, and e', where

~ r . _ ,n~m~

m~ 2

Figure 12.2 shows the profile trace, the spectrum and the cumulative integrals of a typical surface profile from a self-polishing copolymer surface after 9 months at sea. Although surface profiles are 'non-stationary', the power in the spectrum diminishes

450 400 350

(a) Profile

450

400

350 E E 300

250 v

~ 200

o 150

100

50 0

10

Spectral moments direct method

mo = 762.16 m2 = 2.0840 x 10 -2 m4 = 1.3892 x 10 -5

~, = morn4 = 24.38 mJ

20 30 40 "Frequency" wavenumber y = (2~/k rad/mrn)

(b) Spectrum

50

1.0

~ 0.8

C o ~

._> 0.6

E = 0.4 r

~ 0.2 o Z

0.0

. . . . m6

... . . . m~

. . . . . m; E e

s t ~ " - - - - oooo /~ p

t ." I: /

t oo , i

, ' .-'" iE

/ ,,, / ! ,; 1' :

,," / i

,'" I" ! *'" 1" /

~ '~

. . . . . L . . . . . . . . , " ~ " " �9 . . . . . . , , - - ; 1 ' ' " " ~ " I I

10-1 i i 5 6 7 8 9 1 0 0 2 ~1 4 5 6 7 8 9 1 0 1 i :3 ,1 i a"/i9102 Yc "Frequency" wavenumber y = (2~/k rad/mm) YN

(c) Integrals

Fig. 12.2 (a) Profile, (b) spectrum and (c) cumulative integrals


Power spectral density function (PSDF) - smoothed FFT periodogram Sampling interval = 50.00 lam. 18 degrees of freedom. 1000 points + 24 added zeros.

400~ ~ ! i

300 ~ .A A i . J ~ J-., l _ . ^J j . J , . , I f " - - - '

L . ' v , y . . . . , ] - -V ~,~Oli . . . . . 11 l ' ~ ' ' ' ' ' ' " I

0 5 10 15 20 25 30 35 40 45 50 ~<E=200 Length scale (mm)

~ 150 ,~

,,

, i . . . . . . .

0 Yc 10 20 3'0 40 5'0 .... 6'0 YN "Frequency" wavenumber,y = (2n/Z.) rad/mm

(a) Trend removal

Power spectral density function (PSDF)- smoothed FFT periodogram Sampling interval = 50.00pm. 18 degrees of freedom. 1000 points + 24 added zeros.

r . . . . . . . . . . . . . . . . . .

150

I .

125 -- '%L.~ t - .~ :~ , .~_:~.~:: l . , . - ,--- : , . ~.=-~

, I

100 E l_a]~Jl. ,^ ..J .._ ~ I JL . . L= ~ l .,l~_l AJ._,~. L '~L L ,, ,~ = :r,, 1~ '~'' ' ' ' ' * - ' ' ~ " ~ " , " , , ' ~ " ' " 1 ~

75 !1 6 5 ' 10 is 20 25 3'0 .... 35 " 40 45 50

~ . Length scale (mm) ~ ~~

25 i

0 . . . . . - . . . . . . . , . . . . . . . . . . i 0 Yc 10 20 30 40 50 60 YN

"Frequency" wavenumber y = (2n/Z,) rad/mm (b) Moving average

Fig. 12.3 (a) Trend removal; (b) moving average filters; (c) weighted moving average; (d) infinite impulse response (IIR) fi lter

sharply at the high wave-number end. It should be noted that reducing the digitizing interval does not alter this trend. Since m~z represents the variance of the profile slope distribution, then the plot of m~ should give an indication of those wavelengths that contribute to hydrodynamic drag. It will be seen that at low wave-numbers, m~ is insignificant, and in fact it does not begin to acquire any appreciable value until a wavelength of about 2 mm. It then grows almost linearly with log(y) from a wavelength

12.6 Measures of texr:ure 271

Power spectral density function (PSDF) - smoothed FFT periodogram Sampling interval = 50.00 pm. 18 degrees of freedom. 1000 points + 24 added zeros.

0 Yc 10 20 .... 30 4() . . . . . . . 5'0 60 YN

= l o o " v ' :'v-,'" g 5 - r l 0 " ~ , ,~ ; v- 0 30 35 50

Length scale (ram)

"Frequency" wavenumber y = (2niL) rad/mm (c) Weighted moving average

Power spectral density function (PSDF) - smoothed FFT periodogram Sampling interval = 50.00pro. 18 degrees of freedom. 1000 points + 24 added zeros.

0 0 5 10 15 20 25 30 35 40 45 50 ! , _ _

50 Length scale (mm)

25

0 " ,~ , , a J I _ ~ _ i

0 Yc 10 20 30 40 50 60 YN "Frequency" wavenumber y = (2niL) rad/mm

(d) Infinite impulse response (IIR) filter

Fig. 12.3 (cont.)

of about 1 mm. On the other hand, m'4, which represents profile curvature, remains insignificant until the spectrum has attained 80% of its power at a wavelength of about 1 mm, whereafter it increases sharply. This suggests a number of things. First, it is probably safe to adopt a long wavelength cut-off of 10 or even 5 mm for this type of surface, since at longer cut-offs the height parameters will be dominated by power from wavelengths that have negligible slope and are therefore not hydrodynamically


significant. Second, m~z would appear to be a stable parameter which can be used with confidence at any short wavelength cut-offs provided that the cut-off is accurately specified. Third, surface features at wavelengths of 125 ~m and less contribute very little to the power in the spectrum and can probably be ignored as being so small that they are submerged in the viscous sublayer. Fourth, curvature is only significant at similarly high wave-numbers and can probably be ignored altogether for the same reason. Finally, both m4 and e are unstable parameters and are best avoided.

This analysis also lends some force to the current trend of looking for a single, stable texture parameter based on average slope or wavelength and to avoid multi-parameter measures of hydrodynamic texture. It will also be appreciated that, whatever standards are eventually adopted in this field, the accurate specification of cut-offs and standardization of filters is absolutely essential. Therefore, filtering is now briefly considered.

12.7 Filtering and filter assessment

Surface metrologists have traditionally been content to rely for short wavelength filtering upon specification of the digitizing interval. The stylus radius can also act as a short wavelength filter. In the case of a bench stylus machine having a truncated pyramid diamond stylus, typically 3/~m x 8 ttm, the measurement of a painted surface at a digitizing interval of 25 or 50/xm is likely to have an attenuating effect on the short wavelengths. It has recently been shown that even the 'hull roughness analyser' ball stylus (1.56 mm diameter) removes very little of the short wavelength power from painted surfaces because the slopes are so small (see Byrne et al. 1982). It is common practice to adopt the selection of a digitizing interval as the only short wavelength filter used, although there may be arguments in favour of using a three-point moving average as well.

At the long wavelength end, mechanical filters (skids) were and are still used. For the electrical bench stylus machines of the 1960s the 2CR filter was commonplace and is still in use. Its cut-off characteristics are precisely defined in BSl134. Filtering by limiting the length of profile and fitting a single least squares reference line through each cut-off length is the simplest and most common digital method of long wavelength filtering. Unfortunately it does not work very well (see Fig. 12.3(a)). As an improvement to this method, engineers have tended to favour the use of polynomial centre-lines. The problem here is that it is difficult to know what order of polynomial to use. Also, the polynomial function will hardly ever be an intrinsic function of the long wavelengths in the profile. Recent work at Newcastle University has tended to favour methods taken from signal analysis. Moving average filters are the easiest to apply, either unweighted (the simple 'Boxcar' moving average) or weighted with a suitable function (often sin(x)/x). The disadvantage of these methods is that a large window width is required and thus a large length of carefully collected profile at the beginning and end of each trace is effectively thrown away. Computational complexity can also be a problem and here the simple Boxcar moving average filter gains by an order of magnitude.

One particular method which we have found lends itself well to surface metrology work is a cascaded phase-corrected bilinear z-transform filter described by Beauchamp and Yuen (1979), known more concisely as the Infinite Impulse Response (IIR) Filter. This is a recursive filter and uses very few ordinates. It thus benefits from both narrow window width and comparative computational simplicity. Phase distortion is avoided by

I Z 8 Final commenrs 273

passing the profile through the filter twice, forwards and backwards. Only first and second-order filters are implemented as higher orders are achieved by cascading.

Spectral analysis is extremely useful for evaluating the performance of filters. By carrying out a Fourier transform of both the filtered profile and the rejected waveform, the cut-off characteristics can be examined closely and the filter carefully adjusted. In fact, some of the more sophisticated filters produced by signal processing technology incorporate recursive analysis of the Fourier transform of the signal into the filter itself. This level of sophistication is not thought to be necessary for surface metrology work and fixed filters are almost certainly more than adequate. For fixed filters, spectral analysis can be used to help in choosing attenuation rate, window width and to examine phenomena such as side lobes and leakage.

Figure 12.3 shows a comparison of the application of four filters to the profile of Fig. 12.2. The first consists of fitting straight lines through each cut-off length only. It can be seen how extraordinarily inefficient this method is. The simple Boxcar moving average works surprisingly well but suffers a little from leakage. The weighted moving average has the best performance of all, but at the expense of a very large window width. The IIR filter has the best combination of performance and window economy and is reasonably simple to program. Whichever method of filtering is adopted, its parameters should be carefully specified and rigidly adhered to.

12.8 Final comments

This study has shown how the ideas of spectral analysis have been transferred in recent years to a discipline which traditionally did not view its data as random. The problems of wide-bandedness and non-stationarity demonstrate, yet again, that to transfer the concepts of spectral analysis to realistic applications requires a lot of effort. One also has to consider the role and application of the technique in the context of the subject matter, a factor borne out in all the case studies presented.

The importance of cut-off frequency and its unstabilizing effect upon m4 have been demonstrated. The same problem occurs in water wave spectra generated from time series, where the high frequency cut-off can significantly affect the high frequency tail of the spectra and hence the higher order moments. Spectral analysis has also been used here to assess the appropriateness of differing filtering techniques.

The state of the art in terms of internationally accepted ideas can be determined by reading the proceedings of the International Towing Tank Conference held in Madrid in September 1990. Certainly it is now accepted that a correlation between added drag and ship hull surface statistics does exist, and it is important to understand this if ships are to be operated economically.

Appendix I: Mathematics revision

The main text assumes a knowledge of elementary mathematics, including calculus, statistics and probability. The appendix explains and revises some of the results which are crucial to the main argument and which the reader may have forgotten since a first acquaintance, or even managed to avoid meeting during his or her mathematical studies. Calculus and elementary statistics are not, however, included. We expect most readers will be frequent users of the former, but if you are not the book The Essentials o f Engineering Mathematics by A. Jeffrey will help you bring your calculus up to standard in a relaxed manner. A good background in the necessary statistics and probability can be obtained by reading A Basic Course in Statistics by Clarke and Cooke (1992).

Arithmetic series

An arithmetic series is a sum of terms which differ by a constant amount d. If we write a for the first term and S,, for the sum of n terms, then

S ~ = a + ( a + d ) + ( a + 2 d ) + . . . + ( a + ( n - 1 ) d )

To find S,, in terms of a, n and d is straightforward once you realize the 'trick' of writing it in the opposite order and adding corresponding terms which will all give 2a + ( n - 1)d. That is

Sn = a + (a + d ) + (a + 2d) + . . . + ( a + ( n - 1)d)

s . = (a + ( n - 1)d) + (a + ( n - 2)d) + (a + (n - 3)d) + . . . + a

which add to give

2S,, = (2a + (n - 1)d)n

Therefore

(2a + (n - l ) d ) Sn = n

2

Since (2a + (n - 1)d) is the sum of the first and last terms, S,, is the average term multiplied by the number of terms.

Geometric series

A geometric series is a sum of terms which differ by a constant factor ratio r. If we write a for the first term and S,, for the sum of n terms, then

Sn = a + ar + ar 2 + . . . + ar ~-l

To find S,, in terms of a, r and n we multiply both sides of the above equation by r and subtract term by term, to eliminate all terms but the first and last. That is

S,, = a + ar + ar 2 + . . . + a m - l

274

Taylor series 275

and

rSn = ar + ar 2 + . . . + a m - l + ar n

Subtraction gives

Sn - rS , = a - at"

Hence

a ( 1 - r n) ' l - -

1 - r

If the terms of a general series decrease to zero as n tends to infinity there is a possibility that their sum to infinity may be finite. The expression for S,, shows that this will be the ease for a geometric series whenever the modulus of r is strictly less than 1. Then

a S ~ -

1 - r

Despite their mathematical ability, this result was not established by the Greeks, and Zeno's paradox about Achilles and the tortoise was never properly explained. In modern units, Achilles runs ten times as fast as the tortoise, who travels at 1 km hr -I, and gives the tortoise a 1 km start. After Achilles has covered this 1 km the tortoise will have progressed by 0.1 km. After Achilles covers this 0.1 km the tortoise has progressed by 0.01 km and so on. The conclusion that Achilles never catches the tortoise is obviously false. He does so after

1 + 0 . 1 + 0 . 0 1 + 0.001 + . . . = 10/9 km

Harmonic series It should be noted that the infinite series

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + . . .

does not converge. The terms tend to zero as n tends to infinity so it is sensible to enquire about convergence. Finite sums calculated on a computer only increase very slowly and might be wrongly interpreted as evidence of convergence. In fact, the series can be shown to exceed the sum of an infinite number of halves by grouping the terms as below

1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + . . .

> 1 + 1/2 + 1/2 + 1/2 + . . .

Taylor series

The objective is to find a polynomial approximation to some function f ( x ) . The approximation is constructed to be exact at some point a and a good approximation for values of x near to a. It requires that f ( x ) and its derivatives be evaluated for x equal to a, so f ( x ) has to be differentiable a number of times equal to the degree of the polynomial. Suppose l (x) is a linear approximation to f ( x ) as shown in Fig. A1. It is convenient to write l (x) in the form

l (x) = Co + c l ( x - a)

The requirement that l (x) is exact at a can be written as

l(a) = f ( a )

276 A p p e n d i x h Ma themat i cs rev is ion

Y ,

f(a)

y = I(x) ,, y = q(x)

I I

. . . . . . |

a r

X

Fig. A.1 Linear and quadratic approximations to a function about the point (a, f(a))

This implies that Co must equal f(a). A second reasonable requirement is that l(x) and f (x) should have the same gradient when x equals a. That is

l'(a) =f ' (a)

This implies that cl must equal f ' (a) . Now consider a quadratic approximation q(x), also shown in Fig. A1. Writing

q(x) = Co + Cl (x - a) + c2(x - a) 2

equating q(x) with f(a) and the first two derivatives of q(x) with those off(x) evaluated at a leads to

Co = f(a), cl = f '(a) and c2 = f"(a)/2!

If we continue in this manner we obtain

+ f"(a) +if(a) f (x) =f (a ) + f ' ( a ) ( x - a) 7 . ( x - a) 2 + ' ' " n! ( x - a)"

For any given function f, the right-hand side of the above relationship will only tend towards f (x) as n tends to infinity if x is sufficiently close to a. The interpretation of 'sufficiently close' varies from function to function, and for some crucially important functions such as sine, cosine and exponential, the right-hand side always converges to them. In fact, the Taylor expansions we need are precisely those of sine, cosine and exponential about 0. Replacing a with 0 and evaluating the appropriate derivatives in the Taylor series expansion leads to the following results which are valid for all values of x

X 2 X 3

e " = l + x + - - + - - + . . . 2! 3!

X 3 X 5

sinx = x - - - + - - - . . . 3! 5!

X 2 X 4

cosx = 1 - ~ + - - - . . . 2! 4!

Even and odd functions

An even function is a function which has the same value for both positive and negative values of the argument. Thus, if y = f (x) is the function then it is an even function if and only if

f (-x) =f(x) The geometric interpretation of this property is that the graph of an even function is symmetric about the axis x =0 . Examples are cosx and x 2.

Complex numbers 277

An odd function is a function which changes sign, but not amplitude, as the sign of the argument changes. Thus, y = f (x ) is an odd function if and only if

f ( - x ) = - f (x )

Graphically, f ( - x ) can be obtained by rotating f ( x ) by a half turn about the origin. Examples are sinx and x.

Complex numbers The solution of a quadratic equation by 'completing the square' gives the standard formula for the roots of a quadratic equation. That is if

ax2 + bx + c = O

then

x = ( - b + ( b E - 4ac)U2)/(2a)

In many physical applications, for example lightly damped vibrations of a second-order system, the discriminant ( b 2 - 4ac) is negative. The advantages of defining a number j as the square root of - 1 are very wide ranging. In particular the quadratic equation will always have two roots, possibly one repeated. If b 2 is less than 4ac they can be written as,

( - b +_ j (4ac - bE)U2)/(2a)

Any number of the form

x + j y

where x and y are real numbers is called a complex number. Real numbers can be thought of as complex numbers with y equal to zero. Complex numbers satisfy the usual rules of algebra together with the definition

j2 = - 1

It follows that if zl and z2 are two complex numbers defined by

zl = xl + jYl and z2 = x2 + jY2

then

and

azl + bz2 = (axl + bx2) + j(ayl + by2)

zlz2 = (XlX2--YlY2) + j(xly2 + xzYl )

upon application of the definition of j2. It also follows that if A is a real number then

Azl = Axl + jay1

Before demonstrating division, it is useful to define the conjugate of a complex number z~, written z~ and defined by

z~ = x l - j y l

Thus, defining z2 = z~' we note that zlzT = x 2 + y2 is a real number. Now, division can be defined in terms of multiplication, that is

ZI Z1Z~ m

Z2 Z2Z~ - Az~z~

where A = 1/(z2z~) is a real number. The real numbers can be represented by points on a line and complex numbers can be

represented by their position in a plane, as shown in Fig. A2.

278 Appendix h Mathematics revision

Imaginary axis 4 - 3 - 2 -

e Z 1 - t ) I I I I 1 J

- 4 - 3 - 2 -1 0 1 2 3 4 --1 - - Real axis

�9 z* - 2 - - - 3 - -

- 4

Fig. A.2 The complex number z = - 3 + 2j and its conjugate

A typical point, Zl, could also be described in polar form by its distance from the origin r and an angle 0, conventionally measured from the positive real axis with anticlockwise taken as positive. The distance r is known as the m o d u l u s of the number, written Izll. Upon noticing that

IZl) = ziz

and that

z~ = r cos0 + j r sin 0

we are now in a position to justify the result which we have relied on for much of the main text. That is

e j o = cosO + j sin O

an immediate consequence of which is

Z 1 = re j~

When 0 is zero we note that zl = r; that is, zl is real. For 0 = ~r/2 we have zl = j r and so Zl is purely imaginary. It follows that for each increment of 7r/2 in 0 we, in fact, rotate z~ one quarter turn. We return to this observation below.

We assume that Taylor series remain valid for complex numbers, and in particular that

e z - 1 + z + z2/2! + z3/3! + . . .

cos z = 1 - z2/2! + z 4 / 4 ! - . . .

and

sin z = z - z3/3! + z S / 5 ! - . . .

Substituting j0 for z in the series for e z, and remembering that j2 equals - 1 , leads to the result: e j e = cos O + j sin 0. A famous special case gives Euler 's Equation

e J r = 0

which unites the five essential numbers of mathematics, 0, 1, ~r, e and j in one elegant identity. The formula for the multiplication of complex numbers is much easier when they are expressed in polar form. If zl and z2 are r e j ~ and pe j* respectively, their product is

z l z 2 = rpe j( 0+ ~)

If we choose zl = j, that is, r = 1 and 0 = ~r/2 then z l z 2 = pe j ( ~ ~r and so, from our earlier observations, we can deduce that multiplication by j corresponds to rotation of z2 through a quarter turn.

Example 279

Generating pseudo-random numbers Imagine a 'roulette' wheel with ten equal segments and a croupier who always varies the initial conditions. This perfect wheel could produce a sequence of digits from the set { 0 , . . . , 9} such that each digit is equally likely to result from any spin of the wheel. Notice that this definition implies that the outcome for a spin is independent of the outcomes of all other spins. Pseudo-random digits are digits generated by some mathematical algorithm which cannot be distinguished, by empirical tests, from those produced by the 'perfect' roulette wheel. They are, in fact, entirely deterministic, but if we are unaware of the algorithm we can use them as if they are genuine random numbers. The design and testing of such algorithms is a subject in itself. The following example gives an indication of their form but should not be used for serious work.

Example Let u0 equal some arbitrarily chosen prime number and define

ui+l = 91 ui (modulo 105)

The (modulo 10 s) means that we take the remainder after division by 105, that is the least significant five digits. You should check that if you start from 59 you would obtain the following sequence of numbers:

59 5369

88579 6O689 22699 65609 70419 08129

o . ~

and so on. The leading digits 8, 6, 2, 6, 7, 0 can be taken as a sequence of pseudo-random digits. The

numbers 0.88579, 0.60689, 0.22699 . . . . can be used as a sequence of random numbers from a uniform distribution defined on the interval [0,1]. You should ponder why the sequence 885796068922699... cannot be considered a sequence of pseudo-random digits.

The following algorithm is recommended by Clarke and Cooke in their book A Basic Course in Statistics

ui+l = 16807 ui (modulo 231- 1)

Whilst this is not convenient for hand calculations it can easily be programmed. However, for any intensive simulation studies it would be advisable to use some professionally written software, such as a NAG routine, which will have been thoroughly researched and tested.

If random numbers from a uniform distribution defined on [0,1] are available it is relatively straightforward to transform them to a random sequence from any distribution with CDF F. Suppose that the random variable X has the distribution F. Let a and b be arbitrary numbers in the domain of F. Then

P r ( a < X < b ) = F(b) - F(a)

Let R have a uniform distribution on [0,1]. Then, since F(a) and F(b) must be between 0 and 1

Pr(F(a) < R <F(b)) = F ( b ) - F(a)

Therefore, F(X) has the same distribution as R and F-I(R) has the same distribution as X. It follows that if r~, rE . . . . is a sequence of uniformly distributed random numbers, then

F-~(rl), F-l(r2), . . . is a sequence of random numbers from F. This is shown graphically in

280 Appendix h Mathematics revision

'x't . . . . . . . .

r ,m_ !

I I

I I I

0 F-l(r) r

X

Fig. A.3 Generating random numbers from a distribution with CDF F, given uniformly distributed random numbers r

Fig. A.3. If F has an inverse function which can be expressed as a formula the transformation is straightforward. For the exponential distribution

r = F ( x ) = 1 - e - x~

and so x = [ - In(1 - r)]/A

For the standard normal distribution there is no exact formula for �9 or ~ - ] . Possible approaches are to use algebraic approximations, or to interpolate in a numerical table of ~. The usual solution is a compromise which involves using different approximations for different parts of ~. Yet another approach to the problem is to rely on the Central Limit Theorem and take averages of at least ten consecutive uniform random numbers. This requires at least ten times the number of uniform random numbers as normal deviates required. It would not do to take a moving average if independent normal deviates are required!

Impulse responses A unit impulse function is the limit as A tends to zero of a rectangle of width A and height l/A, see Fig. A.4. From the mathematical point of view we are modelling the impact of a hammer blow on a

Fig. A.4 A unit impulse

structure by the Dirac delta, 8(0, discussed in Section 3.10. The response of a linear system to a unit impulse is known as the impulse response, denoted by h ( t ) in this book. Typical impulse responses for stable first-order and second-order systems are shown in Figs A.5 (a) and (b). You should note that if the system is stable then it will return to its equilibrium position as time tends to infinity.

Any input x ( t ) can be described as a sequence of juxtaposed near impulses of width A as shown

Impulseresponses 281

1.0

0.9 0.8 0.7

~0.6 E ~ 0.5

~. 0.4 i5 0.3

0.2

0.1

0.0 1 2 3 4 5 6 7 8 9 10 Time (s)

, 6 . . . . .

0.5

0.4

~ 0.3

.~. 0.2 a

0.1

0.0 ..

. . . . r . j ~ - ~ 1 2 a 4 5 ~ i ~ ~ 10

Time (s)

Fig. A.5 (a) Response of )~+ y = u; (b) ~+ )~+ y = u to unit impulse at t ime 0

in Fig. A.6. The output y(t) caused by the near impulse of strength x(~-)A at some past time u is approximated by

h(t- ,r)x(r)A

x(t) 4 3 2

1 0

0 1 2 t Time (s)

Fig, A,6 The function x(t) as a sequence of impulses of width A and height x(t). Their strength is therefore x(t)A

282 Appendix I: Mathematics revision

_

=.,5 t " -

| 4 - E ~ 3 -

a 1 0 "

0 1 2 Time (s)

Fig. A.7 Output y(t) caused by input x(t)

Since it is a linear system the output y(t) caused by the entire sequence of impulses is

y(t)--- ~ h( t - z )x ( z )A l"<t

Figure A.7 illustrates this. In the limit as A tends to zero this becomes the integral

I' y(t) = h ( t - z)x(r) dr w ~

This integral relationship is often written (see Chapter 6) as

y(t) = h ( t - z)x(z) d~" - - o o

where h(u) is conventionally taken as 0 for negative arguments. This convention is in agreement with our common-sense intuition that a passive physical system will not react to a force before it is applied.

Appendix I1: Inflows to the Font reservoir

The data provided relate to Example 3.1.

The following data are monthly effective inflows (m 3 s -l) to the Font reservoir, Northumberland, UK (Grid Reference NZ049938) from January 1909 until December 1980. They were made available by the Northumbrian Water Authority. A full description is given in Example 3.1. The time series should be read along rows, i.e. rows are years and columns are the months starting with January 1909.

0.423 0.524 1.375 0.694 0.352 0 .281 0.100 0.127 0.144 0.576 0.247 0.949 0.503 1.241 0.313 0.226 0 .171 0.034 0.260 0.385 0.089 0.242 0.853 0.846 0.710 0.370 0.674 0.278 0.080 0.596 0.334 0.453 0.030 0.458 0.932 1.070 1.129 1.087 0.579 0.147 0.097 0.929 0.337 0.911 0.400 0.923 0.602 0.822 1.307 0.504 0.784 0.544 0.662 0.110 0.077 0 .041 0.089 0.334 0.278 0.157 0.822 0.602 0 .721 0.150 0.092 0.293 0.077 0.050 0.046 0.174 1.402 2.158 1.200 1.617 0.932 0.232 0.145 0.095 0.148 0.186 0.116 0.171 0.330 2.102 1.132 1 .002 2.484 0.474 0.494 0.247 0.464 0.154 0.296 0.855 1 .326 1.094 0.382 1 .067 0.597 1 .079 0.198 0.049 0.038 0.846 0.223 0.260 0.281 0.568 0.816 0.439 0.287 0.223 0.293 0.266 0.077 0.047 0.522 0.509 0.281 0.494 1.026 0.674 0.627 0.355 0.242 0.027 0 .041 0.062 0.055 0.408 0.529 1.141 0.852 0.534 0.497 0 .941 0.532 0.070 0.118 0.171 0.046 0.293 0.263 1.070 1.313 0.304 0.145 0.058 0 .041 0.024 0 .021 0.390 0.278 0.266 0.571 0.316 0.970 0.818 0.423 0.358 0.097 0.247 0.198 0.118 0.675 0.242 0.214 0.822 0.405 1 .165 0.526 0.394 0.207 0.058 0.269 0.284 0.229 0.322 0.638 0.763 0.911 0.298 0.180 0.290 0.594 0.562 0.597 0.213 0.370 0.627 0.376 0.568 0.491 1 .149 0.585 0.483 0.627 0.070 0.044 0 .311 0 .391 0.343 0.422 1.129 0.878 0.989 0 .311 0.183 0.237 0 .541 0.573 0.272 0.287 0.585 1 .204 0.231 0.473 0.255 0.665 0.370 0.204 0.284 0.597 0.582 0.715 0.520 0.794 0.656 1.215 0.530 1 .807 0.165 0.083 0.574 0.115 0.163 0.076 0.535 0.910 0.464 0.769 0.377 0.180 0.156 0.133 0.027 0.139 0.281 0.034 0.183 0.736 1.183 0.739 0.949 0.609 0.412 0.154 0.040 0.260 0.529 0.587 0.358 0.969 0.671 0.473 0.910 0.340 0.715 0.269 0.794 0.385 0.597 0.263 0.121 1.195 0.408 0.576 0.160 0.609 0.596 0.609 0.092 0.124 0.080 0.217 1.262 0.412 0.665 0.352 0.759 1.005 0.159 0.278 0.058 0.056 0.047 0.061 0.316 0.770 0.311 0.656 0.180 1 .594 1.564 0.385 0.144 0.204 0.447 0.150 0.866 1.711 1.493 0.843 1.121 0.875 0.831 0.148 0.312 0.059 0 .071 0.299 0.988 1.317 0.902 1.614 1.061 1 .496 0.614 0.148 0.400 0 .151 0.168 0.468 0.213 1.195 0.547 0.680 1 .758 1 .546 1 .002 0 .571 0.110 0.106 0.109 0.449 0.455 0.547 1.724 0.988 0.409 0.299 0.076 0.148 0.553 0.174 0.100 0.137 1.064 1 .063 1.417 2.064 0.599 1 .138 0.220 0.189 0.079 0.201 0.290 0.199 1.555 1 .020 0.644 0.177 1 .152 1.058 0.394 0.139 0 .061 0.724 0.035 0.076 0.322 0.938 0.503 1.047 1 .227 1.431 0.516 0 .171 0.040 0.035 0.047 0.030 0.266 0.914 0.201 0.450 0.517 1 .023 0.205 0.062 0.067 0.100 0.198 0 .281 0.597 0 .171 0.432 1.029 0.488 0.077 0.083 0.544 0.208 0.189 0.529 0.299 0.408 0.651 0.361 0.313 0.386 0.251 0 .321 0.248 0 .101 0.287 0.047 1.228 0.680 1.326 0.603 0.988 1.149 0.038 0.177 0.435 0.214 0.059 0.044 0.137 0.644 0.226 0.385

283

284 Appendix//: Inflows to the Font reservoir

0.263 0.265 0.461 0.043 0.015 0.159 0.071 0.760 0.856 0.092 1.142 0.541 0.503 0.154 1.946 0.950 0.393 0.107 0.417 0.018 0.273 0.021 0.272 0.479 2.170 0.544 0.086 0.196 0.142 0.140 0.027 0.718 0.498 0.272 1.195 0.367 0.340 0.350 0.225 0.110 0.032 0.043 0.047 0.032 0.046 0.488 0.718 0.943 0.387 1.110 0.234 0.211 0.154 0.037 0.180 0.263 0.617 0.382 1.097 0.453 0.872 0.681 0.698 0.489 0.565 0.083 0.032 0.245 0.046 0.056 1.714 0.435 0.470 0.419 0.364 0.174 0.062 0.058 0.041 0.260 0.132 0.893 0.627 0.822 0.573 0.508 0.071 0.336 0.293 0.351 0.080 0.068 0.516 0.148 0.498 0.281 0.550 0.841 0.420 0.104 0.618 0.098 0.142 0.899 0.498 1.233 0.794 0.603 0.680 0.301 1.005 0.125 0.171 0.024 0.166 0.189 0.021 0.044 0.113 0.804 1.177 1.520 0.331 0.306 0.121 0.810 0.831 2.853 0.889 0.517 0.330 0.843 0.489 0.745 0.499 0.100 0.091 0.029 0.119 0.578 0.333 0.328 0.392 0.297 0.200 0.632 0.588 0.461 0.268 0.552 0.540 0.455 0.354 0.490 0.187 0.774 0.573 0.296 0.095 0.264 0.053 0.062 0.185 0.007 0.303 0.040 1.023 0.753 0.682 0.633 0.496 0.190 0.167 0.058 0.184 0.108 0.152 0.574 0.438 0.375 1.018 0.619 0.146 0.426 0.200 0.061 0.251 0.203 0.145 0.560 0.519 0.547 0.783 0.259 0.481 0.495 0.413 0.056 0.102 0.501 0.665 0.206 0.781 0.722 0.155 0.127 1.502 0.637 0.088 0.247 0.207 0.685 0.41.4 0.158 1.454 0.690 0.529 0.417 0.954 0.502 0.080 0.236 0.042 0.141 0.090 0.083 0.200 0.727 0.258 0.446 0.650 0.135 0.103 0.066 0.317 0.124 0.214 0.437 0.491 0.699 0.494 0.801 0.269 0.964 0.373 0.193 0.037 0.570 0.321 0.701 0.582 1.175 0.603 0.470 0.267 0.285 0.71.7 0.103 0.165 0.526 0.266 0.771 0.584 0.548 0.345 0.494 0.475 0.446 0.389 0.132 0.429 0.232 1.028 0.441 0.587 0.465 0.862 0.347 0.436 0.330 0.996 0.451 0.097 0.109 0.263 0.076 0.937 1.096 1.496 1.116 0.699 0.808 0.094 0.056 0.146 0.203 0.029 0.301 0.637 0.582 0.886 0.316 0.670 0.423 0.282 0.470 0.230 0.520 0.087 0.097 0.535 0.183 1.351 1.284 0.586 0.433 0.234 0.272 0.151 0.146 0.066 0.051 0.160 0.347 0.388 0.161 0.123 0.153 0.413 0.079 0.199 0.220 0.075 0.289 0.098 0.588 0.453 0.633 0.519 0.167 0.094 0.061 0.088 0.082 0.267 0.473 1.048 0.755 0.787 0.432 0.519 0.464 0.464 0.176 0.059 0.118 0.211 0.155 0.193 0.285 0.332 0.344 0.845 0.380 0.290 0.116 0.043 0.040 0.531 1.488 0.355 0.618 1.190 1.416 0.491 0.418 0.290 0.550 0.038 0.055 0.061 0.167 0.463 0.698 0.889 1.240 0.713 0.559 0.510 0.049 0.064 0.318 0.130 0.087 0.296 1.600 1.038 0.816 1.830 0.726 0.621 0.142 0.036 0.139 0.075 0.359 0.787 0.753 0.539 0.938 1.055 0.250 0.068 0.567 0.245 0.250 0.116 0.470 0.906 0.855

Appendix II1: Chi-square and F-distributions

The need to use these distributions first arose in Section 3.7.3.

Chi-square distribution The distribution of the sum of v independent squared standard normal random variables is taken as the definition of a chi-squared distribution with v degrees of freedom. It is written as ~ .

It follows that if we have a random sample of size n from a normal distribution, with mean ~ and standard deviation or, then

i= 1 " ~ n

If the, generally unknown, mean # is replaced by its sample estimator, the modified result can be shown to be

i=1 O r

That is

( n - 1)S 2 " " , ~ n - 1

This is one reason for defining s 2 with a divisor of n - 1 degrees of freedom. The distributional result is sensitive to the assumption of normality.

Confidence intervals for or2, and or, are constructed from the following argument.

Pr ~-1,1-~/2 < - - - 7 " - < ~- ' '~ /2 = 1 - a

pr~(n-!)S<or2<(n-1)S2_ 1 L ~ . - 1 . , , / 2 A ~ . - , , 1 - o / 2 J = 1 - a

Hence, a ( 1 - a) 100% CONF for or2 is given by (see Figure A.8)

- 1 ,a12 ~ n - 1, l-a12

Taking square roots gives a ( 1 - a) 100% CONF for or.

285

286 Appendix IIh Chi-square and F-distributions

Fig. A.8 Tail area of chi-square distribution

Example III. I In an experiment to study the effects of acceleration on passengers in Metro trains, the acceleration at which loss of balance occurred was measured for a random sample of 12 adults from the Metro staff. The sample mean, g, and standard deviation, s, were 1.62 m s -2 and 0.36 m s -2 respectively. If a normal distribution of such accelerations is assumed, a 90% CONF for the variance in the corresponding population, or2, is

[(11) (0"36)2 (11) (0"36)2 ] = [0'0724 0"3116] 19.68 ' 4.575

The 90% CONF for or is [0.27, 0.56]. The corresponding population is all adult Metro staff. Any inferences made about the

population of Metro travellers would be subjective.

F-distribution The ratio of two independent )(2 random variables divided by their degrees of freedom ul, u2 has an F-distribution with ul, v2 degrees of freedom, that is

,~V2/y 2 --- F . , , . ~

It is important to note that if F ~- F , , . , 2 then 1 / F - - . F , 2 , , , , see Example III.2. Also

E[F] = v21(v2- 2) for 2 </,'2 The variation of F with vl and v2 is illustrated in the figure.

f(x) l

,

, ,, I

0 1 rx Fig. A.9 F-distributions

Relationship with the t-distribution

Example 111.2 287

[t"-'12 = ' o./~n ~" Fl.n-I

Comparison of variances If S 2, S 2 are independent estimates of the variances o.2, o.2 of normal populations, based on random samples of size nA and nB, then

S2/& S2/~ "" Fn A- 1 ,n B- 1

An important application of this identity is testing the null hypothesis

If H~ is true

S~/S~--F.~_,,.,_,

Example 111.2 A laboratory tests the manufacturers' claims about the replicability of pressure measurements made with their transducers. Samples of 18 pressure transducers from manufacturer A, and 12 pressure transducers from manufacturer B were bought from a range of stockists in order to approximate random samples.

To construct a 90% confidence interval for the ratio of the standard deviation of the population of transducers from A to that for transducers from B (O.A/O.B) we adopt the following argument.

By definition

( S21O.2 A ) Pr FlT,ll,0.95 < ~ < F17,11,0.0s = 0.90

and therefore

pr(S2\s 2 __o.2B S"~A $2 ) F 1 7 , 1 1 , 0 . 9 5 < ~ A < F17,11,o.o5 = 0 . 9 0

Now s~ and s~ equal (0.1931) 2 and (0.2657) 2 respectively, and Fi7,11,o.05 equals 2.71 (interpolating Tables of the F-distribution). To find the lower 5% point we apply the reciprocal relationship noted earlier, namely if

then

s2/4 ) Pr S2AII~A~-<FII,17,0.05 =0.95

(s~/~ ) Pr S21o.6 > l/Fll,t7,o.o5 = 0.95

It follows that

1

- = - - - - =0.414 F17,11,o.95 "- 11F11.17,o.o5 2.415

288 Appendix IIh Chi.square and F-distributions

and so the 90% confidence interval for o2a/o2 A is

[0.7838, 5.131]

It follows that a 90% confidence interval for 0rA/tr a is

[0.44, 1.13]

Appendix IV: The sampling theorem

In Section 4.5.1 we discussed the equivalence of the discrete Fourier transform of an infinite sequence and the Fourier transform presented in Section 4.4.1. The proof of this intuitive but a reasonable result is now presented. Let x ( t ) represent the continuous signal. We define a function

oo

i(t)= ~ 8(t-n) ~'-- --O0

so that the sampled signal xs(t) is given by xs ( t ) = x ( t ) i ( t )

If we now use the convolution theorem we obtain the result

x~(.,) = ~ _~ x ( . , - 0 ) t ( 0 ) d0

We now require an expression for I ( O ) , the Fourier transform of i(0. It is proved by Lighthill (1959) that

I (0 )=27r ~] 8( to-2cm)

We will be content with showing that the result is plausible. We start by considering the Fourier transform of the function

L

Z n(t-- n) n = - L

which tends to i (O as L tends to infinity. This Fourier transform is

//.(to)= ~, 8(t-n)e-J"dt -oo n=-L

and using the definition of the delta function this simplifies to L

e - j ~n

n = - L

Introducing a different dummy counter we may rewrite this expression as

2L e-j oo( r - L )

r=0

We now use the result for a geometric progression, and express this sum as

IL( to) = e j o,L (1 -- e -j ,o2L) (1 --e -j' ')

289

290 Appendix IV: The sampling theorem

Next we define tok as

tok = 2r

where k is an integer. If k is not a multiple of 2L then

IL(~,,) = 0

because its numerator is zero and its denominator is non-zero. It remains to investigate what happens if k is a multiple of 2L. To do this we start by assuming that k = (m + e)2L for some integer m and let e ~ 0. In this case

IL(tok) = e j2~m+0 ( 1 - e-J2'~m+Ot2L)) (1 - -e -j2"n(m+ e))

= ej2,n'e (1 -- e - j2 ~.~L )

(1 - e-J2'~9

In the limit as e ~ 0 , use of a Taylor expansion for the exponential function, leads to a limit of 2L. Therefore

0 k 4= 2mL IL(tok)= EL k = 2 m L

As L increases indefinitely so the tok become more closely packed and tend to cover the whole frequency range. Therefore, noting that

t o2mL = 21rm

suggests that oo

l(to) = A ~ 8( to- 21m) i l - " - -O0

where A is some constant. A more sophisticated argument proves A = 2r We can now revert to the main argument.

Substituting for I(0) in the expression for Xs(to) gives

x~(~o) = ~ x(~o- o)2,~ E 8(o- 2,~n) dO

Hence, again using the definition of ~

Xs(to) = Z X( to - 2"n'n)

is the required result, which is known as the Sampling theorem. If we look at this expression we can see that Xs(to) will equal X(to) if and only if X(to) = 0 for to

outside the range -~r to ~r; that is, there is no contribution for n different to zero.

Appendix V: Wave tank data

The data provided relate to Example 5.1.

The following 397 data were obtained by sampling the signal from a displacement probe, situated at the centre of a wave tank, at 0.1 s intervals. The time series should be read along rows.

367 407 -255 -515 -500 -342 -188 77 494 737 375 -221 -313 -301 -311 -109 - 1 0 150 178 47

47 767 -74 -594 -541 -133 148 116 169 417 295 -317 -472 -266 -12 314 241 375 59 -550

-439 -121 367 478 4 -315 - 3 194 -45 50 136 -42 -296 -394 -145 209 536 116 136 -167

-244 18 -33 -204 -20 120 -89 -83 176 160 166 94 -65 -311 -430 -398 199 659 488 22

-258 -400 -162 -196 29 721 118 -356 -340 1 166 138 93 93 -148 -326 -95 279 -10 -99 96 227 18 61 -125 -460 -337 59 211 73 95 119 - 1 -123 -56 -26 92 125 -362 -324

116 438 235 -120 -209 36 -132 -201 192 310 -116 -282 -172 -41 204 34 -12 276 167 -328 -260 163 -62 -204 -107 172 469 50 -96 - 6 2

-40 342 -205 -488 -363 138 323 189 120 64 326 -71 -306 -150 17 -112 -296 -238 190 578 268 -33 12 -148 -132 -163 -389 -81 328 19

-32 268 380 -26 -245 -309 -289 30 125 170 141 - 7 -114 -29 134 104 -65 -365 -128 221 76 -29 113 64 32 296 -201 -513 -339 167

167 255 271 122 -500 -416 129 195 9 105 343 -136 -522 -196 480 159 -77 -54 -107 -102

-14 324 137 -312 -236 -28 239 176 -24 29 -41 -184 -331 76 276 -111 -171 -53 226 4

-114 171 354 -133 -345 -218 -307 -270 93 574 622 45 -162 -51 12 -312 -517 -265 -33 474 529 -97 -160 127 191 -281 -315 -20 24 -120

-34 260 178 164 322 -293 -522 -468 28 425 365 41 -114 6 -117 -134 -173 -19 180 239

-56 93 -94 -340 -362 189 756 600 -120 -622 -752 -526 115 875 607 -44 -349 -478 -321 30

253 511 30 -249 -262 -56 46 115 183 2 15 -85 -234 -245 72 441 282 -92 -237 -222

-37 90 -20 -78 -56 139 530 -68 -437 -351 94 351 124 0 46 48 -366 -429 -78 396

235 172 -161 -186 10 283 58 -152 -160 -131 -52 38 116 124 -48 -19 276 -29 -331 -382

-158 162 316 375 119 -174 -120 -173 -41 - 5 0 -251 -262 154 391 266 231 177

291

Appendix VI: Sampling distribution of spectral estimators

In this appendix we derive, in a somewhat informal manner, the distributional results assumed in Section 5.3.4.

Discrete whi te noise

We start by justifying the distributional result for discrete white noise {Zt }. Suppose we have a realization of N data from the process. The unsmoothed one-sided spectrum, obtained by scaling the periodogram ordinates, has N/2 independent ordinates at frequencies ranging from 0 to ~r in steps of 2zdN. Each ordinate is based on two degrees of freedom. Intuitively, this is because N data are shared equally between N/2 ordinates, but a more formal explanation is that it is a consequence of the fact that the finite Fourier series is a multiple regression with uncorrelated 'explanatory' variables. If we now think about a two-sided spectrum, we expect the variance of the process, (~z, to be evenly distributed over the interval [-I t , r So

E[ UCzz(Wk)] = (rE/E~r (A6.1)

for any w and, in particular, for Wk evenly spaced at intervals of 2zdN. If the discrete white noise is Gaussian, the sampling distribution of the ordinates UCzz(tOk) will be proportional to a chi-squared distribution with two degrees of freedom. That is

2UCzz(tok)

~/27r ---X2 2 (A6.2)

A linear stochastic process

Define a stochastic process {Xt } in terms of discrete white noise. In particular, consider

X t -- EhkZ t_k

where {hk} is the discrete impulse response function. Referring to Section 6.3, we have

rxx(~) = IH(~)12rzz(~) and the corresponding result for the sample is

UCx~(,o) - IH(o,)I' UCzz(,O) This is equivalent to

UCxx(W) UCzz(~) =

IH(,o)l 2

and by comparison with Equation (A6.2)

(A6.3)

(A6.4)

(A6.5)

292

Smoothedspectralestimators 293

2UCxx(tOk) 0+~2 (A6.6)

o-21H(to)[2/ETr

But from Equations (A6.4) and (A6.1)

rx ,~ , ) = ~lH(+o)lE/2~r

and we have

2UCxx(tok) "+ ~2 (A6.7) rxx(~k)

It is a standard result that the mean and variance of a chi-squared distribution with v degrees of freedom are v and 2v, respectively. Therefore, from Equation (A6.7)

Var[UCxx(a,)] = 2 • 2 x F2xx(a,)/22 =

Smoothed spectral estimators From the definitions in Section 5.3.2, and the remark following Equation 4.15 of Section 4.4.3, our smoothed estimator is

Cxx(to) = f UCxx(to) W(to- 0) dO

If we now discretize the integral in steps of 80 = 2~r/N, to achieve independence of the ordinates

cxx(,o) ucxx(,,,)w(,,,- o) 8o

We next make use of the standard result for the variance of a sum of independent variables to obtain

Var[ Cxx(O)] = Var[ UCxx(to)] ~ wE(,,,- O) 8~

21r = Var[UCxx(to)] --~ ~ W2(o - O) 80

2~r W2 = Var[ UCxx(to)] N f (0) dO

Finally, using Parseval's theorem (Table 4.1 of Section 4.5.4),

2"trfW2(O) dO = ~ w~

and

Var[Cxx(to)] = ._ w,, Var[ UCxx(to) ] N

Since it can be shown that Cxx(to) also has a chi-squared distribution, the degrees of freedom are half the variance. The reduction in variance by a factor of ~ , ~ / N corresponds to degrees of freedom r where

v = 2 N / ~, w~,

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Index

active vibration control 253-260 spectral analysis in 255 technique for 254-255

added resistance gradient 222-227 angular distribution of energy 200-206 'apparent quantities' 174 ARIMA models, for discrete random

processes 51-62 autoregressive processes 53-57 discrete white noise (DWN) 51 Gaussian/non-Gaussian process 60-61 MA vs AR process 61-62 moving average process 51-53 p, d, q processes 57-60 random walk 51

ARIMA (p, d, q) processes 57-60 Arithmetic series 274 ARMA model parameter estimation 64-70

AR(1) 66-66 AR(2) 66 MA(1) 67-70

ARMA process ordering 70-72 AR(1) parameters 64-66 AR(2) parameters 66 AR processes 61-62 autocorrelation function 48 autocorrelation function estimation 64 autocovariance function 48, 83-84

bias 62-64 calculation of 161-162 estimation 62

autocovariance function, and sample spectrum 111-114

lag windows, alternative 121-123 comparisons 123-127

smoothing and 115-119 spectral estimators 119-121

autoregressive integrated moving average processes, see ARIMA (p ,d ,q) processes

autoregressive processes 53-57

backward shift operator 57 bandwidth 125-127 barge mooring 221 Bartlett window 119-123,164

and other windows 123-126 Bessel functions 192

bivariate cumulative distribution function 18 bivariate distribution 9-35

continuous 15-27 expectation for 24-27 modelling of 18-20

discrete 9-15 expectation for 12-15 modelling of 10-12

normal 31-35 bivariate normal density function, perspective

drawings for 32-33 bivariate normal distribution 31-35

regression lines for 34-35 bottom standing wave monitoring system

188-189 Bernoulli, D. 91 Box-Cox transform 61 Box-Pierce statistic 72 Boyle's law 143 Burg, J.P. see maximum entropy method and

also maximum likelihood method

Cartwright-Longuet-Higgins distribution 169, 175

central limit theorem 28, 31 chi-square distribution 127,285-286 coefficient of variation of the variances (CVV)

219-221 coherence 144

formal definition of 148 complex numbers 277,279 conditional probability density function 19 conditional probability mass function 11-12 confidence intervals 35-36

for a spectrum 127 continuous bivariate distributions 15-27

expectation for 24-27 geometric interpretation of 20 modelling of 18-20 sample correlation for 20--23

continuous random processes, classification of 40

continuous time random process models 83-87 autocovariance function 83-84

estimation 84 Dirac delta 83 linear processes 86-87 mean, estimation 84

303

304 Index

continuous time random process models (cont.) Wiener process 84-85

continuous variables 9 convolution integrals 104-105

for infinite sequences 106-107 corrected sum of products 22 correlation 14

sample 20-23 correlograms 49, 70, 71, 76, 163

cross- 152-153 and surface metrology 263

covariance 13-14 sample 22-23

cross-amplitude spectrum 148 cross-correlation function 147 cross-correlograms 152-153 cross-covariance function 143,145 cross-spectral density simulation 198-206 cross-spectral function estimation 152-157

cross-correlograms 152-153 linear system transfer functions 153-155

cross-spectrum 143,147-148 applications 150-152 estimation 152-157

cumulative distribution function 18

DanieU window 127 De Moivre, A. 60 'deseasonalization' 41-47 D F T , see discrete Fourier transform differencing 58--60 Dirac delta function 83,103 discrete bivariate distributions 9-15

expectation for 12-15 modelling of 10-12

discrete Fourier transform 131-134,243 infinite sequence 105-108 inverse 130-131 Nyquist frequency 106 signal description and 91 see also finite Fourier series

discrete processes, in system relationship identification 145-148

covariance 145-147 cross-spectrum 147-148

discrete random process- ARIMA models 51--60 classification of 40 moments of 47-49 spectrum 114

discrete variables 9 discrete white noise (DWN) 51,292 dispersion equation 171 drilling platforms 3 DWN, see discrete white noise

energy spectrum 152

encounter frequency and encounter frequency spectrum 171

ensemble, in discrete random process 47 continuous-case averaging 48 discrete-case averaging 48

ergodic in the mean process 50 ergodicity 39, 49-51 even functions 276-277 even-function signal 99 expectation -

for continuous variables 24 definition of 12 for discrete variables 12-14

fast Fourier transform 130-135 inverse discrete 131-134

F-distribution 286 FFT, see fast Fourier transform filter assessment 272-273 filtering 272-273 finite Fourier series 91-97 leakage 95-97 Parsevars theorem 94-95

first difference operator 58 Fisher's transformation 35 floating wave monitoring system 188 fluid drag 264-267 Fourier, J.B.J. 2-3, 91 Fourier line spectrum 94 Fourier series 97-100, 108--110

finite 91-97 transforms 100-108

Fourier transforms 100-108,243 damped harmonic and 102 double exponential and 102 fast 130-135 pulse and 102

free dampled oscillation 165 frequency representation, of random signals,

see random-signal frequency representation

frequency response function, see transfer function

frequency, of signals- finite, Fourier series 91-97

discrete signal 91-94 leakage 95-97 Parseval's theorem 94-95

Fourier series 97-100 finite 91-97

Fourier transforms 100-108 convolution integrals 104--105 discrete 105-108 generalized 103-104

Gaussian process 47, 60-61 and non- 60-61

Index 305

geometric series 274-275 Gibb's phenomenon 97

Hamming window 238-239 Hanning window 122 harmonic series 275 harmonic signals 2 heave transfer function 178-183 high-resolution spectral estimations 136-140 Hilbert transform 215 histograms 15-16 Hooke's law 144 hull roughness 261-273

IDFT, see inverse discrete Fourier transform infinite impulse response 272-273 infinite sequence, and discrete Fourier

transforms 105 convolution integral results for 106-107

impulse responses 280-282 inverse discrete Fourier transform 131-134

Jaynes' principle 136-137

kinematic excitation number 135 Knott, C.G. 109

lag windows 119-126 alternative 121-122 Bartlett 122-123 comparisons of 125-126 Parzen 123 Tukey 122-123

leakage, and Fourier series 95-97 light emitting diodes 144-145,236-237 linear dynamics systems, in system relationship

identification 148--150 linear processes, in discrete time 86-87 linear stochastic process 292-293 linear systems transfer functions 153-155 line spectrum 111

MA(1) parameters 67-68 magnification factor variation, and damping

165-166 MA processes 61-62 marginal distribution 9 marginal probability density function 19-20 maximum entropy method 110,

136-138 maximum likelihood method 110, 136,

138-140, 196 spectral analysis formulations and 197-198

mean 13 regression towards the 35 stationary in the 50

measurement, and system relationships 143-160

MEM, see maximum entropy method MLM, see maximum likelihood method moored structures 208--232

equations of motion and 212-213 generated time-series analysis 219-221 modelling of 211-212 probabilistic method of simulation 218-219 quadratic transfer function 215-216 random sea simulation 216--218 sensitivity analysis 221 surge motion, wave damping and 221-232 time-dependent wave force 213-215

mooring cable characteristics 221 motion analysis, SELSPOT, six degrees of

freedom 239-240 motion equations, moored structures 212-213 moving average method 42 moving average processes 51-53 multiple regression method 43 multivariate data 8 multivariate normal distribution 36

NAG routine 134 narrow banded spectrum 175 negative frequency 93,106 negative encounter frequency 171 Newman method (QTF) 215-216 normal distribution 175

bivariate 31-35 multivariate 36

Nyquist frequency 106

odd-function signal 99 odd functions 276-277 one-sided spectrum 120, 162, 173, 175,180

Parseval Des Ch6nes, M.A. 91 Parseval's theorem 94-95 partial autocorrelation function 71 Parzen window 123

and other windows 123-126 PDF, see probability density function p , d , q processes 57-60 periodograms 110-114

fast Fourier transform 130-135 random process spectrum 114 sample spectrum, and autocovariance

function 111-114 segment averaging 129-130 smoothing of 127-129 spectrum estimation 127-136 surface metrology and 263

Pierson-Moskowitz spectrum 176,221 population correlation coefficient 14, 35-36 populations 8

306 Index

pressure-volume relationship 143 probability density functions 16, 18-20, 25-27

conditional 19 marginal 19

probability mass function 10-12 process bandwidth 263-264 pseudo-random number generation 279-280

quadratic transfer function 215-216

random components 41 random process spectrum 114 random processes, classification of 40

see a lso time varying signals random sampling 8-9

simple 8 stratified 8

random sea simulation 216-218 random-signal frequency representation

109-142 spectrum definition 110-115 spectrum estimation 115-135

from periodogram 127-135 from sample autocovariance function

115-127 random variables, linear functions of 27-31 random walk 51 Rayleigh distribution 168--169, 175 realization 109 regression, multiple 43 relative frequency density 16-17 response amplitude operator 152,167-168 response spectrum 152

calculation of 163-170 roughness height 262-263

sample autocovariance function, see

autocovariance function sample covariance 22-23 samples 9 sampling theorem 289-290 Schuster, A. 109 seasonal effects 41--47 seasonal indices 41 sea surface simulation 216-218 second-order stationarity 49 segment averaging 129-130 SELSPOT motion analysis 235-242

calculation of 240-242 data acquisition 240-242 results of 245-248 six degrees of freedom 239-240

Shannon and Weaver theory 136 ship resistance 261-273 signal frequency, see frequency, of signals signature analysis 3 significant response calculation 173-179

simple random sampling 8 simulation, in time varying signals 73-74 smoothed spectral estimators 293 smoothing, of Fourier transforms 115-121

autocovariance-based spectral estimators 119--121

need for 115-119 periodogram 127-129

spectral analysis, reasons for studying 1-7 spectral confidence intervals 127 spectral estimator sampling distribution

292-293 spectral estimators, high-resolution 136--140

maximum entropy method 136-138 maximum likelihood method 138-140

spectral moments 175 spectral width parameter 175 spectral windows 119-126 spectrum calculation 162-163

response 163-170 spectrum estimation, from sample

autocovariance functions 115-127 smoothing requirement 115-121

spectrum, and moving observers 170-173 spectrum, one-sided, see one-sided spectrum spectrum, sample, and autocovariance

function 111-114 spreading function 185-186 standard deviation 13 standard normal random variable 29 standardizing method 42--43 stationarity 49

definition of 49 second-order 49 strict 49 use of correlograms 49

strictly stationary stochastic processes 49 sum of products, corrected 22 surface metrology 262-263 surface topography 264-267 surge motion, and wave damping 221-232 system relationship identification 143-160

tanker mooring 221 see a lso moored structures

taylor series 275-276 t-distribution 287 texture, measures of 267-272 three-dimensional histograms 16 time-dependent wave force determination

213-215 time irreversibility, in random processes 60 time series 40--41 time series acquisition 233--252 time varying signals-

ARIMA models 51--60 autoregression 53

discrete white noise 51 Gaussian/non-Gaussian processes 60 moving average process 51-53 p , d , q p r o c e s s e s 57-59 random walk 51

discrete random processes 47--49, 51-73 ARIMA models 51--62 moments 47-49 parameter estimation 62-73

ergodicity 49--51 models for 83-90

autocovariance 83-84 Dirac delta 83 linear processes 86-87 mean 84 white noise in continuous time 85 Wiener process 84-85

reasons for studying 40--41 seasonal effects 41-47

moving average method 42 multiple regression method 43 standardizing method 42-43 trend estimation 43--47

simulations 73-74 stationarity 49

transfer function 149, 152 transient motion 165 trend 41

estimation 43--44 logistic 44 modified exponential 44 polynomials 44

Tukey-Hanning window 122-123 Tukey window 122-123

and other windows 123-126 two-sided spectrum 175

Index 307

variables, relationship between 8-38 bivariate normal distribution 31-35 continuous bivariate distributions 15-27 discrete bivariate distributions 9-15 multivariate normal distribution 36 population correlation coefficient 35-36 random, linear functions 27-31

variance, of the distribution 13 variance function 48 variances, comparison of 287-288 vibration control 253-260 Volterra series 213 volume, and pressure 143

waterfall display 134-135 wave damping, and surge motion 221-232 wave directionality see spreading function wave directionality monitoring 184-207

cross-spectral density simulation 198-199 results of 199-206

Loch Ness data 194-197 mathematical approach 189-192

applications 192-193 maximum likelihood method and 196-198 probe arrangements 193--194 technical problems in 187-189

wave encounter frequency 171 white noise, in continuous time 85

see also discrete white noise (DWN) wide band active vibration controllers 253-260 Wiener, N. 110 Wiener process 84-85 windows, see lag windows

zeros, addition to record 134-135

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