sealevel.pdf

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Sea-Level ScienceUnderstanding Tides, Surges, Tsunamisand Mean Sea-Level Changes

Sea levels change for many reasons and on many timescales, and extreme sealevels can result in catastrophic coastal flooding, such as the Katrina stormsurge in 2005 or the Sumatra tsunami in 2004. As global sea level rises, andcoastal populations increase, understanding sea-level processes becomes keyto plan future coastal defence effectively.

Ocean tides, storm surges, tsunamis, El Nio and the sea-level rise causedby climate change are among the processes explained in this book. Building onDavid Pughs classic graduate-level book Tides, Surges and Mean Sea-Level,this substantially updated and expanded full-colour book now incorporatesmajor recent technological advances in the areas of satellite altimetry andother geodetic techniques (particularly GPS), tsunami science, measurementof mean sea level and analyses of extreme sea levels. The authors, both leadinginternational experts, discuss how each surveying and measuring techniquecomplements others in providing an understanding of present-day sea-levelchange and more reliable forecasts of future changes.

Giving the how and the why of sea-level change on timescales from hoursto centuries, this authoritative and exciting book is ideal for graduate studentsand researchers working in oceanography, marine engineering, geodesy,marine geology, marine biology and climatology. It will also be of key interestto coastal engineers and governmental policy-makers.

David Pugh is a marine science consultant, also holding positions asVisiting Professor at the University of Liverpool and Visiting Scientist at theNational Oceanography Centre (NOC). His research specialises in tides,surges, mean sea level, coastal management and climate change, together withmarine economics and the history of sea level. After a career in science andscience management with the UK Natural Environment Research Council,Dr Pugh served as President of the Intergovernmental OceanographicCommission (IOC) of UNESCO, 20037. He had previously been Director ofthe Permanent Service forMean Sea Level and Founding Chairman of the IOCGlobal Sea Level network, GLOSS. Dr Pugh has authored two books andrecently co-edited Troubled Waters: Ocean Science and Governance(Cambridge University Press, 2010) published for the 50th anniversary of theIOC. He has been awarded an OBE for services to marine sciences.

Philip Woodworth is an Individual Merit Scientist in the NaturalEnvironmental Research Council based at the NOC in Liverpool, and also aVisiting Professor at the University of Liverpool. He has been Director of thePSMSL and Chairman of GLOSS. Dr Woodworth has published extensively

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on tides, sea-level changes and geodesy, including co-editing UnderstandingSea-Level Rise and Variability (Wiley Blackwell, 2010), and has been involvedin each IPCC research assessment. His awards include the Denny Medal ofIMAREST, the Vening-Meinesz Medal of the European Geosciences Union,the 50th Anniversary Medal of the IOC, and a minute share in the 2007Nobel Peace Prize awarded to the IPCC. He was awarded an MBE in 2011 forservices to science.

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Sea-Level ScienceUnderstanding Tides, Surges, Tsunamisand Mean Sea LevelDavid Pugh and Philip WoodworthUniversity of Liverpool

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University Printing House, Cambridge CB2 8BS, United Kingdom

Published in the United States of America by Cambridge University Press, New York

Cambridge University Press is part of the University of Cambridge.

It furthers the Universitys mission by disseminating knowledge in the pursuit ofeducation, learning, and research at the highest international levels of excellence.

www.cambridge.orgInformation on this title: www.cambridge.org/9781107028197

David Pugh and Philip Woodworth 2014

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First edition published as Tides, Surges and Mean Sea-Level: A Handbookfor Engineers and Scientists, 1987, by David Pugh (Wiley Blackwell, allrights reverted to author)Second edition published 2014

Printed and bound in the United Kingdom by

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

Library of Congress Cataloguing in Publication data

ISBN 978-1-107-02819-7 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,accurate or appropriate.
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Contents

Preface page vii

List of acronyms ix

List of symbols xi

1 Introduction 1

1.1 Background 1

1.2 Early ideas and observations 1

1.3 Tidal patterns 3

1.4 Meteorological and other non-tidal changes 7

1.5 Some definitions of common terms 8

1.6 Basic statistics of sea levels as time series 11

2 Sea-level measuring systems 17

2.1 The science of measurement 17

2.2 Datum definitions 20

2.3 Coastal instruments 22

2.4 Open-sea gauges 30

2.5 Data reduction 31

2.6 Data sources 33

3 Tidal forces 36

3.1 Gravitational attraction 36

3.2 The tidal forces: a fuller development 40

3.3 The MoonEarthSun system 44

3.4 Tidal patterns 49

3.5 Extreme tidal forces 53

4 Tidal analysis and prediction 60

4.1 Non-harmonic methods 61

4.2 Harmonic analysis 62

4.3 Response analysis 78

4.4 Analysis of currents 82

4.5 Time zone conversion 86

4.6 Stability of tidal parameters 87

4.7 Tidal predictions 89

5 Tidal dynamics 97

5.1 The real world 97

5.2 Long-wave characteristics 99

5.3 Ocean tides 105

5.4 Shelf tides 111

5.5 Radiational tides 122

5.6 Internal tides 124

5.7 The yielding Earth 126

5.8 Are tides changing? 129

6 Shallow-water and coastal tides 133

6.1 Introduction: some observations 133

6.2 Hydrodynamic distortions 133

6.3 Representation by higher harmonics 136

6.4 Tidal currents 139

6.5 Tidal asymmetry 142

6.6 Tides in rivers 144

6.7 Energy budgets 149

7 Storm surges, meteotsunamis and othermeteorological effects on sea level 155

7.1 Introduction 155

7.2 The depth-averaged (2-D) equations 155

7.3 Storm surges 156

7.4 Statistics of tidal residuals 164

7.5 Seiches 165

7.6 Meteotsunamis 170

7.7 Wave set-up and surf beat 172

7.8 Air pressure-related changes of sea level

in the world ocean 173

8 Tsunamis 189


8.2 Why tsunamis happen 192

8.3 Tsunami propagation across

the ocean 199

8.4 Coastal shoaling and runup 203

8.5 Tsunami signals in sea-level and bottom

pressure data 206 v

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8.6 Sea-level and related technologies for

tsunami monitoring 207

8.7 Tsunami further reading 215

9 Sea-level changes in space 223


9.2 The International Terrestrial Reference

Frame 223

9.3 The Global Positioning System 224

9.4 DORIS 227

9.5 Satellites and Mean Sea Surface 227

9.6 Satellites and the geoid 233

9.7 Models of the MSS, geoid

and MDT 240

9.8 A comment on epochs 243

9.9 Towards a global vertical datum 243

10 Mean sea-level changes in time 252


10.2 Sea-level data 252

10.3 Mesoscale variability in sea level 254

10.4 The seasonal cycle of MSL 256

10.5 Pole tide 259

10.6 Nodal tide 261

10.7 Air pressure-related sea-level

variability 262

10.8 Large-scale patterns of interannual

variability 264

10.9 Long-term changes in sea level 268

10.10 Understanding sea-level change 276

10.11 Future rise in mean and extreme sea

levels 280

11 Sea-level changes in time to do withthe solid Earth 296


11.2 Techniques for measuring vertical land

movement 296

11.3 Glacial Isostatic Adjustment 301

11.4 Tectonic sea-level changes 303

11.5 Man-made crustal movements 307

11.6 Geophysical fingerprints of sea-level

change 308

11.7 Coastal processes 310

12 Sea-level applications 318

12.1 Design parameters 318

12.2 Extreme conditions 319

12.3 Coastal defences 327

12.4 Lagoons and channels 329

12.5 Power generation 331

12.6 Emersionsubmersion probabilities 335

12.7 Flood warning systems 337

12.8 Economics of coastal defences 341

13 Sea level and life 345


13.2 The Moon and us 345

13.3 Intertidal life 346

13.4 Human development 351

13.5 The sea-level present 354

13.6 The sea-level future 355

Appendix A Basic hydrostatic and hydrodynamicequations 361A.1 The hydrostatic equation 361A.2 Conservation of mass 361A.3 The horizontal momentum

equations 361Appendix B Currents 363

B.1 Analysis of currents 363B.2 Current dynamics 365

Appendix C High and low water times andheights from harmonicconstituents 368

Appendix D Theoretical tidal dynamics 370D.1 Long progressive wave, no

rotation 370D.2 Standing waves 372D.3 Long waves on a rotating

Earth 373D.4 Co-tidal and co-amplitude

lines 374Appendix E Legal definitions in the coastal

zone 376Glossary 380Index 389

Contents

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Preface

We spendmuch of our time studying sea-level science, awide-ranging and constantly fascinating subject. Weanalyse data, read and write papers, and present findingsat conferences where there are people in the same sea-level community as us. However, every so oftenwe get tomeet other people who have been exposed to this subjectin a more personal way: someone who lost relatives inthe 1953 North Sea storm surge, another who lost every-thing more than once in Bangladesh floods, a colleaguewho survived the 2004 Sumatra tsunami.

We remember at a conference of sea-level expertsin the Maldives some years ago a small boy holding ahomemade poster declaring Down with sea-level rise,as he feared for the future of his country. Concernabout possible global warming and sea-level rise hasrarely been expressed as simply or as effectively. Theseexamples remind us that the results of our work areimportant, not just for the scientific papers that areproduced, but also for many practical reasons, whichsomehow we find reassuring.

This book is an integrated account of sea level andthe physical reasons why it is endlessly changing: tides,weather effects, tsunamis, long-term climate change,and even changes in the solid Earth. The chapterscover many fields: oceanography, geology, geodesy,climate change, coastal engineering, data managementand others.

It takes as its starting point David Pughs 1987Tides, Surges and Mean Sea Level, which is now longout of print, and significantly out of date. That bookwas published at a time of renaissance for sea-levelscience a rebirth driven by the technology of satel-lites and ever more powerful computers; and by fun-damental public concerns about the effects of climatechange and potential increased coastal flooding. Theseconcerns have been reinforced by recent catastrophictsunami and storm surge events.

This new account has roughly three components.The first component consists of six chapters that followthe 1987 books treatment of tides: instruments, forces,analysis and dynamics. In the second component,

spanning Chapters 7 to 11, we review the major newdevelopments in sea-level science: weather effects, tsu-namis, satellites and geodesy, and global sea-levelchanges related to climate change. Our discussion ofthe latter can be read alongside the recently publishedFifth Assessment Report of the Intergovernmental Panelon Climate Change, which provides evenmore facts andfigures on sea level and climate.

In the third component, containing the final twochapters, we discuss more generally how humankindhas been affected by changes in sea level in the past,and seeks to make practical arrangements for changesin the future. It is undoubtedly the case that changes insea level affect the way we live our lives today, and theywill become increasingly important in the future. Sea-level science matters to us all.

AcknowledgementsWe are grateful for the help ofmany scientific colleaguesand friends, whowere kind enough to comment on earlyversions of each chapter and provide valuable advice.Particular thanks go to Trevor Baker, John Hunter,Alexander Rabinovich and RichardRay, whose expertiseon ocean and earth tides, sea-level extremes and tsuna-mis was so freely made available to us.

Several of our colleagues at the NationalOceanography Centre were imposed upon to readdraft chapters or advise on others. Special thanks goto Angela Hibbert, Miguel Angel Morales Maqueda, JoWilliams, Simon Williams and Judith Wolf.

We have also appreciated specific guidance andhelp from Thorkild Aarup, Yasser Abualnaja, OleAndersen, Isabel Goncalves Arajo, Richard Bingley,David Blackman, David Cotton, Roland Gehrels,Jonathan Gregory, Ivan Haigh, Lee Harris, KevinHorsburgh, John Howarth, Chris Hughes, PaulHughes, Antony Joseph, Mark Lawless, David Long,MarkMerrifield, GlennMilne, Andy Plater, Rui Ponte,Reiner Rummel, Mikis Tsimplis, Javier Valladares, IanVassie, Ric Williams and Chris Wilson. We are also

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grateful for help and advice from our colleagues inthe Permanent Service for Mean Sea Level: LesleyRickards, Simon Holgate, Svetlana Jevrejeva, MarkTamisiea, Andy Matthews, Kathy Gordon and LizBradshaw.

Other colleagues provided us with top copies offigures from their work or helped us find photographs.In some cases, the figures provided were unpublished

ones, as we have acknowledged appropriately in thecaptions.

Robert Smith and Kate Davis have advised on andprepared many of the figures. We acknowledge the useof the Generic Mapping Tools package for others.

David Pugh thanks the King Abdullah Universityof Science and Technology for their hospitality whilesections of this book were prepared.

Preface

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Acronyms

ACC Antarctic Circumpolar CurrentADCP Acoustic Doppler Current

ProfilerAMO Atlantic Multidecadal OscillationAMOC Atlantic Meridional Overturning

CirculationAO Arctic OscillationAOGCM Atmosphere Ocean General

Circulation ModelAR4 IPCC Fourth Assessment ReportAR5 IPCC Fifth Assessment ReportBM Bench MarkBP Before Present or Bottom

PressureBPR Bottom Pressure RecorderCGPS Continuous GPSCM Centre of MassDART Deep-ocean Assessment and

Reporting of TsunamiDD Double Differencing (GPS data

processing method)DORIS Doppler Orbitography and

Radiopositioning Integrated bySatellite

DNA Deoxyribonucleic acidECDIS Electronic Chart Display and

Information SystemEGPS Epochal or Episodic GPSEKE Eddy Kinetic EnergyENSO El NioSouthern OscillationEnvisat Environmental Satellite of the

European Space AgencyEOF Empirical Orthogonal FunctionEOP Earth Orientation ParameterERS-1, -2 European Remote Sensing

satellite-1 and -2ESA European Space AgencyEUMETSAT European Organisation for the

Exploitation of MeteorologicalSatellites

FBM Fundamental Bench Mark

GCN GLOSS Core NetworkGCOS Global Climate Observing

SystemGEOSS Global Earth Observation System

of SystemsGEV Generalised Extreme ValueGFO GeoSat Follow-on SatelliteGGOS Global Geodetic Observing

System (of the InternationalAssociation of Geodesy)

GIA Glacial Isostatic AdjustmentGLONASS Global Orbiting Navigation

Satellite SystemGLOSS Global Sea Level Observing

System (of the IntergovernmentalOceanographic Commission)

GNSS Global Navigation SatelliteSystem

GOCE Gravity Field and Steady-StateOcean Circulation ExplorerSatellite

GOOS Global Ocean Observing SystemGPS Global Positioning SystemGRACE Gravity Recovery and Climate

Experiment SatelliteGTS Global Telecommunications

SystemHAT Highest Astronomical TideIAG International Association of

GeodesyIB Inverse BarometerICESat Ice, Cloud and Land Elevation

SatelliteIERS International Earth Rotation

ServiceIGS International GNSS ServiceInSAR Interferometric Synthetic

Aperture RadarIOC Intergovernmental

Oceanographic CommissionIOD Indian Ocean Dipole

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IPCC Intergovernmental Panel onClimate Change

ITRF International TerrestrialReference Frame

IWO Initial Withdrawal of the OceanJCOMM WMO/IOC Joint Technical

Commission for Oceanographyand Marine Meteorology

LAT Lowest Astronomical TideLEO Low Earth OrbitLGM Last Glacial MaximumLIB Local Inverse BarometerLOD Length Of DayMDT Mean Dynamic TopographyMH[L]W Mean High [or Low] WaterMH[L]WN Mean High [or Low] Water

NeapsMH[L]WS Mean High [or Low] Water

SpringsMSL Mean Sea LevelMSS Mean Sea SurfaceMTL Mean Tide LevelNAM Northern Annular ModeNAO North Atlantic OscillationNLSW Non-Linear Shallow-Water

equationsNOC National Oceanography Centre

(UK)OTL Ocean Tidal LoadingPDO Pacific Decadal OscillationPGR Post Glacial Rebound (now

usually referred to as GIA)POT Peak Over ThresholdPPP Precise Point Positioning (GPS

data processing method)

PSMSL Permanent Service for Mean SeaLevel

PTWC/S Pacific TsunamiWarning Center/System

RLR Revised Local Reference data setof the PSMSL

SAM Southern Annular ModeSLR Satellite Laser Ranging or Sea

Level RiseSSH Sea Surface HeightSOI Southern Oscillation IndexSST Satellite-to-Satellite Tracking or

Sea Surface TemperatureSWH Significant Wave HeightTAR IPCC Third Assessment ReportTEC Total Electron ContentTG Tide GaugeTGBM Tide Gauge Bench MarkTIGA TIde GAuge benchmark

monitoring project of the IGSTNT Trinitrotoluene (a ton of TNT

being a measure of explosiveenergy)

TOPEX/Poseidon TOPography EXperiment/Poseidon radar altimeter satellite

UHSLC University of Hawaii Sea LevelCenter

UNESCO United Nations Educational,Scientific and CulturalOrganisation

VLBI Very Long BaselineInterferometry

WAIS West Antarctic Ice SheetWMO World Meteorological

Organization

List of acronyms

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Symbols

a Earth radiusAl, As right ascensions of the Moon and Sunc wave speed: c ffiffiffiffiffiffigDp in shallow waterCl, Cs hour angles of the Moon and SunCa speed of sound in airCe speed of electromagnetic waveD water depthdl, ds declinations of the Moon and Sunel, ee eccentricity of lunar and Earth orbitsf Coriolis parameter f = 2ws sin F a form factor that describes the relative

importance of diurnal and semidiurnaltides at a particular location

fn nodal amplitude factor for harmonicconstituent n

Fs, Fb surface and bottom stresses in the Xdirection

g gravitational accelerationG universal gravitational constantgn phase lag of harmonic constituent n on

the local Equilibrium Tide; relative to theEquilibrium Tide at Greenwich thesymbol used is Gn (usually expressed indegrees)

gC, gAC phases of clockwise and anticlockwisecomponents of current

gux, gvx phases of Cartesian current componentsGs, Gb surface and bottom stresses in the Y

directionh geocentric mean ecliptic longitude of the

SunHn amplitude of harmonic constituent n of

tidal levelsHo the amplitude of a Kelvin wave at the

coasti, j general integersl length variableL length of ocean basinme, ml, ms mass of Earth, Moon, SunN ascending nodal lunar longitudeO(t) observed series of sea levels

p longitude of lunar perigeep0 longitude of solar perigeeP general pressure variablePA atmospheric pressure at the sea surfacePz pressure at depth zQC, QAC amplitudes of clockwise and

anticlockwise components of currentsq current speedr distance, variously definedRl, Rs lunar and solar distances from

the EarthR(t) residual non-tidal component of

sea levels geocentric mean ecliptic longitude

of the Moont timeT(t) tidal component of sea levelu, v current components in the X and Y

directionsun nodal phase factor for harmonic

constituent nVn nodal astronomical phase angle of

harmonic constituent n in theEquilibrium Tide, relative to theGreenwich meridian

W wind speedx, y, z coordinates of a pointX, Y, Z Cartesian coordinate system. Z is

positive vertically upwardsZ(t) mean sea level dimensionless ratio variously definedl, s ecliptic latitudes of the Moon and Sun a general angular measure displacement of water level from the

mean direction to which current and wind

flow, clockwise from northl, s true ecliptic longitudes of the Moon

and Sun seawater densityA air density

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n angular speed of constituent n, usuallyin degrees per mean solar hour

s, b unresolved surface and bottom stresses standard deviation of a time series First Point of Aries latitude of a point on the Earths surfacen angular speed of constituent n0 to 6 angular speeds of astronomical variables

(see Table 3.2)

s angular speed of the Earths rotation onits axis relative to a fixed celestial point(s = 0 + 3 = 1 + 2)

Equilibrium tidal potential

Harmonic constituents are shown in heavy type thus:X2, to denote their vector property (Hx, gx).

Overbars denote time-averaged values.

List of symbols

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Chapter

1 Introduction

Prospero: . . .ye that on the sands with printless footDo chase the ebbing Neptune and do fly himWhen he comes back

Shakespeare, The Tempest

Sea levels are always changing, for many reasons.Some changes are rapid while others take place veryslowly. The changes can be local, or extend globally.This book is about the science of these changes.

In this first chapter we outline what constitutes sea-level science. A brief account of the development ofscientific ideas is followed by an outline of how sealevels are affected by a wide range of physical forcesand processes. Finally we give some basic definitions,and discuss the fundamental statistics of sea levels astime series.

1.1 BackgroundLiving by the sea has many benefits. Statistics showthat about half the global population lives within 100km of the sea. Most of the worlds largest cities are onor near the ocean. Ninety per cent of all global trade iscarried by sea. The coast offers possibilities of bothtrade and travel, and increasingly of water-based rec-reation. Natural geological processes have often con-spired to create flat and fertile land near to the presentsea level, to which people are drawn or driven tosettle.

There are risks. Throughout history, humankindhas adjusted and coped with changing sea levels: theebb and flow of the tides, storm flooding and, for somevulnerable places, the dangers of being inundated by atsunami. However, as our cities and our patterns ofcoastal development become more intricate, popu-lated and interdependent, we become more and morevulnerable to disasters. The rural response of drivingcattle to higher ground for the duration of a flood ismuch easier than the urban complexity of rebuildingcomplete sewage and transport systems. In extreme

cases flooding, with disastrous long-term consequen-ces, may destroy the delicate infrastructure of coastalcities.

Books dealing with the science of sea levels andtidal phenomena are comparatively rare. However,unified treatments of general interest are found inolder specialist books [1, 2, 3], and in more recentpublications [4, 5, 6]. Accounts are also found inmore general books on oceanography, especially thesecond volume of Defants Physical Oceanography [7].Defant and some other experts have also written morepopular accounts [8], which are useful introductions,though sometimes hard to find.

1.2 Early ideas and observationsThe link between the Moon and tides was known fromvery early times. Sailors had a very practical need fordeveloping this understanding, particularly for theirnear-shore navigation in the small ships of those times.A more scientific explanation of the links betweentides and the movements of theMoon and Sun evolvedmuch later. Many eminent scientists have beeninvolved in this scientific development.

Even 2000 years ago, historical records show animpressive collection of observed tidal patterns [9].However, the ideas advanced by the philosophers ofthat time, and for the following 1600 years, to explainthe connection between the Moon and the tides wereless valid. Chinese ideas supposed water to be theblood of the Earth, with tides as its beating pulse,with the Earth breathing causing the tides. Arabicexplanations supposed the Moons rays to be reflectedoff rocks at the bottom of sea, thus heating andexpanding the water, which then rolled in waves

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towards the shore. One poetic explanation invoked anangel who was set over the seas: when he placed hisfoot in the sea the flow of the tide began, but when heraised it, the tidal ebb followed. During this longperiod there was a decline in critical thought, so thatthe clear factual statements by the classical writerswere gradually replaced by a confusion of supposedfacts and ideas. One notable exception was theVenerable Bede, a Northumbrian monk, whodescribed around AD 730 how the rise of the wateralong one coast of the British Isles coincided with a fallelsewhere. Bede also knew of the progression in thetime of high tide from north to south along theNorthumbrian coast.

Johannes Kepler (15961650), while developinglaws to describe the orbits of the planets around theSun, suggested that the gravitational pull of the Moonon the oceans might be responsible for tides. IsaacNewton (16421727) took this idea much further.Almost incidentally to the main insights of hisPrincipia published in 1687 (the fundamental laws ofmotion, and the concept of universal gravitationalattraction between bodies), Newton showed whythere are two tides a day, and why the relative positionsof the Moon and Sun are important. His contempo-rary, Edmond Halley (16561742, Figure 1.1), madesystematic measurements at sea and prepared a map oftidal streams in the English Channel. Halley hadencouraged Newtons work, paid for the publicationof Principia himself, and prepared an account of thetides based on Newtons theories [10]. Many otherscientists extended and improved Newtons funda-mental understanding, but it remains the basis for alllater developments.

Daniel Bernoulli (17001782) published ideasabout an Equilibrium Tide, which we shall look at indetail in Chapter 3. The Marquis de Laplace (17491827) developed theories of a dynamic ocean responseto tidal forces on a rotating Earth, and expressed themin periodic mathematical terms. Thomas Young(17731829), while developing his theory on thewave characteristics of light, showed how the propa-gation of tidal waves could be represented on charts asa series of co-tidal lines.

The first operational automatic tide gauge andstilling-well system for measuring sea levels wasinstalled at Sheerness in the Thames Estuary in 1831,to provide continuous sea-level data. These measure-ments in turn stimulated a new enthusiasm for tidalanalysis and the regular publication by British

authorities of annual tidal predictions to assist mari-ners to plan safer navigation. Even before the officialtables, tidal predictions were published commercially,sometimes based on undisclosed formulae, for exam-ple those of the Holden family in northwest England[11].

Lord Kelvin (18241907) showed in detail howtides could be represented as the sum of periodicmathematical terms, and promoted a machine(Figure 4.12) that applied this idea for tidal predic-tions. He also developed mathematical equations forthe propagation of tidal waves on a rotating Earth, in aform known as Kelvin waves. In 1867 the Coast Surveyof the United States took responsibility for the annualproduction of official national tide tables for theUnited States. By the beginning of the twentieth cen-tury, most major maritime countries around the worldbegan to prepare and publish regular annual officialtide tables.

Meanwhile, other factors that influence sea-levelchanges were being investigated. James Clark Ross

Figure 1.1 Edmond Halley (16561742) assisted in the publicationof Newtons Principia, the basis for tidal science, and also led the firstsystematic tidal survey, of currents in the English Channel. TheRoyal Society.

Introduction

2


(180062) made sea-level measurements when trap-ped in the ice during the Arctic winter of 18489, andconfirmed the already-known link between higheratmospheric pressures and lower sea levels. EarlierRoss had helped establish Tide Gauge Bench Marksin Tasmania and the Falkland Islands, as datums forscientific mean sea-level studies during his voyage ofexploration in the Southern Ocean. Establishing thesefundamental fixed datum levels was done on the adviceof the German geophysicist Alexander Von Humboldt(17691859).

Harris [9] gives an extensive late-nineteenth-centuryhistorical account of early tidal ideas;Wheeler [12] givesa contemporary hydraulic engineering perspective.More recently, Cartwright [13] gives a comprehensiveanalysis of the scientific history of tides. A more generaldiscussion of sea-level science and its place in the overalldevelopment of marine science is given in Deacon [14];Reidy [15] describes the role of the British Admiralty intidal science and its application.

1.3 Tidal patternsBefore the development of appropriate instrumenta-tion, sea-level observations were confined to the coastand were not very accurate. Modern measuring sys-tems, many of which will be described in the nextchapter, have enabled a systematic collection of sea-level data which shows that regular water movementsare a feature on all the shores of the oceans and of theiradjacent seas. These regular tidal water movements areseen as both the vertical rise and fall of sea level, and asthe horizontal ebb and flow of the water.

The tidal responses of the ocean to the forcing ofthe Moon and Sun are very complicated and tidalfeatures vary greatly from one site to another. Thetwo main tidal features of any sea-level record are thetidal range, measured as the height between successivehigh and low levels, and the period, the time betweenone high (or low) level and the next high (or low) level.Figure 1.2a, which shows the tides for March 2043 atfive sites, clearly illustrates this variability. Figure 1.2bshows the lunar variables for the same month. Thedetails of the relationships between the tides and themovements of the Moon and Sun are developed inChapter 3. In this section we describe the observedsea-level variations at these five sites and relate them tothe astronomy in a more general way.

We can now look in detail at Figure 1.2a. In most ofthe worlds oceans the dominant tidal pattern is similar

to that shown for Bermuda in the North Atlantic, andforMombasa on the African shore of the Indian Ocean.Each tidal cycle takes an average of 12 hours 25minutes,so that two tidal cycles occur for each transit of theMoon (every 24 hours 50 minutes). Because each tidalcycle occupies roughly half of a day, this type of tide iscalled semidiurnal. Semidiurnal tides have a range thattypically increases and decreases cyclically over a 14-dayperiod. The maximum ranges, called spring tides, occura few days after both new and full Moons (syzygy, whenthe Moon, Earth and Sun are in line), whereas theminimum ranges, called neap tides, occur shortly afterthe times of the first and last quarters (lunar quadra-ture). The relationship between tidal ranges and thephase of the Moon is due to the additional tide-raisingattraction of the Sun, which reinforces the Moons tidesat syzygy, but reduces them at quadrature. The astro-nomical cycles are discussed in detail in Chapter 3, butFigure 1.2b shows that when the Moon is at its max-imum distance from the Earth, known as lunar apogee,semidiurnal tidal ranges are less than when theMoon isat its nearest approach, known as lunar perigee. Thiscycle in theMoons motion is repeated every 27.55 solardays. Maximum semidiurnal ranges occur when springtides (syzygy) coincide with lunar perigee [3], whereasminimum semidiurnal ranges occur when neap tides(quadrature) coincide with lunar apogee. Globally,semidiurnal tidal ranges increase and decrease atroughly the same time everywhere, but there are sig-nificant local differences. The maximum semidiurnaltidal ranges occur in semi-enclosed seas. In the MinasBasin in the Bay of Fundy (Canada), the semidiurnalNorth Atlantic tides at Burncoat Head have a meanspring range of 12.9 m. Equally large ranges are foundin Ungava Bay, northeast Canada (see Chapter 5). Themean spring ranges at Avonmouth in the BristolChannel (United Kingdom) and at Granville in theGulf of St Malo (France) are 12.3 m and 11.4 m respec-tively. In Argentina the Puerto Gallegos mean springtidal range is 10.4 m; at the Indian port of Bhavnagar inthe Gulf of Cambay it is 8.8 m; and the Korean port ofInchon has a mean spring range of 8.4 m. More gen-erally, however, in the main oceans the semidiurnalmean spring tidal range is usually less than 2 m.

Close examination of the tidal patterns atBermuda and Mombasa in Figure 1.2a shows that atcertain times in the lunar month the high water levelsare alternately higher and lower than the average.This behaviour is also observed for the low waterlevels, the differences being most pronounced when

1.3 Tidal patterns

3


the Moons declination north and south of the equa-tor is greatest. The differences can be accounted forby a small additional tide with a period close to oneday, which adds to one high water level but subtracts

from the next one. In Chapters 3 and 4 we shalldevelop the idea of a superposition of several partialtides to produce the observed sea-level variations atany particular location.

Karumba

2

1

0

1

2(a)

San Francisco

1

0

1

Mombasa

2

1

0

1

2

Sea

Lev

el (

m)

Bermuda

0.5

0.0

0.5

Courtown

1

0

1

10 15 20 25 30March 2043

5

Figure 1.2 (a) Tidal predictions for March 2043 at five sites that have very different tidal regimes. At Karumba, Australia, the tides are diurnal, atSan Francisco, United States, they are mixed, whereas at both Mombasa, Kenya, and Bermuda, semidiurnal tides are dominant. The tides atCourtown, Ireland, are strongly distorted by the influence of the shallow waters of the Irish Sea.

(b) The lunar characteristics responsible for these tidal patterns. Solar and lunar tide-producing forces combine at new and full Moon to givelarge spring tidal ranges every 14.76 days. Lunar declination north and south of the equator varies over a 27.21-day period. Solar declination iszero on 21 March. Lunar distance varies through perigee and apogee over a 27.55-day period. The thumbnail cartoons show the physics of thevariations (lunar phase, declination and distance) discussed in detail in Chapter 3.

Introduction

4


In the case of the tide at San Francisco, the tideswith a one-day period, which are called diurnal tides,are similar in magnitude to the semidiurnal tides. Thiscomposite type of tidal regime is called a mixed tide,the relative importance of the semidiurnal and thediurnal components changing throughout themonth, as plotted in Figure 1.2a. The diurnal tidesare biggest when the Moons declination is greatestbut reduce to zero when the Moon is passing through

the equatorial plane, where it has zero declination. Thesemidiurnal tides are biggest after new and full Moon;but unlike the diurnal tides, they do not reduce to zerorange, being only partly reduced during the period ofneap tides.

In a few places the diurnal tides are much largerthan the semidiurnal tides. Karumba in the AustralianGulf of Carpentaria is the example shown inFigure 1.2a. Here the tides reduce to zero amplitude

Lunar Fraction Illuminated

0.00

0.25

0.50

0.75

1.00(b)

Fra

ctio

n

New Moon First Qtr Full MoonLast Qtr

Synodic or Lunar month = 29.5306 days

Lunar Declination

30

20

10

0

10

20

30

Dec

linat

ion

(Deg

rees

)

Nodical month = 27.2122 days

Lunar Orbit

3.6

3.8

4.0

4.2

Dis

tanc

e (x

105

km

) APOGEE

PERIGEE

APOGEE

Anomalistic month = 27.5546 days

March 2043

10 20 300

Figure 1.2 (cont.)

1.3 Tidal patterns

5


a day or so after the Moons declination is zero,increasing to their largest values when the Moon is atits greatest declination, either north or south of theequator (Figure 1.2b). Diurnal tides are also found inpart of the Persian Gulf, the Gulf of Mexico and part ofthe South China Seas. The Chinese port of Pei-Hai hasthe worlds largest diurnal tidal range, with a differ-ence of 6.3m between the highest and lowest predictedtidal levels.

Astronomical forces acting on the major oceans ofthe world generate and energise the tides. Thesehydrodynamics are discussed in detail in Chapter 5.From the oceans the tides spread as waves to thesurrounding shallower shelf seas.

The tidal ranges on the relatively shallow con-tinental shelves are usually larger than those of theoceans. However, very small tidal ranges areobserved in some shallow areas, often accompaniedby curious distortions of the normal tidal patterns.Figure 1.2a shows the tides for Courtown on theIrish coast of the Irish Sea, where the range variesfrom more than a metre at spring tides to only afew centimetres during neap tides. When the rangeis very small, careful examination shows that fourtides a day occur. These effects are due to thedistorted tidal propagation in very shallow water.Shallow-water distortions, which are discussed indetail in Chapter 6, are also responsible for thedouble high water feature of Southampton tidesand for the double low waters seen at Portland,both in the English Channel, where semidiurnaltides prevail. Double low waters also occur alongthe Dutch coast of the North Sea fromHaringvlietsluizen to Scheveningen, where theyare particularly well developed at the Hook ofHolland. Double high waters are also found atDen Helder in the North Sea and at Le Havre inthe English Channel.

Tidal currents, often called tidal streams, have sim-ilar variations. Semidiurnal, mixed and diurnal cur-rents occur, usually having the same characteristics asthe local changes in tidal levels, but this is not alwaysso. For example, the currents in the Singapore Straitare often diurnal in character but the elevations aresemidiurnal. The strongest currents are found in shal-low water or through narrow channels, which connecttwo seas, such as the currents through the Straits ofMessina between the island of Sicily and the Italianmainland.

Currents in narrow channels are constrained toflow along the channel axis, but in more open waters

all directions of flow are possible. During each tidalperiod the direction usually rotates through a com-plete circle while the speeds have two roughly equalmaximum and two equal minimum values. Figure 1.3shows the distribution of tidal currents over one semi-diurnal cycle at the Inner Dowsing light tower in theNorth Sea. Each line represents the speed and direc-tion at a particular hour. Because eachmeasurement ofcurrent is a vector described by two parameters, theirvariations are more complicated to analyse thanchanges of sea level.

This book is concerned with movements of theseas and oceans. Two other geophysical phenom-ena that have tidal characteristics are of interest. Intropical regions there is a small but persistent 12-

High water

H + 1

H + 5

H + 6H + 3

H + 4

H + 2

H-5

H-4

H-3

H-2

H-1

H-6

0.2 0.4 0.6 0.8 1.00

Scale in m/s

East

GeographicNorth

West

South

Figure 1.3 Spring tidal current curve, Inner Dowsing light tower,North Sea. Each line represents the velocity of the current at eachhour from 6 hours before to 6 hours after local high water (forexample, H+2 is the current 2 hours after high water). Semidiurnaltides are strongly dominant. The distance from the centre representsthe speed, and the bearing gives the direction of water flow. Forexample, at high water the current is 0.5 m/s to the southeast.

Introduction

6


hour oscillation of the atmospheric pressure(Section 5.5) with typical amplitude of 1.2 milli-bars, which reaches its maximum values near 10:00hours and 22:00 hours local time. This produces asmall tidal variation of sea levels. Also, accuratemeasurements of the Earths gravity field and ofthe tilt of the Earths crust show vertical move-ments of the land surface relative to the centre ofthe Earth that are tidally induced. These Earthtides may have extreme amplitudes of tens ofcentimetres.1 For scientific computations of tideson a global scale the Earth tides must be includedbecause of their effects on ocean dynamics.However, for practical purposes only the observedmovements relative to the land need be considered.

1.4 Meteorological and othernon-tidal changesThe regular and predictable pattern of the tides ismodified to a greater or lesser extent by irregularfactors, the principal ones being the atmospheric pres-sure and the wind acting on the sea surface. Figure 1.4ashows how the regular semidiurnal pattern of sealevels at Venice was modified by the weather over afew days in November 2002. These irregular slowchanges, known as storm surges, are plotted in thesame diagram. The coincidence of a spring tide and alarge increase in the levels due to the weather caused anexceptionally high total level. Figure 1.4b shows thetide gauge record for Hurricane Katrina flooding inLouisiana in August 2005; the green line is the 1.4 msurge, which persisted for less than a day.

Historically there have been many disastrouscoastal floodings caused by the coincidence of largemeteorologically induced surges and large or evenmoderately high tides (see Table 12.5). Thus, inNovember 1885, New York was inundated by highsea levels generated by a severe storm that also causedflooding at Boston. More than 6000 persons weredrowned in September 1900 when the port ofGalveston in Texas was overwhelmed by waters thatrose more than 4.5 m above the mean high water level,as a result of hurricane winds blowing at more than50 m/s for several hours. Also in the Gulf of Mexico,

Hurricane Katrina in August 2005 caused severedamage and many lives were lost in Louisiana(Figure 1.4b). Even these disasters were surpassed bythe Bangladesh tragedy of 12 November 1970, whenwinds of 60 m/s raised sea levels by an estimated 9 m.Flooding extended over several low-lying islands,drowning hundreds of thousands of people. Severestorms, floods and loss of life occurred in the sameregion in both 1876 and 1897, and more recently in1960, 1961 and 1985. In October 1999, 10,000 peoplewere killed in Orissa, India, by a 78 m surge. Alongthe south coast of Burma, cyclone Nargis caused atleast 140,000 deaths in May 2008.

The physics of storm surge generation by air pres-sures and winds is covered in Chapter 7. The largestsurges occur when hurricane winds blow for a long timeover large expanses of shallow water. Whilst tropicalstorms cause the most extreme local flooding, stormsat higher latitudes can also produce very large surgesover wide areas. In January 1953, catastrophic floodingon both the English and Dutch coasts of the North Seaoccurred as a result of a depression tracking into theNorth Sea across the north of Scotland. This surge,which exceeded 2.6 m at Southend, coincided withslightly less than average spring tides, otherwise evenmore damage would have occurred. The coastal regionsof the North Sea are prone to this type of flooding; therewere previous events in 1929, 1938, 1943, 1949, and aparticularly severe case in Hamburg in 1962. Similarly,along the Atlantic coast of the United States, the AshWednesday storm of 7 March 1962 flooded many low-lying barrier islands, causing millions of dollars of dam-age. Recently, Hurricane Sandy, the largest Atlantichurricane on record, in October 2012 caused floodingand major damage along the whole eastern seaboard ofthe United States, with flooding of streets, tunnels andsubway lines and power losses in and around New York.

These dramatic extremes and the resulting coastalflooding are rare events, but there is always a contin-uous background of sea-level changes due to theweather, which raise or lower the observed levels com-pared with the levels predicted. At higher latitudesthese effects are greater during the stormy wintermonths. Knowledge of the probability of occurrenceof these extreme events is an essential input to the safedesign of coastal defences and other marine structures,as discussed in Chapter 12.

The tsunamis generated by submarine earthquakesare another cause of rare but sometimes catastrophicflooding, particularly for coasts around the PacificOcean. Tsunamis are sometimes popularly called

1 Measured coastal sea levels are actually those of the verticalmovement of the sea surface relative to the local land level, whichmay itself be moving relative to the centre of the Earth(Chapter 11).

1.4 Meteorological and other non-tidal changes

7


tidal waves but this is misleading because they are notgenerated by tidal forces nor do they have the periodiccharacter of tidal movements (see Chapter 8). Near theearthquake source the tsunami amplitudes are muchsmaller; the amplification occurs in the shallow coastalwaters and is enhanced by the funnelling of the wavesin narrowing bays. The coast of Japan is particularlyvulnerable to this type of flooding, and the term tsu-nami is of Japanese origin. In March 2011 a majortsunami caused extensive damage to the Fukushimanuclear power plant. Tsunamis can travel long distan-ces across the oceans: the tsunami generated by themajor earthquake in the Indian Ocean on 26December 2004 caused damage thousands of milesfrom the source, and was detected worldwide. In theAtlantic Ocean tsunamis are comparatively rare, butthe flooding after the 1755 Lisbon earthquake is welldocumented. Tsunamis often set up local oscillationsof semi-enclosed sea and basins, called seiches, which

are discussed in Section 7.5; more commonly seichesare triggered by winds or internal ocean tides.

1.5 Some definitions of commontermsIt is now appropriate to define more exactly some ofthe common tidal and non-tidal terms, as they will beused throughout this book; the Glossary contains pre-cise summaries.

Throughout this book, we define sea level as: thelevel of the sea after averaging out the short-termvariations due to wind waves.2

The first important distinction to make is betweenthe popular use of the word tide to signify any change

Punta della Salute, Venice

12 13 14 15 16 17 18 19 20November 2002

0.0

0.5

1.0

1.5

2.0(a)W

ater

Lev

el (

m)

Grand Isle, Louisiana

22 23 24 25 26 27 28 29 30 31 1 2 3 4August 2005 September

0.0

0.5

1.0

1.5

2.0(b)

Wat

er L

evel

(m

)Figure 1.4 Observed (red), tidal(blue) and meteorological (green)variations of sea level: (a) anextratropical surge at Venice,November 1966; (b) a tropical surge,Louisiana (Hurricane Katrina), 2931,August 2005.

2 This is sometimes called still water level.

Introduction

8


of sea level, and the more specific scientific use of theword to mean only the regular periodic variations.

Although any definition of tides will be somewhatarbitrary, it must emphasise this periodic and regularnature of the motion, whether it be of the sea surfacelevel, currents, atmospheric pressure or Earth move-ments. We define tides as periodic movements that aredirectly related in amplitude and phase to some peri-odic geophysical force. The dominant geophysicalforcing function is the variation of the gravitationalfield on the surface of the Earth, caused by the regularmovements of the MoonEarth and EarthSun sys-tems. Movements due to these forces are termed grav-itational tides. This is to distinguish them from thesmaller movements due to regular meteorologicalforces, which are called either meteorological tides ormore usually radiational tides because they occur atperiods directly linked to the solar day. It can beargued that seasonal changes in the levels and in thecirculation of seawater due to the variations of climateover an annual period are also regular and hence tidal.

Any sequence of measurements of sea level orcurrents will have a tidal component and a non-tidalcomponent. The non-tidal component, which remainsafter analysis has removed the regular tides, is alsocalled the residual or meteorological residual.Sometimes the term surge residual is used, but morecommonly the term surge or storm surge is used for aparticular event during which a very large non-tidalcomponent is generated, rather than to describe thewhole continuum of non-tidal variability (seeChapter 7).

Periodic oscillations are described mathematicallyin terms of amplitude and a period or frequency:

X t Hx cos x gx 1:1

where X(t) is the value of the variable quantity at time t,Hx is the amplitude of the oscillation, x is the angularspeedwhich is related to the periodTx byTx= 2=x (xis measured in radians per unit time), and gx is a phaselag relative to some defined time zero.

For scientific tidal studies Hx may have units ofmetres (rarely feet) for levels, or metres per second(m/s) for currents. In its simple form Equation 1.1 canonly represent the to-and-fro currents along the axis ofa channel. If the direction is variable as in Figure 1.3,then Equation 1.1 may define the flow along a definedaxis (X). The speed and direction of the total flow iscompletely specified if the currents along a second axis(Y) at right angles to the first are also defined.

Tidal high water is the maximum tidal levelreached during a cycle. The observed high waterlevel may be greater or less than the predicted tidallevel because of meteorological effects. Similarly lowwater is the lowest level reached during a cycle. Thedifference between a high water level and the next lowwater level is called the range, which for the simpleoscillation defined by Equation 1.1 is 2Hx, twice theamplitude.

Ocean tides and most shelf sea tides are dominatedby semidiurnal oscillations, for which several descrip-tive terms have been developed through both popularand scientific usage (see Figure 1.5a). Spring tides aresemidiurnal tides of increased range, which occurapproximately twice a month as a result of the Moonbeing new or full. The age of the tide is an old term forthe lag between new or full Moon and the maximumspring tidal ranges. The average spring high water leveltaken over a long period (maybe a year) is calledMeanHighWater Springs (MHWS) and the correspondinglyaveraged low water level is called Mean Low WaterSprings (MLWS). Formulae are available for estimat-ing these useful parameters directly from the results oftidal analyses (Section 4.2.6 and the Appendices).Neaptides are the semidiurnal tides of small range that occurnear the time of the first and last lunar quarters,between spring tides. Mean High Water Neaps andMean Low Water Neaps are the average high and lowwaters at neap tides. These too may be estimateddirectly from tidal analyses.

Where the tidal regime is mixed the use of MHWSand other semidiurnal terms becomes less appropriate.Mean HighWater andMean LowWater are frequentlyused. Mean Higher High Water (MHHW) is the aver-age level over a long period of the higher high waterlevel that occurs in each pair of high waters in a tidalday. Where only one high water occurs in a tidal day,this is taken as the higher high water.Mean Lower LowWater (MLLW) is the average of the lower level in eachpair of low water levels in a tidal day. Mean LowerHigh Water and Mean Higher Low Water are corre-sponding terms that are seldom used.

These terms become increasingly difficult to deter-mine exactly, where the tide is dominated by diurnaloscillations (Figure 1.5b). Mean High Water (MHW)and Mean Low Water (MLW) averaged for all highand low levels respectively may be computed asdatums for survey work, but difficulties arise whenconsidering how to treat the multiple small high andlow levels that occur during the time when the diurnal

1.5 Some definitions of common terms

9


range falls to near zero, as for example at Karumba inFigure 1.2a. Diurnal tides show a semi-monthly varia-tion in their range from a maximum to zero and backto a maximum range again because of the monthlycycle in the lunar declination. Some authorities havesuggested that these diurnal modulations should becalled Diurnal Spring and Diurnal Neap Tides [3],but this usage is uncommon, being usually reservedfor the semidiurnal range changes related to the phaseof the Moon.

Levels defined by analysis of long periods of sea-level variations are used to define a reference levelknown as a tidal datum. These datums are often usedfor map or chart making, or for referring subsequentsea-level measurements. For geodetic surveys theMean Sea Level (MSL) is frequently adopted, beingthe average value of levels observed each hour over aperiod of at least a year, and preferably over about 19years, to average over the cycles of 18.61 years in thetidal amplitudes and phases (see Chapter 3), and toaverage out weather. Present United Kingdom MeanSea Levels are now approximately 0.2 m higher thanthe 191521 mean level, adopted as the National MeanSea Level, land datum. In Section 9.9 we discuss theevolution of the way in which these vertical datums aredefined and measured.

The Mean Tide Level (MTL), which is the averagevalue of all high and low water heights, is sometimescomputed for convenience, where a full set of hourlylevels is not available. Before automatic tide gauges,MTL was the only readily calculated mean value. Ingeneral, MTL will differ slightly from the MSL becauseof tidal distortions in shallow water. Chapters 10 and11 will discuss MSL definition, computations andvariations in detail, and show how satellite missionshave greatly extended our understanding of globalsea levels.

Lowest Astronomical Tide (LAT), the lowestlevel that can be predicted to occur under any combi-nation of astronomical conditions is recommended aChart Datum by the International HydrographicOrganization. LAT is adopted for charts prepared bythe British Hydrographic Department. The CanadianHydrographic Department has adopted the level oflowest normal tides. The United States NationalOcean Service (NOS) used a Chart Datum that isdefined in some areas of mixed tides as the mean ofthe lower of the two low waters in each day (MLLW);in some semidiurnal tidal areas as mean low watersprings (MLWS); and elsewhere as the lowest possiblelow water. MLLW is now adopted by NOS as thestandard datum for all locations (see Appendix E).

(a) Highest astronomical tide

Mean high water neaps

Mean sea level (2000)Ordnance Datum Newlyn

Mean low water springs

Lowest astronomical tide Admiralty chart datum

Mean low water neaps

Mean high water springs

Met

res

(to

OD

N)

3.0

2.0

1.0

0.0

1.0

2.0

3.0

1 2 3 5

Hourly levels (*1000)

4

Highest astronomical tide

Mean lower low water

Chart datumLowest astronomical tide

Mean higher high water

Mean sea level

Met

res

(to

Cha

rt D

atum

)

3.0

5.0(b)

1.0

2.0

4.0

0.0

1 2 3 4Hourly levels (*1000)

Figure 1.5 Frequency distribution of hourly tidal levels (a) at Newlyn over an 18-year period relative to Ordnance Datum Newlyn and (b)atKarumba over a single year (1982), relative to Chart Datum. For the semidiurnal Newlyn tides there is a distinct but slightly asymmetric doublepeak in the distribution, but for the diurnal Karumba tides there is a single skewed peak.

Introduction

10


Hydrographers prefer to refer levels on their chartsto an extreme low water Chart Datum, below whichthe sea level seldom falls, in order to give navigatorsthe assurance that the depth of water shown on charts isthe minimum available whatever the state of the tide.Throughout the world, published predictions of tidallevels are given relative to the local Chart Datum so thatby adding them to the water depth shown on the chart,the navigator knows the total depth available for hisvessel. Because the Chart Datum is defined in terms ofthe local tidal characteristics, it is not a horizontal plane.Geodetic datums, however, are theoretically horizontalplanes to within the accuracy of the survey that estab-lished them. As a result, the amount by which ChartDatum falls below Survey Datum is not constant, beinggreatest where the tidal range is greatest.

Although for practical navigation requirementstidal predictions are normally published as levelsabove Chart Datum, for scientific and engineering pur-poses it is normally more convenient to consider levelsrelative to a mean sea-level datum. The tidal levelsplotted in Figure 1.2a are relative to local mean sea level.

If a long series of hourly tidal predictions are exam-ined for the frequency with which each level occurs,certain values are found to be more probable thanothers. Highest and Lowest Astronomical Tides occurvery seldom whereas, in the case of semidiurnal tides asillustrated by a histogram of 19 years of Newlyn pre-dictions in Figure 1.5a, themost probable levels are nearMean High Water Neaps and Mean LowWater Neaps.Water levels are changing relatively quickly when pass-ing through the mean level and so this appears as aminimum in the frequency histogram, whereas itpauses or hangs at the high water and low water levels.As a general rule, one of the two humps has amaximumthat is noticeably greater than the other especially inshallow-water regions. Figure 1.5b shows the corre-sponding distribution for a year of predicted hourlytides at Karumba where diurnal tides prevail. Notethat the double-humped distribution characteristics ofsemidiurnal tides have been replaced by a continuousand slightly skew distribution. During the single yearanalysed at Karumba the diurnal tides failed to reacheither the Highest or Lowest Astronomical Tide levels.This was because the range of lunar declinations, whichhas an 18.61-year cycle, was relatively small in 1982.Several of the other tidal terms defined here are alsoshown relative to the distributions of Figure 1.5. Instatistical terms these distributions of levels are calledprobability density functions.

1.6 Basic statistics of sea levelsas time seriesThe full process of tidal analysis will be considered inChapter 4, but there are certain basic statistical ideasthat may be used to describe tidal patterns withoutapplying elaborate analysis procedures. The probabil-ity density functions of tidal levels shown in Figure 1.5are one useful way of representing some aspects of sea-level and tidal ranges. In this section we consider basicideas: the mean and the standard deviation, and alsothe variance and spectral analysis of a series of sea-levelmeasurements, made over a chosen period of time.

For most purposes it is useful to regard theobserved sea level as the combined result of threemain factors:

Observed sea level mean sea level tidal level residual level

These factors will be considered in turn in this book.The general representation of the observed level

X(t) that varies with time may be written:

Xt Z0t Tt Rt 1:2where Z0(t) is the MSL, which changes slowly withtime,

T(t) is the tidal part of the variation,R(t) is the residual component.Measurements of sea levels and currents are tradi-

tionally recorded as a series of hourly values; record-ings every 15 or every 6 minutes are also common. Forsatisfactory tidal analysis observations should extendover at least a lunar month, 709 hours, or better over ayear of 8766 hours. Suppose that we have M observa-tions of the variable X(t) represented by x1, x2, . . . xM,then the mean is given by the formula:

x 1M

x1 x2 x3 xm 1:3

or, in conventional notation,:

x 1M

XM1xm

Every value of the variable differs from the mean of theseries of observations by an amount that is called itsdeviation. The deviation of the mth observation isgiven by:

em xm x

1.6 Basic statistics of sea levels as time series

11


Note that the average value of emmust be zero becauseof the way in which the mean value x is calculated. Abetter way of describing the extent by which the valuesof x vary about the mean value x is to compute thevariance s2, the mean of the squares of the individualdeviations:

s2 1M

XM1

xm x2 1:4

which must always have a positive value. The squareroot of the variance, s, is called the standard deviationof the distribution of x about x.

A further extremely useful technique called Fourieranalysis represents a time series in terms of the distri-bution of its variance at different frequencies. Thebasic idea of Fourier analysis is that any time seriesmay be represented as the sum of a series of sines andcosines of frequencies that are multiples of the funda-mental frequency:

2MDt

X t Z0 t XM=2m1

Am cosmt XM=2m1

Bm sinmt

1:5where the coefficients Am and Bmmay be evaluated byanalysis of M values of X(t) sampled at constantintervals t. Z0 is the average value of Z0(t) over theperiod of observations. An alternative form ofEquation 1.5 is:

Xt Z0 XM=2m1

Hm cos mt gm 1:6

where

Hm ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2m B2m

qgm arctanBm=Am

Hm and gm are the amplitude and phase lag of the mthharmonic constituent of the function X(t). Note thatthe phase lags and angular speeds must be expressedconsistently in terms of radian or degree angularmeasure; see Section 4.2 for a discussion of the usualtidal notations.

The variance of this function about the mean valueZ0 is given by squaring the terms within the summa-tion symbol of Equation 1.6, and averaging over theperiod of observation. If this long multiplication is

carried through it is found that all of the cross-productterms are zero, except for terms of the form:

H2m cos2mt gm

the average value of which, over an integral number ofcycles, is 12H

2m. The total variance in the series is there-

fore given by:

12

XM=2m1

H2m

We shall see in Chapter 4 that this powerful statisticalresult, which states that the total variance of the seriesX(t) is the sum of the variance at each harmonicfrequency is very important for tidal analyses.

The methods of tidal analysis to be described inChapter 4, which enable separation of the tidal andnon-tidal components of the series X(t), have a con-dition that the two components are statistically inde-pendent. This means that the sum of their individualvariances gives the variance in the total observedseries:XM

k1 X kDt Z0 2

XMk1T

2 kDt XM

k1 kDt Total variance Tidal variance Non-tidal variance

Furthermore, the variance of the tides may be com-puted as the sum of the variance in each frequencyelement. The important conclusion is that the totaltidal variance in a series of observations is the diurnalvariance plus the semidiurnal variance plus the varianceat other higher and lower frequencies. In shallow waterthere may be tidal energy at higher frequencies becausesome semidiurnal tidal energy is shifted, mainly toquarter-diurnal and sixth-diurnal tidal periods. Theconcentrations of tidal energy in groups of similarperiod or frequency, which we discuss in detail inChapter 4, are called tidal species. The main species arethe daily (diurnal) and twice-daily (semidiurnal) groups.

We now consider how the tidal regime at a partic-ular location may be characterised by the way in whichthe variance is distributed among the different species.Table 1.1 summarises the distributions of variance forfour coastal sites. In Honolulu the total variance issmall, and contains nearly equal contributions fromthe diurnal and semidiurnal species. Tides from higherspecies due to shallow-water distortions are negligiblebecause Honolulu is adjacent to the deep ocean. AtMombasa the diurnal tides are larger than atHonolulu, but the tidal curves are dominated by the

Introduction

12


large semidiurnal tides. There is a narrow continentalshelf region adjacent to the Mombasa coast that gen-erates a small variance at higher frequencies. AtNewlyn the diurnal tides are very small and the semi-diurnal tides are very large. Shallow-water tides are asignificant factor, due to the effects of the extensivecontinental shelf that separates Newlyn from theAtlantic Ocean. Courtown has an exceptional tidalregime: the annual and other long period changes oflevel are unusually large, but the semidiurnal tides aresmall for a continental shelf site.

Mombasa and Honolulu have the smallest non-tidal or surge effect in the observed levels. The largestsurge residuals are found in the higher latitudes wherestorms are more severe, and in regions of extensiveshallow water.

The frequency distribution of variance shown inTable 1.1 can be taken a stage further. Figure 1.6 showsthe results of a frequency distribution of the varianceat the whole range of frequencies for a year of hourlysea-level measurements at New York. Theoretically,given a year of 8766 hourly observations, 4383 sepa-rate frequency components could be determined, butsuch a fine resolution gives a spectral plot, which isnoisy and irregular. A more satisfactory presentationis obtained by averaging the variance over severaladjacent harmonic components. In the case illustrated,the values are averaged over 60 elements, and plottedagainst a logarithmic vertical scale to accommodatethe great range of values obtained. The significance ofthe semidiurnal and diurnal tides is clear. Most of thenon-tidal variance is contained in the low frequencies,equivalent to periods of 50 hours or longer. Althoughweak, the fourth-diurnal tides stand out clearly abovethe background noise and there is a spectral peak in thevicinity of the third-diurnal tides near 0.12 cycles per

hour. The averaged values of variance plotted withinthe diurnal and semidiurnal species conceal a finestructure of variance concentration at a few particularfrequencies, which will be considered in more detail inChapter 3. The increase in spectrum energies at lowfrequencies is known as reddening.

Figure 1.7 is a very schematic timespace map ofthe main factors affecting sea level. It is drawn in termsof the timescales and the distance scales over whichthese factors operate. The approximate ranges of thevariations associated with each effect are shown; theshapes plotted are only indicative, but note that tidaleffects appear as narrow lines at times of one day andhalf a day. These are the diurnal and semidiurnal tides.Over long geological times, to the right of the diagram,many tectonic processes have changed land and sea

Table 1.1 Distributions of variance at representative sea-level stations. Long period tidal variations include annual, semi-annual andmonthly changes. Units are cm2.

Tidal

Long period Diurnal Semidiurnal Shallow-water Non-tidal Total

Honolulu Hawaii 9 154 157 0 35 355

Mombasa Kenya 5 245 7,555 2 19 7,826

Newlyn England 17 37 17,055 100 191 17,400

Courtown Ireland 116 55 284 55 222 732

4

6

8

10

12

14

16

18

20

Log

(Pow

er S

pect

ral D

ensi

ty m

m2

hour

)

0.0 0.1 0.2 0.3 0.4 0.5

Frequency (cph)

New York 2008

Figure 1.6 Distribution of variance in observed levels over one year(2008) at New York. The biggest peak (0.08 cph) is for the semidiurnaltidal changes of level, but diurnal tidal changes are also apparent.There are smaller peaks of variance in the fourth-diurnal and othertidal species due to distortions of the tide in shallow water.


13


levels; in the left bottom corner, over much shorterperiods of seconds, there are local wind waves. Theone-year timescale is often taken as the boundarybetween short- and long-termMSL changes. The max-imum values on the X and Y axes are the age andcircumference of the Earth, respectively.

Figure 1.8 shows on a global scale the way in whichthe sea-level variations due to the various factors arespatially distributed. Tidal levels (a) are most variable(red areas) in limited continental shelf areas; air pres-sure and wind variations are greatest at higher lati-tudes (b); the seasonal cycle (c) is most notable intropical areas and in the Gulf Stream and Kuroshiocurrents; low-frequency changes (d) are also in regionsof strong permanent currents; and high-frequencychanges are notable at higher latitudes. Figure 1.8e isan important example of a regional sea-level anomalypattern.

The measurement of variance is convenientlyrelated to the energy contained in a physical systemand descriptions of tidal dynamics (see Section 6.7) areoften expressed in terms of energy fluxes and energybudgets. The variance of a series of sea levels may berelated to the average potential energy of the watermass above and below a regional MSL (see

Chapter 12). In a similar way, for current speeds thevariance may be related to the average kinetic energyof the water movements.

The purpose of sea-level analysis is to represent thevariations in terms of a few significant parameters. InChapter 3 and Chapter 4 we discuss the availabletechniques for separating the regular tides from theother effects in detail. However, our ability to predictfuture sea-level changes depends first on the collectionof high-quality measurements for analysis. In the nextchapter we shall consider some of the basic principleson which these measuring systems operate and someof their individual advantages and disadvantages.

Through the twentieth century a series of scientificand technical advances has brought us to the currentstate of being able to map and to model ocean andshelf tides, and weather effects in great detail, usingsatellite altimeters and the processing power ofmodern computers. The results of various Earth-defining satellite altimetry and gravity missions arediscussed in Chapter 9. Today, one of the highestpriorities is to understand and reliably anticipatechanges in mean sea level (Figure 1.8f) and floodrisks, particularly those that may be due to globalclimate change.

40000

10

100

1000

10000

10001001010 101 1 1 112 1m 100m 4.5 bn

minutes hours years

Leng

th s

cale

(km

)

Time scale

days

Seichesand edge

waves0.03 m

Diurnal tides 0.11 m

Tropical surges

5 m

Extra-tropical Surges

1 m

Semidiurnal tides110 m

Tsunamis110 m

Rossby waves0.1 m

Present day ocean climate

changes 0.2 m

Seasonaleffects0.2 m

Global sea level changesGlacial cycles 100 m

Block

Crustalmovements

1000 m

Regional

Plate

Figure 1.7 A conceptual timespace map of reasons for sea-level changes. The effects vary from wind waves over a few seconds to geologicalmovements over millions of years.

Introduction

14


References1. (1) Darwin, G. H. 1911. The Tides and Kindred

Phenomena in the Solar System (3rd edition). London:John Murray. (2) Marmer, H. A. 1926. The Tide. New

York: D. Appleton and Company. (3) Dronkers, J. J.1964. Tidal Computations in Rivers and Coastal Waters.Amsterdam: North Holland Publishing Company. (4)Lisitzin, E. 1974. Sea Level Changes. Amsterdam:Elsevier. (5) Forrester, W. D. 1983. Canadian Tidal

(a) Tidal Amplitude (cm)

0 30 60 90 120 150

(b) RMS Daily Air Pressure (mbar)

0 3 6 9 12 15

(c) Annual Amplitude (cm)

0 2 4 6 8 10

(d) RMS Variability (cm)

0 3 6 9 12 15

(e) ENSO Anomaly (cm)

20 16 12 8 4 0 4 8 12 16 20

(f) Secular Trend (mm/yr)

10 8 6 4 2 0 2 4 6 8 10

Figure 1.8 Thumbnails of variability in sea level: (a) Amplitude of the tide defined as the sum of the amplitudes of the M2, S2, O1 and K1constituents.

(b) Rms variability (s in Equation 1.4) of daily mean values of air pressure (in mbar) which corresponds to rms variability (in cm) of daily sea leveldue to the inverse barometer effect.

(c) Amplitude of the annual cycle of sea level around the world.(d) Rms variability of sea level (not including the variability due to the seasonal cycles or long-term trends).(e) A regional example of the west to east transfer of sea level in the equatorial Pacific due to the El NioSouthern Oscillation (in this case for

January 1998).(f) Trends in sea level around the world over two decades from 1992.


15


Manual. Ottawa: Department of Fisheries and OceansCanada.

2. Doodson, A. T. and Warburg, H. D. 1941. AdmiraltyManual of Tides. London: His Majestys StationeryOffice.

3. Wood, F. J. 2001. Tidal Dynamics. Volume 1: Theoryand Analysis of Tidal Forces. Palm Beach, FL: TheCoastal Education and Research Foundation (CERF).

4. Pugh, D. T. 2004. Changing Sea Levels. Cambridge:Cambridge University Press.

5. Parker, B. B. 2007. Tidal Analysis and Prediction.NOAA Special Publication NOS CO-OPS 3.Washington, D.C.: U.S. Department of Commerce,National Oceanic and Atmospheric Administration,National Ocean Service.

6. Church, J. A., Woodworth, P. L., Aarup, T. andWilson, W. S. 2010. Understanding Sea-Level Rise andVariability. Chichester: Wiley-Blackwell.

7. Defant, A. 1961. Physical Oceanography: Volume II.Oxford: Pergamon Press.

8. (1) Defant, A. 1958. Ebb and Flow. Ontario:Ambassador Books. (2) Macmillan, D. H. 1966. Tides.London: CR Books. (3) Redfield, A. C. 1980. The Tides

of the Waters of New England and New York. WoodsHole: Woods Hole Oceanographic Institution. (4)Hicks, S. D. 2006. Understanding Tides. Washington,D.C.: U.S. Department of Commerce, National Oceanicand Atmospheric Administration, National OceanService.

9. Harris, R. A. 18971907. Manual of Tides: Appendicesto Reports of the U.S. Coast and Geodetic Survey.Washington, D.C.: Government Printing Office.

10. Cook, A. 1998. Edmond Halley: Charting the Heavensand the Seas. Oxford: Oxford University Press.

11. Woodworth, P. L. 2002. Three Georges and oneRichard Holden: the Liverpool tide table makers.Historic Society of Lancashire and Cheshire, 151, 1952.

12. Wheeler, W. H. 1893. Tidal Rivers. London: Longmans,Green and Co.

13. Cartwright, D. E. 1999. Tides: A Scientific History.Cambridge: Cambridge University Press.

14. Deacon, M. 1997. Scientists and the Sea, 16501900.Aldershot: Ashgate. (First published 1971, London:Academic Press.)

15. Reidy, M. S. 2008. Tides of History. Chicago: Universityof Chicago Press.

Introduction

16

E:/PUGH//9781107028197C02.3D 17 [1735] 14.12.2013 7:55AM

Chapter

2 Sea-level measuring systems

When you can measure what you are speaking about, and express it in numbers, you knowsomething about it.

Lord Kelvin

2.1 The science of measurementThe ocean is its own uncontrollable laboratory and theoceanographer who measures the properties of the seais an observational rather than an experimental scien-tist. Sea levels can be measured in situ, or by altimetry-satellite remote sensing. Technically the necessity ofmaking in situ measurements of sea level presentsmany challenges in terms of the logistics of travel tothe site, for deployment of the equipment, and for itssafe and reliable operation in a frequently hostileenvironment.

This chapter summarises methods of measuringchanges of sea levels over tidal and longer periods.The special requirements of tsunami monitoring arefurther discussed in Chapter 8. Measurements of cur-rents are not included here (but see Section 4.4 onanalyses of currents) as they are covered in manygeneral oceanographic textbooks. Measurements ofsea level by satellite altimetry, which are closely linkedto orbit computations, mean sea level (MSL), and theshape of the Earth, are discussed extensively inChapter 9.

When we talk about sea-level measuring systems,we mean much more than the sensor that detects thelevel of the sea surface. A full system includes struc-tures, sensors, recording devices, data transmission,checking procedures, data banking, and arrangementsfor users to access data on demand. Here we look at allthese components of end-to-end systems.

Traditionally gauges that measure sea level havebeen called tide gauges even though the variationstheymeasure include many influences, including tides.The term is so deeply embedded in popular, hydro-graphic, and even scientific usage, that we use the

terms here synonymously. The term sea-level gaugeis to be encouraged.

One of the major requirements in the measure-ment of variations of sea levels is the ability to resolvethe longer period variations from shorter period var-iations due to short period wind-induced waves. Thesewaves often have amplitudes that are much greaterthan the accuracy demanded for the long period aver-aged values. It is not unusual to require measurementsof tidally changing sea level to an accuracy of 0.01 m inthe presence of waves of 1.0 m amplitude. The designerof measuring systems must overcome the problem ofaveraging these waves in a way that produces no resid-ual distortion.

Measuring systems may be classified as eithercoastal or offshore measuring devices.

Coastal measurements of sea level have a longhistory; many countries operate networks of gaugesat selected sites, for flood warning, and to monitorMSL changes to assess the long-term risks of floodingfrom the sea. The British network of more than 40gauges is an example of a network that transmits thedata automatically to a central point for checking andeventual archiving in a special data bank. The NationalOcean Service of the United States operates a similarsystem with more than 250 gauges. Several nationalsystems are linked through the IntergovernmentalOceanographic Commission of UNESCO Global SeaLevel Observing System (GLOSS), which consists of300 key nationally operated gauges, to monitor longperiod changes in global sea level. GLOSS is part of theGlobal Ocean Observing System (GOSS). Many seriesof sea-level observations extend over several decades,and a few of them, which extend over more than ahundred years, are among the longest available series 17


of ocean measurements. Compared with many othertypes of ocean measurements, the equipment formeasuring sea level is cheap and easy to maintain.

Measurements of sea levels and currents away fromthe coast aremore difficult because there are no obviousfixed reference points. They are also far more expensivebecause ships are needed to deploy and recover theequipment. Even a modest specially equipped researchvessel costs upwards ofUS$20,000per day to operate. Asa result of these practical and financial difficulties, fewmeasurements offshore extend over more than a year,and even measurements over a month have becomepracticable only since the development of automaticdigital recording equipment. Tsunami warning systemscan include input from offshore sensors.

When planning a measuring programme to mon-itor sea levels and currents in a particular area, severaldesign factors must be considered. Where shouldmeasurements be made? For how long? To what accu-racy? Is datum stability and connection to a nationallevelling datum necessary for the coastal levels?Ultimately, can these requirements be met withinacceptable cost limits? The range of equipment nowavailable extends from relatively cheap, imprecise,short-term meters to accurate, long-term, internally

recording, and inevitably expensive, equipment soldin a specialised international market. For a relativelycrude local hydrographic survey the former may suf-fice, but for scientific studies the superior instrumentsand accuracies are essential. With a little care, targetaccuracies of 0.01 m in levels and 0.03 m/s in currentsare possible. Unfortunately it is easy to make badmeasurements, which appear to have this accuracy,but which are later found to contain unidentified sys-tematic or random errors when carefully checked.

When we measure any quantity we are trying todescribe it in terms of numbers; we may say that it isbig, but by comparison with what other entity? Wemay say that it has a value of 43, but this is meaninglesswithout properly specified units. We may say that ithas a value of 43 metres, but it would be more preciseto specify the value as 43.26 metres. It may also bemore accurate to give this more precise value, but onlyif the instruments and procedures used are genuinelycapable of achieving that resolution. Oceanographersgenerally prefer to use SI units; factors for convertingamong these units are given in Table 2.1.

It is important to realise that all measurements arecomparative, and that no measurement is perfectlyaccurate. The comparative nature of measurements

Table 2.1 Conversion factors to SI units

Multiply by

Levels

Conversion to metres

From feet 0.3048

fathoms 1.8288

Pressures

Conversion to pascals(newtons per square metre)

From dynes per square centimetre 0.1000

bars 100000

millibars 100

pounds per square inch 6894.8

Speeds

Conversion to metres per second

From feet per second 0.3048

knots 0.5144

kilometres per day 0.0116

Sea-level measuring systems

18


implies a system of units and standards established byinternational agreement and testing. The metre isdefined as the length of the path travelled by light ina vacuum in 1/299 792 458 of a second, and the secondis defined in terms of radiation from the caesium-133atom. Secondary standards are adequate for marinephysical measurements but it is important that meas-uring systems are regularly and carefully calibratedagainst these. An important distinction must bemade between the absolute accuracy of a measure-ment, which is its accuracy compared to an absoluteinternational standard or datum, and the relative accu-racy or sensitivity of the measuring system, which isthe accuracy with which the difference between indi-vidual measurements in a set has been determined. Forexample, a tide gauge may measure sea-level changesto 0.01 m, but because of inaccurate levelling or poormaintenance, its accuracy relative to a fixed datummay be in error by 0.05 m.

Although the accuracy of a single reading of sea levelmaybeaccurate atbest toonly0.01m, themeanof severalhourly values is potentially much more accurate, pro-vided that the error in each reading has a random prob-ability distribution. If this distribution is statisticallynormal the mean of n independent readings is moreaccurate than a single reading by a factor of the squareroot of n. On this basis, 100 readings would reduce theerror from 0.01 m to 1 mm. It would be unrealistic toclaim any greater accuracy than this for mean monthlyandannual sea levels (seeChapter10), becauseofuniden-tifiable systematic errors.Themain reasonwhyerrors areunlikely to be less than 1 mm is that the original singlereadings will also contain systematic errors due, forexample, to incorrect gauge calibration. Errors can onlybe reduced by sound observational techniques.

Access to real-time sea-level data is very importantfor several reasons. One major benefit is that any faultsthat occur in the tide gauge equipment can be identi-fied and fixed sooner, resulting in better long-term sea-level information. A second benefit is that the datanow become available to users who need real-timeinformation for many kinds of operational applica-tions including flood warning and port operations.

Tide gauge data are commonly transmitted to cen-tral locations, and are available on-line in near realtime through national and international data centres atlow cost and with high reliability. Telemetry methodsdepend on the applications and cost. Mobile telephonyis available for local communications, and satellitesystems such as Iridium provide worldwide telephone-

based links. Fixed landline and satellite broadband arealso affordable options. Many tide gauge agencies willalso have access to one or more of the geostationarymeteorological satellites, which provide complete lon-gitudinal coverage, but latitude coverage is limited toabout 75 north and south. The loggers of a tide gaugecan communicate with one of these satellites using adata collection platform within allocated time slots.Tsunami warning systems have been designed to usethis form of satellite telemetry. Most data that passthrough the geostation

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