Spectral Methods for the Estimation of the Effective Elastic Thickness of the Lithosphere

472 Pages·2022·17.394 MB·English

by  Jonathan Kirby

Advances in Geophysical and Environmental
Mechanics and Mathematics
Jonathan Kirby
Spectral Methods
for the Estimation
of the Effective
Elastic Thickness
of the Lithosphere
Advances in Geophysical and Environmental
Mechanics and Mathematics
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Jonathan Kirby
Spectral Methods
for the Estimation
of the Effective Elastic
Thickness of the Lithosphere
JonathanKirby
SchoolofEarthandPlanetarySciences
CurtinUniversity
Perth,WA,Australia
ISSN 1866-8348 ISSN 1866-8356 (electronic)
AdvancesinGeophysicalandEnvironmentalMechanicsandMathematics
ISBN 978-3-031-10860-0 ISBN 978-3-031-10861-7 (eBook)
https://doi.org/10.1007/978-3-031-10861-7
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ForRuth
Preface
Theeffectiveelasticthickness—commonlyknownbyitsmathematicalsymbol,T —
e
describes an important property of the lithosphere, namely its flexural rigidity, or
resistancetoflexurebyloading.Butbesidesdescribingthisparticularbehaviourof
theEarth,T hasbeenshowntoplayavitalroleintectonicevolution,withzonesof
e
lowT deformingmuchmorereadilythanregionswithhighelasticthickness,such
e
as the ancient cratons that core the continents. However, its estimation has proven
difficult and even controversial. One of the main reasons for this is that there are
three approaches one can take to find its value, and while these methods do not
yield wildly different answers, often they do not agree. One approach is thermo-
rheologicalmodelling,whichcombinesknowledgeofthepropertiesofrocksunder
variousconditionswithinformationabouttheregionaltemperatureandstateofstress,
andfromthesedeterminestheintegratedstrengthofthelithosphereandthence T .
e
The other two approaches both use the topography and gravity field of the Earth.
The method known simply as forward modelling fits predictions of the gravity or
topographyfromthin,elasticplatemodelstoactualobservationsofthesedatasets,
with T being one of the primary controlling factors. Inverse modelling finds the
e
statisticalrelationshipbetweengravityandtopographyinthefrequencydomain—
embodiedinthecoherenceandadmittance—andestimatesT byinversionofthose
e
spectralmeasuresagainstaplatemodel.Itisthethirdmethodthatisthesubjectof
thisbook.
To date, while the topics of T and plate flexure have formed a small part of
e
broad-contentgeophysicstextbooks,therehasbeenonlyonebookdedicatedtothem:
2001’sIsostasyandFlexureoftheLithospherebyTonyWatts.Thisexcellentvolume
containsalmosteverythingoneneedstoknowabouttheeffectiveelasticthickness,
but while it has a whole chapter on spectral methods of T estimation, it does not
e
informthereaderonhowtocalculatethecoherenceandadmittance.Thatiswhere
mybookcomesin.Correctestimationofthegravityandtopographyspectraiscrucial
inordertoavoidbiasintheresultingT estimates,andtherehasbeenmuchdebatein
e
theliteratureoverthepasttwodecadesastohowthisshouldbedonemostaccurately.
I first came across the subject of T estimation during my Ph.D. at Edinburgh
e
University,andalthoughmythesiswasprimarilygeodesy,Icouldn’tresistincluding
vii
viii Preface
ashortchapteronsomeT estimatesImade,essentiallybecauseIfoundthesubject
e
so captivating. I loved the way it brought together such diverse topics as spectral
estimation, geology, algebra and calculus, differential equations and finite differ-
ence approximations, civil engineering, signal processing, fractal geometry, plate
tectonics,mechanicsandelasticitytheory,andallthisjusttofindoutwhathappens
whenyoupopavolcanoonthecrust.Butthereinlaytheproblem.Inordertogetup
tospeedonallthesedifferentsubjects,oneisthrownintoablizzardofjournalsand
textbooksinphysics,geophysics,geology,engineeringandmathematics.Itisboth
dauntingandtime-consuming,andthenotesImadewereallovertheplace.Soone
day,Isetabouttidyingthemup,andintheprocesswroteabook,thekindofbookI
wouldhavewantedwhenIwasaPh.D.student.
Mybookisintendedtocontaineverythingonemightneedwhenembarkingon
studiesinT estimationusingspectralmethods,sothatonedoesn’talwaysneedto
e
consultmyriadothersources.Itrytoderiveequationsfromfirstprincipleswherever
possible,togiveafullerunderstandingofthesubject.Ialsofocusonthepractical
applicationofthetheory,whichisoftenhalfthebattle.Thebookisdividedintothree
parts. Part I is short, non-mathematical and sets the scene, briefly describing what
wemeanbythelithosphere’seffectiveelasticthicknessandwhyitisimportant.This
istheonlypartwherefurtherreading,whilenotessential,isprobablyagoodidea,
especiallyifoneisnewtothesubject.
PartIIconcernsspectralestimation.Manygeophysicstextbooksandjournalarti-
clesglossoverthissubject,saying‘...thentaketheFouriertransformand...’,butitis
actuallythemostimportantpartofT estimation:getthespectrumwrongandyou’ll
e
get T wrong. So I spend a considerable number of pages describing how Fourier
e
transformsintheoryareverydifferenttoFouriertransformsinpractice.PartIIthen
continueswithachaptereachonmultitaperspectralestimationandthecontinuous
wavelettransform,nowthetwomostpopulartechniquesofT estimation.Thereis
e
a chapter devoted to the admittance and coherence and their computation, and to
mapprojections,asthisisalsoanimportant—andoft-neglected—aspectofplanar
spectral estimation of ellipsoidally coordinated data. Importantly, Part II is stand-
alone,writtenwithhardlyanymentionofT .Forthisreason,itwillbeusefultoany
e
scientistwhoneedstheFouriertransformbutisn’tterriblywell-versedinitsuse(and
abuse).
PartIIIgetstothecoreofthesubject,whichisreallyT estimationusingDonald
e
Forsyth’s load deconvolution method. But since the observables are gravity and
topography,eachgetsachaptertoitself.The‘topographychapter’concernselasticity
andflexure;the‘gravitychapter’dealswithpotentialtheoryandphysicalgeodesy.
Thenthereisthecapstonechapteronloaddeconvolution,supplementedbyachapter
onitspracticalapplicationtoreal-worlddata.Butreal-worlddataarenottheonly
kind, so I have included a chapter on generating synthetic gravity and topography
modelsfromtheflexureofathin,elasticplate—modelsthatcanbeusedtotestthe
accuracyofthespectralanalysismethods.
Thebookisaimedprimarilyatgeologistsandgeophysicistsofadvancedunder-
graduateorpostgraduatelevelandhigher,andistargetedspecificallyatthosewishing
toestimateT .However,becausemostofitscontentisderivedfromfirstprinciples,
e
Preface ix
andIhavetriednottoassumetoomuchpriorknowledge,itmayalsobenefitthose
whowishtolearnaboutmultitaperspectralestimationorthewavelettransform,or
elasticityorthefinitedifferencemethod,forexample,andhavenointerestintheEarth
sciencesatall.Forbestresults,thereaderisencouragedtoreadTonyWatts’sbook,
andalsoareviewarticleIwrotein2014forTectonophysics,whichsummarisesand
chartsthehistoryof T estimationusingspectralmethods.Thedifferencebetween
e
thisbookandthatarticleisthattheformerintroducesthesubjectfromscratchwhile
the latter was intended more as a useful reference so that established researchers
couldfindwhodidwhatandwhenwithoutwastingtime.
Whilewritingthisbookhasbeenasoloeffort,itwouldnotexistwereitnotforthe
collaborationsandconversationsIhavehadwithcolleaguesaroundtheworld.Ithank
themsincerely.ForemostamongsttheseisChrisSwain,whomonegrantreviewer
once described as a ‘sage geophysicist’. He is that indeed, and I have enjoyed our
worktogetheroverthepast20yearsimmensely.Ihavealsobeenluckyenoughto
spend time with some of the other leading researchers in the field, notably Tony
Lowry,MartaPérez-Gussinyé,FrederikSimonsandMarkWieczorek;Ithankthem
fortheirwisdom,knowledgeandcompany.Andofotherswhohaveinfluencedmy
thinkingandthushadanindirectinputintothisbook,IwouldliketothankAlberto
Jiménez-Díaz, my Ph.D. supervisor Roger Hipkin and his colleague Roger Banks
atEdinburgh,andtheunfortunatelylateSimonHolmes,who—onememorableday
inVesuvio—stronglyencouragedmetoreadPercivalandWalden(1993).Andfor
theirmoredirectinput,IthankTonyLowryforprovidinghisyieldstrengthenvelope
code,whichIusedtogenerateFigs.1.12–1.15,TonyWattsforsupplyingFig.1.16
and Frederik Simons for assistance with a tricky derivation in Sect. 3.3.2. I would
alsoliketoacknowledgetheGenericMappingTools(GMT)software[Wesseletal.
(2019)GeochemGeophysGeosystvol.20],whichIusedtoplotmostofthefigures.
Andfinally,Ithankmywife,Ruth,whosepatienceisnolessthangodlike.
Perth,Australia JonathanKirby
December2021
Contents
PartI Context
1 Isostasy,FlexureandStrength .................................. 3
1.1 Isostasy ................................................ 3
1.1.1 Beginnings ..................................... 3
1.1.2 Pressure ........................................ 7
1.1.3 Airy-HeiskanenIsostasy .......................... 9
1.1.4 Pratt-HayfordIsostasy ............................ 11
1.2 FlexuralIsostasy ........................................ 12
1.2.1 RegionalSupport ................................ 13
1.2.2 Crust,Mantle,LithosphereandAsthenosphere ....... 15
1.3 TheSignificanceofT ................................... 20
e
1.3.1 PlateStrength ................................... 20
1.3.2 T —CausesandEffects ........................... 26
e
1.4 WhatThisBookDoesNotCover .......................... 28
1.5 Conventions ............................................ 30
1.6 Summary .............................................. 30
1.7 FurtherReading ......................................... 31
References .................................................... 32
PartII Spectra
2 TheFourierTransform ........................................ 37
2.1 Introduction ............................................ 37
2.1.1 Dimensionality .................................. 37
2.1.2 Harmonics ...................................... 38
2.1.3 ContinuousSignalsVersusDiscreteSequences ....... 38
2.2 FourierTheory .......................................... 39
2.2.1 FourierSeries ................................... 40
2.2.2 TheContinuousFourierTransform ................. 42
2.2.3 Amplitude,PowerandPhaseSpectra ............... 43
2.2.4 SignalTranslation ............................... 45
xi