Geomathematics: Modelling and Solving Mathematical Problems in Geodesy and Geophysics

467 Pages·2022·86.024 MB·English

by  Volker Michel

GEOMATHEMATICS
Geomathematics provides a comprehensive summary of the mathematical prin-
ciples behind key topics in geophysics and geodesy, covering the foundations of
gravimetry,geomagnetics,andseismology.Theoremsandtheirproofsexplainwhy
physicalrealitiesingeosciencearethelogicalmathematicalconsequencesofbasic
laws.Thebookalsoderivesandanalyzesthetheoryandnumericalaspectsofestab-
lishedsystemsofbasisfunctionsandpresentsanalgorithmforcombiningdifferent
types of trial functions. Topics cover inverse problems and their regularization,
the Laplace/Poisson equation, boundary-value problems, foundations of potential
theory, the Poisson integral formula, spherical harmonics, Legendre polynomials
andfunctions,radialbasisfunctions,theBiot–Savartlaw,decompositiontheorems
(orthogonal, Helmholtz, and Mie), basics of continuum mechanics, conservation
laws,modellingofseismicwaves,theCauchy–Navierequation,seismicrays,and
travel-time tomography. Each chapter ends with review questions, with solutions
forinstructorsavailableonline,providingavaluablereferenceforgraduatestudents
andresearchers.
volker michel isamathematicsprofessorattheUniversityofSiegen,wherehe
foundedthegeomathematicsgroup.Heisalsoaneditor-in-chiefoftheInternational
Journal on Geomathematics and a member of the editorial boards of the Journal
of Geodesy and Mathematics of Computation and Data Science. Previous works
include Lectures on Constructive Approximation (2013) and Multiscale Potential
Theory (2004) with Willi Freeden. Michel has organized several conferences on
inverseproblemsandgeomathematics.

GEOMATHEMATICS
Modelling and Solving Mathematical Problems
in Geodesy and Geophysics
VOLKER MICHEL
UniversityofSiegen
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,
NewDelhi–110025,India
103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108419444
DOI:10.1017/9781108297882
©CambridgeUniversityPress2022
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2022
PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-108-41944-4Hardback
Additionalresourcesforthispublicationatwww.cambridge.org/9781108419444.
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Tomyparents
RitaMichel(1938–2015)
andWalterMichel(1937–2019)

Contents
1 Introduction page1
2 RequiredMathematicalBasics 4
2.1 SomeImportantDefinitions 4
2.2 AShortCourseonTensors 7
2.3 Derivatives:NotationsandMore 11
2.4 SomeTheoremsonIntegration 13
2.5 SelectedTopicsofFunctionalAnalysis 17
2.6 CurvesandSurfaces 45
3 OnGravitation,HarmonicFunctions,andRelatedTopics 50
3.1 TheGravitationalPotential 50
3.2 SomeFundamentalPropertiesofHarmonicFunctions 70
3.3 Boundary-ValueProblems,Green’sFunction,andLayerPotentials 79
3.4 TheSphereasaParticularBoundary 115
3.5 ABriefExcursiontoOtherDimensions 137
Exercises 139
4 BasisFunctions 142
4.1 SphericalAnalysis 142
4.2 SphericalHarmonicsandLegendrePolynomials 151
4.3 VectorSphericalHarmonics 168
4.4 TensorSphericalHarmonics 183
4.5 OntheMultitudeofTrialFunctions 191
4.6 SlepianFunctionsontheSphere 194
4.7 RadialBasisFunctionsandSobolevSpaces 211
4.8 BasisFunctionsonthe3DBall–Briefly 232
4.9 ABestBasisAlgorithm 237
Exercises 263
vii
viii Contents
5 InverseProblems 264
5.1 Example1:DownwardContinuation 264
5.2 Example2:InverseGravimetry 266
5.3 BasicTheoryofInverseProblemsandTheirRegularization 271
5.4 BestBasesforIll-PosedInverseProblems 305
Exercises 322
6 TheMagneticField 325
6.1 TheGoverningEquations 325
6.2 TheGaußRepresentationandInnerandOuterSources 327
6.3 The Spherical Helmholtz Decomposition and the Mie
Representation 329
6.4 InternalandExternalFieldsundertheMieRepresentation 344
Exercises 349
7 MathematicalModelsinSeismology 350
7.1 ContinuumMechanicswithFocusonElasticity 350
7.2 SpecificsandSimplificationsfortheElasticBodyEarth 374
7.3 PropagationofBodyWavesandNormalModes 386
7.4 AnInverseProbleminSeismology 406
Exercises 423
AppendixA HintsfortheExercises 425
AppendixB QuestionsforUnderstanding 428
References 433
Index 447
1
Introduction
Ichbehaupteaber,dassinjederbesonderenNaturlehrenursovieleigentliche
Wissenschaftangetroffenwerdenkönne,alsdarinMathematikanzutreffenist.
[ButImaintainthatineveryspecialnaturaldoctrineonlysomuchscienceproper
istobemetwithasmathematics.]
ImmanuelKant,1786
EratosthenesofCyrene,wholivedbasicallyinthethirdcenturyBC,wasoneofthosefirst
mathematicianswhoseknowledgeandabilitiesattheseearlystagesofhumancivilization
wereremarkable.Besideshismethodforseekingprimenumbers,heparticularlyalsocon-
tributed to the measurement of the Earth by, for example, determining its circumference.
Inthisrespect,hemighthavebeenthefirstgeomathematician,oratleastoneofthefirst.
Manymorefollowedhim,anddefinitelyCarlFriedrichGaußmustbementionedhere,as
he can be seen as the greatest genius in mathematical history. His works and their influ-
ence are widespread in mathematics, and they are also of essential importance in various
applications,inparticularand(intheauthor’spossiblybiasedpointofview)firstofallin
Earthsciences,especiallygeomagneticsandpotentialtheory.Theawareness,whichreaches
back to classical antiquity, that mathematics is the foremostly required skill and toolbox
forunderstandingtheobjectsandprocessesthatsurroundushasbeenpreserveduptothe
present.Ithasnicelyandmoregenerallybeenputinanutshellbytheprecedingquotation,
whichisfromKant(1786);fortheEnglishtranslation,seeKant(1883).Overthecenturies,
Earthsciencesandmathematicshavebothadvanced.Whiletheachievementsatthetimeof
Eratosthenesandhisfellowsarenowadayspartsofthecurriculaatschools,manymodern
challengesingeosciencesareequallychallengestotwenty-firstcenturymathematics.
The importance of mathematics for the understanding of the entire phenomenologies
whichareassociatedtotheEarthwasrecognizedandhighlightedbytheinitiativeMathe-
maticsofPlanetEarth(MPE),whichwaslaunchedbyUNESCOin2013.Sincethen,the
interestingeomathematicshasgrownextensivelyandmanypublicationshaveoccurredin
thewakeofMPE.Certainly,theEarthisascomplexasitisversatile.Therefore,onesingle
bookcannotcoverthewholemathematicswhichmodelsprocessesoccurringintheEarth,at
itssurface,andintheatmosphere.Thisnewbookthatyouarecurrentlyreadingconcentrates
on specific topics: gravitation, magnetics, and seismology, though also these fields could
easily fill books of the same size on their own. Moreover, this book is not only devoted
to the modelling of these physical areas, but also to the mathematical foundations which
1