Handbook of Mathematical Geodesy: Functional Analytic and Potential Theoretic Methods

938 Pages·2018·37.58 MB·English

by  Willi Freeden

Geosystems Mathematics
Serieseditors
W.Freeden
Kaiserslautern,Germany
M.Z.Nashed
Orlando,Florida,USA
Thisseriesprovidesanidealframeandforumforthepublicationofmathematical
key technologiesand their applicationsto geo-scientific and geo-related problems.
Current understanding of the highly complex system Earth with its interwoven
subsystems and interacting physical, chemical, and biological processes is not
only driven by scientific interest but also by the growing public concern about
the future of our planet, its climate, its environment, and its resources. In this
situation mathematics provides concepts, tools, methodology, and structures to
characterize, model, and analyze this complexity at various scales. Modern high
speed computersare increasinglyenteringallgeo-disciplines.Terrestrial,airborne
as well as spaceborne data of higher and higher quality become available. This
fact has not only influenced the research in geosciences and geophysics, but also
increased relevant mathematical approaches decisively as the quality of available
datawasimproved.
Geosystems Mathematics showcases important contributions and helps to
promote the collaboration between mathematics and geo-disciplines. The closely
connectedseriesLectureNotesinGeosystemsMathemacticsandComputingoffers
theopportunitytopublishsmallbooksfeaturingconcisesummariesofcutting-edge
research,newdevelopments,emergingtopics, andpracticalapplications.AlsoPhD
theses may be evaluated, provided that they represent a significant and original
scientificadvance.
Editedby
• WilliFreeden(UniversityofKaiserslautern,Germany)
• M.ZuhairNashed(UniversityofCentralFlorida,Orlando,USA)
Inassociationwith
• Hans-PeterBunge(MunichUniversity,Germany)
• RoussosG.Dimitrakopoulos(McGillUniversity,Montreal,Canada)
• YalchinEfendiev(TexasA&MUniversity,CollegeStation,TX,USA)
• AndrewFowler(UniversityofLimerick,Ireland&UniversityofOxford,UK)
• BulentKarasozen(MiddleEastTechnicalUniversity,Ankara,Turkey)
• JürgenKusche(UniversityofBonn,Germany)
• LiqiuMeng(TechnicalUniversityMunich,Germany)
• VolkerMichel(UniversityofSiegen,Germany)
• NilsOlsen(TechnicalUniversityofDenmark,KongensLyngby,Denmark)
• HelmutSchaeben(TechnicalUniversityBergakademieFreiberg,Germany)
• OtmarScherzer(UniversityofVienna,Austria)
• FrederikJ.Simons(PrincetonUniversity,NJ,USA)
• ThomasSonar(TechnicalUniversityofBraunschweig,Germany)
• Peter J.G. Teunissen, Delft University of Technology, The Netherlands and
CurtinUniversityofTechnology,Perth,Australia)
• Johannes Wicht (Max Planck Institute for Solar System Research, Göttingen,
Germany).
Moreinformationaboutthisseriesathttp://www.springer.com/series/13389
Willi Freeden • M. Zuhair Nashed
Editors
Handbook of Mathematical
Geodesy
Functional Analytic and Potential
Theoretic Methods
Editors
Willi Freeden M. Zuhair Nashed
Geomathematics Group Department of Mathematics
TU Kaiserslautern University of Central Florida
Kaiserslautern, Germany Orlando, FL, USA
ISSN 2510-1544 ISSN 2510-1552 (electronic)
Geosystems Mathematics
ISBN 978-3-319-57179-9 ISBN 978-3-319-57181-2 (eBook)
https://doi.org/10.1007/978-3-319-57181-2

Library of Congress Control Number: 2018940865

Mathematics Subject Classification (2010): 86A30, 86A20, 31A25, 47A52, 65A20, 60J45

© Springer International Publishing AG, part of Springer Nature 2018
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are
believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors
give a warranty, express or implied, with respect to the material contained herein or for any errors or
omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.

Cover illustration: designed by Prof. Dr. Schreiner, Buchs

Printed on acid-free paper

This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company
Springer International Publishing AG part of Springer Nature.
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
HandbookofMathematical Geodesy
Geosystems Mathematics,v–vi
(cid:2)c SpringerInternational PublishingAG,partofSpringerNature2018
Contents
Preface .................................................................. vii
W. Freeden
Introduction ........................................................ ix
W. Freeden, T. Sonar, and B. Witte
Gauss as Scientific Mediator Between Mathematics and Geodesy
from the Past to the Present ........................................ 1
M. Augustin, S. Eberle, and M. Grothaus
An Overview on Tools from Functional Analysis .................... 165
W. Freeden and M.Z. Nashed
Ill-Posed Problems: Operator Methodologies of Resolution
and Regularization .................................................. 201
W. Freeden and H. Nutz
Geodetic Observables and Their Mathematical Treatment
in Multiscale Framework ............................................ 315
F. Sanso`
The Analysis of the Geodetic Boundary Value Problem:
State and Perspectives .............................................. 459
M. Grothaus and T. Raskop
Oblique Stochastic Boundary Value Problem ........................ 491
M. Augustin, W. Freeden, and H. Nutz
About the Importance of the Runge–Walsh Concept
for Gravitational Field Determination ............................... 517
W. Freeden, H. Nutz, and M. Schreiner
Geomathematical Advances in Satellite Gravity
Gradiometry (SGG) ................................................ 561
M. Gutting
Parameter Choices for Fast Harmonic Spline Approximation ........ 605
vi Contents
W. Freeden and M. Zuhair Nashed
Inverse Gravimetry as an Ill-Posed Problem in
Mathematical Geodesy .............................................. 641
C. Blick, W. Freeden, and H. Nutz
Gravimetry and Exploration ........................................ 687
W. Freeden, V. Michel, and F.J. Simons
Spherical Harmonics Based Special Function Systems and
Constructive Approximation Methods ............................... 753
C. Gerhards
Spherical Potential Theory: Tools and Applications ................. 821
C. Gerhards, S. Pereverzyev Jr., and P. Tkachenko
Joint Inversion of Multiple Observations ............................ 855
S. Leweke, V. Michel, and R. Telschow
On the Non-Uniqueness of Gravitational and Magnetic
Field Data Inversion (Survey Article) ............................... 883
Index .................................................................... 921
HandbookofMathematical Geodesy
Geosystems Mathematics,vii–viii
(cid:2)c SpringerInternational PublishingAG,partofSpringerNature2018
Preface
Geodesy, as most other disciplines, spans activities ranging from theoretical to
applied border lines. In the twenty-firstcentury, geodesy is strongly influenced by
two scenarios: First, the technological progress, in particular, space observation
has opened fundamentally new methods of measurements. Second, high speed
computers have led to a strong “mathematization”. As a consequence, geodesy
is in great shape. However, the width and depth of new geodetic challenges will
simultaneously require basic analysis and understanding of all technologically as
well as mathematically driven components. These requirements are inextricably
necessaryto providefuture improvementsindiversefields ofgeodeticallyinvolved
public concern for our planet such as climate environment, expected shortage of
natural resources, etc.
This “Handbook of Mathematical Geodesy” deals with mathematics as the
key technology for modeling purposes and analysis of today’s geodetic measure-
ments and observations. It supplies deep modern and cutting-edge mathematical
knowledgeas transfer methodology fromthe reality space ofmeasurements to the
model space of mathematical structures and solutions, and vice versa. Essential
interest is laid in studying the gravitational field usually in macroscopic sense,
where the quantum behavior of gravitation may not be taken in account. More-
over,ingeodeticallyreflectedEarth’sgravitywork,velocitiesthatareencountered
areconsiderablysmallerthanthe speedofthe light.Asaconsequence,Newtonian
physics can be safely used.
In detail, this Handbook is concerned with the following selection of topical
areas:
• functional analysis and geodetic functional models
• constructive polynomial, spline and wavelet approximations
• mathematical treatment of geodetic observables and multiscale integrated
concepts
• geodeticboundaryvalueproblemsandobliquestochasticderivativeproblems
• Runge–Walshmono-andmulti-poleexpansionsongeodeticreferencesurfaces
such as sphere, ellipsoid, telluroid, geoid, real Earth’s surface
• regularizationmethods of ill-posed and inverse problems
• gravimetric and gradiometric (multiscale) modelling.
The objective of the handbook is twofold: on the one hand it serves as a
self-consistent collection of newsworthymaterial at the graduate-student level for
viii Preface
allmembersofthemathematicalcommunityinterestedinanyofthediverseprob-
lemsrelevantintoday’sgeodesy.Ontheotherhand,thebookrepresentsavaluable
reference for all geodesists facing innovative modeling supplies involving recently
measureddatasets intheir professionaltasks.For both groupsthe Handbook pro-
vides important perspectives and challenges in crossing the traditional frontiers.
The Handbook consolidates the current knowledge by providing summaries
andconcepts asa guideforgeodetictransferfromrealityspace(“measurements”)
to virtuality space (“models”). All in all, the work is an authoritative forum of-
fering appropriatemathematical means ofassimilating,assessing,andreducing to
comprehensible form the flow of measured data and providing the methological
basis for scientific interpretation, classification testing of concepts, modeling, and
solution of problems.
The editors wish to express their particular gratitude to the people who not
only made this handbook possible, but also made it extremely satisfactory:
• The contributors to the handbook, who dedicated much time, effort, and cre-
ative energy to the project. The handbook evolved continuously throughout
the recruitment period, as more and more facets became apparent, many
aspects were entirely new at the time of recruitment.
• Thefolks at Birkh¨auser, particularlyClemensHeine,whoinitiatedthewhole
work and gave a lot of encouragement and advice.
• Helga Nutz, Geomathematics Group of the University of Kaiserslautern, for
reading most of the proofs and giving valuable comments.
Thank you very much for all exceptional efforts and support in creating a work
offeringexcitingdiscoveriesandimpressiveprogress.Wehopethatthe“Handbook
of Mathematical Geodesy” will stimulate and inspire new research achievements
in geodesy as well as mathematics.
February 2017 Willi Freeden, Kaiserslautern
M. Zuhair Nashed, Orlando
HandbookofMathematical Geodesy
Geosystems Mathematics,ix–xiv
(cid:2)c SpringerInternational PublishingAG,partofSpringerNature2018
Introduction
Willi Freeden
In natural extension to the classical definition due to F.R. Helmert [2], geodesy is
the science that deals with the measurement and modeling of the Earth, including
its gravity field. So, the basis of geodetic science is its measurements, i.e., scalar
numbers, vectors, tensors such as distances, angles, directions, velocities, acceler-
ations. In this respect, the relevance of the gravityfield manifests itself in twofold
sense: from the need to handle heights and from the determination of the Earth’s
shape. Consequently, geodesy realizesa physical rather than a geometricalunder-
standing of height by observing that a point is higher than another if water flows
from the first to the second. In other words, “geometric” obligations do not allow
tobeseparatedfromphysicalones.The gravityfieldisstillpresent,asthe driving
force.
Nowadays, geodesy as a measuring discipline is in greatshape. In fact, com-
puterfacilitiesaswellasmeasurementandobservationmethodsopennewresearch
areas and opportunities. However, it is geodetic trademark to present measured
values always together with a suitable modeling procedure for interpretation and
an appropriate knowledge and estimation about reliability and accuracy. Follow-
ingR.Rummel[6],thisdiligencedemonstratesthegeodesistsroleasnotaryofthe
Earth.Asanevidentconsequence,however,thisnotarialroleexplainsthatgeodesy
is more than a discipline concerned only with measurements. Inherently, mathe-
matics is implied as key technology bridging the real world of measurements and
thevirtualworldofhandlingdatasets,modelinggeodeticquantitiesandprocesses,
and providingillustrations and interpretations. Once more, the resultof measure-
mentsarenumbers,vectors,tensors,i.e.,rawmaterial.Mathematicalhandlingand
approximationofdatasetsaswellasmodelingtechniquesarenecessarytoconnect
the“realityspace”withthe“virtualityspace”.Inthissense,amodelrepresentsthe
result of the transfer, it intends to be an image of the reality, expressed in math-
ematical language, so that an interaction between abstraction and concretization
is involved. The mathematic’s world of numbers and structures contains efficient
tokensbywhichweareabletodescribetherule-likeaspectofarealproblem.This
description includes a simplification by abstraction: essential properties of, e.g.,
a certain geodetic problem are separated from unimportant ones and a solution
scheme is set up. The “eyefor similarities”enables mathematicians to recognizea