Handbook of mathematical geodesy: functional analytic and potential theoretic methods PDF
Preview Handbook of mathematical geodesy: functional analytic and potential theoretic methods
Geosystems Mathematics Willi Freeden M. Zuhair Nashed Editors Handbook of Mathematical Geodesy Functional Analytic and Potential Theoretic Methods 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. Sans`o 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
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