Kai Borre Mathematical Foundation of Geodesy Kai Borre (Editor) Mathematical Foundation of Geodesy Selected Papers of Torben Krarup With 15 Figures PROF.DR.KAI BORRE Aalborg University Danish GPS Center Niels Jernes Vej 14 9220 Aalborg Denmark e-mail: [email protected] Library of Congress Control Number: 2006923696 ISBN-10 3-540-33765-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33765-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, 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. Cover design: Erich Kirchner Typesetting: camera-ready by the editor Production: Christine Jacobi Printing: Krips bv, Meppel Binding: Stürtz AG, Würzburg Printed on acid-free paper 30/2133/cj 5 4 3 2 1 0 Torben Krarup ∗ March 2, 1919 † November 24, 2005 VI Laudatio for Torben Krarup Levallois Medal by Helmut Moritz AcommitteeconsistingofallIAGPastPresidentsunanimouslyrecommended to award at this General Assembly the Levallois Medal to Dr.h.c. Torben Krarup. I have the honor to present the laudatio. Thisisaneasyandpleasanttask.EasybecauseKrarup,sinceabout1969, is generally recognized as the authority on physical geodesy. The name of “least-squarescollocation”is inseparablyconnected with Torben.I amproud thatthisideagoesbacktothetimewhenIwasthechairmanofanIAGStudy Group on Mathematical Methods in Physical Geodesy of which he was the most active and inspiring member. His“LettersonMolodensky’sProblem”senttothemembersofthisStudy Groupbecameaninfluentialinstrument(theinfluencereachedasfarastothe famous Swedish mathematician Lars H¨ormander) although he could not be persuaded to publish them. In this he followed Carl Friedrich Gauss (“Pauca sed matura”). He shows that one can influence the history of geodesy even without participating in the current paper industry. Born March 2, 1919 in Odder (Denmark), he studied first mathematics and physics and then geodesy in Copenhagen, finishing 1952. At the Danish Geodetic Institute he was instrumental in geodetic computations, having ac- tively participated in the construction of the electronic computer GIER built at the Geodetic Institute around 1960. So he was a pioneer also in this field. Torben Krarup is a wonderful person: kind, gentle, helpful, unselfish and, above all, inpeccably honest. He likes to spread his ideas in discussions with friends and in letters in a generous way. I do not know a person more worthy of the Levallois Medal. In the name of the international geodetic community, I congratulate you on this honor and thank you, Torben, for your great contributions to geodesy and for your equally great friendship. IhavethegreatpleasureandhonourtohandovernowtheLevalloisMedal and the related certificate to you, Torben Krarup. The certificate reads: The International Association of Geodesy awards the Levallois Medal to Torben Krarup In recognition of distinguished service to the Association and the science of geodesy in general. XXII IUGG/IAG general Assembly Birmingham, UK, July, 1999 Signed. Klaus-Peter Schwarz, IAG president Preface This book contains contributions from one of the outstanding geodesists of our time. From start to finish the reader will find wonderful insights and be stimulated by the manner in which the interdependence of various subjects emerge from these writings. The influence of Torben Krarup’swork has been tremendous, and by publishing this volume his achievements are made avail- able to everyone. Most of the material in the book has been hard to obtain. The idea of collectingitdatesbacktothe1980’s,whenHelmutMoritzaskedmeifitwere possibletopublishTorbenKrarup’sso-calledMolodenskyletters.Atthattime the difficulties of getting this project started prevented me from trying it. With the coming of TEX and my closer relationship with the publishing worldIsaw new possibilities.So I decided to create pdf-versionsof allpapers that never found their way into a periodical. Often I scanned the typed pa- pers,andopticalcharacterrecognitionsoftwaresecuredasurprisinglycorrect result. Formulas and tables had to be entered in LATEX and the figures were created by means of the powerful MetaPost. This last step took place on my travels abroad where I spent hundreds of hours in hotel rooms in Finland, France, Greece, Italy, Lithuania, Norway, Sweden, and the USA working on this manuscript. The next step was to scan all printed and published material. That job waseasier,butalsomorecomprehensivebecauseofthelargenumberofpages. Finallyallthematerialwascompiledintoabook.Theprojectwasundertaken because I felt strongly that having all of Torben’s writings accessible in one volume would be of great benefit to the geodetic community. The volumecontains a varietyof mathematicaltechniques useful for geodesy. One of Torben’s hallmarks is his broad knowledge of mathematics combined with a rare innovative ability to put relevant topics and concepts together. Modernstudentsofgeodesycanlearnalotfromhisselectionofmathematical tools for solving actual problems. He himself somewhere in the book says: ...Most people seem to be more interested in presenting new ideas than in VIII Preface transforming good ideas into reality. Torben’s writings are mathematically well founded and professionally relevant. It is rare nowadaysto find scientific papers that fulfill both requirements. Torben Krarup was very reluctant to publish. He found it difficult to put his ideas on paper. Typically, papers were re-written more than once, until a formulation was found that satisfied him. Sometimes this did not happen and papers were eventually put away.This painstaking process of finding the right expression often led to compact formulations which some readers find difficult to digest. To some extent he writes for the chosen few. He wants the reader to be actively involved, and therefore he puts high demands on him. Soitmayappearcontradictorythathebecamedisappointedwhenonlyafew people understood his main ideas after the writing had been so laborious. It is more important to be understood than to understand by yourself—he once remarked on this theme. It took a few years before someone understood his collocation theory. It was seen as a framework for theoretical investigations, not as a procedure for creating numbers. Oncehetoldmethatyounevershouldpublishmorethanathirdorahalf ofwhatyouknow.ThepapersthatIhavefollowedcloselyreallyliveuptothis principle. His paper on Helmert Geometry underwent at least three different derivations, all fundamentally different. Only very creative mathematicians can do this. I have co-authored papers with Torben that never passed the final test. In my opinion they were excellent and I remember my disappointment when he did not agree to publication. Because of this, Torben has left numerous unfinished manuscripts. They are not included in this volume because the paperspublishedherecoverallrelevantsubjects towhichhe hascontributed. As a colleague, he was very generous about sharing ideas. I think he did this with a twofold aim: To educate students and colleagues to use the best possiblemathematicaltools,andalsotostartaprofessionaldiscussiononthe subject matter. Never in my life have I experienced more fatiguing moments than those together with him. Most often they were tremendously fruitful! IbelieveTorben’sfinestmathematicalskillishisintuition.Oftenheknows theresultbeforeithasbeenderivedandproved.Mymostrecentmemoryisof theprolongedprocessofestablishingtheHelmertGeometry.Manycolleagues would ask for the result in the form of algorithms. He denied to give them awayashewaslookingforageometricdescriptionoftheproblemratherthan a set of algorithms. His curiosity was the driving force, the result a valuable supplementary gain. TorbenKrarup’sinterestingeodeticproblemscanbecomparedtoasculp- tor’s attention to each work. His interest declines when the solution emerges. The artist’s interest lies in the problem, not in the complete work. In this respect Torben Krarup is an artist. Faced with the life work of a great scientist one asks if there is an over- all structure in the topics and problems treated. Let me try to answer this question by looking at some of his major contributions. Preface IX In the late 1950s Torben worked in a team dealing with computational problems in geodesy. As an outcome of this work the possibility of building a transistorizedcomputer arose.Krarupcontributed much to its logic.It was called GIER and became an important tool for the study of the propagation of rounding errors in geodetic networks. This project also led to a better understandingoftheorganizationandstructureofgeodeticnetworkproblems. WiththepublicationofAContributiontotheMathematical Foundationof Physical Geodesy (1969) Torben created a well founded mathematical frame for a description of the reality of physical geodesy. The booklet introduced a new concept, namely collocation. It took a considerable time before the message was understood by a number of the leading geodesists. It actually happenedwhenHelmutMoritzbecamethepromotor.Anumberofgeodesists considered it as an algorithm for performing comprehensive computations; however,itwasmeantasacommonreferenceframefordescriptionofgeodetic observations,sothat they couldbe correctedproperlyfor the influence ofthe gravity field of the Earth. It was not unexpected that a paper dealing with details on how to treat geodetic observationequations followed.A finalversionofIntegrated Geodesy was published in 1978. The core concept is the definition of a local frame which is based on the reference potential. The classical geodetic observation equations are linearized in this frame and are used in an iterative collocation process. The iteration stops when a minimum condition is fulfilled. In parallel to this development Krarup was also occupied with a rigor- ous formulation of the problem of physical geodesy: Given the potential field of the Earth and some discrete geodetic observations like distances, angles, absolute gravity, height differences, deflections of the vertical etc., determine the surface of the physical Earth. By means of partial differential equations Krarup succeeded in a stringent formulation of a free value boundary prob- lem with oblique derivatives. The problem description was a reformulationof Molodensky’s problem of 1945. In 1973, after ten years of thinking, he wrote the celebrated Molodensky letters which were distributed to the members of astudygrouponthetopic.Perhapsthis washismostimportantcontribution to our science. Models for combining discrete geodetic measurements and the continuous gravity field of the Earth were very much on Torben’s mind. Therefore it was a natural continuation of his activities when he formulated the geodetic elasticity theory. In 1974 he published Foundation of a Theory of Elasticity For Geodetic Networks which is based on the fundamental duality between applied and numerical mathematics. During the later yearsTorbenKrarupandI workedoncertaingeneraliza- tions of the Helmert transformation. The draft paper Helmert Geometry—or the Importance of Symmetry looks at different aspects of the linear part, i.e. the translation,and the non-linear part, i.e. the rotationand change of scale. Itturnsoutthatsingularvaluedecompositionprovidesbetterinsightintothe problem than the classical solution. X Preface Over more than a generation Torben Krarup has continuously been in- trigued by central problems in geodesy. He succeeded in formulating them in a relevant mathematical framework. But equally important, he succeeded in solving the mathematical problems so formulated. Allow me to add a comment on publishing conditions then and now. As a state geodesist at the Geodetic Institute, Torben Krarup did not suffer from the contemporary pressure on publishing. When you have genuine and new resultsyoupublish.Cuttingandpastingpaperswasnotapartofhisworld.If he republished, it happened in order to correct earlier mistakes. He was very honest to himself and to the scientific community in which he worked. Whenhedecidedtopublishitwasmostofteninnon-refereedjournalsand reports. So when circumstances made it a necessity to go through the referee system he was often annoyed and angry. In many cases the result was not to publish. During the editing of the material contained in this book, I have come across situations where another editor certainly would have changed the pre- sentation. Only unpublished material has been properly copy-edited. It has led to minor changes, but always with due respect to Torben’s definite style. Acknowledgements AsubstantialfinancialsupportfromDepartmentofCommunicationTechnol- ogy, Aalborg University is gratefully acknowledged. MycolleaguesPetrHolotaandKlaus-PeterSchwarzhelpedinvariousways during the work. I highly appreciate this help. I’m grateful that Springer has agreed to publish the present volume. Throughout the process I have appreciated the smooth collaboration with Dr. Christian Wischel. ThisbookissetwithTEX,LATEX2ε,andAMS-TEXusingComputerMod- ern fonts. The masterful page layout and numerous intricate TEX solutions are due to Frank Jensen. John D. Hobby’s MetaPost should also be men- tioned. Most figures were created by combining Donald Knuth’s METAFONT and PostScript from Adobe. Using it is a sheer joy. Fjellerad, Kai Borre February 2006 [email protected] Contents 1 Linear Equations . . . . . . . . . . . . . . . . . . . . . 1 2 The Adjustment Procedure in Tensor Form . . . . . . . 17 3 The Theory of Rounding Errors in the Adjustment by Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 A Contribution to the Mathematical Foundation of Physical Geodesy . . . . . . . . . . . . . . . . . . . . . 29 5 A Remark on Approximation of T by Series in Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 On the Geometry of Adjustment . . . . . . . . . . . . . 93 7 Remarks to the Discussion Yesterday . . . . . . . . . . 101 8 Letters on Molodenskiy’s Problem . . . . . . . . . . . . 105 9 On the Spectrum of Geodetic Networks . . . . . . . . . 135 10 Mathematical Geodesy . . . . . . . . . . . . . . . . . . 153 11 Foundation of a Theory of Elasticity for Geodetic Networks . . . . . . . . . . . . . . . . . . . . . . . . . 159 12 Integrated Geodesy . . . . . . . . . . . . . . . . . . . . 179