Solving algebraic computational problems in geodesy and geoinformatics: the answer to modern challenges PDF
Preview Solving algebraic computational problems in geodesy and geoinformatics: the answer to modern challenges
Joseph L. Awange and Erik W. Grafarend Solving Algebraic Computational Problems in Geodesy and Geoinformatics The Answer to Modern Challenges Joseph L. Awange Erik W. Grafarend Solving Algebraic Computational Problems in Geodesy and Geoinformatics The Answer to Modern Challenges With 79 Figures DR.-ING. JOSEPH L. AWANGE DEPARTMENT OF GEOPHYSICS KYOTO UNIVERSITY KITASHIRAKAWA OIWAKE-CHO SAKYO-KU KYOTO-SHI 606-8502 JAPAN E-mail: [email protected] PROF. DR.-ING. HABIL. ERIK W. GRAFAREND DEPARTMENT OF GEODESY STUTTGART UNIVERSITY GESCHWISTER-SCHOLL-STRAßE 24D 70174 STUTTGART GERMANY E-mail: [email protected] Library of Congress Control Number: 2004114234 ISBN 3-540-23425-X 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, broad- casting, 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 springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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: E. Kirchner, Heidelberg Production: A. Oelschläger Typesetting: Camera-ready by the Authors Printing: Mercedes-Druck, Berlin Binding: Stein + Lehmann, Berlin Printed on acid-free paper 32/2132/AO 5 4 3 2 1 0 Thisbookisdedicatedtomyco-author,Ph.D.supervisorand mentorProfessorDr.-Ing.habil.Dr.h.c.mult.ErikW.Grafarend. JosephL.Awange September 2004 Preface Whilepreparingandteaching‘IntroductiontoGeodesyIandII’toun- dergraduatestudentsatStuttgartUniversity,wenoticedagapwhich motivatedthewritingofthepresentbook:Almosteverytopicthatwe taughtrequiredsomeskillsinalgebra,andinparticular,computeralge- bra!Frompositioningtotransformationproblemsinherentingeodesy andgeoinformatics,knowledgeofalgebraandapplicationofcomputer algebra software were required. In preparing this book therefore, we haveattemptedtoputtogetherbasicconceptsofabstractalgebrawhich underpinthetechniquesforsolvingalgebraicproblems.Algebraiccom- putational algorithmsuseful for solving problemswhich requireexact solutionstononlinearsystemsofequationsarepresentedandtestedon variousproblems.Thoughthepresentbookfocusesmainlyonthetwo fields,theconceptsandtechniquespresentedhereinarenonethelessap- plicabletootherfieldswherealgebraiccomputationalproblemsmight beencountered.InEngineeringforexample,networkdensificationand robotics apply resection and intersection techniques which require al- gebraicsolutions. Solutionofnonlinearsystemsofequationsisanindispensabletask in almost all geosciences such as geodesy, geoinformatics, geophysics (justtomentionbutafew)aswellasrobotics.Theseequationswhich require exact solutions underpin the operations of ranging, resection, intersectionandothertechniquesthatarenormallyused.Examplesof problemsthatrequireexactsolutionsinclude; • three-dimensional resection problem for determining positions and orientation of sensors, e.g., camera, theodolites, robots, scanners etc., VIII Preface • coordinate transformation to match shapes and sizes of points in differentsystems, • mappingfromtopographytoreferenceellipsoidand, • analyticaldeterminationofrefractionanglesinGPSmeteorology. Thedifficultyinsolvingexplicitlythesenonlinearsystemsofequations has led practitioners and researchers to adopt approximate numeri- calprocedures;whichoftenhavetodowithlinearization,approximate startingvalues,iterationsandsometimesrequirealotofcomputational time. In-order to offer solutions to the challenges posed by nonlinear systemsofequations,thisbookprovidesinapioneeringwork,theappli- cationofring andpolynomial theories,Groebner basis,polynomial re- sultants,Gauss-Jacobicombinatorial andProcrustesalgorithms.Users faced with algebraic computational problems are thus provided with algebraictoolsthatarenotonlyaMUST,butessentialandhavebeen out of reach. For these users, most algebraic books at their disposal haveunfortunatelybeenwritteninmathematicalformulationssuitable tomathematicians.Westrivetosimplifythealgebraicnotionsandpro- videexampleswherenecessarytoenhanceeasierunderstanding. Forthoseinmathematicalfieldssuchasappliedalgebra,symbolic computations and application of mathematics to geosciences etc., the book provides some practical examples of application of mathemati- calconcepts.Severalgeodeticandgeoinformaticsproblemsaresolved in the book using methods of abstract algebra and multidimensional scaling.Theseexamplesmightbeofinteresttosomemathematicians. Chapter 1 introduces the book and provides a general outlook on themainchallengesthatcallforalgebraiccomputational approaches. Itisamotivationforthosewhowouldwishtoperformanalyticalsolu- tions.Chapter2presentsthebasicconceptsofringtheoryrelevantfor those readers who are unfamiliar with abstract algebra and therefore prepare them for latter chapters which require knowledge of ring ax- ioms.Numberconceptfromoperationalpointofviewispresented.Itis illustratedhowthevarioussetsofnaturalnumbersN,integersZ,quo- tientsQ,realnumbersR,complexnumbers CandquaternionsHare vitalfordailyoperations.Thechapterthenpresentstheconceptofring theory. Chapter 3 looks at the basics of polynomial theory; the main object used by the algebraic algorithms that will be discussed in the book.Thebasicsofpolynomialsarerecapturedforreaderswhowishto refreshentheirmemoryonthebasicsofalgebraicoperations.Starting with the definition of polynomials, Chap. 3 expounds on the concept Preface IX ofpolynomialringsthuslinkingittothenumberringtheorypresented inChap.2.Indeed,thetheoremdevelopedinthechapterenablesthe solution of nonlinear systems of equations that can be converted into (algebraic)polynomials. HavingpresentedthebasicsinChaps.2and3,Chaps.4,5,6and 7 present algorithms which offer algebraic solutions to nonlinear sys- tems of equations. They present theories of the procedures starting withthebasicconceptsandshowinghowtheyaredevelopedtoalgo- rithms for solving different problems. Chapters 4, 5 and 6 are based on polynomial ring theory and offer an in-depth look at the basics of Groebner basis,polynomial resultants andGauss-Jacobi combinatorial algorithms.Usingthesealgorithms,userscandeveloptheirowncodes tosolveproblemsrequiringexactsolutions. InChap.7,theGlobalPositioningSystem(GPS)andtheLocalPo- sitioningSystems(LPS)thatformtheoperationalbasisarepresented. TheconceptsoflocaldatumchoiceoftypesE∗ andF∗ areelaborated andtherelationshipbetweenLocalReferenceFrameF∗andtheglobal referenceframeF•,togetherwiththeresultingobservationalequations arepresented.Thetestnetwork“StuttgartCentral”inGermanythat we use to test the algorithms of Chaps. 4, 5 and 6 is also presented in this chapter. Chapter 8 deviates from the polynomial approaches to present a linear algebraic (analytical) approach of Procrustes that has found application in fields such as medicine for gene recognition andsociologyforcrimemapping.Thechapterpresentsonlythepartial Procrustes algorithm. The technique is presented as an efficient tool forsolvingalgebraicallythethree-dimensionalorientationproblemand thedeterminationofverticaldeflection. FromChaps.9to15,variouscomputationalproblemsofalgebraic naturearesolved.Chapter9looksattherangingproblemandconsiders boththeGPSpseudo-rangeobservationsandrangingwithintheLPS systems,e.g.,usingEDMs.Thechapterpresentsacompletealgebraic solutionstartingwiththesimpleplanarcasetothethree-dimensional ranging in both closed and overdetermined forms. Critical conditions where the solutions fail are also presented. Chapter 10 considers the Gauss ellipsoidal coordinates and applies the algebraic technique of Groebnerbasistomaptopographicpointsontothereferenceellipsoid. Theexamplebasedonthebalticsealevelprojectispresented.Chapters 11 and 12 consider the problems of resection and intersection respec- tively. X Preface Chapter13discussesamodernandrelativelynewareaingeodesy; the GPS meteorology. The chapter presents the theory of GPS mete- orology and discusses both the space borne and ground based types of GPS meteorology. The ability of applying the algebraic techniques toderiverefractionanglesfromGPSsignalsispresented.Chapter14 presentsanalgebraicdeterministicversiontooutlierproblemthusdevi- atingfromthestatisticalapproachesthathavebeenwidelypublicized. Chapter15introducesthe7-parameterdatumtransformationproblem commonlyencounteredinpracticeandpresentsthegeneralProcrustes algorithm.SincethisisanextensionofthepartialProcrustesalgorithm presentedinChap.8,itisreferredtoasProcrustesalgorithmII.The chapter further presents an algebraic solution of the transformation problem using Groebner basis and Gauss-Jacobi combinatorial algo- rithms.ThebookiscompletedinChap.16bypresentinganoverview ofmoderncomputeralgebrasystemsthatmaybeofusetogeodesists andgeoinformatists. ManythankstoProf.B.Buchbergerforhispositivecommentson ourGroebnerbasissolutions,Prof.D.Manochawhodiscussedthere- sultantapproach,Prof.D.Coxwhoalsoprovidedmuchinsightinhis twobooksonrings,fieldsandalgebraicgeometryandProf.W.Kellerof StuttgartUniversityGermany,whosedoorwasalwaysopenfordiscus- sions. We sincerely thank Dr. J. Skidmore for granting us permission to use the Procrustes ‘magic bed’ and related materials from Myth- web.com. Thanks to Dr. J. Smith (editor of Survey Review), Dr. S. J. Gordon and Dr.D. D. Lichti for granting us permission to use the scanner resection figures appearing in Chap. 12. We are also grate- fultoChapmanandHallPressforgrantinguspermissiontouseFig. 8.2 where malarial parasites are identified using Procrustes. Special thankstoProf.I.L.Drydenforpermittingustorefertohisworkand allthehelp.ManythankstoMsF.WildforpreparingFigs.11.7and 12.7.Weacknowledgeyoureffortsandvaluabletime.Specialthanksto Prof. A. Kleusberg of Stuttgart University Germany, Prof. T. Tsuda of Radio Center for Space and Atmosphere, Kyoto University Japan, Dr. J. Wickert of GeoForschungsZentrum Potsdam (GFZ) Germany and Dr. A. Steiner of the Institute of Meteorology and Geophysics, UniversityofGraz,Austriaforthesupportintermsofliteratureand discussionsonChap.13.ThedatausedinChap.13wereprovidedby GeoForschungsZentrumPotsdam(GFZ).Forthese,theauthorsexpress theirutmostappreciation. Preface XI Thefirstauthoralsowishestoexpresshisutmostsincerethanksto Prof.S.TakemotoandProf.Y.FukudaofDepartmentofGeophysics, KyotoUniversityJapanforhostinghimduringtheperiodofSeptember 2002 to September 2004. In particular Chap. 13 was prepared under the supervision and guidance of Prof. Y. Fukuda: Your ideas,sugges- tions and motivation enriched the book. For these, we say “arigato gozaimashita”–Japaneseequivalenttothankyouverymuch.Thefirst author’sstay atKyotoUniversitywassupported byJapan Societyof PromotionofScience(JSPS):Theauthorisverygratefulforthissup- port.ThefirstauthorisgratefultohiswifeMrs.Naomi Awange and histwodaughtersLucy andRuth whoalwaysbrightenedhimupwith their cheerful faces. Your support, especially family time that I de- niedyouin-ordertopreparethisbookisgreatlyacknowledged.Naomi, thanksforcarefullyreadingthebookandcorrectingtypographicaler- rors.However,theauthorstakefullresponsibilityofanytypographical error. Last but not least, the second author wants to thank his wife UlrikeGrafarend,hisdaughterBirgit andhissonJens forallsupport overthesemanyyearstheywerefollowinghimatvariousplacesaround theGlobe. Kyoto(Japan)andStuttgart(Germany) JosephL.Awange September2004 ErikW.Grafarend
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