Solving Algebraic Computational Problems in Geodesy and Geoinformatics

344 Pages·2005·2.66 MB·English

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

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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