Table Of ContentGeosystems Mathematics
Willi Freeden
M. Zuhair Nashed
Michael Schreiner
Spherical
Sampling
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 • Michael Schreiner
Spherical Sampling
Willi Freeden M. Zuhair Nashed
Mathematics Department Mathematics Department
University of Kaiserslautern University of Central Florida
Kaiserslautern, Germany Orlando, Florida, USA
Michael Schreiner
Institute for Computational Engineering
University of Applied Sciences
of Technology NTB Buchs
Buchs, Switzerland
ISSN 2510-1544 ISSN 2510-1552 (electronic)
Geosystems Mathematics
ISBN 978-3-319-71457-8 ISBN 978-3-319-71458-5 (eBook)
https://doi.org/10.1007/978-3-319-71458-5
Library of Congress Control Number: 2018938096
Mathematics Subject Classification (2010): 33C55, 35Q75, 62D05, 65D05
© Springer International Publishing AG, part of Springer Nature 2018
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Preface
The objective of sampling in Euclidean spaces is to deal with the reconstruction
of a (bandlimited) function (or more generally by a function in an appropriate
Hilbert space) by an infinite sum in terms of the function itself at a discrete set
of points. Sampling is a keystone of signal processing, the most common form
of Euclidean approaches is Shannon-type lattice point sampling of a bandlimited
signal. By virtue of Shannon-type lattice point sampling, perfect reconstruction
of a signal from lattice point samples is possible when the samples are taken at a
certain rate related to the bandwidth.
Aparticularformofrecovery,thatisofimportanceinmanybranches,issampling
withinasphericalcontext,brieflycalledspherical sampling.Geomathematicallyit
reflectsboththeapproximateshapeoftheEarth’ssurfaceandthetypicalsatellite
geometry of a low Earth orbiter. Application of spherical sampling theory perme-
ates many areas of geosciences, such as signal analysis, geoexploration by gravita-
tionaland/ormagneticdata,seismicreflectiontomographyandimaging,radaras
wellaslasertechnology,downwardcontinuationofsatellitedata,andmanyothers.
These aspects naturally make spherical sampling theory an offspring of spherical
interpolationandapproximation.Ourinterestisinreconstructionanddecomposi-
tionformulascorrespondingtodifferenttypesofpointsetsonspheresandvarious
observables often naturally characterized by (rotation-invariant) pseudodifferen-
tial operators. Large structured point sets, e.g., latitude-longitude distributions
occurinobservationalmethodsbysatellites,scatteredpointsystemsoftenarethe
result of terrestrial measurements reflecting, e.g., topographic features, which do
notallowageometricallyregularmeasurementpattern.Itturnsoutthattheclose
relationshipbetweensphericalsamplingandreproducingkernelHilbertspacethe-
ory has its manifestations in various polynomial, i.e., spherical harmonics, spline,
and wavelet concepts and settings.
This textbook on spherical sampling mainly is a result of cooperative research
on scientific problems on the sphere during the last two decades. The work deals
with a synopsis of recent contributions in spherically reflected sampling under geo-
mathematicallysignificantaspects.Themajorpurposeistogiveaconsistentand
unifiedoverviewofdiverseresultsandvariousdevelopmentsinthefieldoftoday’s
spherical sampling, particularly those arising in mathematical geosciences such as
mathematical geodesy and geophysics. Although the book often refers to original
contributions, we have tried to make it accessible to (graduate) students and sci-
entists not only from mathematics but also from geosciences and geoengineering.
Theessentialobjectivesaretoprovideanadditiontothelibraryofanyindividual
v
vi Preface
interested in spherical sampling theory and to show how advances in this theory
lead to new discoveries in mathematical, geodetic, geophysical as well as other
scientific branches, for example, neuro-medicine.
To summarize, this book substantially represents a collection of valuable material
inthediversesamplingareas,however,inaconsistentandunifiedsetup.Thegoal
oftheworkistoconvincemembersfromvariousgeodisciplines(andprobablyalso
medical sciences) that spherically oriented sampling provides a rich mathematical
cornucopiathathasmuchtooffertoalargepaletteofapplications.Theefficiency
of our sampling methods is demonstrated by representative examples, typically
originated from geodetic challenges.
Many people generously devoted considerable time to make valuable suggestions
andcomments.Weparticularlywishtothankallco-authorsofearlierpublications
consistently integrated in the work. Many thanks also go to C. Gerhards, Vienna,
Austria, M. Gutting, Siegen, Germany, V. Michel, Siegen, Germany, H. Nutz,
Kaiserslautern, Germany, A. Plattner, Fresno and Princeton, USA, R. Rummel,
Munich, Germany, O. Scherzer, Vienna, Austria, F. Simons, Princeton, USA.
February, 2018
Kaiserslautern, Orlando, Buchs W. F., M. Z. N., M. S.
In memory of
Margret Freeden (1949 - 2017)
in acknowledgment of her helpful support and inspiring encouragement
About the Authors
Willi Freeden
Studies in mathematics, geography, and philosophy at the RWTH Aachen, 1970
philosophicum 1971 Diplom in mathematics, 1972 Staatsexamen in mathemat-
ics and geography, 1975 PhD in mathematics, 1979 Habilitation in mathematics,
1981/1982visitingresearchprofessorattheOhioStateUniversity,Columbus(De-
partment of Geodetic Science and Surveying), 1984 professor of mathematics at
theRWTHAachen(InstituteofPureandAppliedMathematics),1989professorof
technomathematics(industrialmathematics),1989FellowoftheInternationalAs-
sociation of Geodesy (IAG), 1994 head of the Geomathematics Group, University
of Kaiserslautern, 2002-2006 Vice-President for Research and Technology of the
University of Kaiserslautern. Author, editor, and coeditor of 15 books. Published
over190papers,severalexpositorypapers,andbookchapters.2009editorinchief
of the Springer “GEM International Journal on Geomathematics”, 2010 editor in
chief of the Springer “Handbook of Geomathematics”, 2013 editor of the (Ger-
man) Springer-Spektrum “Handbuch Tiefe Geothermie”, 2014 editor in chief of
the Birkh¨auser book series “Geosystems Mathematics”,2014 editor inchief of the
Birkh¨auser lecture notes “Geosystems Mathematics and Computing”, 2015 editor
inchiefofthesecondeditionoftheSpringer“HandbookofGeomathematics”,2018
editor in chief of the (German) Springer-Spektrum “Handbuch Oberfl¨achennahe
Geothermie, 2018 editor in chief of the (German) Springer-Spektrum “Handbuch
derGeod¨asie”,2018editorinchiefoftheBirkh¨auser“HandbookofMathematical
Geodesy”, 2018 member of the editorial board of more than fifteen international
journals. 1996 member of the German Geodetic Commission of the Bavarian
Academy of Sciences, Munich. Organized over 30 international conferences and
mini-symposia.
M. Zuhair Nashed
S.B. and S.M. degrees in electrical engineering from MIT and PhD in Mathe-
matics from the University of Michigan. Served for many years as Professor at
ix
x AbouttheAuthors
Georgia Tech and the University of Delaware. Held visiting professor positions
at the University of Michigan, University of Wisconsin, AUB, and KFUPM, and
distinguished visiting scholar positions at various universities worldwide. Recip-
ient of the Lester Ford Award of the Mathematical Association of America, the
Sigma Xi Faculty Research Award and sustained Research Award in Science, Dr.
ZakirHusainAwardoftheIndianSocietyofIndustrialandAppliedMathematics,
and several international awards. Fellow of the American Mathematical Society
(InauguralClassof2013).Publishedover140papersinmathematicsandphysics,
30 papers in applied sciences and engineering, and 20 expository papers and book
chapters.Author,editor,coeditorof11books.Editoroftwojournalsandmember
of editorial board of about 30journals,includingfourSpringerjournals,2010editor
in chief of the Springer “Handbook of Geomathematics”, 2014 editor in chief of
the Birkh¨auser book series “Geosystems Mathematics”,2014 editor inchief of the
Birkh¨auser lecture notes “Geosystems Mathematics and Computing”, 2015 editor
inchiefofthesecondeditionoftheSpringer“HandbookofGeomathematics”,2018
editor in chief of the Birkh¨auser “Handbook of Mathematical Geodesy”, plenary
lectures at meetings of 10 mathematical and engineering societies, and over 400
invited hour talks at conferences and colloquia. Organized over 30 international
conferences and mini-symposia.
Michael Schreiner
Studies in industrial mathematics, mechanical engineering, and computer science
at the University of Kaiserslautern, 1991 Diplom in industrial mathematics, 1994
PhDinmathematics,2004Habilitationinmathematics.1997–2001researcherand
project leader at the Hilti Corporation in Schaan, Liechtenstein, 2002 Professor
for Industrial Mathematics at the University of Applied Sciences NTB, Buchs,
Switzerland. 2004 Head of the Department of Mathematics of the University of
Applied Sciences NTB, Buchs, since 2004 also lecturer at the University of Kai-
serslautern.Since2009foundingdeanandheadoftheInstituteforComputational
Engineering at the NTB in Buchs, where currently 18 professors and researchers
areemployed.ThecorecompetenciesoftheInstituteforComputationalEngineer-
ing are mathematical modeling and simulation, data sciences, and optimization.
Published over 40 papers and book chapters and three books. Member of the
editorial board of two journals.
