Gravity Inversion and Integration: Theory and Applications in Geodesy and Geophysics

391 Pages·2017·6.05 MB·English

by  Lars E. Sjöberg

Lars E. Sjöberg
Mohammad Bagherbandi
Gravity
Inversion and
Integration
Theory and Applications in Geodesy
and Geophysics
Gravity Inversion and Integration
ö
Lars E. Sj berg Mohammad Bagherbandi
(cid:129)
Gravity Inversion
and Integration
Theory and Applications in Geodesy
and Geophysics
123
LarsE. Sjöberg Mohammad Bagherbandi
Division of Geodesy andSatellite Division of Geodesy andSatellite
Positioning Positioning
RoyalInstitute ofTechnology RoyalInstitute ofTechnology
Stockholm Stockholm
Sweden Sweden
and
Department ofIndustrial Development,
ITandLandManagement
University of Gävle
Gävle
Sweden
ISBN978-3-319-50297-7 ISBN978-3-319-50298-4 (eBook)
DOI 10.1007/978-3-319-50298-4
LibraryofCongressControlNumber:2016963159
©SpringerInternationalPublishingAG2017
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Preface
This textbook provides a comprehensive overview of gravity integration and
inversion, which contributes to physical geodesy and geophysics, and it identifies
classicalandmoderntopicsforstudyingtheEarth.Itdiscussesboththeoreticaland
practical aspects, e.g., for the determination of a precise geoid model besides
presenting ample worked examples. Physical geodesy terminology is used
throughoutthebook.TheunprecedentedknowledgeoftheEarth’sgravityfieldand
its temporal variation are progressively capturing the attention of the geosciences
for many reasons. As a result of recent dedicated satellite missions, knowledge
oftheglobaltoregionalgravityfieldhasreachedextraordinarylevelsofqualityand
resolution. The modeling of the Earth’s mass distributions in the crust and its
interior, as well as the temporal changes/transports of such masses, is most
important in studying geodynamics. The enhanced knowledge of the 3D-layered
structureoftheEarthwillimproveourcapabilitytounderstand,monitorandpredict
geophysical processes, which potentially threaten our technically developed soci-
ety. Today, thanks to the development of atomic clocks, the original idea of
A. Bjerhammar from 1975 of chronometric leveling for direct measurement of
geopotential differences is emerging as a fascinating new tool in geodetic and
geophysical applications.
TheaimofthisbookistoprovidestudentsattheM.ScandPh.D.levels,aswell
as researchers, basic and some in-depth knowledge about the current and recent
theoryandapplicationofgravityforgeodesyandgeophysics,asseenmainlyfroma
geodesist’s perspective and with an emphasize on theory. Physical geodesy is
treated rather generally, and a main goal is to provide the reader with a theory
(the KTH method) for determining “the 1-cm geoid” (including both geoid and
quasigeoid methods), while the geophysical applications of gravity are limited to
thedeterminationofcrustaldepthanddensitycontrastatthecrust/mantleboundary,
and stress and viscosity in the upper mantle, as well as some simple examples of
howtoestimatethemassanddepthofsomeotherlarge-massstructuresintheEarth
from combinations of the geopotential, gravity, and gravity gradients. Temporal
changes of the gravity field are treated with emphasis on long-term trends, while
periodic changes are more sparsely utilized.
v
vi Preface
Anothermotivationbehindthisbookis,tohonourthecentennialanniversaryof,
and,to some extent, announce and revive some research ideas originatingwith the
deceased A. Bjerhammar (1917–2011) at the Royal Institute of Technology
(KTH) in Stockholm, Sweden. Bjerhammar is well-known for his research in
physicalgeodesyandgeodeticstudiesontheFennoscandialand-upliftphenomenon
anditsrelationtotheregionalgravityfieldanduppermantleviscosityintheregion,
as well as for being a pioneer in developing a theory for linear adjustment of
erroneous observations using generalized matrix algebra. Unfortunately, much of
his research canonlybefoundindisparatepapers presentedinvariousjournals,at
conferences and as internal reports from KTH. As the first author of this book has
been a M.Sc and Ph.D. student of Bjerhammart from 1969 to 1975, his co-worker
until1980,andhissuccessorsince1984,thisbooktosomeextentsoughttoreflect
and further continue his research ideas, as well as on other ideas developed and
partly published in papers, etc.
ThesecondauthorwasaPh.D.studentatKTHduring2007–2011andapostdoc
2011–2014, with the prime author as his supervisor on the determination of the
crustaldepthanddensitycontrastfromEarthgravitationalmodelsusinganisostatic
model as a specialty in his research. He currently performs multidisciplinary
researchandisactiveinseveraldirections,suchasingeophysics,geodesyandland
surveying (applied geodesy). His main research interest is to develop and study
relations between geodesy and geophysics especially study on temporal changes
of the Earth’s gravity field and glacial isostatic adjustment.
Stockholm, Sweden Lars E. Sjöberg
March 2017 Mohammad Bagherbandi
Acknowledgements
WeacknowledgethatProfessorArtuEllmannandAssociateProfessorJonasÅgren
spent part of their summer vacation in 2016 to read a draft version of this book.
Their valuable comments significantly improved the final manuscript.
Wethankthefollowingpublishersandjournalsforgrantingpermissionstoreuse
some previously published material of our own: Springer for articles in Acta
Geodaetica et Geophysics Hungary, Pure and Applied Geophysics, Surveys in
Geophysics, Encyclopedia of Geodesy and Journal of Geodesy, Studia Geodaetica
etGeophysica,SurveysinGeophysicsandalsodeGruyterforarticlesinJournalof
GeodeticScienceandJournalofAppliedGeodesy,and,finally,Elsevierforarticles
in Earth Science Review and Physics of the Earth and Planetary Interiors.
Thanks also go to Petra van Steenbergen, Executive Editor in Earth Sciences,
Geography and Environment, and Hermine Vloemans, project coordinator, at
Springer for their help to prepare the book.
Finally we acknowledge the Division of Geodesy and Satellite Positioning at
Royal Institute of Technology (KTH) for providing a good working environment
and the opportunity to complete this book.
vii
Contents
1 Introduction.... .... .... ..... .... .... .... .... .... ..... .... 1
1.1 Contents of the Book. ..... .... .... .... .... .... ..... .... 1
1.2 The Subject Field.... ..... .... .... .... .... .... ..... .... 2
1.3 The Development of the Subject Field Before the Last
Millennium Shift .... ..... .... .... .... .... .... ..... .... 5
1.4 Recent Developments in Gravimetric Theory and Data ..... .... 7
1.4.1 Development of Gravimetric Data... .... .... ..... .... 7
1.4.2 Development of Theory .. .... .... .... .... ..... .... 8
1.5 Reference System, Reference Frame and Datum.. .... ..... .... 10
1.5.1 More on Reference Systems ... .... .... .... ..... .... 14
1.5.2 Different Types of Reference Systems.... .... ..... .... 15
1.5.3 Major Geodynamical Effects on Reference Frames... .... 17
1.5.4 Geodetic Reference System 1980 ... .... .... ..... .... 22
References.. .... .... .... ..... .... .... .... .... .... ..... .... 23
2 Basic Mathematics... .... ..... .... .... .... .... .... ..... .... 27
2.1 Least Squares Adjustment Theory .... .... .... .... ..... .... 27
2.1.1 Adjustment by Elements.. .... .... .... .... ..... .... 28
2.2 Least Squares Collocation... .... .... .... .... .... ..... .... 32
2.2.1 Discrete Collocation . .... .... .... .... .... ..... .... 32
2.2.2 Continuous Collocation... .... .... .... .... ..... .... 33
2.3 Coordinate Systems .. ..... .... .... .... .... .... ..... .... 35
2.4 Legendre’s Polynomials .... .... .... .... .... .... ..... .... 41
2.5 Spherical Harmonics.. ..... .... .... .... .... .... ..... .... 43
2.5.1 Spectral Filtering and Combination.. .... .... ..... .... 46
2.6 Ellipsoidal Harmonics. ..... .... .... .... .... .... ..... .... 56
2.7 Fundamentals of Potential Theory .... .... .... .... ..... .... 57
2.7.1 Basic Concepts and Formulas.. .... .... .... ..... .... 57
2.7.2 Laplace’s and Poisson’s Equations .. .... .... ..... .... 60
2.7.3 Laplace’s Equation and Its Solution in Spherical
Coordinates... ..... .... .... .... .... .... ..... .... 61
ix
x Contents
2.7.4 Gauss’ and Green’s Integral Formulas ... .... ..... .... 62
2.7.5 Boundary Value Problems. .... .... .... .... ..... .... 65
2.8 Regularization... .... ..... .... .... .... .... .... ..... .... 66
2.8.1 Tikhonov Regularization.. .... .... .... .... ..... .... 69
2.8.2 Wiener Filtering .... .... .... .... .... .... ..... .... 72
2.8.3 Spectral Smoothing.. .... .... .... .... .... ..... .... 74
2.8.4 Spectral Combination .... .... .... .... .... ..... .... 74
2.8.5 Optimum Regularization.. .... .... .... .... ..... .... 76
2.8.6 Spherical Harmonic Analysis .. .... .... .... ..... .... 78
2.8.7 Comparison... ..... .... .... .... .... .... ..... .... 79
2.8.8 Concluding Remarks. .... .... .... .... .... ..... .... 80
Appendix: Answers to Exercises.. .... .... .... .... .... ..... .... 80
References.. .... .... .... ..... .... .... .... .... .... ..... .... 81
3 Classical Physical Geodesy..... .... .... .... .... .... ..... .... 83
3.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 83
3.2 Basic Concepts in Physical Geodesy .. .... .... .... ..... .... 84
3.2.1 The Gravity Field ... .... .... .... .... .... ..... .... 84
3.2.2 The Gravity Field of the Level Ellipsoid.. .... ..... .... 85
3.2.3 The Disturbing Potential, Geoid and Gravity Anomaly.... 89
3.2.4 Harmonic Expansion of the Gravity Field. .... ..... .... 92
3.3 Integral Formulas in Physical Geodesy. .... .... .... ..... .... 93
3.3.1 Poisson’s Integral ... .... .... .... .... .... ..... .... 94
3.3.2 Stokes’ Formula .... .... .... .... .... .... ..... .... 94
3.3.3 Hotine’s Formula ... .... .... .... .... .... ..... .... 96
3.3.4 Vening Meinesz’ Integrals .... .... .... .... ..... .... 97
3.3.5 The Vertical Gradient of Gravity ... .... .... ..... .... 98
3.3.6 The Inverse Vening Meinesz Formula.... .... ..... .... 99
3.3.7 The Geoid-from-Deflection Formula. .... .... ..... .... 101
3.3.8 Gradiometry Formulas on the Sphere .... .... ..... .... 102
3.4 Practical Considerations (DITE, DWC, SITE, PITE) .. ..... .... 106
3.4.1 The Free-Air Correction .. .... .... .... .... ..... .... 106
3.4.2 The Bouguer Correction .. .... .... .... .... ..... .... 107
3.4.3 The Direct Topographic Effect (DITE) ... .... ..... .... 108
3.4.4 The SITE, Co-geoid and the PITE .. .... .... ..... .... 108
3.5 Height Systems.. .... ..... .... .... .... .... .... ..... .... 110
3.5.1 Geopotential Numbers.... .... .... .... .... ..... .... 110
3.5.2 Orthometric Heights . .... .... .... .... .... ..... .... 111
3.5.3 Normal Heights..... .... .... .... .... .... ..... .... 112
3.5.4 Normal-Orthometric Heights... .... .... .... ..... .... 113
Appendix 1: Closed-Form Kernels .... .... .... .... .... ..... .... 115
Appendix 2: Solutions to Exercises.... .... .... .... .... ..... .... 116
References.. .... .... .... ..... .... .... .... .... .... ..... .... 118
Contents xi
4 Modern Physical Geodesy. ..... .... .... .... .... .... ..... .... 119
4.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 119
4.2 The Quasigeoid, Surface Gravity Anomaly and Disturbance . .... 124
4.3 Geoid Determination by Spherical Harmonics ... .... ..... .... 126
4.4 The Modified Stokes’ Formula... .... .... .... .... ..... .... 128
4.4.1 General Modification of Stokes’ Formula. .... ..... .... 128
4.4.2 Remove-Restore Techniques... .... .... .... ..... .... 131
4.4.3 Modifications Reducing the Truncation Error.. ..... .... 132
4.4.4 The Least Squares Modification of NL;M and NL;M... .... 135
1 2
4.4.5 Satellite Only Low Degree Modifications. .... ..... .... 140
4.4.6 Modifications with High-Degree EGMs .. .... ..... .... 142
4.5 Summary of Modified Stokes’ Formula Techniques... ..... .... 143
4.6 The Modified Hotine Formula ... .... .... .... .... ..... .... 144
References.. .... .... .... ..... .... .... .... .... .... ..... .... 145
5 Corrections in Geoid Determination . .... .... .... .... ..... .... 149
5.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 149
5.2 Topographic Corrections.... .... .... .... .... .... ..... .... 150
5.2.1 The Topographic Potential and Gravity Anomaly.... .... 151
5.2.2 The Indirect Effect on the Geoid.... .... .... ..... .... 152
5.2.3 The Combined Effect on the Geoid.. .... .... ..... .... 153
5.2.4 Zero- and First-Degree Effects . .... .... .... ..... .... 153
5.2.5 The Topographic Bias by a Strict Formulation. ..... .... 155
5.2.6 The EGM Analytical Continuation Error (EACE).... .... 159
5.2.7 The Topographic Bias in the Modified Stokes’
Formula.. .... ..... .... .... .... .... .... ..... .... 164
5.2.8 Lateral Topographic Density Variations .. .... ..... .... 166
5.3 The Downward Continuation Correction ... .... .... ..... .... 167
5.3.1 The Dwc Effect on the Original Stokes’ Formula.... .... 167
5.3.2 The Dwc Effect for the Modified Stokes’ Formula... .... 170
5.4 Atmospheric Corrections ... .... .... .... .... .... ..... .... 171
5.4.1 The IAG Approach.. .... .... .... .... .... ..... .... 171
5.4.2 The KTH Approach . .... .... .... .... .... ..... .... 172
5.5 Ellipsoidal Corrections ..... .... .... .... .... .... ..... .... 174
5.5.1 Components of the Ellipsoidal Correction
of Stokes’ Formula.. .... .... .... .... .... ..... .... 175
5.5.2 The Ellipsoidal Correction as a Harmonic Series
and a Stokes’ Integral.... .... .... .... .... ..... .... 176
5.6 Corrections in Quasigeoid Determination... .... .... ..... .... 179
References.. .... .... .... ..... .... .... .... .... .... ..... .... 179