Borre Mathematical Foundation of Geodesy

Selected Papers of Torben Krarup
1. Auflage 2006
ISBN: 978-3-540-33767-6
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 416 Seiten

This volume contains selected papers by Torben Krarup, one of the most important geodesists of the 20th century. The collection includes the famous booklet 'A Contribution to the Mathematical Foundation of Physical Geodesy' from 1969, the unpublished 'Molodenskij letters' from 1973, the final version of 'Integrated Geodesy' from 1978, 'Foundation of a Theory of Elasticity for Geodetic Networks' from 1974, as well as trend-setting papers on the theory of adjustment.

Kai Borre has been a professor of geodesy at Aalborg University since 1976. His professional interests are geodesy and satellite based positioning and navigation. He is the (co)author of seven textbooks and has published numerous scientific reviewed papers. He is also an editor of GPS Solutions. Dr. Borre is renowned for having written several hundred Matlab scripts for positioning. He is honorary doctor of the Vilnius Gediminas Technical University.

Autoren/Hrsg.

Borre, Kai

Inhaltsverzeichnis

1;Preface;7
2;Contents;11
3;1 Linear Equations;13
3.1;Introduction;13
3.2;Direct Methods: The General Case;13
3.3;Positive Definite Symmetric Matrix of Coefficients;22
3.4;References;27
4;2 The Adjustment Procedure in Tensor Form;28
5;3 The Theory of Rounding Errors in the Adjustment by Elements of Geodetic Networks;33
5.1;References;37
6;4 A Contribution to the Mathematical Foundation of Physical Geodesy;39
6.1;Exposition of the Ideas;39
6.2;I. Covariance and Collocation;44
6.3;II. The Least-Squares Method in Hilbert Spaces;56
6.4;III. Hilbert Spaces with Kernel Function and Spherical Harmonics;67
6.5;IV. Application of the Method;87
6.6;Appendix. Proof of Runge’s Theorem;93
6.7;Acknowledgements;99
6.8;References;99
7;5 A Remark on Approximation of T by Series in Spherical Harmonics;101
8;6 On the Geometry of Adjustment;103
8.1;Appendix;108
9;7 Remarks to the Discussion Yesterday;111
9.1;Remarks;113
10;8 Letters on Molodenskiy’s Problem;114
10.1;I. The Simple Molodenskiy Problem;114
10.2;II. The Mushroom Problem;125
10.3;III. A Mathematical Formulation of Molodenskiy’s Problem;128
10.4;IV. Application of the Prague Method on the Regular Molodenskiy Problem;139
10.5;References;143
11;9 On the Spectrum of Geodetic Networks;144
11.1;Abstract;144
11.2;Introduction;144
11.3;The Canonical Form of the Adjustment Problem;145
11.4;The Spectral Density of the Discrete Laplacian;147
11.5;The Spectral Distribution Function N(.);152
11.6;On the Smoothness and Roughness of the Eigenvectors of the Normal Equation Matrix;155
11.7;Green’s Formula for Trigonometric Networks;156
11.8;Difference and Differential Equations Corresponding to Networks With Various Types of Observation;158
11.9;Acknowledgment;160
11.10;References;160
12;10 Mathematical Geodesy;161
13;11 Foundation of a Theory of Elasticity for Geodetic Networks;166
13.1;1;166
13.2;2;167
13.3;3;169
13.4;4;173
13.5;5;175
13.6;6;179
13.7;7;182
13.8;References;185
14;12 Integrated Geodesy;186
14.1;Introduction;186
14.2;I;189
14.3;II;195
14.4;III;204
14.5;The Concept of Solution in Integrated Geodesy;207
14.6;The Challenge of Integrated Geodesy;211
14.7;References;213
15;13 On Potential Theory;214
15.1;Preface;214
15.2;1. Homogeneous Polynomials in Rq;214
15.3;2. Harmonic Polynomials;220
15.4;3. Series in Spherical Harmonics;233
15.5;4. Harmonic Functions on Bounded Sets;237
15.6;5. Harmonic Functions on Complements to Bounded Sets;244
15.7;6. Harmonic Functionals and the Green Transform;246
15.8;7. Regular Hilbert Spaces of Harmonic Functions— Duality 1;252
15.9;8. The Fundamental Kernels— Duality 2;256
15.10;9. Runge’s Theorem— Duality 3;259
16;14 La Formule de Stokes Est-Elle Correcte?;263
16.1;Commentaires sur le papier de W. Baranov;263
16.2;Reponse de W. Baranov;264
16.3;References;265
17;15 Some Remarks About Collocation;266
17.1;Abstract;266
17.2;Introduction;266
17.3;1. Definition of Exact Collocation and of Least-Squares Collocation — The Main Theorem for Least- Squares Collocation;268
17.4;2. An Upper Bound for the Approximation Error of a Functional in Least- Squares Collocation;272
17.5;3. The Kernel;275
17.6;4. Discussion;276
17.7;Bibliographical Notes;277
17.8;References;277
18;16 Apropos Some Recent Papers by Willi Freeden on a Class of Integral Formulas in the Mathematical Geodesy;278
18.1;I. Two Types of Approximation;278
18.2;II. The Joy of Recognition;283
18.3;III. Instead of a Conclusion;285
18.4;References;285
19;17 S- Transformation or How to Live Without the Generalized Inverse— Almost;286
19.1;I;286
19.2;II;291
20;18 Integrated Geodesy;294
20.1;The Leading Principles of Integrated Geodesy;294
20.2;The Observation Equations;298
20.3;The Observation Equations Illustrated by Examples;300
20.4;References;304
21;19 A Measure for Local Redundancy — A Contribution to the Reliability Theory for Geodetic Networks;305
21.1;1. Motivation;305
21.2;2. The Model;306
21.3;3.;307
22;20 A Convergence Problem in Collocation Theory;312
22.1;Summary;312
22.2;Introduction;312
22.3;1. Continuous Collocation;313
22.4;2. Discrete Collocation;319
22.5;Remarks;325
22.6;References;325
23;21 Non-Linear Adjustment and Curvature;326
23.1;1;326
23.2;2;328
23.3;3;330
23.4;Epilogue;333
23.5;References;334
24;22 Mechanics of Adjustment;335
24.1;Non-Linear Adjustment;335
24.2;Partial Adjustment;338
24.3;Conclusion;340
24.4;Notes;341
24.5;References;341
25;23 Angelica Returning or The Importance of a Title;342
25.1;1;343
25.2;2;344
25.3;3;345
25.4;References;347
26;24 Evaluation of Isotropic Covariance Functions of Torsion Balance Observations;348
26.1;Abstract;348
26.2;1. Introduction;348
26.3;2. Basic equations;349
26.4;3. Change of covariances caused by a rotation of the local coordinate system;351
26.5;4. Evaluation of isotropic covariances;353
26.6;5. General covariance expressions;357
26.7;6. Conclusion;358
26.8;References;358
27;25 Contribution to the Geometry of the Helmert Transformation;360
27.1;Abstract;360
27.2;Introduction;360
27.3;1;361
27.4;2;363
27.5;3;369
27.6;4;369
27.7;5;371
27.8;References;375
28;26 Letter on a Problem in Collocation Theory;376
28.1;1;376
28.2;2;377
28.3;3;378
28.4;References;378
29;27 Approximation to The Earth Potential From Discrete Measurements;379
29.1;Introduction;379
29.2;1. Smoothing;380
29.3;2. A Lemma;384
29.4;3. Smoothing collocation in the continuous case;385
29.5;4. . . .;386
29.6;Conclusion;392
29.7;Bibliographical Notes;392
29.8;References;393
30;28 An Old Procedure for Solving the Relative Orientation in Photogrammetry;394
30.1;Abstract;394
30.2;1. Introduction;394
30.3;2. The Problem in Terms of Homogeneous Coordinates;395
30.4;3. Determination of the Rotation Matrix;397
30.5;4. The Adjustment and the Iterative Procedure;400
30.6;5. Practical Experiences with the Procedure;402
30.7;References;410
31;A Bibliography for Torben Krarup;412