Borre Mathematical Foundation of Geodesy

Selected Papers of Torben Krarup
1. Auflage 2006
ISBN: 978-3-540-33767-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 416 Seiten

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