Solving Algebraic Computational Problems in Geodesy and Geoinformatics (eBook)

Preface 6
Contents 11
1 Introduction 16
2 Basics of Ring Theory 22
2.1 Some Applications to Geodesy and Geoinformatics 22
2.2 Numbers from Operational Perspective 23
2.3 Number Rings 27
2.4 Concluding Remarks 31
3 Basics of Polynomial Theory 32
3.1 Polynomial Equations 32
3.2 Polynomial Rings 34
3.3 Factoring Polynomials 37
3.4 Polynomial Roots 37
3.5 Minimal Polynomials 39
3.6 Polynomials with Real Coefficients 39
3.7 Concluding Remarks 43
4 Groebner Basis 44
4.1 The Origin 44
4.2 Basics of Groebner Basis 46
4.3 Buchberger Algorithm 53
4.4 Concluding Remarks 60
5 Polynomial Resultants 61
5.1 Resultants: An Alternative to Groebner Basis 61
5.2 Sylvester Resultants 62
5.3 Multipolynomial Resultants 64
5.4 Concluding Remarks 68
6 Gauss-Jacobi Combinatorial Algorithm 70
6.1 Estimating Unknown Parameters 70
6.2 Combinatorial Approach: The Origin 71
6.3 Linear and Nonlinear Gauss-Markov Models 74
6.4 Gauss-Jacobi Combinatorial Formulation 77
6.5 Combinatorial Solution of Nonlinear Gauss-Markov Model 81
6.6 Concluding Remarks 88
7 Local versus Global Positioning Systems 90
7.1 Positioning Systems 90
7.2 Global Positioning System (GPS) 91
7.3 Local Positioning Systems (LPS) 92
7.4 Test Network Stuttgart Central 98
7.5 Concluding Remarks 100
8 Partial Procrustes and the Orientation Problem 102
8.1 Motivation 102
8.2 Procrustes: Origin and Applications 104
8.3 Partial Procrustes Solution 108
8.4 Practical Applications 112
8.5 Concluding Remarks 117
9 Positioning by Ranging 118
9.1 Applications of Distances 118
9.2 Ranging by Global Positioning System (GPS) 120
9.3 Ranging by Local Positioning Systems (LPS) 135
9.4 Concluding Remarks 159
10 From Geocentric Cartesian to Ellipsoidal Coordinates 160
10.1 Mapping Topographical Points onto Reference Ellipsoid 160
10.2 Mapping Geometry 163
10.3 Minimum Distance Mapping 166
10.4 Concluding Remarks 177
11 Positioning by Resection Methods 178
11.1 Resection Problem and its Importance 178
11.2 Geodetic Resection 181
11.3 Photogrammetric Resection 206
11.4 Concluding Remarks 210
12 Positioning by Intersection Methods 212
12.1 Intersection Problem and its Importance 212
12.2 Geodetic Intersection 213
12.3 Photogrammetric Intersection 227
12.4 Concluding Remarks 229
13 GPS Meteorology in Environmental Monitoring 230
13.1 Satellite Environmental Monitoring 230
13.2 GPS Remote Sensing 236
13.3 Refraction (Bending) Angles 240
13.4 Algebraic Analysis of some CHAMP Data 247
13.5 Concluding Remarks 256
14 Algebraic Diagnosis of Outliers 258
14.1 Outliers in Observation Samples 258
14.2 Algebraic Diagnosis of Outliers 260
14.3 Concluding Remarks 271
15 Transformation Problem: Procrustes Algorithm II 272
15.1 7-Parameter Datum Transformation and its Importance 272
15.2 Algebraic (Analytic) Determination of Transformation Parameters 275
15.3 Concluding Remark 304
16 Computer Algebra Systems (CAS) 306
16.1 General and Special Purpose CAS 306
16.2 Some CAS Software Useful in Geodesy and Geoinformatics 307
16.3 Concluding Remarks 313
Appendix 315
Appendix A-1: Definitions 315
Appendix A-2: C. F. Gauss Combinatorial Formulation 317
References 320
Index 338