The book presents modern and efficient methods for solving Geodetic and Geoinformatics algebraic problems using computer algebra techniques of Ring, polynomials, Groebner basis, resultants, Gauss-Jacobi combinatorial and Procrustes algorithms. Although these problems are traditionally solved by approximate methods, this book presents alternative algebraic techniques based on computer algebra tools. This new approach meets such modern challenges as resection by laser techniques, solution of orientation in robotics, transformation and bundle block adjustment in geoinformatics, densification of engineering networks, analytical solution for GPS-meteorology and many other problems. For mathematicians the book provides some practical examples of abstract algebra application and multidimensional scaling.
Author(s): Joseph L. Awange, Erik W. Grafarend
Series: Food Engineering Series
Edition: 1
Publisher: Springer
Year: 2005
Language: English
Pages: 344
Contents......Page 11
Preface......Page 6
1 Introduction......Page 16
2.1 Some Applications to Geodesy and Geoinformatics......Page 22
2.2 Numbers from Operational Perspective......Page 23
2.3 Number Rings......Page 27
2.4 Concluding Remarks......Page 31
3.1 Polynomial Equations......Page 32
3.2.1 Polynomial Objects as Rings......Page 34
3.2.2 Operations “Addition” and “Multiplication”......Page 36
3.4 Polynomial Roots......Page 37
3.6.1 Quadratic Polynomials......Page 39
3.6.2 Cubic Polynomials......Page 41
3.6.3 Quartic Polynomials......Page 42
3.7 Concluding Remarks......Page 43
4.1 The Origin......Page 44
4.2 Basics of Groebner Basis......Page 46
4.3 Buchberger Algorithm......Page 53
4.3.1 Mathematica Computation of Groebner Basis......Page 58
4.4 Concluding Remarks......Page 60
5.1 Resultants: An Alternative to Groebner Basis......Page 61
5.2 Sylvester Resultants......Page 62
5.3 Multipolynomial Resultants......Page 64
5.3.1 F. Macaulay Formulation......Page 65
5.3.2 B. Sturmfels’ Formulation......Page 67
5.4 Concluding Remarks......Page 68
6.1 Estimating Unknown Parameters......Page 70
6.2 Combinatorial Approach: The Origin......Page 71
6.3 Linear and Nonlinear Gauss-Markov Models......Page 74
6.4 Gauss-Jacobi Combinatorial Formulation......Page 77
6.5 Combinatorial Solution of Nonlinear Gauss-Markov Model......Page 81
6.6 Concluding Remarks......Page 88
7.1 Positioning Systems......Page 90
7.2 Global Positioning System (GPS)......Page 91
7.3 Local Positioning Systems (LPS)......Page 92
7.3.1 Local Datum Choice in an LPS 3-D Network......Page 93
7.3.2 Relationship between Global and Local Level Reference Frames......Page 95
7.3.3 Observation Equations......Page 97
7.4 Test Network Stuttgart Central......Page 98
7.5 Concluding Remarks......Page 100
8.1 Motivation......Page 102
8.2.1 Procrustes and the Magic Bed......Page 104
8.2.2 Multidimensional Scaling......Page 105
8.2.3 Applications of Procrustes in Medicine......Page 106
8.3.1 Conventional Formulation......Page 108
8.3.2 Partial Derivative Formulation......Page 111
8.4.1 Three-dimensional Orientation Problem......Page 112
8.4.2 Determination of Vertical Deflection......Page 116
8.5 Concluding Remarks......Page 117
9.1 Applications of Distances......Page 118
9.2.1 The Pseudo-ranging Four-Points Problem......Page 120
9.2.2 Ranging to more than Four GPS Satellites......Page 129
9.2.3 Least Squares versus Gauss-Jacobi Combinatorial......Page 132
9.3.1 Planar Ranging......Page 135
9.3.2 Three-dimensional Ranging......Page 146
9.4 Concluding Remarks......Page 159
10.1 Mapping Topographical Points onto Reference Ellipsoid......Page 160
10.2 Mapping Geometry......Page 163
10.3 Minimum Distance Mapping......Page 166
10.3.1 Grafarend-Lohse's Mapping of T[sup(2)] → E[sup(2)][sub(a,a,b)]......Page 169
10.3.2 Groebner Basis' Mapping of T[sup(2)] → E[sup(2)][sub(a,a,b)]......Page 171
10.4 Concluding Remarks......Page 177
11.1 Resection Problem and its Importance......Page 178
11.2.1 Planar Resection......Page 181
11.2.2 Three-dimensional Resection......Page 188
11.3 Photogrammetric Resection......Page 206
11.3.1 Grafarend-Shan Möbius Photogrammetric Resection......Page 207
11.3.2 Algebraic Photogrammetric Resection......Page 208
11.4 Concluding Remarks......Page 210
12.1 Intersection Problem and its Importance......Page 212
12.2.1 Planar Intersection......Page 213
12.2.2 Three-dimensional Intersection......Page 218
12.3 Photogrammetric Intersection......Page 227
12.4 Concluding Remarks......Page 229
13.1 Satellite Environmental Monitoring......Page 230
13.2.1 Space Borne GPS Meteorology......Page 236
13.2.2 Ground based GPS meteorology......Page 238
13.3 Refraction (Bending) Angles......Page 240
13.3.1 Transformation of Trigonometric Equations to Algebraic......Page 242
13.3.2 Algebraic Determination of Bending Angles......Page 244
13.4 Algebraic Analysis of some CHAMP Data......Page 247
13.5 Concluding Remarks......Page 256
14.1 Outliers in Observation Samples......Page 258
14.2 Algebraic Diagnosis of Outliers......Page 260
14.2.1 Outlier Diagnosis in Planar Ranging......Page 262
14.2.2 Diagnosis of Multipath Error in GPS Positioning......Page 266
14.3 Concluding Remarks......Page 271
15.1 7-Parameter Datum Transformation and its Importance......Page 272
15.2.1 Groebner Basis Transformation......Page 275
15.2.2 Gauss-Jacobi Combinatorial Transformation......Page 280
15.2.3 Procrustes Algorithm II......Page 285
15.2.4 Weighted Procrustes Transformation......Page 299
15.3 Concluding Remark......Page 304
16.1 General and Special Purpose CAS......Page 306
16.2.1 MATLAB......Page 307
16.2.2 MAPLE......Page 311
16.2.4 REDUCE......Page 312
16.3 Concluding Remarks......Page 313
Appendix A-1: Definitions......Page 315
Appendix A-2: C. F. Gauss combinatorial approach......Page 317
References......Page 320
C......Page 338
G......Page 339
J......Page 340
P......Page 341
R......Page 342
T......Page 343
Z......Page 344
