Linear Algebra, Geodesy, and GPS

frontcover......Page 1
Table of Contents......Page 5
Preface......Page 9
The mathematics of GPS......Page 13
1.1 Vectors ......Page 19
1.2 Lengths and Dot Products ......Page 27
1.3 Planes ......Page 36
1.4 Matrices and Linear Equations ......Page 44
2.1 The Idea of Elimination ......Page 53
2.2 Elimination Using Matrices ......Page 62
2.3 Rules for Matrix Operations ......Page 70
2.4 Inverse Matrices ......Page 81
2.5 Elimination = Factorization: A = LU ......Page 91
2.6 Transposes and Permutations ......Page 103
3.1 Spaces of Vectors ......Page 117
3.2 The Nullspace of A: Solving Ax = 0 ......Page 125
3.3 The Rank of A: Solving Ax = b ......Page 138
3.4 Independence, Basis, and Dimension ......Page 150
3.5 Dimensions of the Four Subspaces ......Page 162
4.1 Orthogonality of the Four Subspaces ......Page 173
4.2 Projections ......Page 181
4.3 Least-Squares Approximations ......Page 190
4.4 Orthogonal Bases and Gram-Schmidt ......Page 200
5.1 The Properties of Determinants ......Page 213
5.2 Cramer\'s Rule, Inverses, and Volumes ......Page 222
6.1 Introduction to Eigenvalues ......Page 227
6.2 Diagonalizing a Matrix ......Page 237
6.3 Symmetric Matrices ......Page 249
6.4 Positive Definite Matrices ......Page 253
6.5 Stability and Preconditioning ......Page 264
7.1 The Idea of a Linear Transformation ......Page 267
7.2 Choice of Basis: Similarity and SVD ......Page 274
8.1 Heights by Least Squares ......Page 291
8.2 Weighted Least Squares ......Page 296
8.3 Leveling Networks and Graphs ......Page 298
8.4 Graphs and Incidence Matrices ......Page 304
8.5 One-Dimensional Distance Networks ......Page 321
9.1 The Normal Distribution and  X2 ......Page 325
9.2 Mean, Variance, and Standard Deviation ......Page 335
9.3 Covariance ......Page 336
9.4 Inverse Covariances as Weights ......Page 338
9.5 Estimation of Mean and Variance ......Page 342
9.6 Propagation of Means and Covariances ......Page 344
9.7 Estimating the Variance of Unit Weight ......Page 349
9.8 Confidence Ellipses ......Page 353
10.1 Getting Around Nonlinearity ......Page 359
10.2 Geodetic Observation Equations ......Page 365
10.3 Three-Dimensional Model ......Page 378
11.1 Gram-Schmidt on A and Cholesky on A T A ......Page 385
11.2 Cholesky\'s Method in the Least-Squares Setting ......Page 388
11.3 SVD: The Canonical Form for Geodesy ......Page 391
11.4 The Condition Number ......Page 393
11.5 Regularly Spaced Networks ......Page 395
11.6 Dependency on the Weights ......Page 407
11.7 Elimination of Unknowns ......Page 410
11.8 Decorrelation and Weight Normalization ......Page 416
12.1 Rank Deficient Normal Equations ......Page 421
12.2 Representations of the Nullspace ......Page 422
12.3 Constraining a Rank Deficient Problem ......Page 424
12.4 Linear Transformation of Random Variables ......Page 429
12.5 Similarity Transformations ......Page 430
12.6 Covariance Transformations ......Page 437
12.7 Variances at Control Points ......Page 439
13.1 Free Stationing as a Similarity Transformation ......Page 447
13.2 Optimum Choice of Observation Site ......Page 450
13.3 Station Adjustment ......Page 454
13.4 Fitting a Straight Line ......Page 457
14.1 Positioning by GPS ......Page 463
14.2 Errors in the GPS Observables ......Page 469
14.3 Description of the System ......Page 474
14.4 Receiver Position From Code Observations ......Page 476
14.5 Combined Code and Phase Observations ......Page 479
14.6 Weight Matrix for Differenced Observations ......Page 481
14.7 Geometry of the Ellipsoid ......Page 483
14.8 The Direct and Reverse Problems ......Page 486
14.9 Geodetic Reference System 1980 ......Page 487
14.10 Geoid, Ellipsoid, and Datum ......Page 488
14.11 World Geodetic System 1984 ......Page 492
14.12 Coordinate Changes From Datum Changes ......Page 493
15.1 Baseline Computation and M-Files ......Page 497
15.2 Coordinate Changes and Satellite Position ......Page 498
15.3 Receiver Position from Pseudoranges ......Page 503
15.4 Separate Ambiguity and Baseline Estimation ......Page 504
15.5 Joint Ambiguity and Baseline Estimation ......Page 510
15.6 The LAMBDA Method for Ambiguities ......Page 511
15.7 Sequential Filter for Absolute Position ......Page 515
15.8 Additional Useful Filters ......Page 521
16.1 Random Processes in Continuous Time ......Page 531
16.2 Random Processes in Discrete Time ......Page 539
16.3 Modeling ......Page 543
17.1 Updating Least Squares ......Page 559
17.2 Static and Dynamic Updates ......Page 564
17.3 The Steady Model ......Page 568
17.4 Derivation of the Kalman Filter ......Page 574
17.5 Bayes Filter for Batch Processing ......Page 582
17.6 Smoothing ......Page 585
17.7 An Example from Practice ......Page 590
The Receiver Independent Exchange Format ......Page 601
Glossary ......Page 617
References ......Page 625
Index of M-files ......Page 631
Index ......Page 633