Linear Algebra, Geodesy, and GPS by Gilbert Strang and Kai Borre Wellesley-Cambridge Press Box 812060 Wellesley MA 02181 fax 617 253-4358 phone 781 431-8488 email gs@math.mit.edu 640 pages (1997) hardcover ISBN 0-9614088-6-3 http://www-math.mit.edu/~gs TABLE OF CONTENTS Preface...................................................ix The Mathematics of GPS..................................xiii Part I Linear Algebra 1 Vectors and Matrices....................................3 1.1 Vectors.............................................3 1.2 Lengths and Dot Products...........................11 1.3 Planes.............................................20 1.4 Matrices and Linear Equations......................28 2 Solving Linear Equations...............................37 2.1 The Idea of Elimination............................37 2.2 Elimination Using Matrices.........................46 2.3 Rules for Matrix Operations........................54 2.4 Inverse Matrices...................................65 2.5 Elimination = Factorization: A = LU................75 2.6 Transposes and Permutations........................87 3 Vector Spaces and Subspaces...........................101 3.1 Spaces of Vectors.................................101 3.2 The Nullspace of A: Solving Ax = 0................109 3.3 The Rank of A: Solving Ax = b.....................122 3.4 Independence, Basis, and Dimension................134 3.5 Dimensions of the Four Subspaces..................146 4 Orthogonality.........................................157 4.1 Orthogonality of the Four Subspaces...............157 4.2 Projections.......................................165 4.3 Least-Squares Approximations......................174 4.4 Orthogonal Bases and Gram-Schmidt.................184 5 Determinants..........................................197 5.1 The Properties of Determinants....................197 5.2 Cramer's Rule, Inverses, and Volumes..............206 6 Eigenvalues and Eigenvectors..........................211 6.1 Introduction to Eigenvalues.......................211 6.2 Diagonalizing a Matrix............................221 6.3 Symmetric Matrices................................233 6.4 Positive Definite Matrices........................237 6.5 Stability and Preconditioning.....................248 7 Linear Transformations................................251 7.1 The Idea of a Linear Transformation...............251 7.2 Choice of Basis: Similarity and SVD...............258 Part II Geodesy 8 Leveling Networks.....................................275 8.1 Heights by Least Squares..........................275 8.2 Weighted Least Squares............................280 8.3 Leveling Networks and Graphs......................282 8.4 Graphs and Incidence Matrices.....................288 8.5 One-Dimensional Distance Networks.................305 9 Random Variables and Covariance Matrices..............309 9.1 The Normal Distribution and X2...................309 9.2 Mean, Variance, and Standard Deviation............319 9.3 Covariance........................................320 9.4 Inverse Covariances as Weights....................322 9.5 Estimation of Mean and Variance...................326 9.6 Propagation of Means and Covariances..............328 9.7 Estimating the Variance of Unit Weight............333 9.8 Confidence Ellipses...............................337 10 Nonlinear Problems....................................343 10.1 Getting Around Nonlinearity......................343 10.2 Geodetic Observation Equations...................349 10.3 Three-Dimensional Model..........................362 11 Linear Algebra for Weighted Least Squares.............369 11.1 Gram-Schmidt on A and Cholesky on A T A..........369 11.2 Cholesky's Method in the Least-Squares Setting...372 11.3 SVD: The Canonical Form for Geodesy..............375 11.4 The Condition Number.............................377 11.5 Regularly Spaced Networks........................379 11.6 Dependency on the Weights........................391 11.7 Elimination of Unknowns..........................394 11.8 Decorrelation and Weight Normalization...........400 12 Constraints for Singular Normal Equations.............405 12.1 Rank Deficient Normal Equations..................405 12.2 Representations of the Nullspace.................406 12.3 Constraining a Rank Deficient Problem............408 12.4 Linear Transformation of Random Variables........413 12.5 Similarity Transformations.......................414 12.6 Covariance Transformations.......................421 12.7 Variances at Control Points......................423 13 Problems With Explicit Solutions......................431 13.1 Free Stationing as a Similarity Transformation...431 13.2 Optimum Choice of Observation Site...............434 13.3 Station Adjustment...............................438 13.4 Fitting a Straight Line..........................441 Part III Global Positioning System (GPS) 14 Global Positioning System.............................447 14.1 Positioning by GPS...............................447 14.2 Errors in the GPS Observables....................453 14.3 Description of the System........................458 14.4 Receiver Position From Code Observations.........460 14.5 Combined Code and Phase Observations.............463 14.6 Weight Matrix for Differenced Observations.......465 14.7 Geometry of the Ellipsoid........................467 14.8 The Direct and Reverse Problems..................470 14.9 Geodetic Reference System 1980...................471 14.10 Geoid, Ellipsoid, and Datum.....................472 14.11 World Geodetic System 1984......................476 14.12 Coordinate Changes From Datum Changes...........477 15 Processing of GPS Data................................481 15.1 Baseline Computation and M-Files.................481 15.2 Coordinate Changes and Satellite Position........482 15.3 Receiver Position from Pseudoranges..............487 15.4 Separate Ambiguity and Baseline Estimation.......488 15.5 Joint Ambiguity and Baseline Estimation..........494 15.6 The LAMBDA Method for Ambiguities................495 15.7 Sequential Filter for Absolute Position..........499 15.8 Additional Useful Filters........................505 16 Random Processes......................................515 16.1 Random Processes in Continuous Time..............515 16.2 Random Processes in Discrete Time................523 16.3 Modeling.........................................527 17 Kalman Filters........................................543 17.1 Updating Least Squares...........................543 17.2 Static and Dynamic Updates.......................548 17.3 The Steady Model.................................552 17.4 Derivation of the Kalman Filter..................558 17.5 Bayes Filter for Batch Processing................566 17.6 Smoothing........................................569 17.7 An Example from Practice.........................574 The Receiver Independent Exchange Format.................585 Glossary.................................................601 References...............................................609 Index of M-files.........................................615 Index....................................................617