GilbertStrang,KaiBorre-Linearalgebra,geodesy,andGPS(1997).pdf

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Table of Contents

Preface

The mathematics of GPS

Part I: Linear Algebra

1 Vectors and Matrices

1.1 Vectors

1.2 Lengths and Dot Products

1.3 Planes

1.4 Matrices and Linear Equations

2 Solving Linear Equations

2.1 The Idea of Elimination

2.2 Elimination Using Matrices

2.3 Rules for Matrix Operations

2.4 Inverse Matrices

2.5 Elimination = Factorization: A = LU

2.6 Transposes and Permutations

3 Vector Spaces and Subspaces

3.1 Spaces of Vectors

3.2 The Nullspace of A: Solving Ax = 0

3.3 The Rank of A: Solving Ax = b

3.4 Independence, Basis, and Dimension

3.5 Dimensions of the Four Subspaces

4 Orthogonality

4.1 Orthogonality of the Four Subspaces

4.2 Projections

4.3 Least-Squares Approximations

4.4 Orthogonal Bases and Gram-Schmidt

5 Determinants

5.1 The Properties of Determinants

5.2 Cramer's Rule, Inverses, and Volumes

6 Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues

6.2 Diagonalizing a Matrix

6.3 Symmetric Matrices

6.4 Positive Definite Matrices

6.5 Stability and Preconditioning

7 Linear Transformations

7.1 The Idea of a Linear Transformation

7.2 Choice of Basis: Similarity and SVD

Part II: Geodesy

8 Leveling Networks

8.1 Heights by Least Squares

8.2 Weighted Least Squares

8.3 Leveling Networks and Graphs

8.4 Graphs and Incidence Matrices

8.5 One-Dimensional Distance Networks

9 Random Variables and Covariance Matrices

9.1 The Normal Distribution and X2

9.2 Mean, Variance, and Standard Deviation

9.3 Covariance

9.4 Inverse Covariances as Weights

9.5 Estimation of Mean and Variance

9.6 Propagation of Means and Covariances

9.7 Estimating the Variance of Unit Weight

9.8 Confidence Ellipses

10 Nonlinear Problems

10.1 Getting Around Nonlinearity

10.2 Geodetic Observation Equations

10.3 Three-Dimensional Model

11 Linear Algebra for Weighted Least Squares

11.1 Gram-Schmidt on A and Cholesky on A T A

11.2 Cholesky's Method in the Least-Squares Setting

11.3 SVD: The Canonical Form for Geodesy

11.4 The Condition Number

11.5 Regularly Spaced Networks

11.6 Dependency on the Weights

11.7 Elimination of Unknowns

11.8 Decorrelation and Weight Normalization

12 Constraints for Singular Normal Equations

12.1 Rank Deficient Normal Equations

12.2 Representations of the Nullspace

12.3 Constraining a Rank Deficient Problem

12.4 Linear Transformation of Random Variables

12.5 Similarity Transformations

12.6 Covariance Transformations

12.7 Variances at Control Points

13 Problems With Explicit Solutions

13.1 Free Stationing as a Similarity Transformation

13.2 Optimum Choice of Observation Site

13.3 Station Adjustment

13.4 Fitting a Straight Line

Part III: Global Positioning System (GPS)

14 Global Positioning System

14.1 Positioning by GPS

14.2 Errors in the GPS Observables

14.3 Description of the System

14.4 Receiver Position From Code Observations

14.5 Combined Code and Phase Observations

14.6 Weight Matrix for Differenced Observations

14.7 Geometry of the Ellipsoid

14.8 The Direct and Reverse Problems

14.9 Geodetic Reference System 1980

14.10 Geoid, Ellipsoid, and Datum

14.11 World Geodetic System 1984

14.12 Coordinate Changes From Datum Changes

15 Processing of GPS Data

15.1 Baseline Computation and M-Files

15.2 Coordinate Changes and Satellite Position

15.3 Receiver Position from Pseudoranges

15.4 Separate Ambiguity and Baseline Estimation

15.5 Joint Ambiguity and Baseline Estimation

15.6 The LAMBDA Method for Ambiguities

15.7 Sequential Filter for Absolute Position

15.8 Additional Useful Filters

16 Random Processes

16.1 Random Processes in Continuous Time

16.2 Random Processes in Discrete Time

16.3 Modeling

17 Kalman Filters

17.1 Updating Least Squares

17.2 Static and Dynamic Updates

17.3 The Steady Model

17.4 Derivation of the Kalman Filter

17.5 Bayes Filter for Batch Processing

17.6 Smoothing

17.7 An Example from Practice

The Receiver Independent Exchange Format

Glossary

References

Index of M-files

Index

LINEAR ALGEBRA, GEODESY, AND GPS GILBERT STRANG Massachusetts Institute of Technology and KAI BORRE Aalborg University

Library of Congress Cataloging-in-Publication Data Strang, Gilbert. Linear algebra, geodesy, and GPS I Gilbert Strang and Kai Borre. Includes bibliographical references and index. ISBN 0-9614088-6-3 (hardcover) 1. Algebras, Linear. 2. Geodesy-Mathematics. 3. Global Positioning System. I. Borre, K. (Kai) TA347.L5 S87 1997 5 26' .1' 0 15125-dc20 II. Title. 96-44288 Copyright @1997 by Gilbert Strang and Kai Borre Designed by Frank Jensen Cover photograph by Michael Bevis at Makapu'u on Oahu, Hawaii Cover design by Tracy Baldwin All rights reserved. No part of this work may be reproduced or stored or transmitted by any means, including photocopying, without the written permission of the publisher. Translation in any language is strictly prohibited-authorized translations are arranged. Printed in the United States of America 87654321 Other texts from Wellesley-Cambridge Press Wavelets and Filter Banks, Gilbert Strang and Truong Nguyen, ISBN 0-9614088-7-1. Introduction to Applied Mathematics, Gilbert Strang, ISBN 0-9614088-0-4. An Analysis of the Finite Element Method, Gilbert Strang and George Fix, ISBN 0-9614088-8-X. Calculus, Gilbert Strang, ISBN 0-9614088-2-0. Introduction to Linear Algebra, Gilbert Strang, ISBN 0-9614088-5-5. Wellesley-Cambridge Press Box 812060 Wellesley MA 02181 USA (617) 431-8488 FAX (617) 253-4358 http://www-math.mit.edu/-gs email: gs@math.mit.edu All books may be ordered by email.

TAB·LE OF CONTENTS Preface The Mathematics of GPS Part I Linear Algebra 1 2 3 4 5 Vectors and Matrices 1.1 1.2 1.3 1.4 Matrices and Linear Equations Vectors Lengths and Dot Products Planes Solving Linear Equations 2.1 2.2 2.3 2.4 2.5 2.6 The Idea of Elimination Elimination Using Matrices Rules for Matrix Operations Inverse Matrices Elimination= Factorization: A= LU Transposes and Permutations Vector Spaces and Subspaces 3.1 3.2 3.3 3.4 3.5 Spaces of Vectors The Nullspace of A: Solving Ax = 0 The Rank of A: Solving Ax= b Independence, Basis, and Dimension Dimensions of the Four Subspaces Orthogonality 4.1 4.2 4.3 4.4 Orthogonality of the Four Subspaces Projections Least-Squares Approximations Orthogonal Bases and Gram-Schmidt Determinants 5.1 5.2 The Properties of Determinants Cramer's Rule, Inverses, and Volumes v ix xiii 3 3 11 20 28 37 37 46 54 65 75 87 101 101 109 122 134 146 157 157 165 174 184 197 197 206

vi 6 7 Table of Contents Eigenvalues and Eigenvectors 6.1 6.2 6.3 6.4 6.5 Introduction to Eigenvalues Diagonalizing a Matrix Symmetric Matrices Positive Definite Matrices Stability and Preconditioning Linear Transformations 7.1 7.2 The Idea of a Linear Transformation Choice of Basis: Similarity and SVD Part II Geodesy 8 9 leveling Networks 8.1 Heights by Least Squares 8.2 Weighted Least Squares 8.3 8.4 8.5 Leveling Networks and Graphs Graphs and Incidence Matrices One-Dimensional Distance Networks The Normal Distribution and x2 Random Variables and Covariance Matrices 9.1 9.2 Mean, Variance, and Standard Deviation 9.3 9.4 9.5 9.6 9.7 9.8 Covariance Inverse Covariances as Weights Estimation of Mean and Variance Propagation of Means and Covariances Estimating the Variance of Unit Weight Confidence Ellipses 10 Nonlinear Problems 10.1 Getting Around Nonlinearity 10.2 Geodetic Observation Equations 10.3 Three-Dimensional Model 11 Linear Algebra for Weighted least Squares 11.1 Gram-Schmidt on A and Cholesky on AT A 11.2 Cholesky's Method in the Least-Squares Setting 11.3 SVD: The Canonical Form for Geodesy 11.4 The Condition Number 11.5 Regularly Spaced Networks 11.6 Dependency on the Weights 11.7 Elimination of Unknowns 11.8 Decorrelation and Weight Normalization 211 211 221 233 237 248 251 251 258 275 275 280 282 288 305 309 309 319 320 322 326 328 333 337 343 343 349 362 369 369 372 375 377 379 391 394 400

12 Constraints for Singular Normal Equations 12.1 Rank Deficient Normal Equations 12.2 Representations of the Nullspace 12.3 Constraining a Rank Deficient Problem 12.4 Linear Transformation of Random Variables 12.5 Similarity Transformations 12.6 Covariance Transformations 12.7 Variances at Control Points 13 Problems With Explicit Solutions Free Stationing as a Similarity Transformation 13.1 13.2 Optimum Choice of Observation Site 13.3 Station Adjustment 13.4 Fitting a Straight Line Part Ill Global Positioning System (GPS) 14 Global Positioning System Positioning by GPS 14.1 14.2 Errors in the GPS Observables 14.3 Description of the System 14.4 Receiver Position From Code Observations 14.5 Combined Code and Phase Observations 14.6 Weight Matrix for Differenced Observations 14.7 Geometry of the Ellipsoid 14.8 The Direct and Reverse Problems 14.9 Geodetic Reference System 1980 14.10 Geoid, Ellipsoid, and Datum 14.11 World Geodetic System 1984 14.12 Coordinate Changes From Datum Changes 15 Processing of GPS Data 15.1 Baseline Computation and M -Files 15.2 Coordinate Changes and Satellite Position 15.3 Receiver Position from Pseudoranges 15.4 Separate Ambiguity and Baseline Estimation 15.5 Joint Ambiguity and Baseline Estimation 15.6 The LAMBDA Method for Ambiguities 15.7 Sequential Filter for Absolute Position 15.8 Additional Useful Filters 16 Random Processes 16.1 Random Processes in Continuous Time 16.2 Random Processes in Discrete Time 16.3 Modeling Table of Contents vii 405 405 406 408 413 414 421 423 431 431 434 438 441 447 447 453 458 460 463 465 467 470 471 472 476 477 481 481 482 487 488 494 495 499 505 515 515 523 527

viii Table of Contents 17 Kalman Filters 17.1 Updating Least Squares 17.2 Static and Dynamic Updates 17.3 The Steady Model 17.4 Derivation of the Kalman Filter 17.5 Bayes Filter for Batch Processing 17.6 Smoothing 17.7 An Example from Practice The Receiver Independent Exchange Format Glossary References Index of M-files Index 543 543 548 552 558 566 569 574 585 601 609 615 617

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