Understanding Least Squares Estimation and Geomatics Data Analysis

Contents......Page 3
Preface......Page 11
Website......Page 15
Introduction......Page 16
Significant Digits of Observations......Page 17
Concepts of Observation Model......Page 19
Concepts of Stochastic Model......Page 21
Needs for Adjustment......Page 27
Introductory Matrices......Page 31
Covariance, Cofactor, and Weight Matrices......Page 37
Problems......Page 49
Introduction......Page 53
Model Equations Formulations......Page 54
Taylor Series Expansion of Model Equations......Page 58
Propagation of Systematic and Gross Errors......Page 69
Covariance Propagation......Page 72
Error Propagation Based on Equipment Specifications......Page 81
Heuristic Rule for Covariance Propagation......Page 86
Problems......Page 90
Statistical Distributions & Hypothesis Tests......Page 94
Introduction......Page 95
Probability Functions......Page 96
Sampling Distribution......Page 105
Joint Probability Function......Page 110
Concepts of Statistical Hypothesis Tests......Page 111
Tests of Statistical Hypotheses......Page 113
Problems......Page 130
Adjustment Methods & Concepts......Page 132
Traditional Adjustment Methods......Page 133
The Method of Least Squares......Page 140
Least Squares Adjustment Model Types......Page 145
Least Squares Adjustment Steps......Page 147
Network Datum Definition and Adjustments......Page 149
Constraints in Adjustment......Page 152
Comparison of Different Adjustment Methods......Page 159
Problems......Page 173
Parametric Least Squares Adjustment......Page 176
Parametric Model Equation Formulation......Page 177
Typical Parametric Model Equations......Page 183
Basic Adjustment Model Formulation......Page 192
Linearization of Parametric Model Equations......Page 193
Derivation of Variation Function......Page 199
Derivation of Normal Equation System......Page 201
Derivation of Parametric Least Squares Solution......Page 202
Stochastic Models of Parametric Adjustment......Page 204
Weight-constrained Adjustment Model Formulation......Page 210
Problems......Page 215
Parametric Least Squares Adjustment......Page 218
Introduction......Page 219
Basic Parametric Adjustment Examples......Page 220
Stochastic Properties of Parametric Adjustment......Page 255
Application of Stochastic Models......Page 256
Resection Example......Page 262
Curve-fitting Example......Page 267
Weight Constraint Adjustment Steps......Page 273
Problems......Page 285
Confidence Region Estimation......Page 288
Mean Squared Error and Mathematical Expectation......Page 289
Population Parameter Estimation......Page 293
General Comments on Confidence Interval Estimation......Page 306
Error Ellipse and Bivariate Normal Distribution......Page 307
Error Ellipses for Bivariate Parameters......Page 311
Problems......Page 322
Introduction......Page 324
Preanalysis of Survey Observations......Page 326
Simple One-dimensional Network Design......Page 335
Simple Two-dimensional Network Design......Page 338
Simulation of Three-dimensional Survey Scheme......Page 353
Problems......Page 360
Concepts of 3D Geodetic Network Adjustment......Page 362
Three-dimensional Coordinate Systems and Transformations......Page 363
Parametric Model Equations in Conventional Terrestrial System......Page 367
Parametric Model Equations in Geodetic System......Page 370
Parametric Model Equations in Local Astronomic System......Page 374
General Comments on Three-dimensional Adjustment......Page 378
Adjustment Examples......Page 380
Nuisance Parameters......Page 389
Needs to Eliminate Nuisance Parameters......Page 390
Nuisance Parameter Elimination Model......Page 391
Sequential Least Squares Adjustment......Page 403
Sequential Least Squares Adjustment Model......Page 407
Problems......Page 427
Post-Adjustment Data Analysis & Reliability Concepts......Page 430
Introduction......Page 431
Post-adjustment Detection and Elimination of Non-stochastic Errors......Page 432
Global Tests......Page 435
Local Tests......Page 438
s Approach to Local Test......Page 439
Concepts of Redundancy Numbers......Page 441
s Data Analysis Approach......Page 444
Concepts of Reliability Measures......Page 448
Network Sensitivity......Page 452
Problems......Page 458
Least Squares Adjustment of Conditional Models......Page 461
Conditional Model Equations......Page 462
Conditional Model Adjustment Formulation......Page 469
Stochastic Model of Conditional Adjustment......Page 480
Assessment of Observations and Conditional Model......Page 483
Covariance Propagation for Derived Parameters from Conditional Adjustment......Page 484
Simple GNSS Network Adjustment Example......Page 490
Simple Traverse Network Adjustment Example......Page 494
Problems......Page 500
Least Squares Adjustment of General Models......Page 503
General Model Equation Formulation......Page 504
Linearization of General Model......Page 507
Variation Function for Linearized General Model......Page 510
Normal Equation System and the Least Squares Solution......Page 511
Steps for General Model Adjustment......Page 512
General Model Adjustment Examples......Page 513
Stochastic Properties of General Model Adjustment......Page 522
Horizontal Circular Curve Example......Page 526
Adjustment of General Model with Weight Constraints......Page 534
Problems......Page 548
Introduction......Page 552
Minimal Datum Constraint Types......Page 553
Free Network Adjustment Model......Page 554
Constraint Model for Free Network Adjustment......Page 557
Summary of Free Network Adjustment Procedure......Page 560
Datum Transformation......Page 568
Problems......Page 575
Introduction......Page 579
Static Mode Filter......Page 583
Dynamic Mode Filter......Page 586
Kalman Filtering Examples......Page 591
Kalman Filter and the Least Squares Method......Page 615
Problems......Page 618
Introduction......Page 620
Elements of Least Squares Collocation......Page 623
Collocation Procedure......Page 624
Covariance Function......Page 625
Collocation and Classical Least Squares Adjustment......Page 628
Semivariogram Model and Modeling......Page 631
Kriging Procedure......Page 634
Comparing Least Squares Collocation and Kriging......Page 639
Extracts from Baarda Nomogram......Page 641
Standard Statistical Distribution Tables......Page 644
Tau Critical Values Table for Significance Level α......Page 654
D.1 Azimuth Observable......Page 657
D.2 Total Station Direction Observable......Page 659
D.3 Horizontal Angle Observable......Page 660
D.5 Zenith Angle Observable......Page 661
D.6 Other General Rules for Partial Differentials......Page 662
E.1 Matrix Lemmas......Page 664
E.2 Generalized Inverses and Pseudo-inverses......Page 666
Abbreviations......Page 671
Refs......Page 672
Index......Page 676