Understanding Least Squares Estimation and Geomatics Data Analysis

Content: Preface xiii Acknowledgments xvii About the Author xix About the Companion Website xxi 1 Introduction 1 1.1 Observables and Observations 2 1.2 Significant Digits of Observations 2 1.3 Concepts of Observation Model 4 1.4 Concepts of Stochastic Model 6 1.4.1 Random Error Properties of Observations 6 1.4.2 Standard Deviation of Observations 8 1.4.3 Mean of Weighted Observations 9 1.4.4 Precision of Observations 10 1.4.5 Accuracy of Observations 11 1.5 Needs for Adjustment 12 1.6 Introductory Matrices 16 1.6.1 Sums and Products of Matrices 18 1.6.2 Vector Representation 20 1.6.3 Basic Matrix Operations 21 1.7 Covariance, Cofactor, and Weight Matrices 22 1.7.1 Covariance and Cofactor Matrices 26 1.7.2 Weight Matrices 27 Problems 34 2 Analysis and Error Propagation of Survey Observations 39 2.1 Introduction 39 2.2 Model Equations Formulations 40 2.3 Taylor Series Expansion of Model Equations 44 2.3.1 Using MATLAB to Determine Jacobian Matrix 52 2.4 Propagation of Systematic and Gross Errors 55 2.5 Variance-Covariance Propagation 58 2.6 Error Propagation Based on Equipment Specifications 67 2.6.1 Propagation for Distance Based on Accuracy Specification 67 2.6.2 Propagation for Direction (Angle) Based on Accuracy Specification 69 2.6.3 Propagation for Height Difference Based on Accuracy Specification 69 2.7 Heuristic Rule for Covariance Propagation 72 Problems 76 3 Statistical Distributions and Hypothesis Tests 81 3.1 Introduction 82 3.2 Probability Functions 83 3.2.1 Normal Probability Distributions and Density Functions 84 3.3 Sampling Distribution 92 3.3.1 Student\'s t-Distribution 93 3.3.2 Chi-square and Fisher\'s F-distributions 95 3.4 Joint Probability Function 97 3.5 Concepts of Statistical Hypothesis Tests 98 3.6 Tests of Statistical Hypotheses 100 3.6.1 Test of Hypothesis on a Single Population Mean 102 3.6.2 Test of Hypothesis on Difference of Two Population Means 106 3.6.3 Test of Measurements Against the Means 109 3.6.4 Test of Hypothesis on a Population Variance 111 3.6.5 Test of Hypothesis on Two Population Variances 114 Problems 117 4 Adjustment Methods and Concepts 119 4.1 Introduction 120 4.2 Traditional Adjustment Methods 120 4.2.1 Transit Rule Method of Adjustment 122 4.2.2 Compass (Bowditch) Rule Method 125 4.2.3 Crandall\'s Rule Method 126 4.3 The Method of Least Squares 127 4.3.1 Least Squares Criterion 129 4.4 Least Squares Adjustment Model Types 132 4.5 Least Squares Adjustment Steps 134 4.6 Network Datum Definition and Adjustments 136 4.6.1 Datum Defect and Configuration Defect 138 4.7 Constraints in Adjustment 139 4.7.1 Minimal Constraint Adjustments 140 4.7.2 Overconstrained and Weight-Constrained Adjustments 141 4.7.3 Adjustment Constraints Examples 143 4.8 Comparison of Different Adjustment Methods 146 4.8.1 General Discussions 158 Problems 160 5 Parametric Least Squares Adjustment: Model Formulation 163 5.1 Parametric Model Equation Formulation 164 5.1.1 Distance Observable 165 5.1.2 Azimuth and Horizontal (Total Station) Direction Observables 165 5.1.3 Horizontal Angle Observable 168 5.1.4 Zenith Angle Observable 169 5.1.5 Coordinate Difference Observable 169 5.1.6 Elevation Difference Observable 169 5.2 Typical Parametric Model Equations 170 5.3 Basic Adjustment Model Formulation 179 5.4 Linearization of Parametric Model Equations 180 5.4.1 Linearization of Parametric Model Without Nuisance Parameter 180 5.4.2 Linearization of Parametric Model with Nuisance Parameter 184 5.5 Derivation of Variation Function 186 5.5.1 Derivation of Variation Function Using Direct Approach 186 5.5.2 Derivation of Variation Function Using Lagrangian Approach 187 5.6 Derivation of Normal Equation System 188 5.6.1 Normal Equations Based on Direct Approach Variation Function 188 5.6.2 Normal Equations Based on Lagrangian Approach Variation Function 189 5.7 Derivation of Parametric Least Squares Solution 189 5.7.1 Least Squares Solution from Direct Approach Normal Equations 189 5.7.2 Least Squares Solution from Lagrangian Approach Normal Equations 190 5.8 Stochastic Models of Parametric Adjustment 191 5.8.1 Derivation of Cofactor Matrix of Adjusted Parameters 192 5.8.2 Derivation of Cofactor Matrix of Adjusted Observations 193 5.8.3 Derivation of Cofactor Matrix of Observation Residuals 194 5.8.4 Effects of Variance Factor Variation on Adjustments 196 5.9 Weight-constrained Adjustment Model Formulation 197 5.9.1 Stochastic Model for Weight-constrained Adjusted Parameters 200 5.9.2 Stochastic Model for Weight-constrained Adjusted Observations 201 Problems 202 6 Parametric Least Squares Adjustment: Applications 205 6.1 Introduction 206 6.2 Basic Parametric Adjustment Examples 207 6.2.1 Leveling Adjustment 207 6.2.2 Station Adjustment 215 6.2.3 Traverse Adjustment 223 6.2.4 Triangulateration Adjustment 235 6.3 Stochastic Properties of Parametric Adjustment 242 6.4 Application of Stochastic Models 243 6.5 Resection Example 249 6.6 Curve-fitting Example 254 6.7 Weight Constraint Adjustment Steps 260 6.7.1 Weight Constraint Examples 261 Problems 272 7 Confidence Region Estimation 275 7.1 Introduction 276 7.2 Mean Squared Error and Mathematical Expectation 276 7.2.1 Mean Squared Error 276 7.2.2 Mathematical Expectation 277 7.3 Population Parameter Estimation 280 7.3.1 Point Estimation of Population Mean 280 7.3.2 Interval Estimation of Population Mean 281 7.3.3 Relative Precision Estimation 285 7.3.4 Interval Estimation for Population Variance 288 7.3.5 Interval Estimation for Ratio of Two Population Variances 290 7.4 General Comments on Confidence Interval Estimation 293 7.5 Error Ellipse and Bivariate Normal Distribution 294 7.6 Error Ellipses for Bivariate Parameters 298 7.6.1 Absolute Error Ellipses 299 7.6.2 Relative Error Ellipses 305 Problems 309 8 Introduction to Network Design and Preanalysis 311 8.1 Introduction 311 8.2 Preanalysis of Survey Observations 313 8.2.1 Survey Tolerance Limits 314 8.2.2 Models for Preanalysis of Survey Observations 314 8.2.3 Trigonometric Leveling Problems 316 8.3 Network Design Model 322 8.4 Simple One-dimensional Network Design 322 8.5 Simple Two-dimensional Network Design 325 8.6 Simulation of Three-dimensional Survey Scheme 340 8.6.1 Typical Three-dimensional Micro-network 340 8.6.2 Simulation Results 342 Problems 347 9 Concepts of Three-dimensional Geodetic Network Adjustment 349 9.1 Introduction 350 9.2 Three-dimensional Coordinate Systems and Transformations 350 9.2.1 Local Astronomic Coordinate Systems and Transformations 352 9.3 Parametric Model Equations in Conventional Terrestrial System 354 9.4 Parametric Model Equations in Geodetic System 357 9.5 Parametric Model Equations in Local Astronomic System 361 9.6 General Comments on Three-dimensional Adjustment 365 9.7 Adjustment Examples 367 9.7.1 Adjustment in Cartesian Geodetic System 367 9.7.1.1 Solution Approach 369 9.7.2 Adjustment in Curvilinear Geodetic System 371 9.7.3 Adjustment in Local System 373 10 Nuisance Parameter Elimination and Sequential Adjustment 377 10.1 Nuisance Parameters 377 10.2 Needs to Eliminate Nuisance Parameters 378 10.3 Nuisance Parameter Elimination Model 379 10.3.1 Nuisance Parameter Elimination Summary 382 10.3.2 Nuisance Parameter Elimination Example 383 10.4 Sequential Least Squares Adjustment 391 10.4.1 Sequential Adjustment in Simple Form 393 10.5 Sequential Least Squares Adjustment Model 395 10.5.1 Summary of Sequential Least Squares Adjustment Steps 400 10.5.2 Sequential Least Squares Adjustment Example 404 Problems 415 11 Post-adjustment Data Analysis and Reliability Concepts 419 11.1 Introduction 420 11.2 Post-adjustment Detection and Elimination of Non-stochastic Errors 421 11.3 Global Tests 424 11.3.1 Standard Global Test 425 11.3.2 Global Test by Baarda 426 11.4 Local Tests 427 11.5 Pope\'s Approach to Local Test 428 11.6 Concepts of Redundancy Numbers 430 11.7 Baarda\'s Data Analysis Approach 433 11.7.1 Baarda\'s Approach to Local Test 435 11.8 Concepts of Reliability Measures 437 11.8.1 Internal Reliability Measures 437 11.8.2 External Reliability Measures 440 11.9 Network Sensitivity 441 Problems 447 12 Least Squares Adjustment of Conditional Models 451 12.1 Introduction 452 12.2 Conditional Model Equations 452 12.2.1 Examples of Model Equations 453 12.3 Conditional Model Adjustment Formulation 459 12.3.1 Conditional Model Adjustment Steps 464 12.4 Stochastic Model of Conditional Adjustment 470 12.4.1 Derivation of Cofactor Matrix of Adjusted Observations 470 12.4.2 Derivation of Cofactor Matrix of Observation Residuals 471 12.4.3 Covariance Matrices of Adjusted Observations and Residuals 472 12.5 Assessment of Observations and Conditional Model 473 12.6 Variance-Covariance Propagation for Derived Parameters from Conditional Adjustment 474 12.7 Simple GNSS Network Adjustment Example 480 12.8 Simple Traverse Network Adjustment Example 484 Problems 490 13 Least Squares Adjustment of General Models 493 13.1 Introduction 494 13.2 General Model Equation Formulation 494 13.3 Linearization of General Model 497 13.4 Variation Function for Linearized General Model 500 13.5 Normal Equation System and the Least Squares Solution 501 13.6 Steps for General Model Adjustment 502 13.7 General Model Adjustment Examples 503 13.7.1 Coordinate Transformations 503 13.7.1.1 Two-dimensional Similarity Transformation Example 503 13.7.2 Parabolic Vertical Transition Curve Example 508 13.8 Stochastic Properties of General Model Adjustment 512 13.8.1 Derivation of Cofactor Matrix of Adjusted Parameters 512 13.8.2 Derivation of Cofactor Matrices of Adjusted Observations and Residuals 513 13.8.3 Covariance Matrices of Adjusted Quantities 514 13.8.4 Summary of Stochastic Properties of General Model Adjustment 515 13.9 Horizontal Circular Curve Example 516 13.10 Adjustment of General Model with Weight Constraints 524 13.10.1 Variation Function for General Model with Weight Constraints 524 13.10.2 Normal Equation System and Solution 525 13.10.3 Stochastic Models of Adjusted Quantities 526 Problems 538 14 Datum Problem and Free Network Adjustment 543 14.1 Introduction 543 14.2 Minimal Datum Constraint Types 544 14.3 Free Network Adjustment Model 545 14.4 Constraint Model for Free Network Adjustment 548 14.5 Summary of Free Network Adjustment Procedure 551 14.6 Datum Transformation 559 14.6.1 Iterative Weighted Similarity Transformation 565 Problems 566 15 Introduction to Dynamic Mode Filtering and Prediction 571 15.1 Introduction 571 15.1.1 Prediction, Filtering, and Smoothing 574 15.2 Static Mode Filter 575 15.2.1 Real-time Moving Averages as Static Mode Filter 575 15.2.2 Sequential Least Adjustment as Static Mode Filter 577 15.3 Dynamic Mode Filter 578 15.3.1 Summary of Kalman Filtering Process 581 15.4 Kalman Filtering Examples 583 15.5 Kalman Filter and the Least Squares Method 607 15.5.1 Filtering and Sequential Least Squares Adjustment: Similarities and Differences 608 Problems 610 16 Introduction to Least Squares Collocation and the Kriging Methods 613 16.1 Introduction 613 16.2 Elements of Least Squares Collocation 616 16.3 Collocation Procedure 617 16.4 Covariance Function 618 16.5 Collocation and Classical Least Squares Adjustment 621 16.6 Elements of Kriging 624 16.7 Semivariogram Model and Modeling 624 16.8 Kriging Procedure 627 16.8.1 Simple Kriging 628 16.8.2 Ordinary Kriging 629 16.8.3 Universal Kriging 631 16.9 Comparing Least Squares Collocation and Kriging 632 Appendix A Extracts from Baarda\'s Nomogram 635 Appendix B Standard Statistical Distribution Tables 639 Appendix C Tau Critical Values Table for Significance Level   649 Appendix D General Partial Differentials of Typical Survey Observables 653 Appendix E Some Important Matrix Operations and Identities 661 Appendix F Commonly Used Abbreviations 669 References 671 Index 675