# Understanding Least Squares Estimation and Geomatics Data Analysis

# Understanding Least Squares Estimation and Geomatics Data Analysis

ISBN: 978-1-119-50144-2

Oct 2018

720 pages

$100.99

## Description

**Provides a modern approach to least squares estimation and data analysis for undergraduate land surveying and geomatics programs**

Rich in theory and concepts, this comprehensive book on least square estimation and data analysis provides examples that are designed to help students extend their knowledge to solving more practical problems. The sample problems are accompanied by suggested solutions, and are challenging, yet easy enough to manually work through using simple computing devices, and chapter objectives provide an overview of the material contained in each section.

*Understanding Least Squares Estimation and Geomatics Data Analysis* begins with an explanation of survey observables, observations, and their stochastic properties. It reviews matrix structure and construction and explains the needs for adjustment. Next, it discusses analysis and error propagation of survey observations, including the application of heuristic rule for covariance propagation. Then, the important elements of statistical distributions commonly used in geomatics are discussed. Main topics of the book include: concepts of datum definitions; the formulation and linearization of parametric, conditional and general model equations involving typical geomatics observables; geomatics problems; least squares adjustments of parametric, conditional and general models; confidence region estimation; problems of network design and pre-analysis; three-dimensional geodetic network adjustment; nuisance parameter elimination and the sequential least squares adjustment; post-adjustment data analysis and reliability; the problems of datum; mathematical filtering and prediction; an introduction to least squares collocation and the kriging methods; and more.

- Contains ample concepts/theory and content, as well as practical and workable examples
- Based on the author's manual, which he developed as a complete and comprehensive book for his Adjustment of Surveying Measurements and Special Topics in Adjustments courses
- Provides geomatics undergraduates and geomatics professionals with required foundational knowledge
- An excellent companion to
*Precision Surveying: The Principles and Geomatics Practice*

*Understanding Least Squares Estimation and Geomatics Data Analysis* is recommended for undergraduates studying geomatics, and will benefit many readers from a variety of geomatics backgrounds, including practicing surveyors/engineers who are interested in least squares estimation and data analysis, geomatics researchers, and software developers for geomatics.

## Related Resources

### Student

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 nMethods 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 α 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