Acknowledgments xxiii

About the Companion Website xxv

1 Introduction 1

1.1 Introduction 1

1.2 Direct and Indirect Measurements 2

1.3 Measurement Error Sources 2

1.4 Definitions 3

1.5 Precision versus Accuracy 4

1.6 Redundant Observations in Surveying and Their Adjustment 7

1.7 Advantages of Least Squares Adjustment 8

1.8 Overview of the Book 10

Problems 10

2 Observations and Their Analysis 13

2.1 Introduction 13

2.2 Sample versus Population 13

2.3 Range and Median 14

2.4 Graphical Representation of Data 15

2.5 Numerical Methods of Describing Data 18

2.6 Measures of Central Tendency 19

2.7 Additional Definitions 19

2.8 Alternative Formulas for Determining Variance 22

2.9 Numerical Examples 25

2.10 Root Mean Square Error and Mapping Standards 29

2.11 Derivation of the Sample Variance (Bessel’s Correction) 32

2.12 Software 33

Problems 34

Practical Exercises 37

3 Random Error Theory 39

3.1 Introduction 39

3.2 Theory of Probability 39

3.3 Properties of the Normal Distribution Curve 42

3.4 Standard Normal Distribution Function 44

3.5 Probability of the Standard Error 47

3.6 Uses for Percent Errors 49

3.7 Practical Examples 50

Problems 53

Programming Problems 55

4 Confidence Intervals 57

4.1 Introduction 57

4.2 Distributions Used in Sampling Theory 59

4.3 Confidence Interval for the Mean: t Statistic 64

4.4 Testing the Validity of the Confidence Interval 67

4.5 Selecting a Sample Size 67

4.6 Confidence Interval for a Population Variance 69

4.7 Confidence Interval for the Ratio of Two Population Variances 70

4.8 Software 73

Problems 75

5 Statistical Testing 81

5.1 Hypothesis Testing 81

5.2 Systematic Development of a Test 84

5.3 Test of Hypothesis for the Population Mean 86

5.4 Test of Hypothesis for the Population Variance 88

5.5 Test of Hypothesis for the Ratio of Two Population Variances 91

5.6 Using Software 94

Problems 95

6 Propagation of Random Errors in Indirectly Measured Quantities 99

6.1 Basic Error Propagation Equation 99

6.2 Frequently Encountered Specific Functions 104

6.3 Numerical Examples 105

6.4 Software 109

6.5 Conclusions 111

Problems 111

Practical Exercises 114

7 Error Propagation in Angle and Distance Observations 115

7.1 Introduction 115

7.2 Error Sources in Horizontal Angles 116

7.3 Reading Errors 116

7.4 Pointing Errors 118

7.5 Estimated Pointing and Reading Errors with Total Stations 119

7.6 Target Centering Errors 120

7.7 Instrument Centering Errors 122

7.8 Effects of Leveling Errors in Angle Observations 126

7.9 Numerical Example of Combined Error Propagation in a Single Horizontal Angle 128

7.10 Using Estimated Errors to Check Angular Misclosure in a Traverse 130

7.11 Errors in Astronomical Observations for Azimuth 132

7.12 Errors in Electronic Distance Observations 137

7.13 Centering Errors When Using Range Poles 138

7.14 Software 139

Problems 140

Programming Problems 143

8 Error Propagation in Traverse Surveys 145

8.1 Introduction 145

8.2 Derivation of Estimated Error in Latitude and Departure 146

8.3 Derivation of Estimated Standard Errors in Course Azimuths 148

8.4 Computing and Analyzing Polygon Traverse Misclosure Errors 148

8.5 Computing and Analyzing Link Traverse Misclosure Errors 154

8.6 Software 158

8.7 Conclusions 159

Problems 159

Programming Problems 163

9 Error Propagation in Elevation Determination 165

9.1 Introduction 165

9.2 Systematic Errors in Differential Leveling 165

9.3 Random Errors in Differential Leveling 169

9.4 Error Propagation in Trigonometric Leveling 173

Problems 177

Programming Problems 179

10 Weights of Observations 181

10.1 Introduction 181

10.2 Weighted Mean 183

10.3 Relationship Between Weights and Standard Errors 185

10.4 Statistics of Weighted Observations 186

10.5 Weights in Angle Observations 187

10.6 Weights in Differential Leveling 188

10.7 Practical Examples 189

Problems 192

11 Principles of Least Squares 195

11.1 Introduction 195

11.2 Fundamental Principle of Least Squares 196

11.3 The Fundamental Principle of Weighted Least Squares 198

11.4 The Stochastic Model 199

11.5 Functional Model 199

11.6 Observation Equations 201

11.7 Systematic Formulation of the Normal Equations 203

11.8 Tabular Formation of the Normal Equations 205

11.9 Using Matrices to Form the Normal Equations 206

11.10 Least-Squares Solution of Nonlinear Systems 209

11.11 Least-Squares Fit of Points to a Line or Curve 213

11.12 Calibration of an EDM Instrument 216

11.13 Least-Squares Adjustment Using Conditional Equations 217

11.14 The Previous Example Using Observation Equations 219

11.15 Software 221

Problems 221

12 Adjustment of Level Nets 227

12.1 Introduction 227

12.2 Observation Equation 227

12.3 Unweighted Example 228

12.4 Weighted Example 230

12.5 Reference Standard Deviation 233

12.6 Another Weighted Adjustment 234

12.7 Software 237

Problems 240

Programming Problems 244

13 Precisions of Indirectly Determined Quantities 245

13.1 Introduction 245

13.2 Development of the Covariance Matrix 245

13.3 Numerical Examples 249

13.4 Standard Deviations of Computed Quantities 251

Problems 254

Programming Problems 256

14 Adjustment of Horizontal Surveys: Trilateration 257

14.1 Introduction 257

14.2 Distance Observation Equation 259

14.3 Trilateration Adjustment Example 261

14.4 Formulation of a Generalized Coefficient Matrix for a More Complex Network 268

14.5 Computer Solution of a Trilaterated Quadrilateral 268

14.6 Iteration Termination 272

14.7 Software 274

Problems 276

Programming Problems 281

15 Adjustment of Horizontal Surveys: Triangulation 283

15.1 Introduction 283

15.2 Azimuth Observation Equation 284

15.3 Angle Observation Equation 286

15.4 Adjustment of Intersections 288

15.5 Adjustment of Resections 293

15.6 Adjustment of Triangulated Quadrilaterals 298

Problems 303

Programming Problems 310

16 Adjustment of Horizontal Surveys: Traverses and Horizontal Networks 313

16.1 Introduction to Traverse Adjustments 313

16.2 Observation Equations 314

16.3 Redundant Equations 314

16.4 Numerical Example 315

16.5 Minimum Amount of Control 322

16.6 Adjustment of Networks 322

16.7 χ 2 Test: Goodness-of-Fit 330

Problems 331

Programming Problems 341

17 Adjustment of GNSS Networks 343

17.1 Introduction 343

17.2 GNSS Observations 344

17.3 GNSS Errors and the Need for Adjustment 347

17.4 Reference Coordinate Systems for GNSS Observations 347

17.5 Converting Between the Terrestrial and Geodetic Coordinate Systems 350

17.6 Application of Least Squares in Processing GNSS Data 353

17.7 Network Preadjustment Data Analysis 356

17.8 Least Squares Adjustment of GNSS Networks 363

Problems 369

Programming Problems 382

18 Coordinate Transformations 383

18.1 Introduction 383

18.2 The Two-Dimensional Conformal Coordinate 384

18.3 Equation Development 384

18.4 Application of Least Squares 386

18.5 Two-Dimensional Affine Coordinate Transformation 389

18.6 The Two-Dimensional Projective Coordinate Transformation 392

18.7 Three-Dimensional Conformal Coordinate Transformation 394

18.8 Statistically Valid Parameters 400

Problems 404

Programming Problems 410

19 Error Ellipse 411

19.1 Introduction 411

19.2 Computation of Ellipse Orientation and Semiaxes 413

19.3 Example Problem of Standard Error Ellipse Calculations 418

19.4 Another Example Problem 420

19.5 The Error Ellipse Confidence Level 421

19.6 Error Ellipse Advantages 423

19.7 Other Measures of Station Uncertainty 426

19.8 Check Points 433

Problems 434

Programming Problems 435

20 Constraint Equations 437

20.1 Introduction 437

20.2 Adjustment of Control Station Coordinates 437

20.3 Holding Control Station Coordinates and Directions of Lines Fixed 443

20.4 Helmert’s Method 446

20.5 Redundancies in a Constrained Adjustment 451

20.6 Enforcing Constraints through Weighting 451

Problems 453

Practical Problems 455

21 Blunder Detection in Horizontal Networks 457

21.1 Introduction 457

21.2 A Priori Methods for Detecting Blunders in Observations 458

21.3 A Posteriori Blunder Detection 460

21.4 Development of the Covariance Matrix for the Residuals 462

21.5 Detection of Outliers in Observations: Data Snooping 464

21.6 Detection of Outliers in Observations: The Tau Criterion 466

21.7 Techniques Used in Adjusting Control 468

21.8 A Data Set with Blunders 469

21.9 Some Further Considerations 477

21.10 Survey Design 479

21.11 Software 481

Problems 482

Practical Problems 487

22 The General Least-Squares Method and Its Application to Curve Fitting and Coordinate Transformations 489

22.1 Introduction to General Least-Squares 489

22.2 General Least-Squares Equations for Fitting a Straight Line 489

22.3 General Least-Squares Solution 491

22.4 Two-Dimensional Coordinate Transformation by General Least-Squares 495

22.5 Three-Dimensional Conformal Coordinate Transformation by General Least-Squares 501

Problems 503

Programming Problems 507

23 Three-Dimensional Geodetic Network Adjustment 509

23.1 Introduction 509

23.2 Linearization of Equations 511

23.3 Minimum Number of Constraints 517

23.4 Example Adjustment 517

23.5 Building an Adjustment 524

23.6 Comments on Systematic Errors 526

23.7 Software 529

Problems 531

Programming Problems 534

24 Combining GNSS and Terrestrial Observations 535

24.1 Introduction 535

24.2 The Helmert Transformation 537

24.3 Rotations Between Coordinate Systems 541

24.4 Combining GNSS Baseline Vectors with Traditional Observations 542

24.5 Another Approach to Transforming Coordinates Between Reference Frames 546

24.6 Other Considerations when Localizing a Survey 549

24.7 Using Total Station Pseudo-Observations 551

24.8 Building a Stochastic Model 556

24.9 Least-Squares Adjustment using Pseudo-Observations 560

24.10 Software 562

Problems 563

Programming Problems 566

25 Analyses of Adjustments 567

25.1 Introduction 567

25.2 Basic Concepts, Residuals, and the Normal Distribution 567

25.3 Goodness-of-Fit Test 571

25.4 Comparison of GNSS Residual Plots 574

25.5 Use of Statistical Blunder Detection 576

Problems 577

26 Terrestrial Laser Scanning and Uncertainty Estimation 579

26.1 Introduction 579

26.2 Laser Scanner Coordinates 580

26.3 Beam Divergence and Incidence Angle Effects 583

26.4 Range and Angular Uncertainty 586

26.5 Error Propagation of Laser Scanner Measurements 589

26.6 Registration and Its Uncertainty 593

26.7 Point Cloud Uncertainty 596

26.8 Survey Planning 599

26.9 Conclusions 607

26.10 Software 608

Problems 608

27 Plane, Three-Dimensional Objects, and Elevation Model Estimation with Point Clouds 615

27.1 Introduction 615

27.2 Plane Fitting with Point Clouds 616

27.3 Sphere and Cylinder Estimation with Point Clouds 628

27.4 RANSAC Estimation 645

27.5 Polynomial and Elevation Estimation with Point Clouds 645

27.6 Software 649

Problems 649

28 Computer Optimization 657

28.1 Introduction 657

28.2 Storage Optimization 658

28.3 Direct Formation of the Normal Equations 660

28.4 Cholesky Decomposition 662

28.5 Forward and Backward Solutions 662

28.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix 664

28.7 Spareness and Optimization of the Normal Matrix 664

Problems 670

Programming Problem 671

Appendix A Introduction to Matrices 673

A.1 Introduction 673

A.2 Definition of a Matrix 673

A.3 Size or Dimensions of a Matrix 674

A.4 Types of Matrices 675

A.5 Matrix Equality 676

A.6 Addition or Subtraction of Matrices 677

A.7 Scalar Multiplication of a Matrix 677

A.8 Matrix Multiplication 677

A.9 Computer Algorithms for Matrix Operations 681

A.10 Use of the Matrix Software 683

Problems 685

Programming Problems 687

Appendix B Solution of Equations by Matrix Methods 689

B.1 Introduction 689

B.2 Inverse Matrix 689

B.3 The Inverse of a 2 × 2 Matrix 690

B.4 Inverses by Adjoints 692

B.5 Inverses by Elementary Row Transformations 693

B.6 Example Problem 696

B.7 Eigenvalues and Eigenvectors 698

Problems 700

Programming Problems 702

Appendix C Nonlinear Equations and Taylor’s Theorem 703

C.1 Introduction 703

C.2 Taylor Series Linearization of Nonlinear Equations 703

C.3 Numerical Example 705

C.4 Using Matrices to Solve Nonlinear Equations 706

C.5 Simple Matrix Example 707

C.6 Practical Example 708

C.7 Concluding Remarks 710

Problems 711

Programming Problems 712

Appendix D The Normal Error Distribution Curve and Other Statistical Tables 713

D.1 Development for Normal Distribution Curve Equation 713

D.2 Other Statistical Tables 721

Appendix E Confidence Intervals for the Mean 733

Appendix F Companion Website 739

F.1 Introduction 739

F.2 File Formats and Memory Matters 740

F.3 Software 740

F.4 Using the Software as an Instructional Aid 743

Appendix G Answers to Selected Problems 745

Bibliography 751

Index 757