Adjustment Computations: Spatial Data Analysis, 5th Edition

PREFACE xv

ACKNOWLEDGMENTS xix

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 6

1.7. Advantages of Least Squares Adjustment 8

1.8. Overview of the Book 9

Problems 10

2 Observations and Their Analysis 12

2.1. Introduction 12

2.2. Sample versus Population 12

2.3. Range and Median 13

2.4. Graphical Representation of Data 14

2.5. Numerical Methods of Describing Data 17

2.6. Measures of Central Tendency 17

2.7. Additional Definitions 18

2.8. Alternative Formula for Determining Variance 21

2.9. Numerical Examples 22

2.10. Derivation of the Sample Variance (Bessel’s Correction) 26

2.11. Software 28

Problems 29

Practical Exercises 32

3 Random Error Theory 33

3.1. Introduction 33

3.2. Theory of Probability 33

3.3. Properties of the Normal Distribution Curve 36

3.4. Standard Normal Distribution Function 38

3.5. Probability of the Standard Error 41

3.6. Uses for Percent Errors 43

3.7. Practical Examples 44

Problems 46

Programming Problems 48

4 Confidence Intervals 49

4.1. Introduction 49

4.2. Distributions Used in Sampling Theory 51

4.3. Confidence Interval for the Mean: t statistic 55

4.4. Testing the Validity of the Confidence Interval 58

4.5. Selecting a Sample Size 59

4.6. Confidence Interval for a Population Variance 60

4.7. Confidence Interval for the Ratio of Two Population Variances 61

4.8. Software 64

Problems 66

5 Statistical Testing 70

5.1. Hypothesis Testing 70

5.2. Systematic Development of a Test 73

5.3. Test of Hypothesis for the Population Mean 74

5.4. Test of Hypothesis for the Population Variance 76

5.5. Test of Hypothesis for the Ratio of Two Population Variances 79

5.6. Software 82

Problems 83

6 Propagation of Random Errors in Indirectly Measured Quantities 86

6.1. Basic Error Propagation Equation 86

6.2. Frequently Encountered Specific Functions 91

6.3. Numerical Examples 92

6.4. Software 96

6.5. Conclusions 98

Problems 98

Practical Exercises 102

7 Error Propagation in Angle and Distance Observations 103

7.1. Introduction 103

7.2. Error Sources in Horizontal Angles 103

7.3. Reading Errors 104

7.4. Pointing Errors 106

7.5. Estimated Pointing and Reading Errors with Total Stations 107

7.6. Target-Centering Errors 108

7.7. Instrument-Centering Errors 110

7.8. Effects of Leveling Errors in Angle Observations 113

7.9. Numerical Example of Combined Error Propagation in a Single Horizontal Angle 116

7.10. Using Estimated Errors to Check Angular Misclosure in a Traverse 117

7.11. Errors in Astronomical Observations for Azimuth 119

7.12. Errors in Electronic Distance Observations 124

7.13. Software 125

Problems 126

Programming Problems 130

8 Error Propagation in Traverse Surveys 131

8.1. Introduction 131

8.2. Derivation of Estimated Error in Latitude and Departure 132

8.3. Derivation of Estimated Standard Errors in Course Azimuths 134

8.4. Computing and Analyzing Polygon Traverse Misclosure Errors 134

8.5. Computing and Analyzing Link Traverse Misclosure Errors 140

8.6. Software 144

8.7. Conclusions 145

Problems 145

Programming Problems 150

9 Error Propagation in Elevation Determination 151

9.1. Introduction 151

9.2. Systematic Errors in Differential Leveling 151

9.3. Random Errors in Differential Leveling 154

9.4. Error Propagation in Trigonometric Leveling 159

Problems 162

Programming Problems 164

10 Weights of Observations 165

10.1. Introduction 165

10.2. Weighted Mean 167

10.3. Relation between Weights and Standard Errors 169

10.4. Statistics of Weighted Observations 169

10.5. Weights in Angle Observations 171

10.6. Weights in Differential Leveling 171

10.7. Practical Examples 173

Problems 175

11 Principles of Least Squares 178

11.1. Introduction 178

11.2. Fundamental Principle of Least Squares 179

11.3. Fundamental Principle of Weighted Least Squares 181

11.4. Stochastic Model 182

11.5. Functional Model 183

11.6. Observation Equations 184

11.7. Systematic Formulation of the Normal Equations 186

11.8. Tabular Formation of the Normal Equations 188

11.9. Using Matrices to Form Normal Equations 189

11.10. Least Squares Solution of Nonlinear Systems 192

11.11. Least Squares Fit of Points to a Line or Curve 195

11.12. Calibration of an EDM Instrument 199

11.13. Least Squares Adjustment Using Conditional Equations 200

11.14. The Previous Example Using Observation Equations 202

11.15. Software 203

Problems 204

12 Adjustment of Level Nets 210

12.1. Introduction 210

12.2. Observation Equation 210

12.3. Unweighted Example 211

12.4. Weighted Example 214

12.5. Reference Standard Deviation 216

12.6. Another Weighted Adjustment 218

12.7. Software 221

Problems 223

Programming Problems 227

13 Precisions of Indirectly Determined Quantities 228

13.1. Introduction 228

13.2. Development of the Covariance Matrix 228

13.3. Numerical Examples 232

13.4. Standard Deviations of Computed Quantities 233

Problems 236

Programming Problems 239

14 Adjustment of Horizontal Surveys: Trilateration 240

14.1. Introduction 240

14.2. Distance Observation Equation 242

14.3. Trilateration Adjustment Example 244

14.4. Formulation of a Generalized Coefficient Matrix for a More Complex Network 250

14.5. Computer Solution of a Trilaterated Quadrilateral 251

14.6. Iteration Termination 255

14.7. Software 256

Problems 258

Programming Problems 264

15 Adjustment of Horizontal Surveys: Triangulation 266

15.1. Introduction 266

15.2. Azimuth Observation Equation 266

15.3. Angle Observation Equation 269

15.4. Adjustment of Intersections 271

15.5. Adjustment of Resections 276

15.6. Adjustment of Triangulated Quadrilaterals 282

Problems 287

Programming Problems 296

16 Adjustment of Horizontal Surveys: Traverses and Horizontal Networks 298

16.1. Introduction to Traverse Adjustments 298

16.2. Observation Equations 298

16.3. Redundant Equations 299

16.4. Numerical Example 300

16.5. Minimum Amount of Control 306

16.6. Adjustment of Networks 307

16.7. χ2 Test: Goodness of Fit 315

Problems 316

Programming Problems 326

17 Adjustment of GNSS Networks 327

17.1. Introduction 327

17.2. GNSS Observations 328

17.3. GNSS Errors and the Need for Adjustment 330

17.4. Reference Coordinate Systems for GNSS Observations 331

17.5. Converting between the Terrestrial and Geodetic Coordinate Systems 334

17.6. Application of Least Squares in Processing GNSS Data 337

17.7. Network Preadjustment Data Analysis 340

17.8. Least Squares Adjustment of GNSS Networks 346

Problems 352

Programming Problems 366

18 Coordinate Transformations 368

18.1. Introduction 368

18.2. Two-Dimensional Conformal Coordinate 368

18.3. Equation Development 369

18.4. Application of Least Squares 371

18.5. Two-Dimensional Affine Coordinate Transformation 374

18.6. Two-Dimensional Projective Coordinate Transformation 377

18.7. Three-Dimensional Conformal Coordinate Transformation 380

18.8. Statistically Valid Parameters 386

Problems 390

Programming Problems 396

19 Error Ellipse 397

19.1. Introduction 397

19.2. Computation of Ellipse Orientation and Semiaxes 399

19.3. Example Problem of Standard Error Ellipse Calculations 404

19.4. Another Example Problem 406

19.5. Error Ellipse Confidence Level 407

19.6. Error Ellipse Advantages 409

19.7. Other Measures of Station Uncertainty 412

Problems 413

Programming Problems 415

20 Constraint Equations 416

20.1. Introduction 416

20.2. Adjustment of Control Station Coordinates 416

20.3. Holding Control Fixed in a Trilateration Adjustment 421

20.4. Helmert’s Method 424

20.5. Redundancies in a Constrained Adjustment 429

20.6. Enforcing Constraints through Weighting 429

Problems 431

Practical Exercises 434

21 Blunder Detection in Horizontal Networks 435

21.1. Introduction 435

21.2. A Priori Methods for Detecting Blunders in Observations 436

21.3. A Posteriori Blunder Detection 438

21.4. Development of the Covariance Matrix for the Residuals 439

21.5. Detection of Outliers in Observations: Data Snooping 442

21.6. Detection of Outliers in Observations: The Tau Criterion 444

21.7. Techniques Used In Adjusting Control 444

21.8. Data Set with Blunders 446

21.9. Further Considerations 453

21.10. Survey Design 455

21.11. Software 457

Problems 458

Practical Exercises 462

22 General Least Squares Method and Its Application to Curve Fitting and Coordinate
Transformations 464

22.1. Introduction to General Least Squares 464

22.2. General Least Squares Equations for Fitting a Straight Line 464

22.3. General Least Squares Solution 466

22.4. Two-Dimensional Coordinate Transformation by General Least Squares 470

22.5. Three-Dimensional Conformal Coordinate Transformation by General Least Squares 476

Problems 478

Programming Problems 482

23 Three-Dimensional Geodetic Network Adjustment 483

23.1. Introduction 483

23.2. Linearization of Equations 485

23.3. Minimum Number of Constraints 490

23.4. Example Adjustment 490

23.5. Building an Adjustment 499

23.6. Comments on Systematic Errors 499

23.7. Software 502

Problems 503

Programming Problems 507

24 Combining GPS and Terrestrial Observations 508

24.1. Introduction 508

24.2. Helmert’s Transformation 510

24.3. Rotations between Coordinate Systems 513

24.4. Combining GPS Baseline Vectors with Traditional Observations 514

24.5. Another Approach to Transforming Coordinates between Reference Frames 518

24.6. Other Considerations 521

Problems 522

Programming Problems 524

25 Analysis of Adjustments 525

25.1. Introduction 525

25.2. Basic Concepts, Residuals, and the Normal Distribution 525

25.3. Goodness-of-Fit Test 528

25.4. Comparison of Residual Plots 531

25.5. Use of Statistical Blunder Detection 533

Problems 534

26 Computer Optimization 536

26.1. Introduction 536

26.2. Storage Optimization 536

26.3. Direct Formation of the Normal Equations 539

26.4. Cholesky Decomposition 540

26.5. Forward and Back Solutions 542

26.6. Using the Cholesky Factor to Find the Inverse of the Normal Matrix 543

26.7. Spareness and Optimization of the Normal Matrix 545

Problems 549

Programming Problems 549

Appendix A Introduction to Matrices 550

A.1. Introduction 550

A.2. Definition of a Matrix 550

A.3. Size or Dimensions of a Matrix 551

A.4. Types of Matrices 552

A.5. Matrix Equality 553

A.6. Addition or Subtraction of Matrices 554

A.7. Scalar Multiplication of a Matrix 554

A.8. Matrix Multiplication 554

A.9. Computer Algorithms for Matrix Operations 557

A.10. Use of the MATRIX Software 560

Problems 562

Programming Problems 564

Appendix B Solution of Equations by Matrix Methods 565

B-1. Introduction 565

B-2. Inverse Matrix 565

B-3. Inverse of a 2 × 2 Matrix 566

B-4. Inverses by Adjoints 568

B-5. Inverses by Elementary Row Transformation  569

B-6. Example Problem 573

Problems 574

Programming Problems 575

Appendix C Nonlinear Equations and Taylor’s Theorem 576

C.1. Introduction 576

C.2. Taylor Series Linearization of Nonlinear Equations 576

C.3. Numerical Example 577

C.4. Using Matrices to Solve Nonlinear Equations 579

C.5. Simple Matrix Example 580

C.6. Practical Example 581

C.7. Concluding Remarks 583

Problems 584

Programming Problems 585

Appendix D Normal Error Distribution Curve and Other Statistical Tables 586

D.1. Development of the Normal Distribution Curve Equation 586

D.2. Other Statistical Tables 594

Appendix E Confidence Intervals for the Mean 606

Appendix F Map Projection Coordinate Systems 612

F.1. Introduction 612

F.2. Mathematics of the Lambert Conformal Conic Map Projection 613

F.3. Mathematics from the Transverse Mercator 616

F.4. Stereographic Map Projection 619

F.5. Reduction of Observations 621

Appendix G Companion Web Site 625

G.1. Introduction 625

G.2. File Formats and Memory Matters 626

G.3. Software 626

G.4. Using the Software as an Instructional Aid 630

Appendix H Solutions to Selected Problems 631

BIBLIOGRAPHY 636

INDEX 639