# Adjustment Computations

## Spatial Data Analysis

6. Edition December 2017

720 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-38598-1**

The definitive guide to bringing accuracy to measurement, updated and supplemented

Adjustment Computations is the classic textbook for spatial information analysis and adjustment computations, providing clear, easy-to-understand instruction backed by real-world practicality. From the basic terms and fundamentals of errors to specific adjustment computations and spatial information analysis, this book covers the methodologies and tools that bring accuracy to surveying, GNSS, GIS, and other spatial technologies. Broad in scope yet rich in detail, the discussion avoids overly-complex theory in favor of practical techniques for students and professionals. This new sixth edition has been updated to align with the latest developments in this rapidly expanding field, and includes new video lessons and updated problems, including worked problems in STATS, MATRIX, ADJUST, and MathCAD.

All measurement produces some amount of error; whether from human mistakes, instrumentation inaccuracy, or environmental features, these errors must be accounted and adjusted for when accuracy is critical. This book describes how errors are identified, analyzed, measured, and corrected, with a focus on least squares adjustment--the most rigorous methodology available.

* Apply industry-standard methodologies to error analysis and adjustment

* Translate your skills to the real-world with instruction focused on the practical

* Master the fundamentals as well as specific computations and analysis

* Strengthen your understanding of critical topics on the Fundamentals in Surveying Licensing Exam

As spatial technologies expand in both use and capability, so does our need for professionals who understand how to check and adjust for errors in spatial data. Conceptual knowledge is one thing, but practical skills are what counts when accuracy is at stake; Adjustment Computations provides the real-world training you need to identify, analyze, and correct for potentially crucial errors.

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 / 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 / 18

2.7 Additional Definitions / 19

2.8 Alternative Formula for Determining Variance / 22

2.9 Numerical Examples / 24

2.10 Root Mean Square Error and Mapping Standards / 28

2.11 Derivation of the Sample Variance (Bessel's Correction) / 31

2.12 Software / 32

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 / 50

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 / 63

4.4 Testing the Validity of the Confidence Interval / 66

4.5 Selecting a Sample Size / 67

4.6 Confidence Interval for a Population Variance / 68

4.7 Confidence Interval for the Ratio of Two Population Variances / 70

4.8 Software / 72

Problems / 75

5 Statistical Testing 79

5.1 Hypothesis Testing / 79

5.2 Systematic Development of a Test / 82

5.3 Test of Hypothesis for the Population Mean / 84

5.4 Test of Hypothesis for the Population Variance / 85

5.5 Test of Hypothesis for the Ratio of Two Population Variances / 89

5.6 Software / 92

Problems / 93

6 Propagation of Random Errors in Indirectly Measured Quantities 97

6.1 Basic Error Propagation Equation / 97

6.2 Frequently Encountered Specific Functions / 102

6.3 Numerical Examples / 103

6.4 Software / 107

6.5 Conclusions / 109

Problems / 109

Practical Exercises / 112

7 Error Propagation in Angle and Distance Observations 113

7.1 Introduction / 113

7.2 Error Sources in Horizontal Angles / 113

7.3 Reading Errors / 114

7.4 Pointing Errors / 116

7.5 Estimated Pointing and Reading Errors with Total Stations / 117

7.6 Target-Centering Errors / 118

7.7 Instrument Centering Errors / 120

7.8 Effects of Leveling Errors in Angle Observations / 123

7.9 Numerical Example of Combined Error Propagation in a Single Horizontal Angle / 126

7.10 Using Estimated Errors to Check Angular Misclosure in a Traverse / 127

7.11 Errors in Astronomical Observations for Azimuth / 130

7.12 Errors in Electronic Distance Observations / 135

7.13 Centering Errors When Using Range Poles / 136

7.14 Software / 137

Problems / 138

Programming Problems / 141

8 Error Propagation in Traverse Surveys 143

8.1 Introduction / 143

8.2 Derivation of Estimated Error in Latitude and Departure / 144

8.3 Derivation of Estimated Standard Errors in Course Azimuths / 146

8.4 Computing and Analyzing Polygon Traverse Misclosure Errors / 146

8.5 Computing and Analyzing Link Traverse Misclosure Errors / 152

8.6 Software / 156

8.7 Conclusions / 157

Problems / 157

Programming Problems / 161

9 Error Propagation in Elevation Determination 163

9.1 Introduction / 163

9.2 Systematic Errors in Differential Leveling / 163

9.3 Random Errors in Differential Leveling / 166

9.4 Error Propagation in Trigonometric Leveling / 171

Problems / 174

Programming Problems / 177

10 Weights of Observations 179

10.1 Introduction / 179

10.2 Weighted Mean / 181

10.3 Relationship Between Weights and Standard Errors / 183

10.4 Statistics of Weighted Observations / 184

10.5 Weights in Angle Observations / 185

10.6 Weights in Differential Leveling / 186

10.7 Practical Examples / 187

Problems / 190

11 Principles of Least Squares 193

11.1 Introduction / 193

11.2 Fundamental Principle of Least Squares / 194

11.3 The Fundamental Principle of Weighted Least Squares / 196

11.4 The Stochastic Model / 197

11.5 Functional Model / 197

11.6 Observation Equations / 199

11.7 Systematic Formulation of the Normal Equations / 201

11.8 Tabular Formation of the Normal Equations / 203

11.9 Using Matrices to Form the Normal Equations / 204

11.10 Least Squares Solution of Nonlinear Systems / 207

11.11 Least Squares Fit of Points to a Line or Curve / 211

11.12 Calibration of an EDM Instrument / 214

11.13 Least Squares Adjustment Using Conditional Equations / 215

11.14 The Previous Example Using Observation Equations / 217

11.15 Software / 219

Problems / 219

12 Adjustment of Level Nets 225

12.1 Introduction / 225

12.2 Observation Equation / 225

12.3 Unweighted Example / 226

12.4 Weighted Example / 229

12.5 Reference Standard Deviation / 231

12.6 Another Weighted Adjustment / 233

12.7 Software / 236

Problems / 238

Programming Problems / 242

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 / 250

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 / 269

14.6 Iteration Termination / 273

14.7 Software / 274

Problems / 276

Programming Problems / 282

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 / 312

16 Adjustment of Horizontal Surveys: Traverses and Horizontal Networks 313

16.1 Introduction to Traverse Adjustments / 313

16.2 Observation Equations / 313

16.3 Redundant Equations / 314

16.4 Numerical Example / 315

16.5 Minimum Amount of Control / 321

16.6 Adjustment of Networks / 322

16.7 2 Test: Goodness of Fit / 330

Problems / 331

Programming Problems / 342

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 / 354

17.7 Network Preadjustment Data Analysis / 356

17.8 Least Squares Adjustment of GNSS Networks / 363

Problems / 369

Programming Problems / 386

18 Coordinate Transformations 389

18.1 Introduction / 389

18.2 The Two-Dimensional Conformal Coordinate / 389

18.3 Equation Development / 390

18.4 Application of Least Squares / 392

18.5 Two-Dimensional Affine Coordinate Transformation / 395

18.6 The Two-Dimensional Projective Coordinate Transformation / 398

18.7 Three-Dimensional Conformal Coordinate Transformation / 401

18.8 Statistically Valid Parameters / 407

Problems / 411

Programming Problems / 418

19 Error Ellipse 419

19.1 Introduction / 419

19.2 Computation of Ellipse Orientation and Semiaxes / 421

19.3 Example Problem of Standard Error Ellipse Calculations / 426

19.4 Another Example Problem / 428

19.5 The Error Ellipse Confidence Level / 429

19.6 Error Ellipse Advantages / 431

19.7 Other Measures of Station Uncertainty / 435

Problems / 441

Programming Problems / 442

20 Constraint Equations 443

20.1 Introduction / 443

20.2 Adjustment of Control Station Coordinates / 443

20.3 Holding Control Station Coordinates and Directions of Lines Fixed in a Trilateration Adjustment / 449

20.4 Helmert's Method / 452

20.5 Redundancies in a Constrained Adjustment / 458

20.6 Enforcing Constraints through Weighting / 458

Problems / 460

Practical Problems / 463

21 Blunder Detection in Horizontal Networks 465

21.1 Introduction / 465

21.2 A Priori Methods for Detecting Blunders in Observations / 466

21.3 A Posteriori Blunder Detection / 468

21.4 Development of the Covariance Matrix for the Residuals / 470

21.5 Detection of Outliers in Observations: Data Snooping / 472

21.6 Detection of Outliers in Observations: The Tau Criterion / 474

21.7 Techniques Used in Adjusting Control / 476

21.8 A Data Set with Blunders / 477

21.9 Some Further Considerations / 485

21.10 Survey Design / 487

21.11 Software / 489

Problems / 490

Practical Problems / 496

22 The General Least Squares Method and Its Application to Curve Fitting and Coordinate Transformations 497

22.1 Introduction to General Least Squares / 497

22.2 General Least Squares Equations for Fitting a Straight Line / 497

22.3 General Least Squares Solution / 499

22.4 Two-Dimensional Coordinate Transformation by General Least Squares / 503

22.5 Three-Dimensional Conformal Coordinate Transformation by General Least Squares / 509

Problems / 511

Programming Problems / 515

23 Three-Dimensional Geodetic Network Adjustment 517

23.1 Introduction / 517

23.2 Linearization of Equations / 519

23.3 Minimum Number of Constraints / 524

23.4 Example Adjustment / 525

23.5 Building an Adjustment / 533

23.6 Comments on Systematic Errors / 534

23.7 Software / 537

Problems / 538

Programming Problems / 543

24 Combining GNSS and Terrestrial Observations 545

24.1 Introduction / 545

24.2 The Helmert Transformation / 547

24.3 Rotations between Coordinate Systems / 551

24.4 Combining GNSS Baseline Vectors with Traditional Observations / 552

24.5 Another Approach to Transforming Coordinates between Reference Frames / 556

24.6 Other Considerations / 559

Problems / 560

Programming Problems / 563

25 Analysis of Adjustments 565

25.1 Introduction / 565

25.2 Basic Concepts, Residuals, and the Normal Distribution / 565

25.3 Goodness of Fit Test / 568

25.4 Comparison of GNSS Residual Plots / 572

25.5 Use of Statistical Blunder Detection / 574

Problems / 574

26 Computer Optimization 577

26.1 Introduction / 577

26.2 Storage Optimization / 578

26.3 Direct Formation of the Normal Equations / 580

26.4 Cholesky Decomposition / 581

26.5 Forward and Back Solutions / 583

26.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix / 584

26.7 Spareness and Optimization of the Normal Matrix / 586

Problems / 590

Programming Problems / 590

Appendix A Introduction to Matrices 591

A.1 Introduction / 591

A.2 Definition of a Matrix / 591

A.3 Size or Dimensions of a Matrix / 592

A.4 Types of Matrices / 593

A.5 Matrix Equality / 594

A.6 Addition or Subtraction of Matrices / 595

A.7 Scalar Multiplication of a Matrix / 595

A.8 Matrix Multiplication / 595

A.9 Computer Algorithms for Matrix Operations / 598

A.10 Use of the Matrix Software / 601

Problems / 603

Programming Problems / 605

Appendix B Solution of Equations by Matrix Methods 607

B.1 Introduction / 607

B.2 Inverse Matrix / 607

B.3 The Inverse of a 2 × 2 Matrix / 608

B.4 Inverses by Adjoints / 610

B.5 Inverses by Elementary Row Transformations / 611

B.6 Example Problem / 616

Problems / 617

Programming Problems / 618

Appendix C Nonlinear Equations and Taylor's Theorem 619

C.1 Introduction / 619

C.2 Taylor Series Linearization of Nonlinear Equations / 619

C.3 Numerical Example / 620

C.4 Using Matrices to Solve Nonlinear Equations / 622

C.5 Simple Matrix Example / 623

C.6 Practical Example / 624

C.7 Concluding Remarks / 626

Problems / 627

Programming Problems / 628

Appendix D The Normal Error Distribution Curve and Other

Statistical Tables 629

D.1 Development for Normal Distribution Curve Equation / 629

D.2 Other Statistical Tables / 637

Appendix E Confidence Intervals for the Mean 649

Appendix F Map Projection Coordinate Systems 655

F.1 Introduction / 655

F.2 Mathematics of the Lambert Conformal Conic Map Projection / 657

F.3 Mathematics from the Transverse Mercator / 659

F.4 Stereographic Map Projection / 662

F.5 Reduction of Observations / 663

Appendix G Companion Website 669

G.1 Introduction / 669

G.2 File Formats and Memory Matters / 670

G.3 Software / 670

G.4 Using the Software as an Instructional Aid / 674

Appendix H Answers to Selected Problems 675

BIBLIOGRAPHY 681

INDEX 685