Adjustment Computations: Spatial Data Analysis

An update to a classic in the field of surveying, this is one of the few books that deals with the important issue of error in spatial data. Originally written for surveyors, it has expanded over the years to encompass the needs of new spatial technologies as they've been introduced (GPS, GIS) and new analytical techniques as they find acceptance. This Fifth Edition offers new screenshots to guide students through the exercises, more problems, more worked solutions in the solutions manual, as well as PowerPoint slides from the author's lectures.

Table of Contents:
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