Adjustment Computations

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 / 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