Adjustment Computations

CHARLES D. GHILANI, PhD, is Professor of Engineering in the Surveying Engineering program at The Pennsylvania State University.

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