Adjustment Computations

PREFACE. ACKNOWLEDGMENTS. 1 INTRODUCTION. 1.1 Introduction. 1.2 Direct and Indirect Measurements. 1.3 Measurement Error Sources. 1.4 Definitions. 1.5 Precision versus Accuracy. 1.6 Redundant Measurements in Surveying and Their Adjustment. 1.7 Advantages of Least Squares Adjustment. 1.8 Overview of the Book. Problems. 2 OBSERVATIONS AND THEIR ANALYSIS. 2.1 Introduction. 2.2 Sample versus Population. 2.3 Range and Median. 2.4 Graphical Representation of Data. 2.5 Numerical Methods of Describing Data. 2.6 Measures of Central Tendency. 2.7 Additional Definitions. 2.8 Alternative Formula for Determining Variance. 2.9 Numerical Examples. 2.10 Derivation of the Sample Variance (Bessel's Correction). 2.11 Programming. Problems. 3 RANDOM ERROR THEORY. 3.1 Introduction. 3.2 Theory of Probability. 3.3 Properties of the Normal Distribution Curve. 3.4 Standard Normal Distribution Function. 3.5 Probability of the Standard Error. 3.6 Uses for Percent Errors. 3.7 Practical Examples. Problems. 4 CONFIDENCE INTERVALS. 4.1 Introduction. 4.2 Distributions Used in Sampling Theory. 4.3 Confidence Interval for the Mean: t Statistic. 4.4 Testing the Validity of the Confidence Interval. 4.5 Selecting a Sample Size. 4.6 Confidence Interval for a Population Variance. 4.7 Confidence Interval for the Ratio of Two Population Variances. Problems. 5 STATISTICAL TESTING. 5.1 Hypothesis Testing. 5.2 Systematic Development of a Test. 5.3 Test of Hypothesis for the Population Mean. 5.4 Test of Hypothesis for the Population Variance. 5.5 Test of Hypothesis for the Ratio of Two Population Variances. Problems. 6 PROPAGATION OF RANDOM ERRORS IN INDIRECTLY MEASURED QUANTITIES. 6.1 Basic Error Propagation Equation. 6.2 Frequently Encountered Specific Functions. 6.3 Numerical Examples. 6.4 Conclusions. Problems. 7 ERROR PROPAGATION IN ANGLE AND DISTANCE OBSERVATIONS. 7.1 Introduction. 7.2 Error Sources in Horizontal Angles. 7.3 Reading Errors. 7.4 Pointing Errors. 7.5 Estimated Pointing and Reading Errors with Total Stations. 7.6 Target Centering Errors. 7.7 Instrument Centering Errors. 7.8 Effects of Leveling Errors in Angle Observations. 7.9 Numerical Example of Combined Error Propagation in a Single Horizontal Angle. 7.10 Use of Estimated Errors to Check Angular Misclosure in a Traverse. 7.11 Errors in Astronomical Observations for an Azimuth. 7.12 Errors in Electronic Distance Observations. 7.13 Use of Computational Software. Problems. 8 ERROR PROPAGATION IN TRAVERSE SURVEYS. 8.1 Introduction. 8.2 Derivation of Estimated Error in Latitude and Departure. 8.3 Derivation of Estimated Standard Errors in Course Azimuths. 8.4 Computing and Analyzing Polygon Traverse Misclosure Errors. 8.5 Computing and Analyzing Link Traverse Misclosure Errors. 8.6 Conclusions. Problems. 9 ERROR PROPAGATION IN ELEVATION DETERMINATION. 9.1 Introduction. 9.2 Systematic Errors in Differential Leveling. 9.3 Random Errors in Differential Leveling. 9.4 Error Propagation in Trigonometric Leveling. Problems. 10 WEIGHTS OF OBSERVATIONS. 10.1 Introduction. 10.2 Weighted Mean. 10.3 Relation between Weights and Standard Errors. 10.4 Statistics of Weighted Observations. 10.5 Weights in Angle Observations. 10.6 Weights in Differential Leveling. 10.7 Practical Examples. Problems. 11 PRINCIPLES OF LEAST SQUARES. 11.1 Introduction. 11.2 Fundamental Principle of Least Squares. 11.3 Fundamental Principle of Weighted Least Squares. 11.4 Stochastic Model. 11.5 Functional Model. 11.6 Observation Equations. 11.6.1 Elementary Example of Observation Equation Adjustment. 11.7 Systematic Formulation of the Normal Equations. 11.8 Tabular Formation of the Normal Equations. 11.9 Using Matrices to Form the Normal Equations. 11.10 Least Squares Solution of Nonlinear Systems. 11.11 Least Squares Fit of Points to a Line or Curve. 11.12 Calibration of an EDM Instrument. 11.13 Least Squares Adjustment Using Conditional Equations. 11.14 Example 11.5 Using Observation Equations. Problems. 12 ADJUSTMENT OF LEVEL NETS. 12.1 Introduction. 12.2 Observation Equation. 12.3 Unweighted Example. 12.4 Weighted Example. 12.5 Reference Standard Deviation. 12.6 Another Weighted Adjustment. Problems. 13 PRECISION OF INDIRECTLY DETERMINED QUANTITIES. 13.1 Introduction. 13.2 Development of the Covariance Matrix. 13.3 Numerical Examples. 13.4 Standard Deviations of Computed Quantities. Problems. 14 ADJUSTMENT OF HORIZONTAL SURVEYS: TRILATERATION. 14.1 Introduction. 14.2 Distance Observation Equation. 14.3 Trilateration Adjustment Example. 14.4 Formulation of a Generalized Coefficient Matrix for a More Complex Network. 14.5 Computer Solution of a Trilaterated Quadrilateral. 14.6 Iteration Termination. Problems. 15 ADJUSTMENT OF HORIZONTAL SURVEYS: TRIANGULATION. 15.1 Introduction. 15.2 Azimuth Observation Equation. 15.3 Angle Observation Equation. 15.4 Adjustment of Intersections. 15.5 Adjustment of Resections. 15.6 Adjustment of Triangulated Quadrilaterals. Problems. 16 ADJUSTMENT OF HORIZONTAL SURVEYS: TRAVERSES AND NETWORKS. 16.1 Introduction to Traverse Adjustments. 16.2 Observation Equations. 16.3 Redundant Equations. 16.4 Numerical Example. 16.5 Minimum Amount of Control. 16.6 Adjustment of Networks. 16.7 x2 Test: Goodness of Fit. Problems. 17 ADJUSTMENT OF GPS NETWORKS, 17.1 Introduction, 17.2 GPS Observations, 17.3 GPS Errors and the Need for Adjustment, 17.4 Reference Coordinate Systems for GPS Observations. 17.5 Converting between the Terrestrial and Geodetic Coordinate Systems. 17.6 Application of Least Squares in Processing GPS Data. 17.7 Network Preadjustment Data Analysis. 17.8 Least Squares Adjustment of GPS Networks. Problems. 18 COORDINATE TRANSFORMATIONS. 18.1 Introduction. 18.2 Two Dimensional Conformal Coordinate Transformation. 18.3 Equation Development. 18.4 Application of Least Squares. 18.5 Two Dimensional Affine Coordinate Transformation. 18.6 Two Dimensional Projective Coordinate Transformation. 18.7 Three Dimensional Conformal Coordinate Transformation. 18.8 Statistically Valid Parameters. Problems. 19 ERROR ELLIPSE 369 19.1 Introduction. 19.2 Computation of Ellipse Orientation and Semiaxes. 19.3 Example Problem of Standard Error Ellipse Calculations. 19.4 Another Example Problem. 19.5 Error Ellipse Confidence Level. 19.6 Error Ellipse Advantages. 19.7 Other Measures of Station Uncertainty. Problems. 20 CONSTRAINT EQUATIONS. 20.1 Introduction. 20.2 Adjustment of Control Station Coordinates. 20.3 Holding Control Station Coordinates and Directions of Lines Fixed in a Trilateration Adjustment. 20.4 Helmert's Method. 20.5 Redundancies in a Constrained Adjustment. 20.6 Enforcing Constraints through Weighting. Problems. 21 BLUNDER DETECTION IN HORIZONTAL NETWORKS. 21.1 Introduction. 21.2 A Priori Methods for Detecting Blunders in Observations. 21.3 A Posteriori Blunder Detection. 21.4 Development of the Covariance Matrix for the Residuals. 21.5 Detection of Outliers in Observations. 21.6 Techniques Used in Adjusting Control. 21.7 Data Set with Blunders. 21.8 Some Further Considerations. 21.9 Survey Design. Problems. 22 GENERAL LEAST SQUARES METHOD AND ITS APPLICATION TO CURVE FITTING AND COORDINATE TRANSFORMATIONS. 22.1 Introduction to General Least Squares. 22.2 General Least Squares Equations for Fitting a Straight Line. 22.3 General Least Squares Solution. 22.4 Two Dimensional Coordinate Transformation by General Least Squares. 22.5 Three Dimensional Conformal Coordinate Transformation by General Least Squares. Problems. 23 THREE DIMENSIONAL GEODETIC NETWORK ADJUSTMENT. 23.1 Introduction. 23.2 Linearization of Equations. 23.3 Minimum Number of Constraints. 23.4 Example Adjustment. 23.5 Building an Adjustment. 23.6 Comments on Systematic Errors. Problems. 24 COMBINING GPS AND TERRESTRIAL OBSERVATIONS. 24.1 Introduction. 24.2 Helmert Transformation. 24.3 Rotations between Coordinate Systems. 24.4 Combining GPS Baseline Vectors with Traditional Observations. 24.5 Other Considerations. Problems. 25 ANALYSIS OF ADJUSTMENTS. 25.1 Introduction. 25.2 Basic Concepts, Residuals, and the Normal Distribution. 25.3 Goodness of Fit Test. 25.4 Comparison of Residual Plots. 25.5 Use of Statistical Blunder Detection. Problems. 26 COMPUTER OPTIMIZATION. 26.1 Introduction. 26.2 Storage Optimization. 26.3 Direct Formation of the Normal Equations. 26.4 Cholesky Decomposition. 26.5 Forward and Back Solutions. 26.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix. 26.7 Spareness and Optimization of the Normal Matrix. Problems. APPENDIX A: INTRODUCTION TO MATRICES. A.1 Introduction. A.2 Definition of a Matrix. A.3 Size or Dimensions of a Matrix. A.4 Types of Matrices. A.5 Matrix Equality. A.6 Addition or Subtraction of Matrices. A.7 Scalar Multiplication of a Matrix. A.8 Matrix Multiplication. A.9 Computer Algorithms for Matrix Operations. A.9.1 Addition or Subtraction of Two Matrices. A.9.2 Matrix Multiplication. A.10 Use of the MATRIX Software. Problems. APPENDIX B: SOLUTION OF EQUATIONS BY MATRIX METHODS. B.1 Introduction. B.2 Inverse Matrix. B.3 Inverse of a 2 x 2 Matrix. B.4 Inverses by Adjoints. B.5 Inverses by Row Transformations. B.6 Example Problem. Problems. APPENDIX C: NONLINEAR EQUATIONS AND TAYLOR'S THEOREM. C.1 Introduction. C.2 Taylor Series Linearization of Nonlinear Equations. C.3 Numerical Example. C.4 Using Matrices to Solve Nonlinear Equations. C.5 Simple Matrix Example. C.6 Practical Example. Problems. APPENDIX D: NORMAL ERROR DISTRIBUTION CURVE AND OTHER STATISTICAL TABLES. D.1 Development of the Normal Distribution Curve Equation. D.2 Other Statistical Tables. D.2.1 2 Distribution. D.2.2 t Distribution. D.2.3 F Distribution. APPENDIX E: CONFIDENCE INTERVALS FOR THE MEAN. APPENDIX F: MAP PROJECTION COORDINATE SYSTEMS. F.1 Introduction. F.2 Mathematics of the Lambert Conformal Conic Map Projection. F.3 Mathematics of the Transverse Mercator. F.4 Reduction of Observations. APPENDIX G: COMPANION CD. G.1 Introduction. G.2 File Formats and Memory Matters. G.3 Software. G.4 Using the Software as an Instructional Aid. BIBLIOGRAPHY. INDEX.