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
