Preface xvii
Acknowledgments xxiii
About the Companion Website xxv
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 19
2.7 Additional Definitions 19
2.8 Alternative Formulas for Determining Variance 22
2.9 Numerical Examples 25
2.10 Root Mean Square Error and Mapping Standards 29
2.11 Derivation of the Sample Variance (Bessel's Correction) 32
2.12 Software 33
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 49
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 64
4.4 Testing the Validity of the Confidence Interval 67
4.5 Selecting a Sample Size 67
4.6 Confidence Interval for a Population Variance 69
4.7 Confidence Interval for the Ratio of Two Population Variances 70
4.8 Software 73
Problems 75
5 Statistical Testing 81
5.1 Hypothesis Testing 81
5.2 Systematic Development of a Test 84
5.3 Test of Hypothesis for the Population Mean 86
5.4 Test of Hypothesis for the Population Variance 88
5.5 Test of Hypothesis for the Ratio of Two Population Variances 91
5.6 Using Software 94
Problems 95
6 Propagation of Random Errors in Indirectly Measured Quantities 99
6.1 Basic Error Propagation Equation 99
6.2 Frequently Encountered Specific Functions 104
6.3 Numerical Examples 105
6.4 Software 109
6.5 Conclusions 111
Problems 111
Practical Exercises 114
7 Error Propagation in Angle and Distance Observations 115
7.1 Introduction 115
7.2 Error Sources in Horizontal Angles 116
7.3 Reading Errors 116
7.4 Pointing Errors 118
7.5 Estimated Pointing and Reading Errors with Total Stations 119
7.6 Target Centering Errors 120
7.7 Instrument Centering Errors 122
7.8 Effects of Leveling Errors in Angle Observations 126
7.9 Numerical Example of Combined Error Propagation in a Single Horizontal Angle 128
7.10 Using Estimated Errors to Check Angular Misclosure in a Traverse 130
7.11 Errors in Astronomical Observations for Azimuth 132
7.12 Errors in Electronic Distance Observations 137
7.13 Centering Errors When Using Range Poles 138
7.14 Software 139
Problems 140
Programming Problems 143
8 Error Propagation in Traverse Surveys 145
8.1 Introduction 145
8.2 Derivation of Estimated Error in Latitude and Departure 146
8.3 Derivation of Estimated Standard Errors in Course Azimuths 148
8.4 Computing and Analyzing Polygon Traverse Misclosure Errors 148
8.5 Computing and Analyzing Link Traverse Misclosure Errors 154
8.6 Software 158
8.7 Conclusions 159
Problems 159
Programming Problems 163
9 Error Propagation in Elevation Determination 165
9.1 Introduction 165
9.2 Systematic Errors in Differential Leveling 165
9.3 Random Errors in Differential Leveling 169
9.4 Error Propagation in Trigonometric Leveling 173
Problems 177
Programming Problems 179
10 Weights of Observations 181
10.1 Introduction 181
10.2 Weighted Mean 183
10.3 Relationship Between Weights and Standard Errors 185
10.4 Statistics of Weighted Observations 186
10.5 Weights in Angle Observations 187
10.6 Weights in Differential Leveling 188
10.7 Practical Examples 189
Problems 192
11 Principles of Least Squares 195
11.1 Introduction 195
11.2 Fundamental Principle of Least Squares 196
11.3 The Fundamental Principle of Weighted Least Squares 198
11.4 The Stochastic Model 199
11.5 Functional Model 199
11.6 Observation Equations 201
11.7 Systematic Formulation of the Normal Equations 203
11.8 Tabular Formation of the Normal Equations 205
11.9 Using Matrices to Form the Normal Equations 206
11.10 Least-Squares Solution of Nonlinear Systems 209
11.11 Least-Squares Fit of Points to a Line or Curve 213
11.12 Calibration of an EDM Instrument 216
11.13 Least-Squares Adjustment Using Conditional Equations 217
11.14 The Previous Example Using Observation Equations 219
11.15 Software 221
Problems 221
12 Adjustment of Level Nets 227
12.1 Introduction 227
12.2 Observation Equation 227
12.3 Unweighted Example 228
12.4 Weighted Example 230
12.5 Reference Standard Deviation 233
12.6 Another Weighted Adjustment 234
12.7 Software 237
Problems 240
Programming Problems 244
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 251
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 268
14.6 Iteration Termination 272
14.7 Software 274
Problems 276
Programming Problems 281
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 310
16 Adjustment of Horizontal Surveys: Traverses and Horizontal Networks 313
16.1 Introduction to Traverse Adjustments 313
16.2 Observation Equations 314
16.3 Redundant Equations 314
16.4 Numerical Example 315
16.5 Minimum Amount of Control 322
16.6 Adjustment of Networks 322
16.7 ¿ 2 Test: Goodness-of-Fit 330
Problems 331
Programming Problems 341
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 353
17.7 Network Preadjustment Data Analysis 356
17.8 Least Squares Adjustment of GNSS Networks 363
Problems 369
Programming Problems 382
18 Coordinate Transformations 383
18.1 Introduction 383
18.2 The Two-Dimensional Conformal Coordinate 384
18.3 Equation Development 384
18.4 Application of Least Squares 386
18.5 Two-Dimensional Affine Coordinate Transformation 389
18.6 The Two-Dimensional Projective Coordinate Transformation 392
18.7 Three-Dimensional Conformal Coordinate Transformation 394
18.8 Statistically Valid Parameters 400
Problems 404
Programming Problems 410
19 Error Ellipse 411
19.1 Introduction 411
19.2 Computation of Ellipse Orientation and Semiaxes 413
19.3 Example Problem of Standard Error Ellipse Calculations 418
19.4 Another Example Problem 420
19.5 The Error Ellipse Confidence Level 421
19.6 Error Ellipse Advantages 423
19.7 Other Measures of Station Uncertainty 426
19.8 Check Points 433
Problems 434
Programming Problems 435
20 Constraint Equations 437
20.1 Introduction 437
20.2 Adjustment of Control Station Coordinates 437
20.3 Holding Control Station Coordinates and Directions of Lines Fixed 443
20.4 Helmert's Method 446
20.5 Redundancies in a Constrained Adjustment 451
20.6 Enforcing Constraints through Weighting 451
Problems 453
Practical Problems 455
21 Blunder Detection in Horizontal Networks 457
21.1 Introduction 457
21.2 A Priori Methods for Detecting Blunders in Observations 458
21.3 A Posteriori Blunder Detection 460
21.4 Development of the Covariance Matrix for the Residuals 462
21.5 Detection of Outliers in Observations: Data Snooping 464
21.6 Detection of Outliers in Observations: The Tau Criterion 466
21.7 Techniques Used in Adjusting Control 468
21.8 A Data Set with Blunders 469
21.9 Some Further Considerations 477
21.10 Survey Design 479
21.11 Software 481
Problems 482
Practical Problems 487
22 The General Least-Squares Method and Its Application to Curve Fitting and Coordinate Transformations 489
22.1 Introduction to General Least-Squares 489
22.2 General Least-Squares Equations for Fitting a Straight Line 489
22.3 General Least-Squares Solution 491
22.4 Two-Dimensional Coordinate Transformation by General Least-Squares 495
22.5 Three-Dimensional Conformal Coordinate Transformation by General Least-Squares 501
Problems 503
Programming Problems 507
23 Three-Dimensional Geodetic Network Adjustment 509
23.1 Introduction 509
23.2 Linearization of Equations 511
23.3 Minimum Number of Constraints 517
23.4 Example Adjustment 517
23.5 Building an Adjustment 524
23.6 Comments on Systematic Errors 526
23.7 Software 529
Problems 531
Programming Problems 534
24 Combining GNSS and Terrestrial Observations 535
24.1 Introduction 535
24.2 The Helmert Transformation 537
24.3 Rotations Between Coordinate Systems 541
24.4 Combining GNSS Baseline Vectors with Traditional Observations 542
24.5 Another Approach to Transforming Coordinates Between Reference Frames 546
24.6 Other Considerations when Localizing a Survey 549
24.7 Using Total Station Pseudo-Observations 551
24.8 Building a Stochastic Model 556
24.9 Least-Squares Adjustment using Pseudo-Observations 560
24.10 Software 562
Problems 563
Programming Problems 566
25 Analyses of Adjustments 567
25.1 Introduction 567
25.2 Basic Concepts, Residuals, and the Normal Distribution 567
25.3 Goodness-of-Fit Test 571
25.4 Comparison of GNSS Residual Plots 574
25.5 Use of Statistical Blunder Detection 576
Problems 577
26 Terrestrial Laser Scanning and Uncertainty Estimation 579
26.1 Introduction 579
26.2 Laser Scanner Coordinates 580
26.3 Beam Divergence and Incidence Angle Effects 583
26.4 Range and Angular Uncertainty 586
26.5 Error Propagation of Laser Scanner Measurements 589
26.6 Registration and Its Uncertainty 593
26.7 Point Cloud Uncertainty 596
26.8 Survey Planning 599
26.9 Conclusions 607
26.10 Software 608
Problems 608
27 Plane, Three-Dimensional Objects, and Elevation Model Estimation with Point Clouds 615
27.1 Introduction 615
27.2 Plane Fitting with Point Clouds 616
27.3 Sphere and Cylinder Estimation with Point Clouds 628
27.4 RANSAC Estimation 645
27.5 Polynomial and Elevation Estimation with Point Clouds 645
27.6 Software 649
Problems 649
28 Computer Optimization 657
28.1 Introduction 657
28.2 Storage Optimization 658
28.3 Direct Formation of the Normal Equations 660
28.4 Cholesky Decomposition 662
28.5 Forward and Backward Solutions 662
28.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix 664
28.7 Spareness and Optimization of the Normal Matrix 664
Problems 670
Programming Problem 671
Appendix A Introduction to Matrices 673
A.1 Introduction 673
A.2 Definition of a Matrix 673
A.3 Size or Dimensions of a Matrix 674
A.4 Types of Matrices 675
A.5 Matrix Equality 676
A.6 Addition or Subtraction of Matrices 677
A.7 Scalar Multiplication of a Matrix 677
A.8 Matrix Multiplication 677
A.9 Computer Algorithms for Matrix Operations 681
A.10 Use of the Matrix Software 683
Problems 685
Programming Problems 687
Appendix B Solution of Equations by Matrix Methods 689
B.1 Introduction 689
B.2 Inverse Matrix 689
B.3 The Inverse of a 2 × 2 Matrix 690
B.4 Inverses by Adjoints 692
B.5 Inverses by Elementary Row Transformations 693
B.6 Example Problem 696
B.7 Eigenvalues and Eigenvectors 698
Problems 700
Programming Problems 702
Appendix C Nonlinear Equations and Taylor's Theorem 703
C.1 Introduction 703
C.2 Taylor Series Linearization of Nonlinear Equations 703
C.3 Numerical Example 705
C.4 Using Matrices to Solve Nonlinear Equations 706
C.5 Simple Matrix Example 707
C.6 Practical Example 708
C.7 Concluding Remarks 710
Problems 711
Programming Problems 712
Appendix D The Normal Error Distribution Curve and Other Statistical Tables 713
D.1 Development for Normal Distribution Curve Equation 713
D.2 Other Statistical Tables 721
Appendix E Confidence Intervals for the Mean 733
Appendix F Companion Website 739
F.1 Introduction 739
F.2 File Formats and Memory Matters 740
F.3 Software 740
F.4 Using the Software as an Instructional Aid 743
Appendix G Answers to Selected Problems 745
Bibliography 751
Index 757
