Adjustment Computations: Spatial Data Analysis

Adjustment Computations......Page 3
Contents......Page 7
Preface......Page 17
Acknowledgments......Page 21
1.1. Introduction......Page 25
1.3. Measurement Error Sources......Page 26
1.4. Definitions......Page 27
1.5. Precision versus Accuracy......Page 28
1.6. Redundant Observations in Surveying and Their Adjustment......Page 30
1.7. Advantages of Least Squares Adjustment......Page 32
1.8. Overview of the Book......Page 33
Problems......Page 34
2.2. Sample versus Population......Page 36
2.3. Range and Median......Page 37
2.4. Graphical Representation of Data......Page 38
2.6. Measures of Central Tendency......Page 41
2.7. Additional Definitions......Page 42
2.8. Alternative Formula for Determining Variance......Page 45
2.9. Numerical Examples......Page 46
2.10. Derivation of the Sample Variance (Bessel’s Correction)......Page 50
2.11. Software......Page 52
Problems......Page 53
Practical Exercises......Page 56
3.2. Theory of Probability......Page 57
3.3. Properties of the Normal Distribution Curve......Page 60
3.4. Standard Normal Distribution Function......Page 62
3.5. Probability of the Standard Error......Page 65
3.6. Uses for Percent Errors......Page 67
3.7. Practical Examples......Page 68
Problems......Page 70
Programming Problems......Page 72
4.1. Introduction......Page 73
4.2. Distributions Used in Sampling Theory......Page 75
4.3. Confidence Interval for the Mean: t statistic......Page 79
4.4. Testing the Validity of the Confidence Interval......Page 82
4.5. Selecting a Sample Size......Page 83
4.6. Confidence Interval for a Population Variance......Page 84
4.7. Confidence Interval for the Ratio of Two Population Variances......Page 85
4.8. Software......Page 88
Problems......Page 90
5.1. Hypothesis Testing......Page 94
5.2. Systematic Development of a Test......Page 97
5.3. Test of Hypothesis for the Population Mean......Page 98
5.4. Test of Hypothesis for the Population Variance......Page 100
5.5. Test of Hypothesis for the Ratio of Two Population Variances......Page 103
5.6. Software......Page 106
Problems......Page 107
6.1. Basic Error Propagation Equation......Page 110
6.2. Frequently Encountered Specific Functions......Page 115
6.3. Numerical Examples......Page 116
6.4. Software......Page 120
Problems......Page 122
Practical Exercises......Page 126
7.2. Error Sources in Horizontal Angles......Page 127
7.3. Reading Errors......Page 128
7.4. Pointing Errors......Page 130
7.5. Estimated Pointing and Reading Errors with Total Stations......Page 131
7.6. Target-Centering Errors......Page 132
7.7. Instrument-Centering Errors......Page 134
7.8. Effects of Leveling Errors in Angle Observations......Page 137
7.9. Numerical Example of Combined Error Propagation in a Single Horizontal Angle......Page 140
7.10. Using Estimated Errors to Check Angular Misclosure in a Traverse......Page 141
7.11. Errors in Astronomical Observations for Azimuth......Page 143
7.12. Errors in Electronic Distance Observations......Page 148
7.13. Software......Page 149
Problems......Page 150
Programming Problems......Page 154
8.1. Introduction......Page 155
8.2. Derivation of Estimated Error in Latitude and Departure......Page 156
8.4. Computing and Analyzing Polygon Traverse Misclosure Errors......Page 158
8.5. Computing and Analyzing Link Traverse Misclosure Errors......Page 164
8.6. Software......Page 168
Problems......Page 169
Programming Problems......Page 174
9.2. Systematic Errors in Differential Leveling......Page 175
9.3. Random Errors in Differential Leveling......Page 178
9.4. Error Propagation in Trigonometric Leveling......Page 183
Problems......Page 186
Programming Problems......Page 188
10.1. Introduction......Page 189
10.2. Weighted Mean......Page 191
10.4. Statistics of Weighted Observations......Page 193
10.6. Weights in Differential Leveling......Page 195
10.7. Practical Examples......Page 197
Problems......Page 199
11.1. Introduction......Page 202
11.2. Fundamental Principle of Least Squares......Page 203
11.3. Fundamental Principle of Weighted Least Squares......Page 205
11.4. Stochastic Model......Page 206
11.5. Functional Model......Page 207
11.6. Observation Equations......Page 208
11.7. Systematic Formulation of the Normal Equations......Page 210
11.8. Tabular Formation of the Normal Equations......Page 212
11.9. Using Matrices to Form Normal Equations......Page 213
11.10. Least Squares Solution of Nonlinear Systems......Page 216
11.11. Least Squares Fit of Points to a Line or Curve......Page 219
11.12. Calibration of an EDM Instrument......Page 223
11.13. Least Squares Adjustment Using Conditional Equations......Page 224
11.14. The Previous Example Using Observation Equations......Page 226
11.15. Software......Page 227
Problems......Page 228
12.2. Observation Equation......Page 234
12.3. Unweighted Example......Page 235
12.4. Weighted Example......Page 238
12.5. Reference Standard Deviation......Page 240
12.6. Another Weighted Adjustment......Page 242
12.7. Software......Page 245
Problems......Page 247
Programming Problems......Page 251
13.2. Development of the Covariance Matrix......Page 252
13.3. Numerical Examples......Page 256
13.4. Standard Deviations of Computed Quantities......Page 257
Problems......Page 260
Programming Problems......Page 263
14.1. Introduction......Page 264
14.2. Distance Observation Equation......Page 266
14.3. Trilateration Adjustment Example......Page 268
14.4. Formulation of a Generalized Coefficient Matrix for a More Complex Network......Page 274
14.5. Computer Solution of a Trilaterated Quadrilateral......Page 275
14.6. Iteration Termination......Page 279
14.7. Software......Page 280
Problems......Page 282
Programming Problems......Page 288
15.2. Azimuth Observation Equation......Page 290
15.3. Angle Observation Equation......Page 293
15.4. Adjustment of Intersections......Page 295
15.5. Adjustment of Resections......Page 300
15.6. Adjustment of Triangulated Quadrilaterals......Page 306
Problems......Page 311
Programming Problems......Page 320
16.2. Observation Equations......Page 322
16.3. Redundant Equations......Page 323
16.4. Numerical Example......Page 324
16.5. Minimum Amount of Control......Page 330
16.6. Adjustment of Networks......Page 331
16.7. χ2 Test: Goodness of Fit......Page 339
Problems......Page 340
Programming Problems......Page 350
17.1. Introduction......Page 351
17.2. GNSS Observations......Page 352
17.3. GNSS Errors and the Need for Adjustment......Page 354
17.4. Reference Coordinate Systems for GNSS Observations......Page 355
17.5. Converting between the Terrestrial and Geodetic Coordinate Systems......Page 358
17.6. Application of Least Squares in Processing GNSS Data......Page 361
17.7. Network Preadjustment Data Analysis......Page 364
17.8. Least Squares Adjustment of GNSS Networks......Page 370
Problems......Page 376
Programming Problems......Page 390
18.2. Two-Dimensional Conformal Coordinate......Page 392
18.3. Equation Development......Page 393
18.4. Application of Least Squares......Page 395
18.5. Two-Dimensional Affine Coordinate Transformation......Page 398
18.6. Two-Dimensional Projective Coordinate Transformation......Page 401
18.7. Three-Dimensional Conformal Coordinate Transformation......Page 404
18.8. Statistically Valid Parameters......Page 410
Problems......Page 414
Programming Problems......Page 420
19.1. Introduction......Page 421
19.2. Computation of Ellipse Orientation and Semiaxes......Page 423
19.3. Example Problem of Standard Error Ellipse Calculations......Page 428
19.4. Another Example Problem......Page 430
19.5. Error Ellipse Confidence Level......Page 431
19.6. Error Ellipse Advantages......Page 433
19.7. Other Measures of Station Uncertainty......Page 436
Problems......Page 437
Programming Problems......Page 439
20.2. Adjustment of Control Station Coordinates......Page 440
20.3. Holding Control Fixed in a Trilateration Adjustment......Page 445
20.4. Helmert’s Method......Page 448
20.6. Enforcing Constraints through Weighting......Page 453
Problems......Page 455
Practical Exercises......Page 458
21.1. Introduction......Page 459
21.2. A Priori Methods for Detecting Blunders in Observations......Page 460
21.3. A Posteriori Blunder Detection......Page 462
21.4. Development of the Covariance Matrix for the Residuals......Page 463
21.5. Detection of Outliers in Observations: Data Snooping......Page 466
21.7. Techniques Used In Adjusting Control......Page 468
21.8. Data Set with Blunders......Page 470
21.9. Further Considerations......Page 477
21.10. Survey Design......Page 479
21.11. Software......Page 481
Problems......Page 482
Practical Exercises......Page 486
22.2. General Least Squares Equations for Fitting a Straight Line......Page 488
22.3. General Least Squares Solution......Page 490
22.4. Two-Dimensional Coordinate Transformation by General Least Squares......Page 494
22.5. Three-Dimensional Conformal Coordinate Transformation by General Least Squares......Page 500
Problems......Page 502
Programming Problems......Page 506
23.1. Introduction......Page 507
23.2. Linearization of Equations......Page 509
23.4. Example Adjustment......Page 514
23.6. Comments on Systematic Errors......Page 523
23.7. Software......Page 526
Problems......Page 527
Programming Problems......Page 531
24.1. Introduction......Page 532
24.2. Helmert’s Transformation......Page 534
24.3. Rotations between Coordinate Systems......Page 537
24.4. Combining GPS Baseline Vectors with Traditional Observations......Page 538
24.5. Another Approach to Transforming Coordinates between Reference Frames......Page 542
24.6. Other Considerations......Page 545
Problems......Page 546
Programming Problems......Page 548
25.2. Basic Concepts, Residuals, and the Normal Distribution......Page 549
25.3. Goodness-of-Fit Test......Page 552
25.4. Comparison of Residual Plots......Page 555
25.5. Use of Statistical Blunder Detection......Page 557
Problems......Page 558
26.2. Storage Optimization......Page 560
26.3. Direct Formation of the Normal Equations......Page 563
26.4. Cholesky Decomposition......Page 564
26.5. Forward and Back Solutions......Page 566
26.6. Using the Cholesky Factor to Find the Inverse of the Normal Matrix......Page 567
26.7. Spareness and Optimization of the Normal Matrix......Page 569
Programming Problems......Page 573
A.2. Definition of a Matrix......Page 574
A.3. Size or Dimensions of a Matrix......Page 575
A.4. Types of Matrices......Page 576
A.5. Matrix Equality......Page 577
A.8. Matrix Multiplication......Page 578
A.9. Computer Algorithms for Matrix Operations......Page 581
A.10. Use of the MATRIX Software......Page 584
Problems......Page 586
Programming Problems......Page 588
B-2. Inverse Matrix......Page 589
B-3. Inverse of a 2 × 2 Matrix......Page 590
B-4. Inverses by Adjoints......Page 592
B-5. Inverses by Elementary Row Transformations......Page 593
B-6. Example Problem......Page 597
Problems......Page 598
Programming Problems......Page 599
C.2. Taylor Series Linearization of Nonlinear Equations......Page 600
C.3. Numerical Example......Page 601
C.4. Using Matrices to Solve Nonlinear Equations......Page 603
C.5. Simple Matrix Example......Page 604
C.6. Practical Example......Page 605
C.7. Concluding Remarks......Page 607
Problems......Page 608
Programming Problems......Page 609
D.1. Development of the Normal Distribution Curve Equation......Page 610
D.2. Other Statistical Tables......Page 618
Appendix E Confidence Intervals for the Mean......Page 630
F.1. Introduction......Page 636
F.2. Mathematics of the Lambert Conformal Conic Map Projection......Page 637
F.3. Mathematics from the Transverse Mercator......Page 640
F.4. Stereographic Map Projection......Page 643
F.5. Reduction of Observations......Page 645
G.1. Introduction......Page 649
G.3. Software......Page 650
G.4. Using the Software as an Instructional Aid......Page 654
Appendix H Solutions to Selected Problems......Page 655
Bibliography......Page 660
Index......Page 663