Understanding Least Squares Estimation and Geomatics Data Analysis

719 Pages·2018·7.31 MB·English

by  John Olusegun Ogundare

Understanding Least
SquaresEstimationand
Geomatics Data Analysis

Understanding Least Squares Estimation
and Geomatics Data Analysis
John Olusegun Ogundare, PhD, PEng
Instructor of Geomatics Engineering
Department of Geomatics Engineering Technology
School of Construction and the Environment
British Columbia Institute of Technology (BCIT) – Burnaby, Canada
Thiseditionfirstpublished2019
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LibraryofCongressCataloging-in-PublicationData
Names:Ogundare,JohnOlusegun,author.
Title:Understandingleastsquaresestimationandgeomaticsdataanalysis/
JohnOlusegunOgundare.
Description:1stedition.|Hoboken,NJ:JohnWiley&Sons,2018.|Includes
bibliographicalreferencesandindex.|
Identifiers:LCCN2018033050(print)|LCCN2018042480(ebook)|
ISBN9781119501404(AdobePDF)|ISBN9781119501442(ePub)|
ISBN9781119501398(hardcover)
Subjects:LCSH:Estimationtheory.|Leastsquares.
Classification:LCCQA276.8(ebook)|LCCQA276.8.O342018(print)|
DDC519.5/44–dc23
LCrecordavailableathttps://lccn.loc.gov/2018033050
CoverDesign:Wiley
CoverIllustration:©naddi/iStockphoto
Setin10/12ptWarnockbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
v
Contents
Preface xiii
Acknowledgments xvii
AbouttheAuthor xix
AbouttheCompanionWebsite xxi
1 Introduction 1
1.1 Observables and Observations 2
1.2 Significant Digits of Observations 2
1.3 Concepts of Observation Model 4
1.4 Concepts of Stochastic Model 6
1.4.1 Random ErrorProperties of Observations 6
1.4.2 Standard Deviation of Observations 8
1.4.3 Mean of Weighted Observations 9
1.4.4 Precision of Observations 10
1.4.5 Accuracy of Observations 11
1.5 Needs for Adjustment 12
1.6 Introductory Matrices 16
1.6.1 Sums and Products of Matrices 18
1.6.2 Vector Representation 20
1.6.3 Basic Matrix Operations 21
1.7 Covariance, Cofactor, and Weight Matrices 22
1.7.1 Covariance and Cofactor Matrices 26
1.7.2 Weight Matrices 27
Problems 34
2 AnalysisandErrorPropagationofSurveyObservations 39
2.1 Introduction 39
2.2 Model Equations Formulations 40
2.3 Taylor Series Expansion of Model Equations 44
2.3.1 Using MATLAB to Determine Jacobian Matrix 52
vi Contents
2.4 Propagation of Systematic and Gross Errors 55
2.5 Variance–Covariance Propagation 58
2.6 Error Propagation Based on Equipment Specifications 67
2.6.1 Propagation forDistance Based on Accuracy Specification 67
2.6.2 Propagation forDirection (Angle) Based on Accuracy
Specification 69
2.6.3 Propagation forHeightDifference Based on Accuracy
Specification 69
2.7 Heuristic Rule for Covariance Propagation 72
Problems 76
3 StatisticalDistributionsandHypothesisTests 81
3.1 Introduction 82
3.2 Probability Functions 83
3.2.1 Normal Probability Distributions and Density Functions 84
3.3 SamplingDistribution 92
3.3.1 Student’s t-Distribution 93
3.3.2 Chi-squareand Fisher’s F-distributions 95
3.4 Joint Probability Function 97
3.5 Concepts of Statistical Hypothesis Tests 98
3.6 Tests of Statistical Hypotheses 100
3.6.1 Test of Hypothesis on a Single Population Mean 102
3.6.2 Test of Hypothesis on Difference of Two Population Means 106
3.6.3 Test of Measurements Against the Means 109
3.6.4 Test of Hypothesis on a Population Variance 111
3.6.5 Test of Hypothesis on Two Population Variances 114
Problems 117
4 AdjustmentMethodsandConcepts 119
4.1 Introduction 120
4.2 Traditional Adjustment Methods 120
4.2.1 Transit Rule Method of Adjustment 122
4.2.2 Compass (Bowditch) Rule Method 125
4.2.3 Crandall’s Rule Method 126
4.3 The Method of Least Squares 127
4.3.1 Least Squares Criterion 129
4.4 Least Squares Adjustment Model Types 132
4.5 Least Squares Adjustment Steps 134
4.6 Network Datum Definition and Adjustments 136
4.6.1 Datum Defect and Configuration Defect 138
4.7 Constraints inAdjustment 139
4.7.1 MinimalConstraint Adjustments 140
Contents vii
4.7.2 Overconstrained and Weight-Constrained Adjustments 141
4.7.3 Adjustment Constraints Examples 143
4.8 Comparison of Different Adjustment Methods 146
4.8.1 General Discussions 158
Problems 160
5 ParametricLeastSquaresAdjustment:ModelFormulation 163
5.1 Parametric Model Equation Formulation 164
5.1.1 Distance Observable 165
5.1.2 Azimuth and Horizontal (Total Station) Direction Observables 165
5.1.3 Horizontal Angle Observable 168
5.1.4 Zenith Angle Observable 169
5.1.5 Coordinate Difference Observable 169
5.1.6 Elevation Difference Observable 169
5.2 Typical ParametricModel Equations 170
5.3 Basic Adjustment Model Formulation 179
5.4 Linearization of Parametric Model Equations 180
5.4.1 Linearization of Parametric Model Without Nuisance
Parameter 180
5.4.2 Linearization of Parametric Model withNuisance Parameter 184
5.5 Derivation of Variation Function 186
5.5.1 Derivation of Variation Function Using Direct Approach 186
5.5.2 Derivation of Variation Function Using Lagrangian Approach 187
5.6 Derivation of Normal Equation System 188
5.6.1 Normal Equations Based on Direct Approach
Variation Function 188
5.6.2 Normal Equations Based on Lagrangian Approach Variation
Function 189
5.7 Derivation of Parametric Least Squares Solution 189
5.7.1 Least Squares Solution from Direct Approach
Normal Equations 189
5.7.2 Least Squares Solution from Lagrangian Approach Normal
Equations 190
5.8 Stochastic Models of Parametric Adjustment 191
5.8.1 Derivation of Cofactor Matrix of Adjusted Parameters 192
5.8.2 Derivation of Cofactor Matrix of Adjusted Observations 193
5.8.3 Derivation of Cofactor Matrix of Observation Residuals 194
5.8.4 Effects of Variance Factor Variation on Adjustments 196
5.9 Weight-constrained Adjustment Model Formulation 197
5.9.1 Stochastic Model for Weight-constrained Adjusted Parameters 200
5.9.2 StochasticModelforWeight-constrainedAdjustedObservations 201
Problems 202
viii Contents
6 ParametricLeastSquaresAdjustment:Applications 205
6.1 Introduction 206
6.2 Basic Parametric Adjustment Examples 207
6.2.1 Leveling Adjustment 207
6.2.2 Station Adjustment 215
6.2.3 Traverse Adjustment 223
6.2.4 Triangulateration Adjustment 235
6.3 Stochastic Properties of Parametric Adjustment 242
6.4 Application of Stochastic Models 243
6.5 Resection Example 249
6.6 Curve-fitting Example 254
6.7 Weight Constraint Adjustment Steps 260
6.7.1 Weight Constraint Examples 261
Problems 272
7 ConfidenceRegionEstimation 275
7.1 Introduction 276
7.2 Mean Squared Error and Mathematical Expectation 276
7.2.1 Mean Squared Error 276
7.2.2 Mathematical Expectation 277
7.3 Population Parameter Estimation 280
7.3.1 Point Estimation of Population Mean 280
7.3.2 Interval Estimation of Population Mean 281
7.3.3 Relative Precision Estimation 285
7.3.4 Interval Estimation forPopulation Variance 288
7.3.5 Interval Estimation forRatio of Two Population Variances 290
7.4 General Comments on Confidence Interval Estimation 293
7.5 Error Ellipse and Bivariate Normal Distribution 294
7.6 Error Ellipses for Bivariate Parameters 298
7.6.1 Absolute Error Ellipses 299
7.6.2 Relative Error Ellipses 305
Problems 309
8 IntroductiontoNetworkDesignandPreanalysis 311
8.1 Introduction 311
8.2 Preanalysisof Survey Observations 313
8.2.1 Survey Tolerance Limits 314
8.2.2 Models forPreanalysisof Survey Observations 314
8.2.3 Trigonometric Leveling Problems 316
8.3 Network Design Model 322
8.4 Simple One-dimensional Network Design 322
8.5 Simple Two-dimensional Network Design 325
Contents ix
8.6 Simulation of Three-dimensional Survey Scheme 340
8.6.1 Typical Three-dimensional Micro-network 340
8.6.2 Simulation Results 342
Problems 347
9 ConceptsofThree-dimensionalGeodeticNetworkAdjustment 349
9.1 Introduction 350
9.2 Three-dimensional Coordinate Systems and Transformations 350
9.2.1 Local Astronomic Coordinate Systems and Transformations 352
9.3 ParametricModelEquationsinConventionalTerrestrialSystem 354
9.4 Parametric Model Equations inGeodetic System 357
9.5 Parametric Model Equations inLocal Astronomic System 361
9.6 General Comments on Three-dimensional Adjustment 365
9.7 Adjustment Examples 367
9.7.1 Adjustment in Cartesian Geodetic System 367
9.7.1.1 Solution Approach 369
9.7.2 Adjustment in Curvilinear Geodetic System 371
9.7.3 Adjustment in Local System 373
10 NuisanceParameterEliminationandSequentialAdjustment 377
10.1 Nuisance Parameters 377
10.2 Needs to Eliminate Nuisance Parameters 378
10.3 Nuisance Parameter Elimination Model 379
10.3.1 Nuisance Parameter Elimination Summary 382
10.3.2 Nuisance Parameter Elimination Example 383
10.4 Sequential Least Squares Adjustment 391
10.4.1 Sequential Adjustmentin Simple Form 393
10.5 Sequential Least Squares Adjustment Model 395
10.5.1 Summary of Sequential Least Squares Adjustment Steps 400
10.5.2 Sequential Least Squares Adjustment Example 404
Problems 415
11 Post-adjustmentDataAnalysisandReliabilityConcepts 419
11.1 Introduction 420
11.2 Post-adjustment Detection and Elimination of Non-stochastic
Errors 421
11.3 Global Tests 424
11.3.1 Standard Global Test 425
11.3.2 Global Test by Baarda 426
11.4 Local Tests 427
11.5 Pope’s Approach toLocal Test 428
11.6 Concepts of Redundancy Numbers 430