 # Understanding Least Squares Estimation and Geomatics Data Analysis

Standards Information Network (Verlag)
• 1. Auflage
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• erschienen am 10. Oktober 2018
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• 720 Seiten

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978-1-119-50144-2 (ISBN)

Provides a modern approach to least squares estimation and data analysis for undergraduate land surveying and geomatics programs Rich in theory and concepts, this comprehensive book on least square estimation and data analysis provides examples that are designed to help students extend their knowledge to solving more practical problems. The sample problems are accompanied by suggested solutions, and are challenging, yet easy enough to manually work through using simple computing devices, and chapter objectives provide an overview of the material contained in each section. Understanding Least Squares Estimation and Geomatics Data Analysis begins with an explanation of survey observables, observations, and their stochastic properties. It reviews matrix structure and construction and explains the needs for adjustment. Next, it discusses analysis and error propagation of survey observations, including the application of heuristic rule for covariance propagation. Then, the important elements of statistical distributions commonly used in geomatics are discussed. Main topics of the book include: concepts of datum definitions; the formulation and linearization of parametric, conditional and general model equations involving typical geomatics observables; geomatics problems; least squares adjustments of parametric, conditional and general models; confidence region estimation; problems of network design and pre-analysis; three-dimensional geodetic network adjustment; nuisance parameter elimination and the sequential least squares adjustment; post-adjustment data analysis and reliability; the problems of datum; mathematical filtering and prediction; an introduction to least squares collocation and the kriging methods; and more. * Contains ample concepts/theory and content, as well as practical and workable examples * Based on the author's manual, which he developed as a complete and comprehensive book for his Adjustment of Surveying Measurements and Special Topics in Adjustments courses * Provides geomatics undergraduates and geomatics professionals with required foundational knowledge * An excellent companion to Precision Surveying: The Principles and Geomatics Practice Understanding Least Squares Estimation and Geomatics Data Analysis is recommended for undergraduates studying geomatics, and will benefit many readers from a variety of geomatics backgrounds, including practicing surveyors/engineers who are interested in least squares estimation and data analysis, geomatics researchers, and software developers for geomatics.
 Auflage: 1. Auflage Sprache: Englisch Verlagsort: Newark | USA Verlagsgruppe: John Wiley & Sons Inc Zielgruppe: Für höhere Schule und Studium | Für Beruf und Forschung Dateigröße: 19,01 MB Schlagworte: ISBN-13: 978-1-119-50144-2 (9781119501442) weitere Ausgaben werden ermittelt
JOHN OLUSEGUN OGUNDARE, PHD, is a professional geomatics engineer and an instructor in the Department of Geomatics at British Columbia Institute of Technology (BCIT), Canada. He has been in the field of geomatics for over thirty years, as a surveyor in various geomatics engineering establishments in Africa and Canada and as a geomatics instructor or teaching assistant in universities and polytechnic institutions in Africa and Canada. Dr. Ogundare is also the author of Precision Surveying: The Principles and Geomatics Practice (Wiley, 2015).
• Intro
• Title Page
• Copyright Page
• Contents
• Preface
• Acknowledgments
• About the Author
• About the Companion Website
• Chapter 1 Introduction
• 1.1 Observables and Observations
• 1.2 Significant Digits of Observations
• 1.3 Concepts of Observation Model
• 1.4 Concepts of Stochastic Model
• 1.4.1 Random Error Properties of Observations
• 1.4.2 Standard Deviation of Observations
• 1.4.3 Mean of Weighted Observations
• 1.4.4 Precision of Observations
• 1.4.5 Accuracy of Observations
• 1.5 Needs for Adjustment
• 1.6 Introductory Matrices
• 1.6.1 Sums and Products of Matrices
• 1.6.2 Vector Representation
• 1.6.3 Basic Matrix Operations
• 1.7 Covariance, Cofactor, and Weight Matrices
• 1.7.1 Covariance and Cofactor Matrices
• 1.7.2 Weight Matrices
• Problems
• Chapter 2 Analysis and Error Propagation of Survey Observations
• 2.1 Introduction
• 2.2 Model Equations Formulations
• 2.3 Taylor Series Expansion of Model Equations
• 2.3.1 Using MATLAB to Determine Jacobian Matrix
• 2.4 Propagation of Systematic and Gross Errors
• 2.5 Variance-Covariance Propagation
• 2.6 Error Propagation Based on Equipment Specifications
• 2.6.1 Propagation for Distance Based on Accuracy Specification
• 2.6.2 Propagation for Direction (Angle) Based on Accuracy Specification
• 2.6.3 Propagation for Height Difference Based on Accuracy Specification
• 2.7 Heuristic Rule for Covariance Propagation
• Problems
• Chapter 3 Statistical Distributions and Hypothesis Tests
• 3.1 Introduction
• 3.2 Probability Functions
• 3.2.1 Normal Probability Distributions and Density Functions
• 3.3 Sampling Distribution
• 3.3.1 Student´s t-Distribution
• 3.3.2 Chi-square and Fisher´s F-distributions
• 3.4 Joint Probability Function
• 3.5 Concepts of Statistical Hypothesis Tests
• 3.6 Tests of Statistical Hypotheses
• 3.6.1 Test of Hypothesis on a Single Population Mean
• 3.6.2 Test of Hypothesis on Difference of Two Population Means
• 3.6.3 Test of Measurements Against the Means
• 3.6.4 Test of Hypothesis on a Population Variance
• 3.6.5 Test of Hypothesis on Two Population Variances
• Problems
• Chapter 4 Adjustment Methods and Concepts
• 4.1 Introduction
• 4.2 Traditional Adjustment Methods
• 4.2.1 Transit Rule Method of Adjustment
• 4.2.2 Compass (Bowditch) Rule Method
• 4.2.3 Crandall´s Rule Method
• 4.3 The Method of Least Squares
• 4.3.1 Least Squares Criterion
• 4.4 Least Squares Adjustment Model Types
• 4.5 Least Squares Adjustment Steps
• 4.6 Network Datum Definition and Adjustments
• 4.6.1 Datum Defect and Configuration Defect
• 4.7 Constraints in Adjustment
• 4.7.1 Minimal Constraint Adjustments
• 4.7.2 Overconstrained and Weight-Constrained Adjustments
• 4.7.3 Adjustment Constraints Examples
• 4.8 Comparison of Different Adjustment Methods
• 4.8.1 General Discussions
• Problems
• Chapter 5 Parametric Least Squares Adjustment: Model Formulation
• 5.1 Parametric Model Equation Formulation
• 5.1.1 Distance Observable
• 5.1.2 Azimuth and Horizontal (Total Station) Direction Observables
• 5.1.3 Horizontal Angle Observable
• 5.1.4 Zenith Angle Observable
• 5.1.5 Coordinate Difference Observable
• 5.1.6 Elevation Difference Observable
• 5.2 Typical Parametric Model Equations
• 5.3 Basic Adjustment Model Formulation
• 5.4 Linearization of Parametric Model Equations
• 5.4.1 Linearization of Parametric Model Without Nuisance Parameter
• 5.4.2 Linearization of Parametric Model with Nuisance Parameter
• 5.5 Derivation of Variation Function
• 5.5.1 Derivation of Variation Function Using Direct Approachvariation functiondirect approach
• 5.5.2 Derivation of Variation Function Using Lagrangian Approachvariation functiondirect approach
• 5.6 Derivation of Normal Equation System
• 5.6.1 Normal Equations Based on normal equation systemDirect Approachvariation functiondirect approach Variation Function
• 5.6.2 Normal Equations Based on normal equation systemLagrangian Approach Variation Functionvariation functiondirect approach
• 5.7 Derivation of Parametric Least Squares Solution
• 5.7.1 Least Squares Solutionleast squares solution from Direct Approachvariation functionLagrangian approachdirect approach...
• 5.7.2 Least Squares Solutionleast squares solution from Lagrangian Approach Normal Equationsvariation functionLagrangian ap...
• 5.8 Stochastic Models of Parametric Adjustment
• 5.8.1 Derivation of Cofactor Matrix of Adjusted Parametersadjusted parameters
• 5.8.2 Derivation of Cofactor Matrix of Adjusted Observationsadjusted observations
• 5.8.3 Derivation of Cofactor Matrix of Observation Residuals
• 5.8.4 Effects of Variance Factor Variation on Adjustments
• 5.9 Weight-constrained Adjustment Model Formulation
• 5.9.1 Stochastic Model for Weight-constrained Adjusted Parameters
• 5.9.2 Stochastic Model for Weight-constrained Adjusted Observations
• Problems
• Chapter 6 Parametric Least Squares Adjustment: Applications
• 6.1 Introduction
• 6.2 Basic Parametric Adjustment Examples
• 6.2.1 Leveling Adjustment
• 6.2.2 Station Adjustment
• 6.2.3 Traverse Adjustment
• 6.2.4 Triangulateration Adjustment
• 6.3 Stochastic Properties of Parametric Adjustment
• 6.4 Application of Stochastic Models
• 6.5 Resection Example
• 6.6 Curve-fitting Example
• 6.7 Weight Constraint Adjustment Steps
• 6.7.1 Weight Constraint Examples
• Problems
• Chapter 7 Confidence Region Estimation
• 7.1 Introduction
• 7.2 Mean Squared Error and Mathematical Expectation
• 7.2.1 Mean Squared Error
• 7.2.2 Mathematical Expectation
• 7.3 Population Parameter Estimation
• 7.3.1 Point Estimation of Population Mean
• 7.3.2 Interval Estimation of Population Mean
• 7.3.3 Relative Precision Estimation
• 7.3.4 Interval Estimation for Population Variance
• 7.3.5 Interval Estimation for Ratio of Two Population Variances
• 7.4 General Comments on Confidence Interval Estimation
• 7.5 Error Ellipse and Bivariate Normal Distribution
• 7.6 Error Ellipses for Bivariate Parameters
• 7.6.1 Absolute Error Ellipses
• 7.6.2 Relative Error Ellipses
• Problems
• Chapter 8 Introduction to Network Design and Preanalysis
• 8.1 Introduction
• 8.2 Preanalysis of Survey Observations
• 8.2.1 Survey Tolerance Limits
• 8.2.2 Models for Preanalysis of Survey Observations
• 8.2.3 Trigonometric Leveling Problems
• 8.3 Network Design Model
• 8.4 Simple One-dimensional Network Design
• 8.5 Simple Two-dimensional Network Design
• 8.6 Simulation of Three-dimensional Survey Scheme
• 8.6.1 Typical Three-dimensional Micro-network
• 8.6.2 Simulation Results
• Problems
• Chapter 9 Concepts of Three-dimensional Geodetic Network Adjustment
• 9.1 Introduction
• 9.2 Three-dimensional Coordinate Systems and Transformations
• 9.2.1 Local Astronomic Coordinate Systems and Transformations
• 9.3 Parametric Model Equations in Conventional Terrestrial System
• 9.4 Parametric Model Equations in Geodetic System
• 9.5 Parametric Model Equations in Local Astronomic System
• 9.6 General Comments on Three-dimensional Adjustment
• 9.7 Adjustment Examples
• 9.7.1 Adjustment in Cartesian Geodetic System
• 9.7.1.1 Solution Approach
• 9.7.2 Adjustment in Curvilinear Geodetic System
• 9.7.3 Adjustment in Local System
• Chapter 10 Nuisance Parameter Elimination and Sequential Adjustment
• 10.1 Nuisance Parameters
• 10.2 Needs to Eliminate Nuisance Parameters
• 10.3 Nuisance Parameter Elimination Model
• 10.3.1 Nuisance Parameter Elimination Summary
• 10.3.2 Nuisance Parameter Elimination Example
• 10.4 Sequential Least Squares Adjustment
• 10.4.1 Sequential Adjustment in Simple Form
• 10.5 Sequential Least Squares Adjustment Model
• 10.5.1 Summary of Sequential Least Squares Adjustment Steps
• 10.5.2 Sequential Least Squares Adjustment Example
• Problems
• Chapter 11 Post-adjustment Data Analysis and Reliability ConceptsSensitivity
• 11.1 Introduction
• 11.2 Post-adjustment Detection and Elimination of Non-stochastic Errors
• 11.3 Global Tests
• 11.3.1 Standard Global Test
• 11.3.2 Global Test by Baarda
• 11.4 Local Tests
• 11.5 Pope´s Approach to Local Test
• 11.6 Concepts of Redundancy Numbers
• 11.7 Baarda´s Data Analysis Approach
• 11.7.1 BaardaBaarda´s Approach to Local Test
• 11.8 Concepts of Reliability Measures
• 11.8.1 Internal Reliability Measures
• 11.8.2 External Reliability Measures
• 11.9 Network Sensitivity
• Problems
• Chapter 12 Least Squares Adjustment of Conditional Models
• 12.1 Introduction
• 12.2 Conditional Model Equations
• 12.2.1 Examples of Model Equations
• 12.3 Conditional Model Adjustment Formulation
• 12.3.1 Conditional Model Adjustment Steps
• 12.4 Stochastic Model of Conditional Adjustment
• 12.4.1 Derivation of Cofactor Matrix of Adjusted Observationsadjusted observations
• 12.4.2 Derivation of Cofactor Matrix of Observation Residuals
• 12.4.3 Covariance Matrices of Adjusted Observations and Residuals
• 12.5 Assessment of Observations and Conditional Model
• 12.6 Variance-Covariance Propagation for Derived Parameters from Conditional Adjustment
• 12.7 Simple GNSS Network Adjustment Example
• 12.8 Simple Traverse Network Adjustment Example
• Problems
• Chapter 13 Least Squares Adjustment of General Models
• 13.1 Introduction
• 13.2 General Model Equation Formulation
• 13.3 Linearization of General Model
• 13.4 Variation Function for Linearized General Model
• 13.5 Normal Equation System and the Least Squares Solution
• 13.6 Steps for General Model Adjustment
• 13.7 General Model Adjustment Examples
• 13.7.1 Coordinate Transformations
• 13.7.1.1 Two-dimensional Similarity Transformation Example
• 13.7.2 Parabolic Vertical Transition Curve Example
• 13.8 Stochastic Properties of General Model Adjustment
• 13.8.1 Derivation of Cofactor Matrix of Adjusted Parametersadjusted parameters
• 13.8.2 Derivation of Cofactor Matrices of Adjusted Observations and Residualsadjusted observations
• 13.8.3 Covariance Matrices of Adjusted Quantitiesadjusted observations
• 13.8.4 Summary of Stochastic Properties of General Model Adjustment
• 13.9 Horizontal Circular Curve Example
• 13.10 Adjustment of General Model with Weight Constraints
• 13.10.1 Variation Function for General Model with Weight Constraints
• 13.10.2 Normal Equation System and Solution
• 13.10.3 Stochastic Models of Adjusted Quantities
• Problems
• Chapter 14 Datum Problem and Free Network Adjustment
• 14.1 Introduction
• 14.2 Minimal Datum Constraint Types
• 14.3 Free Network Adjustment Model
• 14.4 Constraint Model for Free Network Adjustment
• 14.5 Summary of Free Network Adjustment Procedure
• 14.6 Datum Transformation
• 14.6.1 Iterative Weighted Similarity Transformation
• Problems
• Chapter 15 Introduction to Dynamic Mode Filtering and Prediction
• 15.1 Introduction
• 15.1.1 Prediction, Filtering, and Smoothing
• 15.2 Static Mode Filter
• 15.2.1 Real-time Moving Averages as Static Mode Filter
• 15.2.2 Sequential Least Adjustment as Static Mode Filter
• 15.3 Dynamic Mode Filter
• 15.3.1 Summary of Kalman Filtering Process
• 15.4 Kalman Filtering Examples
• 15.5 Kalman Filter and the Least Squares Method
• 15.5.1 Filtering and Sequential Least Squares Adjustment: Similarities and Differences
• Problems
• Chapter 16 Introduction to Least Squares Collocation and the Kriging Methods
• 16.1 Introduction
• 16.2 Elements of Least Squares Collocation
• 16.3 Collocation Procedure
• 16.4 Covariance Function
• 16.5 Collocation and Classical Least Squares Adjustment
• 16.6 Elements of Kriging
• 16.7 Semivariogram Model and Modeling
• 16.8 Kriging Procedure
• 16.8.1 Simple Kriging
• 16.8.2 Ordinary Kriging
• 16.8.3 Universal Kriging
• 16.9 Comparing Least Squares Collocation and Kriging
• Appendix A Extracts from BaardaBaarda´s Nomogram
• Appendix B Standard Statistical Distribution Tables
• Appendix C Tau Critical Values Table for Significance Level a
• Appendix D General Partial Differentials of Typical Survey Observables
• D.1 Azimuth Observable
• D.2 Total Station Direction Observable
• D.3 Horizontal Angle Observable
• D.4 Distance Observable
• D.5 Zenith Angle Observable
• D.6 Other General Rules for Partial Differentials
• Appendix E Some Important Matrix Operations and Identities
• E.1 Matrix Lemmas
• E.2 Generalized Inverses and Pseudo-inverses
• Appendix F Commonly Used Abbreviations
• References
• Index
• EULA

# Preface

Paradigm changes are taking place in geomatics with regard to how geomatics professionals function and use equipment and technology. The rise of "automatic surveying systems" and high precision Global Navigation Satellite System (GNSS) networks are changing the focus from how data are captured to how the resultant (usually redundant) data are processed, analyzed, adjusted, and integrated. The modern equipment and technology are continually capturing and storing redundant data of varying precisions and accuracies, and there is an ever-increasing need to process, analyze, adjust, and integrate these data, especially as part of land (or geographic) information systems. The methods of least squares estimation, which are the most rigorous adjustment procedures available today, are the most popular methods of analyzing, adjusting, and integrating geomatics data. Although the concepts and theories of the methods have been developed over several decades, it is not until recently that they are gaining much attention in geomatics professions. This is due, in part, to the recent advancement in computing technology and the various attempts being made in simplifying the theories and concepts involved. This book is to complement the efforts of the geomatics professionals in further simplifying the various aspects of least squares estimation and geomatics data analysis.

My motivation to write this book came from three perspectives: First, my over 15 years of experience in teaching students in the Diploma and Bachelor of Geomatics Engineering Technology (currently, Bachelor of Science in Geomatics) at the British Columbia Institute of Technology (BCIT). Second, my over 10 years as a special examiner and a subject-matter expert for Canadian Board of Examiners for Professional Surveyors (CBEPS) on coordinate systems and map projections, and advanced surveying. Third, as an expert for CBEPS on least squares estimation and data analysis. As a subject-matter expert, I have observed after reviewing syllabus topics, learning outcomes, study guides, and reference and supplementary materials of CBEPS Least Squares Estimation and Data Analysis that there is a definite need for a comprehensive textbook on this subject.

Currently available undergraduate-level books on least squares estimation and data analysis are either inadequate in concepts/theory and content or inadequate in practical and workable examples that are easy to understand. To the best of my knowledge, no specific book in this subject area has synergized concepts/theory and practical and workable examples. Because of this, students and geomatics practitioners are often distracted by having to go through numerous, sometimes irrelevant, materials to extract information to solve a specific least squares estimation problem. Because of this, they end up losing focus and fail to understand the subject and apply it efficiently in practice. My main goal in writing this book is to provide the geomatics community with a comprehensive least squares estimation and data analysis book that is rich in theory/concepts and examples that are easy to follow. This book is based on Data Analysis and Least Squares Estimation: The Geomatics Practice, which I developed and use for teaching students at BCIT for over 15 years. It provides the geomatics undergraduates and professionals with the foundational knowledge that is consistent with the baccalaureate level and also introduces students to some more advanced topics in data analysis.

Compared with other geomatics books in this field, this book is rich in theory/concepts and provides examples that are simple enough for the students to attempt and manually work through using simple computing devices. The examples are designed to help the students extend their knowledge to solving more practical problems. Moreover, this book assumes that the usually overdetermined geomatics measurements can be formulated generally as three main mathematical models (general, parametric, and conditional), and the number of examples can be limited to the adjustment of these three types of mathematical models.

The book consists of 16 chapters and 6 appendices. Chapter 1 explains survey observables, observations and their stochastic properties, reviews matrix structure and construction, and discusses the needs for geomatics adjustments.

Chapter 2 discusses analysis and error propagation of survey observations, including the application of the heuristic rule for covariance propagation. This chapter explores the concepts and laws of systematic error and random error propagations and applies the laws to some practical problems in geomatics. The use of interactive computing environment for numerical solution of scientific problems, such as Matrix Laboratory (MATLAB) software, is introduced for computing Jacobian matrices for error and systematic error propagations.

In Chapter 3, the important elements of statistical distributions commonly used in geomatics are discussed. The discussion includes an explanation on how statistical problems in geomatics are solved and how statistical decisions are made based on statistical hypothesis tests. The chapter introduces the relevant statistical terms such as statistics, concepts of probability, and statistical distributions.

Chapter 4 discusses the differences among the traditional adjustment methods (transit, compass and Crandall's) and the least squares method of adjustment, including their limitations, advantages, and properties. The concepts of datum definition and the different constraints in least squares adjustment are also introduced in this chapter.

Chapter 5 presents the formulation and linearization of parametric model equations involving typical geomatics observables, the derivation of basic parametric least squares adjustment models, variation functions, and normal equations and solution equations. This chapter also discusses the application of variance-covariance propagation laws in determining the stochastic models of adjusted quantities, such as adjusted parameters, adjusted observations, and observation residuals. The discussion ends with an explanation of how to formulate weight constraint parametric least squares adjustment models, including the solution equations and the associated stochastic models.

In Chapter 6, the concepts of parametric least squares adjustment are applied to various geomatics problems, which include differential levelling, station adjustment, traverse, triangulation, trilateration, resection, and curve fitting. The general formulation of parametric model equations for various geomatics problems, including the determination of stochastic properties of adjusted quantities and the adjustment of weight constraint problems, is also discussed in this chapter.

Chapter 7 discusses the confidence region estimation, which includes the construction of confidence intervals for population means, variances, and ratio of variances, and the construction of standard and confidence error ellipses for absolute and relative cases. Before these, some of the basic statistical terms relating to parameter estimation in geomatics, such as mean squared error, biased and unbiased estimators, mathematical expectation, and point and interval estimators, are defined.

Chapter 8 discusses the problems of network design and pre-analysis. In this chapter, different design variables and how they relate to each other, including their uses and importance, are discussed. The chapter also presents the procedures (with numerical examples) for performing simple pre-analysis of survey observations and for performing network design (or simulation) in one-, two- and three-dimensional cases.

Chapter 9 introduces the concepts of three-dimensional geodetic network adjustment, including the formulation and solution of parametric model equations in conventional terrestrial (CT), geodetic (G), and local astronomic (LA) systems; numerical examples are then provided to illustrate the concepts.

Chapter 10 presents, with examples, the concepts of and the needs for nuisance parameter elimination and the sequential least squares adjustment.

Chapter 11 discusses the steps involved in post-adjustment data analysis and the concepts of reliability. It also includes the procedures for conducting global and local tests in outlier detection and identification and an explanation of the concepts of redundancy numbers, reliability (internal and external), and sensitivity, and their applications to geomatics.

Chapters 12 and 13 discuss the least squares adjustments of conditional models and general models. Included in each of these chapters are the derivation of steps involved in the adjustment, the formulation of model equations for different cases of survey system, the variance-covariance propagation for the adjusted quantities and their functions, and some numerical examples. Also included in Chapter 13 are the steps involved in the adjustment of general models with weight constraints on the parameters.

Chapter 14 discusses the problems of datum and their solution approaches and an approach for performing free network adjustment. It further describes the steps for formulating free network adjustment constraint equations and explains the differences between inner constraint and external constraint network adjustments and how to transform adjusted quantities from one minimal constraint datum to another.

Chapter 15 introduces the dynamic mode filtering and prediction methods, including the steps involved and how simple...

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