# Understanding Least Squares Estimation and Geomatics Data Analysis

# Understanding Least Squares Estimation and Geomatics Data Analysis

ISBN: 978-1-119-50144-2

Oct 2018

752 pages

$100.99

Product not available for purchase

## Description

**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.

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Preface

Acknowledgements

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 Hypotheses 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 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. Over-constrained 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 Approach

5.5.2. Derivation of Variation Function using Lagrangian Approach

5.6. Derivation of Normal Equation System

5.6.1. Normal Equations based on Direct Approach Variation Function

5.6.2. Normal Equations based on Lagrangian Approach Variation Function

5.7. Derivation of Parametric Least Squares Solution

5.7.1. Least Squares Solution from Direct Approach Normal Equations

5.7.2. Least Squares Solution from Lagrangian Approach Normal Equations

5.8. Stochastic Models of Parametric Adjustment

5.8.1. Derivation of Cofactor Matrix of Adjusted Parameters

5.8.2. Derivation of Cofactor Matrix of Adjusted 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 Levelling 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

9.7.2. 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 Concepts**

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. Baarda'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 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

13.7.2 Parabolic Vertical Transition Curve Example

13.8. Stochastic Properties of General Model Adjustment

13.8.1. Derivation of Cofactor Matrix of Adjusted Parameters

13.8.2. Derivation of Cofactor Matrices of Adjusted Observations and Residuals

13.8.3. Covariance Matrices of Adjusted Quantities

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 Modelling

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

Index