Optimization and Design of Geodetic Networks

A Review of Network Designs: Criteria, Risk Functions, Design Ordering.- 1. Classification.- 2. Objective Functions.- 3. Solution Methods.- References.- B. Zero Order Design: Generalized Inverses, Adjustment, the Datum Problem and S-Transformations.- 0.1 Introduction.- 0.2 Notations and Preliminaries.- 1. Generalized Inverses, a Geometric Approach.- 1.1 Characterization of a Set of Linear Equations.- 1.2 A Unique Characterization of an Arbitrary Generalized Inverse.- 1.3 Right - and Left Inverses.- 1.4 An Arbitrary System of Linear Equations and Arbitrary Generalized Inverses.- 1.5 Transformation Properties and Some Special Types of Generalized Inverses.- 1.6 Summary.- 2. On S-Transformations.- 2.1 Introduction.- 2.2 Coordinates and Datum Definitions.- 2.3 S-Transformations.- 2.4 The Relation with Generalized Inverses.- References.- C. First Order Design: Optimization of the Configuration of a Network by Introducing Small Position Changes.- 1. Introduction.- 2. Gaußt-Markof Models Not of Full Rank.- 3. Projected Parameters.- 4. Datum Transformations.- 5. Choice of the Datum for a Free Network.- 6. Choice of a Criterion Matrix for a Free Network.- 7. First Order Design Problem by Introducing Small Position Changes.- 8. Criterion Matrix for the Optimization.- 9. Optimization Problem.- 10. Quadratic Programming Problem.- 11. Linear Complementary Problem.- 12. Solution of the Linear Complementary Problem.- References.- D. Second Order Design.- 0. An Example.- 1. Three SOD-Approaches.- 1.1 Direct Approximation of the Criterion Matrix, Approach i).- 1.2 Iterative Approximation of the Criterion Matrix, Approach ii).- 1.3 Direct Approximation of the Inverse Criterion Matrix, Approach iii).- 1.4 Diagonal Design.- 1.5 Approximation Quality.- 1.6 Modification of Approach iii).- 2. Solution Methods.- 2.1 Least-Squares Solution.- 2.2 Linear Programming.- 2.3 Nonlinear Programming.- 3. Mean Least-Squares Approximation, Comparison of the Three Approaches.- 4. Directions in the SOD-Problem.- 4.1 Elimination and Group Weights.- 4.2 Elimination and Individual Weights.- 4.3 Correlated Angles.- 4.4 Extracted Khatri-Rao-Product and Individual Weights.- 4.5 Comparison.- 4.6 Three-Step-Strategy.- 5. Defect Analysis of the Final Equation.- 5.1 Defect Analysis for Distances.- 5.2 Free Distance Networks.- 5.3 Defect Analysis for Directions.- 5.4 Free Direction Networks.- 6. Direct Creation of the Final Equation.- 6.1 Individual Weights.- 6.1.1 Distances.- 6.1.2 Directions.- 6.1.3 Mixed Network.- 6.2 Group Weights.- 6.3 Common Weights for Sets of Directions.- 7. Examples.- 7.1 Example 1.- 7.2 Example.- References.- E. Third Order Design.- 1. THOD as Instrument in FOD and SOD.- 2. Mathematical Model for Network Densification.- 3. THOD with Criterion Matrices.- References.- F. Numerical Methods in Network Design.- 1. Introduction.- 2. Optimal Design Problems.- 2.1 Precision Criteria.- 3. Network Design Strategies.- 3.1 Computer Simulation.- 3.2 Analytical Methods.- 3.2.1 Generalized Matrix Algebra.- 3.2.2 Linear Programming.- 3.2.3 Non-Linear Programming.- 4. Conclusions.- Appendix A: Linear Programming.- Appendix B: Generalized Matrix Algebra.- Appendix C: Least Squares Techniques.- References.- G. Some Additional Information on the Capacity of the Linear Complementarity Algorithm.- 1. Introduction.- 2. Inequality Constrained Least-Squares Approximation.- 3. The Linear Complementarity Algorithm.- 4. Examples.- References.- H. Quick Computation of Geodetic Networks Using Special Properties of the Eigenvalues.- I. Introduction.- 2. Iterative Procedures.- 3. Properties of the Conjugate Gradient Method.- 4. Acceleration of the Convergence by an Approximation with Finite Elements.- 5. Survey of Formulae.- 5.1 Conjugate Gradient Method.- 5.2 Approximation with Finite Elements.- References.- I. Estimability Analyses of the Free Networks of Differential Range Observations to GPS Satellites.- 1. Introduction.- 2. Types of Rank Deficiencies.- 3. Rank Deficiencies of Free Networks Based on Differential Range GPS Observations.- 3.1 Determination of Station and Satellite Coordinates.- 3.2 Determinations of Station, Satellite and Non-Geometric Parameters.- 4. Estimability Analysis.- 4.1 Patterns of Observations for Moving Stations.- 4.2 General Criteria of Estimability for Subnetwork Design.- 5. Numerical Adjustment.- 6. A-Priori Information in GPS Satellite Networks.- 7. Effect of A-Priori Constraints on the Adjustment Results.- 8. Summary and Conclusions.- References.- J. Optimization Problems in Geodetic Networks with Signals.- 1. Introduction.- 2. Data Analysis and Signals.- 3. Geodetic Networks with Signals.- 4. Different Approaches for the Adjustment of Observations Depending on Signals.- 4.1 The Deterministic Approach.- 4.2 The Model Function Approach.- 4.3 The Stochastic Approach.- 4.4 Hybrid Approaches.- 5. Zero Order Design with Signals.- 5.1 General Remarks.- 5.2 Three-dimensional Networks.- 6. Deformable Networks.- 7. Estimability Problems.- 8. Other Optimization Problems.- 8.1 General Remarks.- 8.2 First Order Design.- 8.3 Second Order Design.- 8.4 Third Order Design.- Appendix: Observation Equations of Three-dimensional Networks.- References.- K. Fourier Analysis of Geodetic Networks.- 0. Introduction.- I. Spectral Methods in Geodesy.- 1.1 Fourier Techniques in Interpolation Methods.- 1.1.1 Step Function "Interpolation".- 1.1.2 Piecewiese Linear Interpolation.- 1.1.3 Quadratic Spline Interpolation.- 1.1.4 Cubic Spline Interpolation.- 1.1.5 Higher and Highest Order Spline Interpolation.- 1.2 Fourier Techniques in Physical Geodesy.- 2. Distributions and Fourier Transforms.- 3. Leveling Lines, Leveling Networks.- 4. Traverse, Trilateration Networks.- References.- L. Continuous Networks I.- 0. Introduction.- 1. Continuous Networks of First Derivative Type.- 1.1 Networks on a Line.- 1.1.1 The Fixed Network.- 1.1.2 The Free Network.- 1.2 Networks on a Circle.- 1.3 Variance - Covariance Function of Estimable Quantities.- 1.4 Higher Dimensional Networks.- 2. Continuous Networks of Second Derivative Type.- 3. Discrete versus Continuous Geodetic Networks.- References.- M. Continuous Networks II.- 0. Introduction.- 1. Elementary Examples: A Single Line Leveling.- 2. On the Conditions for a Continuous Approximation of Network with some exceptions.- 3. A Planar Circular Leveling Network.- Appendix: A Numerical Comparison Between a Discrete Network and its Continuous Analogue.- References.- N. Criterion Matrices for Deforming Networks.- 0. Introduction.- 1. Deformation Measures and Their Finite Element Approximation.- 2. The Datum Problems in Estimating Deformation Measures.- 3. Criterion Matrices for Deformation Measures.- 4. Datum Transformation of a Criterion Matrix and the Comparison of Real Versus Ideal Dispersion Matrices by Factor Analysis.- 4.1 Datum Transformation of a Criterion Matrix.- 4.2 Canonical Comparison of an Ideal Versus a Real Variance- Covariance Matrix.- 4.2.1 The Eigenvalue Problem for the Matrix A.- 4.2.2 The Eigenvalue Problem for the Matrix B.- 4.2.3 The Eigenvalue Problem of General Type.- 4.3 Observational Equations of a Deforming Network.- References.- O. A Criterion Matrix for Deforming Networks by Multifactorial Analysis Techniques.- 1. Optimal Versus Improved Design.- 2. Essential Eigenvector Analysis.- 3. Procrustean Transformation.- References.- P. The Analysis of Time Series with Applications to Geodetic Control Problems.- 0. Foreword.- 1. Notations and Preliminaries.- 1.1 The Object of our Analysis.- 1.2 Prerequisites on Stochastic Processes.- 1.3 Stationarity.- 1.4 The Estimation of the Autocovariance Function.- 1.5 The Estimation of the Spectral Density.- 2. The Hilbert Space Setting.- 2.1 Basic Definitions.- 2.2 Establishing the Spectral Representation of the Time Series.- 2.3 The World Decomposition Theorem.- 2.4 Causality and Analytical Properties of the Spectral Functions.- 2.5 The General "Linear" Prediction Problem.- 3. The Autoregressive - Moving Average Processes.- 3.1 Definition of ARMA (p,g) Models.- 3.2 The Covariances of ARMA Processes.- 3.3 The Spectral Densities of ARMA Processes.- 3.4 The Yule-Walker Estimates and Forecasts.- 3.5 Examples.- 3.6 The Maximum Likelihood and "Least Squares" Estimates.- 3.7 Model Testing.- References.- Q. Quality Control in Geodetic Networks.- 0. Introduction.- 1. Model Assumptions and Estimation.- 2. Hypothesis Testing.- 3. Reliability.- 4. Precision.- References.- R. Aspects of Network Design.- 0. Introduction.- 1. The Datum Problem for Criterion Matrices.- 2. The Fundamental Design Problems.- 3. The Canonical Formulation of the Second Order Design Problem with Respect to an S-System.- 4. Review of Optimization Principles.- 5. The "Choice-of-Norm" Problem for Network Optimization.- 6. Transformation of the Quadratic Program into a Linear Complementarity Problem.- 7. The Optimal Design within Mixed Linear Models.- 8. The Second Order Design and Third Order Design Problem within the Mixed Model.- 9. The Second Order Design Problem within Mixed Models Admitting a Singular Covariance Matrix = ?ee=?2P+e.- Appendix 1: Criterion Matrices Reflecting Homogeneity and Isotropy.- Appendix 2: Computational Rules for Matrix Products.- Appendix 3: A Review of Reliability.- References.