Optimization and Design of Geodetic Networks

During the period April 25th to May 10th, 1984 the 3rd Course of the International School of Advanced Geodesy entitled "Optimization and Design of Geodetic Networks" took place in Erice. The main subject of the course is clear from the title and consisted mainly of that particular branch of network analysis, which results from applying general concepts of mathematical optimization to the design of geodetic networks. As al­ ways when dealing with optimization problems, there is an a-priori choice of the risk (or gain) function which should be minimized (or maximized) according to the specific interest of the "designer", which might be either of a scientific or of an economic nature or even of both. These aspects have been reviewed in an intro­ ductory lecture in which the particular needs arising in a geodetic context and their analytical representations are examined. Subsequently the main body of the optimization problem, which has been conven­ tionally divided into zero, first, second and third order design problems, is presented. The zero order design deals with the estimability problem, in other words with the definition of which parameters are estimable from a given set of observa­ tions. The problem results from the fact that coordinates of points are not univocally determined from the observations of relative quantities such as angles and distances, whence a problem of the optimal choice of a reference system, the so-called "datum problem" arises.

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.