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Optimization of geodetic networks is classified in different orders. (1) Zero-order design solves the geodetic datum problem of how to transform geodetic measurements into absolute Cartesian coordi...

Optimization of Geodetic Networks

In the last 5 years, Lake Victoria water level has seen a dramatic fall that has caused alarm to water resource managers. Since the lake basin contributes about 20% of the lakes water in form of discharge, with 80% coming from direct rainfall, this study undertook a satellite analysis of the entire lake basin in an attempt to establish the cause of the decline. Gravity Recovery And Climate Experiment (GRACE), Tropical Rainfall Measuring Mission (TRMM) and CHAllenging Minisatellite Payload (CHAMP) satellites were employed in the analysis. Using 45 months of data spanning a period of 4 years (2002–2006), GRACE satellite data are used to analyse the variation of the geoid (equipotential surface approximating the mean sea level) triggered by variation in the stored waters within the lake basin. TRMM Level 3 monthly data for the same period of time are used to compute mean rainfall for a spatial coverage of .25°×.25° (25×25 km) and the rainfall trend over the same period analyzed. To assess the effect of evaporation, 59 CHAMP satellite’s occultation for the period 2001 to 2006 are analyzed for tropopause warming. GRACE results indicate an annual fall in the geoid by 1.574 mm/year during the study period 2002–2006. This fall clearly demonstrates the basin losing water over these period. TRMM results on the other hand indicate the rainfall over the basin (and directly over the lake) to have been stable during this period. The CHAMP satellite results indicate the tropopause temperature to have fallen in 2002 by about 3.9 K and increased by 2.2 K in 2003 and remained above the 189.5 K value of 2002. The tropopause heights have shown a steady increase from a height of 16.72 m in 2001 and has remained above this value reaching a maximum of 17.59 km in 2005, an increase in height by 0.87 m. Though the basin discharge contributes only 20%, its decline has contributed to the fall in the lake waters. Since rainfall over the period remained stable, and temperatures did not increase drastically to cause massive evaporation, the remaining major contributor is the discharge from the expanded Owen Falls dam.

The Falling Lake Victoria Water Level: GRACE, TRIMM and CHAMP Satellite Analysis of the Lake Basin

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- 01 Introduction- 02 Notations and Preliminaries- 1 Generalized Inverses, a Geometric Approach- 11 Characterization of a Set of Linear Equations- 12 A Unique Characterization of an Arbitrary Generalized Inverse- 13 Right - and Left Inverses- 14 An Arbitrary System of Linear Equations and Arbitrary Generalized Inverses- 15 Transformation Properties and Some Special Types of Generalized Inverses- 16 Summary- 2 On S-Transformations- 21 Introduction- 22 Coordinates and Datum Definitions- 23 S-Transformations- 24 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- 11 Direct Approximation of the Criterion Matrix, Approach i)- 12 Iterative Approximation of the Criterion Matrix, Approach ii)- 13 Direct Approximation of the Inverse Criterion Matrix, Approach iii)- 14 Diagonal Design- 15 Approximation Quality- 16 Modification of Approach iii)- 2 Solution Methods- 21 Least-Squares Solution- 22 Linear Programming- 23 Nonlinear Programming- 3 Mean Least-Squares Approximation, Comparison of the Three Approaches- 4 Directions in the SOD-Problem- 41 Elimination and Group Weights- 42 Elimination and Individual Weights- 43 Correlated Angles- 44 Extracted Khatri-Rao-Product and Individual Weights- 45 Comparison- 46 Three-Step-Strategy- 5 Defect Analysis of the Final Equation- 51 Defect Analysis for Distances- 52 Free Distance Networks- 53 Defect Analysis for Directions- 54 Free Direction Networks- 6 Direct Creation of the Final Equation- 61 Individual Weights- 611 Distances- 612 Directions- 613 Mixed Network- 62 Group Weights- 63 Common Weights for Sets of Directions- 7 Examples- 71 Example 1- 72 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- 21 Precision Criteria- 3 Network Design Strategies- 31 Computer Simulation- 32 Analytical Methods- 321 Generalized Matrix Algebra- 322 Linear Programming- 323 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- 51 Conjugate Gradient Method- 52 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- 31 Determination of Station and Satellite Coordinates- 32 Determinations of Station, Satellite and Non-Geometric Parameters- 4 Estimability Analysis- 41 Patterns of Observations for Moving Stations- 42 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- 41 The Deterministic Approach- 42 The Model Function Approach- 43 The Stochastic Approach- 44 Hybrid Approaches- 5 Zero Order Design with Signals- 51 General Remarks- 52 Three-dimensional Networks- 6 Deformable Networks- 7 Estimability Problems- 8 Other Optimization Problems- 81 General Remarks- 82 First Order Design- 83 Second Order Design- 84 Third Order Design- Appendix: Observation Equations of Three-dimensional Networks- References- K Fourier Analysis of Geodetic Networks- 0 Introduction- I Spectral Methods in Geodesy- 11 Fourier Techniques in Interpolation Methods- 111 Step Function "Interpolation"- 112 Piecewiese Linear Interpolation- 113 Quadratic Spline Interpolation- 114 Cubic Spline Interpolation- 115 Higher and Highest Order Spline Interpolation- 12 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- 11 Networks on a Line- 111 The Fixed Network- 112 The Free Network- 12 Networks on a Circle- 13 Variance - Covariance Function of Estimable Quantities- 14 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- 41 Datum Transformation of a Criterion Matrix- 42 Canonical Comparison of an Ideal Versus a Real Variance- Covariance Matrix- 421 The Eigenvalue Problem for the Matrix A- 422 The Eigenvalue Problem for the Matrix B- 423 The Eigenvalue Problem of General Type- 43 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- 11 The Object of our Analysis- 12 Prerequisites on Stochastic Processes- 13 Stationarity- 14 The Estimation of the Autocovariance Function- 15 The Estimation of the Spectral Density- 2 The Hilbert Space Setting- 21 Basic Definitions- 22 Establishing the Spectral Representation of the Time Series- 23 The World Decomposition Theorem- 24 Causality and Analytical Properties of the Spectral Functions- 25 The General "Linear" Prediction Problem- 3 The Autoregressive - Moving Average Processes- 31 Definition of ARMA (p,g) Models- 32 The Covariances of ARMA Processes- 33 The Spectral Densities of ARMA Processes- 34 The Yule-Walker Estimates and Forecasts- 35 Examples- 36 The Maximum Likelihood and "Least Squares" Estimates- 37 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

Optimization and design of geodetic networks

This monograph contains a thorough treatment of methods for solving over- and under-determined systems of equations, e.g. the minimum norm solution method with respect to weighted norms. The considered equations can be nonlinear or linear, and deterministic models as well as probabilistic ones are considered. An extensive appendix provides all necessary prerequisites like matrix algebra, matrix analysis and Lagrange multipliers, and a long list of reference is also included.

Linear and Nonlinear Models : Fixed Effects, Random Effects, and Mixed Models

From the contents: Space Geodetic Measurements and Analysis, Geodetic Reference Frames, Earth Rotation.- Geodetic Boundary Value Problems, Gauss-Listing Geoid, Molodensky Quasi-Geoid.- Geodetic Data Analysis, Mathematical Statistics, Mathematical Cartography, Geodetic System Analysis.- Geophysical Geodesy.

Erik W. Grafarend

Papers

Optimization of Geodetic Networks

The Falling Lake Victoria Water Level: GRACE, TRIMM and CHAMP Satellite Analysis of the Lake Basin

Optimization and design of geodetic networks

Linear and Nonlinear Models : Fixed Effects, Random Effects, and Mixed Models

Geodesy-The Challenge of the 3rd Millennium