Algebraic Geodesy and Geoinformatics, Methods and Applications - 2009

Preface

This electronic supplement gives computational examples illustrating the different topics in the book. The algorithms areimplemented in Mathematica system providing symbolic, as well as numeric computation methods.

This electronic guide gives a practical approach to the application of CAS (Computer Algebra System), especially Mathemat-

ica to nonlinear geodetical computations of mainly algebraic types. The users will find a great amount of examples explain-ing and illustrating the different techniques, as well as the solutions of different problems of nonlinear types.

The organization of this material is as follows:

In the PART I, from Chapter 1 to 9, symbolic-numeric methods are presented to solve mainly nonlinear, polynomialsystems. In this regard:

è Symbolic methods like Dixon resultant and Groebner basis can be applied to determined polynomial systems having less than 10 unknowns.

è Global numerical technique like linear homotopy can solve determined systems of considerably higher dimensions.

è Overdetermined systems can be sometimes transformed into determined systems via algebraic least square solution (ALESS) and solved by techniques mentioned above.

è Gauss -Jacobi combinatorial algorithm extended to nonlinear systems is an other effective method to solve overdetermined systems, especially when the solution of the corresponding determined subsets can be achieved by symbolic method in simple form.

è Extended Newton -Raphson method is a very general numerical technique to solve over-, under-, and determined systems, respectively. Although this is basically a local numerical method employing singular value decomposition (SVD). It is very robust, therefore the solution of one of the corresponding determined subsystems of the problem can give a good initial guess for this method.

è Procrustes method that provides a global numerical solution for coordinate transformation problems is also presented for overdetermined systems.

In the PART II, from Chapter 10 to Chapter 20, the applications of these algorithms to typical geodetical problems areintroduced. These include:

è Positioning in 2D and 3D, by Ranging, Resection and Intersection, including global (GNNS) and local (LPS) positioning systems, Photogrammetric Resection and Intersection.

è Geocentric Cartesian versus Ellipsoidal coordinates, including Minimum Distance Mapping.

è Datum transformation problems, for Helmert transformation with 7, and affine transformation with 9 parameters in 3D.

è GPS Meteology in Environment Monitoring.

è Algebraic diagnosis of Outliers, including the cases of planar ranging and multipath error analysis in GNNS positioning

The fast spreading of multi-core machines makes parallel computations potentially usual for everybody. Gauss-Jacobi

combinatorial method as well as linear homotopy are parallel algorithms by nature. This feature is illustrated in someexamples, since Mathematica supports parallel computation in a simple, transparent way.

We also present some Mathematica programs to carry out the computer solution of some of these practical real worldgeodetical problems.

This supplement therefore can be useful not only as a teaching material for many different types of courses in ComputationalGeodesy and Geoinformatics from bachelor up to PhD degree level, but also as a useable resource of computation tech-niques for researchers and engineers in the different fields of Geoscience, as well.

The Mathematica commands are mostly transparent, although novice users will need to consult with the Help system ofMathematica, which is very didactical and easy readable material. In addition, the excellent book, Mathematica Navigator,by Ruskeepaa (2009) is strongly recommended.

The best way for following and understanding these materials is to read the book in parallel.This material can be considered as a Practical Guide for Solving Nonlinear Problems in Geodesy and Geoinformatics via

Mathematica.

The computations were carried out on Lenovo W500 laptop computer with Intel Core 2 Duo T9600 2.8 GHz processor and4 GB RAM, running under XP 64 bit operation system.

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