Awange / Grafarend / Paláncz Algebraic Geodesy and Geoinformatics

2. Auflage 2010
ISBN: 978-3-642-12124-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 377 Seiten, eBook

While preparing and teaching ‘Introduction to Geodesy I and II’ to undergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taught required some skills in algebra, and in particular, computer algebra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we have attempted to put together basic concepts of abstract algebra which underpin the techniques for solving algebraic problems. Algebraic computational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds, the concepts and techniques presented herein are nonetheless applicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robotics apply resection and intersection techniques which require algebraic solutions. Solution of nonlinear systems of equations is an indispensable task in almost all geosciences such as geodesy, geoinformatics, geophysics (just to mention but a few) as well as robotics. These equations which require exact solutions underpin the operations of ranging, resection, intersection and other techniques that are normally used. Examples of problems that require exact solutions include; • three-dimensional resection problem for determining positions and orientation of sensors, e. g. , camera, theodolites, robots, scanners etc.

Zielgruppe

Research

Autoren/Hrsg.

Awange, Joseph L.

Awange, Joseph

Grafarend, Erik W.

Paláncz, Béla

Zaletnyik, Piroska

Inhaltsverzeichnis

Algebraic symbolic and numeric methods.- Basics of ring theory.- Basics of polynomial theory.- Groebner basis.- Polynomial resultants.- Linear homotpy.- Solutions of Overdetermined Systems.- Extended Newton-Raphson method.- Procrustes solution.- Applications to geodesy and geoinformatics.- LPS-GNSS orientations and vertical deflections.- Cartesian to ellipsoidal mapping.- Positioning by ranging.- Positioning by resection methods.- Positioning by intersection methods.- GNSS environmental monitoring.- Algebraic diagnosis of outliers.- Datum transformation problems.

Leseproben

"1 Introduction (p. 1-2)

1-1 Motivation

A potential answer to modern challenges faced by geodesists and geoinformatics (see, e.g., Sect. 1-3), lies in the application of algebraic computational techniques. The present book provides an in-depth look at algebraic computational methods and combines them with special local and global numerical methods like the Extended Newton-Raphson and the Homotopy continuation method to provide smooth and efficient solutions to real life-size problems often encountered in geodesy and geoinformatics, but which cannot be adequately solved by algebraic methods alone.

Algebra has been widely applied in fields such as robotics for kinematic modelling, in engineering for o set surface construction, in computer science for automated theorem proving, and in Computer Aided Design (CAD). The most wellknown application of algebra in geodesy could perhaps be the use of Legendre polynomials in spherical harmonic expansion studies. More recent applications of algebra in geodesy are shown in the works of Biagi and Sanso [77], Awange [14], Awange and Grafarend [41], and Lannes and Durand [259], the latter proposing a new approach to di erential GPS based on algebraic graph theory. The present book is divided into two parts.

Part I focuses on the algebraic and numerical methods and presents powerful tools for solving algebraic computational problems inherent in geodesy and geoinformatics. The algebraic methods are presented with numerous examples of their applicability in practice. Part I can therefore be skipped by readers with an advanced knowledge in algebraic methods, and who are more interested in the applications of the methods which are presented in part II.

1-2 Modern challenges

In daily geodetic and geoinformatic operations, nonlinear equations are encountered in many situations, thus necessitating the need for developing efficient and reliable computational tools. Advances in computer technology have also propelled the development of precise and accurate measuring devices capable of collecting large amount of data. Such advances and improvements have brought new challenges to practitioners in fields of geosciences and engineering, which include:

• Handling in an efficient and manageable way the nonlinear systems of equations that relate observations to unknowns. These nonlinear systems of equations whose exact (algebraic) solutions have mostly been difficult to solve, e.g., the transformation problem presented in Chap. 17 have been a thorn in the side of users. In cases where the number of observations n and the number of unknowns m are equal, i.e., n = m, the unknown parameters may be obtained by solving explicitly (in a closed form) nonlinear systems of equations."