Algebraic Geodesy and Geoinformatics

"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."