Role of algebra in modern day Geodesy

Abstract

Algebra, in particular the “Abelian Group” and the “Semi Group” (also known as “Monoid”) axioms, which form a “ring with identity”, are employed to define the “polynomial ring”. Polynomial ring theory enables the solution of geodetic observations that can be converted into (algebraic) polynomials. The advantages of algebraic approaches are that they provide exact solutions to problems requiring closed form approaches (e.g. solving for geocentric coordinates from Helmerts projection through minimum distance mapping) and also act as tools to control iterative procedures. As a motivation, we present several examples of geodetic problems solved algebraically. These examples include; nonlinear analysis of bending angles in GPS-Meteorology, transformation of geocentric Cartesian coordinates into ellipsoidal, densification problems etc. The overriding advantage of the algebraic approach is the removal of the requirement of approximate starting values; they are non-iterative and enable detection of outliers.