Getchell’s method for conversion of Cartesian-geocentric to geodetic coordinates – Its properties and Newtonian alternative

Abstract

Abstract The paper concentrates on the iterative Getchell’s method (formulated in 1972) and its alternative Newtonian implementation for conversion of Cartesian geocentric coordinates into geodetic coordinates. The same basic equation formulated in the Getchell’s method is used in both cases. The equation has a stable form in the whole range of argument (latitude) variation ⟨ − π / 2 , π / 2 ⟩ \langle -\pi /2,\pi /2\rangle . The original Getchell’s method (somehow “forgotten”) has a simple geometric interpretation and its applications turn out to be particularly effective. Many studies on iterative algorithms usually omit theoretical proofs of convergence replacing them with conclusions based on numerical examples. This paper presents theoretical proofs of algorithms convergence both for the Getchell’s method and the Newton procedure. The convergence parameter and numerical error of results were estimated in each case. Numerical tests were carried out for a set of points distributed on the Earth’s space, also for extreme h values. For typical practical applications of the Getchell’s method, sufficiently accurate results are obtained after 1–3 iterations, while in the Newton procedure already after one iteration, assuming the same numerical error and initial conditions. The accuracy of the geodetic coordinates determinations meets all practical requirements with some margin. For example an absolute numerical error for latitude is approx. 0.4 · 10 − 13 0.4\cdot {10^{-13}}  [rad] i. e. about 0.00026 mm in the length of the meridian arc. The proposed methods were compared with other methods (algorithms), including in terms of stability and non-singularity in the entire usable space of the Earth, but excluding the near geocenter, which has no practical significance. Both the modification of the Getchell method and its Newtonian alternative are very good determined in this area (in the Earth’s poles, the final solution is directly the starting value of iterative algorithms). The discussed algorithms were implemented in the form of procedures in DELPHI language.