λ-Spheres as a New Reference Model for Geoid: Explicit Solutions of the Direct and Inverse Problems for Loxodromes (Rhumb Lines)

In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter lambda. These surfaces, which are called lambda-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrization that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter lambda for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian m(p) (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature.

Automated summary

In this paper, a new reference model is presented to approximate the shape of the Earth. The model, called lambda-spheres, is based on the concept of deformed spheres with the parameter lambda. The explicit solutions for loxodromes on these surfaces are obtained, and the direct and inverse problems for loxodromes are formulated and solved. The proposed reference model is evaluated and compared with the existing ellipsoidal and spherical models using the values of the semi-major axis and quarter-meridian of the Earth.