I am currently experimenting with this tool by the ICGEM to validate my own implementation of the GGM05C gravity model.
However, the tool by the ICGEM requires input in ellipsoidal height, while I am using directly spherical coordinates to calculate the gradient of gravity potential at any given point.
This document states in page 18 that Physical Geodesy, by Hofmann-Wellenhof and Moritz, contains expressions for the transformation between ellipsoidal and spherical coordinates. However, after searching for these in the book, I was not able to find them.
So, can anybody point to a source about or explain how to convert between spherical coordinates r, lambda and phi (respectively the distance to the center of masses of Earth, the longitude and the geocentric latitude) to ellipsoidal coordinates h, lambda and phi (height above ellipsoid, longitude and geodetic latitude; from what I have read, the longitude value will still be the same, which makes sense)?