I know that the relationship of geoid height
N and ellipsoid height
h to orthometric height
H is defined by
H = h - N, but what if I have some position in Earth-Centered-Earth-Fixed (ECEF) coordinates?
Specifically, I am looking to relate the height of the geoid (for some particular position on the geoid) to the earth-centered height. (Using WGS84 here.)
Earth Centred Earth Fixed (ECEF) coordinates is a Cartesian system typically defined by the EPSG as:
Z axis corresponding with the earth’s rotation axis positive northwards, the X axis through the intersection of the prime meridian and equator, and the Y axis through the intersection of the equator with longitude 90°E.
In ECEF coordinates therefore there is no real concept of height in the meaning of distance above/below an equipotential surface. It is however possible to represent an orthometric point height in an ECEF XYZ coordinate:
His the orthometric height and
Nis derived from the respective geoid model. Typically the geoid model is a gridded surface that must be interpolated at the given location.
Inverse equations are also given in the same section, to convert from ECEF XYZ to lat/lon/h to perform the reverse procedure. After the ellipsoidal height
h has been determined, use the geoid height to subtract and determine orthometric height: