Height of the geoid in ECEF coordinates

Question

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.)

Answer 1

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:

1. Determine the ellipsoidal height by h=H+N where H is the orthometric height and N is derived from the respective geoid model. Typically the geoid model is a gridded surface that must be interpolated at the given location.
2. Convert lat/lon/h to ECEF XYZ using the equations in section 4.1.1 of the EPSG Geomatics Guidance Note 7, inserting the relevant parameters for the ellipsoid used.

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: H=h-N.