N the Geoid-ellipsoid separation

What is the Geoid?

The geoid is an equipotential surface of the Earth's gravity field, which closely approximates mean sea level and is by definition perpendicular to the direction of the gravity vector at all points. Since the mass distribution of the Earth is not uniform and the direction of gravity changes accordingly, the resultant shape of the geoid is irregular.

The ellipsoid is a mathematical surface obtained by rotating an ellipse about its semi-minor axis. The dimension and orientation of the ellipse are chosen "best fit" the ellipsoid to the geoid over a given area, which in this case is over the whole earth.

In practical terms, the geoid-ellipsoid separation at a given point is considered to consist of two components. The first is the long wavelength component, which is derived from global geopotential models (it was OSU91A for AUSGeoid93 and is now EGM96 for AUSGeoid98). This is by far the larger of the two components. The second component is the short wavelength contribution, which is evaluated using terrestrial gravity anomalies and digital elevation data. The sum of the two components at a given point results in a determination of the full geoid-ellipsoid separation.

The Geoid-ellipsoid separation

The reference surface for heights is traditionally taken as Mean Sea Level (MSL). In Australia this height reference is known as the Australian Height Datum (AHD). The geoid is a surface of equal gravity potential which closely approximates mean sea level.

However, heights derived from GPS are relative to the GPS reference ellipsoid (WGS84). The separation between the geoid and an ellipsoid is known as the geoid-ellipsoid separation, or N value. As an N value is relative to a specific ellipsoid, extreme care must be taken to ensure that the N value used refers to the correct ellipsoid.

N-values diagram of Geoid-ellipsoid separation, AUSgeoid, Geodesy

Diagram showing the Geoid-ellipsoid separation

In the examples below, both the N value and the ellipsoidal height refer to the same ellipsoid (usually WGS84 when working with GPS-derived ellipsoidal heights).

Although the Australian Height Datum (AHD) and Mean Sea Level (MSL) are generally thought to aligned with and the geoid, it has recently been found that there is actually a north/south slope across Australia of about 1 meter between the geoid and MSL. Applying Geoid-ellipsoidal separation "differentially" eliminates this problem, but if applied in an "absolute sense" the MSL/AHD-Geoid difference may be evident. Future Australian versions of the Geoid will include AHD-Geoid differences thus eliminating this problem.

Comparison examples

Example 1
In an absolute sense N is used as follows:
H = h - N
NOTE: N is negative in this example
h  =  623 m
N  =  -(-15) m
H  =  638 m (derived)
Example 2
In a relative (baseline) sense where the change in N is used:
(H2 - H1) = (h2 - h1) - (N2 - N1)
ie
diff H = diff h - diff N
H2 = H1 + diff H
H1  =  636.5 m (known)
h1  =  623 m
h2  =  581 m
N2  =  -17 m
N1  =  -15 m
diff h  =  581 - 623 = -42 m
diff N  =  -17 - 15 = -2
diff H  =  (-42) - (-2) = -40 m
H2  =  636.5 + (-40) = 596.5 m
Where:
H  = height with respect to the geoid (MSL)
h  = height with respect to the ellipsoid
N  = Geoid-ellipsoid separation (N value)
NOTE: If the geoid is above the ellipsoid, N is positive. If the geoid is below the ellipsoid, N is negative.