Geoid and Height Correction Models

2.4.5.1 Geographic3D to GravityRelatedHeight

Although superficially involving a change of dimension from three to one, this transformation method is actually one-dimensional. The transformation applies an offset to the ellipsoidal height component of a geographic 3D coordinate reference system with the result being a gravity-related height in a vertical coordinate reference system. However the ellipsoidal height component of a geographic 3D coordinate reference system cannot exist without the horizontal components, i.e. it cannot exist as a one-dimensional coordinate reference system.

Geodetic science distinguishes between geoid-ellipsoid separation models and height correction models. Geoid separation models give the height difference between the ellipsoid and the geoid surfaces. Height correction models give height difference between ellipsoidal a particular vertical datum surface. Because a vertical datum is a realisation

of the geoid and includes measurement errors and various constraints, a vertical datum surface will not exactly coincide with the geoid surface. The mathematics of the application of these models is identical and for the purposes of the EPSG Dataset they are considered to be one method.

The correction value ζ5 _{is interpolated from a grid of height differences and the interpolation requires the}

latitude and longitude components of the geographic 3D coordinate reference system as arguments.

If h is the ellipsoidal height (height of point above the ellipsoid, positive if up) in the geographic 3D CRS and H is the gravity-related height in a vertical CRS, then

H = h – ζ

Note that unlike the general convention adopted for offsets described in 2.4.1, geoid separation and height correction models conventionally use the true mathematical convention for sign.

The EPSG Dataset differentiates between the formats of the gridded height files and distinguishes separate coordinate operation methods for each file format. The coordinate operation method may also define the interpolation technique to be used. However the density of grid nodes is usually sufficient for any reasonable interpolation technique to be used, with bi-linear interpolation usually being applied.

Reversibility

The reverse transformation, from gravity-related height in the vertical coordinate reference system to the ellipsoidal height component of the geographic3D coordinate reference system, requires that a horizontal position be associated with the gravity-related height. This is indeterminate unless a compound coordinate reference system is involved (see the Geographic3D to Geographic2D+GravityRelatedHeight method described below). Geographic3D to GravityRelatedHeight methods therefore are not reversible.

2.4.5.2 Geographic3D to Geographic2D+GravityRelatedHeight

This method transforms coordinates between a geographic 3D coordinate reference system and a compound coordinate reference system consisting of separate geographic 2D and vertical coordinate reference systems. Separate operations are made between the horizontal and vertical components. In its simplest form it combines a Geographic 3D to 2D conversion and a Geographic3D to GravityRelatedHeight transformation (see sections 2.2.2 and 2.4.5.1 above). However, complexities arise (a) for the forward transformation if the source 3D and target 2D geographic coordinate reference systems are based on different geodetic datums, or (b) in the reverse transformation of height from compound to geographic 3D.

Horizontal component

If the horizontal component of the compound coordinate reference system and the geographic 3D coordinate reference system are based on the same geodetic datum, this operation is simply the Geographic 3D to 2D conversion described in section 2.2.2 above except that for the reverse case (2D to 3D) no assumption is required for the ellipsoidal height as it will come from the operation for the vertical part.

If the horizontal component of the compound coordinate reference system and the geographic 3D coordinate reference system are based on different geodetic datums then any of the geographic to geographic transformations discussed in section 2.4.4 above, including those using geocentric methods (sections 2.4.3 and 2.4.4.1), may be used.

Vertical component

The forward transformation from geographic 3D to vertical component of the compound system uses the

5 _{Geodetic science recognises several types of gravity-related height, differentiated by assumptions made about the}

gravitational field. A discussion of these types is beyond the scope of this document. In this document the symbol ζ is used to indicate the correction to be applied to the ellipsoid height.

Geographic3D to GravityRelatedHeight method described in section 2.4.5.1 above. Then: H = h – ζ

where, as before, h is the ellipsoidal height (height of point above the ellipsoid, positive if up) in the geographic 3D CRS, H is the gravity-related height in the vertical CRS part of the compound CRS and ζ is the correction from ellipsoidal height to gravity-related height from the gridded data.

The reverse transformation, from vertical component of the compound system to geographic 3D system, requires interpolation within the grid of height differences. However the latitude and longitude arguments for this interpolation must be in the geographic 3D coordinate reference system, as the nodes for the gridded data will be in this system. Therefore the reverse operation on the horizontal component of the compound system must be executed before the reverse vertical transformation can be made. Then:

h = H – –ζ

2.4.5.3 Geographic2D with Height Offsets

(EPSG Dataset coordinate operation method code 9618)

This method used in Japan is a simplified form of the general Geographic3D to Geographic2D+GravityRelatedHeight method described above. It combines the geographic 2D offset method described in section 2.4.4.3 above with an ellipsoidal height to gravity-related height value A applied as a vertical offset.

ϕWGS84 = ϕTokyo + dϕ

λWGS84 = λTokyo + dλ

hWGS84 = HJSLD + A