IOGP

EPSG guidance note #7-2, http://www.epsg.org

2021-02-05

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided. This is a multi-step transformation involving the transformation of a geographic 3D CRS to a compound CRS comprising of a geographic 2D CRS horizontal component and a 1D vertical CRS. It is a requirement of the method that the geographic 2D CRS is the horizontal subset of the geographic 3D CRS. Only the height is affected by this transformation; the geodetic latitude and longitude are not. The height involves the application of a geoid height difference interpolated at a point in a "geoid model" consisting of grid of latitude and longitude with the height of the geoid above the ellipsoid at each grid node. The forward transformation from geographic 3D φλh to compound (φλ + H) involves the following sequence of steps: 1) apply a Geographic3D to 2D conversion (method 9659, §4.1.4) to convert the geographic 3D coordinates φλh to geographic 2D coordinates referenced the interpolation CRS φλ. 2) use φλ to interpolate the geoid model for N. 3) then H = h - N The compound CRS coordinates (φλ + H) are from steps 1 and 3. Reverse transformation from compound (φλ + H) to geographic 3D φλh: 1) Use the horizontal component of the compound CRS (φλ) to the interpolate geoid model for N. 2) h = H + N 3) In theory, convert φλ to φλh (using the reverse of method 9659, §4.1.4). Theoretically this is indeterminate and requires an assumption about ellipsoidal height. But here use h from step 2, which is the correct value. In practice, geographic 3D coordinates φλh are obtained by combining φλ from step1 with h from step 2. The combination of the coordinate tuples replaces the Geographic2D to 3D conversion.

For coordinate transformation: GDA2020 to GDA2020 + AHD height (1), code 9466: For a point at 36°54'01"S, 144°46'46"E (36.9003°S, 144.7794°E), with GDA2020 ellipsoidal height of 50.000 metres, to find its AHD height: Using the interpolation CRS GDA2020, code 7844, for interpolation, obtain the offsets at each of the surrounding grid nodes: NW corner 36.9000°S, 144.7667°E, geoid height = 34.267m NE corner 36.9000°S, 144.7833°E, geoid height = 34.293m SE corner 36.9167°S, 144.7833°E, geoid height = 34.205m SW corner 36.9167°S, 144.7667°E, geoid height = 34.185m Interpolation for 36.9003°S, 144.7794°E gives geoid height N = 34.285m Then AHD height H = h - N = 50.000 - (34.285) = 15.715m. For the reverse transformation of compound CRS GDA2020 36°54'01"S, 144°46'46"E + AHD height 34.285m to GDA2020 geographic 3D, first use the horizontal component of the compound CRS, interpolation CRS code 7844, to interpolate for the geoid height N at 36°54'01"S, 144°46'46"E as in the forward transformation. N = 34.285m. Then ellipsoidal height h = H + N = 34.285 + 15.715 = 50.000m. GDA2020 geographic 3D coordinates are 36°54'01"S, 144°46'46"E, 50.000m.