EPSG

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

2021-11-04

Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.

This transformation involves the application of a height difference interpolated from a hydroid surface correction model. The model provides the offset of the tidal surface from the ellipsoid, positive when the tidal surface is above the ellipsoid, of a specified geographic 3D CRS (the source CRS) at nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation. The grid is referenced to a specific geographic CRS (the horizontal component of the source CRS) and interpolation must be made in the latitude and longitude of this system. Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (ζ) of the target datum above the ellipsoid of the source Geographic 3D CRS. Because it is applied in the target CRS in the depth domain, the height correction model relationship to ellipsoidal height has its sign reversed. Then: D = ζ – h In a marine context depth of the seabed below the tidal surface is derived from observed water depth and observed three-dimensional position including ellipsoidal height, both measurements reduced to a common vessel reference point, and the ellipsoid height of the tidal surface interpreted from the hydroid model. Then: D = (Dobs – hobs) + ζ where D is the depth below the tidal surface (for example, depth below LAT) Dobs is an observed depth below the vessel reference point hobs is an observed ellipsoidal height of the vessel reference point ζ is the height of the tidal surface above the reference ellipsoid interpolated from the hydroid model The method is not reversible because the depth is 1-dimensional and not associated with a horizontal CRS.

For coordinate transformation: ETRS89 to CD Norway depth: The Interpolation CRS is ETRS89 geographic 2D, CRS code 4258. A point in the geographic 3D coordinate reference system ETRS89, code 4937, with: latitude φ = 60°00'05.4"N = +60.0015° longitude λ = 4°59'45.6"E = +4.9960° ellipsoidal height h = 50.000m has an observed water depth Dobs of 12.00m This is converted to the geographic 2D coordinate reference system ETRS89, code 4258, as: point for interpolation: φ = +60.0015° λ = +4.9960° To find its CD Norway depth, first obtain the ellipsoid height offset (ζ) at each of the surrounding hydroid model grid nodes: NW node φ = +60.005° λ = +4.990° ζ = +43.885m NE node φ = +60.005° λ = +5.000° ζ = +43.887m SE node φ = +60.000° λ = +5.000° ζ = +43.882m SW node φ = +60.000° λ = +4.990° ζ = +43.880m Use bi-linear interpolation in ETRS89, code 4258, to obtain the ellipsoid height offset (ζ) at the observation point: ζ = +43.8827m which rounded to three decimal places gives ζ = +43.883m Then the depth D below CD Norway = (Dobs – hobs) + ζ = (12.00 − 50.000) + 43.883 = +5.883m.