11.3.9 GPS Heighting

GEOID HEIGHTS AND THE TRANSFORMATION PROCESS

Three types of coordinate transformation can be identified:

• Cartesian ( X,Y,Z ) <--> Geodetic ( ,,h )
• Geodetic ( ₁,₁,h₁ ) <--> Geodetic ( ₂,₂,h₂ )
• Cartesian ( X₁,Y₁,Z₁ ) <--> Cartesian ( X₂,Y₂,Z₂ )

The Figure below summarises the various transformation options involving the coordinate conversions referred to above.

Coordinate transformation options.

In the context of GPS surveying Datum 1 can be identified as the GPS datum and Datum 2 as the local geodetic datum. GPS provides data directly in the Cartesian system on a global geocentric datum (for example, WGS84). Data can be easily transformed into the geodetic coordinate system ( ,,h ) using eqn (11.1-14) without the need for any geoid height information.

Geoid height mapped on the global datum ellipsoid must be used to transform to ( ,,H ). On the other hand, conventional geodetic networks such as the AGD and AHD provide the coordinate triad ( ,,H ), and in order to relate them to either ( ,,h ) or ( X,Y,Z ) requires geoid height mapped on the local datum ellipsoid. (Note that to perform a similarity transformation, the two sets of coordinates must be Cartesian.)

Hence a prerequisite for determining, or applying, a set of parameters to transform from Datum 1 to Datum 2 (and visa versa) is the conversion of local ( ,,H ) to ( X,Y,Z ). How can this be done? Only the astro-geodetic and the geometric technique of geoid determination is capable of directly mapping the geoid on the local datum reference ellipsoid.

In the Table in section 11.3.8 it was suggested that a geopotential-based geoid model (provided by spherical harmonic coefficients) is adequate for this purpose. The procedure would therefore be:

• Use eqn (11.3-2) to evaluate geoid height N₁ on Datum 1 (that is, on the WGS84 ellipsoid).
  
• Assume h₁ = N₁ and transform ( ₁,₁,h₁ ) to ( ₂,₂,h₂ ) using the Molodensky formula for transforming ellipsoid height (it is not necessary to transform the latitude and longitude components):

(11.3-8)

where:

a	is the semi-major axis of the WGS84 ellipsoid,
b	is the semi-minor axis of the WGS84 ellipsoid, and
X, Y, Z	are the offsets of the WGS84 ellipsoid relative to the local ellipsoid origin. In the case of the AGD, the values are 116.0m, 50.47m and -141.69m (section 11.2.6).
a	is the difference between the semi-major axis of the WGS84 ellipsoid and the local ellipsoid, and hence in the case of the AGD a = 23m.
f	is the difference between the flattening of the WGS84 ellipsoid and the local ellipsoid, and in the case of the AGD f = 0.812045-7.
	where e is the eccentricity of the WGS84 ellipsoid.

 

• Apply h to N₁ to obtain N₂, the quantity required to convert H₂ to h_2Z.

This is the procedure used to determine the quantities in the Table in section 11.3.1.

Under what circumstances can a knowledge of geoid height be dispensed with entirely? This condition is satisfied if there are only three common points used to determine the transformation parameters, as the geoid slope (relative to the an ellipsoid surface) can be accommodated by some of the transformation parameters.

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© Chris Rizos, SNAP-UNSW, 1999