VCV MATRIX OF COORDINATE "OBSERVATION" |
The VCV matrix for the coordinates in both networks reflects the precisions of the individual coordinate components, as well as the correlations between the different coordinate components. In the case where one network is GPS-determined, and the other is a conventionally defined terrestrial network, there are several unique problems.
VCV of Terrestrial Networks
In practice the input VCV of the coordinate set used as data in the determination
of the transformation parameters is full because the ( X,Y,Z
) coordinates of the points are all correlated. However, rarely does a surveyor
have access to the VCV information of the terrestrial control points (if
the points are part of the national geodetic network containing say 5000
stations, the VCV information of interest is contained within the full 15000x15000
VCV matrix of parameters produced by the 3-D adjustment of all the geodetic
observations!). Hence the input VCV is often assumed to be diagonal,
with very approximate variances on the diagonals.
As input to a 3-D transformation parameter adjustment, only the horizontal ground coordinates from a conventional 2-D ellipsoidal adjustment are available and these must be combined with ellipsoidal heights, determined from sum of the orthometric height (from levelling) and geoid-ellipsoid separation (Figure in section 11.1.11). It is generally assumed that these heights are not correlated with the horizontal coordinate components. The ellipsoidal coordinates and the VCV matrix must be converted to the Cartesian equivalents using eqns (11.1-10) to (11.1-13).
Finally, in both GPS and terrestrial network adjustments the coordinates of one point are conventionally assigned zero variance (the so-called "minimally constrained solution"). For terrestrial networks, only a portion of the ground net is used as input for the transformation parameter determination. Apart from the problem of assembling the VCV matrix, the subnet may be a considerable distance from the fixed datum point (for example, the Johnston origin station in the case of the AGD) and hence the variances of coordinates of the points within the subnet will usually be very large. Formulae are available for converting VCV matrices so that they have minimum trace, or zero variances for some point(s) within the selected portion of the network (HARVEY, 1986).
VCV of GPS Networks
Compared to the problems listed above: (a) obtaining the VCV information,
(b) constructing a 3-D VCV, and (c) modifying the VCV to reflect a more
convenient datum station; the VCV for a GPS network is generally easily
extracted and manipulated. The VCV is already in 3-D form, and the fixed
station is usually within the network to be transformed (and is often selected
to be one of the common points from which the transformation parameters
are derived). As usual, there is the problem that the elements of the VCV
matrix of the GPS coordinates represent only precision estimates. However,
if the GPS network is the result of a multi-session secondary network adjustment
(section 9.3.1), the VCV matrix may already
have been scaled or modified in some way, so that the precisions are more
realistic (that is, closer to what may be considered the accuracy of the
network).
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© Chris Rizos, SNAP-UNSW, 1999