INTERPRETING SOLUTION VCV MATRICES |
The solution VCV matrix is influenced by:
Any interpretation of the VCV must be made with an awareness of
these factors. The output VCV matrix of the estimated parameters
( Q
) is quoted in the same reference system
as the parameters, that is, as Cartesian components in the WGS84
system. Conventional geodetic adjustments are essentially two-dimensional,
utilising ellipsoidal components ( 
, 
) or
local topocentric components ( East, North ), with the vertical
parameters being estimated in a separate procedure. In order to
aid interpretation of the results of a GPS adjustment, it may
therefore be useful to modify the Cartesian formulation to an
ellipsoidal or topocentric coordinate formulation. The process
of changing the VCV from the Cartesian form to the equivalent
ellipsoidal and topocentric form is described in HARVEY
(1994) and in (section 11.1.5).
Figure 1 is an illustration of the structure of a 2-D VCV matrix
(the rows and columns correspond to the horizontal components
and their correlations).
Error Ellipses
In all subsequent discussions relating to error figures, attention will be restricted to the horizontal precisions as exemplified by the error ellipses. In the case of planimetric coordinates, the VCV matrix of the coordinates of a point extracted from the whole VCV (Figure 1) is:
 |
(9.1-7) |
where r is the coefficient of correlation = 
, 
E
and N are the standard deviations of the
parameters, and EN is the covariance between the parameters.
Click
here
Figure 1. VCV components of a 2-D survey network.
(HARVEY, 1994)
The "absolute" or "point" error ellipse gives
the confidence region of the coordinates of a single point, independent
of any other points in the adjustment. To define the error ellipse
it is necessary to define the size and orientation of the semi-major
and semi-minor axes of the ellipse. The bearing 
for
which 2x is maximised is:
 |
(9.1-8) |
The formulae for the semi-major and semi-minor axes are:
 |
(9.1-9) |
The absolute error ellipses will vary with the choice of origin
of a network, as illustrated in figure below. When the origin
of the minimally constrained network is located approximately
30km to the south-east of the network (about 15-20km in east-west
and north-south extent), the error ellipses are larger (Figure
2a) than if the origin station is located at PM43494, in the centre
of the network (Figure 2b).
Figure 2a. Effect of datum selection on absolute 2-D error ellipses:
Canobolas T.S. datum
(approximately 30km S.E. of centroid of figure).
The error ellipses in Figure 2 are also referred to as "standard"
error ellipses. The probability of a point being in the ellipse
for the 2-D case is 39%. It is common practice to draw 95% confidence
ellipses at 2.45 times their "standard" size.
Often it is more important to obtain estimates of the precision of the relative positions of two points, rather than of their absolute positions. These estimates can also be found from the VCV matrix of the coordinates.
Consider two points A and B, the relevant section of the VCV matrix is:
 |
(9.1-10) |
Then
| DEDE = 2EA - 2. EAEB
+ 2EB |
(9.1-11a) |
| DNDN = 2NA - 2. NANB
+ 2NB |
(9.1-11b) |
| DEDN = NBEB
+ NAEA - NBEA
- NAEB |
(9.1-11c) |
Figure 2b. Effect of datum selection on absolute 2-D error ellipses:
PM43494 datum
(approximate centroid of network).
The orientation and lengths of the axes of the semi-major axis
can then be obtained from eqns (9.1-8) and (9.1-9), in the same
way as for the standard point ellipses. Figure 3 shows the relative
or line error ellipses between points in a minimally constrained
network. The relative error ellipses are unaffected by
the choice of origin.
In the case of a GPS baseline, the semi-major axis of the (relative)
error ellipse would be expected to be oriented approximately east-west,
and the error ellipse to be nearly circular
if the ambiguities had been resolved (although is
a little weaker than ), but more elongated in the
case of double-difference ambiguity-free solutions.
Figure 3. Example of relative 2-D error ellipses.
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© Chris Rizos, SNAP-UNSW, 1999