2. THE RELIABILITY OF THE GROUND-CONTROL POINTS:
In general boundary values can be calculated from
Vi = Thi 2.1
In 2.1' the following quantities can be distinguished
V1: resulting boundary value
cs: vector representing the used alternative hypothesis
N°: N = c Pr! — AG, A )Pc where
: weight matrix of observations
: design matrix
ge variance covariance matrix of unknowns resulting from
the adjustment
: non-centrality parameter of a Fisher distribution
> Q p» t3
o
Now 2.1 can be simplified, when using the conventional alternative
hypothesis (see [1]). That means that each time a boundary value is
calculated only one error in the observation data is assumed. Such a
hypothesis is represented by a c-vector which is a unit vector.
If one wants to obtain the boundary value of observation number 10,
the c-vector consists of zero's at all places except on the tenth,
where a "1" is placed
f S
»1oth place 2.2
ce Oy Dee OO
Qe
\ J
As in the second phase of the adjustment, that is the already mentioned
pseudo least squares adjustment, the x- and y-coordinates of the
ground control points act as observations, it can be easily seen that
the boundary values of these coordinates can be calculated according
to 2.1 using c-vectors as shown in 2.2.
The question arises if it is also possible to obtain boundary values
of the coordinates which act as S-base, whereas they are not an
observation in the adjustment problem.
A complicated solution is to carry out the whole adjustment again but
with respect to another S-base. Since boundary values are independent
with respect to an S-transformation, the four missing ones can be
calculated in this way. A quicker and more elegant procedure is the
following. Assume that points | and 2 act as S-base for all co-
ordinates and their variance covariance (v.c.) matrix. Then we are
facing the problem of obtaining boundary values of Xj»y,, X, and Yo“
As said earlier this problem could be solved by carrying out the
whole adjustment using another S-base for example points 3 and 4
(in this case the c-vectors of points 1 and 2 are again unit vectors
as in 2.2).
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