Global Positioning System Reference
In-Depth Information
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[16
Lin
—
0.9
——
No
PgE
Figure 4.13
Changing minimal constraints.
sta
tion 1 fixed. This results in a degenerated ellipse (straight line) at station 1 and a
re
gular ellipse at station 3. In the second case, we hold one of the coordinates of station
3 fi
xed. The result is a degenerated ellipse at station 3 and a regular ellipse at station 1.
Th
e ellipses of standard deviation change significantly due to the change in minimal
co
nstraints. Clearly, if one were to specify the quality of a survey in terms of ellipses
of
standard deviation, one must also consider the underlying minimal constraints.
Fi
gure 4.13 also shows that the adjusted coordinates for stations 1 and 3 differ in
bo
th cases, although the internal shape of the adjusted network 1-2-3 is the same.
The inner constraint solution, which is a special case of the minimal constraint
so
lutions, has the property that no individual coordinates need to be held fixed.
Al
l coordinates become adjustable; for
s
stations of a plane network, the vector
x
a
co
ntains 2
s
coordinate parameters. The ellipses reflect the geometry of the network,
th
e distribution of the observations, and their standard deviations. Section 4.8 contains
th
e theory of inner constraints. The elements for drawing the ellipses are taken from
th
e cofactor matrix (4.203) and Equation (4.212) gives the adjusted parameters. A first
ste
p is to find a matrix
E
that fulfills
AE
T
[16
O
according to (4.177). The number of
ro
ws of
E
equals the rank defect of
A
. For trilateration networks with distances and
an
gles we have
=
···
10
···
10
···
E
=
···
01
···
01
···
(4.387)
···
−
y
i
x
i
···
−
y
k
x
k
···
Four constraints are required for triangulation networks that contain only angle ob-
servations. In addition to fixing translation and rotation, triangulation networks also
require scaling information. The
E
matrix for such networks is
