Global Positioning System Reference
In-Depth Information
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are given. In all cases, the distances are of the same length and observed with the same
accuracy. The angle observations are 180° and measured with the same accuracy but
are changed by a common factor for each solution. If we declare the solutions with
the smallest ellipses in Figure 4.11 as the base solutions with observational standard
deviation of
σ
a
, respectively. The shape of
the ellipses elongates as the standard deviation of the angles increases.
σ
a
then the other solutions use 2
σ
a
and 4
Ex
ample 2:
This example demonstrates the impact of changing network geometry
us
ing a resection. Four known stations lie exactly on an imaginary circle with radius
r
.
The coordinates of the new station are determined by angle measurements, i.e., no
di
stances are involved. For the first solution, the unknown station is located at the
ce
nter of the circle. In subsequent solutions it location moves to 0
.
5
r
,0
.
9
r
,1
.
1
r
,
an
d 1
.
5
r
from the center while retaining the same standard deviation for the angle
ob
servations in each case. Figure 4.12 shows that the ellipses become more elongated
th
e closer the unknown station moves to the circle. The solution is singular if the new
sta
tion is located exactly on the circle.
[16
Lin
—
-3.
——
Nor
PgE
[16
Figure 4.12
Impact of changing network geometry.
Example 3:
Three cases are given that demonstrate how different definitions of the
coordinate system affect the ellipses of standard deviation. All cases refer to the
same plane network using the same observed angles and distances and the same
respective standard deviations of the observations. A plane network that contains
angle and distance observations requires three minimal constraints. Simply holding
three coordinates fixed imposes such minimal constraints. The particular coordinates
are constants and are not included in the parameter vector
x
a
, and, consequently,
there are no columns in the
A
matrix that pertain to these three coordinates. Inner
constraints offer another possibility of defining the coordinate system.
Figure 4.13 shows the results of two different minimal constraints. The coordinates
of station 2 are fixed in both cases. In the first case, we hold one of the coordinates of
