Global Positioning System Reference
In-Depth Information
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[16
Lin
—
-
——
No
PgE
Figure 4.15
Weighting approximate coordinates to define the coordinate system.
ra
tio of the variances of the approximate coordinates over the average variance of the
ob
servations.
The weighted parameter approach is also a convenient way of imposing minimal
co
nstraints. Only a subset of three approximate coordinates needs to be weighted in
ca
se of a plane angle and distance network.
[16
4.
13 KALMAN FILTERING
Le
ast-squares solutions are often applied to surveying networks whose network points
re
fer to monuments that are fixed to the ground. When using the sequential least-
sq
uares approach (4.140) to (4.144), the parameters
x
are typically treated as a time
in
variant. The subscript
i
in these expressions identifies the set of additional obser-
va
tions added to the previous solution that contains the sets 1
1. Each set
of
observations merely updates
x
, resulting in a more accurate determination of the
fix
ed monuments.
We generalize the sequential least-squares formulation by allowing the parameter
ve
ctor
x
to change with time. For example, the vector
x
might now contain the
th
ree-dimensional coordinates of a moving receiver, the coordinates of satellites,
tro
pospheric delay of signals, or other time-varying parameters. We assume that the
dy
namic model between parameters of adjacent epochs follows the system of linear
eq
uations
≤
i
≤
i
−
x
k
(
−
)
=
Φ
k
−
1
x
k
−
1
+
w
k
(4.389)
We have used the subscript
k
, instead of
i
, to emphasize that it now indicates the
epoch. The matrix
Φ
k
−
1
is called the parameter transition matrix. The random vector
