Global Positioning System Reference
In-Depth Information
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was achieved using satellite techniques and that an independent method for verifying
this accuracy was available. The Orange County densification demonstrated the use
of least-squares to quality control large vector data sets.
8.1 GPS VECTOR NETWORKS
In the case of two receivers observing, carrier phase processing gives the vector
between the stations, expressed in the reference frame of the ephemeris, and the 3
3
covariance matrix of the coordinate differences. The covariance matrix of all vector
observations is block-diagonal, with 3
×
×
3 submatrices along the diagonal. In a session
solutions, in which case R receivers observe the same satellites simultaneously, the
results are (R
1 ) covariance matrix.
The covariance matrix is still block-diagonal, but the size of the nonzero diagonal
matrices is a function of R .
Like any other survey, a GPS survey that has determined the relative locations of a
cluster of stations should be subjected to a minimal or inner constraint adjustment for
purposes of quality control. For example, the network should not contain unconnected
vectors whose endpoints are not tied to other parts of the network. At the network
level, the quality of the derived vector observations can be assessed, the geometric
strength of the overall network can be analyzed, internal and external reliability can be
computed, and blunders may be discoverable and removable. For example, a blunder
in an antenna height will not be discovered when processing a single baseline, but it
will be noticeable in the network solution if stations are reoccupied independently.
Covariance propagation for computing distances, angles, or other functions of the
coordinates should be done, as usual, with the minimal or inner constraint solution.
The mathematical model is the standard observation equation model, i.e.,
1 ) independent vectors, and a 3 (R
1 )
×
3 (R
[30
Lin
1.0
——
Lon
PgE
[30
a =
f ( x a )
(8.1)
where
a contains the adjusted observations and x a denotes the adjusted station
co ordinates. The mathematical model is linear if the parameterization of receiver
po sitions is in terms of Cartesian coordinates. In this case the vector observation
be tween stations k and m is modeled simply as
x km
x k
x m
=
y km
y k
y m
(8.2)
z km
z k
z m
The relevant portion of the design matrix A for the model (8.2) is
x k y k z k
x m
y m
z m
100
10 0
010 0
A km =
10
(8.3)
001 0
0
1
 
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