Global Positioning System Reference
In-Depth Information
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4.11.3 ChangingWeights of Observations
Th
is method, although not based on rigorous statistical theory, is an automated method
w
hereby blunders are detected and their effects on the adjustment minimized (or
ev
en eliminated). The advantage that this method has, compared to previous methods,
is
that it locates and potentially eliminates the blunders automatically. The method
ex
amines the residuals per iteration. If the magnitude of a residual is outside a de-
fin
ed range, the weight of the corresponding observation is reduced. The process of
re
weighting and readjusting continues until the solution converges, i.e., no weights
ar
e being changed. The criteria for judging the residuals and choice for the reweight-
in
g function are somewhat arbitrary. For example, a simple strategy for selection of
th
e new weights at iteration
k
+
1 could be
if
v
k,i
>
3
p
k,i
e
−|
v
k,i
|
/
3
σ
i
σ
i
[16
=
p
k
+
1
,i
(4.381)
if
v
k,i
≤
1
3
σ
i
Lin
—
0.1
——
Sho
PgE
where
σ
i
denotes the standard deviation of observation
i
.
The method works efficiently for networks with high redundancy. If the initial
approximate parameters are inaccurate, it is possible that correct observations are
deweighted after the first iteration because the nonlinearity of the adjustment can
cause large residuals. To avoid unnecessary rejection and reweighting, one might not
change the weights during the first iteration. Proper use of this method requires some
experience. All observations whose weights are changed must be investigated, and
the cause for the deweighting must be investigated.
[16
4.12 EXAMPLES
In the following, we use plane two-dimensional networks to demonstrate the geom-
etry of adjustments. As mentioned above, the geometry of a least-squares adjust-
ment is the result of the combined effects of the stochastic model (weight matrix
P
—representing the quality of the observations) and the mathematical model (design
matrix
A
—representing the geometry of the network and the spatial distribution of the
observations). For the purpose of these examples, it is not necessary to be concerned
about the physical realization of two-dimensional networks. The experienced reader
might think of such networks as being located on the conformal mapping plane and
all that it takes to compute the respective model observations. However, it is entirely
sufficient here to stay simply within the area of plane geometry.
We will use the observation equation model summarized in Table 4.1. Assume
there is a set of
n
observations, such as distances and angles that determine the points
of a network. For a two-dimensional network of
s
stations, there could be as many
as
u
2
s
unknown coordinates. Let the parameter vector
x
a
consist of coordinates
only, i.e., we do not parameterize refraction, centering errors, etc. To be specific,
x
a
contains only coordinates that are to be estimated. Coordinates of known stations are
=
