Global Positioning System Reference
In-Depth Information
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The respective elements for the base satellite ambiguities in the design matrix are,
again, 1 and
1.
One must identify (R
1 ) independent double-difference functions in
network solutions. In session networks that contain a mixture of long and short base-
lines, it might be important to take advantage of short baselines because the respective
unmodeled errors (troposphere, ionosphere, and possibly orbit) are expected to be
small. Fixing the ambiguities to integers adds strength to the solution. This additional
strength gained by fixing the ambiguities of a short baseline may also make it possible
to fix the ambiguities for the next longer baseline, even though the ambiguity search
algorithms might not have been successful without that constraint. The technique is
sometimes referred to as “boot-strapping” from shorter to longer baselines. A suitable
procedure would be to take baselines in all combinations and order them by increasing
length and identify the set of independent baselines, starting with the shortest.
There are several schemes available to identify independent baselines and observa-
tio ns. Hilla and Jackson (2000) report using a tree structure and edges. Here we follow
th e suggestion of Goad and Mueller (1988) because it highlights yet another useful
ap plication of the Cholesky decomposition. Assume that matrix D of (7.83) reflects
th e ordering suggested here; i.e., the first rows of D refer to the double differences of
th e shortest baseline, the next set of rows refer to the second shortest baseline, and so
on . We write the cofactor matrix (7.88) as
1 )(S
[26
Lin
* 1 ——
Lon
PgE
0 DD T
0 LL T
Q = σ
= σ
(7.101)
w here L denotes the Cholesky factor (A.94). The elements of the cofactor matrix Q
are
[26
q ij
=
d i (k) d j (k)
(7.102)
k
w here d i (k) denotes the i th row of the matrix D . It is readily verified that the i th and
j t h columns of Q are linearly dependent if the i th and j th rows of D are linearly
de pendent. In such a case Q is singular. This situation exists when two double
di fferences are linearly dependent. The diagonal element j of the Cholesky factor
L will be zero. Thus, one procedure for eliminating the dependent observations is
to carry out the computation of L and to discard those double differences that cause
a zero on the diagonal. The matrix Q can be computed row by row starting at the
to p; i.e., the double differences can be processed sequentially one at a time, from
th e top to the bottom. For each double difference, the respective row of L can be
co mputed. In this way, the dependent observations can be immediately discovered
an d removed. Only the independent observations remain. The process ends as soon
as (R
1 ) double differences have been found.
If all receivers observe all satellites for all epochs, this identification process needs
to be carried out only once. The matrix L , since it is now available, can be used to
decorrelate the double differences. The corresponding residuals might be difficult to
interpret, but could be transformed to the original observational space using L again.
1 )
×
(S
 
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