Global Positioning System Reference
In-Depth Information
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receiver
k
while observing satellite
q
between the epochs
i
−
1 and
i
. The cycle slip
is denoted by
. Because the double differences are a function of observations at one
epoch, all double differences starting with epoch
i
are offset by the amount
∆
. Only
one of the triple-differences is affected by the cycle slip, because triple differences are
differences over time. For each additional slip there is one additional triple-difference
outlier and one additional step in the double-difference sequence. A cycle slip may
be limited to just one cycle or could be millions of cycles.
This simple relation can break down if the receiver software attempts to fix the slips
internally. Assume the receiver successfully corrects for a slip immediately following
the epoch of occurrence. The result is an outlier (not a step function) for double
differences and two outliers for the triple differences.
There is probably no best method for cycle slip removal, leaving lots of space for
optimization and innovation. For example, in the case of simple static applications,
one could fit polynomials, generate and analyze higher-order differences, visually
inspect the observation sequences using graphical tools, or introduce new ambiguity
parameters to be estimated whenever a slip might have occurred. The latter option is
very attractive in kinematic positioning.
It is best to inspect the discrepancies rather than the actual observations. The
observed double and triple differences show a large time variation that depends on
the length of the baseline and the satellites selected. These variations can mask small
slips. The discrepancies are the difference between the computed observations and
the actual observed values. If good approximate station coordinates are used then the
discrepancies are rather flat and make even small slips easily detectable.
For static positioning, one could begin with the triple-difference solution. The
affected triple-difference observations can be treated as observations with blunders
and dealt with using the blunder detection techniques given in Chapter 4. A simple
method is to change the weights of those triple-difference observations that have par-
ticularly large residuals. Once the least-squares solution has converged, the residuals
will indicate the size of the cycle slips. Not only is triple-difference processing a
robust technique for cycle slip detection, it also provides good station coordinates,
which, in turn, can be used as approximations in a subsequent double-difference
solution.
Before computing the double-difference solution, the double-difference observa-
tions should be corrected for cycle slips identified from the triple-difference solution.
If only two receivers observe, it is not possible to identify the specific undifferenced
phase sequence where the cycle slip occurred from analysis of the double difference.
Consider the double differences
∆
[18
Lin
—
0.0
——
Nor
PgE
[18
=
ϕ
1
−
ϕ
2
−
ϕ
1
ϕ
2
ϕ
1
p
12
−
(5.46)
for stations 1 and 2 and satellites 1 and
p
. The superscript
p
denoting the satellites
varies from 2 to
S
, the total number of satellites. Equation (5.46) shows that a cycle
slip in
ϕ
1
or
ϕ
2
will affect all double differences for all satellites and cannot be
separately identified. The slips
1
1
1
2
cause the same jump in the double-
difference observation. The same is true for slips in the phase from station 1 to satellite
∆
and
−∆
