Global Positioning System Reference
In-Depth Information
8.4.1.4 Pseudorange (Code) Smoothing
Thus far in this description of GPS interferometry, two distinct sets of DDs have
been created. The first is based on differencing the low noise (less than 1 cm) but
ambiguous carrier phase measurements; the second set is formed from the unambig-
uous but noisier (1-2m) pseudorange (code) measurements. The two sets of mea-
surements can be combined using a variety of techniques to produce a
smoothed-code DD measurement. This is extremely important since the baseline
vector b determined from the smoothed-code DDs provides an initial solution esti-
mate for resolving the carrier-cycle integer ambiguities. Based on [25], a comple-
mentary Kalman filter is used to combine the two measurement sets. The technique
uses the average of the noisier code DDs to center the quieter carrier-phase DDs,
thereby placing a known limit on the size of the integer ambiguity.
The filter equations are as follows:
(
)
DD
=
DD
+
+
DD
DD
s
s
cp
cp
n
n
1
n
n
1
pp q
=
+
+
n
n
1
(
)
1
kppr
=
+
(8.23)
n
n
n
(
)
DD
+
=
DD
+
k
DD
DD
s
s
n
pr
s
n
n
n
n
(
)
+
p
=−
1
k
p
n
n
n
The first line of (8.23) propagates the smoothed-code DD to the current time epoch
( n ) using the estimate of the smoothed-code DD from the previous epoch ( n 1)
and the difference of the carrier-phase DD across the current and past epochs. The
estimate ( DD s + ), which is based on averaging the DD pr (code) difference, centers the
calculation; the DD cp (carrier-phase) difference adds the latest low-noise informa-
tion. Note that differencing two carrier-phase DDs across an epoch removes the
integer ambiguity; hence, the propagated smoothed-code DD ( DD s )remains
unambiguous. The estimation-error variance ( p ) is brought forward (line two)
using its previously estimated value plus the variance of the carrier-phase DD mea-
surement q. The Kalman gain is next calculated in preparation for weighting the
effect of the current code DD measurement. Line three shows that as the variance
on the code DD r approaches zero, the Kalman gain tends to unity. This is not sur-
prising since the higher the accuracy of a measurement (the smaller the variance),
the greater is its effect on the outcome of the process. Lines four and five of (8.23)
propagate the estimate of the smoothed-code DD ( DD + ) and estimation-error vari-
ance to the current epoch ( n ) in preparation for repeating the process in the next
epoch ( n 1). DD (to be used in the next epoch) involves the sum of the current
value of the smoothed-code DD (just predicted) and its difference from the current
code DD (just measured) weighted by the Kalman gain. Intuitively, if the prediction
is accurate, then there is little need to update it with the current measurement.
Finally, the estimation-error variance p is updated. The update maintains a careful
balance between the “goodness” of the code and that of the carrier-phase DDs
based on whether the Kalman gain approaches unity or zero or lies somewhere in
between.
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