NavCom SF-3050 A Computationally Efficient Ambiguity Resolution - Manual (Page 2)

2
INITIAL CODE SOLUTION
The primary motivation in the initial code solution is to
obtain the most accurate code solution possible with as
few measurement epochs as possible. Since the baseline
distances over which the ambiguities can be easily
resolved are typically less than 15 kilometers, it is
assumed that the ionospheric refraction effects can be
modeled at both base station and user with sufficient
accuracy that the residual ionospheric errors are almost
negligible. This means that the largest errors, which will
affect the differential code solution, will be the multipath
effects upon the dual-frequency code measurements. The
combination of code measurements which has the
minimum multipath error (or at least very close to the
minimum) is the frequency-weighted sum of the code
measurements. This combination also has the advantage
that it is exactly matched in ionospheric refraction effects
by the wide-lane difference carrier-phase measurement.
This means that it can be smoothed by the wide-lane
carrier-phase measurement with an increasing time
constant, i.e. the multipath error will become smaller and
smaller as the satellite is continuously tracked.
Because the base-station receiver is typically put in
operation long before the user receiver, our choice of
generating the code and carrier phase corrections at the
base-station receiver allows the base-station code
measurements to be smoothed such that most of the
multipath error is removed before the code corrections are
generated. This means that the multipath noise
contributed by the base station receiver is largely
eliminated. This represents a substantial advantage over
the normal practice of sending raw base-station
measurements to the user.
The frequency weighted combined measurement is:
2 1
2 2 1 1
f f
P f P f
P
+
+
=
(1)
where f represents the frequency and P the pseudorange
code measurement
.
The carrier-phase measurement which is affected by the
same amount of ionospheric refraction error is:
w w
N C λ φ φ ) (
2 1
− + =
(2)
where N is the integer ambiguity, φ is the carrier-phase
measurement and λ the wavelength .
By setting equation (1) equal to equation (2) and solving
for the wide-lane ambiguity, N
w
, a new value for the
floating ambiguity can be computed each epoch.
) (
) ( 2 1
2 1
2 2 1 1 φ φ
λ
− −
+
+
=
w
w f f
P f P f
N (3)
This value for the wide-lane floating ambiguity value can
be improved each epoch by computing the recursive
average over all epochs up to the present time. This
smoothed value of N
w
can then be substituted back into
equation (2) to give a smoothed carrier-phase
pseudorange:
w w
N C
λ φ φ
) (
2 1
− + =
(4)
One can use the process represented by equations (1)
through (4) at both the base-station receiver and at the
user receiver. At the base station the smoothed carrier-
phase pseudorange is differenced with the true range to
yield a pseudorange correction which, after removing the
common clock effect, is sent to the user receiver. At the
user receiver the correction is applied to the combined
frequency-weighted code measurement as represented by
equation (1). A wide-lane carrier-phase correction is also
generated at the base station by arbitrarily selecting a
whole-lane integer in equation (2) and differencing it with
the true range. The whole-lane error, which is made by
arbitrarily selecting the ambiguity value at the base station
is absorbed into the whole-lane ambiguity value at the
user receiver. After clock removal, this carrier-phase
correction is sent to the user receiver and applied to the
wide-lane carrier-phase measurement as represented by
equation (2). The significant advantage to this process, as
indicated above, is that, due to the separate smoothing, the
multipath corruption of the base station measurements is
usually largely removed even before the user receiver is
turned on. This means that the initial smoothed code
measurements at the user receiver are more accurate and a
more accurate code solution will be obtained.
At the user receiver, the corrected and smoothed value of
equation (4) for each satellite is now used in a least-
squares process to obtain an initial code solution.
WIDE-LANE SEARCH
While the process described above, i.e. equation (4), gave
us a set of floating (non-integer) ambiguity values for
each satellite, a better set (fewer degrees of freedom) can
be obtained from the initial code solution. Also we note
that given a set of integer ambiguities, a second set,
differing from the first by a constant integer value for
each satellite, will result in the same identical spatial
position but with a different clock value. Because our
clock solution is of no interest and can be selected
arbitrarily, this means that we can constrain one of the
ambiguity values arbitrarily. In our process, we choose to
use the highest elevation satellite as the reference satellite
and force its ambiguity value to zero. Using the initial
code position solution, we can compute a range, R, to
each satellite and set it equal to the measurement
represented by equation (2). Solving the result for the
ambiguity value and subtracting the ambiguity value of
the highest elevation satellite gives: