Global Positioning System Reference
In-Depth Information
of sorting out the integer ambiguities is the next area of interest. To do so requires
that the carrier-phase DD measurement be examined in light of its constituent parts.
The following equation so illustrates:
(
)
$
DD
=
φ
+
n
+
R
+
S
λ
(8.31)
cp
DD
b
n
is the
unknown DD ambiguity,
R
b
is the inherent receiver channel bias plus residual prop-
agation delays,
S
is the noise due to all sources (e.g., receiver, multipath), and the use
of
where
φ
DD
is the DD fractional phase from the receiver measurements,
$
converts the DD to units of length. Strictly speaking, multipath is not noise. It
does, however, add a noise-like uncertainty to the DD measurement, which, unfor-
tunately, cannot be uniquely separated at a given instant in time from other noise
sources. To solve this dilemma, multipath is simply included with the noise.
Equation (8.31) can be reexpressed using the smoothed-code DDs and with the
knowledge that the uncertainty in the sources on the right-hand side of the equation
is bounded. The terms
λ
ρ
DD
and
n
. This follows from the
knowledge that the smoothed-code DDs are accurate to within 1-2m, their inherent
noise level. This noise level is equivalent to
φ
DD
and
n
are replaced with
$
±
11 wavelengths at L1 and allows the
n
integer ambiguity to be bounded; hence
ρ
DD
, then, repre-
sents the geometric distance (in carrier-cycles) of the smoothed-code DD within the
noise bound. The equation now appears as follows:
−≤≤+
11
11
. The term
~
(
)
DD
=
ρ
+
n
+
R
+
S
λ
(8.32)
s
DD
b
The resolution of
n
can now be attacked using the residuals from the least-
squares solution developed as (8.28). This equation is expanded and shown next:
(
~
(
~
)
)
[
qDD
s
=
q
ρ
++++
nRS q
ρ
+++
nRS
1
DD
1
b
1
2
DD
2
b
2
(8.33)
1
1
2
1
)
]
(
~
)
(
~
+
q
ρ
+
n
+++
RS q
ρ
+++
nRS
λ
=
η
3
DD
3
b
3
4
DD
4
b
4
3
3
4
4
where the
q
r
are the elements of the least-squares residual vector and
n
r
represents
a wavelength ambiguity number associated with the applicable DD. Ideally, the
value of , the measurement inconsistency, would be zero, but this could only be
true in the presence of noiseless measurements and resolved carrier-cycle integer
ambiguities.
In any particular epoch, the values for
q
remain constant—the residual of the
least squares solution does not change until another set of measurements is taken,
the DDs are computed and smoothed, and the QR factorization is completed. In
modern receivers, great effort is expended to minimize interchannel biases; the same
holds true for receiver noise. This leaves multipath as the major component of noise.
Fortunately, code multipath, over time, behaves in a noise-like fashion, although
not necessarily tending to a zero mean [34]. It is worthwhile, then, to consider
(8.33) with emphasis on the component that is constant from epoch to epoch,
knowing that the other sources of error will be mostly random or small over an
extended period of time. This component is the unknown carrier-cycle integer
ambiguity in each of the smoothed-code DDs. If the ambiguity can be removed from
the DD, then the only remaining error sources are noiselike and will approach zero
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