Global Positioning System Reference
In-Depth Information
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γ
f
are given by (5.14) and (5.15). The geometry of the
epoch solution is implicit in the covariance matrix. For computing the epoch co-
variance matrix, we assume that the observations
The coefficients
β
f
and
b
are not correlated. We further
assume that th
e s
tandard deviation of the carrier phases
σ
1
,ϕ
and
σ
2
,ϕ
are related as,
σ
2
,
Φ
= σ
1
,
Φ
√
α
f
, and that the standard deviations of the pseudorange and the linear
carrier phases follow
k
= σ
P
/
σ
Φ
for both frequencies. With these assumptions the
Σ
b
consists of diagonal elements
k
2
,
α
f
k
2
,1,
α
σ
2
Φ
covariance matrix
.
Applying the law of variance-covariance propagation (4.34), the covariance matrix
of the parameters becomes
f
, and a scalar
Σ
b
A
−
1
T
A
−
1
Σ
x
=
(7.24)
If we set
k
equal to 154, which corresponds to the ratio of the L1 frequency and the
P-code chipping rate, and use
[24
σ
1
,
Φ
=
0
.
002 m, then the standard deviations and the
correlation matrix are
σ
ρ+∆
,
σ
2
,N
=
Lin
—
3.0
——
No
PgE
σ
I
,
σ
1
,N
,
(
0.99 m, 0.77 m, 9.22 cycL
1
, 9.22 cycL
2
)
(7.25)
1
−
0
.
9697
−
0
.
9942
−
0
.
9904
1
0
.
9904
0
.
9942
C
x
=
(7.26)
1
0
.
9995
1
[24
The standard deviations for the integer ambiguities have been converted to cycles
in (7.25). Striking features of the epoch solution are the equality of the standard
deviation for both ambiguities and the high correlation between all parameters. Of
particular interest is the shape and orientation of the ellipse of standard deviation for
the ambiguities. The general expressions (4.304) to (4.308) can be applied to the third
and fourth parameters. The ellipse can be drawn with respect to the perpendicular
N
1
and
N
2
axes, which carry the units L1 cycles and L2 cycles. The computations show
that the ellipse almost degenerates into a straight line with an azimuth of 45°, the
semiminor and semimajor axes being 0.20 and 13.04, respectively.
A standard procedure for breaking high correlation is reparameterization by means
of an appropriate transformation. For example, consider the transformation
z
=
Zx
(7.27)
100 0
010 0
001
Z
=
(7.28)
1
001 0
−
