Global Positioning System Reference
In-Depth Information
The slope is a geometric parameter that can be directly computed from the specific satellite-
user geometry, based on the following equations, in the horizontal and vertical planes
respectively:
2
2
KK
+
σ
1
i
2
i
i
(20)
Hslope
=
i
1
P
ii
K
σ
3
1
i
i
(21)
Vslope
=
i
P
ii
=⋅ ⋅ ⋅ ⋅ is the weighted pseudo-inverse of the design matrix, where
W the inverse of the covariance matrix, while PGK
T T
KGWG GW
1
where
(
)
= ⋅ . The geometric contribution to the
slope is given by the K and P matrices.
4.2 RAIM protection levels
The Protection Levels in the vertical and horizontal planes can be described by the following
equations (Walter & Enge, 1995), for the vertical and horizontal cases, respectively:
VPL
=
max{
V
}
T N P
(
,
)
+
k P σ
(
)
(22)
FD
slope
fa
md
V
(23)
HPL
=
max{
H
}
T N P
(
,
)
+
k P σ
(
)
FD
slope
fa
md
H
where:
V slope and H slope are the satellite error slope in the vertical and horizontal planes
T(N,P fa ) is the test statistic threshold, and it is a function of the number of satellites (N)
and the desired probability of false alarm (P fa ). Given the probability of false alarms, the
threshold can be found by inverting the incomplete gamma function:
1
2
T
−−
sa
1
−=
1
P
e
s
ds
(24)
fa
Γ
()
a
0
where a is the number of degrees of freedom divided by two, or, in terms of the number of
measurements N and unknowns M:
NM
(25)
a
=
2
k(P MD ) is the number of standard deviations corresponding to the specified Probability
of Missed Detection. The smaller the P MD value, the higher the number of standard
deviations should be considered, since longer tails for the Gaussian distribution should
be taken into account.
σ and σ are the standard deviations of the error in the position domain in the
vertical and horizontal planes.
It should be noted that, when using RAIM, it is common to allocate the whole Integrity Risk,
and so the whole Pmd is confined to only one plane (vertical or horizontal) according to the
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