Global Positioning System Reference
In-Depth Information
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The
A
matrix has as many rows as there are satellites observed, which typically
includes all satellites in view. The superscript
i
is a sequential identification for the
satellite and not necessarily equal to the PRN number. The symbol
e
k
denotes the
1
3 row vector defined in (5.48). It contains the direction cosines for the vector
from the nominal station location to satellite. The clock error parameter is often set
to
×
cd
t
k
, making the elements in the second column of (7.45) unity. The cofactor
matrix for the adjusted receiver position and receiver clock is
ξ
k
=
q
x
q
xy
q
xz
q
xt
Q
x
=
A
T
A
−
1
q
y
q
yz
q
yt
=
(7.46)
q
z
q
zt
sym
q
t
[25
Si
nce it is more convenient to interpret results in the local geodetic coordinate system
(w
) (consisting of the coordinates northing
n
, easting
e
, and up
u
), we transform the
co
factor matrix (7.46) using (2.112) and (2.113) of Section 2.3.5.3. The result is
Lin
—
0.2
——
Nor
PgE
q
n
q
ne
q
nu
Q
w
=
q
e
q
eu
(7.47)
sym
q
u
The DOP factors are functions of the diagonal elements of (7.46) or (7.47). Table
7.2 shows the various dilution factors: vertical dilution of precision (VDOP) for the
height, horizontal dilution of precision (HDOP) for horizontal positions, positional
dilution of precision (PDOP), time dilution of precision (TDOP) and geometric dilu-
tion of precision (GDOP). GDOP is a composite measure reflecting the geometry of
the position and time estimates.
The DOPs can be computed in advance, given the approximate receiver location
and a predicted satellite ephemeris. The DOPs are useful for finding the best subset
of satellites if a receiver has only four or five channels. Even though most receivers
today observe all satellites in view, the DOPs are still useful to identify a temporal
weakness in geometry in kinematic applications, in particular in the presence of signal
obstruction.
[25
TABLE 7.2 DOP Expressions
VDOP
=
√
q
u
HDOP
=
√
q
n
+
q
e
q
u
=
q
x
+
PDOP
=
√
q
n
+
q
e
+
q
y
+
q
z
TDOP
=
√
q
t
q
n
+
GDOP
=
q
e
+
q
u
+
q
t
c
2