Global Positioning System Reference
In-Depth Information
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each epoch, a relatively inexpensive quartz crystal clock in the receiver is sufficient
rather than an expensive atomic clock. The basic requirement, however, is that there
are four satellites visible at a given epoch. This visibility requirement is a key factor
in the design of the GPS-type of constellation that assures global coverage is available
at any time.
Modifications of the basic point positioning solution can be readily envisioned.
For example, for applications on the ocean it might be possible to determine the
ellipsoidal height accurately from the height above water and the geoid undulation.
Equations (7.42) can be expressed in terms of ellipsoidal latitude, longitude, and
height using transformations (2.66) through (2.68) and the ellipsoidal height can be
considered a known quantity. Therefore, at least in principle, pseudoranges of three
satellites are sufficient to determine positions at sea. Other variations are possible.
Point positioning accuracy depends on the accuracy of the navigation message
and the satellite constellation used. In practice, one prefers to observe not just four
satellites but all satellites in view in order to achieve redundancy and better geometry.
See the discussion on dilution of precision (DOP) below. The achievable accuracy is
therefore subject to change as receiver technology keeps improving and the broadcast
ephemeris gets more accurate. The modernization of GPS will, of course, have a
major positive impact. Dual-frequency users can use the ionospheric-free function in
Equation (7.42) and, therefore, eliminate the effect of the ionosphere. While single-
frequency users can use the C/A-code pseudoranges, they unfortunately depend on
the ionospheric model to reduce the impact of the ionosphere on the solutions.
[25
Lin
—
-
——
Lon
PgE
7.4.1 Linearized Solution and DOPs
[25
It
has become common practice to use DOP factors to describe the effect of receiver-
sa
tellite geometry on the accuracy of point positioning. The DOP factors are simple
fu
nctions of the diagonal elements of the covariance matrix of the adjusted parame-
ter
s, derived from the linearized model. In general,
σ = σ
0
DOP
(7.43)
where
is a
one-number representation of the standard deviation of position and/or time. When
computing DOPs, the pseudorange observations are considered uncorrelated and of
the same accuracy; i.e., the weight matrix is
P
σ
0
denotes the standard deviation of the observed pseudoranges, and
σ
=
I
. If the ordered set of parameters is
x
T
=
[
dx
k
dy
k
dz
k
d
t
k
]
(7.44)
th
en the design matrix follows from (7.42) after linearization around the nominal
sta
tion location
x
k,
0
,
e
k
c
e
k
c
A
=
(7.45)
e
k
c
.
.
