Global Positioning System Reference
In-Depth Information
drms
=
σσ
x
2
2
(7.61)
y
For a zero-mean random variable such as
d
R
, one has drms =
R
2
, and the
drms corresponds to the square root of the mean value of the squared error (hence,
its name). From (7.47), one immediately has
Ed
(|
|
)
drms
=
HDOP
⋅ σ
UERE
(7.62)
and the drms can be computed from the values of HDOP and
σ
UERE
. The probability
that the computed location is within a circle of radius drms from the true location
depends on the ratio
σ
S
/
σ
L
for the 1
ellipse. If the two-dimensional error distribu-
tion is close to being circular (
σ
S
/
σ
L
≈
1), the probability is about 0.63; for a very
elongated distribution (
σ
S
/
σ
L
≈
0), the probability approaches 0.69. Two times the
drms is given by
2
drms
=⋅
2
HDOP
⋅ σ
UERE
(7.63)
and the probability that the horizontal error is within a circle of radius 2 drms
ranges between 0.95 and 0.98, depending on the ratio
σ
L
. The 2-drms value is
commonly taken as the 95% limit for the magnitude of the horizontal error.
Another common metric for horizontal errors is
circular error probable
(CEP),
defined as the radius of a circle that contains 50% of the error distributions when
centered at the correct (i.e., error-free) location. Thus, the probability that the mag-
nitude of the error is less than the CEP is precisely one-half. The CEP for a
two-dimensional Gaussian random variable can be approximated by the formula
σ
S
/
(
)
CEP
≈
059
.
σσ
L
+
(7.64)
S
assuming it is zero mean. For a derivation of this and other approximations, see
[33].
The CEP can also be estimated in terms of drms and, using (7.62), in terms of
HDOP and
σ
UERE
. This is convenient since HDOP is widely computed in GPS appli-
cations. Figure 7.7 presents curves giving the probability that the magnitude of the
error satisfies |
d
R
|
σ
L
.
(The horizontal error is assumed to have a zero-mean two-dimensional Gaussian
distribution.) For
k
equal to 0.75, one obtains a probability in the range 0.43 to
0.54. Hence, one has the approximate relation
≤
k
drms as a function of
k
for different values of the ratio
S
/
CEP
≈
075
.
drms
=
075
.
⋅
HDOP
⋅
σ
UERE
(7.65)
It is interesting to note that for
k
=
1.23, the probability that |
d
R
|
≤
k
drms is
roughly 0.78, almost independent of
σ
L
. The probabilities associated with several
other values of
k
are summarized in Table 7.5.
As an application of these formulations, for an average global HDOP of 1.0 and
with
σ
S
/
1.4m, estimates for the CEP, the 80% point, and the 95% point for the
magnitude of the horizontal error are given as follows:
σ
UERE
=
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