Global Positioning System Reference
In-Depth Information
Universal Transverse Mercator Grid System (UTM). The reference ellipsoid is
the International Ellipsoid of 1924. The mapping is defined for the whole Earth.
It was a desire to limit the variation of the scale
m
so the total mapping of the
Earth is divided into 60 sections that are named zones. Each zone covers 6
◦
in
longitude, and they are numbered from 1 to 60. Number 1 covers 180
◦
-174
◦
We s t ,
number 2 covers 174
◦
-168
◦
West, and so on. The scale at the central meridian is
m
0
= 0.9996. The UTM system is limited by the parallels 84
◦
North and 80
◦
South. The polar regions are mapped stereographically.
The coordinates are called
northing N
and
easting E
. Each zone has its own
coordinate system. The central meridian has a false easting (FE) of 500,000 m and
the equator has a false northing (FN) of 0 m for points on the northern hemisphere
and a false northing of 10,000,000 m for points on the southern hemisphere. This
simple arrangement leaves all coordinates positive.
The transformation of geographical coordinates (
ϕ,λ
) into UTM coordinates
(
N
E
) and reversely appears often in practice. We point to the
M
-files
geo2utm
and
utm2geo
:
[N, E] = geo2utm(phi, lambda, zone)
[phi, lambda] = utm2geo(N, E, zone)
,
8.9
Dilution of Precision
The covariance matrix
x
as described in (8.25) contains information about the
geometric quality of the position determination. It is smaller (and
x
is more accu-
ˆ
rate) when the satellites are well spaced.
The covariance matrix
x
is a 3 by 3 matrix in case we estimate
(
X
,
Y
,
Z
)
and
a 4 by 4 matrix in case we estimate
. It is positive definite, so its
inverse exists and is likewise positive definite. We introduce a local coordinate
system with origin at
(
X
,
Y
,
Z
,
cdt
)
(
X
,
Y
,
Z
)
and with axes parallel to the original ones. In
T
the local system, let the point
x
=
(
x
,
y
,
z
)
lie on a surface described by the
quadratic form
x
T
−
1
ˆ
c
2
x
=
(8.39)
x
−
1
ˆ
or equivalently, if
is diagonal,
x
x
2
c
2
y
2
c
2
z
2
c
2
x
+
y
+
z
=
1
.
(8.40)
σ
σ
σ
−
1
ˆ
If
is nondiagonal, it can be brought on diagonal form in a rotated coordinate
x
system.
The surface is an ellipsoid because
−
1
ˆ
is positive definite. It is the
confidence
x
ellipsoid
of the point. Let 1
−
α
be the probability that the correct position falls
95,
c
2
is given as
2
3
within the ellipsoid. With 1
81. In MATLAB
this number is computed as
chi2inv(0.95,3)
. The magnification factor
c
of the
confidence ellipsoid is
c
−
α
=
0
.
χ
−
α
=
7
.
,
1
=
√
7
.
81
=
2
.
80.
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