Global Positioning System Reference
In-Depth Information
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2.3.5.2 Reparameterization
Often the geodetic latitude, longitude, and height
are preferred as parameters instead of the Cartesian components of (x). One reason
for such a reparameterization is that humans can visualize changes more readily in
latitude, longitude, and height than changes in geocentric coordinates. The required
transformation is given by (B.16).
−
+
λ
−
+
λ
λ
(M
h)
cos
sin
ϕ
(N
h)
cos
ϕ
sin
cos
ϕ
cos
dϕ
d
d
x
=
−
+
λ
+
λ
λ
dh
(M
h)
sin
sin
ϕ(N
h)
cos
ϕ
cos
cos
ϕ
sin
(M
+
h)
cos
ϕ
0
sin
ϕ
dϕ
d
=
J
dh
(2.106)
[48
Th
e expressions for the radius of curvatures
M
and
N
are given in (B.7) and (B.6).
The matrix
J
must be evaluated for the geodetic latitude and longitude of the point
under consideration; thus,
J
1
(ϕ
1
,
Lin
—
6.1
——
Sho
PgE
λ
2
,h
2
)
denote the transformation
matrices for points
P
1
and
P
2
, respectively. Substituting (2.106) into (2.105), we
obtain the parameterization in terms of geodetic latitude, longitude, and height:
λ
1
,h
1
)
and
J
2
(ϕ
2
,
dϕ
1
d
λ
1
dh
1
···
dϕ
2
d
d
α
1
[48
=
d
β
1
ds
[
G
1
J
1
:
G
2
J
2
]
(2.107)
λ
2
dh
2
To achieve a parameterization that is even easier to interpret, we transform the
differential changes in geodetic latitude and longitude parameters
(dϕ, d
)
into cor-
responding changes
(dn, de)
in the local geodetic horizon. Keeping the geometric
interpretation of the radii of curvatures
M
and
N
as detailed in Appendix B one can
fu
rther deduce that
λ
=
M
+
h
0
0
dϕ
d
dϕ
d
d
w
=
0
(N
+
h)
cos
ϕ
0
dh
H
(ϕ, h)
dh
(2.108)
0
0
1
[
dn de du
]
T
intuitively related to the “horizontal” and
“vertical” and because the units are in length, the standard deviations of the param-
eters can be readily visualized. The matrix
H
is evaluated for the station under con-
=
The components
d
w
