Geography Reference
In-Depth Information
Fig. 21.1.
Gauss surface normal coordinates, curvilinear geodetic datum transformations, “global”
{
L,B,H
}
versus “local”
{
l,b,h
}
{
L, B, H
}
into Cartesian coordinates
{
x, y, z
}
and
{
X,Y,Z
}
and vice versa, namely by means of
(
21.13
)-(
21.17
). In addition, we introduce the curvature radii
{
n
(
b
)
,m
(
b
)
}
and
{
N
(
B
)
,M
(
B
)
}
of
a
1
,a
2
A
1
,A
2
. While the direct transformation (
21.11
) is given in a closed form, the inverse
transformation (
21.12
) (following
Grafarend and Lohse 1991
) is a complicated closed form, in
particular, for
b
(
x, y, z
)
,B
(
X,Y,Z
)and
h
(
x, y, z
)
,H
(
X,Y,Z
). The height functions
h
(
l,b
)and
H
(
L, B
) have been computed with respect to a set of functions which are orthonormal with respect
to an ellipsoid-of-revolution by
Grafarend
(
1992a
). Note that one set of ellipsoidal coordinates
{
E
and
E
does not cover the Earth's surface completely, namely due to the pole
singularity of these coordinates (Fig.
21.1
).
l,b,h
}
and,
{
L, B, H
}
{
l,b,h
} →{
x, y, z
}
,
{
L, B, H
} →{
X,Y,Z
}
,
(21.11)
{
x, y, z
} →{
l,b,h
}
,
{
X,Y,Z
} →{
L, B, H
}
.
(21.12)
The main idea of the applied datum transformation of curvilinear coordinates as being outlined
in Box
21.3
is the following. In the global coordinate system, ellipsoidal coordinates
{L, B, H}
are available, for example, from a survey by means of the Global Positioning System (GPS,
GLONASS, PRARE), but in the local coordinate system, only ellipsoidal longitude, ellipsoidal
latitude are accessible. It has to be emphasized that, due to the older separation of “horizontal
control” and “vertical control”, the ellipsoidal height
h
(
l, b
) in the local coordinate system is not
available. It is for this reason that, by means of (
21.18
)and(
21.19
), we have only formulated the
curvilinear datum transformation close to the identity for ellipsoidal longitude, ellipsoidal latitude
{
,
respectively. A closed form expression was derived only for ellipsoidal longitude
l
.TheTaylorseries
expansion up to second order in terms of datum parameters of type translation, rotation, and
scale as well as of variation of semi-major axis
δA
and of squared first eccentricity
δE
2
is outlined
by (
21.20
). The detailed results of the linearization are collected in Box
21.4
. In particular, we
end up with the linear system of first order
y
=A
x
+ hit, where hit means “higher order terms”,
introducing
l
l, b
}
in the local reference system as a function of
{
L, B, H
}
and
{
t
x
,t
y
,t
z
,α,β,γ,s
}
t
x
,t
y
,t
z
,α,β,γ,s,δA,δE
2
as the
unknown vector
x
. The Jacobi matrix A is rigorously computed by the chain rule J
23
J
1
,where
the Jacobi matrix J
1
contains the derivatives of Cartesian coordinates with respect to the datum
−
L, b
−
B
as the given vector
y
and
{
}
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