Geography Reference
In-Depth Information
The system (
21.23
) of linear equations
y
=A
x
+ hit is
characterized by two pseudo-observed ellipsoidal longitude
and ellipsoidal latitude differences. In case we have access
to the ellipsoidal form parameter variation
δA
and
δE
2
,
we are left with two equations for seven datum param-
eters per station point. Obviously, in order to determine
the seven datum parameters, we need at least four station
points with the data
{l,b}
and
{L, B, H}
available. But,
in general, we are left with an adjustment problem when
the data are accessible at four or more station points. Since
ellipsoidal longitude and ellipsoidal latitude
{l,b}
are ele-
ments of
E
a
1
,a
2
, the distance between the adjusted points
and the given data points
{l,b}
has to be minimized. The
distance along a geodesic as outlined in Box
21.5
originates
from a series expansion of the minimal geodesic on
.A
zero order approximation is the Euclidean distance known
as the method of least squares. (For a review of robust dis-
tance functions for those pseudo-observations given on a
circle
E
a
1
,a
2
1
2
r
of radius
r
, we refer to
Chan and
He 1993
). Here, we have chosen the zero order approxima-
tion, the method of least squares
S
r
or on a sphere
S
2
= min leading
to
x
=(A
T
PA)
−
1
A
T
P
y
as the best approximation of the
datum parameters.
y
−
A
x
In the first step of the partial least squares solution, we have given the prior information of
the rotation parameters as well as the scale parameter a large weight. Accordingly, we have
solved for the translation parameters
x
1
;=[
t
x
,t
y
,t
z
] exclusively. The second step is split up into
a forward and a backward one. First, we remove the data
x
1
(translation parameters) of best
approximation from the reduced pseudo-observed data, namely
y
a
9
δE
2
). Second, in
the 2nd partial least squares solution, we associate to the prior information of the scale parameter
a large weight. Indeed, we compute the rotation parameters
x
2
:= [
α,β,γ
] exclusively from
γ
.
The third step is split up again into a forward and a backward one. First, we remove the data
x
2
(rotation parameters) of best approximation from the reduced pseudo-observed data, namely
y
1
−
A
2
x
2
=:
y
2
. Second, in the 3rd partial least squares solution, we finally compute the scale
parameter
x
3
=
s
exclusively from
y
2
.
−
(
a
8
δA
−
a
1
,a
2
Box 21.5 (The distance between the points
{
l,b
}
and
{
l
0
,b
0
}
on
E
along a minimal
geodesic).
(
b
a
1
(1
− e
2
)
2
s
2
=
b
0
)
2
l
0
)
2
cos
2
b
0
+
−
e
2
sin
2
b
0
)
2
+(
l
−
e
2
sin
2
b
0
1
−
(1
−
e
2
)
2
cos
b
0
sin
b
0
e
2
)cos
b
0
sin
b
0
b
0
)
3
(1
−
l
0
)
2
(1
−
+3(
b
−
−
(
b
−
b
0
)(
l
−
+
e
2
sin
2
b
0
)
3
e
2
sin
2
b
0
(1
−
1
−
8sin
2
b
0
+
e
2
sin
2
b
0
(25
21 sin
2
b
0
)]
+(
b − b
0
)
4
(1
−
e
2
)
e
2
[4
−
−
−
(21.29)
e
2
sin
2
b
0
)
4
4(1
−
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