Geoscience Reference
In-Depth Information
Recall that Eq. (5-49) is now a system of linear equations for point
i
.For
n
common points, the design matrix
A
is
⎡
⎣
⎤
⎦
A
1
A
2
.
A
n
A
=
.
(5-51)
In detail, for three common points the design matrix is
⎡
⎤
100
X
1
0
−
Z
1
Y
1
⎣
⎦
010
Y
1
Z
1
0
−
X
1
001
Z
1
−Y
1
X
1
0
100
X
2
−
Z
2
Y
2
0
A
=
010
Y
2
Z
2
0
−
X
2
,
(5-52)
001
Z
2
−
Y
2
X
2
0
100
X
3
0
−
Z
3
Y
3
010
Y
3
Z
3
0
−X
3
001
Z
3
−
Y
3
X
3
0
which leads to a slightly redundant system. Least-squares adjustment yields
the parameter vector
δ
p
and the adjusted values by (5-45), (5-46), (5-47).
Once the seven parameters of the similarity transformation are determined,
formula (5-41) can be used to transform other than the common points.
For a specific example, consider the task of transforming GPS coordi-
nates of a network, i.e., global geocentric WGS 84 coordinates, to (three-
dimensional) coordinates of a (nongeocentric) local system indicated by the
subscript LS. The GPS coordinates are denoted by (
X, Y, Z
)
GPS
and the lo-
cal system coordinates are the plane coordinates (
y, x
)
LS
and the ellipsoidal
height
h
LS
. To obtain the transformation parameters, it is assumed that the
coordinates of the common points in both systems are available. The solution
of the task is obtained by the following algorithm:
1. Transform the plane coordinates (
y, x
)
LS
of the common points into
the ellipsoidal surface coordinates (
ϕ, λ
)
LS
by using the appropriate
mapping formulas.
2. Transform the ellipsoidal coordinates (
ϕ, λ, h
)
LS
of the common points
into the Cartesian coordinates (
X, Y, Z
)
LS
by (5-27).
3. Determine the seven parameters of a Helmert transformation by using
the coordinates (
X, Y, Z
)
GPS
and (
X, Y, Z
)
LS
of the common points.
