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squares, for example,
Young
(
1994
), (ii) ridge type regression (Tychonov regularization), for exam-
ple,
Saleh and Kibria
(
1993
), and (iii) truncated or total least squares, for example,
Fierro and
Bunch
(
1994
). Indeed, in the analysis of the curvilinear datum transformation Jacobi matrix for
regional geodetic networks, we identified a spectral condition number (the ratio of the largest and
smallest eigenvalue) of the order of 10
9
. For some reasons, we have accordingly chosen partial
least squares in analyzing the datum parameters from horizontal control, exclusively. Our results
are presented in Sect.
21-1
.
Section 21-2.
In contrast, Sect.
21-2
is devoted to the synthesis of ellipsoidal longitude and ellipsoidal latitude
known in the local reference system and to be transformed into the global reference system or,
equivalently into Gauss-Krueger coordinates or UTM coordinates from local to global reference.
A real data example is given.
Section 21-3.
Finally, Sect.
21-3
three reviews the error propagation in the analysis and synthesis of a curvilinear
datum transformation.
21-1 Analysis of a Datum Problem
Datum transformation, datum parameters (translation, rotation, scale). General conformal
group, special orthogonal group. Jacobi matrix, chain Jacobi matrix.
Analysis
is understood as the determination of
datum parameters
between two sets of curvilinear
coordinates of identical points which cover
R
3
equipped with an Euclidean metric. The datum
parameters like translation
t
, rotation R, and scale 1 +
s
characterize a
datum transformation
(“Kartenwechsel”) which leaves (as a passive transformation) angles and distance ratios equiv-
ariant. In its linear variant, they are the parameters of the conformal group C
7
(3), the seven-
parameter transformation in
3
. The general conformal group C
10
(3), in contrast, includes three
parameters more, as outlined in
Grafarend and Kampmann
(
1996
) , for instance. As soon as we
cover
R
3
by Cartesian coordinates (say
in
the global reference system), we arrive at the forward transformation (
21.1
) of datum type versus
the backward transformation (
21.3
) of datum type. Actually, (
21.1
)and(
21.3
) are datum trans-
formations of
R
{
x, y, z
}
in the local reference system versus
{
X,Y,Z
}
4
covered by homogeneous coordinates. All datum transformations are written
R
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