21-2 Synthesis of a datum problem 463
A real case study is the following. For the analytic part, i. e. the determination of local to global
transformation parameters, we have chosen 60 points from the German and European GPS reference
network (DREF 91, EUREF), where global coordinates {X, Y, Z} given with respect to the GRS 80
datum have been determined. The same points are equipped with official local coordinates {x, y, z}
in the Gauss–Krueger system given with respect to the Bessel ellipsoid and Potsdam datum. In the
above Figs. 21.2 and 21.3, we have listed these data being transformed into ellipsoidal coordinates
{L, B, H} and {l, b, h}, respectively. (Ellipsoidal longitude l and ellipsoidal latitude b of local type
(“Rauenberg”, Bessel ellipsoid-of-revolution, a 1 = 6377397.155 m, e
2 = 0.006674372) versus ellipsoidal longitude L and ellipsoidal latitude B, ellipsoidal height H of global type (DREF ITRF 91,
GRS 80, A 1 = 6378137 m, E
2 = 0.0066943800229) of German 1st order stations.) No heights are
available in the local system. Column 10 shows the impact of the incremental semi-major axis δA
and squared first eccentricity δE
2 on the pseudo-latitudes in the partial least squares process. The
estimated transformation parameters according to the partial least squares process as described before
are presented in Table 21.1.
Table 21.1. Analysis of datum parameters of type translation, rotation, and scale. This analysis is based
upon the pseudo-observations presented in Figs. 21.2 and 21.3.
t x = −610.144 m ,
t y = −21.658 m ,
t z = −421.401 m ,
α = −1
.0396
β = −0
.1859
γ = −1
.2712
s = −0.519 485 × 10
−6
p
y − Aˆ x 2 /(n − 7) (longitude) = 0
.03117710 ,
p
y − Aˆ x 2 /(n − 7) (latitude) = 0
.02640297
A =
=
⎡
⎣
−
sin L
(N +H) cos B
cos L
(N +H) cos B
0
[(1−E
2 )N +H] sin B cos L
(N +H) cos B
[(1−E
2 )N +H] sin B sin L
(N +H) cos B
−
sin B cos L
M +H
−
sin B sin L
M +H
cos B
M +H
[N
2 (E
2 −1)−MH] sin L
M (M +H)
[N
2 (1−E
2 )+MH] cos L
M (M +H)
· · ·
· · ·
−1
0
0
0
0 −
E
2 N sin B cos B
M +H
−
E
2 N sin B cos B
A 1 (M +H)
−
[E
2 M sin
2 B+2(1−E
2 )N ] sin B cos B
2(M +H)(1−E 2 )
⎤
⎦
∈ R
2×9 .
(21.30)
21-2 Synthesis of a datum problem
Datum transformation, datum parameters (translation, rotation, scale). Synthesis matrix, local heights.
Jacobi matrix, chain Jacobi matrix.
Synthesis is understood as the determination of curvilinear global coordinates of points from curvilinear
local coordinates based upon given datum parameters. Since the datum parameters are close to the
identity, Box 21.6 collects the formulae for the computation of ellipsoidal longitude L and ellipsoidal
latitude B sought for in the global reference system from known ellipsoidal longitude l and ellipsoidal
latitude b and given datum parameters {t x t y , t z , α, β, γ, s, δa, δe
2
} by a Taylor series expansion. In
particular, Box 21.7 highlights the computation of the synthesis matrix B. Since in the local reference
system pseudo-observations of ellipsoidal heights h(l, b) are not available, in general, we here study
by Box 21.8 the impact of local heights on the synthesis matrix B, namely by the decomposition
B = B 0 + hB 1 .
A real case study is the following. For the analytic part, i. e. the determination of local to global
transformation parameters, we have chosen 60 points from the German and European GPS reference
network (DREF 91, EUREF), where global coordinates {X, Y, Z} given with respect to the GRS 80
datum have been determined. The same points are equipped with official local coordinates {x, y, z}
in the Gauss–Krueger system given with respect to the Bessel ellipsoid and Potsdam datum. In the
above Figs. 21.2 and 21.3, we have listed these data being transformed into ellipsoidal coordinates
{L, B, H} and {l, b, h}, respectively. (Ellipsoidal longitude l and ellipsoidal latitude b of local type
(“Rauenberg”, Bessel ellipsoid-of-revolution, a 1 = 6377397.155 m, e
2 = 0.006674372) versus ellipsoidal longitude L and ellipsoidal latitude B, ellipsoidal height H of global type (DREF ITRF 91,
GRS 80, A 1 = 6378137 m, E
2 = 0.0066943800229) of German 1st order stations.) No heights are
available in the local system. Column 10 shows the impact of the incremental semi-major axis δA
and squared first eccentricity δE
2 on the pseudo-latitudes in the partial least squares process. The
estimated transformation parameters according to the partial least squares process as described before
are presented in Table 21.1.
Table 21.1. Analysis of datum parameters of type translation, rotation, and scale. This analysis is based
upon the pseudo-observations presented in Figs. 21.2 and 21.3.
t x = −610.144 m ,
t y = −21.658 m ,
t z = −421.401 m ,
α = −1
.0396
β = −0
.1859
γ = −1
.2712
s = −0.519 485 × 10
−6
p
y − Aˆ x 2 /(n − 7) (longitude) = 0
.03117710 ,
p
y − Aˆ x 2 /(n − 7) (latitude) = 0
.02640297
A =
=
⎡
⎣
−
sin L
(N +H) cos B
cos L
(N +H) cos B
0
[(1−E
2 )N +H] sin B cos L
(N +H) cos B
[(1−E
2 )N +H] sin B sin L
(N +H) cos B
−
sin B cos L
M +H
−
sin B sin L
M +H
cos B
M +H
[N
2 (E
2 −1)−MH] sin L
M (M +H)
[N
2 (1−E
2 )+MH] cos L
M (M +H)
· · ·
· · ·
−1
0
0
0
0 −
E
2 N sin B cos B
M +H
−
E
2 N sin B cos B
A 1 (M +H)
−
[E
2 M sin
2 B+2(1−E
2 )N ] sin B cos B
2(M +H)(1−E 2 )
⎤
⎦
∈ R
2×9 .
(21.30)
21-2 Synthesis of a datum problem
Datum transformation, datum parameters (translation, rotation, scale). Synthesis matrix, local heights.
Jacobi matrix, chain Jacobi matrix.
Synthesis is understood as the determination of curvilinear global coordinates of points from curvilinear
local coordinates based upon given datum parameters. Since the datum parameters are close to the
identity, Box 21.6 collects the formulae for the computation of ellipsoidal longitude L and ellipsoidal
latitude B sought for in the global reference system from known ellipsoidal longitude l and ellipsoidal
latitude b and given datum parameters {t x t y , t z , α, β, γ, s, δa, δe
2
} by a Taylor series expansion. In
particular, Box 21.7 highlights the computation of the synthesis matrix B. Since in the local reference
system pseudo-observations of ellipsoidal heights h(l, b) are not available, in general, we here study
by Box 21.8 the impact of local heights on the synthesis matrix B, namely by the decomposition
B = B 0 + hB 1 .