Bernese GPS Software Version 5.0 - Bernese GNSS Software

10.6 Coordinate and Velocity Related Auxiliary Programs
The program computes Helmert transformation parameters between the coordinates in the
second input file to those in the first (reference) file. The coordinates of the reference file
may be propagated to the epoch of the second file by specifying an input velocity file. Note,
that only stations with flags in both coordinate files are used for the comparison because
flags indicate coordinates that are the result of an analysis run. Coordinate sets without
flags are ignored.
The program may run in two different modes that can be selected in option “System of
transformation” in panel “HELMR1 2: Options for Helmert Transformation”:
• Transformation in geocentric system
If coordinates from large regional or global networks are compared the option should
be set to XYZ. In this case the estimated transformation parameters refer to the
geocentric cartesian coordinate system:
X ′ = (1 + κ) · Rz(rz) · Ry(ry) · Rx(rx) · (X + X0) (10.1)
where X0 is the translation vector, Ri(ri) the matrix of a rotation by an angle ri
around the axis i, and κ the scaling parameter. The transformation parameters are
estimated by minimizing the sum of squared differences between the coordinates from
the second input file, transformed according to Eqn. (10.1), and the coordinates in
the reference file.
If no translation parameters are estimated the two sets of coordinates are first reduced
to their respective barycenter in order to allow a proper interpretation of the estimated
rotation parameters.
• Transformation in local system
If the option is set to NEU the estimated transformation parameters refer to the
local horizon system in the barycenter of the second coordinate set. This option is
particularly useful for local networks. The following algorithm is used:
(1) The barycenter of the second coordinate set is computed according to
¯X = 1
n ·
n
xi
i=1
where xi are the geocentric coordinates of the n stations involved.
(2) With the help of the ellipsoidal coordinates ¯ φ and ¯ λ of the barycenter ¯ X both
coordinate sets are transformed to the local horizon system using
Xlocal =
1 0 0
0 −1 0
0 0 1
· Ry( ¯ φ − 90 ◦ ) · Rz( ¯ λ − 180 ◦ ) · (X − ¯ X)
(3) Finally the Helmert transformation parameters in the equation
X ′ local = (1 + κ) · Rz(rz) · Ry(ry) · Rx(rx) · (Xlocal + X0) (10.2)
are estimated by minimizing the square sum of the difference between the coordinates
of the second set, transformed according to (10.2), and the reference
coordinate set.
Bernese GPS Software Version 5.0 Page 233