THE SIMILARITY TRANSFORMATION MODEL |
In the geodetic context, the general transformation model in eqn
(11.1-1) is often referred to as the Bursa-Wolf model.
When this model is invoked for small networks, the rotation parameters are
highly correlated with the translation parameters. (The reader can convince
themselves of this by considering, for example, a rotation about the Z-axis
of a network on the Greenwich meridian; the effect is almost indistinguishable
from a translation of the network along the Y-axis.) An alternative formulation
that avoids this correlation "problem" is the Molodensky-Badekas
model (HARVEY, 1986):
 |
(11.1-2) |
where Xm=
XAi/n, Ym=YAi/n,
Zm=ZAi/n are the coordinates of the centroid
of the network. Alternatively Xm, Ym, Zm
may be selected to be the coordinates of one of the points in network A.
Although the translation parameters are different, the rotation matrix and
the scale factor are the same as for the Bursa-Wolf model.
Rotation Matrices
The rotation matrices about the X-, Y-, and Z-axes are:
 |
The most common combined rotation matrix is: R = Rz(
).Ry(
).Rx(
),
leading to:
 |
For small rotations this matrix may be approximated by:
 |
(11.1-5) |
where , , and are the rotation angles in radians about the X-, Y-,
and Z-axes respectively. The small angles assumption is usually valid for
rotation angles up to 10". The rotation angles depend on the baseline
vectors (that is, the relative positions) and not on the absolute coordinates.
Thus it does not matter where the origin of coordinates is because the estimated
rotation angles will be the same. For this reason the same rotation matrix
is used in the Bursa-Wolf model as in the Molodensky-Badekas model.
Scale factor
A scale factor can be visualised as follows. Imagine a network drawn on the surface of an inflatable sphere. As the sphere is inflated, the points of the network spread apart from each other, and from the centre of the sphere (Figure in section 11.1.3). The inflation of the sphere is equivalent to the application of a scale factor greater than unity.
Multiplication of a set of rectangular Cartesian coordinates by a scale factor is identical to multiplying the corresponding baseline lengths by the same scale factor. Hence the scale factor can be determined from either the 3-D site coordinates or from the baseline lengths. Thus, as with the rotation angles, the origin of the coordinates has no affect on the results.
In the case of ellipsoidal coordinates the longitude is not affected by a scale change but the geodetic latitude does change slightly. For example, a 1ppm scale change will change geodetic latitudes by less than about 0.0007" (2cm). However, the effect on ellipsoidal height is significant. For example, a 1ppm scale change will produce a change in height of about 6.4 metres.
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© Chris Rizos, SNAP-UNSW, 1999
