4.3 The Burša-Wolf type datum parameters

The Burša-Wolf type parametrization method (called after the Czech Milan Burša and the German Helmut Wolf) handles not only the difference of the positions of the geometric centers of the datum ellipsoids, but also the orientation differences and the small scale variations as one or both datum’s size differs indeed from the ideal size of the selected ellipsoid (Fig. 18). The transformation is expressed for the geocentric Cartesian coordinates as input and out data, as follows:

(4.3.1)

This is a special case of the spatial Helmert similarity transformation for very small rotation angles (a few or a few tens of arc seconds), with the possible simplifications. In this equation, the dX, dY and dZ are the same as in case of the abridging Molodensky transformations (but, as it shown below, cannot be handle in that without analysis!), εx, εy and εz are the rotations along the coordinate axes and k is the scale factor. If there are no rotations and the scale difference is zero, the Equation (4.3.1) becomes the same to Equation (4.2.1).

Shift-rotate an ellipsoid to a new position and orientation

Fig. 18. The Bursa-Wolf transformation handles both the shift and the orientation differences between the two datum ellipsoids.

In fact, there are two different sign convention of the non-diagonal matrix members in Equation (4.3.1). If these signs are used like in the Equation (4.3.1), it is called coordinate frame rotation, which means that the coordinate axes are rotated around the fixed position vector. However, if all the signs of the non-diagonal members in the matrix of (4.3.1) are reversed, this convention is called position vector rotation, as this vector is rotated in the fixed coordinated frame.

Neither of the above conventions is an accepted standard. The United States, Canada and Australia use the ‘coordinate frame rotation’, while in Europe the ‘position vector rotation’ is mostly preferred. The international draft ISO19990 also proposes this latter one, however because of the U.S. refusal its international acceptation is questioned. We have to know that as most GIS software packages are developed in the U.S., Canada and Australia, the ‘coordinate frame rotation’ is a quasi-standard in them, while most European meta-data are published according to the ‘position vector rotation’ convention. If we are provided a Burša-Wolf type parameter set for a datum, first try to use it assuming the ‘coordinate frame rotation’, and if the results are obviously erroneous, switch all the signs of the rotation parameters.

Similarly to the abridging Molodensky transformation, the Burša-Wolf formula is commutative. It is possible to express the resultant of two transformations by simply summarize their respective parameters. This perhaps surprising statement can be easily understood mathematically:

The Equation (4.3.1) after two, successive transformation can be expressed in form

x’=dx2+(1+k2)A2[dx1+(1+k1)A1x]

(4.3.2)

where dx1 and dx2 are the two shift vectors, k1 and k2 are the two scale factors, and A1 and A2 are the rotation matrices, x is the input position vector and x’ is the result. Organizing this can be expressed in form

x’=dx2+(1+k2)A2dx1+(1+k2)(1+k1)A1A2x

(4.3.3)

where the dxr,kr and Ar parameters of the resultant transformation are

dxr=dx2+(1+k2)A2dx1

(4.3.4)

kr=k1+k2+k1k2 ≈ k1+k2

(4.3.5)

Ar=A1A2 A1+A2

(4.3.6)

The approximation of (4.3.5) can be immediately understood in cases when the scale factors are in order or 1-10 part per million (ppm). The approximation of (4.3.6) is a bit more difficult, we should accomplish the matrix multiplication, omitting the resulted members falling to the range of the squares of the rotation angles and the scale factor. The right side of the Equation (4.3.4) is the second transformation done to the dx1 shift vector. Omitting the effect of the very small scale factor, it is

dxe=dx2+A2dx1≈ dx1+dx2

(4.3.7)

As the shift vector is usually much more short that the position vectors (n*100 meters compared to the Earth’s radius), this approximations fits well to the practice. The three-dimensional error of this simplification is in the order of some centimeters while its horizontal component is even smaller. So, the linear commutation can be applied in the practice for the Burša-Wolf transformation, too.