Bursa–wolf transformation model
The Bursa-Wolf method assumes a similarity three dimensioned relationship between two consistent sets of Cartesian coordinate through seven parameters :
- three translations (DX, DY, DZ)
- three rotations around X, Y, Z axis respectively
(Î,y, w)
- a scale change ( DL )
Fig. 1 shows graphic representation of the transformation model.
Fig.1 Graphic representation of a 7 Parameter similarity Transformation
(a: Translations b: Rotations. c: Scale and d: Total)
If U, V and W represent the Cartesian components of a station in reference frame 1 say EVEREST and X, Y, Z represent the Cartesian component of same stations in reference frame number 2 say WGS - 84, the transformation can be expressed as:
where R represents a 3 x 3 rotation matrix and defined as
R= R1 (e) R2 (y) R3 (w)
If all three angles are small the above rotation matrix can be written in its simplified form by setting sine of an angle equal to the angle itself, cosine of the angle equal to 1 and the product of sines equal to zero.
Thus after simplification the above matrix will appear as
The transformation equation (4-1) can now be written as
Method of estimation of transformation parameters
A point physically identifiable on the surface of Earth which has been assigned coordinates in at least two separate systems of coordinates is termed as collocated station. The Cartesian coordinates of sufficient number of collocated stations (U, V, W, X, Y, Z) can be used as observations in a least square adjustment for the seven transformation parameters. The model in symbolical form can be written as :
F ( L, X ) = 0 .......................................................(4-4)
Where
L = observations (U, V, W, X, Y, Z)
X = parameters (
DX,
DY,
DZ,
DL,
w,
y,
e)
By arranging the equations (4-3) into this form will result in
DX + U + w V -
yW +
DL U + w
DL V -
y DL W - X = 0
............................. (4-5a)
DY + V - w U +
eW +
DLV - w
DL U +
eDLW - Y = 0
.........................................(4-5b)
DZ + W +
yU -
eV +
DL W +
y DL W -
eDL V - Z = 0
.....................................(4-5c)
These three equations represent the functional relationship between any two closely oriented, closely scaled, ortho normal cartesian coordinates systems. Since the observations (6 cartesian components per station) have systematic and other errors with them, the usual combined least squares procedure of minimizing the weighted sum of residuals squared (VTPV) is followed.
