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Relationship between TRS's

The standard relation of transformation between two reference systems is an Euclidian similarity of seven parameters: three translation components, one scale factor, and three rotation angles, designated respectively, T1, T2, T3, D, R1, R2, R3, and their first times derivations : 1, 2, 3, , 1, 2, 3. The transformation of coordinate vector X1, expressed in a reference system (1), into a coordinate vector X2, expressed in a reference system (2), is given by the following equation:

 (1)

with :

It is assumed that equation (1) is linear for sets of station coordinates provided by space geodetic technique (origin difference is about a few hundred meters, and differences in scale and orientation are of 10-5 level). Generally, X1, X2, T, D, R are function of time. Differentiating equation (1) with respect to time gives :

 (2)

D and R are of 10-5 level and is about 10 cm per year, the terms D1 and R 1 are negligible which represent about 0.0 mm over 100 years. Therefore, equation (2) could be writen as :

 (3)