Relationship between TRS's

$$ \global\def\fig#1{\scriptsize\textcolor{E83E8C}{fig.#1}}$$

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 : $\dot{T1}$, $\dot{T2}$, $\dot{T3}$, $\dot{D}$, $\dot{R1}$, $\dot{T2}$, $\dot{R3}$. The transformation of coordinate vector $X_1$, expressed in a reference system ($\fig1$), into a coordinate vector $X_2$, expressed in a reference system ($\fig2$), is given by the following equation:

$$ X_2 = X_1 + T + DX_1 + RX_1 \newline~\newline with :\newline~\newline T = \begin{pmatrix} T1\\ T2\\ T3\\ \end{pmatrix} ~~~~~ and ~~~~~ R = \begin{pmatrix} 0 & -R3 & R2 \\ R3 & 0 & -R1 \\ -R2 & R1 & 0 \end{pmatrix} \newline~ \fig1 $$

It is assumed that equation ($\fig1$) 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 ($\fig1$) with respect to time gives :

$$ \dot{X_2} = \dot{X_1} + \dot{T} + \dot{D}X_1 + D\dot{X_1} + \dot{R}X_1 + R\dot{X_1} \newline~ \fig2 $$

$D$ and $R$ are of 10-5 level and is about 10 cm per year, the terms $D\dot{X_1}$ and $R\dot{X_1}$ are negligible which represent about 0.0 mm over 100 years. Therefore, equation ($\fig2$) could be writen as :

$$ \dot{X_2} = \dot{X_1} + \dot{T} + \dot{D}X_1 + \dot{R}X_1 \newline~ \fig3 $$