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# Transformation between Terrestrial Frames

Fundamentals
Title Transformation between Terrestrial Frames
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

From elemental linear algebra, all transformations between two Cartesian coordinate systems can be decomposed in a shift vector $\left ( \Delta \mathbb{\mathbf X}=[\Delta x, \Delta y, \Delta z] \right )$, three consecutive rotations around the coordinate axes ($\theta_1$, $\theta_2$, $\theta_3$), and a scale factor ($\alpha$). That is, they can be described by the following equation, which involves 7 parameters:

$\mathbb{\mathbf X}_{TRF2}=\Delta\mathbb{\mathbf X}+\alpha \; \mathbb{\mathbf R}_1[\theta_1] \; \mathbb{\mathbf R}_2[\theta_2] \; \mathbb{\mathbf R}_3[\theta_3] \; \mathbb{\mathbf X}_{TRF1} \qquad\mbox{(1)}$

where:

$\begin{array}{l} \mathbb{\mathbf R}_1[\theta]=\left [ \begin{array}{ccc} 1 & 0 & 0\\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta \\ \end{array} \right ] \;;\;\; \mathbb{\mathbf R}_2[\theta]=\left [ \begin{array}{ccc} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0\\ \sin \theta &0 & \cos \theta \\ \end{array} \right ]\\ \\ \mathbb{\mathbf R}_3[\theta]=\left [ \begin{array}{ccc} \cos \theta & \sin \theta &0\\ -\sin \theta & \cos \theta & 0\\ 0 & 0 & 1\\ \end{array} \right ] \end{array} \qquad\mbox{(2)}$

Adopting the convention used by IERS, the previous equation (1) can be written as follows:

$\left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF2}} = \left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF1}} + \left ( \begin{array}{c} T_1\\ T_2\\ T_3\\ \end{array} \right ) + \left ( \begin{array}{ccc} D & -R_3 & R_2\\ R_3 & D & -R_1\\ -R_2 & R_1 & D\\ \end{array} \right ) \left ( \begin{array}{c} x\\ y\\ z\\ \end{array} \right )_{_{TRF1}} \qquad\mbox{(3)}$

where $T_1$, $T_2$, $T_3$ are three translation parameters, $D$ is a scale factor and $R_1$, $R_2$ and $R_3$ are three rotation angles.

Transformation parameters from ITRF2000 to past ITRFs are listed in table 4.1 of IERS Conventions (2003) [Denis et al., 2004].[1]

## References

1. ^ [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32. IERS Convention Center., Frankfurt am Main.