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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 $\displaystyle{ \left ( \Delta \mathbb{\mathbf X}=[\Delta x, \Delta y, \Delta z] \right ) }$, three consecutive rotations around the coordinate axes ($\displaystyle{ \theta_1 }$, $\displaystyle{ \theta_2 }$, $\displaystyle{ \theta_3 }$), and a scale factor ($\displaystyle{ \alpha }$). That is, they can be described by the following equation, which involves 7 parameters:

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

where:

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

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

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

where $\displaystyle{ T_1 }$, $\displaystyle{ T_2 }$, $\displaystyle{ T_3 }$ are three translation parameters, $\displaystyle{ D }$ is a scale factor and $\displaystyle{ R_1 }$, $\displaystyle{ R_2 }$ and $\displaystyle{ R_3 }$ 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.