If you wish to contribute or participate in the discussions about articles you are invited to contact the Editor

Transformation between Terrestrial Frames

From Navipedia
Jump to navigation Jump to search

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

[math]\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)} }[/math]


[math]\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)} }[/math]

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

[math]\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)} }[/math]

where [math]\displaystyle{ T_1 }[/math], [math]\displaystyle{ T_2 }[/math], [math]\displaystyle{ T_3 }[/math] are three translation parameters, [math]\displaystyle{ D }[/math] is a scale factor and [math]\displaystyle{ R_1 }[/math], [math]\displaystyle{ R_2 }[/math] and [math]\displaystyle{ R_3 }[/math] 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]


  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.