Let us suppose to have magnetic field data sampled in the RTN reference system. If the large-scale mean magnetic field is oriented in the direction, we will look for a new reference system within the RTN reference system with the axis oriented along the mean field and the other two axes lying on a plane perpendicular to this direction.
Thus, we firstly determine the direction of the unit vector parallel to the mean field, normalizing its components
so that is the orientation of the first axis, parallel to the ambient field. As second direction it is convenient to choose the radial direction in RTN, which is roughly the direction of the solar wind flow, . At this point, we compute a new direction perpendicular to the plane
Consequently, the third direction will be
At this point, we can rotate our data into the new reference system. Data indicated as in the old reference system, will become in the new reference system. The transformation is obtained applying the rotation matrix
to the vector , i.e., .
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