ESA Science & Technology - Triangulation method to estimate a magnetic discontinuity normal

Variance ellipsoids derived from the magnetic field data for a Directional Discontinuity (DD) observed simultaneously by each Cluster spacecraft. From top left to bottom right Cluster 1 (black), Cluster 2 (red), Cluster 3(green) and Cluster 4 (blue). The ellipsoids are shown in GSE* coordinates.

The Minimum Variance Analysis (MVA) normal estimate is indicated by a thin black arrow. The triangulation normal is shown as a thick yellow arrow, partly transparent. The λ₂/λ₃ value and the angle between the MVA normal and the triangulation normal are indicated in the top left corner of each ellipsoid.

For Cluster 1, 2, and 4, the ellipsoids have the form of flat surfboards (large value of λ₂/λ₃). For Cluster 3, the ellipsoid has the shape of a cigar. In this case, it is difficult to distinguish between the direction of minimum and intermediate variance, and the resultant normal could be anything perpendicular to the direction of maximum variance. As a consequence the MVA normal deviates from the triangulation normal.

*GSE coordinates: the X axis points from the centre of the Earth to the centre of the Sun; the Y axis lies in the ecliptic plane and points in the opposite direction of the Earth's orbital motion; the Z axis completes a right-handed orthogonal coordinate system and is parallel to the ecliptic pole.