Some comments on the descriptions of the polarization states of waves

Summary. Procedures are developed for specifying the polarization characteristics of n-dimensional waves, and in particular three-dimensional waves of geophysical interest. We show that when a wave is in a pure state or is totally polarized, all the polarization information can be represented by a single vector u in an n-dimensional unitary space. Simple measures of the degree of polarization of the wave are constructed from the characteristic equation of the spectral matrix S. These measures are functions only of the scalar invariants of S and consequently S need not be diagonalized. If S represents a purely polarized wave, the unitary vector which contains the polarization information about the wave can be obtained directly from S using any 2n– 1 equations of n2 possible equations. By multiplying by a phase-factor this unitary vector can be written in the form u=r1+ir2 where r1 and r2 are orthogonal vectors in a real space. For an elliptically polarized wave, r1 and r2 locate the major and minor axes of the ellipse, and the ellipticity is given by the ratio of their magnitudes. The polarization parameters of ULF magnetic waves at the Earth's surface are computed from one set of five equations (n= 3) and compared with parameters calculated using established techniques.