Descriptions of the Polarization States of Vector Processes: Applications to ULF Magnetic Fields (1973)

In recent years a wide variety of methods has been used to describe the polarization characteristics of ultra low frequency (10−3 to 1 Hz) magnetic fields. This paper gives a more complete outline of some of the descriptions derived from the spectral matrices of n-variate stochastic processes. The matrices are expanded in three different, standard sets of matrices in order to add some simplification to the interpretation of the polarizations. One set is composed of n2 trace-orthogonal, hermitean matrices and leads directly to a generalization of the Stokes parameters and the degree of polarization for n-variate processes. The second set is developed from the dyad expansion, which in particular cases is analogous to the spectral decomposition of the matrix. The third set is composed of n commuting idempotent matrices and proves to be the most useful set when the stochastic process is not strictly polarized. Finally, two examples of digital records of ULF magnetic fields are analysed to illustrate some of the limitations of the methods, and to indicate the biases which are inherent in numerical analyses. The expansions of the spectral matrix which are given in this paper can lead to more simplified and objective descriptions of the polarization states of vector processes. In selecting strictly polarized waves, the degree of polarization P can be determined directly from the Stokes parameters, and then only those waves for which P ≈ 1·0 can be chosen for analysis. The interpretation of the polarization states of these waves is simplified by diagonalizing the real part of S, and if the wave is quasi-monochromatic, the parameters of the polarization ellipse can be computed directly from the elements of the transformed matrix. Conversely, if n = 3 and only the vector perpendicular to the plane of an elliptically polarized wave is required, then this vector can be computed directly from S without first transforming the matrix. Interpretation of polarization states when P is less than 1·0 is more difficult, and in these cases it is probably easiest to expand S in the set of commuting, idempotent matrices. Although the general form of the idempotent expansion can be very complicated, interpretation of the cases n = 2 and n = 3 can be considerably simplified by diagonalizing the real parts of the idempotent matrices. By following this procedure, the information in D1 can always be represented by a two-square matrix. This procedure also gives an indication of whether a1D1+a2D2(n= 3) can be considered to be the matrix of a plane wave. The descriptions of the polarization states which are outlined in this paper should be particularly useful in analysing the frequency-dependent polarization characteristics of irregular waveforms, or waves accompanied by broad band noise. Some examples of ULF magnetic fluctuations for which the descriptions should be very useful are continuous emissions, which may in some cases have structured components (see, e.g. Jacobs 1970), Pi2 micropulsations accompanying geomagnetic substorms, and the irregular Pc5's which occur during the recovery phase of substorms (see, e.g., McPherron et al. 1972). Objective descriptions of the polarization states of these waves will certainly lead to a better understanding of the physical processes which are involved, will allow more direct comparisons of the data with pertinent theories, and will simplify the comparison of data obtained through different experiments.