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Descriptions of the polarization states of vector processes: applications to ULF magnetic fields
Geophysical Journal of the Royal Astronomical Society, 34 (4): 403--419 (декабря 1973)
DOI: 10.1111/j.1365-246X.1973.tb02404.x
Аннотация
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
a1 D1+a2 D2(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.
