## 1. Introduction

[2] Global electromagnetic resonances were predicted by *Schumann* [1952] and are referred to as the Schumann resonances. Radio waves of extremely low frequencies (ELF) ranging from a few Hz to a few tens of Hz originate from global lightning strokes. An electromagnetic pulse radiated by a discharge circles the planet a few times, and resonance peaks appear in the power spectra at frequencies of nominally 8, 14, 20, 26 Hz, etc. Oscillations in a spherically uniform and isotropic Earth-ionosphere cavity are associated with a degeneracy of the eigenvalues: there is a set of different eigenfunctions corresponding to a single eigenvalue or resonance frequency [*Wait*, 1962, 1965; *Bliokh et al.*, 1968, 1971, 1977, 1980; *Sentman*, 1995]. The eigenfunctions have different azimuthal indices *m*∈[−*n*, +*n*], where *n* denotes the resonance mode number, so that there are (2*n* + 1) different eigenfunctions corresponding to a single resonance frequency, and we speak about a (2*n* + 1)–fold degeneracy. This frequency degeneracy is similar in many respects to the energy degeneracy that occurs among the electron states of atomic and molecular systems, so the same concepts and language may be employed.

[3] The presence of the geomagnetic field turns the ionospheric plasma into an anisotropic (locally gyrotropic) medium characterized by a tensor dielectric constant. The superposition of a geocentrically centered geomagnetic dipole magnetic field onto an otherwise spherically symmetric scalar conductivity reduces the symmetry to that of a cylinder with symmetry axis parallel to the geomagnetic dipole axis. The presence of the magnetic field violates the equivalence of the eastward and westward propagation (in magnetic coordinates), and the waves corresponding to distinct eigenfunctions acquire different phase speeds. The degeneracy is therefore removed, the resonance frequencies are split, and elliptical polarization appears [*Wait*, 1962; *Large and Wait*, 1966, 1967a, 1967b, 1968; *Bliokh et al.*, 1968, 1971, 1977, 1980; *Sentman*, 1987, 1989, 1995; *Nickolaenko and Hayakawa*, 2002].

[4] Attempts to detect the line splitting have been undertaken in many experiments. As far as we know, the first systematic investigation utilized the vertical electric field component [*Bliokh et al.*, 1971; *Bormotov et al.*, 1971]. An attempt was undertaken by *Tanahashi* [1976] to observe the fine structure in the power spectra of horizontal magnetic field using a Z transform analysis method. Polarization of horizontal magnetic field was addressed in the works of *Sentman* [1987, 1989] and *Labendz* [1998].

[5] *Sentman* [1989] reported on the different polarization characteristics of the magnetic component of the Schumann resonances computed separately for the two orthogonal horizontal magnetic axes. It was shown that large *Q* bursts often exhibited strong elliptical polarization, with a gradual rotation of the major axis of the polarization ellipse occurring over the duration of the event. This behavior is consistent with the presence of eastward and westward traveling waves of different frequencies. As mentioned above, the differences in the frequencies of eastward and westward traveling waves is an expected signature of the normal modes associated with the side multiplets (*m* = ±1) in the presence of a magnetized ionosphere, which led *Sentman* [1989] to conclude that the observed behavior was in fact due to such line splitting. While this interpretation appears to have been correct, the observations of only two components of the magnetic field at a single site did not permit unambiguous resolving of the full set of multiplets, three in the case of the lowest *n* = 1 mode at 8 Hz.

[6] The basic computational difficulty in detecting the line splitting by using the fine structure of Schumann resonance peaks when computed from either the electric field or a single axis of magnetic field is a consequence of the considerable width of the resonance lines. It is about 2 Hz at the 8 Hz basic mode, so that the *Q* factor is typically equal to 4. Estimates indicate that the highest splitting may reach a few tenths of a Hz [*Bliokh et al.*, 1968, 1977, 1980], which is much smaller than the natural width of the peak; therefore the resonance pattern is barely perturbed by splitting. For this reason, *Tanahashi* [1976] turned to the Z transform [see *Marple*, 1987] instead of FFT in an attempt to sharpen the sublevels of the resonance line. The Z transform method tries to compensate the attenuation by applying the basic functions growing in time. However, it is highly susceptible to noise, and tends to create spectral structure that depends on the order of the model, making it difficult to distinguish real peaks from model-dependent artifacts. There exist several additional natural effects that create fine structure unrelated to the removal of degeneracy. First, a scalloped profile is always observed owing to the phase interference arising among modes excited by randomly occurring global lightning strokes, which result in a highly nonstationary spectrum for spectra computed over short timescales. Averaging raw FFT spectra reduces these fluctuations, but these never vanish completely. Additional mechanisms leading to spectral structure are: Alfven ionospheric resonance observed just below the Schumann resonance [*Belyaev et al.*, 1990] and possible impact of magnetospheric ULF signals [*Bliokh et al.*, 1969].

[7] To completely resolve the line splitting problem requires that the number of independent measurements equal or exceed the order of the degeneracy. Since there are only three components in the TM_{0} fields (two magnetic and one electric), one must therefore sample the fields at different spatial locations to unambiguously resolve the various subfields. The most effective locations for such measurements would be at a combination of locations corresponding to the various peaks and nodes of the underlying eigenfunctions. However, observations at sites separated by distances that are an appreciable fraction of the characteristic spatial scales of the modes should be adequate to obtain unambiguous results, and form the basis of the present work.