## 1. Introduction

[2] The electromagnetic field radiated by a source in the Earth-ionosphere cavity can be expressed in terms of a superposition of transverse waveguide modes with relative amplitudes depending on the nature of the source. At extremely low frequencies (ELF), the modal representation of the fields is especially convenient because only the zeroth-order TM mode (the TEM mode) is nonevanescent in the waveguide. For a laterally homogeneous waveguide where the conductivity is a function only of altitude, the propagation equations can be separated into a one-dimensional differential equation pertaining to the vertical structure of the TEM mode and a two-dimensional equation describing the lateral propagation in the waveguide. Each of the separated equations contains a frequency-dependent propagation parameter *S*_{0}(*f*) that represents, mathematically, the eigenvalue of the TEM mode and, physically, the normalized complex horizontal wave number. The real and imaginary parts of *S*_{0}(*f*) are related, respectively, to the lateral phase velocity *V*_{PHASE}(*f*) and the attenuation rate *α*(*f*) via the equations

where *c* is the free-space speed of light.

[3] The waveguide is, of course, not laterally homogeneous, with the day-night difference in the electron density being the most obvious example of a lateral inhomogeneity. The separation of variables procedure described above is therefore not strictly valid. However, since the scale lengths for lateral variations of the ionospheric conductivity are much larger than those for vertical variations, it seems reasonable to treat the waveguide locally as horizontally stratified. With this assumption, the propagation parameter *S*_{0}(*f*) calculated on the basis of the local conductivity profile is not a constant, but a function of the lateral coordinates. With *S*_{0}(*f*) as an input parameter, the two-dimensional propagation equation then describes the lateral propagation in a nonuniform waveguide.

[4] The process of calculating *S*_{0}(*f*) is simplified by the fact that the solution of the one-dimensional equation depends on the details of the conductivity in two limited, well separated characteristic layers of the ionosphere, and not on the details of the region in between. For frequencies at the low end of the ELF band (3 to 100 Hz), the lower layer lies in the isotropic portion of the ionosphere below approx. 75 km, while the upper layer is in the anisotropic ionosphere above approx. 85 km. The lower and upper layers determine, respectively, the altitude dependence of the vertical electric and horizontal magnetic fields. This simplification is the basis of a noniterative analytic procedure proposed by *Greifinger and Greifinger* [1978, 1979] to obtain analytic expressions for *S*_{0}(*f*) and for the altitude-dependent electromagnetic fields for an isotropic ionosphere [*Greifinger and Greifinger*, 1978], and for a high magnetic latitude anisotropic ionosphere with idealized daytime and nighttime electron density profiles [*Greifinger and Greifinger*, 1979]. For both cases, *S*_{0}(*f*) has the form

where *H*_{C}(*f*) and *H*_{L}(*f*) are the complex characteristic altitudes of the lower and upper layers, respectively.

[5] The real part *h*_{C}(*f*) of *H*_{C}(*f*) is approximately the altitude (in the 45 to 75 km range at ELF) at which the conduction and displacement currents are equal. The TEM electric field is predominantly vertical below *h*_{C}(*f*) and falls off rapidly to a small horizontal component above *h*_{C}(*f*). The layer in which “refraction” takes place is also the region in which the Joule dissipation associated with the vertical currents has a maximum. The imaginary part of *H*_{C}(*f*) is a measure of the vertical extent of the dissipation maximum.

[6] The determination of the upper characteristic altitude *H*_{L}(*f*) is complicated by the profound effect of the geomagnetic field on the propagation. Although numerical techniques have been developed to calculate *H*_{L}(*f*), there are no published analytic approximations that apply to realistic profiles at all magnetic latitudes. Until such a model is developed, the real part *h*_{L}(*f*) of *H*_{L}(*f*) can be regarded as the effective magnetic height of the waveguide [*Mushtak and Williams*, 2002]. The existence of an imaginary part of *H*_{L}(*f*) is a consequence of both the escape of “Whistler mode” energy from the waveguide [*Madden and Thompson*, 1965; *Greifinger and Greifinger*, 1979; *Mushtak and Williams*, 2002] and collisional heating in the upper layer.

[7] Further insight into the unique role played by the two complex characteristic altitudes *H*_{C}(*f*) and *H*_{L}(*f*) is provided by the two-dimensional transmission line model of the waveguide suggested by *Madden and Thompson* [1965] and developed by *Kirillov et al.* [1997], *Kirillov and Kopeykin* [2002] to analyze two-dimensional propagation in the spherical waveguide. The model is based on the fact that the relationship between the TEM electric and magnetic field components as described by Maxwell's equations has the same form as the relationship between the voltage and current in a transmission line (actually, in the case of a spherical waveguide, not a line but a two-dimensional surface [*Madden and Thompson*, 1965]). The shunt admittance of the “line” is the analogue of and the series impedance is the analogue of *H*_{L}(*f*). The normalized complex propagation constant of the “line”, proportional to the square root of the product of the shunt admittance and series impedance, is equal to *S*_{0}(*f*) given by (3). The propagation equation for the “line”, the two-dimensional telegraph equation (TDTE) [*Kirillov et al.*, 1997; *Kirillov and Kopeykin*, 2002], is then analogous to the two-dimensional propagation equation obtained from the separation of variables described above. The use of the subscripts *C* and *L* for the lower and upper characteristic altitudes is motivated by the fact that *H*_{C}(*f*) and *H*_{L}(*f*) describe, respectively, the capacitive and inductive properties of the analogous transmission line. Similarly, the lower and upper ionospheric layers whose conductivities determine *H*_{C}(*f*) and *H*_{L}(*f*) will be referred to as the ELF-C layer and ELF-L layer, respectively.

[8] As mentioned above, the process of obtaining an analytical expression for the upper characteristic altitude, *H*_{L}(*f*), is complicated [*Greifinger and Greifinger*, 1979, 1986; *Kirillov*, 1993; *Kirillov and Kopeykin*, 2003]. It is the subject of ongoing research and will not be addressed further in this paper. On the other hand, the lower characteristic altitude can be approximately determined from the ELF-C layer conductivity profile *σ*(*h*) as an integral [*Greifinger and Greifinger*, 1978; *Sentman*, 1990; *Kirillov*, 1993]

where *h* is the altitude, ɛ_{0} is the permittivity of free space, and a time dependence exp(−*i*2*πft*) has been assumed.

[9] A simple analytic expression for the lower characteristic altitude is obtained by using a single exponential to represent *σ*(*h*), the conductivity profile. For this profile, the real part of *H*_{C}(*f*) is the frequency-dependent altitude at which the conduction and displacement currents are equal, and the imaginary part is a constant equal to *ζ*, where *ζ* is the scale height of the exponential. Suggested by *Greifinger and Greifinger* [1978] as the simplest approximation of the ELF-C layer conductivity profile, the single-exponential model for calculating *H*_{C}(*f*) has been widely used in ELF studies [*Sentman*, 1990; *Nickolaenko and Hayakawa*, 2002]. For some purposes, this simple model is adequate.

[10] However, as had been well known for many years before the concept of the two characteristic layers was developed, there is a “knee”-like transition in the ELF-C layer conductivity profile from the lower, ion-dominated portion of the profile to its upper, electron-dominated portion that is important to take into account for an adequate description of the Schumann resonance (SR) quality factors [*Galejs*, 1962; *Wait*, 1964; *Cole*, 1965; *Madden and Thompson*, 1965; *Jones*, 1967; *Williams et al.*, 2006]. To address this feature of the profile, *Mushtak and Williams* [2002] considered a very simple conductivity model (known as the “knee” model) of the lower ionosphere of a uniform waveguide. The model uses exponentials of different scale heights to represent the ion and electron conductivity profiles, with an abrupt transition from the ion to the electron conductivity at the altitude at which their values are equal. In the “knee” model, the “height integral” (4) can be evaluated analytically, and in contrast to the results of the single-exponential model, both the imaginary and real parts of *H*_{C}(*f*) are frequency dependent.

[11] Although the model is very useful in helping to explain the frequency behavior of the Schumann resonance (SR) quality factors [*Mushtak and Williams*, 2002], it has an important drawback. The model is constructed for a laterally uniform ionosphere with globally averaged properties. It is not completely clear how the idealized laterally invariant “knee” model relates to the very different daytime and nighttime conductivity profiles, each of which shows a “knee”-like transition from ions to electrons, but with significantly different values of the “knee” altitude and conductivity. The modeling of the nonuniform waveguide requires separate daytime and nighttime conductivity profiles generating separate values of the characteristic altitudes. This process is obviously necessary to explain numerous phenomena that relate specifically to the existence of lateral asymmetry, from the dusk/dawn amplitude variations [*Mushtak et al.*, 1998; *Greifinger et al.*, 2005] to global resonant phenomena [*Mushtak et al.*, 1999; *Williams et al.*, 2006], to reactions of the Earth-ionosphere waveguide to extraterrestrial factors [*Price and Mushtak*, 2001; *Sátori et al.*, 2005], to refractive [*Mushtak et al.*, 1998, 2002] and polarization [*Mission Research Corporation*, 2000] effects. This process is also necessary to obtain the global average of the ratio *H*_{L}(*f*)/*H*_{C}(*f*) that determines the SR frequencies and frequency-dependent *Q* factors.

[12] The goals of the present study are as follows: (1) to fit the variables of physically realistic analytic conductivity models of the ELF-C layer to available daytime and nighttime aeronomical data using a best fit criterion and (2) to use the parameterized models to obtain analytic approximations of the real and imaginary parts of the lower characteristic altitude (4) which are reasonably accurate over the entire frequency range of interest (3 to 100 Hz) and simple enough to be used for parametric analysis.