Improved observations in the Schumann resonance frequency range have expanded interest in improved models for the Earth-ionosphere waveguide, particularly models that incorporate day-night asymmetry. Aeronomical data are used here to construct daytime and nighttime analytic conductivity models of the lower characteristic layer, one of two vertically separated regions governing ELF propagation in the waveguide. The models have the form of double-exponential functions that represent the “knee”-like transition from ion- to electron-dominated conductivity in a physically realistic manner. On the basis of these profiles, analytic approximations of the lower complex characteristic altitude are obtained for the frequency range 3–100 Hz. The approximations are accurate and simple to use for parametric analysis.
 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 S0(f) that represents, mathematically, the eigenvalue of the TEM mode and, physically, the normalized complex horizontal wave number. The real and imaginary parts of S0(f) are related, respectively, to the lateral phase velocity VPHASE(f) and the attenuation rate α(f) via the equations
where c is the free-space speed of light.
 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 S0(f) calculated on the basis of the local conductivity profile is not a constant, but a function of the lateral coordinates. With S0(f) as an input parameter, the two-dimensional propagation equation then describes the lateral propagation in a nonuniform waveguide.
 The process of calculating S0(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 S0(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, S0(f) has the form
where HC(f) and HL(f) are the complex characteristic altitudes of the lower and upper layers, respectively.
 The real part hC(f) of HC(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 hC(f) and falls off rapidly to a small horizontal component above hC(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 HC(f) is a measure of the vertical extent of the dissipation maximum.
 The determination of the upper characteristic altitude HL(f) is complicated by the profound effect of the geomagnetic field on the propagation. Although numerical techniques have been developed to calculate HL(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 hL(f) of HL(f) can be regarded as the effective magnetic height of the waveguide [Mushtak and Williams, 2002]. The existence of an imaginary part of HL(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.
 Further insight into the unique role played by the two complex characteristic altitudes HC(f) and HL(f) is provided by the two-dimensional transmission line model of the waveguide suggested by Madden and Thompson  and developed by Kirillov et al. , Kirillov and Kopeykin  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 HL(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 S0(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 HC(f) and HL(f) describe, respectively, the capacitive and inductive properties of the analogous transmission line. Similarly, the lower and upper ionospheric layers whose conductivities determine HC(f) and HL(f) will be referred to as the ELF-C layer and ELF-L layer, respectively.
where h is the altitude, ɛ0 is the permittivity of free space, and a time dependence exp(−i2πft) has been assumed.
 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 HC(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  as the simplest approximation of the ELF-C layer conductivity profile, the single-exponential model for calculating HC(f) has been widely used in ELF studies [Sentman, 1990; Nickolaenko and Hayakawa, 2002]. For some purposes, this simple model is adequate.
 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  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 HC(f) are frequency dependent.
 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 HL(f)/HC(f) that determines the SR frequencies and frequency-dependent Q factors.
 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.
2. Form of the ELF-C Layer Conductivity Profile
 The ELF-C layer is the ionospheric stratum whose conductivity profile determines the lower characteristic altitude HC(f) defined by (4). Since we are interested in a range of frequencies between specified values fMIN and fMAX, what is needed to evaluate the height integral (4) is a conductivity profile that extends from a conductivity value that is well below 2πɛ0fMIN to one that is well above 2πɛ0fMAX. For frequencies between 3 and 100 Hz, the appropriate conductivity range is approximately 10−11 to 10−7 S/m. The corresponding altitude range depends on the details of the profile, but is approximately 30 to 75 km (see section 4), a region in which the ionosphere at low ELF can be treated as isotropic.
 The conductivity consists of an electron and an ion component. The ion component is actually the sum of the conductivities of all the ions (both positive and negative), lumped together to form a single effective ion conductivity. Typical generic profiles of the electron and effective ion conductivities, obtained from the work of Reid , are shown in Figure 1 along with the profile of the total conductivity. It can be seen from Figure 1 that it is reasonable to represent the individual components of the conductivity in the layer by simple exponentials with different scale heights. This gives for the total conductivity a four-variable double-exponential model of the form
where hC0 is the “knee” altitude, defined as the altitude at which the electron and ion conductivity components are equal, σC0 is the conductivity of each component at the “knee” altitude, and ςi and ςe are the scale heights of the ion-dominated and electron-dominated portions of the profile, respectively. The parameterization of the model using available aeronomical data is described in the next section.
3. Conductivity Profiles of the ELF-C Layer
 For a number of reasons [Danilov, 1970; Reid, 1986; Hargreaves, 1992], information about the lower D region of the ionosphere (which overlaps the ELF-C layer in our terminology) is significantly more meager and is being accumulated much more slowly than information about higher ionospheric regions. Indeed, nighttime and daytime conductivity profiles shown by Cole and Pierce [1965, Figure 8] and Madden and Thompson [1965, Figures 6 and 7] are similar to those considered to be representative by Reid [1986, Figure 3.10] twenty years later. All the profiles clearly show the “knee”-like transition from the ion-dominated to the electron-dominated conductivity of the lower ionosphere.
 In this section we construct daytime and nighttime conductivity models, in the form given by (5), using representative experimental data available in the literature after two additional decades of aeronomical research. The reference conductivity profiles that were used when constructing the models are shown in Figure 2. The ion conductivity reference profile comes from direct measurements, while the electron conductivity reference profiles are computed from electron density data. The collision frequency model used in the computations is that given by Pasko et al. [1997, Figure 1].
3.1. Reference Profile of the Ion Conductivity Component
 The results of these measurements are shown in Figure 2. The horizontal bars indicate the minimum-to-maximum conductivity range estimated at each altitude from all of the above experimental data sets.
3.2. Reference Profiles of the Electron Conductivity Component
 One reference daytime electron conductivity profile (open circles in Figure 2) has been computed from the results of direct low-latitude measurements of electron density made by Gupta  in rocket experiments in the 60 to 80 km altitude range using the D.C. Langmuir technique. A second reference conductivity profile (open stars in Figure 2) was computed from the well-established International Reference Ionosphere model [Heliospheric Physics Laboratory, 2001] for the low-latitude electron density. Both profiles show essentially similar altitude behavior, including the characteristic “ledge” above 70 km discussed in detail by Gupta .
 The daytime and nighttime conductivity profiles obtained by fitting the four-variable double-exponential model (5) to the reference profiles are shown in Figure 2. The estimates of the model variables are shown in Table 1. Included in Table 1 is the “knee” frequency fC0 defined by
Note that fC0 (day) is well below all the Schumann resonance (SR) frequencies, while fC0 (night) is approximately equal to the frequency of the lowest SR mode.
Table 1. Double-Exponential Approximations of the ELF-C Layer Conductivity Profiles
Variables of Double-Exponential Model
“Knee” Frequency fC0, Hz
4. Characteristic Altitude of the ELF-C Layer
 The lower complex characteristic altitude HC(f) is given by the height integral (4) with the conductivity profile given by (5). For purposes of comparison with other conductivity models, in particular the widely used single-exponential profile, it is convenient to write HC(f) in the form adopted by Mushtak and Williams :
The quantity ζCEFF(f) is called the “effective scale height” of the ELF-C layer. (For a single-exponential conductivity model, the “effective scale height” is equal to the altitude-independent scale height of the exponential.) Before evaluating the height integral to obtain hC and ζCEFF, it is instructive to examine the integrand.
The real part of (f,h) is equal to ∣(f,h)∣2, and the imaginary part is equal to (f,h)∣(f, h)∣2. Thus Re (f,h) and Im (f,h) represent the altitude profiles of the normalized electric energy density and Joule dissipation rate, respectively [Madden and Thompson, 1965; Greifinger and Greifinger, 1978; Sentman, 1990]. The profiles are shown in Figures 3, 4, and 5 for both daytime and nighttime conditions for the following frequencies: 8 Hz (approximately the resonant frequency of Schumann mode 1), 32 Hz (approximately the resonant frequency of Schumann mode 5 considered, under most conditions, to be the last mode providing reliably identified resonant parameters), and 100 Hz, the upper limit of the ELF communication band.
 For each frequency, the dissipation peak in Figures 4 and 5 is located at the altitude h0(f) for which (f, h0) = 1. At that same altitude, the electric energy density (shown in Figure 3) is half its value at the ground. For large relative frequencies ( 1), an approximation for h0(f) is readily obtained. The equation (f,h0) = 1 can be written in the form
and p is the scale height ratio
For large values of , the solution of (11) is approximately
 For single-exponential models, the real part hC(f) of the lower characteristic altitude is equal to h0(f). Although for a double-exponential model, hC(f) is not equal to h0(f), it can be anticipated from the shape of the profiles shown in Figure 3 that h0(f) provides a reasonably good estimate of hC(f) for frequencies larger than the “knee” frequency fC0.
 From Figures 4 and 5, it can be seen that both the thickness of the dissipation layer and the location of the dissipation peak depend on frequency. As the frequency increases and the peak migrates upward, the thickness of the dissipation layer decreases. At the frequency of 8 Hz, the relative frequency is approximately equal to 1 under nighttime conditions. The location h0(f) of the peak is a few kilometers below the “knee” altitude hC0 , and the vertical extent of the dissipation layer can be seen to be several kilometers. At 100 Hz, the peak is approximately 5 km above the “knee”, and the thickness of the layer is just a few kilometers. The quantity ζCEFF(f), which is a measure of the vertical extent of the dissipation layer, can be expected to show particularly strong frequency dependence under nighttime conditions.
4.2. Approximation of the Height Integral
 To obtain an approximation of the height integral (4) with the double-exponential conductivity profile (5), we make use of the fact that > 1, and hence the parameter β defined by (15) is less than 1 for the entire frequency range of interest under daytime conditions, and for frequencies greater than the lowest SR mode at night. The integral can be expressed in terms of a power series with β as the expansion parameter. The details of the computational procedure are described in Appendix A. The results to second order in β are as follows:
 The results for the approximations (16)–(19) of hC(f) and ζCEFF(f) for the parameters presented in Table 1 are shown in Figures 6 and 7, and compared with those obtained from numerical integration of (4). The approximations can be seen to be very accurate over the whole range of frequencies from the lowest SR frequency to the top of the communication band, for both daytime and nighttime conditions. The strong dependence of ζCEFF(f) on frequency, particularly at night, is apparent from Figure 7. Note that ζCEFF(f) is significantly larger than ζe even for the frequencies in the communication band well above the “knee” frequency.
 The exponential conductivity models of Nickolaenko and Hayakawa (hereinafter referred to as NH) are shown in Figure 2. The quantities hC(f) and ζCEFF(f) obtained from these models have the form
where ζ is the scale height of the exponential, hREF is a reference height, and fREF is equal to . In the models proposed by NH, fREF is 10 kHz and hREF is 89 km for both daytime and nighttime conditions, while the day and night values of ζ are 4.5 and 3.5 km, respectively.
 From Figure 6, it can be seen that the NH nighttime values of hC(f) differ very little from those obtained from the double-exponential model, but the daytime results differ significantly. The explanation of these facts is very simple. For each frequency, hC(f) is determined by the conductivity profile within a region extending from a couple of conductivity scale heights below h0(f) to a couple of scale heights above h0(f). For the frequencies of interest, the range of conductivities contributing to the determination of hC(f) is approximately 10−10 to 10−8 S/m. From Figure 2, it can be seen that under nighttime conditions the NH straight-line conductivity profile is a reasonable match to the double-exponential one over the “heart” of the relevant conductivity range, while the daytime match is poor. It can also be seen from Figure 2 that the daytime double-exponential profile in this conductivity range could be reasonably approximated by a single exponential that, if properly parameterized, would give good estimates of hC(f) for the frequency range of interest.
 As far as ζCEFF(f) is concerned, any single-exponential model generates a value equal to the assumed scale height ζ of the exponential, and is therefore independent of frequency. The imaginary part of the height integral that generates ζCEFF(f) is more sensitive to the details of the conductivity profile over a larger conductivity range than is the real part that generates hC(f). Approximations of the conductivity profile in the ELF-C layer that may be adequate to provide good estimates of hC(f) may not do so for ζCEFF(f), and any single-exponential model of the layer is an example of such an approximation.
 The “knee” model presented by Mushtak and Williams  (hereinafter referred to as MW) is one that pertains to a uniform Earth-ionosphere waveguide with globally averaged daytime and nighttime as well as latitude-dependent properties. To model the “knee”-like transition from the ion-dominated to the electron-dominated portion of the lower ionosphere, MW use two exponentials with different scale heights to represent the ion and electron conductivities, as is also the case for the present (GMW) double-exponential model. However, instead of representing the conductivity as the sum of the two components, MW adopt a discontinuous scale height model that neglects the contribution of the electrons to the conductivity below the “knee”, and the contribution of the ions above the “knee”. This idealization of the more realistic double-exponential model allows for an analytic expression of HC(f) for all frequencies.
 There is no point in comparing the GMW daytime and nighttime values of hC(f) and ζCEFF(f) with those obtained by MW for a uniform waveguide. However, it is useful to compare the GMW results with those obtained by replacing the exponential sum in the integrand of the height integral with the MW discontinuous scale height approximation, using the same parameters for hC0, σC0, ζe, and ζi. The comparison profiles are shown in Figure 2, for both daytime and nighttime conditions. The analytic expressions for hC(f) and ζCEFF(f) for the MW model are given by
The overall similarity of the results for hC(f) and ζCEFF(f) (shown in Figures 6 and 7) for the MW and GMW models is certainly not surprising. This could be anticipated from the overlap of the conductivity profiles shown in Figure 2. It is of interest to examine the differences. The differences are a consequence of the underestimation by the MW model of the total conductivity that accompanies the neglect of the electrons below the “knee” and the ions above the “knee”. The effect on the real part of the integrand in (4) is an overestimation in the “knee” region (altitudes near the “knee” height hC0), which results in an overestimation of hC(f) for frequencies near the “knee” frequency. For large relative frequencies ( 1), ≪ 1 in the “knee” region for both models, and this region plays only a minor role in determining hC(f). As the frequency increases, the characteristic altitude hC(f) becomes close to h0(f) for both models.
 The situation is different for ζCEFF(f). The results for the two models agree well for frequencies near the “knee” frequency, but begin to diverge at ∼ 2. The MW model underestimates ζCEFF(f) for values 1, an underestimation that persists through the communication band at night. The explanation lies in the altitude and frequency dependence of the imaginary part of the integrand. The integrand is approximately equal to for altitudes below h0(f), and rapidly vanishes for altitudes above h0(f). For values of ≫ 1, h0(f) is above the “knee” altitude hC0, and the MW model neglects the contribution to ζCEFF(f) of the integrated ion conductivity between hC0 and h0(f) as well as the integrated electron conductivity between the ground and hC0. This results in a different frequency dependence of ζCEFF(f) for the two models at high relative frequencies. The difference in ζCEFF(f) as a function of the scale height ratio p and frequency f can be demonstrated by comparing (18) and (19) with (23) for ≫ 1: GMW model:
The coefficient of the frequency-dependent term is the same for the two models, but the frequency dependence is different. The decay with frequency of [ζCEFF(f) − ζe] is more rapid for the MW model.
 The numerical differences between the MW and GMW results for hC(f) and ζCEFF(f) are fairly small—at most a few percent for hC(f) and no more than 10–12% for ζCEFF(f). The GMW double exponential is the preferred conductivity model since it is physically realistic and is a fit to available aeronomical data. The approximations of hC(f) and ζCEFF(f) in (16)–(19) are accurate and easy to use for parametric analysis: the coefficients of β and β2 are frequency-independent and depend only on the scale height ratio p. However, the approximations (16)–(19) of the height integral only apply to frequencies greater than or equal to the “knee” frequency. For applications of the double-exponential conductivity model to frequencies below the “knee” frequency, the approximations (22) and (23) obtained from the MW “knee” model provide good estimates of hC(f) and ζCEFF(f), and there is no need for alternatives.
 The aeronomical data used in this study were obtained mostly at low and middle latitudes. As more data are accumulated at all latitudes under various ionospheric conditions, the parameters of the double-exponential models can be modified accordingly. In the meantime, what is needed to complete the modeling of the ELF propagation parameters in the Earth-ionosphere waveguide is the analysis of propagation in the upper layer that generates the upper characteristic altitude HL(f), an analysis that includes the important effects of the geomagnetic field, and daytime and nighttime conductivity models appropriate to the upper layer [Greifinger and Greifinger, 1978, 1979, 1986; Kirillov, 1993; Kirillov and Kopeykin, 2002, 2003]. The electron densities above 85 kilometers that are needed to generate the daytime and nighttime conductivity profiles for the upper characteristic layer cannot be obtained by extrapolating the lower layer profiles to higher altitudes. This would not only be wrong but also unnecessary, since there is abundant information available in the well-established International Reference Ionosphere model (for example, in its IRI-2001 version).
Appendix A:: Calculation of the Lower Complex Characteristic Altitude With the Double-Exponential Conductivity Profile
 To calculate the complex characteristic altitude HC, it is convenient to rewrite the height integral (4) with a change of variable as
and the parameters , p, and β are defined in (10), (13), and (15), respectively.
The integrand of (A7) is expanded in powers of β (β < 1 for the frequency range of interest). The coefficients of βn (n = 1, 2, 3,…) are integral representations of Beta functions. The function g(β, p) is calculated to first order in β which, combined with (A5) and (A6), provides approximations for hC(f) and ζCEFF(f) to second order in this parameter. The results are as follows:
 With the scale height ratio p ∼ 5 (see Table 1), the quantity is small, and sin() and cos() can be replaced by and 1, respectively. The simplified forms of (A8) and (A9) are given by (17) and (19) in section 4.
 The authors are grateful to S. Gupta and Steven Cummer for information and ideas concerning ionospheric data. The first author would like to thank Carl Greifinger for his invaluable support. Research support for E.R.W.'s contribution is derived from the Physical Meteorology Section of the National Science Foundation, with assistance from R. Rogers and A. Detwiler.