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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, D16107, doi:10.1029/2005JD006944, 2006

Distinguishing ionospheric models using Schumann resonance spectra Earle R. Williams,1 Vadim C. Mushtak,1 and Alexander P. Nickolaenko2 Received 2 December 2005; revised 2 December 2005; accepted 25 April 2006; published 25 August 2006.

[1] A number of models for Schumann resonance (SR) behavior in uniform

approximations of the real Earth-ionosphere cavity now populate the literature. These models are treated in terms of variously formulated propagation parameters: as the complex eigenvalue of the propagation problem, as the complex incident angle’s sine, as the phase velocity and attenuation rate, or as a pair of complex characteristic altitudes. This study compares a priori theoretical propagation parameters with the corresponding quantities recovered from calculated Schumann resonance spectra by means of Lorentzian spectral fits. To estimate the ultimate accuracy of the recovery procedure, the influence of source-receiver separation is excluded by assuming a globally uniform distribution of lightning sources. The comparisons show a practically acceptable agreement, within several percent, agreement between recovered and a priori parameters for all models studied. When judged against real Schumann resonance observations, these results shed light on problems with certain models. More importantly, the results reaffirm the ability of procedures based on SR observations to resolve global features of the ionosphere’s state and structure. Citation: Williams, E. R., V. C. Mushtak, and A. P. Nickolaenko (2006), Distinguishing ionospheric models using Schumann resonance spectra, J. Geophys. Res., 111, D16107, doi:10.1029/2005JD006944.

1. Introduction [2] A number of recent developments, the expanded interest in global (including climate) change, the discovery of sprites and jets in the mesosphere, and the unprecedentedly enhanced potential of digital signal processing and archival, have all contributed to a resurgence in the study of the Earth’s Schumann resonance (SR) phenomenon [Schumann, 1952; Galejs, 1972; Bliokh et al., 1980; Nickolaenko and Hayakawa, 2002]. Accompanying this resurgence is an increased interest in more accurate models of propagation parameters for the Earth-ionosphere waveguide within the extremely low frequency (ELF) range. Numerous analytical and numerical models for uniform approximations of the waveguide have been developed [Madden and Thompson, 1965; Jones, 1967; Ishaq and Jones, 1977; Greifinger and Greifinger, 1978; Nickolaenko and Rabinowicz, 1982, 1987; Sentman, 1996; Fullekrug, 2000; Mushtak and Williams, 2002; Yang and Pasko, 2005] since the discovery of the SR more than a half-century ago, with quantitative predictions for the wave phase velocity and attenuation rate, or alternatively, for the waveguide’s modal frequencies and quality factors. Despite the great progress in modeling as well as in collecting experimental observations, relatively little work has been devoted to extracting propagation parameters from the observations toward testing and distinguishing various theoretical models. One important 1 Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 2 Usikov’s Institute for Radio-Physics and Electronics, National Academy of Sciences of the Ukraine, Kharkov, Ukraine.

Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JD006944

exception in this context is the recent work by Mushtak and Williams [2002] who showed that the Q-factor’s dependence on frequency in Schumann resonance spectral observations is in good agreement with an analytical ‘‘knee’’ model, providing results close to those obtained by Jones [1967] and Ishaq and Jones [1977] when realistic profiles of the ionospheric conductivity are considered. [3] The present study is concerned with the use of the Lorentzian spectral form in fitting SR power spectra for the purpose of extracting fundamental propagation parameters at the corresponding modal frequencies. The main objectives are threefold: (1) to estimate the ultimate accuracy of the Lorentzian method in recovering the theoretically assumed (postulated) propagation parameters; (2) to test the applicability of the Lorentzian fits procedure to distinguish between various models of propagation parameters; and (3) to judge the adequacy of presently available models on the basis of real SR observations. Taking as an example a waveguide with a spherically symmetrical (uniform) structure, it will be shown that the Lorentzian method is a reliable extracting instrument that can be profitably used to recover the waveguide’s global physical properties and to make clear distinctions between different propagation models.

2. General Relationships for Waveguide Propagation at ELF [4] Depending on the task to be considered, it is advisable to present the ELF fields propagating within the spherically symmetrical Earth – ionosphere cavity either via the modal expansion or via the zonal harmonic series representation (ZHSR) [Wait, 1962; Galejs, 1972]. In what follows, we use

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the ZHSR for the spectral vertical component of the electric field on the Earth’s surface under the exp(+i2pft) time convention [Jones, 1970; Nickolaenko and Hayakawa, 2002]: iM ð f Þ nð f Þ½nð f Þ þ 1 4p2 e0 a2 fHEFF ð f Þ 1 X ð2n þ 1ÞPn ðcos qÞ ; nðn þ 1Þ nð f Þ½nð f Þ þ 1 n¼0

Er ð f Þ ¼

ð1Þ

where M(f) is the current moment of a vertical electric dipole simulating a typical lightning source, e0 is the dielectric constant of vacuum, a is the Earth’s radius, HEFF(f) is the effective height of the ionosphere, q is the source – observer angular distance, and n(f) is a complex propagation parameter identifying mathematically the eigenvalue of the propagation problem’s radial operator for the only globally propagating TEM mode [Hynninen and Galyuck, 1972; Bliokh et al., 1980]. Generally, the effective height of the ionosphere is presented by the lower characteristic altitude in the Greifinger and Greifinger [1978] formalism and, as such, is frequency dependent. [5] Representation (1) is widely used in numerical computations for the reason that it allows accelerating the convergence of the series [Connor and Mackay, 1978; Bliokh et al., 1980; Jones and Burke, 1990]. An additional advantage of this representation, to be used below, is a possibility to integrate analytically power and cross-spectra over a uniform distribution of sources [Raemer, 1961; Bliokh et al., 1977; Sentman, 1996]. [6] Five different models of ELF propagation parameters for a uniform Earth-ionosphere cavity are evaluated in this study. Methodologically, these models are formulated as various sets of frequency-dependent propagation parameters in one of the following forms: (1) the zero-order complex eigenvalue n( f ) of the radial operator and the effective height of the ionosphere [Hynninen and Galyuck, 1972; Bliokh et al., 1977, 1980]; (2) the incident angle’s complex sine S(f) and the effective height of the ionosphere [Wait, 1962; Galejs, 1972]; (3) the phase velocity ratio c (c representing the free space velocity of rPHASE( f ) VPHASE light) and attenuation rate a(f) [dB/Mm]; or (4) a pair of complex characteristic altitudes H0( f ) and H1( f ) presenting, in a condensed form, the ionosphere’s properties within two relatively isolated ionospheric layers responsible for ELF propagation [Greifinger and Greifinger, 1978; Mushtak and Williams, 2002]. The propagation parameters listed above are related to each other as: S2ð f Þ ¼

nð f Þ½nð f Þ þ 1 ðkaÞ2

ð2Þ

(with k 2pf c denoting the free space wave number, and a being the Earth’s radius); rPHASE ð f Þ ¼ Re S ð f Þ; að f Þ ¼ j0:182 f ImS ð f Þj;

S2ð f Þ ¼

H1 ð f Þ H0 ð f Þ

[Wait, 1962; Galejs, 1972; Greifinger and Greifinger, 1978; Nickolaenko and Rabinowicz, 1982; Sentman, 1996; Mushtak and Williams, 2002]. [7] Besides, the waveguide can be characterized by its resonant properties formulated as the complex eigenfrequencies Fn fn + ijn (n = 1, 2, 3, . . .) presenting the poles of the corresponding modal members in (1), or as pairs of the modal frequency fn and quality factor Qn defined as

Qn ¼

fn ðn ¼ 1; 2; 3; . . .Þ 2jjn j

ð5Þ

[Madden and Thompson, 1965; Galejs, 1972; Bliokh et al., 1980]. If a model is formulated via the complex eigenvalue, the equation for the eigenfrequencies is nðFn Þ½nðFn Þ þ 1 nðn þ 1Þ ¼ 0;

ð6Þ

while for models formulated via the complex sine the equation turns into

2 2pFn a S ðfn Þ nðn þ 1Þ ¼ 0 c

ð7Þ

with an obvious transformation via (4) for models formulated in terms of complex characteristic altitudes. Equation (7) can be reduced to a transcendental equation for the modal frequencies

fn ¼ fnð0Þ

Re S ðfn Þ jS ðfn Þj2

ð8Þ

;

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c where fn(0) 2pa nðn þ 1Þ are the resonance frequencies of a hypothetical waveguide with sharply bounded and perfectly conductive walls [Schumann, 1952; Jones, 1964]. After equation (8) is solved, the imaginary parts of the eigenfrequencies are expressed from (7) as

jn ¼ fnð0Þ

Im S ðfn Þ jS ðfn Þj2

;

ð9Þ

while for the quality factors follows, from (5), (8), and (9), a well-known [Galejs, 1972; Mushtak and Williams, 2002] expression

ð3Þ

ð4Þ

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Qn ¼

Re S ðfn Þ 2jIm S ðfn Þj

ð10Þ

and vice versa, on the basis of the key equation (7), the values of the complex sine (and therefore those of the phase

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velocity and attenuation rate) can be calculated from the waveguide’s resonant characteristics as Re S ðfn Þ ¼

fnð0Þ Im S ðfn Þ ; fn 1 þ 4Q1 2

ð11Þ

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recorded at widely separated sites [Bliokh et al., 1977], and from Q-burst spectra, Nickolaenko [1984] derived a linear formulation for the frequency dependence of the eigenvalue:

n

nð f Þ ¼ Af þ B: Re S ðfn Þ Im S ðfn Þ ¼ : 2Qn

ð12Þ

[8] As in an earlier work by Sentman [1996], a particular emphasis has been placed in this study on the modal resonance frequencies and the quality factors, rather than on the real and imaginary parts of the complex eigenvalue. The reasons are simple: first, only the former quantities can be directly extracted from SR spectra by means of the Lorentzian technique described below (section 4). Second, that is the extracted resonant characteristics that can be directly recalculated, via (3), (11), and (12), into physical properties of the waveguide, namely into the values of the phase velocity and attenuation rate at the corresponding modal frequencies.

3. Theoretical Models of Propagation Parameters [9] The five analytical models to be tested are briefly discussed below along with an indication of the literature sources of the specific values of their variables. 3.1. Ishaq and Jones [1977] Model [10] The values of phase velocity and attenuation rate were computed via full-wave numerical calculations in a uniform cavity [Jones, 1967] with a large (exceeding 20) number of layers presenting a profile of ionospheric conductivity assumed to be globally representative. The values of these propagation parameters obtained by Ishaq and Jones [1977] are based on this earlier work by Jones [1967] and not on SR observations, as it is commonly believed. In a later study by Burke and Jones [1992], where the analysis of large transient excitations of the Earthionosphere waveguide was used to determine the eigenvalue n( f ), it was reported that Ishaq and Jones’s [1977] results are ‘‘in agreement with the more comprehensive current results.’’ This finding is important, as it confirms the expectation that propagation parameters derived from the so-called ‘‘background’’ and ‘‘transient’’ Schumann resonances should, at least in principle and under certain methodological limitations, agree. [11] The analytical formulations for the propagation parameters’ frequency dependences derived by Ishaq and Jones [1977] from their full-wave computations are c ¼ 1:64 0:1759 ln f þ 0:0179 ln2 f ; VPHASE ð f Þ

að f Þ ¼ 0:063f 0:64 :

1 This formulation, with A = 16 i100 and B = 13, was later applied for interpreting both background Schumann resonance signals and Q-bursts [Nickolaenko, 1994, 1997]. In the present study, a slightly modified version of the linear model, in better agreement with the observed quality factors, is tested, with the first coefficient changed 1 . to A = 16 i70

3.3. Two-Exponential Model [Greifinger and Greifinger, 1978] [13] Greifinger and Greifinger [1978] developed an elegant analytic theory for ELF propagation parameters focused on two layers of pronounced energy dissipation known from the earlier studies by Madden and Thompson [1965] and Cole [1965]. The two-exponential version of this theory (representing the portion of the conductivity profile in each layer as an exponential) was developed, and successfully applied, for submarine communications at frequencies 45 Hz and up, i.e., beyond the SR frequency range that is the focus of the present study. Nevertheless, Nickolaenko and Rabinowicz [1982, 1987] and independently Sentman [1990, 1996] and Fu¨llekrug [2000] adopted this model for theoretical studies within the SR frequency range. In its simplified version [Sentman, 1996], this model has been used with four variables, two frequency-independent real parts h0 and h1 of the complex characteristic altitudes and two scale heights, V0 and V1, characterizing the exponential portions of the conductivity profile within the lower and upper characteristic layers, respectively. The scale heights selected by Sentman [1990] were equal to 5.0 km for both dissipation layers, with the frequency-independent characteristic altitudes h0 = 50 km and h1 = 90 km. For this model to be in accordance with the consistent twoexponential approach of Greifinger and Greifinger [1978], corresponding logarithmic frequency dependences need to be ascribed to the real parts of the characteristic altitudes [Nickolaenko and Rabinowicz, 1982; Kirillov, 1993; Mushtak and Williams, 2002], after which the complete formulation of the two-exponential model is

ð13Þ

ð14Þ

3.2. Linear Model [Nickolaenko, 1984] [12] On the basis of comparing Q-factors extracted from the Schumann resonance power spectra, from cross-spectra

ð15Þ

f pV0 * H0 ð f Þ ¼ h0 þ V0 ln ; þi 2 f*

ð16Þ

f pV1 i H1 ð f Þ ¼ h0* V1 ln ; 2 f*

ð17Þ

where f * is an arbitrary reference frequency, and h*0 and h*1 are the real parts of the characteristic altitudes calculated from the corresponding physical conditions at the reference frequency [Greifinger and Greifinger, 1978; Mushtak and Williams, 2002]. The specific values of the variables used in this study are close to those suggested by Sentman [1996]:

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Table 1. Comparisons of Theoretically Calculated and Recovered Schumann Resonance Characteristics (Modal Frequencies and Quality Factors) in Various Models of Propagation Parameters Model

Quantities

Mode I

Mode II

Mode III

Mode IV

Mode V

Ishaq and Jones [1977]

Theoretical fn/Qn Recovered fn/Qn Deviations Theoretical fn/Qn Recovered fn/Qn Deviations Theoretical fn/Qn Recovered fn/Qn Deviations Theoretical fn/Qn Recovered fn/Qn Deviations Theoretical fn/Qn Recovered fn/Qn Deviations

7.71/4.08 7.83/4.02 +1.6%/1.5% 7.94/5.83 8.00/6.18 +0.8%/+6.0% 7.71/4.15 7.80/3.98 +1.2%/4.1% 7.66/6.30 7.69/6.17 +0.4%/2.1% 7.75/4.01 7.86/3.88 +1.4%/3.2%

13.98/4.86 14.11/4.74 +0.9%/2.5% 13.90/5.83 13.97/6.08 +0.5%/+4.3% 13.97/4.25 14.06/4.07 +0.6%/4.2% 13.65/6.41 13.71/6.28 +0.4%/2.0% 13.97/5.20 14.10/5.16 +0.9%/0.8%

20.24/5.43 20.38/5.25 +0.7%/3.3% 19.85/5.83 19.93/5.98 +0.4%/+2.6% 20.31/4.31 20.39/4.12 +0.4%/4.5% 19.66/6.47 19.72/6.26 +0.3%/3.2% 20.11/6.02 20.23/5.91 +0.6%/1.8%

26.52/5.90 26.67/5.68 +0.6%/3.7% 25.81/5.83 25.89/5.95 +0.3%/+2.1% 26.75/4.35 26.84/4.22 +0.3%/3.0% 25.71/6.52 25.78/6.27 +0.3%/3.8% 26.24/6.61 26.36/6.46 +0.5%/2.3%

32.81/6.31 32.96/6.12 +0.5%/3.0% 31.77/5.83 31.85/5.81 +0.3%/0.3% 33.28/4.38 33.42/4.30 +0.4%/1.8% 31.82/6.55 31.88/6.26 +0.2%/4.4% 32.39/7.07 32.50/6.87 +0.3%/2.8%

Linear Two-Exponential Single-Exponential ‘‘Knee’’

V0 = V1 = 5 km, and h*0 = 50 km, h*1 = 93.5 km at the reference frequency f * = 8 Hz. 3.4. Single-Exponential Model [Greifinger and Greifinger, 1978] [14] Nickolaenko and Rabinowicz [1982, 1987] adopted the even more simplified single-exponential version of the Greifinger and Greifinger [1978] technique to calculate Schumann resonance characteristics expected within the waveguide on the planet Venus. A single-scale profile was derived also for the Earthionosphere cavity as a reference model for checking the modeling validity of this approach. [15] Fullekrug [2000] also made use of the single-exponential version of the Greifinger and Greifinger [1978] formalism. Appealing to the conductivity profile shown by Tran and Polk [1979] and by Polk [1982, Figure 34] for the height range 55 to 90 km, Fu¨llekrug [2000] modeled the profile by a single exponential with the scale height V0 = 3.22. The propagation parameters are then formulated, in terms of two complex characteristic altitudes, as H0 ð f Þ ¼ h0* þ V0 ln

f pV0 þi ; 2 f*

H1 ð f Þ ¼ ReH0 ð f Þ 2V0 lnð2kV0 Þ i

pV0 ; 2

ð18Þ

sion of the long-known [Cole, 1965; Jones, 1967] ‘‘kneelike’’ (when shown in the semilogarithmic scale) vertical transition from ion-dominated to electron-dominated conductivity, while still retaining the theory’s analyticity and the formalism of the two characteristic dissipation altitudes. This model also attempts a phenomenological treatment of the wave energy escaping from the waveguide (an effect especially pronounced under nighttime conditions), following earlier modeling effort by Madden and Thompson [1965]. Mathematically, the ‘‘knee’’ model is formulated as " 2 # f 1 fkn þ ðVa Vb Þ ln 1 þ ; Re H0 ð f Þ ¼ hkn þ Va ln f fkn 2 ð20Þ

Im H0 ð f Þ ¼

3.5. ‘‘Knee’’ Model [Mushtak and Williams, 2002] [16] Mushtak and Williams [2002] sought to improve the application of the general Greifinger and Greifinger [1978] theory to the Earth-ionosphere waveguide by explicit inclu-

ð21Þ

f pV1 ð f Þ * ; Im H1 ð f Þ ¼ Re H1 ð f Þ ¼ h1 V1 ð f Þ ln ; ð22Þ 2 f1*

ð19Þ

where f * is an arbitrary reference frequency, h*0 is the real part of the lower characteristic altitude calculated from the corresponding physical condition (namely, the conduction current being equal to the displacement current) at the reference frequency [Greifinger and Greifinger, 1978; Mushtak and Williams, 2002], and k = 2pf c . The set of the specific values for the variables used in this study is adopted from Fu¨llekrug [2000]: f * = 8.202 Hz, h*0 = 48.68 km, V0 = 3.22 km.

Va p fkn ; ðVa Vb Þ arctan f 2

V1 ð f Þ ¼ V1* þ b1

1 1 ; f f1*

ð23Þ

where fkn is related to the ionospheric conductivity skn at the ‘‘knee’’ altitude hkn (symbolizing the vertical transition from the ion-dominated to electron-dominated conductivity) as fkn = skn/(2pe0); Vb and Va are the scale heights of the exponential functions approximating the conductivity profile within the lower characteristic layer below and above the hkn altitude, respectively; V*1 is the scale height within the upper characteristic layer related to an arbitrary reference frequency f* 1, and b1 is a coefficient responsible for frequency dependence (23) of the effective scale height V1( f ) within the upper layer. The specific values of the variables used in this study are adopted from the ‘‘Empirical – SR Band’’ model in the work of Mushtak

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Table 2. Comparisons of the Theoretically Calculated and Recovered Values of Propagation Parameters (the Phase Velocity Ratio and the Attenuation Rate) Related to the Corresponding Recovered Modal Frequencies From Table 1 Model

Quantities

Mode I

Mode II

Mode III

Mode IV

Mode V

Ishaq and Jones [1977]

Theoretical rPHASE/a Recovered rPHASE/a Deviations Theoretical rPHASE/a Recovered rPHASE/a Deviations Theoretical rPHASE/a Recovered rPHASE/a Deviations Theoretical rPHASE/a Recovered rPHASE/a Deviations Theoretical rPHASE/a Recovered rPHASE/a Deviations

1.354/0.235 1.334/0.236 1.5%/+0.5% 1.326/0.166 1.317/0.155 0.7%/6.3% 1.354/0.232 1.339/0.239 1.2%/+3.0% 1.376/0.153 1.370/0.155 0.4%/+1.7% 1.347/0.238 1.327/0.245 1.4%/+2.6%

1.300/0.343 1.287/0.349 0.9%/+ 1.8% 1.313/0.278 1.306/0.273 0.5%/1.9% 1.297/0.390 1.288/0.405 0.7%/+3.8% 1.337/0.260 1.332/0.265 0.4%/+1.6% 1.303/0.320 1.291/0.321 0.9%/+0.3%

1.272/0.434 1.263/0.446 0.7%/+2.9% 1.299/0.393 1.295/0.393 0.4%/0.1% 1.262/0.543 1.256/0.566 0.5%/+4.1% 1.314/0.364 1.309/0.375 0.3%/+3.1% 1.283/0.392 1.275/0.397 0.6%/+1.4%

1.255/0.515 1.248/0.533 0.6%/+3.5% 1.290/0.509 1.286/0.509 0.3%/+0.2% 1.237/0.695 1.232/0.713 0.4%/+2.7% 1.297/0.467 1.293/0.484 0.3%/+3.7% 1.271/0.460 1.265/0.470 0.5%/+2.1%

1.244/0.590 1.238/0.607 0.5%/+2.8% 1.284/0.624 1.280/0.639 0.3%/+2.3% 1.218/0.846 1.213/0.858 0.4%/+1.4% 1.283/0.568 1.280/0.593 0.2%/+4.4% 1.262/0.528 1.257/0.541 0.3%/+2.6%

Linear Two-Exponential Single-Exponential ‘‘Knee’’

and Williams [2002]: fkn = 10 Hz, hkn = 55 km, Vb = 8.3 km, Va =2.9 km, f *1 = 8 Hz, h* 1 = 96.5 km, V* 1 = 4 km, b1 = 20 km Hz.

4. Theoretical Computations of Resonant Characteristics and Propagation Parameters [17] The theoretical values of the resonant characteristics (modal frequencies fn and quality factors Qn for SR modes I to V) calculated for uniform waveguides with propagation parameters defined in the above five models are presented in the ‘‘theoretical’’ rows of Table 1. (The ‘‘recovered’’ values in the table are considered and discussed later, in section 6). The eigenfrequencies of the Ishaq and Jones [1977] model (13) –(14) have been computed by solving transcendental equation (8) with the complex sine calculated from the model’s parameters via (3). The eigenfrequencies of the linear model (15) have been determined by directly solving (6) that, in the limits of this model, turns into a complex quadratic equation. The eigenfrequencies for the two-exponential (16) – (17), single-exponential (18) – (19), and ‘‘knee’’ (20) – (23) models have been computed from transcendental equation (8) with the values of the complex sine calculated via (4). The quality factors shown in Table 1 have been computed from the eigenfrequencies via (5) for the linear model and from the complex sine’s values at modal frequencies via (10), using, when necessary, (4), for the rest of the models. [18] The theoretical values of the physical characteristics (phase velocity ratios rPHASE(fn) and attenuation rates a(fn) at the theoretical modal frequencies fn for SR modes I to V) for uniform waveguides with propagation parameters defined in the above models are shown in the ‘‘theoretical’’ rows of Table 2. (The ‘‘recovered’’ values in the table are discussed later, in section 6). These characteristics have been computed from the theoretical eigenfrequencies (Table 1) via relationships (11), (12), and (3).

5. Spectral Computations [19] It is a well-known mathematical and experimental fact that the form of Schumann resonance power spectra,

and hence the apparent resonant and propagation parameters recovered from the spectra, depend on the sourcereceiver geometry, regardless of what model is assumed [Balser and Wagner, 1962; Ogawa et al., 1968; Jones, 1969; Galejs, 1970; Ogawa and Murakami, 1973; Bliokh et al., 1977, 1980; Sentman, 1996; Heckman et al., 1998; Nickolaenko and Hayakawa, 2002; Ogawa, 2002]. The distance dependences of apparent resonant characteristics are the most explicit for individual sources (lightning flashes modeled by vertical electric dipoles). The distance dependence becomes less pronounced, but still readily apparent, in scenarios in which thousands of flashes occurring on continental scales contribute to the observed signal. In such a scenario, the resonant characteristics, being global parameters of the Earth-ionosphere waveguide, are reflected in SR observations in a rather distorted form. One of the ELF challenges is developing procedures for recovering these characteristics (and, therefore, the waveguide propagation parameters) as accurately as possible. [20] The objective of the present study is estimating the ultimate accuracy of the widely used Lorentzian fitting procedure (section 6) when recovering propagation parameters from ‘‘experimental’’ Schumann resonance data simulated by theoretically calculated power spectra in various models. It is surmised that the ultimate accuracy is achieved by eliminating the source-receiver factor assuming an idealized situation with a globally uniform distribution of sources [Raemer, 1961; Bliokh et al., 1977, 1980; Sentman, 1996; Nickolaenko and Hayakawa, 2002]. In this hypothetical situation, every observation site, irrespective of its geographical position, collects and records the same spectrum, which provides the desired global invariant. Mathematically, such a methodological idealization can be realized only for electric power spectra, since magnetic spectra have been shown to lack the necessary physical and mathematical convergence in the case of a uniform distribution of sources [Sentman, 1996; Nickolaenko and Hayakawa, 2002]. [21] The electric spectrum obtained by spatially integrating over the uniform distribution and statistically averaging an ensemble containing a series of individual signals (1), as

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Figure 1. Normalized theoretical spectra for the five models of ELF propagation parameters presented in section 3.

is formulated [Raemer, 1961; Bliokh et al., 1977, 1980; Sentman, 1996; Nickolaenko and Hayakawa, 2002] as

nð f Þ½nð f Þ þ 1 2

hjE 2 ð f Þji ¼ AhjM ð f Þj2 i

fHEFF ð f Þ

1 X ð2n þ 1Þ n ð n þ 1 Þ nð f Þ½nð f Þ þ 1j2 j n¼0

ð24Þ

where A is a frequency-independent factor (and not so important for the present study), and the angular braces denote statistical averaging over the ensemble of individual signals. When deriving (24), the orthogonality of the Legendre polynomials is being exploited, illustrating an additional advantage of the field’s representation via the zonal harmonics series (1). [22] From (24), the electric field power spectra were computed for each of the five models presented in section 3; to exclude edge effects when applying the recovering procedure (section 6), as many as 25 terms had been summed in the series figuring in (24). Depending on the model, the starting point was either the eigenvalue n( f ) (the linear model), or the complex sine S(f) (the Ishaq and Jones [1977] model), or the pair of complex characteristic altitudes (single- and two-exponential as well as ‘‘knee’’ models) recalculated into the eigenvalue on the basis of general relationships presented in section 2. In the models consistently formulated via the complex characteristic altitudes, the effective height HEFF( f ) is equal to the lower characteristic altitude [Mushtak and Williams, 2002], while the Ishaq and Jones [1977] and linear models do not formally define the frequency dependence of this parameter. For this reason, in the two latter models the effective height was included in the frequency-independent factor A in (24).

[23] The computed power spectra for all five models are shown in Figure 1. For methodological reasons, the spectra are normalized in such a manner that the apparent peak power of the third resonance (as the center one within the frequency range under consideration) in each model is equal to one relative unit. When computing the spectra, the hjM( f )j2i factor was supposed to be frequency-independent (the ‘‘white noise’’ scenario); in practical computations, this dependence can also be taken into account [Jones and Kemp, 1971]. The bold curves in Figure 1 show the ±25% deviations from the power spectrum in the Ishaq and Jones [1977] model considered as a reference one for two important reasons: first, it is based on a realistic (even though averaged) conductivity profile; second, its variables had been obtained by means of direct full-wave, not approximate, computations [Jones, 1967]. At first glance, the spectra from different models are remarkably similar: all curves are found within the ±25% interval almost everywhere except in the vicinity of SR I, where the single-, two-exponential and linear, but not the ‘‘knee,’’ models noticeably spill out of the interval.

6. Lorentzian Procedure and Results of Simulations [24] In this study, following Sentman’s [1987] suggestion, the frequency behavior of each SR mode is assumed approximately described by an individual Lorentzian function (for short, a Lorentzian) Pn ð f Þ ¼ Pnmax

h

1 þ 2Qn

1

f fn

i2 ; 1

ð25Þ

where Pmax n , fn, and Qn are the nth mode’s spectral intensity, center frequency, and quality factor, respectively. (In this

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formulation, the quality factor presents the ratio of two visual parameters of representation (25), namely that of the center frequency to the half-power width of the resonance curve.) This assumption is hardly a novel one, since the very earliest studies of SR spectra and procedures for extracting the modal quality factors followed this approach [Balser and Wagner, 1962]. The improvement here is that by assigning an independent Lorentzian to each resonant mode, the well-known effect of modal overlap (resulting from the strongly damped nature of the waveguide at ELF) can be reasonably accounted for. For this purpose, the theoretical spectrum is presented, treated and fitted to experimental spectra as a sum of individual Lorentzians: hjE ð f Þj2 i ¼

N X

Pn ð f Þ;

ð26Þ

n¼1

where N is the number of individual Lorentzians to be summed, a parameter selected on the basis of the frequency range considered and the desired level of compensating the edge effects (in the present study, this number was selected as 10). Sentman [1987] recommended this improved fitting method a quarter century after Balser and Wagner [1962], when digital (and therefore more suitable for sophisticated signal processing) spectra were first appearing. Tens of thousands of SR background spectra automatically collected at a Rhode Island ELF site have now been fit by this method [Heckman et al., 1998; Mushtak and Williams, 2002]. [25] The Lorentzian form (25) has a well-studied physical basis; for instance, the natural lines of atomic and molecular spectra are formulated in an analogous form. As discussed by Feynman et al. [1963], (25) presents a solution of the damped single-dimensional harmonic oscillator problem in the case of very weak damping, the usual situation in atomic physics. As applied to the Schumann resonance range, the form (25) is approximate in at least two ways: (1) the SR modes are not weakly damped (it is well-known from experiments that the quality factors of the Earth-ionosphere cavity are of order 3 – 8 rather than several tens or hundreds in other applications), and (2) the SR phenomenon occurs in a three-dimensional damped oscillator, conforming to Maxwell’s equations, rather than to the single-dimensional system with a lumped damping parameter. [26] The nature of approximations occurring when the Lorentzian approach is applied for recovering the Schumann resonance characteristics can be presented in a mathematically (and physically) visual way if the frequencydependent (assuming the ‘‘white-noise’’ scenario) factor of spectrum (24) is reformulated as # 1 2 X ðn Þ hjE ð f Þji 2 þ A ðfÞ ; HEFF ð f Þ f 2 n¼1 E 1

2

"

ð27Þ

where ðn Þ

AE ð f Þ

2ð2n þ 1Þf 2 jf 2 Y2n ð f Þj2

Yn ð f Þ

fnð0Þ ; Sð f Þ

;

ð28Þ

ð29Þ

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c and fn(0) 2pa nðn þ 1Þ are ‘‘the natural resonance frequencies of the cavity [. . .] with a sharply bounded, perfectly conducting ionosphere’’ [Jones, 1964]. If members (28) of series (27) are, in their turn, reformulated as ðn Þ

f2

2ð2n þ 1Þ

jf Yn ð f Þj jf þ Yn f j2 ( ) 2ð2n þ 1Þ f2 ¼ ImY2n ð f Þ jf þ Yn ð f Þj2

AE ð f Þ ¼

2

1þ

h

ReYn ð f Þ ImYn ð f Þ

1

f ReYn ð f Þ

i2 ; 1

ð30Þ

it can be seen that four simplifications of the Lorentzian approximation (25) – (26), in comparison with the exact formulation (24) now recast as (30), are (1) the effective height of ionosphere HEFF( f ) is assumed to be frequencyindependent; (2) the zeroth member 2

1

2

fHEFF ð f Þ

of initial series (24) is omitted; (3) the frequency dependences of the factors f2 jf þ Yn ð f Þj2

(related to the system’s poles in the negative frequency domain) are neglected, the factors themselves, along with Im Y2n(f), being fixed at the corresponding resonant frequencies f = fn, and the now frequency-independent expressions within the braces denoting the corresponding spectral intensities Pmax n ; (4) the Yn(f) functions in the rightmost factor’s denominator of (30) are also fixed at the corresponding resonant frequencies as Yn(fn) = fn(1 + i2Q1 n ), a relationship that follows from these functions’ definition (29) and relationships (8), (9), and (10) for the complex resonant frequencies of the system. [27] The logarithmically weak dependence of the lower characteristic altitude on frequency reasonably justifies simplification 1: in an averaged uniform model of the Earth-ionosphere waveguide [Mushtak and Williams, 2002] this parameter changes by less than 10% over the first four resonant modes; in a nonuniform day/night model [Greifinger et al., 2005], its change also does not exceed 10% under both daytime and nighttime conditions. The nonresonant character of the zeroth term [which we associate with the steady excitation of the ‘DC’ global circuit in accordance with Wait’s [1962] interpretation] and of the terms related to the poles in the negative frequency domain justify simplifications 2 and 3, respectively. Simplification 4 means, physically, an assumption of a ‘‘limited’’ frequency dispersion of the system: the phase velocity depends on the mode’s number but is assumed to be frequency independent and equal to its value at the corresponding modal frequency in the Lorentzian approximation for the given mode. (An analogous limitation of the frequency dispersion can be found, for instance, in Sentman’s [1995] ‘‘resonant’’ expres-

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Figure 2. Comparison of theoretical (filled symbols) and recovered (clear symbols) quality factors for SR modes I to V in the five models presented in section 3.

sions for the electric and magnetic fields for a trivialized two-exponential model with frequency-independent characteristic altitudes.) [28] To achieve this study’s objective, each of the ‘‘experimental’’ spectra shown in Figure 1 were fit with a sum of ten Lorentzians (25), with ten sets {Pmax n , fn, Qn} estimated to obtain finally the resonant characteristics for SR modes I to V. The individual modes were fitted first, after which the procedure was performed globally for the entire spectrum (26) until the deviations became sufficiently small. (Some details of the least-squares fitting procedure, applied in practice for processing SR observations, are discussed by Mushtak and Williams [2002].) The extracted modal frequencies and quality factors are presented in the rows labeled ‘‘Recovered fn/Qn’’ in Table 1, along with the deviations of the extracted resonant characteristics from the postulated (theoretical) ones. An analogous comparison for recovered propagation parameters, recalculated from the extracted resonant characteristics on the basis of the general relationships of section 2, is presented in Table 2.

7. Discussion [29] The degrees of agreement between the recovered and postulated properties of the model waveguides are shown in the ‘‘deviations’’ rows of Table 1 (for resonant characteristics) and Table 2 (for propagation parameters). In general, the agreement is practically acceptable for all five SR modes and for all five models under consideration. Regarding specifically the modal frequencies (and, correspondingly, the phase velocity ratios), all but a few pairs of values agree to better than one percent. As to the quality factors (Table 1) and, correspondingly, attenuation rates (Table 2), the Q-factors’ deviations are consistently greater than in the frequency comparisons, but are still practically acceptable, with the worst-case values (excluding SR I in the linear, the

least adequate, model) lower than 5%. Noteworthy however is the systematic sign of the deviations, a feature likely attributable to the deviations of the symmetrical forms of Lorentzian terms (25) from the more complicated asymmetrical forms of exact modal terms (30) due to simplifications 1– 3 and, chiefly, 4 discussed in the previous section. [30] The extracted quality factors are visually compared with the postulated ones in Figure 2, while the recovered propagation parameters, recalculated, for uniformity, as the complex sine S( f ) (equations (11) and (12)), are compared with their theoretical values in Figure 3. It can be seen from both figures that the general trends of the quality factor with frequency in any given model are faithfully reproduced in the spectral fits (when considering Figure 3, it will be recalled that Qn ¼

ReS ðfn Þ ; 2ImSn ðfn Þ

see section 2). At the same time, the most notable distinctions between the models lie exactly in the Q-factor’s dependence on the modal number.

8. Conclusions [31] The simple Lorentzian procedure has been shown, in the limits of the modeling approach, to provide a promising technique for extracting SR resonant characteristics and recovering ELF propagation parameters from SR spectra. Under idealized conditions of both uniform waveguide and uniform source distribution, accuracies of about a few percent or better are achieved for all the parameters and for all the models examined. At the same time, clear-cut distinctions among these models in the behavior of waveguide dissipation, the distinctions, not readily apparent in the initial spectra (Figure 1), are revealed in the extracted

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Figure 3. Comparison of theoretical (filled symbols) and recovered (clear symbols) values of the complex sine as an ELF propagation parameter for SR modes I to V in the five models presented in section 3. characteristics (Tables 1 and 2, Figures 2 and 3), making the Lorentzian procedure also applicable for estimating the adequacy levels of various models of propagation parameters. [32] These general conclusions, while purely theoretical in the present study, have been practically confirmed by Mushtak and Williams [2002] when comparing the observed frequency behavior of the Q-factor with modeling predictions. Namely, it was shown that the Q-factor’s dependence on the mode number is too flat, in comparison with experimental findings, for any model that ignores the realistic structure of the lower ionosphere. The most important feature of this structure is the pronounced vertical transition from the ion-dominated to electron-dominated conductivity in the lower D region. The importance of this transition for ELF propagation has been repeatedly emphasized and discussed since the 1960s [Galejs, 1962; Wait, 1964; Cole, 1965; Madden and Thompson, 1965; Jones, 1967], but up to now most analytical models do not incorporate this key feature. The models by Ishaq and Jones [1977] and Mushtak and Williams [2002], even though constructed in methodologically different ways (numerical and analytical, respectively), are in a reasonable agreement with each other in a theoretical sense (Tables 1 and 2, Figures 1, 2, and 3) and with experimental findings in a practical sense [Mushtak and Williams, 2002]. Recent findings by Yang and Pasko [2005] using yet another modeling approach, confirm this ability to distinguish between different models. [33] The distinguishing efficiency of the Lorentzian procedure has been recently demonstrated in even more pronounced manner by Satori et al. [2005]. On the basis of long-term SR observations collected at ELF sites in the USA, Hungary, and the Antarctic during two solar cycles, systematic variations of SR characteristics with solar activity have been revealed by means of various processing techniques. It was the above-discussed Lorentzian procedure that, being applied at the MIT site in West Greenwich

(Rhode Island), provided the most informative set of experimental data suitable for estimating the ionospheric dynamics with solar activity. In the application of the ‘‘knee’’ model (section 3.5) for interpreting the observed variations, it was shown that the estimates of the model’s variables for the upper characteristic layer (which is much more subject to the solar cycle modification than the lower one) demonstrate a pronounced, statistically significant and physically consistent contrast between solar minimum and solar maximum. Recently, Schumann resonance observations carried out on the basis of the Lorentzian technique had been successfully used to test theoretical predictions for the day-to-night change in the lower characteristic altitude [Greifinger et al., 2005]. The cited results not only reaffirm the earlier findings that some important features of the ionospheric structure can be linked with and therefore tested on the basis of experimental SR spectra [Madden and Thompson, 1965; Jones, 1967; Ogawa and Murakami, 1973; Bliokh et al., 1977; Tran and Polk, 1979], but also demonstrate a prospect, provided properly chosen models and correspondingly planned experiments, to distinguish between different states of the ionosphere (solar min/max, day/night, etc.). [34] Acknowledgments. This study could not have been completed without an invaluable contribution of Robert Boldi (MIT Lincoln Laboratory), an expert on Lorentzian fits. Discussions with Martin Fu¨llekrug, Phyllis Greifinger, David Jones, Toshio Ogawa, Davis Sentman, Gabriella Sa´tori, Alexander Shvets, and Victor Pasko are greatly appreciated. The authors are grateful to Physical Meteorology Program of the U.S. National Science Foundation (ATM-0337298) and the Twinning Foundation for supporting this work and to Kelly Robbins for efficient assistance.

References Balser, M., and C. Wagner (1962), On frequency variations of the Earthionosphere cavity modes, J. Geophys. Res., 67, 4081. Bliokh, P. V., Yu. P. Galyuck, E. M. Hynninen, and A. P. Nickolaenko (1977), On the resonance phenomena in the Earth-ionosphere cavity (in Russian), Isvestiya Vuzov, Radiofiz., 20, 501.

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Bliokh, P. V., A. P. Nickolaenko, and Y. F. Filippov (1980), Schumann Resonances in the Earth-Ionosphere Cavity, Peter Perigrinus, New York. Burke, C. P., and D. Ll. Jones (1992), An experimental investigation of ELF attenuation rates in the Earth-ionosphere duct, J. Atmos. Terr. Phys., 54, 243. Cole, R. K., Jr. (1965), The Schumann resonances, J. Res. Natl. Bur. Stand., 69D, 1345 – 1349. Connor, J. N. L., and D. Mackay (1978), Accelerating the convergence of the zonal harmonic series representation in the Schumann resonance problem, J. Atmos. Sol. Terr. Phys., 40, 977. Feynman, R. P., R. B. Leighton, and M. L. Sands (1963), Resonance, in The Feynman Lectures on Physics, pp. 1 – 5, chap. 23, Addison-Wesley, Reading, Mass. Fu¨llekrug, M. (2000), Dispersion relation for spherical electromagnetic resonances in the atmosphere, Phys. Lett. A, 275, 80. Galejs, J. (1962), A further note on terrestrial extremely low frequency propagation in the presence of an isotropic ionosphere with an exponential conductivity-height profile, J. Geophys. Res., 67, 2715. Galejs, J. (1970), Frequency variations of Schumann resonances, J. Geophys. Res., 75, 3237. Galejs, J. (1972), Terrestrial Propagation of Long Electromagnetic Waves, Pergamon Press, New York. Greifinger, C., and P. Greifinger (1978), Approximate method for determining ELF eigenvalues in the Earth-ionosphere waveguide, Radio Sci., 13, 831. Greifinger, P., V. Mushtak, and E. Williams (2005), The lower characteristic ELF altitude of the Earth-ionosphere waveguide: Schumann resonance observations and aeronomical estimates, paper presented at 6th International Symposium on Electromagnetic Compatibility and Electromagnetic Ecology, IEEE, St. Petersburg, Russia. Heckman, S., E. Williams, and R. Boldi (1998), Total global lightning inferred from Schumann resonance measurements, J. Geophys. Res., 103, 31,775. Hynninen, E. M., and Y. P. Galyuck (1972), The field of a vertical electric dipole over the spherical Earth’s surface below the vertically inhomogeneous ionosphere (in Russian), in The Problems of Diffraction and Wave Propagation, vol. 11, pp. 109 – 115, Leningrad State Univ., Leningrad, Russia. Ishaq, M., and D. L. Jones (1977), Method for obtaining radiowave propagation parameters for the Earth-ionosphere duct at E.L.F., Electr. Lett., 13, 254 – 255. Jones, D. L. (1964), The Calculation of the Q factors and frequencies of Earth-ionosphere cavity resonances for a two-layer ionosphere model, J. Geophys. Res., 69, 403. Jones, D. L. (1967), Schumann resonances and ELF propagation for inhomogeneous, isotropic ionospheric profiles, J. Atmos. Terr. Phys., 29, 1037. Jones, D. L. (1969), The apparent resonant frequencies of the Earth-ionosphere cavity when excited by a single dipole source, J. Geomagn. Geoelectr., 21, 679. Jones, D. L. (1970), Numerical computations of terrestrial ELF electromagnetic wave fields in the frequency domain, Radio Sci., 5, 803. Jones, D. L., and C. P. Burke (1990), Zonal harmonic series expansions of Legendre functions and associated Legendre functions, J. Phys. A. Math. Gen., 23, 3159 – 3168. Jones, D. L., and D. T. Kemp (1971), The nature and average magnitude of the sources of transient excitation of Schumann resonances, J. Atmos. Terr. Phys., 33, 557 – 566. Kirillov, V. V. (1993), Parameters of the Earth-ionosphere waveguide at extremely low frequencies (in Russian), in The Problems of Diffraction and Wave Propagation, vol. 25, pp. 35 – 52, St. Petersburg State University, St. Petersburg, Russia. Madden, T. R., and W. Thompson (1965), Low frequency electromagnetic oscillations of the Earth-ionosphere cavity, Rev. Geophys., 3, 211.

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Mushtak, V. C., and E. R. Williams (2002), ELF propagation parameters for uniform models of the Earth-ionosphere waveguide, J. Atmos. Sol. Terr. Phys., 64, 1989. Nickolaenko, A. P. (1984), On the influence of the localized ionosphere non-uniformities on ELF radio propagation (in Russian), Izvestiya Vuzov Radiofiz., 27, 1227. Nickolaenko, A. P. (1994), ELF radio propagation in a locally non-uniform Earth- ionosphere cavity, Radio Sci., 29, 1187. Nickolaenko, A. P. (1997), Modern aspects of Schumann resonance studies, J. Atmos. Sol. Terr. Phys., 59, 805. Nickolaenko, A. P., and M. Hayakawa (2002), Resonances in the EarthIonosphere Cavity, 380 pp., Springer, New York. Nickolaenko, A. P., and L. M. Rabinowicz (1982), Possible Global Electromagnetic Resonances on the Planets of the Solar System, Plenum, New York. (Translated from Kosmicheskie Issledovaniya, 20, 82 – 88, 1982). Nickolaenko, A. P., and L. M. Rabinowicz (1987), Applicability of Ultralow-Frequency Global Resonances for Investigating Lightning Activity on Venus, pp. 239 – 243, Plenum, New York. (Translated from Kosmichseskie Issledovaniya, 25, 301 – 306, 1987). Ogawa, T. (2002), Polarity of Q-bursts, J. Atmos. Electr., 22, 35. Ogawa, T., and Y. Murakami (1973), Schumann resonance frequencies and the conductivity profiles in the atmosphere, Contrib. Geophys. Inst. Kyoto Univ., 13, 13. Ogawa, T., Y. Tanaka, and M. Yasuhara (1968), Diurnal variations of resonance frequencies in the Earth-ionosphere cavity, Contrib. Geophys. Inst. Kyoto Univ., 8, 15. Polk, C. (1982), Schumann resonances, in Handbook of Atmospherics, edited by H. Volland, CRC Press, Boca Raton, Fla. Raemer, H. R. (1961), On the spectrum of terrestrial radio noise at extremely low frequencies, J. Res. Natl. Bur. Stand., 65D, 581. Satori, G., E. Williams, and V. Mushtak (2005), Response of the Earthionosphere cavity resonator to the 11-year solar cycle in X-radiation, J. Atmos. Sol. Terr. Phys., 67, 553, 562. Schumann, W. O. (1952), On the radiation free self-oscillations of a conducting sphere, which is surrounded by an air layer and an ionospheric shell (in German), Zeitschrift Naturforschung, 7a, 149. Sentman, D. D. (1987), Magnetic elliptical polarization of Schumann resonances, Radio Sci., 22, 595. Sentman, D. D. (1990), Approximate Schumann resonance parameters for a two-scale height ionosphere, J. Atmos. Terr. Phys., 52, 35. Sentman, D. D. (1995), Schumann resonances, in Handbook of Atmospheric Electrodynamics, vol. 1, edited by V. Holland, CRC Press, Boca Raton, Fla. Sentman, D. D. (1996), Schumann resonance spectra in a two-scale height Earth- ionosphere cavity, J. Geophys. Res., 101, 9474. Tran, A., and C. Polk (1979), Schumann resonance and electrical conductivity of the atmosphere and lower ionosphere, J. Atmos. Terr. Phys., 41, 1241. Wait, J. R. (1962), Electromagnetic Waves in Stratified Media, Pergamon Press, New York. Wait, J. R. (1964), On the theory of Schumann resonances in the Earthionosphere cavity, Can. J. Phys., 42, 575. Yang, H., and V. P. Pasko (2005), Three-dimensional finite difference time domain modeling of the Earth-ionosphere cavity resonances, Geophys. Res. Lett., 32, L03114, doi:10.1029/2004GL021343.

V. C. Mushtak and E. R. Williams, Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. ([email protected]) A. P. Nickolaenko, Usikov’s Institute for Radio-Physics and Electronics, National Academy of Sciences of the Ukraine, Kharkov, 61085, Ukraine.

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