## 1. Introduction

[2] The electromagnetic field generated by about 2000 thunderstorm cells permanently active around the Earth, which produce approximately 50 lightning events every second [*Christian et al.*, 2003], is trapped in the Earth's atmosphere between the conducting ground and the ionosphere. The resonances of this enormous cavity are called Schumann resonances in honor to the work of *Schumann* [1952], who predicted the existence of these resonances, which were experimentally detected several years later by *Balser and Wagner* [1960]. Besides the use of Schumann resonances as a tool for monitoring the variations of the global thunderstorm activity and of ionospheric properties, there is a renewed interest in the phenomenon of Schumann resonances nowadays, for mainly two reasons. First, there exists a clear correlation between annual variations of the first Schumann resonance intensity and the tropical temperature over a 6-year span, which suggests the possibility of using these frequencies as a global thermometer for the tropics of the Earth [*Williams*, 1992]. Second, information from Earth's Schumann resonances may be of interest in the research of terrestrial-like electrical activity on other celestial bodies such as Mars [*Schwingenschuh et al.*, 2001a] or the Saturnian moon Titan [*Fulchignoni et al.*, 1997].

[3] Schumann frequencies are associated with magnetic modes transverse to *r* (TM^{r}). For a lossless cavity, these modes are determined by the roots of the following transcendental equation [*Morente et al.*, 2003]:

where *j*_{n} and *y*_{n} are the spherical Bessel functions and the prime denotes the derivative with respect to the function argument, *r*_{0} is the Earth's radius, *h* is the height of the ionosphere above the surface, and *k* is the wave number, i.e., *k* = 2π/λ. For *r*_{0} = 6370 km and *h* = 60 km, the above equation provides the following first six resonance frequencies, *f* = *ck*/2π: 10.54, 18.26, 25.83, 33.34, 40.83, and 48.32 Hz. These frequencies are in the extremely low frequency (ELF) range because their wavelengths are of the order of the Earth's diameter. Note that the values depend on *r*_{0} and *h*, even though the *h* dependence is very small. However, the observed Schumann frequencies are given by 7.8, 14, 20, 26, 33, and 39 Hz, respectively, i.e., by values noticeably lower than the theoretical predictions. It is assumed by most authors that this discrepancy is due to neglecting the effects of electric losses on the resonance frequencies. For the system under study, there exist losses due to propagation through an atmosphere with an electric conductivity which increases with height, together with losses associated with the fact that the ionosphere and the terrestrial surface are good but imperfect conductors. As it happens with other physical systems, for example, a damped harmonic oscillator, the inclusion of the damping effects not only reduces the amplitude as time proceeds, but also lessens the resonance frequency; the larger the dissipative constant, the larger the reduction.

[4] Strictly speaking, Schumann frequencies correspond to peaks in the spectrum of the global energy stored between the Earth's surface and its ionosphere. In this sense, Schumann frequencies are a property of the system as a whole and therefore do not depend on the position. Nevertheless, the experimental determination of this total energy is inaccessible, and Schumann frequencies are actually obtained through a local measurement of the electric or magnetic field resonances. As it will be shown in the next section by considering an electric circuit which largely replicates the electrical properties of the terrestrial atmosphere, the resonances in the current or voltage of inductors or capacitors for a lossy linear circuit do not necessarily match the resonance of the global circuit, these local resonances being a function of the dissipative parameters defining the energy losses in the system. In addition, it is worth noting that excitation will be explicitly considered in the method, which causes that the frequencies of maximum amplitude in the electric or magnetic field spectrum are not eigen but forced oscillations, which is a more realistic model for this oscillating system. Translating these results to the electromagnetic cavity defined by the Earth's atmosphere when excited by a temporal impulse signal to generate forced oscillations, a dependence on height of the peaks in amplitude of the electromagnetic field Fourier transform spectrum is expected, since the conductivity, which affects the frequency displacement, is also a function of the altitude.

[5] A realistic model for the determination of the Schumann resonances which includes loss effects related to the nonideal behavior of the dielectric between the conducting surfaces as well as to the fact that the ground and the ionosphere possess a high but finite conductivity, cannot be treated analytically and requires the application of numerical methods. The transmission line matrix (TLM) is a numerical tool that models electromagnetic field phenomena by means of circuit concepts and has been successfully used in the simulation of especially complex electromagnetic systems [*Christopoulos*, 1997; *Portí et al.*, 1998]. *Morente et al.* [2003] carried out a numerical modeling of the natural cavity formed by the Earth's surface and the lower ionosphere by means of the TLM method, and the results obtained correspond very well with the experimental ones. Furthermore, a good correlation between the widely accepted two-scale-height ionospheric model [*Greifinger and Greifinger*, 1978; *Sentman*, 1990] and the TLM results has also been observed. Therefore the previous work indicates that this numerical tool is useful in predicting the Schumann frequencies of other planets or moons as well, without making it necessary to fit the conductivity profile to particular schemes such as the two-layer model does. The versatility of the TLM method as a numerical tool allows the study of the dependence of the Schumann resonances for the electromagnetic field on the observation point. This study, for an atmosphere excited with an impulse signal, reveals that the maxima in the amplitude of the magnetic field Fourier transform spectrum remain practically constant for heights below 70 km, while they decrease between 1.5 and 4 Hz in the altitude range from 70 to 100 km, corresponding to a mean reduction of about 10%. It is important to emphasize that these results do not contradict previous ones of *Greifinger and Greifinger* [1978] and *Sentman* [1990], who found constant resonance eigenfrequencies, because they examined not forced but eigen oscillations, and hence they could not obtain a height-dependent frequency.