Schumann resonance frequencies are the resonance frequencies for an electromagnetic field inside the natural conducting cavity formed by the Earth's surface and its ionosphere. The spectrum peaks for the total electromagnetic energy stored inside the cavity define a unique value for each Schumann resonance. Because of the impossibility of measuring the total energy stored in the atmosphere, the experimental determination of Schumann resonances is carried out by a local measurement of the magnetic or electric field spectrum. However, peak frequencies in the amplitude of the electromagnetic field Fourier spectrum of a lightning discharge excited atmosphere may depend on height because this field depends on the atmosphere conductivity and this physical magnitude varies with height. Therefore it is reasonable to expect a slight shift between the peaks corresponding to the electric or magnetic field spectrum at a given height and those obtained by considering the total energy stored. In this paper the Earth's electromagnetic cavity is numerically modeled by using the transmission line matrix (TLM) numerical method. The altitude profile of the magnetic field spectrum remains constant from ground to 70-km height; however, a decay between 1.5 and 4 Hz is observed for heights between 70 and 100 km, which represents a medium reduction of 10%.
 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 , who predicted the existence of these resonances, which were experimentally detected several years later by Balser and Wagner . 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].
 Schumann frequencies are associated with magnetic modes transverse to r (TMr). For a lossless cavity, these modes are determined by the roots of the following transcendental equation [Morente et al., 2003]:
where jn and yn are the spherical Bessel functions and the prime denotes the derivative with respect to the function argument, r0 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 r0 = 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 r0 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.
 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.
 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.  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  and Sentman , who found constant resonance eigenfrequencies, because they examined not forced but eigen oscillations, and hence they could not obtain a height-dependent frequency.
2. Resonance in Electrical Circuits
 Those frequencies which allow the electromagnetic energy to be stored in a given electromagnetic cavity bounded by good conductors are known as system resonance frequencies. However, a noticeable difference of resonant cavities when compared with low-frequency resonant circuits is that the cavity possesses an undefined, although discrete, number of resonant modes or frequencies instead of a single or reduced number of resonance frequencies. The electromagnetic cavity in the Earth's atmosphere is excited by lightning activity, and the main lightning stroke is modeled by a current wave represented by two or more exponential functions of time [Nickolaenko and Hayakawa, 2002]. The circuit shown in Figure 1 represents a lossy linear system which shares several features with the Earth's cavity. This circuit is excited by a current source and is formed by the superposition of n stages in a similar manner as some planetary atmosphere models do [Schwingenschuh et al., 2001b]. Each stage introduces an inductance L, a capacitance C, and a resistance Ri, which are analogous to the magnetic permeability, the electric permittivity, and the atmospheric conductivity, σ, respectively. For frequencies below 50 Hz, which include the first six Schumann resonance frequencies, the atmosphere's permeability and permittivity are practically those corresponding to vacuum, μ0 and ɛ0, respectively, and hence the parameters L and C are identical for all the stages in the circuit, while the resistance is different for each stage as it happens with the atmospheric conductivity. The inductor, capacitor, and resistor are parallel connected at each circuit stage, a greater conductivity being obtained as the resistance in that stage becomes smaller. Since the conductivity increases with altitude, we assume that Ri > Ri+1 (Figure 1). It is interesting to note that a lossless circuit would be one with infinite resistance in all its stages, because in this case all resistor branches would in fact be open circuited, i.e., no current would circulate through them.
 Let us note I = I0ejωt, with j the imaginary unity, in order to study the circuit behavior at frequency ω. The total voltage V and impedance Z are defined by the contribution of all the circuit stages, i.e.,
The impedance of stage i is given by
which may be expressed in polar or Steinmetz () notation as
 By taking the derivative of the Zi module with respect to ω, it is easily shown that the maximum amplitude of ∣Zi∣ is reached at a frequency
 Since the result does not depend on Ri, the maximum impedance at all the stages will occur at the same frequency ω0, and because of the relation between voltage and current phasors, Vi = ZiI0, all the partial voltages Vi, and therefore the total voltage V as well, reach their maximum value also at this frequency. Hence ω0 is the resonance frequency for the circuit since it represents the frequency where the maximum energy transfer from the source to the rest of the circuit appears.
 Upon defining the damping factor at stage i by
and using equation (5), the relationship between the impedance Zi and the resistance Ri may be expressed as
The current through the resistance Ri is given by the phasor
and by means of equation (7) it is straightforward to show that the amplitude of is also maximum at the resonance frequency ω0 of the circuit.
 On the other hand, the current through the inductor at stage i is given by
and reaches its maximum amplitude at the frequency
Thus the resonance frequency (ωL)i for the inductors differs for each stage and is lower than the global circuit resonance, which means that the magnetic field reaches its maximum amplitude at a frequency different from ω0. The shift between (ωL)i and ω0 is proportional to δi and grows with increasing the index stage i, since Ri > Ri+1. Both resonance frequencies only coincide for a lossless system, in which δi equals zero and no current circulates through the resistive branches because they are open-circuited.
 Finally, the current passing through the capacitor at stage i can be expressed as
which reaches its maximum amplitude at the frequency
showing that there is also a difference between the resonance frequency (ωC)i of the capacitors and the resonance frequency ω0 of the global circuit.
 Summarizing, when dissipative elements are to be considered, the resonant linear system studied above has several magnitudes which resonate at frequencies different from the global circuit. In particular, the current through the capacitor and inductor, and therefore the magnetic fields associated to these currents, resonate at a different frequency from the global circuit voltage or the partial voltages at each stage do. Thus it is proved that quantities involved in a lossy linear system may resonate at different frequencies and that these frequencies deviate from the global system resonance at which the maximum energy transfer happens. The shift between the global and local resonance frequencies is proportional to the dissipative factor describing these local losses. This behavior is also expected in the terrestrial electromagnetic cavity, and it is therefore not surprising to find resonance frequencies associated with electric and magnetic field measurements that depend on height. These measured resonances will differ from the global resonance frequency of the system, where the discrepancies become more important with increasing height because the conductivity is a growing function of altitude.
3. The Earth's Cavity TLM Model: Numerical Results
 The TLM method is a powerful simulation tool widely used in the analysis and design of antennas, microwave devices, and electromagnetic compatibility structures with arbitrary geometry. The TLM method sets up a temporal and spatial iterative process, which allows the temporal evolution of the six electromagnetic field components to be obtained. It is based on the construction of a three-dimensional transmission line network, formed by interconnecting unitary circuits termed nodes or cells. Voltages and currents in this mesh behave similarly to the electromagnetic field in the original system [Christopoulos, 1995]. An efficient numerical model of the electromagnetic fields in the terrestrial cavity is given by Morente et al. , which provides numerical Schumann resonance frequencies for the Earth at ground level of 8.3, 15.8, 22.0, 29.4, 35.3, and 43.8 Hz, in good agreement with experimental values. The average electric and magnetic field profiles obtained with this model also show a very similar shape when compared with that obtained using the two-scale-height ionospheric model.
 The TLM mesh used is 20 × 18 × 1 nodes wide in the r, ϑ, and ϕ directions. The mesh models an atmosphere with a height of 100 km, which means that Δr = 5 km and Δϑ = 10°, and an optimum time step of Δt = 2.46 × 10−6 s is chosen to minimize dispersion errors. A wideband excitation is introduced in the TLM node located at point (3, 9, 1) near the equatorial plane, by imposing an impulse signal formed by an initial positive unitary voltage pulse for transmission lines 1–6 at this node and opposite voltage pulses for lines 7–12. The center of the excitation node is at 12.5-km altitude, which approximately corresponds to the region where electrical activity occurs. A total of 4.8 × 106 time step calculations have been carried out, which allows the signal to cover the Earth's perimeter more than 88 times. The boundary conditions imposed for the ϕ direction make the system and excitation to present rotational symmetry around the z axis (for details, see Morente et al. ). The conductivity profile of the atmosphere has been taken from Schlegel and Füllekrug , which is based on an electron density profile from the international reference ionosphere for afternoon equinox conditions. It should be noted that no additional parameters are required to achieve a good agreement with experimental values, in contrast to the semianalytical two-scale-height ionospheric model. The nodes located at points (i, 8, 1), with i = 1, …, 20, i.e., points along the radial line at ϑ = 75°, have been chosen as output points.
Figure 2 shows the Hϕ magnetic field spectrum, obtained via a discrete Fourier transform of the time-dependent series, for a lossless resonant electromagnetic cavity with dimensions similar to the terrestrial cavity at three different altitudes: close to ground level, at an intermediate altitude, and near the outer conducting sphere. Since it corresponds to a lossless system, the three Fourier transforms are almost coincident and therefore the peaks in the spectrum at different altitudes have identical values, which in addition coincide with the first Earth's resonances reported by Schumann . The frequency step is Δf = (4.8 × 106 Δt)−1 = 0.085 Hz, and the high-frequency aliasing error is negligible because the bandwidth needed to obtain the first six Schumann frequencies is very far from the Nyquist critical frequency, (2Δt)−1 ≈ 0.2 MHz. Figure 3 shows the Hϕ magnetic field Fourier transform spectrum for a more realistic model including the conductivity profile of the Earth's atmosphere and external surfaces with large but finite conductivity. This amplitude spectrum represents basically the system response to a lightning source, the actual excitation being composed of a continual series of randomly electric discharges. Now the peaks in the Fourier amplitude transform are smoother and lower than in the lossless case due to the damping taken into account in the model. Near ground level and at an intermediate altitude, the maxima in the magnetic field spectrum have similar frequency values, but a noticeable shift to lower frequencies can be observed in the maxima of the magnetic field Fourier amplitude for high altitudes, where conductivity effects are more important.
 Regarding the time attenuation factor of these signals, the atmospheric conductivity increases from 10−14 S/m near ground level to 10−3 S/m just below 100 km high, as shown in Figure 4. Therefore, at 25 Hz, in the middle of the frequency range of interest, the ratio between the displacement and conductivity currents, ω/σ, is 1.4 × 105 at ground level and 1.4 × 10−6 at 100 km high. The atmosphere behaves as a dielectric with low losses at ground level to become a good conductor for higher regions where the electromagnetic signal damps in the form e−βt due to electric losses. In the dielectric zone at ground level, the attenuation factor β is given by β ≈ σ/2ɛ ≈ 5.6 × 10−4 s−1, whereas for the conducting zone, at an altitude of 100 km, β is given by β ≈ ≈ 9.4 × 104 s−1 [Balanis, 1989]. In the intermediate zone, around 60 km high, ωɛ > σ, and the attenuation factor is β ≈ σ/2ɛ ≈ 10 s−1, which is roughly the medium value that is obtained in the TLM temporal series, common at all the altitudes, because the electromagnetic signal is constantly crossing regions with large and small attenuations factors.
Figure 4 also illustrates the profiles of the first six maxima in the Hϕ magnetic field spectrum. This spectrum is obtained via a Fourier transform of the time-dependent field with the entire time series as periodic input. An independent spectrum estimation has also been calculated by applying a unit window from 103 to 106 time steps. This window excludes the initial part of the time series, more related to the source signal than to the resonant system response. Both windows provide similar solutions for the frequencies at which Hϕ is maximum. The conductivity profile used is also included in this figure. It is seen that the peak frequencies are almost constant below about 70 km but start a slight decrease above this altitude where the conductivity becomes significant. At 100 km the difference in the peak frequencies with respect to ground level is found to be between 1.5 and 4 Hz, which corresponds to a mean reduction of around 10%. The agreement between these two Fourier transform options proves that the obtained frequency shifts are actual resonance frequencies which are altitude dependent and that changes are not merely artifacts of the discrete Fourier analysis applied to time truncated signals. Despite the change of the magnetic field peak frequencies with altitude not being very large, it represents a minimum shift of 17 Δf for the first resonance and 46 Δf for the sixth one, which makes the possibility of this dependency being a numerical error associated with the Fourier transform calculation unlikely. Therefore, if Schumann frequencies are experimentally determined through the local resonances of the magnetic field, then our results predict an altitude dependence of these frequencies which is expected to be observable at least above ∼70 km. The peak frequencies of the electric field, however, cannot be obtained, since the high level of the continuous electric field masks the low-frequency region where the Schumann resonances are located. Finally, shifts in the resonance frequencies are a priori expected when the zenithal coordinate varies and the distance from the source changes, since the different characteristic modes that are excited have different vertical structures and decay at different rates away from the source, so that the sum of them must have somewhat different properties at different distances from the source. Nevertheless, the TLM analysis has not reported these shifts, which means that forced resonances are only weakly dependent on the distance from the source.
 Measurements concerning the height dependence of the Schumann frequencies at Earth have been reported by Ogawa et al. , who analyzed data obtained aboard a balloon at 20- and 26-km altitude. They found no difference in the frequencies for the first and second modes, but the third resonance frequency at these heights was 0.2 Hz lower than on ground. In any case, the observation heights were relatively low and further measurements at higher altitudes are needed in order to confirm the numerical prediction mentioned above.
 The magnetic field Fourier transform spectrum in the natural electromagnetic cavity bounded by the Earth's surface and the ionosphere has been obtained using the TLM method when the system is excited by an electromagnetic impulse. Almost constant peak frequencies of the azimuthal component of the magnetic field spectrum for altitudes below 70 km are found, while a gradual decay for heights between 70 and 100 km is observed. Depending on the mode, the decrease ranges between 1.5 and 4 Hz at 100 km when compared with ground level values and is caused by the relatively high conductivity at higher altitudes. While the resonance frequencies of the cavity as a whole, for which the total energy becomes maximum, do not depend on altitude, the experimental determination of Schumann frequencies related with the maxima in the amplitude of the electromagnetic field Fourier transform spectrum is sensitive to the conductivity profile and thus will change with altitude.
 This work was supported in part by project HU2001-0017, between Austria and Spain, and by project 512002 of the WTZ-Programme of the ÖAD. The authors thank Ana Espinosa for her help in preparing the manuscript.
 Arthur Richmond thanks Yu P. Maltsev for assistance in evaluating this paper.