The study of the propagation of extremely low frequency (ELF) waves is essential for electromagnetic sounding investigations planned for some of the future Martian missions. Future surface stations will have the possibility of continuously recording low-frequency electromagnetic field fluctuations. Natural electromagnetic waves produced near the surface by electrostatic discharges in dust storms (dust devils) or by geological activity can be trapped in the resonant cavity formed by the surface and lower ionosphere as it occurs on the Earth. Low-frequency electromagnetic waves can also travel along the magnetic field lines of the recently discovered magnetic anomalies from the magnetosphere to the surface and may produce resonant structures in the cavity. The structure of the resonant frequencies, also called Schumann frequencies, is mainly determined by the geometry of the cavity and by the global electrical conductivity of the ionosphere/atmosphere. Measurements of Schumann frequencies by surface stations can be used for remote sensing of the electrical conductivity of the lower ionosphere/atmosphere. We present a numerical model of electromagnetic wave propagation based on the transmission line modeling (TLM) method with the aim of calculating the resonance frequencies on Mars and their dependence on solar activity and various possible ionization sources like meteoroids. The model has been previously validated by application to the terrestrial case. The numerical results obtained for the Earth are very close to the experimental ones, which supports our predictions on Mars. Our model can be used to study the global atmospheric conductivity using future real ELF measurements by surface stations or even balloons on Mars.
 The upper ionosphere of Mars has been explored by several spacecraft: Mariner, Mars, Viking and recently Mars Global Surveyor [Zhang et al., 1990; Acuña et al., 1999; Ness et al., 2000]. Our present knowledge mainly comes from the electron density measurements obtained by radio occultation experiments and also from the concentration of the most abundant ions obtained by the Retarding Potential Analyser instrument on board the Viking landers [Hanson et al., 1977]. Despite the relatively high number of electron density profiles for various values of the solar zenith angle, no special attention has been paid to the ionospheric layers below 110 km because the occultation technique is not effective at these altitudes [Parrot et al., 1996]. Therefore the sounding of the lower part of the ionosphere must be performed during the descent of probes or by ground-based instruments. Several instruments and techniques for electromagnetic sounding of the lower ionosphere have been proposed [Parrot et al., 1996; Menvielle et al., 2000; Witasse et al., 2001].
 The lowest resonant modes possible for such a system are extremely low frequency waves, since their wavelengths are of the order of magnitude of the spatial dimensions of the cavity. For Mars (r0 = 3390 km), Schumann frequencies are thus expected to be below 100 Hz.
 On Earth, the Schumann resonance modes are excited by globally distributed electrical discharges, while on Mars, despite the Martian atmosphere shows some weather activity as clouds, winds, dynamics and global flows, the requirements for lightning generation as it occurs in terrestrial thunderstorms are not satisfied. This fact, however, does not rule out the possibility of natural ELF wave generation on Mars and some alternative sources have been proposed. Hydromagnetic waves propagating or generated outside the cavity could propagate through the ionosphere and reach the ground as electromagnetic waves with reasonably high transmission coefficients, as suggested for the Earth by Abbas . Experimental [Eden and Vonnegut, 1973] and theoretical [Farrell et al., 1999] investigations of dust grain electrification have found that both glow and filamentary discharges on Mars could be generated in dust storms. There is no evidence of recent plate tectonics on Mars, but if it occurs, even if there are no resurfacing movements, it should also produce electromagnetic waves as the terrestrial earthquakes do [Parrot, 1995]. This issue is still under debate and the reader is referred to a recent publication and references therein [Siingh et al., 2005].
 Schumann spectra depend on ground and atmospheric conductivity, especially at low altitude levels where space orbiters cannot perform sounding by radio occultation techniques and variations in the electron concentration change the resonant modes. Therefore ELF detectors located on the Martian surface or lodged in balloons may be used as a tool to study ionospheric variations produced by changes in the solar activity and by the effect of dust storms or meteoric showers in the lower ionosphere.
 For the Earth, a clear correlation between annual variations of the first Schumann resonance intensity and the terrestrial tropical temperature over a six year period has been discovered, which suggests the possibility of using these frequencies as a global thermometer for the Earth tropics [Williams, 1992]. For Mars a dependence of dust devil occurrence on temperature variations could not yet be established.
Sukhorukov , on the basis of the two-layer semianalytical model of the Martian ionosphere's conductivity, estimated the Schumann frequencies on Mars to be f1 ∼ 13–14 Hz, f2 ∼ 24–26 Hz, f3 ∼ 35–58 Hz. This model has been widely used on Earth and produces good results after some calibration. However, the atmospheric conductivity profiles are quite different at Earth and Mars, therefore the calibrating procedure used on Earth may not be valid on Mars. Recently, Pechony and Price  calculated the Schumann resonance peak frequencies for Mars and found f1 = 8.6 Hz, f2 = 16.3 Hz, and f3 = 24.4 Hz, which are substantially lower than estimates by Sukhorukov .
 We calculate the resonance frequencies of the Martian resonator by using the numerical transmission line modeling (TLM) method, which has been successfully used in the simulation of especially complex electromagnetic systems [Portí et al., 1998a], and concentrate on the information one can deduce from it. About the discussion of the quality factor of such resonators and the information it contains the interested reader is referred to the literature [e.g., Nickolaenko and Hayakawa, 2002; Jones, 1964].
2. TLM Numerical Method
 The TLM method has been widely used in the analysis and design of antennas, microwave and electromagnetic compatibility structures and other electromagnetic systems with arbitrary geometry. The numerical algorithm is based on the construction of a three-dimensional transmission line network that is analogous to the original system in the electromagnetic sense [Christopoulos, 1995]. This global network is formed by interconnecting elemental transmission line circuits, termed nodes. Each node is a group of specially connected transmission lines which substitutes a portion of the medium. Its electrical characteristics describe the electromagnetic features of the medium, and a node is designed in such a way that the voltage and current pulses traveling through the lines resemble the propagation of the original electromagnetic field in the actual medium.
 Although the node design is quite complicated, the basis of the TLM algorithm is quite simple. Time and space are both discretized. Each small volume of discretized space is substituted by a TLM node. At each time step, a set of voltage pulses, represented by a column matrix Vi, is incident at each node in the TLM global network and a set of scattered voltage pulses Vs is generated, which is related to the incident one through the scattering matrix S in the form Vs = SVi. The scattered pulses become incident to adjacent nodes at the next time step and the process is repeated.
 Time synchronism must be imposed for the nodes so that all the voltage pulses in the mesh are simultaneously incident at all node centers at each discrete time, with Δt the time needed by any pulse passing from one node center to an adjacent one.
 The time evolution of voltage pulses at each node was calculated by following the TLM algorithm described in detail by Christopoulos , Morente et al. , and Portí et al. [1998b]. The results of the TLM method for calculating Schumann peak frequencies have been successfully compared with experimental measurements on Earth during quiet and disturbed ionospheric conditions [Morente et al., 2003a]. The numerical results obtained in the terrestrial simulation closely agree with experimental values and vindicate the confidence in the predictions for Mars. The method was also applied to the Saturnian moon Titan to study its lower ionosphere [Morente et al., 2003b].
3. Martian Ionospheric Resonator
 The Martian ionosphere has seasonal and diurnal variations and, theoretically, also shows a dependence on meteoroid incoming flux. The crust magnetic field of Mars is of rather complicated local appearance [Langlais et al., 2004] and has not been considered for the ELF wave propagation study. The concentration of ions expected in the lower atmosphere [Molina-Cuberos et al., 2002] is not high enough to affect the electrical conductivity. Therefore atmospheric conductivity is directly calculated from the electron density and the collision frequency between electrons and CO2, the major atmospheric constituent. The collision frequency was evaluated on the basis of work by Banks and Kockarts [1973, p. 188] and the temperature and pressure profiles used are taken from work by López-Valverde et al. .
Figure 1 shows the Martian atmospheric conductivity for various conditions as a function of altitude and for comparison, the terrestrial conductivity profile for solar quiet conditions is also included [cf. Morente et al., 2003a]. Martian profiles corresponding to high (f10.7 = 166) and low (f10.7 = 69) solar activity were calculated by merging the measurements of Mariner 6 and Viking 1, respectively, with a theoretical model for the lower ionosphere which considers the ionization by high-energy photons and cosmic rays [Molina-Cuberos et al., 2002]. The profile labeled “meteoroids” is calculated for Viking 1 conditions (day, solar minimum) and is based on a theoretical model of meteoroids [Molina-Cuberos et al., 2003]. The nighttime profile was assembled by considering the ionization due to precipitation of magnetotail electrons [Haider, 1997], meteoroids and cosmic rays.
 Electromagnetic waves propagating through the ionosphere are continuously reflected from the upper layers with high conductivity. Therefore waves are concentrated at the lower part of the atmosphere, where the conductivity is much lower. The reflection coefficient is given by
where Z0, Z are the vacuum and medium impedance and ω, ε0 and σ are the angular frequency, vacuum permittivity and the conductivity, respectively [Morente et al., 2003a].
Figure 2 shows the real part of the reflection coefficient for 10 Hz and 100 Hz, which is obtained upon inserting the conductivity value for the corresponding height. This represents an upper limit since we thereby assume vacuum wave propagation up to this height.
 The real part reflection coefficient is very close to −1 at 100 km, so the atmosphere at this altitude is a good conductor, the electromagnetic waves have been completely reflected and the exterior border of the electromagnetic cavity can be placed at this altitude.
3.1. Lossless Analytical Model
 The first approximation to the calculation of Schumann frequencies can be analytically done by assuming two spherical perfectly conducting layers and neglecting the electrical losses. The surface forms the lower layer and the upper one is located at the altitude level where electromagnetic waves are completely reflected, for example, 100 km. For such spherical resonators the resonant modes with lowest frequencies correspond to transversal magnetic modes TMr, that is, to the roots of the transcendental equation [Broc, 1950; Morente et al., 2003a]
where jn and yn are the spherical Bessel functions, the prime denotes the derivate with respect to the functions' argument kr, r0 is the Martian radius and h is the altitude of the ionosphere. For each value of n, the above equation has multiple roots, the first of them are summarized in the second column of Table 1.
Table 1. Schumann Frequencies on Mars for Lossless Model
3.2. Lossless and Damped Models With the TLM Method
 The numerical model is tested by applying it to the lossless cavity. Since a full three-dimensional TLM model demands too much memory and time to calculate, a two-dimensional azimuthally symmetric model is used; a TLM mesh in the radial and zenithal variables was implemented, with 10 × 18 × 1 nodes (corresponding to 100 km × 180°) in the r, θ, and ϕ directions. A wideband excitation is imposed at the node (3, 6, 1), which corresponds to 30 km of altitude and 60° North for simulating a (vertical) electrical discharge. For the particular situation modelled, an output point was chosen in order to simulate the signal for ground magnetic sensors, like MagNet on board NetLander [Menvielle et al., 2000]. The Hϕ component at the node (1, 8, 1) is recorded for an FFT analysis. The optimum time step is 4.9 × 10−6 s, obtained by reducing the numerical dispersion [Morente et al., 1995]. A total of 1.2 × 106 time step calculations was carried out, which allows the signal to cover the Martian perimeter more than 80 times and therefore to establish a quasi-permanent regime.
Table 1 compares the system resonances of the lossless TLM model (column 3), with the analytical values (column 2). The agreement of the results proves the suitability of the TLM method for dealing with Schumann resonances. The differences range from ≤0.1% for f1 to 2.2% for f6.
 Now, a more realistic approximation is implemented by including the conductivity profiles shown in Figure 1, together with a finite conductivity at the boundaries. Concerning the Martian ground, estimations of its conductivity range from 10−7 S/m [Cummer and Farrell, 1999] to 10−5 –10−4 S/m [Sukhorukov, 1991], while for the ionosphere at 100 km, the conductivity values are close to 10−3 S/m. Both boundaries are considered in the TLM method as load complex impedances connected to the corresponding link lines. The reflection coefficient for a pulse reaching these loads can be derived from equation (1) by taking into account the relation = ±(1/ + i/) and expressed as [Morente et al., 2003a]
Figure 3 shows the discrete Fourier transform of the time-dependent magnetic field Hϕ, at the output point for the different atmospheric conductivity profiles. The losses associated with the atmospheric electric conductivity, finitely conducting ionosphere and ground reduce the peak frequencies when compared to the case of the lossless cavity. The spectral peaks are smoother and some of them remain merged with adjacent peaks, following the general rule: the higher the conductivity value, the flatter the spectral response. The conductivity structure of the Martian atmosphere at low levels makes the second and third resonance come close to each other forming a single peak at 21–25 Hz (depending on the ionospheric model). This is the result of the interference of two adjacent modes, and because of the numerical technique used in our study, a posterior separation of the individual contributions cannot be performed easily. Therefore we only can show the peak frequency of the merged structure in Table 2. The peak merging is easily observed if the conductivity is decreased (in the simulation) by some orders of magnitude (not shown in Figure 3). Similarly, the fourth, fifth and sixth modes, produce a very smooth-shaped peak at around 60 Hz. This fact does not occur on Earth because the atmospheric conductivity at low altitude is 2 orders of magnitude lower than on Mars and therefore Schumann frequencies are more pronounced.
Table 2. Schumann Peak Frequencies on Mars for Damped Models
Table 2 shows the resonance peak frequencies for the damped cases and those calculated by Sukhorukov  using the two-layer semianalytical model and by Pechony and Price  using an improved model of fitting the ionospheric conductivity. We found the first resonance 2 Hz lower than Sukhorukov and f2 is included in the 24–26 Hz range predicted by Sukhorukov. The peak frequencies of Pechony and Price  are substantially lower than our results, which might be due to their 1 order of magnitude higher conductivity in the lower part of the ionosphere.
 By inspection of Figure 3 and Table 2 we can appreciate that, for solar maximum (f10.7 = 166) to solar minimum (f10.7 = 69) conditions, the peak frequencies f1 and f2 change by about 0.5 Hz. The presence of a meteoroid layer at around 80 km increases the conductivity at these levels and therefore the effective height of the Mars-ionospheric cavity decreases, producing a shift of the Schumann resonances to higher frequencies. This variation is more noticeable for f2 than for f1. If only the nighttime profile is considered, the electron density decreases, reflection occurs at higher levels and the whole Schumann spectrum is shifted to lower frequencies.
 The TLM numerical method has been applied for predicting the Schumann frequencies on Mars. Because of higher atmospheric conductivity near the ground (in comparison with Earth; see Figure 1), the resonances are very smooth and have a fundamental mode with a peak frequency of approximately 11 Hz, depending on the solar conditions. The resonance frequencies are very sensitive indicators of the global conductivity profile. Therefore experimental measurements of Schumann resonances by ground sensors are a valid tool for remote sensing of the lower ionosphere and could contribute to the detection of sporadic meteoroid layers in the ionosphere.
 This work was supported by project HU2001-0017 of Austria and Spain and project 15/2002 of the WTZ-Programme of the ÖAD. This work was partially supported by the Ministerio de Educatión y Ciencia of Spain under project FIS2004-03273 and was cofinanced with FEDER funds of the European Union. The authors thank the referees for their constructive comments to improve the document.