The space formed by the ground and ionosphere is known to act as a resonator for extremely low frequency (ELF) waves. Lightning discharges trigger this global resonance, which is known as Schumann resonance. Even though the inhomogeneity (like day-night asymmetry, local perturbation, etc.) is important for such a subionospheric ELF propagation, the previous analyses have been always made by some approximations because the problem is too complicated to be analyzed by any exact full-wave analysis. This paper presents the first application of the conventional numerical FDTD method to such a subionospheric ELF wave propagation, in which any kind of inhomogeneity can be included in this analysis. However, the present paper is intended to demonstrate the workability of this method only for a uniform waveguide (without day-night asymmetry), by comparing the results from this method with those by the corresponding analytical method.
 Recently, there has been again great interest in subionospheric ELF waves because of two important findings. The first one is the suggestion by Price and Rind  and Williams  that global warming, which is a very important issue for human beings, can be effectively monitored by the intensity of the Schumann resonance (SR). SR is a global resonance phenomenon in the spherical Earth-ionosphere cavity as shown in Figure 1, which is triggered by lightning discharges in the equatorial thunderstorm-active regions [Bliokh et al., 1980]. Especially in the initial phase of SR study in 1960s and 1970s there were published a lot of papers on the SR observations for the study of global lightning activity [e.g., Polk, 1969; Sao et al., 1973; Ogawa et al., 1969; Jones and Kemp, 1970, 1971] and also on the study of ionospheric electron density [Jones, 1969]. The second reason is closely associated with the finding of optical phenomena (red sprites, etc.) in the mesosphere and lower ionosphere, and it is found that this mesospheric optical phenomenon is associated with strong ELF signals (called ELF transients) [e.g., Boccippio et al., 1995; Huang et al., 1999; Hobara et al., 2001; Nickolaenko and Hayakawa, 2002; Hayakawa et al., 2003, and references therein]. This ELF transient signal is one of the important tools for the study of those mesospheric optical phenomena and then the electrodynamic coupling between the atmosphere and ionosphere.
 There have been published a few monographs dealing with the subionospheric ELF propagation [e.g., Wait, 1962; Galejs, 1972; Bliokh et al., 1980; Nickolaenko and Hayakawa, 2002]. Schumann  was the first to predict the presence of resonances in the spherical Earth-ionosphere cavity and suggested a mathematical formulation of the propagation problem at ELF. A great simplification is the presence of only a single globally propagating, zero-order TEM mode [Madden and Thompson, 1965], while all the higher-order modes attenuate severely at distances of several waveguide effective heights. Despite this simplification, the complex structure of the lower ionosphere imposes an intricate three-dimensional electrodynamical problem that cannot be reduced to practical techniques without certain additional simplifying assumptions. This is the reason why several fundamental observed properties of SR cannot be well explained [Bliokh et al., 1980; Nickolaenko and Hayakawa, 2002], and please look at the latest monograph on ELF [Nickolaenko and Hayakawa, 2002].
 Here we briefly review the progress in the ELF propagation modeling which is the subject of this paper. Since 1980 there have been done a lot of theoretical efforts especially in the ELF propagation modeling closely related with the reflection mechanism of ELF waves in the lower ionosphere. Greifinger and Greifinger  suggested the use of two contiguous exponential profiles with different scale heights, and Sentman  has used their model to estimate the ELF propagation characteristics. Barrick  has attempted quick and simple estimates of the ULF/ELF dipole field strength in the cavity. Furthermore, Mushtak and Williams  have made the extensive comparison of ELF propagation parameters for different ionospheric density profiles even though they have assumed the uniform cavity model. The factors making the problem very complete are (1) radial (vertical) inhomogeneity of the ionosphere [Mushtak and Williams, 2002], (2) day-night asymmetry, and (3) local ionospheric perturbations, etc.
 In this paper we propose a new application of one computational method (so-called FDTD (finite difference time domain) method) to our very complicated ELF propagation problem. This FDTD method is based on the time domain numerical method for Maxwell's equations by finite difference principle, which is becoming very popular and useful in the field of computational electrodynamics [e.g., Kunz and Luebbers, 1993]. We have used this conventional FDTD for our ELF cavity model in which we have used the realistic conductivity profile. The variations of the SR intensities at fundamental and higher harmonics on several observation points obtained numerically by the FDTD method are compared with the corresponding analytical results in order to validate the potential use of this FDTD method in ELF problem.
2. Three-Dimensional FDTD Analysis
 The above discussion has led us to the requirement for a three-dimensional (3-D) finite-difference time domain (FDTD) application to our ELF propagation problem in a spherical coordinate system with azimuthal dependence. Fortunately, this 3-D FDTD code has been developed by Holland , which can be used in our application. Figure 2 (right panel) shows the spherical coordinate system (r, θ, ϕ) and location of a unit cell. The left panel of Figure 2 illustrates the location of field-evaluation points on a unit cell. As shown in Figure 2, we build a cell entitled I, J and K along each of three axes (r, θ, and ϕ). In the r direction I = 1 is the starting grid (r = a (the Earth's radius) 6.4Mm) and this ground is assumed to be a perfect conductor. I = Nr is the outermost radial boundary, is taken to be r = a + 120km and to be a perfect conductor. In the θ direction J = 1 is the North Pole (θ = 0°) and J = Nθ is the South Pole (θ = 180°). The coordinate ϕ indicates the longitude in such a way that K = 1 and K = Nϕ are the same point, but the latter has encircled the globe. On this condition the six electromagnetic field components are expressed by Holland .
 The Maxwell's equations to be solved are expressed as follows:
where E is the electric field, H is the magnetic field, ε0, μ0 are the dielectric constant and permeability of free space, σ is the conductivity which is a function of space, and Jext is the current source of the system (this is a lightning discharge in our case).
 Here we write down the field updating equations in the spherical coordinates as follows. In the region of I = 1 ∼ Nr−1, J = 2 ∼ Nθ−1 and K = 2 ∼ Nϕ, the field updating equation for Er component is given by
The corresponding equation for Eθ is expressed as follows:
and this equation pertains to I = 1 ∼ Nr−1, J = 2 ∼ Nθ−1 and K = 2 ∼ Nϕ. Then, the field updating equation for Eϕ component is as follows:
for I = 2 ∼ Nr−1, J = 1 ∼ Nθ−1 and K = 2 ∼ Nϕ. The differences of Δr, Δθ and Δϕ are the corresponding grid sizes in the r, θ, ϕ directions, and Δt is time step.
 The field updating equation for three magnetic field components (Hr, Hθ, and Hϕ) are given as follows:
for I = 2 ∼ Nr, J = 1 ∼ Nθ−1 and K = 1 ∼ Nϕ−1.
for I = 1 ∼ Nr−1, J = 2 ∼ Nθ−1 and K = 1 ∼ Nϕ−1.
for I = 1 ∼ Nr−1, J = 1 ∼ Nθ−1 and K = 1 ∼ Nϕ−1.
 The field components are on a modified Yee cell with unit vectors , , and , and the corresponding indices I, J, and K. Spatial locations are given in terms of positions R0(I), θ0(J), and ϕ0(K) and points lying midway between these locations are given by R(I), θ(J), and ϕ(K).
3. Modeling of Conductivity Profile and Current Source, and Some Numerical Examples
 In order to verify the usefulness of this 3-D FDTD computation in the ELF propagation, we show some numerical results only for a uniform cavity without any day-night asymmetry, because we can compare our own computational results with previous analytical computation by Nickolaenko and Hayakawa  and Nickolaenko et al. . The most important point for this ELF propagation modeling is what kind of conductivity profile we take [e.g., Mushtak and Williams, 2002; Nickolaenko and Hayakawa, 2002]. We assume the following exponential height profile of the atmospheric conductivity, which is expressed by
where z is the height above the ground, g is the scale height (g = 4.5km), and Gh is the reference altitude of the vertical exponential conductivity profile (Gh = 89km). ωG is taken as 2π × 10kHz. This is considered to be one of the most typical conductivity profiles as shown in Nickolaenko and Hayakawa . Next we assume a lightning discharge current in the conventional form consisting of two exponentials by Bruce and Golde , but we adopt a model with much slower time variation than the actual lightning in order to validate the horizontal grid size of 250km.
where I0 is the amplitude and t is in second. Assuming this current source, we have performed the FDTD computations by means of the formulations mentioned above, in which the grid size in r is 2km, and the grid sizes in θ and ϕ are π/72 (about 250km). This problem refers to the propagation of ELF transients.
Figure 3 illustrates a comparison of the computational results by the two methods on the temporal evolution of horizontal magnetic field for three different distances (d = 5, 10, and 15Mm, indicating the distance in the θ direction) when the current source is located at the equator (θ = π/2, ϕ = 0°). The pulse is generated at t = 0 and we can understand the propagation of a pulse in the waveguide. Figure 3 indicates the presence of the direct wave and its corresponding antipodal wave. The first three peaks corresponding to their propagation distances indicate the direct wave, while the latter three peaks are corresponding to its antipodal wave. We now compare the results by the two computational methods (the analytical solutions in broken lines and our present FDTD computations in solid lines). The most important comparison is the position of the peak. The position of the first peak for each propagation distance is found to be in good agreement between the two. You will see some discrepancy between the two, which are the negative sharp peak and subsequent fast fluctuations. This discrepancy is definitely associated with the filtration effect due to the difference method itself [see Nickolaenko and Hayakawa, 2002, p. 166]. Due to the finite grid size in time in the FDTD method, we cannot analyze the fields at frequency above a certain frequency. Consequently, our FDTD computational result on the waveforms should be fed to a low-pass filter, which would be compared with the observational data. This figure suggests that with the increase in propagation distance, the higher frequency components tend to decay so that we have a spread in waveform for larger propagation distances. The horizontal magnetic field of the antipodal wave is opposite in sign to that of the direct wave, so that the signal for d = 20Mm is zero because of the complete cancellation of the direct and antipodal waves (though not shown as a figure). An important characteristic from this figure is that we expect severe damping between d = 5Mm and 10Mm, but negligible damping between d = 10Mm and 15Mm. As the conclusion, we have found a good similarity between these two figures by the two methods.
Figure 4 shows the next comparison of the results on the wave impedance by the two methods (analytical and our FDTD computations). In our FDTD analysis we first perform the time domain analysis and then we transform it into the frequency domain. Because the wave impedance does not depend on the source characteristics, we can concentrate on comparing the propagation characteristics by two methods. The solid lines refer to our FDTD computations, while the broken lines, the analytical solutions. Three different propagation distances (source-observer distance) are chosen, d = 5, 10 and 15Mm (from the top to the bottom in Figure 4). The left panels indicate the real part of the wave impedance, and the right panels, the imaginary part. We have found that the wave impedance calculated by our FDTD method seems to be very consistent with the analytically calculated wave impedance. Thus, the computational results of ELF propagation in this frequency range using our FDTD code are reasonably acceptable as compared with the previous analytical calculation, and our FDTD method seems to be useful for any ELF propagation problems.
4. Effect of Source-Observer Distance on SR Intensity
 In order to further verify the use of our FDTD method in ELF propagation, we have computed the amplitude of SR for different source-observer distances (d = 5, 10, 15 and 20Mm) when the current source and an observer is both located at the equator (θ = π/2). The height profile of conductivity is assumed to be the same of night-profile (equation (8)). We assume a lightning discharge current in the form of two exponentials (equation (9)), and we change this equation into random time interval pulses to simulate the source for SR. We use the following equation:
where t is in second and tn is the starting time of each discharge. I0 is assumed to be random (positive and negative lightning discharges) and tn is assumed to vary at random in order to simulate the random occurrence of causative lightning discharges, because a uniform time interval between pulses leads to a periodical modulation of the spectrum.
Figure 5 presents the power spectra of electric and magnetic fields at the observer with different distances. The frequency range of our interest is from 4Hz to 40Hz as plotted along the abscissa, and the power of the electric and magnetic fields are shown along the ordinate in a relative unit. The upper panel refers to the vertical electric field component, and the lower refers to the horizontal magnetic field component. The frequency spectra computed are found to depend on the distance between the source and observer. That is, the size and width of the SR peaks and their position along the frequency axis depend on this distance. As for the first fundamental mode (around 8Hz), the upper panel of Figure 5 shows that there exist the corresponding peaks around 8Hz in the electric field for different propagation distances (d = 5, 15 and 20Mm) except for a particular distance of d = 10Mm. That is, we notice no significant peak around 8Hz for d = 10Mm in the electric field, but there is observed the peak for the second harmonic mode in the electric field. No corresponding peak is expected for the second mode for d = 10Mm, but we find an extremely small amplitude peak in the lower panel of Figure 5. Since the pulse with opposite sign is known to interfere with the direct signal at the antipode (d = 20Mm), there is no resonance spectrum in the magnetic field so that we have not plotted the result for this distance in the lower panel of Figure 5. In the magnetic field spectra (in Figure 5) we can find out clearly significant peaks at the frequencies of 8Hz, 14Hz and 20Hz.
 We compare these computational results with the previous analytical result [e.g., Sentman, 1995; Nickolaenko and Hayakawa, 2002]. Their analytical results indicate that the peaks corresponding to the first mode appear for d = 5, 15 and 20Mm in the electric field, with the same characteristic for the magnetic field (except for d = 20Mm). Additionally, there have been observed the peaks corresponding to the second mode for d = 10 and 20Mm in the electric field and for d = 5 and 15Mm in the magnetic field. These characteristics of the appearance of peaks in the analytical calculation are consistent with our results. Therefore our results are reasonably acceptable as compared with the previous analytical calculation, and our FDTD method seems to be also useful for SR problem.
5. Conclusions and Future Application of This FDTD Analysis to Much More Complicated Models
 In the previous section we have shown a few examples of our FDTD analysis for the subionospheric ELF propagation only for a uniform cavity, and a comparison with the previous analytical solution has indicated that our FDTD application would be of great potential in our future studies. As you see from the above discussion, the field updating mechanism is exactly the same as that of the high-frequency FDTD [Kunz and Luebbers, 1993], and it seems that there is no special treatment in the ELF wave analysis.
 Here we comment briefly on the distinction of our FDTD method from the previous approach by the 2-D transmission line [Madden and Thompson, 1965; Kirillov, 1996]. The transmission line circuit theory by Madden and Thompson  is based on the numerical solution of the same differential equation as in our FDTD method, but they use the fundamental assumption of the TEM mode of propagation, though Kirillov's  method allows not only numerical but also analytical solutions. When we have a very complicated cavity structure including day-night asymmetry and local perturbations, even evanescent or highly attenuated modes can contribute to the fields. We do not know whether this TEM mode assumption is completely acceptable or not, but we can conveniently use our FDTD method even in this situation. We can list what we will be able to do in this ELF field by means of our FDTD analysis.
 As for the ELF propagation itself (such as the propagation of ELF transients (Q bursts, slow tails)), we can elucidate the reflection mechanism of these ELF waves in the lower ionosphere in details by comparing our FDTD results with the previously proposed approximations [Greifinger and Greifinger, 1978; Mushtak and Williams, 2002].
 Then, as for Schumann resonance, there are several possibilities for the application of our FDTD method, for example: (1) the full consideration of day-night asymmetry in order to well explain the diurnal variation of Schumann resonance intensity and frequencies, and (2) the local perturbation (such as the perturbation at high latitudes (auroral regions)). We will perform our FDTD analyses for these above mentioned questions in order to obtain further understanding for the problems in the study of the SR and ELF transients.