## 1. Introduction

[2] 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* [1990] and *Williams* [1992] 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.

[3] 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* [1952] 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].

[4] 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* [1979] suggested the use of two contiguous exponential profiles with different scale heights, and *Sentman* [1990] has used their model to estimate the ELF propagation characteristics. *Barrick* [1999] has attempted quick and simple estimates of the ULF/ELF dipole field strength in the cavity. Furthermore, *Mushtak and Williams* [2002] 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.

[5] 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.