## 1. Introduction

[2] The ultimate goal of this study is to establish a method to draw the world map of lightning activity by observing Schumann resonance spectra collected at a few ground-based observatories. As the first step of the study, this paper treats the reconstruction of lightning distribution as a function of distance from the observation point by solving an inverse problem.

[3] Schumann resonance is the global extremely low frequency (ELF) phenomenon that occurs in the spherical shell between the Earth's surface and the ionosphere [*Schumann*, 1952; *Sentman*, 1995; *Nickolaenko and Hayakawa*, 2002]. The resonant frequencies are about 8, 14, 20, etc. Hz, corresponding to the first, second and third modes. Schumann resonance has been investigated recently as an indicator of the global lightning activity [*Nickolaenko et al.*, 1996, 1998, 1999]. However, it is not properly estimated without account of the dependence of the Schumann resonance data on the distance between the source lightning and the observation point. At the present time the most powerful method to observe lightning distribution is the satellite-based observation, e.g., lightning image sensor (LIS) [*Christian et al.*, 1999]. However, the satellite-based observation cannot detect instantaneous distribution of all lightnings although it is possible to specify very precise location of lightnings. In this paper, however, we set apart the precision in location, but try to identify instantaneous distribution of the global lightning activity by resolving the ELF spectra which carry information both on the source locations and the propagation paths. The global lightning activity observation by resolving the inverse problem would demand minute costs in comparison with the use of satellites, or any other techniques. This paper demonstrates results of our attempt to formulate the method based on recent developments of the computational electromagnetics.

[4] There are two issues to be considered in order to accomplish our purpose. The first of them is to calculate precisely the cavity response to a single lightning stroke, including as many realistic properties as possible. The second one is to recover properly the source distribution over the Earth's surface from the resonance data, especially in the presence of the noise.

[5] The Schumann resonance calculations were so far based on the effective propagation parameters [*Sentman*, 1995; *Nickolaenko and Hayakawa*, 2002], or the conductivity profiles assumed to be uniform all over the globe [*Jones*, 1967; *Bliokh et al.*, 1977; *Mushtak and Williams*, 2002]. This conventional method had an advantage to calculate the response quickly, with the loss of realistic simulation. Now we adhere to a more realistic model, and therefore calculate Schumann spectra by the frequency domain finite difference (FDFD) method, which makes it possible to introduce empirical profiles, and we utilize the International Reference Ionosphere 2000 (IRI 2000) [*Bilitza*, 2001, 2003] and the mass spectrometer incoherent scatter (NRLMSISE-00) model [*Hedin*, 1991; *Picone et al.*, 2002], as widely accepted electron and neutral density profile models. Time domain calculation, e.g., the finite difference time domain (FDTD) method [*Simpson and Taflove*, 2004] and the transmission line matrix (TLM) method [*Morente et al.*, 2003], is appropriate for calculation of ELF transient, but for the present case it is not appropriate because the launched waves propagate for long time in the closed region so that it takes long computational time. This paper deals with a spherical shell cavity which has fixed vertical electron and neutral density profiles uniform in horizontal directions, because the model is conditioned in part by the lack of computational resource to carry out the complete three-dimensional (3-D) calculations. This approach reduces the importance to utilize the FDFD method, but the particular results would imply the necessity to realize the 3-D FDFD calculation.

[6] The second problem, that is, reconstruction of the lightning source distribution based on Schumann resonance spectra, was treated by *Shvets* [2001], who exploited the model of effective boundaries. In this paper, we adopt this method entirely, but with the results calculated by the FDFD method in spherical coordinates. The numerical experiments are given to estimate the accuracy of reconstruction when the initial data contain some noise, and the approach presented here is applied to the observed data in order to identify lightning activity centers. There still exist some problems to be overcome for better identification, which is discussed in section 3.2.