#### 2.1. Direct Problem

[6] The direct problem in this study is to compute responses of the Earth-ionosphere cavity to a lightning source, and we use the finite difference method in the frequency domain for the formulation. The time domain method is not appropriate because in order to obtain resonant spectra by taking the Fourier transform of the computed time domain results, it is necessary to calculate the electromagnetic fields until the ELF fields launched by lightning vanish, and the computational time becomes very long. It is straightforward from the geometry to model the Earth-ionosphere cavity in the spherical coordinates, and the azimuthal symmetry is introduced in order to simplify the calculation because of the limitation of our computational resources. Figure 2 shows the analysis model in this study. The coordinates *r* and θ are discretized by Δ*r* and Δθ and are represented by *i*Δ*r* (*i* = 1, 2, ⋯, *N*) and *j*Δθ (*j* = 1, 2, ⋯, *M*), respectively. The current source segment with the length Δ*l* and with the intensity *I* is located at (*i* = , *j* = 1), and is directed to +*r* (radially outward). The above conditions on the source are essential to the azimuthal symmetry. The finite difference expression of the Helmholtz equation is given by

where

and *k*_{0}, *δ*_{m,n}, *R*(*i*), and *S*(*j*) are the wave number of light in a vacuum, the Kronecker's delta, *R*(*i*) = *r*_{i} + (*i* − 1)Δ*r*, and *S*(*j*) = sin {(*j* − 1)Δθ}, respectively.

[7] The dielectric constant _{r}(*i*, *j*) is determined by a modeled atmospheric conductivity and plasma parameters in the lower ionosphere. In the present analyses, we adopt the Earth-ionosphere cavity with the spherical symmetry; that is, the ionospheric parameters vary with only the altitude because there are no cases such that the ionospheric geometry has the azimuthal symmetry with respect to lightning strokes. The atmospheric conductivity from the ground up to an altitude of 60 km is given by [*Volland*, 1984; *Holzworth*, 1995]

where

At altitude higher than 60 km, the medium is characterized directly as the plasma:

where *ω*_{pe}, *ω*, *ν*_{en}, and *ν*_{ei} are the electron plasma frequency, the angular wave frequency, and the collision frequencies between electrons and neutral particles and between electrons and ions, respectively. Here *ν*_{en} and *ν*_{ei} are given by [*Holzworth*, 1995; *Richmond*, 1995]

where , , and *N*_{O} are the concentrations (in m^{−3}) of N_{2}, O_{2}, and N_{O}. *R*_{e} is *T*_{e}/(300 K), and *T*_{e} is temperature of electrons in degrees Kelvin. All parameters associated with electron and neutral particles are derived by IRI-2000 [*Bilitza*, 2001, 2003] and NRLMSISE-00 [*Hedin*, 1991; *Picone et al.*, 2002].

[8] In the analysis, the parameters are chosen as follows: *r*_{i} = 6370 km, which denotes the radius of the Earth, and *r*_{o} = 6500 km, which is the level where the perfect conductor is assumed to be located, which is practically a sufficient altitude for numerical analyses of the ELF waves observed in the lower atmosphere because most of the fields are reflected and dissipated in the lower ionosphere and cannot penetrate into the ionosphere at an altitude higher than 130 km. The numbers of discretization are taken as *N* = 33 and *M* = 2001, i.e., Δ*r* = 3.94 km and *r*_{i}Δθ = 10.0 km. We found that these parameters formed the largest cell in order to model the atmosphere and the lower ionosphere and to guarantee the computational convergence of the field intensity on the Earth's surface with about 1% precision. In the computation, the generated sparse matrices (66033^{2} elements, only 0.4% of them being nonzero ones) are directly solved without any preconditioning by utilizing UMFPACK [*Davis and Duff*, 1999], the solver of asymmetric sparse linear systems.

[9] Detailed results are given by *Ando et al.* [2005], so in this paper we show the numerical results of the spectra with the different distances between the source and the observer. The measurable fields in actual experiments are those on the Earth's surface, and the calculated spectra are obtained by computing the fields, *H*_{ϕ}(, *j* + ), with sweeping the frequencies. The calculated spectra are given in Figure 3. It is observed that the shapes of the curves are close to the experimental one in Figure 1 and that the resonant frequencies are close to the experimental ones. One may notice a discrepancy between the computed and the measured spectra, which is because the measured spectra have a decreasing trend with the increase in frequency. This phenomenon is explained as the so-called 1/*f* noise, which exists in many physical processes in nature [*Mandelbrot*, 1999], and the spectrum of noises due to the natural sources actually has 1/*f* dependence [*Lanzerotti and Southwood*, 1979].

[10] These agreements justify the finite difference computation for the Schumann resonance by using IRI-2000 and NRLMSISE-00. The different curves correspond to the different distances *d* between the source and the observer. Schumann resonances are observed from Figure 1, and it is obvious that the spectra depend on the distance. Those curves will form a set of basis functions for the inverse problem to deduce lightning distribution.

#### 2.2. Inverse Problem

[11] Here we formulate the inverse problem to decompose a synthesized spectrum into a set of elemental spectra with different source-observer distances. First, it is necessary to consider limitations imposed by the ELF field measurement. The used quantity for the inversion is the horizontal magnetic field because the measurement of electric field requires highly skilled techniques, and the data may include more noise. Although two orthogonal magnetic fields in the horizontal plane are being measured, we utilize them not as two separate components but as one combined horizontal field. One may think that the separate use of two components would allow us to deduce not only the distance between the source and the observation point but also the direction of the source location. However, the separate use is dangerous because the installation of magnetic field sensors does not have sufficient precision with respect to directions. This fact would cause huge errors, identifying lightning very far from the observation point. Moreover, the period sufficient to detect Schumann resonances contains hundreds (or thousands) of lightning strokes, so the estimation of the direction of arrival (DOA) is made only in the sense of their vectorial sum. It is clear from this point of view that the DOA by using Schumann resonances is practically useless. Consequently, we use the horizontal magnetic field, ∣*H*(*f*)∣, for the inverse problem, and the solution gives us only the source-observer distance and the intensity. Only the information on the distances is obtained in comparison with the observation of two components, but it is reasonable if we consider the possibilities of the errors for one-site observation.

[12] The basis function set for the inverse problem is given by the finite difference method described in section 2.1. Consider that the observation point is put at a point in the Earth's surface. We divide the entire interval, i.e., the distance from the observation point to the antipode, which is 20 Mm, into Λ sections with equal intervals. From the reciprocity of the fields with respect to the positions of the source and the observer, the averaged response from lightning in the *λ*th section is given by

where Δ = 20/Λ (Mm) and *H*_{ϕ} is the field calculated by the finite difference method and is redefined here as a function of the following arguments: *ζ* and *f*_{k} are the source-observer distance and the discretized frequency, respectively.

[13] The measured field *H*(*f*_{k}) is considered to include two types of noise. One is the white noise, and the other is the 1/*f* noise mentioned earlier. In order to take into account the former, it is necessary to consider the fitting with nonuniform weighting because the data with larger fluctuation are less reliable. The well-known method for determining the weights is to adopt the standard deviation of the data for fitting [*Press et al.*, 1992]. Therefore we define

where *σ*_{k} is the standard deviation of ∣*H*(*f*_{k})∣^{2} around the time under consideration, and we also define *A*_{kλ} ≡ *a*_{λ}(*f*_{k})/*σ*_{k}. The fitting problem is to find the solution of the system of linear equations:

where *b* = [*b*(*f*_{1}) *b*(*f*_{2}) ⋯ *b*(*f*_{K})]^{T}, **A** = [{*A*_{kλ}}], and [·]^{T} is the transpose of the matrix. The unknown coefficients, *x* = [*x*_{1}*x*_{2} ⋯]^{T}, represent the lightning distribution. The latter noise can be considered by introducing additional basis functions *a*_{kλ} = 1/(*f*_{k})^{p(λ−1−Λ)}, (*λ* = Λ + 1, ⋯, Λ + *L*).

[14] The proper solution of this inverse problem under noisy conditions is given by introducing Tikhonov's regularization [*Shvets*, 2001]. The solution is given by minimizing the functional:

where *α* is called the regularization parameter. The minimization is readily reduced to the problem of finding *x* for which the gradient **∇**Π is zero, that is,

where **I** is the identity matrix. Here the physically meaningful solution of equation (19) is given by the nonnegative least squares [*Lawson and Hanson*, 1974].

[15] It remains to choose the regularization parameter *α*. In the earlier analyses [*Ando and Hayakawa*, 2004; *Ando et al.*, 2005], empirical approaches were used. In this study, we introduce the GCV method [*Golub et al.*, 1979] for more reasonable estimation. The GCV is known as the method to estimate *α* without knowledge of the error norm with respect to the exact data and is used successfully for practical applications [*Iwama et al.*, 1995; *Franchois and Pichot*, 1997]. The GCV score as a function of *α* is given by

where

and Trace[**X**] stands for taking the trace of the matrix **X**. The *α* to minimize the GCV score, *α*_{GCV}, is obtained by simple search in the appropriate range of *α*.