Radio Science

Finite difference analyses of Schumann resonance and reconstruction of lightning distribution



[1] This paper deals with the computation of extremely low frequency propagation associated with the Schumann resonance phenomena and the reconstruction algorithm for source lightning location on the basis of measured Schumann resonance data. The finite difference equations are derived in terms of discretized magnetic fields in the spherical coordinates, introducing the azimuthal symmetry for simplicity. The most reliable electron and neutral density models in the atmosphere and the ionosphere can be used to describe extremely low frequency wave propagation. A linear inverse problem for the reconstruction is formulated using the computed spectra as a set of basis functions to identify lightning distributions with respect to the distances from any observatories to the global thunderstorm centers. Numerical experiments allow us to evaluate properties and precision of the solution in the absence or in the presence of noise in the initial spectral data. The inverse problem is applied to the experimental data collected at a field site in Japan, and distribution of global lightning activity is reconstructed for the data covering the period from March to December 1999. The reconstructed data show a reasonable set of distances from the observatory to well-known global thunderstorm centers, and they indicate the seasonal drift of lightning activity. The problems that were involved in solving the inverse problem are also discussed.

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.

2. Finite Difference Method

2.1. Derivation of Finite Difference Equations

[7] The problem geometry and relevant coordinates are shown in Figure 1. The analysis region lies between the inner perfect conductor with the radius ri, corresponding to the Earth, and the outer one with the radius ro. It will be demonstrated later by numerical results that the absorbing boundaries are not necessary if we keep a sufficient altitude in the analysis region, because electromagnetic fields in the ELF band under consideration are reflected and dissipated enough until this altitude. So that, we can truncate the analysis region, that is, we put the outer conductor at the altitude of about 130 km. The analysis region is discretized by Δr and Δθ in the r and θ coordinates, respectively. The current source with the length Δl and the intensity I is located at (r, θ) = (ri + Δr/2, 0), and directed radially. If the medium in the analysis region is independent of ϕ, i.e., εr(r, θ), then the problem is symmetric over the azimuth, or ∂/∂ϕ = 0. As a result, points in the discretized coordinates are represented by (r, θ) = (ri + {i − 1}Δr, {j − 1}Δθ) ≡ (i, j), where i runs from 1 to N + 1 with the definition ro = ri + N Δr, and j does from 1 to M + 1 with MΔθ = π. In the present configuration, only TE mode with respect to ϕ direction is excited; that is, Er, Eθ, and Hϕ are nonzero components which are assigned on staggered points of cells as Yee's algorithm [Yee, 1966]:

equation image

The finite difference expression of the equations, × × Hk2H = × J, are given in terms of the magnetic field:

equation image

The coefficients are given by

equation image
equation image
equation image
equation image
equation image
equation image

where k0, δm,n, R(i), and S(j) are the wave number of light in vacuum, the Kronecker's delta, R(i) = ri + (i − 1) Δr, and S(j) = sin{(j − 1)Δθ}, respectively, and

equation image
equation image

We introduce the dielectric constant, εr(i, j), later. The electric field is given from the magnetic field by the straightforward finite difference procedure as follows:

equation image
equation image

For j = 1 and M + 1, Er has a special form:

equation image
equation image
Figure 1.

Analysis model of a uniform spherical shell cavity.

2.2. Conductivity Profile

[8] The Earth-ionosphere cavity where Schumann resonances take place is dissipative. The losses appear due to the presence of charged particles which collide with neutral molecules and other charged particles. The air slab is characterized in electrodynamics by its conductivity profile where contribution from ions plays an important role, especially in the lower atmosphere [Reid, 1986]. The conductivity dependence on the altitude above the ground is approximated by [Volland, 1984; Holzworth, 1995]

equation image


equation image

and z is the altitude in kilometers. Ai and Bi indicate the magnitude and the scale height of the conductivity in the lower atmosphere, respectively. We use this profile up to the altitude of 60 km where the electronic conductivity starts to play the major role. Relevant conductivity is included into the following complex dielectric constant:

equation image

where ωpe, ω, νen and νei are the electron plasma frequency, the angular wave frequency, and the collision frequency between electrons and neutral particles, and between electrons and ions, respectively. νen and νei are given by [Holzworth, 1995; Richmond, 1995]

equation image
equation image

where Nequation image, Nequation image, and NO are the concentration (in m−3) of N2, O2, and NO. Re is Te/(300 K), and Te is temperature of electrons in K. The concentrations and temperatures of electrons, ions, and neutrals are given by the IRI model [Bilitza, 2001, 2003] and the NRLMSISE-00 model [Hedin, 1991; Picone et al., 2002]. The temperature of electrons is assumed to be equal to the one of neutrals in the D region of the ionosphere, thus the neutral temperature obtained by NRL-MSISE is used as Te at the altitude lower than 120 km. At the altitude from 60 km to 80 km, the electron density is modeled by Friedrich and Torkar [1992, 1998] and Friedrich et al. [2001], which is given as an optional IRI function. They are based primarily on experimental results and on theoretical consideration as interpolation. We can specify any arbitrary latitude and longitude. In the IRI the vertical resolution is 1 km and the temporal resolution is 1 hour. We examined that calculated results had no difference when ±1-hour profiles are used. In NRL-MSISE, it is possible to specify arbitrarily both of spatial and temporal parameters. Consequently, there is no restriction in using these models in the present analysis.

[9] If we define the equivalent conductivity in terms of εr in equation (17), given by

equation image

where equation image stands for the imaginary part, the conductivity profile is obtained as shown in Figure 2. Here we used the electron density profile at the point of 44°20′N 142°15′E at 0000 UT (0900 LT) and 1200 UT (2100 LT) on 1 March 1999. In the computation, we used equation (17) directly as the parameter of media.

Figure 2.

An example of conductivity profile.

[10] The real electron density profiles are known to depend on latitude, longitude, and time. However, we assume here that it is the function of only altitude, and the profile at the observation point is used. The result will be slightly different from those obtained for a nonuniform profile, which is expected from our previous work [Ando and Hayakawa, 2004] although the analyzed geometries are different. Advantages of our FDFD method do not appear clearly by introducing this simplification because conventional methods enable us to calculate the response with the same configuration even though the formulas will be much more complicated due to many layers involved in the computations. The present computation thus serves as a preliminary examination for the validity of the FDFD method.

2.3. Numerical Results

[11] Some numerical examples are demonstrated in this section. The electron profile used is the one on 1 March 1999, at 0000 UT, and at 44°20′N 142°15′E, which simulate the situation of our observation point of Moshiri, Hokkaido, Japan [Hobara et al., 2000], where we have actually been measuring the Schumann spectra for the last 5 years. The parameters for calculation are chosen as follows: ri = 6370 [km], Δr = 5 [km], riΔθ = 10.0 [km], and ro = 6370 + 130 [km]. Relatively finer discretization is necessary in the radial direction because of modeling of the electron profile.

[12] Figure 3 shows the amplitude of calculated magnetic field distribution as a color map when the source current segment is I Δl = 1 [A · m]. The color scale is shown on the side of the picture. The source is located on the top surface of the Earth in this figure. For visualization, we magnify the altitude of the analysis region by a factor of 20 in Figure 3.

Figure 3.

Numerical result of distribution of magnetic field intensity.

[13] Frequencies used in this calculation are 8 Hz and 20 Hz which almost correspond to the first and the third resonant frequencies in this resonator, respectively, as observed from Figure 3. It is evident from this figure that the absorbing boundary condition is not necessary in this analysis because the fields decay rapidly at the altitude of about 80 km, and the reflection at the outer boundary does not affect any numerical results.

[14] Measurable field is the one on the Earth's surface. Thus Schumann resonant spectrum is obtained by collecting the surface field at the observation point with sweeping frequencies in the computation.

3. Reconstruction of Lightning Distribution

[15] Computing resonant spectra allows us to reconstruct the lightning distribution by solving an inverse problem from measured data. In this section, we describe the reconstructing method and apply it to measured data. Although it is desirable to reconstruct the distribution fully over the entire Earth's surface, we here consider only the distance between the observation and source points, that is, one-dimensional distribution because only one site data are treated at the present stage.

3.1. Inverse Problem

[16] Our objective is to reconstruct the lightning distribution ranging from the observation point to its antipode. This inverse problem was formulated and examined by Shvets [2001]. We use the method similar to his, except for the set of basis functions with an improvement.

[17] We divide the region (20 Mm) by L with the interval Δ = 20/L [Mm], and let the unit response from the jth region lightning be aj(fi) in terms of computed magnetic fields Hϕ(ri, θ) at the discretized frequency fi, indicated by h(θ, fi):

equation image

where aj(fi)s consist only of calculated spectra, and the unit response is the square of the magnetic field for the case where a source lightning with the unit current segment, I Δl = 1 [A · m] occurs once. Letting the measured horizontal magnetic field H(fi), we obtain the measured response with taking into account the measurement inaccuracy and the distortion in lower frequencies (up to 5 Hz) by local weather:

equation image

where σi is the standard deviation of the measured ∣H(fi)∣2. This response is assumed to be a linear combination of Aijaj(fi)/σi with unknown coefficients xjs which correspond to a mean intensity of lightning over the jth region. Thus

equation image

which is rewritten into a matrix form:

equation image


equation image
equation image
equation image

where K is the number of discrete frequencies of measured data (the discretization and the number of frequencies are determined by the measurement system), and the superscript T denotes transpose.

[18] The distribution of lightning activity can be reconstructed by solving the least squares problem of equation (24); that is, minimizing χ2 = ∥bA · x2. Introducing Tikhonov's regularization yields the smoothing functional, Π(x), given by [Tikhonov and Arsenin, 1977]

equation image

where α and Λ(x) are, respectively, the regularization parameter and the stabilizing functional which is chosen as follows:

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The proper solution is the x to minimize Π(x) and corresponds to the condition that the gradient of equation (28) turns into zero, that is, the solution must satisfy

equation image

in the sense of the nonnegative least squares [Lawson and Hanson, 1974; Shvets, 2001].

[19] We must now choose the regularization parameter α, which is determined empirically. At first, we postulate a lightning distribution, x0, arbitrarily, then calculate the noiseless response, b. The reconstructed result, x is obtained by solving the above inverse problem. The relative error is estimated in terms of the difference norm, ∥x0x∥/∥x0∥. An example is shown in Figure 4. The postulated lightning distribution depicted in Figure 4a by boxes is generated randomly, and the relevant response is calculated as shown in Figure 4b (solid line). After solving the inverse problem we reconstruct the distribution and obtain spectra as shown in Figures 4a and 4b in dotted line and dots, respectively. The two regularization parameters α = 1.12 × 10−50 and 1.12 × 10−47 are used in the reconstruction, and it is clear that the former case gives an excellent agreement with the original distribution. Figure 4c shows the relative error estimated for the regularization parameter α ranging from 1 × 10−50 to 1 × 10−46. The error approaches zero at around α = 1 × 10−50.

Figure 4.

Numerical experiment of the inverse problem.

[20] Here we consider the choice of α in the presence of noise because experimental data suffer from noise. The biggest noise is the one from power lines, and it can be avoided by excluding the corresponding frequency from the bandwidth of consideration. It is considered that interference from the experimental instrument is not observed. Any other systematic noise which has the characteristic modeled spectra can be removed by introducing the spectra into the function set for reconstruction. Here we include the 1/fn noise. Consequently, the noise to be considered here is nonsystematic one, and it is characterized by the white Gaussian distribution. The white noise with the maximum magnitude Anb∥ is added to b, and the error is then evaluated as a parameter of the noise magnitude An. Numerical experiments are performed for the same distribution as above.

[21] Figures 5a–5c depict the original spectra and those spoiled by noise in the range An = 0.01 up to 0.05. In addition, the reconstructed results from the spectra with noise are shown in Figures 5d–5f, respectively. It is apparent from Figure 5 that even small noise results in crucial degradation of recovered distribution. The discrepancy increases rapidly with increase of noise magnitude in the case of α = 1.12 × 10−47, so that a relatively large α must be chosen in order to avoid degradation.

Figure 5.

Spectra with noise and reconstructed results.

[22] Figure 6 shows relative error of the example as a function of the regularization parameter α and the noise amplitude An, as a gray-scale map. The minimum errors at every An are connected by the broken line in the figure. The regularization parameter α providing the minimum error moves to larger value when the noise increases. It is difficult to estimate the noise contribution in actual data, besides the frequency dependence is not flat. Thus we cannot fix the value α at this moment to be used for the reconstruction. The general rule is that a larger value of α must be used for data with higher nonsystematic noise level.

Figure 6.

Relative error of the example case as a function of α and An.

[23] The precision of distance reconstruction can be estimated from Figures 5d–5f. Deviations as great as 1 Mm might be expected in the reconstructed distribution.

[24] Here we have examined the influence of noise with up to 5% amplitude, but the observed data seem to include at largest 3% noise, and mostly about 1%. Therefore the fictitious peaks which are seen at 16 Mm in Figures 5e and 5f, possibly appear, but the magnitude is, at largest, about 20% of the major peak.

3.2. Application to Measured Data

[25] In this section the reconstruction method is applied to the real data which were collected at Moshiri (44°20′N 142°15′E), from March to December 1999. The measurement setup is described in the work of Hobara et al. [2000]. The parameters to obtain the inverse problem solutions are: α = 1 × 10−46 − × 10−43, Δr = 5 [km], riΔθ = 10 [km], L = 40, and K = 111–151 which correspond to the frequency range from 2.5 [Hz] to 30.0–40.0 [Hz] with the discretization 0.25 [Hz]. The discretization frequency is determined as follows; We measure data with the sampling frequency of 2 kHz. At first we collect 28 s data, i.e., 56000 points, in every 1 min. Every 8000 points (4 s) data are separately Fourier-transformed, and then, taken the average at every frequency (0.25 Hz) over seven spectra. The resulting data is defined as the spectrum at the 1 min. Finally, ten spectra are averaged, so that one spectrum with the discretized frequency of 0.25 Hz is obtained every 10 min.

[26] The variations in K and α are conditioned by noise, and smaller K and larger α are chosen when the spectrum includes higher nonsystematic noise, and particular values are adapted manually. The standard deviations are calculated for the data measured over ±1 hour interval from the given time. Figure 7 shows typical examples of spectra with little and much systematic noise and their fitted results. It is observed that the spectrum in Figure 7b is more contaminated by noise both in lower frequencies and in the vicinity of 50 Hz. The noise in lower frequencies are likely caused by rain, weather, etc. The one in 50 Hz is radiation from the commercial power supply, and it is considered that it expands its wing when there exists additional noise in power line. For those cases the frequency range for fitting should be narrowed to avoid the influences of noise, and for the case of quiet spectra, however, we use as wide range as possible to obtain more information from spectra.

Figure 7.

Examples of measured spectra of Schumann resonance and fitted results. (a) Quiet spectrum (7 March 1999, 2040 UT). The frequency range for fitting is from 2.5 to 40.0 [Hz] (K = 151), and α = 1 × 10−43. (b) Noisy spectrum (8 August 1999, 0520 UT). The frequency range for fitting is from 5.0 to 35.0 [Hz] (K = 121), and α = 3 × 10−46. Note that the noise is a systematic one, so that it is not necessary to use larger α here.

[27] The noise and the errors from the difference between the real and computed spectra affect the reconstructed lightning distribution. Thus we take the average of the reconstructed data over the month in order to diminish such affection and to extract monthly tendency of lightning activity. The calculated results are shown in Figures 8 and 9 as color maps of daily (averaged) variation. The measurements have different numbers of samples caused by the local interference. The total number of available spectra is summarized in n equivalent days, where the tabulated fraction arises from comparison with the total number of days in each month. Note that the result for November has extremely small amount of data.

Figure 8.

Reconstructed daily variation of 1-D lightning distribution (March through August). Here n is the equivalent days of available spectra.

Figure 9.

Reconstructed daily variation of 1-D lightning distribution (from September). Here n is the equivalent days of available spectra.

[28] There are three main regions where the lightning activity is intense at about 5, 10, and 13 [Mm] from the observatory and they are observed throughout the measurement period. The one at 5 [Mm] is identified as the lightning distributed around the southeast Asia because of distance from the observation point and the active time (about 0800 UT), and the regions at 10 and 13 [Mm] can be expected as the one from American region and the one from Africa (it was reported that active times are about 1500 UT at Africa and 2300 UT at America [Nickolaenko and Hayakawa, 2002]). It is possible that the region at 13 Mm consists both of lightnings in America and Africa because the active regions at 10 and 13 Mm show very similar daily variation in Figures 8d–8f and data observed by satellites (optical transient detector and lightning image sensor) also indicate that American lightning region ranges from North to South America. However, it is difficult to specify the more detailed localization because those areas are positioned almost at an equal distance from our observatory.

[29] Seasonal variation of active lightning region is observed such that in March, April, and December the active region ranges approximately from 13 [Mm] to 16 [Mm], and in June through September, however, the one from 10 [Mm] to 13 [Mm] becomes active. It seems that the lightning activity is located mainly in the Northern Hemisphere in June through September, but the distribution moves to the southern one in another half of the year. It is also quite reasonable that the result in August shows the strongest activity.

[30] The results show us the problems of this method, which are discussed here. Some figures show three active regions from 10 to 16 Mm (for example, the ones in March and April), but some do only two. We cannot conclude immediately that the number of active regions varies depending on month, because the variable parameters of K and α are likely to influence the results. For spectra with less noise, larger K, i.e., wider bandwidth, are used for fitting, and vice versa. It seems that fewer spectra among a function set are enough to make good fitting to the measured data with narrower frequency band while it is necessary to have many spectra to fit the wideband measured data. This effect may appear in the figures, because the data in March and April actually include less noise. Also, the variation of α influences the results, e.g., too small α is possible to split a large peak into two peaks as discussed often in inverse problems. To solve this problem, we need to improve the following points. The first is the improvement of the measurement system against noise. As we mentioned in Figure 5, the existence of large noise affects reconstructed results. Thus the acquisition of data with less noise is essentially important. Also, using high-performance PCs allows us to collect data with high sampling frequency, which is useful to process data to reduce the effect of noise. In July 2004, we have installed a new measurement system with low-noise sensors, and the data with much higher quality are being accumulated, which will enhance the validity of the present method drastically. The second point is the appropriate choice of parameters for reconstruction and the regularization, i.e., K and α. Some methods were reported concerning the choice of α for the case that data include noise with an unknown norm [Golub et al., 1979], and the consideration will be important in the future work.

[31] Figures 8 and 9 show that the lightning distributions from America and Africa are much more active than Asian one. Here is also a problem to be overcome, and we cannot conclude it again as the results show. We confirmed that the difference between the measured and the fitted spectra, i.e., the goodness of fit, varied depending on the measured time of the spectra. It is considered that the used function set is not appropriate according to time because we used a uniform electron density profile for computation. More precise identification will require 3-D computation with fully nonuniform profile.

[32] We can see that a small peak exists at 1 Mm in Figures 8d–8f and Figure 9a. It is difficult to identify this peak by only one observatory. It is possible to be due to lightnings in Japan, or due to noise. Inclusion of data observed at different sites will provide preciseness and more information on it.

4. Conclusion

[33] The finite difference method for Schumann resonance analysis has been developed in the spherical cavity with a uniform profile of electron density. The atmospheric and ionospheric parameters are derived from IRI 2000, and NRLMSISE-00 models, to calculate realistic Schumann resonance spectra. The identification of lightning activity distribution has been successful, and the distribution is obtained with respect to distance from the observation point. We have applied the present method to the observed data at Hokkaido, Japan, and the recovered results show some reasonable characteristics: (1) mainly three active regions exist at a distance of 5, 10, and 13 Megameter from Hokkaido, Japan, (2) the distribution moves prominently nearer in the Northern Hemisphere summer, and farther in winter, (3) by the active time and the distances, the one at 5 Mm is identified as southeast Asian region, and the ones at 10 and 13 Mm are likely to be a mixed response from middle and southern America, and Africa, and (4) in the Northern Hemisphere summer the lightnings become very active.


[34] The authors would like to thank Y. Hobara, N. Iwasaki, and T. Hayashida for having installed and maintained the measurement system collecting data for this work. One of our authors (M. H.) is grateful to the Mitsubishi Foundation for its support.