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Keywords:

  • generalized cross validation;
  • global lightning distribution;
  • Schumann resonances

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[1] In this paper, generalized cross validation is implemented to solve an inverse problem to identify global lightning distribution from measured Schumann resonance. The inverse problem to identify the lightning distribution as a function of the distance from the observer is formulated in terms of Schumann resonance spectra calculated by the finite difference method. Tikhonov's regularization is introduced for proper inversion, and the optimal parameter for the regularization is evaluated by using the generalized cross validation as an objective estimation. We compared the results obtained in the present paper with the data collected by the Lightning Image Sensor during December 1999, and this comparison shows a better agreement than the previous results.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[2] Drawing lightning maps of the globe is an important subject in atmospheric electricity, electric power engineering, meteorology, and environmental science, so many attempts have been performed as ground- and satellite-based observations, e.g., the Lightning Image Sensor (LIS) [Christian et al., 1999] and the U.S. National Lightning Detection Network (NLDN) [Cummins et al., 1998]. However, instantaneous global lightning maps have not been obtained yet: The satellite cannot survey all lightning over the globe at a given moment, and the ground observatories are not deployed so as to cover the entire region. The goal of this study is to demonstrate the possibilities of establishing a method for identifying the lightning distribution by solving an inverse problem with measured extremely low frequency (ELF; 3 Hz to 3 kHz) electromagnetic spectra.

[3] ELF waves launched by lightning strokes propagate several times around the globe because of low dissipation and show resonances in the spherical cavity between the Earth's ground and the lower ionosphere, which is called the Schumann resonance [Schumann, 1952; Sentman, 1995; Nickolaenko and Hayakawa, 2002] with the resonant frequencies close to 8, 14, 20 Hz, and so on, as shown in Figure 1. It is readily expected that Schumann resonance would contain information on the position and intensity of the source lightning and on the propagation paths, and actually, many reports were published to explain the resonant frequencies, the intensity, and their daily and annual variations in association with lightning activities. However, the explanations of the Schumann resonant spectra were not convincing enough because the conventional analyses were based on much simplified models of the conductivity profile of the lower ionosphere, e.g., the two-exponential model [Greifinger and Greifinger, 1978], while the real parameters, e.g., electron density, neutral densities, and temperatures, in the lower ionosphere have very complicated profiles in comparison with the conventional models.

image

Figure 1. Example of measured Schumann resonance. The data were measured at Moshiri (44°20′N, 142°15′E), Hokkaido, Japan, on 15 August 2004. The resonances up to the sixth one are observable. The sharp peak at 50 Hz is radiation from power lines. The intense response at the frequency less than 5 Hz is not a resonance but natural noise from other mechanisms.

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[4] Recently, the authors' group reported a more rigorous approach to identify global lightning distribution from Schumann resonance, which consisted of the finite difference method and the inverse problem with Tikhonov's regularization [Ando and Hayakawa, 2004; Ando et al., 2005]. Some papers reported identification of lightning strokes by using ELF waves [Huang et al., 1999; Füllekrug and Constable, 2000], but those methods use ELF transients, prominent signals launched by strong lightning strokes, and successfully identify intense strokes. Our method uses the Schumann resonance, which becomes apparent after the Fourier transform. The method using Schumann resonances can cover many strokes which happen about a hundred times per second but cannot identify individual strokes, while the methods using ELF transients have the opposite features. The finite difference method allows us to consider much more realistic electron and neutral particle density profiles in the calculation of Schumann spectra, and we actually implemented the two models obtained from experimental data, the International Reference Ionosphere (IRI) 2000 [Bilitza, 2001, 2003] and the mass spectrometer incoherent scatter (NRLMSISE-00) [Hedin, 1991; Picone et al., 2002]. Measured Schumann spectra are considered to be synthesized from the responses to lightning sources distributed in some areas, and we formulated the inverse problem to decompose the measured data into elemental responses to separated sources; that is, lightning distribution is reconstructed from Schumann resonance observations. Tikhonov's regularization method is introduced to obtain proper solutions of the inverse problem, while the choice of the regularization parameter was empirically made in the previous reports [Ando et al., 2005].

[5] In this paper, we introduce the generalized cross-validation (GCV) method to estimate appropriate regularization parameters in the inversion [Golub et al., 1979]. For verification of the validity of the suggested approach, we make comparison between our results and the data collected by the LIS during December 1999. The results reconstructed by the present method are improved with respect to the shape of monthly distribution, though there are some prominent discrepancies, which will be discussed in section 3.

2. Formulation of Lightning Activity Reconstruction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

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 Δr and Δθ and are represented by iΔr (i = 1, 2, ⋯, N) and jΔθ (j = 1, 2, ⋯, M), respectively. The current source segment with the length Δl and with the intensity I is located at (i = equation image, 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

  • equation image

where

  • equation image
  • equation image
  • equation image
  • equation image
  • equation image
  • equation image
  • equation image
  • equation image

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

image

Figure 2. Analysis model of a uniform spherical shell cavity. The radii of inner and outer spheres are ri and ro, respectively.

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[7] The dielectric constant equation imager(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]

  • equation image

where

  • equation image

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

  • equation image

where ωpe, ω, νen, and νei 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 νen and νei are given by [Holzworth, 1995; Richmond, 1995]

  • equation image
  • equation image

where equation image, equation image, and NO are the concentrations (in m−3) of N2, O2, and NO. Re is Te/(300 K), and Te 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: ri = 6370 km, which denotes the radius of the Earth, and ro = 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., Δr = 3.94 km and riΔθ = 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 (660332 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, Hϕ(equation image, j + equation image), 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].

image

Figure 3. Examples of numerical calculation of the Schumann resonance.

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[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 λth section is given by

  • equation image

where Δ = 20/Λ (Mm) and Hϕ is the field calculated by the finite difference method and is redefined here as a function of the following arguments: ζ and fk are the source-observer distance and the discretized frequency, respectively.

[13] The measured field H(fk) 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

  • equation image

where σk is the standard deviation of ∣H(fk)∣2 around the time under consideration, and we also define Aaλ(fk)/σk. The fitting problem is to find the solution of the system of linear equations:

  • equation image

where b = [b(f1) b(f2) ⋯ b(fK)]T, A = [{A}], and [·]T is the transpose of the matrix. The unknown coefficients, x = [x1x2 ⋯]T, represent the lightning distribution. The latter noise can be considered by introducing additional basis functions a = 1/(fk)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:

  • equation image

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,

  • equation image

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

  • equation image

where

  • equation image

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

3. Results and Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[16] The data used for the inversion were measured at Moshiri (44°20′N, 142°15′E), Hokkaido, Japan, in December 1999. An example of the measured spectra is shown in Figure 4. The discretization of frequency is 0.25 Hz. The details of the data acquisition and the experimental setup are described by Hobara et al. [2000]. For the fitting, we use the data ranging from 2.50 to 36.0 Hz, i.e., f1 = 2.50, f2 = 2.75, ⋯, fK = 36.00, where K = 135, because there exist atmospheric and 1/f noises below 2.5 Hz, while the range above 36 Hz is contaminated by radiation from power lines. Λ is chosen as 40 because it gives enough fine resolution of the source-observer distance. The values p = 0.1 and L = 61 are used as the parameters of the basis functions corresponding the 1/f noises. These values of p and L are determined empirically so as to fit the reconstructed spectra to the measured ones.

image

Figure 4. Measured and fitted spectra. The experimental data were collected during 1500–1510 UT on 2 December 1999.

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[17] Figure 4 also shows the fitted result with αGCV = 10−43.365. It is found that a reasonable fitting is obtained. This extremely small value of the regularization parameter is caused by the norm of the basis functions which are the responses from the unit current segment and are used without normalization.

[18] In order to verify the validity of the present method, we compare the result with the satellite-based LIS observations. We collect all flashes detected over the month and sort them according to regions with the equal interval, 100 km, i.e., 200 divisions. The LIS does not collect the worldwide instantaneous lightning distribution as mentioned earlier, but the data summed up over 1 month show the tendency of general distribution of the month. Figure 5 shows the statistics of all flashes in December 1999, including events occurring in both day and night. The reconstructed result by the present method with αGCV, over the whole period of December 1999, is also plotted by the thick solid line in Figure 5. The dashed line indicates the reconstructed result by the constant α = 1 × 10−42, which is chosen according to the previous method [Ando et al., 2005]. The reconstructed results are, of course, multiplied by an appropriate constant for the comparison with the LIS.

image

Figure 5. Comparison of the reconstructed lightning distribution with the LIS observations.

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[19] It is found from Figure 5 that the results obtained on the basis of the GCV method show a distribution similar to that of the LIS except a few points. First, both results have a big peak of lightning flashes over the interval from 12 to 14 Mm. The peaks in our result are shifted by 0.5 ∼ 1 Mm away from those of the LIS. In our result, there exists a valley between two peaks, while the one in the LIS observations is not so apparent. The peaks at 9 Mm coincide with each other.

[20] Obvious discrepancies are found at 5 Mm and at 17 Mm. In the former point our result shows a sharp and high peak in comparison with that of the LIS, and in the latter one, on the other hand, no lightning is deduced by the present method. The discrepancies are likely to be caused by the following reasons. The first is the quality of the acquired data. For example, only 8000 points of data were used for the Fourier transform with the sampling frequency 2 kHz for which an insufficiently wide frequency range and few spectrum data are available for the inversion. The second one is the modeling and the calculation of the direct problem. In this paper, we have introduced the azimuthal symmetry because of the limitation of our computational resources and performed two-dimensional calculation. In this case, the computed spectra as basis functions are not precisely obtained because of the lack of the complete modeling of the conductivity profiles of the lower ionosphere.

[21] The reconstructed result with constant α = 1 × 10−42 shows worse agreement than the present result. First, the peaks at 12 and 15 Mm have drastically different intensities. The peak at 9 Mm does not appear clearly. The erroneous disappearance of the peak at 17 Mm is the same as in the present method. Only one better point is that the error at 5 Mm is not so large as the present one.

4. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[22] The reconstruction of lightning activity distribution by the Schumann resonance has been proposed in this paper. The direct problem is formulated in terms of the finite difference method in the frequency domain to calculate the Schumann resonance spectra generated by lightning discharges. The inverse problem under noisy conditions is formulated by introducing Tikhonov's regularization. The generalized cross-validation method has been implemented to choose the appropriate regularization parameter. The result obtained by means of the proposed method shows an improved agreement with the LIS observations in comparison with the previous result [Ando et al., 2005]. The discrepancies of the comparison have been discussed, and further tasks for the enhancement of this study are as follows: (1) The quality of the acquired measured data should be improved, and (2) three-dimensional computation should be performed for taking into account the day-night asymmetry of the lower ionospheric height.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References

[23] This work is partially supported by a Grant-in-Aid for Young Scientists (B) (18760212) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the JFE 21st Century Foundation, TEPCO Research Foundation, and SECOM Science and Technology Foundation. The authors would like to thank Y. Hobara, N. Iwasaki, and T. Hayashida for installing and maintaining the measurement system collecting data for this work. The authors would like to acknowledge A. P. Nickolaenko and A. V. Shvets for fruitful discussion.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulation of Lightning Activity Reconstruction
  5. 3. Results and Discussion
  6. 4. Conclusion
  7. Acknowledgments
  8. References
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