Schumann resonance (SR) observations performed simultaneously by a global network consisting of three stations (Lekhta (Karelia, Russia), Moshiri (Hokkaido, Japan), and West Greenwich (Rhode Island, United States)) during almost 1 year were used for mapping world thunderstorm activity. A two-stage inverse problem is solved for locating lightning sources distributed over the Earth's surface from the SR background signals. The first stage consists of inversions of the SR magnetic field power spectra to the distributions of lightning intensity by distance relative to an observation point. The obtained distance profiles of intensity of sources are used as tomographic projections for reconstructing a spatial distribution of sources in the second stage. We have suggested the use of source distance profiles obtained from the spectra of outputs of two orthogonal magnetic antennas operating at each observatory as separate tomographic projections. It is shown that the implementation of additional information on the azimuthal distribution of sources, provided by angular selectivity of magnetic sensors, significantly improves the quality of global lightning mapping under the condition of a limited number of observation stations. Maps of the global lightning distributions constructed by the result of inversions of SR spectra show that the most active regions vary zonally on the seasonal time scale and meridionally on the diurnal time scale being connected mainly with continental areas in the tropics.
 Lightning discharges are the source of electromagnetic radiation in a very wide spectral range, but the bulk of energy is concentrated in the extremely low and very low frequency ranges (ELF and VLF). Radio waves in the lower ELF band, radiated by lightning discharges, are trapped between conductive shells that are the Earth's surface and the lower boundary of the ionosphere [e.g., Nickolaenko and Hayakawa, 2002]. This Earth-ionosphere waveguide provides round-the-world propagation at ELF with very low attenuation less than 1 dB/Mm [Bernstein et al., 1974] that allows for a unique possibility for nearly real-time global monitoring of the lightning activity and properties of the Earth-ionosphere cavity from one or a few receiving stations.
 The spectra of natural electromagnetic fields observed as resonant peaks at frequencies 8, 14, 20, 26, and 32 Hz, represent a global electromagnetic phenomenon known as Schumann resonances (SRs) that are primarily connected with the worldwide lightning activity [Schumann, 1952; Balser and Wagner, 1960, 1962; Polk, 1969]. As it follows from theoretically calculated SR spectra and experimentally measured ELF transients, the amplitudes of peaks at basic resonant frequencies are modulated due to the superposition between direct and antipodal waves, propagating toward each other [Jones and Kemp, 1970]. This modulation appears as a result of arrival delay between the direct and antipodal waves at an observation point and, in such a way, form the SR spectrum of specific shape determined mainly by the source-to-observer distance.
 The distance signatures in SR spectra and wave impedance [Kemp and Jones, 1971; Burke and Jones, 1995; Huang et al., 1999; Nickolaenko and Hayakawa, 2002] combined with the direction finding technique provides a single-site location of super powerful lightning discharges. In addition, multistation observations based on triangulation or/and the “time of arrival” method are used to study the temporal and regional variations of lightning occurrences and their relation to sprite activity and climate variability [Füllekrug and Constable, 2000; Sato et al., 2008; Nakamura et al., 2010]. However, due to the essential overlapping between successive pulses that form the natural ELF background, these techniques can be applied only to very strong events exceeding the background level by a few times that occur comparatively rarely and relate only to a small part of the total lightning activity.
 The problem of analysis of the background SR signals was usually solved by an application of different kinds of models of lightning distribution and its diurnal and seasonal variability elaborated on the results of meteorological [World Meteorological Organization (WMO), 1956] and space [see, e.g., Christian et al., 2003; Hayakawa et al., 2005; Nickolaenko et al., 2006] observations. These solutions provide source intensity of the global thunderstorm centers associated with tropical continental regions in South America, Africa, and Southeast Asia [see Nickolaenko and Hayakawa, 2002, and references therein]. They assume that positions and dimensions of the source areas are known from a chosen model.
 Our purpose is to develop techniques to make SR observations an all-sufficient tool for monitoring global changes in both the Earth-ionosphere properties and source distributions with only a few observation stations that are provided by the global character of the SR phenomenon. Such an approach can be useful not only for terrestrial studies but also for the other celestial bodies with likely lightning activity [see, e.g., Simões et al., 2008, and references therein].
 To infer the global lightning distribution from the background SR signal we proposed a technique of inversion of measured power field spectra into the distribution of sources' intensity in reference to an observation station based on distance signatures in SR spectra [Shvets, 2001; Ando et al., 2005a; Ando and Hayakawa, 2007]. Application of a tomography procedure to a set of such distance profiles of lightning intensity related to a network of observation stations allows us to obtain source distribution over the Earth's surface [Shvets, 2000; Shvets et al., 2009].
 This paper is devoted to the further improvement of our newly developed technique for the global lightning mapping [Shvets et al., 2009] to implement additional information on the azimuthal distribution of sources provided by a separate use of signals received by orthogonal magnetic components in the tomographic procedure. Three-station SR records are used to analyze spatial variability of the global lightning on diurnal and seasonal time scales on the basis of the constructed maps. We also discuss methods for preliminary data processing to minimize rejecting data from our measurement results spoiled by different kinds of interferences affecting field sensors.
2. Equipment and Data Processing
 Results of simultaneous observations of vertical electric (Ez) and two orthogonal magnetic horizontal (Hew, Hns) components collected during the years of 1999–2000 at three stations in the world are used to study the temporal and regional variation of the worldwide lightning activity. Measurements of power density spectra of ELF electromagnetic background signals, calibrated in absolute units, were carried out at the following three places: (1) Lekhta, Karelia, Russia (geographic coordinates, 33.9°E, 64.4°N), (2) Moshiri, Hokkaido, Japan (142.25°E, 44.3°N), and (3) West Greenwich, Rhode Island (71.6°W, 41.6°N).
 The digital data bank collected at the site of Rhode Island contains 12 min average power spectra of the three field components in the frequency range from 5 to 55 Hz. Every experimental spectrum is fitted to a sum of eight Lorentzians, characterizing the first eight normal modes of SRs. A modal frequency, a resonance quality factor, and a peak power spectral density characterize each Lorentzian, so that each spectrum is characterized by only 24 parameters. Two other data banks consist of 10 min average power spectra recorded with frequency resolution 0.1 Hz and 0.25 Hz in the frequency ranges of 4–40 Hz and 1–870 Hz at Lekhta and Moshiri, respectively. More details on the equipment and preliminary data processing techniques applied at all the stations are described by Heckman et al. , Hobara et al. , Ando et al. [2005b], Belyaev et al. , and Shvets et al. .
 The field spectra recorded at Rhode Island has been previously edited in MIT to make clear the data bank from spectra spoiled by local weather conditions and man-made noise near the observatory. Nevertheless, further examination of the data quality has shown a necessity in additional data selection. The same problem occurred with data from other two stations. Taking into account a huge volume of data (more than 400,000 of 10 min spectra recorded at three stations during a year), it was practically impossible to edit data “manually.” Therefore, we have developed an automated method to select good SR spectra.
 For our study, we use the frequency range from 5 to 35 Hz that cover the first five SR modes typically recognized in the spectra collected at all the three stations. Testing for resonance properties of a spectrum, we calculate a model Lorentzian curves Ln(f) with unit amplitude and fixed values of the peak frequencies Fn = [8, 14.3, 20.5, 26.5, 32] Hz, and the quality factors Q = [6, 6, 6.5, 7, 8] of the first five resonance modes in the frequency range ΔFn = [Fn−3. Fn+3] Hz. These parameters (excluding amplitude) correspond to, on average, experimentally observed values. Then we analyze the angles between the vectors of model Lorentzians Ln(f) and those representing each resonant peak Sn(f) in the SR spectrum to decide whether an experimental spectrum is rejected or not. The angle between two vectors representing the two discrete series characterizes discrepancy between their shapes. The angle between two vectors is calculated in a standard way through the scalar product: An = ArcCos(Rn), where Rn = (Ln,Sn)/∣Ln∣/∣Sn∣ is a cross-correlation coefficient between the model Lorentzian Ln(f) and the corresponding part Sn(f) of the spectrum. A rule for selection of a spectrum is proposed as follows: the maximal value of An calculated for each five resonance mode must not exceed 40 and the mean value of all five angles must not exceed 22°.
 Examples in Figure 1 demonstrate the result of application of this method to the selection of SR spectra. Shown thin curves in the graphs are the tested experimental spectra. The bold curves represent the reference Lorentzians, scaled for the convenience of presentation, which we compare with the experimental spectra. The values of the calculated discrepancy angles for each mode are shown near the corresponding resonant peaks on the graphs. Figure 1 (left) demonstrates a case of rejected spectrum: the mean value Amean and the value of Amax indicated on the graph exceed the threshold levels (22° and 40°, respectively). An example of good spectrum for further analysis passed through the selection procedure is shown in Figure 1 (right).
Figure 2 shows an example of working this automatic selection technique with an extended set of raw SR spectra, recorded at Moshiri during the day of 3 March 2000. The upper row of the graphs represents an overlapping of all the spectra of two magnetic and electric components passed through the selection procedure. The total number (144 of 10 min spectra per day) and the number of rejected spectra are shown in parentheses in the ordinates' labels. The middle row shows the rejected spectra. Daily variations of spectral densities of the three field components resulted from the selection procedure are shown in the bottom row of graphs. We can see from this example that the electric component is the most contaminated by spoiled spectra as compared with magnetic ones, which is a typical situation for any observatory site. Additional algorithms and software have been developed to suppress narrowband noises in the spectra that appear occasionally at approximately 25 Hz and 16.7 Hz originated from nearby railway automatic control circuits and to reject bad spectra (even if they passed through the automatic procedure) manually.
 As the result of selection procedure, data banks of monthly average 1 h power density spectra of the three field components in the frequency range [5 ∼ 35] Hz that cover first five SR modes and data banks of 1 h average spectra for every day have been elaborated for all the three observation stations. A survey in Figure 3 demonstrates daily runs of mean power densities, calculated over the entire frequency range, for the orthogonal magnetic components 〈Hew2〉, 〈Hns2〉 and the total magnetic power 〈H2〉 = 〈Hew2〉, 〈Hns2〉 averaged for every month from August 1999 to May 2000.
3. Inversion of SR Spectra Into Distance Profiles of Source Intensity
 Our approach to solve the first-stage inverse problem is based on a spherically symmetrical model of the Earth-ionosphere cavity. The spectra of the vertical electric and horizontal magnetic field components produced by a point vertical electric dipole are defined by the current moment Idl(ω) of a lightning discharge, the complex propagation parameter ν(ω) and an angular source-observer distance θ [Jones and Kemp, 1970]:
where Pn is a Legendre polynomial, a is the Earth's radius (6375 km), ω = 2πf is the angular frequency (f, frequency), ɛ0 is the permittivity of vacuum, and i = . The effective height of the ionosphere h is an average value (77 km) of the day-to-night variations of the lower ionosphere (from 68 to 86 km) estimated from the VLF/LF measurements [Smith et al., 2004]. We assume a linear frequency dependence for the dimensionless complex eigenvalue elaborated from the analysis of experimental spectra of the background SR signals [Nickolaenko and Hayakawa, 2002, chap. 4]:
 We divide the whole distance range [0,π] between an observation point and its antipode to N = 40 intervals on the Earth's surface. The distance intervals correspond to areas limited by narrow circular stripes around the observation point, which cover the entire sphere. The frequency responses h(ω,ν,θ) and e(ω,ν,θ) of the Earth-ionosphere cavity at an observation point are defined by the distance θ to the middle of the stripe which a source belongs to. We assume that lightning discharges occur with equal probability and independently from each other on a long enough period of the observation that is much longer than the discharge length. The moments of discharge occurrences and the number of discharges are independent statistical variables. Poisson's law describes the distribution of number of pulses in a succession with statistical properties defined above [see, e.g., Middleton, 1960]. On the above assumptions, the power spectral densities of the field components formed by Poisson successions of electromagnetic pulses from lightning return strokes distributed around the Earth's surface, are expressed as expansions in a series of distance-dependent basis functions [Shvets, 2001]:
where θi is the angular distance from an observer to the middle of the ith stripe and triangle brackets denote averaging over an ensemble of the power spectral densities of the measured field components. The vector of expansion coefficients, M (ω) = x measured in (A·m)2 s−1, represents power spectral densities of current moment squared produced by the pulse Poisson successions of lightning discharges with average current moment squared and effective occurrence rates x (events per second) within segments on the Earth's surface corresponding to the full set of distance intervals. The expansion coefficients Mew and Mns representing partial “ew” and “ns” distance profiles of source intensity depend on the azimuthal distribution of sources with reference to an observatory owing to directional selectivity of magnetic sensors, while their sum equals to the full distance profile: Mew + Mns = M [Shvets, 2001].
 We assume that a lightning return stroke is described by the exponential model with typical values of the peak current I0 = 20 kA, time constant τ = 150 μs and the channel height dl = 5 km [e.g., Nickolaenko and Hayakawa, 2002, chapter 3]. These parameters provide a flat spectral density of the current moment in the SR frequency band: Idl (ω) = I0τdl. For this case Mi = xi ∣I0τ dl∣2 = xi∣Q dl∣2 is interpreted as a change of the total charge moment squared within ith distance interval per second and Q is an average charge transferred by lightning return strokes [Heckman et al., 1998].
 The field power spectra equations (3)–(5) discretized by frequency can be written in a compact matrix form:
In this equation, b is a vector of the measured field's power spectrum, A is a matrix of the model field spectra produced by an elementary vertical dipole, calculated for a set of source-observer distances Aij = ∣I0τ dl∣2 · ∣e (ωj, θi)∣2 or Aij = ∣I0τdl∣2 · ∣h(ωj, θi)∣2, and x is the vector of distance distribution of an effective flash rate. The condition number of matrix A is of the order of 1016–1017, then the system equation (6) is solved by minimization in the least squares sense of the functional regularized by Tikhonov :
where α is a small positive regularization parameter. This problem is solved by the nonnegative least squares algorithm [Lawson and Hanson, 1974]. Determination of the regularization parameter is based on the following preestimated discrepancy of reconstruction δ [Tikhonov and Arsenin, 1979]:
where x0 is an initial approximation of the unknown distance profile that can be estimated, e.g., by solving equation (7) without regularization. The dimensionless constant k was chosen empirically in the range of 5 × 10−5–5 × 10−4.
 Different kinds of noises in experimental measurements, errors in determination of the ELF propagation parameters, unknown frequency dependence of a source spectrum, and a problem of choice of the regularization parameter α in the inverse problem solution were investigated by Shvets , Ando et al. [2005a], and Ando and Hayakawa . It was shown by these studies, in particular, that serious errors in the reconstructed distance profiles are connected with “slow” trends in the experimental field spectra introduced by the source spectrum Idl (ω), if it is not flat in the SR frequency range, or/and by uncorrected frequency response of the receiving channels.
4. Tomography Reconstruction of a Global Lightning Distribution by Partial Distance Profiles of Source Intensity
 Full distance profiles of lightning intensity obtained from a network of observatories could be used to estimate the spatial structure of lightning distribution by finding an intersection area between them on the Earth's surface. An example of such a procedure applied to the experimental data is demonstrated in Figures 4 and 5. The distance profile defines flash rates (number of discharges per second) within elementary circular stripes of 5° width on the Earth's surface, centered at an observation point, corresponding to a full set of distance intervals for a chosen observatory. Having reconstructed the distribution by distance we can plot it as stripes on a map filled with colors corresponding to source's intensity within them. The distance profiles resulted from inversions of SR spectra from the three stations are shown in Figure 4 with respect to each corresponding observation point (from left to right, Moshiri, Rhode Island and Lekhta). The intersection area is found by the calculation of geometric mean between the three distance profiles presented in Figure 4 that bound the most probable regions with lightning activity. An application of geometric mean allows for a raw estimation of the distribution of lightning intensity within the intersection areas as can be seen from Figure 5a. It is obvious that increasing the number of observation stations will improve results of the location. A more accurate approach consists in the application of tomography reconstruction by a set of projections obtained as different “views” on the sources' structure represented by distance profiles of source intensity relative to the observation points. The map of the global lightning distribution resulted from tomographic reconstruction by full projections using the technique described in our previous paper [Shvets et al., 2009] is shown in Figure 5b. We can see that solution of the tomography task limits sources to compact regions connected mainly with continents emphasizing especially the most intense areas in the geometric mean map.
 As modeling had shown [Shvets, 2000], the lightning activity concentrated in tropical continental regions can be mapped with our tomography procedure quite accurately by a network of seven stations placed in the northern hemisphere. Our recent study by Shvets et al.  have demonstrated reconstruction of global lighting mapping by the three-station network (Moshiri, Lekhta, and West Greenwich) that allows tracking the diurnal redistribution of lightning activity between the most active continental areas in tropics.
 Further numerical study of the tomography procedure performed in the present work shows that the network of those three observation stations could reconstruct only relatively simple structures of source distributions consisting of a few compact active areas on the Earth's surface. Complication of the spatial lightning source structure can lead to the appearance of lots of artifacts that show significant activities after the tomographic reconstruction at places with the absence of sources in the initial source distribution.
 To implement additional information into the tomography procedure we have developed a new technique that employs the properties of directional sensitivity of two orthogonal magnetic antennas, applied as a rule for SR measurements at most every observation station. This technique allows facilitating determination of the spatial distribution of sources with a limited number of observation stations.
 Advantages of this new approach are demonstrated by a comparison between results of reconstruction of a complicated reference lightning distribution performed with omnidirectional data (power spectra of the total horizontal magnetic field used in our previous paper [Shvets et al., 2009] and reconstruction with separate use of the distance profiles resulted from the inversion of spectra of the orthogonal magnetic components.
 Formulation of the problem of tomography reconstruction by partial projections in application to the global lightning study is as follows. The SR observation network consists of K stations, each equipped by two orthogonal magnetic sensors and distributed on the globe. A network of ordered points M(ϕj,γj,sj) with geographical latitudes ϕj and longitudes γj forms an unknown L vector s that determines the spatial distribution of source intensity over the Earth's surface. For the sake of simplicity, we use the grid of cross points between meridians and parallels 5 degrees of latitude and longitude apart. N vectors xkew and xkns representing intensity profiles of lightning, resulted from the inversion of SR spectra of Hew and Hns magnetic components (“ew” and “ns” profiles), respectively, measured at kth station of the network. Vector x = [x1ew, x1ns, x2ew, x2ns … xkew, xkns]T (T means transpose) is a concatenation of the distance profiles obtained at all observation stations. Then we can construct the following system of linear algebraic equations that establishes the connection between spatial source distribution s and the set of projections x as follows:
where the system matrix W has 2KN rows and L columns:
The elements of the system matrix are determined as follows:
αkj is the azimuth (measured clockwise from the north direction) of Mjth point as it is seen from kth observation point. The squared cosine and sine of the corresponding azimuthal angle determine directional sensitivities of east-west and north-south oriented magnetic antennas to a source at the point Mj with geographical coordinates ϕj, γj. Triple indexation for the elements of vector x indicates their belonging to subvectors xkew and xkns that correspond to the “ew” profile (n = 1) and “ns” profile (n = 2) from kth observation station.
 Sufficient reducing of the dimension of the system can be reached by bounding the set of points on the surface by areas formed by intersections between circular stripes with nonzero intensities, drawn for each observation point. The presence of zero elements in s allows removing corresponding columns in W in equation (11), reducing in such a way the task dimension and required computing power consumption. To facilitate calculations we also rejected from consideration subpolar regions with latitudes higher than 75° where we, a priori, do not expect any noticeable lightning activity.
 Solution of the above equations system uses the nonnegative least squares algorithm by [Lawson and Hanson, 1974]; it minimizes the following functional:
Possibilities of tomographic reconstruction of the global lightning distribution by the two methods with application of three observatories were examined by numerical modeling. Results of modeling reveal that a network of the three stations allows for reliable quality of reconstruction a spatial source distribution for only quite simple configuration of active areas. It could consist of one to three relatively compact regions filled with sources. More complicated source distributions can seriously affect the results of reconstruction. As is demonstrated below, the application of partial distance profiles essentially improves the quality of reconstruction.
 Lightning distribution in October evaluated by the data from space observations by OTD and LIS instruments (http://thunder.msfc.nasa.gov/) during 1995–2002 shown in Figure 6a served as a reference map for our modeling. It represents discharge rates in relative units in 5° × 5° cells. Calculated from this October OTD/LIS distribution, “ew” (blue curve), “ns” (green curve), and full (red curve) source distance profiles related to the observation points shown in Figure 6d are used for tomographic reconstructions.
 Results of reconstruction of the reference October OTD/LIS source distribution employing both the full and partial distance profiles are presented on the maps in Figure 6b and Figure 6c correspondingly. For smoothing, we use the convolution with five-point Hamming window by rows and columns of resulted source distributions to regularize resulted solutions. Bold triangles on the maps designate positions of the observation stations. The color bars in the lower left corners of the graph show the scale of the reconstructed source's intensities. Meridional and zonal totals of lightning intensity in 5° bands calculated by the reference and reconstructed distributions are shown in the graphs docked to the maps at the top and right sides, respectively. Correlation coefficients between the reference and reconstructed two-dimensional distributions and between corresponding latitudinal and longitudinal profiles are shown in the left upper corners of the graphs.
 We can observe essential improvement in the case of using extended information on the azimuthal sources' distribution provided by partial distance profiles. In particular, the correlation coefficient between the reference and reconstructed source distributions grows from 0.45 to 0.81 for the cases of tomography reconstruction by full and partial profiles. In addition, correlation coefficients for longitudinal and latitudinal source intensity distributions are found to increase from 0.42 and 0.86 to 0.94 and 0.98, respectively.
5. Diurnal and Seasonal Variations of the Global Lightning Distribution
 Solutions of the inverse problem for the global lightning mapping were obtained from 1 h power SR spectra averaged for every hour of a day during every month since August 1999 until May 2000. The evolution of reconstructed lightning activity on the globe in December 1999 is demonstrated on the maps constructed for every hour of a day shown in Figure 7. The color scale shown denotes an effective discharge rate within 5° × 5° cells. We should note that the obtained values of the effective discharge rate are obtained under the assumption that an average discharge transfers a charge of 2 C from 5 km height that corresponds to the short exponential pulse approximation with typical time constant 100 μs and peak current 20 kA (see section 3). Maps in Figure 7 demonstrate the gradual redistribution of lightning activity during a day with tendency of amplification of activity in the vicinity of the evening terminator (shown by red curve on the maps) mainly over land areas in the tropics.
Figure 8 shows diurnal changes of integral lightning activity, averaged over the period of the analyzed data in longitudinal segments covering the world thunderstorm centers (130°W–30°W, America; 25°W–70°E, Africa; 85°E–160°E, Asia), which are plotted in the local time calculated for the central points of the corresponding segments. In addition to the dominant peaks of activity at 16–17 h of local time, before the local sunset, the second minor modes placed near the local sunrise can be discerned in the diurnal dependences. We can mark an essential level of lightning activity during the first half of a local day compared with diurnal dependencies of lightning activity in a world thunderstorm center provided by WMO data [WMO, 1956] shown in Figure 8 by the curve with crosses. Belyaev et al.  revealed the existence of a minor nighttime maximum of activity in Africa also on basis of analysis of Poynting vector spectra at SR band.
 In addition to the confirmed diurnal redistribution of lightning activity between the world thunderstorm centers, probing another feature of global lightning dynamics consisting of the latitudinal seasonal drift can be used to test the results of SR spectra inversions. Average distributions of lightning activity in the latitudinal direction during summer (August 1999), fall (September–November 1999), winter (December 1999 to February 2000), and spring (March–May 2000) are shown in Figure 9. We can observe a drift of the peak activity from the northern hemisphere in summer to the southern hemisphere in winter. Such a behavior is found to be in agreement with results of previous studies of SR, locating sources of intense ELF transients, and meteorological observations.
 The annual map of lightning distribution is presented in Figure 10. It is constructed by averaging the results of inversions by 1 h monthly average SR spectra accumulated at the three stations, shown by labeled crosses during the period from August 1999 to May 2000. Corresponding longitudinal and latitudinal distributions obtained by the integration of intensities in 5° longitude and latitude bands are shown at the top and at the right side of the map. Three main peaks in the longitudinal distribution are apparent to correspond to Central America, Central and South Africa and the third peak covers Southeast Asia. At the same time, we can observe a lack of activity in the midlatitudes (higher than 30° for both hemispheres) and a shift of the “Asian” maximum to Hindustan in comparison with those obtained from space observations [Christian et al., 2003]. In particular, midlatitude activity in the southern hemisphere appears to exceed lightning activity in the midlatitudes of the northern hemisphere during summer months as it is seen from Figure 9. In addition, we can observe minor “hot spots” in the oceans.
 The lightning activity reconstructed is found to be concentrated within a relatively narrow range (approximately 30° as a width at a half level of the peak amplitude) around the equator. This seems to be different from the results of the space observations by OTD/LIS instruments that show a corresponding width of about 75° [Christian et al., 2003; Hayakawa et al., 2005]. At the same time, results of global location of intense discharges performed by Sato et al.  and Yamashita et al.  during a year demonstrate the latitudinal distribution of almost the same extension as we can see from our SR inversions. This fact can be explained in terms of the latitudinal dependence of lightning current moments; namely, lightning discharges in the tropics are much stronger than those in the middle and high latitudes and they are the main source of excitation of SR. This hypothesis deserves further experimental and theoretical study to clarify results of the inverse problem solution.
6. Summary and Conclusion
 A two-stage inverse problem is solved for the SR background signals measured at a network of observation stations aimed for mapping global lightning activity. The first stage consists of inversions of the SR magnetic field power spectra to the distributions of lightning intensity by distance to every observation station. The obtained distance profiles of sources' intensity are used as tomographic projections for reconstruction of spatial distribution of sources on the second stage.
 The numerical modeling shows that the number of stations sufficient for accurate reconstruction of complicated global lightning distributions obtained from OTD/LIS space observations can be reduced to three stations in the world with an application of our newly developed technique. We have suggested the use of source distance profiles obtained from the spectra of outputs of two orthogonal magnetic antennas operating at each observatory, as separate tomographic projections. Implementation of the additional information on azimuthal distribution of sources to the tomography reconstruction procedure improved significantly the quality of global lightning mapping under the condition of a limited number of observation stations. We consider that such an approach is a universal way to improve the accuracy of reconstruction as much as possible in the inverse problem.
 For the future real-time inversion, we need to perform the automatic data selection, so that the way of selecting good or bad data from abundant data of SR records is of extreme importance in future. We have suggested one way in this paper, which would be an important sales point in the future multistation coordinated SR observations.
 Results of reconstructions of SR measured at three stations, Lekhta (Russia), Moshiri (Japan), and West Greenwich, Rhode Island (United States) indicate that the dynamics of global lightning distribution is similar to those obtained from earlier SR, space, and meteorological observations on diurnal and seasonal time scales. Finally we have to repeat that this inversion method is based on a uniform cavity without day/night asymmetry, so that it will be an interesting work to investigate this terminator effect as suggested by Satori et al. .
 The authors would like to thank E.R. Williams of MIT for having kindly provided the SR data observed at Rhode Island. This work was supported by NiCT (R & D promotion scheme funding international joint research) Japan, to which we are grateful. A.V.S. is grateful to SVBL (Satellite Venture Business Laboratory) of University of Electro-Communications for offering an opportunity for collaborative research in the University.
 Robert Lysak thanks the reviewers for their assistance in evaluating this paper.