We present model computations of global electromagnetic resonance (Schumann resonance) based on the monthly worldwide distribution of thunderstorm activity collected by the Optical Transient Detector satellite. To obtain diurnal and seasonal variations of electromagnetic signals produced by the thunderstorms worldwide, we apply masks that “activate” definite parts of global thunderstorms and move around the planet with the Sun. Three types of masks are discussed: The first one triggers the activity at the dayside of the globe; the second one activates its afternoon sector; and the third one switches on the thunderstorms within a hot spot, a circular area of a fixed radius. We perform computations of Schumann resonance spectra and demonstrate that numerical models are a useful tool for studying and interpreting resonance features.
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 Studies of global electromagnetic (Schumann) resonance are a prosperous part of the Earth sciences, since the phenomenon is driven by electric activity of thunderstorms and thus allows one to monitor the dynamics of global thunderstorm activity. The latter is of a great interest: its variations might be connected to global climate change. The lightning strokes occur where developed atmospheric convection exists. Global warming will change the heat balance in the troposphere and may cause a redistribution of thunderstorms worldwide. Recently, lightning flashes have been monitored from space by optical detectors [Christian et al., 2003]. The records are processed and averaged over the annual and monthly periods, producing maps reflecting the long-term dynamics in the spatial distribution of thunderstorms. Since the satellite circles the planet, it is impossible to directly acquire the short-scale temporal variations of thunderstorm activity, say, on the diurnal scale. Such variations are inferred from the electromagnetic data collected by the ground-based observatories: measurements of the fair weather field electricity (the static vertical electric field between the ground and the ionosphere) and records of global electromagnetic (Schumann) resonance.
 Schumann resonance observation involves passive remote sensing of the global environment. The fact that the signal is only detected, not generated, is a great advantage of the method. However, this is the reason why interpretation of experimental data is one of the most complicated problems. There are approaches based on the formally correct solution of the inverse electromagnetic problem: deducing the global distribution of thunderstorm activity from Schumann resonance records [Shvets, 2001; Ando et al., 2005]. To avoid resolving the complexities of the inverse electromagnetic problem, the direct solutions are often used in interpretations. These establish “effective” models for the sources of resonance oscillations, which generally belong to two classes [Nickolaenko and Hayakawa, 2002]. The first class implies that thunderstorms are distributed in a compact area that circles the globe during the day. Temporal variations of the source intensity and the width of area covered by thunderstorms are usually found from comparison with experiment. Other models consider lightning activity distributed in a few fixed centers, or even thunderstorms covering the globe. The diurnal drift arises from temporal variations in the activity at individual areas.
 We present here a model of a new generation. It is based on the monthly averaged maps of the global distribution of thunderstorms obtained by the Optical Transient Detector (OTD) [Christian et al., 2003]. Special masks produce the diurnal variations in thunderstorm activity, agreeing with concepts of modern climatology. Algorithms with accelerated convergence are used for computing the field components in the uniform Earth-ionosphere cavity relevant to an instant distribution of lightning strokes. Thus the model variations are obtained of Schumann resonance spectra on the diurnal and seasonal scales. Since the model is based on observations, we expect that realistic properties of radio signals will be obtained. Such an approach allows for carrying out the numerical simulations with controllable properties of cavity and sources for an arbitrary observatory with the subsequent data comparison. We demonstrate below what kind of alterations in the Schumann resonance signal take place at a given observatory.
2. Description of the Model
 As the model input, we use the OTD map of the annually mean flash rate with 2.5 × 2.5° spatial resolution. Description of the spaceborne measurements and the results obtained by OTD can be found in the detailed work by Christian et al. . The maps showing global lightning distribution are available at http://ghrc.msfc.nasa.gov. Data reveal the well-known features of global lightning activity [Christian, 2003; Christian et al., 2003]. Thunderstorms tend to concentrate over the land. Lightning activity in the Northern Hemisphere is higher than in the Southern Hemisphere. Global thunderstorm distribution has a stable southern boundary placed at about 40°S. The seasonal drift of the northern boundary is apparent together with changes in the east-to-west extension of lightning flashes.
 A moving OTD spacecraft presents only averaged distribution of lightning strokes characterized by the long-term variations. Yearly, seasonal and monthly distributions are obtainable, but variations on the diurnal scale are not available. Therefore additional suppositions are necessary for obtaining diurnal variations in the level of global lightning activity.
 We introduce the models based on conventional geophysical concepts: tropical thunderstorms arise from deep convection during definite periods of local time. The storms occur predominantly in the afternoon sector of the dayside of the globe. The models incorporate the motion of the day-night interface and its arrival to different continents. In the simplest, the dayside (DS) model, the lightning in a given cell of the monthly OTD map occurs when it occupies the dayside of the globe. Activity stops at night. To formally introduce such a model, we accept that the day-night interface is associated with a unit step function
being equal to zero when an arbitrary point is found in the night hemisphere (the condition x ≥ 0 is held) and to unity at the dayside of the globe (x ≤ 0).
 Earth's rotation combined with the east-west asymmetry of continents causes daily alterations in the cumulative number of lightning flashes. The north-south asymmetry results in annual variations, as the activity drifts northward or southward owing to the tilt of the Earth's rotation axis. Thus variations on the daily and seasonal scales arise in association with the distribution of continents over the globe [Christian et al., 2003]. We must note that the concept is not new, however “activation” of spatial distribution of lightning strokes has not yet been applied in computations of Schumann resonance signals.
 We present in Figure 1 a diagram showing the masks mentioned in the Introduction and used in computations. The top left plot shows the sample initial global OTD map, presenting the potential activity. The dayside model (DS) activates the strokes at the sunlit side of the globe. A realistic model accounts for the time required for the development of atmospheric convection, cloud formation and electric charge separation. This is why in computations the activation of strokes in the DS model is delayed by 3 hours.
 In the afternoon (A) model, the strokes become active in the afternoon sector of the dayside hemisphere. The top right plot of Figure 1 demonstrates this model for the UT noon on 22 December (the winter solstice). The terminator and the afternoon sector move to the left with time, and activity shifts westward, from Africa to America causing changes in the source-observer geometry and in the cumulative number of active strokes.
 The hot spot (HS) model activates strokes within a circular area of a given radius around the center of the day hemisphere (under-the-Sun point). Obviously, the HS model turns into the DS model when its radius is equal to 6 hours. The bottom plots in Figure 1 illustrate the HS models of 3 hour radius. Shown are noon on 22 December (the bottom left plot) and 1800 UT on 22 June (bottom right plot). We observe that active thunderstorms shift to the Northern Hemisphere in summer. At 1800 UT (the bottom right map), the strokes in the Northern America are responsible for the activity at the time. The spot circles the globe and “opens” or “closes” particular centers. Because of the tilt of the Earth rotation axis, the latitude of the HS center varies from month to month sweeping from −23.5° in the winter to +23.5° in the summer solstice. Longitudinal position of the noon point also varies with season in accordance with the ellipticity of the Earth's orbit; this shift is described by a special “time” equation [Nickolaenko and Hayakawa, 2002]. All seasonal factors were accounted for in the computations. A lag must be introduced of maximum thunderstorm activity against the maximum of solar illumination. We postulate that delay is equal to 3 hours; that is, the thunderstorm maximum is shifted eastward by 45° from the under-the-Sun point. (The shift is not shown in Figure 1).
 All three models exploit the same assumption: the electromagnetic impact of thunderstorms is directly proportional to the cumulative number of lightning flashes at a given moment. This implies that the average stroke radiates the same amount of ELF energy regardless of its geographical position and time of day. Such a supposition is physically appropriate, since the lightning stroke is a breakdown of the insulating air slab, and the air content and mechanisms of breakdown remain the same all over the planet.
 Application of globally distributed strokes in the Schumann resonance computations is not a great novelty. The global annual World Meteorological Organization  map was already used in Schumann resonance modeling by Ogawa and Murakami . The distinction of our work is in applying the recent optical data collected in orbital observations rather than the acoustic detection of thunderstorm days at a network of weather observatories, and in the modern software developed for SR computations since those times.
3. Computation Procedure
 As the first step, we compute the geographic coordinates of the center of the dayside hemisphere. The distance from the center to the day-night boundary is accepted to be exactly 10 Mm, or to 90°. In fact, the center of the solar disk occupies the horizon at this distance, provided that atmospheric refraction is absent. We could introduce the day-night interface for the upper edge of the Sun and account for the regular atmospheric refraction [Nickolaenko and Hayakawa, 2002]. In this case, the angular distance from the under-the-Sun point would increase to 90.85° or to 10.0944 Mm. Such an extension is negligible.
 The boundary of the HS model was found in a similar way, but for the smaller radius of the active zone. After the particular mask is formed, we compute the net number of strokes as a function of universal time by summing them over the cells of OTD map corresponding to the active area. Such computations were performed for varying time of day to give the expected diurnal variations of the planetary thunderstorm activity.
3.1. Diurnal Variations of the Net Number of Lightning Flashes
Figure 2 shows diurnal variations of the cumulative number of flashes for the three models: dayside (DS) model (left plot), afternoon (A) model (middle plot), and the hot spot (HS) model of 3 hour (45°) radius (right plot). The universal time is shown in hours along the abscissa and the net number of strokes is depicted along the ordinate. Computations show that daily variations depend on the mask. In particular, they alter by the factors of 2.5, 5, and 7 for the DS, A, and HS models correspondingly. The variations are noticeably greater than those observed in variations of Schumann resonance intensity or in the vertical electrostatic field.
 Lightning activity in all models increases by a factor of 2 in the months of boreal summer. The feature was already attributed to the south-to-north asymmetry in the continent distribution over the globe [Christian et al., 2003]. The HS model predicts the major contribution from thunderstorms in North America in summer, which is in agreement with Schumann resonance observations. Diurnal patterns in autumn and spring occupy an intermediate position in the plots, and these are very close to each other. Such a result might be considered as a trivial one; however the reciprocity indicates that computations are correct.
 A definite “podium” (background level) is a noticeable feature of cumulative flash number, especially in the DS model. Three global thunderstorm centers in Africa, America and Southeast Asia are seen in diurnal variations, although variations in the DS model are weak and less structured than in afternoon (A) and HS models. Substantial advantage of the model data presented here is their quantitative rather than a qualitative character. In particular, diurnal variations from a few tens of percents to an order of magnitude were ascribed to the global thunderstorm activity in literature. Since the present models exploit recent and more accurate global thunderstorm data, one may expect that results of computations correspond to the electromagnetic records in a better way than before.
 Plots of the middle plot of Figure 2 (the afternoon model) have more pronounced alterations in the level of thunderstorm activity than in the DS model. Substantial contribution comes from Africa, which corresponds to observations. A conceptual drawback of the model is that the cells at different latitudes equally contribute to cumulative activity. Such a simplification ignores the height of the Sun over the horizon, which seems to be physically unjustified. Therefore we introduce the hot spot model.
 Results computed with the HS mask are given in the right plot of Figure 2. We present the data for the 3 hour radius of the hot spot. The size corresponds to estimates of the characteristic area occupied by the global thunderstorm activity [Nickolaenko and Rabinowicz 1995; Nickolaenko et al., 1998]. The westward daily motion of the mask activates different areas, and variations become sharp and discrete for the HS model.
3.2. Second Statistical Moments of Schumann Resonance Signals
 Here ak(ω) denotes the amplitude of the current moment of the kth stroke of lightning; ek(ω) and hk(ω) are the complex spectra of elementary electric and magnetic pulses:
where a is the Earth radius and h is the effective ionosphere height; ν(f) is the complex propagation parameter defined with the help of the following linear model: ν(f) = (f − 2)/6 − f/70; θk and Bk are the source angular distance and its azimuth (for the particular pulse). The exponential factor accounts for the phase shift connected with the pulse arrival time tk to the observer. Horizontal axis X is directed along local parallel from west to east, the Y axis points from south to north, and the azimuth B is measured clockwise from the direction to north. We apply the zonal harmonic series representation for the Legendre and associated Legendre functions with accelerated convergence when computing the elementary fields ek(ω) and hk(ω) [Nickolaenko and Hayakawa, 2002]. Explanations must be given concerning the infinite limits in the above sums. In fact, the summation stops when indices k reach the value satisfying the condition (t − tk) ≤ 0 (the future cannot influence the reality). Owing to the temporal decrease of individual pulses, the “early” strokes become progressively smaller. In realistic models of Schumann resonance signal, the lower limit of summation is chosen so that a few seconds are included prior to the current time moment.
 We assume that individual pulses form a Poisson random process with the pulse flow density L [events per second], and parameters of individual signals are independent. To obtain the second statistical moments (the power and cross spectra), we take products of the sums (1) with corresponding complex conjugates and average the result over the ensemble of realizations. Only diagonal terms (with coincident subscripts) remain in the second statistical moments of the Poisson process owing to averaging over pulse arrival times [Middleton, 1960]. This fact is often regarded as “pulsed energies are summed”. The resulting power spectra of different field components of Schumann resonance oscillations acquire the following form [Nickolaenko and Hayakawa, 2002]:
 It is easy to generalize the above relations and obtain expression for the cross spectrum of two orthogonal horizontal magnetic field components:
 The above formulas correspond to an arbitrary distribution of lightning strokes in space, and we apply them in computations based on the OTD maps for a given observatory position. The strokes occur all over the globe distributed in OTD cells of 2.5 × 2.5° size. The cells are numbered with index “p,” and the pth cell has the distance θp from the observer and is characterized by the wave arrival azimuth Bp. Since the distance θp is ascribed to all the strokes occurring at the pth cell, our description is of an approximate character. For a given time of day, a cell is activated or deactivated by one of the three model masks. Hence contribution from the given active cell ‘p’ into the field power (2) must be multiplied by the total number Ap of strokes observed in a given cell; that is, the substitution must be made 〈ak2〉 → Ap in the above formulas.
 We use these formulas to show how the summation is performed over the OTD map:
The real part of complex Poynting vector (5) describes both the cumulative power flow from the global thunderstorms and the unambiguous average value of wave arrival angle for a given field site at a given time. Thus its amplitude measures the current intensity of global thunderstorms and its orientation shows the direction from the centroid of lightning distribution.
4. Results of Computations
4.1. Schumann Resonance Data
 We present the daily averaged Schumann resonance spectra in Figures 3 and 4computed for the afternoon and the hot spot models for the Lehta (Lekhta) observatory (64°N, 37°E). Each plot contains twelve curves that depict daily averaged spectra for every month of a year.
Figure 3 shows the power spectra of individual field components. Combinations of spectra demonstrate a substantial increase in the field power during the summer, especially in the HS model. The asymmetric resonance lines in the power spectra are a distinctive feature of the HS model. Spectra in the intermediate seasons are very close to each other. This could be expected, as the source intensities shown in Figure 2 are also close for these seasons. In general, seasonal behavior seen on Figure 3 has much in common with that in Figure 2.
Figure 4 surveys the daily averaged spectra of the real part of the Poynting vector, which is the electromagnetic power flow. The general level of the Poynting spectra also increases during summer, especially in the HS model. Spectra of the intermediate seasons are again very close to each other.
 According to the OTD data, the average power flow along the geographic parallel (PoyX, or the east-west component) changes its direction with the season (from east to west in summer and from west to east in autumn and winter), reflecting the seasonal redistribution of activity between Asian and American sectors. Seasonal alterations in the direction of power flow along the meridian (PoyY, or south-to-north component) are absent, coinciding with observations [Belyaev et al., 1999].
4.2. Dynamics of Global Thunderstorm Activity Present in the Schumann Resonance Signal
 Let us consider what dynamics of global thunderstorm activity might be inferred from the model Schumann resonance data. We use the cumulative intensity of Schumann resonance as the estimate of the global lightning activity. Numerical experiments have shown that the level of thunderstorm activity might be found from the complete intensity of resonance oscillations, i.e., from the power spectra of the vertical electric or full horizontal magnetic field components integrated in the frequency band covering three resonance modes or more [Nickolaenko, 1997]. An alternative technique to suppress the source-to-distance dependence was suggested by Fraser-Smith et al. , and it is currently used at the Arrival Heights station (Antarctica), when the signal is monitored between the first and the second resonance modes, i.e., around 11 Hz frequency.
 We present in Figure 5 the variations of cumulative Schumann resonance power (HS model) integrated in the frequency band from 4 to 26 Hz. Computations were made for several observatories. The following sites were used: Arrival Heights, Antarctica (78°S, 167°E), Rhode Island, USA (42°N, 72°W), Mitspe Ramon, Israel (31°N, 35°E), Nagycenk, Hungary (48°N, 17°E), Moshiri, Japan (44°N, 142°E), and Lehta, Russia (64°N, 34°E). Figure 5 consists of individual plots each presenting a field site, while the bottom plot depicts variations of the cumulative thunderstorm activity of the HS model used in computations. Schumann resonance intensity at Arrival Heights observatory (Antarctica) is weak, and we multiplied it by two.
 The abscissa in Figure 5 shows the time, it is divided into 12 intervals each corresponding to a month. An interval pertinent to each month depicts the 24 hour variations of Schumann resonance intensity. Thus Figure 5 presents a succession of 12 diurnal patterns expected at the given field site. The plots have much in common. All of them show the summertime increase in the Schumann resonance signal connected with changes in the global lightning activity. On the diurnal scale, the level of signal is higher during the UT day than at UT night. However, there is only a general reciprocity: individual curves do not coincide in detail. In particular, activity is overestimated or underestimated at particular areas when observed by different field sites. For example, activity in America is overestimated by the Rhode Island observatory (USA). Similarly, the activity in Asia is overestimated by the Moshiri observatory (Japan). Such a behavior reflects an uncompensated impact of the source proximity to the observer. It was significantly reduced by the frequency integration, but there are remnants left that affect the data.
 As Figure 5 demonstrates, there is a general similarity among the data. Therefore we may imply that an appropriate averaging procedure of the globally collected data will result in a reliable planetary index of thunderstorm activity similar to the planetary indices of geomagnetic activity [Füllekrug and Fraser-Smith, 1997; Nickolaenko, 1997]. Ambiguity and nonlinearity of mutual activity – intensity links are clearly seen in the Lissajous plots demonstrated in Figure 6. Here, we plot the same data by using the source intensity as the argument and the Schumann resonance intensity at a given observatory as the function. Figure 6 shows the Lissajous plots for Arrival Heights (left plots) and for Mitspe Ramon (right plots). Cumulative data (summed over three months for the fixed time of day) are given for four seasons: winter (December, January, February), spring (March, April, May), summer (June, July, August), and autumn (September, October, November). The thunderstorm activity is shown on the abscissa relevant to the bottom plot in Figure 5. The Schumann resonance intensity is shown on the ordinate. The left plots depict the Lissajous figures computed for the polar observatory in the Southern Hemisphere, and the right plots correspond to the site at the middle latitudes of the Northern Hemisphere. At these positions, the seasonal and diurnal motion of thunderstorms occurs in “antiphase” relative the observatory.
 The Lissajous figure must be a straight line in the ideal case of direct proportionality of data, i.e., in the case when Schumann resonance intensity at a site directly reflects variations of the global thunderstorm activity. As model computations show, this is almost the case for the Arrival Height (AH) high-latitude observatory. Patterns in Figure 6 are close to straight lines all though the year. Deviations increase in spring and autumn, but dependence is linear in summer and winter. The middle latitudes observatory Mitspe Ramon (MR) is closer to the global thunderstorms, and its plots always deviate from the line.
 Deviations of electromagnetic data from the source intensity put a natural limit to the accuracy of reconstruction of the level of global lightning activity by using Schumann resonance records at a given place. We have an ideal situation in the model computations: the known source distribution and postulated intensity variation. Numerical experiment shows that electromagnetic intensity deviates from that of the source. It tends to follow the changes in the global thunderstorm activity; however, the effect of source proximity is always present, which modifies the pattern. Special calibrations might be employed to the particular electromagnetic data that compensate deviations arising from the source motion, calibrations that are based on a particular model and depend on the season and time of day [Pechony and Price, 2006].
 Since Schumann resonance intensity records reflect general variations of the global thunderstorms, temporal patterns must be similar when recorded at different observatories. We demonstrate validity of this statement by presenting results in Figure 7. Here, the patterns of global thunderstorm activity and of the Schumann resonance intensity (Ez component) are shown at widely separated observatories to demonstrate their reciprocity. Five sites are presented in Figure 7: Arrival Heights (78°S, 167°E) – Antarctica, Nagycenk (48°N, 17°E) – Hungary, Moshiri (44°N, 142°E) – Japan, Mitspe Ramon (31°N, 35°E) – Israel, and Rhode Island (42°N, 72°W) – USA. The distribution of sites provides a really global coverage. In fact, there are a few nonmentioned observatories working at present, however, their coordinates are close to those listed above (especially when measured in Schumann resonance wavelengths), so that relevant maps practically coincide. Plots in Figure 7 are in fact the 3D combination of Figure 5 curves.
 While the structure of diurnal patterns varies from station to station, these variations are imbedded in a frame (transition from white to dark inking emphasized by the contour line) having similar form worldwide and changing in the same fashion as the global thunderstorm activity. There are deviations in the start and end time of intense activity at different observatories (about an hour), but the general behavior is similar all over the globe and corresponds to thunderstorm activity. The model data in Figure 7 show that cumulative power of Schumann resonance corresponds to the level of global thunderstorm activity, as the resonance is a global phenomenon, which allows us to monitor the global lightning activity from a single observation point.
 We must recall that the uniform Earth-ionosphere cavity model was used, and no day-night asymmetry of the ionosphere was included into the model. When thunderstorms become active (the relevant threshold is shown in Figure 5 by the horizontal line), intensity of resonance oscillations increases worldwide, and this is observed in Figure 7 as transition from the dark to white shading. Local factors govern the “interior” details arising with the “local day”, and these details seriously deviate from one observatory to another, reflecting variations in the observer-thunderstorms geometry (closed contours in the white area of maps in Figure 7).
5. Discussion and Conclusions
 Models introduced in this work allowed us to present temporal variations of the activity at a particular cell of the global OTD map. The mask moving in time corresponds to the general concept that activates different thunderstorm centers appropriately. The activity lasts longer in summer, which is conditioned by purely geographic factors. The north-south asymmetry in the continent outline combined with the east-west asymmetry causes the variations. North American continent broadens to the north. Its west coast extends approximately 60° to the west relative to the west coast of South America. Similar widening is valid to Asia, but has a smaller extent. The global thunderstorm activity starts earlier and continues to later hours in summer thus increasing the duration of the high level of Schumann resonance oscillations observed worldwide. Seasonal alterations in the thunderstorm duration coincide with the varying day length for the middle latitudes in the Northern Hemisphere.
 The following major results were obtained in the study:
 1. Numerical modeling of Schumann resonance was performed on the basis of the OTD global distribution of thunderstorm activity. Application of the OTD map in Schumann resonance computations allows for interpreting the behavior of the field.
 2. The approach is a controllable numerical experiment with the known properties of the field sources and the resonance system. Results of computations confirm that cumulative intensity of Schumann resonance oscillations tends to reflect the diurnal and seasonal variations in the global thunderstorm activity.
 3. Observations at the polar and subpolar observatories are connected with a smaller impact of the source distance on the resonance intensity.
 4. The model described offers a way for evaluating the influence of source motion on the acquired electromagnetic data. It may be used in developing the data processing algorithms that reduce the factor of the source motion. Applications of OTD data provide reasonable estimates for the accuracy of monitoring of the global electric activity by using Schumann resonance and for finding an optimal geographical configuration of the network of Schumann resonance observatories.
 We would like to thank V. C. Mushtak for the constructive criticism of the paper. The work of one of the coauthors (A.P.N.) was supported by the STCU grant under project 2070. We use the version 1.0 gridded satellite data produced by the NASA LIS/OTD Science Team (principal investigator, Hugh J. Christian, NASA Marshall Space Flight Center). The data are available from the Global Hydrology Resource Center (http://ghrc.msfc.nasa.gov).