A technique suggested by Sentman and Fraser (1991) for separating the global and the local contributions to the observed Schumann resonance (SR) field power variations is applied to simulated SR field power calculated in a uniform cavity. The spatial/temporal distribution of lighting strokes included in the computations is based on satellite lightning data. It is shown that the local intensity modulation function calculated with a uniform model is similar to the function obtained by Sentman and Fraser with experimental data. Since the local modulation function computed from a uniform model simulation cannot be induced by ionosphere day-night asymmetry, the local modulation function cannot represent diurnal ionosphere height variation. It is shown that the local modulation function depends primarily on the source-receiver distance geometry and that Sentman and Fraser have developed an efficient technique of removing source-receiver distance effects from SR records.
 Schumann resonances (SR) are excited in a waveguide formed by the Earth's surface and the lower ionosphere. Any change in the waveguide parameters would inevitably influence waves propagating in the waveguide. As the lower ionosphere serves as an upper boundary of the waveguide in the SR band, changes in the lower ionosphere conductivity influence SR parameters. Aside from variations with altitude, ionosphere conductivity has also lateral variations. There are substantial differences between the day and night hemispheres above about 50 km, where ionization is due to solar rather then cosmic ray radiation [Polk, 1982]. Influence of this asymmetry on Schumann resonances concerned researchers ever since the first interest in SR arose.
 The interest in the influence of the day-night asymmetry of the ionosphere conductivity on SR field power arose with a new strength in the 1990s after publication of a work by Sentman and Fraser  (hereinafter referred to as SF). SF's work was inspired by observations reported by Keefe et al.  performed simultaneously in Kingston, Rhode Island, and Brannenburg, Germany. These observations showed that diurnal variations of the HEW component at the first resonance mode were similar for the two widely separated stations in local time (LT) rather than in universal time (UT). Observations by Keefe et al.  clearly suggested that SR intensities may be modulated by local time effects. The goal of the SF work was to separate the global and the local contributions to the observed field power variations using records obtained simultaneously at two stations. On the basis of energy flux conservation arguments, SF related the local contribution, the local time modulation function, exclusively to ionosphere height variation. Their work convinced many scientists of the importance of the ionospheric day-night asymmetry and inspired numerous experimental studies.
 Experimental observations [e.g., Melnikov et al., 2004; G. Satori, personal communication, 2005] seemed to support SF's conclusions, showing significant SR field amplitude variations across the day-night terminator. Melnikov et al.  presented long-term records of diurnal and seasonal variations of the SR EZ amplitude at Mitzpe Ramon (Israel) and Nagycenk (Hungary) stations, showing a general intensification of the daytime amplitudes compared to the nighttime amplitudes with a distinct gradient around the time of the local terminator. Because of the apparent correspondence between time of amplitude increase/decrease and time of local sunrise/sunset, Melnikov et al.  suggested that the terminator has a significant effect on the observed diurnal and seasonal variations of the SR amplitudes. Nevertheless, Melnikov et al.  and O. Pechony et al. (Importance of the day-night asymmetry in the Schumann resonance records, submitted to Radio Science, 2006, hereinafter referred to as Pechony et al., submitted manuscript, 2006) have shown that similar diurnal and seasonal patterns of SR amplitudes can be obtained in the uniform cavity, i.e., with no systematic ionosphere variations invoked. This suggests that the observed variations are governed primarily by the variations in the source-receiver geometry and that the effect of the ionosphere is secondary.
 The interpretation of the local time modulation function by SF was based on the assumption that the average intensity over the lowest three modes is independent of the distance from the source, neglecting the actual spatial structure of the wavefields. Nevertheless, the source-receiver distance effects are considered to be significant and cannot be completely neglected in favor of the much weaker [Large and Wait, 1968; Bliokh et al., 1980; Nickolaenko, 1986; Rabinowicz, 1988; Nickolaenko and Hayakawa, 2002; Yang and Pasko, 2006; Pechony et al., submitted manuscript, 2006] day-night asymmetry influence. The aim of the present work is to show, using a controlled model experiment, that the local modulation function introduced by SF cannot be associated with variations in the ionosphere height and that SF in fact developed an elegant technique of eliminating the source-receiver distance effect.
2. Local Modulation Function
 We review below the experimental setup and data interpretation. SF performed simultaneous observations at Table Mountain, California, and Learmonth Solar Observatory, Australia (the local times of the two stations differ by ∼9 hours). To compare the diurnal variations between the two stations, an average horizontal magnetic power density was computed over the first three SR modes. Using the total magnetic field density H of the HEW and HNS components removes effects from directional intensity anisotropies present in individual magnetic channels. The averaging over the three SR modes reduces (but not removes) the source-receiver distance effect.
 In order to separate the local and the global contributions, SF suggested the following procedure. The recorded intensity variations P(tU, λ) were presented as a product of two functions: U(tU), representing the universal time dependence, and a local time modulation function L(tL) (SF):
where tU is universal time and tL is local time, both in decimal hours, and λ is east longitude.
 The local time modulation function was introduced in the following form (SF):
where A = Ar + iAi is a complex constant whose modulus and phase describe the magnitude and the local time position of the maximum, respectively, of the diurnal variation. Assuming ∣A∣ ≪ 1, A is evaluated numerically through equation (4) (SF):
over the selected time interval T1 ≤ tU ≤ T2 using intensities P1 and P2 recorded at two stations. The value of A is then substituted in equation (3) to estimate the intensity dependence on local time. The correction function L(tL)−1 obtained by SF is shown in Figure 1.
 In the following analysis, SF ignored the source-receiver distance effect and attributed the obtained local time dependence to the variations in the ionosphere height by assuming that L(tL) = (tL)−1.
3. Model Description
 We repeat the SF procedure of data processing using total magnetic field power H that was computed in a uniform model of the Earth-ionosphere cavity [Pechony and Price, 2004]. The spatial/temporal distribution of lighting strokes included in computations was based on the OTD satellite lightning data [Christian et al., 2003].
 We used a uniform two-dimensional telegraph equation (TDTE) model developed by Kirillov . The TDTE equation has a form of [Kirillov, 1993]
Here u = Erdr is the voltage (in a spherical coordinate system (r, θ, ϕ)), and the exp (−iωt) time dependence is assumed. The inductance L and capacitance C are expressed as L = − μ0HM and C = ɛ0/HE, where HE and HM are the lower and the upper characteristic altitudes of the ionosphere. The real parts of these complex altitudes correspond to the effective “electric” and “magnetic” ELF heights of the waveguide [Greifinger and Greifinger, 1978]. These parameters depend on frequency and incorporate the vertical alterations of the ionosphere electrodynamic properties.
 The electromagnetic field components are computed as ɛ0Er = Cu, Hϕ = − jθ, and Hθ = jϕ at the Earth surface [Kirillov, 1993]. The surface current density j is calculated from iωLj = gradu [Kirillov, 1993]. The voltage u(θ, ϕ) originating from a vertical dipole source P located at the point (θS, ϕS) on the Earth surface can be described by [Kirillov et al., 1997]
 The lower ionosphere conductivity profile is approximated with the “knee” model [Mushtak and Williams, 2002], which accounts for an important intermediate section of the conductivity profile. The characteristic altitudes HE and HM are expressed as [Mushtak and Williams, 2002]
where hkn is a symbolically defined “knee” altitude; ζb, ζa are the scale heights of the exponential functions approximating the conductivity profile below and above hkn, respectively; fkn = σkn/(2πɛ0), where σkn is the conductivity at hkn; and the scale height ζm depends on frequency as ζm (f) = ζ*m + bm (1/f − 1/f*m); here h*m and ζ*m are the real part of the characteristic altitude and the “effective” scale height at an arbitrary frequency f*m [Mushtak and Williams, 2002]. The altitude h*m(f) can be determined from the equation [Kirillov, 1993, Mushtak and Williams, 2002]
where k0 is the free space wave number; ω0 is the electron plasma frequency; ωHz is the vertical projection of the electron gyrofrequency; νe is the electron collision frequency; and ζNe and ζν are the scale heights of the exponential approximations of the electron density and electron collision frequency profiles, respectively. The model described above is actually a uniform version of the partially uniform knee (PUK) model presented by Pechony and Price . The model variables are summarized in Table 1.
 The model used in this work is a uniform model and as such cannot include lateral height variations of the real cavity. Therefore the fields calculated with this model depend only on the source-receiver distance, without accounting for the day-night asymmetry, and hence the local modulation function computed from these fields cannot be influenced by diurnal ionosphere variations, a property unachievable in the real waveguide. By comparing the local modulation function calculated from fields in a uniform cavity with the one obtained from experimental data, it is possible to conclude whether diurnal ionosphere variations are necessary to explain the experimentally obtained local modulation function.
4. Lightning Data
 As the model input we used 5 years (April 1995 to March 2000) of optical transient detector (OTD) lightning data, available at no charge at http://ghrc.msfc.nasa.gov/. The OTD is a space-based optical sensor with a 100° field of view, on an orbit inclined by 70° with respect to the equator [Christian et al., 2003]. The OTD detects lightning flashes during both ambient day and night with a detection efficiency ranging from 40% to 65%, depending upon external conditions (http://thunder.nsstc.nasa.gov/otd/). Although the OTD sensor detects predominantly intracloud lightning discharges (while it is the cloud-to-ground activity that is the dominant SR exciter), SR records modeled using OTD data simulate rather well experimental SR records (Pechony et al., submitted manuscript, 2006).
 To obtain representation of diurnal lightning activity, OTD orbital data were assembled to diurnal data, and the time resolution was reduced to hours. The diurnal data were then averaged for each month, providing diurnal monthly mean (DMM) variations. The DMM data were then averaged over the 5 years of OTD operation. In this way, for each month of the year, 24 maps were obtained, one for each hour, each containing worldwide lightning activity “typical” for a given month at a given hour. Examples of SR records modeled with OTD DMM input can be found in work by Pechony and Price  and Pechony et al. (submitted manuscript, 2006). For all simulations, the same input is used.
Figure 2 presents the correction function L(tL)−1 computed by using the SF procedure but from intensities calculated using the uniform model and OTD data. The complete horizontal H field was simulated within a uniform Earth-ionosphere cavity (i.e., no day-night asymmetry) for the same pair of stations that was used by SF (Learmonth, Australia, and Table Mountain, California). As can be seen, the model results practically coincide with those acquired by SF (Figure 1). Computational data were obtained in the uniform model and therefore cannot bear any trace of diurnal ionosphere variations by definition. Consequently, the local modulation function obtained by SF cannot represent variations in the ionosphere height.
 The constant A (equation (4)) defines the magnitude and the local time position of the minimum and maximum of the local modulation function. A itself depends on the longitude of the two stations and on the source-receiver geometry. The latter defines the difference between the intensities recorded at the two stations from the same source. In general, the greater the difference between the intensities P1 and P2 recorded at the two sites, the greater becomes the amplitude of the correction function (equation (4)). The time of the maximum is determined by both the location of the stations and the recorded intensities P1 and P2 which determine the relative contribution of the cosine and sine parts of Re (Ae) = Ar cos(2πtL) − Ai sin(2πtL) in equation (3).
 The choice of the California-Australia pair by SF resulted in diurnal variation of the correction function that greatly resembles the ionosphere height variation. Had a different pair of sites been selected, completely different results could have been obtained. Figure 3 shows calculations of the correction function obtained from observed data collected in Mitzpe Ramon, Israel, and Rhode Island, Massachusetts. Rhode Island SR data are available at http://stanheckman.com/. It is important to emphasize that the Israel-Massachusetts correction function, shown in Figure 3, was obtained from experimental data and is therefore independent of the OTD data accuracy or the model used to compute the SR fields. The correction function obtained for the Israel-Massachusetts pair is significantly different from that obtained by SF for the California-Australia pair. The correction function for the California-Australia pair has a minimum at 1300–1400 LT for both months and peak-to-peak difference of ∼50% of the mean, while the Israel-Massachusetts pair results in a minimum around 0800–1000 LT and peak-to-peak difference of ∼20–30% of the mean. Such variations in amplitude and minimum/maximum position between different station pairs cannot be addressed to ionosphere height, but they can be explained by variations in the source-receiver geometry.
 SF used the correction function L(tL) to improve the agreement between the SR data recorded at the two stations, Australia and California (via equation (1)). This procedure improved the correlation coefficient r from 0.51 to 0.7 for September and from 0.39 to 0.82 for April. Since the correction function was interpreted as relative ionosphere height, this significant improvement was attributed to accounting for the local time height dependence of the ionosphere. We repeated the SF procedure with high-quality SR data sets collected at Hollister, California (available at http://www.ncedc.org/), and Mitzpe Ramon, Israel, during 15 days of March 1998. The correction function was calculated from intensities computed using the uniform model and OTD data. Figure 4 shows the uncorrected (top) and corrected (bottom) diurnal intensity profiles. The correlation between the data collected in Israel and California (compared in UT) is poor: r = −0.15. After applying the SF correction procedure, the correlation improved drastically to r = 0.73. We corrected experimental field intensities with the correction function calculated from modeled field intensities. Since we used a uniform model, the remarkable improvement in the correlation coefficient cannot be addressed to accounting for the ionosphere height variation but rather to correct for the source-receiver distance effect.
6. Discussion and Conclusions
Sentman and Fraser  developed an efficient technique of removing the local effects (described by local modulation function) from SR intensity records, by using simultaneous observations at two stations. The local modulation function was interpreted by SF as diurnal variation in the local ionosphere height. The SF results stimulated a new interest in contribution of ionosphere day-night asymmetry on observed SR intensity records and inspired extensive research in the field. However, the local modulation function calculated from SR intensities simulated with a uniform model, for the same station pair as the one used by SF, is similar to the local modulation function obtained by SF. Obviously, the local modulation function computed from a uniform model simulation cannot carry any signature of ionosphere height variations, and the only local time dependence preserved in such a model is the source-receiver distance and the relative position of the two stations. Moreover, experimental data collected at sites different from those chosen by SF yielded local modulation function significantly different from that obtained by SF. This disparity cannot be explained by day-night ionosphere height variation, but it can be addressed to the differences in source-receiver distance geometry. Furthermore, the local modulation function calculated from SR intensities simulated with a uniform model was used to correct experimental SR data from two distant sites, drastically improving the correlation between the two data sets in UT.
 The problem of separating the source-receiver distance effect and the source intensity is a long-challenged one. Polk  suggested using the cumulative intensity of three SR modes as a technique to reduce the source proximity effect and to estimate the current source intensity. An alternative method is to measure the ELF signal at the intermediate frequency, 10 Hz, i.e., between the first and second modes [Fraser-Smith et al., 1988]. Another approach, following Nickolaenko's  suggestion, is placing the receiver at the North or South poles, which remain approximately equidistant from the main thunderstorm centers during the day. However, completely removing the source-receiver distance effect and obtaining pure global thunderstorm activity records from SR data remains an open problem.
 The technique developed by Sentman and Fraser  is an efficient method of removing the source-receiver distance effects from SR intensity records and allows us to obtain global thunderstorm intensity variations from SR records performed simultaneously at two stations. The SF technique and numerical modeling can aid in selecting the most promising existing site pairs or to suggest a location for future sites. This is a step further in creating a background for organizing a network for monitoring Schumann resonances and obtaining the estimate of contemporary global thunderstorm activity.
 The authors whish to thank E. Williams for the fruitful discussions which inspired this work. We would also like to thank the MIT Schumann group and the Northern California Earthquake Data Center for gathering and distributing online Schumann resonance data. Our sincere thanks are due to A. P. Nickolaenko for comments and suggestions throughout the course of this work.