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 Electromagnetic wave radiation from an underground current source related to seismic activity is discussed. In order to estimate the ionospheric effects on the electromagnetic waves associated with the seismic activity, ELF waves in the frequency range from 10 Hz to 1 kHz in the ionosphere radiated from a possible seismic current source modeled as an electric dipole located in the lithosphere, are precisely computed by using a full-wave analysis. In this calculation, the ionosphere is assumed to be an inhomogeneous and anisotropic medium, and the Earth's crust is assumed to be a homogeneous and isotropic conductive medium. Especially, the effects of the geomagnetic field on the ionospheric wave propagation are precisely considered. The results of the calculations in the frequency range from 10 Hz to 1 kHz show frequency dependence in spatial distributions of the wave intensities due to the geomagnetic field-aligned whistler propagation in the ionosphere and the Earth-ionosphere waveguide propagation. Wave intensities which could be observed on the ground and in the ionosphere are determined by assuming the magnitude of the current moment of a seismic dipole source. In a possible situation, the current moment is estimated to be about 80 A·m/Hz1/2 which generates a detectable wave magnetic field on the ground just above a seismic source. However, if we try to detect it in the ionosphere, the source current moment must be thousands of times more intense.
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 Electromagnetic phenomena that may be associated with seismic activity have been reported in a wide range of frequencies. These seismic-related electromagnetic phenomena include perturbations of DC electric fields in the atmosphere and the ionosphere due to emanation of soil gases such as radon, abnormal solar terminator transition of subionospheric VLF propagation, and abnormal electromagnetic wave radiations in the DC–HF frequency range, which appear to be directly radiated from an underground source [e.g., Hayakawa and Fujinawa, 1994; Parrot, 1995; Hayakawa, 1999; Hayakawa and Molchanov, 2002]. These phenomena are detected not only near the Earth's surface but also in the ionosphere, so we should consider the effect of the ionosphere to thoroughly evaluate them in theoretical calculations. So far, a number of theoretical calculations considering the effects of the ionosphere have been proposed. As for the disturbances of the DC electric fields in the atmosphere and the ionosphere, Sorokin et al.  presented theoretical model of DC electric field formation in the ionosphere by seismic activity, which considered the effects of oblique geomagnetic fields in the ionosphere. They showed that the horizontal DC electric field in the ionosphere over the seismic region can reach magnitudes of up to 10 mV/m in accordance with satellite observation results. Grimalsky et al.  also numerically solved for the penetration of the electrostatic field into the ionosphere, for the cases of the vertical inhomogeneous and anisotropic ionosphere with the oblique geomagnetic field, and a source located in the lithosphere. Their calculations showed that the electrostatic field in the ionospheric D layer could effectively change the lower ionosphere; the ionospheric conductivity of around 60 km could be changed by 30–70% due to the electrostatic perturbations. Soloviev et al.  computed the VLF Earth-ionosphere waveguide propagation taking into account a 3D local inhomogeneity of the anisotropic ionosphere to present an abnormal solar terminator transition related with the local ionospheric perturbation above a seismic area discussed by Molchanov and Hayakawa .
 On the other hand, as for the electromagnetic wave radiation directly from an underground source, Molchanov  and Molchanov et al.  computed electromagnetic fields below the frequency of 100 Hz on the Earth's surface and in the ionosphere. Molchanov et al.  assumed a cylindrical current source in the lithosphere, and took into account the vertical inhomogeneity of the ground and atmospheric conductivities, and the anisotropic ionosphere with the vertical geomagnetic field. Their calculation results showed that the ULF emissions associated with the seismic activity could be observed only for strong earthquakes (Ms > 5–6) at horizontal distances less than 100–200 km from the epicenter with a low-depth (<30 km) seismic source. ELF (100 Hz to 1 kHz) whistler mode wave excitation in the ionosphere above a seismic source was discussed by Borisov et al. . They suggested that the effect of direct electromagnetic radiation by seismic-related underground sources is quite small. They presented another mechanism, that the seismic-related ELF waves in the ionosphere above a seismic area originate from the general spherics, which are enhanced by ionospheric irregularities (acting as ducts) associated with earthquakes. Such ionospheric irregularities have actually been observed by satellites prior to seismic activity [Molchanov et al., 2006].
 Anomalous ELF/VLF signals in the ionosphere above seismic areas are reported by many satellite observations [e.g., Parrot, 1995; Hayakawa and Molchanov, 2002]. However, there have even been negative results for the seismic-related electromagnetic waves in the ionosphere [e.g., Henderson et al., 1993; Rodger et al., 1996]. In this study, we discuss the electromagnetic waves radiated from an underground current source, to theoretically evaluate the magnitude of the current source giving a detectable wave radiation. To account for these satellite observations of ELF/VLF waves above the seismic zones, we have theoretically calculated intensities of electromagnetic waves in the ionosphere, free space, and on the ground in the frequency range from 10 Hz to 1 kHz, radiated from a current source located in the lithosphere, by using full-wave analysis [Nagano et al., 1975]. We have then evaluated the possibility of direct ELF wave observations in the ionosphere and on the ground by comparing the sensitivities of actual electromagnetic sensors with our calculation results. Molchanov et al.  focused on Alfven and magnetosonic waves in the ionosphere and the magnetosphere at the ULF frequency range (0.01–100 Hz), here we have focused on whistler mode waves in the ionosphere at the ELF frequency range (10 Hz to 1 kHz). Our calculation method thoroughly considers the effects of the magnetized ionosphere with the oblique geomagnetic field by using full-wave analysis. Based on the mathematical technique of the plane wave expansion in the spherical wave [e.g., Nagano et al., 1993; Yagitani et al., 1994], we can calculate the spatial structures and intensities of the electromagnetic spherical waves radiated from a current source in the lithosphere.
 This paper is arranged into five sections. Section 2 describes the mathematical techniques for applying the full-wave analysis to calculate the ionospheric propagation of the electromagnetic waves radiated by a dipole source in an isotropic conductive medium deep beneath the Earth's surface. Section 3 presents the calculation results and a discussion of ELF wave power dependence on frequency, altitude, seismic source depth, conductivity of the lithosphere, and direction of the current source. Section 4 discusses the possibility of observation of the electromagnetic waves associated with seismic activity by comparing the calculated wave intensities with the sensitivity of an electromagnetic sensor onboard a satellite. The final section summarizes the key properties of the waves radiated from the dipole in the lithosphere.
2. Calculation Method of ELF Wave Radiated From a Dipole Source Located in the Lithosphere
 To thoroughly evaluate the ELF (10 Hz to 1 kHz) electromagnetic wave propagation from an underground current source, we should consider the radiation, induction, and static fields in the ionosphere, free space, and the lithosphere. These fields are included in our calculation method based on the full-wave techniques described by Nagano et al.  and Yagitani et al. . The calculation model includes the horizontally stratified lithosphere–free space-ionosphere as shown in Figure 1. The y and z axes point toward the geomagnetic north and the vertical directions, respectively. The x axis is perpendicular to them and the geomagnetic field line exists in the y−z plane. Plasma in the ionosphere is assumed to be “cold” and is treated as a magnetized anisotropic medium. In the full-wave analysis, it is assumed that the inhomogeneity of each medium is only in the vertical direction, so that it is made of many overlapping homogeneous horizontal layers. Each slice of the ionosphere is characterized by electron density and collision frequency between the electrons and neutral particles. Similarly, the lithosphere is also assumed to be a vertically inhomogeneous but isotropic conductive medium beneath the Earth's surface, and is determined by permittivity, permeability, and conductivity. A dipole source is located in one of the lithospheric layers.
 In Figure 1, when there is a small dipole source in the uniform conductive medium in the lithosphere, the electric field E and magnetic field H radiated from it are given by using the Hertz vector Π as follows,
where each parameter means that: ɛ ≡ ɛ0(ɛs−) is the complex electric permittivity with the angular frequency ω, the relative electric permittivity ɛs, the permittivity of free space ɛ0, and the electric conductivity σ, μ ≡ μsμ0 is the magnetic permeability with the relative magnetic permeability μs and the permeability of free space μ0, k is the wave normal vector, r = (x, y, z−h) is an observation point vector with respect to the source depth h, (r = ∣r∣), p ≡ Il/jω is the dipole moment vector with a current vector I of length l. The basic technique of this calculation is to represent the spherical wave (e−jk·r/r) by a large number of plane waves (see equation (9) of Nagano et al. ). Then, by using spherical Bessel function in the calculation coordinate shown in Figure 1, the spherical wave in equation (1) can be expanded as
where kx, ky, and kz are x−, y−, and z−components of the wave normal vector k, respectively. Since this calculation model is based on the multiple horizontally stratified media, the vertical wave normal component is calculated as kz = . From these equations (1), (2), (3), and (4), the electromagnetic fields are represented as,
 Here, equations (5) and (6) mean that a spherical wave at an arbitrary point (x, y, z) can be represented as a synthesis of a large number of elementary plane waves propagating in all directions of wave normal vector, and and in equations (7) and (8) represent the electromagnetic field of each elementary plane wave. The propagation of the elementary plane waves at an arbitrary altitude z is precisely calculated in the horizontally stratified ionosphere, free space, and the lithosphere as the full-wave solution described by Nagano et al. . The full-wave solutions in the ionosphere are derived from the Booker roots for upgoing and downgoing waves of R (whistler) and L modes with the electron density, the collision frequency, and the geomagnetic field vector B0 in the ionospheric layer [see Booker, 1938; Nagano et al., 1975]. The solutions in free space and in the lithosphere are derived from the propagation constants of upgoing and downgoing waves of TM and TE modes, which are determined by the permittivity, permeability, and the electrical conductivity. These solutions are connected by Snell's law in all the horizontal layers [Nagano et al., 1975], thus we can precisely obtain the full-wave solution of each elementary plane wave. The full-wave solution is expressed by using a vector of its horizontal components as ≡ (Ex, −Ey, Z0Hx, Z0Hy)t, where Z0 is the wave impedance in free space and t indicates the transpose of the vector. By synthesizing all of the elementary plane waves, finally the spherical wave field e is reconstructed as follows,
 It is noted that the integrand (representing the elementary plane waves) of equation (9) is evaluated as a function of altitude z by the full-wave analysis. The infinite integration in equation (9) can be replaced as a numerical finite integration, because kz becomes purely imaginary for large kx and ky, where the elementary plane wave becomes an evanescent wave in the vertical direction. It means the integrand exponentially decreases to zero for large kx and ky, so that equation (9) can be simply computed by numerical finite integration as the fast Fourier transform. These numerical integration techniques were used by Yagitani et al. .
3. Ionospheric Penetration Characteristics of ELF Waves Radiated From a Dipole in the Lithosphere
3.1. Frequency and Altitude Dependence of ELF Wave Power
 To identify the ionospheric penetration characteristics of the ELF waves radiated from a dipole source associated with a seismic activity, it is assumed that a horizontal dipole source of length l along the north-south direction is located beneath the Earth's surface with an electric current I. The lithosphere was assumed as an inhomogeneous isotropic medium in the calculation method, but here it is defined as a homogeneous isotropic medium with the relative electric permittivity ɛs = 4.0, the relative magnetic permeability μs = 1.0, and the electric conductivity σ = 10−3 S/m for the simple calculation model. Figure 2 shows the altitude profiles of the electron density N and the collision frequency v in the ionosphere. The ionospheric model of the electron densities are from the International Reference Ionosphere model for Northern Hemispheric midlatitudes in the daytime [Bilitza, 2001]. Table 1 shows the other parameters for our calculation. Here, we present the wave energy in the frequency range from 10 Hz to 1 kHz (10, 30, 60, 100, 300, 600, and 1,000 Hz), in the altitude range from 0 to 300 km (0, 35, 75, 150, and 300 km).
Table 1. Parameters for the Full-Wave Analysis
Geomagnetic dip angle
Layer thickness of each medium
Current moment Il
Depth of dipole source
Conductivity of the ground
Dielectric constant of the ground
x−y plane range
1024 × 1024
x−y plane resolution
 An example of the distributions of the z component of Poynting flux Pz in the geomagnetic meridian plane over the dipole source is shown in Figure 3, where the altitude range is from 0 to 150 km, the horizontal range is from −300 km to +300 km, the frequency is 100 Hz, and the value of 0 dB corresponds to 1 pW/m2/Hz. The horizontal distribution of Pz at the frequency of 100 Hz and at the altitude of 150 km is shown in Figure 4 over the horizontal area of 1000 km × 1000 km. The two line plots in Figure 4 are the Pz profiles along the x axis (east–west) and the y axis (north–south). Here the distribution of Pz in the free space is the result of spherically radiated upgoing wave from the underground dipole with its reflection from the lower ionosphere. The wave gradually propagates along the geomagnetic field line southward in the ionosphere. The reason is that the radiated electromagnetic wave becomes a whistler mode wave in the ionosphere propagating along the geomagnetic field lines. On the other hand, L–O mode waves are reflected around the altitude of 70 km from the wave mode analysis. Though the L–O mode reflection is not clearly observed, the wave interference between the upgoing R mode waves and the reflected downgoing L–O mode waves is seen around the altitude of 70 km in Figure 3. We see that the effect of the geomagnetic field line is important on the distribution and direction of the wave energy flow in the ionosphere in Figures 3 and 4.
 Next, wave power spectra at each altitude, evaluated as the integration of Pz over the horizontal plane (∫∫ Pzdxdy) are plotted in Figure 5, where 0 dB corresponds to 1 pW/Hz. The powers on the ground are 66.5 dB at 10 Hz, 46.3 dB at 100 Hz, and −38.3 dB at 1 kHz, so the power drastically attenuates on the ground with wave frequency. It should be recalled that the wave power radiated from a dipole source placed in the free space is directly proportional to the square of frequency. However, in our case, the power radiated from the dipole in the isotropic conductive medium decreases as frequency increases. These results are due to the effect of the skin depth, where shorter wavelengths suffer from larger attenuation caused by the electrically conductive medium beneath the Earth. The lower frequency waves have longer wavelengths with less attenuation due to the conductivity.
 As for the altitude variation of the power at 10 Hz, it becomes as 66.5 dB, 29.0 dB, 18.7 dB, 18.3 dB, and 18.7 dB at the altitude of 0 km (on the ground), 35 km, 75 km, 150 km, and 300 km, respectively. In the free space, the wave fields radiated from the underground dipole source are mixed with the wave fields reflected from the lower ionosphere. At the same time, part of the wave energy propagates up to the upper ionosphere along the geomagnetic fields. In a steady state, the wave power (as the total wave field of upgoing and downgoing wave fields at the altitude of 35 km) becomes much less than the power on the ground. The power spectra in the ionosphere above 75 km become almost the same in the whole frequency range from 10 Hz to 1 kHz. This is caused by the fact that the radiated waves become whistler mode waves in the ionosphere, which suffer from almost no attenuation above 75 km, as shown in Figure 3.
 The wave power at altitudes over 75 km is almost the same in the whole frequency range. The power spectra at the altitudes of 0, 35, 75, and 150 km calculated in this section will be discussed hereafter.
3.2. Dependence on Source Depth
 The wave power dependence on the dipole source depth from Earth's surface is calculated. The calculation conditions are the same as those in section 3.1 except for the depth of the dipole source h, which is assumed here as 10 km (the same as in section 3.1), 20 km, and 40 km.
Figure 6 shows the power spectrum dependence on the source depth at four altitudes. The power spectra at each altitude decrease as the source depth increases. As is represented by the equations (7) and (8), the source depth affects exponentially the power transmitted upward and kz gives attenuation in the lithosphere and the ionosphere (the latter effect is evaluated in section 3.3). The powers on the ground at 10 Hz become 66.5 dB with h = −10 km, 45.8 dB with h = −20 km, and 8.3 dB with h = −40 km, respectively. Also, the powers at the altitude of 150 km in the ionosphere at 10 Hz become 18.3 dB with h = −10 km, 1.0 dB with h = −20 km, and −33.6 dB with h = −40 km, respectively. We clearly see that the wave power exponentially attenuates with increasing wave source depth. It could be important for estimating where the seismic electromagnetic source is located (inside the ground or near the ground surface) during actual seismic activity.
3.3. Effect of Conductivity in the Lithosphere
 The electromagnetic field attenuates exponentially with conductivity in the lithosphere, so the power dependence on the conductive medium in the lithosphere is evaluated as shown in Figure 7. The calculation conditions are the same as those in section 3.1 except for the conductivity beneath the Earth's surface: here the three conductivities are assumed 10−2, 10−3 (the same as in section 3.1), and 10−4 S/m. The results show that the frequency dependence of power becomes clearly different with conductivity, as expected. For σ = 10−4 S/m, the powers on the ground are calculated as: 78.3 dB at 10 Hz, 76.5 dB at 100 Hz, and 56.4 dB at 1 kHz, respectively.
 From ground-based observations by Tsutsui , it is reported that the electromagnetic pulses associated with earthquakes are detected at frequencies mainly in a few kHz. It is not easy to know the actual crustal structure of the seismic center, but the waves in a few kHz may be able to propagate with little attenuation for the conductivities less than 10−4 S/m from our calculation results. Meanwhile, the spectra of the seismic source pulse would dominate mainly in a few kHz [Ogawa et al., 1985].
3.4. Different Current Sources
 The directional radiation pattern of a dipole source is well known as “a figure eight.” Here we calculate the powers radiated from a horizontal (above mentioned) and a vertical dipole as two extreme conditions in Figure 8, where the black lines are the profiles for the horizontal dipole and the gray lines for the vertical dipole. The calculation conditions are the same with those in section 3.1, except for the direction of current moment.
 The power spectra radiated from a horizontal dipole are greater than the vertical dipole at all altitudes. Such an altitude dependence of the power spectra comes from the radiation patterns of the horizontal and vertical dipoles. For the case of a vertical dipole located inside a conductive medium, its main power spreads horizontally, and very little power would be radiated upward and penetrate into the free space. These results are caused by the difference of direction of current source, and they agree with previous work by Dong et al. .
 Additionally, Fraser-Smith et al.  also calculated amplitudes of ULF/ELF electromagnetic fields from a dipole source located on the seafloor in four cases of vertically and horizontally directed electric and magnetic dipoles. They calculated the horizontal wave attenuation properties from the dipole source. Though our results cannot be simply compared with their calculation results, they showed that the horizontal electric and magnetic dipoles produce much larger wave fields than the case of the vertical electric and magnetic dipoles. Moreover, Molchanov et al.  calculated the ULF (0.01–100 Hz) Poynting flux on the ground radiated from electric and magnetic type sources. Their calculation showed that the ULF Poynting flux on the ground radiated from a magnetic type (loop current) source is a thousandfold greater below 1 Hz than in the case of a vertically directed current source. Our calculation method could also evaluate wave fields radiated from a magnetic dipole. Specifically, we can simply change the Hertz vector Π to the one for the magnetic dipole in equation (1). However, as it would be difficult to form the loop current during seismic activity, we took only the case of the simplest electric dipole in this study.
4. Possibility of Magnetometer Observations on the Ground and in the Ionosphere
 This section discusses the amplitude of the source current in the conductive lithosphere, from the viewpoint of the possible detection of radiated waves based on the sensitivity of an electromagnetic sensor. In previous works, the amplitudes of wave fields were evaluated by using complicated electromagnetic source models related with seismic activity. Gershenzon et al.  presented a dipole model based on electrokinetic, piezomagnetic, and induction effects with magnitude of seismic activity, and suggested that magnetic field observation on the ground requires a strong seismic magnitude over 5 in their model. Another source model based on microfracturing process was discussed by Molchanov et al. . They suggested that a magnetic type source is more efficient than an electrical type source for penetration into the upper ionosphere in the ULF range (<10–20 Hz), and ULF emissions could be observed only for strong earthquakes (Ms > 5–6). It is important to theoretically evaluate the source amplitude related with the seismic magnitude as in previous works, but it is also important to discuss quantitatively the equivalent amplitudes of the electromagnetic source. Therefore, we have estimated the amplitude of the current moment based on the sensitivity of an electromagnetic sensor for wave observation on the ground and in the ionosphere. It is necessary to have a highly sensitive electromagnetic sensor to detect slight variations of ELF emissions radiated from an underground current source. The highly sensitive electric field antennas become the nonrealistic size of the order of wavelength, while the magnetic field antennas such as loop antennas and search coils are generally easier to use and more sensitive in the ELF wave observations. Furthermore, the magnetic fields of whistler mode waves in the ionosphere increase by the factor of the ionospheric refractive index (order of several tens), while the corresponding electric fields decrease. Thus, we assume the case of the ELF magnetic field observation.
 The frequency dependence of magnetic field intensities ∣B∣ on the ground is shown in Figure 9, where each panel is the profile along the north–south direction above the dipole source in the horizontal range from −100 km to +100 km, and the calculation conditions are the same as those in section 3.1. Even for the most intense case at 10 Hz, the magnetic field intensity attenuates significantly with the horizontal distance. The horizontal attenuation ratio is plotted in Figure 10, which is calculated from the gradient of the magnetic field shown in Figure 9. The attenuation ratio near the epicenter (<20 km) increases with frequency. The attenuation ratio on the ground is 18 dB/10 km, even at 10 Hz. Thus, the location of an observatory must be quite close to the point above the seismic source to detect the electromagnetic field associated with the earthquake. Hayakawa et al.  presented that seismic-related ULF emissions on the ground could be observed about 60 km from the epicenter for a seismic magnitude over 6 and the detectable distance of ULF emissions would be extended to about 100 km for an seismic magnitude over 7 from the observations of ULF anomalies. Our calculation results of the horizontal attenuation of ELF magnetic fields on the ground support their experimental results. In ground-based magnetic field measurements of ULF (0.01–10 Hz) and ELF/VLF (10 Hz to 32 kHz) ranges of the Ms 7.1 Loma Prieta earthquake of 17 October 1989, Fraser-Smith et al.  reported that the ELF/VLF data revealed no precursor activity, but the ULF data included some anomalous features, where ULF and ELF/VLF systems were located about 7 km and 52 km from the epicenter, respectively. These results can also be explained by our calculated results of the differences in the frequency and the distance from the epicenter in Figures 9 and 10. When detecting seismic electromagnetic waves on the ground, furthermore, we should distinguish them from other natural electromagnetic noises such as spherics. A subterranean receiving system would have a much better probability of detecting seismic electromagnetic signals with higher signal-to-noise ratios [see Kingsley, 1989; Tsutsui, 2002].
 In order to evaluate the possibility of detecting on the ground and in the ionosphere, magnetic fields associated with seismic activity, we compared the calculated maximum magnetic fields with the magnetic sensitivity of the search coil magnetometer onboard DEMETER satellite [Séran and Fergeau, 2005]. It is assumed that the satellite is at an altitude of between 300 km to 400 km. Although our calculation of the maximum magnetic field strength was done for the altitude of 150 km, the whistler mode waves in the ionosphere should not be attenuated above 75 km, as shown in Figure 5.
Figure 11 shows the magnetic sensitivity of the search coil onboard DEMETER satellite with a thick solid line. This figure also shows the maximum magnetic field in units of pT/Hz1/2 calculated at the altitudes of 0 km and 150 km for the situations described in section 3.1 (the underground conductivity 10−3 S/m, see Table 1 for other parameters), and for the underground conductivity of 10−4 S/m as in section 3.3 which allows the more intense wave radiation into the free space and the ionosphere. The maximum magnetic fields on the ground at all the frequencies are taken just above the dipole source (see Figure 9), while those at the altitude of 150 km in the ionosphere are taken about 30–40 km southward of the point just above the dipole source as predicted from Figure 3. As an example for the lithospheric conductivity σ = 10−4 S/m, the maximum wave intensities of the magnetic fields and electric fields at 100 Hz in the ionosphere (z = 150 km) are 5.2 × 10−4 pT/Hz1/2 and 2.7 × 10−4μ V/m/Hz1/2, respectively. The sensitivities of the search coil and the electric field antenna onboard DEMETER satellite are 0.1 pT/Hz1/2 and 0.1 μV/m/Hz1/2 at 100 Hz, respectively [Berthelier et al., 2006]. Both the magnetic and electric field observations of ELF whistler mode waves are quite difficult in the ionosphere.
 For the case of σ = 10−3 S/m as in section 3.1, the intensities of the magnetic field on the ground below 100 Hz are almost the same as the magnetic sensitivity, so we should be able to observe it if the source current moment is larger than 1 kA·m/Hz1/2. Since the intensity of the magnetic field is proportional to the current moment, if the horizontal dipole source is larger than 1 kA·m/Hz1/2 placed at 10 km underground in the conductive medium of 10−3 S/m, we can detect it at 10 Hz to 100 Hz by the search coil magnetometer on the ground. However, even for the 1 kA·m/Hz1/2 dipole, we cannot observe it in the ionosphere at the altitude of 150 km. To detect it at 10 Hz, the current moment over 2.5 MA·m/Hz1/2 is required at the source, which would not be realistic. In the possible situation described in section 3.3, when the conductivity of the lithosphere is 10−4 S/m and the source depth is 10 km, the magnetic intensity in the frequency range from 10 Hz to 1 kHz could be detected on the ground with the current moment of 1 kA·m/Hz1/2, and the detectable minimum current moment is estimated at about 80 A·m/Hz1/2. Meanwhile, the detectable current moment becomes over 200 kA·m/Hz1/2 if we are going to detect it in the ionosphere. Therefore, we found that direct detection of seismic electromagnetic waves on the ground requires at least an amplitude of the dipole moment of about 80 A·m/Hz1/2 for a simple horizontal electric dipole source, and detection in the ionosphere requires over several hundreds of kA·/Hz1/2. It is not easy to discuss whether these amplitudes of dipole moments are realistic or unrealistic in the case of seismic activity, but it is important that we need the seismic electromagnetic energies which can create the equivalent amplitudes of dipole moment, to directly detect the seismic electromagnetic waves. Molchanov et al.  suggested that direct detection of seismic-related electromagnetic waves in the ionosphere radiated from an underground current source is possible only in the ULF range (<10–20 Hz) for strong earthquakes (Ms > 5–6). The current moment of Ms = 5 corresponds to 12.6 kA·m/Hz1/2 in their source model (see equation (22) of Molchanov et al. ).
 On the other hand, regarding possibilities other than the direct observation of seismic-related ELF/VLF waves in the ionosphere, Molchanov et al.  suggested that the ELF/VLF waves in the ionosphere associated with seismic activity are caused by the nonlinear conversion from Alfven waves in the ULF range, which are directly radiated from a seismic-related underground current source and are more effectively transmitted into the ionosphere than those in the ELF/VLF range. Borisov et al.  suggested another mechanism, which is that ELF emissions in the ionosphere above a seismic zone originate from the general spherics enhanced by the ionospheric irregularities associated with seismic activity (see Introduction). The amplitude of the dipole moment giving the detectable ELF waves in the ionosphere becomes at least several hundred kA·m/Hz1/2, so our results support the importance of these mechanisms instead of direct radiations from an underground current source.
 In order to theoretically evaluate the ELF emissions in the ionosphere above seismic zones, we presented theoretical calculations of the ELF waves radiated from an underground current source. Former advanced work by Molchanov et al.  focused on Alfven and magnetosonic waves in the ionosphere in the ULF (0.01–100 Hz) range, while our study focused on whistler mode waves in the ELF (10 Hz to 1 kHz) range. We have quantitatively discussed the amplitude of the dipole moment giving detectable ELF emissions directly radiated from the current source in the lithosphere by comparing our calculation results with the sensitivity of an actual electromagnetic sensor. Our discussions support the suggestions that observations of the seismic-related ULF emissions are more effective than those of ELF emissions in accordance with the discussions by Molchanov et al. , and other mechanisms instead of direct wave radiations from a current source in the lithosphere are important [Molchanov et al., 1995; Borisov et al., 2001].
 The main results of this study are summarized below:
 First, the spatial distributions of ELF waves radiated from an underground current source can be obtained as shown in Figures 3 and 4. We see that the radiated ELF waves become whistler mode beam waves in the ionosphere.
 Second, wave attenuation in the lithosphere is found to be more dominant than the effect of the ionosphere from Figures 5, 6, 7, and 8. We cannot directly compare our results with former studies, as the calculation assumptions are different, but the tendencies of wave dependency on the altitude, source depth, and direction of the current source are consistent with previous works by Molchanov et al.  and others.
 Last but not least, we have found the amplitude of the electromagnetic source needed to directly observe the radiated electromagnetic waves on the ground and in the ionosphere with a realistic magnetometer. We have quantitatively shown that the amplitudes of the dipole moments from several hundreds of A·m/Hz1/2 to 1 kA·m/Hz1/2 are necessary to directly observe the radiated ELF wave magnetic fields on the ground, while dipole moment amplitudes over 200 kA·m/Hz1/2 are necessary to observe them in the ionosphere. It should be noted that wave emissions from an earthquake epicenter are only one hypothesis to explain the ionospheric perturbations above seismic zones. Further investigation is necessary to identify the actual source mechanism. In any case, our calculations could be applied to the electromagnetic waves radiated from any seismic-related ELF/VLF sources located in the lithosphere.
 The authors thank Roger R. Anderson at University of Iowa, USA, for his many useful discussions and suggestions to improve this paper. The authors also thank Keely Brandon at Kanazawa University, Japan, for her careful reading of this manuscript for revision.