Corresponding author: M. Fujii, Graduate Research Division of Science and Engineering, University of Toyama, 3190 Gofuku Toyama, 930-8555, Japan. (firstname.lastname@example.org)
 It is shown theoretically that if mobile electrical charge exists on the surface of the ground, a ground surface plasma wave is induced by radio waves. If the electrical charges are generated by tectonic stresses acting on crustal rocks prior to major earthquakes, the detection of a ground surface plasma wave could be used as a pre-earthquake electromagnetic phenomenon. The ground surface plasma wave has a dispersion relation, i.e., the relation between frequency and wavelength, similar to that of the free-space plane wave in the atmosphere over the radio broadcast frequency range. It allows for a strong coupling between these two types of waves. This is a mode of electromagnetic wave propagation that has not been previously reported. Numerical analysis demonstrates (1) the propagation of the ground surface plasma wave along a curved surface beyond the line of sight, (2) anomalous scattering by ground surface roughness, and (3) the generation of cross-polarized waves due to the scattering. These results all agree well with radio wave anomalies observed before large earthquakes.
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 Pre-earthquake signals have long been investigated [Uyeda et al., 2009; Freund, 2011, and references therein]. However, some of them may have been evaluated as nonsystematic anecdotal phenomena and have suffered from the difficulty in reproducibility due to the nature of earthquakes as natural phenomena. Recently, the mechanism of the generation of electrical charges in rocks by the stress of earthquakes has been proposed [Freund, 2000, 2002, 2011], with which various electrical phenomena associated with earthquakes are inferred reasonably.
 Among many pre-earthquake phenomena observed so far, anomalously enhanced signals of very-high-frequency (VHF) radio waves around 70 MHz to 90 MHz have been investigated systematically [Moriya et al., 2005, 2009, 2010]; observation stations have been established in a district of northern Japan where earthquakes occur frequently. Detection of such signals and their statistical analysis have been performed to the level that enables the deduction of empirical rules to relate the duration of the radio wave anomalies and the strength of the impending earthquakes. Such radio wave anomalies are reported to last for typically weeks or months prior to large earthquakes, for which reliable detection of the signals has been possible. However, the mechanism of how the radio wave anomalies occur is remained unclear, which has resulted even in doubts in its scientific and social significance.
 The observed phenomena of radio wave anomalies are as follows [Moriya et al., 2005, 2009, 2010; Hayakawa et al., 2007; Fujiwara et al., 2004; Sakai et al., 2001]. (1) Anomalously enhanced VHF radio signals are detected at a radio observatory facility, between which observatory and a radio broadcast station large earthquakes indeed occur. (2) The radio signals are detected beyond the line of sight from radio broadcast stations only before large earthquakes. (3) The anomalous radio signals are received from directions of a small elevation angle below 20 degrees, rather than angles orienting higher ionosphere; this suggest that the anomalies occur near the ground surface, implying an important contribution of the surface charge. All these phenomena can not be explained reasonably by the existing theory of atmospheric, ionospheric, or meteorological phenomena of electromagnetic wave propagation.
 In this paper, I show the anomalies are caused by ground surface plasma wave, which is an oscillation of free charges generated when rocks are stressed before earthquakes. The free charges arise in rocks by mechanical impact or stress, which propagate in rocks and accumulate on the surface of the ground, called “the positive holes” [Freund, 2000, 2002, 2011]. The charges on the ground act like electrons on a metal surface; when illuminated by light, the free electrons couple with light under the condition of phase matching, of which mode of propagation is called “a surface plasmon polariton” [Raether, 1977, 1988]. The surface charges, on ground or on metal, couple with electromagnetic waves below their plasma frequencies and enhance the fields near the surface. The electromagnetic theory of surface waves [Collin, 1991; Russer, 2003] and surface plasmon polaritons [Raether, 1977, 1988] describes very well the observed phenomena of the preseismic radio wave anomalies.
 The phenomena of coupling between the atmospheric radio wave and the surface plasma wave are studied also numerically by the finite-difference time-domain (FDTD) method [Yee, 1966; Taflove and Hagness, 2005]. The numerical analysis demonstrates the coupling of these waves intuitively. The analysis has been carried out with and without a plasma region on the surface of the ground, with and without roughness of the surface; it has clarified the mechanism of the strong scattering of the radio wave from shallow elevation angles by the charges and the roughness on the ground surface.
 On the basis of the presented theory, more accurate methods for the detection of the signals before impending earthquakes will be possible, e.g., where to place the sources of radio waves and where and how to detect them. Detection of the signals in a manner of pinpoint in time and location will further enhance the importance of the observation of such radio wave anomalies.
 Among many electrical phenomena associated with earthquakes, of particular interest is the generation and propagation of electrical charges in rocks described in Freund [2000, 2002, 2011]. It is shown that crustal rocks contain dormant electrical charge carriers in the form of peroxy defects. When rocks are stressed, the electronic charge carriers are released from the defects near possible dislocations in the crystal of the rocks and the succeeding microfracturing before its cracks and failure. The electrical charge is highly mobile, and is known as the positive hole charge carriers from the analogy to semiconductor physics. The stress-activated positive holes then propagate across the grains of rock crystals, soils, and sands, subsequently accumulate on the surface of the ground. The charges stay on the surface till they combine with electrons and disappear. Although not confirmed rigorously, it is anticipated from the experiments that the density of the positive charges can reach 1025 m−3 to 1026 m−3 [Freund, 2002].
 The anomalies are detected as random-noise-like impulse signals; each of the impulse duration is an order of milliseconds [Freund, 2000, 2002, 2011], and the noise-like signals last for typically minutes to hours; the anomalies occur intermittently for weeks to months [Moriya et al., 2010]. Since the surface plasma wave propagates a few hundred kilometers for one millisecond, it is considered reasonable to observe the signals at such distant locations. This section describes the theory of electromagnetic wave propagation under the existence of free charges on the ground surface, and its coupling with the atmospheric propagation of a plane wave.
2.1 Plasma Frequency of Ground Surface Charges
 Free charges can oscillate collectively in the vicinity of the surface of a conductor at a frequency lower than the plasma frequency [Kittel, 1986]
where n denotes the density of charged particles, q is the charge of the particle, ϵ0 = 8.85 × 10−12 F/m is the permittivity of vacuum, and m the mass of the particle.
 The effective mass of the free charges, or the positive holes, has not been measured and is unknown. However, considering the analogy to doped p-type Si, and assuming that the effective mass of the positive holes is larger than the mass of an electron me by several factors, it should be justifiable to take possibly a large value such as 10 × me = 1.67 × 10−30 kg. The charge of the particle is chosen to be the same but positive value as an electron, i.e., q = 1.60 × 10−19 C. The particle density is assumed from the discussion in Freund  to be n = 1025 m−3. Then we get ωp = 4.2 × 1014 rad/s (67 THz), which is in the optical frequency range and much higher than the VHF radio waves of several ten megahertz.
 Even if the charged particles have a larger effective mass by two orders of magnitude, the plasma frequency is in the order of 1013 rad/s (order of 1 THz), still much higher than the VHF radio frequency. As long as the plasma frequency ωp is higher than the outer field frequency, the charged particles behave as plasma, namely, the surface plasma wave is induced by the radio wave.
2.2 Surface Wave and Surface Plasma Wave
 A surface wave is a wave that propagates along the surface of metal or dielectric by means of the difference of the dielectric permittivity of two materials [Russer, 2003] ϵ1 = ϵ0ϵr1 and ϵ2 = ϵ0ϵr2. We suppose region 1 is air, i.e.,
 For region 2 as ground, the relative permittivity is given by
where ϵ′r2 and ϵ″r2 are the real and the imaginary parts of ϵr2, respectively, j is the imaginary unit, ϵ∞ is the relative permittivity of the ground at an ideally high-frequency limit, σ is the conductivity of the ground, and ω is the angular frequency of the radio wave.
 The third term in the right-hand side of (3) is the Drude dispersion function, representing the susceptibility due to the polarization by free electrical charges. This is derived by considering the equations of motion of electrical charges as done for plasma [Kittel, 1986]. The free charges respond to the outer field and oscillate below the plasma frequency (1), with its damping factor Γ. The term of the Drude dispersion is considered only for the location where the surface free charges appear.
 To clearly illustrate the relation between the permittivity and the plasma frequency, as well as the dispersion relation of the surface plasma wave, the plasma frequency of the positive holes is assumed to be only several times larger than the radio broadcast frequency, i.e., ωp = 2π × 109 rad/s. This still allows the charged particles to respond to the radio wave field and thus to act as plasma. The damping factor for the oscillation of the charges assumed to be smaller than ωp, i.e., Γ = 2π × 106 rad/s, which may be justified by considering that the charges are not highly damped, but rather highly mobile [Freund, 2002, 2011]. Rigorous justification of these values is by far difficult due to the lack of exact information about the positive holes. However, these assumptions lead to reasonable results as shown later.
 Then the complex relative permittivity (3) is plotted in Figure 1 for the ground with ϵ∞ = 6, and σ = 10−3 S/m, which have been chosen for wet soil. It is known [Raether, 1988] that the surface plasma wave propagates if ϵ′r2 < 0 and ϵ′r2 > ϵr1. This is satisfied for wide range of frequency and other parameters, e.g., n > 1020 m−3, ϵ∞ ≈ 10, σ < 10−2 S/m.
 The relation between the frequency of operation and the complex propagation constant is given by [Russer, 2003; Raether, 1988]
where α and β are attenuation constant and phase constant, respectively, and c0 the velocity of light in vacuum. The phase constant, or the wave number, β = 2π/λ, with λ the wavelength, is plotted in the dispersion diagram of Figure 2. It is found in Figure 2 that the curve for the surface plasma wave overlaps with the light line of air at low-frequency range, i.e., the surface plasma wave has the frequency and the wavelength close to those of the radio wave in the free space of air. In contrast, if the free charges do not exist on the ground, the mode of propagation becomes a surface wave, not the surface plasma wave. The surface wave has a larger frequency than the wave in air (light line), which means that the surface wave does not couple with the radio wave in the atmosphere. Moreover, the surface wave attenuates strongly and does not propagate for long distance.
3 Numerical Analysis
3.1 Plasma Surface Wave on Flat Surface
 With these theoretical backgrounds, the propagation of radio wave in the atmosphere over the ground is analyzed with the FDTD method for solving Maxwell's equations. In brief, the method solves Ampère's law and Faraday's law alternately to find the time evolution of electromagnetic fields with various material properties and configurations. The constitutive equation for the plasma medium with the relative permittivity (3) is computed together with Maxwell's equations, by means of solving the equation of motion for charged particles with the so-called auxiliary differential equation (ADE) technique [Taflove and Hagness, 2005].
 I have calculated the radio wave propagation first in two-dimensional (2-D) space of longitudinal distance Z in the direction of propagation and the vertical height H as shown in Figure 3. The ground is considered up to the depth of D = 10 m below the surface, and the height of the atmosphere is H – D. The outer boundary is imposed to be free from reflection realized by the perfectly matched layers (PML) [Gedney, 1996]. The plasma region is modelled for the depth of 1 m under ground and longitudinal distance from z = 0.3Z to 0.8Z. The discretization size of the finite-difference analysis was Δ = 0.1 m. The ground and the plasma parameters used were identical to those of Figures 1 and 2. The frequency of the radio wave is 100 MHz, namely ω = 2π × 108 rad/s. The source signal is excited at z = 0 right above the ground with the electric field perpendicular to the ground.
 In order to practically model the beam-like wave from a radio broadcast station, the source excitation region has a certain height about 10 m to 40 m above the ground, depending on the height of the analysis region, and is given a half-Gaussian distribution of the source emitting strength. This allows the control of the broadcast wave emitting direction, preventing the wave from hitting the upper computation boundary and causing unnecessarily small reflections.
 The mode of propagation analyzed in this configuration is transverse magnetic (TM), where the electric field is polarized vertically, namely in parallel with the plane of analysis region (Figure 3), and the magnetic field perpendicular to it. For the propagation of the transverse electric (TE) wave where the electric field is polarized horizontally, the wave behaves differently from the TM wave. However, the dominant mode of the surface wave and the surface plasma wave is TM, effectively analyzed in the present configuration. If the broadcast source is of the TE wave configuration, cross polarized TM wave is yielded by reflection and diffraction of the waves, thus resulting in the same effect as discussed for the TM wave. Cross polarized effects of the TM wave will be considered further in the later section of 3-D analyses.
 First, the wave field is shown for a small configuration of Z = 100 m and H = 40 m. The ground surface locates at height h = 10 m. The plasma region locates from z = 30 m to 80 m. In Figure 4 the wave is radiated from the source at z = 0 into the atmosphere and partly into the ground as the surface wave. The surface wave then attenuates rapidly. When the wave reaches the plasma region, a strong vertical electric field Ex is induced on the surface of the ground (Figure 4c), which is the surface plasma wave. The surface plasma wave propagates along the surface of the ground, when it hits the end of the plasma region, it is scattered again back into the atmosphere and partly into the ground. The vertical electric field Ex is the dominant field in the surface plasma wave. The longitudinal field component Ez in Figures 4b and 4d shows clearer contrast in the propagation and scattering phenomena of the surface wave and the surface plasma wave.
 The propagation over longer distance is then analyzed for Z = 500 m and H = 100 m. The wave goes higher into the atmosphere, propagates nearly in the plane-wave-like mode. It is shown in Figure 5c and 5d that the ground surface plasma wave is induced on the ground surface plasma region of the distance from z = 150 m to 400 m; interestingly, the surface plasma wave seems to become more intense as it propagates. At the end point of the plasma region z = 400 m, the ground surface plasma wave is scattered and turned to a wave radiated into the atmosphere, which should have been received from shallow elevation angles by the researchers.
 On flat surfaces, the surface plasma wave is formed at any distance from the emitting source as long as the atmospheric radio wave is of certain strength. It is not straightforward to predict the limit from what point to what point the surface plasma wave is formed; numerical results up to 500 m distance imply the formation of the surface plasma wave at any locations. Longer distance analyses will be the subject of future research.
3.2 Long Distance Propagation Beyond the Line of Sight on Curved Surface
 The most critical situation of the beyond-the-line-of-sight propagation is analyzed for Z = 1000 m with a convex round ground surface as shown in Figure 6. The curvature radius of the ground surface is set to 6360 m, rather small compared to the real surface of earth due to the computational capacity limitation. In this configuration, the ground surface is at height 10 m for z = 0 and z = 1000 m and at height 30 m for z = 500 m. The plasma region exists from z = 300 m to 800 m along the surface of the ground with 1 m in depth under the ground. From the results of Figures 6a and 6c, it is found that the major part of the wave goes higher up into the atmosphere. However interestingly, for the case the plasma region exists as in Figures 6c and 6d, the surface plasma wave is induced and propagates along the curved surface of the ground beyond the line of sight, till the end of the plasma region at z = 800 m. Without the plasma region as in Figures 6a and 6b, the wave becomes weak near the surface. This should be the origin of the radio wave anomaly detected at distant locations beyond the line of sight when surface charges appear on the ground before earthquakes. Similar effects will be expected when the wave goes beyond hills or mountains, although the detail should be investigated further.
 All these analysis examples are of relatively short distance compared to the actual VHF radio broadcast. It is nevertheless anticipated for longer distance propagation that the plasma surface wave propagates in the similar manner as the previous cases while attenuating more significantly. For the case of 1000 m propagation over the curved ground surface, the signal attenuates approximately by an order of magnitude in voltage. For the actual broadcast in three space dimensions, the attenuation will be larger due to the radiation to all the azimuth angle of 2π. Thus the attenuation will be larger by a factor of 2π, consequently, at most two orders of magnitude in voltage for one kilometer propagation. For longer distance propagation, the radio signal is attenuated by a factor inversely proportional to the distance squared. Considering that the radio tuners typically receive signals as weak as microwatt power level, and that the typical broadcast power of 1 kW, attenuation to 1 μW (power ratio of 109) is reached by the propagation of distance r km, calculated by (20π r2)2 ≈ 109, i.e., r ≈ 10 km to 20 km. It is therefore anticipated that the surface plasma wave can carry radio signals for the distance of more than 10 km beyond the line of sight. Although it depends of course on the density and the area of the free charges, the present theory can after all explains reasonably the various reported anomalous phenomena regarding the radio signals.
3.3 Effect of Surface Roughness and Obstacles
 Another important factor is the roughness of the ground surface. The wavelength of the 100 MHz radio wave is 3 m in air. On the other hand, the vertical height to which the surface plasma wave extends is at least several meters. Disturbances such as trees, rocks, and waters on the ground of the same size as the wavelength may cause certain effects to the wave propagation; disturbances smaller than the wavelength will not have large effects; disturbances much larger than that may affect the distribution of the free charges, hence affect the propagation of the surface plasma wave. Preliminary analyses regarding the effect of different roughness agreed to this general inference (not shown). However, remarkable differences have been found between the cases where the roughness is electrically charged by the positive holes and those cases not charged.
 Similar 2-D analyses as the previous sections have been performed for Z = 500 m propagation by including roughness of some round obstacles near the half distance to Z. The configuration of the obstacles is shown in Figure 7a by the red color from z = 150 m to 400 m; there are three semicircles of radius 2 m placed on the flat ground surface. The ground with the roughness was analyzed with and without the free charges. Without the free charges, as shown in the results of Figures 7b and 7c, the surface plasma wave does not exist on the surface, hence the atmospheric radio wave propagation is not perturbed by the roughness. Contrary, for the case with the free charges on the ground and with the roughness also charged electrically, as shown in the results of Figures 7d and 7e, the surface plasma wave is strongly scattered forward and backward by the roughness.
 Then, the number of obstacles are increased as in Figure 8, where many obstacles of the same size are aligned randomly near z = 250 m to 350 m attached on the ground plasma region of z = 150 m to 400 m. The effect of strong scattering is similar as in the previous case of the roughness in Figure 7. These results suggest for the actual radio wave propagation that strong scattering occurs from the ground roughness under the existence of the surface charges, which should be detected as an anomalous signal coming from a shallow elevation angle.
3.4 Analysis of Guiding Effect and Cross-Polarized Fields in Three-Dimensional Space
 The 2-D analyses in the previous sections deal with fields only of the TM mode, i.e., the electric fields in parallel with the plane of the 2-D analysis region. In order to analyze the cross-polarized field, which is perpendicular both to the source electric field and to the direction of propagation, the analysis is performed in the three dimensions (3-D) with all the electric and magnetic field components taken into consideration. The configuration for the present 3-D analysis is shown in Figure 9. Owing to the limitation of the computer memory, the size of the analysis region is chosen to be distance Z = 100 m, width Y = 30 m, and height H = 25 m, with and without a rectangular plasma region of distance 50 m, width 6 m, and thickness 1 m under the ground.
 The purpose of this analysis is to see how large the generated cross polarized field is after the wave is scattered by the plasma region. However, the earth magnetism is not included in the analysis, thus the magnetic rotation effects such as Faraday rotation or Kerr rotation are not taken into consideration, although they may have certain effects in reality. Longer distance phenomena may be inferred by considering together the present 3-D and the previous 2-D results. The other parameters such as the permittivity, conductivity, plasma frequency and the radio wave frequency are chosen to be identical to the previous 2-D analyses. The broadcast source is given by the Ex field placed near the origin, and the analysis region is surrounded by the PML [Gedney, 1996].
 In the 3-D analysis, the plasma region is a long rectangular region placed in such a way that it is looked obliquely down from the broadcast source point. This configuration enables to analyze if the surface plasma wave is guided obliquely along the elongated plasma region. It is shown by comparing Figure 10a with Figure 11a that the wave field is indeed guided on the ground surface along the plasma region (green rectangular region), despite that the atmospheric radio wave propagates obliquely against the direction from the source point. The side views of the respective fields in Figure 11b and Figure 11b show that the surface plasma wave is induced and propagates in the z-direction, which are similar to, but not so clear as, the 2-D case of Figures 4a and 4c.
 The cross-polarized field components are observed after the radio waves hit the plasma region by comparing the fields in Figures 10c and 10d with those in Figures 11c and 11d, respectively. The cross-polarized field is generated near the surface of the ground, due to the non-plane (non-TEM) wave nature of the guided modes. For the surface plasma wave as well as for the surface wave, longitudinal field components exist. When such field components are scattered by discontinuity of the media, cross-polarized field is yielded even without the Faraday rotation effect.
 When the surface charges exist, the cross-polarized field is much stronger than that without the surface charges. These results are reasonable because the stronger scattering causes larger cross-polarized fields. This fact has been also confirmed in experiments by turning the polarization of the receiving antenna either vertically or horizontally [Moriya et al., 2009]; the cross-polarized scattered wave may be easily discriminated from the main radio wave than the scattered wave of the same polarization, hence it can be an accurate detector of the ground surface charges.
 The dispersion relation of the ground surface plasma wave has been derived and shown to be very close to that of the radio wave in the atmosphere over the broadcast frequency range. Owing to the similarity of their dispersion relation, the ground surface plasma wave is strongly induced by the VHF radio wave when the electrical free charges appear on the ground surface. Consequently, the ground surface plasma wave is scattered by the roughness of the ground surface associated with free electrical charges more strongly than for the case where the roughness is not associated with free electrical charges.
 On the basis of the theoretical and numerical results as summarized in the following, it is concluded that the VHF radio wave is indeed a sensitive detector of the precursory free charges that may appear on the ground surface from stressed crustal rocks prior to earthquakes.
It is in the most suitable frequency range, i.e., the VHF radio wave is not reflected by the ionosphere and the normal propagation distance is limited by the line-of-sight distance. In addition, the frequency of VHF radio waves is considered much lower than the estimated plasma frequency of the ground surface charges, for which the ground surface plasma wave is sensitively induced by the radio wave.
If the polarization of the broadcast radio wave, namely the direction of the electric field, is vertical to the ground surface, it corresponds to the dominant field component of the ground surface plasma wave. When the ground surface plasma wave is scattered by a surface roughness, a cross-polarized (i.e., horizontally polarized) wave is yielded, which may be simply discriminated from the main radio wave. In contrast, if the broadcast radio wave is horizontally polarized, polarization rotation may well occur by similar scattering effects and vertically polarized wave will be yielded. Magnetic rotation effects must have a certain contribution to the cross-polarization effects in reality, which will be the subject of future research.
Radio tuners can receive signals as weak as a microwatt level, thereby detect subtle free charges on the ground surface. Radio waves can cover remotely wider areas of 10 to 100 km square than other methods of measuring the charges locally on the ground. Possibly, the observation methods of various principles better complement each other. Since the ground surface plasma wave scattered by ground roughness directs in any angles of elevation, it can possibly be detected from a high elevation angle by an airplane or satellite, which may enable even wider remote sensing.
 Although the effects of various topographies on the distribution of the free charges and the propagation of the ground surface plasma waves are still necessary to be investigated, the mechanism of the ground surface plasma wave describes very well the reported phenomena regarding the radio wave anomalies prior to large earthquakes.
plasma frequency, rad/m
density of charged particles, 1/m3
electrical charge of particle, C
dielectric permittivity of vacuum, 8.85 × 10−12 F/m
mass of a charged particle, kg
mass of an electron, 1.67 × 10−29 kg
dielectric permittivity of medium 1, F/m
relative dielectric permittivity of medium 1
dielectric permittivity of medium 2, F/m
relative dielectric permittivity of medium 2
real part of relative dielectric permittivity of medium 2
imaginary part of relative dielectric permittivity of medium 2
relative dielectric permittivity of ground at an ideally infinite frequency limit
conductivity of ground, S/m
angular frequency of the radio wave, rad/m
damping factor of Drude dispersion model of plasma region, rad/m
complex propagation constant = α + jβ, rad/m
attenuation constant, Np/m
phase constant or wave number = 2π/λ, rad/m
velocity of light in vacuum, m/s
longitudinal distance of the analysis region, m
vertical height of the analysis region, m
depth of the ground below surface in the analysis region, m
H − D
height of the atmosphere, m
coordinate in the longitudinal direction, m
discretization size for the finite-difference analysis, m
coordinate in the vertical (x) direction or height, m
electric field in the x-direction, V/m
electric field in the y-direction (only for the 3D analysis), V/m
electric field in the z-direction, V/m
width of the analysis region for the 3D analysis, m
estimated propagation distance of the surface plasma wave beyond the line-of-sight, m