N. Gershenzon and G. Bambakidis
Department of Physics Wright State University, Dayton, Ohio U.S.A.
Electromagnetic phenomena preceding and accompanying seismic events continue to attract attention not only as possible earthquake precursors but also as additional parameters for describing the earth’s crust and its dynamics. Before considering seismo-electromagnetic (SEM) phenomena, however, we address the question of whether such phenomena do in fact exist.
In spite of the publication of several hundred articles from the beginning of the twentieth century dealing with the relationship between EM signals and pre-seismic processes in the earth’s crust, there is still no definitive proof that the two are connected. While there are several reasons for this, we think that the main reason is the practical impossibility of repeating the results of field observations. It is necessary to wait a long time, sometimes many years, for a large earthquake to occur with similar parameters in approximately the same location.
A second reason is that areas of high seismic activity usually differ widely in geological and seismic characteristics. In addition such areas generally have a high degree of geological inhomogeneity. Both of these factors make the planning of field experiments and the interpretation and comparison of results difficult.
Thirdly, to capture an SEM signal, one must distinguish it from the background, which consists of both natural and man-made EM noise spanning a broad frequency range.
Finally, various research groups have used different measurements and discrimination techniques and frequency ranges, which limits comparison of their results.
Taken together, the above factors have contributed to a lack of consensus among the many research groups which have investigated the existence of SEM phenomena over the past century. Nevertheless we list here some evidence for a positive answer to the question posed in the opening paragraph.
1. The phenomenon of mechano-electromagnetic transduction has been well studied in the laboratory in a wide variety of solids including rocks [Cress et al., 1987; Kapitsa, 1955; Khatiashvili, 1984; Nitsan, 1977; Ogawa et al., 1985; Parkhomenko, 1971; Schloessin, 1985; Volarovich and Parkhomenko, 1955; Volarovich et al., 1962; Yamada et al., 1989].
2. Creation of an electric field by the passage of a seismic wave through soil has been observed [Eleman, 1965; Ivanov, 1939; Leland and Rivers, 1975; Martner and Sparks, 1959]. These seismo-electric effects have been applied to geophysical prospecting [Kepis et al., 1995; Sobolev and Demin, 1980; Sobolev et al., 1984; Thompson and Gist, 1993].
3. The emission of EM signals has been observed in the field experiments over distances as small as tens to hundreds of meters [Mastov et al., 1983, 1984; Solomatin et al., 1983a, 1983b]:
4. The unusual appearance of local atmospheric light seconds or minutes before, and close to the epicenter of, some earthquakes has been witnessed by many individuals [Derr, 1973; Ulomov, 1971; Yusui, 1968]. Obviously, the appearance of atmospheric light implies the existence of strong electric fields.
5. Some pre-seismic data show unusual EM signals. For example, Fraser-Smith et al. [1990] monitored EM signals 7 km from the epicenter of the strong Loma Prieta earthquake of 1989. Starting forty days before the quake and continuing for several weeks after the quake, anomalous signals were detected. In particular, three hours before the quake, disturbances began which exceeded the background noise by two orders of magnitude.
The above reasons provide a basis for serious consideration of the existence of SEM phenomena. Such phenomena have been discussed in several monographs and reviews [Gokhberg et al., 1995; Hayakawa and Fujinawa, 1994; Johnston, 1989, 1997; Lighthill, 1996; Park, 1996; Park et al., 1993; Rikitake, 1976a, 1976b]. We wish to describe briefly a scenario for them based on 1) analysis of experimental field and laboratory data and 2) modeling of electromagnetic emission which incorporates a mechanical model for pre-seismic deformation developed by us over the past fifteen years [Dobrovolsky et al., 1989; Gershenzon, 1992; Gershenzon and Gokhberg, 1989, 1992, 1993, 1994; Gershenzon et al., 1986, 1987, 1989a, 1989b, 1990, 1993, 1994; Grigoryev et al., 1989; Wolfe et al., 1996]. We have used specific mechanical models of pre-earthquake processes [Dobrovolsky, 1991; Karakin, 1986; Karakin and Lobkovsky, 1985] in some of these articles. But we will not use any of them in this paper because no one model is good enough to describe such complicated processes and developing such a model is not a goal of this paper. Furthermore a specific mechanical model of pre-seismic processes is not needed in constructing a model for SEMS.
The final stage of an earthquake cycle is characterized by stress release processes including foreshocks, the main shock (or swarm), aftershocks, and creep (before and after the earthquake). All of these processes are accompanied by formation of a number of cracks since the Earth's crust consists of extremely brittle materials. So we suppose that the initial source of most SEM anomalies is a localized high density of cracks. Such questions as how these localized regions of high crack density are related to the earthquakes, and how far from the earthquake origin and how long before the earthquake they appear, are not considered here. Part of the mechanical energy released in the formation of these cracks is transformed into electromagnetic energy by a variety of mechanisms. Typical "transducer'' mechanisms in crustal rocks include piezo-magnetic, classical and non-classical piezo-electric, electro-kinetic and induction. Under appropriate but realistic conditions, phenomena such as quasi-static geomagnetic and electrotelluric anomalies, ultra-low-frequency (ULF) magnetic variations, and radio-frequency (RF) emissions, all associated with pre-seismic processes, can be explained and estimated on the basis of known mechanical and electromagnetic parameters of the earth's crust.
A comparison of our estimated values of seismo-electromagnetic signals (SEMS) with reported field measurements leads to the conclusion that the sources of most anomalous SEMS are relatively close to the detector. In other words, the source of the signal is local. We expect that all pre-earthquake processes, including SEMS, are connected indirectly through global shear stresses, in agreement with Kanamori's interpretation of both SEMS and earthquakes as manifestation of regional geophysical processes [Kanamori, 1996].
There is a recognition that clarification of the physical mechanisms of SEMS generation, transmission and reception is needed [Uyeda, 1996]. We address these problems in the sections that follow. In section 2 the spectral density of a mechanical disturbance associated with the appearance of a crack is introduced. Section 3 will describe the various coupling mechanisms between a mechanical disturbance and the resulting electromagnetic disturbance. Formulas will be given which relate the parameters of the mechanical disturbance to the parameters which produce the EM disturbance (e.g. magnetization, polarization, current density). This section also includes a comparison of the strengths of these EM sources. Section 4 connects, via Maxwell's equation, the resulting SEMS to the electrical and mechanical parameters of the crust and the characteristics of the mechanical disturbance. The formulas enable us to estimate the spectral density of the SEMS for each type of source considered here. Section 5 describes briefly the morphological features of several observed SEM phenomena, namely tectonomagnetic variation, electrotelluric anomalies, geomagnetic variation in the ultra-low frequency range, and electromagnetic emission in the RF range. The interpretation of these phenomena is discussed based on our model. The paper concludes with a summary of our results and suggestions for future field experiments.
It is natural to suppose that the origin of SEM anomalies lies in mechanical processes occurring in the earth's crust before an earthquake. The large build-up of mechanical energy during deformation is expended mainly through stress relaxation before, during and after the quake, but a small part goes into EM emission. Note that an important circumstance is that the vibrational spectrum associated with the release of mechanical energy, since it arises, on an atomic scale, from the motion of atoms carrying an electrical charge or a magnetic moment, is related to the spectrum of the EM emission. So if we observe an SEM anomaly of a certain frequency, there must be an associated mechanical disturbance of the same frequency. This does not imply that the corresponding spectral densities are identical, because the electro-mechanical coupling is frequency-dependent. But we would not expect a large EM emission peak in a frequency range where the vibrational spectral density is very small.
![]() |
Figure 1 |
In this figure, the crack begins to grow at t=0. As it opens, a strain pulse propagates away from the crack, reaching point xi at time ti ( i = 1, 2, 3). The pulse grows to a maximum value and, after it passes, the strain relaxes to a steady value. The maximum and steady values fall off with distance from the crack, but the duration of the pulse, Dt, is constant and given by lc/Vc where lc is the crack size and Vc is the crack opening speed (speed of propagation of the crack tip). The quantity Vc is a complicated function of the type of crack, the elastic modulus of the material and the details of the crack formation process, but is less than the velocity of Rayleigh wave [Kostrov, 1975] and independent of crack size, having a characteristic value of order 1 km/sec [Kuksenko et al., 1982].
From Figure 1b, we see that the appearance of a crack gives rise to a seismic impulse. The magnitude of this strain impulse is large near the crack and there remains a residual value at long times. We want to estimate the spectral density of mechanical vibrations associated with this impulse. Assume that the change in strain varies with time according to
![]() | (1) |
where H(r) is the unit step function ( H(r) is zero for r<0 and one for r>0 ), r is the distance from the crack, V is the elastic wave velocity, ec is the average strain change in the vicinity of the crack, e imp(r) is the strain change due to the seismic impulse, and e res(r) is the change in residual strain due to formation of the crack.
The quantity e imp(r) decreases with r as r-1 as the pulse propagates away from the crack. But it also is attenuated by the factor exp[-r/L] due to inelastic processes, where L is the attenuation length. We represent e imp(r) as
![]() | (1a) |
The attenuation length is a complicated function of frequency, rock structure, temperature and pressure, and can range from 10 to 10,000 wavelengths [Carmichael, 1989]. In this paper we shall assume that L lies in the range 10lc to 103lc. The strain e res(r) falls off with distance much more quickly than e imp(r). We assume
![]() | (1b) |
which expresses the fact that the residual strain extends to about three time the crack size.
Now we can find the spectral density for e(t, r) using equations 1, 1a, and 1b. The result is
![]() |
The spectrum includes a zero-frequency spike (last term on the right) and a broad,
flat spectrum from
w=0 up to the frequency
wc=(Dt)-1.
As we will see in Section 4, the existence of a characteristic pulse lifetime
t=L/V adds a narrow, flat portion
from
w=0 up to
w imp=(t)-1.
For
L=102lc, V=3Vc,
and
lc=1 mm,
10-1 m and 10 m, the value of
wc is of order 10
6, 104, and
102 rad/s, respectively, and that
of
w imp is
3104, 300 and 3 rad/s. A result
analogous to the
preceding expression is obtained for the change in volume strain,
q(w, r).
Under natural conditions, practically all rocks contain pore water. In Section 5 we will show that the diffusion of pore water plays a major role in some SEM phenomena.
As shown in equations A6 and A9 of Appendix A, when monochromatic elastic waves propagate through the crust, the pressure change P is proportional to the volume strain q for both high and low frequencies. Thus there is a linear coupling between pore pressure and volume fluctuations during a seismic disturbance. This provides us with a means of estimating the relative contribution of pore water diffusion to SEM signals. In particular it enables us to connect changes in crustal deformation and pore water pressure.
Equations A6 and A9 are strictly true only for a monochromatic disturbance. Let's consider again the stress pulse associated with a crack (Figure 1b). While the pulse is growing, the relation between each elastic wave component and the corresponding pore pressure wave component will be given by equation A6 for high frequencies and equation A9 for low frequencies. After the pulse passes, leaving a constant residual stress, the pore pressure will relax according to the diffusion equation A7, as in Figure 1c.
In this figure, the pore pressure has a characteristic diffusion relaxation time DTD (see equation A7) given by
![]() | (2) |
where
l is the linear extent of the region in the vicinity of the crack which has
appreciable residual
stress
(l3lc), and
D0=K2k0/mvb
is the diffusion
coefficient. The quantities appearing in
D0 are defined in Appendix A.
To estimate the frequency spectrum associated with the pressure change
P(t, r) we use an expression similar to equation 1:
![]() | (3) |
where P0 is the pore water pressure change due to a change of volume strain. We obtain
![]() |
From this result we see that the spectrum consists of three parts, a broad low-intensity region (first term) and two low-frequency regions of much higher intensity, one related to the seismic impulse (second term) and the other to a pore water diffusion process (third term). The ratio of the intensities in the first and third regions is Dt/DTD. Recall that Dt is given by lc/Vc. Together with equation 2 this gives
![]() |
Usually this ratio is much less than one, which means that the process related to water diffusion will dominate in the low frequency range.
From the last two subsections we arrive at the following conclusions. Cracks are a source of two types of mechanical disturbances, a seismic impulse and, in the presence of water-saturated rock, diffusion of pore water. The frequency spectrum of these mechanical disturbances in general will consist of four parts. First there is a zero-frequency spike arising from the residual strain. This spike could potentially be a source of tectonomagnetic anomalies (see Section 5). Second there is a broad spectral range from zero up to high radio frequencies, related to the crack-opening process. The spectral density is nearly flat in this range. Third there is a much narrow spectral range from zero up to low radio frequencies, associated with the seismic impulse generated by the crack. Fourth, there is a low-frequency contribution arising from the diffusion of pore water. This diffusion is driven by the stress associated with crack formation through equation A7. The spectral density of this part of the spectrum exceeds by several orders of magnitude the broad flat part.
In our model, the basic source of the SEM anomaly is a mechanical deformation resulting in the formation of a large number of cracks. The volume density of cracks, n, cannot exceed the value n max, which, in agreement with laboratory data [Zhurkov et al., 1977], is related to the typical crack length, lc, via
![]() | (4) |
If n reaches the value n max, the laboratory sample disintegrates. This implies that n will not exceed n max in the earth's crust. (Some early estimates of the magnitude of SEM anomalies did not take this restriction on n into account [Warwick et al., 1982].) We therefore express the crack density by the relation
![]() | (5) |
where a is the ratio of the actual to the maximum crack density ( 0<a<1 ).
In general, the growth direction of a microcrack is random. But formation of these cracks is a stress-release mechanism, so we would expect that locally there would be some preferred growth direction. Then the average growth direction in a macro-volume undergoing mechanical deformation would be non-zero. In this case the average volume strain, q, and average shear strain, e, are
![]() |
where q and e are the typical volume and shear strains near a crack.
Formulas (1, 3, 4, 5, and 5a) will be the basis for calculating the magnitude of the various types of SEM anomaly (see sections 4 and 5).
In this section we estimate the contribution of possible sources to the EM field. We start by writing Maxwell's equation (in SI units) for an isotropic medium described by electrical permittivity e, magnetic permeability m and electrical conductivity s:
![]() | (6a) |
![]() | (6b) |
![]() | (6c) |
![]() | (6d) |
with constitutive relations
![]() | (7a) |
![]() | (7b) |
![]() | (7c) |
Here c is the speed of light, u is the velocity of the medium, F is the main geomagnetic field, and j0, E0, P0 and M0 are the external current density, electric field, polarization and magnetization, respectively.
In general we can classify possible sources into three groups: active, passive and apparent. Examples of active sources are
By passive sources we mean changes in the electromagnetic parameters s, m and e of the earth's crust due to mechanical processes. An example of a passive source is a change in s in the presence of an external electric field. Such external fields nearly always exist due to magnetospheric/ionospheric geomagnetic variation. Estimation of the effect of passive sources indicates that it is generally much smaller than that of active sources. Therefore we will not consider such sources here, although under special circumstances (e.g. the geometry of local conductivity changes) they may produce anomalies of substantial magnitude [Honkura and Kubo, 1986; Merzer and Klemperer, 1997; Rikitake, 1976a, 1976b].
An example of an apparent source would be the apparent electrotelluric field arising from a change, for whatever reason, of the chemical composition of pore water during measurement of the field, since such a measurement is done using electrodes placed just below the surface [Miyakoshi, 1986]. Another example is a change in the apparent local orientation of a detector due to local displacement of the crust. This would give rise to an apparent change in the EM field in the radio frequency range. Sometimes the amount of local displacement needed to give an observed effect is very small. The existence of apparent sources depends on the type of detector and how it is installed, so it is difficult to discuss them in general. However one should keep their existence in mind when interpreting electromagnetic anomalies.
We want to compare the contribution of different active sources. For ease of calculation we assume that the crust is homogeneous and static, i.e. s, e and m are constants independent of location and time. This makes the use of equations 6 (a-d) and 7 (a-c) convenient. We can eliminate H, B and D and obtain a single equation for the electric field E in terms of its sources:
![]() | (8) |
The right-hand side of this equation contains the sources. M0, P0, j0, E0 and u, which are connected to the mechanical state of the crust through various mechano-electromagnetic coupling mechanisms. In order to estimate and compare the effects of the various sources, we need to write down what these coupling mechanisms are.
Some minerals in the earth's crust show residual magnetism due to ferromagnetic inclusions (e.g. titano-magnetite). This residual magnetism was "frozen in'' at the time of formation by the paleogeomagnetic field. The deformation of this type of rock leads to changes in M0 due to changes in the orientation of the inclusions [Kapitsa, 1955; Kern, 1961; Stacey, 1964; Stacey and Johnston, 1972]. These changes are in general a complicated function of the applied stress, the size of the inclusions and the microstructure of the rock, but the following expression is widely used in calculations [Hao et al., 1982; Zlotnicki and Cornet, 1986]:
![]() | (9) |
where c|| is the stress sensitivity in the direction parallel to the axial load, sij is the stress tensor, I is the reference magnetization and repeated indexes are summed. In terms of the strain tensor eij and the shear modulus ms, we can write
![]() | (10) |
One of the most commonly occurring minerals, quartz, shows this effect, namely, the occurrence of an electric polarization under the influence of mechanical stress [Parkhomenko, 1971; Volarovich and Parkhomenko, 1955]. Usually, quartz grains are randomly oriented and show very weak piezo-electricity in the aggregate. But because of residual orientation of the quartz grains some rocks will show a larger piezo-electric effect [Ghomshei and Templeton, 1989], although still perhaps two to three orders of magnitude less than in monocrystalline quartz [Bishop, 1981]. The piezo-electric properties of rocks have been widely studied in the context of geophysical prospecting [Kepis et al., 1995; Neyshtadt et al., 1972; Sobolev and Demin, 1980; Sobolev et al., 1984]. Changes in polarization can be related to the strain tensor via
![]() | (11) |
where
Dkij is the piezo-electric modulus. For
making estimates, we consider only the contribution
from
i=j=1. The magnitude of
Dkij depends on the scale of the mechanical
disturbance. If this scale is of the order of the diameter of a typical quartz
grain (~0.5 mm) then
D Dk11
- 2.3
10-12
C/N;
if the scale is much larger than this we will take
D
-10-14 C/N
[Bishop, 1981].
It is possible for the polarization to change for reasons other than the classical piezo-electric effect. All real crystals contain extended defects (dislocations). In dielectric materials, dislocations usually are charged because point defects are associated with them. Under static conditions the charge around the dislocation is neutralized by point defects of opposite charge (the Debye-Huckel cloud). Under the influence of an applied stress the dislocation can move. At nearly all temperatures of interest an unpinned dislocation will move much more quickly than its associated Debye-Huckel cloud and therefore will carry a net charge along with it. This will give rise to charge transport and polarization of the material. This effect was discovered by Stepanov [Stepanov, 1933]. Its magnitude is a complicated function of dislocation density, density and type of point defects, temperature and pressure. Because rocks have an extremely high density of dislocations and point defects, pinning effects will make dislocation movement almost impossible at normal temperatures in these materials; at greater depths where the temperature and pressure are greater this effect could be important [Slifkin, 1996]. There are some mechanisms related to movement or polarization of point defects which also lead to the appearance of an electric field. One of these effect is the so called "pressure stimulated current'' [Varotsos and Alexopoulos, 1986]. Another mechanism for polarization of the crust is directly associated with crack formation [Cress et al., 1987; Khatiashvili, 1984; Nitsan, 1977; Ogawa et al., 1985; Schloessin, 1985; Warwick et al., 1982; Yamada et al., 1989]. It is well known that the appearance of a crack in almost any material produces effects such as light bursts, electron streams and broad-band electromagnetic emission up to x -ray frequencies [Deryagin et al., 1973; Finkel et al., 1985; Gershenzon et al., 1986]. These phenomena are due to the electric field (up to 108 V/m) associated with a buildup of high electric charge density on the new surfaces of the emerging crack. This effect has been studied by several groups and is another type of non-classical piezo-electric effect. It is almost impossible to estimate theoretically the contribution from this mechanism since there are many different effects which accompany crack formation and many unknown parameters, but it could be important in high-frequency SEM phenomena.
As mentioned above, under natural conditions rocks practically always contain pore water. At the pore boundary there exists an electric double layer due to the difference in the electrochemical potentials of water and rock. Deformation of the earth's crust leads to changes in pore pressure and, as a consequence, diffusion of the pore water in a direction opposite the pressure gradient. The motion of this water layer next to the pore boundary results in charge transport, i.e. an electric current, parallel to the boundary. This phenomenon was discovered in the mid-nineteenth century and was first used in a geophysical context by Frenkel [1944] to explain the observations of Ivanov [1939]. More recently it has been used in explaining and modeling SEM phenomena [Ishido and Mizutani, 1981; Mizutani and Ishido, 1976; Mizutani et al., 1976] and in geophysical prospecting [Maxwell et al., 1992; Sobolev and Demin, 1980; Thompson and Gist, 1993; Wolfe et al., 1996]. The relation between the electro-kinetic current density j0 and pore pressure P can be written as
![]() | (12) |
where C is the coefficient of the streaming potential. The pressure is related to the volume strain by equation A1 (Appendix A).
Movement of conducting crustal material in the geomagnetic field gives rise to an induced electric field u x F and current j=su x F. There are two possible sources of this movement; one is the deformation and movement of rock material itself and the other is diffusion of pore water due to volume deformation. In the first case the velocity u of the crustal material is related to the strain tensor eij by
![]() | (13) |
where V is the velocity of elastic waves. This formula reflects the fact that any change in strain is propagated through the crust at the seismic velocity.
In the case of induction due to pore water movement the effective velocity u can be estimated by Darcy's law,
![]() | (14) |
where P is given in equation A9 and the parameters k0, m and mv are defined in Appendix A.
Now that we have expressed all the terms on the right-hand side of equation 8 in terms of electromechanical coefficients and strain changes, we can compare different kinds of sources. Let's consider the terms
![]() | (15a) |
![]() | (15b) |
![]() | (15c) |
![]() | (15d) |
which are piezo-magnetic, piezo-electric, electro-kinetic and induction sources,
respectively. We have two
types of induction sources, a source A
I1 due to deformation and movement of rock material
itself and a
source A
I2 due to diffusion of pore water. It is easy to
show that the second term in relation 15d is small
compared with the first term, and we shall neglect it. Replacing
t with
Vclc
and
r with
1lc,
we can find the ratios of the strengths of the various transducer mechanisms using
equations 9 through 14 and A9:
![]() |
![]() |
Table 1 shows the results of comparing these strengths for the values of D, C, lc and k0 given in Table 1a. The values of the remaining coefficients appearing in these ratios may be obtained from Table A1 (see Appendix A). We see that the strength of a piezo-magnetic source has the same order of magnitude as that of piezo-electric pure quartz. Monocrystalline quartz occurs in nature only as small grains ( < 1 mm), which have practically no preferred orientation ( D=D1 ). For such small grains a piezo-electric source would be important only in the radio-frequency range. If the piezo-electric grains were to have a preferred orientation on a macro-scale ( D=D2 ), we would expect their source strength to be at least two orders of magnitude weaker than a piezo-magnetic one.
Referring again to Table 1, comparison of piezo-magnetic and electro-kinetic sources shows that under some conditions, namely high (but still reasonable) values of electrical conductivity s and electro-streaming potential C, an electro-kinetic source could have a strength of the same order of magnitude as a piezo-magnetic or piezo-electric source. For low conductivity ( s<10-3/W-m ) the piezo-magnetic source exceeds the electro-kinetic source.
The three types of sources considered above have the same dependence on the source size; for this reason the latter does not enter into the comparison of their relative strengths. In the case of an induction source of type 1 (AI1 ), the induction source strength relative to the strength of any of the other three sources depends on the size of the source. We can see from Table 1 (first column/fourth row) that the piezo-magnetic source essentially exceeds the induction source AI1 on small to medium size scales; the induction source AI1 becomes important only for a source size greater than 1 km and high electrical conductivity ( s>10-1/W-m ).
Both electro-kinetic sources and induction sources of the second type (AI2 ) arise from diffusion of
pore water, so they occur together. We see that the induction source AI2 is always much less than the
electro-kinetic source. Even if the water layer in the crust has an extremely high
permissibility
( k010-10 m
2 ) the electro-kinetic source exceeds AI2 by two to three
orders of magnitude. For usual values of permissibility ( k0 = 10-12 to
10-16 m2 ),
the difference in source strength ranges from four to nine orders of magnitude.
The preceding considerations lead us to the following conclusions. For small sources, the most important mechano-electromagnetic processes considered here are the piezo-magnetic, electro-kinetic and piezo-electric. For large sources, only piezo-magnetic and electro-kinetic processes are important. The induction effect is important only on a very large size scale, when the source size is comparable to the earthquake focal zone.
Comparison of the strengths of the various types of sources using scaling arguments as in Section 3 gives us some superficial information about relative strengths. More accurate results would require solving the system of equations 6(a-d), 7(a-c), and the appropriate equation connecting the magnetization or polarization to the mechanical state of the crust (cf. equations 9-14). In general such solutions cannot be obtained analytically without some simplification.
The goal of this section is to present a set of asymptotic formulas for estimating the magnitude of the electric and magnetic fields at the earth's surface over a broad frequency range, ranging from quasi-static to radio frequencies, generated by mechanical disturbances in the earth's crust. We will also present the results of calculations of the spectral density of an EM source produced by an impulse mechanical source due to various mechano-electromagnetic coupling mechanisms.
First we consider the electromagnetic fields to be weak enough that they do not influence
the
mechanical state, i.e. the stress and strain tensors are independent of
E and
H.
We also assume that the size of the source is much smaller than the distance
from the source to the field point (dipole approximation).
We are ignoring higher-order multipoles; their effect is normally negligible unless
the
dipole moments vanish. Introducing a distribution of effective electric and magnetic
dipoles characterized by
P and
M
, respectively,
the
system of equations 6(a-d) can be written as follows
[Gershenzon et al., 1993]:
![]() | (16a) |
![]() | (16b) |
![]() | (16c) |
![]() | (16d) |
where
![]() | (17a) |
![]() | (17b) |
and the bar symbol indicates an integration over volume, i.e.,
J0 J0 dV,
etc. We also have
![]() | (17c) |
and
![]() | (17d) |
As expressed in equation 17d, we can always consider J0 as the sum of two contributions, a rotational part J0 rot associated with magnetic dipoles and an irrotational part J0 irrot associated with electric dipoles.
![]() |
Figure 2 |
![]() |
Figure 3 |
We shall estimate the value of these distances for typical values of
s, e and
m.
In this paper we shall take
e=3e0
and
m=m0
for
the crust, where
e0 and
m0 are the values in vacuum.
Most frequencies
f=w/2p
of interest lie in the range
10-3 Hz
The expressions for the fields close to the surface for
monochromatic electric and magnetic dipoles
[Banos, 1966]
are given in Appendix B. At the interface, all these expressions will contain
the complex exponential factor
eik1h, which
includes the attenuating
effect of the crust. Introducing the parameter
the exponential can be written,
For
w
For distances
r>rA, the formulas of Banos [1966]
could be used.
However, as we have seen, in the quasi-static range and part of the ULF range,
these formulas do not apply because
rA exceeds the range of interest. Even
at higher frequencies, when
rA becomes small, we need a formula for
r
For a static magnetic dipole, we have
E=0 and
where
R is the distance from the source.
For a static electric dipole embedded in a conducting half-space, we use
if the dipole is oriented horizontally along the
x axis, and
if the dipole is vertical. These formulas were obtained in the point dipole
limit of expressions given by Edwards [1975] and
Gokhberg et al. [1985].
Now consider the case
w>s/e,
for which the Banos
formulas do not apply. For this case we can ignore the conductivity of the
crust (as long as the source depth is less than the skin depth) and use formulas
appropriate to a dipole in vacuum. For distance
r less than
rB (i.e. less
than the vacuum wavelength/
2p ), the following formulas
[Landau and Lifshitz, 1971]:
for an electric dipole, and
for a magnetic dipole.
For distances
r greater than
rB, we have [Landau and Lifshitz, 1971]
for an electric dipole, and
for a magnetic dipole.
Once we have a solution for the monochromatic source the standard procedure for solving
the
time-dependent problem is to integrate the monochromatic solution over all frequencies,
weighted by the
spectral density of the source. In order to estimate the parameters of the EM emission
it is useful to consider
an impulse dipole source, since the mechanical disturbances have an impulse structure
(see Section 2).
Using equations 17(a-d), 10-14, 1, 1(a-b) and 3, we can express the electric
and magnetic dipoles for all
the source types considered here. For a piezo-magnetic source, the magnetic dipole
has the following form:
where
and the integral is over the volume
V crack = lc3 associated with
a crack.
The magnetic and electric dipoles for an induction source have essentially the
same form as equation 21a, except that the third term is absent and
MI and
PI appear in place of
MM. The electric dipole
for an electro-kinetic source has the form shown in equation 21b:
where
The electro-kinetic source appears as a discontinuity in the electro-kinetic properties
of the
medium across a planar boundary, so the last two terms in equation 21b are expressed
as
surface integrals rather than volume integrals. See Appendix C for a more detailed
explanation.
For a piezo-electric source, the electric dipole can be expressed as
where
and
n(r) is a unit vector in the direction of the local
dipole moment
in the volume element
dV at point
r in the crust. This local moment
has magnitude
| PE| attenuated by the factor
(lc/r)e-r/L.
The magnitudes of
MM, MI, PI,
PK and
PE will be determined later in
this section.
Now we can find the associated spectral densities. For a piezo-magnetic source we
have,
where
t=L/V is the lifetime of the seismic
impulse. The same expression (without the last term) also gives
the spectral density for the magnetic and electric dipoles of an induction source.
Using equation 21b we can find the spectral density associated with an electro-kinetic
source. We
obtain
Figures 4b
and 4c
show
P
Finally, we consider the spectral density associated with a piezo-electric source.
The orientation
of piezo-electric grains are usually random, but in some cases may be partially ordered.
We need to consider
both cases. First, suppose there is a preferred orientation
n= PE/PE.
Using equation 21c we find
Figure 4d
shows this spectral density. As before, curve 1 is for the crack only. The presence
of the seismic
impulse (curves 2-4) increases the magnitude of the maximum and shifts it to lower
frequencies.
For the case of random orientation we have
In obtaining this result we assumed that the contribution from grains located at
distance
r from the crack is proportional to
[N(r)]1/2, where
N(r) is the number
of such grains, given by approximately by
4pr2/lc2.
Figure 4e
shows that
including the seismic impulse increases the low frequency part of the spectrum.
Comparison of the spectra of the various sources considered here (Figures 4(a-e))
shows that piezo-magnetic and electro-kinetic sources activate the low-frequency
EM modes while a piezo-electric source activates the
high-frequency modes. Note that in most cases the contribution of an acoustic impulse
generated by a crack
is much larger then the contribution of the crack itself.
Even for these simplified spectral densities, however, an exact solution for the
radiation fields
would require a more accurate solution to the monochromatic problem, over all frequencies,
than is afforded
by using the asymptotic formulas. Attempting such a solution would lie outside the
scope of this paper.
Nevertheless, based on our results so far, we can arrive at some qualitative conclusions
concerning the shape
and behavior of an electromagnetic pulse propagating along the earth's surface.
For
w max
For
w max>wh,
which is the usual case, attenuation will be small
in the frequency range
0<w<wh
but will be significant in the range
w>wh.
The shape of the radiation pulse will be altered by
this dispersion. With increasing distance from the hypocenter, the pulse will broaden
and become oscillatory
in addition to becoming weaker.
where
q and
e are the volume and shear strains, respectively.
For either type of strain, the spatial
dependence can be considered to be some linear combination of a unipolar and a bipolar
square pulse
(Figure 5).
Thus for a shear strain,
where
e up and
e bp are the magnitude of the unipolar
and bipolar shear strains
respectively. A similar expression can be written for the volume strain
q.
Based on the expressions for
e(w, r)
and
q(w, r)
and
on the formulas of this section and the previous section (equations 21(a
From this table we see that a piezo-magnetic source produces a magnetic dipole for
a unipolar
pulse, while a piezo-electric source produces an electric dipole for a unipolar pulse.
An electro-kinetic
source produces an electric dipole for a unipolar pulse volume strain. An induction
source produces a
magnetic dipole for a bipolar pulse and an electric dipole for a unipolar pulse.
Now we have all the formulas necessary to estimate the electromagnetic
field at the earth's surface
due to a mechanical disturbance in the earth's crust. The relevant equations and
the various cases to which
they apply are summarized in Table 3.
So far in this section we have considered the temporal and spatial distribution of
the
electromagnetic field from an impulse source. To compare with measurements we must
consider how the
real EM field at the detector is related to what the detector records. The latter
depends on the measurement
technique and the detector parameters.
Usually any detector of electromagnetic emissions will have two distinct parameters,
the
frequency channel and the acquisition time. We characterize the frequency channel
by a filtering function
g(w) and denote the acquisition time by
DT. If we denote by
Em the
mean electric field recorded by the detector, and by
Ef (t) the filtered electric
field at the detector at time
t ( 0
and
where
E(w) is the fourier component (i.e. spectral
density) at frequency
w of the source electric field at the location
of the detector. Using equations 23
and 24 and doing the integration over
t, we can express
Em as
For simplicity we take
i.e. a frequency window of width
Df = Dw/2p centered
at frequency
f0 = w0/2p.
We suppose that
Df
We will assume the duration
Dt of the impulse to be small enough that
(Dt)-1
Let's consider two quite different cases. In the first case,
(DT)-1
while in the second case we get
Suppose that during the time
DT
The preceding discussion can be carried through for the magnetic field, with completely
analogous
results. The first case results in
and the second case in
We thus obtain two different formulas for the measured field
Em (or
Bm ).
If
Df = Dw/2p = 103 Hz
and
DT = 10 s, the relative
magnitudes of
Em (or
Bm ) from the two expressions (26) and (27) (or (28)
and (29))
is
(DfDT)1/2=100.
The reason for this difference requires some
clarification. In equations (26) (or (28)), the detector accumulate the energy
throughout the time
DT. If the source is emitting in an impulse
mode,
with a pulse width
Dt
For both
Em and
Bm, we will see later in Section 5 that the two cases
imply two different
measurement techniques, and the probability of detecting an observable SEM anomaly
may depend critically
on which technique is used.
During the past century a wealth of SEM data has accumulated which has been interpreted
as having
some relation to pre-seismic processes. We want now to apply our model to the interpretation
of some of
this data. The search for SEM anomalies has spanned a wide frequency range from quasi-static
(periods of
weeks or months) up to radio frequencies ( < 50 MHz). Both magnetic and electric fields have been
measured, using detectors below ground as well as above ground. A partial list of
the types of anomaly
which have been reported includes
We will estimate and compare with reported data the magnitude of the various types
of SEM anomaly.
This type of field experiment has a very long history. Occurrences of very
large anomalies were observed centuries ago
[Rikitake, 1976a, 1976b].
These early effects were reported to be comparable in magnitude to
the geomagnetic field. However, as Rikitake has jokingly commented, the magnitude
of reported anomalies
seems to have fallen off exponentially with time, so the present day anomalies are
some four orders of
magnitude smaller, i.e. of order 1-10 nanotesla (nT). The typical duration time,
T, varies from days to
month, but there have been reports of
T as small as minutes
[Johnston, 1989;
Moore, 1964;
Mueller and Johnston, 1998;
Shapiro and Abdullabekov, 1982;
Shapiro et al., 1994;
Skovorodkin et al., 1978].
In those cases where tectonomagnetic anomalies are followed by quakes, the elapsed
time before the quake is comparable to
T. Anomalies have been detected at distances
up to 30 times the size of their origin. The general
hypotheses put forth to explain these anomalies are based either on the piezo-magnetic
effect
[Sasai, 1980, 1991;
Stacey, 1964;
Stacey and Johnston, 1972;
Zlotnicki and Cornet, 1986]
or the electro-kinetic effect
[Fitterman, 1978, 1979a;
Mizutani and Ishido, 1976;
Mizutani et al., 1976].
First, let's estimate the size of the anomalous magnetic field assuming it originates
from piezo-magnetism.
Suppose we have some region of the crust, of linear extent
r, located at distance
R from the point of measurement, and that this region has piezo-magnetic
properties.
Suppose further that during the duration time
T of the anomaly a number
N of cracks
form in this region, with
N given approximately by (see equation 5)
The residual strain field around an isolated crack extends out to about three times
the crack length
(see equation 1b). This leads to the appearance of a magnetic dipole, for the
ith crack, of moment
where
ti is the time of appearance of the ith crack (measured
from when the anomaly begins) and the
components of
M0 are given by equation 10. From equations 18a and
31
the cumulative effect is to create a magnetic field
B whose magnitude is given approximately by
For simplicity we assume that all cracks have the same length and orientation
and have the same value of
R. Then from equations 4, 10 and 30 the magnetic
field, after all
N cracks have formed, becomes approximately
From this we see that the size of the magnetic anomaly is independent of crack size
but
proportional to the coefficient
a or the average shear strain
e,
and strongly dependent on the ratio ( r/R ). The latter dependence means
that
the anomaly is stronger when it arises in a region close to the observation
point. It also means that sources close to the observation point ( r/R
The model based on the electro-kinetic effect was first proposed by
Mizutani and Ishido [1976].
These authors noted the relation between local geomagnetic anomalies and the level
of the
water table during the Matsushiro earthquake swarm. Fitterman developed this model
further
[Fitterman, 1978, 1979a, 1981].
The magnetic field associated with a system of currents generated by an electro-kinetic
source has some characteristic features. It is known
[Fitterman, 1978]
that in any homogeneous medium with an arbitrary distribution of pressure variations
due to pore water, the geomagnetic effect is zero. That is, the electric current
due
to the motion of the pore water is exactly cancelled by the current arising from
the electric field which is created. A non-zero effect occurs only in an inhomogeneous
medium, and its magnitude depends only on the degree of inhomogeneity. However, not
all types of inhomogeneity will produce a geomagnetic effect on the earth's surface.
For example, it can be shown that when the inhomogeneity is only in the vertical
direction,
i.e. in a horizontally stratified crust, there is no above-ground geomagnetic effect
[Fitterman, 1978;
Gershenzon, 1992].
But there will be an effect below the surface, and
it will be greatest at the boundary between strata. So geomagnetic effects of an
electro-kinetic nature arising
from geodynamic processes would be observed on the earth's surface only if there
were inhomogeneities in
the horizontal direction (e.g. faults or inclusions).
For the estimation of the magnitude of magnetic field disturbances, therefore, it
makes sense to
consider, as the simplest model, two media (medium 1 and medium 2) separated
by a vertical boundary. In
this case the magnitude of the magnetic field disturbance at the intersection of
the boundary with the surface
can be estimated by the simple formula [Fitterman, 1979a,
1979b]
where
f is a dimensionless geometrical factor weakly dependent on the size of the
boundary.
For realistic cases
f does not exceed 20
[Gershenzon and Gokhberg, 1992].
We shall use
f =10. The pressure variation
P arises from the appearance of
multiple cracks in some regions. Expressing
P in terms of
q , the
change in the volume strain, via equation A9, we obtain
B for various values of
q, s1,
s2 and
C1-C2 (Table 4). This table shows
that the magnetic disturbance becomes significant (~ 1 nT) only for
changes in volume strain greater than 10-6 and conductivities greater than
10-2/W-m.
We have not considered changes in water permeability or changes in the streaming
potential,
C,
accompanying a pre-seismic process. These changes could affect the size of the geomagnetic
anomaly.
Changes in
C are difficult to calculate because they are related to changes in the composition
of the pore
water. Changes in water permeability likewise are difficult to calculate but might
be significant when the
porosity is small and
q is large. The seasonal runoff of ground water
in mountainous areas has been
observed to give rise to a large electro-kinetic current and magnetic field. For
example, a pore pressure
difference of 10
6 Pa over a horizontal distance
DL of 100 m gives a pressure
gradient
dP/dx of 104 Pa/m, which results in a magnetic field
B = CmsDLdP/dx ranging from 10 to 100 nT
for
s=10-1/W-m
and
C = 10-7 to 10-6 V/Pa. A change of 10% in this field due to a pre-seismic
process would give rise to an easily observable anomaly of 1 to 10 nT.
So both mechanisms, under appropriate conditions, can produce fields strong enough
to be
comparable to observed anomalies. However we emphasize that one of these conditions
is that the distance
from the point of measurement to the source be comparable to the size of the source.
We also note that there is a simple way of distinguishing which of these two types
of sources,
piezo-magnetic or electro-kinetic, is responsible for an observed magnetic anomaly,
based on the temporal
behavior of the latter. If the anomaly behaves approximately like a step function,
i.e. shows a residual field,
then it is piezo-magnetic in origin. If it behaves approximately as a unipolar pulse
with no residual field, it is
electro-kinetic in origin.
The search for earthquake precursors in the electrotelluric field extends back about
eighty years.
This search has been conducted in many areas of high seismic activity worldwide (Japan
[Fukutomi, 1934;
Miyakoshi, 1986;
Noto, 1933;
Ozima et al., 1989;
Shiratori, 1925;
Uyeda, et al., 2000],
the former Soviet Union
[Sobolev et al., 1981],
Greece [Lighthill, 1996;
Varotsos and Alexopoulos, 1984a, 1984b,
1987;
Varotsos et al., 1993],
China [Qian et al., 1990;
Raleigh et al., 1977],
Bulgaria [Ralchovsky and Komarov, 1987],
USA [Sornette and Sornette, 1990]).
Several hundred cases have been reported which relate such anomalies to
strong nearby quakes. In general, the morphological features of these anomalies are
defined by the region of
occurrence and method of detection. The magnitude of the disturbances ranges from
a fraction of a millivolt
to a few tens of millivolts; it depends on the quake magnitude and distance from
the epicenter but in many
cases the magnitude is independent of the measurement line length. The duration of
an anomaly can range
from minutes to weeks and is only weakly dependent on quake magnitude and distance
from epicenter.
The nature of these anomalies is not well known. There exist models based on the
classical and
non-classical piezo-electric effects discussed in Section 3B
[Gokhberg et al., 1985;
Slifkin, 1996;
Sobolev and Demin, 1980;
Varotsos and Alexopoulos, 1986],
and the electro-kinetic effect (Section 3C)
[Bernard, 1992;
Dobrovolsky et al., 1989;
Fitterman, 1979b;
Gershenzon and Gokhberg, 1989, 1993].
Sometimes the occurrence of anomalies has been ascribed to changes in the conductivity
of the crust
[Meyer and Pirjola, 1986].
In some cases they have been explained in terms of changes in the chemical composition
of pore
water around one or both of the measurement electrodes, giving rise to an apparent
electrotelluric signal
[Miyakoshi, 1986].
Such chemical changes could alter the
z -potential of the double electric layer at the
electrode-water interface.
We ask how pre-seismic processes relate to the appearance of an electrotelluric field
at distances
of tens to hundreds of kilometers from the epicenter. There are at least two alternative
explanations.
The first one, preferred by most experimenters, supposes that somehow a large current
is
generated in the focal area and is distributed over a great distance to the detector.
This explanation has not
been confirmed in most cases. For example, Yoshimatsu [1957]
used two parallel lines 100 m and 1500 m in length and the signal appeared
only
in the shorter line, whereas we would expect it to appear in both lines if the
above explanation were true. Noto [1933] used two lines
perpendicular
to each other and the anomalous signals appeared in only one of them, which is
again difficult to explain in terms of a large-scale current distribution, even
in the presence of inhomogeneities in the conductivity. Miyakoshi [1986]
described anomalies as close as 3 km from the epicenter of a 5.2 magnitude quake.
He used three short
lines, each 30 m long, in a geophysical tunnel, and two perpendicular lines
on the ground, each 600 m long.
He saw anomalous signals only in one of the short lines. They clearly exceeded normal
levels and lasted
about two months. If these signals were electrotelluric in origin, they certainly
could not be explained in
terms of a large-scale current system. The so-called
DV/L test has been introduced to
eliminate noise for
short lines on VAN and similar networks. But there are examples where this test does
not work on such
networks
[Gershenzon and Gokhberg, 1993;
Uyeda, et al., 2000].
In order to explain the appearance of SES generated by pre-seismic processes in the
focal area but
located hundreds of km from this area, Varotsos et al. [1998]
proposed the existence of a special pencil-like region of high conductivity
in the crust,
extending from the dipole and terminating at the field point.
Although regions of strong inhomogeneity are not unusual in the crust, we think the
existence of this special
pencil-like form extending for several hundreds of km and terminating very
close to the detector is
unrealistic. Calculations by Bernard [1992] also support
this conclusion.
The inadequacy of the explanation in terms of a large-scale current system originating
in the focal
area leads us to the second alternative, namely, a large-scale mechanical stress
field which can, under certain
condition, produce electrotelluric anomalies locally. Then the morphological
features of the anomalies
represented by the several examples cited in the previous paragraph can be explained
in terms of the electro-kinetic
effect
[Bernard, 1992;
Dobrovolsky et al., 1989;
Gershenzon and Gokhberg, 1989, 1993].
As mentioned before, the current distribution created by an electro-kinetic source
in a homogeneous medium is
zero on a scale larger than the individual grains. Non-zero currents will appear
only on the boundary
separating two homogeneous components of an inhomogeneous medium. Across this boundary,
the electric
potential will be discontinuous, and two electrodes placed on opposite sides of the
boundary will show a
voltage practically independent of the distance between the electrodes. This voltage
will depend on the
strength of the electro-kinetic source and the electrical characteristics of the
two components. This would
explain why the anomalous signals are selective and why the strength of an anomalous
signal is sometimes
independent of the length of its measurement line.
The typical pore pressure relaxation time depends on the size and shape of the disturbed
region
and can range from minutes to hours, days or weeks, which coincides with the time
range over which
electrotelluric anomalies are observed to occur. No single mechano-electromagnetic
mechanism involving
the crystal matrix of the rocks, e.g. piezo-magnetic or piezo-electric (classical
and non-classical), could act
over such large time scales, because the formation time for such disturbances is
only 1 second or less, i.e. of
the order of the linear extent
l of the region divided by the elastic wave velocity. Once formed, such
disturbances dissipate with a time constant
msl2
which is even smaller.
Let's calculate the magnitude of the electrotelluric field from an electro-kinetic
source. The
maximum effect occurs when the two electrodes span an inhomogeneity. In this case
the potential difference
Df can be expressed
by
In Table 5
we tabulate
Df for a range
of
q and
C1-C2.
Table 5
shows that strains exceeding the tidal value (10
-8 ) can be detected,
but only under the following conditions:
In spite of the fact that no macroscopic electric current will exist in a homogeneous
medium, an
electric field can exist and is measurable. In a homogeneous medium the potential
between points 1
and 2 is
Df=C(P1-P2).
Normally, if the electrodes are at the same depth
the difference between
P1 and
P2, and hence
Df, is negligibly
small.
But if the electrodes are at different depths (for example, one above and one
below the water table) a significant potential can arise.
Consider a simple example. Suppose at
t =0 a step-function change in volume strain (and
resulting pore pressure) occurs in the vicinity of two electrodes placed in a homogeneous
medium.
If
P1=P2 after the change, then
Df remains zero.
But if the environment around the two electrodes is slightly
different, leading, for example, to different water permeabilities or different distances
between the electrode
and the water table level, the pressures
P1 and
P2 will relax back to their equilibrium values at different
rates, leading to a spike in
P1 and
P2 at some time
t>0. In such a case the duration of the anomaly does not
depend on the source but on the local environmental differences between the electrodes.
The theoretical
shape of the spike coincides with the shape usually observed, in the SES reported
by the VAN group. In
addition, when several signals are observed their duration is usually the same, in
agreement with this simple
model.
The above example indicates that if, instead of offsetting the electrodes horizontally,
they are
offset vertically above and below the water table, the electrotelluric anomaly in
a homogeneous medium
might be maximized. In such an experiment it was shown that a good correlation existed
between the
electrotelluric field and tidal deformation
[Gershenzon et al., 1990].
So under some conditions this method of measuring the electrotelluric field can be
extremely
sensitive. The use of a short vertical baseline also serves to reduce background
noise. This method does have the disadvantage that it cannot detect an
electrotelluric disturbance unless the electrodes are in a region where the volume
strain is changing.
We have shown how placing the electrodes within a source
of mechanical disturbances can yield
detectable electrotelluric fields, even when the source is as weak as tidal deformation.
Now we consider the
situation where the electrodes are placed outside the source. For purposes of estimating
the effect we use
equation 18b, which gives the electric field associated with a static electric
dipole.
Setting
h =0 and
y =0 for simplicity, and using
PKV from Table 2 (with
q in place of
q ) for the dipole moment
P
The field varies as
l2, the area of the vertical inhomogeneity, and inversely as the
cube of the
distance
R between source and detector. In Figure 6 we show how
E varies with
R (logarithmic scale) for
q equal to 10-4 and for a range of
l (10 m to 10 km), based on the above formula.
The range of applicability of the dipole approximation limits
R to values larger than about
3l, so the graphs
in Figure 6
do not extend down to values of
R which are below this limit. Even for a source comparable in size
to the origin of a large earthquake (1 km
For a small source ( l =10 and 100 m) the field is detectable at distances close to the source,
as seen
in Figure 6.
The graphs for
l =1 km and
l =10 km correspond to co-seismic effects. These are negligibly
small, which is consistent with their absence in electrotelluric field measurements
far from the epicenter.
Here the term "co-seismic'' refers to EM signals generated at the focal area during
the main shock, and not to
seismic waves.
Consideration of the possible physical mechanisms for electrotelluric anomalies leads
to the
conclusion that the electro-kinetic effect is the most appropriate candidate.
A few observations of anomalous electromagnetic emission in the ultra-low frequency
(ULF)
range (10-2 -10 Hz) have been reported. In 1964 two articles
appeared dealing with the detection of
anomalous magnetic impulses before an earthquake
[Breiner, 1964;
Moore, 1964].
The duration of these impulses indicated the disturbance was in the ULF range. The
report by
Fraezer-Smith et al., [1990]
stimulated more interest in this field. This article reported ULF disturbances in
the geomagnetic field before
and after the magnitude 7.1 Loma Prieta earthquake. These disturbances appeared more
than one month
before the quake and were detected over several frequency channels. The most intense
disturbance, in the
range 0.1 to 0.2 Hz, began approximately three hours before the quake and exceeded
background by two
orders of magnitude. Analysis of the data indicated that it was not due to magnetic
fluctuations in the earth's
upper atmosphere nor to quake-induced detector motion. The authors concluded that
the intense ULF
disturbance was probably a magnetic precursor. Anomalies in the ULF range have also
been reported in
other seismically active regions
[Fujinawa and Takahashi, 1998;
Hayakawa et al., 1996;
Kopytenko et al., 1993].
Several models have been developed to explain the geomagnetic phenomena associated
with the
Loma Prieta quake. The first was based on the induction effect
[Draganov et al., 1991].
However, it was shown by others that this model requires unrealistic conditions,
such as
very large changes in the volume strain and a pore water pressure exceeding lithostatic
values
[Fenoglio et al., 1995;
Gershenzon and Gokhberg, 1994].
On the other hand, using the same parameters as in the induction model, the electro-kinetic
effect
has been shown in Section 3 and in the paper by
Gershenzon and Gokhberg [1994]
to give a magnetic disturbance three to five orders of magnitude greater than the
induction effect.
The model using the electro-kinetic effect was developed by
Gershenzon and Gokhberg [1994].
It was shown that multiple cracks appearing in a sub-surface layer could account
for the observed effect. This
model was able to explain the frequency dependence as well as the magnitude.
Fenoglio et al., [1995]
presented a model, also based on the electro-kinetic effect, with a more developed
mechanical component,
i.e. they showed that the source of the pore pressure gradient could be "failure
of faults containing sealed
compartments with pore pressures ranging from hydrostatic to lithostatic levels''.
A similar idea was
proposed earlier by Bernard [1992]. The pressure gradient
in this case would be much larger than for the
changes in the volume strain considered by
Gershenzon and Gokhberg [1994].
The model of Fenoglio et al. [1995]
could account for the observed effects at great distances from the mechanical source.
Merzer and Klemperer [1997]
proposed a model based on local conductivity changes in the earth's crust. In the
presence
of normal external geomagnetic variation, conductivity changes in the crust could
lead to local changes in
the magnitude of the geomagnetic variation. However, the proposed geometry of the
region necessary to
account for the effect, i.e. an infinite horizontal elliptic cylinder of diameter
several kilometers and
instantaneous changes in the conductivity of several orders of magnitude, extending
throughout this region,
seems to us to be unrealistic. If the region of high conductivity were not infinite,
the electric field arising in
this region, and the resulting geomagnetic disturbance, would be much less.
Molchanov and Hayakawa [1998]
developed a model based on the formation of multiple cracks in the focal area during
the earthquake
formation stage. Since crack formation is accompanied by electric charge separation
(due to various
mechanisms) an EM field will appear. The low frequency part of this field could reach
the surface. We find
that the energy associated with charge separation during crack formation is distributed
over a broad
frequency range and only a small part occurs in the ULF region. Our model also considers
the formation of
multiple cracks (not necessarily restricted to the focal area), and we find that
the electro-kinetic effect is a
mechanism which provides considerably more energy in the ULF range, because it is
controlled by the
diffusion of water with a diffusion time comparable to the period of ULF emissions.
where the active region during the acquisition time
DT is assumed to be a sphere of radius
r.
Figure 7
shows the variation of
Bm with distance
R from the source, for four different combinations of crack size and
frequency. It is known that the crack size defines the spectral density of the signal
through the diffusion
relaxation time
DTD and seismic impulse
life time
t. From Figure 7 we see that a 1m crack could account for
the disturbance in the 5 to 10 Hz channel associated with the Loma Prieta quake,
and a 20 m crack could
account for the 0.01 to 0.02 Hz channel (see
Gershenzon and Gokhberg [1994]
for a more extensive discussion). In summary, Figure 7 shows that the Loma Prieta
data could be accounted for by a local source lying within a water saturated layer
and located within 1 km of the detector.
Over thirty years ago, Vorobyev proposed that earthquakes are the result of an electromagnetic
storm within the crust. Although this hypothesis must be rejected on theoretical
grounds, since the electrical
conductivity in the crust is too high for charge separation to occur over the large
distances required,
experiments to detect RF emission at the surface before and during a quake have been
performed for several
years in Uzbekhistan by Vorobyev and collaborators
[Mavlyanov et al., 1979;
Vorobyev et al., 1975, 1976].
The same type of experiments were also conducted in Carpathia
[Sadovsky et al., 1979] and the Caucasus
[Gokhberg et al., 1979].
A Russo-Japanese collaboration resulted in further experiments in Japan
[Gokhberg et al., 1982].
The authors reported intense electromagnetic emission (EME) at 81 kHz beginning
1.5 hours before a major
quake (magnitude 6.1). The EME activity increased steadily with time until the main
shock occurred, when
it decreased abruptly. Twenty minutes later a large aftershock occurred and the EME
activity decreased
abruptly again, falling to the background level. This experiment initiated similar
experiments in the RF
range by several groups in various countries. One group in Japan reported twenty
such occurrences over a
two-year period
[Oike and Yamada, 1994;
Oike and Ogawa, 1986].
This group used a frequency of 163 kHz because of the low background activity
at this
frequency at their location. The associated quakes were all
centered in Japan, had magnitudes greater than 6.0, and the EME was from four to
twelve times background.
Some evidence of EM emission relating to earthquakes in America also have been reported
[Tate and Daily, 1989;
Warwick et al., 1982].
Anomalies at 3 and 10 kHz and 41 and 54 MHz were observed before several
earthquakes in Greece
[Eftaxias et al., 2000, 2001;
Varotsos et al., 1996b].
Further references to field experiments in the RF range may be found in
Gokhberg et al. [1995].
It is difficult to compare the results of the several groups working in this area
because of
differences in detectors, frequency ranges, and discrimination techniques. Nevertheless
there are some
common morphological features of the phenomenon:
EME noise in the RF range can have several sources, some natural (e.g. atmospheric
lightning)
and some man-made (e.g. commercial radio broadcasting, industrial activities). These
sources are all hard to
eliminate, because multiple reflections from the ground and the ionosphere can cause
a signal to propagate
over large distances.
We now discuss possible models for EME anomalies. The simplest model identifies the
EME
source with the origin of the quake, and requires a mechano-electromagnetic transducer
of some kind which
is active only during the precursor period. A serious flaw in this model is that
it requires an attenuation
length for RF signals which is on the order of the typical depths at which quakes
originate. It is easy to show
that the attenuation length in the crust is much less than this.
Another model was developed by Gokhberg et al. [1985].
It involves the creation of a large-scale
electrical current, of very low frequency, which extends to the earth's surface.
The electric field associated
with this current loop, though weak, could extend into the atmosphere and produce
a lightning discharge at
high altitude, where the ionization potential becomes small. The lightning discharge
could be a source of
secondary emission in the RF range. The idea that atmospheric processes could be
a source of EME has also
been considered by Fujinawa et al. [1997].
Sadovsky et al. [1979] suggested that electromagnetic
anomalies were related to changes in the
electrical characteristics of the environment about the measurement point. These
changes could be related to
pre-seismic processes, while the source of the electromagnetic fields could simply
be background noise.
Malyshkov et al. [1998] proposed a model in which the
earth's crust constantly emits EM signals
associated with background elastic waves. These signals would normally be part of
the background EM
noise, but a change in these signals, i.e. an EME anomaly, could occur when an elastic
wave propagates
through the altered mechanical state of an earthquake preparation region (i.e. an
altered stress state or an
altered density of microcracks). However this model is based on laboratory experiments
in the ultrasound
range (kHz to MHz). It is difficult to see how an EM signal in the RF range could
be produced by the low-frequency
seismic waves (10-2 to 10 Hz) observed in field experiments.
From our viewpoint, the model developed here provides a natural framework for discussing
the
above RF phenomena. We will calculate the magnitude of the anomalies and discuss
the circumstances
under which they can be detected. But first we consider two different techniques
which are applied in field
experiments for monitoring SEM anomalies in the RF range. The first technique (simple
averaging) is the
usual one, in which all signals received during the acquisition window
DT are averaged, and leads to
equations (26) and (28). The second technique (which we shall call impulse averaging)
is based on the supposition
that the useful signals (as opposed to noise) are pulsed rather than continuous.
In this technique only filtered
signals above a present threshold are considered. In this case, the time between
impulses (the "dead time'')
does not influence the measurement. This technique corresponds to equations (27)
and (29) when
DT is large enough to count one pulse or
a cluster of closely spaced pulses but small enough to exclude the dead time.
As we mentioned at the end of Section 4, the ratio of
Em (or
Bm ) calculated using pulse averaging to
Em (or
Bm ) using simple averaging is of order
(DfDT)1/2,
which can be very large for reasonable
values of
Df and
DT (here
DT is the acquisition time for simple averaging).
This means that impulse averaging can be much
more sensitive than simple averaging in detecting RF anomalies, in agreement with
RF field experiments
[Gokhberg et al., 1986]. In the remainder of this section
we shall present results based on the impulse
averaging technique.
By use of the equations and figures presented here, we see that our model can explain
the
appearance of RF anomalies of order 10-100
m V/m under the following conditions.
This model cannot explain reported anomalies as large as 100 mV/m
[Mavlyanov et al., 1979;
Sadovsky et al., 1979;
Vorobyev et al., 1975, 1976].
Such anomalies may be related to atmospheric phenomena (lighting activity)
connected with pre-seismic processes
[Fujinawa et al., 1997].
A. Major seismo-electromagnetic phenomena can be described by the following
model. The final phase of
the pre-earthquake process is accompanied by the formation of multiple cracks. Cracks
appear not only in
the focal area but also in a neighborhood. This occurs because regional geodynamic
processes are connected
by the global shear stress. The appearance of a crack creates, in its neighborhood,
a mechanical disturbance
over a broad frequency range. In general, the spectral density of the disturbance
consists of two parts: a
zero-frequency spike related to a change in the residual strain, and a broad region
from zero up to the MHz
range, related to an impulse-like process. In a crust saturated with water (the usual
case), a crack will also
cause a change in the pore water pressure, and the spectral density of this response
is much less broad,
extending from zero up to the range of kHz. Localized mechanical or pore pressure
changes give rise to
electromagnetic emission by a variety of mechano-electromagnetic transducer mechanisms.
The emission
will have a spectral density which spans the same range as its source.
B. The major known mechano-electromagnetic mechanisms
which may be applied to the earth's crust,
namely piezo-magnetic, piezo-electric, electro-kinetic and induction, have been considered
and are
compared in Table 1. From this table one can see that a piezo-magnetic source
has strength comparable to
that of a pure quartz piezo-electric source. In the case of high conductivity in
the earth's crust, the strength
of an electro-kinetic source and a piezo-magnetic source are also comparable. The
magnitude of the
induction effect is usually very small compared to the other three, but it could
be comparable to them for a
source of size 1 km or more.
C. Since the dimensions of the EM source are, in most real cases, much smaller
than the distance from
source to detector, all sources are represented by a magnetic or electric dipole.
Formulas 21 and 22 and
Table 2
express the magnitude of the electric and magnetic dipoles in terms of the parameters
of both the
earth's crust and the mechanical disturbances. The mechanical parameters important
in calculations, namely
strain spectral densities
e(w, r)
and
q(w, r),
pore water pressure spectral
density
P (w, r), crack density
n, and average shear and volume strain changes
e and
q, are given in equations 1c, 3a, 4, 5 and
5a. The relation between volume strain and pore water pressure is
given by equation A9.
Expressions 18(a-d) can be used for calculating the magnetic and electric fields
in the so-called static zone
and expressions B1-B32 in the near, intermediate and far zones for
s/we>1.
When
s/we<1,
expressions 19(a-d) and 20(a-d) can be used in the
near and far zones, respectively. In order to connect the "detected''
(filtered and averaged) EM field to the real field one uses expressions 26, 28 and
27, 29.
Collectively, all the above formulas represent the necessary relationships for estimating
the measured EM
field, at the detector, for a wide range of frequencies and at various distances
from the source. Through
them, the physical parameters of both the earth's crust and of a localized disturbance
are connected to the
measured EM field.
D. The magnitude and morphological features of major SEM phenomena have been
interpreted on the basis
of the model developed here.
1. Tectonomagnetic anomalies can be described either by the piezo-magnetic or
electro-kinetic effect. In
rocks containing titano-magnetite, residual strain as a result of the cumulative
action of crack formation in a
localized area can produce magnetic field variations of the order of nT. The same
order of magnetic
variation can be produced in water-saturated rocks of high electrical conductivity
(s
2. Of the mechanisms considered in this paper, only the electro-kinetic mechanism
can explain the main
features of electrotelluric field anomalies, namely duration, magnitude, and high
degree of selectivity.
3. Geomagnetic variation in the ULF range may be explained on the basis of the
electro-kinetic effect,
when the detector is located about a water-saturated layer which has a comparatively
high conductivity.
While the piezo-magnetic effect could also contribute to the variation, the electro-kinetic
effect is a more
likely mechanism. The main reason for this is the difference in the spectral density
of the mechanical
disturbances associated with each. The energy of a piezo-magnetic source is distributed
widely from zero up
to radio frequencies. The electro-kinetic source is usually much narrower, with correspondingly
much more
energy in the ULF range.
4. The most powerful source of EME in the RF range is the piezo-electric effect
due to the presence of
quartz grains in the crust. The magnitude of the piezo-magnetic effect in the RF
range is 2-3 orders of
magnitude less.
E. Calculations based on the model presented here show that the source of all
types of EM anomalies
considered here should be local, i.e. close to the detector but not necessarily
in the focal region, in order to
be observed. The maximum distance from source to detector depends on the type of
anomaly and on the
detector, and can range from several hundred meters to several kilometers. One of
the best experimental
confirmations of this statement is the fact that there are no SEMS during earthquakes
(excluding co-seismic
signals accompanying seismic waves). An earthquake itself is a huge mechanical disturbance,
much bigger
than the disturbances we expect during the pre-seismic time. So it should, and probably
does, produce large
SEMS from all the mechanisms we have discussed. But the magnitude of these signals
decreases as an
inverse power of the distance and at 10 km or more from the focal region it
should be negligible in most
cases. That is why "global'' models, which suppose that the source of SEM anomalies
is at the earthquake
origin, have to introduce some additional (and sometimes unrealistic) suppositions
to explain how the signal
can still be detected at several hundred kilometers from the focus. Even with these
suppositions, the
experimental fact is that no SEMS are observed during a quake.
F. Some general recommendations for field experiments can be made based on the
model described here.
Since all sources should be close to the detector to be observed, the placement of
detectors is critical and
depends on the frequency range and on whether an electric or magnetic measurement
will be made. For
example, the nearby presence of rock containing titano-magnetite is required for
monitoring tectono-magnetic
variation. The existence of a nearby water-saturated layer provides a necessary condition
for
electrotelluric anomalies as well as geomagnetic variation in the ULF and quasi-static
ranges. The best
condition for the appearance of anomalies in the RF range is the nearby presence
of quartzite or granitic rock
and, for their detection, a low crustal conductivity about the detector is also necessary.
Because all sources
become active only under a mechanical disturbance, the presence of an active geophysical
structure such as a
fault is necessary.
Since magnetic variation from an electro-kinetic source can apear only in the presence
of a vertical
inhomogeneity, the detector should be placed close to it. For monitoring an electrotelluric
anomaly the best
setting of the electrodes is across the vertical plane defining the inhomogeneity.
The anomaly is enhanced if
the water table is close to the surface, provided the electrode separation is not
much less than the depth of the
table. An electrotelluric anomaly can also be observed without a vertical inhomogeneity
if the electrodes are
displaced vertically, with one above the water table and the other below it.
Since ULF and RF anomalies are expected to have a pulse-like character, “impulse
averaging” is
the best technique for monitoring them. This technique avoids the “dead time” between
impulses, and
requires a small acquisition time. On the other hand, the acquisition time should
be large enough to record
the entire impulse (or a cluster of multiple impulses). The choice of optimal threshold
value is also
important for this type of averaging, since it should be low enough to record small
impulses but high enough
for a good signal-to-noise ratio. Following the above recommendations would involve
investigating, at the
detector location, both the composition of the crust and the noise level in the frequency
range of interest.
G. Since our results indicate that SEMS arise from a local source, the usual
triangulation technique cannot be
used to locate a distant seismic focus. At the heart of our model, however, earthquakes
and SEMS are
connected by a global stress field. This connectedness means that the possibility
still exists for locating a
seismic focus using SES. One way is by statistical analysis of these two types of
events (SES and
earthquakes) using a detector network
[Varotsos et al., 1996a, 1996b].
Such an analysis may allow one to establish a connection between a group of SES measurement
points and a group of seismic areas.
H. EM emission is a secondary effect of local changes in the stress field of
the crust. The question arises,
why not measure the stress changes directly? After all, it is known that the latter
can be measured with more
accuracy. There is at least a twofold answer to why measurements of EME can be useful
and can give
additional (and sometimes the only) information about stress changes. First, EME
measurements can give
information about remote stress field changes, while direct stress field measurement
give information only at
the immediate vicinity of the detector. Second, the latter type of measurement is
usually more expensive than
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s
10-1/W-m.
Figure 3
shows the dependence of
rA, rB and
rC on frequency for three
different values of
s. We are interested in distance less than 1000 km.
From the figure we see that the Banos formulas cannot be used in the quasi-static
regime ( f<10-2 Hz) except for large conductivities
( s>10-1/W-m.)
and for distances
r>10 km, since
r must be greater than
rA and here
rA=10 km for
s=10-1/W-m
and
f=10-2 Hz. In the ultra-low
frequency (ULF) range ( 10-2 Hz
rB
rC, the displacement
current in the crust approaches the conductive current, and the Banos formulas
no longer apply.
e-(w/wh)1/2.
wh
the attenuation will make the fields negligible at the
interface. For
w
wh
there is no appreciable attenuation and
the radiation propagates along the surface away from the hypocenter as an
inverse power of the distance.
B. Asymptotic Formulas for
r > |(w2me + iwms)-1|
h. In this zero-frequency
approximation, any electric field produced by the magnetic dipole is ignored.
(18a) (18b) (18c) (18d) C. Asymptotic Formulas for w > s/e
(19a) (19b) (19c) (19d) (20a) (20b) (20c) (20d) D. Spectral Density of Electric and Magnetic Dipoles
(21a) (21a') (21b) (21b') (21c) (21c') (22a)
Figure 4a shows
M
Figure 4
(w)
for various value of
L. Curve 1 ( L=lc=1 mm)
is the contribution of only the microcrack and corresponds to the first term in equation 22a.
It consists of a zero-frequency spike (not shown in the figure because of the
logarithmic scale) and a broad, flat spectrum extending from zero up to
wc=(Dt)-1,
with magnitude
MMp1/2Dt. The
contribution of the impulse (curves 2-4) increases the spectral
density magnitude by the factor
(L/lc)2 and shifts the spectrum to a
lower
frequency range
(0<w<w imp
t-1).
(22b) (w)
for several values of
L, including
the cases
L=lc=0.1 m (Figure 4b) and
L=lc=10 m (Figure 4c). In both
cases, curve 0 represents the contribution of the crack itself and
corresponds to the first term in equation 22b. Including
the diffusion of
pore water (third term) results in curve 1. Including the contribution of the
impulse (second term) results in curves 2-4. The impulse contribution
scales as ( L/lc ). From these figures we see that the contribution
of the
diffusion and impulse terms exceeds that of the crack itself at low frequencies.
For a small crack (Figure 4b) the impulse contribution exceeds the
diffusion contribution, but for a large crack (Figure 4c) the diffusion term
dominates.
(22c) (22d) wh, attenuation is very
small, as noted above.
In particular, the shape of the radiation pulse observed at the surface will be
similar to that of the dipole pulse and, therefore, of the mechanical
pulse producing it. Its intensity will fall off as some inverse power of the distance
from the hypocenter.
E. Magnitude of Magnetic and Electric Dipole Moments
Now let us determine the electric and/or magnetic dipole associated with various
mechanical
sources. To do this we need to assume that the strain tensor is a given function
of space and time. This
tensor can always be represented as the sum of a pure volume strain (diagonal components
only) and a pure
shear strain (off-diagonal components only). For example, in an isotropic medium,
Figure 5
[H(y
+ lc/2) -
[H(z
+ lc/2) - H(z - lc/2)],
,
b
,c
), 17(a-d), 10-14
and A9), the magnetic and electric
dipoles for various sources, after integrating over the spatial coordinates, are
presented in Table 2 in terms of the parameters of the mechanical disturbance.
F. Relation Between Measured and Actual Fields at the Detector
(23) (24) (25) f0.
This is the usual case experimentally.
w0; thus we expect the spectral
density
E(w) to be a slowly varying function of
w in the neighborhood of
w0.
This will be used in integrating equation 25.
Dw
w0,
while in the second case
Dw
(DT)-1<w0. In the first
case, integration of equation 25 yields
(26a) (27a) Dt there are
N electric impulses
randomly distributed in time which appear at the measurement point. Then it is easy
to show that the average field from
N equal impulses scales as
N1/2, as a
consequence of their random nature. In this case equations 26a and 27a yield
(26) (27) (28) (29) DT, there is a considerable amount of
"dead time'', during which the detector receives no signal. This reduces the
average signal considerably. In equations (27) and (29) we suppose that
Dw
(DT)-1<w0, which means that the detector is receiving
a signal
throughout its acquisition time. This is why
DT does not appear in equations 27
and 29 and a much lager signal is measured (for an impulse source).
5. Interpretation of Field Experiments
A. Tectonomagnetic Anomalies
(30) (31) (32) 1 )
have a
higher probability of being observed compared to distant sources ( r/R ranging
from 1/10 to 1/30). We expect this statement to be valid
even when the density of cracks in nearby regions is low (small
a or
e )
compared, for example, to the high crack density near the focal point of a distant
quake. Estimating the strain
e to be 10
-4 to 10
-5 ,
setting
r/R=1, and using values of
ms, c||
and
I from Tables A1
and 1, equation 32 gives
B
0.3-3 nT, which is comparable
to the observed data.
(33) B. Electrotelluric Anomalies
(34)
This model explains in a natural way the strong selectivity of electrotelluric anomalies.
associated
with an
electro-kinetic source, we have
Figure 6
(35) 10 km), equation 35 gives a field at
a distance of 100 km from the
source which is three to five orders of magnitude less than the reported anomalies.
Incorporating the effects
of inhomogeneities in the conductivity in the crust could increase the field by several
times, but it is highly
unlikely to increase it by three to five orders of magnitude.
C. Electromagnetic Emission in the Ultra-Low Frequency
Range
So from the point of view of our model, geomagnetic anomalies in the ULF range could
appear as
the result of the action of multiple cracks via the electro-kinetic effect. Consider,
for example, a water-
saturated crustal layer existing underneath the detector, with a number
N of cracks appearing in the layer due
to pre-seismic processes. The magnitude of the ULF field,
Bm, could be calculated
using equations 28, 19b and 18c (with
h=y=0 ), Table 2 (with
l=lc ), and with
Figure 7
D. Electromagnetic Emission in the RF Range
Figure 8 contains the results of calculations of
Em (equation 27a) in the neighborhood of
f =10
5 Hz generated by one microcrack, due to piezo-electric (Figure 8a) and
piezo-magnetic (Figure 8b) sources. From the figure we see that
the magnitude of
Em from a piezo-electric source is several orders greater
than that from a
piezo-magnetic source. This would seem to contradict the result of Section 3,
where
it was shown that the strengths of these two types of source are comparable. Actually
there is no contradiction. The energy in a piezo-magnetic source is distributed over
a
much broader frequency range than that for a piezo-electric
source (see Figures 4a and 4d). For the high end of the RF range, this means that
a piezo-electric source gives
a much stronger signal, while at the low end the contribution of a piezo-magnetic
source could dominate.
Figure 8
From laboratory experiments and calculations it is apparent that the emission of
one microcrack
with typical size 1mm produces a very weak signal at distances greater than several
meters (see Figure 8).
From the standpoint of a field experiment this means that emission from a single
microcrack will be below
the threshold of the detector. However during the failure of a large portion of rock
(i.e. larger then typical
grain sizes) a large number of microcracks are formed in a short period of time.
Their combined effect could
produce a signal large enough to exceed detector threshold. Figure 9 shows the result of calculating
Em using
equation 27, based on a piezo-electric source. In using equation 27 we
suppose that during the acquisition time
N microcracks appear, where
N=ar3
n max and the failure region is assumed to be
a cube of side
r. From this figure we see that, at a given distance
R, Em increases with decreasing conductivity
s up to a value corresponding
to
s0=we. For
s<s0,
Em is independent of
s.
The quantity
Em has an inverse power law dependence on
R; this power law changes from
zone to zone. For
s>s0
(cf. Figure 2b)
the dependence is
R-3 for the
static and near zones,
R-1 for the intermediate zone, and
R-2 for the far zone.
For
s<s0
(cf. Figure 2c)
the power law is
R-3 for the near zone
and
R-1 for the far zone.
Figure 9
s0
10-4(W
m)-1 for
f =100 kHz).
Figure 10 shows
Em calculated for a 1 km source (a typical size for
an earthquake focal region). At
distances greater than 10 km, the field is virtually undetectable. This means
that processes at the focus
cannot be responsible for RF anomalies occurring at a large distance from it.
Figure 10
6. Discussion and Conclusions
10-2/W-m) by changes in pore water pressure. These
two mechanisms can be distinguished by the temporal behavior of the
magnetic variation (step-function behavior corresponding to the piezo-magnetic mechanism
and unipolar
behavior corresponding to the electro-kinetic mechanism).
Acknowledgments
The authors would like to thank their colleagues Paul J. Wolfe, Kostas Eftaxias
and Nikos Bogris
for several helpful conversations. Special thanks go to Ms. Barbara O'Brien for typing
the manuscript. This
research was supported in part by NATO Grant CRG970028.
References
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