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., J^{0}_{} J^{0} dV, etc. We also have
(17c) |
and
(17d) |
As expressed in equation 17d, we can always consider J^{0} as the sum of two contributions, a rotational part J^{0}_{ rot} associated with magnetic dipoles and an irrotational part J^{0}_{ 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=3e_{0}
and
m=m_{0}
for
the crust, where
e_{0} and
m_{0} 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 e^{ik1h}, which includes the attenuating effect of the crust. Introducing the parameter
the exponential can be written,
For ww_{h} the attenuation will make the fields negligible at the interface. For ww_{h} there is no appreciable attenuation and the radiation propagates along the surface away from the hypocenter as an inverse power of the distance.
For distances
r>r_{A}, 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
r_{A} exceeds the range of interest. Even
at higher frequencies, when
r_{A} becomes small, we need a formula for
r
For a static magnetic dipole, we have E=0 and
(18a) |
where R is the distance from the source.
For a static electric dipole embedded in a conducting half-space, we use
(18b) |
(18c) |
if the dipole is oriented horizontally along the x axis, and
(18d) |
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 r_{B} (i.e. less than the vacuum wavelength/ 2p ), the following formulas [Landau and Lifshitz, 1971]:
(19a) |
(19b) |
for an electric dipole, and
(19c) |
(19d) |
for a magnetic dipole.
For distances r greater than r_{B}, we have [Landau and Lifshitz, 1971]
(20a) |
(20b) |
for an electric dipole, and
(20c) |
(20d) |
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 piezomagnetic source, the magnetic dipole has the following form:
(21a) |
where
(21a') |
and the integral is over the volume V_{ crack} = l_{c}^{3} 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 M_{I} and P_{I} appear in place of M_{M}. The electric dipole for an electrokinetic source has the form shown in equation 21b:
(21b) |
where
(21b') |
The electrokinetic source appears as a discontinuity in the electrokinetic 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 piezoelectric source, the electric dipole can be expressed as
(21c) |
where
(21c') |
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 | P_{E}| attenuated by the factor (l_{c}/r)e^{-r/L}. The magnitudes of M_{M}, M_{I}, P_{I}, P_{K} and P_{E} will be determined later in this section.
Now we can find the associated spectral densities. For a piezomagnetic source we have,
(22a) |
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.
Figure 4 |
Using equation 21b we can find the spectral density associated with an electrokinetic source. We obtain
(22b) |
Figures 4b and 4c show P^{}(w) for several values of L, including the cases L=l_{c}=0.1 m (Figure 4b) and L=l_{c}=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/l_{c} ). 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.
Finally, we consider the spectral density associated with a piezoelectric source. The orientation of piezoelectric 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= P_{E}/P_{E}. Using equation 21c we find
(22c) |
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
(22d) |
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 4pr^{2}/l_{c}^{2}. 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 piezomagnetic and electrokinetic sources activate the low-frequency EM modes while a piezoelectric 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}w_{h}, 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.
For w_{ max}>w_{h}, which is the usual case, attenuation will be small in the frequency range 0<w<w_{h} but will be significant in the range w>w_{h}. 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.
Figure 5 |
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 ^{} , 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.
From this table we see that a piezomagnetic source produces a magnetic dipole for a unipolar pulse, while a piezoelectric source produces an electric dipole for a unipolar pulse. An electrokinetic 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
E_{m} the
mean electric field recorded by the detector, and by
E_{f} (t) the filtered electric
field at the detector at time
t ( 0
(23) |
and
(24) |
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 E_{m} as
(25) |
For simplicity we take
i.e. a frequency window of width Df = Dw/2p centered at frequency f_{0} = w_{0}/2p. We suppose that Df f_{0}. This is the usual case experimentally.
We will assume the duration Dt of the impulse to be small enough that (Dt)^{-1}w_{0}; thus we expect the spectral density E(w) to be a slowly varying function of w in the neighborhood of w_{0}. This will be used in integrating equation 25.
Let's consider two quite different cases. In the first case, (DT)^{-1}Dww_{0}, while in the second case Dw(DT)^{-1}<w_{0}. In the first case, integration of equation 25 yields
(26a) |
while in the second case we get
(27a) |
Suppose that during the time DTDt 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 N^{1/2}, as a consequence of their random nature. In this case equations 26a and 27a yield
(26) |
(27) |
The preceding discussion can be carried through for the magnetic field, with completely analogous results. The first case results in
(28) |
and the second case in
(29) |
We thus obtain two different formulas for the measured field E_{m} (or B_{m} ). If Df = Dw/2p = 10^{3} Hz and DT = 10 s, the relative magnitudes of E_{m} (or B_{m} ) 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 DtDT, 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}<w_{0}, 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).
For both E_{m} and B_{m}, 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.
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