4. The Electromagnetic Field of an Impulse Dipole

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

We consider our source to be embedded at depth h in a homogenous static conducting half-space (the crust) with conductivity s, dielectric constant e and magnetic susceptibility m. We want to find the EM fields near the air-crust interface as a function of the azimuthal angle f and the distance r from the coordinate origin O (see Figure 2a).

A. Asymptotic Formulas for r > |(w²me + iwms)^-1| and w < s/e

Figure 3

The next simplification is to consider a monochromatic source. The radiation due to a monochromatic point dipole in a conducting half-space is a classical problem first considered by Sommerfeld. The full results may be found in the monograph by Banos [1966] and we shall use them here. For points near the interface the solution has a particularly simple asymptotic form. The specific form of the solution depends on whether the field point is in the so-called near zone, intermediate zone, or far zone. These zones are characterized by distance from the hypocenter, frequency and the electromagnetic parameters of the medium. In terms of the distance parameters r_A, r_B and r_C, we have r_A r r_B in the near zone, r_B r r_C in the intermediate zone and r r_C in the far zone. The distances r_A, r_B, and r_C are defined by

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 7 Hz and the conductivity of most rocks is in the range 10^-5s10^-1/W-m. Figure 3 shows the dependence of r_A, r_B and r_C 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 r_A and here r_A=10 km for s=10^-1/W-m and f=10^-2 Hz. In the ultra-low frequency (ULF) range ( 10^-2 Hz 4 Hz, either the near zone or intermediate zone formulas will apply. For RF frequencies greater than 10⁴ Hz the near, intermediate and far zone formulas can be used for almost all cases except for low conductivities and high frequencies, i.e. when w>s/e. Then r_A r_B r_C, the displacement current in the crust approaches the conductive current, and the Banos formulas no longer apply.

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,

e^ik₁h = e^i(w/w_h)1/2 e^-(w/w_h)1/2.

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.

B. Asymptotic Formulas for r > |(w²me + iwms)^-1|

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 rA. In this situation we estimate the fields using the static or zero-frequency limit. This will be valid for frequencies whose associated skin-depth d is much greater than h: d=(2/wms)^1/2 h. In this zero-frequency approximation, any electric field produced by the magnetic dipole is ignored.

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].

C. Asymptotic Formulas for w > s/e

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.

D. Spectral Density of Electric and Magnetic Dipoles

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_c³ 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

Figure 4a shows M(w) for various value of L. Curve 1 ( L=l_c=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 w_c=(Dt)^-1, with magnitude M_Mp^1/2Dt. The contribution of the impulse (curves 2-4) increases the spectral density magnitude by the factor (L/l_c)² and shifts the spectrum to a lower frequency range (0<w<w _impt^-1).

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 4pr²/l_c². 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 _maxw_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.

E. Magnitude of Magnetic and Electric Dipole Moments

Figure 5

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,

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,

+ H(x - l_c/2))][H(y + l_c/2) -

- H(y - l_c/2)][H(z + l_c/2) - H(z - l_c/2)],

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.

F. Relation Between Measured and Actual Fields at the Detector

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 ( 0DT ), then

(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 f₀ = w₀/2p. We suppose that Df f₀. This is the usual case experimentally.

We will assume the duration Dt of the impulse to be small enough that (Dt)^-1w₀; 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.

Let's consider two quite different cases. In the first case, (DT)^-1Dww₀, while in the second case Dw(DT)^-1<w₀. 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 = 10³ 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<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).

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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