The homogenous half-space model can be illustrated by setting either the conducting sheet depth to d=0 or σ= σ₀, in Fig. 1. The vertical magnetic dipole source (small horizontal loop) has a magnetic moment IA and is located at z=-h on the z axis of a cylindrical coordinate system (ρ, ϕ, z), where I is the current through the loop wire and A is the area formed by the circular loop. The earth media has a conductivity of σ₀ and an intrinsic propagation constant of γ0=μω(jσ0−ωεe), where j is the square root of -1, ω the operating angular frequency, μ the permeability of the earth or air, and ε_e the earth dielectric constant. The displacement currents are usually very small and will be neglected for all TTE frequencies compared to the conduction currents, so approximation γ0≈jωμσ0 is reasonable. The vertical H-field at the surface can be derived from a magnetic Hertz vector with only a z component, along with the application of appropriate boundary conditions [8]. It can also be obtained by setting the conductivities σ₁ and σ₂ both equal to σ, or the thickness h₁ of the upper layer can be considered to vanish and σ₂ = σ in a 2-layer model [8]. This gives:

where

and

In the equations above, b₀=IA/(2πh³) is the value of H-field on the axis a distance h above a loop in free space, Q_homo(ω, σ₀, h) represents the attenuation factor due to the conductive earth. J₀ is the first order of Bessel function of the first kind, and k₀ = (λ² + γ₀²)^1/2.

As we will see later, one result from applying the homogeneous model to the data of the 94 mines is that the apparent conductivity appears to decrease as the mine depth increases, which is contradictory to the model itself. However, this dependency might be explained by the presence of a thin highly conducting layer at the surface of the earth. In the thin sheet model, as depicted in Fig. 1, the highly conducting thin sheet represents the overall effect of possible surface metal structures, such as cables, pipes, cased bore holes, etc., as well as the relatively high conductivity at the surface which usually contains more dissolved salt and mineral substances. Note that d≪h. Wait and Spies have derived the H-field from the electric vector potential [8]. Following their work, here we use magnetic Hertzian potential to obtain the vertical H-field in the air based on potential theory.

For the earth layer (z<0) and free space (z>0), the magnetic Hertzian potential Π* satisfies the wave equation except at the exciting source:

where

is the intrinsic propagation constant. For free space (z>0), σ₁ = 0, hence, γ₁ ≈ 0.

In a cylindrical coordinate system, the fields in the half-space (z<0) can be expressed in terms of Hertzian potential Π0∗, in which 0 denotes the medium 0 (earth):

and the fields in the free space (z>0) can be expressed in terms of Π1∗:

Note that the magnetic Hertzian potential Π* has only a z component; i.e., Π* = Π*ẑ. The magnetic Hertzian potential in each region is listed below:

R₀(λ) and T₁(λ) are unknowns and can be determined by the application of the boundary conditions. The boundary condition here requires that at the layer interface the azimuthal E-field is continuous and the tangential H-field is discontinuous by the amount of longitudinal current per unit length carried by the thin sheet:

The surface vertical H-field in (7) then is given by:

in which

For free space, k₁ = (λ² + γ₁²)^1/2 = λ. Equation (13) then can be rewritten as:

To avoid exponential attenuation when the signal passes through the highly conducting thin sheet, the value of γ₀d in this model is required to be small enough (γ₀d < 1). Similarly as in (1), Q_thin(ω, σ₀, h, σ, d) here represents the attenuation factor due to the conductive earth and the thin sheet. An interesting feature of the attenuation factor Q_thin(ω, σ₀, h, σ, d) in (14), is that the dependence on σd is algebraic rather than exponential. This type of algebraic dependence is typical of thin conducting sheets regardless of the geometry [8].

Since Q in (1) and (12) monotonically decreases with σ₀ and/or σ, an apparent conductivity value σ_a can be determined by assuming reasonable input values for the homogeneous half-space and thin sheet model. The procedure is to compute Q_thin(ω, σ₀, h, σ, d) using reasonable values of σ₀ and σd, and then equate the magnitude of Q_thin to that of a homogeneous half-space and determine the value of σ_a:

For the TTE tests at the 94 mines, four transmission frequencies-630, 1050, 1950, and 3030 Hz-were used at each site. The magnetic moment, M=NIA (N is the number of turns of wire), for the in-mine transmitting loop was recorded and calibrated. Corrections have also been made on the overburden depth h for all the mine sites to account for possible horizontal offsets from the point directly above the transmitters [7]. After normalizing the surface vertical H-field to the corresponding M, the apparent earth conductivity, σ_a, can then be obtained by solving (1).

The resulting apparent earth conductivity distribution with overburden depth interval is listed in Table 2. The computed conductivity values for the mine with the least overburden depth appear to be a large outlier compared to the rest of the data and were excluded from further consideration. Also, because of the large expected uncertainty in the conductivity estimates for large Q, the data for about 25% of the 94 mines in which |Q|>0.5 were excluded. By examining Table 2, we can see that the estimated conductivity values tend to decrease with increasing depth by a factor of 25-30 over the depth range at each given frequency. This number is in contrast to the factor of 10 obtained by Durkin based on the data of 27 mines [10]. The estimated apparent conductivity in Table 2 also shows a dependence on frequency that decreases by a factor of ∼3 over the frequency range at each given depth interval. Analyzing only 27 out of 94 mine data in [10], does not quite predict the magnitude of decrease in the apparent conductivity with depth and frequency. However, we would not expect the conductivity to decrease with increasing frequency or depth for the homogenous model. Another observation based on Table 2, is that the standard deviation of the conductivity distribution in each depth interval tends to decrease with overburden depth for all frequencies. This suggests that deep coal mines have smaller and more evenly distributed conductivity values than shallow coal mines.

It may be possible to explain the behavior of the conductivity on frequency and depth as seen in the previous section by the addition of a thin, highly conducting layer at the surface of the earth [11]. A shallow mine would then have a more weighted contribution from the high conducting surface layer than a deep mine, and the apparent conductivity would decrease with greater depth. The estimated apparent conductivities of those mines with |Q|<0.5 based on the thin sheet model are calculated and sorted into several overburden depth intervals, and then plotted with depth interval as in Fig. 2 for various σd values and various frequencies. As mentioned earlier, although the conductivity of the thin sheet can be very high, choosing of product value σd is not arbitrary. The value of γ₀d, hence the value of d, is required to be small enough to avoid exponential attenuation as the signal travels through the highly conducting thin sheet. One interesting finding is that the apparent conductivity has a greater dependency on the overburden depth at low frequency than at high frequency, as shown in Fig. 2; i.e., the derivative of the exponential curve fit has a greater value at low overburden depth than at greater depths. The exponential curve-fitting equations in these plots provide a good prediction for the depth dependency of the conductivity. This model also predicts the magnitude of decrease in the apparent conductivity with frequency as seen in Table 2, by the appropriate choice of value of σd.

While EM measurements of both transmitting and receiving antennas are needed for the models described above, the Q-factor model or attenuation factor model requires only two parameters: the estimated earth conductivity σ₀ and the conductivity thickness product σd of the thin sheet. Through the use of (15), we can calculate the apparent conductivity. The apparent conductivity values based on this model are plotted in Fig. 3. Again the value of γ₀d, hence the value of d, is required to be small enough for the reason mentioned in the thin sheet model above. By comparing the exponential coefficients in the fitting functions as shown in Figs 2 and 3, the depth dependency of this model when σd = 20 is very close to that of the thin sheet model. Furthermore, this approach can predict the dependency of apparent conductivity on frequency, as well as on depth, by choosing the value of σd appropriately.

The conductivity of overburdens above mines in U.S. coal fields can be characterized as a function of overburden depth and operating frequency. The overburdens consist of a large number of horizontal layers of different materials and thicknesses. For any given overburden depth, we can expect overburden characteristics such as conductivity to vary from location to location within the coal fields. Hence, we can develop a statistical approach of sampling a representative number of mines within each of the depth intervals of interest in order to characterize the overburden conductivity, and the corresponding variability about the average, as a function of depth and operating frequency.

In the regression model, overburden apparent conductivity is considered to be related to depth and frequency in an unknown pattern. Up to 94 data points were obtained as a result of field tests conducted at each of four frequency levels. With the assumption that the estimated values of apparent conductivity represent a random sample from a normal distribution with a mean dependent upon both frequency and depth and variance independent of both frequency and depth, a regression model can be obtained to describe the dependency of the apparent conductivity on both frequency and depth as shown in (17). In this model, σ_a is the apparent conductivity to be estimated, and a, b, and c are the regression coefficients to be determined from the mine data. The model values are given in Table 3. The random mine selection process was used to ensure that all measurements can be described by a log-normal probability law. The Cumulative Distribution Function (CDF) of the estimated apparent conductivity (based on the homogenous model) for all frequencies and depths is shown in Fig. 4. From the CDF plot, about 95% of conductivity values fall below 0.5 S/m, and about 60% of them fall below 0.1 S/m. It is worthy to mention that since the values in Table 3 (including standard error) is averaged over all the frequencies and overburden depths (as shown in Table 2), this regression model predicts apparent conductivity more reasonably for low frequencies and shallow mines than for high frequencies and deep mines:

In the regression model as described in (17), the coefficient of frequency is of the same order as that of the depth, so the frequency dependency cannot be ignored in the apparent conductivity estimation in contrast to previous analyses [10].

The apparent conductivity of a specific mine with given overburden depth and operating frequency can then be estimated based on this model, with the related coefficients to fit (17) as shown in Table 3.