Bureau of Mines Report of lnvestigations/l984 

Apparent Earth Conductivity Over 

Coal Mines as Estimated 

From Through-the- Earth 

Electromagnetic Transmission Tests 

By John Durkin 

UNITED STATES DEPARTMENT OF THE INTERIOR 
UNITED STATES DEPARTMENT OF THE INTERIOR 

William P. Clark, Secretary 

BUREAU OF MINES 

Robert C. Horton, Director 
Library of Congress Cataloging in Publication Data: 

Durkin, John 

Apparent earth conductivity over coal mines as estimated from 

through-the-earth electromagnetic transmission tests. 

(Keport of investigations / United States Department of the Interior, Bureau of Mines ; 8869) 

Bibliography: p. 16. 

Supt. of Docs. no.: I 28.23:8869. 

1. Earth resistance-Measurement. 2. Electromagnetic wavesTransmission-Measurement. 3. Coal mines and mining-United States. 

I. Title. 11. Series: Report of investigations (United States. Bureau 

of Mines) ; 8869. 

QC809.E 15D87 1984 55 1.1'4 83-600385 

CONTENTS 

. 

Page 

Abstract ....................................................................... Introduction ................................................................... Earth conductivity measurement techniques ...................................... TTE transmission tests ......................................................... Mine selection ............................................................... Test description and data .................................................... Earth conductivity estimates ................................................. Study of conductivity data ................................................... Thin conducting sheet earth model ............................................ Regression model .............................................................. Summry ........................................................................ References ..................................................................... Appendix A.--Names and locations of mines visited during field tests ........... Appendix B.--Abbreviations and symbols used in this report ..................... ILLUSTRATIONS 

1 . Uncertainty in conductivity normalized to the uncertainty in Q for a depth 

of 200 m and a frequency of 3. 030 Hz ...................................... 2 . Cumulative distributions of conductivity estimates segregated by frequency . 3 . Geometry for a finite magnetic loop of current buried in a conducting halfspace earth containing a highly conducting thin sheet at the surface ...... 4 . Regression line and 95-pct confidence interval for log(0) versus depth ..... 5 . Surface vertical magnetic field versus depth by frequency for an underground transmitter with a magnetic moment of 1 A/m ........................ TABLES 

1 . Mine depth and physical arrangement for TTE transmission tests ............. 2 . Surface vertical magnetic signal and underground transmit moment. at different frequencies ........................................................ 3 . Q values and apparent earth conductivity values as derived from a halfspace earth model. at different frequencies ............................... 4 . Mean and standard deviation of conductivity values for each frequency within different depth intervals for estimates having Q < 0.5 ................. 5 . Conductivity and Q values for different depths and frequencies for the thin 

sheet model of the earth with ad = 20 and a1 = mho ................... 6 . Regression results for log(0) versus mine depth and transmit frequency ..... 7 . Regression results for log(0) versus mine depth ............................ 
APPARENT EARTH CONDUCTIVITY OVER COAL MINES AS ESTIMATED FROM 

THROUGH-THE-EARTH ELECTROMAGNETIC TRANSMISSION TESTS 

By John Durkin ' 

ABSTRACT 

Electromagnetic narrow-band signals were transmitted through the earth 

at 27 coal mines located throughout the United States. From those Bureau of Mines tests, apparent earth conductivity values were derived 

based upon a homogeneous half-space model of the earth. The derived 

conductivity values were found to be inversely proportional to the 

transmitted frequency and mine depth. A linear regression model relating the logarithm of the conductivity to the mine depth was formulated, 

and the results indicate that the mine depth can be an adequate predictor of the apparent earth conductivity above coal mines. Apparent 

earth conductivity was found to decrease with mine depth. 

l ~lectrical engineer, Pittsburgh Research Center, Bureau of Mines, Pittsburgh, PA. 
INTRODUCTION 

Earth conductivity (or earth resistance) has been the topic of many studies 

over the years. Earth resistance is the 

resistance of the earth to the passage of 

an electrical current. For the most 

part, the earth is not a good conductor 

of current, and in comparison with metallic conductors, it is extremely poor. 

The earth was once used as a conductor 

for power distribution, and for safety 

considerations, the earth resistance was 

a matter of considerable importance. 

Surveying the conductivity structure of 

the earth has been a frequent practice 

for different applications. This work 

has enabled the civil engineer to determine the depth of bedrock or the 

geologist to find the depth of the water 

table. Conductivity surveying has been 

found useful in detecting and locating 

ore bodies or mineral lodes and has 

recently become popular amnng treasure 

hunters in locating lost items. 

There are many factors that influence 

the conductivity of the earth; the principal ones are type of soil, moisture 

content, chemical composition and concentration of salts dissolved in the contained water, and temperature. These 

factors then give rise to earth conductivity variations both in time and 

location. Therefore, the only meaningful 

way of estimating the value of the earth 

conductivity is to measure it. 

EARTH CONDUCTIVITY MEASUREMENT TECHNIQUES 

There are numerous techniques for making earth conductivity measurements. 

These techniques can be broadly classified into three distinct approaches. The 

first approach calls for removing a 

sample of the media and making a conductivity measurement. In the second 

approach, a drillhole is made and instrumentation to make the conductivity measurement is then lowered into the hole. 

The third approach involves probing the 

earth electromagnetically with a transmitter and receiver where conductivity 

information is contained in the received 

s ignal. 

The choice of the proper approach depends mostly on the manner in which the 

conductivity information will be applied. 

In this report, the application is for 

predicting the influence of the conducting earth on a through-the-earth (TTE) 

communications system. The third approach, electromagnetic probing, appears 

to be the most closely related to this 

application. 

A number of techniques are available 

for this type of probing, including the 

well-known four-electrode method (1)2 in 

which an electrical potential is applied 

between probes on the surface, and the 

resulting potential between another 

spaced pair is observed. Other techniques employ current-carrying coils of 

wire as magnetic dipole probes (2). In 

addition, several passive techniTues are 

employed that make use of the natural 

electric and magnetic fields existing at 

the earth's surface (3). Informative 

summary texts can be found in Keller and 

Frischknect (4, chapters 4-7) or Grant 

and West (5): Excellent theoretical 

background work can be found in Wait (3). 

There is also an Institute of ~1ectri;al 

and Electronics Engineers guide that 

describes the application and interpretation of these techniques (6). - 

In addition to all of these techniques, there is another basic approach 

that is more aligned with applications 

for TTE communications. This technique 

can be described as transmission path 

lunderlined numbers in parentheses ref er to items in the list of references 

preceding the appendixes. 
attenuation tests. By transmitting a can be inferred fromthe received signal 

signal of known level and frequency from strength. This approach was taken in 

a transmitter located within the earth to this study, and the results are discussed 

a receiver located on the surface, infor- in the next section. 

mation on the signal's path conductivity 

TTE TRANSMISSION TESTS 

MINE SELECTION 

Since experimental results were intended to be representative of a large 

number of active mines, each with a 

unique physical and operational characteristic, it was necessary to utilize 

some type of sampling process to determine specific test locations for the 

program. By involving statistical sampling theory in the selection process, it 

was possible to (1) assure the validity 

of estimated performance measures subsequently derived from test results, (2) 

increase the precision of these estimates, and (3) reduce or estimate the 

possibility of built-in biases in interpreting test results. 

The choice of the mines could have been 

done on a random basis, in which case 

each mine would have an equal probability 

of being selected; however, it was decided that the sampling plan should consider two other points. First, since TTE 

Eastern Field Operations Center, which 

contained data on the maximum overburden 

depth at each mine. The compiled listing 

included the mine name, address, maximum 

overburden depth, number of miners employed, and MSHA identification number. 

All mines were subsequently strat if ied 

according to maximum mine overburden 

depth values into 15 different depth intervals. Total miners employed at all 

mines contained within each depth interval were also tabulated. Although the 

sample could have been allocated so that 

the ratio of the number of mines sampled 

to the total number of mines would be 

constant within each interval, it was 

decided to vary the sampling fraction 

based upon the first consideration above: 

that a greater variability in results was 

anticipated at greater depths. 

In determining the number of mines to 

be sampled in a given depth interval, a 

technique known as optimum allocation was 

used. This technique is based upon the 

commnications are believed to be very principle that larger samples are reeffective at small depths and less effec- quired in depth intervals that exhibit 

tive at deeper depths, the distribution greater variability. This principle can 

of mines selected should take into be expressed as follows: 

account the greater variability in test 

results anticipated at greater depths. NhSh 

Also, within each depth interval of mine nh = ws (1) 

selection, emphasis should be placed on 

selecting the mines having the largest 

number of miners employed, thus giving 

rise to a selection that covers the most 

miners possible. 

The population considered was identified from a computer listing of 1,222 

active mines in the United States. This 

computer listing was constructed by the 

Bureau of Mines Pittsburgh Research Center (PRC) from two independent computer 

data bases, one from the Mine Safety and 

Health Administration (MSHA), which contained data on the number of miners at 

each mine, and one from the Bureau's 

where nh = sample size for the h-th depth 

interval, 

Nh = total number of mines in the 

h-th depth interval, 

Sh = variance of the characteristic 

being measured in the h-th 

stratum (e.g., the estimated 

probability of successful 

transmission at a specific 

frequency), 

and n = total sample size. 
By estimating the relative variability 

Sh, the number of mines to be tested in a 

given depth interval was determined. 

Also, as stated above, the sample selection within each depth interval was based 

on the number of miners employed at each 

mine. 

The important features of the sampling 

procedure used 'in this study are summarized below. 

Each mine had a chance of being selected for this test. 

The chance (i.e., the probability of 

selection) was known beforehand and was 

based on the relative size of the mine in 

terms of miners employed. 

The selection process was random. 

All depth intervals were represented. 

Test results could be used to make 

valid inferences about all mines. 

However, as in any field test program 

of this size, what is initially hoped for 

and planned and what is actually performed are of ten two different things. 

Although a reasonable attempt was made to 

visit the mines selected, necessity, mine 

availability, and practical travel schedule constraints introduced deviations 

from the originally selected sample of 

mines. However, it is believed that 

these deviations do not significantly 

affect or bias the results derived from 

the data. Care was taken to adhere to 

the spirit and form of the original 

selection list while coping with the 

realities imposed on the practical implementation of such an extensive field measurement study. 

At the completion of the test program, 

measurements had been performed at 27 

mine sites well distributed throughout 

the United States coalfields. The names 

of the mines tested can be found in 

appendix A. 

TEST DESCRIPTION AND DATA 

The primary objective of this field 

test was to obtain a data base on the 

transmission loss incurred by a signal as 

transmitted through the earth. At each 

mine site, a minimum of two test personnel were used, one underground and one on 

the surface, positioned approximately 

above the underground individual. 

Underground, a loop of wire of known 

length, in a configuration that can be 

described loosely as being between a circle and a rectangle, was deployed around 

one or two coal pillars. The ends of 

this wire loop were attached to a small 

and lightweight electromagnetic (EM) 

narrow-band transmit ter. This transmitter produces a pulsed square wave output voltage with an on-time of 0.1 s and 

an off -time of 0.9 s, and a frequency of 

630, 1,050, 1,950 or 3,030 Hz. 

On the surface, recordings were made of 

received signals. The signal recordings 

were returned to the laboratory where 

signal levels were reduced using a fast 

Fourier transform (FFT). 

Table 1 gives the number of the 

mine tested, mine depth, underground 

antenna area, offset parameter D (which 

is equal to the ratio of the distance 

between the vertical axis of the underground antenna and the observation 

point to the mine depth), and parameter 

A (which is equal to the ratio of the 

radius of the underground antenna, with 

the assumption that the antenna may be 

represented as a circle, to the mine 

depth) . 
TABLE 1. - Mine depth and physical in depth may be larger for shallow mines 

arrangement for TTE transmission if the surface position is not near a 

tests survey point. 

Table 2 gives the mine tested, transmitter magnetic moment (M), and received 

surface vertical magnetic signal (Hz). 

The antenna current, which was used in 

calculating the magnetic moment, was measured by placing a l- resistor in 

series with the transmit antenna and measuring the voltage across it. 

The mine depths were provided by the 

mining companies when the tests were performed, This information is obtained 

from topographical maps that are very 

accurate, and the depth estimate of the 

in-mine transmitter can be obtained 

within an uncertainty of approximately 21 

pct when the surface position is near a 

survey point, However, owing to the 

natural variation in the terrain above 

coal mines, this estimate of uncertainty 

Mine 

1 ....... 

2 ....... 

3 ....... 

4 ....... 

5 ....... 

6....... 

7 ....... 

8 ....... 

9 ....... 

lo...... 

ll...... 

12...... 

13...... 

14...... 

15...... 

16...... 

17...... 

18...... 

19...... 

20...... 

21.....m 

22...... 

23...... 

24...... 

25...... 

26...... 

27...... 

The underground antenna length was measured using a conventional tape-measuring 

rule, and the estimated uncertainty in 

this measurement is expected to be much 

less than 21 pct. Using the mine maps 

coupled with the approximate deployment 

configuration and knowledge of the length 

of the loop, the underground antenna area 

was calculated, and this estimate is 

believed to be accurate within f10 pct. 

Transmit 

antenna 

area, m2 

279 

195 

929 

126 

335 

543 

418 

547 

520 

256 

522 

223 

282 

565 

28 

335 

929 

455 

130 

232 

455 

344 

238 

557 

204 

79 

132 

Mine 

depth, m 

143 

58 

473 

116 

122 

279 

131 

201 

128 

198 

88 

165 

171 

198 

2 1 

79 

366 

152 

183 

153 

80 

365 

140 

219 

86 

67 

46 

It should also be mentioned that the 

position of the underground transmitter 

was determined by measuring the horizontal signal field at different locations and, by finding the null in the 

field, finding the underground location 

by triangulation, This location measurement technique had the added benefit of 

assuring that the transmitted signal was 

actually traversing the earth medium and 

was not conducted along pipes, wires, or 

other conductors, 

The A parameter was defined as the 

ratio of the radius of the underground 

loop to the mine depth. However, the 

underground loop was deployed into a 

configuration that was between a circle 

and a rectangle. Therefore, the estimate 

of the radius, which assumes the antenna 

D 

0 

.29 

0 

.06 

0 

0 

.11 

-01 

.14 

.11 

.03 

.14 

0 

.02 

.48 

.04 

.03 

0 

0 

.16 

0 

.03 

.02 

.OO 

.35 

-14 

1.70 

length represents the circumference of 

a circle, is in error. In later estimates of conductivity, this point will 

need to be considered if the A value 

becomes large, becoming less of a dipole 

situation, 

A 

0.07 

.14 

.04 

.05 

08 

.05 

.09 

.07 

.lo 

.05 

.15 

.05 

.06 

.07 

.14 

.13 

.05 

.08 

.04 

-06 

*15 

.03 

.06 

.06 

.09 

.07 

.14 

Assuming that the transmit current measurement is accurate, the accuracy of the 

estimate of the magnetic moment is related to the accuracy of the antenna area 

estimate, discussed earlier. It is concluded that the uncertainty in the magnetic moment is the same as the uncertainty of the area estimate, 210 pct. 

The signal strength measurements were 

made by playing the tapes into an 

FFT instrument. Signal averaging was 

done with 128 independent samples. The 
TABLE 2. - Surface vertical magnetic signal (Hz) (in decibels re 1.0 p~/m-~~~) 

and underground transmit moment (M) (inampere-meters squared), at different 

frequencies 

Mine 

ND No data. 

in£ ormat ion needed was the rootmean-square (RMS) value of the signal 

during transmit time. Since the transmitted signal had a duty cycle associated 

with it, this had to be first measured 

for each transmitter and used with the 

FFT measurement to derive the on-time 

signal level. It was then found that the 

95-pct confidence interval for the signal 

strength measurement was -1.44 to 1.56 

dB. From these field tests, data on 

earth transmission loss can be obtained. 

Implicit in these transmission loss 

data is information on apparent earth 

conductivity. It is then the objective 

of this study to derive estimates of 

apparent earth conductivities from the 

signal transmission data. 

EARTH CONDUCTIVITY ESTIMATES 

As an electrostatic EM signal is transmitted TTE, it experiences attenuation 

from two sources. The first is the 

attenuation from geometric spreading, 

which falls off as the distance cubed. 

The second is the attenuation caused by 

the conducting earth, which can be used 

in TTE transmission experiments to gather 

information on the earth conductivity. 
The vertical magnetic signal received 

on the surface from an underground transmitter can be expressed as 

where Hz = surface vertical magnetic 

field, A/m, 

M = magnetic mount, ~-m~ 

(I = transmit current, A, 

N = number of turns of the 

transmit antenna), 

d = mine depth, m, 

and Q = attenuation factor of the 

conducting earth. 

The functional form of Q depends upon 

the conductivity model of the earth. The 

value of Q would be 1 for a nonconducting 

earth and, in general, decreases with increases in earth conductivity, Therefore, though Q can be used to determine 

the value of the earth conductivity, the 

appropriate model must be used. Choosing 

a particular model and using the data 

from the field tests, conductivity values 

can be obtained, and these values, in 

turn, can be studied to verify the choice 

of the model. If the model appears acceptable, a statistical study of the conductivity data can be done to obtain 

information on expected or variations in 

earth conductivity, 

Wait and Spies (7) give the functional - 

form of Q for a homogeneous half-space 

model of the earth as 

where H = (opow)1/2d, 

0 = earth conductivity, mho/m, 

po = free space permeability, H/m, 

w = angular frequency , i-ad/s , 

d = mine depth, m, 

A = ratio of the loop radius to 

mine depth, 

and D = ratio of the offset to mine 

depth. 

From the field test values given in 

tables 1 and 2, apparent earth conductivity values were found using this 

form of Q, These values are given in 

table 3 for each mine and for each 

frequency tested. Also given in this 

table are Q values derived from equation 2, As discussed in the previous 

section, an expected uncertainty existed 

in the measurement of the signal, magnetic moment, and mine depth. Assuming 

the simplest case of a coaxial dipole 

measurement scheme (A = D = 0), an uncertainty expression for Q can be derived 

over small regions of Q where linearity can be assumed. This can be written 

as 

From the earlier stated values of expected error in these parameters, the 

resultant expected error in Q would then 

be approximately 220 pct. 

exp i id^)^'^] Jo(Dx)dx, (3) 
TABLE 3. - Q values and apparent earth conductivity values (0) as derived 

from a half-space earth model, at different frequencies 

Mine 

l......... 

The error in estimated conductivity is 

related to this error in Q. However, 

this relationship is nonlinear and is 

dependent upon frequency, depth, and the 

value of Q, A typical relationship, one 

that will be of most interest, is shown 

in figure 1 where the error in conductivity estimate is normalized to the error 

in Q for a frequency of 3,030 Hz and a 

mine depth of 200 m. Relationships for 

other frequencies and depths are similar 

to that shown in this figure, As can be 

630 Hz 1,05 

CI Q 0 

0.589 0.045 0.366 

2......... 

3.. ....... 4 ......... 

5......... 

6*e*e**ee. 

7......... 

8. ........ 9......... 

lO*e..e... 

ll........ 

12e*e..... 

13. ....... 14........ 

15........ 

16.. ...... 17........ 

18........ 

19........ 

20........ 

21........ 

22........ 

23........ 

24. ....... 25........ 

26........ 

27........ 

ND No data. 

seen from figure 1, estimates of conductivity can be found with the same uncertainty in Q for values of Q around 0.3. 

For small values of Q, a situation that 

exists for the majority of the field 

data, the uncertainty in conductivity 

estimation is less than the uncertainty 

for the Q measurement (<20 pct) . For 

values of Q greater than 0,3, the uncertainty of the conductivity estimate is 

greater than the uncertainty of the Q 

estimate 020 pct). 

1.961 

.024 

.I76 

.210 

ND 

ND 

.039 

el10 

ND 

.290 

ND 

.080 

ND 

9.476 

.494 

.020 

1,171 

ND 

ND 

.633 

ND 

,108 

.031 

.282 

.693 

.001 
ATTENUATION FACTOR (Q) 

FIGURE 1. - Uncertainty in conductivity normalized to the uncertainty in Q for a depth of 

200 m and a frequency of 3,030 Hz. 

STUDY OF CONDUCTIVITY DATA 

natural contour of the land between survey points would not influence the depth 

estimate. 

Considering these points, it was decided to choose a subset of the conductivity estimates that were obtained at 

mines exceeding a certain depth and where 

the Q estimates were less than some given 

value. The choice made was to consider 

only those mines where d > 100 m and Q 

< 0.5. It is felt that, with the choice 

of these data points, the conductivity 

estimate would be valid with an upper 

uncertainty of approximately 223 pct. 

An observation of the data from table 3 

shows that the conductivity estimate at 

most mines tends to decrease with frequency, which contradicts the homogeneous 

half-space assumption. If all of the 

mine data for which d > 100 m and Q < 0.5 

are pooled and segregated by frequency, 

approximate curves for the cumulative 

distributions of the conductivity data 

(fig. 2) also show that the conductivity 

tends to decrease with frequency. 

PERCENTAGE 

E 

\ 

The preceding section described how f 

apparent earth conductivity estimates ,- 

were made based on a homogeneous half- 5 I 

space model of the earth. The data provide estimates for different frequencies 5' 

and for mine sites having different 

depths. Conductivity estimates obtained 

for Q values greater than 0.3 must be 

checked for the possibility of large 

errors. Also, as discussed in the section "Test Description and Data," depth 013 

- 1,050 Hz - 

- -

I - 

- - - 

I - 

- 

- - 

- - 

1,950 HZ 

I I 

4 estimates contained errors on the order 5 6 7 

PROBITS 

of I pct, and this error value might be 

valid only in cases where the mine depth Fl GURE 2. - Cumu lat ive distributions of conwas large enough (d > 100 m) that the ductivity estimates segregated by frequency. 
Besides the frequency dependency, the 

conductivity also appears to show a dependency on depth. This is illustrated 

in table 4 where mean and standard deviation values of the conductivity are given 

for different depth intervals. The conductivity appears to decrease with mine 

depth. It can be seen from table 4 that 

for a given frequency, the apparent conductivity decreases by about a factor of 

10 over the depth range, with slightly 

smaller differences for an increase in 

frequency. Also, it can be seen that in 

a given depth interval, the apparent conductivity decreases by a factor of 2 with 

frequency over the frequency range. 

Neither the frequency nor depth dependency of the conductivity values would be 

expected for a homogeneous half-space 

model of the earth. Geyer (8) - found that 

conductivity was generally higher near 

the surface above a number of mines. A 

high conductivity surface layer could 

explain the trend of apparent conductivexplain the frequency dependence of the 

conductivity data. 

An alternate earth conductivity model 

proposed by Durkin3 and studied by Shope 

(9) proposes an earth model having a 

txin, highly conducting surf ace layer 

over a lower conducting layer in which 

the transmit loop is buried, followed by 

a higher conducting half-space. Results 

by Shope (9) indicate that this model may 

explain bzth the frequency and depth 

dependency of the conductivity data, but 

studies are continuing. A related model 

proposed by Hill (10) consists of a 

magnetic dipole sourcrburied in an earth 

consisting of a highly conducting thin 

sheet located over a homogeneous halfspace. This model is interesting because 

it is simple, shows the conductivity 

depth dependency, and when the upper conductivity layer is considered extremely 

thin, demonstrates the general trend in 

the frequency dependency found in the 

conductivity data. 

ity with depth for the data reported 

here, since the weighted contribution of 3~urther information on this model is 

this layer would be proportionally higher available upon request from J. Durkin, 

the closer the transmitter was to the Pittsburgh Res. Cent., BuMines, Pittsmeasurement point. However, the exist- burgh, PA. 

ence of this surface layer would not 

TABLE 4. - Mean and standard deviation of conductivity values (in mho 

per meter) for each frequency within different depth intervals for 

estimates having Q < 0.5 

ND No data. 

Freq, Hz 

630....... 

1,050..... 

1,950..... 

3,030..... 

All freq.. 

201-300 m 

Mean I Std 

100-150 m 

Mean I Std 

Over 300 m 

Mean I Std 

0.238 

.I68 

3 

.091 

.I61 

151-200 m 

Mean I Std 

dev 

0.201 

.I17 

.063 

0070 

-132 

0.125 

.I23 

.074 

.076 

.094 

dev 

0.064 

.047 

,025 

.038 

.044 

ND 

0.034 

.033 

0025 

.032 

dev 

ND 

ND 

0.008 

ND 

.005 

0.022 

.017 

,014 

.011 

.Ol6 

dev 

0.003 

.OO2 

.001 

ND 

-005 
In the next section, following the work 

by Hill (lo), a homogeneous conducting 

earth model- containing a thin sheet is 

studied for its influence on apparent 

conductivity estimates for transmission 

with a finite loop transmitting antenna. 

THIN CONDUCTING SHEET EARTH MODEL 

Figure 3 shows a vertical magnetic loop 

buried in a homogeneous earth containing 

a thin conducting sheet at the surface. 

The homogeneous earth has a conductivity 

ao, and free space permeability po is 

assumed everywhere. The thin conducting 

sheet has a small thickness d and a 

conductivity-thickness product ad. The 

propagation constant-distance product 

(yd) of this thin sheet is assumed small. 

Displacement currents are neglected 

everywhere. This model is similar to the 

metal-cased borehole problem studied by 

Wait (11). - 

In the half-space, z < 0, the magnetic 

vector potential F satisfies the Helmholtz equation everywhere except at the 

source, and is given by 

where yl=(ioplol) 1 /2* 

For z > 0, F satisfies Laplace's 

equation. 

In the half-space, the EM fields can be 

derived from F by 

where E is the elective field and H is 

the magnetic field. 

For the region z > 0, the EM fields are 

derived from F by 

FIGURE 3. - Geometry for a finite magnetic loop 

of current buried in a conducting half-space earth 

containing a highly conducting thin sheet at the 

surface. (Symbols are identified in the text and 

in appendix B.) 
The specific expressions for Fo and F1 are given by Wait (3) - as 

- ip0wa21 " J (~a) A 

FO - 2 lo - T(x) exp (-klho-k z kl ) J~ (Xp 

where the wave number (k) = (X2 + y2)1/2, 

and R(x) and T(x), the reflection and transmission terms, are unknown. 

The boundary conditions are such that the tangential electric fields are continuous, and from Ampere's law, the tangential magnetic field is discontinuous by the 

amount of the longitudinal current per unit length carried by the thin sheet. Also, 

recalling that yd is very small, it can be assumed that 

and 

Using these boundary conditions and equations 14 and 15, it follows that 

From equation 13 and noting that ko = A, the vertical magnetic field above the earth 

is given by 

It is convenient to write H1, in the following form with x = Xh0, where ho is the 

depth of the buried loop: 

where 

" J (Ax) x3 = lO h' x + (xz + i~~)~~~ + iHS exp (-Zx) exp I-(x2 + i~2)1/2} Jo(Dx)dx, (21) 
where H = (wpOa1)1'2 h 0 9 

and s = ad(wpo/al)1/2, 

Q is dimensionless and approaches zero 

as H, A, D, and Z approach zero, and 

it can be considered a correction factor 

to the field of a static magnetic 

dipole, For S equal to zero (ad = O), 

Q reduces to a homogeneous half -space 

equation, 

In general, the integral in equation 21 

must be evaluated numerically, The case 

of an observer at the surface (Z = 0) 

directly above the source (D = 0), with 

an underground dipole source (A = 0), is 

of particular interest, By setting the 

products ad and 01, Q(H,S) values can be 

found for the field data, Further, by 

setting 

where Q(Ha,O) represents the homogeneous half-space factor, the corresponding 

Ha value can be found and the apparent 

half -space conductivity can be found 

from 

These conductivity values can then be 

compared with those found from the field 

data shown in table 3 in order to check 

the appropriateness of the thin sheet 

model, 

The choice of the ad product is not 

entirely arbitrary, It must be realistic 

and still maintain the assumption made in 

the formulation of the thin sheet model, 

The assumption stated earlier is that yd 

is small to avoid the exponential attenuation factor of the signal as it traverses the thin layer, If this product 

is set equal to yd = 0.1, then it places 

a constraint on the product ad, 

Typically in the literature, conductivity measurements on earth material rarely 

exceed 1 mho/m, If this value is considered an upper bound and the highest 

frequency of interest is 3,030 Hz, then 

it follows that 1 1 ma, = 0.155/m and d 

= 0,647 m and a = 0,647 mho, 

Computer results using this ad product 

with various values of a1 showed a 

decrease in apparent conductivity with 

depth but an increase in conductivity for 

an increase in frequency at a given 

depth, 

A different choice of ad having a value 

of 20 with al = mho does show the 

trend of conductivity with depth and frequency as reflected in the field data, 

These results are shown in table 5, A 

comparison of the conductivity and Q values of table 5 with those shown in tables 

3 and 4 shows that the thin sheet model 

does predict the qualitative behavior of 

the experimental values, but does not 

predict the magnitude of decrease in 

apparent conductivity with depth, Also, 

it should be noted that the choice of ad 

= 20 may not be realistic and may need 

further study, 

TABLE 5, - Conductivity (a) and Q values for different depths 

and frequencies for the thin sheet model of the earth with 

ad = 20 and a1 = mho 

Freq, Hz 

630, ,,,,,, 

1,050..... 

1,950..... 

3,030..... 

175 m 250 m 

a 

0,074 

,063 

,048 

,039 

a 

0,114 

,100 

,081 

,066 

125 m 400 m 

Q 

0,226 

,141 

,077 

,050 

Q 

0,306 

,197 

,110 

,072 

a 

0,040 

,032 

,024 

,018 

a 

0,167 

,154 

,129 

,108 

Q 

0,147 

,089 

,049 

.03 1 

Q 

0,397 

,265 

-151 

,099 
REGRESSION MODEL 

Since the apparent conductivity shows a 

depth and frequency dependency, it may be' 

possible to obtain a linear model that 

would be acceptable for making statements 

of expected values of apparent conductivity for a given frequency and mine depth. 

Such an approach was taken through two 

parameter regression studies. 

Several linear regression models were 

considered. The model found to best fit 

the behavior of the data is one in which 

the log of the apparent conductivity is 

linearly related to frequency and depth. 

This is shown by 

log(a) = a + Bl (depth) 

+ $2(f req) + E. (24) 

The parameters a, B1, and B2 are numbers 

that are estimated from the data. The 

parameter E represents a random variable 

that is normally distributed, with expected value of zero and uniform variance 

across the independent variables. Regression results for this model are shown 

in table 6. Data from mine 1 were deleted because they appeared as a large 

outlier to the rest of the data. Also, 

because of the expected uncertainty of 

the conductivity estimates, all data for 

Q > 0.5 or d < 100 m were excluded. 

TABLE 6. - Regression results for log(a) 

versus mine depth (in meters) and 

transmit frequency (in hertz) 

Number observed................ 42 

Intercept...................... -0,45818 

Depth slope.................... -0.00276 

Frequency slope................ -0.00012 

Correlation coefficient........ 0.88 

Estimated standard error.. ..... 0.1794 

Comparing the two slopes of table 6, it 

can be seen that the expected conductivity value appears to be more sensitive to 

the depth parameter. This is consistent 

with the results of table 4, as previously discussed. However, for small 

depths and high frequencies this would 

not be the case. 

Earth conductivity estimates are often 

used for estimating the transmission loss 

an EM signal incurs as it is transmitted 

TTE. As such, earth conductivity estimates can be used with an earth model for 

making predictions of transmitted signal 

strengths. For small depths, the signal 

loss will be small and exact knowledge of 

the earth conductivity value would not be 

as important as it is at larger depths. 

After reviewing the results of table 6, 

it can be argued that possibly the 

conductivity frequency dependency can 

be ignored and the depth parameter can 

be used as the sole predictor of 

conductivity. 

A single parameter regression model was 

formed relating the log of conductivity 

to mine depth, and the results are shown 

in table 7. This model was formed without the data from mines 1 and 3 and the 

single data point from mine 7 because 

they appeared as large outliers to the 

balance of the data. Also, as be£ ore, 

all data for which Q > 0.5 and d <. 100 m 

were excluded. The regression line for 

this model is shown in figure 4 along 

with a 95-pct confidence interval (CI). 

As can be seen from figure 4, a prediction of apparent earth conductivity can 

be obtained from a depth measurement with 

an uncertainty of around +20 pct, 

TABLE 7. - Regression results for log(a) 

versus mine depth (in meters) 

Number observed.. .............. 37 

Estimated intercept............ -0.45131 

Estimated slope.. ............ .. -0.00393 

Correlation coefficient........ 0.87 

Estimated standard error. ...... 0.1655 

Standard deviation depth. ...... 74.27 

The regression analysis assumes normally distributed residuals with a homogeneous variance across the independent 

variable. These assumptions were checked 

and found to be adequately met. 
is tested against 

The F-test statistic for this regression analysis was found to be F 

= 112. Comparin this with the critical 

F35 ,1,0 -05 F=~~~*~, H0 cannot be rejected and the linear model can be 

accepted. 

From this study, it can be concluded 

that depth can be a good predictor of 

conductivity. However, since the objective of this study is to be able to predict the expected signal loss for TTE 

comnrtnications, the related uncertainty 

of expected received signal strength from 

an underground transmitter for this earth 

conductivity regression model nust be 

determined. 

DEPTH, rn 

FIGURE 5. - Surfacevertical magnetic field versus depth by frequency for an underground transmitter with a magnetic moment of 1 A/m. 
SUMMARY 

Field data from a number of narrow-band 

EM TTE communication tests were studied, 

and apparent earth conductivity estimates 

were made based on a homogeneous halfspace earth model. The derived conductivity estimates were found to decrease 

with the transmitted frequency and mine 

depth. This result is not consistent 

with the assumption of a homogeneous 

half-space earth. 

A different earth model was studied 

that contained a thin sheet of high conductivity in a lower conductivity homogeneous earth. Results of studies of 

this model show that it can predict the 

qualitative behavior of the conductivity 

estimates with frequency and depth, but 

falls short of predicting the quantitative results. 

Linear regression analysis was performed, and it was found that the apparent earth conductivity could be estimated 

from a linear combination of the transmitted frequency and mine depth. Further 

regression studies demonstrated that an 

adequate predictor of the apparent earth 

conductivity could be the mine depth 

alone. The major value of this work is 

that an adequate statistical model of the 

apparent earth conductivity has been 

found, which can be used in future studies of TTE communications. 

REFERENCES 

1. Stefanescu, S., and Re Me Schlumberger. (Electric Potential Distribution 

Around a Point in the Earth.) J. Phys. 

et Radiation, v. 1, 1930, pp. 75-83. 

2. Wait, J. R. Mutual Electromagnetic 

Coupling of Loops Over a Homogeneous 

Ground. Geophysics, v. 20, No. 3, July 

1955, pp. 630-637. 

3. . Electromagnetic Waves in 

Stratified Media. Pergamon, 1962, 

372 pp. 

4. Keller, G. V., and F. C. Frischknecht. Electrical Methods in Geophysical Prospecting. Pergamon, 1966, 486 pp. 

5. Grant, F. S., and G. F. West. Interpretation Theory in Applied Geophysics. McGraw-Hill, 1965, 525 pp. 

6. Institute of Electrical and Electronics Engineers. Guide for Radio Methods of Measuring Earth Conductivity. 

IEEE Standard 356, 1974, 18 pp. 

7. Wait, J. Re, and K. P. Spies. SubSurface Electromagnetic Fields of a Circular Loop of Current Located Above 

Ground . IEEE Trans. Antennas and 

Propag., v. AP-10, No. 4, July 1972, 

pp. 520-522. 

8. Geyer, Re G., G. V. Keller, and 

T. Ohya. Research on the Transmission of 

Electromagnetic Signals Between Mine 

Workings and the Surface (contract 

H0101691). BuMines OFR 61-74, 1974, 

67 pp.; NTIS PB 237 852. 

9. Shope, S. Me Electromagnetic Surface Fields Due to a Magnetic Dipole 

Buried in a Three-Layered Earth. BuMines 

RI 8702, 1982, 22 pp. 

10. Hill, D. A. Theoretical Noise and 

Propagation Models for Through-the-Earth 

Communication (contract 50113058, Natl. 

Telecommun. and Inf. Admin.). Unpubl. 

rep., 1982, 45 pp.; for inf., contact J. 

Durkin, Pittsburgh Res. Cent., BuMines, 

Pittsburgh, PA. 

11. Wait, J. R. Electromagnetic 

Shielding of Sources Within a Metal-Cased 

Borehole. IEEE Trans. Geosci. Electron., 

v. GE-15, No. 2, Apr. 1977, pp. 108-111. 

12. Lagace, Re L., J. Me Dobbie, T. E. 

Doerfler, We S. Hawes, and R. H. Spencer. 

Detection of Trapped Miner Electromagnetic Signals Above Coal Mines (contract 

50188037, Arthur D. Little, Inc.). BuMines OFR 99-82, 1980, 258 pp.; NTIS 

PB 82-244732. 
APPENDIX Am--NAMES AND LOCATIONS OF MINES VISITED DURING FIELD TESTS 

Mine 

1 

Depth, ft Name and location 

470 Youghiogheny and Ohio Coal, Allison; Beallsville, Belmont County, 

OH. 

190 Peabody Coal Co., Alston No. 4; Centertown, Ohio County, KY. 

1,550 Jim Walter Resources, Blue Creek No. 3; Adger, Jefferson County, 

AL 

38 1 Cal Glo, No. 21; Siler, Knox County, KY. 

400 Eastover Mining Co., Highsplint No. 4; Highsplint, Harlan County, 

KY. 

915 U.S. Steel, Gary No. 2; Wilcoe, McDowell County, WV. 

430 U.S. Steel, Gary NO. 9; Filbert, McDowell County, WV. 

658 Gateway Coal Co., Gateway Mine; Clarksville, Greene County, PA. 

420 Allied Chemical Corp., Harewood; Boomer, Fayette County, WV. 

650 Alabama By-products Corp., Mary Lee NO. 1; Goodsprings, Walker 

County, ALo 

289 Monterey Coal Co., Monterey NO. 1; Carlinville, Macoupin County, 

IL. 

540 Youghiogheny and Ohio Coal, Nelm's NO. 2; Hopedale, Harrison 

County, OH. 

560 Consolidation Coal, Oak Park NO. 7; Cadiz, Harrison County, OH. 

650 Old Ben Coal Co., Old Ben NO. 26; Sesser, Franklin County, IL. 

70 Owl Creek Corp., Sue-Jan; St. Charles, Hopkins County, KY. 

260 Peter Cave, Mine NO. 1; Lovely, Martin County, KY. 

1,200 Plateau Mining Co., Star Point NO. 2; Wattis, Carbon County, UT. 

500 Pontika, NO. 1; Lovely, Martin County, KY. 

600 North American Coal, Powhatan NO. 1; Powhatan Point, Belmont 

County, OH. 

500 North American Coal, Powhatan NO. 3; Powhatan Point, Belmont 

County, OH. 

260 Peabody Coal Co., Sinclair NO. 2; Drakesboro, Butler County, KY. 

1,200 Kaiser Steel, Sunnyside NO. 1; Sunnyside, Carbon County, UT. 

460 Zeigler Coal Co., Mine NO. 4; Johnston City, Williamson County, 

IL. 

720 Helen Mining Co., Helen Mine; Homer City, Indiana County, PA. 

282 Bureau of Mines, Lake Lynn; Fairchance, Fayette County, PA. 

220 5-4 Cave; University Park, Centre County, PA. 

150 Woodward Cave; University Park, Centre County, PA. 
APPENDIX Be--ABBREVIATIONS AND SYMBOLS USED IN THIS REPORT 

NOTE.--This list does not include the unit of measure abbreviations listed at the 

front of this report. 

a loop radius 

A ratio of radius of underground 

antenna to mine depth 

CI confidence interval 

d mine depth 

D ratio of distance between vertical axis of underground 

antenna and observation point 

to mine depth 

E electric field 

EM electromagnetic 

F vector potential 

FFT fast Fourier transform 

H magnetic field 

ho depth of buried loop 

Hz surface vertical magnetic 

signal 

I current 

R 

RMS 

T 

TTE 

wave number 

magnetic moment 

attenuation factor of conducting 

earth 

reflection coefficient 

root-mean-square 

transmission coefficient 

through the earth 

field measurement point 

ratio of distance of field measure 

ment point above earth to mine 

depth 

propagation constant 

integration constant 

free space permeability 

earth conductivity 

angular frequency 

fiU.5. GPO. 1984-705-02015014