Bureau of Mines Report of lnvestigations/l983 

Above-the-Earth Field Contours 

for a Dipole Buried 

in a Homogeneous Half-Space 

By Steven M. Shope 

UNITED STATES DEPARTMENT OF THE INTERIOR 
Report of Investigations 8781 

Above-the-Earth Field Contours 

for a Dipole Buried 

in a Homogeneous Half-Space 

By Steven M. Shope 

UNITED STATES DEPARTMENT OF THE INTERIOR 

James G. Watt, Secretary 

BUREAU OF MINES 

Robert C. Horton, Director 
This publication has been cataloged as follows: 

Shope, Steven M 

Above-rhe-earth field contours for a dipole buried in a homogeneous half-space. 

(Bureau of Mines report of investigations ; 8781) 

Inclu~lrs bihli~gra~hical references. 

Supt. of Docs. no.: 128.23:8781. 

1. Mine rescue work-Equipment and supplies. 2. Mine communication systems. 3. Electromagnetic fields. 4. Magnetic dipoles. I. 

Title. 11. Series: Report of invesrigarions (United Scares. Bureau of 

Mines) ; 8781. 

TN23.U43 [TN297] 622s [622'.81 82-600362 

7 
CONTENTS 

- Page 

Abstract....................................................................... 

Introduction................................................................... 

Field solutions........................................,....................... 

Numerical evaluation........................................................... 

Contouring ..................................................................... Conclusions.................................................................... 

Nomenclature................................................................... 

Appendix.--Field solution derivation........................................... 

ILLUSTRATIONS 

1. Normalized vertical magnetic field (Q) versus H.......................... 

2. Normalized vertical magnetic field contours.............................. 

3. Normalized volume of vertical magnetic field contour versus H............ 

A-1. Dipole buried in a homogeneous half-space......................... ....... 

TABLE 

1. Normalized lobe volumes of all contours graphed.... ...................... 9 
ABOVE-THE-EARTH FIELD CONTOURS FOR A DIPOLE 

BURIED IN A HOMOGENEOUS HALF-SPACE 

By Steven M. Shape 

ABSTRACT 

An essential element of the Bureau of Mines electromagnetic location 

and communication system for trapped miners is a quantitative knowledge 

of the surface and above-surface electromagnetic fields created by an 

underground transmitter. The field solutions given in this report use 

approximations in which the displacement currents are negligible for 

low frequencies. The earth is represented by a homogeneous half-space 

model in which a dipole source is immersed. The vertical magnetic 

field equipotential contours at and above the surface are graphically 

mapped. The volumes of the regions bounded by these contours are directly related to the geometrical zones of signal detectability. 

Knowledge of the detectability zones will enable the proper search 

strategies to be developed for airborne detection of trapped miners. 

'~lectrical engineer, Pittsburgh Research Center, Bureau of Mines, Pittsburgh, Pa. 
INTRODU CTION 

Ip response to the 1969 Coal Mine 

Health and Safety Act, the Bureau of 

Mines has maintained a research effort 

aimed at increasing the survivability of 

miners trapped underground following a 

disaster. A principal element of this 

effort is to develop a practical and effective means to detect, locate, and communicate with miners who cannot evacuate 

the mine immediately following a disaster. In 1970, the National Academy of 

~n~ineerin~2 reported that an electromagnetic (EM) communications technique may 

be used for this purpose. Since the 1970 

report was issued, the Bureau has sponsored a continued effort toward developing an EM communication and location system for postdisaster use. These programs 

have reached the stage where practical 

voice frequency (VF) signaling hardware 

has been developed for use by trapped 

miners .3 

The Bureau is now in the process of developing suitable guidelines for the 

eventual implementation of such a system 

in the mining community. Included in 

such guidelines will be search strategies 

to be utilized by surface rescue teams to 

detect, locate, and rescue trapped 

miners. At mines where large surface 

areas need to be searched, airborne 

equipment will be employed to determine 

the general locale of the trapped miner's 

position. Surface personnel will then 

provide a more accurate location of the 

signal source. Therefore, a necessary 

component in formlating search strategies is a quantitative knowledge of the 

surface and above-the-surface EM field 

'~ational Academy of Engineering. Mine 

Rescue and Survival (contract 50190606). 

BuMines OFR 4-70, 1970, 81 pp.; NTIS PB 

191 691. 

3~eneral Instrument corp. Development 

and Prototype Production of a Trapped 

Miner Signaling Transmitter/Transceiver 

(contract J0395017). Final Rept., 1980, 

80 pp. Available for consultation at 

Pittsburgh Research Center, Bureau of 

Mines, Pittshrgh, Pa. 

characteristics produced by the trapped 

miner's loop antenna. 

Previous research has shown the relationship between probability of detection 

and signal strength, noise, receiver 

operating characteristics, depth, magnetic moment, and other factors.4 It is assumed that search strategies will be developed that will enable rescue teams to 

define a satisfactory probability of 

detection for a particular disaster. 

Knowledge of noise statistics along with 

equipment and operator characteristics 

will determine the minimum signal 

strength required to achieve this particular detection probability. 

The "zone of detectability" is defined 

as the geometrical volume in which the 

surface and above-surface field strengths 

of an EM signal are greater than or equal 

to the minimum value required for the 

desired probability of detection. Knowledge of this zone would enable rescue 

teams to search the surface in an effective and efficient manner. 

This report analyzes the surface and 

above-the-earth fields (in particular Hz, 

the vertical magnetic field) and notes 

the effect of the electrical properties 

of the mine overburden on the field 

distributions. The fields are presented 

as two-dimensional cross-sectional views 

of the field strength contours mapped on 

a normalized Z-D (Z, vertical; D, horizontal) graph. Z and D are the vertical 

and horizontal dimensions normalized to 

the depth of the mine. The volume generated by rotating these contours 360' 

about the Z axis at D = 0 is also provided. This is valid because of the rotational symmetry of the vertical magnetic field. 

4~rthur D. Little, Inc. Detection of 

Trapped Miner Signals Above Coal Mines 

(contract J0188037). Final Rept., 1980, 

288 pp. Available for consultation at 

Pittsburgh Research Center, Bureau of 

Mines, Pittsburgh, Pa. 
FIELD SOLUTIONS 

The electromagnetic field structure for 

an oscillating magnetic antenna has been 

thoroughly investigated by Wait and Campbe1lO5 Their derivations include line 

sources, dipoles, finite-sized loops, and 

other kinds of signal sources, and have 

also considered the effects of layering 

in the earth half-space. 

In particular, Wait and Campbell have 

presented the surface fields due to a 

magnetic dipole buried in a homogeneous half-space. Their solution is similar to that developed in the appendix of 

this report; the only difference results 

from the use of a different coordinate 

origin in the model. Without claim to 

originality, the Bureau of Mines derivation has been included to supplement the 

technical discussion in the main text of 

this report. Throughout this report, the 

rationalized MKSA (meter, kilogram, second, ampere) unit system is employed. 

As derived in the appendix, the solution for the normalized magnetic field, 

defined as Q, on or above the surface is 

expressed by 

NUMERICAL EVALUATION 

The above expression for Q is evaluated where x is a new variable of integration: 

by numerical integration, specifically, 

12-point Gaussian quadrature. Before x = Ah. 

this is done, however, the integrand is 

simplified if the dimensional variables The expression for Q now appears as 

are normalized with respect to the depth 

of loop burial, h. The following dimen- -(X~+~H~) 1/2+x(1-Z) 

sionless, transformed variables are J,(XD)~X . introduced: X+(X~+~H~) 

D = plh, Z = z/~, and H = (uuu)"~~. 

H is not subscripted, but it is assumed 

to apply to the electrical properties of 

the earth half-space; H above the earth 

is zero. The factor H may be associated 

with the depth of loop burial divided by 

skin depth. The wave number k in region 

1 (fig. A-1, appendix) becomes 

Q has been evaluated at D = 0 and Z = 1 

(coaxial at the surface). and is shown 

graphically in figure 1 as a function 

of H. The rapid decrease in Q as H increases is readily seen in this figure. 

Physically, this may be interpreted as 

the effect the electrical properties 

of the half-space have upon the surface 

field strength. For a fixed burial 

depth, H is inversely related to the skin 

depth; thus, Q falls off proportionally 

to skin depth as seen in figure 1. 

CONTOURING 

Q is a multivalued function that is 

not readily adaptable to conventional 

5~ait, J. R., and L. L. Campbell. The 

Fields of an Oscillating Magnetic Immersed in a Semi-Infinite Conducting Medium. J- Geophys. Res., V. 58, No. 2, 

June 1953. 

contouring methods. The contours presented here were generated by first creating a data grid of Q(DI,Zj) values for 

D = 0 to 10 and Z = 1 to 10. Each grid 

element was obtained by numerically evaluating the integral expression for Q. 

The increment of the D coordinate was 

0.010, and that of the Z coordinate was 
H 

FIGURE 1. - Normalized vertical magnetic field (Q) versus H. 

0.080. This resulted in a data grid containing over 125,000 elements for each 

value of H. For a particular Q value to 

be contoured, denoted Q,, the data grid 

values Q(Di,Zj) were searched for an inflection about Q,. This procedure alleviated problems associated with the fact 

that Q is multivalued. For this report, 

the following Q, values were contoured: 

Q = 0.001, 0.005, 0.010, 0.050, 0.100, 

for 10 values of H: 

The contours were then mirrored into 

the (-D,+Z) quadrant since the surface and above-surface fields nust be 

symmetrical about the Z axis. In figure 

2, the Q contours are graphically presented. Because of the size of the increment in the data grid, the resolution 

of the contours may be slightly limited 

in some of the figures; however, the general shape is very apparent. Note that 

the graphs in figures 2F through 25 have 

a larger scale vertical axis. 

In this set of contour figures, the Q 

contours are composed of primary and secondary lobes. For the free-space condition, H = 0, the lobes are very distinct. 

However, as Q, increases, the secondary 

lobes disappear. Also, as H increases, 

the primary and secondary lobes become 

distorted and less distinct, and the lobe 

size diminishes. This last result is 

physically intuitive since the attenuative effects of the earth would limit 
FIGURE 2. - Normalized vertical magnetic field contours. A, H = 0.00; B, 

H : 0.100. 
- NORMALIZED OFFSET, D 

FIGURE 2. - Normalized vertical magnetic field contours-Continued. C, 

H .= 0.500; o, H = 0.800. 
NORMALIZED OFFSET, D 

FIGURE 2. - Normalized verticol magnetic field contours-Continued. E, 

H = 1.00; F, H = 2.00. 
NORMALIZED OFFSET, D 

NORMALIZED OFFSET, D 

FIGURE 2. - Normalized vertical magnetic field contours-Continued. G, H = 4.00; ti, H = 6.00; 1, 

H = 8.00; J, H = 10.0. 
the field pattern. To obtain a general 

idea of the decrease in size, the 

approximate volumes of each contour were 

calculated by rotating them about the 

Z axis. Volume, as used here, is dimensionless (since Z and D are also dimensionless) and may be thought of as a 

normalized volume. Since the normalization is to depth cubed, actual volume 

values could be readily calculated for 

any particular mine depth. The volumes 

are listed in table 1 and plotted on a 

semilog plot versus H in figure 3. 

s his graph shows that the lobe volume 

for a particular Q contour may be fitted 

to the expression 

Vol, = constant 

where c is a constant and has been found 

empirically to be approximately -113. 

TABLE 1. - Normalized lobe volumes of all contours graphed 

Q contour........ 

H = 0.00: 

Primary........ 

Secondary ...... Total....... 

H = 0.100: 

Primary........ 

Secondary...... 

Total....... 

H = 0.500: 

Primary........ 

Secondary.. . . . . Total....... 

H = 0.800: 

Primary........ 

Secondary...... 

Total....... 

H = 1.00: 

Primary........ 

Secondary...... 

Total....... 

H = 2.00: 

Primary........ 

Secondary...... 

Total....... 

H = 4.00: 

Primary........ 

Secondary.. . . . . Total....... 

H = 6.00: 

Primary........ 

Secondary..... . Total....... 

H = 8.00: 

Primary........ 

Secondary.. . . . . Total.. . . . . . H = 10.0: 

Primary........ 

Secondary.. . . . . Total....... 

0.001 

401.900 

210.200 

612.200 

409.200 

198.200 

607.900 

636.700 

.OOO 

636.700 

470.600 

.OOO 

470.600 

376.700 

.OOO 

376.700 

139.600 

.OOO 

139.600 

30.830 

.OOO 

30.830 

8.670 

.OOO 

8.670 

2.540 

.OOO 

2.540 

.624 

.OOO 

.624 

0.005 

78.860 

17.840 

96.700 

78.120 

18.290 

96.410 

106.600 

1.580 

108.900 

101.900 

.OOO 

101.900 

90.730 

.OOO 

90.730 

41.400 

.OOO 

41.400 

9.600 

.OOO 

9.600 

2.380 

.OOO 

2.380 

.442 

.OOO 

.442 

.028 

.OOO 

.028 

0.100 

2.67 

.OO 

2.67 

2.67 

.OO 

2.67 

2.47 

.OO 

2.47 

2.18 

.OO 

2.18 

1.95 

.OO 

1.95 

1.000 

.OOO 

1.000 

.I58 

.OOO 

.I58 

.OOO 

.OOO 

.OOO 

.OOO 

.OOO 

.OOO 

.OOO 

.OOO 

.OOO 

0.010 

38.520 

2.950 

41.470 

38.290 

3.030 

41.320 

42.800 

.983 

43.830 

45.400 

.OOO 

45.400 

42.470 

.OOO 

42.470 

22.100 

.OOO 

22.100 

5.140 

.OOO 

5.140 

1.090 

.OOO 

1.090 

.lo9 

.OOO 

.lo9 

.OOO 

-000 

.OOO 

0.050 

6.500 

.000 

6.500 

6.490 

.OOO 

6.490 

5.870 

.OOO 

5.870 

5.060 

.OOO 

5.060 

4.530 

.OOO 

4.530 

2.890 

.OOO 

2.890 

.555 

.OOO 

.555 

.050 

.OOO 

.050 

.OOO 

.OOO 

.010 

.OOO 

.OOO 

.OOO 
FIGURE 3. - Normalized volume of vertical magnetic field contour versus H. 

CONCLUSIONS 

Several normalized vertical magnetic 

field (Q) contours for various overburden 

characteristics (H) have been presented 

graphically. The general shape of the 

contours remains consistent for all values of Q contoured; however, the vertical 

and horizontal ranges rapidly diminish 

with increasing Q, as would be expectd. 

The multilobe characteristic of the 

smaller value Q contours vanishes for Q 

larger than about 0.010. 

The overburden characteristics, as expressed by H, along with mine depth (h), 

have been previously measured at a variety of mines across the United States. 

This data base has enabled a statistical 

description of the distribution of H and 

h for all U.S. coal mines.6 Utilizing 

this, along with receiver detectability 

data, efficient airborne search strategies can be developed for detecting a 

trapped miner's signal as rapidly as possible. As an example, one may visualize 

from the contour plots that the radial 

range on the surface of the multilobe 

contours (e.g., Q = 0.001) is up to 16 

times the mine's depth. However, since 

there are primary and secondary lobes, 

one must be careful not simply to use a 

6~ork cited in footnote 4. 
signal maximum as an indication of the 

trapped miner's coaxial axis (D = 0). If 

a probability of detection has been defined requiring a surface or above-thesurface field strength, Hz min, then the 

minimum detectable Q can be expressed as 

Hz min 

Q = I -b 1. 

where b is the free-space vertical magnetic field strength. This assumes prior 

knowledge of the magnetic moment underground, the mine depth, and overburden 

conductivity. This minimum detectable Q 

will define the physical region of signal 

detectability, which may be determined 

from the contours developed in this 

report. 

NOMENCLATURE 

A ratio of loop radius to mine depth M magnetic moment 

a radius of circular current loop Q normalized vertical magnetic field 

b free-space vertical magnetic field 

strength 

D horizontal surface offset divided by 

mine depth 

d horizontal surface offset 

E electric field strength 

H ratio of skin depth to mine depth 

Hz vertical magnetic field strength 

h depth of loop burial (mine depth) 

I current in loop 

mth order of Neumann function 

vertical height above surface divided 

by mine depth 

vertical height above surface 

dielectric permittivity 

magnetic permeability 

magnetic vector potential 

ohmic conductivity 

amplitude coefficient 

angular frequency 

Jm mth order Bessel function 

k wave number 
APPENDIX.--FIELD SOLUTION DERIVATION 

The regions below the earth's surface following expression derived from Maxwell 

are treated as a homogeneous half-space equations: 

processing an ohmic conductivity as expressed by u and dielectric permittivity E - =p a (i x ii*), 

E. The magnetic permeability p of all 

regions in the problem is assumed to be 

that of free space, p = po'. - -* a ii* a2i* . ii=-V(v.n)+p~- +pc7 at at 

The trapped miner's antenna is considered a circular current loop of radius a, 

oscillating at an angular frequency w. 

The model is shown in figure A-1. The 

loop is located at a depth h below the 

surface and possesses a magnetic moment, 

M, entirely in the direction. The 

geometry of the situation lends itself to 

the use of a circular cylindrical coordinate system with an origin at the loop 

axis. The loop axis divides the halfspace into regions 1 and 2; the two regions are electrically identical, and the 

purpose of this additional boundary is to 

introduce the source term into the boundary conditions. 

The solution of this problem is enhanced by using magnetic vector potentials, ii*, which are related to the 

electric and magnetic fields by the 

Region 0 

Air 

/Earth ............................ =h 

It follows that lI* must be a solution of 

the Helmholtz equation: 

Assuming lI* has a e-i wt time dependence 

and making the substitution 

the above differential equation becomes 

v2ii: - y;ii: L 0, 

where the subscript, i, denotes the geometrical region of the model. The vector 

potential %* is a vector quantity processing only a z component. Solutions 

to the above are of the form 

+ sin m$] [e kiz + e-k z I. 

The J,(Ap) and Y,(Ap) terms are Bessel 

Region I functions of the first and second kind, 

order m, respectively. The separation 

constant m is not restricted to integer 

-------- z=o values. The angular homogeneity allows 

Loop no angular field variations, thus, m = o. 

Reg~on 2 Also, the solution must remain finite 

as p + o, requiring the Y,(Ap) term to 

be excluded. The above solution requires Z the following separation constant 

,P(P. +, 2) k relationship: 

k: = A2 + y:. 

A P The solution takes on the form 

FIGURE A-1. - Dipole buried in a homogeneous 

half-space. ni * = J,(A,,) e +kg z 
where ki is the complex wave number. 

Since the plane wave propagation associklz and e 

-kiz ated with e need not be 

of equal amplitudes, the introduction 

of Pt(A) and Yi(A) generalizes the 

solution: 

and the complete solution is the superposition of all solutions: 

Since ii* has only a component, the 

electric field has only a + component and 

is of the form; 

an* El = i,,%, 

and the magnetic field has only p and 

components: 

In order to obtain explicit expressions 

for the components of E and H, an explicit form for must first be obtained, 

which requires determination of the coefficients Y? (A) and Pi (A) for each region i. This is accomplished by applying 

the boundary conditions requiring continuity of tangential E and H at all boundaries except that at the source. The 

tangential E and H fields are 

The set of boundary conditions requires 

It can be seen in the third equation that 

Hp is discontinuous across the source 

boundary by an amount equal to the 

surface current density j+,(p) (Ampere's 

law). Since the current loop antenna has 

been modeled as an infinitesimally thin 

ring of current, of radius a, j+(p) will 

be the source term for the field. Also, 

by inspection, the coefficients '+:(A) and 

'?:(A) are equated to zero since there are 

no sources below z = o or above z = h. 

The source term j+,(p) may be mathematically expressed as a two-dimensional 

current ring, using the Dirac delta 

function: 

j+(p) = IG(p-a) 

where I is the current. This may be represented by a Fourier-Bessel integral: 

where the expansion coefficient f(A) is 

defined as 

After integrating, 

Thus, the source term becomes 

where n = 1 has been chosen to keep the 

Fourier-Bessel representation of the 

current density in the same functional 

space as that of the Hp representation. 

Although the details are not included, 

the four boundary value equations have 
been solved simultaneously to yield an 2 - -* a2 

explicit expression for Yo(X): Hz, = Yi a22 

IaJ1 (Xa)kle (h-kl )h For the region above the surface, y$ = o; 

Y;(X) = thus, X+k a2 Hzo = - - using the dipole approximation that, for az2 n : 

small ha, 

Jl(Xa) 

ha +l. 

However, the finite size of the loop may 

have an influencing effect on the surface 

and above-surface field structures in the 

limiting case of shallow loop burial and 

large loop dimensions. By disallowing 

the above approximation, the solutions 

may be adapted to account for the loop 

size. However, the dipole approximation 

is greater than 98 pct accurate for A < 

0.10, where A is defined as the ratio of 

loop radius to mine depth. Hill and Wait 

have presented a detailed discussion of 

the finite-sized loop problern.l The rest 

of this derivation will assume the loop 

to be a dipole. 

Then, Y,(X) becomes 

As shown previously, the vertical magnetic field is defined as 

ill, D. A. and J. R. Wait. Analytical Investigations of Electromagnetic 

Location Schemes Relevant to Mine Rescue 

(contract H0122061, Inst. Telecommunication Sci.). BuMines OFR 25-75, 1974, 

147 pp. 

ZrU.S. GOVERNMENT PRINTING OFFICE: 1983605-015152 

In the main text of this report, the surface and above-the-surface vertical magnetic field are referred to as Hz with no 

region zero subscript. 

The results presented are normalized to 

the field strength in free space for a 

coaxial source located at a distance h. 

The normalized factor is contained in b 

and explicitly defined as 

INA b =-, 

271 h3 

where INA = M, the magnetic moment; I is 

the current in the loop of area A and of 

N turns. Hz may be defined as 

Thus, the loss mechanisms of the halfspace and field strength diminishing due 

to observations above z = h are contained 

in Q. If the half-space were free space, 

then Q = 1 at z - h. Q is explicitly expressed as