Bureau of Mines Report of lnvestigations/l982 

Electromagnetic Surface Fields 

Due to a Magnetic Dipole 

Buried in a Three-Layered Earth 

By Steven M. Shope 

UNITED STATES DEPARTMENT OF THE INTERIOR 
Report of Investigations 8702 

Electromagnetic Surface Fields 

Due to a Magnetic Dipole 

Buried in a Three-Layered Earth 

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, S. M. (Steven M.) 

Elrcrromapnrric surface fields due to a magnetic dipole in a threelayered earth. 

(Keporr of inresrigations ; 8702) 

Bibliography: p. 20. 

S~tpr. of Docs. no.: I 28.2F870L 

1. Mine rescue work-Equipmrnt and supplies. 2. Elecrromag 

neric waves-Transmission. 3. Mine communication sysrcms. I. 

United Stares. Bureau of Mines. 11. Tirle. 111. Series: Report of invesrigarions (llnited Scares. Hureau of Mines) ; 8702. 

TN23.U43 [TNZYi'I 622s [622'.81 82-600172 AACK2 
CONTENTS 

Abstract ....................................................................... Introduction ..................................................................... Acknowledgments ................................................................ Definition of the problem ...................................................... Derivation of field solutions .................................................. Explicit field forms ...................................................... Numerical evaluation ...................................................... Limiting cases ............................................................ Case B: Two-layered half-space ...................................... Case C: Two-layered half-space ...................................... Case D: Homogeneous half-space ...................................... Conclusions .................................................................... References ..................................................................... Appendix.--Boundary equation set ............................................... 

ILLUSTRATIONS 

1 . Magnetic dipole buried in three-layered earth. case A ...................... 2 . Vertical component of normalized magnetic field for various field layer 

parameters. case A ....................................................... 3 . Magnetic dipole buried in a two-layered earth. case B ...................... 4 . Magnetic dipole buried in a two-layered earth. case C ...................... 5 . Magnetic dipole buried in a homogeneous half-space. case D ................. 6 . Normalized Hz for a dipole buried in a two-layered earth. case B ........... 7 . Normalized Hz for a dipole buried in a two-layered earth. case C ........... 
ELECTROMAGNETIC SURFACE FIELDS DUE TO A MAGNETIC 

DIPOLE BURIED IN A THREE-LAYERED EARTH 

By Steven M. Shope ' 

ABSTRACT 

The Bureau of Mines electromagnetic trapped miner location and communications system requires a thorough understanding of through-the-earth 

electromagnetic wave propagation. An earth model incorporating a magnetic dipole buried in a three-layered earth has been developed. The 

dipole source is located in the second subsurface layer. By application of proper limiting values, the three-layered model is reduced to 

two different two-layered models and eventually to the homogeneous 

half-space model. The solutions are in the form of infinite integrals. 

A numerical analysis was carried out and a computer program written to 

evaluate the surface magnetic fields. Numerical values for the magnetic field at the point above the source are presented in a variety of 

models. 

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

The Bureau of Mines electromagnetic 

trapped miner location and communication 

program requires a thorough understanding 

of subsurface wave propagation since the 

effective design of such a system requires a quantitative knowledge of both 

the propagation characteristics of the 

earth, and the radiational characteristics of the subsurface magnetic dipole. 

To gain this understanding of the propagation, it is necessary to combine this 

theoretical analysis with the results of 

field measurements. 

Subsurface transmission experiments 

have been conducted at a diverse collection of coal mines throughout the United 

States under Bureau contract (8).2 This 

has provided the Bureau with a iarge data 

base upon which the analytical propagational models could be validated. The 

data from these field tests present an 

opportunity to determine the electrical 

conductivity structure at each mine. The 

data consist of the surface vertical magnetic field strength at four frequencies. 

The receiver and source location were 

fixed, with the receiver on the surface 

directly above the source. Recent analysis of these data by the Bureau and a 

Bureau contractor (1) has suggested that 

a model employing tFe dipole embedded in 

a homogeneous half-space does not sufficiently describe the experimental data. 

It is the purpose of this discussion to 

formulate analytical models that include 

a layered subsurface. This will accoinmodate the surface field effects due to major overburden stratification. The development begins with the formulation of 

a three-layered model from Maxwell's 

equations. The source is located in the 

second layer. By applying proper limiting values to appropriate variables, the 

three-layered model reduces to less complex stratifications. 

A series of computer programs were 

written that numerically evaluate the integrals which represent the solutions. 

The programs allow fields to be computed 

at arbitrary points on or above the 

earth's surface. In this report results 

for a number of examples are given. In 

each case the vertical magnetic field on 

the surface directly above the source is 

compared with the field strength that 

would result for a homogeneous earth with 

the electrical conductivity of the source 

layer. Field strengths were computed 

directly above the source to allow initial comparison with field data. 

ACKNOWLEDGMENTS 

This problem was suggested by John University, and David Hill of the U.S. 

Durkin of the Bureau of Mines' Pitts- Department of Commerce, Institute of 

burgh Research Center. The author wishes Telecommunications Sciences, Boulder, 

to thank Roy Greenfield of the Depart- Colo., for their assistance in the prepament of Geosciences, Pennsylvania State ration of this document. 

DEFINITION OF THE PROBLEM 

The half-space model representing the 

earth is comprised of three regions 

of varying conductivity. The boundaries 

are located at z - h, b, and -c, as seen 

in figure 1. For the purpose of this 

2~nderlined numbers in parentheses refer to the list of references preceding 

the appendix. 

derivation, each region is horizontally 

infinite, isotropic, and homogeneous. 

The electrical characteristics of each 

region as expressed by o and E, the conductivity and dielectric constant, respectively, are known to be frequency 

independent for most geological materials 

aside from certain metallic ores and 
frequencies above 1 GHz. Also, the spe- Air .7=h cific magnetic permeability / will "Eort h '//////////////////////////A 

be assumed unity for the subsurface 

materials (2, - - 4). 

The buried antenna will be considered z=b 

a circular current loop posse2sing a magnetic moment entirely in the z direction. --------- 0 z=o 

The geometry of this problem lends itself Loop 

to the use of a circular cylindrical co- I=-c 

ordinate system, with an origin at the 

loop axes. As noted in figure 1, the 

radial loop axis divides the second layer 

into region 2* and 2-. The two regions 

are electrically identical, and the purpose of this additional boundary is to 

readily include the source by a boundary 

conditional at this interface. 

Throughout this paper, the rationalized FIGURE 1. - Magnetic dipole buried in a threeMKSA (meter, kilogram, second, ampere) layered earth, case A. 

unit system is employed. 

DERIVATION OF FIELD SOLUTIONS 

The proper field solutions will be appropriate boundary conditions and 

obtained from Maxwell's equations via charge distribution. Maxwell's equations 

Hertz potentials and application of the for a homogeneous conducting space are 

The analysis of certain electromag- showed it possible to define an elecnetic fields is often facilitated by tromagnetic field in terms of a single 

use of potentials. The derivation of vector quantity i* by the following 

field solutions for the problem at hand relationships: 

will utilize Hertzian potentials. Hertz 

and 
-* 

II nust be a solution to the following second-order differential equation: 

2-* a%* a ii* vn -llE- -po-=~. a t2 at 

-* 

Without any loss of generality, n can be assumed to be a harmonic function with a 

time dependence *of e-IWt. Denoting the solution in each region by subscript i, 

i = 0, 1, 2, 3, n must satisfy 

-* 

v2ii: + w (WE, + lo,) n, = o 

in each region. 

The above will simplify to 

2-* - y ,2n; P 0 v El 

with the substitution of 

7: - w (WE, + io,). 

Solutions to this equation are obtained by the separation-ofvariables technique 

with solutions of the form 

-* (X2 + y12)1/2z - (X2 + y12)1/2z 

n, = [J, CAP) + Y, (A~)][cos~$ + sin ql 1;. (9) 

+ e 

-* 

Immediately, the symmetry of the problem will reduce this solution. Since nI must 

remain finite as p+O, the Neumann functions Y, (Xp) must be excluded from the solution owing to their infinite behavior at p = 0. The angular homogeneity of all regions dictates no angular field variation; thus, 

,* 

- an = o for all i. 

a+ 

This demands periodic functions; hence only integral values are permitted for m. The 

above condition is satisfied for 

hence 

m cos(m +) - m sin (m+) ; 

m=O. 

-* 

The above conditions reduce Ill to the general form 
-* 

Since lIi has only a i component, the vector no tation will be eliminated. Allowing 

the substitution kt2 = X2 + y:, the solution takes the well-known form 

kl can be associated with a complex wave number. Since the fields propagating in the 

tz and -z direction may not be of equal amplitudes, the introduction of amplitude coefficients Yt(X) and Yi(A) will generalize the solution: 

The complete form is the superposition of all solutions: 

Since A is a continuum over the region o + m, the summation may be replaced by an 

integral: 

n, : j J, {y t~)e-~i~ + YT (A)eklz)dX . (15) 

It was previously shown that the electric and magnetic fields could be expressed in -* terms of IIi. Since ii? has only a component, the electric field is 

- an: - 

El - iw- $ , a P 

and the magnetic field is 

It is 2oted that the electric field has 

only a @ component, A whil: the magnetic 

field contains only p and z terms. These 

are the expected field orientations pro; 

duced by a magnetic dipole with a z 

moment . To obtain explicit expressions for E 

and H, one must first derive an explicit 

expression for It:. This requires determination of the coefficients Yi (A) and 

YT (A) for the region i of observation. 

This will be accomplished by proper application of boundary conditions at all 

boundaries.- The boundary condition that 

tangential E and H must be continuous 

will be sufficient to solve for all unknowns. F~om the prevtous expressions 

for E and* H, only the -@ component of E 

and the p component of H are needed to 

apply the boundary conditions. Expressing these fields as 
and 

and utilizing the identity ~ 

the above expressions for E$ and Hp reduce to 

m 

E+i = o! AJI CAP) {Y: (~)e-~l' + yf (A)ek~zl dA (21) and 

P = dm ki A JI CAP) Y (A)e-kiz - y+ (A)ekiz)dA , (22) 

where 'Yi (A) and Yf (A) are as-yetundetermined expansion coefficients. ~ 

The set of boundary conditions requiring tangential and H to be continuous across 

all boundaries follows: 

H~3 - H~2- = 0 @z=-c (23) 

It can be noted in equation 27, at the modeled as an infinitesimally thin ring 

2+, 2- boundary, that Hp is discontinuous of current, j+(p) will be the source term 

across this boundary by an amount equal for the field. No bound charges are preto the surface current density j+ (p). sent anywhere in the model; thus, there 

Since the current loop (antenna) has been will be no electric source terms present. 
The above set of 8 equations contains 10 unknown coefficients: 

T T 7 i T 

Yo (A), Y (A), Y2+ (A), y2- (A), Y3 (A) 

1 

However, some immediate simplifications Before going further, an explicit excan be made: YT (A) : 0 since there are pression for the current density, j l(p), 

no sources below z = -c; also Y$ (A) - 0 existing at z = o must be obtained. The 

since there are no sources above z = h. loop antenna may be described as a twoThis reduces the set to eight equations dimensional ring of current, expressed 

and eight unknowns. mathematically as 

which can be represented by a Fourier-Bessel integral: 

where the expansion coefficient f(A) is defined as 

Owing to the delta function, the integral is readily evaluated to be 

f (A) = Ia Jn(Aa) . (35) 

Thus the source term may be represented by the following Fourier-Bessel integral: 

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. 

The set of eight equations generated 

by the boundary conditions 23-30 is given 

in the appendix (equations A-1-A-8). 

For the purpose of this paper, the surface will be the only region for which 

fields are explicitly derived. Hence, an 

* 

expression for & must be obtained. This 

requires an explicit form for the coefficient %',(A), which entails simultaneous 

evaluation of the eight equations in the 

appendix. The lengthly process associated with this has not been included in 

this discussion; however, the final form 

of Y, (A) is included in the appendix. * Utilizing this, lIo may be explicitly expressed, from which the surface electromagnetic fields will follow. 
Explicit Field Forms 

The form of the vertical magnetic field in region i is 

a2 nZI -y2 -- n: , a22 

which in region o is 

I -koz 

H = [ - *$ Jo (ip) u; (i)e dl . 

Likewise, the form of the horizontal magnetic field is 

an: 

-apaZ* 

which yields the following in region o: 

By including of the expression for Y; (A) into the above integrals, a set of complete solutions may be had for the vertical and radial magnetic fields. The integral 

forms for Hz, and Hpo are 

and 

- koX Jl(~p) IaJl(Xa)T12 [I +R23-2k2~] 
where the Rl and TI terms are the amplitude reflection and transmission coef f icients, respectively, defined as 

kl - kJ 

and 

2kl 

Rlj ' kl +kj Tij = kl +kJ 

Recall that y is defined as 

For low frequencies (tl MHzj, the ohmic currents will be much greater than any 

displacement currents; thus y can be approximated by I 

Obviously yo - o; hence ko - A. 

Utilizing a well-known convention (61, - the fields may be expressed as 

and 

- M 

where b = 2nh3 and M is the magnet- condition). P, and can be viewed as 

ic moment, INA. For observations- di- the transmission loss vdue to the presence 

rectly above the buried loop, b is of a conducting earth. Thus Q and Po 

the vertical magnetic field strength may be written as 

for a nonconducting earth (free space 
10 

and 

The above terms in double brackets may be replaced by 1 for small a, this is the dipole approximation.3 For J1(Xa), when Xa << 1, 2Jl(Aa) + 1. 

X a 

Numerical Evaulation 

The above forms simplify for numeri- depth (h). The following new variables 

cal evaluation if the dimensional vari- are introduced: 

ables are normalized with respect to 

D = p/h B = b/h C = c/h Z = z/h 

Also, 

H is the ratio of source depth to skin The conductivities of regions 1 and 3 

depth. In the low-frequency limit the are normalized to the conductivity of the 

skin depth becomes infinite; thus H = 0, source layer, region 2. These normalized 

and Qo has the value for free space. For conductivities are defined as 

a fixed source depth, H is proportional 

to the square root of frequency. 

and 

%he function Jn(x) may be expanded as follows: 

J (XI = x" x2 + x" 

n 2 nr (n+l) 2 (2n+2) 2 4(2n+2)(2n+4) 

- . . . 1 (50) 
It follows that the wave numbers may be expressed as 

and 

Introducing a new variable of integration, 

the Qo and Po expressions for a dipole source become 

and 
Since the coefficients Rlj and TIJ are 

dimensionless ratios, they remain numerically unchanged by the variable 

transformations. The infinite integrals 

contained in Q, and P, have been numerically evaluated by means of 12-point 

gaussian quadrature. 

The results of the integration are presented graphically, as Q, versus H for 

various numeric values of the parameters 

'12s '231 B, and C. These graphs may be 

seen in figure 2. Only the fields observed coaxially above the source have 

been presented, D = 0; here the radial 

fields are zero. The dashed curve on 

each plot is Q versus H for a homogeneous 

half-space with conductivity a2. Comparison of the other curves with the dashed 

one shows how the more complicated structures affect the variation of Qo with H 

and how the variation differs from that 

of the homogeneous half-space model. 

As seen in the graphs, the presence of 

layering may actually produce larger 

fields than a homogeneous half-space 

would. This is in part due to the reflecting layer below the source. 

Limiting Cases 

From the three-layered half-space solution, field expressions for less complex 

layering can be derived by inspection. 

Figures 1 and 3-5 characterize the cases 

for which solutions will be obtained; 

"case A is the three-layer-earth model 

already discussed. These limiting solutions are obtained either by reducing a 

layer's vertical dimension to zero or by 

letting the conductivity of two adjacent layers be identical; for example, 

02 = "3. 

Case B: Two-Layered Half-Space 

In this model the half-space has two 

layers, with the interface located at 

z = b. In this case, the solution is 

obtained by allowing a3 = a2 ; thus SZ3 +1 

or C + m in the three-layered expression. 

The dipole solutions are 

-kl (h+b) ko (h-z) 

h3 XZT0 lT1ze e J,(XP)~X (59) 

QO = TO/m 

-kl (h+b) ko (h-z) 

h3 X2T01T12e e J1 (Xp)dX 

Po - ~~l 

The above expressions agree with comparison with their work can be made, 

an earlier derivation of a dipole buried the following coordinate transformation 

in a two-layered earth by Wait and would have to be made: z + z + h, and 

Spies (2). However, before a direct b + h - b. 
FIGURE 2. - Vertical component of normalized 

case A. 

magnetic field for vorious field layer parameters, 
n n 

FIGURE 2. - Vertical component of normalized magnetic field for various field layer parameters, 

case A-Continued. 
KEY 

FIGURE 2. - Vertical component of normalized magnetic field for various field layer parameters, 

case A-Continued. 

After a similar variable transformation, as done in case A, the above two expressions become 

(x2+iHZS2 (1+B) 12 x(1-2) 

1 ~~~12~01~ Ja(xD)e dx 

QO = .Z. (61) 

and 
Air z=h Air 

 art h '///////////////////////////, 7zZi .........................

2 ' il 

--------- ---------- 2'0 -------A- 0 z=o 

Loop Loop 

FIGURE 3. - Magnetic dipole buried in a twolayered earth, case B. 

Air 

h'~a rt h '/////////////////,'////////h ' 

---------z=o 

Loop 

FIGURE 4. - Magnetic dipole buried in a twoloyered earth, case C. 

presence of a reflector below the loop, 

it is possible to have surface fields 

larger than the homogeneous half-space 

situation. Conversely, the fields may 

also be smaller, depending upon the 

parameters of layer thickness and 

conductivity. 

Case C: Two-Layered Half-Space 

This model is comprised of an earth 

containing two layers, with the layer interface located at z = -c. The layer located below the loop should serve as a 

reflector, causing larger surface fields 

than would be expected with a homogeneous 

half-space. 

FIGURE 5. - Magnetic dipole buried in a homo- This model is derived from the threegeneous half-space, case D. layered expression by allowing al = a2, 

hence SI22 +1, or by setting B = 1. The 

Graphs of the numerical integration may case C solutions for the vertical and rabe seen in figure 6. Even without the dial magnetic fields are 
and 

J1(Ap)dA 

h3 Po = - jm 2 0 ? - 

After transformation of variables to facilitate numerical evaluation 

FIGURE 6. - Normalized H,for a dipole buried in a two-layered earth, case B. 
18 

and 

Graphs created by the evaluation of the 

previous expressions may be seen in figure 7. Again, depending upon the value 

of the lower layer's conductivity and 

position, the surface field may be larger 

or smaller than the homogeneous halfspace case. By examining the graphs, one 

can see that when 03 is greater than u2, 

the surface field decreases. The phase 

of the reflected field is reversed from 

that of the primary field; thus, there is 

a degree of cancellation at the surface. 

Case D: Homogeneous Half-Space 

As a final example, the homogeneous 

half-space expression may be derived from 

the case A (three-layered half-space) 

expressions. This is done by simply combining the lid ting factors introduced in 

the previous two cases; that is, SlZ2 - 1 

and s~~~ = 1. 

The case D solutions for the vertical 

and radial magnetic surface fields are 

and 

-k2h k0(h-2) 

x 3e J1(Xp)e d X 

Po = h3 Jrn (X+k2) 

As in the previous examples, the variable of integration may be changed to facilita te numerical integration: 

-(x+iH2) ~(1-Z) 

and 

The above expressions for Qo have been cases; thus, no separate graph of the 

evaluated and superimposed as a dashed homogeneous half-space model is included. 

line on the graphs of the previous three 
10 

9 Curve MrloMe 

U ; 523 ; 0.0 

8 Y - 523: 5 W ; 523:50 X : 523 ;I00 

7 

6 

0 .5 

4 0: 

.3 

.2 

I 

023456789lO 

H 

0 

9 curve mlda 

U : 523. 0.0 V . 523. .5 W . 523 = 5.0 X : S23 .I00 

7 

6 

0 5 

4 

.3 

.2 

I 

23456189l0 

H 

H H 

FIGURE 7. - Normalized Hz for a dipole buried in a two-layered earth, case C. 

CONCLUSIONS 

This discussion has shown the electromagnetic surface field solutions for a 

magnetic dipole buried in a three-layered 

earth as derived from Maxwell's equations. Two approximations were made, the 

first being that the displacement currents are negligible as compared with the 

ohmic currents; this is certainly valid 

in the frequency range of interest (less 

than 5 kHz). The other assumption is 

that the loop radius is small compared 

with the burial depth and may be considered a dipole source. 

Even in instances when this may not be 

valid, the approximation was made late in 

the formulation and the final solutions 

may be readily adapted to account for a 

finite-size loop. 

As seen in the three cases involving stratified half-space models, the effects of layering produce nontrivial 

changes in the surface fields as compared 

with the homogeneous half-space model. 

The magnitude of these effects is wholly 

dependent upon the characteristics of 

the layers as expressed by conductivity, 

thickness, frequency, and position. Not 

only are the surface fields diminished, 

but in some instances the stratification 

may cause larger fields than expected. 

The mechanisms responsible for altering 
the fields appear to be attenuation, reflection (transmission) effects of the 

interfaces, and constructive (destructive) addition of fields due to phase 

shifts caused by interface reflections. 

The programs developed will be useful for several purposes--interpreting 

data from past field tests, planning 

search procedures, developing location 

procedures, and predicting the signal to 

noise ratios for calculation of signal 

detection probabilities. However, the 

primary objective of this work is to 

compare existing and ongoing experimental 

data with the various theoretical models. 

From this, it may be found that VF (voice 

frequency) through-the-earth propagation 

may be best represented by one particular 

model in a majority of cases. 

REFERENCES 

1. Arthur D. Little, Inc. Technical 

Support of Through-the-Earth EM Transmission Measurement Program. U.S. Bureau of 

Mines, Final Report, Contract 50188037, 

1980. (Report is available from the author at the Bureau of Mines, Pittsburgh 

Research Center, Pittsburgh, Pa.) 

2. Cooper, R. The Electrical Properties of Salt Water Solutions Over the 

Frequency Range 1-4000 MHz. J. IEEE, 

v. 93, 1946, p. 69. 

3. Jackson, J. Classical Electrodynamics. John Wiley & Sons, Inc., New 

York, 1962, pp. 269-333. 

4. Lange, I. N., and S. M. Shope. 

Microwave Properties of Drilling Fluids. Geophysics, v. 46, No. 3, 1981, 

pp. 322-332. 

5. Stratton, J. A. Electromagnetic 

Theory. McGraw-Hill Book Co., New York, 

1941, pp. 28-32. 

6. Wait, J. R. Evaluation of the Surface Fields for a Buried Magnetic Dipole 

Source. Air Force Cambridge Research 

Laboratories, Contract PRO-Y-71-872, Sci. 

Rept. 52, Feb. 5, 1971, pp. 1-9. 

7. Wait, J. R., and K. P. Spies. 

Electromagnetic Fields of a Small Loop 

Buried in a Stratified Earth. IEEE 

Trans. on Antennas and Propagation, 

v. AP-19, No. 5, 1971, pp. 717-718. 

8. Westinghouse Electric Corp. Reliability and Effectiveness Analysis of the 

USBM Electromagnetic Location System for 

Coal Mines. U.S. Bureau of Mines, Final 

Report, Contract J0166060, 1978. (Report 

is available from the author at the Bureau of Mines, Pittsburgh Research Center, Pittsburgh, Pa.) 
APPENDIX.--BOUNDARY EQUATION SET 

The set of eight equations generated by application of tangential continuity boundary conditions follows: 

By simultaneous evaluation of the above eight expressions, an explicit form for the 

coefficient of interest Y;(A) may be found. Explicit forms of Pi(A) and Pf(A) have 

also been presented, which would be valuable in developing subsurface field expressions; this, however, is beyond the scope of this paper. 

2kl (h-b) 2k2 (b+c) 2kl (h-b) 

(ko +kl) (R23 ] + [%1.12 + 
It may be noted that 

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