A minerals research contract report 

May 1982 

Contract J0113058 

National Telecommunications and 

Information Administration 

Institute for Telecommunication Sciences 

BUREAU OF MINES * UNITED STATES DEPARTMENT OF THE INTERIOR 

Minerals Health and Safety Technology 
The views and conclusions contained in this document 

are those of the authors and should not be interpreted 

as necessarily representing the official policies or 

recommendations of the Interior Department's Bureau 

of Mines or of the U.S. Government. 
502Tl-(01 REPORT DOCUMENTATION / 1. "m"T 2. I. R.cipi.nt~* kL...i.," NO. I 

i7.1f~~?~ill and 3.R. Wait 

PAGE ( I 

Theoretical Noise and Propagation Models for 

Through-the-earth Communication 

National Telecommunications & Information Admin. 

Institute for Telecommunication Sciences 

325 Broadway 

Boulder, CO 80303 

12. s~nloriw on.ni2 

2401 E. Street N.W. 

I. R.W* D.t. 

May 1982 L 

16, abslnd (Limit: 200 words) 

A survey of the literature on ELF and VLF atmospheric noise has been 

made. Most measurements have concentrated on the vertical electric field 

and the horizontal magnetic field rather than the vertical magnetic field 

which is of interest in transmission between horizontal loops. The 

limited vertical magnetic field data indicates that the vertical component is generally 10 to 20 dB below the horizontal component. 

Three laterally inhomogeneous earth models have been analyzed for conversion of horizontal to vertical magnetic field. The rough surface 

model examines surface effects, the inhomogeneous thin sheet model treat 

volume effects, and the buried cylinder model treats localized man-made 

effects. All three models predict the possibility of significant conversion from horizontal to vertical magnetic field. 

I A thin sheet model has also been used to examine transmission from a 

buried magnetic dipole source. It it found that the predicted fields (co 

17. ~cum.nt rn.lpis . ~.uwor. 

more nearly model the measured data than does the homogeneous half-space 

model. 11111) 

Atmospheric noise; conductivity; mine communication; vertical magnetic 

dipole; vertical magnetic field. 

1% Av.il.bility St.t.m.nt 

Release Unlimited 

19. %curtly Class (This Rego*) 

Unclassified 21. No. Of Pam, 

45 

" p",*Cl; FYi'e't 22. Price 
TABLE OF CONTENTS 

Page 

ABSTRACT 

1. INTRODUCTION 

2. REVIEW OF ELF AND VLF NOISE 

2.1 Introduction 

2.2 Atmospheric Noise 

2.3 Vertical Magnetic Noise 

3. ROUGH SURFACE SCATTERING 

3.1 Introduction 

3.2 Formulation 

3.3 Perturbation Solution 

3.4 Quasi-static Approximation 

3.5 Conclusions 

4. SCATTERING BY AN INHOMOGENEOUS THIN SHEET 

4.1 Introduction 

4.2 Formulation 

4.3 Solution for cn 

4.4 Vertical Magnetic Field 

4.5 Surface Impedance 

4.6 Conclusions 

5. SCATTERING BY A BURIED CLYINDER 

5.1 Introduction 

5.2 Formulation for the Fields 

5.3 Derivation of the Cylinder Current 

5.4 Numerical Results 

5.5 Conclusions 

6. EFFECT OF A THIN CONDUCTING SHEET ON THE FIELDS OF A BURIED 

MAGNETIC DIPOLE 

6.1 Introduction 

6.2 Formulation 

6.3 Vgrtical Magnetic Field in Air 

6.4 Apparent Earth Conductivity 

6.5 Conclusions 

7. CONCLUSIONS AND RECOMMENDATIONS 

8. REFERENCES 

iii 
FOREWORD 

This report was prepared by the National Telecommunications 

and Information Administration, Institute for Telecommunication 

Sciences, Systems Technology and Standards Division, Boulder. 

Colorado 80303 under USBM Contract Number 50113058. The contract 

was initiated under the Minerals Health and Safety Technology 

Program. It was administered under the technical direction of 

U.S. Bureau of Mines with John Durkin acting as Technical Project 

Officer. A.G. Young was the contract administrator for the 

Bureau of Mines. This report is a summary of the work recently 

completed as a part of this contract during the period June 1, 

1981 to May 31, 1982. This report was submitted by the authors 

on May 3, 1982. 
THEORETICAL NOISE AND PROPAGATION MODELS 

FOR THROUGH-THE-EARTH CO~ICATION 

D. A. Hill* and J. R. Wait** 

ABSTRACT 

A survey of the literature on ELF and VLF atmospheric noise 

has been made. Most measurements have concentrated on the vertical 

electric field and the horizontal magnetic field rather than the 

vertical magnetic field which is of interest in transmission between horizontal loops. The limited vertical magnetic field data 

indicates that the vertical component is generally 10 to 20 dB 

below the horizontal component. 

Three laterally inhomogeneous earth models have been analyzed 

for conversion of horizontal to vertical magnetic field. The rough 

surface model examines surface effects, the inhomogeneous thin 

sheet model treats volume effects, and the buried cylinder model 

treats localized man-made effects. All three models predict the 

possibility of significant conversion from horizontal to vertical 

magnetic field. 

A thin sheet model has also been used to examine transmission 

from a buried magnetic dipole source. It is found that the predicted fields more nearly model the measured data than does the 

homogeneous half-space model. 

Key Words: Atmospheric noise; conductivity; mine communication; 

vertical magnetic dipole; vertical magnetic field. 

1. INTRODUCTION 

During the past decade, a great deal of theoretical and experimental 

research has been conducted on communication with miners trapped within the 

mine workings. Numerous antenna configurations and earth conductivity models 

have been proposed and analyzed. Many of the results are contained in a recent 

summary report (Wait and Hill, 1980a). Also, the systems aspects of electromagentic detection of trapped miners have recently been studied (Lagace, et 

al., 1980). 

A weakness in attempting to compute detection probabilities for trapped 

miners using horizontal loop antennas is the lack of a good data base for 

* The author is with the Institute for Telecommunication Sciences, National 

Telecommunications and Information Administration, U.S. Department of 

Commerce, Boulder, Colorado 80303. 

**The Author is with the Department of Mectrical Engineering, University of 

Arizona, Tucson, Arizona 85721, and is a consultant to NTIA~ITS, Boulder, 

Colorado 80303. 
magnetic noise for the low frequencies (ELF and VLF) which are required in 

transmission through the earth (Lagace, et al., 1980). The primary purposes of 

this report are to review the current state of knowledge of magnetic noise at 

ELF and VLF and to examine theoretical models for the vertical component of 

magnetic noise. Indeed, it is the vertical component of magnetic noise which 

is the interfering component in transmission between horizontal loops. 

Our secondary objective is to analyze earth conductivity models in order 

to explain some trends in measured data for transmission between horizontal 

laops. An analysis of transmission data gathered at numerous coal mines has 

indicated that the usual homogeneous half-space model for the earth is inadequate for explaining the depth and frequency dependence of transmission loss 

(Durkin, 1981). 

The organization of this report is as follows: In Section 2, the past 

work on ELF and VLF noise is reviewed with an emphasis on atmospheric noise. 

A lack of measured data underlies the need for more measurements and a better 

understanding of vertical magnetic noise. Sections 3-5 analyze three laterally 

inhomogeneous earth models and the generation of vertical magnetic noise from 

the dominant horizontal component. Section 3 analyzes rough surface scattering 

and includes the effect of finite earth conductivity. Section 4 analyzes an 

inhomogeneous thin conducting layer which is representative of volume scattering 

effects. Section 5 analyzes scattering by a buried cylinder which could represent a buried pipe, rail, or wire. In Section 6, the effect of a thin surface 

layer of high conductivity on the surface fields of a buried horizontal loop 

are analyzed and compared with measured data from coal mines. Finally, Section 

7 contains the overall conclusions and recommendations. 

2. REVIEW OF ELF AND VLF NOISE 

2.1 Introduction 

The range of frequencies of interest in through-the-earth detection of and 

communication with trapped miners is roughly from 10 Hz to 6 kHz. Thus both 

the ELF (3 Hz - 3 kHz) and VLF (3 kHz - 30 kHz) bands are included. Also both 

surface and subsurface noise are important because both uplink and downlink 

transmission are desirable. Although many noise measurements reported in the 
literature have been made with vertical whips to obtain the vertical electric 

field in air, we are interested in magnetic noise for loop-to-loop applications. 

During the 19701s, the National Bureau of Standards made wideband noise 

measurements both within and above numerous coal mines (Bensema, 1972; Bensema, 

1977). The majority of these measurements were made under operational conditions, and man-made noise was generally dominant. Two of the prominant features 

of man-made noise in an operational mine are the 60 Hz harmonics from power 

lines and wideband impulses from machinery. For mine rescue situations, the . mine will be shut down, and a narrow-band signal located between the 60 Hz 

harmonics will be transmitted (Lagace, et al., 1980). Consequently, man-made 

noise is not expected to be significant and will not be discussed further here. 

Atmospheric noise from lightning is the dominant interferer and will be discussed in detail in the following section. 

2.2 Atmospheric Noise 

Atmospheric noise produced by lightning can usually be split into nearfield noise from local thunderstorms and background noise from distant thunderstorms which propagates via the earth-ionosphere waveguide. A good description 

of the characteristics of lightning and atmospheric noise is given by A. D. 

Watt in Chapter 5 of his book on VLF Radio Engineering (Watt, 1967). The noise 

from local thunderstorms tends to be randomly polarized and highly impulsive. 

Since search operations would probably be temporarily halted during periods of 

high noise activity from local storms (Lagace, et al., 1980), we will consider 

here only atmospheric noise from distant thunderstorms. 

The most extensive reference on atmospheric radio noise is probably CCIR 

Report 322 (1964). It contains worldwide contour maps of atmospheric noise 

levels at 1 MHz for 4-hour intervals for 3-month periods. The maps are in local 

time, and 24 maps are required to cover the 6 time blocks and 4 seasons. With 

each map curves of frequency dependence are given, but the lower limit of each 

curve is 10 kHz. Thus the maps are not directly useful for our frequency 

range. However, Stone and Maxwell (1965), have generated worldwide noise maps 

in Universal Time for 10 kHz from CCIR Report 322. Again 24 maps are given to 

cover the 6 time blocks and 4 seasons. Watt has also reproduced the same maps 

in Chapter 5 of VLF Radio Engineering, (1967). In addition Watt (1967) has 
published frequency dependence curves for the range from 1 kHz to 100 kHz. 

Thus the 10 kHz maps can be extrapolated downward to cover much of our frequency range. Watt (1967) has also published curves of the upper decile, i.e., 

the noise value which is exceeded 10% of the time, for the four seasons for 

frequencies from 1 kHz to 100 kHz. 

Although the 10 kHz maps of Maxwell and Stone (1965) are still the best 

VLF atmospheric noise maps available, they suffer from the weaknesses that they 

are primarily obtained by downward extrapolation in frequency and they do not 

account for the known non-reciprocal earth-ionosphere waveguide attenuation 

rates. In addition, it is necessary to do further extrapolation downward in 

frequency from the curves of Watt (1967) to reach our frequency range (below 

6 kHz). Fortunately there were some atmospheric noise measurements at ELF 

performed during the 1960's and the 1970's because of the interest in long 

distance communication to submarines for Project Sanguine (later Seafarer). A 

special issue of the IEEE Transactions on Communications was devoted to ELF 

communications and, Project Sanguine in 1974 (Wait, 1974), and it contains 

several references to ELF noise. Earlier measurements and noise curves were 

also given by Maxwell and Stone (1963) and Maxwell (1967). The measurements of 

vertical electric field with vertical whips and horizontal magnetic field with 

vertical loops indicated that the E/H ratio was approximately that of free 

space (377 ohms) except between 1 kHz and 5 kHz where the ratio is somewhat 

greater. The region between 1 kHz and 5 kHz is where the dominant earthionosphere waveguide mode switches from the TEM (transverse electromagnetic-) 

mode at ELF to the first order TM (transverse magnetic) mode at VLF (Watt, 

1967; Wait, 1962). A result of this transition is that the overall attenuation 

reaches a peak and the atmospheric noise is at a minimum in the vicinity of 

2 kHz (Maxwell, 1967). 

A recent review of ELF noise by Soderberg (1981) gives an extensive bibliography and lists presently active ground stations with capability for measurement of ELF noise. Most published data of ELF or VLF atmospheric noise give 

either the vertical electric field or the horizontal magnetic field because 

they are the dominant components of both the TEN or the first order TM modes in 

the earth-ionosphere waveguide (Wait, 1962). The vertical magnetic field which 

is of interest in our application will be discussed in the following section. 
2.3 Vertical Magnetic Noise 

Although most magnetic noise measurements have considered only the horizontal component, the National Bureau of Standards (Adams, et al., 1974) and 

the U. S. Bureau of Mines (Lagace, et al., 1980) have made some measurements of 

the vertical component using horizontal loops. The National Bureau of Standards (Adams, et al., 1974) measured both horizontal and vertical magnetic 

noise in a remote area in Utah with no power lines in the vicinity, and they 

were able to observe strong atmospheric noise above the cutoff of the first 

order TM mode. The cutoff frequency was approximately 1.7 kHz at night and 

3.5 kHz during the day. The vertical component tended to be 10 to 15 dB below 

the horizontal component which is consistent with earlier dip angle measurements by Ward, et al. (1966). The U.S. Bureau of Mines measured vertical 

magnetic noise at four frequencies, 630, 1050, 1950, and 3030 Hz, above 27 

mines, and the findings are sumarized in (Lagace, et al., 1980). The general 

trends are in agreement with the published data on horizontal magnetic atmospheric noise, but the levels are lower. Further magnetic noise measurements 

are proceeding in order to learn more about the relationship of vertical to 

horizontal noise and the depth dependence with is important in downlink transmission. 

Although the vertical magnetic field is not normally measured in the 

magneto-telluric method of geophysical prospecting (Keller and Frischknecht, 

1966), the presence of a vertical magnetic field has been noted as an indicator 

of lateral variation in earth conductivity (Sims and Bostick, 1969; Word, et 

al., 1970). At frequencies above 1 Hz, thunderstorms are the primary source of 

energy used in magneto-tellurics (Keller and Frischknecht, 1966). It is only 

below 1 Hz that micropulsations of the earth's magnetic field account for the 

magneto-telluric field. 

A general analysis of inhomogeneous earth models is very complicated 

(Word, et al., 1970), but the physics of the generation of vertical magnetic 

noise can be understood from fairly simple models. In Sections 3-5, we analyze 

three cases of generation of vertical magnetic noise by inhomogeneous earth 

models. In each case we make the following simplifying assumptions which are 

commonly made in magento-tellurics (Keller and Frischknecht, 1966): (1) for a 

given spectral component, that varies as eiot, the exciting field is spatially 

uniform and linearly polarized, (2) earth curvature is neglected, and (3) the 
earth conductivity varies only in the vertical and one horizontal direction. 

Specifically we assume that the incident field has only a y-directed electric 

field E and an x-directed magnetic field Hx and that the earth conductivity is 

Y 

invariant in the y direction. Thus we can treat two-dimensional geometries 

with E-polarization for which only Ey, Hx, and H are nonzero. The vertical 

Z 

magnetic field H is a consequence of lateral variation of earth conductivity Z 

and would be zero for a simple horizontally-stratified earth. Actually an Hpolarized solution with Hy, Ex, and EZ is also possible, but is not of interest 

to us in this report because it does not contain a vertical magnetic field 

component. In sections 3-5, we will be particularly interested in the ratio of 

H to H and the spatial and frequency dependence of H . Z X Z 

3. ROUGH SURFACE SCATTERING 

3.1 Introduction 

In this section we consider a conducting earth model where the earth is 

homogeneous, but the air-earth interface is slightly rough. Our perturbation 

analysis is similar to earlier analyses of periodic sea waves (Barrick, 1971; 

Wait, 1971b; Rosich and Wait, 1977), but here we treat the finite conductivity 

of the earth directly rather than employing the impedance boundary condition. 

In this way we are able to examine the fields in the earth which are relevant 

to the donwlink communication application. 

We make the usual magneto-telluric assumption that the incident field is a 

plane wave at normal incidence (Mann, 1964), and we assume that the surface is 

periodic and two-dimensional. Also we treat only electric polarization (E Y' 

Hx, and H are nonzero) because we are primarily interested in the generation z 

of the vertical magnetic field (H ) by rough surface scattering. Z 

3.2 Formulation 

The geometry of the two-dimensional surface is shown in Figure 1. Initially the surface, z = s(x), is periodic with period L, and s(x) can be 

written as a Fourier series 
where Po = 0 and Pm = P * for a real surface with zero mean height. The 

-m 

region z < s(x) is free space with permittivity E~, and the region z > s(x) is 

earth with permittivity E and conductivity o. Free space premeability po is 

assumed everywhere. 

Figure 1. Geometry of the two-dimensional periodic rough surface, 

Z = S(X). 

The electric field in free space E can be written as a discrete spectrum OY 

of plane waves: 

y C z-i2wmxIL E =e-yoz+Ca m eom 

OY m 

112 

112 where yo = i~(!.i~~~) , cm = [+($r] 
and exp(iot) time dependence is assumed. The term exp(-y z) is the inci0 

dent plane wave normalized to unit magnitude and the summation is the discrete 

spectrum of plane waves where the amplitudes am are unkown at this point. Cm 

is chosen so that E satisfies the scalar Helmholtz equation OY 

The magnetic field components in free space, Hox and HoZ, are obtained from 

E by Maxwell's curl equation OY 

1 a E 

H = and Hoz - - -1 OY , OX iw p az iwuo ax 

0 

The electric field in the earth E is written as a discrete sum of transY 

mitted plane waves 

and amplitudes bm are unknown. Again Dm is chosen so that E satisfies the 

Y 

scalar Helmholtz equation 

The magnetic field components in the earth, Hx and H , are given by z 

1 a E a E 

H 

x =- iwuo 

Y 

az 

and H 

z = iwu0 

-1 2 

ax 

In writing (2)and (5), we are invoking the Rayleigh hypothesis (Strutt, 

1896) which states that only outgoing waves are needed in the trough regions. 

According to Millar (1971), this is valid for sinusoidal surfaces when the 

slope is less than .448 everywhere. This is not a significant limitation for 

our perturbation method which assumes small slope. 
The boundary conditions, imposed at the surface z = s(x), are that the 

tangential electric and magnetic fields are continuous across the interface 

and 

By substituting (2) and (5) into (8) and (4) into (9), the following expressions are obtained 

and 

where no = (F~~/E~) 112 

In obtaining (ll), we have canceled the square root denominators in (9). 

3.3 Perturbation Solution 

In general the solution of (10) and (11) for am and b must be obtained 

m 

numerically (Chuang and Kong, 1981), but for sufficiently small surface slope 

and height we can utilize a perturbation solution (Barrick, 1971; Wait, 1971; 

Rosich and Wait, 1977; Mann, 1964; Strutt, 1896). Specifically, we assume 
that ds/dx, y s, and ys are small first-order quantities. The unknowns a and 

0 m 

bm are expanded in a perturbation series to first-order 

and (12) 

b = bm (0) + bm + , . . m 

The perturbation series in (12) could be carried to any arbitrary order (Rosich 

and Wait, 1977), but a first-order result is sufficient to yield information 

concerning the vertical magnetic field. 

If we substitute (12) into (10) and (11) and expand the exponential 

factors involving s in a power series, then we can equate zero order quantities 

to obtain 

and a (0) = (0) = 0 for m + 0. 

0 bm 

The results in (13) are simply the plane wave reflection and transmission 

coefficients for a planar interface. Substitution of (13) into (2), (4), (5), 

and (7) yields the zero-order plane wave fields 

(0) 

(0) = -bo -7'2 

and Hx 

TI 

Note that the zero-order solution does not generate any vertical magnetic field 
By equating first-order terms in the expansions of (10) and (ll), we 

obtain 

which agrees with an earlier result of Rice (1951). An interesting special 

case is to let the surface be a pure cosine of height D 

Then the Fourier coefficients in (1) are 

P =P 1 -1 = D/Z 

and Pm = 0 for m # - + 1. 

Then (16) reduces to 

and a 

m = b = o for m z + 1. 

m 

From (4j, (7), and (IS), the first-order vertical magnetic field components, H 

OZ 

and H (I), are given by Z 

The factor, sin(Zsx/L), in (19) indicates that the first-order vertical magnetic field is proportional to the surface slope dsldx. 
3.4 Quasi-static Approximation 

In the frequency range of interest here (ELF and VLF), the following 

approximations are valid 

/yly0I >> 1 and y = (iwuoo) 112 (2'3) 

By substituting (20) and (13) into (14), we obtain the zero order magnetic 

field 

and 

Thus the zero-order magnetic field at z = 0 is twice the incident field. and we 

define it according to 

By substituting (20) and (22) into (19), we obtain the quasi-static 

approximation for the vertical magnetic field 

where 

The vertical magnetic field expression in (23) has a simple interpretation. Ho 

is the zero-order horizontal magnetic field at the interface. The factor, 

-(2nD/L)sin(2nx/L), is the slope of the surface, dsldx. The exponential factors 

account for the propagation of the m = 1 mode away from the interface and are 
approximately equal to unity for small 121. The factor F(yL) accounts for the 

imperfect conductivity of the earth. For large ly IL (perfect conductivity), 

F(yL) = 1. For small ly [L (poor conductivity), F(yL) = 0. 

The dependence of the magnitude and phase of F on ly lL is shown in Figure 

2. It is readily seen that lyl~ must be quite large before F approaches 

-2 

unity. To consider some typical parameters, let f = 1 kHz, o = 10 S/m, 

and L = 1 km. Then lyl~ = 8.9, and fromFigure 2, IF/ 0.59. 

Figure 2. Magnitude and phase of F as a function of ly IL. 

As a check for perfect conductivity, when z is small (23) can be written 

When this is combined with the zero order magnetic field in (21), the result is 

that the normal component of magnetic field is zero at the surface. This is a 
required boundary condition for a perfect conductor, and it agrees with 

Ishimaru's result (Ishimaru, 1978) for a perfectly conducting rough surface. 

3.5 Conclusions 

We have derived a first order result for the fields in the presence of a 

two-dimensional rough surface. The vertical magnetic field has been examined 

in detail and is found to have a simple physical interpretation. For a perfectly conducting earth, the vertical magnetic field is proportional to the 

surface slope times the unperturbed horizontal magnetic field. The effect of 

finite conductivity is a reduction by the simple factor F(yL) as given by (24). 

Only the cosine surface was considered in detail here, but the general 

result in (16) for arbitrary Fourier coefficients would allow the treatment of 

arbitrary surface profiles by superposition. Hughes and Wait (1975) have 

treated a two-layer earth with an arbitrary interface profile by perturbation. 

Also we could extend our treatment to arbitrary incidence in the y-z plane, but 

then the solution becomes more compicated because the E- and H- polarizations 

are coupled. 

4. SCATTERING BY AN INHOMOGENEOUS THIN SHEET 

4.1 Introduction 

In the previous section we examined the generation of the vertical magnetic field by surface roughness, In this section we analyze a simple model 

which illustrates the effect of volume irregularities. The model consists of a 

thin conducting sheet at an arbitrary depth in an otherwise homogeneous half 

space. The conductivity-thickness product of the thin sheet is allowed to vary 

in the x direction as indicated in Figure 3. Since the structure is invariant 

in the y direction, we have a two-dimensional problem with two possible polarizations. Only the electric polarization is of interest here, and the nonzero 

field components are E y, Hx, and HZ. 

To simplify the problem even further, we assume that the tangential magnetic field at the interface is a constant. 
Figure 3. Geometry for a thin conducting sheet in a homogeneous half 

space. 

The constant magnetic field assumption is frequently made in magneto-telluric 

problems (Rankin, 1962; Geyer, 1972; Reddy and Rankin, 1973; Wait and Spies, 

1974), and the validity of this assumption has recently been examined by Hohman 

(1981). The primary advantage of the assumption is that only the fields in the 

earth (z > 0) need to be considered. However, the model yields information on 

the fields at the surface in air because of the required continuity at the 

earth surface. 

4.2 Formulation 

The geometry of the conducting half space is shown in Figure 3. The 

conductivity of the half space is oo, and the conducting sheet of thickness d 

is located at a depth h. The sheet conductivity u(x) is taken to have the 

periodic form 
where 6 = 2nlL and L is the period. Free space permeability p is assumed 

0 

everywhere, and d is assumed to be small compared to the layer skin depth. 

The y component of electric field E satisfies the scalar Helmholtz 

Y 

equation 

where y = (iwpouo) 112 

and displacement currents are neglected. The magnetic field components H and 

X 

H are derived from E : 

Y Y 

1 a E aE 

H= -2 and H=LY 

x iwp az z iw p ax 

0 0 

Given the periodicity of o(x), the fields are periodic in x and are of the 

2 general form exp(kF z - iBnx), where r,, = (y + 02n2)112 with Re(rm) > 0. 

n 

The fields above and below the thin sheet can be written in the specific 

form 

and 

where K = iwp IT the n summations are over all integers, and a bn, and c 

n o n' n' n 

are unknown coefficients. From the boundary condition on H at z = 0 in (26) 

X 

we obtain 
1,n=O 

where 6n = { O,n#O 

Also the continuity of E at z = h yields Y 

From (32) and (33) we can solve for an and b in terms of c : 

n n 

The remaining boundary condition is that, in accordance with Ampere's law, the 

discontinuity in H at z = h is equal to the current density carried by the 

X 

thin sheet: 

Substituting (30) and (31) into (35), we obtain 

Using (34) and replacing cos(6x) by its exponential form, we can rewrite (36): 
Since (37) must be satisfied for all x, each Fourier coefficient must be zero: 

From (38) it is seen that the effect of the x variation of o(x) is to couple 

all of the cn coefficients together and to introduce higher harmonics (n#O). 

For the special case of no x variation (A=O), (38) has a simple solution 

In the following section, we examine the solution of (38) for A # 0. 

4.3 Solution for cn 

For n 0, we can write (38) in the following form 
Ad -r h andp =-~e n . n 2n 

We can rewrite (40) in the following form: 

By incrementing n by unity, repeated use of (41) yields a continued fraction 

for c /c~-~. In particular, for n = 1 we have the following continued fraction 

n 

The continued fraction in (42) converges quite rapidly. Once c /c is computed 10 

from (42), co is computed from (38) for n = 0: 

Then all higher order c 's can be computed from (40) by recurrence. Because of 

n 

the symmetry of the geometry, c = c and only positive n's need be considered. 

-n n' 

The convergence of (42) is illustrated in Tables 1 and 2. The magnitude 

and phase of c /c are given as a function of N the number of terms in the 10 T' 

denominator of (42). Only a few terms are needed for convergence, and the 

convergence improves as h is increased. The value of Ad (1s) in Tables 1 and 2 

is fairly large. For small values of Ad, the convergence is even faster and 

I cl/col is smaller. 
TABLE 1. Magnitude of cl/co 

0 20m 50m lOOm 

f = lldlz, Ad = IS, od = 0, oo = 5 x S/m, L = lkm. 

TABLE 2. Phase of cl/co 

f = lkHz, Ad = lS, od = 0, oo = 5 x ~/m, L = lkm. 

\ h 

1 

2 

3 

4 

5 

For small values of od and Ad, we can obtain approximate expressions for 

c and c From (42), c /c to lowest order is 

0 1' 10 

0 2 Om 5 Om lOOm 

-135.0" -139.6" -145.0" -150.5" 

-131.0" -136.3" -142.5' -148.7" 

-131.3' -136.5" -142.5' -148.7' 

-131.2" -136.5" -142.5" -148.7' 

-131.2' -136.5" -142.5" -148.7" 

where n = iwuo/~ . 
For small od and small h, (44) reduces to 

For the same approximations in (39), co and c reduce to 1 

Thus co is approximately the homogeneous half-space value, H and c is pro- 0' 1 

portional to Ad. 

4.4 Vertical Magnetic Field 

From (29) - (31), the vertical magnetic field HZ is given by 

The primary contribution to HZ comes from the n = 1 term in the summation, and 

H can be approximated by Z 
It is easily seen that HZ is continuous at z = h and that the maximum IH I 

Z 

occurs at 

where m is any integer. 

The maximum value of 1~~1 as a function of depth is shown in Figures 4 and 5. 

The maximum always occurs at z = h, and the field has exponential decay for z > h 

and cosh(rlz) dependence for 0 < z < h. The h dependence is shown in figure 4, 

and the frequency dependence is shown in Figure 5. The presence of a constant 

component (ad 0) in the conducting layer always lowers the IH*~ values. 

Figure 4. 

Z (m) 

The maximum value of 1~~1 versus depth for various laytjr 

depths. Parameters: f = 1 kHz, Ad = IS, o o = 5 x 10 s/m, 

and L = I km. 
Figure 5. The maximum value of 1 Hz 1 versus depth for various fre-- 

quencies. Parameters: h = 50 m, Ad = lS, o = 5 X 10 

3 

0 

S/m, and L = I km. 

A further approximation for HZ can be obtained from (46) and (48): 

In addition, if Irl/h and 10Iy lare small, (50) reduces to 
4.5 Surface Impedance 

In many geophysical applications, the ratio of the orthogonal electric and 

magentic fields tangential to the earth's surface is measured. This ratio is 

the surface impedance (Keller and Frischknecht, 1966) and is defined by 

From (30) and (52), Z can be written 

S 

m 

zs = - (Kn/Ho) (an-bn) ~,cos(Bnx) , n=o 

and an and bn are given by (34). Thus Z is even in x, and H is odd. 

S Z 

For the special case of od and Ad equal zero, (53) reduces to 

This is just the intrinsic impedance of the half space. 

4.6 Conclusions 

The effect of a thin conducting layer with lateral variation on the fields 

in a homogeneous half space has been analyzed. A vertical magnetic field is 

found to be generated, and its maximum value occurs at the layer depth as shown 

in Figures 4 and 5. The lateral dependence of HZ is approximately proportional 

to the derivative of the layer conductivity. Thus H is a good indicator of 

Z 

lateral variation of earth conductivity as has been noted previously in magneto-tellurics (Sims and Bostick, 1969; Word, et al., 1970). 
Although the generation of vertical magnetic fields was of primary interest 

in this analysis, expressions were derived for all field components and for the 

surface impedance. This model is simpler than most models for volume irregularities because the thin sheet approximation allows an analytical solution 

whereas numerical solutions are generally required for inhomogeneous half-space 

geometries (Stodt, et al., 1981). 

5. SCATTERING BY A BURIED CYLINDER 

5.1 Introduction 

In the previous two sections we examined the generation of vertical magnetic fields by periodic surface or volume irregularities. In this section we 

consider an infinitely long cylinder in an otherwise homogeneous half space. 

The cylinder could represent a man-made object such as a wire, rail, or pipe 

(Wait and Hill, 1974; Wait and Hill, 1977b) or a natural object such as a long 

ore body. 

5.2 Formulation for the Fields 

The geometry of the buried cylinder in a homogeneous half space is shown 

in Figure 6. The circular cylinder of radius a is centered at x = 0 and z = h, 

the conductivity is ow, and the permeability is pw. The half-space conductivity 

and permeability are oo and po, and the displacement currents are neglected. 

As in the previous section, we assume that the tangential magnetic field 

at the surface is a constant (Wait and Spies, 1974): 

The electric field outside the cylinder can be written in the following form 

where n = (i~"o/~o) 112 Y = (i!Jwooo) 112 

2 2 112 2 112 

r = [X + (z-h) 1 , P = [x2 + (z+h) 1 
Figure 6. Geometry for a circular cylinder in a conducting half 

space. 

and K is a modified Bessel function of first order (Abramowitz and Stegun, 0 

1964). The longitudinal current I carried by the cylinder is unknown at this 

point. The first term in (56) represents the incident plane wave and the 

second term represents the field of the cylinder plus its image. 

The horizontal magnetic field Hx is obtained by differentiating (56): 

It is easily seen from (57) that the boundary condition in (55) is satisfied 

-1 Also (57) has the proper r singularity for small r. 

The vertical magnetic field HZ is also obtained by differentiating (56): 
Thus H is zero in the absence of the clyinder (I=O) and is also zero directly z 

above or below the cylinder (x=O). 

A quantity of interest in magneto-tellurics (Keller and Frischknecht, 

1966) is the surface impedance: 

From (56), Zs is given by 

The first term in (60) is the intrinsic impedance of the half space, and the 

second term is the cylinder contribution which is a maximum overhead (x=O). 

5.3 Derivation of the Cylinder Current 

To obtain an expression for the current I, we apply the usual axial impedance condition at the edge of the cylinder (Wait and Hill, 1974): 

Solution of (61) for I yields 

The term involving Ko(2yh) yields the interaction of the cylinder with the 

interface and has sometimes been neglected (Word, et al., 1970). 

The explicit expression for the axial impedance Za is (Wait, 1959) 
where yw = (iwv o and I. and I1 are modified Bessel Functions (Abramowitz 

W W 

and Stegun, 1964). For small 1 yal , (63) reduces to 

For large /ywal, (63) reduces to 

5.4 Numerical Results 

A computer program was written for the current I as given by (62), for the 

vertical magnetic field H as given by (58), and for the surface impedance Z as 

Z S 

given by (60). Results for IHZ/Hol and IZ I are shown as a function of x in 

S 

Figures 7-9. The curves are shown only for positive x because they are even in 

x. In each case the cylinder parameters are a = 2.5 cm, o = lo6 S/m, and 

W 

vw = 100~~ which could be representative of a rail or pipe. 

In Figure 7 results are shown for the cylinder near the surface (h=2m). 

At the surface, [H I peaks at about x = 2m whereas Z has a minimum directly Z S 

overhead, x = 0. Physically the minimum in Z is caused by a "shorting out" of 

S 

the horizontal electric field E . Also note that the peak in lHzl corresponds Y 

to the maximum slope in I Z I as predicted by (58). For larger depths, 1~~1 is 

S 

seen to be much smaller and to peak at larger values of x. 

In Figure 8, the cylinder is moved to a depth of 90m. At that depth, the 

cylinder has only a small effect on the surface impedance. However, a significant vertical magnetic field is still produced in the vicinity of the cylinder. 

In Figure 9, surface results (z=0) are shown for three frequencies. The 

decrease in H with frequency is due to the increase in the cylinder impedance Z 

Z which is predicted by the approximate expression in (65). This results in a a 

decrease in cylinder current with increasing frequency. Again the peak in 

/H~ I and in the slope of 1 Zsl occurs at about x = 2m. 

5.5 Conclusions 

The simple model of a conducting cylinder in an otherwise homogeneous half 

space has been analyzed. The axial current carried by the cylinder produces 
Figure 7. Vertical magnetic field and surface impedance for a cylinder 6 

near the surface. Param'ters: a = 2.5 cm, o = 10 S/m, pw = -9 W 100 po, h = 2 m, o = 10 S/m, and f = 1 kHz. 
X (m) 

Figure 8. Vertical magnetic field and surface impedance for a cylinder 

at depth. Parameters: same as Figure 7 except that h = 90m. 
Figure 9. Frequency dependence for the vertical magnetic field and the 

surface impedance f r a cylinder near the surface. Pa ameters: 8 -5 

a= 2.5 cm, o w =10 S/m, uw=lOOuO, h= Zm, 0=10 S/m, 

and z = 0. 
strong secondary electric and magnetic fields in the vicinity. The peak in the 

vertical magnetic field at the earth surface coincides with the maximum rate of 

change in the surface impedance. Numerical results have been presented both 

for a cylinder near the surface and at a large depth. 

One logical extension would be to treat multiple cylinders in a half space 

in order to represent an ensemble of wires, pipes, rails, etc. The primary 

difference in the formulation would be that the axial impedance condition for 

the cylinder current in (61) w~uld be replaced by a simultaneous set of equations 

involving all the cylinder currents (Wait, 1977). Also it is easy to modify 

the axial impedance of the cylinder as given by (63) to include a concentric 

dielectric layer to account for insulation or corrosion of the cylinder (Wait 

and Hill, 1977). 

6. EFFECT OF A THIN CONDUCTING SHEET ON THE FIELDS OF A BURIED MAGNETIC DIPOLE 

6.1 Introduction 

The subject of transmitting from a buried magnetic dipole or small loop to 

the surface has been thoroughly studied with respect to mine communication and 

miner location (Wait, 1971a; Large, et al., 1973). Complications such as earth 

layering (Wait and Spies, 1971a; Wait and Spies, 1971b) and loop size and shape 

(Wait and Hill, 1980b) have also been analyzed. A recent attempt to infer 

earth conductivity from multi-frequency transmission at a large number of coal 

mines has indicated that a homogeneous half space is not a good earth model 

(Durkin, 1981). When a homogeneous half-space model is used to interpret the 

transmission data, the apparent earth conductivity generally decreases with 

both depth and frequency (Durkin, 1981). 

Although it is possible to model the transmitted field using purely 

statistical theory (Lagace, et al., 1980), we attempt here to analyze an earth 

model which can explain the physics involved. Our model consists of a highly 

conducting thin sheet located over a homogeneous half space as shown in Figure 

10. This is actually a special case of the thin sheet model conisdered in 

section 4 and can be obtained by setting both h and Ad equal to zero. Although 

we selected this thin sheet model primarily because it appears to fit the 

measured transmission data (Durkin, 1981), it actually has some further justification from earlier conductivity measurements which indicated an increased 

conductivity near the surface (Geyer, et al., 1974). 
Figure 10. Geometry for a vertical magnetic dipole (small horizontal 

loop) in a conducting half space with a thin conducting 

sheet at the surface. 

6.2 Formulation 

The geometry for the thin sheet model and the buried vertical magnetic 

dipole source is shown in Figure 10. The half space has conductivity oo, and 

free space permeability p is assumed everywhere. The conducting sheet has a 

0 

thickness d (which is small compared to the layer skin depth) and a conductivitythickness produce ud. Displacement currents are neglected everywhere. 

The vertical magnetic dipole source (small horizontal loop) has a moment 

IA and is located at z = -h on the z axis of a cylindrical coordinate system 

pz. The fields can be derived from a magnetic Hertz vector with only a z 

component. In the half space, z < 0, the Hertz vector F satisfies the fol0 

lowing Helmholtz equation everywhere except at the source 
where yo = (iWpoUo) 112 

Above the half space, z > 0, the Hertz vector F1 satisfies Laplace's equation 

In the half space, the fields are derived from Fo by (Wait and Spies, 

1971) 

a Fo and E = - 

O$ ap 

Above the half-space, the fields are derived from F1 by 

1 a 2~1 H =-- ip iwpo apaz ' 

a F1 and E = - 

a~ 

The specific expressions for F and F are (Wait and Spies, 1971) m 0 1 

and m - 

2 2 112 where u = (1 + yo) 0 
and R(X) and T(X) are unknown. 

The boundary conditions are that the tangential electric field is continuous at z = 0 and that the tangential magnetic field is discontinuous by the 

amount of longitudinal current per unit length carried by the thin sheet: 

and 

By substituting the field expressions, (76) and (77) become 

Simultaneous solution of (78) and (79) yields 

An interesting feature of the transmission coefficient T(h) in (81) is that the 

dependence on od is algebraic rather than exponential. This type of algebraic 

dependence is typical of thin conducting sheets regardless of the geometry 

(Wait and Hill, 1977a). 

6.3 Vertical Magnetic Field in Air 

From (72), (75), and (81), the vertical magnetic field above the earth HIZ 

is given by 
It is convenient to write H in the following normalized form by setting lz 

g = Ah (Wait, 1971: Wait and Spies, 1971) 

where 

H = h, D = plh, Z = z/h, and T = od(w /o )'I2. Q is dimensionless 

0 0 

and approaches unity as H, Z, and D approach zero. It can be considered a 

correction to the field of a static magnetic dipole. For T equal to zero 

(od = O), Q in (83) reduces to the previous result for a homogeneous half space 

(Wait, 19 71). 

In general the integral in (84) must be evaluated numerically. The case 

of an observer at the surface (Z = 0) directly above the source (D = 0) is of 

particular interest, and numerical results for the magnitude of Q versus H are 

shown in Figures 11 and 12. The curves of IQl versus H for fixed values of T 

in Figure 11 can be thought of as curves of IQ I versus depth h for fixed values 

of w and od. Note that Q approaches unity as H approaches zero regardless of 

the value of T. The curves of IQl versus H for T proportional to H (T = oH) in 

Figure 12 can be thought of as curves of IQI versus w for fixed values of h and od. 

We present both formats for the variation of 191 versus H because the 

variation of field strength with both depth and frequency are of particular 

interest in mine communication and miner location (Lagace, et al., 1980; Durkin, 

1981). The results for T = 0 in Figure 11 and for a = 0 in Figure 12 agree 

with previous results for the homogeneous half space (Wait, 1971). 

6.4 Apparent Earth Conductivity 

Durkin (1981) has presented apparent earth conductivity data which is 

based on loop transmission data taken at a number of U.S. coal mines. The 
H 

Figure 11. Magnitude of Q versus H for various values of T. 
Figure 12. Magnitude of Q versus H for T proportional to H. T = aH. 
depths varied over a wide range, but were always known. Four transmission 

frequencies, 630, 1050, 1950, and 3030 Hz, were used at each site. By normalizing the data, it was possible to determine the value of JQ/ from (83) for 

D = 0. Since 1 Q 1 has a monotonic decrease with H, it was possible to determine 

an apparent value H by assuming a homogeneous half space (T = 0). From H an 

a a 

apparent conductivity value oa is simply 

The surprising feature of the results was that o tended to decrease with both 

a 

depth and frequency. 

Since measured rock conductivities (Parkhomenko, 1967) do not show a 

decrease with increasing frequency at ELF, it appears that the effect must be 

caused by the spatial conductivity variation. The thin sheet model which we 

have analyzed here is probably the simplest model which can predict both the 

depth and frequency effects. 

To illustrate the effects, we can compute Q(H,T) for the actual values of 

o and od and equate the magnitude of Q to that of a homogeneous half space 0 

(T = 0) : 

The apparent conductivity is then determined from H by (85). Table 3 shows 

a 

values of apparent conductivity determined from (85) and (86) for three depths 

at the four frequencies used in the Bureau of Mines measurements (Durkin, 1981). 

We assumed values of o = S/m and ad = 10s. Note that the apparent con0 

ductivity decreases with both depth and frequency. Other calculations for 

different values of o and od showed the same trends. Also shown in Table 3 

0 

are the mean values of apparent conductivity averaged over the various mine 

sites. Since the depths were different at each mine, the experimental mean 

values are ~oughly averaged over depth. Note that the experimental values tend 

to fall between our calculated values of h = 100 m and 200 m. 

Although our thin sheet model does predict the qualitative behavior of the 

experimental values, it does not quite predict the magnitude of decrease in 

apparent conductivity with frequency. However, an examiniation of raw data 

rather than mean values has shown that the depth behavior is easily predicted 

by the thin sheet model. 
Table 3. Apparent conductivity, 0 (S/m) . 0d = 10s and Go = loF3 S/m. a 

Experimental 

100 m 200 m 400 m Mean* 

* (Durkin, 1981) 

6.5 Conclusions 

The effect of a thin conducting sheet on the fields of a subsurface vertical magnetic dipole has been analyzed. The modification of the field expressions due to the conductivity thickness product Ud shows up only in the 

algebraic part of the reflection and transmission coefficients in (80) and 

(81). The particular effect on the vertical magnetic field at the surface is 

shown in Figures 11 and 12. 

This earth model is particularly simple because it requires only two 

parameters, the earth conductivity u and the conductivity thickness product 0 

nd. Although more complicated layered or laterally inhomogeneous earth models 

could be analyzed, this simple thin sheet model is capable of at least qualitatively predicting the frequency and depth dependence of apparent conductivity 

deduced from transmission measurements (Durkin, 1981). Specific results are 

given in Table 3. 

As far as modeling the true conductivity structure at operational coal 

mines, it may be that the thin sheet model is actually good for representing 

surface conductors, such as cables, pipes, etc., as well as increased surface 

conductivity (Geyer, et al., 1974). More detailed earth conductivity and 

transmission measurements would be required to obtain a more sophisticated 

knowledge of earth conductivity structure and of the effect of man-made conductors which can either enhance (Wait and Hill, 1975) or degrade transmission. 
7. CONCLUSIONS AND RECOMMENDATIONS 

This report consists primarily of a review of vertical magnetic noise, 

analyses of the conversion from horizontal to vertical magnetic noise, and an 

analysis of magnetic dipole transmission through a thin sheet model for the 

earth. Each section is self-contained and contains specific conclusions. In 

this section we present some general conclusions regarding noise and propagation models. 

As far as the status of atmospheric noise at ELF and VLF, the thunderstorm 

sources are well known and well mapped (CCIR, 1964), and farily good noise maps 

have been produced for VLF and ELF (Watt, 1967; Maxwell and Stone, 1965) by 

extrapolating downward in frequency from the CCIR atmospheric noise maps (CCIR, 

1964). The main difficulty is that the past maps as well as ELF noise measurements for Project Sanguine (Wait, 1974) apply only to horizontal magnetic noise 

and not to the vertical component of interest. More recent vertical magnetic 

noise measurements by Westinghouse and the U.S. Bureau of Mines (Lagace, et 

al., 1980) provide a good beginning, but further measurements of both horizontal and vertical components both at the surface and at depth would be useful. 

Such noise measurements would help to understand the physics of vertical magnetic noise generation as well as enlarging the data base. Also, accompanying 

in situ earth conductivity measurements would provide further insight. 

Three laterally inhomogeneous earth models were analyzed in sections 3-5. 

It was found that conversion from horizontal to vertical magnetic noise can be 

caused by rough surface scattering (section 3), volume scattering (section 4) 

or by long man-made conductors (section 5). For surface scattering, the vertical magnetic field will be strongest at the surface where the slope is 

maximum and will decay both above and below the surface. For volume scattering, 

the vertical magnetic field is strongest at volume irregularities and decays 

both above and below. In all cases, the lateral variation of surface impedance 

as measured in magneto-tellurics (Keller and Frischknecht, 1966) correlates 

strongly with the vertical magentic field. 

In section 6, an earth model with a thin conducting sheet was analyzed and 

was shown to agree at least qualitatively with the depth and frequency effects 

measured for magentic dipole transmission (Durkin, 1981). The model is attractive because it contains only two parameters, earth conductivity and conductivity-thickness product of the layer, And because the calculations are no more 

difficult than those of the commonly used homogeneous half-space model. 
The optimum goal of developing a single earth model tb predict both the 

conversion from horizontal to vertical magnetic noise as well as the observed 

frequency and depth dependence of magnetic dipole transmission has not yet been 

reached. It is clear that lateral variation is necessary to predict the generation of vertical magnetic noise, but lateral variation may not be required to 

explain the transmission data as indicated in section 6. Probably a composite 

earth model containing both vertical and lateral variation could explain both 

the noise and transmission data. The simplest such model would be the inhomogeneous thin sheet which was analyzed in section 4 for plane wave incidence. 

It would be useful to analyze the same model for magnetic dipole transmission to 

see what effect the lateral variation has on transmission. 

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u.s.GoVERNMENTPRlNTlNG OFFICE: 1982-5788821257 REGION No8