ANALYTICAL INVESTIGATIONS OF ELECTROMAGNETIC LOCATION 

SCHEMES RELEVANT TO MINE RESCUE 

PART I - EXECUTIVE S-Y 

Prepared by David A. Hill 

Institute for Teleconarmnication Sciences 

Office of Telecommunications 

U.S. Department of Commerce 

Boulder, Colorado 80302 

PART I1 - COLLECTED REPRINTS - ANALYTICAL INVESTIGATIONS OF 

ELECTROMAGNETIC LOCATION SCHEMES RELEVANT TO MINE 

RESCUE 

Prepared by James R. Wait 

Cooperative Institute for Research in Environmental Sciences 

University of Colorado/National Oceanic and Atmospheric Administration 

Environmental Research Laboratories 

U.S. Department of Commerce 

Boulder, Colorado 80302 

2 December 1974 

Summary Report, Contract H0122061 

Prepared for 

Bureau of Mines 

Pittsburgh Mining & Safety Research Center 

U.S . Bureau of Mines 

4800 Forbes Avenue DO NOT REMOVE 

Pittsburgh, PA 15213 FROM PMSRC FIG 
ANALYTICAL INVESTIGATIONS OF ELECTROMAGNETIC LOCATION SCHEMES 

RELEVANT TO MINE RESCUE AND EMERGENCY COMMUNICATIONS 

Preface 

This document consists of an executive summary and a reprinted 

collection of pub1 ications that have emanated from the U.S. Bureau 

of Mines Contract No. H0122061 with the Institute of Telecomnunications 

Sciences of the Office of Telecanmunications. Most of the material is 

based on preliminary reports that were submitted during this phase of 

the contract period (1971-1974). The Bureau of Mines contract monitors 

have been Howard Parkinson and Dr. James A. Powell of the Pittsburgh 

Mining and Safety Research Center. Valuable guidance was also received 

from John D. Murphy, Research Supervisor in Pittsburgh. Also, we are 

grateful to the National Oceanic and Atmospheric Administration and 

the Office of Telecanmunications of the U.S. Department of Comnerce 

who helped support this research. 

It is stressed that these pubZications do not represent the 

official policy of the U.S. Bureau of Mines or any other agency of 

the U.S. Government. 
ANALYTICAL INVESTIGATIONS OF ELECTROMAGNETIC LOCATION 

SCHEMES RELEVANT TO MINE RESCUE 

PART I - EXECUTIVE SUMMARY 

Prepared by D.A. Hill 

ABSTRACT - This volwne contains a swnnary of the results of past analyses 

on U.S. Bureau of Mines Contract No. lo.0122061 which are relevant to 

electromagnetic location in mine rescue situations. The analytical 

results indicated that detection of a trapped miner equipped with a 

special transmitter is feasible in most situations. Location of the 

miner by surface measurements is also generally feasible within certain errors. The feasibility of miner detection and the accuracy of 

miner Location have been examined theoretically for a wide variety 

of situations. Agreement with experimental results is generatty 

good in cases where measurements are available. 

Personnel involved in the investigation include J.R. Wait 

(Principal Investigator), D.A. Hill (Co-Principal Investigator), 

K.P. Spies, A.Q. Howard, and Lana Slusher (Administrative Assistant). 

TABLE OF CONTENTS 

1. INTRODUCTION 

P 2. CW TRANSMISSION WITH LOOP ANTENNAS 

2.1 Horizontal Loop (Vertical Magnetic Dipole) 

2.1.1 Homogeneous Earth 

2.1.2 Two-Layer Earth 

2.2 Vertical Loop (Horizontal Magnetic Dipole) 

3. CW TRANSMISSION WITH LINEAR ANTENNAS 

3.1 Infinite Line Source 

3.1.1 Flat Homogeneous Earth 

3.1.2 Two-Layer Earth 

3.1.3 Curved Earth 

3.2 Finite-Length Line Source 

3.2.1 Downlink Transmission 

3.2.2 Uplink Transmission 

4. PULSE TRANSMISSION WITH LOOP ANTENNAS 

4.1 Horizontal Loop 

4.1.1 Vertical Magnetic Dipole 

4.1.2 Finite Size Loop 

4.2 Vertical Loop (Horizontal Magnetic Dipole) 

5. PULSE TRANSMISSION WITH LINEAR ANTENNAS 

5.1 Infinite Line Source 

5.2 Finite-Length Line Source 
6. SCATTERING BY OBSTACLES 

6.1 Two-Dimensional Geometry 

6.2 Three-Dimensional Geometry 

6.2.1 Cylindrical Obstacle 

6.2.2 Spherical Obstacle 

6.2.3 Prolate Spheroidal Obstacle 

7. CONCLUSIONS AND RECOMMENDATIONS 

PART I1 - Collected Reprints - ANALYTICAL INVESTIGATION OF ELECTROMAGNETIC 

LOCATION SCHEMES RELEVANT TO MINE RESCUE (follows Part I) 

1. INTRODUCTION 

The purpose of this report is to organize and summarize the results and 

conclusions of past analytical investigations of electromagnetic location 

schemes which are relevant to mine rescue. Mathematical details are included 

in Part I1 that contains a complete collection of reprints of publications 

which have emanated from U.S. Bureau of Mines Contract No. H0122061. References to these specific publications are identified by an asterisk whenever 

they appear in Part I. 

For convenience, the contents of Part I have been organized according to 

application 

the antenna types, 

of the 

sources, 

specific configuration 

and geometries 

to 

involved. 

electromagnetic location is disHowever, the possible - 

cussed in each case. Also, related application of the analytical results by 

other investigators to electromagnetic location is included wherever appropriate. Location here is defined in the broad sense so that electromagnetic 

detection and through-the-earth coarmunications are also applications which 

are included. A brief review of analytical techniques related to propagation 

in the earth was recently given by Wait (1974*). 

2. CW TRANSMISSION WITH LOOP ANTENNAS 

Transmitting antennas that have proved successful in location tests have 

consisted of either single turn or multiple turn loops usually deployed in a 

horizontal plane on mine floors. Normally, the loop is sufficiently small 

that it can be treated as a magnetic dipole. The magnetic moment is NIA 

where N is the number of turns, I is the loop current, and A is the loop area. 

2.1 Horizontal Loop (Vertical Magnetic Dipole) 

The null location method utilizes a loop antenna w~rich is deployed by the 

miner in the horizontal plane and excited by a CW transmitter at a relatively 

low frequency. As shown in Fig. 1, a null in the horizontal magnetic field 

exists directly above the transmitting loops. A small loop receiving antenna 

can be used to search for the null at the earth's surface, and the performance 

of an actual system has been evaluated experimentally in both coal and hardrock 

mines (Large et al, 1973; Farstad, 1973a; Olsen and Farstad, 1973). Essentially the same method has also been used to survey underground quarries 

(Gabillard et al, 1973). 
2.1.1 Homogeneous Earth 

The surface magnetic fields of a small buried horizontal loop (vertical 

magnetic dipole) have been examined analytically by Wait (1971a*,b*) for a 

homogeneous half-space model of the earth. He has shown that the location 

of the loop can be determined from the complex ratio of the horizontal to 

vertical magnetic field at a point on the surface provided that the frequency 

is sufficiently low that the fields have a static-like behavior. Even if the 

usual null search were used, the information contained in the complex ratio 

might be useful in reducing the time required in the search for the null in 

the horizontal magnetic field. The above formulation and numerical results 

can also be applied to downlink transmission by application of reciprocity. 

Although the transmitted field strength is normally computed for a 

specified loop current I, the power required to maintain the specified current 

is also of importance. In order to calculate the required power, the input 

impedance of the loop is required. Wait and Spies (1973a*) have calculated 

the input impedance of a loop above a homogeneous earth and related this 

impedance to the power requirements for a downlink communication system. 

2.1.2 Two-Layer Earth 

Wait and Spies (1971a*) have also considered the effect of earth layering 

on the location configuration in Fig. 1 by computing the complex magnetic field 

ratio at the surface when the vertical magnetic dipole is located in a twolayer earth. The null in the horizontal magnetic field is unaffected, but the 

structure of the fields away from the null is considerably modified unless the 

P frequency is sufficiently low. 

2.2 Vertical Loop (Horizontal Magnetic Dipole) 

Wait (1972a*) has also considered the surface magnetic fields of a 

buried vertical loop (horizontal magnetic dipole). The primary advantage of 

the horizontal magnetic dipole in location is that the overhead null occurs 

in the vertical rather than the horizontal magnetic field as shown in Fig. 2. 

Consequently, the atmospheric noise which has a smaller vertical component is 

less of a problem. The disadvantages of a h?rizontal magnetic dipole are that 

the surface null is a line rather than a point and that a vertical loop configuration may be more difficult for a trapped miner to implement (Farstad, 

1973b). 

Another reason that the analytical solution for the horizontal magnetic 

dipole is useful is that it can be combined with the vertical magnetic dipole 

solution to yield the solution for a magnetic dipole at an arbitrary orientation with respect to the earth surface. Consequently, the effect of a 

tilted tunnel or earth surface on location can bc estimated. Such effects 

have been examined both analytically and experimentally by Olsen and Farstad 

(1973). 

3. CW TRANSMISSION WITH LINEAR ANTENNAS 

The horizontal wire antenna has been shown experimentally to be effective 

for both downlink (Geyer, 1973; Olsen and Farstad, 1973) and uplink (Farstad, 

e 

1973~) transmission. One disadvantage of the horizontal wire antenna is that 

some type of grounding is required at the ends in order to allow sufficient 

current flow. However, Farstad (1973~) has successfully demonstrated the 

use of roof bolts for grounding in uplink transmission. 
3.1 Infinite Line Source 

The two-dimensional infinite line source model has analytical advantages 

over the more realistic finite line source considered later. The two-dimensional model is valid when the wire is sufficiently long and the observer is 

not located near either end. 

3.1.1 Flat Homogeneous Earth 

The subsurface fields of a line source on a homogeneous half space have 

been analyzed by Wait and Spies (1971b*), and numerical values have been computed for a wide range of parameters. The complex ratio of the vertical to 

horizontal magnetic field has been shown to be diagnostic of the position of 

the receiver relative to the source. A location scheme involving two line 

sources, one with variable excitation, has been described by Wait (1971~"). 

By changing one line current, a null in the vertical magnetic field can be 

swept through the earth. Where the miner detects a null in the vertical 

field, he signals to the surface. Such signaling could perhaps be done 

seismically by a harmer blow. 

3.1.2 Two-Layer Earth 

The subsurface fields of a line over a two-layer earth have also been 

analyzed by Wait and Spies (1973b). The numerical calculations reveal that 

the subsurface field structure can be considerably modified by the layering. 

3.1.3 Curved Earth 

The feature of a curved earth has been treated analytically (Wait, 1971d; ? 

Wait and Wilkerson, 1972) by treating the problem of radiation of a line source 

at the surface of a circular cylinder. The radius of the cylinder is chosen 

to match the radius of curvature of the local topography. The calculations 

indicate that small curvatures have little effect on the subsurface fields, 

but that large curvature affects both field structures and magnitudes. 

3.2 Finite Length Line Source 

In order to handle the finite length line source analytically, the antenna 

is subdivided into short pieces, and the total fields are summed numerically. 

The antenna is assumed to carry constant current which is normally a valid 

assumption for insulated antennas grounded at the ends. 

3.2.1 Downlink Transmission 

The formulation of the subsurface magnetic field has been simplified 

for efficient computation for the case of a line source located on the 

surface of a homogeneous half-space (Hill and Wait, 1973a*). Calculations 

reveal that for a cable length roughly twice the obser- .r depth, the fields below 

the cable center are essentially those of an infinite line source. This has 

an important practical implication in that nothing is to be gained in field 

strength by making the cable longer. However, a longer cable will result 

in greater volume coverage. The subsurface electric fields of the same 

configurations have also been computed (Hill and Wait, 1973b*). The electric 

fields are important when reception is with a grounded cable rather than a 

loop (Geyer , 1973). 
3.2.2 Uplink Transmission 

The horizontal wire antenna has also been shown to be useful in uplink 

transmission where roof bolts can sometimes provide convenient grounding 

points. In order to account for possible tilts in either the tunnel or the 

earth surface, the case of a tilted finite line source as shown in Fig. 3 

has been analyzed (Hill, 1973a*). Calculations of the magnetic field components at the surface were made for a wide variety of parameters. These are 

the components of interest in miner detection and location when small loops 

are used for reception. Measurements by Farstad (1973~) for a horizontal 

wire in a hardrock mine have demonstrated good signal detection for antenna 

depths greater than 3000 ft. In fact, the location of the overhead null in 

the vertical field, that may be useful in location, was also shown to be 

feasible in rough terrain. Calculations indicate that the infinite wire 

result is not reached until the cable length is several times greater than 

the cable depth. Thus, the cable should generally be made as long as possible 

to achieve maximum signal strength. 

4. PULSE TRANSMISSION WITH LOOP ANTENNAS 

It is also possible to pulse or shock excite a loop antenna for use in 

electromagnetic location. In this case the loop current is a pulse waveform 

rather than a CW or time-harmonic waveform. The transmitter could be a single 

battery-switch combination or a more sophisticated waveform generator. 

4.1 Horizontal Loop 

The geometry of interest in location is the same as that shown in Fig. 1 

for the buried transmitting loop. An overhead null exists in the horizontal 

magnetic field as it did in the CW case, and a pulse system has been tested 

using the null technique (Sturgill, 1973). However, the waveshape distortion 

which occurs in propagation to the surface contains information on the loop 

location including the depth. No experimental attempt has yet been made to 

utilize the waveshape information. The remainder of this chapter summarizes 

the analytical results available. All the results assume a homogeneous halfspace model for the earth and an earth conductivity which is independent of 

frequency. 

4.1.1 Vertical Magnetic Dipole 

As in the CW case, the solution simplifies when the loop is sufficiently 

small to be treated as a magnetic dipole. The case of an impulsive or delta 

function loop current was treated first (Wait and Hill, 1971*) because it is 

the most basic transient excitation. Calculations of the vertical and horizontal magnetic field waveforms at the surface wcre performed for a wide 

variety of parameters. It was shown that the waveshapes contain information 

on the loop location and that a knowledge of earth conductivity is an aid in 

interpreting the waveshape information. Similar results for step-function 

excitation have been obtained by integration of the impulse response (Wait 

and Bill, 1972a*). These waveforms contain an equal amount of location 

information. An exponential excitation is also of interest since it is the 

current waveform which results from discharge of a capacitor into a resistive 

c loop. Responses for exponential excitation have been obtained from impulse 

responses by convolution (Wait and Hill, 1972b*). The waveshapes are influenced by the time constant of the exponential, but the location information 

is preserved. 
T The possibility of passive detection and location has also been analyzed 

for the configuration shown in Fig. 4 (Hill and Wait, 1973c*). The transmitting loop at the surface sends out a pulse which excites a current in the scattering loop which is set up by the trapped miner. This current reradiates, and 

the receiving loop (or loops) at the surface hopefully detects this reradiated 

field. Calculations reveal that the reradiated signal contains location information but is of very low strength. A mare practical system might allow the 

miner to modulate the loop impedance in some manner while some sophisticated 

signal processing is employed at the surface to increase the signal to noise 

ratio. This idea has never been explored theoretically or experimentally, 

but there still appears to be some interest in passive detection (Farstad, 

1973b). The obvious advantage is that no power is required by the miner. 

4.1.2 Finite Size Loop 

It is often desirable to make the transmitting loop quite large in order to 

increase signal strength, particularly in the downlink case where a large 

area is normally available. In such cases, the usual magnetic dipole approximation may not be valid and the finite loop must be taken into consideration. 

Calculations of the transient magnetic fields (both uplink and downlink configurations) have been made for various size loops by Wait and Hill (1972c*). 

In general, the response waveforms are more spread out and less peaked as the 

loop size is increased. 

4.2 Vertical Loop (Horizontal Magnetic Dipole) 

As in the CW case, there are two main reasons for analyzing the pulsed 

horizontal magnetic dipole. First, it may be a useful source for location 

because it has an overhead null in the vertical magnetic field for which 

atmospheric noise is less of a problem. Second, the solution can be combined with that of the vertical magnetic dipole to yield the solution for a 

magnetic dipole at an arbitrary angle to the earth-air interface. The configuration which has been analyzed (Hill, 1973b*) is that shown in Fig. 2, 

and the loop current was an impulse. The surface magnetic field waveforms 

were computed and were found to contain information on loop location. 

5. PULSE TRANSMISSION WITH LINEAR ANTENNAS 

As with loops, only a small amount of experimental data is available for 

transmission of pulses with horizontal wire antennas (Geyer, 1973). In this 

chapter we summarize the limited analytical results available for downlink 

transmission of pulses with line sources. Similar results could be obtained 

for uplink transmission if the information becomes necessary. 

5.1 Infinite Line Source 

The simplified two-dimensional model of an infinite line source on a 

homogeneous half-space has been analyzed by Hill and Wait (1974a*). The 

subsurface electric and magnetic field waveforms were computed for an impulsive source current, and the waveforms were generally found to be stretched 

out and attenuated as the observer moves away from the source. If desired, 

results for other current waveforms could be obtained by convolution. 

5.2 Finite-Length Line Source 

A finite-length line source on a homogeneous half-space has also been 

considered (Hill and Wait, 1973b*). The subsurface electric field components 

were calculated for a step-function current. The horizontal electric field 
P corresponds to the component measured by Geyer (1973) with a grounded cable 

receiver, and at least quantitative agreement was obtained for the waveshape. 

6. SCATTERING BY OBSTACLES 

Generally the overburden is not homogeneous, and inhomogeneities can distort fields and cause errors in source location. Layered earth models which 

were mentioned earlier mainly provide insight into field strength effects. 

Such models provide no prediction in source location errors because their 

horizontal uniformity does not produce any shift in the overhead null. To 

examine source location errors, models which are nonuniform in at least one 

horizontal direction are required. 

6.1 Two-Dimensional Geometry 

Even two-dimensional geometries can be quite complicated analytically 

because of the effect of the air-earth interface on the scattering by a 

cylindrical inhomogeniety. A problem which has been successfully treated by 

Wait (1972b*) and Howard (1972*) using different methods is illustrated in 

Fig. 5. A line source is located at the surface, and a circular cylindrical 

inhomogeniety (which could represent a pipe, rail, or elongated ore body) 

causes a distortion of the subsurface field. Such distortion could affect 

the feasibility of the location scheme outlined by Wait and Spies (1971b*) 

which relies on the complex ratio of the vertical to horizontal magnetic 

field to determine position. The solutions of Wait (1972b*) and Howard (1972*) 

agree quite closely in a common range of validity, and their numerical results 

P indicate significant location errors for certain ranges of parameters. 

6.2 Three-Dimensional Geometry 

Three-dimensional geometries are of such complexity that either an approximate or a highly numerical treatment is required. A given configuration can 

be made three-dimensional by the introduction of a finite source (such as a 

magnetic dipole) even though the inhomogeneity may be two-dimensional (such 

as a long pipe). For example, Stoyer (1974) has recently treated the effect 

of a half-layer on the fields of a buried vertical magnetic dipole. The 

geometry is shown in Fig. 6, and the fields are three-dimensional even though 

the overburden is two-dimensional. Stoyer's calculations reveal that significant location errors can occur unless the dipole is located far away from the 

layer boundary. The remainder of this chapter will deal with various approximate techniques applied to a number of different inhomogenieties. 

6.2.1 Cylindrical Obstacle 

The effect of an infinite circular cylinder oq the surface magnetic fields 

of a buried vertical magnetic dipole has been examined by Hill and Wait (1973, 

d*). The geometry is as shown in Fig. 7. The conductivity of the cylinder 

is arbitrary, but the frequency is assumed to be sufficiently low that currents 

in the overburden can be neglected. Calculations reveal that significant 

location errors can result if the cylinder is sufficiently close to the interface. A rather complicated treatment of the problem which considers overburden currents and the air-earth interface effect has been presented by 

Howard (1973). 

P 
The effect of a finite-length cylinder has also been considered in an 

T 

approximate treatment. The geometry is similar to that of Fig. 7 except that 

the infinite cylinder is now of finite length and the small loop source is 

replaced by either a finite-length line source (Wait and Hill, 1973*) or a 

long parrow loop (Hill and Wait, 1972*). Calculations again reveal significant location errors when the cylinder is near the surface. Also, the calculations reveal that the cylinder must be extremely long before the results 

approach those of an infinite cylinder. 

6.2.2 Spherical Obstacle 

The effect of a spherical obstacle (such as an ore zone) on the surface 

magnetic fields of a buried vertical magnetic dipole has also been examined 

(Hill and Wait, 1973e*), and overburden currents are included in the treatment. The geometry is the same as that in Fig. 7 except that the cylinder is 

replaced by a sphere. The calculations reveal that location errors caused 

by small spheres are small. 

The above treatment has been specifically applied to the calculation of the 

the shift in the null of the horizontal magnetic field when the sphere is 

near the surface (Hill, 1974*). Such a treatment could apply to man-made 

obstacles such as vehicles or machinery. The calculations reveal some secondary nulls in some cases, but the null shifts are still small for vehicle-size 

obstacles. Farstad (1973~) has observed experimentally that the presence of 

a van has only a very localized effect on the surface magnetic fields. 

6.2.3 Prolate Spheroidal Obstacle 

The effect of a prolate spheroidal obstacle on the surface magnetic 

fields of a buried vertical magnetic dipole has also been examined (Hill and 

Wait, 1974b*). The geometry is the same as that in Fig. 7 except that the 

cylinder is replaced by a prolate spheroid. The prolate spheroid is a useful 

shape to analyze because the axial ratio can be varied to obtain shapes 

ranging from a sphere to an infinite cylinder. Unfortunately, the mathematical difficulties only allow the case where overburden currents are neglected 

and the spheroid is perfectly conducting to be handled conveniently. Calculations indicate that the strength of the anomalous fields increases as 

the length of the spheroid is increased. However, the location errors are 

still small unless the obstacle is close to the interface and in the vicinity 

of the source loop. 

7. CONCLUSIONS AND RECOMMENDATIONS 

The conclusions and recommendations given here are generally in agreement with those of the Working Group on Electromagneti Location Techniques 

(Farstad, 1973b) at the Thru-the-Earth Electromagnetics Workshop, Golden, 

Colorado. 

The homogeneous half-space earth model is well understood for both loop 

and horizontal wire transmitting antennas. Extensive numerical results for 

field strengths and profiles have been generated for a wide range of parameters as indicated in chapters 2 and 3. Some results for transient waveforms 

have also been generated for a few configurations as noted in chapters 4 and 

5. The CW results are generally in good agreement with the available experi- n 

mental results. However, there are almost no experimental pulse data for 

comparison with the theory since pulse location techniques have not been 

pursued. 
The effects of terrain and conductivity anomalies pose difficult analytical problems. Some progress has been made for specific anomalies as discussed 

in chapter 6. Comparison with experimental results is difficult, and careful 

documentation of actual field environments and anomalies would be useful to 

theoretical investigators. One structure which merits some theoretical consideration is the hoist shaft which is normally found in hardrock mines. 

Farstad (1973~) has observed large location errors and large signal enhancements apparently due to a guiding effect along the hoist shaft which will generally be present in hardrock mines. Some analysis would be desirable in order 

to learn how to utilize the guiding effect for detection and communication and 

to minimize the effect on location accuracy. 

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* Hill, D.A. (1973b), Transient signals from a buried horiaontal magnetic 

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* Wait, J.R. and D.A. Hill (1973), &citation of a hanogeneous conductive cyzinr der of finite le~th bya prescribed axial current distribution, Radio Science, 

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* Wait, J.R. and K.P. Spies (1971a), Electromagnetic fields of a sma22 Zoop 

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over a conducting ground, Electronics Letters, 2(15), 346-347. 

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source on a two-layer earth, Radio Science, - 8(8,9), 805-810. 

* Wait, J .R. and R.E. Wilkerson (19721, The subsurface magnetic fields produced 

by a line.currat source on a na-flat earth, Pure and Appl. Geophys., 95 

(111). 150-156. 
WLL POIN FQR HORIUmrAh 

MA6nffP& FEUD 

FIGURE 1 MAGNETIC FIELD LINES FOR A HORIZONTAL LOOP SOURCE (VERTICAL EUWETIC DIPOLE) 
NULL LINE FOR VERTICAL 

MAGHETIC FIELD 

1 AIR 

MAGNETIC FIELD LINES 

FIGURE 2 MGNETIC FIELD LINES FOR A VERTICAL LOOP SOURCE (HORIZONTAL 

MAGNETIC OIWLE) 
EARTH 

GROUNDED CABLE 

FIGURE 3 TILTED GROiiWDEB CABLE SOURCE FOR UPLINK TRANSMISSlON 
TRANSMITTING 

LOOP 

RECEIVING 

LOOP 

EARTH 

FIGURE 4 PASSIVE DETECTION CONFIGURATION 
LINE SOURCE AIR //////-////m 

EARTH 

CIRCULAR CYLINDRICAL &) INHOMOGENIETY 

OBSERVER 

FIGURE 5 TWO-DlMNSIONAL GEOMETRY FOR A CYLINDRICAL ANOMALY 
AIR OBSERVER 

EARTH, u, I LAYER, 

0 LOOP SOURCE 

FIGURE 6 HALF-LAYER OVERBURDEN MODEL WITH A VERTICAL MAGNETIC DIPOLE 

SOURCE 
OBSERVER 

CIRCULAR EARTH 

CYLINDER @ 

0 LOOP SOURCE 

FIGURE 7 CIRCULAR CYLIWR AllOFfALY WITH A VERTICAL MAGNETIC 

DIPOLE ScMCE 
ANALYTICAL INVESTIGATIONS OF ELECTROMAGNETIC LOCATION 

SCHEMES RELEVANT TO MINE RESCUE 

PART I1 - COLLECTED REPRINTS 

Prepared by J.R. Wait 
TABLE OF CONTENTS 

c:18itel~icr for Locatfng an 0noil.Zating Magnetic Dipole Buried irt ihl, F:arth, I 

by J.R. Wait, Proc. IEEE (Letters), Vol. 59, No. 6, 1033-1035, June 1971. 

Electromagnetic Induction Technique for Locating a Buried Source, by 3 

J.R. Wait, IEEE Trans. Geosci. Electr., Vol. GE-9, No. 2, 95-98, April 1971. 

Electromagnetic Fields of a &all Loop Buried in a Stratified Earth, by 7 

J.R. Wait and K.P. Spies, IEEE Trans. (Communications) Antennas and 

Propagation, Vol. AP-19, No. 5, 717-718, September 1971. 

Locating an OsciZZating Magnetic Dipole in the Earth, by J.R. Wait, 

Electronics Letters, Vol. 8, No. 16, 10th August 1972. 

The Electromagnetic Response of a Buried Sphere for Buried-Dipole 11 

Excitation, by D.A. Hill and J.R. Wait, Radio Science, Vol. 8, Nos. 8,9, 

August-September 1973. 

Effect of a Spherical Scatterer on EM Smce Location, by D.A. Hill, 17 

Reprint of Preliminary Report to U.S. Bureau of Mines on Contract No. 

H0122061, with the Institute for Telecommunication Sciences, Office of 

Telecommunications, U.S. Department of Commerce, Boulder, CO, monitored 

by Dr. James A. Powell. Principal Investigator - Dr. James R. Wait, 

14 January 1974. 

Transient Signals from a Buried Magnetic Dipole, by J.R. Wait and D.A. Hill, 23 

Journal of Applied Physics, Vol. 42, No. 10, 3866-3869, September 1971. 

Transient Magnetic Fields Produced by a Step-Function-Excited Loop Buried 27 

in the Earth, by J.R. Wait and D.A. Hill, Electronics Letters, Vol. 8, 

No. 11, 1 June 1972. 

Transient Electrmagnetic Fields of a Finite Circular Loop in the Presence 2 9 

of a Conducting Half-Space, by J.R. Wait and D.A. Hill, Journal of Applied 

Physics, Vol. 43, No. 11, 4532-4534, November 1972. 

Transient Signals From a Buried Horizontal Magnetic Dipole, by D.A. Hill, 32 

Pure and Applied Geophysics, Vol. 111 (19731X). 

Subsurface Electromagnetic Fields of a Line Source on u Conducting Half- 41 

Space, by J.R. Wait and K.P. Spies, Radio Science, Vol. 6, Nos. 8,9, 781- 

786, August-September 1971. 

The Effect of a Buried Conductor on the Subsurface Fields for Line Source 4 7 

Ezcitation, by J.R. Wait, Radio Science, Vol. 7, No. 5, 587-591, May 1972. 

The Electromagnetic FieZds of a Subterranean Cylindrical Inhomogeneity 52 

Excited by a Line Source, by A.Q. Howard, Jr., Geophysics, Vol. 37, No. 6, 

975-984, December 1972. 

The Sub-Surface Magnetic Fields Produced by a Line Current Source on a 62 

Non-Flat Earth, by J.R. Wait and R.E. Wilkerson, Pure and Applied Geophysics, 

Vol. 95, 150-156, (1972/III). 
TABLE OF CONTENTS (cont'd) 

InfZuace of Earth Curvature on the Subsurface Electromagnetic 

Fields of a Line Source, by J.R. Wait, Electronics Letters, Vol. 7, 

No. 23, 18 November 1971. 

Subsurface Electranagnetic Fields of a Line Source on a Two-Layer Earth, 

by J.R. Wait and K.P. Spies, Radio Science, Vol. 8, Nos. 8,9, 805-810, 

August-September 1973. 

Diffusion of Electrmagnetio Pulses into the Earth from a Line Source, 

by D.A. Hill and J.R. Wait, IEEE Trans. AP, Vol. 22, No. 1, 145-146, 1974. 

Array Technique for Electromagnetic Positional Determination of a Buried 

Receiving Point, by J.R. Wait, Electronic Letters, Vol. 7, No. 8, 

22 April 1971. 

Low-Frequency Impedance of a Ciroular Loop Over a Conducting Ground, by 

J.R. Wait and K.P. Spies, Electronics Letters, Vol. 9, No. 15, 26th July 

1973. 

Subsurface EZectrmagnetic Fields of a Grounded Cable of Finite Length, by 

D.A. Hill and J.R. Wait, Canadian Journal of Physics, Vol. 51, No. 14, 

1534-1540, 1973. 

Subsurface Electric Fields of a Grounded Cable of Finite Length for both 

Frequency and Time Domain, by D.A. Hill and J.R. Wait, Pure and Applied 

Geophysics, Vol. 111, 2324-2332 (1 973/X). 

Electromagnetic Surface Fields of an Inclined hried Cable of Finite 

Length, by D.A. Hill, Journal of Applied Physics, Vol. 44, No. 12, 

5275-5279, December 1973. 

&citation of a Hmogeneous Conductive Cylinder of Finite Length by a 

Prescribed Axial Current Distribution, by J.R. Wait and D.A. Hill, 

Radio Science, Vol. 8, No. 12, 1169-1176, December 1973. 

Electrmagnetic Response of a Conducting Cylinder of Finite Length, by 

D.A. Hill and J.R. Wait, Geofisica Internacional, Vol. 12, No. 4, 

245-266, October 1972. 

Perturbation of Magnetic Dipole Fields by a Perfectly C.~nducting 

F'roZate Spheroid, by D.A. Hill and J.R. Wait, Radio Sc~mce, Vol. 9, 

No. 1, 71-73. January 1974. 

Perturbation of Magnetic Dipole Field by a Finitely Conducting CirrmZar 

Cylinder, by D.A. Hill and J.R. Wait, Rivista Italiana di Geofisica, 

Vol. XXII, No. 5/6, 421-424, 1973. 

F'ropagation Under the Earth's Smface (A Review), by J.R. Wait, Paper 

presented by Dr. S.F. Mahmoud at the Symposium on Electromagnetic Wave 

Theory, International Union of Radio Science, London, 9-12 July 1974; 

this is an updated version of review given at the Bureau of Mines 

sponsored Workshop on Thru-the-Earth Electromagnetics at the Colo. 

School of Mines. Golden, Colo., 13-15 August 1973. 
Proc. IEEE (Letters), Vol. 59, No. 6 

1033-1035, June 1971 

Criteria for Locating an Oscillating Magnetic Dipole 

Buried in the Earth 

- . . .. 

that th. comp1.i ratio of the hor#zont.l and th. veniul fldd component. b. mwwrd. The limitation durn to the finit. condunivoty of 

th. overburdmn is 81.0 s0nsid.r.d and shown to b. 8 not insurmountable problem. 

Following a disaster in a coal mine. there is a need to locate personnel 

who may be trapd . . or lost. The possibility that they can be round by an 

electromagnetic direction-finding scheme is intriguing. Unfortunately, the 

waves must propagate through a conductive overburden. Apart from the ~ ~. 

resulting attenuation. there is a distortion of the field veclorswhich can be 

be expected to produce errors. However, as indicated m a recent report' 

on this subjm. the cKsct ofths condvctive environment c !n be minimized 

ifa currentcarrying loop source is used at a sufficiently I.'* frequency. Such 

a method may be a u~ful complement to the highly promising seismic 

location technique being developed elsewhere.' 

In this letter, we consider an analytical model of the direction-finding 

scheme and provide same quantitative eritena that may form the basis 

of working models. Also, hopefully, the results should provide useful 

information to other groups or individuals who aregiving this matter their 

attention. 

Conceptually, the simplest case is wherc we seek to determine the Ition of a small horizontal loop of area A carrying a total current I at an 

Manuscript mcrirrdlanuary 22. 1971; -vised February 10. 1974. 

' Mine Relrvr ondSurr~wl Tlchnipurs. Nnlronal A~adnnyof Enllnlcrine Waahmpcon. 

DC.. 1969. 

'D. B. Lam, Westinphounc Georerarch Labs., Boulder. Colo.. nrirnm communlation. 
I , Observer 

i . _ _ _ . - 

source LOOP '+ 

antular frequency o. Thus we are interested in the detailed strunure of 

the fields ~raduccd on the rt~rfacz with il view lo specifying meaningful 

measurements. To simplify the analysis, we represent theearth ssa homogeneous half-space of conductivity a and prmeability p. Here we assume 

p=p,, where po is the permeability of free space. Also, because of the low 

frequencies involved. we can neglect all displacement currents in the prob 

Iem. The situation is illustrated in Fig. I where the earth's autface is the 

plane r=O and the loop is lwtcd al I= -k on the axis of a cylindriil 

coordinate system (p, 4, :I. 

The mapctic field components H, and H, produd on the surface 

at p=dcanbecaleulatd from intcgralerpr8ssions that havea well-known 

form? The numerical task, however, is not trivial sine the distanos involved may be comparable with the effective wavelength or skin depth in 

the conductor. Thus the otherwise erhaustive asymptotic and near-field 

derivations scattered throughout the literature could not be used directly. 

Here we rcsm to a numerical evaluation of the integrals but utilize the 

static (i.e., o-0) results and other spsial cass as a check. Details ofthe 

aoalysis and computations are to be given clacwhere. 

The desired field mmponmts at p=d and i=0 are given explicitly as 

where P and Q arc convenient normalized quantitia. The latter are to be 

obtained from 

x3 L] + (x' + iH1)* cxp [-(XI + iH2)*] {d (3) 

where Jo and J, m Bavl functions, H =(opw)*h, and D= plh. 

The ratio of thc magnitudes of H, and H, at the observing station is 

simply lPllQl or IPlQl. This ratio is plotted in Fig. 2 as a function of the 

distance puamtcr D. Not surprisingly, this function rises from n sharp 

null at D-0 to a maximum that is near D= 1.4 for the small H values in 

accordance with the expected static behavior ofthe dipole 1i.e.. a static or 

dc theory prodins Q=O at D-&). This pronounad maximum is blumd 

as H incrc~scs and to a frrst order. it shihs to smaller valua of D. Nevntheless, the cum- in Fig. 2 indicate that, for a single mcssuremsnt of 

IHJH,( or [P/Q(, two admissible values of D can be ascertained. 

The phuc differma between H, and H, is a quantity of interest since 

it can be measured at the okver station without any phase reference 

Fig. 2. The msgna#udsollhr ratno ofthe honzontnl and lhc vertical mawtic f~cld. 

from the soura. This quantity is of course equal to the argument of the 

ratio PlQ. It is plotted in Fig. 3 as a function of D for the same range of H. 

Clearly, we act that, for sufficiently small H. the phass is near either 0 or 

183". depending on whether D< 1.4 or D> 1.4, rcapclively. This fan permits us to eliminate the ambiguity from the sing* measurement of the 

amplitude ratio alluded to in the foregoing Thus, in principle, we can 

deduce the appropriate value of D from the measurement of the complex 

PIQ. Of course, if h is known, m then can hnd the desired radial wordinate p. 

In our discussion hew we have required that H be a small parameter m 

if the conductive effects in the overbukden are to be minimized. In tmns of 

actual qyntitia. this can be written 

H = z&/ h,. x LO-' 2 O.M-/Z, 

' J. R. WUL. "EkUom-tk IOU- in low&.." in lllruvl Thw. PC. &Collin 

.nd luster. Ed. New Yort: MeOmv-HilI.1969. oh. 24. 

where 0. is the earth conductivity in millimhcx per meter, / is the frequency in hertz, and ht. is the burid depth h eapmwd in kilometers. For 
example. tf o,= I. /= 100. and h..=0.2. we sn that Hk0.17 wh~ch tr 

.~n,,,t~~~l! small that the fields should have a aalic-like behavior. 

The results given here. while not at all exhaustive. indicate lhal directton-finding possibilities definitely exist for a buried magnetic diwlr 

source. provided the field structure can be measured a1 a single obseruation station on thesurface. Conversely. wecan rcversc the roles oftherourcc 

and the observer and locate lhe rccenring apparatus in the earth and transma from agrounded loopan the surfaccTt~elatter may bea more practical 

scheme from the operalional point of vlew. since the lo-be-rescued miners 

would lhen need only to ulilizea cross-l<wpreceiver with su~tableswitching 

prav~sions lo measure botb the field ratios and the phase differences. Also. 

ofcourse. the verltcal loop lie.. horizontal axis1 can he oriented to give the 

radial direclaon toward the source For feasibility tests, however. it is 

recommended that the loop transmitter he buried and the field quantities 

be measured on the surface. From an experimental point of view, this 

would snm to have major advantages in the scope of the measurcmenls 

and the variety of silualions to be examined. 

Ana1)llcal ,ludna on thtr sub)cct are conllnutng' Among some of the 

problems bclng J8uud arc IhecRm ofa urattfied and nununlform over. 

hurden. other dipole and line-source cnnfigurations, and the transient 

solutions br pulsed dipole sour=. The work bean a close simllarity lo 

earlter investigations in geophysical exploration using elalromagnctic 

methods.' The configurations arc sufficiently diffcrcag however. that the 

previous numerical dam cannot be used dimly. 

A~KNOWLEW~MENT 

The author wishes to thank K. P. Spies for his numerical analysis and 

general assistance in this task. 

1~rne.s R. W~tr 

Inst. Tclsammun. Sci. 

Boulder, Colo. 80302 

' ~hrunudr. arr bclngcunod out undsr~erk S ofs contract fmm @he US. Bvmu of 

~ina to UKO~*~ orTclrcommutliulions. ' c. V. Koller md F. Fnrhkmsht. Elrm,r.l Mahds in Glophysiml Prospctmq. 

Oxford. Endand: hemon. 1967. 

IEEE Trans. Geosci. Electr., Vol. GE-9, No. 2, 

95-98 (April 1971) 

Electromagnetic Induction Technique for Locating - 

a Buried Source 

JAMES R. WAIT, FELLOW, IEEE 

Abslracl-The Bald structure of a buried vertical oscillating mag- ing this tltesis are currently 1,eiltg sl,onsoreri hy the 

netic dipole is examined. POI an 0b.ervsr On the earth's surface, it [J. S, Bureau of 3!lilles. In this paper, \vitllnut ;l,Iy alailll is indicated that the Bnite conductivity of the earth will modify the 

geometrical character of the vertical and horuontd magnetic field to originality, I wisll to present the i,;lsic tlleory of ;I 

components. H~~~~~~, this efiect is small if burill depth is tecllniquc that ntny itold sotne prntl~ise. , . less than an electrical akin depth. I he specific ~~llfi~llr;lt~~t~ corlsidererl is ;i sr11;111 wirr 

loop contained in a h~~rizott~el plane and 1,uried at s,,~tlc 

INTROINJ~TION depth h 11c11c;ttll tlte rartlt's surf:lce. 'l'llc: r,l,jectivc itvrc 

T IIl:l<E 15 A possibility that elertrotnagtretic is to ex;lt~ti~tclte slrucfttre of tl~i: clcc.tro~n;~x~~c~i<- fic.l<l 

P \vavcs c;ut be elttployed to locate coal 111i11ers on tlte earth's surf;u:e prc~lur:od iry it11 ;~llc.rlt;ttirt~ curiollo\\-ing ;r disaster. A numl~er of studies support- rent ir~jectccl into II1e loop. 'I'l~e sitt~iitiolt is ilitrstr:l~ell 

in Fig. 1 with rusl~c*:~. to a c~vlirtdri~.;tl i.c,ortlizl:~~<. svs1t.111 

(p, 6, e). The cartll is reprcsclttc:,l ;is :I IIOIII~,~<:II~,,IIS M;,nuw-ript received Jaw~ary 11. 1971. 

The attthor is with the Iostitote for l'elecon~musication Sciences, half-space of cottductivity a. All dis11l;u:enient curret~ts 

Onice 01 Telwomrnunicatiot~s, Boulder. Colo. 80302. can be neglected because we may assunle that all linear 
leET TRANSACTIONS Cm OeoaClENCE ELECTRONICS, APRIL 1911 

Fig. I. Ceumnetry of buried Imp. 

distances involved are small compared with the freespace wavelength. 

FORMULATION AND SOLUTION 

Considering the loop, whose areaxturns product is 

IA, as a vertical magnetic dipole, we can derive the 

resultant magnetic fields in the air (i.e., s>O) from the 

following [I]: 

ifiowH, = d'F/(apds) (1) 

and 

irwH, = d'F/aza (2) 

where p, is the permeability of free space, w is the 

angular frequency, and Fis ascalar function. Astraightlorward boundary value analysis leads to the following 

integral form [I]: .A* --h 

.e e 

where u = (X'+ya)), y8=wpow, and Jo is a Beesel function of order zero. 

(:sing (1) and (2) along with (3). we find that the 

fields at the location (p, 8) for a > 0 are 

and 

H. - bOQ (5) 

where 

and 

and the normalizing factor is - 

bo = IA/(2rha). (8) 

By changing the variable of integration to x =Ah, the 

integrals P and Q are expressible in the form 

and 

where H= (apow)l". D =p/h, and Z=e/h are dimensionless parameters. Equations (9) and (10) are in a form 

suitable for nulnerical integration, and a computer program has been prepared to accomplish this task. Before 

discussing the results, some special limiting cases will 

be mentioned. 

When the conductivity a and/or the frequency w are 

sufficiently low, the parameter H is ei7ectively zero. 

Then, without difficulty, the integrals P and Q are 

expressible in the closed forms 

and 

whereR= [Da+(Z+l)a]l~a. These simple results for the 

normalized magnetic fields P and Q correspond to static 

conditions. 

The other limiting case of interest is wllen tlie roriductivity a and/or tlie frequency w are sufiriently high 

that the integrals P and Q nlay be approxinialed in an 

asytnptotic sense. I*:ssentially, tllis ;~n~ounts to arguing 

that when lypl >>I, the exponential fartor exp( -uh) irl 

(3), (6), and (7) may be adequately approxinlatrd hy 

exp(-yh). Then, treating both H and D as large parameters, we find that 

P - - ei.14(3/H)P4 exp [-e"/'H] (13) 

and 

Q - + e"/'(O/H')Vh exp I--ei'14Nj. (14) 

A preciae description of the validity of these aeyrnptotic 

forms is subtle, but on coniparimn with numer~cal integration on tlie exact forms, it appears that they are 

useful when 110th tl and HD are >>I. 

Another interesting check is to let h be identically 

zero. 'Then for z = 0, we I~avt. [I 1 
fl where we have n~de UK of a known identity. 'Shell, 

wain for or - 0, we have 

R, - aF/ap - iwIA -.-- 

2r yap' 

3 

u 

(17) IPI !ik 

01 

where o I 

a - 1 - [l + yp + (y'p'/3)]~". (18) 0 -- 

0 01 10 15 I0 I1 30 

D 

If we I >" and the approximate Fig. 2. Horizontal component ol normalized magnetic held cm (Leontovich) boundary condition [I], we have surlace as Iunctiun of distance parameter 1). 

This is consistent with (1-3) given above if we note that 

the right-hand side of (20) is lim~,~ boP. Also, we can 

obtain an exact expression for H. at s=O by using 

ip~~. = - (~/~)/(a/ap)p~+ and (17); thus H, 101 

= - IAp/(2qYp")@ where 

Again, if IypI >>I. @1, and (21) is consistent with (14). 

As a matter of theoretical interest, we have one final 01 

special case which deserves mention. Specifically, we 

01 take p = 0, which means the observer is directly above r- the source loop. Then, of course, P=O, but Q is finite. l!Lz-l Oo os 10 15 ,a is -.-.jo 

If, in fact e =0, we have D 

Fig. 3. Vertic:tl component 01 normi~lized in;tgneti~. lield r,u .~trlitre 

Q = Q. = haI'A*b + A)-l~"h~. (22) as functio~~ 01 distance paritmeter I). 

0 

Now. where KO is a modified Bessel function of order zero. 

IJsing only the well-known identities K/(Z) = - K,(Z) (U + A)-' = (I( - A)(wa - A2)-I = - A), and K,'(Z) = - [Ko(Z)+Z-IKI(Z)], it turns out that 

and therefore (24) can be reduced to the compact form 

We can now verify that 

. - (24) ah' ah6 Jk' ah 

where 

and 

(26) 

n 

These latter two integrals are known to be 

and 

where x=yh=dZH. 'l'llis provides an alternative 

method to compute the field directly illrr~ve the sour1.r 

dipole. 

As indicated al~ove, P and Q are 1111: norn~alizccl I.OIIIponents of the horizontal and the verti1.;11 i~~;rgi~ctic fi,.l(l 

on the surfiice for the 1,uried loop sourre. 'l'l~e III;IUII~- 

tudes I PI and 1 Q/ plotted as a function of tl~c! i~r,r~~,:llized distimce D (or p/h) are illustrated in I'igs. L :tlirl 3, 

respectively. 'rlie par;lnleler on the <:llrvc. is the i~or~~l;rlized depth paranleter II (or (ap,w) "'h). 'l'his c;rr~ I,(. ,.;,Irulated from the si~~~l)le forn~ul;; I1 =0.0X6(a,.,,..h,,J')'~'h,,.. 

where amr,,h. is the earth ror~durtivity ir~ n~illii~il~os 1x.r 

meter, f is the frequency in hertz, ;lrld hk., is the l,uri;~l 

depth in kilon~eters. I'or example, ifa ,.,., h,, = I. j= I():, 

and ht,=0.2, we have N=0.17. In this case. we ran 

make the ratl~er i~nportant observation tl~at the c-l~;~racter of the I PI versus D curves in Fig. 2 are essentially 
the same as for the cane H=O. Thus, the conductivities 

of the overburden are probably negligible for this configuration of the receiving loop provided the burial 

depths are not more than 0.5 km and the operating 

frequencies are in the low audio range. It is significant, 

I.owever, that the vertical magnetic field component 

as indicated by the I QI versus D curve in Fig. 3 has 

a significant modification in the region near D = 42. 

Here, the static field (i.e.. for H-0) has a null which 

is smeared out wen for small values of the parameter H. 

This effect, however, is still not detrimental to the 

operation of a detection scheme. 

While there are many other aspects of a practical 

scheme for using the induction field for source location, 

we feel that this approach is worthy of critical examination. Furthermore, by applying some simple notions q 

of reciprocity, we can adapt the present results to the 

situation when the source loop is located on the surface 

and the receiving loop is located at depth. Other factors 

receiving our attention are the influence of an inl~on~ogeneoun overburden and the effect of irregular terraill. 

The author wishes to thank K. P. Spies lor his assistance with the numerical calculations and Dr. D. B. 

Large for several stimulating discussions. 

KEFERI~NCES 

[I] J. R. Wait, "El 81lctir fieldsol sc,urces in Ik*sy media." in 

~nlcnno ~eor~t~niY~k. E. Collin and I:. Zurker, Eds. New 

York: ~c~raw-Ail, 1969, ~h. 24,pp.468-4kl. 
IEEE Trans. Antennae and Propagation, vol. AP-19, No. 5, 

(Communications) 71 7-71 @ (Sept. 1971) 

r'. 

Electmmyaetic Fields of a Small Loop 

Burled in a Stratified Earth 

AbrhacI-The deld itruchue of a burled vertical osctllatlng 1.0 -P 

nugnetlc dlpolo L onmlnod. Tho earth li represented a8 a twohvoz h.Lf-.~.ce. md the nonrce In located in the bottom semi- \ ? "I 

i&te region. .~iing aumeriul integration, the magnitude of 

the ntlo of the horizontal to the vdul nupnetlc Bold L enmined 

for m ob.nrer on the earth's aurlace. It L ahown that at sufficiently 

low frequsnciea, the ddd ntio & not apprhbly modllled by the 

Wta coodactirlty md the geometry 01 the Iaysrw. 

Flu. 1. Magnetle dlPOls located within two-layer stratified earth. 

Communication to the earth's surtaes from a buried transmit,ter 

may be dl11 in emergency nituations such as after a mine dinmter 

[I]. Unfortunately, the mnduclive lmses in the intervening rock 

will weverely attenuate the signal unless extremely low frequencien 

are employed. Ah, dinrtiol~ finding of the buried nourn, from t.he 

surfam obarver, in inhibited by the wnductive overburden. 111 

thk mmmuniation, we'wish to wnaider an analytical model 

which may shed mme light on this pmblem. 

As indicated in Fig. 1, we consider a vertical magnetic dipole or 

small bop (with vertical axis) located at a depth h beneath the lp/Ol 

surf- d a flat &h. The current in the Imp ia taken to vary as 

exp (id). To simulate the enact of an inhomoge~~eoua earth, we 

adopt a two-layer model. The upper layer has a conductivity 0, 

and a thirk~~cas b The semi-infinite lower layer, which contain* the 05 ..--H/I~ #I 

muroe dipok. b a wnd~tctivity a. A cyli~~driral m,rdi~lata .... a7 ""4 ,\\ 

system in chwn M illtutnted in Fig. 1 with the obnerver 

lorated at Plp.2) on or above the auth'a stwfnce at, r - 0. It, i* our ... . ~ 

05 10 15 20 25 $11 $5 40 objective to evaluate the ma/netic field compunalta H, nnd H, at n ~6,s) for a spsrikd total current I in the~snmll loop of wen A - 

beatedata - -h. Plg. 1. Pletd ratlo us narlm for varloua vsluna of I1 - lv,#w)~~zh sndBII1 -... ~~, ~~ - bihforflxed . ,.. ~~ ~ . K - In,in.l~!r , 5 

The formal mlutwn lor the pmblem M stated was given some 

time am - r27. - dtho~uh ie the origiaal derivation, it ww w~imed -. 

that Uls magnetic dipole wan in the air above a two-layerst~mlified IW .- ~- 

dh. Clsuly, the mlution b the ~me lorn w lor the problem 

pd hen. Ah, M indicated before, all dinplacement current8 

can be nqlkcled if the lm-spaca wavelength is much greater than 

the signifipsnt dimensions of the problem. We mume this is the cw 

h. Ah, to simplify matten, we sssi~me that both of the wnductive layen have magnetic permssbilitia which ars t,he name 

M for free apace. Thua, without difficulty, we find thst I PIP I 

H, - b8 (1) 

and 

H. - bA (2) 

.. . .. when b. - IAl(2rh8) irt a nuitsble normalizing factor, while P and 02 HIH it5 

Q are the dimennionlm f/dld qwntitiw. Under ths ~*anlpli<~na 

dated pmio~uly, we have the intrd formul~ I1 

P - [ hsT(h) exp (-A@,(*) rR (3) Flu. 8. VInl11 rntlo fvr V&FI~,IIIT v<t"~ uf 11 aw~d li <UP IIXWI II,II - 

1,11t - o,n. 

and 

wh @lll(-e 61f Lha II~IIIKVI~IN i* IWMII~IUIny 1110 IIXIII,III.~~II,~~ ri~,I#,r it, 1111. 

lllllrleral#rr 111 11111 ~.~~JII-IIIIIIII *i~lf% 111 1:). AINO, WI. 1,018. II~,~I 1,101~11 

() %r.s flll,l~1.il,l~n t>f (Im ~Ii~n~~n~it~nIutw IB~~~,&I~~#~~+.I.x It Inlrllwllllb, 

r11 - (rt&o~)~~~l!,, 1) - p/lt, Z -r ~111, INI~~I ti -- A ~II~II~I 

~(h) - I~,II~IIII~II~!ruyrlrnl II~L* I#~.~!II w1.il11.18 111r I.VIIIIIIIIIIIK I' 111111 0, I,IIII (" ru~~lll~ bnvs II~U~II i~ld!a.i~o~vtl for II wit111 rltt~yt, 111 11~1. 1~~tt~~n1to~1+~rtt L:I ) 

[I - (:s<&-z) exp (-2uLb)] A I~IIILI~I,~~~ t~f ~pwi,tl lt~l~*rvrt, i~ III~I rl~ttyl~il~ltl~' '11 II~,, 1111111 11,/11. 

1r1. a - O fit1 1l1n t%#~rll~'w 111111111.11. ')Pur ~tt~~y!~jl~~O,, 11~1~ ~,LII~, I" 
Flp. 4. meld ntlo fw variuu. valua of H fw llxed K - (et/ot)'d* - 5 

and B :H - bfI - 0.6. 

rult ta verify that, for z = 0, (3) and (4) reduce to 

P - (3/2)DR' 

and 

Q = (1/2)[3P - R'] 

rhere R = (W + 1)"'. The curw in Fip. 24 are mnsistent with 

lhese static field formulw for the smJl vdues of H. In particular. 

we LR U~athe 111t11 for Q at D = fi kds to an bfinity for 1 PIQ I. 

Thii pmno~mnd pe&, however, is mmewhrt blurml when H 

is diRerent fmyn zero. In faet, the behavior of the ratio ( P/Q I becomes modified drastically for the - v.lw of H. 

To mltsider a spnrifir exampde, we choone f - o/2r = 10' Hz, 

o, lo-' mblnl, and b - 200 m; thus, H = 0.17. In thi cane, 

the upper i..~ of thickness (BIH) X 200 rn, hsa a conductivity 

R' x 10-* rnbulnl. As indided in Fi 2 and 3, the preaenw of 

thia mwkin# Iayer deterbratea the ided static-like behavior of 

the field ratio I P/Q 1. 

Using the atmlytienl results from this study, it is pogsible to 

elucidate the cRert of the stratification in the overburden and to 

pmvide improved rriteria for dimtion finding techuiques. The 

results given here .ue ronsideml only LS a11 example of the many 

possibilities that exist. 

Jurss K. WAIT 

KENNETH P. SPIES 

Inst. Telecomrnun. &i. 

Ofha, of Telecornrnun. 

U. 8. Dep. Commerce 

Boulder, Colondo 80302 

RSIBRENCFB 
Electronics Letters, Vol. 8, 

No. 16, 10th August 1972 

LOCATING AN OSCILLATING MAGNETIC 

DIPOLE IN THE EARTH 

AInd,xlr snns: Mar~r,lcfi,lds. Mining 

The merits 'or a harizon!al mrgnrlic Jipolc in a source. localion rchemr arc conndcd. Thr rrrvlls hare polr#14r applicalionr in hurlin@ ltappd mincn blloving di<r*tr.n. 

In a previous communication.' we pointed out the porrihilily 

rhdt. following a disaster, miners could located by an 

electromagnetic technique. The basis of the method war to 

ohrcrvt the structure ofthe vcnor magnetic ficld on the suriace 

produced by a buried magnelic dipole whore location was lo 

h determined. Specifically. we considered the source to be s 

vmically orirnted magnctic dipolc (i.e. a small harirontitl 

loop) buried at a depth h in the conducting halfspace. It uss 

indicasd that thc radial &ordinale of the obxrvcr, relativc 

to the roum, could be awcrtained from measurements of the 

horizontal and vertical magnetic fields al the surface. In such 

a whcmc. it is necessary that the trapped miner(s) have 

availaMc a means toexcitc a bopd wire withsutlicient audiofrsgucncy cumnt to be detmed on the surface. 

- I 

Fig. 1 H~ontal magnetic dipah localad at depth h in a 

ronducliq ha#sPICs 

Currrnl field invesligation~ ni the em. location ehemr 

have shown that the null of the horizontal magnetic field 

directly over a buried venienl magnetic dipole is a relatively 

simple and useful criterion that may tlave promise in a 

practical rhun:t Unfonu?ptely, in this care. the rscived 

magnet& field must mmpcte with atmospheric noix that 

always has signilkant horizontal components of magnetic 

field. This immediately ruggals that a better scheme would be 

D 

Fig. 2 Funstton \MI. proporBon#l lo IH, . lor observe, at 

(0. y. 0) as lunc!ron 01 0 = ylh lm varmur valuer 01 

H r lollo w)' h 

ROmnd Iron ELECTflONICS LETTERS IOLh Au#usl l972 

lo employ n llnlc~untul n1.~:,11~1;~-Jl1,.llc ruulr.,. 88, \~1,,~1, i.,,,- 

the null over lhc source uxttv. ;I: (!:c \crtic:O ~,t.,?,,~.~,,.,~~)d 

component. The ~eccircil :~ri~~v.i,ineric nolhr. 8.. I'.~;I il,,,~ii 

reduced. \Pith Ill#\ moti\.;lll,~~. I,< cunridcr llctu ill.: ..~l:~~,:,,~ 

olthcsurf.~ce lirlcts proiliwctl br .i burled hori,~,>,i.)l i!, .i:,,i,li 

dipole (LC. o rrn.811 loop co~iii#i.r.<l in o rerlii..tl ~p!.i~~.~ .\I,,,, 

as we ~1~111 in~Ii~.!lc, lherc arc ;t:!ts,r ic;ttures ,,I' (I,, . <,\:I,: ,,,,+ 

lion that may h.nc some o .r 

Theanalylici~l nlodel is inili~..tlr~l in I'ig. I, ui>rle ,,i ~i:is,,,~ 

a Carasi;bn ro-ordinate sytlcn~ (v. y, ;), with ilir sp,,ci. : .. (I 

correspondsng lo a l~nmi~~i~~~~n~~~ nonmagne:li c.!~!i, ,,r 

conductivily 0. The source is a srni~ll loop or fc!lnc ,,:.tic., I 

crying a I I 1 I t I r . '111,. 

cquivaknt rnagnr~i~dipoleso~~~~c is Located at: ~ -.I: 3)~ 11:~. 

:axis and is oricurcd in the ydincclion. As bcfilrc.' ar. s.,,, 

neglect dl dtrplaccmenl culre8;ir hause nl :lit !a5! 

frequencies invol\cd. 

Theresul1;lnl ~~~agncticficld cornponcnlr 11,. I/, :>r>.l If. ;ti .I 

paint P on lhr \~,rf.~te r - 0 can hc citlct~lated il-olil irl:~?~,,! 

cxprcrrions lh.tt arc wcll knoan.' Tla laltcr a,< c\.t:u.~trli 

numerically. Cvch ll~ough Ccnnin special carescan lx.c\l,ic\~c~l 

in clowd form.' 

First of all. we notc that IIic fi~ltli in the rcgiotl . , 1) ~.,n 

be &rived ftom a scalar polential rn via 

8=-~ . . . . . . . . . . . 11, 

where 

where p = (xz t I.')' and u = O.'+ in~s w)'. If the condocrivily 0 and/or the ircquency cu are suniciently lo^, eqn. I is 

nducd to the simple static form 

IA d a = - -- - fJ0(ip) crp {-i(h+~))al. 

4n Jy0 

On carrying out the diRcrcnliations in eqnr. 1 and 2 and 

changing the integration variable to dimensionless form. \be 

D 

FIQ. 3 Function IN!. propordona1 lo JU:l. lor obrcrrcr a! 

10. y. 0) as a Iunct,un 01 O = ylh 

readily oblain convenient explicit forms for the fields fur 

I ;> 0. Thus . 

H. = I1Alf2nhl'lCS~(TIDl-M1 . . . . 141 

H, = - IIA/l2nlt1';l~S'lD~ +L' M; . . . 151 

H, = lIAI(Znhl'I.CN . . . . . . . . (6) 

Vol 8 No. 16 
whcm 

C - (pl~) CM h 

S - (xlp) = sin 40 

and 

D = (xa+y2)'Ih = plh 

'The dimcnrionlrrr integral functions M. N and T uc defined 

b" 

when H = (ope 61)+ h and Z = zlh. 

The cnmspording static forms for the buic inM. N. and T W the ekmntary hnctions 

The following csJa arc ofspcdal interest. Ifx = 0 (i.~. S - 0 

"ci. we find that 

H. = 0 1 

On the other hand, if y = 0 (i.c. C - 0 -). wevc thu 

A general compulet program har ko written for the intewals 

M, N, and TID that will yidd numerical results for any value 

Of the parameters D, 11 and Z. To illustrate the situation hen, 

we plot IMI. IN1 and ITIDI as funelionr of Din Figs. 2. 3 sod 

4, rcrpcttrely. for the case Z = 0 and various value of H. 

The appropriate static khaviour is evident for the cuc 

H = 0. In general. we nolc that H - 0.086fc.f)'h,, where 

0. is the earth conductivity in millisiemens pr metm.fb the 

fIWcw in hertz and h, is the bud depth in kihM. 

For rumple, if a. - 1. f - Im and h.. - 0.2, we ICE that 

H - 0.17. 

The functional hehaviour of (NI indicated in Fig. 3 show, 

that, in moral, tho vcrdcsl component 11, ha a null Jircctly 

over the source. This would appcar lo k a useful diagnostic 

pmpay. On the other hand. the functional dcpndencer of 

M and T provide Ycondy criteria that could also be 

upldte.3. 

lb mulls decrikd hen arc part of analytical stud; 

rupportcd by the US Bureau of Mines under r contract 

monitored by H. E. Partin. I am grateful to K. P. Spie 

for performing the enkulations.' 

L wm 20th June 1972 

hvlroumenld Resrarch Lobomtwk~ 

NotbnaI Ocwjc & Atmsphm'e Adn*ristrotrion 

US Drprlmcnl of Commerce 

Boulder. Cob. dO302, USA 

RefI w~n. r. I.: 'Critrri. lor louting M r ..,'lacins m.mnis dipdo ~~~Cdintbeurth'.Ploc.lur.a~r.~~~~on. fil.. 1971.9.pp. 1031- ,",a 2 wm. r. I.. and c*u.mrr. r. r.: The fislds or an aarilletinr MW~~ dimlc imm& in rrni.infinlO conductmn nudium'. 

I. C4P)V.. Re,.. 1953. U PP. 161-118 (In rqn. 22, b.' ia the .Y"w.lOl ,h0"16 be Zn'., 
Radio Science, Volume 8, Numbers 8, 9, pages 813-818, Ausust-September I973 

The electromagnetic response of a buried sphere 

for buried-dipole excitation 

David A. Hill 

Insrirute for Telecommunicorion Sciences, Ofice of Telecommunications, 

US Department of Commerce, Boulder, Colorado 80302 

James R. Wait 

Cooperative Institute for Research in Environmental Sciences, Universiry of Colorado. 

Boulder, Colorado 80302 and National Oceanic and Atmospheric Administration, 

Environmental Research Lnboratories, US Department of Commerce, 

Boulder, Colorado 80302 

(Received April 4, 1973.) 

The effect of a spherical inhomogeneity on the surface fields of a buried magnetic dipole 

soum is determined from the reradiated fields of induced electric and magnetic dipole 

moments. For small spheres wherc the theory is valid, the errors introduced in source location 

arc shown to be small. The resulta have pmible application to rescue operations following 

wal mine disasters. 

INTRODUCTION 

The feasibility of locating a buried horizontal-loop 

source from surface measurements of the vertical 

and horizontal magnetic-field components has been 

investigated by Wait [1971]. For sufficiently low 

frequencies, the magnetic fields have a statidike 

behavior, and a single observation of the ratio and 

relative phases of the vertical and horizontal components is sufiicient for location when the earth is 

homogeneous. However, when inhomogeneities are 

present, the surface fields wiU be m&ed, and 

source location may be more &cult. 

In order to obtain a quantitative idea of the surface field modiications, we consider a spherical conducting ore zone as a perturbation to the home 

geneous half-space. The corresponding rigorous 

problem of scattering by a buried sphere has been 

treated by D'Yaknnov [19591. Unfortunately, his 

solution is restricted to azimuthally symmetric excitation, and even then the solution is not in a 

convenient computational form. However, if the 

sphere is electrically small and is located at a sufficient distance from both the loop source and the 

interface, the scattered fields can be identified as 

the secondary fields of induced dipole moments. 

The latter are equal to the product of the incident 

fields and the polarkabilities of the sphere. 

Wait [I9681 has used this induced dipole approach for scattering by a small sphere above a 

conducting half-space. The method has the advantage that it is easily generalized to scatterers 

of other shapes for which both the electric and magnetic polarizabiiities are known, such as spheroids 

[van de Hulst, 19571. This concept has also been 

considered by Ward [I9671 in the context of electre 

magnetic detection of massive sulphideore bodies 

from airborne platforms. In hi applications, however, the induced electric dipole moments for hi7 

spheres and disk models were not used. This W:I\ 

justified for the low audio frequencies used in ex 

plorat:, ? systems such ae AFMAG. 

The geometry is illustrated in Figure 1, whicl~ 

shows both the Cartesian coordinate syntem (x, y, Z) 

and the cylindrical coordinate system (@, +, z). Thc 

half-space z < 0 is free space, and, except for thc 

sphere, the lower half-space z > 0 has conductivity 

U, permittivity c, and free-space pemeability p,. The 

sphere of radius a is located at (&, 0, a), and has 

conductivity u8, permittivity r., and permeability p,, 

The small source loop of area A is located at 
HILL AND WAIT 

Fig. 1. Geometry for source loop and buried sphere with 

induced dipole moment. 

(0, 0, h) and carries a current I which varies as 

exp (id), where a is the angular frequency. Since 

A is small, the source is equivalent to a vertical 

magnetic dipole. For an observer at the surface (z 

= 0)) the vertical and horizontal components in 

the absence of the scattering sphere, Hd and Hzc, 

have been given by Wait [1971]: 

Here, P and Q are the dimensionless quantities: 

where u = (9 + iH2)1'a, Jo and JI are zero- and 

iirst-order Bessel functions, H = (~~)''~h, and 

D = p/h. The result in (2) assumes that displacement currents are negligible, and we retain this assumption throughout since we are interested in low 

frequencies. Wait [I9711 has shown that for H 

sufficiently small, a single measurement of the complex ratio Hd/H: is adequate to determine D and 

consequently p if h is known. 

INDUCED DIPOLE MOMENTS 

Since both types of dipole moments are induced, 

both the unperturbed elcctric and magnetic fields 

at the sphere center (h, 0, &) must be calculated. 

The unperturbed magnetic field has both r 2nd x 

components, Hao and Hzo, and the unperturkd electric field has only a y component, EU0. These components can be obtained from Wait [1951; 19711, 

Bmios [1966], or Ward [1967], and written in the 

following form which is similar to (1): 

H/ = (1~/2ah~)Q' 

H,' = (-IA/Z~~~)PO (3) 

E: = (-iHZ/A/4noh') YO 

where 

Note that Z must be positive, but X can be either 

positive or negative. The evaluation of the integrals ' 

will be discussed later. 

Although the distances involved here are much The induced magnetic dipole moments, m, and 

less than a freespace wavelength, they are not m,, are given by the product of the magnetic 

polarizability and the incidcnt magnetic field [Wait, necessarilv small com~ared to a wavelenah - in the - -- - --. 

lower half-space. Consequently, the induced electric IYbu; 19681 

dipole moment must be considered as well as the 

m. = -2ra'[3(i~ + ~N)]H: induced magnetic dipole moment. This is in con- (4) 

trast to the case in which the source and the scat- m, = -2rins[3(.\f f ~N)]H.O 

terer are above ground [Wait, 19681. where I 

2p.(sinh a - a cosh a) $- po(sinh u - a cosh a f aZ sinhor) -3(M f iN) = - p.(sinh a - a cosh a) - p,(sinh u - a cosh a + a'sinh a) 
where 

ol = (iwp.a.)"'a 

We observe that the function 3(M + iN) has the 

following high-frequency limit 

lim [3(M + IN)] = 1 

191-- (5) 

If me assume that o8 >> o, then the y-directed 

induced electriocurrent moment (Ids), is given by 

[van de Hulsi, 19571 

(Ids), = 4unaa~: (6) 

which is valid even if is not large [Wait, 19601. 

This electric-current moment has dimensions of current times length and differs by a factor of io from 

the electric-dipole moment which has dimensions 

of charge times length. 

Several assumptions are implicit in the induced 

dipole-moment method used here to calculate the 

reradiated fields. First, the sphere radius a must 

be small in comparison to a wavelength in the lower 

half-space [Stratton, 19411. Second, the sphere 

radius must be small compared to the geometric 

mean of the source and observer distances [Wait, 

19601. These conditions insure that higher-order 

multipoles are not important. Third, the sphere 

radius must be small compared to the distance between the sphere and the interface so that interactions between the sphere and the interface are not 

important. Actually, it is possible to compute wrrection terms which take into account interactions 

with the interface, and it has been done for a buried 

cylinder [Wail. 19721. However, the correction 

terms are more complicated for a sphere because of 

coupling between the electric and magnetic modes 

[Hill, 19701. At any rate, the interaction in the 

static limit has been shown to be unimportant for 

a sphere-interface separation of at least two sphere 

radii [Hill and Wait, 19721. 

which can bc wriilc~i ;IS tllc st1111 ~11 (II~CC c,711 

tributions. 

I/," - H,' + ti," + H,' 

(7) 

ff." = H" + H," + H.' 

Hsv and H." are the reradiated fields of the vertical 

magnetic moment m, HBa and HaA are the reradiated 

fields of the horizontal magnetic moment m, and 

Hze and HSe ate the reradiated fields of the electric 

moment (Ids),. 

The fields of the vertical magnetic moment can 

be written in the following form [Wait, 19711 

H,' = (rn2/2r/?@' 

(8) 

HSm = (- III~/~T/~'')/~' 

where 

Since we are observing the fields on the x axis, we 

redefine D to be x/h and let it be either positive 

or negative. 

The fields of a horizontal magnetic moment have 

been given by Wait [1972], and for this geometry 

are given by 

where 

1, [s(lJ -- X)] r1.v 

SCATIERED FIELDS , (( J,,[d 1) - XjJ From the induced dipole moments in (4) and 

(6), it is possible to comuute the scattered fields - ! J - I/IJ XJJJI .I 

at any point On the How- The fields of the m<,,,lc,,[ ;,rc ,,y 

ever, to simp* the discussion, we restrict our ob 

Wait [I9611 and Bafios [1Yh6J, and cirn hc wriltc~a servation point to the x axis. This, of course, in- in the following form 

cludes the point directly above the sphere (x = .Q) 

where the distortion of the surface fields is expected 

to be greatest. The scattered magnetic field on the H.' = ( [-(ld$)r]/2rh21 .r' 

x axis has only z and x components, H.8 and H,: H.' = [[-(lds)vl/2rh'; 7" (10) 
HILL AND WAIT 

where 

S' = lm [Is2 exp (-uZ)I/(s + u)) - X)] dr 

- (sJJs(D - X)ll/(D - X))) ds 

By substituting (3), (4), (6), (8), (9). and 

(10) into (7), the ha1 expressions for the scattered field are obtained 

netic moment goes to 0 in the static limit only if 

the sphere is nonmagnetic (& = This can be 

seen by examining the behavior of 3(M + iN) in 

(4). This point has also been emphasized by Ward 

[19591. 

The total fields observed on the surface are the 

sum of the incident and scattered fields 

H.' = ( [-I ~(a/h)'l/2?rh~I where H.' and H,' are given by (I), and Hea and 

H.' are given by (1 1 ) . [3(M + iA')(a0Q' + P'N') - IH'( Ps')] . . 

Hz' = ([1~(a/h)~]/2sh') NUMERICAL RESULTS 

. [3(M + iN)(e0P' - P'M') -I- IH'(Y'T')] Before performing a numerical evaluation of the 

integrals required for the magnetic components in 

Note that the contribution from the electric moment ) and ), it is useful to obtain the static limits 

goes to O as the frequency goes to O because of the integrals by letting H approach 0. These limits vanishes. However, the contribution from the mag- , useful because they serve as a check on 

general formulation and because H is sometimes 

quite small in realistic situations. By setting H = 0, 

the following static limits denoted by subscript s 

are obtained [Wheelon, 19681: 

Y." x/1x2 + (1 - Z)Z]"' 

P,' = [3(D - X)Zl/(2I(D - X)' + z2)'"] 

Q.' = 122' - (D - X)']/[Z[(D - X)' + z']"~) 

I If these static results are subtracted from the 

I I 1 i I I I I vergence of the remaining numerical integral is im- general integral forms given earlier, then the con0.1 -2 - 1 0 1 2 proved if H is not too large. Of course, for H = 0, 

D the integrands vanish and only the static terms re 

Fig. 2. Effect of sphere location on IH.*/x.'l. H = 0.2, main. Computer programs were written for the nun/h = 0.3, z = 0.5, la[ = 5.0, and + = r, merical integrals, and the resultant quantities of 
BUR1 

F 

interest, HH and arg (H.'/HSt), were examined as a function of the various parameters. 

Results for a nonmagnetic sphere (ps = m) are 

shown in Figures 2 and 3. Parameter values are 

H = 0.2, Z = 0.5, a/h = 0.3, la/ = 5, and X = 0 

and 0.5. In a typical example where j = 100 Hz 

and v = I@' mhos m-', h would be 225 m. The 

amplitude curve in Figure 2 with no sphere present 

agrees with the previous result of Wait [1971]. 

With the sphere directly above the source (X = O), 

the curve is still symmetrical about D = 0. The 

peak magnitudes are increased somewhat, but the 

peak locations are only slightly shifted. When the 

sphere is no longer above the source (X = 0.5), 

symmctry is lost and the peak for negative D increases while the peak for positive D decreases. 

The peak locations are again only slightly shifted. 

The phase curves in Figure 3 indicate that the 

greatest phase deviations occur close to the sphere 

location (D small for X = 0 and D > 0 for X = 

0.5). 

Results for smaller spheres and spheres farther 

from the source indicate smaller deviations. Another 

interesting result is that deviations were smaller 

fl for larger values of la1 due to the fact that the scattered field becomes more nearly real. Results were 

not obtained for larger spheres because the simplified treatment used here is not expected to be valid 

if a/h becomes too large. 

CONCLUDING REMARKS 

A method for calculating the perturbation of the 

surface fields of a buried magnetic dipole has been 

developed. The method utilizes the reradiated fields 

of induced electric and magnetic dipole moments 

and is valid when the sphere radius is small compared with the source distance, the interface distance, and the wavelength in the earth. 

A typical example shows that the magnitude and 

phase of the ratio of horizontal to vertical magnetic 

field can be noticeably altered by the presence of a 

sphere which is located at half the source depth 

and has a radius of 0.3 times the source depth. However, the variation of the surface fields as a function 

of position remains quite similar, and only small 

errors in source locatlon should result. 

Larger deviations can be expected when the 

sphere is larger or closer to the source or the interface, but the simplified theory presented here is 

r' not valid under such conditions. The study of this 

case is n worthwhile extension. 

Fig. 3. Effect of sphere location on arg (H.'/H,'). 

REFERENCES 

Baios, A. (1966), Dipole Radiation in fhe Presence of a 

Conducting Half-space, PP. 195-230, Pergamon. New 

York. 

D'Yakonov, B. P. (1959), The diffraction of elecuomagnetic waves by a sphere located in a half-space, 

Bull. Acad. Sci. USSR. Geophys. Ser., 11, 1120-1125. 

Hill. D. A. (1970). Electromagnetic scattering concepts 

applied to the detection of targets near the ground, 

Ph.D. dissertation, pp. 90-100, Ohio State Univer\ity. 

Columbus, (,"lo. 

Hill, D. A.. hnd I. R. Wait (1972), Elcctromawnrtic bcat~ 

tering of a small spherical obataclc near thc protrn<l. 

Can. I. Phys.. 50(3), 237-243. 

Stratton, I. A. ( 1941 ), Elec!rmu~neric Tlter,ry, pp 5/,3 

573. McGraw-Hill, New York. 

van dc Hulst, H. C. (1957). Lbhf ~correrln~ hy ~,,,.ii 

Pufliclcs, PP. xxx-xxx, John Wiley, New York. 

Wait. J. R. (1951). The magnetic dipole over the h~,~izontally stratified earth, Can. I. Phyr, 29(6), 577-592. 

Wair 1. R. (1960). On the electromagnetic rclponac of a 

conducting sphere to a dipole field, Cer,phy.vic,r, ZSf?), 

649-658. 
-- HILL AND WAlT 

Wait. 1. R. (1961). The electmmagnctic fields of a horizontal dipole in the presena of a amducting half-ap~ee, 

Can. I. Phys., 39(7), 1017-1028. 

Wait, I. R. (1968). Ekctromag~tic induction in a small 

conduaing sphere above a resistive half-space, Rodlo 

Sci., 3(10), 1030-1034. 

Wait, J. R. (1971), Criteris for locating an oscillating 

magnetic dipolo buried in the earth, Proc. IEEE, 59(6), 

1033-1035. 

Wait, J. R. (1972). The effect of a buricd conductor on 

the subsu~£ace fields for linc lam ucitation, Radio Sn'., 

7(5), 587-591. 

Ward. S. H. (1959). Unique determination of conductivity. 

susceptibility, sizc, and depth in multifrcquency elcctrcmagnetic exploration, Geophysics, 24. 531-546. 

Wd, S. H. (19671, Electromagnetic theory for geophyaieal 

applications, in Mining Geophysics, vol. 2, edited by 

D. A. Haaacn, W. R Heinricks, Jr., R. C. Holmen, R. E 

MacDougall, O. R. Rogers, J. S. S. Sumner, and S. H. 

Ward, pp. 10-196, Society of Exploration Geophysicists, 

Tulsa, Okla 

Won, A. D. (1968). Tablnr of Summablc Series and 

Integrals Inuoluing Bern1 Function, pp. 68-92, HoldenDay, San Francisco. Calif. 
:'reilminary deport to U.S. Bureau of Mines on Contract No. H0122061, with 

the Institute for Telecommunication Sciences, Office of Telecommunications, 

U.S. Department of Commerce, Boulder, CO, monitored by Dr. James A. Powell 

F Principal Investigator - Dr. James R. Wait 14 January 1974 

Effect of a Spherical Scatterer on M Source Location 

by 

David A. Hill 

Institute for Telecommunication Sciences 

Office of Telecommunications 

U.S. Department of Commerce 

Boulder, Colorado 80302 

ABSTRACT 

Calculations are made to determine the shift in the overhead 

null of a buried vertical magnetic dipole caused by a spherical 

scatterer located near the earth surface. A table of results is 

given to illustrate the effect of various parameters, and a closed 

form expression is given for a special case. 

INTRODUCTION 

The distortion of surface fields of a buried vertical magnetic dipole 

(horizontal loop) has been examined for several scattering bodies: spheres 

[Hill and Wait, 1973a1, prolate spheroids [Hill and Wait, 1973b1, and infinite 

cylinders [Hill and Wait, 1973~1. Although the entire magnetic field profile 

is of interest in source location, the shift in the null of the horizontal . . 

magnetic field is a limiting factor in location accuracy. Consequently, we 

examine here the null shift caused by a highly conducting body near the earth 

surface. The conducting body could be natural or man-made (vehicle, machinery, 

etc.). 

Since the past treatments of the prolate spheroid and infinite cylinder 

have a more restricted realm of validity, we choose to utilize the previous 

work on the conducting sphere [Hill and Wait, 1973al. The previously developed 

computer program is applicable with only minor modifications, and the specific 

numerical parameters of interest have been furnished by Dr. James A. Powell 

(private communication. September 7, 1973). 
GEOMETRY AND PARAMETERS 

The geometry of the source and scatterer are shown in Fig. 1. The sphere 

is centered at x = x and y = 0. Although the general analysis is for an 0 

arbitrary sphere depth zo, here we set zo = a so that the sphere is tangent 

to the surface. This is as high as we can position the sphere since the previous analysis is for a buried sphere. We omit the analysis and assumptions 

here since they have been covered previously [Hill and Wait, 1973al. Some 

error will result from positioning the sphere close to the interface (since 

interaction between the sphere and the interface have been neglected), but 

this error will be small for sufficiently low frequencies. The permeability 

of both the half-space and the sphere are taken to be that of free space. 

The frequency of use is taken to be 2000 Hz. Several sphere masses and 

densities have been suggested, but only the largest resultant sphere radius 

of 11.7' is used here. This corresponds to a mass of 100 tons and a density 

of 30 lbs/ft3. We present results in English units as requested, but all 

calculations arc actually performed in MKS units. The sphere conducLivity is 

taken to be 10~mhos/rn which corresponds to that of cast iron. The resultant 

electrical size of the sphere la1 is given by 

la1 = 4- a = 448. 

Consequently, the sphere appears to be almost a perfect cobductor at 2000 Hz. 

Two depths, h = 500' and 1000', are assumed for the transmitter depth. 

Two values of the horizontal displacement of the sphere, x = 10' and loo', 0 

are also assumed. For earth conductivity o, 10-~mhos/m and 10-~rnhos/m are 

assumed. Consequently, we are considering a total of eight cases. Other 

values of parameters have been examined, but the resultant null shifts are too 

small to be of interest. 
NUMERICAL RESULTS 

In order to locate the null in the horizontal magnetic field, the x - 

component Hx was computed along the x - axis. Although the primary field 

H' has a null at x = 0, the total field is complex and in general there is 

X 

no value of x for which H = 0. Consequently, we are actually searching for 

X 

local minima of H which occur at x = x . The values of xln for eight cases 

X m 

discussed earlier are given in Table 1. 

Note that for xo = lo', there are actually three minima, and one of them 

occurs for negative x Of course, the largest error in source location would 

m ' 

occur for the maximum lx, 1 which in these cases is on the order of 30'. Results 

arc not shown for larger values of xo and smaller values of a because the 

resultant values of IxmI are extremely small. 

SPECIAL LOW-FREQUENCY CASE 

Since a partial profile of H is required to obtain the resultant null X 

shift for each set of parameters, a great many calculations are required to 

determine the effect of various parameters on the null shift. Consequently, 

it is desirable to obtain an expression which gives the dependence on some of 

the parameters explicitly. This can be done if the frequency is sufficiently 

low to neglect currents in the overburden and if the sphere conductivity is 

suiiicicntly high to nppcar pcrfcctly conducting. Mnthcmnticnlly, t11csc 

conditions are equivalent to the followng inequalities 

(W DO 0)'' h <k 1 and (w po a >> 1. 

Under these conditions, the magnetic fields take on a static behavior, 

and the fields scattered by the sphere are essentially those of an induced 

P- mn~netic dipole. To aimpli.fr cvcn further, we let zo.= O in Pi):. I. Cons<.- 

qucntly only tllc horleontnl induced l~~ng~lctlc dlpolc IIIOIII~I,~ px COI~L~ ~I>(LL<.:: LO 
the horizontal magnetic field H on the x axis. The horizontal n~;l;;n~,tic X 

field incident at the sphere center KO is given by X 

where IA is the magnetic moment of the transmitting loop. The magnetic polarizability of the sphere is -ma3. Consequently, the x component of the induced 

magnetic dipole moment is given by 

On the x axis, Hx can now be written as the sum of the primary and 

secondary fields: 

In order to find the null shift xm, we set H = 0: 

X 

In general, (4) can be solved by plotting the function versus x . However, if 

m 

x is sufficiently large, then the following two approximations are valid: 0 

X 

- m << 1 and - xm 

x h << 1 

0 

If (5) is applied to (4), we have a closed form solution for x . m' 
Note that the null is moved toward the sphere due to the sgn(xo) factor. 

This approximate expression has been found to check with the computer solut~on 

in appropriate test cases. 

CONCLUSIONS 

From the results in Table 1 and Eqn. 6, we can draw some conclusions 

about null shifts caused by scatterers near the surface. First, the null 

shift is only weakly dependent on the transmitting antenna depth and the overburden conductivity. Second, the null shift is roughly proportional to the 

volume of the scattering body (proportional to a3). The null shift decreases 

rapidly as the horizontal displacement xo increases. Consequently, even a 

moderate size scatter (11.7' radius sphere) has a significant effect only 

when it is close to the null point. This is consistent with measqements by 

Farstad (1973) on the effect of a vehicle on the surrounding magnetic field. 

REFERENCES 

Farstad, A.J. (1973), Elcctromagnetic location experiments in n dcc,[). llnrdro~n 

mine", prepared by Westinghouse Georesearch Laboratory, for thc U.S. hurcuu 

of Mines. 

Hill, D.A. and J.R. Wait (1973a), "The electromagnetic response of a buried 

sphere for buried dipole excitation", Radio Science, 8, (8,9), 813-818. 

Hill, D.A. and J.R. Wait (1973b). "Perturbation of magnetic dipole fields by 

a perfectly conducting prolate spheroid", Preliminary report to U.S. 

Bureau of Mines. 

Hill. D.A. and J.R. Wait (1973~). "Perturbation of magnetic dipole fields by 

a finitely conducting circular cylinder", Preliminary report to U.S. 

Bureau of Mines. 
TABLE AND FIGURE CAPTIONS 

Figure 1 Geometry for the source loop and a spherical scatterer. 

Table 1 Effect of various parameters on the null shift. 

Figure 1 

TABLE 1 
Journal of Applied Physics, Vol. 42, 

No. 10, 3866-3869, September 1971 

Transient Signals from a Buried Magnetic Dipole 

James R. Wait and David A. Hill 

Coopemlive Institutqfor Hcsc.nrch i?t the Envirmmenlnl Sricnc~s. Unisr,rsily of Columdo. Roulrl,.v, <'olow,do HO:,~:: 

(I<ecelvcd 28 Jsnunry 1971) 

The solution for a pulse-excited magnetic dlpolo In cnrriml through for n conduatinji h:ilrspace m<xlol of the earth. All displacement curmnts arc ncglccbd so thc solu0on is not valid 

at very small times in the transient rcawnso. I1 is shown that Olc wavcfonns of tho mnLmr:lir ~.. ~~ 

field 'omponcnt~ on the surface depend bolh on the conduclivity 01. Lhr r::trtl~ owl lha. grom<.lrical parsmeters of the pmblem. The results hsvo possible applicattun to electn,magn~!tnc. 15:- 

naling and direction finding for a buried source. 

I. INTRODUCTION 

One method of signaling to the surface from a terminal buried in the earth is to shock excite an underground loop of wire. Because of the Ohmic 

conduction in the overburden, we can expect only 

that the low-frequency spectral components will 

reach the bxrface. Also, because of the dispersion, 

the radiated transient signal will undergo considerable distortion in waveform. It is of interest to examine this situation in a quantitative fashion. To facilitate the analysis, we shall adopt a homogeneous 

half-space model for the earth. The unground loop 

buried at depth h is represented by a magnetic dipole 

with a vertical ads. The situation is illustrated in 

Fig. 1. 

The tlme-harmonic problem 01 a magnetic dipole in 

the presence of a half-space is very well known. A 

great deal of attention has been given to the numerical evaluation of the integrals for the range of parameters of interest in geophysical prospecting. ' In 

such cases, the linear distances involved are very 

small compared with the wavelength in air, but not 

necessarily with the wavelength in the conductor. 

Except for certain second-order effects, this means 

that all displacement currents can be neglected at 

the outset. In spite of this simplification, the integrals in the time-harmonic problem are nonelementary and numerical integration is the order of the 

day. I" However, we wish to point out that the pdse 

solution of the problem can he carried through in a 

relatively simple fashion. This fact was demonstrated earlier for a buried electric dipole,' but the 

method we adopt here for the magnetic dipole seems 

more straightforward. 

11. TIME-HARMONIC FORMULATlON 

As indicated above, the formal integral solution of 

the time-harmonic problem is well !mown. ' Thus, if 

the time factor is eiY', the magnetic field components H, and H, in the air (i.e., z >O) are given by 

H,= bo(iw) P(iw) 

and 

where bo(iw) is a normalizing quantity which is proportional to the current in the small loop, while 

P(iw)= hS S0*F(iw, A, h)hs e-"J,(hp) dA, 

Q(iw)=hs cF(iw, A, h)A3e-".J,(x~) dh, 

(3) 

where Jo and Jl are Bessel functions of order zero 

and order one, respectively. The function of i w, A, 

and h in the integrand of (3) is 

(4) 

For a loop of area A, turns N, and current I(iw), wc 

have 

We now consider the fact that the loop current (assumed uniform) is lob(/), where b(1) is the unit impulse function. We then wish to calculate the time-. 

dependent voltages v,(t) and v,(l) induced in a small 

receiving loop oriented in the p and z directions, respectively. If the receiving loop has area A, and 

turns N,, these voltages are obtained as follows: 

a 

v,(t) =Aa,/J h,(i), 

Observer 

.ip,zl 

Transient 

Source Loop 

FIG. 1. Geometry for the aituaiion 
TRANSIENT SIGNALS FROM A BURIED MAGNETIC DIPOLE 

a 

u.(O =Aiv,W , h,(t), (6) relation4 to perform the inversion before applying 

the derivative operation. 

where Combining (12) and (l4), we see that the deslred 

he(/) -- S-lh0(s)P(v), time-depcndent wavefornls are 

(7) h,(l) = S-'ho(s)~(s), 

and (15) 

h0(?) =A~(2nh~)-'1~. (8) 

Here w is the magnetic permeability of the whole do=- hs ed- awh2/41) qnaw)i/zlsIz I" 6-$ +A*) eke 

space. 2-' denotes the Inverse Laplace transform 

operator and s is the transform variable which can 

be idenbfied farmally with iw in the time-harmonic x exp - - Job) dA, 

problem. Thus, consolidating the above develop- ( :;) 

ments , we have where it is understood that the right-hand side is 

zero if I < 0. For a gen~ralocation (p, a) of the obANA,.N r 4) =+ ~(t), server in the air, the integrals are of a form which 

can be evaluated numerically without undue difficulty. 

(9) In particular, we note that for t >O, the integrands 

are highly damped as A approaches infinity along the 

real axis. Also, the convergence is improved if z >o, 

where corresponding to an observer in the air above the 

ground. 

p(i) = S-~SP(S), (10) 

q(/) : S-~SQ(S), (11) IV REDUCTION TO WORKING FORM 

or, equivalently, For the important special case z = 0, the integrals in 

(15) can be treatcd analytically. The integrals to 

p(1) = hS 1; [&-'sF(s, A, h)]~~e-~d~(xp)dA, contend with are all of the form 

(12) 

q(f) = hs lo-[$[e-ls~(s, A, h)]~e'".J0(hp)dA, A,,,= ~~r-~" J,(@ dk, (16) 

and where a=f/ow, m=3,4, and n=0,1. Using the ascending series definition of the Bessel function 

S"SF(S, A, h) = E-'S[(A~ + U&S)~I~ + All J,(hp), we can deduce without difficulty that 

xexp[- (Az+UPS)'^^^]. 

(17) 

The inverse transform can be evaluated by performing the following sequences of operations: (18) 

E-'SF(S, A, h)= (UP)-'E-~[(A~ + ups)112- A] (19) 

x exp[- (x' + al~s)~~'h] 

- 

r(n + 2) (20) 

n=o 

xexp[- (Az + o~~s)'~'h] Actually, the latter two relations can be expressed 

in terms of derivatives of the modified Bessel function I,. For example, if we start with the known 

form5 

ah hat xexp Ao.o= h(~/a)"zex~(- d/8~)lo(pz/80), (21) 

we see that 

1 

(22) 

xexp ( a$= ;;). (14) and 

Here, we have used a standard Laplace transform O2A0 0 As,,==. (23) 
J.R. WAIT AND D.A. HILL 

and 

FIG. 2. Waveform of the time derivative of Wle horizontal magnetic field on the stirface for an impulsive current 

in the buried source loop. 

r These can be further reduced to a complicated combination of exponential factors and modified Bessel 

functions lo and I,. Thus, we have 

and 

where T= U/h2=t/a&h2 and D=p/h. Here, we note 

that X and Y are dimensionless functions of the rime 

parameter T and the distance parameter fl. As a 

consequence, p(1) and q(f) have dimensions of sec-" 

which is consistent with our formulation. 

The series expansions indicated in (26) and (27) arc 

rapidly convergent provided T is not zero and I) is 

not too large compared with unity. In fact, if LJ= 0 

(corresponding to locating the observer right over 

the source loop), we see that X(0, T) = 0 and 

(28) 

Equivalent forms of (26) and (27) involving modified 

Bessel functions are found, from (2Zb) and (23b), to 

be 

where the arguments of the modified Bessel functions are p8/8a. 

Using the results (17)-(20), the waveform functions 

defined by (15) are given by 

P(/) = - [4n'12(oph2)zl-1 '(D, T) (24) 

and 

q(/) = - [4n"a(~ph2)2~1 Y(D, T), 

where 

FIG. 3. Waveform of the time dcriv;~livc: I,T Llnc: v,.rl.it::tl 

magnetic ficld on lhv ~ur(:u:e. 
TRANSIENT SIGNALS FROM A BURIED MAGNETIC DIPOLE 

where the arguments of 7, and I, are d/8~. The 

functions X and I' are plotted in Figs. 2 and 3 vs the 

time parameter. To facilitate the presentation, the 

cube roots of X and Y are shown as a function of Ton 

a linear scale. Various values of the parameter 

D(= p/h) are chosen, including the value D= 0, which 

corresponds to the observer directly underneath the 

source loop. 

We may observe that the waveforms shown in Figs. 

2 and 3 have a characteristic shape which is significantly dependent upon D. Thus, it is not inconceivable that the shape can be used as a criterion to locate the source from an observer on the surface. 

We note that the normalized time axis T may be read 

directly in msec, if 4nohZ: 10'; if, for example, o 

1 mhos/m, we find that h= 283 m. 

We stress that in our formulation, all displacement 

currents have been neglected. This means that our 

results are only valid when t is somewhat greater 

than €/o sec.' If, for example, u= 10" and E/<,= 10 

for the half-space, we have c/o=- 8.85x10" sec. 

Further work on this subject should allow for the iuhomogeneity and possible layering of the conducting 

half-space. Also, without additional difficulty, we 

can allow for the finite duration of the current pulse 

in the source loop and the transient response of the 

loop receiving antennas. 

'J. R. Wait, J. Geophys. 20, 630 (1955). 

'J. H. Coggon and H. F. Morrison, d. Gcophys. 35, 476 

(1970). 

'J.R. Wait, Appl. Sci. Res. (Sec. B) 8, 213 (1960). 

'Here, we use the classical result 

e-texp(- s113 =i(nt3)-'/'exp (-1/4t) 

along with a simple change of variable. 

=A. D. Wheelon, Tables of Surnmable Series and Integmls 

Involving BeSsel PuncPias (Holden-Day. Los Angeles, 

1968). p. 81, No. 1.104. 
Electronics Letters, Vol. 8, No. 11, 

1 June 1972 

TRANSIENT MAGNCTIC FIELDS PRODUCED 

BY A STEP-FUNCTION-EXCITCD I.O(IP 

BURIED IN THC CAfiTlI 

In a prevlour paper.' uc dcritvd erpresrion~ for thc tr;tnrtcnl 

cln.lrun,~gnelr tkldi of mlsll wore luvp h~trtcJ in the 

Earth and rricird hy :u, tnlpal\~re currvnt. llcrr. nu we tllir 

form.tltmm to r.~lcul~te lllu li?ld~ on Ibe .LII~.ICC 111 IIIC I:.ITIII 

when ;he currcal in thr. loop varnrr hr~th t~vv .A ;L \~cp f\tnctt<>n. 

The model wed kforr. Is duwnhud h1.1c11y IIUTP. As 

indic;!tcd in Fig. I. bc adopt a h~,n,~~sf~~c~~~~~ h;,lfsp;~ce 

madcl for the Earlh ntt11 conductivity o, 1 lhc loop huried 

at depth 13 is reprc,mtrd b?: a rn.~gnclu d~polc ui~ll ;! \cruc;il 

axis. In lcrnms of a r?ltndrical co-ordin.tlc r!\tcfn (/r, 6,;). 

the Canh'i rurfxe ir ; - 0 and thc honiofeneot~~ IIilrth of 

conductivity n wcupncr the region : < 0. The loop 15 1kw;gted 

a1 ; = -18 M the ; a~ir. The quantilirs of inl~.rc,t are the 

voltapes i;.(r) and ,-,(,I induced in sm;,ll rcvr.i\ing loops 

oriented in the radial and \rrtrcal dir~tion.;. resperlt\elg. In 

the pcvious pcpr. ue dcri\ed clored-form e\prrsrionr for 

lhcv voltage \$a\eforrnr $r hen lhccurrent n;tr I,,6(0 Amperes, 

uhercdtrl 15 the unit impulre and 10 is a conrt;lnt. Numerical 

rcrulll for l!lc impulse-rrsponse unreiormi p(0 and c,ltl ucre 

8i.m for the special casr nhcrc (he obwr\er \\its located at 

thc Earth's rurldce ; = 0. The results h;ne ;ilso been 

gencrali*J lor n sourcc luop 01 Iinlte size.' I? lhis cart. the 

invcrw lran,fo,mmr required numerical inle$rat#on. In what 

foilour hcrc. ue uill restri~.t attcnliun lo ihc nv;ign~ta~-dipob 

observer 

.Ip,Zl 

approximuion. so that thc d8mensions 1, and p should be at 

lcart livc lime$ the rldius n of the source loop. Ar belore, we 

ncglut all d~splar~mcnt currents. 

H'c now consider that the ~urrsnl in ihc sourcc loop is 

I(tln(l1ampercr. xhcre t~(11 = I for I > 0 and 0 for r < 0. 

Then, by an inpplication of tlic con\olution theorcm, the 

corrcrwnding \oltage responrn arc 

where 110 = 4nx 10-'Hm. A = no' = area of 5ourie loop 

li.c magnctlc d~pole) of ralllus o. .V = numhcr of ttlror in 

roue tlmp. A. = are* of reccl\lnS loop (of infinltc\inlal \ur) 

and S. = nulnkr of turn, of recctrbng loop5. Lxpliclt 

erprcrrions ucrc gibe" before' for the 1unil$ons pr:) and 

qlr). For what follnus, it IS conbcnient to reurite cqn. I ar rollou-r: 

-- 

vOlU henrys 

Rqinnd lrrm ELECTRONICS L ETTERS Vol. 8 No. 11 

hhcnc -t . zlo~~,,l:'l ' :tncl 1) I>;/, A, r~~~~~~li~~~:!lvnc, 

lllc fullctic~ll\ \ .111\1 ). UCm ~.\illc~.#lrtf Ihy 11t>ll> .~~~.ilyti~,~l .itti 

I I I I ti. .II,LI~ filr ,pt.~i~,ct~ 

h,lm for 11,). ac ~ncCcl 10 perforln .I ir~rtl,er iutcgr;ltinil .I. 

indicclted hy CQIII. 2. 

Wllm llle SOIIICV CLIrrCnt is it SICI) fu11ctiOn. /(I) = 1. t,(!). 

\\c scc tl,:,t 

u here . ' '"*r 1972 Yo18 No. 71 

uhere T = I /IoI, b*). 

Using numrrical urtcgration. thc dimensiunle,~ funct~olr\ s. 

and Y, arc plotled in Figs. ZA and 28 as functtdn~ of 7' ftrr 

T 

iig. 2A W,v~fonn o/ Ihe voll8go m the rrcetrrng loop wdtlo 

radcally 0t;eienled arlr 

u\rral value, of I1 Ar inda'alcd. ~l>u,c \~cl' runiu,,n ,,. wnx\ appro;trl8 irru k,r,l.arpe bdluer tljc llmc p.tr.~#,iv!~ 8 

7: Th~s can c;trtly he ~lcn~on~tn~l~~d ,tc~,~lyc~iillly. 41~~1 11411.. 11 

is a ~hcck on tile vucur.~cy c8f ihc rnu,lccrl<i<l 1111~1.r.t11~111 

\VC 5tCC3V lf~dl, 0" <Ntr f<>~,7tt~l,#~#<l!?, ,811 ~!#'.p~.~~~~?k#'O

currcnls havc hrcn n~~nlecled. Thn rtx;an% tlt.jt w,r rc~lr., 

arc only valid $bl8~.n I 15 ~omcuhdl &:~c.$ter 111211 r 'o%u,tttcl. 

11. for cxam~le. n Ill-' and 1.14, ~ It1 163, tlzc lh.~ll-~~~~~

ne have c/o 2 H NS r 10 ',, 
Fig. 2e Waveform of the voltage in the receiving loop with 

verticallv oriented axis 

- 

it is pn.;sible 1hi11 the \\;I\C~~F~I L..II~ 11c t~scd ;I. ,I 1,..1,1~1, 

detected critrrlon of so~lrcc loci~llon tiw :I sttlglc ol~~cr\it~~,~~i 

point on the surface. We nolc 11ii11 ~hc ti111c irlr.;ilnctcl 1 ,.,r~ 

k read directly in rnill~scc~~nd\ il'Jnnh: I(!': tf, Cc>r c~~l~l~~~lc. 

n = 10 ' Slm, we see lhal h 2x3 111. 1'118, l'cil%~i>ili~y <>~~ttcll 

a scheme in nine rescue opri~lic~~is need, 1,) hc e~;1011,1e~l. 

JAMES H. WAll' 311, ,lfui, 107-' 

E~n,ironm~ntul Rusrurch Luhoruturir., 

National Oce-onic & Almu.sphhpric. Arln~i,ri.str-rjrio,, 

US DDvportment ut Commerce 

Botrlder, Colo. 80302. USA 

DAVID A. HILL 

Insfifrife for Telecommunication Scie~~crr 

Ofice of Telecommunicofion.s 

US Deportment of Commerce 

Boulder, Colo. 80302, USA 

References 

I WAIT. I. K.. and HILL. D. A.: 'Transient signals from a huricd magnetlc 

dipolc'. J. Ap~l. Php.. 1971.42 Pp. 3866-3869 

2 WAIT. J. n.. and HILL. D. n.: 'Transient electromagnetic fields of a 

finite circular loop in the Presence Of a conducling half-space', ;hi,/., 

1972,43 (to be published) 
lnurnsl of Applied Physics, Vol. 43, No. 11, 

4532-4534, November 1972 

Transient electromagnetic fields of a finite circular loop in the presence of a 

r conducting half-space 

James R. Wait and David A. Hill 

Cooperative lnstifute for Research in Environments1 Sciences. Univesity of Colorsdo. Boulder, Colorado 80302 

(Received 27 December 197 1) 

A previous solution for a pulse-excited dipole is extended to allow for the finite size of the buried source 

loop. Numerical results for the above surface fields are presented. We ah consider the situation when 

the finite source Imp is located on the surface and the okrver is buried. It is suggested that the 

waveform of the rscived pulse signals can be used for direction-finding and source location. 

I. INTRODUCTION where X and Yare functions of the dimensionless aaIn a previous paper' we presented a theoretical analysis rameters D=p/h, T= t/(owha), Z=z/h, and A =a/h. 

for the transient magnetic fields produced by a pllse- From (7) and (8) in the earlier paper1 it follows that 

excited magnetic dipole buried in the earth. All dis- X 2 

placement currents were neglected, so the solution was y= W(- &)l-(l +X - A) 9 

not valid at very small times in the transient response. 

It was shown that the magnetic field waveforms on the " "(-'T){J,(~D) J1(xD'f(x~) exp(- xz) dx, (9) 

surface depended on the conducitivity of the earth and 

the geometrical parameters of the problem. The re- where AxA) = ZJ,(xA)/(xA). sults have possible application to mine rescue operations. We consider here a number of new aspects of Ill. SOME RESULTS FOR BURIEDSOURCE LOOPS 

this aroblem. 

The case of immediate aractical interest is when the 

II. FORMULATION 

An important extension of the theory pertains to the use 

of a source loop of finite size.= The situation is indicated in Fig. l(a) where we have located a horizontal 

single-hrn loop of radiue a a1 depth h. For an impulsive uniform current 1,6(f), we then wish to calculate 

the waveform of the field at an observer point P, at a 

radial coordinate p on the surface. The theory given 

before is almost directly applicable to this situation. In 

particular, we again consider the time-dependent voltages v,(f) and v,(t) induced in small receiving loops, of 

area A, and N, turns, oriented in the p (radial) and the 

z (vertical) loops, respectively. Then it follows that 

where A=naa, 

p(t) =F [P"-'sF(s, A,h)]A3e-".J,(Ap)AA)dk, (3) 

q(t)=~f ~'sF(s, A,h)]hSe-".J0(Ap)AA)dA, (4) 

with 

AAa) =ZJ,(Aa)/(Aa) 

observer is located at the earth's surface (i.e., Z=O). 

The waveform functions X and Y have been calculated 

using numerical integration for this case and for a 

range of values of T, D, and A. These results for X are 

shown in Figs. 2(a) and 2(b) where the abscissa is the 

time parameter T. The curves in Fig. 2(a) are for A 

=0, and the results are in agreement with those given 

before' for the magnetic dipole case. The influence of 

having a nonzero value of A is seen in Fig. 2(b). While 

the curves have a qualitative similarity, there is a sigand 

ST1sF(s, A, h) =T's[(x' + ~jls)'" +A]-' 

The inverse Laplace transform indicated above was 

treated explicitly before.' But now the added complica- '. tiona is the functionfiha) that can only be replaced by 1 

if a becomes vanishingly small (i.e., the magnetic dipole limit). 

Following the normalization used before, we write 

p(t) = -[~T'/~(O~)~~~]-'X(D, T, Z, A), (7) 

q(t) = - [4n'1z(~~)2h4]~'Y(D, T, 2, A), (8) 

FIG. 1. (a) Burled circular loop excited by an impulsive 

current. 0 Circular source loop on surface. 
3 0 

TRANSIENT ETdECTROMAGNETIC FIELDS OF A CIRCULAR LOOP 

FIG. 2. Normalized impulse response of voltage in amall vertical loop on surface (with axis oriented in radial direction) for 

buried source loop. - 

nificant difference between the two cases. The agree- formula. To correct the earlier development,' we note 

ment between the results based onthe analytical formu- that the factor multiplying I, in (22') should be 

la for X (for A =0) and the numerical integration results (for A =O) gi4es confidence to the curves. @/or - p4/8a2 

The corresponding waveform functions Y for Z=0 are and the factor '1 in (30) be 

shown in Figs. 3(a)-3(c) for the same range of the pa- LF/T -D'/8T2. 

rameters. The case for A =O here disagreed with the 

numerical results given before,' but this was traced to The earlier numerical results1 for Y, when corrected 

an algebraic errot in the derivation of the working for D> 0, agreed to within five digit accuracy with the 
v 

(h) 

FIG. 4. Normalized impulse response of voltage in small 

buried vertical loop (with evis oriented in radial direction) 

for source loop on surface. 

numerical integration data shown in Figs. 3(a) and 3(b). 

IV. SOURCE LOOP ON SURFACE 

A situation of even more practical interest is when the 

SouPCe loop of radius a is Located on the surface and the 

r' fields are to be observed within the earth. The situation 

is indicated in Fig. l(b). We again consider that the 

source current is I&), and then the time-dependent 

voltages uD(t) and u,(t) induced in small receiving loops 

at P are given by 

up(') = - [AA,N,~~/(2rrh~)]m(t), (10) 

f,'(t).= [nLL,.A',i>/,, '(2nlP)l,l(t). (I 1 I 

The wnvefornl I,,(/) is 1111. su111r III~I.~ 11s io III,. rt.,.il,t.,>- 

cnl situation mentio~l~d nbovt.. l'l~os, ,[(I) is :IK:I~~I ,I,.- 

fined by (4). 011 the other hand, ,I,(/) is to hr obt:~i~l(.,l 

from 

(12) 

where now z, corresponds to the elevation of the source 

loop above ground and h is the depth of the observer 

Evaluating the inverse transform, as indicated before, 

we get 

Putting this in dimensionless form, we find that 

m(t)= -[4n'la(ouPh']-'v@, T, Z, A), 

where 

where D=p/h, A=a/h, T=t/(oph2), and z=z&. 

Using ntImerical integration, the waveform function v 

for z=o is shown plotted in Figs. 4(a) and 4(b) for the 

same range of the parameters. 

V. CONCLUDING REMARKS 

The waveforms p(t), q(t), and m(t) shown here all depend significantly on the parameter D, which is the 

ratio of the observer's radial coordinate p to the depth 

h. This ir true for both finite and infinitesimal loops. 

Also, r~. .late that the transient coupling between a 

finite horizontal source loop on the surface and an infinitesimal buried receiving horizontal loop is the samo 

as for the reciprocal situation where the finite source 

loop is buried. On the other hand, the transient coupling between the horizontal finite source loop and a 

small vertical receiving loop depends on whether the 

source loop is on the surface or buried. In the latter 

case, there is a difference between the up-link and the 

down-link situation. We stress, however, that this is 

not inconsistent with the principle of electromagnetic 

reciprocit~, since the mutual impedance for a given 

configuration does not depend on which loop transmits. 

'I. R. Wait and D. A. Hill, I. Appl. Phys. 42, 3866 (1971) 

'1. R. Wait. Geophysics 19. 290 (1954). 
Pure and Applied Geophysics (PAGEOPH), Vol. 111, 

2264-2272, (19731X) 

Transient Signals From a Buried Horizontal 

Magnetic Dipole 

Ssmmory-The impulse response iora horizontal magnetic dipole buried inaconducting half-space 

is obtained inclosed fornl by an inversr Fouricr transform of the known timc harmonic solution. Since 

displacement currenrsare neglated, the rcsponse is not valid for very short limes. The aavcforms of the 

magnetic field components at the surface have geometrical dependences which may haveapplication to 

source location. 

In a previous communication (WAIT and HILL [I]) it was pointed outrhat a method 

of signaling to the surface was to shock excite an underground l~orizontal loop of wire 

(vertical magnetic dipole). It was indicated that the source location could be determined 

either by searching for the null in tlie vertical magnetic field or by observing the wciveshapes, which contain sufficient ir~formation, at arbitrary locations. However, if thc 

loop is notquite horizontal, then theetTectivemagneticdipole moment has an additional 

horizontal component which changes both the null location and waveshapes. Consequently, it is of intcrest to obtain the pulse solution for a horizontal magnetic dipole so 

that the solution for arbitrary orientation can be obtained by superposition. Also, the 

horizontal dipole itself may be useful in source location since it has a null in the vertical 

field directly overhead. 

The time harmonic solution is well known. but even in the qu;1si-st3tic case wherc 

displ;tcement currents arc negligible the solution requires a numerical intcg~uion. 

However, for the ptilsc solution weare ablc r~~evc~luate the rcquircd integr;~lsan;~lyticaIly 

and obtain a closed-form solution. This hit5 also been done et~rlier for tlie huricd 

electric dipole (WAIT [?I). 

The starting point is the well-known time harmonic solutioli (Whn and CA~IPBCLI. 

[31 and \\'AIT [J]), at~d the geometry is shown in Fig. I. The 1i;tlf-space (: -: 0) is hontogcneous, isotropic, al~d nonmagnetic. nnd h3s co~~ducti\,ity u. The sourcc is ;i smoll .i,op 

') Department of Commerce, OTIITS. Boulder. Colorado 80302. USA 
of arra A and N turns cili.r.,ing a current I wllicll v:irics ;IS exp(ito/). .l'l~i. cquiv:tlcnt 

n~agneticdipoles~urce islocated at z = -h on thez-axis and is oricntcd ii~ tl~c po\iti\c~ 

direction. Displacement currents are negligible at the low frequencies of interest here. 

Figure 1 

Horizontal magnetic dipole in a conducting halfispace 

If theobservation pointislocated at theinterface, then the magnetic fieldcomponents 

(H,, H,, Hz) are given by 

IAN H,(o) = - CS 

2~h' 

IAN 

H,(o) = - CN(o), 2xh3 

where C = (yip) = cos#o, S = (xl~) = sin40, and D = p/h. The dimensionless integral 

functions M, N, and Q are defined by 

(A' + i~p,)'/~ exp[-(RZ + ioap,)'/z h] F(2, a,) - 

A+ (Az + iw~p,)"~ 
)lo is the permcubility of free space, and Jo and J, are the zero-ordcr and first-order 

Hcssel functions. 

In order to obtain the impulse response, iet'the loop current equal Qod(t), where Q, 

is a constant with dimensions of charge. The' transierit voltdges induced in small 

receiving loops at the surface(u,,u,,u,j, where the subscript refers to the direction of the 

loop axis, are given by the following 

where h,, h,, and h, are the time-dependent magnetic field components which can be 

obtained from the inverse Fourier transform of the time harmonic maggetic field 

components given by (I). Consequently, the time-dependent voltages can Imwritten: 

where M', N', and Q' are the time derivatives of the inverse Fourier transforms of M, 

N, and Q. By the differentiation rule they are given by 

a'(!) = F-'[io M(o)]. (5) 

R'(!) = F-'[ioN(w)], 

@(I) .= F-'[iwQ(w)], 

where F-I denotes inverse Fourier transform. 

Since the only frequency dependence in M, N, and Q is contained in thefunction 

F(l.,o), the inverse Fourier transforms in (5) can be rewritten 

rn 

@(t) = hz P'(1, t) J,(Ap) rl dl., 0 
where 

Po., I) = F .'[icoF()., IO)] 

Consequently, the remaining problem is to evaluate f"(i,t). By rationalizing thc 

denominator of F(l.,u) ano apply~ng the differentiation rule, we arrive at the followi~ig: 

F-'{exp[-(AZ + im~p,)~~~ h]). 

The following transform pair can be derived from classical results plus the shifting 

theorem : 

GOhexp 

F-'{exp[-(P + imp, a)"' h]} = 

2G (8) 

wher2:the time function is zero for negative t. Since this will always be the case, all 

folloWing time-dependent quantities will apply only for positive I. By colnbining(7) and 

(8) and carrying out the derivative operations, we obtain the following 

aoh1 PI 

exp 

E(n, I) = ( 4. 'JPJ l"p;h3 3apoh - 

2 v\/sapo tW2 4t2 21 

Thus, the received voltages are now given by (4), (6) and (9). Only the A integrations 

remain to be carried out, and this will be done in the following section. 

3. Reduction to working form 

In order to evaluate the functions in (6), we need to evaluate integrals which are all 

of the form: 

(10) 0 

whiiedt t/(opo); m - 1,2,3; and n - 0, I. The following integrals have known forms 

(WHEELON [5]): 
where I, is the zero-order modified Bessel function. The remaining integrals can be 

derived by differentiation: 

exp (g) - 2a2 (I -d). 

where 1, and I, are zero- and first-order modified Bcssel functions of argument pa/(8u). 

It is noy possible to substitute the results of (9), (1 I), and (12) into (6) to obtain the 

required time responses. However, it is first convenieqt to introduce a normalized time, 

T= t/(cp0h2). By also using the earlier parameter, D'= plh; we can now write the time 

responses in their following normalized forms: 
/ 

Dexp 

@(I) = 

In order to express the received voltage in (4) in the simplest form, we can substitute 

(13) into (4) and rewrite the result in the final form: 

where 

exp 

-%(TI = 8&~~/~ (" ")[fi(l - TI" 4T 2 3)(l 2 4T D~) (10 - 1,) 
Note that Klias unitsofvoltsand the ~iormalizedfunctions M., N., Q.aredimensionless. 

The forni of (14) is especially convenient because the response magnitude is determined 

by K, the wavesh2pesaredetermiried by the normalized functions, and theeffective time 

constant is op,h2. 

4. Nionerical results 

For a general observation point on the surface, the three received voltages are 

functions of the three normalized waveforms, M., N., and @.. However, for special 

cases on the x- and y-axes, the results simplify. For instance, on the positive y-axis, wc 

have 

u,(t) = K -M,,(T) 

ux(')l vz(t) (15) 

A plot of M. which is proportional to v, in this case is shown in Fig. 2 for various values 

of D. Note that the waveform stretches out and changes sign as D increases. This sign 

Figure 2 

Normalized waveform, M.(T), lor various values of D 

change 1s not surprising since it also occurs in the static case. A plot of N. which is 

proportional to u, is shown in Fig. 3 for various values of D. The waveform is zero for 

D - 0 because of the overhead null in the vertical magnetic field. The waveform 

increases in magnitude for small D to a maximum value and then begins to fall or8 ind 

stretch out for larger D. 

On the x-axis, we have the following result: 
Figure 3 

Normalized waveform. N,,(T), for various values of D 

Figure 4 

Normalized waveform. Q.(T), for variir,~s values of D 

A plot of Q, which is proportional to u, in this case is shown in Fig. 4 for various values 

of D. As D increases, Q. is seen to dr-rease in magnitude and stretch out. 

5. Co !ding remarks 

A closed form solution has been obtained for the impulse response of a buried 

horizontal magnetic dipole with the restriction that displacement currents are negligible. 

This means that the results are valid only for t somewhat greater than E/U. However, 

this restriction is not serious since such timesare orders of magnitude smaller than those 

of interest here. 
Althou~h the prime motivation for obtaining the horizontal dipole solution is to 

allow a solution for arbitrary dipole orientation, the horizontal dipole (vertical loop) 

possibilities of use in direction finding. For instance, it may be possible to search for 

the null in the vertical magnetic field or to deduce the location from the waveshapes 

which dcpend on the source-receivcr geometry asshown in Figs. 2-4. Although only tlie 

impulse response wuvcforms aregiven here it is possible toohtain responses to arbitrary 

current pulses by convolution as was done for the vertical magnetic dipole case (WAIT 

and HILL [6]. and W~nand HILL [7]. 

The author would like to thank Dr. J. R. WAIT for his useful comments. 

[I] J. R. WAIT and D. A. HILL, Trmient ~igt~als/TOm n brrriedmwnetic &pie, J. Appl. Phys. 42 (10) 

(1971). 3866-3869. 

[21 J. R. WAIT. Propagationof electrom~pneticpt~Ises in o homogeneous conducting earth, Appl. Sci. Ra. 

8 (B) (1960). 213-253. 

131 J. R. W~l~and L. L. CAMPBELL, Thefields~fon~scillati~ mwneticdipoie immerredinaseml-iw/rnite 

cond~rrtine tnedir~m. I. Geophys. Res. 58 (2) (1953). 167-178. 

(41 J. R. Whrr. Lorotiwan ~scillating m~netlcdipule in theearth, Electronics Letters 8 (1 6) (1972). 401. 

(51 A. D. WII~LLON, Tabie.7 of S~ilanmable Serie~ wd Integrals lfrcolvi~ Brssel Functions (Holden-Day. 

San Francisco 1968). pp. 80-81. 

1-51 J. R. WAIT and D. A. HILL. Transient magneticfields prodrrred by a slep/ro~ctio,r excitedloop bnried 

in the earth. Electronics Lettcm 8 (1 1) (1972a). 294-295. 

[7] J. R. WAIT and D. A. HILL, Electromag~etic surfacefieldsproduecdby a pule-excited loop buriedin 

the earth. J. Appl. Phys. 43 (10) (1972). 3988-3991. 

(Received 19th December 1972) 
Radio Science, Volume 6, Numbers 8, 9, pages 781-786, August-September 1971 

Subsurface electromagnetic fields of a line source on a 

conducting half-space 

James R. Wait and Kenneth P. Spies 

Institute for Telecommunicarion Sciences 

OBfcr of Telecommunications, Boulder, Colorado 80302 

(Received January 15, 1971.) 

An infinite line source or current.carrying cable on the surface of a Rut homogeneous earth 

is considered. The integral fornlula, for the case of negligible displacement currents, is re 

duced either to tabt~latcd functions or to a form suitable for numerical integration. A t:,ble of 

numerical values of these basic integrals is presented. We also show that, in principle, the 

relative location of the line source can be deduced from measurements of the subsurface field 

The results have possible application to mine rescue operations. 

INTRODUCTION ~OWS [Waif, 19621: 

In certain aspects of applied geophysics [Grant E = (-ip&1/2r)G(x, y; x', y') and West, 19661, one is interested in the nature of (1) 

the electromagnetic fields within the earth for a where WI = 4rr lo-' is the permeability free 

source on the surface. F~~ examDle, if a communica- Space. Then the 'Green's function,' for Y >,.0, is a. , , 

tion link is to be established from a sender on the G(x, y; x,, ,,,) = - x,)s + (y - y,)21~/21 earth's surface to a receiving point in a mine or other 

subterranean terminal, it is necessary to choose a - Ko[Y[(x - x')' f O) + ~')~1"*1 

transmitting antenna and an operating frequency -u(v+u') 

that are aooronriate for the task. With this obiective + 2 ( e- cos [X(x - x')] CIA (2) .. . in mind, we shall consider a two-dimensional model 

which hopefully provides insight to this problem. The 

earth is regarded as a homogcneous conducting halfspace in this formulation. We also consider the possibility that the structure of the measured subsurface 

field can be employed to deduce the relativc location of the line source on the surface. 

FORMULATION OF THE GREEN'S FUNCTION 

We begin with the fields of a uniform line current 

I at (x', y') in a nonmagnetic homogeneous halfspace (y < 0) of conductivity m as indicated in Figure 1. Everything is considered to be invariant in 

the z direction, so the posed problem is entirely twodimensional. The fields are taken to vary with time 

as exp (id), and the (angular) frequency is sufficiently low that all displacement currents are negligible compared with the conduction currents. The 

JO UtA 

where u = (iZ + rZ)''Z and r? = in*,.,. The first 

modified Bessel function on the right-hand side of 

(2) can be identified as thc Grecn's function of thc 

line sourcc for an infinite medium, whilc the sccond 

term is associated with an 'image' of thc linc souroc 

at x = x' and y = -y'. For thc region y < 0, we havc 

G(x, y; x', y') = 2 lmez coi (h(x - x')] dA (3) 

It is not difficult to verify that G has the propcr singularity ..r the source, and that G and aG/dy arc continuous at the interface y = 0. Also, we scc that 

(V2 - y2) G = 0 for y > 0, while V'C; = 0 for 

y < 0. Furthermore, if Re u > 0, and noting that A 

is also real, the solutions for G bchavc prupcrl) it, 

lyl 4 m. In establishing the correctness of (2) :ind 

(31, one makes use of the integral reprc\cnt:~lion 

ecectric fieid at P(x, y) has only a z component I Y:Y' 

K,[y(x2 + yZf"] = 1 C-- co\ hx dX (4) E(.x, y) and it can be conveniently written as fol- il u 

This key formula is also uscd in thc clcvclopn~cnl\ 

Coprrtht @ 1971 by thc American Geophysiul Unbr,. which follow. 
Similarly, if we use the standard form 

7///////l//////rfl/lI//IlIII/II/X 

P(x,Y) h (10) 

/ we have 

I(x:il ah h (1 1) 

Thus, making use of (9) and (ll), we see that (7) 

is equivalent to 

i 

Fig. 1. Current line source in the presence of a homo. " "3 (12) 

geneous half-space. By utilizing the Bessel function relations 

SUDSURFACE MAGNETIC FIELDS FOR LINE dK,(a)/da = - K,(a) (1 3) 

CURRENT ON SURFACE 

dK,(a)/da = - [Kdrr) -b a-'K,(a)l (14) 

A situation of special interest is when the line we can express (12) in the form 

source is on the earth's surface and we observe the 

resultant magnetic field H at a subsurface point H/ = - [I/(2nh)l A&) (15) 

P(x, h), as indicated in Figure 2. Thus, in (I), we take where 

x' = y' = 0 and use ikwH. = -aE/ay and 

~POU H, = aE/ax to obtain AoU = 2 Kob) + + !)*I(.) 

e-" cos Xx dA (5) - rr-'(2 + 2a + a2)e-' (16) 

where a = = (i~~~)'/~h. At sufficiently low free-'* sin Ax dA (6) quencies (i.e., a + 0), we see that A,,(=) approaches 

1, and we recover the expected static form. On the 

while, of course, Hz = 0. While the evaluation of (5) other hand, the high-frequency limit (i.e,, a + and (6) for the general case requires numerical in- leads to 

tegration, the important special case for x = 0 is 

amenable to analytical treatment. Then the magnetic A&) - (2na)"$-0 (1 7) 

field is purely horizontal, (i.e., H, = 0) and is re 

duced as follows: 

dA 

(7) 

where we note that (u - A) (u + A) = y*. Now a 

special case of (4) is 

(8) H~ 

thus Y 

lm d2 ue-"' dA = Ko(yh) (9) Fig. 2. Line source on surface and field components to 

be observed at subsurface point. 
- 

P which is a factor of 2 greater than the corresponding I' I xh 

result for an infinite medium. In this asymptotic limit, H, = HO. = % c-~' sin rih = - ,-, 2xh h- + .r' the air-ground interface has the effect of doubling 

the field of the line source. (22) In the general case, for numerical work, it is convenient to express (15) and (16) in terms of dimen- Then, the corresponding forms of ( 18) icnd ( 19) 

sionless parameters H = (O~,U)'/~/I and X = x/h. are simply 

Thus, on changing the integration variable to s = ~h. 

we have A(0. X) = A, = (1 + x2)-' (23) 

I B(0, X) = BO = X(l + xZ)-' H. = -- A(H, X) Zrh 

(24) 

Another useful limiting case is to allow the conI ductivity Or frequency or both to be sufficiently high H, = += B(H. X) that 1~x1 can be regarded as a large parameter. Some 

where consideration shows that this justifies the reolace 

merit of exP (-uh) by exp (-yh) in the integrands 

" (s' + IH')"' of (5) and (6). Then also A(H,X)=2/ (s + r~')'''-ts , 

cos hx -"h exp I-(sl + i~~)"~] COJ (sX) ds (18) F = 2 l- -- "+he A - l- 2 fi 

B(H,X)=2Jm (s , + rf12)"% ,'-- - -1 -7h 

y2x2 (25) 

'exp [-(" + iH')"'] sin (JX) ds (I9) where we have used previously developed results 

The corresponding form for the electric field is - TABLE 1. Field calculations for a line source on a conE = (-ipwl/2r)~(~, * ducting half-space at zero frequency limit (H = 0) --- - - - where *O 

0.0 

no 1.003.. 

'" ', 

*. , 9.90, .I 0. *a0 . a .. .~, 

F(H. X) = G(x, Y; 0, 0) 0.2 '9.625.l 7.90, .2 

,.el,. l 

I.YO0.i 

0.3 e.,,.., Z.7.Z.L 5. DO0 I0 

0.4 8.611 . x 11.8.1 

1311.0 

0. 1.000.1 ..OOO. l 

3 -00 0 

(20) 0.6 1. Bl, . , 1.1z. / 

:oo"+" 

o. 7 6.7,L.t ..Lea - I 

"b,*p 

0.8 6.098. l 1.8il.j 

t ,:?+* 

0.9 I.%S.I 4.97:. z I 21YIY 

1.0 I. ooo - I *.ow . , :.,/,*o 

. exp I-(s' + i~')"~] cos SX ds t., I.5Ij. l lSi7. l 

,.""c.d 

1.2 4.098 . l 4.91s. I 

*"i: . 1 

1.3 I.?,,., I. Sil . l 

I. i,, . ! 

t.4 1.11.. 1 .,lo 2 

: ,,a: . , Because these integrals recur frequently in elec- ,.6 ,.n,,., ..ex$. , 7.lhl.1 

2.6 2.809 . 1 4, 

".",,, . > 

tromagnetic induction problems involving buried 1.7 :.,,I . , +,:a ~ 

~.:.,, . , 1." >.,%., ..>.-. , ira>. l 

8.9 2. /'9 - 1 I I:>. I 

. jo I terminals, tables of the integrals A, B, and F have 2.0 l.000. l ..O"O . , .. zm,. , 2.1 I.8.8 . I 1.112 . 1 

.,#"O. , been prepared. Except for the case X = 0, numerical . . ,712 .L ,7n., + 7,,2 . , 1.3 IllY. 1 ,,t5,. , 4 >.. , integration is necessary. First of all, however, we dis- 21 1.47" ., 5. <.a. , 4. ,,, ~ > 

2.. l3lP -3 1141.1 

,,,,.. 

1.' I'*S. 1 I.!?, . I 4 ,,,,,.; cuss some limiting situations which are useful for >. 7 ,.,o*_, >,>., . , .>/ ~ , >. s I,,,., %.,I,./ 

>,,3 . , checking the numerical work. 2.7 >.ea3., 3.0.: . , : ,,~ 1.0 ,.',OO . , I""0. I , 49, 

>. , ,,.a>, .: ..,,::. , ' $3 

3.2 8. .,,, . ' i I,, . I 

, . LIMITING FORMS 3. 3 *.to _' ' 775.) 

, ,.A 7, *LA . 1 I. 791 . I 

:.,., , A very simple limiting form is when the frequency 5. 5 1.fll.l L.G.Z., z l*, , I.. I. 8'3. I z, ..>, . t , ..,, , I. I '.*YI.' 1 / tends to zero. Then y2 + 0 and then (5) and (6) re- I..,', .l >.s ',,,,.' 1 '4, . , , ,I,, . I 

duce to 1.7 L. #re. r I.40' ., / ILI , 4.0 1. IU . l L. $6' .I '1 , I. L ..1,,.2 2 I*<. I .,,,, , 4.2 . 111 . 1. 1 (-1 . 1 ,./, , 4.3 '.1%1.' 2. ,,,A . , I 1.1 . , - I /I? d. < 1.V11-2 1 I*, . 1 A ,,,, . , H. = H,, = -- 3.3 *.,"' . l '.,,.., <,d,, . , 4.6 ..lll.Z I.",,., 

.."A . 1 

4.7 <%11_2 I.",'. I 1.,1.., 

*." 1. i* -2 ,.,,, . , I,,.., 

(21) ..- I. 9s.. l 1.71.1 '.,,., . , I .I*. ' , ~, 1Y.I .I 

. .,. , 
[Wait, 19621. Thus, for the asymptotic limit HX 

--f *, 

I 

B - (5.) 7 exp [-(i)"'~] (28) 

In a similar fashion, wc rcadily find thc corrspond- In particular, we scc that 

ing asymptotic forms A/U - (i)lnHx/2 (29) 

TABLE 2. Field calculations for a line source on a conducting half-space - - ". ".,& 
TABLE 2. (continued) 

i*.lllel .,. . ~,Q~O.~OO 

8. CSC~CC, 

I.2ri-aol -Ir?.OL L.l,,.*l, 

i.1P1~IYI .1.1.11 *.O>l..l, 

L.ri59.1Di -171.15 ._#so ...> .o, I .,l(. . -7. I.lllt0#, 

. -9. ..l.,.l.. i.111-00, 111.11 

7.0 2.5 ..lllla.. 1.1111-001 LI..l, 1.iil.60i 111..1 

x.llz-onl I11.LL 

1.<1.-001 ,.._% l...,.nD. 82 x.. I.l*..llP. 

5 2. >.~OL.SOO . ' I.L,'.O#a X.1I1~YI1 161.6, ,.i,,lDil 

1.11a-601 (0.1. ,.5x,.oao 

..LZ..LISI i°.LI 1...7.111e 

1.111-101 1.... i..01.O"# 

3.71.-001 5. 1.5. l.oll ..ISi.001 51.1, s.ss:.n~o ..liiOP, .&,b. ~.~o~.ooo I.illill 0 itb ..., 1.:11.111 11.61 ,.,z,,@$a :.111.0111 >-... I.,,,. CII. 

l.llS.001 ,,..I I.li,...o 

I.*el-a6, 6.3 I.I...#O0 

I.bll.l)l ?..A1 

1.111-10, ll.01 

I.I.:-YoI I".". 

9.. 16.2. ..16t.nO( 

8.111.00. li.ll i ,.,,., L.l.l.ml ,,.,. ,.ai*...l 

I.(sOaO. L1.1. .,:.5.00B 

*.6.I.YLII l0.b. ..ll...oI 

1.411-an. 21.1. stil..*PI 1.110-(a" i..PL ..c,>.a.. 2.7(ll.l0. li.5, l.l.,.OE. ?.*ll.OL)I 10.6. l.,.l.O1l 

IUI.II. 11.'. l.#bdIal. 

8.i*l.O#. ,I.?. I.lb(it.YI 

2.2li.OP. 16.Z. l.,,l."llo 

1.liO.LIOI S9.6. a.Jz,~coo 1.111-011 .1.1* 

2.lll1l.iDl 61.111 7 .,.l.OLli 

S.l.6.PB. 6..(I l ..,I. 3." rill ..l. L.~ll.iO. 6%. °. 7 .l,,.c(. 1.1..1-011 6,..0 7 ..ae.aa, J.I(I~IYL 'i.3) 7 ..17.c011 

6.00 

o.o> 

0.6. 

S.06 

0,oa 

o..o 

0.15 

v.29 

o.,n 

(1..* 

I.Sl 

0.bB 

I.70 

o.,r 

m..o 

0.&5 

..sa 

I.,, 

1.80 

i.0, ..,* 

j.2, 

,.a0 

..2> 

t.xo 

I.dS 

5.3. 

*.,8 

t.8. 

LAO 

x..> 

x.<. 

,.,a 

I.,. 

c.30 

..,I 

8.7. 

I.,. 

..,a 

I.LI 

%.6Z 

I.#* 

1.61 

L.., 

..r* 

c.7, 

2.u 

2.85 

,.s// 

s.99 

z.uo. 

,.lr,l,ll 5 011111.1 

. OC~.S~, E.5xb.saz ch l,v.5. from 0.1 to 10. Here, the amplitudes of the .a 1.311-102 ll...l 

2....-B.. t.9.., quantities and their phases arc given. For the smallcr .,.*& >.,*:.49> 

.1.11 i..S*.YLI) LI8.i. 

.. . . H values, it is immediately evident that the rcsult~ I5D.l 8.L1I.I". 131.62 

-%.a6 7 ..,. .~ >? ,,L,l.iii .,.,, ..,.. . ,, ,,,. are consistent with the zero-frcqucncy valucs in T:lhlc .O.'I i.,,.~, 7 li5.5' 

p.*05.040 0.b. L.,, ~t~.~c 1. Also, if IIX is sufficiently large, the complex val. 2.. I.,, lll.,, 

,.I. . . I.illu 2 ll..Om . WS of A. B, and A/B do appear to approach their 1.11 a.LIY.#I &*.*> 

. ,..,. l.o,l~eO1 ,.,,,.,,, ,,.,, lltll respective asymptotic limits. On a point of tllco- I...$ 1.. Ii~lD. 61 ..l 

I..ha . I..llII~l . . ,... , retical interest, we note that the asymptotic rcsults 1.117 l.YIP-I6I LI.01 

.I.$. 

v:.s% I_id,.YOI X.,V~-DO~ GI.,. 

,9.,0 arc rcr~lly only valid at quitc 1;irgc o[ tllc so..1o ....aJ.3 ., ,a,j .... paran~ctcrs .Y and H. Nevcrtl~clcss, they do serve as I*../. I.", ,., 11 ,#.,? 

.. , ..., q.llI,IYI ..: .., ,.,.... lea* a good cll~ck on the numcric;~l integration, sincc I..li.Y11 1,.1? 1.,l..*"i il.*I 

i.lbO.OnB ! .,,..,,, ...,, ll.llI ...*.,. ,.,. C1l ",, iil.ll ,,.,, this is just when the inlegrands lhnvc tllc grr.;ttcst 11.16 3.1, 110. 1.... 

s*.-z s..~:-ra. bt... numhcr of oscillations over the r:lnpc or intror:ntion *& ,, . ".~."", ~. ", 

A 

This 

~ ~~~~ - - . . . . . . - > . . . . . . , . . . 

' par;~mcter in thc abovc forniulation is 11. 

be cx~rcssed in a fairlv cimnle fnrm n, r..l ... . .I.(, 0 L,.*l . ~~- - , ...... I..I .".,,, Y., ,"I- 

:;:: ;:::;:;;: ;;:;; lows: 53.L1 1.lIi.IlQi LI.il 10.15 1..1~.111 E5,qq 

.1.71 3. '..O: jl.16 1.11I.11. 6%. $9 

a5.1. ,.llD.001 LI.ll 

H = !u~w)''~h = 0.086(u,,,/)"?h ,,,, 

...ss I.YO.-rl. I..1I ...I. e..~~~,a. SS.~. if ,, i, the conductivity cxprcsscd in mrnh,, ...a 6 ?.eL.~ao. ,.,. 1, I ...r, 2.711.DO1 65.lil is the lrequcncy in Hz, and /I,,. is the buri;ll CI~.,,: 1, ;,I ~ -, '. .,' krn. Thus, for cxamplc, if rr,,, = I, f :: IfjO, ;I!I,I 

NUMERICAL RESULTS 171io, = 0.2, we have H = 0.17. A5 wc can hcc lr~

TIic numcric:~l values of A, B, 2nd A/B for the the tables, :his valuc of I1 is sullicicntly snl;lIl iir;,t 

zero frcqt~cncy liniit (i.e., H = 0) arc listed in Table thc behavior 01 the functions A and II i5 \cry ~illliI lor .A' ranging from 0 to 5. Of course, in this static lar to that for the static casc (i.c., zcn, rrl.quL,,rcy) 

limit, both A and B are purely rcal. where H = 0. 

Thc nunicrical results for the complex values of We stress th;it thc prcscnt an:lly\i\ rc,i~lt,. :,r,: A, B, and A/B are given in Table 2 for H ranging applicable only to ;I homo~cnc<~u~ Il;l!r-\l,:lc~ Ii~oI~, 1 
for the earth. Ncverthclcss, this should serve as a 

guide for further work on tlic suhjcct. 

121NAI. REMARKS 

A graphical presentation of the ratio IA/BI is 

given in Figure 3. As is indicated, the ordinate is 

the magnitude of the ratio between the horizontal 

and the vertical magnetic field at the observer for 

a line source on the surface. The abscissa X or x/h 

is the normalized horizontal coordinate (i.e., x = 0 

for the observer at a distance h directly beneath the 

source). The phase of the quantity A/B is shown in 

Figure 4. In each case, the H values range from 0 

to 10. 

Both the amplitude and phase of the quantity 

A/B are measurable quantities if the observer is 

equipped with a crossed loop receiving antenna and 

suitable switching facility. If the value of H is known 

(or if it can be assumed to be cffectively zero), the 

value of X should, in principle, be estimated from 

the measured ratio IA/BI. Even if H is not known, 

Pig. 3. Ratio of thc magnitudes of the horizontal and 

vertical magnetic field components plotted as a function of 

the horizontal coordinate. 

Fig. 4. Phase diffcrcnce between the horizontal and thc 

vertical niagnetic field components plotted is n function of 

the horimnti~l coordin:tte. 

the combined use of Figure 3 and Figure 4 should 

permit an unambiguous choice for X. Though operational procedures for carrying out this step have 

not been worked out, the approach seems promising. 

The possibility of using an array of line sources 

on the surface is also being considered. At the expense of a more complicated source configuration 

with scanning capabilities, the observer would need 

only to signal the occurrence of a null. The up-link 

communication in this case could well he a seismically propagating impulse generated by a hammer 

blow. 

REFERENCES 

Grant, F. S.. and C. F. West (1966), Itzlrrprcrarion Theory 

in Applied Geophysicr, McGl w-Hill, New York. 

Wait, 1. R. (1962), Elrcfro,iiugnclic Wdves in Srmrifird 

Media, chap. 2, Pergamon, New York. 
h#o Science, Volume 7, Numkr 5, pw 587-591, May 1972 

The effect of a buried conductor on the subsurface fields for line 

source excitation 

Jama R. Wait' 

Cooperatiue Institute for Research in the Environmental Sciences, Univcrsify of Colorado, 

Boulder, Colorodo 80302 

(Received November 12. 1971.) 

An approximate analytical method is pmsented for treating electromagnetic scattering from 

a slender cyliodrical conductor buried in a homogeneous conductina half-s~acc. The excitation 

is by a uniform line source of cumnt located 0; the surface. he-solutioi is achieved by an 

iterative method that accounts for the interaction of the induced current in the cylinder and 

the air-earth interface. It is shown that the cylindrical inhomogeneity will distort the profile 

of the subsurface magnetic field as the horizontal displacement is varied. 

INTRODUCITON 

In a previous paper [Wair utui Spies, 19711 we 

considered an in6nite line source or current-carrying 

cable on the surface of a flat homogeneous earth. 

The integral representation for the subsurface fields 

was reduced to tabulated functions. The results were 

of interest in mine communication problems, particularly during emergency conditions, for transmission paths through the overburden. The possibility 

that an array of such lie sources could be used in 

direction-finding should not be overlooked [Wait 

19711. In the present paper, we wish to consider an 

inhomogeneous earth model that provides insight to 

the general problem. Specifically, we assume that the 

uniform half-space model adopted earlier [Waii and 

Spies, 19711 has a cylindrical conductor embedded 

within it. This is a possible means to account for the 

presence of conductive rails, pipes, or ore veins that 

would modify the total field at the subsurface observing point. In order to simplify the problem, we 

consider a twc-diiensional model in which the 

buried cylindrical inhomogeneity is taken to be 

parallel to the surface line source. 

Electromagnetic scattering from buried cylinders 

has been considered by Pmry ond Ward [I9711 

and HohmMn [I9711 using integral equation formulations that require elaborate numerical techniques to 

achieve usable results. A m&ed integral equation 

Consultaot, 06ice of Tele~mmunications. Bouldtr, 

Colorado. 

hnkbt O 1971 by the American Geopbydcxl Umim 

technique for such problems has recently been developed (A. Q. Howard, Jr., personal communication, 1971). Also, we should point out that 

D'Yakonov [I9591 has actually published a formally 

exact solution to the case for a buried cylinder of 

circular cross section. Unfortunately, his results are 

not in a form to obtain numerical data. Instead, we 

shall adopt an approximate analytical method that 

is particularly suitable when the buried conductor 

has a small cross section which is the case for a 

buried metallic pipe or rail. 

We refer the reader to the previous paper [Wait and 

Spies, 19711 for the formulation of the necessary 

two-dimensional Green's function G(x', y'; x, y) for 

the homogeneous nonmagnetic half-space of conductivity a. As before, we neglect all displacement 

currents, and the time factor is exp (lot). Thus, the 

prop?, tion constant in the half-space is -y -(iap,w)"'. 

We now consider the influence of an idealized 

inhomogeneity in the half-space for a current I on 

the surface. As indicated in Figure 1, the halfapace 

is homogeneow for y > 0 except for a cylindrical 

region of conductivity 6, radius a, centered at (x', y'). 

In ordcr to simplify the analysis, we now consider 

the radius a to be ~ufliciently small that the currents 

induced in the cylinder are effectively ~yrnmctrical. 

As indiited in the Appendix, this is an adequtlk 

approximation provided Iyal << 1 even though the 

exciting field is nonuniform. We note this does not 

pb a restriction On the parameter 9a where 9 - 

(B~W)~'. In fact, we wish to allow for the possibility 
An adequate form for the external impedance is 

X 

z, - 7 Ko(v) ( 1 

2ra K,(ya) 2s log - + 0.116 

va ) (4) 

since we have already adopted the premise that 

lyal << I (see Appendix). 

The total field E, at P(x, h) is now regarded as the 

sum of the direct field due to the line source I at 

(0, 0) and the secondary or scattered field from the 

cylinder at (x', y'). Thus, to within the first-order 

Y approximation 

Fig. 1. Geometry for the buried cylinder in a conducting -ipI half-space. ET'"(x, h) = --- 2r G(x, Y; 0.0) 

that the cylindrical region has a high conductivity such 

as might exist for a metallic pipe or track buried in 

the earth. 

In the first approximation, we argue that the field 

E(xl, y'), which excites axial currents in the cylinder, 

is given by 

E(x', Y') = (-iwI/Zr)G(x', y'; 0, 0) (1) 

Then the corresponding axial current I. "' is given by 

I."' -- (2TLE(x', y') (2) 

where Z is the effective series impedance per unit 

length for excitation by a uniform axial field. In 

analogy to the development in the Appendix, we see 

that Z = Z. + 2, which is the sum of the internal 

and external impedances, Z, and Z., respectively. 

Without difficulty, we find that 

z, = [fll0(9a)l/12naI1(9a)l (3) 

where I, and I, are modified Bessel functions of order 

zero and one, respectively, and 9 = ifioo/9 is the 

intrinsic impedance of the material inside the cylinder. 

Here, we note that in the low-frequency limit (i.e., 

o -t 0), Z. approaches (raZd)-', which is the directcurrent resistance per unit length of the cylinder. 

On the other hand, at high frequencies where 

I9al >> 1, Z, is well approximated by Z. CY 9/(2ra) = 

ipow/(2rfa) which, of course, vanishes in the perfect 

conductivity limit. Eddy current losses within the 

cylinder are accounted for by (3), and the numerical 

calculations can be facilitated by expressing I, and I, 

in terms of Thomson's ber and bei functions and their 

derivatives: 

I~[(I)"~Z] = ber Z + i hei Z 

and 

i"'~,[(t)"~~] = ber' Z + I beit Z 

which automatically satisfies the boundary condition 

at the air-ground interface if we employ the form of 

the Green's function given in Wait and Spies [197l]. 

To obtain a second order approximation I."', we 

correct I."' in (5) to account for the interaction with 

the interface. Thus 

where 

where r2a' = ip,o/(2rZ) and G is the appropriate 

Green's function. Using h >> a, we find that 

exp (--2uy') 6 - K0(2~y1) + 2 l- + dl (7) 

where p = 2?y'. Carrying o~t the derivative operations leads to 

The corresponding second-order approximated field 

at P(x, h) is thus 

This in turn can be employed to calculate the thirdorder approximated current When the inter- 
.- . ~ ~ 

active process is contiaued indehitely, we arsive at 

I. N Z'EI(X', ye) (11) 

where 

.[I - rhaE + (r20a€)2 - (r'a'G)l + + . ..I (12) 

which is formally summed to give 

Using (ll), we now obtain the final form of the 

pemntwl total field at P(x, h) : 

and 

I 

HT, = - [B(H, Xj 2rh 

- QF(H1. X')( K,[exp (i%/4)R,] exp (h/4)HC, 

- K,[exp (:7r/4)~,1 exp (iu/4)HC, 

+ B(H + H'. X - X')) 1 (20) 

where 

- QG(xl. y'; 0. O)G(x, h; x'. y')] (14) (X - x') 

= [(x - xl)' + (h + y')1]1/2 

where 

S, = (h - Y') 

a = raaa/(l + Taa%) (15) , 1 112 [(x - XI)' + (h - Y ) 1 

The corresponding magnetic field components are S - (h - Y') . r 1 l/S - [(x - x')' + (h + Y ) I 

AN ILLUSTRATIVE EXAMPLE 

The explicit forms given by (18), (19), and (20) can 

- 0G(x1, y'; 0, 0) ] (16) be wed to ca~date the resultant fields at the sub 

a~ .-& surface observation point at depth x for any specified 

location of the cylindrical conductor. Of particular HT,(x, h) N -- interest is the vertical component H,, of the magnetic 

field since this is readily detected by a rudimentary 

- QG(xl, y'; 0, 0) aw, Y: XI, Y') (17) loop contained in the horizontal plane. In Figures 2 ax and 3, we illustrate the normalized field magnitude 

Using the integral forms A(H, X), B(H, X), and 

F(H, X) defined in Wait and Spies [1971], we can 0.6 01~~

express (14), (16), and (17) as follows: H-0.1 

- iwI 

Er = 2* [RH. X) 

- DRH'. X1)[F(H -I- H'. X - X') 

-I Hfi = - IA(H, X) 2xh 

- QRH'. X')[ K,[exp (i*/4)RL1 exp (ir/4)Hh 

%/h 

- KJexp (ir/4)R.l exp (iw/4)HS, pis. 2. Normalized vertical magnetic Aeld at a dep(h h + A(H + HI, x - X')II (19) for H = 0.1. 
Fig. 3. Normalized vertical magnetic field at a depth h 

for H = 0.5. 

2rh IH,,I/l as a function of the relative horizontal 

displacement, x/h, of the 0bse~er. For this example, 

we located the buried metallic cylinder at x = h/2 and 

y = h/2. In Figures 2 and 3, we choose H = (u~,o)"~h 

to be 0.1 and 0.5 respectively, while A = Ira( = 

(up,p)"'a is 0.0, lo-', and 10-"or each figure. Also, 

we assume that U/u = 10'. 

First of all, we note the curves for A = 0 in Figures 

2 and 3, corresponding to a homogeneous half-space, 

are perfectly symmetrical with the expected null of 

the vertical magnetic field directly beneath the line 

source. However, when A > 0, the asymmetry becomes evident, and the null is shifted. In this sense, 

the buried conductor distorts the subsurface field. 

In order to attach a more physical meaning to the 

curves in Figures 2 and 3, we note first of all that the 

parameter H = (~j)''~h,, X 8.6 X lo-', when a, 

is the overburden conductivity expressed in mmhos/m, 

f is the frequency in Hz and h,, is the depth of the 

observer expressed in km. Thus, if a, = 1 and 

f = 100, we see that H = 0.1 and 0.5 corresponds to 

h,, = 0.116 and 0.58, respectively. Then the parameter values A = lo-' and lo-' correspond to a buried 

cylinder radius of 0.116 and 1.16 m, respectively, with 

a conductivity of 10' mhos/m. Of course, by a suitable 

scaling of the parameters, other possibilities exist. 

Also, with limited numerical effort, many other cases 

can be treated. 

CONCLUDING REMARKS 

The analytical method given in this paper permits 

quantitative estimates to be made of the effect of an 

idealized inhomogeneity in the conducting half-space. 

The method is admittedly not usable if the buried 

cylinder is very near the air-earth interface. Also, 

it is restricted to cylinder radii that are electrically 

small in terms of the wavelength in the external 

media. In other words, the parameter A should be 

small compared with 1. Recent numerical comparisons with integral equation formulations indicate, 

however, that the method works well for A as large 

as 0.1 (A. Q. Howard, Jr., personal communication, 

1971). 

APPENDIX 

The method used to estimate the scattered field 

from a cylinder buried in the half-space employs 

certain concepts which can be clearly illustrated in 

the problem [Wait, 19591 of scattering from a cylinder 

of electrical constants 8, i, and p,, located in an infinite 

medium of electrical constants u, 6, and p,. 

The situation is illustrated in Figure 4, in which we 

take the line current I at (p,, #.) to be parallel to the 

axis of the cylinder. Without repeating any details of 

the derivation, we can write down an exact expression 

for the electric field E. external to the cylinder in the 

following form: 

where 

and E.' is the scattered or secondary field. We know 

that 

where 

where 

and where 7 = ip,o/y, 9 = ip,o/9, and 9 = 

[i@ + i~o)ppo]"z, 

If we now insist that lyal << 1 (without any restriction on fa), we can approximate the Bessel functions 

of argument yo by the leading term of their series 

expansions. Also, if additionally, we allow p >> a, only 
the n = 0 term in (22) need be retained. For this 

dominant term, we see that 

where 

are the external and the internal impedances of the 

cylinder as employed in the context of the previous 

derivation. Thus, (22) is expressible in the meaningful form 

where 

is the total induced current in the cylinder. These 

results are consistent with the analysis for the halfspace configuration provided we recognize that the 

appropriate Green's function here is simply the modified Bessel function KO(~~). 

Acknowledgments. I am indebted to Dr. David A. Hill 

for his cogent comments on an earlier version of this paper. 

Also, I wish to thank AUcn Q. Howard, Jr.. for carrying out 

Fig. 4. Prototype problem of line source in presence of 

cylinder in an infinite medium. 

the numerical calculations and for his collaboration on related integral equation approaches to the problem. The 

march was supported by the U. S. Bureau of Mines. 

REFERENCES 

D'Yakonov. B. P. (1959). The diffraction of electromnpnctic 

waves by a circular cylinder in a homogeneous half-space. 

Bul. Acad. Scl. USSR, Geophys. Ser., no. 9, 950-955. 

Hobmaan, 0. W. (19711, Elcctcomagnctic scattering by 

conductom in the earth near a line source of current. 

Geophysics, 36, 101-131. 

Parry, 1. R., and S. H. Ward (1971), EM scattering from 

cylinders of arbitrary cmss section in a conductive balfspace, Geophysics, 36.67-100. 

Wait, J. R. (19591, Electromagnetic Radiation from Cylindrical Structures, Pergamon, Oxford. England. 

Wait. J. R. (1971). Array technique for electromagnetic 

~ositional determination of a buried receivina ooint. - Elect. * 

Lett. 7(8), 186-187. 

Wait. J. R.. and K. P. Soiw . (1971). Subsurface electr* . . . magnetic fields of a line source on a conducting halfspace, Radio Sci. 6(8/9). 781-786. 
<iliOPIIYSICS. VOL. 37. NO. 6 (1)I:CEMBliR IV72). P. YiCYS4. 8 I'lGS. 

THE ELECTROMAGNETIC FIELDS OF A SUBTERRANEAN 

CYLINDRICAL INHOMOGENEITY EXCITED BY A LINE SOURCE1 

The anomalous fields from a huried cylindrical 

inltomogeneity in an otherwise uniform halfspace are analyzed. The problem is rendered two- 

(limensional by assuming that the uniform line 

source of current is parallel to the subsurface 

cylinder. The multipole scattered field coefficients 

are obtained from the numerical solution to the 

associated singular Fredholm integral equation of 

the second kind. The horizontal magnetic field 

amplitude, the vertical magnetic field phase, and 

the amplitude and phase of the ratio of horizontal 

to vertical magnetic fields are shown to he diagnostic of the location of the inhomogeneity. The 

results have possible applications to electromagnetic location in mine rescue operations and to 

geophysical prospecting. 

INTRODUCTION 

In a number of situations in applied geophysics 

we are interested in the electromagnetic response 

of an inhomogeneou s half-space. A recent special 

issue of GEoPHYsrC s (Ward and Morrison, 1971) 

xas devoted to this general problem. In particular, papers by Parry and Ward (1971) and Hohmann (1971) considered the effect of a huried inhomogeneity. We extend Hohmann's analysis 

here using a similar formulation. However, our 

technique is more analytical, and thereby we 

avoid the difficulties associated with the pointmatching method. Of course, in doing so, we must 

pay with some loss in generality; namely, the inhomogeneity must be describable in a separable 

coordinate system. In the point-matching technique, used by Hohmann and others, poorly conditioned large matrices must he inverted to solve 

for the coefficients of the scattered field. 

Actually, D'Yakonov (1999) rather ingeniously 

found a solution to the problem we pose as a limiting case of a boundary value problem involving 

two nonconcentric cylinders. His final results, 

however, appear difficult to treat numerically. 

An approximate method, accaunt- 

 in^ for interactions between the induced monopoles and the air interface, has been proposed 

recently (Wait 1970). It is useful for treating 

cylindrical scatterers of small cross-section. As 

indicated below, this technique (described as 

"Wait's method") serves as a goo4 fheck of the ; . ,,, numerical method. 

THE INTBGRAL EQUATION, GREEN'S PWNCTION, 

AND UNPlUlTtIRBeD.PlEj&?S 

In the analysis to follow, we give a complete but 

concise description of the mode-mas?) solution to 

the two-dimknsional Fredholm integral equation 

of the second kind for our problem. The interested 

reader is referred to Howard (1972) for background material and more detail on the application of this technique to problems of geophysical 

interest. 

We refer to Fi~ureia. lor the geometry. In the 

following, a polar coordinate system (p, +) and a 

cartesian system (x, y) will be used interchangeably. In either case the coordinate origin is at the 

center of thecylinder. The total unknown electric 

field, denoted by Ez, can he expressed in terms of 

an integral equation (Howard, 1972). For the 

case of TE polarization and assuming a time varition e-"I, the relation is given by 

E2(*) = El(*) + iu(e: - ti)(iur 

(1) 

~a2r&$ SoGdp'p'~z(r')~(r, rl), 

t Manuscript received hy the Editor 1:ebruary 7, 1972; revised manuscript received July 14, 1972. 

'Institute for Telecommunication Sciences, Ohice of Telecommunications, U. S. Department of Commerce, 

Boulder, Colorado 80302. 

@ 1972 by the Society of Exploration Geophysicists. All rights reserved. 
IJic. In. Buried cylinder geometry. 

Y 

I (x . -?) I Observer 

FIG. Ib. Special geometry for numerical r~arnples, 

where the Green's function G(f, 8) is given by 

(Wait and Spies, 1971) 

81 - 80 a h(1) = -c"'*,L?a = (ko - f>a, and 

81 + 8a 

a 2 

81 = (k~ - l)', 

for 

'I 

c: = tj + +ia,/o, j = 0, 1, 2, k, = 02p&j', 

Irn 80 2 0, Im 81 2 0, 

and E,(d is the field due to an electric line source 

at (xr, h/2) and in the absence of the cylinder; for 

y<l1/2, we know that (Wait and Spies, 1971) 

i-= ei(=r-=)t+zO~(l) I(h/Z)-#l 

- dl. (3) 

Equation (I) can be bol\-e<l h! intrwlucinji .i 

complete representation for &(v) that is \.iili,l iw ($1 $0. ~hus, 

The circular c)-lindrical symmetr)- can be incurporated into the problem by expressing both 

G(f, 3') and El(<) in terms of their complex angle 

representations. This is effected by makin: the 

change of integration variable t=kl coa$ in equations (2) and (3) to obtain 

x =o. (I+o)~@, (9 

where 

k1 sin * - (A - Ci cos? *)& --- e%X~A sin q, 

and = (2% + yl)i, p' = + Y'*)~, 

cos Q = z/p, sin Q = y/p. 

The contour c is a simple open curve in the complex* plane which comes from -im in tbr strip 

6 <Re ($) <r+6 and recedes to +i- in the strip 

6-*<Re ($) <6, where 6=arg (k~). Similarly, 

pa = [(h/2I1 + xi,jl, 

COS 60 = XT/PO, (8) 

sin 60 = B/ 2pa, 

and 6 ia dehnr(l in lfi) 
The singular term in G(8, 2') can be expanded 2' 

So e-."r eo. * Yf)+"i'dQ using the addition theorem for cylinder functions. (- 

- 2~e'"''~-*i~'J~(k~p') (13IL 

Hr1(kl1 e - ) to obtain 

where p,= max (p, p'), p<= min (p, p'). 

Because the integration with respect to S' in . [ h(q)e-&Vr O"" '*-")+kl~ "'(V++)dq 

the nonsingular part of G(f, z') converges uni- .IU 

iormly withrespect to f and *',provided y+yf<h, (14) 

and, similarly, the sum in (9) converges uniformly = - 

in r and 1'. the order of inte~ration in the former 2(ki - k?) --- - 

and integration and summation in the latter can S h(q)Cikw "8 (*+O+tm*dq, legitimately be interchanged. Hence, in particular, 

we have 

where 

~~o"d+1~o>Ut2(rf)~:"(kl~ 4 z - 5'1 ) K, = a[kJ,,tr(Raa)J,(k~a) 

- k~J~(k~a)J,+~(kla)]. (15) 

= 2 a,e'**(H:'(ky) Combining results (11) and (14) yields, for the 2 ,--- ssond term in the right-hand side of (I), 

~op~'~m(kl~f)~m(ki~f)d~' (lo) .- ,a(*; - ~;)(i~~~) 

+ J,(~IP) Jo2.d+l S:ptdi~~~:&(z. zf)

~'P~J~(~~P')H~)(~~P')~P'} . = +m am{edw[~m(k2p) + ir/2~~(klp)Q~] --e 

! ' ..! This becomes, after doing the integrations and i (16) 

- - -""I21 simplifying, 2 ,e 

= -- (11) Substituting (4) lor the left-hand side of (1) and 

k: - ki "--- (16) for the first and second terms on the rightir .I hand side of (i) results in a matrix equation for 

(J.(kXp) f Jn(kl~)Qrn. , the coefficienfs a,. (Note that the first term on the 

;' .', .. right-hand' side of (16) cancels the unknown 

E,(z).] We have A. where 

11 

(1) Q, = a[kJm+l(k~a)Hm (k~a) 

'1) (12) i+" - kd,(kta)~,+~(k~a)]. 

+; --m C["~P~m(*lp)-~m.-~~..rl) The angle integration over the nonsingular term 07) 

in the Green's function can be effected using the ,It is not dismlt to show that this relation is 

result (GrBbner and Hofreiter, 1961) for arbitrary complex Y. 
'To proceed, multiply both sides of (17) b>- 

r;r,e and intcgrntc 4 from 0 to 2a. Sote that all 

oi the pertinent integrals needed \\.ere discusscit 

in the steps that lead to (16). Factoring the comnlon factor J.(kp) which results in each tern: 

!gives 

This, upon slight manipulation, becomes 

5 L,am=T., tt=O, +_I, t_2,. . ., (18) I--- 

where 

L., = (rQ,6., - K,ei(n-"""I.,), (19) 

and where 

QI. = [k~al,+t(k.a)lfd (kla) 

(1) 

- klaJm(k2a) Il,+t(kln)] 

and 

li, = [k~d.+:(k.a)~,(k,a) 

In (18) I,$, and T. are integrals which must be 

evaluated numerically. In the quasi-static regime 

(i.e. when displacement currents can be neglected), T, is given by 

. [cos (.~~s)~;"(r) + i sin (~~)~,*(s)]ds, 

. / I, 

gz(s) = '11 [s + (xi - if/,)\ $ . , + I-> 4- (.v .- if/,;) 'IP'/, 

and 

? 5 ', 

L(s) = [s + (s? - iN,,).] .> , - (-8 + (5' - i//i)~["[. 

Similarly, ac find that 

2je-i(n8+n1ri4 

1"- = -- (Ho).M+,h , n, 

I dse-""-'Mi)' 

(8' - i~a' - s ,n+n 

. 

(s' - iH:)' + s 

6. (s). (21) 

An interesting limiting case of (18) is tlic linrit 

when Lo-b~ [note here that the terms in (IS) 

must not be interpreted in their quasi-static 

forms]. In this limit the interface vanishes; then 

I,=0 and the matrix equation becomes diagonal. 

Furthermore, T. in this limit becomes 

It folhr\vs that 

This is, in fact, the classical x~littion lor the linr 

source excitation of a loss) cylinder (Wait, 1950j. 

In general, the matrix erluatior~ for ttlu c.orRi~ 

cients in (18) is not diagonal al~rl must hc: .r,lverl 

by truncatinq and then invrrtiny 1111: uctntplzx 

matrix L,,. For electrically sntall cylir~~lcr:. .uch 

that lk,ol?l (hut with no restriction or, k,~, 

only a few terms in the multipole ftclrl ~.x,r~n~iott 

(4) are rcquircd Since the matrix L., is al!rt<,ht 

rliaponally dominant, tllc trunr:atior, 1rroca:sb ib rlol 

critical. This pl~ysically mean, Lllat tit<, ~r~,,<l<i ,I,, 

not couplestrongly. h'umcriciilly. llti, i, t,r,rrwout 
in calculations which show that, for 1 k,al <I, 

the coefficients G, as obtained from a 5x5 and 

9x9 matrix, agree to five significant figures. 

Another check which was performed was to see 

if the electric field is continuous at the cylinder 

boundary. The results show that the continuity is 

good to within 5 percent of the incident field 

strength evaluated on the boundary. 

FIELD EXPRESSIONS 

We now assume the coefficients o, are known. 

The electric field at any point in the lower halfspace OI <h/2) is then determined from (1). Thus, 

we find the total electric field E* is 

Now, EI(z), the z component of the field in the 

absence of the cylinder, is given by 

where H,=(a,pw)~(h/2-~), XI=(*-x)/ 

(h/2-y). The term G, which can be interpreted 

as the contribution to the scattered field that is 

independent of interface interactions, is 

Lwhere 

The term E;, which accounts for multiple interface interactions and combines with Ef to propedy match the boundary conditions at the interface, is given by' 

Z m, 

where 

~jc;.",,4 

6 = - 

(Hz)"' 

Thisdam osition of the scattered field is mainly of 

heuregtic si&nce; the cddtnts o, also depend 

upon the interfa. 

X [cos (~,s)~,"(s) + i sin (~?s)~!(s)]ds 

and 

2 Itn 

g~a(s, Hn) = a ([s - (s - HE) 1 2 2 : m 

- +[-s - (s - iHJ ] 1, 

where Hz=(o,pow)+(h-y), X;=x/(h-y). 

Similar results apply to the magnetic field components. Thus, from (22), and Maxwell's equations, the x component is 

H* = H*, + H:, + H:,, (25) 

where 

i 

H*, = - - -(t - .) 

11 D coq (Xls)e(.2-*~?)'(s2 - ;a1) 

d s ( 2 6 ) 

s + (sZ - iHl,)t 

and 

r +m i* im cos $ dl=- Ke [--~:"(kd 2 -- wap 

(27) 

+ YI sin $~:"~(k.d], 

where 

and 

where 

ICOS (x,s)~"(s, HI) (28) 

+ i sin (~~s)~y(s, H2)jds. 

The analogous results for they compooents are 

H" = H~, + fi:, + H:.. (29) 
- 

where 

--I 

H,, = - 

(30) 

-- as, 

I. +* iw im sin + (1, mi,= -- n,K,c [- ifm (~IP) 

2 ,--, OPOP 

I (31) 

+ YI sin +~f,!,"'(k~p) , 

r.24 z 2: * SF(* -.*a) (s- [S -iH3 ] 

. [cos (x,s);(s, H*) (33) 

+i sin (x,s){(s, Hz)]ds. 

This completes the description of the field c(t111- 

pooents. 

NUMERICAL RESULTS 

In ccrtain suggcstcd location schemes (\\':lit 

and Spirs, 1971), the ratio of 1lorizont:rl to vcrtical nragnetic hclds (IIJB" for both nmpli tude and 

phase) is of interest. Both amplitude and phase of 

H,/Hv are measurable quantities if the observer 

has a crossed magnetic loop antenna. The bnsii 

model to be considered consists of thc symmetric 

case ST= -h/2 y= -/1/2 (refer to Figure 1b) lor 

a range of x= -3h to 3h. The figures arc defined 

in terms of numerical distances H=(uxpow)~/~, 

X=x/c/lt, A=(mpoo)+a, and the conductivity 

contrast I(-uz/uL. The abscissa in each case is X. 

The ordinate for Figures Za, 3a, and cla is the 

amp!itude of HJtl,, while the ordinate of Figures 

Zb, jh, and 4b is the phase of II JB, in degrees. 

In each figure, the homogeneous case A =O is included for comparison. In Figures 2a and 2b, the 

numerical parameters are II=O.l, A-0.01, and 

K= 10'. These ~~unllers give an induction number 

of 31.7. Both the present integal equation apx in Meters 

FIG 3a. Amplitude comparison for ratio of horizontal to vertical magnetic fields versus S for I1 ;;0.1, 

K=107,and A =0.01. 
preach and Wait's method (mentioned above) are 

plotted. Note that in this case the two methods 

are coincident. These figures correspond to the 

situation arising from the occurrence of a metallic 

pipe or track in the vicinity of the EM link. The 

remaining figures use H= 2.0 and K = I@. This 

corresponds to a receiver dZ skin depths below 

the surface. These cases are suitable for considering the field modification due to a larger naturally 

occurring inhomogeneity. Figures 3a and 3b apply 

to the case A =&I. The isotropic scatter term 

(n=O) is still seen to be the only significant contribution. Figures 4a and 4h consider the case 

A =O.j. The inhomogeneity is now large enough 

to require the excitation of the nonisotropic 

modes, which explains the discrepancy in amplitude (the corresponding phase is not plotted). 

The first thing to note in the figures is the singularity in each instance for X= -$ for the homogeneous case. This corresponds to x=m; i.e., 

the point directly under the source where, according to (17), HuL vanishes. Similarly, at this 

point the phase of the ratio (HG/HN) in the homogeneous case shifts by 180 degrees [see (15) 

and (I?')]. The oscillation(s) to the right of the 

first peak in the amplitude plots is related to the 

vanishing of the isotropic term in Ht, in (17) at 

X=O where it changes sign. This effect is thus 

more pronounced when the inhomogeneity is 

large (for larger values of A). 

The horizontal coordinate of an inhomogeneity 

for the examples considered can be inferred by a 

steep gradient in the amplitude ratio and an extremum in the corresponding phase ratio. 

In all figures, the integral equation method was 

truncated such that /nl, iml <N=2 (5 modes). 

In a case identical to the data for Figures 4a and 

4h, except that A =0.5, the calculation was also 

run for N=4. The coefficients a, changed at most 

in the fifth significant figure. The mode amplitudes O+L were in this case approximately 30 percent of themagnitude of ao (hence the discrepancy 

in amplitude--the corresponding phase is not 

plotted). The truncation error is less critical than 

x in Meters 

Pm. 2b. Phase comparison for ratio of horizontal to vertical magnetic fields versus X far H=0.1, 

K=l@, and A =0.01. 

0 

2 

0 -40 

Q) 

n 

.- c -80 

-, -120 

I 

\% I -160 

- -200 YO 

a, -240 V) 

0 

a -280 

-320 

---.-.-.-A ., -A Homogeneous - 'A 

- - 1 q - - 

- - 

- \ \.Ad 11 A \ h ,A.A.~*A-* - 

- - 

Ik, al=0.01 A Wait 7 A 

- H - = 0.1 

i P 

Howard i - m2/m1 = lo7 \ f 

2 - 

I I 

%e./ ' I 

-3 -2 - 1 0 1 2 3 
X = x/h 

FIG. 33. Amplitude comparison for ratio of horLontal to vertical m*gnetic fields rersusX for R=2, 

K=lOq andA=0.1. 

X = x/h 

FIG. 3b. Phw comparison for ratio of horizontal to vcrtical magnetic fields versus .Y for 11 =2, 

K= lv, and A =0.1. 

10 

-30 

a, 

-70 a, 

L3 

c -110 

-- 

T.-150-*16 

I 

\, - 190 

I 

- + -230 

0 

a, -270 m 

f3 if -310- 

-350 

1 I I I I 

- - 

- [kt a1 ~0.1 

H = 2.0 A Wait - 

- CT~/~~= lo3 Howard - 

-b\ 

A-~>.-, 

- -*A\. Homogeneous - 

- "1. - 

l - - i - 

A/~-*-*- 

I I '~-.k.~*~ ,,,..A-o 

A..A-. 

-3 -2 - 1 0 1 2 3 
H= 2.0 f 7- C~/C,=IO~ 

\ . I" 6- A A 

A 

% 5- 

Howard A 

4- 

3 

C 

'Z 3 - a 

E 

a 2- 

1 - Homogeneous - 

I I I I o1 -3 -2 -1 0 1 2 3 

X = x/h 

FIG. la. .lmplitude ratio of horizontal to vertical magnetic fields versus X for II=2, 

K= 10: and A =0.5. 

0 I I I I I 

% -40 - - w 

-80 - a, - 

0 

C - .- 

-120 - 

Homogeneous 

w -280 H= 2.0 

V) 

0 

-360 -3 -2 -1 0 1 2 3 

X = x/h 

1:~. B,. Phase ratio of horizontal to vertical nlagnetic fields vecsos X for II=2, 

K= 1V, and A =0.5. 
in problems that have edge singulirities. This is 

-use the neglected termsfnr Im/ >IV, /*I SN 

in (j), ahere 3' is the truncation integer, are small 

if N>>I k,aI. [See .\ppendix .\ (Hom.ard, 1972).] 

A modified mode-match method in conjunction 

with an integral equation has been shown to be 

effective in dealing with the problem of scattering 

from a buried cylindrical inhomogeneity. The 

method of Wait (1970), which treats the case of 

small inhomogeneities in a much more direct manner, has been demonstrated to be in complete 

agreement with the mode-match treatment. The 

success of the latter can he attributed to the fact 

that all hut one of the integrations occurring in 

the formalism, including the singular component 

in the kernel of the integral equation, can he done 

in dosed form. Application of the mode-match 

method to scattering from penetrable objects results in a matrix equation for the mode amplitudes. The number~f modes required in the 

Fourier Bessel series description of the scattered 

field depends on the argument of the Bessel functions occurring therein. Since the Bessel functions 

of the hrst kind are maximum where the order is 

approximately equal to the argument, the number 

of terms needed in the mode series is proportional 

to I klol, where a is the radius of the inhomogeneity. Thus, the mode-match matrix for good solution must have approximately I knal' elements. 

In contrast, for twc-dimensional problems, the 

point-match technique requires the field to be 

matched at a number of points proportional to 

the cross-sectional area of the inhomogeneity, 

and, thus, the matrix for the point-match method 

is of order I klal'. The reduction of the matrix size 

is clearly an a<lv;tnt:i$t~ it! rcd~lc~n): ctvnl>ut.tliot~ 

tinte. .\Is0 imporl:t~cl in rtrlucing nvllpulcr linw 

was the factorinp of tbr tnairix clt~rnt~n~s. Ihr 

semianalytic approach also yirlds i~rll.l,rl,;~v~~tl 

matrices in contrast to tltc prohlcnts which sometimes occur in point-matching schcntcs. 

ACKNOWLEDGMENTS 

The author thanks Dr. J. R. Wait for suggesting the problem and for helpful discussion in the 

course of this research. This research was sponsored in part by the United States Bureau of 

Mines. 

REFERENCES 

D'Yakonov, B. P , 1959, The diffraction oi clectn,. 

magnetic waves by a circular cylinder in a homoge- neous half-space: Bull. Acad. Sci. U.S.S.K., Geuphys. 

Ser., No. 9, p. 950-955. 

Grobner, W., and Hofreiter, N., 1961, IntegraltaJ, Zweiterteil, Bestimmte Integrale: Vienna, Springer- Verlae. ~ .~ o. 65. "., ~~ 

Aohmann, C. W.. 1971. Electrl nrarnrric scdltrrin,: try 

conducturs in the earth near a line source of current: 

Gmnhssin v .Ib, no. I. u 101-131. 

HOW&~,-A. Q., J;., 197i. Mode mtch solutions to 

electromagnetic scattering problems in a conducting 

half-space environment: University of Colorado. 

Ph.D: thesis. 

Parry, J. R., and Ward, S. H., 1971, Electromagnetic scattering from cylinders of arbitrary cross section in a conductive half-space: Geophysics, v. 36, no. 1, p. 67-120. 

Wait, J. R., 1959, Electromagnetic radiation from 

cylindrical structure: Pergamon Press, Oxford. ---- 1972, The effect of a buried conductor on the 

sub-surface fields for line source excitation: Radio 

Science, v. 7, no. 5, p. 587-591. 

Wait, J. R., and Spies, K. P., 1971, Sub-surface elenromagnecc fields of a line source on a conducting halfspace: Radio Sci., v. 6, no. 8/9, p. 781-786. 

Ward S. II., and Momson, H. F., (Editors), 1971, 

~p&al issue on Electromagnetic Scattering: Geophysics, v. 36, no. 1. 
Reprint from the Review 

PURE AND APPLIED GEOPHYSICS (PAGEOPH) 

Formerly 'Geotsica pura e applicata' 

Vol. 95. 19721111 BIRKHAUSER VERLAO BASEL Pages 150-156 

The Sub-surface Magnetic Fields Produced by 

a Line Current Source on a Non-Flat Earth 1) 

By JAMES R. WAIT') and R. E. WILKERSON~) 

Summary - A simple cylindrical model is employed to estimate the effect of non-fiatness of the 

ground on the sub-surface electromagnetic field from a current-carrying cable on the surface. It is 

shown that, if the surfacecurvature is suffrcientlysmall, the fieldsin thecylinder model are very similar 

to those for the conducting half-space model of the earth employed earlier. The results can be used 

to provide estimates of expected errors in electromagnetic direction-finding of a buried receiving 

terminal. 

I. Introduction 

In a previous communication []]a), we reported on the analysis and calculations 

of the sub-surface fields in a conducting half-space produced by a line current located 

on the surface. Similar calculations have since been brought to my attention [2], 131. 

In our work, we were interested in the application of the basic Green's function to 

scattering from inhomogeneities such as longitudinal conductors. The latter can 

he expected to have a major effect on modifying the structure of the received field at 

the buried terminal. This could pose a serious limitation on the direction-finding 

capability of a simple crossed loop receiving device. 

Here, we would like to consider an alternative and much simpler approach that 

should give needed insight to the effect of non-flatness on the ground surface. Thus, 

we adopt a cylindrical model chosen so that the curvature of the surface is matched 

to the local topography rather than the mean curvature of the earth (for present 

purposes, the latter is zero!). 

2. A classical preliminary problem 

The basic prototype problem is the classical one of a uniform line current source 

located outside an infinitely long homogeneous cylinder of radius R, [4]. The situ:, tn 

') The research reported here was supported in part by the Mine Safety Center, U.S. Bureau 

of Mines. 

2) Institute for Telecommunication Sciences, Office of Telecommunications, Boulder, Colorado, 

80302, U.S.A. 

Numbers in brackets refer to References, page 156. 
SubSurfaa Magnetic Fields F'roduad on a Non-Flat Earth 

is illustrated in Figure I where we have chosen a cylindrical coordinate system (Q, 4. z). 

The external medium e>Ro is free space with permittivity E, and the internal mediu~ii 

e<Ro has a conductivity a and permittivity 6. The line source with current I amps, 

located at (eo, 4,, z), varies harmonically as exp(i w I) where eotRo. 

Figure 1 

The prototype cylinder problem 

For e>Ro, we have only a z-component of the total electric field that can be 

written 

E, = E: + E: (1) 

as a sum of a primary and secondary parts. We know that 141 

where yo =i(eo p,)"' w and po =4 n x 10- '. Using a standard addition theorem for 

modified Bessel functions, this, for RO<e<eo, can be written 

where €.=I for m=O, =2 for m=1,2, 3, ... . Then, it follows that the secondary field 

for e > Ro will have the form 

where A, is a coefficient yet to be deterr-ined. 

The next step is to write down a suii,,!e form for the total internal field. Thus, for 

the sourceless region e<Ro, we have 
- 

J.R.Wait and R.E. Wilkerson 

where y = (i a po w - E po w2)L1Z and where B, is also an unknown coefficient. 

The boundary conditions for the present problem requires that E, and aE,/dp be 

continuous at e=Ro. This leads to the system 

where the prime indicates differentiation with respect to the argument of the Bessel 

function. On eliminating A, from (6) and (7), we get 

where we have used the Wronskian relation 

Km(x) IJx) - KA(x) I,,,(.-#) = l/x. 

Letting eo -r Ro, we see that 

This constitutes a statement of the exact solution of the problem. 

For the present application, we can exploit the condition ]yo Rol 4 1. Then, using 

the fact that for x4 I, Ki(x)/K,(x)- -m/x, (9) reduces to 

Inserting this into (5). we have the following explicit expression for the internal fields 

for a line source on the surface of the cylinder 

Corresponding expressions for the magnetic fields are 

and 
Vol. 95, 1972/Ul) Sub-Surface Mapnetic Fields Produced on a Non-Flat Earth 

3. Application to a topographic feature 

We now apply the prototype solution to the situation depicted in Figure 2. A 

Cartesian coordinate system (x, y, z) is chosen with the line source I located at the 

origin. The observer at P has coordinates (x, y, z) in the Cartesian system, and 

coordinates (Q, 4, z) in the cylindrical system where the latter has an origin at Q. The 

line Q is the center of curvature of the cylindrical surface, and it has Cartesian coordinates (x,, yo, 2). [Note that the z-axes of the two coordinate systems are coincident 

but in opposite directions.] 

Figure 2 

Geometry for a line current source on the cylindrical surface and the observer at buried location P 

The quantities of interest are the orthogonal components of the magnetic field at 

the observer P expressed in the Cartesian system. Clearly, these are obtained as follows: 

H,=Hocos4 - H,sin+ (14) 

where H, and H, are given by (12) and (13). For numerical work and comparisons 

with earlier studies (e.g. [I]), it is desirable to define two dimensionless field parameters 

A and B as follows: 

2nh A=-- H,=A,cos4 -A,sin+ I (16) 

and 

where 

A, = - (2 n h/l) H, and A+ = - (2 n hll) Ho 
1. R. Wait and R. E. Wilkerson 

From (12) ana (13), the latter are given explicitly as follows: 

m 

and 

where 

is the power series definition of the modified Bessel function. Of course, (18) and (19) 

are only valid if @5Ro. 

In the case where displacement currents in the earth are negligible (i.e., where 

E use), we see that y- (i a p, w)"'. Then the modified Bessel functions of order m 

can be expressed in terms of Thompson's ber and bei function of order m. Specifically, 

Im(Z e'"14) =e-'(ri2'm (berm Z+i bei. Z) where Zg 1 y RoI - (a po w)"~ Ro is real. 

Some needed geometrical relations follow immediately from Figure 2. First, we 

note that yo =(R;-X;)'~', then P T=yo-h and Q T=x-x,. Consequently 

[(~./h)' - Wh)']"' - I} 4 = arc tan { blh, .~ - (xo,~) ~ 

Also, it is seen that 

2 112 e = [(P 7')' + (Q T) I 

and thus 

where y h=e"I4 Hand H=(a po w)'/~ h. Furthermore, because P is an interior 

point, p<R, and (23) tells us that (x-X,)%R;-(h-yo)'. 

4. Discussion of some typical numerical results 

Numerical calculations of the normalized field functions A and B have been carried 

for a range of the dimensionless parameters H= ly hl, R = ly RoI, X=x/h and Xo =xo/h. 

In order to illustrate the results here, we plot IAl and IBI as a function of the dimensionless distance parameter X=x/h for R=5, 10, and oo. For the curves in Figure 3a, 
Vol. 95, 1972IIII) Sub-Surface Magnetic Fields Produced on a Non-Flat Earth 

we have xo=O and H=0.2, in Figure 3b, we have xo=0.5h and H=0.2, and in Figure 

3c, we have x, =O and H= 1 .O. The qualitative behavior of the curves is very similar in 

all threecases. The effect of finite curvature of the air-ground interface is to increase the 

level of the normalized horizontal field (i.e., IAI), while the maximum response of the 

vertical field (i.e., IBI) is lowered somewhat. Actually, the curves designated R = co are 

based on the results for the conducting half-space or flat earth model as described 

before [I]. As we see, if the radius of curvature of the surface is sufficiently large, it 

C X=x/h 

Figures 3, a, b and c 

Normalized magnetic fields as a function of horizontal distance for geometry of Figure 2 
-- - 

J. R. Wait and R. E. Wilkerson 

appears that the conducting cylinder results approach those for the half-space. This 

is a reassuring check on the numerical data, since the method of calculation is vastly 

different in the two cases. 

To deal with a specific example, we choose the ground conductivity a to be 1 millimho/m and the operating frequency to be 100 Hz. Then the parameter H=0.20 if the 

observer depth h=225 m. while the parameter H=1.0 if h=1.12 km. Thus, if R=10, 

the radius of curvature R, of the surface is 11.2 km, while if R=5, we have R, =5.6 

km. These correspond to moderately rough terrain features. 

The theory and the numerical data available for this problem should be useful for 

making various estimates of the expected errors introduced by the non-flatness of the 

ground on the direction of the magnetic field at a buried observer. In particular, the 

phase angle between the complex phasors A and Bat the observer position will also 

depend on the radius of curvature R,. This factor and other interesting questions are 

reserved for future work on this subject. 

[I] 1. R. WAIT and K. P. SPIES. Subsurface electromagnetic fields of a line source on conducting havspace, Rad~o Sci. 6(AugustSeptember 1971), 781-786. 

I21 D. B. LARGE, Private wmmunication (Westinghouse Georesearch Laboratory, Boulder, Colo. 

Jan. 1971). 

[31 G. W. HOHMANN, Elecrrom~netrc scattering by conducfors h fhe earth near a line source of 

rurrenl. Geophys. 36. No. 1 (1971). 

[41 J. R. WAIT, Electronqgnefic radiation from cylindrical structures (Pergumon Press, Oxford. 

England, 1959). 

(Received 26th July 1971) 
Electronics Letters, Vol. 7, No. 23 

18 November 1971 

INFLUENCF OF EARTH CURVATURE I * mI.tr,,l st,) urtd - 4.) 

ON THE. SUBSURFACE ELECTROMAGNETIC ,,--r . - . . . (11 no Y,. yKi)1- .,ii.K~l 

FIELDS OF A LINE SOURCE 14 

There BIT a number of problems inapplied geophysics' where 

an clstromagnetic soum field is set up by a long insulated 

cable carrying an allernating current. Most recently, an array 

of such cablcs has seen propowd as a mechanism to provide a 

modulated field at a ruburf*re polnt for rrrrplcun by tnppd 

m;ncna If the rlrucrum of the rubrurfdcc ncld can be pre. 

dieted. them is a good porstbtllly, ullh a simple recc~vong 

dcvicv. that the lwallon of personnel follaw6ng a mine 

dtrasar can be amrta~ncd. We have undcnakcn to craminr 

this qua1.m from a thmmt~cal poml of vlew. lo the pmnl 

kttcr, the spcc~fic problem of the influcnn of local unh 

cumstun ir taken up. Thns 13 an extensfion of an ~arliu study 

where the rubrudacc field of a ublc on a homogeneous Rat 

urth was treated.' 

A general Wlulion for the elcctroma@nctic *Ids of a line 

sumnt source. on a nonRat, homogcncour earth is not 

available. On the other hand. main 2-dimensional mnfigurations are ~~nnble lo numerical treatment 8s dixusd 

in ur uorllcnt ppcr by Hohmann.' We use a similar model 

Fig. 1 Geomeny 01 the 2-dimNioml model lor a current line 

.~ 8awrs M 1)M sudau d # local& wlindric#l IOPOOI&-P~~ fe8ture 

here which has tho vinuc of relative simplicity. SpsiREnlly. 

.ve wnsider a cumnl line source on the rurfacc of P homogcneopr earth that has the 1-1 ehararrcristics of a circular 

cylinder. The riluation is illurlratnl in Fig. I, wherc we have 

designated the line roum by I and the subsurface receiving 

pdint by P. The quantities of intern1 are the magnetic-field 

components H. and N, at P. in the Cartesian frame (I, y..). 

Tor a uniform rourec current I cxp (I-). 

To obtain r tncrablc \olut~on. w assume that the crrcncc 

ofthe problem un be premed #I ue adopt the %olulion for a 

uniform lane sourn wrdl8cl to a 'large' homoacneour cyltnder 

of conductnity o Thc r.td.ur of the cylmdcr or KO. and 11 is 

orntrcd at Q ulth m-ordtnatc\ (x..).). Alu,. ur wn aehncvc 

rome furthcr slmpl CII) :hy nf~l~xt~ng all d~rpldccmcnt currents 

bath m the a.r and in lhc c8,nd~cung earth Clcarly. thlr utll 

bs valid when all significant dimcnrions am small compared 

with a free-space wavelength. The ~lulion for lhtr problem 

and the underlying asrvmptionr were dercribed previously.' 

ad it is mnvcnient. as an inlcdiate slcp. to crpias the 

raultr for the magnellc fields at P in tcrms of cylindrical 

cc-ordinala (p. 4.r) centred at Q. Thus 

@'0 

u 

1 2 3 4 5 

x = x/n 

Flp. 2 Magnet~c~wava lrll produced by the 11nc source at Ow 

rubrdace poml P for vattour valuer of the psramercrs 

R = 1rR.I and X, = xlh, as e funel,on of X = xlh. lor 

H- l*nl-0.r 

Reprinted from ELECTRONICS LETTERS Vol. 7 NO. 23 
where ;. = (io~r~w)'. lra = 4n x 10.' and l.(i!) is the modikd 

Bnvl hlnclioo oi inlugrr order ,wand (complcxl argument 2. 

The desired Cdrtrrian cornponmlr are now obtained from 

11, = 11, car 4- 11, sin d . . . . . . . (3) 

and 

tl,=-Il,~in9-11.cos6 . . . . . . (4) 

nf pitnirubr interesl is the rave-lilt ralio IH../N,I. This is a 

quanlily uluch ran tu. muasurcd earily by a single rolatablc 

loop antennn at P. Usmg rqns. I. 2. 3 and 4. lhis ratio ir 

calculvted and plt~tlcd in Figs. 2. 3 and 4 as a funclion of !he 

prrmwtcr .\' = v!l, for tiscd valuer of thc psramelcn 

H = I = ( 1 SO = xUIh 

nnd 

R = l:.RoI = (nlrocul' Ro 

On cach Figure. uc show the 'idcdl' rcrponse function for 

11 = 0 corrtrpundinp to ire-spa= conditions. Alro, using 

the carlirr rcrulls' for a halfspacc or flat earth, the rcsponw 

functions for R = r are shown in Fipr. 2. 3 and 4. In 

addition. the rclalivc location of the centre curvature of the 

cylindrical air-crnh interface is spcc#fied by the mtlo 

. .Yo .; .r.iR. As indicated. lhis parameter is either 0 or 0.5. 

Finally. uc note that the paramctcr 11 = /;*I determines the 

electrical dcplh of thc observer bncath the level of the linc 

source.. For a typical operating frequency of IWHz and a 

ground conductivity o = I0"Sim. we earily calculale that 

the valucr H = 0.1, 0.2 and 1.0 correspond to actual depths 

of 113. 225 and 1125m. rerprrlively. Under the same 

conditions, the assigned valuer R = S and 10 correspond 

to actual radii of curvature of 5.6 and 11.3 km. Thc latter 

correspond to moderately rough terrain features. 

The mults shown in Figs. 2. 3 sod 4 indicate that the 

wsvc-tilt ratio is influenced not only bs the finite mnductivnty 

of the ovcrburdcn, but also by the curvature of the air-earth 

interface. The departurer from the ideal rnponw function 

(i.c. for H = 01 increase progres~vcly with the eleclrical 

depth parameter 11. Alro. such dcpartum am cmpharivd 

by the finite curvature of the surlacc. These lactr should be 

borne in mind when contemplating the uw of clstromagnctic 

wavn for a detection scheme in mine rescue operations. 

- Furlher work on the subject should allow for the 3- 

dimmional nature of thc problem by considering curvature 

in both principal plan-. Also. without formal mathcmallal 

diliiculty, stratilkalion of the overburden could k ircatcd. 

Thcse are worthy subjects for further analytical sludy if the 

rmpe alclcclromagnelic tcchniyus in mine rerue operalions 

is to be fully realixd. 

I m cxlremely grateful la R. E. Wilkewn for performing 

the numerical calculations of the wave till for finite vvlucr 

of R. 

Inslilrtrr for Te'lrrommwicorio,~ Scirncs 

US Deporrmenl of Commerce 

Mdrr. Colo. 80302. USA 

Rclwrnnr 

I o.r*r. r, 3.. ."a wnr, a. r.: '1nt.rpr.l.lion theary in applied mphnicr' (Ml0r.w-Hill. 19661 

2 whn. ,. I.: Array trshncquc lor etcclromagnctic pn%it!onml dclcrmin.tioncla buried rccrivin~p01t11'. Lleclro1.L111.. 1971,7, pp. 18& ..- ,., 

3 w*r, r. I.. and irlrr. r. r.: 'Su+urlacc dccttommprtic liclds of s line sourn on 8 conducung hnll.rpncc'. R?<Bo S<i. 1971. 6 

4 YWYINY. 0. w.: 'E~CSIIO~RJIICIIC scr11erlng hy ~ondu~lors in ihc avth near a linc sour- of current'. Gs,olrhv*ar. I9lI.Zb. pp. 101-lil 

5 *.A,? I n.: 'The cylindr~eal om body in ,he rcllnuc of a cohlc 

uny)ni an odllating rumnt'. ihid. 19J2. 17, np. 3711-386 [In n. I3 replreo L~IP, by n~~lp~. and in wn. I4 replace (k,nJ by 

%*t)l 
Rndlo Science, Volume 8, Numbers 8.9, pages 805-810, August-Septernba 1973 

Subsurface electromagnetic fields of a line source 

on a two-layer earth 

James R. Wait and Kemelh P. Spies 

Imtirute for Tclemmmunicarion Sciences, Ofice of Telecommunirati~~nr, 

US Department of Commerce. Bouldcr, Colorado 80302 

(Received January 22, 1973; revised Janunry 29, 1973.) 

A pscvious analysis for a line source on a homogeneous half-space model of the earth 

is extended to allow for the presence of stratification. The formal integral solution is reduced 

to a form suitable for numerical integration. Alao, a number of special eases and limiting 

foms an wnsidered. It is shown that the subsurface field has a structure that can be 

etly modified when the half-space is no longer homogeneous. The results have 

application to the design of eleetrornagnetic techniques in mine rescue operations. 

INTRODUCTION 

Communication from a surface-based transmitter 

and an observer buried in the earth must be established following mine disasters. Because telephone 

wires and other nod means of commuuication 

can be intermpted, it is desirable that the electromagnetic signals propagate directly through the 

earth medium. In such situations, a desirable transmitting facility consists of a long insulated cable 

excited by an audio-frequency electric current. The 

fields at a subsurface point can be detected either 

by a multiturn loop or by the voltage induced in 

wire connecting two electrodes. The latter may be 

metal roof bolts or other metal probes that have 

good contact with the host rock. 

If the length of the transmitter cable is much 

greater than all other physical dimensions, we can 

assume, for purposes of analysis, that the subsurface fields are. the same as those for an infinite 

line source of uniform current. In other words, the 

resultant fields are those produced by induction 

rather than by conduction. Unfortunately, the situation is complicated by the presence of the inhomogeneous overburden that both attenuates and distorts the subsurface fields. In a previous investigation, 

Wa't and Spies [I9711 considered a homogeneous 

conducting half-space model that permitted an explicit calculation of the fields for a tim~harmonic 

source current. Also, Wait I19711 advocated the use 

Gpytight 0 1973 b the Amcrian Geophysial Union. 

of such cable fields in a direction-finding scheme. 

This would permit as subsurface observer to determine his location relative to the surface-based transmitter. In such cases, it may be necessary to employ 

an array of such cables with a capability to allow a 

subsurface point to be scanned in a regular sequence. 

A fundamental limitation to any down-link communication or subsurface location scheme is imposed 

by the conductive inhomogeneous over-burden. With 

this motivation in m~ind, we extend our earlier analysis to a two-layer half-space. In particular, we are 

interested in how tlie structure of the subsurface 

field is mmed when the conductivity of the upper 

layer differs from the conductivity of the lower layer. 

Such a condition is typical in most environments in 

the vicinity of coal mines, although we admit that 

the model is still highly idealized. 

THE INTEGRAL FORMULATION 

.Wi". reference to the rectangular wordinate system &..~wn in Figure 1, a layered nonmagnetic halfspace consists of a homogeneous slab of conductivity 

01 occu?ying the region 0 < y < h,, and a semiinfinite homogenema medium of conductivity ", 

occupying the region y > h,, while a uniform lino 

source current I coincides with the z axis. If the fields 

vary with time as exp(Ll), and the frequency is 

sufficiently low that displacement currents are negligible compared with conduction currents, the nonvanishing field components at a subsurface point 

(x, h) are given by 
- - -- - 

WAIT AND SPIES 

[~XP (-u,h) - R exp [-4(2h1 - h)l I cosXrdA O<h<h, 1 - RRo exp (-2u,h,) (1) 

cosXrdh h, 5 h 

? I' 2 [om (-u,h) + R exp [-u,(2h. - h)] 

r o ul+A 1 - RRo exp (-2ulh,) 1 sink& O<h<h, 

2 (1 + R) exp (-u,h,) exp [-u,(h - h,)] f rl + A [ 1 - RRo exp (-2u,h1) 

1- 5 (-u~h) + R exp [-ul(2hl - h)] I . 1 - RR. exp (-2u1h,) cos Ax dA 0 < h <:h, 

* (3) 

cos Ax dh h, ( h 

where 

R = (u, - u,)/(q + ~2) 

Ro = (u, - L)/(UL + A) 

where 

ui = (AP + y,=)'/= 

yia = iUi~4-J 

The relevant formal theory for this layered geometry is available [Wait, 19701. We remark here 

that (a) the field components satisfy the appropriate 

Helmholtz equation in the layers, (b) the field components are all continuous at the interface y = ht, 

(c) the fields all decay exponentially as y + m, and 

(d) the fields reduce to those for a uniform halfspace if hl + rn and/or if ol + 4. 

For numerical work, it is convenient to express 

these components in terms of the following dimensionless parameters: 

Also, as indicated below, we change the variable 

of integration to a dimensionless s = hh. Then, the 

fields may be written as 

H. = -(I/Zrh)A (4) 

H. = (I/2rh)B (5) 

E. = -(ip@I/Zx)F (6) 

where A, B, and F are dimensionless field functions 

that conform to an earlier analysis for the homogeneous balf-space model. For the configuration being 

considered here, we see that 

(7) 

B = b(s) sin Xs ds (8) 

and 

H = (~,w)"~h where 

I 2(s2 + iH2)"' eXp [-(sa f iHZ)"'] - R exp [-(ZD/H - + i~')"~] 

(s' + iH1)"' + s 1 - RRo exp [-2(D/H)(sS + iH2)'"] 

I 0,HID (10) a(s) = 

2(s2 + iHS)'" (1 - R) exp [-(D/H)(sa + iH2)"' - (1 - D/H)(s' + iKH')"'] 

(s2 + i~9~" + s 1 - RRo exp [-Z(D/H)(S' + i~~)"~] 

DIH 
LINESOURCE UECTROMAGNKnC Pllil.l>S 

2s exp [-(s' + i~')'"] + R exp [-(2DE- l)(sa + i~')'"] 

1 - RRo cxp [-2(D/H)(s1 + iH')"'] 

b(s) = OSH5D (11) 

(1 - R) exp [-(D/H)(s2 $. iH2)'/' - (1 - D/H)(s' + i KH~)"~] 

(sa + iH4"' + s 1 - RRo exp [-2(D/H)(s2 + iHZ)"'] 

I 2 exp [-(sa + i~~)"~] + R exp [-(2D/H - 1)(s2 + iHZ)'"] 

(sa + i~')"~ + s 1 - RRo exp [-2(D/H)(s1 + iHa)'"] 

f(s) = Ib(s)l/s = O<H<D (12) 

(1 + R) exp [--(D/H)(s' + i~~)~/~ - (1 - D/H)(s' + ~KH~)"~] 

1 - RRo exp [-Z(D/H)(S' + iHZ)"'] 

1 DSH 

- - 

(s' + i~')'/' - (s' + 1 KH')'/' origin. In this case, however, it is not difficult to 

R = (sa + i~3'/' + (sl + ~KHZ)'" (I3) show that 

(s2 + i~')"~ - s E, a lim (w log w) --t 0 

R - 

O - (sl + iH3'" + s (14) "-0 

Another special case that is useful for checking Our task is to calculate A. B. F. where X, H, D/H, the numerical data given below is to let h, -, 0, and K are given. Thus, we should recover the results for a homoSPECIAL CASES AND LIMITING FORMS geneous half-space with conductivity rz. For example, if we use the functional notation A = A(H. Fit of all, we note that if K = 1 (i.e., if CI= -), X, K, D) for the general mo-layer case, we see that 

then R = 0 and 

2(sa + i~')"~ A(H. X, K, 0) = A(K"'H, X) (21) 

4) = +xP sexp(s' + ~H~)'/'I (15) where 

b(s) = sf(s) A(H, X) = 2 1 (s2 + iHa)'" 

2s (8% + iHZ)"' + s - - + iH2)1/)l11 + exp I-(s' + IH')"'~ (16) . (exp [-(sz + iH2)"2]) cos s X ds (22) 

thus, A, B, and F reduce, as they should, to the This is one of the integrals considered before (1) 

forms for a homogeneous half-space (1). on the for the homogeneous half-space. The same identity 

other hand, if H = 0 corresponding to the frequency applies to the field functions B and F. 

tending to 0, then R = Ro and A limiting case of the two-layer formulation oc4s) = b(s) = exp (-s) (17) 

f(s) = [exp (-s)Vs 

Therefore 

- < .,, ,, .. . , 

. . A = / [exp (-s)] cos Xs ds = ,/(I + X3 (19) ':.:.::': ' 7 "". .. i'. . . , .:.. .,.. ,., 0 

Y 

In this static limit, the integral for F does not con- Fig. 1. Two-layer model of the earth for excitation by a 

verge, since the integrand behaves as l/s, near the uniform line current on the aurface. 
WAIT AND SPIES 

Fi. 2. Ratio between the horizontal and the vertical 

components of the magnetic field at a subsurface observing 

point located in the lower stratum of a two-layer earth. 

For this figure and for the succeeding ones, X = x/h, H = 

(w)l'P k. D/H = hJh, and K = aJal. For this figure 

D/H = 0.5, and K = 10. 

curs when x + m [Wait, 19701. In this high-fre 

quency limit, the electric field E. varies as xd/' exp 

(-i ko x), where k, is the free-space wave number. 

This fact emerges from a general asymptotic de 

velopment for the far-zone fields of a line source 

on a stratified half-space (3). Provided Ik, XI < 1, 

being consistent with the neglect of displacement currents, E. varies simply as 1-1 in this quasi-static 

asymptotic range. Some results for this case were 

given by Bannister [1968], who was specifically 

interested in the application to the measurement of 

the conductivity of the sea bed. In his case, the 

parameters equivalent to lYixl and Inxl were very 

large so that the fields could be described by their 

quasi-static asymptotic forms. In our contemplated 

application to mine rescue procedures, we would 

seldom be working in a frequency or distance range 

where these asymptotic results could be used. Nevertheless, they do serve as an excellent check on the 

numerical integration results discussed below. 

In the notation used here, the quasi-static asymptotic results are valid if both HX and K1f*HX are 

sufficiently large (i.e., greater than about 5). Then 

we find that 

B = 2Q2~/(i~'p) (24) 

and 

F = 2Q2G/(i~2~2) (25) 

where Q is the commonly encountered wavetilt factor [Wait, 19701 given by 

Q = [1 + r exp (-2i"2~)]/[l - r eXp (-Z~"~D)] 

(26) 

Fig. 3. Ratio between the horizontal and the vertical components of ths magnetic field at a subsurface observing 

point located in the lower stratum of a twc-layer earth. 

D/H = 0.5; K = 0.1. 
LINESOURCE ELECTROMAGNETIC PIEI-DS 

P = ((1 - r)/[l - r exp (-2i1"D)l] 

.exp [-il"D - il'*(Jf - D) ~"'j (31) 

50 for H 2 D. 

DISCUSSION OF RESULTS 

Using numerical integration as discussed before 

[Wmt and Spies, 19711, A, B, and F have been 

evaluated for a selected range of the parameters 

X, H, K, and D. These results are available on 

request from the authors in tabulated form for a 

range of X from 0 to 5 and for all combiiatiws 

of the following values of the parameters: H = 0.1, 

0.2, 0.5, 1.0, 1.5, 0.2, 0.5, 1.0, 2.0; D/H = hl/h 

= 0.5, 1.0, 2.0; K = 0.1, 10. Also, some values 

of A, B, and F were obtained for K = 0.01 and 

100 for a restricted range of the other parameters. 

A derived quantity of special interest is the magnitude of the ratio HJH, at the sub-surface observer point. As we mentioned before (I), this ratio, 

equal to [A/B[, can be measured by two crossed 

Pig. 4. Ratio of the &Id components when the observing 

point is within the upper atraturn. D/H = 2; K = 10. 

and r is a reflection coefficient defined by 

r = lim R = (1 - ~'")/(l + K'") (27) .Q 

Here, G and P are depth-gain functions that correspond to the ratio of the field at depth H to that 

at the surface. These are found to have the form 

G = [exp (-iL/'H) + r 

exp (-2i1/1~ + il"H)l/[l + r exp (-2i1"0)] 

(28) 

P = [exp (-il/'H) - r 

. exp (-2i'"D + il"H)l/[l - r exp (-2i1"D)] 

' (29) 

for 0 < H 5 D. The corresponding forms in the 0.11 I I 1 I 1 

lower region are 0 1 2 " 3 4 3 A 

G = + r)/U + r exp (-2i"'~)l) Fig. 5. Ratio of the Beld components when the observing 

. exp [- I'"D - i"'(~ - D)K"'~ (30) point is within the upper stratum. D/H = 2; K = 0.1. 
WAIT AND SPIES 

loops or by a single loop rotatable about an axis. 

Using the numerical data for A and B, this ratio 

is plotted in Figures 2 and 3 as a function of the 

displacement X for K = 0.1 and K = 10, respectively, for D/H = 0.5, and a range of H values. 

Here and in the figures that follow, the curves are 

broken whenever IAl and/or IBI is less than lW 

(i.e., < -60 db). These results in Figures 2 and 3 

correspond to locating the observer within the substratum. The corresponding results for the case in 

which the observer is within the upper layer are 

shown in Figures 4 and 5, where we choose D/H 

= 2.0. 

The effect of varying the conductivity contrast 

K(= u2/u1) is illustrated in Figures 6 and 7, where 

we choose D/H = 0.5 and 2, respectively, and I~X 

H = 1 (i.e., D = 0.5 and 2). 

In general, IA/BI always has a pronounced peak 

at X = 0 corresponding to the observer directly 

Fig. 6. Ratio of the field components illustrating the effect of conductivity contrast between the layers. H = 1; 

D/H = 0.5. 

0.1 I I I I I I I I I , 0 I 2 3 4 5 

X 

Fig. 7. Ratio of the field components illustrating the effect of conductivity contrast between the layers. H = 1; 

D/H = 2.0. 

beneath the source cable. On examining the curves 

in Figures 2, 3, 4, 5, 6, and 7 we see that the peak 

may be obscured to some extent by the presence 

of the stratification in the overburden. No doubt 

this behavior should be taken into account in any 

feasibility studies of source-location techniques that 

utilize subsurface magnetic-field measurements. 

An important and nontrivial extension of the 

present analysis is to allow for the nouuniformity 

of the thickness of the upper layer. This subject is 

being currently investigated by using a wave-optical 

technique. 

Acknowledg~enfs. We would like to thank Dr. R. G. 

Geyer of the Colorado School of Mines for suggesting this 

problem, and Mrs. Loys Gappa for her help in preparing 

tho manuscript. The =search was supported by the US 

Bureau of Mines. 

REFERENCES 

Bannister, P. R. (1968) Electromagnetic fields within a 

a stratified earth produced by a long horizontal line 

source, Radio Sci., 3(4), 387-390. 

Wait, I. R. (19701, Elecrromagneric Waves in Stratified 

Media, 2nd cd.. chap. 2, pp. 8-63. Pergamon, New York. 

Wait, J. R. (1971), Array technique for electromagnetic 

positional determination of a buried receiving point, Electron. Lett., 7(8), 186-187. 

Wait, I. R., and K. P. Spies (1971), Subsurface electromagnetic fields of a line source on a conducting halfspace, Radio Sci., 6(8/9), 781-786. 
1 I 

IEEE Trans. AP, Vol. 22, No. 1, 145-146, 1974. 

IK'~~!OD~~TIOS 

Expwiun For the sol~>~~i:acc eletric held ~1 a line sttrlrce a-irh 

sn impttlsive current ware derived previu!lsl) [I]. TTowever, ~n 

~pplication to downlink cornmunicstictn and aigilalirrg, the titrio 

dependent voltq~s induced itr srnnll Im,p rceeivrils srttrntlr~~ :,re 

also of interest. The= vr,lt:tjia \vnveformn sre pntporriimal to !l~r 

time. dcrival.ives of l.he ap~xc>pritxte ~naqnetie firtlal compaiirt\ir. 

IIerr. we co~~ridcr thrar r~3putiae w:ivrf~,rnms tui~l ol>t:hin rtttzwt.i~,:\l 

reaailts lor irll the lield cocopo~tents iar ;my iocrsiit~n of tile o11ren.cr 

within the lorver b:rlf-space. 

ELELTI~IC Fixt.~ 

The line sorlrce carries an imp~~lsive eorrmt 108(t) awl is r:oli~ri- 

(ient will, tltc z ,=is. 'Tile s!~I~~~lrf~t~!t: t>Iherver is l6>c:~ied xi. [x,~-lt) 

it, llw c:~rllt (!I < 11;'. :LSSIIOW~! iu 13~~. I. Il >!I1 ~li~l~l~~~:~r~~coir~:~~t.; 

are it<v;lrrtrd, tito iuvrrsc trnti.i<lrm of rlie LIICIIV~ lime-l!:unt,in:i. 

snlrttitm for tlie clectric field [2] ~::trk Ire readily ev;~lic:tlrd. Tliis 

irn1)~llir I.PS~NISC [I] for. 1 > 0 is ri~cn I)! 

I where o is tile conductivity of the half space, rid w is penne;rhiiity 

I oi free space. 

1 In order to erduate (I), we consider irrtegnils of the type 

I The fint irttegrsl is well knorvn, anri thc seaorlrl crtrl Ibe ~vriltv~, in :i 

1 corlvenienl suntntati,,~: lurrn [:$I. Tl>aw 

i (w)'IZ exp (-2?/4m) I, = - - -. 2.,"2 

(1) 

I By ~ubstit,ilti~~g (4) nttd (5) into (I) and carryiog out the d~ilcrentiation, we obtain 

I lo &(l) = Ez(l)27') 4~wI.tc~hE) (6) 

' where the normalized elrctric-field wnvefann is given hy 

DifIusion of Electmmagnetic Pulses into the Earth 

from a Lie Source 

DAVID A. HILT, AND JAMES R. WAIT 

Abstracl--T hesubsurface electromagnetic fields of a line some 

dth a delta function cunent are considered. All displacement 

cments are neglected. The timedependent electric and magnetic 

fields at the buried observer are illustrated. The resulk waveforms all have a leading spiJte followed by a slowly decay& tail of 

opposite - solari-. The results have possible ap~lication to downlink - - - signaling in mine resfoe work. 

hlnnuaeri t rmivcd &ray 11. Lw3. 

D. A. H\I Is uith tlm lostitute ror Tel~cornrnunicztion Seienaes o_rt!#:c orr~il?eommullic;ltionJ. u.s. Devnnmena commt.rca. nou!dcr: <:1)1<1 FU.<0' 

J It \IJ~I ie ullh tbs. Cuot,rr;,lir. In.titu!r lor Ilrrr&rrn in F.n\lma>. rnmlal Srhmrm. Lni\rotly 01 ~'olur;utn. Uuul~lw. C#,lcl a.0102. ansl 

COnlloltal1t tn the institute 1°C l'nl~aumnltin?i~nli~~n Snmca. 001a. or Trl*comn~unicatioru. Us. Department or ('ommemr. Hual<ler. Colo. 

Wu". 

where D ;- z/h, sl:d T = I/jo&'). IIcre, lo hw urlils 111 cl~:tl.,:r, 

while i',, D, nnd T are di~nensionless. 

Actadly, (6) is eqeiv:~lent to thnt uf Wnit'n [I] whrlie rr+ult. 

contains the tabt~lated l)awson'!, intepd [4] rattler th;m arr inti~ii!c 

sum, and also T is deAni~l ilifferentlg. Ilcre, we ch(axe ll111: awrum;,- 

tion lop becnniw it is rel;rlivoly c<,nvenieni. to evrrlttate rlrl r crarzhputer and 3180 it yields the i~lyzript,~tic:xp:in~i(m for large 7'clin:et 1,". 

Wavefornu of 8, for vnriotw vulu<;+ of fl trre shtwn in VI. '2. 

Not,e that 8. is altentintetl rrotl rtratrl!eri out ici I) is iti~:n.,~-acl. 

brd~1~C1~1,: I'.,l:,.ll 

The time derivativ~ of tlte twr, rntrgnt:tie fir:l<l r:rrrriprloc:nln ti, 

and hu can be obtained from hlnxwull'x equnli<rrrr: 

ah.(t) - 1 ,?e,(l) . = 

rll Dl, (8) 

3) - =.-- I ae,(l) 

at Yo JZ (9) 
iEEK ~BANSACPIONJ ON AH'PI:NNA* AN11 *'1II~I'A~IAPIII\, .!.%2111itV t1):1 

Consequently, t11r rulta(tw iad,tt.t~il lit atn:di li>s>l>r aitlr vniwi .I-- ,ar 

y- directed RX~J (y, or 0,) are give11 by - 

Je.(t) "'(1) = .4x - 

ah (10) 

-- ~- 

Je'(t) ""(1) = AN - 

ar (11) 

where A is the loop area 311d N is the number of ttlms. 

By substituting (6) into (10) and (ll), and carrying oltt the x differentiations, we obtain tlie voltage wsveiorm 

(Earth) 

Observer (12) 

Ix.-hl (1:;) 

Fig. I. l'oeduetin~ haif spare rilil line source on surface and where sul~surfaea observer. 

v (D T) s 8 TJ 

4 

2T'I't-o (2k + I)! 

and 

V.(D,T) = T" 

1 - (-tY(k .t I)! +---I (l)T-li2j?k+l (15) 

dl2 ha (2k -t I j! 

Fig. 2. Normalized electric Reid waverorma ar, function or position. where D = z/h, and T = l/(.,&*). 

I 

As defined, V, and V, are dimensiunla~s, while v, and v, irre 

proportional to h-'. In the icnportant special cese of the (I~MTVPI 

directly below the source (D = O), Vv is zero and I= siiuplifie+ to - 

3 I - ..y-* - -r-!,?y-,l' + ~ 7'-4 

2 2 4 'I 

(It0 

6 

The response function^ V, and V,, fur arbitr;~ry D, were awnputed imn~ (12) and (I:<), and al.~? *hewn in Figi. :i ~n,i <. As imlicated, the \vavefurn~s are itrel.rhcd out a* I, increbea. Tile 1ir:ik elf 

V, decrensw with inweaving D while V. incre~ue from Zen, for 

small D before decreaqinp; for iarge:. I). -25 

Fig. 3. >ornralie~d rvltnyo wavafunn lor luov wlth horizuntal axis 811 runction or pmltion. C(l~cl.onlxo l{r:ac~ass 

Yhile the ~wulte giveri here pertain to n delta ionction current, 

the response fur an arhitrmy current can be obluined hy conv<,ltttion. All waveforms shown have a spike fr,lloaed hy a slow ixil, aml . the length of the tail could he a determining factor in pidrewidlhi 

and repetition freqnencih. for downlink corn~nunicstion~ [5]. On 

the other baud, the depeni-nce of.putse shape on position cotttd he 

useful in source location I a buried ohserver. 

ACKS~WLEDOMEST 

We wish to thank fir. Geyer, Colorado Schwl of Llines, for his 

suggestions. and for the opportunity to exchange unpttblislled 

materid. 

RErsREscEs 

111 .I. R. \~ait. :'Transtent. arcitatiutr or tire ~i~rtln hy R iine xnut.~c or e 

c~~rrclil." Pnlc. IEBE ilrtt.). ?"I. fie. ('1,. l2hT- llhW AIL% i!);L. 

Fig, +, > or,,lll tiletl ,or witl, vanical a7 191 --. Elmn~rnoanslic Il-ocrr ~n Slraltlied AlPdin. 2t~i -<I, orr~,~~l. 

hlnction 01 patcion. En land: Pergarhon. 1970. 

I31 1. %I. Ryshik and I. 8. Gradstein. Tablrr 0) Srrirg. Produt.1.~. ntid 

Inlrgrnlr. Reriin. Germans: \mi> Deutsnher V~rlng clrr \liarr,,- 8maft1-n. 10R8. 1,. 180. 

141 A. Abramnnitz and .J. A. sre~als. Edr.. Ilondhnnk 01 \~nlhrrrz,tllrot 

Funcrfnnr. \Vashingtvs. D.C.: Xaliot~uI Ilureilll or Sia~kda~tio, l!l(i4. 

El Fi.3dginrue. L. Obll and .4. J. Farstatl. "Radio tranxmi~ric~n to n?iil 

Worn roderground 6oa1 mines-tileory and mesrurenwttt.'' IEER 

Trnns. Cumnun.. vol. (-0JI 21. I,V. 191-201. 31ar. 1~73. 
Electronic Letters, Vol. 7, No. 8, 

22 April 1971 7 

ARRAY TECHNIQUE FOR ELECTROMAGNETIC 

POSITIONAL DETERMINATION OF A 

BURIED RECEIVING POINT 

Id,.llq trmr: l'vlrron~n~mirntlum lids, SSIw -~ .. . - . . .- . -- . - - - - .- .. ... We d~~rnh a VIICIIW 10 1.1(01. Fmaaxl m . ~u.1 rnm II,IC~,~C. da~tt~r 1.. n.<.t,. 1. xn.te not tnt<d. would 

m.qu,,c ""I, d., its tl"~,l..l "8 rur ur equ4ppll rot" . 1,np trnlrmc dr.= opre*t .,s frqucwk IUI inr low 

ran'. .I( <.llcd lor 

There is mom than a casual intcrest in txploitin~ elrclra 

magnctic waver to localc ininerr following a disostcr. The 

powibility that all convcnlionol means of communication 

are climinalcd should not be ruled out. Howevcr, if the 

miners arc equipped with a simple receiving dcvice, it is 

feasible for then) lo perceive an audio-frequency cleclro- 

~pelic signal from a surface tra!~smilter. Such a downlink 

communication scheme has been demonstrated by the 

Cmrerearch Laboratory af the Weslinghouse Elcclric 

Corporation. They alro envisage an uplink scheme using 

~cismic signals enc crated by hammer blows. A technique for 

utiliring the xismie signals for direction finding andlor source 

location has alro baen considered (as described orally by 

D. B. Large of Wertinghouu). Hers. we would like lo 

su~nt lhat lhcre may be some dircction-hnding capability 

lnin8 a scanning array of clectromagnctic line sources on the 

surf=. If such a method is to be feasible, the receiving 

terminal must bc simple. We outlinc a porsiblc scheme to 

reh*ve this goal. 

The simplest scheme which wc envisage is a parallel pair 

of long currcntcsrrying cables on the earth's surface. If the 

Wc now chwv a rcnangalar co-ordinae system (t. y. :) 

so that thc Earth's surfz~ic is tk plilnc y = 0 nnd the Lint 

cable is coincidCnt with the : ards. Thcn. the srcc,t,d cluhlr is 

hted at r - --d and ). - 0. Ihr rrcezvlng lu~~llioo ir 

depth y - -A, but Its lwttlil,n lclilllvc to lllc cohlc plur 

ir not known: i.e. wc wish lo detcrnlinc thc t co.o~,l~a;lte 

of the ruhsurfa~~ point P tndlcurcd tn k,i~. I. 

A rclativcly simple qu'lnlil~ 10 ~1~1~~1 at tht w~$L!~~.Icc 

point P ir the vertical conlponcnt of the rmtllrant ni;t$netic 

field. If the frequency of llie alternating current III the <ahl-.\ 

is suficicnlly low. the fields a1 1' can be erumsled fnrur *trlic 

theory. As dirwrvd elrwherc. this calls for opcrattna 

frequeneics of the ordcr of IOOHz or Icss. and lrrnltr thc 

depths to lar than several hundred metres for modrratrlv 

conducting overburdens. For purpuscs of this discusvt>n. 

we will employ the static theory, even though there is a more 

rcfincd analysis available', I. 

Under the conditions stated. the vertical magnetlofield 

componml H, at P is given by a simpk adaptation of the 

Biot-Savarc law. Thus. 

where X - xlh and X. = dlh. If the ob~rver is equipped 

with a rudimentary horironlal loop (i.e. vertical axfir) and 

vohagc detector, he is able to note by a viFmic hammer 

blow when the signal has a maximum or a mtn~mum. For 

example. if a is varied in some systematic vattern by the 

surface operator (e.g. a saw-tooth ran.ofone minute). there 

will bc a mint in timc whcn 11. vanishes. For fixed and 

known vaiue of X. one can th;n determine X from eqn. I 

by sr.ting the term in braces equal lo zero. In general, this 

kads to the cubic equation 

.+x. 

X(I+(X+Xo)') = .(I+X'1(X+Xn) . . . (2) 

whnchcan te solveo erplicttly ior the des~rm root a.. 

A graphical interpretation of the rcsultr is ureful at this 

slage. To thin snd, we write qn. 1 in the form 

. H, = (1/(2nh))(Bo(X)-sB0(X+X~)) . . . (3) 

where B.(X) I X(I + X*)-' = (X+ X-')-I. The normalised ficld of the current I in the main cable is B.(X) and the 

normalis4 ficld of the current -111 in the wcondary cable is 

-uBo(X+Xo). Thex functions arc sketched in F)g. 2. x As indicated. the sum of thm is the resultant normalired 

W ZnhH,lI. The latter is obviously mro at some point 

X = X. where X. > 0, provided that 0 < ~r < 1. 

Fig. 1 Cress-secrh~I vidw ol$ifwfion It is easydo w. from eqn. 2 that. if both X and X. 4 1. 

the solution for the root is partjcularly simple. Then 

\ Y X. - .X.I(l-8) . 

urihcd endpoints of thev cabln are ruffrimtly far away. 

lhc kids in the vicinity may bc imagimd lo emanate from 

infinite line roureer. 

T' configuration is illustrated in Fig. I, which is a cross. 

Mional view of the situation. As indicated. a ceurrtni I 

ampem flow in one cable. while a current -mi ampm 

no% in the olkr fabk. Hcrc # is a fector which fan be 

nried by the operator on the surfact. The two cables are 

scpmed by a distanced. Wch is kept constant for a given 

IQ1. 

Rapintad lrm ELECTRONIC LETTERS VOl. 7 NO. 8 

I1 is also evident that the resolving power ol the scheme is 

only satisfanory ?hen X. and X. arc not too large. 

A8 mcntioncd above, thesimple analysis given here is only 

valid when eddy currents induced in thc overburden are 

negligible. At suRfiently low frcquencicr, this will always 

bc thc caw, Of courw, this impom a dilkully wnlh ihe 

rrcrivins equipment. which necdr to he mate unauvc a% the 

In ,ucncy is lowcrcd. Astu;rtly. Ihe Pln~tple of the dclectktn 

~.....~irUillapplr(lMawhm cddy currentv arc not ncyl~ubk. 

It lhcn bccom ncnury lo anulne a halhpu rnok.1 ul lh. 

earth and rep- the kld lunct#nn U,,lrr b 118 nwrr mwrol 

form B(X.NJ. The Iatlcr iq a campkz fuwttun 'I hnlt 

X and !I - #2 hid when d IS thc cleclrocrl rktn dcpth ,,I llu 

overburden. TeMn of BlX. 11) urr ewrn eluvlure.' In 

principle. thy on bc uxd lo dcvlv a dclcetoon ulsmw lur 

the gcneral cau of a condtoct~vc ovcrhurden. 

The d&lcussim has conridcred r 2-diwnroml configurd. 

tion involving naralkl-line source*. In a ssnvlc nppl~c*ttt,n 

ofthe technique, it is only po.sible lo deduce thc value of the 

xm-ordinate of the obxrvcr. The procedure. howsvcr. 

on bc repeated for current line sour- parallel lo thc xsr~s. 

in which caw the zux)rdinatc of tiy obxrvcr could be 

dcurmimd. 

Olhn paribililiu incldc the ux of arrays of line rovron 
opmtinl either at the nmc rwucncy or at rnultlpic 

fraqucnc*.. 1he rc.itb~lity of wch =hema ahot,lJ Iw 

inventimtcd rt 1he.Urlmt opportunity. 

I. R. WAIT 9th March I971 

lnslilutr for Trlrmnmr,nirariun Srjenm 

OBpc of Trkcottm~uu~~tionl 

b/&r, Cdo. 80302. USA 
Electronic Letters, Vol. 9, No. 15, 

26th July 1973 

LOW-FREQUENCY IMPEDANCE OF A 

ClRCULAR LOOP OVER A CONDUCTING 

GROUND 

In wrious lo~hnical rppli~';llions, onc is inlcmtcvl in transmitting WRII powr Tmm a ground-baud loop anlcnnr. I:or 

example. in propod in@c-resuctwhniques.' it is dcsircd to 

conmmunic~lc with n suhsarfacc ohserwr via a lrunsmissian 

path through thc overbunlen. Inn previous rnvlyliwl study,' 

we considered thc rubrurlm rilds (or on inwlulcd circuhr 

Imp of wiro on or just abovc tho Earth's rurlra. We did 

not. h-vcr, cansidcr the power rcquimd to mainlain thc 

spaifmi uniform current I in the wire loop. In this lcttcr. urc 

cornidathis problem. 

As belore. the frcqucncy is assumed to be ruffcicntly low 

for the current round tho Imp to be uniform. Alternatively, 

the fed mints arc sufiicntly numerous lor an adc~uetcl~ 

uniform Nrmnt to be mrintaiakd, even for higher ircqknci& 

Thc quantity of intcrssl hcm is ihc rll impcduncc 2, which is 

ddvrd bv Z - VII. . whcre V is tho tobl vdtaxc intcsratcd . 

mund 1h;loop. . 

The gometry is indicatd in Fie 1, whcre the Earth is 

taken to be a hom~us halfspace of conductivity a lor 

x < 0 with mpt to a cylindricakwrdinate s~stm (p. I. I). 

Thc loop of radius 0 is located at z - k. in the vacuum region 

for I> 0. The singleturn wire making up the loop ha a 

radius c, whcm e< 0. The wim italf i urumsd Lo Lm 

pddly conducting. 

For thc modcl adopted, the elstric Md E. in the swa 

z>Ocan be derived from E,-JFIap, wh F i a du 

fwfion of p and I.Under quasishtic m~nditioiu, the latter 

irnimbYz 

x Imp(-111-hol)+(L-u)(L+u)-' cxp (-l(z+ho))]dL 

. . . (I) 

whueu - (La+ &IIO a)'. ll~~timsfacloradoptcd isex0 (iw). 

and tho angular ircquary m is ~uumed to Lm suffricntly 

IW for all ll!~ dirpl;trmrnl currcnls to h? ncgligiblc. Thc 

nwg~rlic Frnr;thility of thc wlwlc rtwr is lvkcn Lo bc 

,to(= 4nx 10-'Hlml. 

Thc self inlwdamr Z is now nlcubtcd Tmm 

1rwe now writc 

2-ly.rurr(P+S) . . . . . . . . . (3) 

it easily fdlovr that 

r - r j(l,(r)iaerp(-8.x)dr . . . . (4) 

0 

Fig. 1 Side view 0fcicul.r loop or" the eonducf#ng hallrpacs 

#mod#/ ol It18 pound 

m x- (2 +i2Aa)+ ' 

S - x J (J~x))' x+(Xz+iZAz)t exp (- ~x)dx (5) 

a loop radius Am-- 

d skindepth 

c wire radius a,--- a loop radius 

and 

U0+c _ _ 

2x loop height 8-- - a a loop radius 

Htn. of course, P is the (normliwd) primary inductance of 

che Imp, and can be expreucd in lCrM of tho complcle 

elliptic integrals as follows: 

((I- ) k-(k)) . . . . (6) ko 

whcn ka - 2(Bo2+4)-'. Hem, lollowing standard notation, 

and 

.,I 

~(k) - J (I -&¶ sin*^*@ 

0 

In th wncral uw, it spFepn that lhc la~nddry lnflucncc 

chnr~clcr~wJ by S must be cv.tluittd by numcr~wl mn1audt 

For rmdll v~lun of A, us wn write 

S= -U(u)@. . . . . . . . . . (n 

whom A' 
If. in ~Uilion. U .I I. wo can rcPlveo tho eaponcntivl by 

I -fix in qn. Y, ybldiny ~lu~nl inluyrulra 'thm 

Flp. 2s Function fl8) ShoWblg b *mitho b.)uvkw bI d 

e v.lmr 

.s~ll-d..m.i~lm4 rn . Ill-@I*). .- 

b bB.ppo.rut14SL). "lil 

is thc leadins tern, in an expansion in inwns powerr of A. 

This inlcgral has bCen given by Luke;' thi~r 

. . It apgwisn thi. is only II v;tlid corra'i'tn if tlr pndan All 

is wry largo ~"~~~~~.~nxl with I. 

Nuilrriui~l vdtm of I' front cqn. 6 wrc oht,,incd 

Wral Vulucr or U. ur rolkrur: 

U? - lo-.; I' - 13.69 

BO - lo-'; P - 11.59 

Be - lo'.; P - 9.29 

The mwwnding values of thc inductam L of the loop in 

fmeap;,ecarcthcn L = 1,"nl' ht:~~rys. 

Using n~~nicril'nl inlrfir;~tion. the cornplcx qa;dntity s w;, 

cvalu;~l~d* front cqll. 5 ftlr various v;thhcs of U and A. of 

parli~uhr inlcnrl is lhc rmisliv: casltponanl K = lac2 =. 

II~I,W(- lnt S) ohms. The non~talised quantity #IllO C,X, iq 

plolld in Pis. 2A rs il f~llwtiall of A fnr variaar v:thter of U. 

Aa indirulnl. H vurnx linearly will* A' fur A ILYS tll;,,~ about 

0.2. In this nngc, fnr 1110 U v;durx indic;tled. 1110 itpprnximalim S 2 -iF(U) A' is npplicablc. 'To uid the urcr, a pllri 

of lllc h~nctioti IYU) is shown in Fig. 2s. along with the 

limiting asyn~plotcs. - 

The liillililig valueof R for old lending toward$ inlinily I* 

rathcr intcrcsting. If B is sulliciently smitll, wc scu 11,;tt tt,c 

ordinatein Fig. 2r approactm n14. This is cntircly consnstent 

with thc mmpt that thc mistan* of thc loop is cquivnlcnt 

to 2nax(1tom/8) ohms. The factor p0ru18 is idcntnwl 

to the mistance per unit~lcnsth of an infinitely long currcntcanying wirelyinpon orjust abovcthssurf~ofaconducting 

hpltswa.' 

The prcxnl rmull can bc ad in nlilnuling tllc total pouer 

rquircnxts tor dnwnlink ~r,rnlnunicutian system. I.or 

example. if w sh- o = 0.5 km. f = LW) Hir an11 n = 10' 

slm wc sec from Fig. 2n th~r -1m S e 0.25 and thrt the 

wcnponding urlw of the rni~tancc K is 0.10<1. l'hn i5 

complrablc lo thc ohmic mistance of thc loop itxlf if, for 

cumplc, we rclectcd a ringlo-turn copper wirc with a radius 

bltb order of 1.4 m. 

1. I. WAIT 3rd July 1973 

K. h MlU 

I~~lllunfor Trlc~m~m~~t~icution Srlmrrr 

om of lil~~~mmltt~,cyI,",,, 

US Dr~lmr 01 Commm 

dorrkkr, Cdo. 8UM2. USA 

I UIOI. D. D.. IALI.. L.. and rrlrrru, r. 3.: 'Radio tran%rni%sion in and 

horn wdnpravnd coal rnina-lhcory lnd marursnml', IECC 

Trams.. 1911. COM-21. pp. 194-202 

1 WAIT. r. I.. and s~lr4 K. P.: I~h~daco ~Icclmrnugnctic field< or r C~IWI~I loan of cvmnl located sborr rrwnlt'. ibid.. 1972. AhM. 

pp. 520-522. 

3 w,lrrrou, A. n.: Teblc, or svrnnnblc wric, md integrals involving ?+tuMions' (Holdsn-Day. 1968) (c.8. wc lorrnutas 11.308 and Il.'U, 

4 LUW r. L.: .I~U~IS or BIUCI ~uIK~~O~S., (MCG.~~HIII. 1962). 

p. 114 no. M, 

$ WAIT, 8. I.: On lhc impcdsncs of a tons rirc rvrlxndcd ovrr the 

smund'.Py. Inrt. Radio W.. 1961.49, p. 1129. (Notelhat the 3 on 

th. denommaDr or the -r mn, m ihc two c~urt8ons Tor An .bwY be 3") 

ELECTRONICS LETTERS 26th Juh 1973 Val. 9 No. 16 
Canadian Journal of Physics, Vol. 51, No. 14, 

1534-1540, 1973 

Subsurface Electromagnetic Fields of a Grounded Cable of Finite Length' 

DAVID A. HILL AND JAMES R. WAIT 

Insrirurefor Telecommunicarion Scirnccs, gficr of Telecomn~r~nicarionr. 

U.S. Deparrmenr of Commerce and Croperarive Inslir~rre for 

Restarch in Environmrnlol Sciences. Universiry of Colorado, Boulder. Colorado 80302 

Received March IS. 1973 

The subsurface fields of afinite line source or current-camying cable arc examined. Some special cases, 

such as the low-frequency limit, are treated analytically, and some simple working formulas are obtained. 

Thegeneral fieldexpressionsarereducedtosingleintegrals with finitelimits that are evaluated numerically. 

It is shorn that if thc rource cable is sufficiently long, the fields are approximated by those for an infinite 

cable. The results have possible application lo downlink communication and radio location of trapped 

miners. 

Onttudie les champs produ~tsou, lasurface du rol par une source const~tute pdr unc ponnon rectllogne 

lime d'un cible transportant du courant. Cenans cas seciaux, par exemple, la limltr aux ba,e\ frtquences, sont traitts analyfiquement. el auelaues formiles directement a~~licables sont obtenues. Les 

cx~ressions ~CnCrales des com~osantes dc charno son1 riduites a des int6Ges simoles avec des limiter .. ~ . ~~-~- ~~~ 

fin&s, el ces;nt~~rales sont 6valuks num6nqwment. On montrc quc si lc cible source cst suffisamn~ent 

long. les champs ront appmximativemenl les mhes que ceux d'uncihlc infini. Les resultat, on! des rpplC 

cntions pour lcs communications souterraines el la bcaliration par rad~o de mincurs emprisonnis. 

Can. I. Phyr.. 51. 1534 (1973) 

The subsurface fields of an infinite line source 

on a conducting half-space have been examined 

by Wait and Spies (1971). In any real communication link from the surface to an underground 

terminal, the line source or current-carrying 

cable is of finite length. Here, we examine the 

subsurface fields of a finite line source with the 

purpose of determining what cable length is 

sufficient, and when the infinite line source is a 

good approximation. The earth is taken to be a 

homogeneous conducting half-space, and the 

current in the cable is assumed to be constant. 

The latter assumption is well justified at audio 

frequencies when an insulated cable is terminated 

by grounded electrodes that are separated by 

distances small compared with a free-space wavelength (Wait 1952; Sunde 1968). 

The geometry of the line sourcc of length 21 

located on the half-spice, and the observer in 

the half-space, are as shown in Fig. 1. The observer is located at (x, y, z), and the x-directed 

source current I runs from -I to I on the x axis. 

ITraduh par Le journal] 

The current varies as exp(iof), and the angular 

frequency o is sufficiently low that all displacement currents are negligible. The Hertz vector 

contains both x and z components (II, and II,) 

and the contribution from an incremental source 

of length dx'located at x' is given by Wait (1961) 

and Banos (1966): 

I dx' dn, = -- 2za Jm sx J~(~P)A d~ 

Cll Idx'a euz dn, =-- - JO(~P) dl 2m ax S, ~+x 

where 

u = (a2 + y2)'12, yz = iopoa, 

p = b2 + (x - x')~]"~ 

'The r-h reported here was suppomd by the U.S. ~~lx,y,zl 

Bumu of Mina. Pittsburgh Mining alld Safety Research FIG. I. Finite line sourcs on a homupncou, half. 

Ccatsr. space 
HILL AND WAIT SUBSURFACE ELECTROMAGNETIC FIELDS 

po is the permeability of free space, and a is the 

conductivity of the half-space. Here, and in what 

follows, we restrict the observer to the subsurface region z < 0. 

Since a numerical integration on x' will be 

required, it is desirable to eliminate the hintegration. For this quasi-static situation, Wait (1961) 

has shown that the h integration can be performed to yield the result 

where N = Io[(y/2)(R + z)lKo[(r12)(R - dl, 

P = R-' exp (-yR), R = (p2 + z~)'/~, and I, 

and KO are modified Bessel functions of order 

zero. The magnetic field components are derived 

from the curl of the Hertz vector 

131 dH = o curl dn 

By carrying out the curl operation in [3], and 

utilizing the fact that N satisfies the wave equation, (V2 - y2)N = 0, the magnetic field components are obtained: 

The electric field components have been given 

by Wait (1961): 

Idx' a3N C51 dE, = -- 2x0 ayaxaz 

To obtain the resultant fields for the finitelength cable, we integrate [4] and [5] over the 

range of x' from -I to I. 

The components of interest here are H,, H,, 

and Ex, since all other components vanish in the 

vertical plane x = 0. Also, these are the only 

nonzero components everywhere if the line 

source is of infinite length. For normalization 

purposes, it is useful to write the fields in the 

manner 

I 

H, = A(H, Y, X, L) 

where H = (~p,cr)"~h, Y = ylh, X = xlh, and 

L = I/h and h(= -2) is the observer depth. 

Note that A, B, and Fare dimensionless. For the 

case of the infinite line source (L = a), these 

quantities have been tabulated (Wait and Spies 

1971). By comparing [4] and [5] with [6], the 

following integral forms are found for A, B, 

and F: 

The specific expressions for the partial derivatives ofPand N in [7] are given in the Appendix. 

Low-Frequency Limit 

The low-frequency limit is useful both as a 

check on the numerical work and as a simple 

means of determining trends at low frequencies. 

If the frequency approaches zero, then y approaches zero and the Hertz anpouents are 

given by 

Id'' Idx' 1 dn. = Jm eLzJo(hp) dh = - 4no - R 

-I dx' (X - x') dn,= -- 

a p e"~~(~p) 
CAY. J. PHYS. VOL. 51. 1973 

where we make use of the integrals listed by I dx' 1 

Wheelon (1968). It is important to note that dH, = 4a R3 

dn, does not vanish in this limit. This is con- [g] sistent with the notion that the electric fields 

can be derived from a scalar potential cD that is 

given by 

There is also an x comnonent which ~~~. ~-is not ~~.. of -- 

for the static limit. interest here. The quantities in [9] can be inteThe magnetic field components are obtained grated from x' = -I to x' = +I to obtain the 

from the curl operation in [3]. Thus total fields 

From [lo], the normalized quantities A and B, as defined in [6], are found to be 

B(0, Y, X, L) = A[ L- X L+ X 1 + Y2 [(L- xy + 1 + ~~11'" [(L+ x)~ + 1 + YZ]1/2 I 

As L approaches m, the following results are obtained: 

The results are of course independent of X and agree with infinite line source results (Wait and Spies 

1971). 

Results for finiteL are shown in Figs. 2 and 3 for '. = 0. Note that A in Fig. 2 overshoot:^ its final 

value and that L = 1 is sufficient to obtain a horiLu,,tal magnetic field equal to that of the inlinire 

line source. However, the B values in Fig. 3 approach their final values quite slowly, particularly firr 

larger Y. Consequently, a fairly large value ofL is required to obtain a vertical magnetic field nearly 

equal to that of the infinite line source. 

High-Frequency Limit 

The other useful limiting case is for high frequencies where lyyJ is large. In this case, terms involvingP are negligible, and N is asymptotically given by (Wait 1961) 
HILL AND WAIT: SUaSURFACE ELECTROMAGNETIC FIELDS 

FIG. 2. Low-frequency limit of the horizontal magnetic field as a function of Y and L. Final values are dashed. 

FIG. 3. Low-frequency limit of the vertical magnetic field as a function of Y and L. Final values are dashed. 

El31 eYz N-- for z 6 0 

YO . . Consequently, dE, and dH, are given by 

In order to determine dH,, we must include higher-order terms in N. If this is done, the result is 

The x' integrations can be done in [14] and [I51 to yield the total fields: 

E, - -pew- a2 11" [I - x + [(I - x)~ + yZ11/2~ 2no ay2 

- In [-(1 + X) + [(I + x)' + y2]1'2]} 

I 

[I61 H, - --eTz- {ln [I - x + [(I - x)' + Y2]112] 2ny ay2 

- In [-(I + X) + [(I + x)' + y2]112]} 

If I approaches oo, then [I61 reduces to 

Thus, the normalized quantities A, B, and Fare given by 

C181 4 B(H, Y, X, oo) - i~,ys exp (- i'IZH) 

2 

F(H, Y, X, 00) -- i~2~2 exp (-ill%) 
CAN. I. PHYS. VOL.. 51. I973 

: 0 8 [: DI - 

1.0.1 li0.l 

1.1 

0.4 Is01 020 - 

0 11 

0 1 111 0 15 H~2,r=O 

"il. liO 0.10 

7.2 0 10 02 

"~2 

7.2 om I i as u-l, I.0 

0 81145 2341 11145 

L 

FIG. 4. Maenitude of the horizontal mannetic field as a function of Y and L. 

Fu,. 5. ~ablludc of the venlcal magnGc field as a functlon of Y and I.. 

Fa. 6 Magnitude of the horizontal electr~c ficld a- a runctlon of Y and L 

These results agree with those of the infinite line 

source (Wait and Spies 1971). 

We stress that the "high-frequency" limit 

solutions discussed in this section are applicable 

when the horizontal distance, p, is much greater 

than a skin depth in the earth, and much less 

then the free-space wavelength. Thus, the 

formulaa are of limited applicability, although 

they are useful for checking the results of the 

numerical integration. 

General Numerical Results 

For general values of H, a numerical integration over finite limits of the expressions in [7] 

is required. Even though the integrands are 

quite complicated, they are well behaved 

numerically. Consequently, the computer program for A, B, and F is quite fast, and the influence of the various parameters can be studied 

easily. Here, we illustrate some of the important 

features in graphical form. 

Results for H = 2 and X = 0 are shown in 

Figs. 4, 5, and 6. For large L, all quantities 

approach those of the infinite line source (Wait 

and Spies 1971) which is reassuring. Although 

A, B, and Fare now complex, only the magnitudes are plotted since the phases are relatively 

constant with L. 

We note that [A1 in Fig. 4 still has a slight 

overshoot before settling down to its final value 

for large L. A similar trend for IF( is shown in 

Fig. 6. In both cases, L greater than about 1 is 

sufficient to yield field strengths nearly equal to 

that of the infinite line source. However, a larger 

value of L would be required to reach a nearly 

maximum value of IBI as shown in Fig. 5. Consequently, if the vertical magnetic field is 

important, a value for L in the neighborhood of 

2 might be more appropriate, if we wish to 

simulate an infinitely long cable. This could be a 

crucial factor in designing a subsurface location 

scheme that makes use of the well-defined structure of the magnetic fields produced by an 

infinite line source. 

Concluding Remarks 

Analytical expressions for the subsurface 

fields of a finite line source have been derived 

for high-frequency and low-frequency limits. 

For the general case, a single numerical integration along the source is required. Although a 

constant current distribution has been assumed, 

an arbitrary distribution would present no 

problem since the source integration is done 

numerically. Also, multiple grounding of the 

sol7 .: cable and arrays of such cables could be 

treated by simple extensions of the present 

formulation. 

Results for low and medium frequencies indicate that the finite line source fields approach 

those of an infinite line source quite quickly. For 

instance, L - I appears to he sufficient for the 

horizontal electric and magnetic fields, and 

L - 2 is sufficient for the vertical magnetic field 
- HILL. AND WAIT SUBSURFACE ELECTROMAGNETIC FIEL.DS 

It is important to realize that the finite line for finite L implicitly account for the effect of 

source excites currents in the earth by induction the conduction currents injected at the elecand by conduction. The effect of the latter trodes. 

vanishes in the limit of an infinitely long cable In a sequel to this work, we plan to consider 

when the magnetic field is measured at the the fields produced by a finite source cable losubsurface point. The results shown in Figs. 24 cated within a coal mine. 

Appendix 

There are four terms in derivatives of P which are required in 171. The differentiation is straightforward, and the resultant forms are 

ayaz2 - ~3 

zzR{[l 2f-l. X>x'2] +?I R[l- ~2 

The four terms in derivatives of N can be simplified by replacing the derivatives of the modified 

Bessel functions by Bessel functions of zero and first order. This procedure, used previously by Wait 

and Campbell (1953a, b), yields the following expressions which are quite suitable for numerical 

evaluation. 

C201 -- a2N TY - -CAooIoXo + AIIIIKI + AOI~OKI + AIOIIKOI ayaz 2 

where 

a3N y ~=~CBOOIOKO + BIII~KI + BollOK1 + B1ollKo] 

where 

- a4N -- y ayZazz - 2 CCOOIOKO + CIIIIKI + COI~OKI + C~ol~Kol 

where 
CAN. I. PHYS. VOL. 51. 1973 

3zZ(1 - 2y2y2) -- 15zy2 15y2z2 

R2 ~3 +7+- 

3 - y2y2 zy2 12z2 r2v2.(R 

R2 --p--7 :;i] + 

f 

where 

, DII=a 13-7-72 R "( R 

"" 3 

The argument of I, and I, is (y/2)(R + z), and the argument of KO and K, is (7/2)(R - z). 

Acknowledgments sion systems (Dover Publications, New York), Chap. 

V, pp. 140-172. We wish to thank Dr. R. G. Geyer, of the WAIT, J. R. 1952. Can. J. ~hys. 30,512. 

Colorado School of Mines, for his suggestions, - 1961. Can. I. ~hys. 39. 1017. 

and Mrs. Lays A. Gappa for her assistance in WAIT. I. R. CAMPBEL BELL, L. L. 19530. J. Geophys. Res. 

preparing the manuscript. - 58, 21. 19536. J. Geophys. Res. 58, 167. 

WAIT, 1. R. and SPIES, K. P. 1971. Radio Sci. 6, 781 

BANOS, A. 1966. Dipole radiation in the presence ofa con- WHEELON, A. D. 1968. Tables of surnmable series and 

ducting half-space (Pergamon Press. New York). integrals involving Bessel functions (Holden-Day , Snn 

SUNDE. E. D. 1968. Earth conduction effects in transmis- Francisco). p. 72. 
Pure and Applied Geophysics, Vol. 111, 

2324-2332, (1973fX) 

Subsurface Electric Fields of a Grounded Cable of Finite Length 

for both Frequency and Time Domain 

By DAVID A. HILL') and JAMES R. WAIT? 

Summory - The subsurface electric fields of a currentsarryingcable are examined for both steadystate and transient situations. Closed form expressions are obtained for two of the electric field components, and a numerical integration is used to obtain the third component. Waveforms for a stepfunction current excitation are displayed. The results have possible application to pulsed downlink 

communication in mine rescue oprations. 

The time harmonic subsurface fields of a finite length line source or current-carrying 

cable on a conducting half-space have been examined by HILL and WAIT [I]. The 

emphasis was on examining the effects of finite cable length on the subsurface magnetic 

fields. Here we consider the electric field; also we extend the analysis to the time domain 

by treating a step-function current. The nature of the build-up of the subsurface 

electric field is of importance. This rise time dictates what pulse width and pulse repetition frequency can be used in a downlink communication system. 

The half-spece is taken to be homogeneous, and the current in the cable is assumed 

to be uniform. This is justified for an insulated cable with grounded ends for sufficiently 

low frequencies 14, 3, 81. Actually, related problems of the transient subsurface fields 

of an infinitesimal horizontal dipole and transient coupling between surface cables 

have been treated by Wait [5]. The techniques employed here are similar. 

2. Frequency domain solurion 

The line source extends from -I to I on the x-axis, and the observer in the half-space 

(I < 0) is located at (x,y,z) as shown in Fig. I. The x-directed current is lexp(iof), 

and the angular frequency w is sufficiently low that displacement currents are negligible 

') Institute for Telecommunication Sciences, Office of Telecommunications, U.S. Department 

of Commerce, Boulder, Colorado 80302. 

') Cooperative Institute for Research in EnvironmentaJ Sciences, University of Colorado. 
Subsurface Electric Fields 

everywhere. Under these conditions, the electric field components from an incremental 

source of length dx' located at (x',O,O) are given by [6] 

Idx' J3 N 

dE,=--- 2na Jydx Jz' 

where 

OBSERVER 

Figure 1 

Finite line soum on a homogeneous half-space with a subsurface observation point 

o is the conductivity of the half-space, po is the permeability of free space, and I,, and 

KO are zero-order modified Bessel functions. To obtain the fields of the cable, wc 

integrate (1) on x' from -I to I. 

Since dEy and dE, contain a/ax (= -J/axT), the integration can be performed 

easily to yield 

where 

and 

r, = [(x 7 + yZ + z~]~'~. I 
David A. Hill and James R. Wait (Pagwph, 

After carrying ~ut thedifferentiation, E, can be written in the following convenient 

form 

E, = -[1/(2no;12)l E,.(W, L, X, Y), (3) 

where 

E,, = Ri3 exp [-(iW)'12Rl] [I + (iW)li2 R1] - Ri3exp [<iW)llQz] [I + (iW)'12R2], 

W = wpo oh2, L = l/h, X = x/h, Y = y/h, 

R, = r;/h = [(X T L)' + Y2 + 1]1/2, 2 

and h (= -2) is the observer depth. Note that E,,, W, L, X, and Yare all dimensionless 

and that W can be considered a normalized frequency since it is proportional to w. 

The dependence of 1EZ.1 on W is illustrated in Fig. 2 for L = 1 and various values of 

X and Y. In a typical example with o = 10-' mhos/m and h = 100 m, the frequency f 

is given by 

f = W/(2noh hZ) = 1266 W Hz. (4) 

Figure 2 

Frequency dependence of IE..] for various observation points 

Consequently, the frequency range in Fig. 2 is 126.6 Hz to 126.6 kHz for the above 

values of o and It. The phase of E,. is a generally decreasing function of W. 

In order to express E, in a convenient form, the differentiations in (2) are carried 

out and derivatives of the modified Ressel functions are replaced by zero- and firstorder functions. The resultant E, in normalized form is 

E, = -[1/(2noh31 Em( W, L, X, Y), (5) 

where 
Vol. 111, 1913)X) Suburfacc Electric Fields 

the argument of 1, and I, is (iW)"'(R - I)/2, and the argument of KO and K, is 

(iW)'12(R + I)/2. The dependence of IE,,I on W is illustrated in Fig. 3 for L = I and 

various values of X and Y. The phase of E,. is again a decreasing function of W. 

The integration for E, must be done numerically, and some results have been given 

111. However, here we wish to normalize in the following form 

Ex = -[Il(Zxah2)l E,,(W, L, X, Y), (6) 

where 

the argument of I, and I, is y(r + z)/Z, and the argument of KO and K, is y(r - z)/2. 

The x' integration was done numerically, and the resultant dependence of Ex:,.[ on 

W for L = I and various values of X and Y is illustrated in Fig. 4. It is interesting that 

the curves peak near W= 5 before falling off for large W. As in the previous cases, 

the phase is a decreasing function of W. 

A simple special case is the low frequency limit, where the electric field is given by 

the gradient of a scalar potential (61. Thus 

Zdx' dEdc = - P 'A) 

2n0 ax \r ' 
David A. Hill and James R. Wail 

Consequently, the normalized components are given by 

E&=(L- X)Ri3+(L+ X)Ri3, 

All computed results in Figs. 24 agree with these limits as W -+ 0. These limits are 

important because they are the final values of the step responses that are computed in 

the following section. 

Figure 3 

Frequency dependence of IE,.l for various observation points 

Figure 4 

Frequency degendence of IE,.I for various observation points 

3. Step response 

In this case, the cable current is given by IU(t) where U(f) is the unit step functior 

at t = 0. The step responses are given by the following inverse Fourier transform: 

exp (iwt ) 
Vol. 11 1. 1973IX) Subsurfaoe Electric Fields 

where Ex, E,, and E, are given in the previous section. 

The following standard inverse Fourier (or Laplace) transform is useful ln obtaining 

E 

5-'[exp (-a(io)112)/(io)] = erfc [a/(2t112)] U(t), (10) 

where 

erfc (x) = ;;ii3: e-Y2 dy 

is the complement of the error function Tor real argument x. By applying (10) to (2), 

we obtain 

-I a 

() --{re r12 b0 - ref r2 (p~/t)'~]} 1 I I ) 2na Jz 

T 

Figure 5 

Step response i?:" for various observation points 

The unit step U(t) has been omitted in (1 I), and it is understood from hereon that all 

time responses are zero for negative time. We now write (11) in the normalized form 

E:(r) = -[I/(2nahZ)] &(T, L, X, Y), (12) 

where 

erfc [(R1/2) T-lI2] exp [-R:/(4T)] E;" = - 

R? (nT)'I2 R: 

- 

erfc [(R,/2) T-"'1 exp [-RZ/(4T)] + 'c (nT)"'R: 

and T = t/(ajh, h2). 

Curves of I?&, vs. Tare given in Fig. 5 for L = 1 and various values of X and Y. 

Note that, at T= 2, E;" has nearly reached its static limit as given by (8). For typical 

values of o (= mhos/m) and h (= 100 m), the quantity o$,h2 is 0.1258 msec. 

Consequently, T= l corresponds to t = 0.1258 msec. The quantity up,h"as also 

appeared as the effective time constant for the transient fields of a pulsed loop [7]. 
David A. Hill and James R. Wait 

- (Papeoph, 

In determining 4, the following inverse Fourier (or Laplace) transform is useful 

[21: 

exp [-(a2 + b2)/(4t)] R-l{Io[o(iw)ll" Ko[b(iw)l"]) = 

2t 10 (j (13) 

Application of (13) to (2) allows the expression of the impulse response in closed form. 

Consequently, the step response can be written as an integral of the step response: 

&t) = -[l/(2nahZ)] E;"(T, L, X, Y), (14) 

where 

-O.qO u 

PI 1.0 1.5 20 

T 

Figure 6 

Step response 3;" for various observation points 

and 

The integration in (14) was done numerically, and the resultant step responses for 

L = 1 and various values of X and Yare shown in Fig. 6. Again, the final values as 

given by (8) have essentially been reached at T = 2. 

In order to determine E:, the inverse Fourier transform was done numerically as 

follows: 

where rn 

1 Ex.(W,L, X, Y) E&(T.L x, Y) =z J exp (i WT) d~, 

-m iW 
Vol. 111,19731X) Subsurface Eleztric Fields 

and Ex. is given by (6). The resultant curves 9f b?;':, vs. Tare shown in Fig. 7 for L = 1 

and various values of Xand Y. These waveforms, unlike those for b?;. and &:., contni-I 

an overshoot and a slow recovery to the final value. 

Figure 7 

Step response g:. for vwious observation points 

4. Concluding remarks 

Both the time-harmonic and the transient subsurface electric field for a finite length 

cable carrying a step function current have been examined. Closed forp expressions 

have been obtained for they and z components, whereas numerical integration was 

required for the x component. The time-domain waveforms are fairly simple and the 

final values can be obtained from a sca!ar potential. However, the waveforms for the 

x component do contain an overshoot. The time constant of the waveforms is roughly 

up,h2 which is consistent with previous results for loop sources. 

Acknowledgemenrs 

We thank Dr. RICHARD G. GEYER for his useful suggestions, Drs. H. PARKISS(I'II 

and J. POWELL for their encouragement, and their organization, the U.S. Bureau L' 

Mines, for its support, and Mrs. LOYS GAPPA for her help in preparing the manuscript. 

111 . . D. A. HILL and I. R. WA~. Suhsurfoee 'ecfromoanetic - Fields ofa GroundedCable ofFinite Lenrrh. -. unpublished report, January 1973. 

121 G. E. ROBERTS and H. KAUFMAN, Tdle ofLaplace Transforms (W. B. Saunders Co.. Philadel~hia ~ ~ ~ ~ 

1966). 311 pp. 

131 E. D. SUNDE, Earfh Conducfion Efecfs in Trammission Systems (Dover Publications, New York 

1968). pp. 14CL172. 

141 J. R. WAIT, Elecfromagneficfields of currenf-carrying wires in a conduefins medium, Can. J. Phys. 

30 (1952), 512-523. 
David A. Hill and Jama R. Wait 

[5] J. R. WAT, Propagation of cl~fromagnefic pulses in a homogeneous conducting earth, Appl. Sci. 

Res. 8B (IW), 213-253. 

[6] J. R. WAIT, The electromagneticfieids of a horizontaldipole in thepresence of a conducting haw--space, 

Can. 3. Phys. 39(1961). 1017-1028. 

[71 J. R. WAIT and D. A. HILL, Tfanrient signnlsfrom a burieddipole, J. Appl. Phys. 42 (10) (1971), 

38663869, 

[8] J. R. WAIT, Electromagnetic wauepropqgation along aburiedinsuiotedwire, Can. J. Phys. 50 (1972), 

2402-2409. 

(Received 29th Mwh 1973) 
Journal of Applied Physics, Vol. 44, No. 12, 

P 

5275-5279, December 1973 

Electromagnetic surface fields of an inclined buried cable 

p of finite length 

David A. Hill 

ZuIttutefir TeIavmmuninrt6WI kicnw OBca of Telecommunkcltha (I. S Dqarlmenr qfGxnmrm, 

Boulde?. Cdomdo balm 

(Received 5 July 1973; in final form 17 August 1973) 

The magnetic fields on the surface produced by a burid currmtclrrying cdblc of finite length arc 

caasidmd. A homogmeous hrlf-spacs model of the eartb and quiautic conditions us assumed. 

The cdbk is grounded at its ead pints and oriented at an arbitruy an& to the inwhce. While, in 

general, numerical integration is required. the loar-frequency limit may be trsrtcd dytidy. hall 

cable tilts us shown to modify the didon of the surface field but not the magnitude. The mults 

have possible apphtition to uplink mmmunicetion and radio locairn of trapped minu% 

I. INTRODUCTION tinuous across the interface. However, the limit of 

dn: as I approaches zero is seen to be zero. Since the The eubsurface 5elds of both in5nite-lengthLa and x and y derivitives are also zero, the curl is zero and 

flnlte-lengtha Ilne sources have been for the the magnetic 5eld is zero. Consequently, the vertical 

source located at the suriPce. Here, we exnmim the 

fields of a buried MQ-length cable whtch makes an component of the buried source doas not contribute to 

the surface magnetic field (within the quasistatic arbitrary angle wltb the interface. This cue is of approximation). ~ractical interest hcauae either the c&le - mav -. be tilted -- or the earth surfnee mav not be level. llre contrfbution from the horizontal component of 

The earth is taken to be a homogeneaua half-space, 

Md the current in the cable is ~ssumed to be ionstant. 

The constant-current usumption is valid it suifldently 

low frequencies when an ineulated cable ie grounded at 

the end points .c5 Slnce we.are prfmnrily interested in 

reception at the surface with loop antermaa, the deslred 

r- quantities are the three magnetic field compomnts 

evaluated at the interface. 

II. FORMULATION 

The geometry of the buried cable of length 21 carrying a cumnt lexp(iot) is shown in Fig. 1, and the 

angular frequency w is eufilcientiy low that dl displacement currents are negligible. Tl cable is located in 

the xzplane andmakes anangle u withthe z=Oplane. 

Ib center is on the z axis at z = - k, and a general 

source point on the cable hs Cartesian cmtdinates 

G',O,- kt). We introduce 9 as the distance from the 

cable center to the source point. Conaequentlg, the 

source coordinates are glven by 

x'=scosu and kf=k-ssinu. (1) 

The Hertz vector haa both x and z components (n, 

and n,), and in deriving the contribution from an incremental source of length ds it is convenient to resolve the current into x and z (horizontal and vertical) 

components. The contribution from the vertical component is given by Wait4 and Banos': 

mi= Y ds sina/4*)[&'exp(- Y%) -RLexp(- YR)], 

(2) 

where 

P ~=(i~yu)~~', R,,=[k-xfP+?+G+krP~/a, 

R=[Gi-x'P+~+k-k'Pl'la, z<O, kl>O, 

y is the permeability of ires apace, e.nd o is the conducU~l.itg of the half-space. Although @) applies to the 

case of both source and obsemr in the conducting halfit is useful for evaluation of the magnetic fleld 

d (h. .tuhee k - 0-), aince the magnetic fleld is concurrent is glven by Wait': 

where 

u=(Xa +y')'la and p=[(~-xr)l+yl]lf1. 

Wait7 ha6 shown that the X integrations can be carried 

out to yield 

where 

Y 

7, "0 

? Observer I r,y,zl 

FIG. 1. Ttlted Ilne source in a homogweous half-..raa 

Dnid A. Hill: Fiddr of n inclimd burad 4 of finita length 

FIG. 2. Low-frspuency hehnvior of H, as a hmctlon of L and a 

for observer directly above source. Cireles indicab the intersection of the cable with be interfaoe. 

Ids cosa a' mx=2,,exp(- ~h')~b-~), 

Ids cosa a" 

exd- ykl)-b-9. Ju' 

In order to determine dHz, we must include higher order 

P terms in N, and the result is 

a,=- Ids cosa 

2nyZ exp(- ~h')db-~). ay (16) 

In order to simplify the discussion, we again treat the 

case of the level cable (a =O). In this case, the s integrations can be performed to yield the total cable 

fields : 

-Zexp(- yk) aa -[Ln(I-x+p,)-In(-l-%+&)I, 4- Zny au' 

(17) 

FIG. 4. Low-frequency behavior of H,, as a function of L and a. 

If the cable becomes infinite (1 - -), then (17) reduces 

to 

These limits are identical to those for an infinite cable 

at the surface for a subsurface observer.= 

The ''high-frequency" solutions are actually valid 

only for a range where the horizontal distance is large 

compared to a skin depth in the earth and small compared to a free-space wavelength. However, even though 

this solution is somewhat restricted, it is quite useful 

in checking the general numerical results. 

V. GENERAL NUMERICAL RESULTS 

For general values of the various parameters, a numerical integration of the expressions as given in (9) is 

required. Even though the integrands appear quite complicated, they are well behaved numerically, and the 

resultant computer program for Hm, H,, and 4, is 

quite fast. Consequently, the influence of the parameters (H, Y,X,L,n) can be studied easily. 

Results for the static case @=O) with the observer 

directly above the cable center (X= Y=O) are shown as 

a function c' L for various cable angles a in Fig. 2. 

Only the 4 ~nponent 8, is nonzero in this case, and 

FIG. 3. Low-frequency behs-vor of H, aa a function of L and a. FIG. 5. Low-frequency behavior of H, as a function of L and a. 
Dnid A Hill: Fii of m indid burial wb* of fink b@h 

FIG. 6. Behavior of I H,I as a function 01 L md a for H-2. 

the final value for the level case (a =0°) is - 1. Consequently, even at L = 4, the level cable does not approximate an inftnite cable. By examining the curves 

for small L, it is possible to determim where the short 

dipole approximation fails-that is, the point where the 

curves are M longer linear in L. EZaminaUon of the 

curves shows that the linear behavior ends in the 

vicinity of L = 0.3. The various curves terminate at 

values of L where the cable would intersect the suriaca. 

The curves indicate that small tilts (a <zoo) do not 

strongly influence the results. However, further increases in a can change the redts considerably. 

Results for a different observation point (X=O, Y=l) 

are shown in Figs. 3-5. Aa u is increased, H, in Fig. 

3 becomes nonzero. For this component, the dipole approximation is always invalid because it gives eero for 

Hm. in Fig. 4 is seen to approach lte Mnite-linesource value of - 0.5 quite slowly. H,, in Fig. 5 is seen 

to approach its final value of 0.5 somewhat more 

rapidly. Figures 4 and 5 reveal that fairly large values 

of a are again required to significantly change the results from those of the level case. The crossover in the 

curve for a =60q in Fig. 4 is a result of a strong negative contribution from the end of the cable near the 

interface. &ch bhavior has been fad in both H, and 

FIG. 7. Behavior of I H, I ae a flmctlon of L and a for H=2. 

FIG. 8. Behavior of I H, I as a functlon of L and a for H=2. 

H, for other observation points when a becomes large. 

Resulb for H=2 at the same observation point are 

shown in Figs. 6-8. The tilt again introduces nonzero 

values of IH,I in Fig. 6. However, IH,l and IH,,I are 

again relatively unaltered by amall values of or. The 

dipole approximation again fails in the neighborhood of 

L =0.3. The phases are not shown since they remain 

fairly constant with changing L. 

VI. CONCLUDING REMARKS 

Analytical expressions for the surface fields of a 

buried flnite line source have been derived for the high 

and low-frequency limits when the source is level. For 

the general case, a single ncmerical integration along 

the source is required. A constant current distribution 

hna been assumed, but arbitrary distributions or cable 

arrays cwld be handled, since the source integration is 

done numerically. 

Results for low and medium frequencies indicate that 

the infinite-cable result is achieved only for very long 

cables (L > 5). The inllnite-cable result is reached for 

much shorter cables when the cable is at the intsria~e.~ 

At the other extreme, the short dipole approximation 

appears to be valid for only very short cables (L <0.3). 

Consequently, a large range exists where the general 

formulation ia required. When the cable is tilted, the 

short dipole approximation becomzs even worse. For 

instance, the x components of the magnetic field, shown 

in Figs. 3 and 6, are predicted to be zero. 

The effect of cable L.. is shown to be small for the y 

and z components for tilts less than 20'. However, the 

tilt does strongly alter the normally small x component. 

Consequently, small cable nits can be expected to produce only small chaqges in the surface field strength, 

but they may considerably alter the direction of the 

field. 

ACKNOWLEDGMENTS 

me author would like to thank Dr. J.R. Wait and 

Dr. R. G. Geyer for their useful comments. 

APPENDIX 

There are three terms in derivatives of P which are 

required in (Q), and the resultant forms evaluated at 

the interface k = 0) are 
Mi A. Hill: Fi&b of m inclined burW uM. of finio Inn@ 

- 

asp =Ir-x')yh'R-'exp(-yR)(y' +6yaR' +ha -ySR-'-6yaR-2-15yR.S-15R-~)]. 

axayaz 

The four nns in derivatives of N can oe simplified 

+15yRa +15R4), by repladng he derivatives of the modified Bessel 

functions by essel functions of zero and first ~rder.~.'~ 

hn~*exp(- y~)[- ya +y(yPf -3)R-I eylaL= 

This procedu e yields the following expressions at the 

evaluation : 

interface (a 

I 

=O), which are quite suitable for numerical 

+3(2ysy'-1)P+1S~R"+16y'R4], 

a'N II =W[A~,% +A,,I,K, +A,,IA +A,,I,K,] ax ay az aSp =y~aexp(- y~)[y-a +SYR-' +3~. 

ay az where 

I 

where 

B, = - yk'Ra, B,, = yhllT', 

B,, = - ~~(1 +WR-~), B~, = -R~(I - WR~); 

P a' N -jrp a a =$[c.J&, +Cl,IIKl+ Co1I&, +CIJIKoI, 

where 

c, = yheR-'(-3 + y'ja +15QR9, 

c,,=~P[-~ +hnR'(S -yay) +6y'R'-159kaR-'+2yp4R-'(@ -2kG)], 

C,, =R-~[~'R"(~~~P -3) -yaw +ha) -3h"R4(1 -2ya9)+15h'yR-' +15yh"r'- ~~yk'p"(R - k')], 

C,, =~-l[h*~-l(~ajP -3) +yaw +ha) +ShaR"(l -2ya9)+15k'y'R-' - 15ylk'2X4- ya3k'p-2(R +kt)]; 

where 

D,=yk'~+(3 - 15k"R-2- yah"), 

D,, = yk'R-'(- 13 +15hnR-' + yah"), 

Do, =R'l(3R4 +3~'R-'-15h"R'4-15k"R~s- ya +3yahrR-'- yak"R-a - 6yahrnR-'), 

D,,=R4(3Rd -3k'R-a -15k"R'.+15k"Rd- ya -3yak'R-'- yzk"Rq+6~'h"Rs). 

The argument of I, and I, is (Y/~)(R - h'), and the argument of KO and K, is (Y/~)(R +k') 

'J.R. Wait and K.P. Spies, Rdo Boi. 6, 781 (1971). I 'J.R. Walt, Can. J. Phys. 38. 1017 (1961). 

'D. B. urge, L. Ball, and A. J. Farstad, IEEE Trana. Q.D. Whealon. Table# dsvmmable Ssrten and intrpal# In- 

? Commuo. COM-21, 144 (1973). volvfng Bea8el FuncUw (Holden-Day. Ban Frnnolrcu. lHfl111, 

'D.A. Hill and J.R. Walt. Can. J. Phya. 61. 1534 (1973). p. 72. 

'J.R. Wait, Caa J. Phys. 30. 512 (1952). 'J.R. Walt and L.L. Campbell. 3. Why#. Re#. 68, 21 

'E.D. sunde. Earth Cadaction Eflects in Tmnsnfssfon sys- (1953). 

tems (Dover. New York. 1968). pp. 140-172. "J.R. Wait and L. L. Campbell. J. Geophyn. Res. 58, 167 

'A. Banas. Dipole Ru&Um (n the Preswe of a Caducung (1953). 

Ray-space (Pergamon, New York, 1966). 
Rodlo Science, Volume 8, Number 12, papen 1169-1176, December 1973 

Excitation of a homogeneous conductive cylinder of finite length 

by a prescribed axial current distribution 

James R. Wait 

Cooperative i~lstitu?~ for Research in Environmental Sciences, University of Colorado, and 

Environnle~tfal Research Laboratories, National Oceanic and Atmospheric Adrninislra:io~t. 

US Department of Commerce. Boulder, Colorado 80302 

David A. Hill 

Institute for Telecomn~unication Sciences, Ofice of Telecommunications, 

US Departmenr of Commerce, Boulder, Colorado 80302 

(Received June 14, 1973.) 

The boundary value problem posed is a truncated cylindrical region excited by a specified 

distribution of electric current over a concentric cylindrical surface. The end conditions are 

that the total normal current density is zero. The solution is carried through for excitation by 

a symmetrically located axial current filament that is adjacent to the cylinder. Numerical results of the resultant magnetic field are given for the quasi-static situation for a prrfectly 

conducting cylinder. It is indicated that the results depend significantly on the length of the 

cylindrical target. Even for very long cylinders, there is no quantitative similarity with the 

corresponding two-dimensional model of a cylinder of infinite length. 

INTRODUCTION 

The theory of the interaction of electromagnetic 

waves with finite or bounded structures is extremely 

involved except in a few special cases. For example, 

the vector wave solution for the sphere is about the 

only case that is tractable when the incident wave 

is arbitrary and the body dimensions are comparable 

with a wavelength. The class of obtainable solutions for scalar wave problems is somewhat broader, 

but even here we are restricted to separable geometries. 

The need for useable results for finite body scattering has motivated many investigators to utilize 

numerical techniques [Ward and Morrkon, 19711 

such as the moment method [Harrington, 19681. 

Even here the three-dimensional nature of the problem compounds the usual difficulty of excessive 

computation time. one well-known alternative to a 

direct numerical attack is to employ high frequency 

asymptotic techniques [Fox, 19641 such as J. B. 

Keller's geometrical theory of diffraction. 

Here, we should like to point out the possibility 

of getting useful results from a special form of a 

truncated conducting cylinder that is excited by a 

Cowright O 1973 by tbc American Gcophysicnl Union. 

finite axial current element. It is suggested that the 

results to this problem can provide insight to scattering by finite cylinders and, hopefully, they will be 

useful as a check on numerical procedures developed 

for more general three-dimensional structures. Also, 

we shall utilize the low frequency solutions to show 

graphically how the magnetic field varies in the 

vicinity of the cylinder for various positions of the 

adjacen,t current element. 

FORMULATION 

The specific situation of interest is illustrated in 

Figure 1. A conducting cylinder of conductivity o 

and magnetic permeability p occupies the space 

< a and /zl < s in terms of a -ylindrical coordinate 

system (p, 4, z). The regior. . > a external to the 

cylinder has a conductivity oo and magnetic permeability m. To allow for the existence of displacement currents, we can allow both o and go to be 

complex. Over the concentric cylindrical surface 

p = po , we specify that the axial current density 

is a known function K(+, z) amp m-'. Actually, in 

Figure 1 we only show the region for z > 0. 

We now propose to find a solution of the problem under the stipulation that the normal current 

density is zero at the surface z = Is\. This, in turn, 
- -- 

WAl? AND HILL 

Fig. 1. General configuration showing only region for z > 0. 

requires that the normal electric field E, is also zero 

over this surface. On the other hand, since we are 

interested mainly in the situation of symmetry about 

the plane z = 0, we can utilize the fact that E, = 0 

on this plane. Thus, attention may be restricted just 

to the region 0 < z < s, if we remember that the 

solutions for even symmetry are even about z = 0. 

In what follows, we shall adopt a time factor 

exp(iwt). 

We fully appreciate that the posed problem is not 

directly applicable to an isolated finite cylinder, but 

we can assert that the solution of this idealized 

situation should tell us something about the effects 

of truncating the cylindrical structure. If the latter 

is relatively thin and highly conducting, it can be 

expected that the normal current density within the 

structure tends to vanish at the ends. 

FORMAL SOLUTION 

To facilitate the analysis of the posed problem, 

we express the total field in terms of electric and 

magnetic Hertz vectors that have only z components 

n and n*, respectively [Wait, 19591. Thus, for the 

region p < a, the field components are obtained 

from 

H, = (iy2/fiw)(alT/ap) + (vp)(a2n*/a+ az) (4) 

E, = I-7% + (aZ/azZ)~II (5) 

H. = [-7' + (a2/az2)iII* (6) 

where y = (i~p)'~' is the propagation constant for 

the cylinder. The solutions for the region > a 

b2tvc thc snnlc Iortn as (1) thrt\i~ph ((>). I\ttt wt :ttI,I 

3 subscript zero whcrc :tppn,pri;~tc. 

The reqt~ircd fornis of thc solution fur ,, . (i :Ire 

JJ = 2 u ,,,. ~,(o,p)e'"'" cos A,,: (7) " f,, 

and 

n* = x x a,,,*l,(u,p)eim* sin X,z (8) 7, m 

where u,, = (A,?? + yZ)'/'. Here, I,,, is a modified 

Bessel function of the first kind that remains finite at 

p = 0. For single-valuedness in 4, obviously m 

ranges from -m to +a through all integers. Also, 

since E; = 0 at z = s, we note that A,, = (2n t 1) 

(~/2ss) where n = 0, 1, 2, 3, . . . . In addition, we 

readily confirm that E, = 0 at z = 0, in eccordalice 

with our assumptions concerning symmetry. 

In dealing with thc region external to the cylinder, 

we choose the wavc-functions to havc thc follow~ng 

form, for a < ,, < p,,. 

+ c,.,K,(u,p)le~'"%os A,z (9) 

IT* = x x cm .* K~(u,~)~-""* sin 1-1 

" m 

00) 

and, for p > 

where u. = (A2 + Here, K,,, is a modified 

Bessel function of the second kind that assures outgoing waves as -t m. These forms of the solutions 

are chosen to allow for a possible discontinuity in 

H, at the surfzce = po, while Ed, E:, and H: are 

continuous. The latter immediately tells us that 

c,,," = d,,,,.*. 

We next express the prescribed current cxcitat~on 

as a do. e Fourier series in the manner 

K(+, z) = F,.,c~'"~ cos h,,z. (1 3) * "7 

where F,#,, is easily determined. Thc houndery condition at ,, = written as 

is now easily applied. Using the Wronskiiln rcl;tlion 

l,,;(ZK,(Z - IZKZ = I/Z, wc dctcrmine that 
-- 

EXCITATION OF HOMOGENEOUS CONDUCTIVE CILINDER 

where 7,) = ifiw~/~,,. Thus, the coefficient b,,,, is determined entirely by the excitation. 

The boundary conditions at the surface of the 

cylinder arc that E,. H,. E,, and H, are continuous 

at p = a. Application of these leads to four linear 

equations to solve for the four unknown coefficients. 

Omitting algebraic details, this process leads to 

c,,,,' = (- bm../~)(im~./a2)[(uZ/oZ) - I~Yo/~~ (17) 

where 

[(ru'/u)(vl,'/l,~) K,. - Y,,UV,, Kt,'] (1 8) 

Hcre, wc have used thc following abbreviated notation I,, = I,,,(u,,a), I,, = I,,,(u,,a), K,, = K.,,(u"a), 

fr = if,,. v = u,,, and 7 = ip4Jv. The two coefficients 

for the internal fields can be obtained from 

Fig. 2. Specialization to a, 

linear current element adjacent 

to the conducting cylinder. 

where A is the angular width of the strip. These 

elementary integrations yield 

The corresponding value for h,,,,,, is given by (15). 

If A 4 0 corrcsponding to a filamental line current, 

we have 

a,,, = (u2b, "I,, + U~~~.~K.)/D~I, (19) b,,, = (~/~)(~~/~o)Km(u,~o)e""~~(sin A,I)IX.s (24) and The corresponding electric dipole excitation is oba, .* = u2cm,.* K,,/v21, (20) tained by regarding ~,,l as a small quantity whence 

To complete the statement of the formal solution bm ,, 5-- (~l/rs)(no/~o)K,(u.,~n)e'"'" (25) 

of the posed problem, we observe that the Fourier where 1, is the moment, inverse of (1 3) is We shall now consider some aspects of the general 

solution before considering the static limit. Equation 

F, = Is) " K(+, z)r7"" d+ ca i.r dr (21) 16 can be written 

If we specialize this to an axial current I amp in a , = -b/I (26) 

filamental strip of half-length 1 as indicated in Figure where 

u K,, a A",," = 

u K. u u 

2, we can replace (21 ) by In the case of a perfectly conducting cylinder (Y 

m), we see that A,,, = I and, in this case. c.., ,,* = 0 da + A/? 

prn = ' A, z A (22) is a consequence of (17). An important special case 

4.-A/? of (27) is when m = 0. Then, in explicit notation, 
WAIT AND HILL 

- ~ro/~.~o~~~~'~~.~~/~~~~.~~l~ 

+ l(~/u~~)[~~'(~~~)/~~(~~a)l 

- (ro/u.tio)[ KO1(u.a)/ K~(u.a)l I (28) 

In this case, we also note that co,.* = 0. 

EXPLICIT FIELD EXPRESSIONS 

We now consider the explicit forms for the electric Hertz potential n, since this is of prime interest 

in calculating the fields in the principal plane z = 0. 

First of all, we note that this function for p > a 

can be split into a primary and secondary portion 

as follows: 

rI = n.. + n... 

The primary component, for a < p < w, is 

p'"l*o-*l [(sin X.I)/A.s] cos A.z (29) 

For p > po, the corresponding expression is obtained 

by interchanging w and @. Using a well-known addition theorem for modified Bessel functions, the 

summation over m is readily performed to yield 

[(sin X.I)/X.s] cos X.z (30) 

where pa = [p" + - 2p0 COS(+,, - (p)l1lP is the 

radial distance from the observer to the source current element. 

The secondary part of the electric Hertz potential 

in the region p > a is conveniently written 

,-I**-+, I,(u.a) K-(u.p)A,,.e cos h.2 (3 1 ) 

where there is no apparent further simplicity. 

To obtain the magnetic field components, we 

merely perform the differentiations indicated in (3) 

and (4). Restrictinq attention to the plane z = 0, 

it easily follows from (30) that the primary magnetic 

fields are 

H," = LIP, sin (4, - +)/=pd] 

, K,(v.p.Ol(all~ X.1) h.rls., (11) "-11 

where A,, = (2n + 1)~/2s. On the other hmd, the 

secondary fi51ds that follow from (3 1 ) are 

I,(u,a) K,(u,p)ime S~I*~-+) [(sin X,l)/hnsl (34) 

and 

I,(u.a)u. ~~'(u.p)e'"'"-" [(sin h.l)/X,sl (35) 

Using (32) through (35), the total fields are obtained exactly from H, = Hpgr + HP",' and H, = 

H4vr + H,"'"' ia the plane z = 0 and for p > a. The 

double series expansions in (34) and (35) arc rather 

cumbersome, hut the convergence is rapid whcn 

lual, lufl), and lu,,vj are not excessively large compared with unity. 

SOME SPECIFIC RESULTS FOR THE 

QUASI-STATIC LIMIT 

Some simplification is achieved in the low frequency or quasi-static case. For example, if (-,,al, 

IroPI, and irqpol are all <<I, we can effectively replace u. by h, everywhere. However, unless lya is 

also small, we must retain the form for u,,. But, of 

course, if the cylinder is very highly conducting, 

such that >>I, the complicated factor A,,,, can 

be replaced by unity in (34) and (35). Thus, in 

the case of a perfectly conducting cylinder and in 

the quasi-static limit, we can use (32) through (35) 

with u, replaced everywhere by A, and A ,,,,,, by unity. 

Some numerical data for this case are given below. 

To present these results, the reader is referred to 

Figure 3, where we have chosen a cartesian coFig. 3. Crorr rection (i.c.. z I 11 

plane) of the cylindrical largct in thc 

presence of the current cable. 
EXCITATION OF HOMOGENEOUS CONDUCTIVE CYLINDER 

ordinate system (x, y, z) with the current element at 

the origin. The field components H,, H, at the 

observer are thus obtained from 

H, = H, cos 6 - H, sin 6 

H. = H, sin + + He cos 6 

We can visualize this situation as the scanning of 

the resultant fields at the earth's surface (y = 0) 

for a buried cylindrical target centered at x = d and 

y = -h. In view of our quasi-static assumptions, we 

have neglected the effects of the induced currents in 

the region external to the cylinder. 

In Figure 4, we plot the normalized horizontal 

field 2nhH,/I as a function of the normalized horizontal coordinate x/h in the transverse plane z = 0. 

Here, we choose a/h = 0.5, d/h = 1, l/s = 0.5, and 

/h = 2, 10, and m. The case s/h :. m is based 

on the solution for the corresponding twodimensional problem of a parallel line source carrying the 

same current I (see appendix A). The corresponding 

results for 2rhH,/I are illustrated in Figure 5. A 

similar set of curves is shown in Figures 6 and 7, 

where the only difference is that d/h = 0. This case 

corresponds to having the cylindrical target located 

directly beneath the current element. 

FINAL REMARKS 

The general features of the response curves in 

Figures 4 through 7 are not surprising. In all cases 

Fig. 4. Normalized horizontal magnetic field as a function of the horizontal displacement for a buried and 

displaced finite cylinder, a/h = 0.5, 

d/h = 1.0, l/s = 0.5, y = 0. 

Fig. 5. Normalized vertical magnetic 

field as a function of the horizontal 

displacement for a buried and displaced finite cylinder, a/h = 0.5, d/h 

= 1.0,1/s = 0.5, y = 0. 

they have the form expected for a secondary source 

of current induced in the target tha! tends to oppose 

the fields of the primary current. Of particular interest is the fact that the half-lengths of the truncated 

cylinder must be greater than a factor of 10 of the 

burial depth h before the magnitude of the response 

Fig. 6. Horizontal field for a finite 

cylinder buried directly beneath the 

source cable. a/h = 0.5, d = Q. 11s = 0.5, y = 0. 
- - 

WAlT AND HILL 

H,.' = (l!iw,,w)(aE.'" ap) 

= 1 I[P - &) cos (6,) - +)I; ?TP.I~Y~~KI(Y~~P,O (AN 

H," = (- I!pifi,w)(c?E.'";d+) 

= [+ IP,, sin (+,, - +)/~~P.,IY,~~,(Y,,P,O (A4)

In the quasi-static case where !YOPBi << 1, these are 

expressible in the simpler form 

and 

H," N IpO sin (+, - +)/2rp2 (A6) 

Using a previously known solution, we can writs 

down the corresponding forms for the secondary 

fields external to the cylinder. The exact expressions 

o for the two magnetic field components for the region 

0 I 2 p > a are [Vhit, 19521 

xlh 4 - 

Fig. 7. Vertical field for a finite H,"' = -(I/~*P) C ~.[K.(YoP~,)~ Km(-toa)l 

cylinder b~ried directly beneath the m--- 

source cable, o/h = 0.5, d = 0, 11s = . I~(~~~)~-~~'*-*~' 

0.5, y = 0. YOPCI K,n'(~,,~) (A71 

and 

*- 

approaches that for an infinite cylinder with the same H,"" = -(I/Zrp) q,fK,(y,p,)/ K,(y,,R)I m--m transverse dimensions. This is an important factor 

to keep in mind when two-dimensional models are . Im(yonk-zm<+-*91 rm K,(YoP) (A81 

being employed to simulate the scattering from finite where 

Structures. 

qm = { -(l/~)[J~'(ya)/I~(ya)l 

APPENDIX A: TWO-DIMENSIONAL MODEL 

For comparison purposes, it is useful to consider 

the purely two-dimensional problem of a uniform line 

current parallel :o a homogeneous circular cylinder 

of radius, a, with conductivity c and permeability fi 

[Wait, 1952 (in equation 13 replace w/pI by nP2/pl, 

and in equation 14 replace (k * n) by n(k * I)), 

19591. This can be considered as the limit of the 

finite cylinder configuration ass+ m. 

Using cylindrical coordinates (p, +, z), the cylinder 

is defined by < a for -m < z < m. The line 

source is located at = and + = where po > a. 

The primary electric field of the line source has only 

a z component, given by 

where 

P. = [PO? + P? - 2pop cos (+a - ~p)]"~ (A2) 

Tlie corresponding magnetic field components are 

Fig. Al. Normalized field as il function of the horizontal displacement for 

a buried infinite cylinder, d/h = 2, 

y = 0. 
EXCITATION OF HOMOGENEOUS CONDUCTIVE CYLINDER 

No Cylinder 

0.9 

Fig. A2. Normalized field as a function of the horizontal displacement for 

a buried infinite cylinder, d/h = 2, 

y = 0. 

x/h -I -2 

1 

No Cylinder 

Fig. A4. Normalized field as a function of the horizontal displacement for 

a buried infinite cylinder, d/h = 1, 

y = 0. 

The quasi-static forms for the secondary fields are 1, the limiting case where we let + 0, we see that then found to be 

qm = -1 and q, = -1, and ll,,.cr 2 -1/(2no). . . ,, 

Then (A10) and (~11) can be summed as geometric series to give 

cos !)I(+ - 4,) + if,,'" (AIO) H,"" (- lj2np) 

and .[I - (a2/~,p) cos (4 - 4d1/[1 - 2(a2/p,,p) 

,~inrn(4-+~) (All) 

where 

Fig. A3. Normalized field as a fvnction of the horizontal displacement for 

a buried infinite cylinder, d/h = 1, 

Y = 0. 

C~J (4 - +,I) + POP)^^ 

and 

Fig. AS. Normalized field as a function of the horizontal displacement for 

a buried infinite cylinder, d/h = 0, 

Y = 0. 
WAIT AND HILL 

No Cylinder 

x/h 

Fig. A6. Normalized field as a function of the horizontal displacement for 

a buried infinite cylinder, d/h = 0, 

y = 0. 

H,"' = (I/2rp) 

I(a2/p0p) sin (+ - +0)1/[1 - 2(a2/p,,p) 

cos (4 - 60) + (a'/~~p)'l 

Again, referring to Figure 3, we consider thc total 

H, and H, components in the plane y = 0. where 

now the buried cylinder (of perfect conductivity) is 

of infinite length. In Figure Al wc show n plot t>i 

2~lzH,/I as a function of x/lr for d/lr = 2, for u/lr -2 

0.9, and all! = 0.1. The corresponding rcsults lor 

ZrhH,/I are shown in Figurc A2. Similar curves are 

shown in Fiyrcs A3 and A4, wherc d/h = 1, and 

finally in Figures A5 and A6 where d/h = 0. 

REFERENCES 

Fox, 1. (Ed.) (1964), Quosi-Optics, pp. 1-25, Polytechnic, 

New York. 

Harrington, R. F. (1968). Field Computalio!~ by Momenr 

Methods, pp. 82-102, Macmillan, New York. 

Wait, 3. R. (1952). A cylindrical ore body in the presence 

of a cable cat.rying an oscillating current, Geophys., 

17(2), 378-386. 

Wait, J. R. (1959), Elecrromogneric Radiorion from Cylinrical Strucrures, pp. 182-183, Pergamon, New York. 

Ward, S. H., and H. F. Morrison (Eds.) (1971). Special 

Issue on Ele~tromagnetic Scattering, Geoplrys., 36 ( I), 

(A14) 1-183. 
Geofisica Internacional, Vol. 12, No. 4, 

245-266, October 1972 

(Received from publisher - June 1974) 

(Submitted - October 1973) - 

REVISTA DE LA UNION GEOFISICA MEXICANA, AUSPlClADA FOR EL INSTITUTO DE 

GEOFlSlCA DE LA UNIVERSIDAD NACIONAL AUTONOMA DL MEXICO 

Vol. 12 Mexico, D. F., lo. de octubn de 1972 N6m. 4 

ELECTROMAGNETIC RESPONSE OF A CONDUCTINGTYrLINDER 

OF FINITE LENGTH 

Lor amp01 abctromyn6tims a dcuh pur ua clllndm hito. excitdo par unr fwnb 

c~rcuk y, pua prop6sitos & com~url6n, medirnm cxcitaci6n do unr fuenla llnarl. Estos 

mlultadot hbl6n a canpuan ma loc &I wo &I dhdm hlinlm. So enmnur qw la 

impmllacu mlatin de la~componentc con timetria acbnutal de La mnknte hinuye 

cuando el dlindro cs finite. Eatc uticulo ha tido motindo por la: convinmntcs summnciaa 

de S.K. Singh mlativu a la poabls impatancia de la cMisnte can rimstria acbnuW 

inducida en cuerpor dargados do mlnonlaq par &unu wnfiplndonoa do Pusntsr sktmm.gnCtiar utilhdns en la explonci6n gofiica. 

ABSTRACT 

The ckctromagmetic fields an calcuLtcd for a truncated cylindea excited by a loop awra and, for comparison, for excitation by a line ma. These multa an .Ira mmpmd with 

the infmik cylinder -. It is found that the relative bpmtmm of the .zhuth& 

symmcfric component of the current ia diminisbad when tho cylinh is truncated This paper 

is prompted by the mgent luggertions of S. K. Sin@ Oonammg the poaibk importma of 

the symmefriul current indud in clongtcd om bodies by ckctromgmtic vrura confgurations used in eaphyncal Pxploration. 

Imrirutc for TekcMlmunication Scirnns, Office of Tekcommrrniurtionr 

** Cmpcntive I~fimfe fn Rcxodr in Envim~lcnlal Scirnas Unhrdty of Cobmdo. 

National Omanic and AlmorpSm.c Admbrhwalon U.S. Depmmenrof Cmnrnem, BarWer, 

Colomda. 
GFDFISICA INTERNACIONAL 

INTRODUCTION 

In a previol~s analysis (Wait and Hill 1973), we formulated the 

boundary value problem of a truncated cylindrical region excited by 

acspecified distributionof electric current over a concentric cylindrical surface. Preliminary numerical results were given for excitation by 

an electric line source of finite length. Here we consider the 

extension for excitation by a long slender loop. Also we examine the 

importance of the azimuthally symmetric component of the current 

induced in the cylinder. The effect of truncation and type of 

excitation on the azimuthally symmetric current component provides 

insight to the scattering mechanism for scatterers of finite length. 

Quantitative information on this subject is important from the 

standpoint of determining the performance of electromagnetic sourct 

Location schemes in mine rescue operations. For example, we can 

anticipate major distortions of the surface fields for a buried source if 

metallii'conducton, such as pipes and rails, are in the vicinity. It is 

also possible that the truncated cylinder could serve as a convenient 

model for an elongated sulphide metallic body that is to be detected 

by a grwnded or non-grounded e.m. exploration technique. 

FORMULATION 

For the sake of simplicity, we consider a perfectly conducting 

cylinder excited by a finite line, source as shown in Figure 1. The 

cylinder occupies the space p < a and lzl < s, and the region external 

to the cylinder (p > a) has conductivity o, and magnetic permeability 

p,. The proble'rn is made tractable by requiring that the normal 

current density be zero at the two horizontal surfaces where lzl = s. 

This condition requires that the normal electric field is also zero at 

lzl = s. Because of this cor ition, we are not actually treating an 

isolated finite cylinder, but we expect the solution to tell us 

something about the effects of truncation. 

If we excite 'the cylinder with a z-directed line current, I exp (iwt), 
GEMISICA INTERNACIONAL 

from z = -P to z = + Q, then the solution is even about z = 0. The PI primary magnetic field components, Hp and H;', are given by (Wait 

and Hill, 1973). 

m 

H? = ]Po sin (bo - @) sin An P C KI ('nod) I+, cos A, Z, (1) =Pd An 

m 

Pr 1 [P - p0 cos (b, - @)I sin An P 

He = C K1 %~d) Un COS An Z 

Ir pd ins 

Where pd = Ip2 + P: - 2 PP, cos (4, - #)I*, 

A, = (2n + 1) n/(2s),' u, =[Xi + 7;lH, 

r; = IWM, . 0, 

and K, is a modified Bessel function of the second kind. The 

secondary magnetic fields, Hr and Hgc, are given by 

sin An Q explimt@, - 011 cos xnz, 

An s 
sin A, P 

exp[im(9, - 0)l cos A,Z, 

AaS 

where I, is a modified Besd function of the fmt kind. The general 

expressions for the secondary fields of a cylinder of finite conduct~vrty are given by Wait and Hill (1973). 

if we comida tw appositely directed line sources to approximate 

a lone slendar loop -, then the line souice solutions for the 

primary and second- fwlda are directly applicable. If the positive 

line source is located at (pz, @:) and the negatrve line source is 

located at Cp;, #,), thar the secondary fields for loop excitation, 

H~C, ye gim, by 

wherc H'" and dw are given by (2). 

P 0 

Since we a& particahrly interested in the behavior of the azirnuth- 
GEOPlSlCA INTERNACIONAL 

ally symmetric (m = 0) term of the secondary field, it is useful to 

examine the behavior of this term analytically as the loop width w 

goes to zero. For the case of broadside excitation fpz = pa the loop 

width w is equal to p 4 where A is the angular separation of the two 

line sources. We examine only the 6 component, since the p compoP,M nent of the secondary field has no m = 0 term. The limit of H as 

w (or A) goes to zero is # 

By noting that a/a #o introduces a factor im in (2), we conclude 

that the m = 0 term is not excited by broadside excitation. 

For the case of endsn excitation (62 = # -), we again examine 

the result as w goes to zero. In this case, the liiit of is given by 

By Examing (5) and (2), we note that the m = 0 term is excited by 

end-on excitation. 

Since the effect of truncation on the secondary fields is of interest, 

it is useful to examine the limit of an infmite cylinder. If we let Q = s 

and let s approach m, then u, approaches yo and A,, approaches 

zero. Consequently, the n summation in (2) reduces to 
m m 

sin A, s -2 1 -1 I 

C AnS = - 

IT 2ntl 2' (6) 

n-0 n=O 

where a summation formula by Wheelon (1968) has been used. By 

applying (6) to (12), the secondary fields for an infinite cylinder, 

and H;": are found to be 

The results in (7) agree with the known two dimensional result (Wait, 

1952; 1959). 

NUMERICAL RESULTS FOR LINE SOURCE EXCLTATION 

It is useful to examine some numerical results for line source 

excitation before examining loop e zitation. The special geometry for 

numerical results is shown in Figure 2 with the line source at the 

origin of a artesian coordinate system x, y, z. The magnetic field 

components H, and Hy are obtained from 
y = Hp sin I$ + H# cos # 

All results are computed for an observer on the x-axis. Consequently, 

the situation could involve scanning the vertical and horizontal 

magnetic fields (H, and H,) at the earth's surface (y = 0) for a 

buried cylinder centered at x = d and y = -h. 

In Figure 3, we show the magnitude of the normalized vertical 

secondary magnetic field 2nh MyCI/I as well as the m = 0 term for 

an infinite cylinder. For all cases in this section, we choose d = 0, a/h 

= 0.5, and ly,hl= 0.5. Since d = 0, the curves are symmetrical about 

x = 0. In Figure 4, we show the magnitude of the normalized 

horizontal magnetic field 2nhlqCI/I as well as the m = 0 term for 

an infinite cylinder. Results in Figures 3 and 4 are obtained from (7) 

and (8). 

In Figure 5, we show the case similar to that of Figure 3, but for 

a truncated cylinder with s/h = 2.0 and P/s = 0.5. Note that the 

relative importance of the m = 0 terms is somewhat reduced here due 

to the truncation of the cylinder. In Figure 6, we show the case 

similar to that in Figure 4, but again for a truncated cylinder with 

s/h = 2.0 and P/s = 0.5. Again the relative magnitude of the m = 0 

term is somewhat reduced due to the t~ncation. Results in Figures 5 

and 6 are obtained from (2) and (8). 

NUMERICAL RESULTS FOR LOOP EXCITATION 

The geometry for loop excitation is also shown in Figure 2. The 

positive line source is located on the x-axis st x = w/2 and the 

negative line source is on the x-axis at x = - w/2. The required 

values of pi, #:, and I$; are given by: 
GEOFISICA INTERNACIONAL 

&, = tan-' [h/(-d + w/2)],+; = tan- ' [h/(-d - w/2)]. (9) 

The results in this section are obtained from (3) and (8). We again 

can visualize the situation as scanning the vertical and horizontal 

magnetic field components at the surface (y = 0) for a buried 

cylinder with a loop excitation. One possible reason for usrng a loop 

is that no groundmg of the cumnt-canying cable is reqwed. 

in Figures 7 and 8, we show the magnitudes of the vertical and 

horizontal sedndary normalized components, 2?rh2 I nFC I/(w I) and 

2n h2 {H?CI/(W I), for an infmite cylinder. Again we haw chosen a/h 

= 0.5 and kohl = 0.5, but we have taken d/h equnl to 0.5 so that the 

m = 0 term will be noazao. We can see from (4) that the m = 0 

tern would be zero for d = 0 because p*, would equal p; . Also, wh~le 

the m = 0 temu are symmetric about x/h = 0.5, the total secondary 

fields arc not. 

In Figurn 9 and 10, we show the same cases for a truncated 

cylinder with s/h = 2.0 and R/s = 0.5, and the results are quite similar 

to those in Eis 7 and 8. The total field whidt would be measured 

includes also a primary vertical component from the loop. 

FINAL REMARKS 

The results for loop excitation in Figures 7-10 are qualitatively 

similar to those for line source excitation in Figures 3-6, but they 

show a reduced importance of the azimuthally symmetric m = 0 

term. Of course the primary fields of the loop are quite different 

and, as a result, the measured vertical component would differ 

considerably from that of a line source. 

Also it has been demonstrated in Figures 5, 6, 9, and 10, that 

truncation of the cylinder results in a reduced magnitude of the m = 0 

term. This is expected on physic reasoning since the symmetric 

cumnt must be zero at the ends, but other currents could flow up 

one side of the cylinder and down the other. As pointed out by 

Singh (1973);it is important to consider the effect of the m = 0 
GEOFISICA INTERNACIONAL 

ten since it may be very significant. It's effect was ignored in an 

early paper (Wait 1952) on the subject, although the general theory 

given there contained it. 

The numerical results shown in Figs. 3 to 10 inclusive are 

restricted to the value kohl = 0.5. This is a typical value of the 

parameter when the surrounding medium is finitely conducting. As 

pointed out by Singh (19731, this parameter should not be assumed 

zero when modelling ore bodies immersed by country rock. However, 

the conclusions and general nature of our results are not affected by 

the precise value of I 7, h I. To illustrate this point, we illustrate the 

effect of finite -yo for the infinite cylinder case and for line source 

excitation only in Figs. 11 and 12 with the pertinent parameters 

indicated on the figure. Conesponding curves for a truncated cylinder 

are illustrated in Figs. 13 and 14. Here the case yo = 0 corresponds to a 

purely static condition. The general theoretical results given by Wait 

and Hill (1973) also allow the effect of finite conductivity of the 

cylinder to be accounted for. We plan to carry out these calculations 

in the near future. 
bop source, / 

Fi~m 2. Spchl geometry for numaim1 results, both line source and Imp excitation. 
F~um 3. Normalized vertical scondary nugnctic fild of an infhlte cyHnder for line source 

excitation. AU numuial rcmlts are for y and z equal zero. 

h 

Figure 4. Normalid harizontnl gcondsry mwetic fild of m inflnite cylinder for line 

source excitation. 
5. Namdizd wrtiul mdsry mametic fxld of a truncated cylinder for fine soura excitation. 
Figun 6. hbnnalizcd horimnhl teonduy magnetic Iiild of a truncated cylinder for line 

saura excitation. 
.4 - infinite cylinder 2nh2 1,sec -by wI I f =.5 

Fira 7. Normnked vertical omndvy magnetic field of an infinite cylinder for loop 

excitation. 
CEOFiSICA IMERNACIONAL 

Fin 8. Normnlized horiwntsl secondary magnetic f~ld of an infmite cylinder for iwp 

excitation. 
.L/[\ K=.5 

all terms 

Fiyn 9. no ma^ mtial secondary wetic field of r mcatsd cyMdcr for loop 

excitation. 
Figure 10. Namrlizcd h&nW rcc- m.pletic fiald of a truncated cylinder for loop 

excitation. 
WC 11. Effect of 7, on the hOti~ontal magmetic field for the infinite cylindcr cau. 
infinite cylinder 

0 I - X 2 3 

FI- 12. Effect of 'yo on the n-tudc of the ratio of vertical to horimntd magnetic field 

(mfinitc cyhdm case). 
GEOFlSlCA INTERNACIONAL 

Pb 13. Effat of 7, on th. horimW mametic field for the truncltcd cylinder mu. 
Fiw 14. Ufst of 7, m fhc mamitude of the ratio of vertical to horizontal mwtic field 

(tnmcatd cylinder QSC). 
GEOFISICA INTF~RNACIONAL 

BIBLIOGRAPHY 

SINGH, S. K., 1973. On axially symmetric electric current induced in a cylmi~: 

under a lie source, Geophysics. 38:971-974. 

WAIT, J. R., 1952. A cylindrical ore body in the presence of a cable carrying an 

oscillating current, Geophysics. 17(2): 378-386. 

WAIT, I. R., 1959. Elechomrrgnetic Radiation from Cylindrical Shuchuec, 

Pergamon Pns, Oxford, England. 

WAIT, J. R. and D. A. HILL, 1973. Excitation of a homogeneous conductive 

cylinder of finite length by a prescribed axial current distribution, Rcdio 

Science, 7 October 1973 (in press). 

WHEELON, A. D., 1968. Tables of SummaMe Series and Integmls Involving 

Bessel Functions, Holden-Day, San Francisco, p. 9. 

ADDITIONAL BIBLIOGRAPHY 

NEGI, I. G., C. P. GWTA, and U. RAVAL, 1972. Induction anomaly due to an 

elongated covered ore-zone excited by a long current carrying cable, Geophys, 

Rasp., 20: 193-211. 

NEGI, I. G. and U. RAVAL, 1969. Negative electromagnetic screening by a 

cylindrical conducting cover, Geophys. 36(6): 944-957. 

SINGH, S. K., 1972. Transient electromagnetic response of a conducting infinite 

cylinder embedded in a conducting medium, Geofis Int. 12(1): 7-21. 

WAIT, J. R., 1973. On electromagnetic induction in elongated ore bodies, 

Ccophy~ (Letter to the Editor) 38: 984-985. 

WAIT, J. R. and R. E. WILKERSON, 1972. The sub-surface fields produced by a 

hecurrent source on a non-flat earth (cylindrical model), Pure and Appl. 

Geophys. 95(III): 150-1 56. 

WARD, S. H. and H. F. MORRlSON (Eds.) 1972. Special issue on Electromagne. 

tic Scattering, Geophys. 36(1): 1-183. 
Rudio Soet,ce. Volume 9, Number 1, pages 71-73. January 1974 

Perturbation of magnetic dipole fields by a perfectly 

conducting prolate spheroid 

David A. Hill and James R. Wail 

lnslilule for Telecommunicalion Sciences, Ofice of Telscommunicalions, 

US Deporrmenr of Commrrce, Boulder, Colorado 80302 

(Received September 6, 1973.) 

The effect of a perfectly conducting prolate spheroid on the surface fields of a buried 

vertical magnetic dipole is examined in the quasi-static case. When the spheroid is in the 

vicinity of the source and near the surface, the erron introduced in source location can be 

significant. The results have possible application to rescue operations following coal mine 

disasters and in similar situations. 

INTRODUCTION c = (b' - o-) * 1/1 (1) 

The feasibility of locating a buried vertical magnetic dipole sourcc (horizontal loop) from surface 

measurements of the vertical and horizontal magnetic field components has been investigated by 

Wait [1971]. For sufficiently low frequencies the airearth interfacc has a negligible effect and a single 

observation of the complex ratio of the vertical and 

horizontal field components is sufficient for source 

location. Inhomogcneitics in the earth distort the 

surface ficlds and may lead to errors in source location. To providc insight into this problem, the effect 

of a spherical inhomogeneity has been investigated 

IHillarzd Wair, 19731. 

Herc we examine thc effect of a perfectly conducting prolate spheroid for thc case where the frequency 

is sufficiently low that currcnts in the surrounding 

medium can bc neglected. Consequently, the situation reduces to a magnetostatic problem, and an exact 

solution is available [Wait, 19601. The prolate 

spheroid is considered to be a reasonable model to 

study the influence of a finite or bounded conductor 

such as a metal pipe, railway conductor, or even an 

ore vein. At least it is not so idealized as a sphere. 

. . 

The prolate spheroidal coordinates are (?, 6, +) and 

the surface of the spheroid is defined by = 7, 

where 

2 -I,% 

7" = [I - (alb) I (2) 

The boundary condition on the spheroid is that 

the normal component of the magnetic field must 

vanish. The resultant solution for the magnetic potential n of a magnetic point charge K located at 

(,' S', 6) is [Wair, 19601: 

J 

where 

FORMULATION R = [(x - + (,. - ,')? + (: - r')l]"2 

The geometry of the spheroid and source dipole 

is shown in Figure 1. The prolate spheroid is 

centercd at the origin of a Cartesian coordinate system (x, y, z), and its major axis is coincident with 

the z axis. The semiminor axis a, semimajor axis b, 

and semifocal dist:rnce c are related by 

P,," and Q,,"' are associated Legendre functions of 

degrec rl and order m of the first and second kind, 

and prime denotes differentiation with rcspect to the 

argument. The definitions of P,,"' and Q,,"' follow 

those of S111yrhe [1968]. 

The masnctic - field H of a directed maenetic L 

Col~~right o 1974 by the ~~~~i~~~ c~D~~~~~~~I tlnion. dipole of strength Kdl is given by 
HILL AND WAIT 

H = -drv(a/ayf)n (4) 

In order to simplify the results, we restrict both the 

source and the observer to the equatorial plane (z 

= z' = 0) as shown in Figure 1. Consequently, H, 

= 0 and H, is given by [Wait, 19601 

where 

Fig. 1. Geometry for source and observer in thc 

equatorial plane of a prolate spheroid (z = 0). 

The imagined earth-air interface is the plane y = y' + s. 

v." = Q."'(v)P."'(q') cos m(4 - +')I [(sin +)/c1(v2 - 1)"2/vl ([(sin +')/c][(~')* - 1]'/'/~'1 

- Q."(v)P."'(v')m sin m(+ - +')([(cos 4)/cl(v2 - I)-'''! ([(sin +')/cl[(ql)' - I]"':~'\ 

+ Q."'(n)P.^(v')m sin m(+ - +')I [(sin +)/cl(v2 - l)-"'l( [(cos +')/clt(vf)' - 11 "'I 

The following similar expression for Hz can be derived by applying (4) to (3): 

H. = (Kd1/4*) -3(x - x1)(y - y')~.' ( 

+ UM f: 2 ~~(-1)" "-0 "-0 

(2n + I)[(n - nl)!/(n + nl)!12[~."'(vo)/~.m'(vrlrl)l 

(6) 

where 

NUMERICAL RESULTS 

For the interesting case when both the source and 

observer are reasonably close to the observer, a 

large number of terms are required to achievc convergence of the summations in (5) and (6). Consequently, a computer subroutine was written for 

P,,"' which utilizes forward recurrencc in n and 111. 

and a subroutine was written for Q,,"' wllicli utilizes 

backward recurrence in n and forward rccurrencc in 

in. The derivatives P,,"" and Q,,"" are easjly obtained 

in terms of P,,"' and Q,,"'. Computed values for P,,"', 

Q."', P."", and Q,,"' checked exactly with tabulated 

w." = ~."'(v)P."'(v') cos nl(4 - 4')t [(cos +)/cl(vz - 1 )"'/?I ([(sin +')/c1[(qf)' - I ]"'/q'f 

- ~."(?)P.~(v')m' cos m(4 - 4')II(sin 4)/cl(qi - I) "'I I [(con 9')/~ll(~')' - I I i 

-I- Q~*(v)P~-'(v')ni sin m(+ - +')II(sin +)/c1(q2 - I) "'1 I[(+#" 91)/,]11qf,' I [",~,~,i 

The prolate spheroidal coordinates required in (5) six-figurc v;ilucb ~N/II;IJI~U/ Iiur,~111~ O/ .$I~III,/,I,~/.,, 

and (6) arc related to the Cartesian coordinates by: 19451. A computcr prcrpr;lnl iljr 11. ;~r~cl /I,, ill 1:) 

r' 4 = tan' (~,:x). = (1/c)(xZ + up + 2)"' :~nd, (6) was thcn wrillun ;lnd L.IIL.C~CCI Ily ~~(IIIIIIII~! 

('I .that the houndi~ry condilic~n wa\ s;~tirliecl ;I! tl~c 

9' = tan-' (Y'I'.~'). v' = (I/c)[(xf)' + (y'f + ~'1'~ surface of the sphcroid (, = ?,,). 
- ~ - ~~+~~-~~~ ~ ~ . ~- DIPO1.E FIEL3 PERTURBATION 

Fig. 2. Ratio of vertical to horizontal magnetic Fig. 3. Ratio of vertical to horizontal magnetic field component for a prolate spheroid which is field component for a buried prolate spheroid; tangent lo the interface; x'/r = 2, y'/s = -0.5, I'/S = 2, ).' = 0, ,I/$ = 0.5. 

;~nd n/.r = 0.5. 

If we imagine that the carth-air interface is locatcd ? = .v + ?' as shown in Figure 1, then H, 

i~nd H, arc thc vcrticnl and horizontal magnetic field 

componcnfs that would hc measured in a sourec location scheme. Sincc thc ratio of vertical to horiznntill componcnfs has bccn suggested as a useful 

quantity 1 Wail. 19711, the ratio H,/H, is shown 

its a function of .r in Figurc 2 for o/s = 2, y'/s = 

-0.5. and x'/,s = 2. Since the spheroid is tangent to 

the interfacc at x = 0, thc vcrtical component H, 

is zero at x = 0. Note that a slightly larger anomaly 

occurs for the longer sphcroid (h/a = 2). Results 

for the same case with thc spheroid buried (?' = 0) 

are shown in Figure 3. The ;tnomaly in H."/H, is 

wtncwhat weaker, but it occurs over a larger horizontal range. 

CONCLUDING REMARKS 

The cxact quasi-static solution for n magnetic 

dipole in thc prcscncc of a perfectly conducting prw 

late spheroid has been used to calculate the perturbation of magnetic field components. Results in 

Figures 2 and 3 indicate that significant errors in 

source location could rcsult when thc sphcroid is in 

the vicinity of ihc sourcc ;tnd closc to the interfacc. 

REFERENCES 

Hill. D. A,. and J. R. Wait (1973). The electromapnctic 

response of a buried sphere for buried dipole excitation. 

Rodio Sci., 8(8), in press. 

National Bureau of Standards (1945). Tablr.s of A,~s~x.ioted 

Lcgr~rdrr Funrri(>,,rr, 300 pp., Columbia University Press, 

New York. 

Smythe. W. R. (1968). Slotir rir~d Drr~urnic Elc~tricity, 3rd 

ed.. pp. 144-152, McGraw-Hill, New York. 

Wait. 1. R. (1960). Some solotions for electromagnetic 

problems involving spheroidal, spherical, and cylindrical 

bodies. I. RPS. NO/. Btw. Sf~nd., 64Bf 1 ), 14-32. (In this 

reference. in equation 12 and the following equations 

-d/dy' should be +dl>?' for a dipole in the positive y 

direction.) 

Wait. J. R. (1971). Criteria .u; locating an oscillating 

magnetic dipole in the earth. P!.oc. IEEE. 59(161, 

1033-1035. 
Rivista Italiana di Geofisica, Vol. XYII, N. 5/b, 

pp. 421-424 (1973) 

Perturbation of Magnetic Dipole Field 

by a Finitely Conducting Circular Cylinde~. 

Introduction. The fields inside the cylinder are a solution of the 

wave equation, and the exterior fields are a solution 

Location of a buried vertical magnetic dipole source of Laplace's equation. The boundary conditions arc 

(horizontal loop) from surface measurements of the that the normal flux and the tangential fields are contivertical and horizontal magnetic field components has nuous at the cylinder boundary. The resultant solution 

been investigated by WAIT (I). The air-earth interface for the magnetic potential Cl of a magnetic point charge 

has a negligible effect at sufficiently low frequencies, K located at (x', y', 2') for both source and observer 

and the source locations can be determined from a outside the cylinder is [WAIT (91: 

Am (k) Km ( I h I p') Km ( I h / p) exp [- ih (e - z')] dh exp [- im (4 - b')] 

single observation of the complex ratio of vertical where 

and horizontal magnetic field components. However, 

inhomogeneities in the earth distort the surface fields R = [(x -z')~ + iy - y')" + (z - ~')2]~'1 , 

and cause errors in source location. Effects of spherical 

[HILL & WAIT (O)] and prolate spheroidal [HILL & 

WAIT (3)] inhomogeneities have been investigated. H~ 

Here we examine the effect of a finitely conducting j t.-- 

H~ f ,Magnetic 

circular cylinder for the case where the frequency is 

sufficiently low that currents in the surrounding medium r- I +.-~x:Y'.o, _. - -- . . - , can be 

reasonable model for long conductors 

neglected. The circular cylinder should 

such as metal 

be a 

po 6 T,P --4 . .. pipes, metal tracks, or even ore veins. Also, the compa- .::.r;.'or cylinder 

rison with the prolate spheroid is of interest in determining the effect of finite length of the scatterer. Fig. I Geornetryfor source and obrarorer in the plane z o. 

Also, of course, this serves as an excellent The r*..~ginrd earth-air inregface ir the y y' ~: S. 

check on the numerical calculations since the two 

representations are basically different. 

Formulation. and 

The geometry of the cylinder and source dipole are a = (US + h7112, y ~ (i o, 1,. o)'r2 , 

shown in Fig. 1. The axis of the cylinder coincides 

with the z axis of a Cartesian coordinate system - I'm (4 I"# (&) = --. I'.,, (a) 

(x, y, 2). The cylinder has conductivity o and permea- I?,," (z) ~ ~ . e lm (a) ' n K, (a) ' 

bility p while the external region has permeability p,. 

All displacement currents are neglected. 

- Here p and + are standard cylindrical coordinates, and 

(*) Institute for Telecommunication Sciences, Office of I, and Km are modified Bessel functions of order m. 

Telecommunications, U. S. Department of Commerce, A time dependence exp (iwt) is assumed. Note that 

Boulder, Colorado 80302, USA. for a perfectly conducting cylinder ( 1 ,I , .-. ,,) , 
I'm ( I h I a) 

(2) Am (h) K:,(I~IT' 

4 

The magnetic field H of a y- directed magnetic 

dipole of strength Kdl for p > a, is given by 

In order to simplify the results, we restrict both the 

source and observer to the equatorial plane (z = z' = 

0). Consequently, Hz = o and H, is given by 

[WAIT i4)] : 

A computer program was written to evaluate Hv and 

Hz in (4) and (5) with the time-saving features described 

above. The case of a perfectly conducting cylinder 

was checked by requiring that the normal magnetic 

field was zero at the cylinder boundary (p -- a). A 

partial check was obtained for finite ya by checking 

Am (h) with previous values [WAIT 

If we imagine that the eartb-air interface is located 

at y = s + y' as shown in Figure I, then H, and H, 

are the vertical and horizontal magnetic field components which would be measured in a source location 

scheme. Since the ratio of vertical and horizontal 

components has been suggested as a useful quantity 

[WAIT (91, H,/H, is shown as a function of x in Figure 2 for 1 y I a = m, a/s = 2, y'ls = - 0.5, and 

x'/s = 2. ALSO shown are results for a prolate spheroid 

of axial ratio bla with the major axis coincident with 

im / h I K', ( / h 1 P) sin O - - Km ( 1 h / F) cos 4 x R6 P 1 

The following similar expression can he derived by the a-axis [HILL & WAIT is)]. Both the spheroids and 

applying (3) to (1): the cylinder are tangent to the interface at x = o, 

Numerical results. 72.0 

For the interesting case where both the source and 

observer are close to the cylinder, a large number of 

terms are required in the m summations in (4) and 

(5); also the h integration must be performed numerically. Consequently, the computer time can become 

excessive. Fortunately, the m summation can be converted to run over positive values only by using the 

following relationships : 

(6) 

I-m (m) = Im (a), K-, (a) = Km (a), A-, (h) = Am (h) . -2 2 

Also, the h integral can be taken over just positive 

values of h since the integrands are even in h. A further -0.5 

saving is made possible by interchanging the order 

of summation and integration since the modified Bessel 

functions required in the summation are quickly generated by recurrence (forward for K, and backward 

for I,). The remaining h integral is well behaved. 

The integrand approaches a constant as h approaches 

zero and decays exponentially as h becomes large as Fig, - Ratio verricnl horizonral mognrtic for 

long as the source and are On the scacte~r (cylindm or prolore spheroid) mhich is rongent ro the 

cylinder. interface. 
Fig. 3 - Ratio of ocrticol to horizontal magnetic field for 

a buried rcatterer (cylinder or prolate spheroid). 

and the vertical component is zero. Results for the 

same case with the scatterer buried (y' = o) are shown 

in Figure 3, and the anomaly in H,/H, is somewhat 

weaker. 

The effen of finite I y I a for p = f10 is illustrated 

in Figure 4 for the same geometry as in Figure 2. 

Note that a cylinder with 1 y 1 a = 10 looks nearly 

Fig. 4 - Ratio of vertical to hon'zontal magnetic field for a 

finiroly conduering cylinder which is parallel to the interfafncr. 

perfectly conducting. The field components are rcal 

for 1 y / a = o where no scattering occurs and for 

I y I a - w where the scattered field is in phase with 

the primary field. For gcneral 1 y 1 o, the scakrcl.cJ 

field is complex, and the nulls in thc total fiel~i arc 

partly filled in. This could make source location more 

difficult. 

Concluding remarks. 

The exact quasi-static solution for a magnetic dipole 

in the presence of a finitely conducting circular cylindcr 

has been use to calculate the perturbation of magnetic 

field components. Results in Figures 2, 3, and 4 indicate that significant errors could result in source location 

when the spheroid is in the vicinity of the source and 

close to the interface. 

Comparison of the anomalies caused by cylinders 

and prolate spheroids in Figures 2 and 3, indicates 

that the infinite cylinder is a fairly good model for 

finite length scatterers when the magnetic dipole source 

and the observer are in the equatorial plane. Such 

was not the case for a finite electric line source excitation 

[WAIT & HILL (9. 

The effect of conduction currents in the overburden 

(but not the interface effect) could be included by 

using the general dyadic Green's function [TAI (")I. 

The effen of the interface is difficult to compute even 

for the two dimensional problem of line source cxcitation [WAIT ('); HOWARD (A)]. However, for thin cylinders, HOWARD(*) has indicated that the interface 

effect in the three dimensional problem might bc bcbt 

handled by an interaction technique in transform 

space with the resultant solution given by a double 

fast Fourier transform. 

REFERENCES 

C) J. R. WAIT: Criteria for locnting an oscillating mag.neric 

dipole in the earth, Proc. IEEE, 59, (16), (1971), 1033-1035. 

- (%) D. A. HILL & J. R. WAIT: The eLcrromrrgnetic rrsponr~ of n buried sphere for buried dipole excitntion, Radio 

Science, 8, (a), (1973), 813-818. - (') D. A. HILL & 

J. R. V IT: Perturbation of magnetic dipole field3 by a perfectly .irdueting prolate spheroid, Radio Science, 9, (I), 

(1974), 71-73. - (9 J. R. WAIT: Some rolutionr /or electromagnetic problems involvzng rphrroidnl, spherical and cyIindricnl bodiss, Jour. Res. Nat'l Bureau of Standards, 

64B, No. I, Jan/Mar (1960), 15-32 (Note that the aryumcnt, 

ha and hp of the Bessel funcrions xhould he I hu ' and 

I hp I). - (=) J. R. WAIT & D. A. HILI.: f<xcitutio)r o/ u 

homogeneous conductive cylinder of finite lon~th by o prrrcrihed 

axial current distribution, Radio Scicncc, 8, f !2), f I y73) 

1169-1176. - (9 C. T. TAI: Dyadic Green's Iun~lintr> in 

electromagnetic theory, Intext Educational Publishera, Scranton, Ch. 7, (1971). - (') J. R. WAIT: The effect of a hurird 

conductor on the sub-surface field, for line source exritari,m, 

Radio Science, 7, (5), (1972), 587-591. - (*) A. Q. HOWAUI,, 

Jr.: The electromagnrtic fields of o ~ubrerrvneaw cylindricrrl 

inhomogeneity excited by a line rourcr, Gcaphyhicr, 37, (6). 

(197~)~ 975-984. - (O) A. Q. HOWARD, Jr.: Fields NJ u mognmc dipole excited buried cylinder, pre~ented at the 'l'hru- 
The-Eanh Electromagnetics Wo*shop, Golden, Colorado, compared with previous results for the prolate spheroid. 

August 15 (1973). When the cylindrical conductor is ncar thc surface in the 

vicinity of the source, significant crrars can result in source 

Summary - The effect of s finitely conducting circular Locarion. The results have possible application to rescue 

cylinder on the surface fields of a buried vertical magnetic operations following mine disasters. 

dipole is examined in the quasi-static regime. Results are 
Paper presented at Symposium on Electromagnetic Wave Theory, International 

Union of Radio Science, London, 9-12 July 1974. 

Propagation Under the Earth's Surface (A Review) 

James R. Wait 

Cooperative Institute for Research in Environmental Sciences 

University of Colorado 

Boulder, Colorado 80302 

In this paper we review theoretical model concepts in through-the-earth transmission of electromagnetic waves. While the subject is very broad in its defined 

scope, we will be conscious of the specific applications to mine rescue and emergency telecommunications in coal mine environments. Without pretending to be 

entirely impartial we include a critique of the limitations of the available analytical models. 

The central problem is to calculate the steady state mutual impedance of a 

four-terminal network that is the black box characterizing the through-the-earth 

path. Problems of secondary importance include the determination of the input 

impedances of the two pairs of terminals and the associated transient responses. 

P The simple equivalent circuit given tells us a lot about the problem that we are 

seeking a solution for. The mutual impedance Zm describes the propagation phenomena involved in transmission from input (source) to the output (receiver). 

and Z2 , on the other hand, are the input impedances. Clearly the latter are 

needed in a total systems study since we have only a finite amount of power available 

at the source and the receiver must be matched in some sense to the output terminals 

of the receiving antenna. 

Most of the past literature deals with the calculation or merely the derivation 

of the mutual impedance Zm for some highly idealized geometrical configuration. 

For example, the source for an ungrounded loop is a magnetic dipole that is assigned 

a fixed magnetic moment. In the most extreme case this magnetic dipole is an infinitesimally small loop (of area dA) with a total circulating current I amps. The 

moment is IdA! Now we can still use this concept for a multi-turn loop if I is the 

total azimuthal current and provided the uniform current assumption is valid. Also, 

of course, we recognize that the finite size of the loop needs to be accounted for 

in some cases. 

In some of the earliest work the fields of this dipole were calculated as if the 

conduction currents in the earth were negligible. Of course, this is a useful beginning. The next refinement was to correct these magneto-static fields by assuming 

that we need only multiply them by exp(-s/b) where s was the transmission range and 

6 was the skin-depth in the earth for the operating frequency being adopted. With 

the benefit of hindsight we can criticize this correction as being grossly peseimistic 

for many cases of interest to mine rescue or to communication to a receiving station 

above or below the source loop. 

fi 

By using the correct electrodynamic forms of the magnetic dipole fieldn in ;ir~ 

homogeneous conductive medium of infinite extent, we can obtain a better estfmiltc <,f 

the relevant value of Zm . For example, if we are transmitting between two smll 

coaxial loops, we can easily show that Zm Zo(l + rs)exp(-rs) where r - (1 + i)6-' and Zo is the static or D.C. coupling limit. 
Now usually the transmission takes place from the earth's surface to a burled 

receiving terminal. Or the converse situation may exist if we are dealing with LIlink communication. In both cases the air-earth interface must be considered. Here 

we can immediately call attention to the classical formulation of Arnold Sommerfeld q 

that dates backo 1909 in its earliest version. This was used as the basis oi many 

analyses in later years but the interest was mainly for the case where the range 

exceeded a free-space wavelength. Here we are interested in the near zone where the 

significant distances are small compared with the wavelength in the air but such distances (i.e. depth or offset) may be comparable with the wavelength in the earth. 

This is "quasi-statics" in the vernacular of the current workers. To obtain field 

estimates we now have to perform some integrations. Fortunately it was found that 

identities in Bessel function theory, recognized by mathematicians of the late 19th 

Century, were ripe for the picking. Thus some closed form field expressions were 

obtained and published in the 1950's for a fairly broad class of such problems. 

Numerical results required the manipulation of modified Bessel functions whose arguments were complex. Fortunately for cases of interest, the phase angles were near 

1114 radians so that the tabulated Thornson's functions could be used. These are 

sometimes called the Kelvin functions and denoted ber, bei, kei, etc. 

With the ready availability of high speed computers, the use of intricate closed 

form field formulas is giving way to direct numerical integration of the Somerfeld 

integrals. This is satisfactory provided the programmer has some limiting checks or 

if he can refer back to some of the earlier work where the more elegant closed-form 

expressions in terms of special functions are used. Also, as has been indicated 

recently for finite-length source elements, the special function representations for 

the dipole source is a convenient starting point. Otherwise double numerical integration is needed! 

As mentioned above, the air-earth interface problem is treated by regarding the 

overburden as a homogeneous half-space. A rather straight-forward extension is the - 

stratified half-space wherein the intermediate interfaces are plane and parallel to 

the air-earth interface. Further extensions involve electric dipoles rather than 

magnetic dipoles. These are appropriate when dealing with grounded electrodes connected by insulated cables. No new basic difficulties are encountered here. 

In the class of problems discussed above, all three of the impedance elements 

'm* '1 and Z2 , are determinable. Thus, in principle, a complete determination 

or prediction of the system performance is possible. This includes down-link communication efficiency estimates of source location accuracy for the models assumed. 

Traditionally, the source current is assumed to vary harmonically in time. For 

dealing with transient excitation or in estimating signalling rate we need to understand the time-domain behavior of the system. Formally this is obtained by Fourier . 

or Laplace transform inversion of the preceding transfer functions. In general, this 

is not a trivial task. However, some rather interesting closed-form results can be 

obtained if the original spatial wave number integration be deferred until after the 

Laplace transforms are inverted. 

Now that the homogeneous and stratified half-space models have been exhausted, 

it is appropriate to consider more realistic situations. For example, the earth's 

surface is not always flat. One approach to allow for this situation, is to adapt 

cylindrical and spherical boundaries. In such cases, the local radius of curvature 

is chosen to match roughly the local topography. Results of calculations from such 

models indicated that transmission ranges were not markedly affected but that source 

location errors could be significant. An even simpler method conceptually was to 

retain the uniform half-space model but to allow the plane interference to be tilted 

at such an angle that the local slope of the terrain was matched. Further analytical - 

work on such models is justified since much insight can be gained without excessive 

numerical effort. Unfortunately, formally exact solutions for such models are strictly 

limited. One class of boundary that could be treated for line source excitation is 

a parabolic or hyperbolic interface. The chree dimensional version involving dipoles 

in the presence of a paraboliodal interface is extremely intricate but it may be 

worth doing. 
One must clearly recognize that the overburden is not a homogeneous slab whether 

r'. it be bounded by plane or uniformly curved interfaces. Thus, the influence of omnipresent inhomogeneities needs to be understood. An extreme case is when metallic 

&racks and,pipes are in the vicinity of the normal transmission path. To treat such 

configurations. two dimensional models have been used. An example is thc line 

source excited buried cylinder of infinite length. This is not a trivial problem 

if one considers the inter-actions between the buried cylindrical inhomogeneity 

and tlie air-earth interface. Two specific analytical techniques have becn used 

to get numerical results for this problem. One is essentially a perturbation 

procedure that involves successive reflections between the cylinder and the plane 

interface, while the other is a sophisticated integral equation procedure. The 

results between those methods agree in a common region of validity. The extension 

to three dimensional versions of these problems is nontrivial. Some progress has 

been made, however, in using perturbation procedures. An example is the buried 

sphere in the presence of a surfcce based dipole source. Some work has also been 

done on cylinders of finite length where certain assumptions were made about the 

nature of the axial induced current flow at the ends of the cylinder. Among other 

things, this analysis showed that predictions based on infinite cylinders may be 

quite misleading. 

Within the scope of these analytical techniques, we encompass both active and 

passive location concepts. In the active method, of course, the unknown source is 

energized by the to-be-rescued party. On the other hand, in the passive method, 

the target to be located is typically a loop of wire or a similar configuration. 

Various geometries, for such problems, have been considered in both the frequency 

and the time domain. 

P The foregoing account of theoretical analyses of electromagnetic induction 

problems is, by no means, claimed to be exhaustive. The selection has been based 

mainly on work that the author is familar with. Many of the source papers are to 

be found in Radw Science, the Journal of the U.S. National Committee of URSI. 

Some recent offerings on this and related subjects from this writer and his 

colleagues are: 

* "Array technique for electromagnetic positional determination of a buried receiving 

pointr', J.R. Wait, Electronic Letters, Vol. 1, No. 8, 186-187, 22 April 1971. 

* "Criteria for locating an oscilkzting magnetic dipole buried in the earth", J.R. Wait, 

Proc. IEEE (Letters), Vol. - 59, No. 6, 1033-1035, June 1971. 

"Etectromagnetic-pulse propagatwn in a simple dispersive mediwn", Electronic 

Letters, Vol. 1, No. 11, 285-286, 3 June 1971, J.R. Wait. 

"Transient excitation of the earth by a line source of currmt", J.R. Wait, - Proc. 

IEEE (Letters), Vol. 59, No. 8, 1287-1288, August 1971. 

*"EZectromagnetic induction technique for heating a source", J.R. Wait, 

IEEE Trans. Geosci. Elect., Vol. GE-9, No. 2, 95-98, April 1971. 

*''Transient signals from a buried magnetic dipoler$ J.R. Wait and D.A. Hill, J. Appl. 

. Vol. 42, NO. 10, 3866-3869, September 1971. 

"In situ measurements of the complex propagation constant in rocks for frequencies 

from I MHz to 10 Mhar', R.N. Grubb and J.R. Wait, Elect. Letters, Vol. - 7, No. 17, 

506-507, 26 August 1971. 

r * "Subsurface electromagnetic fietds of a Zine source on a conducting half-space", 

Radio Sci., Vol. - 6, Nos. 8-9, 781-786, Aug.-Sept. 1971, J.R. Wait and K.P. Spies. 

* rrElectromagnetic fields of a small loop buried in a stratified earthN, J.R. Wait 

and K.P. Spies, IEEE Trans., Vol. AP-19, No. 5, 717-718, September 1971. 
"On radio propagation thmugh the earth", J.R. Wait and J.A. Fuller, IEEE Trans., 

Vol. x, No. 6, 796-798, November 1971. 

"mlytical investigations of electromagnetic wave propagation in the earth 's e 

Crust': J.R. Wait, in The Structure and Physical Properties of the Earth's Crust, 

(ed. J.G. ~eacock), pp. 315-324 (Geophys. Mono Series, Vol. 14, AGU, Washington, 

D.C., 1971. * "Influence of carath curvature on the subsurface electromagnetic fields of a line 

soii~s:c", J.R. Wait, Elect. Letters, Vol. - 7, No. 23, 697-699, 18 November 1971. 

"S;cbsidrfacc electrornagnctic telecomnication - A review", R. Gabillard, P. Degauque, 

and J.R. Wait, IEEE Trans., Vol. COM-19,No. 6, 1217-1228, December 1971. 

'Tmnsient dipole radiation in a conducting medium", J.R. Wait, Revista Brasileira 

de Tecnologia, Vol. 2, No. 1, 29-37, March 1972. 

* "The effect of a buried conductor on the subsurface fields for line source excitation", J.R. Wait, Radio Science, Vol. L, No. 5, 587-591, May 1972. 

"On the theory of transient electromagnetic sounding over a stratified earth", 

J.R. Wait, Can. J. Phys., Vol. 50, No. 11, 1055-1061, 1 June 1972. 

* "The sub-surface magnetic fieMs produced by a line current source on a non-flat 

earth", J.R. Wait and R.E. Wilkerson, PAGEOPH, Vol. 95, No. 111, 150-156, 1972. 

"On calculating transient electromagnetic fields of a small current-carrying loop 

over a homogeneous earth", J.R. Wait and R.H. Ott, PAGEOPH, Vol. 95, No. 111, 157- 

162, 1972. 

"Absorption mode for ELF electromgnetic propagation in the earth-crust waveguideN, 

J.R. Wait, Elect. Letters, Vol. 8, No. 11, 292-294, 1 June 1972. 

* "Transient magnetic field8 proheed by a step-function-excited loop buried in the 

arth", J.R. Wait and D.A. Hill, Electronic Letters, Vol. 8, No. 11, 294-295, - 

1 June 1972. 

* 'fSubsurface electromagnetic fields of a circuZar Zoop of current located above 

ground", J.R. Wait and K.P. Spies, IEEE Trans., Vol. E, No. 4, 520-522. July 1972. 

"ELectromagnetic pulse transmission in homogeneous dispersive rock", J.A. Fuller 

and J.R. Wait, IEEE Trans., Vol. - AP-20, No. 4, 530-533, July 1972. 

"kttenuation of electromagnetic waves in the earth-crust waveguide from ELF to VLFf', 

J.R. Wait and K.P. Spies, Radio Science (Letters), Vol. - 7, No. 6, 689-690, June 1972. 

"EM Coupling of coaxial and coplanar loops in uniform dissipative media", J.A. Fuller 

and J.R. Wait, Proc. IEEE (Letters), Vol. 60, No. 8, 993-994, August 1972. 

* "Locating an oscillating magnetic dipole in the eartht$ J .R. Wait, Electronic 

Letters, Vol. - 8, No. 16, 404-406, 10 August 1972. 

"Theory of electromagnetic reflection from a parallel grid of dielectric coated 

wires buried in the earth", J.R. Wait, Can. J. Phys., Vr . - 50, No. 18, 2149-2157, 

September 1972. 

*'rElectromagnetic surface fields produced by a pulse-excited loop buried in the earth", 

J.R. Wait and D.A. Hill, J. Appl. Phys., Vol. -9 43 No. 10, 3988-3991, October 1972. 

"Klectromagnetic wave propagation along a buried insulated wire': J.R. Wait, Can. 

- J. Phys., Vol. 50, No. 20, 2402-2409, 15 October 1972. 

Argand representations of the mutual electromagnetic coupling of loops on a twolayer earth", J.R. Wait and J.A. Fuller, Geoexplor., Vol. - 10, 221-227, 1972. - 

* "Transient electromagnetic fields of a finite circular Zoop in the presence of a 

conducting half-space': J.R. Wait and D.A. Hill, J. Appl. Phys., Vol. 43, No. 11, 

4532-4534, November 1972. 

"EZectrcmagnetic transient response of a small wire loop buried in a conducting 

earth, D.A. Hill and J.R. Wait, Pure and Appl. Geophys., E(IV), 869-878, 1973. 
"Dipole &citation of ultralaw-frequency electromagnetic wares in the earth crdsz 

wave guider: J.R. Wait and K.P. Spies, JGR, Vol. 77, No. 35, 7118-7120, 10 Dec. 1972. 

r- "High frequency electromagnetic coupling between small coplanar loops over an 

inhomogeneous gracnd", J.A. Fuller and J.R. Wait, Geophys., Vol. 2, No. 6, 99;-1004, 

December 1972. 

'lElectmmagnetic propagation in an idealized earth crust waveguide, Part II", 

J.R. Wait and K.P. Spies, PAGEOPH, Vol. 101, No. IX, 188-193, 1972. 

"li~,.:;::ttz~~t. of ti?rtl~ cloctx*odasr', J.R. Wait, Elect. J.ctlcrti, Vol. - 9. No. 4, 90-91, 

22 Fcbrunry 1973. 

"Efictual electmnagnctic coupling of coaxial loops in a borehole ", J .A. Fuller and 

J.R. Wait, Radio Science, Vol. 8, No. 5, 453-457, May 1973. 

"Transmission Zine theory for an insulated linear antenna in a fluid-or-air-filled 

borehole", J.R. Wait and J.A. Fuller, Appl. Phys., Vol. 1, 311-316, June 1973. 

"Subsurface electromagnetic fields ofa grounded cable of finite length", Can. Journ. 

of Phys., Vol. 2, No. 14, 1534-1540, 1973, D.A. Hill and J.R. Wait. 

* "Low-frequency impedance of a circular loop over a conducting ground", J.R. Wait 

and K.P. Spies, Electronics Letters, Vol. 9, No. 15, 346-348, 26th July 1973. 

* "Electromagnetic transient response of a small wire loop buried in a homogeneous 

conducting earth", D.A. Hill and J.R. Wait, PAGEOPH, Vol. - 105, 869-878, 1973lIV. 

* "The eZectromagnetic response of a buried sphere for buried-dipole excitation", 

D.A. Hill and J.R. Wait, Radio Science, Vol. 8, Nos. 8-9, 813-818, August-Sept. 1973. 

Y ifSubsurface electromagnetic fields of a Zine source on a two-layer earth", J.R. Wait 

P and K.P. Spies, Radio Science, Vol. - 8, Nos. 8-9, 805-810, August-Sept. 1973. 

"Range dependence of the surface impedance for a line-source excited two-layer earth", 

J.R. Wait and K.P. Spies, IEEE Trans., w, No. 6, Nov. 1973. 

u "Excitation of a homogeneous cylinder of finite length by a prescribed axial 

current distributionr', J.R. Wait and D.A. Hill, Radio Science, Vol. 8, No. 12, 

December 1973. 

w l'Sub-surface electric fie- of a grounded cable of finite length for both frequency 

and time domainr', D.A. Hill and J.R. Wait. Pure and Applied Geophysics, (in press). 

* "EZectromagnetic response of a conducting cyytinder of finite length", D.A. Hill 

and J.R. Wait, Geofisica International (Mexico) (in press). 

"ELF propagatwn aZag a horiaontal wipe located above or buried in the earth", 

D.C. Chang and J.R. Wait. IEEE Trans., CS (in press). 

of magnetic dipole field by a finitely conducting circu~ar 

cylinder, D.A. Hill and J.R. Wait, Rivista Italian dl Geofiaica, Vol. 22, No- 5/6, 391-394, 1973. 

Those marked with an asterik are included in this reprint collection. 
LISTING OF PKEL.[I.IINtiRY REPORTS TO 

U.S. BUREAU OF ElINIIS ON CONTRACT NO. H01220hl 

THAT RELATE TO M LOCATIOS AND DOI4N-AND UP-LINK COEPtUNICA'CION 

Electromagnetic transient response of a srnzZZ wire Zoop buried in a honlogeneous 

conducting earth, D.A. Hill and J.R. Wait, 1 July 1972. 

Locating an osciZZating magnetic dipoZe in the earth, J.R. liait, 10 August 1972. 

Electromagnetic surface fields produced by a pulse-excited loop buried in the earth, 

J.R. Wait and D.A. Hill, October 1972. 

Transient magnetic fields producod by a step-function excited loop buried in the 

earth, J.R. Wait and D.A. Hill, October 1972. 

The electromagnetic response of a buried sphere for bwied dipole acitation, 

David A. Hill and James R. Wait, 15 January 1973. 

Subsurface electromagnetic fields of a grounded cable of finite Zength, David A. Hill 

and James R. Wait, 7 February 1973. 

Sub-surface electromagnetic fieMs of a Zine sov2,ce on a two-layer earth, J.R. Wait 

and K.P. Spies, (including Special Report of same title with Numerical Tables), 

22 February 1973. 

Subsurface electric fields of a grounded cable of finite Zength for both frequency - 

and time domain, David A. Hill and James R. Wait, 15 March 1973. 

Note on the surface impedance a-nd wave tilt for a line source excited tuo-Zayer earth, 

James R. Wait, 6 April 1973. 

Electromagnetic surface fieMs of an inclined buried cable of finite length, 

David A. Hill, 20 April 1973. 

Diffusion of ezectromagnetic pulses Cnto the earth from a Zilze source, D.A. Hiil and 

J.R. Wait, 11 May 1973. 

Lou frequency impedance of a circular Zoop over a conducting ground, James R. Wait 

and Kenneth P. Spies, 18 June 1973. 

SuppZementary -tables !relating to Zou frequency irpedance of a circutur loop over a 

conducting ground, James R. Wait and Kenneth P. Spies, 7 Ju 1973. 

Excitation of a.homogeneous conductive cyZinderr of finite Zength by a prescribed 

axial current distribution, James R. Wait and David A. Hill, 10 July 1973. 

State of knowledge of analytical techniques for thru-the-earth electromagnetic wave 

probtems relevant to mine rescue, James R. Wait, 16 July 1973. 

Electromagnetic response of a conducti-lzg cylinder of finite Length, David A. Hill 

and James R. Wait, 1 August 1973. 

Perturb~tion of magnetic dipole fields by a perfectly conducting prolate sphei%oi?, 

David A. Hill and James R. Wait, 22 August 1973. 
CaZcuZat%ons of sub-surface grounded ca3Ze and loop sources for comparison w-itk A forthnoning experimental dnta, James R. Wait and David A. Hill, 19 September 1973. 

Perturbation of magnetic dipole fields by a finitely conducting circuZar cyZinder, 

David A. Hill and James R. Wait, 23 November 1973. 

Effect of a spherical scatterer on EM source location, David A. Hill, 14 January 1970. 

NOTE: Issuances prior to 1 July 1972 are not listed here.