NISTIR 89-3909

CLUTTER MODELS FOR

SUBSURFACE ELECTROMAGNETIC

APPLICATIONS

David A. Hill

National Institute of Standards and Technology

(formerly National Bureau of Standards)

U.S. Department of Commerce

Boulder, Colorado 80303-3328

February 1989

NISTIR 89-3909

CLUTTER MODELS FOR

SUBSURFACE ELECTROMAGNETIC

APPLICATIONS

David A. Hill

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

(formerly National Bureau of Standards)

Boulder, Colorado 80303-3328

February 1989

Sponsored by

U.S. Army Belvoir RD&E Center

Fort Belvoir, VA 22060

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

Ernest Ambler, Acting Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Acting Director

CONTENTS

Page

Abstract 1

1. INTRODUCTION ' 1

2. INHOMOGENEOUS EARTH MODELS 2

2.1 Deterministic Models 2

2.2 Random Discrete Scatterers 3

2.3 A Random Continuum 4

3. MATHEMATICAL METHODS FOR RANDOM MEDIA 5

3.1 Coherent Field 5

3.2 Incoherent Field 6

4. ELECTRIC DIPOLE RADIATION 11

4.1 Far-field Pattern 11

4.2 Total Radiated Power 15

5. MAGNETIC DIPOLE RADIATION 16

5.1 Far-field Pattern 16

5.2 Total Radiated Power 18

6. TRANSMISSION BETWEEN VERTICAL DIPOLES 20

6.1 CW Transmission 20

6.2 Pulse Transmission 22

7. CONCLUSIONS AND RECOMMENDATIONS 24

8 . ACKNOWLEDGMENT 25

9. REFERENCES 25

APPENDIX A - VOLUME INTEGRATION FOR ELECTRIC DIPOLE SOURCE 29

APPENDIX B - VOLUME INTEGRATION FOR MAGNETIC DIPOLE SOURCE 30

APPENDIX C - ANGULAR INTEGRATION FOR SCATTERING CROSS SECTION 30

iii

CLUTTER MODELS FOR SUBSURFACE

ELECTROMAGNETIC APPLICATIONS

David A. Hill

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, CO 80303

Clutter models for subsurface electromagnetic applications are

discussed with emphasis on tunnel detection applications.

Random medium models are more versatile and require less

detailed information than deterministic models. The Born

approximation is used to derive expressions for the incoherent

field, and electric and magnetic dipoles are treated in detail.

When random inhomogeneities are located in the near field of the

dipole source, an electric dipole radiates a larger incoherent

field than a magnetic dipole because of its larger reactive

electric field.

Key words: clutter; coherent field; electric dipole; incoherent

field; magnetic dipole; random medium.

1 . INTRODUCTION

The use of electromagnetic (EM) waves in subsurface applications

continues to increase. Some current applications are geophysical

prospecting [1-3], mine communication [4,5], tunnel detection [6-8], and

mine detection [9,10]. The simplest earth model is a homogeneous medium; a horizontally stratified medium [11,12] is a better earth model for some

applications [13-16].

EM transmission measurements for tunnel detection [7] and scattering

measurements for remote sensing [17] indicate that the real earth is often

inhomogeneous in a fairly complicated manner. When an EM wave propagates

through an inhomogeneous medium, it suffers attenuation and dispersion, and

an incoherent (clutter) signal is produced by scattering. Such effects are

detrimental to tunnel detection [7], but they have not been quantitatively

evaluated. In some applications of geophysical tomography [18-19], random

noise has been added to the received signal to test the sensitivity of the

imaging algorithm. Such tests are useful, but it would be more realistic if

1
we had mathematical models for clutter signals that represent the physics of

an inhomogeneous earth.

A typical borehole- to-borehole example for tunnel detection is shown in

figure 1. The received signal is made up of a direct-path signal, a signal

scattered from the target (in this case a tunnel), and a clutter signal

scattered from the inhomogeneities in the rock. In typical analyses of EM

detection of tunnels [20,21], the clutter signal is not included. However,

the clutter signal (sometimes called geologic noise [22]) is often the main

limitation in tunnel detection. In underground communication or source

location [23], the target signal does not exist, but the direct-path signal

is still contaminated with the clutter signal. This report is primarily

concerned with clutter models for use in the analysis of tunnel detection

and other subsurface applications.

The organization of this report is as follows. Section 2 discusses

types of inhomogeneous earth models, both deterministic and random. Random

media models offer more flexibility, and the mathematical techniques for

analyzing random media are discussed in section 3. The analysis of an

electric dipole radiating in a random medium is contained in section 4.

Numerical results are given for both the far- field pattern and the total

radiated incoherent power. Analogous results are given for a magnetic

dipole source in section 5. Section 6 contains an analysis for transmission

between electric dipoles for both cw and pulse excitation. Conclusions and

recommendations for further work are contained in section 7.

2. INHOMOGENEOUS EARTH MODELS

In this section, we discuss three types of inhomogeneous earth models.

Deterministic models are useful when the details of the local earth medium

are well known, but they require more information than is generally

available for most applications. Random media models are more flexible and

require less information. However, the EM fields propagating in random

media are random variables, and they can be described only statistically.

2.1 Deterministic Models

2
The most useful deterministic model is composed of a homogeneous earth

with a single scatterer whose response can be calculated. In this case the

scattered field is considered to be the clutter signal. For example in the

problem of surface location of a buried magnetic dipole source, location

errors caused by scattering from a sphere [23] or a prolate spheroid [24]

have been evaluated. The geologic noise caused by a more complicated

overburden scatterer in geophysical prospecting has been evaluated

numerically [22].

In these cases the parameters that describe the scatterer (dimensions,

location, constitutive parameters, etc.) are deterministic quantities, and

the scattered fields are deterministic. Thus the scatterer parameters are

chosen from a knowledge of the specific application, and each case yields

only one deterministic result for the clutter signal. Repeated calculations

for different parameters are required for general clutter characteristics,

and such repeated calculations can be tedious. For tunnel detection where

the structure of the rock is not well known, we favor random media models

because statistical properties of the clutter are obtained directly.

Consequently, the remainder of this report will deal with random media

models

.

2.2 Random Discrete Scatterers

Random media can generally be grouped into three categories [25]:

random scatterers, random continua, and rough surfaces. We will not

consider rough surfaces in this report because we assume that volume clutter

is more important than surface clutter in most tunnel detection

applications. However, for cases where the source or receiver is located

near the earth surface or near a subsurface layer with a rough boundary,

rough surface clutter could be important.

Typical examples of discrete scatterers are rain [26] and hail.

Discrete scatterers can be characterized by their bistatic radar cross

A A A section cr^^(o,i), where i is a unit vector in the direction of propagation

A of the incident wave and o is the direction of the scattered wave [25, Sec.

2-1]. (Vectors and dyads are indicated by boldface throughout this report.)

3
For a random, uniform distribution of identical scatterers, the other

important parameter is the number p of scatterers per unit volume.

For a tenuous distribution of scatterers in the far field of the

source, such as a weather radar [26], the single - scattering approximation

yields a fairly simple volume integral expression for the scattered power

[25, Sec. 4-1]. The cases of a dense distribution or a near-field source

are more complicated.

2.3 A Random Continuum

In this report we define a random continuum as a medium whose

permittivity is a random continuous function of position. In some cases,

such as atmospheric turbulence [27], the permittivity is also considered to

be a function of time. In modelling the earth, we assume that the

permittivity e(r) is a random function of position r only.

We write the earth permittivity as a sum of the average value and a

fluctuation

:

e (r) = e Q

[ 1 + e^r) ] , (1)

where is the fluctuation with zero average

< e 1(r)> = 0 (2)

and < > represents the ensemble average [28]. For lossless media, and

are real, but for lossy media we make them complex. If the medium is

statistically homogeneous and isotropic, then the covariance <e^(r^)

is a function only of the difference r , = |r,| = I r. - r~ I : d d 1 z

i

(

r

i

) e^( r

2

) > = ( r d) > ( 3 )

where superscript * denotes complex conjugate.

4
The equivalence between continuous and discrete scattering models has

been studied by Besieris [29]. As we will see in section 3, the scattering

cross section per unit volume can be written in terms of the Fourier

transform of B . Various analytical forms of B (r,) have been used [25, Ch.

e e d

16], and anisotropic forms of B^(r^) have been used for anisotropic media

such as vegetation [30,31]. In this report we use only the isotropic form

B^(r^) because it is simpler, but anisotropic forms might be more realistic

for modelling nonspherical inhomogeneities or fractures.

3. MATHEMATICAL METHODS FOR RANDOM MEDIA

An EM wave propagating in a random medium can be expressed as the sum

of an average field and a fluctuating field. The average field is also

called the coherent field, and the fluctuating field is also called the

incoherent field [25, Sec. 4-4]. For example, we can write a scalar

component of the total time -harmonic electric field E^(r) as

E

u

(r) = <E

u

(r)> + E

fu

(r)

,

(4)

where <E^(r)> is the average (coherent) field and E^ (r) is the fluctuating

(incoherent) field. The time dependence exp(-iwt) is suppressed throughout

this report.

3.1 Coherent Field

When an EM wave propagates through a random medium, the average

(coherent) field is generally attenuated and slowed by scattering. However,

if the randomness is very weak, then the coherent field can be approximated

by the incident field that would propagate in a uniform medium. The

approximation of the coherent field by the incident field is known as the

Born approximation [25, Sec. 4-1].

5
Karal and Keller [32,33] have used a perturbation technique to study

propagation in a random continuum. They have derived a dispersion equation

for the effective wavenumber for plane -wave propagation and have derived

expressions for the effective dielectric constant and conductivity. Roth

and Elachi [34] have used the dispersion equation of Karal and Keller to

generate numerical results for loss tangent as a function of the parameters

of a random continuum with an exponential correlation function.

When the medium has strong fluctuations, perturbation methods are no

longer valid. Dyson's equation for the coherent field [34] is exact in

principle, but various approximate methods [34-39] are required to obtain a

solution

.

3.2 Incoherent Field

By definition the average value of the incoherent field is zero. For

example, from (4) we can see that <E^> = 0. However, if we take the

average of the square of the absolute value of (4)

,

then we can write

< | E | 2

> - | <E >|

2

+ <|E„ | 2

1 >. (5)

u

1 u

1 fu1

2

If we use Ishimaru's terminology [25, Sec. 4-4], then < | | > is the average

2 2

intensity <I>, | <E >| is the coherent intensity 1^, and <|E^J> is the

incoherent intensity I.. Thus (5) could be written

<I> = I + I. . (6)

c 1

Ishimaru's definitions are actually applied to scalar fields, but we can use

the same terms for scalar components E^ of the electric field or the vector

2

field E. If we want the intensities to have units of W/m , then each term

in (5) or (6) should be multiplied by the intrinsic impedance rj of the

6
medium. In most cases we will be interested in the ratio I. /I , and the

l c intrinsic impedance will cancel out.

In a weakly random medium, the coherent intensity is nearly the same as

the intensity in absence of randomness, and the incoherent intensity 1^ can

be considered the intensity of the clutter signal. Thus the main problem in

clutter applications is the evaluation of the incoherent intensity I =

2 <|E

fu l >• The simplest methods for calculating I^ use the Born

approximation [25, Sec. 2-6], and we will use the Born approximation

throughout the remainder of this report. There are more sophisticated

methods, such as the Bethe - Salpeter equation [25], for calculating 1^, but

the earth medium has not been well characterized in tunnel detection, and

there is no point in making the initial calculations overly complicated.

We can start the analysis by writing the source -free wave equation for

the electric field E,

V X V X E - 1 + € 1

)E = 0, (7)

where \i ^ is the permeability of the medium and the permittivity has been

written in the form shown in (1)

.

We now write (7) in the equivalent form,

V X V X E - k^E = k^e E, (8)

2 2 where k^ = u> Mq € q- We can write the total field E as the sum of the

incident field E^ and the scattered field E^:

E = E

Q + E r (9)

The incident field satisfies

7
V x V x Eq k

2

E

0 0

0. ( 10 )

If we subtract (10) from (8), we obtain

V x V x E^ k

o

E

i - k0

£ iE - (ID

We assume that we know the incident field which satisfies (10) . The

scattered field E^ satisfies the wave equation with a source term on the

right side as given by (11). The problem is that the source term in (11) is

not known because E is not known. In the Born approximation [25, 2-6], we

approximate E by the incident field E^ in the source term in (11). Thus the

scattered field approximately satisfies the inhomogeneous wave equation,

V X V X E^ k

o Ei

k

o

e i

E

o

( 12 )

Since the right side of (12) is known, we can write E^ in integral form

using the Green's function approach [40]

Vr) - k0 J Mr')

V

G (r , r

'

)

E

o (r ' } dr' (13)

where G(r,r') is the dyadic Green's function and the integration volume V is

all space. In some remote sensing cases, G(r,r') could be the dyadic

Green's function for a layered medium [29,30], but we consider only a

homogeneous medium (constant k^) for the deterministic portion of the

problem. In this case, G(r,r') is [40]

G(r , r

'

)

zi (l + h

vv) *•

k

o

(14)

8
where if> = expCik^lr - r'|)/|r - r'| and 1 is the unit dyad. For points

where r = r', either G or the integral in (13) must be appropriately

modified [41,42]. We can also write (13) in component form for a scalar

component E^ of E^:

E

lu

(r) - k

0 I

V

e x (r') G (r , r '

)

uv

E0v

(r>) dr' (15)

where G is a scalar component of G and the repeated subscript v indicates

summation over the three values of v. From (15) we can see that <E- > = 0

lu

because <£^> = 0.

2

To evaluate the incoherent intensity, we need to evaluate <|E^ | If

we multiply (15) by its complex conjugate and take the average value, we

obtain

<|£ (r)| 2

> - |k I 4

J ;

B (| r

v V

r "l) G

uv

(r,r')E0v (r’).

G (r , r

uw

)E0w

(r ' ) dr' dr'

(16)

where we have used (3) to introduce into (16) . In the modified Born

approximation [43,44], the Green's function in (16) is replaced by the

effective Green's function for the random medium.

Generally the double volume integral in (16) is difficult to evaluate,

but when the random volume is in the far field of the source and observer,

the integrals simplify considerably [25,27]. For the geometry in figure 2,

Ishimaru [25, Sec. 16.1] writes the received power P^as

A

2

G (i)G (o) A a P “

U

A A A <7(0, i) dV, (17)

V (4tO

Z

R^R2

9
where P^_ is the transmitted power, and are the gains of the

transmitting and receiving antennas, and are the distances shown in

figure 2, V is the volume shown in figure 2, and A is the wavelength. We

assume here that the medium is lossless, but an exponential loss factor can

be included when necessary. The bistatic scattering cross section per unit

A A volume a(o,i) can be derived in terms of B^ in integral form [25, 16.2]

<7 ( 0 , i) - *kQ sin2 *

4> £ (k

s )/2, (18)

where $ (K) = r f B (r,) exp(iK»r,) dr,,

e . „ . 3

J

e d d d

( An) °o

= 2k^ sin(0/2),

A A

and the unit vectors o and i are shown in figure 2. The scattering angle 9 and the polarization angle \ are shown in figure 3.

There are many possible forms for B , but the exponential form is one

which is mathematically convenient [45]

B (r ) = <e?> exp (- r /a) . (19)

e d 1 d

2

This model is characterized by the variance <e^> and the correlation

distance a. The integral in (18) is easily evaluated to yield

4 3 2 2

kn a sin y <e >

y

1

T . (20)

2tt( 1 + 4k^a sin (0/2))

When the correlation distance a is small compared to wavelength A, the

cross section a reduces to

10
o{6) ~ k^a ( 21 ) 3 sin2 * <e^>/(2n)

.

The expression in (21) represents Rayleigh scattering where the scattered

power is proportional to frequency to the fourth power times the correlation

length to the third power.

4. ELECTRIC DIPOLE RADIATION

In the previous section we noted that the double volume integral in

(16) is difficult to evaluate except when the random medium is in the far

field of both the source and receiver. Dorfman [46] has treated the case

where an electric dipole is located within a spherical random medium, and he

was able to evaluate the intensity in the far field. In this section we

elaborate on his treatment, and in section 5 we treat the analogous case for

a magnetic dipole source.

4.1 Far-field Pattern

The geometry for a z-directed, electric- dipole source of moment IL is

shown in figure 4. The randomness of the medium is confined to a spherical

shell, b < r < c, where the permittivity is given by (1). The spherical

shell model is idealized, but it allows us to examine near- field effects for

the transmitting dipole. Following Dorfman we assume that the medium is

lossless, but loss could be added to the model. Since we use the Born

approximation, the coherent far field has only a 6 component of electric

field E n given by [47]

U o

- iw/^ IL sin#

E

06 ~ W exp(ikQ

r) . (22)

The coherent intensity I is y c

11
( 23 )

(w/i

q

IL sin#

)

k

- ' Eod - ——T

(47rr

)

Following Dorfman [46] we approximate B in the form,

B

f (rd

) - <£

2

> P «(r

d

)

,

(24)

where 8 is a Dirac delta function and 2 plays the role of a correlation

length. The form of is physically realistic only when the incident field

is nearly constant over the distance i. The main advantage of (24) is that

it allows the integral in (16) to be evaluated analytically. If the actual

correlation function is r(r,), then i is obtained from

d

1 - Sr ( r d

) dr

d

- (25)

3

For example, if T is an exponential function exp(-r^/a), then f is

3 r r 2 3

i = J exp(-r^/a) dr^ = 47r J exp(-r^/a) r^ dr^ = 87ra . 0

(26)

If we substitute (24) into (16) and carry out the r'' integration, we

obtain

< E

lu

(r)| 2> = k

4 «

2

i>

I

3

f | G (r,r')E. (r')

|

2 ^

1 uv Ov

dr

(27)

where V is the spherical shell volume. When the observation point is in the

far field (large k^r) , the Green's function in (14) or (27) simplifies

considerably, and (27) simplifies to

12
< I E]_0 (r) I (28)

2

>

<t>

2 3 2 VZ&<e^>

(47rr ) ^ f \*V

*

E

Q (r')| dr',

A A

^ where 5 and <f> are unit vectors. In the far field, <|'E^ (r) | > ~ 0.

For an electric dipole source, E^(r') is [47]

E

o (r<) -

-iw/z^IL exp(ik

Q

r') A

_ 2

47rr

lr' [T +

ik r' 2

0 (ik

Q

r')

] cos 6

(29)

+ O' [1 - tT— + —1

T] sin0'}.

lk

0

r fik

0

r’)

If we substitute (29) into (28)

,

we can carry out the volume integration

analytically. The details are included in Appendix A, and the result is

<|E

1S | 2

> - k

3

i

3 <e]> E2

, (30)

<t> <f>

where F = [k (c - b)(cos^/9 + 8 sin^$) + k/(b ^ - c ^)(3 cos^d + 4 sin^0)

v U U

+ k"

3

(b"

3

- c"

3

) (23 cos

2

$ + 4 sin2 0)]/(6O ir) ,

F - [k

Q

(c - b) + Sk'bb" 1

- c'

1

) + 15k

Q

3

(b'

3

- c'

3

)]/(60 k) ,

and Eq = IL)/(47rr). The incoherent intensity I

^ is

h

- < i e

i^i

2> + <i E i/> - k

l p <£ i>

E

o

( h

+

v- (31)

13
The ratio of the incoherent to coherent intensities is also of interest

because it gives a measure of the degradation of the transmitted signal

between a pair of electric dipoles:

I. /I =

3

<e

3

> (F. + F ) /sin2

# . (32)

1 c 0 1 0 <f>

The results in (31) and (32) represent Rayleigh scattering where the

incoherent intensity is proportional to third power of the correlation

3 4 length, 1 . This is similar to (21). However, the dependence is

obtained only when the k c term is dominant in F and F , . This is as

U u (p expected because this is the case where far-field scattering is dominant.

When near- field scattering is important, then the k^ dependence is to some

-3 -3

lower power. When the quasi-static term, kg b , is dominant, the

scattering is independent of k^ . If we let b approach zero, then becomes

infinite. This physically unrealistic result is a result of using the delta

correlation function in (24). If we had chosen a finite correlation

function, then I would have remained finite as b approached zero [46,48].

However, it is difficult to evaluate the volume integral for finite

correlation functions. The results in this section are still valid as long

as b is somewhat greater than i.

Numerical results for the incoherent intensity are shown in figures 5

2 and 6. The results are normalized to CE^ to make them dimensionless, and

3 3 2

the normalization constant C = k^i <e^>. In figure 5, far-field scattering

is dominant, and the maximum occurs in the broadside direction (8 — 90°) as

it does for the coherent component. In figure 6, near-field scattering is

dominant because of the small value of k^b . Here the maximum occurs off the

end of the dipole (

6

= 0°). An interesting feature of both cases is that

the <f> component is independent of 8. The cross -polarized component is

normally zero for far-field, plane-wave scattering in the Born approximation

[49], but not for the near-field case considered here.

14
4.2 Total Radiated Power

The total radiated power can be obtained by integrating the intensity

over a sphere at infinity. The coherent power P^ is

p

o -

27T 7T f f I r sin# d# d<fi . (33)

Zrj o o oc

If we substitute (23) into (33) and carry out the integrations, we obtain

p

o - r) 0 (k0

IL)

2

/(12*). (34)

The incoherent power P^ is

P

1 - 2n n

> J J

I. r sin# d# d<f>

.

(35)

Zr)

0 0 o

1

If we substitute (30) and (31) into (35) and carry out the integrations, we

obtain

p

i - WL> 2 & <•?>

[ . l

. vi. ri -i

9

[5k (c - b) + 5k (b - c ) 360 n

(36)

-3 -3 -3

+ 19kg (b - c )].

The results in (31) and (36) are similar to those of Dorfman [46], but the

coefficients are not the same. It is not possible to check Dorfman'

s

derivation because he does not give any intermediate steps.

The ratio of incoherent to coherent power can be regarded roughly as a

clutter- to - signal ratio and is given by

15
0

1 3,3 2 V <€

1

>

30 7T

1 „ -1

[5k

Q

(c - b) + 5k

Q

^(b

-1 -3 -3

c ) + 19k (b - c )]. (37)

Because we have assumed that the observation point is in the far field of

both the source and the random region, P^/P^ is independent of r. Normally

the ratio of the incoherent to coherent intensity would increase with r if

the entire medium were random [25, Ch. 6].

5. MAGNETIC DIPOLE RADIATION

In this section we extend the results of the previous section to a

magnetic dipole source. A magnetic dipole source (small loop) has been

suggested for the detection of long conductors in tunnels [50],

5.1 Far-field pattern

The geometry for a magnetic - dipole source is the same as that in figure

4 except that the electric-dipole source is replaced by a z-directed

magnetic-dipole source of moment IA. Here we will compute the magnetic

field because we anticipate reception with a magnetic dipole in borehole-toborehole transmission. The coherent far field has only a 6 component of

magnetic field H_ given by [47]

U v

IA sin#

4?rr

exp ( ik

Q r )

.

The coherent intensity I is

c

I c

(k^IA sin0)^

(47rr ) ^

(38)

(39)

16
If we again use the Born approximation and the Green's function

approach [40]

,

we can derive the magnetic field analog to (13) for the

incoherent field H^:

-iwe

Hpr) - —— f c^r') W x E

Q

(r') dr' , (40)

where V* = expCik^lr - r'|)/|r - r'|. As in the previous section, we

approximate B^ by (24) and carry out the delta function integration. In

addition we make the far-field approximation and obtain the magnetic - field

analog to (28)

:

3 3 2

9

coe k i <e,> a a

9 <|H (r)|> = f \6 • [r x E

0

(r')] | dr

4

(W) V

l

In the far field, <|H^ (r)| > ~ 0.

For a magnetic dipole source, E^(r') is [47]

(41)

E

Q

(r') = 4>

w/i

0

k

Q

IA exp(ik

Q

r')

47rr

(1 + s in0 '

. (42)

If we substitute (42) into (41)

,

we can carry out the volume integration

analytically. The details are included in Appendix B, and the result is

<|H

19 | 2

> = k^i 3

H

2

C

e

, (43)

4> 4

>

where G

g - [kQ

(c - b) + - c"

1

)

}

/(12tt) ,

G, = cosh [k A (c - b) + kh(b 2

- c

(p DU

17
and = (w6Qk^IA)/(47rr) . The incoherent intensity I is

h

- <i h

i 4

2> + <iy 2> - k

o i3 <£ h

H

o y + y• < 44)

The ratio of the incoherent to coherent intensities is also of interest

because it gives a measure of the degradation of the transmitted signal

between a pair of magnetic dipoles:

I. /I = kV <e*> (G. + GJ/sinV (45)

i c 0 10 (f>

'

The main difference between the magnetic and electric dipole results is

-3 -3

that the magnetic dipole result in (43) contains no quasi-static b

term. This is so because the primary electric field of the magnetic dipole

/ -3

in (42) contains no quasi-static r' term. Thus the near-field scattering

from random irregularities near the source is much weaker for a magnetic

dipole source, and this is a potential advantage for magnetic dipole (small

loop) antennas.

Numerical results for the incoherent intensity are shown in figure 7.

The maximum scattering occurs off the end of the magnetic dipole (0 = 0°

,

90°). Again we see that there is a cross -polar ized <f> component.

5.2 Total Radiated Power

The total radiated power can be obtained by integrating the intensity

over a sphere at infinity. The coherent power is

P =

0 2

ij 2n n — | J I r sin0 d9 d<f> 0 0

° (46)

If we substitute (39) into (46) and carry out the integrations, we obtain

18
P (47)

Q - (k^IA)

2

/( 127r) .

The incoherent power P^ is

r? 2 tt 7T P

t - f- f f I. r sin# d# &4>. (48)

0 0

If we substitute (43) and (44) into (48) and carry out the integrations, we

obtain

n 2 TAN 2 ,3„3 2

r? (k IA) k I <e

i

>

-1 _i _l P

x - -y— ^ — [k

0

(c - b) + k

0

L

(b - C

1

)].

72 7T (49)

The ratio of incoherent to coherent power can be regarded as a clutterto-signal ratio and is given by

2 3 2

P k i <e

i

>

-i -i _i = ^ - b > +k

0

(b - ^ )] (50)

The power ratio in (50) agrees with the first two terms of the electric

dipole result in (37) . The difference is that the electric dipole result in

-3 -3

(37) contains an additional quasi-static term proportional to k^ (b

-3

c ) . Since the magnetic dipole result in (48) does not contain this term,

a magnetic dipole does not scatter as much power when near-field scattering

is dominant (small k^b) . Numerical results for the power ratios are shown

in figure 8 for both a vertical magnetic dipole (VMD) and a vertical

electric dipole (VED) . For large k^b , the results are the same for the two

dipole types. For small k^b , the incoherent power rises rapidly for the VED

because of the additional quasi-static term in (37).

If we write the two power ratios for the small k^b limit, we obtain

19
(VMD)

,

2 3 2

p

i V <£i>

0

67T

2

19 <e^>

30 7T (VED)

.

( 51 )

It is clear from (51) that the incoherent power increases much faster (b

dependence) for decreasing b for the VED than for the VMD. Also, the power

ratio is independent of frequency for the VED, but is proportional to

frequency squared for the VMD. Both of these comparisons indicate a

possible advantage for the VMD when the source is located within an

inhomogeneous region. (By reciprocity the same conclusions apply when the

receiving antenna is located within an inhomogeneous region.) It is

interesting that similar conclusions hold for near-field conduction losses.

Input impedance calculations for a VMD within a spherical [51] or

cylindrical [52] cavity in a conducting medium show the same b

^ dependence

as (51)

.

Input impedance calculations for a VED close to a conducting half

-3

space [53] show the same b dependence as (49). The frequency dependences

for dipoles in a lossy medium are somewhat different from those in (51)

,

but

even there the VED shows more loss at low frequencies [52].

6. TRANSMISSION BETWEEN VERTICAL DIPOLES

If the transmitting and receiving antennas are located in an infinite

random medium, then the integrals arising from the Born approximation are

difficult to evaluate and might not even converge. However, by neglecting

backscatter and by making far-field approximations, Ishimaru [25, Ch. 6] has

developed some useful approximations. In this section we specialize his

results to two vertical dipoles in a random continuum. In section 6.1 we

consider cw transmission, and in section 6.2 we consider pulse transmission.

6.1 CW Transmission

20
The geometry for transmission between vertical dipoles is shown in

figure 9. Ishimaru writes the ratio of the received coherent power to

the transmitted power P^_ as

P

c P

t

.2 GG

A t r -7

2 2

6

(4tt) L

(52)

where G and G^_ are the gains of the transmitting and receiving antennas,

and e represents attenuation of the medium due to scattering. For a lossy

medium, e would also include absorption loss. For the geometry in figure

9, we have L = d/cosa and G = G^ = (3/2)cosa. The results in this section

apply to both electric and magnetic dipoles because both have the same farfield pattern, and we can rewrite (52) as

P

c P

t

2 2 4

A (3/2) cos o -7

2 2 e

(4tt) d

(53)

An approximate ratio of the incoherent power P^ to the coherent power

P^ is given by Ishimaru:

P./P

c = 7 - poh, (54)

where the total scattering cross section per unit volume per

is given by the

integral of the bistatic cross section over 47r steradians [25]

A A

PO = f °(°’ *) dQ > 47T

(55)

and o is given by (18)

.

For the exponential model of given in (19)

,

o is

given by (20). In general the integration in (55) is- difficult to carry

21
out, but approximate results for the limits of small kn a and large kn a are

u u given in Appendix C. If these results are substituted into (54), the

following results are obtained for the ratio of the incoherent to coherent

power

:

p

(4/3)<ep>(k

0

a)

3

k

Q

L, k

Q

a « 1

^ » ( (56)

C (l/2)< £ 3>(k

0 a)k0

L, k

Q

a » 1.

The approximate results in (56) also agree with Ishimaru's plane-wave

results, and for this weak scattering case the power ratio is proportional

to the path length. As usual the power ratio is also proportional to the

2 variance <e^>.

6.2 Pulse Transmission

Pulse transmission is also of interest in tunnel detection, and arrival

time and pulse dispersion are useful quantities in tomographic imaging [7].

Ishimaru [25, Sec. 6.5] has obtained an approximate result for pulse

propagation between a pair of antennas, and we apply his result to the

geometry in figure 9.

The transmitted pulse has energy and varies with time as the delta

function <$(t). The received coherent intensity I (t) has the same time

variation with a time delay:

h

(t) - P

c E0 5(t - (57)

where c^ is the velocity in the background homogeneous medium and P^ is

given by (53)

.

The received incoherent intensity I^(t) is obtained approximately by a

saddle point evaluation [25, Sec. 6.5]:

22
I.( t ) - 2(,O (58)

1/2

P

c E0Vas c0 exp [ -

(

c q/L)

(

8a^ - pa

^ L)(t L/c 0

)]

The result in (58) applies for t > L/c^, and I^(t) = 0 for t < L/c^. Lor

large particle size D, a and the exponential time constant r are given by

a

P

ttD 2

( 2A ) and T L/(8c

0

a ) 2 2 2

LA /(2tt c D ) . (59)

The total incoherent energy E. in the received pulse is given by

E

i = f I

i

(t) dt « P

c E0 pas L (jt

1/2

/4). (60)

The ratio of the incoherent to coherent energy in the received pulse is

E./E

c = (tt

1/2

/4) pa

s L. (61)

This result is the same as the cw result in (54) except for a constant

factor /4) . For small particle sizes (D « A) , and the exponential time constant

are given by

a »

' 2,77 « 1.12 and r « L/(8c a ) * L/(9c ) . (62)

P (n/2)

Z U P U

In obtaining (62) from Ishimaru's result, we have made the assumption that

the beamwidth for small particle scattering is tt/ 2 . This result for the

decay time constant is quite different from the result in (59)

,

but in both

cases the length of the incoherent pulse is proportional to the path length

L. Thus for either large or small particles, the relative size of the

incoherent pulse grows in both amplitude and length as the path length L

23
increases. This is in agreement with the cw description of the incoherent

wave in Ishimaru [25, Ch . 6],

7. CONCLUSIONS AND RECOMMENDATIONS

This report represents a first effort at modelling clutter in

subsurface electromagnetic applications, particularly for tunnel detection.

While deterministic earth models [22] can be used for modelling clutter, it

appears that random media models [25] are more versatile and require less

information about the inhomogeneous earth. The Born approximation is the

simplest approach to computing the incoherent field that is generated in a random medium, but even with the Born approximation the calculations are

difficult for real antenna applications where the fields are not plane

waves. The main difficulty arises in evaluating the volume integrals of the

type discussed in Section 3.2.

However, useful results can be obtained when only the transmitting

antenna is located close to the random medium, and specific results for

electric and magnetic dipoles are derived in Sections 4 and 5. The pattern

of the incoherent intensity radiated by either type of dipole is different

from that of the coherent intensity. A practical result is that electric

dipoles are more strongly affected by near-field inhomogeneities than are

magnetic dipoles. Similar results have been previously obtained for the

effects of near-field conduction losses for electric and magnetic dipole

sources [ 52 ]

.

There are many unsolved problems in clutter analysis. The isotropic

correlation functions that we have used are not applicable to flat or

fracture irregularities, and fractures are thought to be the most important

scatterers in tunnel detection applications [54], It remains to be seen

whether an anisotropic correlation function or a totally new approach is

required to model fractures. Even wl n clutter is adequately modelled, the

question of how to image or detect tunnels in the presence of clutter

remains. The general problem of imaging in a random medium has been studied

[55], but the specific application of tunnel detection might require special

techniques to enhance the tunnel signal and to reduce the clutter effects.

24
It seems likely that a combination of further analysis coupled with EM

measurements at a well characterized site will yield the most rapid progress

on the overall tunnel detection problem.

8 . ACKNOWLEDGMENT

This research was supported by the U.S. Army Belvoir RD&E Center.

9.

[ 1 ]

[ 2 ]

[3]

[ 4 ]

[5]

[ 6 ]

[7]

[ 8 ]

[9]

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Kaufman, A. A.

;

Keller, G.V. The Magnetotelluric Sounding Method.

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Kaufman, A. A.

;

Keller, G.V. Frequency and Transient Soundings.

Amsterdam: Elsevier Scientific Pub. Co.; 1983.

Murphy, J.N.; Parkinson, H.E. Underground mine communications. Proc

.

IEEE, 66: 26-50; 1978.

Delogne, P. Leaky Feeder and Subsurface Radio Communications.

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Lytle, R.J.; Laine, E.F.; Lager, D.L.; Davis, D.T. Cross -borehole

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44: 1667-1676; 1979.

Olhoeft, G.R.

Third Technical

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Interpretation of hole-to-hole radar measurements.

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Chan, L.C.; Moffatt, D.L.; Peters, L. A characterization of subsurface

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[10] Hill, D.A. Electromagnetic scattering by buried objects of low

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[12] Wait, J.R. Electromagnetic Waves in Stratified Media. New York:

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[25]

[26]

Wait, J.R.; Hill, D.A. Fields of a horizontal loop of arbitrary shape

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Hill, D.A. Radio propagation in a coal seam and the inverse problem.

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Fry, R. ; Schotsch, J.; Stolarczyk, L.G. The radio imaging method (RIM)

a means of detecting and imaging anomalous geologic structures in a

coal seam. 4th Conference on ground control in mining. Morgantown,

WV: July 22-24, 1985.

Hill, D.A. Effect of a thin conducting sheet on the fields of a buried

magnetic dipole. Electromagnetics, 7: 71-79; 1987.

Schaber, G.G.; McCauley, J.F.; Breed, C.S.; Olhoeft, G.R. Shuttle

imaging radar: physical controls on signal penetration and subsurface

scattering in the eastern Sahara. IEEE Trans. Geosci. Rem. Sens., GE24: 603-623; 1986.

Lager, D.L.; Lytle, R.J. Determining a subsurface electromagnetic

profile from high-frequency measurements by applying reconstructiontechnique algorithms. Rad. Sci., 12: 249-260; 1977.

Radcliff, R.D.; Balanis, C.A. Reconstruction algorithms for

geophysical applications in noisy environments. Proc. IEEE, 67: 1060-

1064; 1979.

Lee, T.K.; Kim, S.Y.; Ra , J.W. Resonant scattering of CW

electromagnetic wave by an underground tunnel of circular cross

section. Third Technical Symposium on Tunnel Detection. Golden, CO:

Jan. 12-15, 1988.

Shope , S.; Greenfield, R. Electromagnetic cross-hole tomography for

tunnel detection. Third Technical Symposium on Tunnel Detection.

Golden, CO: Jan. 12-15, 1988.

Eaton, P.A.; Hohmann, G.W. An evaluation of electromagnetic methods in

the presence of geologic noise. Geophys . , 52; 1106-1126, 1987.

Hill, D.A.

;

Wait, J.R. The electromagnetic response of a buried sphere

for buried- dipole excitation. Rad. Sci., 8: 813-818, 1973.

Hill, D.A.; Wait, J.R. Perturbation of magnetic dipole fields by a perfectly conducting prolate spheroid. Rad. Sci., 9; 71-73, 1974.

Ishimaru, A. Wave Propagation and Scattering in Random Media. New

York: Academic Press; 1978.

Battan, L.J. Radar Obsvervation of the Atmosphere. Chicago: Univ. of

Chicago Press; 1973.

26
[27] Tatarskii, V.I. The Effects of the Turbulent Atmosphere on Wave

Propagation. Jerusalem: Israel Program for Scientific Translations;

1971.

[28] Davenport, Jr., W.B.; Root, W.L. Random Signals and Noise. New York:

McGraw-Hill; 1958.

[29] Besieris, I.M. On the applicability of continuous scattering methods

to discrete scattering problems. AGARD Conference Proceedings No. 419,

Scattering and Propagation in Random Media. Rome, Italy: May 18-22,

1988.

[30] Fung, A.K.; Fung, H.S. Application of first-order reformalization

method to scattering from a vegetation- like half -space. IEEE Trans.

Geosci. Electron., GE-15: 189-195; 1977.

[31] Tsang, L. ; Kong, J.A. Wave theory for microwave remote sensing of a half-space random medium with three-dimensional variations. Rad. Sci.,

14: 359-369; 1979.

[32] Karal , Jr., F.C.; Keller, J.B. Elastic, electromagnetic, and other

waves in a random medium. J. Math. Phys . , 5: 537-547; 1964.

[33] Keller, J.B.; Karal, Jr., F.C. Effective dielectric constant,

permeablility , and conductivity of a random medium and the velocity and

attenuation coefficient of coherent waves. J. Math. Phys., 7: 661-670;

1966.

[34] Roth, L.E.; Elachi, C. Coherent electromagnetic losses by scattering

from volume irregularities. IEEE Trans. Ant. Propagat., AP-23: 674-

675; 1975.

[35] Frisch, V. Wave propagation in random media. Probabilistic Methods in

Applied Mathematics, Vol. 1 (A.T. Barucha-Reid, ed.). New York:

Academic Press; 1968.

[36] Rosenbaun, S. The mean Green's function: a non-linear approximation.

Rad. Sci., 6: 379-386; 1971.

[37] Dence , D.; Spence, J.E. Wave propagation in random anisotropic media.

Probabilistic Methods in Applied Mathematics, Vol. 3 (A.T. BaruchaReid, ed.). New York: Academic Press; 1973.

[38] Tsang, L. ; Kong, J.A. Microwave remote sensing of a two-layer random

medium. IEEE Trans. Ant. Propagat., AP-24: 283-288; 1976.

[39] Lee, J.K.; Mudaliar, S. Backscattering coefficients of a half-space

anisotropic random medium by the multiple - scattering theory. Rad.

Sci. , 23: 429-422; 1988.

[40] Tai, C.T. Dyadic Green's Functions in Electromagnetic Theory.

Scranton, PA: Intext Ed. Pub.; 1971.

27
[41] Van Bladel, J. Some remarks on Green's dyadic for infinite space. IRE

Trans. Ant. Propagat., AP-9: 563-566; 1961.

[42] Yaghjian, A.D. Electric dyadic Green's functions in the source region.

Proc. IEEE, 68: 248-263; 1980.

[43] Varvatsis, A.D.; Sancer, M.I. On the renormalization method in random

wave propagation. Rad. Sci., 6: 87-97; 1971.

[44] Stogryn, A. Electromagnetic scattering by random dielectric constant

fluctuations in a bounded medium. Rad. Sci., 9: 509-518; 1974.

[45] Booker, H.G.; Gordon, W.E. A theory of radio scattering in the

troposphere. Proc. IRE, 38: 401-412; 1950.

[46] Dorfman, Y.M. Electric dipole radiation in a randomly nonuniform

medium. Rad. Engr . Electron. Phys .

,

15: 1105-1106; 1970.

[47] Harrington, R.F. Time-Harmonic Electromagnetic Fields. New York:

McGraw-Hill; 1961.

[48] Dokuchaev, V.P.; Ryzhov, Y.A.

;

Tamoikin, V.V. Radiation of an

elementary dipole in a randomly inhomogeneous medium. Radiophysics and

Quantum Electronics, 12: 1182-1188; 1969.

[49] Ulaby, F.T.; Moore, R.K.; Fung, A.K. Microwave Remote Sensing, Vol.

III. Norwood, MA: Artech House; 1986. Sec. 13-3.

[50] Hill, D.A. Magnetic dipole excitation of a long conductor in a lossy

medium. IEEE Trans. Geosci. Rem. Sens., 26: to be published; 1988.

[51] Wait, J.R.; Spies, K.P. A note on the insulated loop antenna immersed

in a conducting medium. Rad. Sci. J. Res. NBS , 68D: 1249-1250; 1964.

[52] Hill, D.A.; Wait, J.R. The impedance of dipoles in a circular tunnel

with an axial conductor. IEEE Trans. Geosci. Electron., GE-16: 118-

126; 1978.

[53] Wait, J.R. Impedance characteristics of electric dipoles over a

conducting half-space. Rad. Sci., 4: 971-975; 1969.

[54] Olhoeft, G.R. U.S. Geological Survey, Denver, CO 80225-0046. Private

communication, 1988.

[55] Ishimaru, A. Imaging in random media. Proc. URSI International

Symposium on Electromagnetic Theory: 59-62. Budapest, Hungary; 1986.

28
APPENDIX A - VOLUME INTEGRATION FOR ELECTRIC DIPOLE SOURCE

The unit vectors in (28) can be written in terms of the Cartesian unit

vectors

:

A A A A

# = x cos0cos# + y sin0cos# - z sin#

,

(Al)

A A A

0 = -x sin0 + y cos 0.

We can write the primed unit vectors in (29) in a similar manner:

A A A A

#' = x cos0'cos#' + y sin0'cos#' - z sin#',

(A2)

A A A A

r' = x cos0'sin#' + y sin0'sin#' + z cos#'.

Using (Al) and (A2)

,

we can evaluate the dot products required in (28)

:

A A

#•#' = cos (0-0 ' ) cos#cos# ' + sin#sin#',

A A

#*r' = cos (0-0 ' )cos#sin# ' - sin#sin#',

(A3)

A A

0«#' = sin(0 ' -0) cos# '

,

A A

0*r ' = sin(0 ' -0)

.

The volume integral in (28) can be written

A r\ C 2 7T 7T A

0 0

/ 1# • E (r')| dr' = f f f |# • E (r')| r' sin#' d#'d0'dr'. (A4)

V ^ b 0 0

" U

0 0

The r' integration involves only powers of r' and can be done analytically.

The 0' and #' integrations involve only trigonometric functions, and they

29
can also be done analytically. When we substitute (Al)-(A3) and (29) into

(A4) and carry out the r' , <f>

' , and 9' integrations, we obtain the final

result in (30)

.

APPENDIX B - VOLUME INTEGRATION FOR MAGNETIC DIPOLE SOURCE

The dot products involving the unit vectors in (41) can be written

#• (rx<£' ) = <y • (#xr) = -<f>' (Bl)

AAA A AA AA

<f>»(rx4>') = r) = 4>'»9.

The dot products in (Bl) are evaluated in the same manner as in Appendix A,

and the results are

-<f>

' »<f> = -cos (<£-<£') , (B2)

4>'*9 = sin(<f> -<f> ' ) cos# .

The volume integral in (41) takes the same general form as that in (A4) , and

again the r ' , <f>' , and 6' integrations can be done analytically. When we

substitute (Bl) , (B2) , and (42) into (41) and carry out the integrations, we

obtain the final result in (41)

.

APPENDIX C - ANGULAR INTEGRATION FOR SCATTERING CROSS SECTION

The first step in the evaluation of (55) is to write the solid angle

integration in spherical coordinates:

2tt tt pa = f f a sin# d# d<f>

,

(Cl)

S

0 0

30
where o is given by (20) and the geometry is shown in figure 3. The sin x

factor can be written in terms of and 9 :

sin^x = 1 - cos^ sin^0. (C2)

For small values of k^a, we can approximate (Cl) in the following form:

P°

.

. 4 3 2 _ k a <€-,> 2 7T 7T J f (1 - cos <f> sin 6) sin0 d9 d<f>

.

2n

0 0

(C3)

The 9 and <j> integrations in (C3) can be done in closed form to yield the

desired final result for small k^a:

4 3 2

pa * (4/3) ktTa <e7>.

s 0 1

(04)

2 When k^a is large, we make small-angle approximations for sin 9 and

2 sin (9/2), and (Cl) can be approximated

P°.

i 4 3 2 0 kn a <e-> 2n A

0 j; 9 d 9 d6

2tt 2 2 2 2

0 0 (1 + k^a 9 )

(C5)

where A is chosen large enough to include the significant portion of the

integrand, but small enough for the small 9 approximations to be valid. The

<f> integration in (C5) contributes a 2n factor. The 9 integration is

evaluated by substitution:

A

;

9 d 9 A ,, .2 2 . 2.2

0 (1 + k^a 9 )

A , p dx

J 2 2 2

0 2(1 + k^a x)

2 2

2k

0

a

(1 - 1 + v

2 2

a 2 k^a A

) (06)

31
We can neglect the second term in the parentheses, and the desired final

result for large k^a is

P°

.

2 2

k

0

a <e i>/ 2 -

32
Clutter

O 0

Figure 1. Typical geometry for tunnel detection in the presence of clutter.

33
Random

Receiver

Figure 2. Geometry for scattering by an elemental volume dV in a random

medium

.

34
A

Figure 3. Geometry for the scattering angle 6 and the polarization angle \

•

A A The unit vectors i and o indicate the propagation directions of

A the incident and scattered waves, and the unit vector

indicates the polarization of the incident wave.

35
Figure 4. An electric dipole source at the center of a random medium with

spherical boundaries.

36
G

Figure 5 . Radiation pattern of the incoherent scattered field for an

electric dipole source.

37
250

200

<[Eil

2

>

CE0

2

150 -

100

50

0

k 0 c

k 0 b

10

0.1

\

\

[— \

\ Component

\

\

9 Component

0° 45° 90° 135° 180°

e

Figure 6. Radiation pattern of the incoherent field of an electric dipole

for a smaller value of k^b

.

38
e

Figure 7. Radiation pattern of the incoherent field of a magnetic

source

.

dipole

39
0.1 1.0 10.0

k 0 b

Figure 8. Ratio of radiated incoherent to coherent power for electric and

magnetic dipole sources.

40
Figure 9

Random

Medium

Geometry for dipole- to-dipole transmission in a random medium.

41

U.S. DEPT. OF COMM. 1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

BIBLIOGRAPHIC DATA

SHEET (See instructions)

NISTIR 89-3909 February 1989

4. TITLE AND SUBTITLE

Clutter Models for Subsurface Electromagnetic Applications

5. AUTHOR(S)

David A. Hill

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7 . Contract/Grant No.

national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

U.S. Army Belvoir RD&E Center

Ft. Belvoir, VA 22060

10- SUPPLEMENTARY NOTES

2 Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABST RACT (A 200- word or less factual summary o f most si gnifi cant information. If document includes a significant

bibliography or literature survey, mention it here)

Clutter models for subsurface electromagnetic applications are discussed with

emphasis on tunnel detection applications. Random medium models are more versatile

and require less detailed information than deterministic models. The Born

approximation is used to derive expressions for the incoherent field, and electric

and magnetic dipoles are treated in detail. When random inhomogeneities are

located in the near field of the dipole source, an electric dipole radiates a

larger incoherent field than a magnetic dipole because of its larger reactive

electric field.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) clutter; coherent field; electric dipole; incoherent field; magnetic dipole;

random medium

13. AVAILABILITY

[TCj Unlimited

For Official Distribution. Do Not Release to NTIS

T Order From Superintendent of Documents, U.S. Government

20402.

[~Xl Order From National Technical Information Service (NTIS),

14. NO. OF

PRINTED PAGES

Printing Office, Washington, D.C.

48

15. Price

Springfield, VA. 22161

USCOMM-DC 6043-P80

*- 0 S GOVERNMENT PRINTING OFFICE: 1989 -674-724/ 5048
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