PILE CC

DO NOT re;

NBSIR 76-851 ~ '

ELECTROMAGNETIC REMOTE SENSING OF

INHOMOGENEQUS MEDIA

Wolfgang A. Bereuter

David C. Chang

University of Colorado

Electromagnetics Division

Institute for Basic standards

National Bureau of Standards

Boulder, Colorado 80302

January 1977

IMBSIR 76-851

ELECTROMAGNETIC REMOTE SENSING OF

INHOMOGENEOUS MEDIA

Wolfgang A. Bereuter

David C. Chang

University of Colorado

Electromagnetics Division

Institute for Basic standards

National Bureau of Standards

Boulder, Colorado 80302

January 1977

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Edward 0. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

CONTENTS

1. INTRODUCTION 1

2. THE REFLECTION OF WAVES BY INHOMOGENEOUS MEDIA 2

3. TRANSFORMATION OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION INTO THE WAVE

EQUATION 3

k. EXPRESSIONS FOR THE SURFACE ADMITTANCE AND THE REFLECTION COEFFICIENT 7

5. NUMERICAL RESULTS AND DISCUSSION 9

ACKNOWLEDGMENTS 11

REFERENCES 11

APPENDIX A — ASYMPTOTIC SOLUTION OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION 12

iii

ELECTROMAGNETIC REMOTE SENSING OF INHOMOGENEOUS MEDIA

Wolfgang A. Bereuter and David C. Chang

This report deals with the electromagnetic response of inhomogeneous

dielectrics; i.e., media whose permittivity is a function of depth. The re- sulting boundary value problem is solved for a large number of permittivity

functions which can model almost any medium of interest.

Since those permittivity profiles are characterized by only a few parameters, they are particularly useful for the inverse problem; i.e., the retrieval

of profiles from the measured electromagnetic response.

It is shown how the non-uniformity of the permittivity changes the response

and how the change is related to the profile characteristics.

Key words: Inhomogeneous dielectrics; profile inversion.

1 . INTRODUCTION

With the advent of more accurate electromagnetic measurement techniques , especially

over large frequency bands, the interest in non-destructive testing methods has considerably

increased. A host of literature is available on theory [1,2]* as well as on applications

[3,^,5] which range from atmospheric sounding to geologic explorations.

The problem of obtaining information about an object using electromagnetic means is one in the theory of inverse scattering. Unlike a conventional radar which determines the

position and shape of a conducting object from the return signal, the kind of remote probing

technique we are interested in concerns the reconstruction of the electrical profile of

vertically-stratified dielectrics from information obtained by illuminating them with

electromagnetic waves

.

Once the scatter-field is successfully "inverted," the electrical properties of the

medium which caused it become known. More specifically, we are interested in the determination of the moisture profile of grain as stored in bins , trucks , etc . Since the

moisture changes the grain's effective permittivity and conductivity, we therefore attempt

to measure the electrical properties of grain using microwave signals. For the particular

application we have in mind, the pile of grain is generally assumed to be several meters on each side, and the operating frequency is typically in the L- and S-band. This means that

the unknown medium is basically a dielectric material at these frequencies, with a loss

tangent in the neighborhood of 0.1. Hence, the remote-sensing method for this problem is

substantially different from the magneto-telluric techniques where only the conductivity

profile is considered [h]. Indeed, in this report we have only treated the case of

dissipationless dielectrics. The extension to lossy dielectrics for which the ratio of the

conductivity profile to the permittivity profile is independent of depth will be dealt with

in a later report

.

*Figures in brackets indicate the literature reference at the end of this paper.
A number of theories dealing with this kind of remote measurement have been developed.

For example : ~

1. the multilayer approach [6] in which one views the medium as a system of uniform

transmission lines, connected in series;

2. the Parameter Optimization Method [5] where one approximates permittivity profiles by polynomials whose coefficients are obtained from an optimization

process involving the numerical integration of the wave-equation ; and •

3- the Synthesis approach [7] which is based on the assumption that the input

admittance is a rational function of frequency.

For details on these and other methods see the collection [l] which also contains an extensive bibliography on the subject. The implementation of these theories, however,

usually requires extensive computer facilities. Yet, in many remote sensing problems it is

desired to process all measurement data in situ in order to obtain the required information

immediately — thus imposing severe limitations on the available machinery. Our objective

is then to develop an optimized microwave sensing method which can describe the profile of

an unknown medium with sufficient accuracy without employing any elaborate numerical computation schemes. In this report we consider mainly the "forward" problem of determining the

reflection coefficient and the surface admittance of a plane wave as a result of electromagnetic interference with a vertically stratified dielectric. A new exact solution for a large class of inhomogeneous media is presented and compared with previously obtained

results. While we shall discuss a conceptually attractive approach to the remote-sensing

problem, the actual inversion based upon this analysis will be given in our next report.

2. THE REFLECTION OF WAVES BY INHOMOGENEOUS MEDIA

Generally speaking, the determination of the reflection of plane waves by media whose

permittivity is a known function of depth is directly related to the problem of finding a solution to the general linear second order differential equation with two boundary values

which incidentally is also encountered in the theory of non-uniform transmission lines [8],

quantum mechanics [9]? and other disciplines of physics.

In the special case of a normally incident linearly polarized plane wave, the electric

field inside the medium must satisfy the well-known wave equation in its time-harmonic form

[3]:

+ k

2

e (x) dx z r E = 0, 0 < x < 1, (l)

y

where x is the normalized depth, ^(x) is the relative permittivity of the medium, k = k

QD (with D being the total depth) is the normalized wave number, and k = wAi e is the free-

• o o o space propagation constant. The time dependence factor exp(jwt) is suppressed; co being the

operating angular frequency. The geometry of the problem is depicted in figure 1. A number of exact solutions have been derived when e (x) is of a special form. For example,

2
when the permittivity obeys an

inverse square law: £r

(x)

exponential law: £r

( x ) = (a2 -b 2 )e ^ + b

2

, (3)

or a -a power law: er

( x ) = a (l + • (M

where a, b, and a are constants, the resulting equation (l) may be transformed into Bessel's

differential equation. Similarly, if the permittivity varies linearly, (l) transforms into

Stokes' differential equation. A detailed discussion of these results may be found in

[6]. We note, however, that these profiles are monotonic with depth, hence are not useful

to model for instance a medium containing a denser slab between to less dense layers.

Epstein and others [10,11] have transformed (l) into the hypergeometric differential

equation and obtained field solutions for a much larger class of profiles; namely, the

so-called Epstein profiles

:

e (x) = a + —

f [(e_-e 1 )(e 5+l) + eJ ; 5 = kx (5) r (eVL) 21 3

where a, e^, e^, and are adjustable parameters. However, a careful examination shows

that except in trivial cases these parameters are not independent of frequency. Consequently, they are not suitable to model inhomogeneous media in any remote-sensing method

which is based upon frequency scanning. Therefore, it is the purpose of this report to

develop exact solutions of the wave-equation within the framework of Epstein's, but with

profiles characterized by genuine constants; i.e., do not vary with the operating frequency.

, x+x (2)

3. TRANSFORMATION OF THE HYPERGEOMETRIC DIFFERENTIAL

EQUATION INTO THE WAVE EQUATION

Starting from the hypergeometric differencial equation (henceforth abbreviated as hg.

DE)

9(1-6) f^r + {c-(a+b+l)9} §§ - abll = 0 , (6) d9^ d6

one obtains upon transforming the independent variable 8 = f (x) , where f is any meromorphic

function, an alternative differential equation of the form

^| +AfL-BH = 0 (7) dx^ dx

3
with

;c-l)-(a+b)f , . f(l-f)

1 f f"

f ~f

1 and

B = ab

f(l-f)

where from now on " ' " denotes the derivative with respect to x.

(8)

Now let II = EG, where E will play the role of the electric field E in our 'model and G

is a function of x yet to be determined. Then (7) transforms into

E" + (2 |p + A)E ' + {f- + f- A - B)E = 0 . (9)

Forcing the linear term to vanish, one obtains the equation

G (c-l)-(a+b)ff(l-f)

f

ILL Hi

f f

'

which determines G(x) in terms of f(x) modulo an integration constant. Also,

(10)

G^ _ A£ AJ_ G h ~ 2

; A

f(l - f)f

f

;

( -_^ (l - 2f) {(c-l)-(a+b)f}

f' 2 f(l-f) ; a+b) + (ii)

If now one defines

rA£ A/

1+ +

2 (12)

or equivalently

,

-1+6 = A

2 +2A'

~(c-l)-(a+b)f

f

,1 2 + 2 (c-l)-(a+b)f

f

, f(l-f) f(l-f)

'f

'

r f f"' f

+ 2 f(l-f)f^-J.2 (l .2f)

{(c _l) _ (a+b)f} _

- 2(a+b) + 2 f f

'

f(l-f

)

Then the function E(x) = H( 6(x) ) /G(x) satisfies the differential equation

(13)

dx<

+ A(x) E = 0 (11+)

with

A(x) = 6(x) - B(x)

.

(15)

k
Now (lU) must be identified with the wave-equation (l). When the inhomogeneous medium

is lossless, e is only a function of depth, but not of frequency. On the other hand, k

Q

is proportional to frequency, and since {lh) is intended to he used for many different

frequencies, the following conditions on A(x) must be imposed:

(I)

(II)

A(x) must have a factor which can be identified with k*

must be a function of x only

(III) A(x) > 0 for 0 < x < 1. It is easily seen that (i) and (II) are satisfied if and only if

c-1 = (k D)c, a = (k Dja. b = (k D)b

o o o

(16)

where D is the total depth of the medium, and c, a, b are independent of frequency and

therefore may be extracted from data at different frequencies containing them; i.e., may be

"sensed" from surface impedance data at frequencies f , f_, , ... f„.

o 1 N Furthermore, to satisfy (II), all terms not containing k

Q

must be eliminated by imposing

conditions on f (x) , which clearly must not contain k^. This can be achieved by setting the

terms in (13) which are linear in (k

QD) and the terms which are constant with respect to

k D to zero : i.e.,

o

2{(c-l)-(a+b)f}

f

2

( j;_ f) 2 tf ,2 (l-f) - f'f(l-f) + ff"(l-f) - f' 2 (l-2f)] -

f»f f

'

+ 2 f

" f'J (IT)

After some manipulations one arrives finally at the following condition for f(x)

1 f"l 2 f

i f'l ' 2 I* f

»

4 + f

* fj (18)

Since f(x) must be independent of k

Q

, (l8) is equivalent to the condition

(c-l)-(a+b) = 0 or c - a - b = C (19)

plus the condition

2 f f

, + f " f'J

0. (20)

Ln order to solve the differential equation (20) for f (x) , let g(x) = 'ZL £1] Then

(21)

So

/ \ _ ^_ _ f'(x) f"(x) gU; " x-r " f(x) " f'(x) (22)

5
where r is an integration constant. Integrating twice yields the most general transformations of the hg. DE" into the wave-equation; namely,

f(x) = q exp

x-r p,q,r (23!

where p,q,r are arbitrary constants. From (15) it then follows that the class of e - profiles this method can handle, henceforth called "permissible profiles," is defined by

the family

e r (x;a,b;p,q,r) = - f02

r a+bl 2 _ f

'

2 2 j f(l-f)

ab

ab q-exp(p/x-r

)

a+bi

2

\2k:

To illustrate the versatility of the class of permissible profiles £ (x) a few of its

members are shown in figure 2. The profiles depicted there are special cases of {2k);

namely,

a = -b

a^q

e (x) =

i4" r p

z exp(y)-q (four parameter family); y - —^- (25)

For the sake of simplicity we will limit ourselves in this report to profiles of the

form (25).

We further note that if they are rewritten as

e r(x) = y

4 (a/p) 2 6(l-6)

-1

(26!

where

= q exp(-y) , y

= :2T)

it follows that the range of the hypergeometric differential equation extends only from

6 = 0 to 1, if the relative permittivity is to remain positive.

The solution to the wave equation (l) with permittivity profiles (2k), therefore, is

given by

Ey

= n(a,b,c;6(x) )G

1

(x) (28)

where II is the solution to the hg. DE (6) and G is calculated from (10) and (23):

G(x) = . (29) x-r

6
k. EXPRESSIONS FOR THE SURFACE ADMITTANCE AND THE REFLECTION COEFFICIENT

With the geometry chosen as in figure 1, the transverse admittance is given by

H

Y(x) =

-1 dE

juu D E dx

o y

(30)

where D is the total depth of the medium, and the reflection coefficient at the air-dielectric

interface "becomes

R =

i - Y

c 1 + Y (31)

where Y is now the surface admittance evaluated at x = 0 and normalized to free-space

o admittance: i.e., Y = Y(0)ti with n - 120 tt ohms.

o o o „ Furthermore, we have shown that the profiles ohtained hy setting a = -b in equation

(2h) can model many media of interest; and for those profiles the hypergeometric function II satisfies a much simpler differential equation containing only one parameter; namely,

d^n an ed-e) (i-e) ^ + a^n = o, o < e < i. a = a(k D)

.

o (32)

The solution of this equation is given in terms of Weierstrass sums as [13]

= AFT? a

'9| + B(1-6)F l-a,l+a;

2; ~ (33)

where A,B are the two arbitrary constants to be determined from the boundary condition

at x = 1. For instance, one of the cases of interest is when the pile of grain is supported

at the bottom by a perfectly conducting plane. In that event, the boundary condition of a vanishing tangential electric field now yields the condition that

A/B = -(l-e)F l-a,l+a:T a ,— a ;

,

for x = 1. Substitution of (28) into (30) finally provides a formal expression for the transverse

admittance as follows:

Y(x) = (x-r)

-2

n e D

o o

v dn

: - (x-r) zs. : s (3h)

As mentioned earlier, the grain pile is typically several meters deep, while the operating

frequencies lie above several hundred megahertz. This means that a is usually in the order

of 10 or larger. Consequently, an approximate expression for the transverse admittance can be derived from the asymptotic evaluation of the hypergeometric functions. In Appendix A

ii - -2 we nave shown that as a •* 00 the asymptotic expansion of II, accurate to the order of a ,

:
is given as

n(e) = a h(e)exp -Jaa(8) - 1 6(0) a

+ A h( G)exp jaa(e) + f 3(e)

EL (35)

where

h(e) = e-lA(i-e) lA

,

a(6) = 2 arccos(e '

) ,

6(e) = - ±g e -1/2

(i-e) _1/2

(2e+i) ,

and A

+

, A are some linear combination of A and B in (33). It is of particular interest

to note that the first term in (35) actually represents a wave propagating in the forward

x-direction, while the second term represents a wave reflected from the boundary at x = 1. In particular, when a perfectly conducting plate is located at x = 1, the boundary condi-

+ tions require that A = -A , and the transverse admittance at x = 0 takes the following

form:

Y = a

° %

7- (B + K)cot (aa(6) + £ 6(e)).

o-' mhos (36)

where

A = -(.25 pr

-2 Ql/2

:i-e

0

) _1

+ r"

= pr :i-e

Q

) -1/2

0,1

= e(x=o,i) ,

and 6' is the derivative of 6 with respect to x. The constants in Y

q as written in (36), which is a function of frequency alone, relate

the profile characteristics quite explicitly to the measurement characteristics if one transforms 6 back to the normalized depth variable x:

n Ik D r

e 1/2

+ . 5 (kD)-VlA

( e -lA )" r o r r

ikoDL + 3Td f

i mhos (37)

where the profile-function e is assumed to be evaluated at x = 0, and L and are its

r 1 optical depth and average degree of concavity, respectively.- i.e.,

L =

J

1

e 1/2

(x)dx

0 r

-1/k, -1/k "

, e ( e ) dx

r r (38)

8
Expression (37) resembles the input admittance for a uniform medium terminated in a perfect

conductor; namely,

Y. = =2- e 1/2 cot(k De1/2

) mhos. (39) in n r or o In fact, this is the limit that Y

q attains when the profile tends to a constant. For uniform media are indeed limiting cases of the permissible profiles (25) and are obtained

as follows : Let a = r

2 and take

2 lim e (x) = ^ ^ = constant,

r 1 - q

Thus it appears that the electromagnetic response of an inhomogeneous medium depends on its

permittivity profile, specifically on the profile's initial value at the surface as well

1/2

as on its overall optical length, i.e., the integral of e (x) , and its concavity, i.e.,

«{<*>.

This demonstrates how the non-uniformity of the profile enters analytically into the

electromagnetic response. In the next section we will investigate the numerical aspect of

these conclusions and discuss some experimental implications.

5. NUMERICAL RESULTS AND DISCUSSION

In 1973 Kuester and Chang [12] developed a computer program to calculate the transverse impedance to analyze propagating modes in dielectric slab guides with arbitrary

permittivity profiles. Based on the invariant embedding principle, it integrates a first

order Riccati differential equation for the impedance numerically across the slab using a fourth-order Runge-Kutta method with error estimation. This program is not only highly

accurate, but also very versatile in that it can handle all possible profiles without

limitations on frequency. This approach, while possibly too complex for inversion purposes,

will be used as a guideline and standard in our investigations.

In table 1 we compare the normalized input admittance for a lossless inhomogeneous

medium as calculated by Kuester and Chang with the asymptotic expression (37). Specifically, the relative permittivity used for this table is profile (e) in figure 3. Since

the medium considered is lossless, the conductive part of the admittance is of course zero in both cases. As expected, the agreement between our asymptotic result and the numerically

"exact" result is excellent everywhere except close to the sharp resonances. Even then,

the frequency where any given resonance occurs, differs only very insignificantly in all our studies.

We note that due to the very nature of the asymptotic formulations of the admittance,

in particular the expansion (A-l) , one should expect increasingly better agreement between

the two results with increasing frequency. However, at higher frequencies the perturbation terms in (27) of order (k

QD) ^ decrease. Hence, a non-uniform medium responds more

and more like a uniform one with dielectric constant equal to the square of the optical

9
Length of the non-uniform medium. From the viewpoint of microwave remote-sensing this means that the constants in Y

q containing information about the profile, i.e., F^

,

e"^\e , e "^Ne , being dependent on (k D) ^ are more difficult to extract

r r r r o x- Q

from measurements made beyond a certain frequency. We also note that the constants in Y

q as expressed by (37) comprise all the information about the profile one can measure, no matter how many additional frequencies one is determined to employ. (All neglected terms of

Y are of order (k D) n for n > 3 which is expected to lie below the noise level of the

o o —

experiment.) Consequently, the maximum number of equations containing the unknown profile

parameters is five. This means that any profile inversion-technique using high-frequency

measurements is limited to profiles characterized by less than five parameters, especially

since at least two of the equations are non-linear.

Next, we will present and discuss the results for the input admittance for a number

of profiles. The permittivity profiles used are shown in figure h. First, we note that the optical length L can easily be determined from the average

separation between adjacent resonant frequencies. Since we have arbitrarily set the

total optical length equal to 1.^99 for all profiles considered in the following, the

separation in resonance frequencies is approximately the same; namely, about 100 kHz.

Now in order to distinguish between different profiles , we have tabulated the shift

of each resonance and anti-resonance frequency (for which the surface admittance is

infinite or zero, respectively, for a lossless dielectric) as well as the resonance and

ant i -resonance frequencies for a uniform medium with the same optical length.

The shifts of resonance and anti-resonance frequency as compared to those of a uniform medium (table 2) may now be interpreted from (37) as being due to the profile's

average concavity (F ) and other functionals of the relative permittivity profile e^(x).

From the above observations it now appears that the proper procedure for the remote

measurement of a permittivity profile could be divided into the following steps

:

(l) From swept frequency measurement, one can determine three consecutive resonances (i.e., Y

q

-> ») at f^, f^, and f_, say. Then the following three

equations in L and F^ are obtained.

k

i L + 2T7 F

i

= V W

where N is a positive integer and N. =N+i-l(i= 1,2,3), k. = 2irf.D(u e 1 1 l o o Solving for L and F yields

:

,1/2

F

1 = 2-^(^-2^+^ )/C where C = (k^k^^/kg) + ( k-j/k -k ) + ( k^-k^^ ) (1+1)

_

L = (2/k

2 - l/k

3 - 1/k^TT/C. (U2)

,2) Since the argument of the cotangent function in (37) is now known, linear equa-

-1/k -1/h " tions in £^(0) and (0)(e (0)) may be determined from the measured value

of Y

q at other non-resonant frequencies.

10
Consequently, all constants in Y

q are measurable within at least 5 frequencies. More

measurements will only yield consistency results, within the accuracy of the experiment.

In our next report, we will show that the profile-constants a, p, q, and r can be determined

from two independent equations containing the "measurable" quantities and a single unknown.

ACKNOWLEDGMENTS

The authors thank Mr. William E. Little, Chief of the Remote Measurement Section, for

initiating this research and continuously taking an active interest in it. We are also

indebted to Mr. Edward F. Kuester for many fruitful discussions and for the assistance in

adapting his computer program for our purposes

.

REFERENCES

[I] Collin, L. Ed., Mathematics of Profile Inversion, NASA Workshop Proceedings (1972).

[2] Mittra, R. , Computer Techniques for Electromagnetics (Pergamon Press, Elmsford, N.Y.

,

1973).

[3] Budden, K.G. , Radio Waves in the Ionosphere (Cambridge Univ. Press, New York, N.Y.

,

1961).

[k] Becher, W.D. and Sharpe, C.B. , A Synthesis Approach to Magnetotellurgic Exploration,

Radio Science, Vol. h, No. 11, 1089-109^ (1969).

[5] Mostafavi, M. and Mittra, R. , Remote Probing of Inhcmogeneous Media Using Parameter

Optimization Techniques, Radio Science, Vol. 7, No. 12, 1105-1111 (1972).

[6] Wait, J.R. , Electromagnetic Waves in Stratified Media (Pergamon Press, Elmsford, N.Y.

,

1970).

[7] Sharpe, C.B. , The Synthesis of Infinite Lines, Quart. Appl. Math., Vol. 21, No. 2,

105-120 (1963).

[b] Kuznetsow, P.I. and Stratonovich, R.L. , The Propagation of Electromagnetic Waves

in Multi-conductor Transmission Lines (Pergamon Press, Elmsford, N.Y. , I96U).

[9] Kemble, E.C., The Fundamental Principles of Quantum Mechanics (Dover Press, New York,

N.Y. , 1937).

[10] Epstein, P.S., Reflection of Waves in an Inhomogeneous Absorbing Medium, Proc. Natl.

Acad. Sci. (USA), Vol. l6, No. 10, 627-637 (1930).

[II] Rawer, K. , Elektrische Wellen in einem geschichteten Medium, Annalen der Physik,

Vol. 35, No. 5, 383-1+16 (1939).

[12] Kuester, E.F. and Chang, D.C., Modal and Coupling Characteristics of Inhomogeneous

Dielectric Slab Waveguides, Electromagnetics Laboratory, Univ. of Colo. (1973).

[13] Erdelyi, A., et al. , Higher Transcendental Functions, Vol. 1, p. 75 (McGraw Hill

Book Co., Inc., New York, N.Y., 1953).

Seifert, H. , Die hypergeometrischen Differentialgleichungen der Gasdynamik, Math.

Annalen, Vol. 120, 75-126 (19^7).

11
APPENDIX A

ASYMPTOTIC SOLUTION OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION

The asymptotic form of the hypergeometric function n with a large parameter may be

obtained as follows.

First, one assumes the general form

11(e) = A

+

h (9)exp[-jaa (6) - £ 6 (6)] + aV ( 6 ) exp [ Jaa ( 9) + £ 3 n (9)] ' (A-l)

o oao jl j_a_L

and then attempts to satisfy the differential equation (32), repeated below for ' con- venience, with the unknown functions.

e(i-e)n" + (i-e)ir + a

2 n = o . (32)

Equating terms containing like powers of the parameter a shows that ct Q

= ot^, $ Q = 3 , and

h

Q = h . We therefore may drop the subscripts and obtain a, 3, and h as the solution of the

following differential equations:

a' = e^d-e)"1

26a'h' + (6a"+a' )h = 0

26a'h3' = h

1 + 9h" . (A-2)

_2 The asymptotic form for II in the range 0 < 6 < 1 and accurate to the order a is then

given by (A-l), with

1/2

ot(0) = 2 arccos 9

3(9) = -

9"1/2 (1-9)'172

(29+1)

h(9) =

9"lA

(l-9)

lA

(A-3)

with the limitation given by the solution of the remaining differential equation, i.e.

-2

the a term in the expansion:

a"

2 (l+9)(l-e)"2

9 _1

/l6 « 1 . (A-U)

Consequently, the asymptotic form (A-l) need~not carry this correction term as long as the

condition {A-k) is satisfied.

To first order, the result (A-2) agrees with [ik] where it is confirmed by a steepest

descent integration.

12
Table 1. Comparison of the input susceptance as calculated by Kuester

and Chang (B ) with expression (37), (B )

.

GHz B

- GHz B

2

.-: -I89.682 -179.298 1.00 -448.563 -443.582

.4i -6.52I -6.523 1. 31 -6.586 -6.586

.42 -1.590 -1.597 1.02 -1.578 -1.579

A3 1-587 1.584 1.03 1.573 1.573

.hh 6.935 6.951 i.o4 6.T64 6.765

.45 -210.695 -201.193 1.05 -470.469 -465.666

.-6 -6.529 -6.530 1.06 -6.589 -6.589

hi — — • ~s - - -1. 593 1.07 -1.578 —j_ .piu

.48 1.583 1.581 1.08 1.573 1.573

.49 6.904 6.914 1.09 6.759 6.759

.50 -231.888 -223.137 1.1: -492. 4l4 -487-752

oi -6.537 -6.538 1.11 -6.592 -6.592

• 52 -1.586 -1.590 1.12 -1.578 -1.578

• 53 I.581 1.579 1.13 1.573 1.573

.51* 6.878 6.885 1.14 6.754 6.754

• 55 -253-222 -245.114 1.15 -514.360 -509.839

• 56 -6.545 -6.544 1.16 -6 . 595 -6.594

• 57 -1.585 -1.587 1.17 -1.577 -1.578

• 58 1-579 1.577 1.18 1.573 1.572

• 59 6.857 6.862 1.19 6.750 6.750

.60 -27^.667 -267.118 1.20 -536.334 -531.928
Table 2. Resonance and anti-resonance frequencies for a uniform medium

and profiles (i) - (j) in figure 4.

Anti-resonance Frequencies

(in MHz)

Profile

uniform 550.0 650.O 750.0 85O.O 950.0 1,050.0

(i) 551. ^ 651.2 751.0 850.8 950.7 1,050.7

(J) 550.5 650A 750.3 850.3 950.3 1.050.2

(k) 552.2 651.8 751.6 851.4 951.3 1.051.1

(1) 550.3 650.2 750.2 850.2 950.1 1,050.1

(m) 551-9 651.7 751.5 851.4 951.2 1.051.1

Resonance Frequencies

(in MHz)

Profile

uniform 500.0 600.0 700.0 800.0 900.0 1,000.0

(i) 502.1 601.8 701.5 801.3 901.2 1,001.1

(J) U99-1 599.3 699.4 799.5 899.5 999.6

(k) 501.2 601.0 700.8 800.8 900.7 1,000.6

(1) 499.9 600.0 700.0 800.0 900.0 1,000.0

(m) 497-5 597-9 698.2 798.4 898.6 998.8

14
H o—> E

r

=0 o o

-> y

y o'

£ = £ o

£ r(x)

Figure 1. Geometry of the problem.

15
Figure 2. Permissible profiles (a)-(d) with parameters as given below:

a P q

r

(a) 13.0 3.5 .135 -.38

(b) 1.8 2.0 2.719 -.25

(c) 1.2 1.2 2.0 -.40

(d) 1.0 1.25 2.3 -.40

16
IT
1.0

._„._-4 \ 1 1

1 4 1 1 i Q.2 0.4 0.6 0.2 l.o

Normalized Depth x -*

Figure h. Permissible profiles (i)-(m) with optical length L = 1.^99 used in

table 2.

a P q r

(i) 0.747 1.25 2.3 -0.4

(j) 2.255 4.0 1.64 -0.7

(k) 0.661 1.4 2.718 -0.3

(1) 20.150 -1.0 1.0 6.0

(m) 7.257 3.5 .135 -0.48

18
NBS-1 14A (REV. 7-73)

U .S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OK REPORT NO. NBSIR 76-851

2. Gov't Acccssio

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

ELECTROMAGNETIC REMOTE SENSING OF INHOMOGENEOUS MEDIA

5. Publication Date January 19 77

6. Performing Organization ( ode 276.07

7. AUTHOR(S)

Wolfgang A. Bereuter and David C. Chang

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Projcct/Task/Work Unit Nc 2762275

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Same as no. 9

13. Type of Report & Period Covered

14. Sponsoring Agency Codi

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary ol most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report deals with the electromagnetic response of inhomogeneous

dielectrics, i.e., media whose permittivity is a function of depth. The

resulting boundary value problem is solved for a large number of permittivity functions which can model almost any medium of interest.

Since those permittivity profiles are characterized by only a few

parameters, they are particularly useful for the inverse problem; i.e.,

the retrieval of profiles from the measured electromagnetic response.

It is shown how the non-uniformity of the permittivity changes the

response and how the change is related to the profile characteristics.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Inhomogeneous dielectrics; profile inversion.

18. AVAILABILITY [X_ Unlimited

I" or Official Distribution. Do Not Release to NTIS

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