70 - 19,318
HILL, David Allen, 1942- ELECTROMAGNETIC SCATTERING CONCEPTS APPLIED TO THE DETECTION OF TARGETS NEAR THE GROUND.
The Ohio State University, Ph.D., 1970 Engineering, electrical
University Microfilms, A XEROX Company , Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED ELECTROMAGNETIC SCATTERING CONCEPTS APPLIED TO THE DETECTION OF TARGETS NEAR THE GROUND
DISSERTATION
Presented in P a rtia l Fu lfillm en t of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University
by
David Allen H i l l , B .E .E ., M.Sc.
********
The Ohio State University 1970
Approved by
' \ ^ Q W V ^ ---- Adviser Department of Electrical Engineering FOREWORD
The remote sensing of d is tin c t objects on the surface of the
earth by electromagnetic waves is a very broad goal to which this
dissertation is addressed. Needless to say, the optimum method
has not yet been found and the study described in these pages can
at most contribute new ideas and analytical results on several
phases of the overall problem. S p e c ific a lly , the major con
tributions presented w ill be briefly summarized below.
New results for dipole radiation above a plane dielectric earth
are obtained, for the case of aperiodic, or impulsive excitation in
a number of special cases. These are simple in mathematical form
and corraborate, apply and extend the work of others. Applications
of resulting formulas are described, so as to determine the form of
the total illuminating wavefront at the dielectric interface and the
multiple interactions between a dipole scatterer (radiator) and the
plane dielectric earth. These results are contained in Chapter I.
The model of a scattering body as several induced e le c tric and magnetic dipoles, valid at s u ffic ie n tly long illum inating wavelengths,
is applied to express the chanaes in induced dipole strengths, as functions of frequency, when such a body is near a plane dielectric or highly conducting earth. The complex resonances of a scat
tering object, described by poles in the complex frequency plane may be influenced by the plane earth in several ways. The pole
locations and pole-shifts produced by d ie le c tric and conducting h a lf
spaces are estimated for conducting spherical and spheroidal scat-
terers in several cases. Including the changes in illumination as well as the complex pole locations, formulas which express the
dependence of backscattered signal upon signal frequency for such
objects above a plane dielectric earth are derived. These results
are presented in Chapter I I .
The use of a small number of harmonically related source fr e
quencies to illuminate a scatterer permits one, upon coherent
detection and recombination of the backscattered signal at all fre
quencies, to produce a characteristic waveform related to the target
size and shape. This technique and in particular the synthesis of
a ramp response waveform by this means has been proposed by others
as a method of target identification. Applying a difference equation which depends only upon the complex poles of the scattering function,
a method is now suggested fo r target id en tific a tio n and discrim i
nation. This difference equation, described by Corrington for a
lumped constant network, is used to predict current and future
values of the transient response from N prior values of the re
sponse, where N is the number of complex poles. In the detection
of a target at the earth's surface, knowledge of the complex poles
of the target in the presence of earth would thus prescribe the
coefficients of the difference equation, which in principle would
be satisfied by the characteristic waveform produced a t any location of source and receiver. This independence of the target recogniti
or detection scheme on direction of source and receiver is ex
tremely important in reducing the number of constants which must
be known for each object. In Chapter II I, a method for applying
the difference equation is proposed and illustrated for spherical
and spheroidal targets. The use of this technique in a linear
predictor-correlator device is also applied to the detection of spherical targets in the presence of simulated vegetative clutter on a conducting ground, using experimental data from a m ultiple frequency anechoic chamber f a c ilit y , with encouraging resu lts. ACKNOWLEDGMENTS
The author would lik e to acknowledge the many helpful ideas and the assistance of his graduate adviser, Professor E.M. Kennaugh.
Dr. D.L. Moffatt was responsible for the experimental data as well as some useful suggestions. A debt of gratitude is also owed to the members of the reading corrmittee, Professors C.E. Warren and
W.H. Peake, for their critique of the original dissertation draft.
The research reported in this dissertation was sponsored in part by the A ir Force Cambridge Research Laboratories, O ffice of
Aerospace Research, under Contract F44620-67-C-0095, and by The
Ohio State University Research Foundation.
v VITA
April 21, 1942 Born - Cleveland, Ohio
June 1964...... B.S.E.E., Ohio University, Athens, Ohio
1964-1966 ...... Acting Instructor, Electrical Engineering Department, Ohio University, Athens, Ohio
June 1966 ...... M.So., Ohio University, Athens, Ohio
1966-present.. Research Associate, ElectroScience Laboratory (formerly Antenna Laboratory), The Ohio State U niversity, Columbus, Ohio
Publications
"The Effects of Irre g u la r Contour on Image Glide-Path Systems," M.Sc. Thesis, Ohio U niversity, 1966.
"Transient Dipole over a Dielectric Half-Space," Proceedings of the Conference on Environmental Effects on Antennas, Boulder, Colorado, July 1969.
vi Fields of Study
Major Field: Electrical Engineering
Studies in Electromagnetic Field Theory. Professors E.M. Kennaugh and R.G. Kouyoumjian
Studies in Radar Systems Professor A.A. Ksienski
Studies in Communications Professor C.E. Warren
Studies in Applied Mathematics Professor S. Drobot
Studies in Classical Physics Professor W.H. Shaffer
vii TABLE OF CONTENTS
Page
FOREWORD...... i i
ACKNOWLEDGMENTS...... iv
VITA...... v ii
LIST OF FIGURES...... x
Chapter
I THE TRANSIENT FIELDS OF DIPOLES IN THE PRESENCE OF A DIELECTRIC HALF-SPACE...... 1
A. Introduction 1 B. Classical Sommerfeld-Weyl-Van der Pol Solution 2 1. Exact frequency-domain forms 2 2. Applications 7 3. Extension to aperiodic excitation 3 C. Vertical Electric Dipole 10 1. On-axis solution in upper medium fo r anisotropic lower medium 10 2. Surface fields for a dipole at the interface of an anisotropic dielectric 17 3. Magnetic field for general observation point in upper medium 21 4. On-axis solution in the lower medium 26 5. Dipole and field point in the lower medium 28 D. Horizontal Electric Dipole in Upper Medium 30 E. Magnetic Dipole 36 1. On-axis solution for vertical orientation 36 2. On-axis solution for horizontal orientation 37 F. Approximate Expressions fo r Return Fields 40
I I EFFECT OF GROUND ON THE DIPOLE MODES OF SCATTERERS...... 51
A. Introduction 51 B. Dipole Modes of Scatterers in Free Space 56
viii Chapter Page
C. Change in Dipole Modes Due to Ground When Dipole Modes do not Interact 56 1. New polarizability 56 2. New poles 61 D. Coupling of Dipole Modes 69 E. Backscattered Field 74
I I I THE DIFFERENCE EQUATION IN TARGET DETECTION AND DISCRIMINATION...... 77
A. Introduction 77 B. The Difference Equation for Transient Responses of Scatterers 78 C. Discrimination of Targets by a Difference Equation Receiver 86 D. Detection of Targets in the Presence of Clutter Using a Difference Equation Receiver 93
IV SUMMARY AND CONCLUSIONS...... 102
REFERENCES...... 105 LIST OF FIGURES
Figure Page
1 Vertical dipole above a conducting half-space ...... 3
2 Vertical electric dipole above a uniaxial anisotropic dielectric half-space ...... 11
3 The scattered electric field on the z-axis as a function of tim e...... 16
4 The magnetic field at the surface of a dielectric half-space ...... 20
5 Isometric view of ct = ..12...... 24 O9 6 Isometric view of ct = 15 ...... 25 0$
7 Isometric view of pH . ct = 18 ...... 26 O9 8 The Hertz vector in the lower medium on the z-axis... 28
9 Vertical electric dipole within a dielectric half-space ...... 31
10 Scattered electric field in a dielectric medium 31
11 Horizontal electric dipole above an isotropic dielectric half-space ...... 31
12 The scattered horizontal electric field on the z-axis as a function of time ...... 34
13 The mutual impedance of a short v e rtic a l dipole above a dielectric half-space ...... 46
14 The mutual impedance of a short horizontal dipole above a dielectric half-space ...... 48
15 The mutual impedance of a short v ertical dipole in a dielectric half-space ...... 50
x Figure Page
16 Backscatter from conducting sphere ...... 54
17 Ramp response of conducting sphere ...... 55
18 Prolate spheroid above a half-space and its induced dipole moment ...... 57
19 Sphere above a half-space with uncoupled e le c tric and magnetic dipole modes ...... 59
20 Dipole moment induced in sphere by dipole source 60
21 New poles for vertical prolate spheroid over perfect conductor ...... 63
22 New poles of vertical electric dipole mode of sphere over dielectric half-space ...... 65
23 New poles of a thin horizontal prolate spheroid above a dielectric half-space, axial ratio = .013...... 66
24 Coupled horizontal electric and magnetic dipole modes of a sphere over a half-space ...... 69
25 New poles of coupled horizontal e le c tric and magnetic dipole modes for a sphere over a perfect conductor ...... 72
26 Vertically polarized plane wave incident on a prolate spheroid over a half-space ...... 75
27 Scattering from a hemispherical boss using image theory ...... 81
28 The dipole modes of a sphere above a conducting half-space excited by vertical polarization ...... 83
29 Ramp response for a sphere on a conducting ground for vertical polarization and a 45° aspect angle ...... 84
30 The correlation between the synthesized ramp response of a sphere on the ground and the predicted waveform ...... 85
31 Ramp response of a thin conducting prolate spheroid...... 86
xi Figure Page
32 Block diagram discriminator using two different time increments ...... 87
33 Correlation of two predicted signals, thin prolate spheroid in to sphere difference equation...... 88
34 Comparison of input waveform with predicted waveforms, spheroid into sphere difference equation...... 89
35 Comparison of input waveform with predicted waveforms, sphere in to thin spheroid d if ference equation * ...... 90
36 Correlation of predicted and input waveforms ...... 90
37 Discrimination against spheres of d iffe re n t sizes by use of the difference equation ...... 91
38 Difference equation receiver for N targets ...... 93
39 Detection of boss with vertical polarization at a 45° aspect angle ...... 95
40 Detection of d iffe re n t size bosses in the same clutter ...... 97
41 Difference equation receiver for the detection o f a target in clutter...... 98
xi i CHAPTER I
THE TRANSIENT FIELDS OF DIPOLES IN THE PRESENCE OF A DIELECTRIC HALF-SPACE
A. Introduction
The transient fields of short dipoles in the presence of half
spaces are of importance in both the source and target features of
the detection problem. If the dipole represents the source, then
the return field is characteristic of the ground return with no
target present. The fields in the vicinity of the interface are
also of importance because they represent the target illumination.
The short dipole can also represent one of the dipole modes of an
excited target, and the return field can then be used in calculating
the effect of the ground on the dipole modes of the target. The
solutions for the four dipole types (vertical and horizontal electric
and vertical and horizontal magnetic) are required because both
electric and magnetic modes are of importance when dealing with
perfectly conducting targets.
The classical dipole solutions in both the frequency and time
domains are summarized firs t. The remaining portion of Chapter I
derives new exact closed-form solutions for impulsive dipole ex
citation in several special cases. While closed-form solutions are
obtained for only lossless half-spaces, the simplicity of the
1 transient solution suggests that a d ire c t numerical treatment in the case of an imperfectly conducting ground may also be simpler fo r impulsive excitation than fo r the harmonic case.
A. Classical Sommerfeld-We.yl-Van der Pol Solution
1. Exact frequency-domain forms
The radiation from short dipoles with time-harmonic current in the presence of ground results in the well known Sommerfeld-type integral solutions. The solutions for vertical and horizontal electric and magnetic dipoles above or within an isotropic con ducting half-space have been given by Banos[1].
For an electric dipole source, the electric and magnetic fields can be obtained from a single electric Hertz vector, _n, by
(1) E_ = vv • jt + k2 n. ,
(2) H = ( jioe +a) V x n .
For a magnetic dipole source, the electric and magnetic fields can be obtained from a single magnetic Hertz vector, i/”, by
(3) E = -juiu0 v x nm,
(4) H = vv • / + k2 nm.
Consider a vertical electric dipole of length ds and current
I exp(jojt) located at a height h in free space above a conducting 3
7 7 7 7 7 7 7 7 7 ^ P
Fig. 1—Vertical dipole above a conducting half-space.
half-space in a cylindrical coordinate system as shown in Fig. 1.
The e le c tric Hertz vector has only a z-component whose value in the upper medium nQZ is given by
I = -ji—-— — [G99 - G91 + k2 V99] , (5) OZ 4ue jco 22 21 22 where 2.2 - J f J U - h j + P
22 J (z -h )2 +
-j f/w o 2 + »2
’21 2 + p2 4
J x 2-M C)2 (z+h) j o(Xp) xdx
V22 * 1
^ J ( e—jcr/a))/ eq .
The term involving G 22 is the free-space dipole radiation, and the terms involving and give the scattered field. The Hertz vector in the lower medium n is given by
ju>yn I ds ^ naz= 3^ V21 *
where
j A 2 -(w/c)2h - J a 2-k2
J o 'Ap) AdA. V21= 2 0 k‘ a2 -(o )/c ) 2 + ( - !A2 - k2
The special case of a lossless lower medium can always be obtained by setting a = 0 in Eq. (5) to yield
ruo (7) 1 c c J k -
Next consider a vertical magnetic dipole of length ds and mag netic current K exp(jcot). The magnetic Hertz vector has only a z- component whose value in the upper medium nj^ is given by 5
„m K ds (8) oz 4irjuju [G22 “ G21 + U22] *
where
eJx2-(ro/c )2 (z+h) J0(xp) xdx. '22 0 ]\2-{ 0)/c )2 + J ~ x 2- k 2
The magnetic Hertz vector in the lower medium is given by
( 9 ) nm = _ .K.-js u \ z 4 tt jo> u0 21’ where ^ J x2-(u j/c )2 h-Jx2- k 2z e 0Q(xp) xdx. 21 Q J\2- ( u / c)2 + J x2 - k
The horizontal dipole case is more complicated because the 4> symmetry is no longer present. For an x directed electric dipole both an x component and a z component are required and in the upper medium are given by
(10) nQX - 4u [G22 - G21 - U22] ,
dW I ds 22 OZ 4tt eQ ju> 3X
where 6
W22 = ^2 Iz [ 2G21 - ( k£ + c2 ) V22] *
In the lower medium the x and y components are given by
- jojp I ds nnx = 2 ^21’ 4tt k:
-jwu0I ds aw21
4, ' k f SX where
Wft, = - f 4 + Irr) V21. *21 = “ + 32
For an x directed magnetic dipole, the x and y components of the
magnetic Hertz vector in the upper medium are
^ nox = 4irjwJ* [G22 “ G21 + kJl V22] ’
m _ K ds 3W22 oz “ 4iTjwy0 ax *
The x and y components in the lower medium are
m _ -juK ds (13) '21’ 4TTyQC
9W m K ds n 21 12 4Trju)y oX One extension to the conducting half-space results given here is the problem of a dipole over a uniaxial anisotropic conducting h a lf space, and Wait[2,3] has given the solution for both vertical and horizontal dipoles. Consider a half-space with vertical and horizontal permittivities, ev and eh, and vertical and horizontal conductivities, a „ and a. . The solution for the case of a vertical electric dipole v h above the half-space has only a z component of the electric Hertz vector which in the upper medium is given by
o k2J x2- ( x / c)2 J A2K-kj; where
2. Applications
The half-space solutions given here have several applications in the problem of detection of a targ et near the ground. F irs t, the dipol solutions are required to compute the unperturbed fields near the inter face when the source is not fa r enough removed to permit the plane wave approximation. Second, the dipole solutions are also required to determine the ground return when no target is present since the plane wave approximation predicts no return. Third, the target can be reore- sented by electric and magnetic dipoles if the frequency is low enough, and the half-space solutions can thus be used to calculate the target return.
Since the evaluation of the integrals in the solutions given here
is quite d ifficu lt, a great deal of work has been done on the various approximations of the Sommerfeld integrals. Norton[4] obtained
numerical results for the vertical electric dipole above a conducting
ground, and Banos[l] includes both asymptotic and quasi-static ap
proximations to the U and V integrals given in Eqs. (5) and (8). His
results are quite complete because the four dipole types located both
above and in the conducting half-space are included as well as more
than one hundred references on the subject. The problem of detection
of a conducting sphere over a conducting ground has been treated by
Wait£5] using a quasi-static analysis. He employs a vertical mag
netic dipole source, represents the sphere by an induced magnetic
dipole, and suggests a two-frequency detection scheme based on the
difference in frequency dependence of the sphere and ground return.
A very complete discussion on the detection of metallic targets in
the ground is given by Grant and West[6]. In order to model the
jungle environment, Wait[7] has examined the fields of a vertical
electric dipole over a conducting ground covered by an anisotropic conducting slab, and has also examined the case of a general stratified anisotropic half-space[7].
3. Extension to aperiodic excitation
The transient solutions which will be obtained later in this chapter are the inverse Fourier transforms of the frequency domain solutions given previously. Only some special cases where exact closed-form solutions result will be covered, but the simplicity in these cases suggests the p o s s ib ility of f a ir ly simple approximations in the general cases. Only lossless half-spaces will be considered although the results obtained w ill be valid for short times even if some conductivity is present.
The transient fields of a vertical electric dipole over a lossless half-space have been discussed by Van der Pol[8], Poritsky[9], Pekeris and Alterman[10], De hoop and Frankena[11], and Brenner[12].
Van der Pol obtains the exact solution for the Hertz vector by taking the inverse transform of the Sommerfeld solution for the case when both the source and the observation point are located on the interface. Poritsky uses a generalization of Weyl's method to obtain an integral solution by expressing the field as a superposition of time-dependent plane waves. Pekeris and Alterman obtain the Hertz vector in both media by an inverse transform method and give some numerical results for the case where the dipole is located at the interface. De Hoop and Frankena utilize a Laplace transform and a two dimensional spatial transform to obtain an integral solution in the upper medium. Working s tr ic tly in the time domain, Bremmer obtains the integral solution in both media when the dipole is at the interface. Frankena[13] obtains the integral solution in the upper medium for a horizontal electric dipole by using a Laplace transform and a two-dimensional spatial transform.
In this chapter, the exact closed-form solutions are obtained for the observation point on the vertical axis (p = 0) for the four dipole types. These solutions give the time-dependent ground return if the dipole is considered to be a source or the interaction of a target with the ground if the target is represented by electric and magnetic dipoles as shown in Chapter II. The solution for a vertical dipole over an anisotropic half-space is also given and could be interpreted as jungle return. The additional case for a v e rtic a l dipole in the lower medium could be useful fo r the case of a buried targ et. The case of both the v e rtic a l e le c tric dipole and the observation point on the interface is applicable to the problem of low angle radiation, and the magnetic field is evaluated numerically for general observation points to add some insight to that given by the special cases.
C. Vertical Electric Dipole
1. On-axis solution in upper medium for anisotropic lower medium
Figure 2 shows a vertical electric dipole located at a height h above a uniaxial nondispersive, anisotropic dielectric half-space with a vertical index of refraction ny and a horizontal index of refraction 11 n^. The field in the upper medium consists of the primary field plus the scattered field which is due to the presence of the half-space.
The field can be obtained from a Hertz vector which has only a z component, and the frequency-domain solution fo r the Hertz vector has been given by Wait[5,6] for general ground constants. When the ground is nondispersive, the solution for the scattered Hertz vector as given in Eq. (1 4 ), ir! , reduces to
- i f ] p 2+ (z+ h)2 e
where ds is the length of the dipole and the current is I exp(jwt).
A z
h
1
Fig. 2--Vertical electric dipole above a uniaxial anisotropic dielectric half-space. 12
In order to obtain the time-dependent solution when the current is I 6 (t ) , i t is necessary to take the inverse Fourier transform of the frequency dependent solution of Eq. (15). Since the time-dependent
Hertz vector ns (t) is real and causal, the inverse Fourier transform oz can be obtained from just the real part of n^z(w) as shown in
Papoulis[14].
z r h R e i£ 2M ] = f [Re noz((u)]cos ut dm, t > 0 (16) n5 ( t ) = ozv 7 0 , t < 0 .
The form of n^.U) is simplified if it is evaluated on the z-axis
(p = 0), and the frequency dependence is simplified if the substi tution X = wa is made in the integral.
- j f (z+h) I ds (17) 4ire joj(z+h) V “> 0
,-jW (V c)2-a2 (z+h) ada ■2nvnh 0 v J ( l / c ) 2- a2 + J( nv/ c )2 - a2 -
The inverse Fourier transform of the first term in Eq. (17) yields a unit step and the inverse transform of the integral term can be ob tained using the inverse transforms of the functions, cos[wj(l/c)2-a2] and e~ ^ a2-^ /c^ (z+h)t which are given in standard transform tables.
Using Eqs. (16) ana (17), the following expression for ^ z(t) is ob
tained: 13
(18)
I ds U(t) »(t - g . ) 4 < t ) = - 4 it e z + h
nv/c JJ. (l/c)d (z+h)/7T Re a da 2V h j|c | [a2-(l/c )2](z+h)2+t2 Lf r f - 2 ♦ - A m .
1/c [t-J(l/c)2 - a2(z+h)] a da + K * 2V n V hil/c )2-a2 +J(^-)2 - where the constant K is necessary because the frequency-dependent ex pression did not include the 6(to) term required to yield the proper constant level in the time-dependent function ^ z(t). The first in tegral in Eq. (18) can be evaluated from standard integral tables, and the second integral can be evaluated by using the following identity for delta functions:
(19) 6 [f(a )]= I 6 (a -a ,.), where a^ are the zeros of f(a). The delta function in Eq. (18) has two roots, but only one is positive and w ill contribute to the in tegral. The final form for fr^U) after the integration is carried out is
_ f c t Id s V h Fz+ hj ' ” 41 z+h •)2 * " v -l 4ire0 (z+h) (>■“ ) If the solution for a general observation point is desired, an ex pression similar to Eq. (18) can be obtained for arbitrary p. This was done by DeHoop and Frankena[ll] for an isotropic half-space.
However, the resulting integrals cannot be evaluated in closed form as th e / were fo r the special case of p = 0 given in Eq. (2 0 ). Con sequently, a numerical integration would be required.
Both the e le c tric and magnetic fie ld s can be obtained from the
Hertz vector by differentiation. The magnetic field on the z-axis is zero, and the electric field has only a z component Eqz( t) which
is given by
(21) Es (t) = —___— ns ( t ) 02 sz2 C2 3t j 02
The differentiation implied in Eq. (21) leads to the following ex- pression for E (t).
+ 2nvnh A(1-T2)-2TA“2(1-T2)
-2TA’ 1 F (1—T2) } + — 2— j £ , (z+h) B 15
1 + T
8 = nvnh T + A» and F = nynh T - A.
By letting T = 1 in Eq. (22), the initial value of (t) is obtained.
ds 4nh(ny - l) nh ~ 1 (23) E=z (' ^ +)= / r22nette0 ( z+h)' Inv ( nh + D ) ‘ nh + 1
The final value is obtained by letting T approach infinity.
I ds "v "h ~ 1 (24) E~LH = oz' 2nt(3(z + h)' V h * 1
I t can be shown that Egz (°°) Is less than or equal to E^2 +) lo r
a ll values of ny and n^, and that E ( t ) is a smoothly decaying z+li function with a time constant roughly equal to A typical sketch
is shown in Fig. 3, and although the expression in Eq. (22) is quite
complicated, the waveform is fairly simple. The limiting cases of
either free space or a perfect conductor for a lower half-space can
be obtained from Eq. (2 2 ). When the half-space reduces to fre e -
space (n = n^ = 1), then the scattered field is zero. When the
indices of refraction go to infinity (ny = n^ = «), the solution
reduces to that obtained by image theory for the perfectly conducting
half-space: z+h I ds i i i i - f ) (25) E „ ( t ) = oz 2 tt eQ (z + fij 0(2 + h) T (z + h)2 16
E*oz
0 I 2 3 4 5 c* z+h
Fig. 3—The scattered electric field on the z-axis as a function of time.
The significance of the on-axis solution as given here is that the waveform for z = h represents the field scattered back to the source. This is indicative of the return from the jungle when modeled by an anisotropic medium. The same waveform could be used to deter mine the interaction between an antenna and the ground as shown la te r in this chapter or the interaction between a scatterer and the ground as shown in Chapter I I . The most logical extension would be an aniso tropic slab over a lossy ground. However, this complicated model would probably require a numerical integration to obtain frequency domain values followed by a Fourier synthesis to obtain the time waveform. 2. j'urface fields for a dipole at the interface of an anisotropic dielectric
The time dependent Hertz vector can also be obtained in closed form fo r the case where both the dipole and the observation point are located at the interface (z = h = 0). Wait's frequency-domain
expression for the total Hertz vector nQZ(“) as given in Eq. (14)
reduces to the following form fo r this case:
I ds n..nui.JIl r°° J.(xp) xdx (26) nn,C«)L _ _ V UJ I = o ------V• h OZ 2tte. j (X) • %
The frequency dependence can be sim plified by the same substitution
X = ua.
- I ds n n. J (wap) ada o ' 27> 2 ^ 0 o nvnhJ ( l / c ) 2 - a2 + J { n j c)2 - a2
In order to take the inverse Fourier transform of Eq. (27), the fol
lowing inverse transform of the zero order Bessel function[15] is
needed.
(28) F"1 [0 (wap)] - ■■ - p .1.- [U(t+aP) - U(t-ap)]. 4 a p ) 2 - t 2
Using Eqs. (16) and (2 8 ), the inverse transform of Eq. (27) can be % taken to yield the time-dependent Hertz vector nQZ(t). The integration in Eq. (29) can be carried out in closed form using standard integral tables.
(3°) noz(t) = 2JeI ds I f "v" 2 2 ,, v h U " ^ v + 1
"v"h nvnh VU - 1 U t - 2 2 V h "h - "v + 1
The final expression in Eq. (30) agrees with the result obtained by
Van der Pol 11J for an isotropic lower half-space.
The electric and magnetic fields can again be determined from the Hertz vector by differentiation. The magnetic field has only a
$ component H (t) given by ^ a2 n_ ( t ) 19 2 2 I ds nvnh "h + 1 V ‘ > 2 2 2 2 , <* - 1) 2irP [ V h ' 1 {V nv+1)
V"h+1) • (t - ^ ) "h 3c ct/p 15/2 P “(t - 1) - u(‘ - n jf ) The time dependent electric field has both z and p components given by 3"oz(t) p- <32> 9p -v 9 n (t) E (t) = — —— O p ' 1 ; 3 Z 3 p The p component cannot be obtained from ii ( t) in Eq. (30) because the z dependence is not known. However, the z component ^ ( t) can be ob tained from Eqs. (30) and (32) and is given by _ 2 2 - I ds 1 v h nh + 1 (33) Eoz(t) c 2 2 , 9 , J /2 - 1 ) 2tte0p^ v h ^nh" v J 1 "v<"h+1> 2 2 (-¥ ) K " S - !> 20 2n; 1 (33) + I 3/2 (cont.) <"v"h + 1) ( " H t f - 3n. 5/2 [(nh+i) ( f T - "v ] I f H n H l L - ) - 1 (,. p W n b l J /«2 „2 . . l J V " v + 1 The most significant feature of the fields given in Eqs. (31) and (33) is the double wavefront. The fir s t delta function travels above the interface in free space at a velocity c, and the second waveform travels below the interface in the dielectric at a reduced velocity c/ny. A typical sketch of H (t) is given in Fig. 4. o 4> Fig. 4—The magnetic field at the surface of a dielectric half-space. 21 Although the result given here is quite restrictive, the effect should be similar for nearly grazing incidence angles. The main con- _9 elusion is that the geometrical terms cancel leaving only a p *■ field as given by Eq. (33), and this will occur for lossy ground as well. Consequently, the plane wave approximation is not useful for grazing incidence, and the dipole solution is required to determine the ex citing fields for a scatterer located above the ground. 3. Magnetic field for general observation point in upper medium The time dependent Hertz vector fo r general source and ob servation points can be written as a definite integral over fin ite limits which must be evaluated numerically. For an isotropic lower medium of refractive index n, De Hoop and Frankena[ll] have derived the integral solution for the scattered Hertz vector in the upper medium which can be written in the following form: (34) where 2 ct(z+h) 1 COS f , and 22 An equivalent expression has been derived by Pekeris and Alterman[10] fo r the dipole at the interface and has also been numerically evaluated for various elevation angles. In order to gain knowledge of the actual field components, i t 'W is necessary to obtain the required partial derivatives of noz(t) in Eq. (1) before performing the numerical integration. The mag- nettc fie ld H ft) is obtained by taking o and t derivatives of 0 'wu (t) <35> C (t) = R. = p1 ds _v c 4 IT R2 2 2n2(n2-l) n Yr Y2 Yi Re 7T Rr 2 " ( 2 »2 Y1 Y2 y 2^ y 1 y2' where "2( f ) - " 2 - 1 R.. = -ct(z+h) p/R, (z+hV + J cos ip, r i ° pRo R? 23 and Yj and y^ are given in Eq. (34). In order to examine the total field Hu f t ) , the primary fie ld H ^ (t) must also be included and is 0 (36) HS“(t)U(p =7 * # ^ [c6(‘ - r ) + ^ u(t'^). where R 1 = J ( z - h)2 + p2 . LI Three isometric views of pHQ^(t) are shown in Figs. 5, 6 and 7 for different values of time with n = 2 and h = 5. The determination of the field in the lower medium is a more d iffic u lt problem, and Pekeris and Alterman[10] have derived the fie ld in the lower medium only for the special case of the dipole at the interface. Although only the magnetic field is shown in Figs. 5, 6 and 7, the z and p components, E and E , are obtained from H , by r oz op 04> (37) E It) = - L (pH“ ) , "02 eQp 3p 0(f)' Consequently, a rough idea of the electric field can be obtained from Eq. (37) at the specific times represented in Figs. 5, 6 , and 7. For very large times, the scattered field is that produced by an image O p of strength (n -l)/(n +1), and the magnetic field is given by 24 (38) » ;(t) = 1 + n2 - 1 J_ n2 + 1 Rjj This behavior is not yet evident for the values of time in Figs. 5, 6 , and 7. ~u **P Fig. 5—Isometric view of pH . ct = 12. 0 Fig. 6 —Isometric view of pH^x, ct = 15. 0 Although i t would also be useful to know the transient fields of a dipole over a lossy ground, the calculation would involve a numeri cal integration to obtain frequency domain values followed by a Fourier synthesis to obtain a time waveform. Wait[16] has worked on the problem assuming a constant conductivity and neglecting dis placement currents, but as he points out the physical validity of this model is questionable. 26 u Fig. 7--Isometric view of pHQ , ct = 18. 4. On-axis solution in the lower medium The time-dependent Hertz vector in the lower medium can also be obtained when the observation point is on the z axis (p =0). The problem is simplified when the lower medium is isotropic (ny = n^ = n), and the frequency dependent Hertz vector iij (u>) for this case which can be obtained from Eq. ( 6 ) reduces to 27 where h is the dipole height and -z is the depth of the observation point in the lower medium. The frequency dependence of Eq. (19) is simplified by the substitution \= wa, and the inverse Fourier trans form can be taken using Eq. (16). The remaining integral can be evaluated in closed form, and the resulting expression for the time % dependent Hertz vector nlz (t) is l+(n2-l) ^ 4 (c t )2 -1 (c t) A sketch of n^ (t) with its in itia l and final values is given in Fig. 8 , and the waveform is a smoothly decaying function with an approximate time constant (h-nz)/c. The magnetic fie ld is zero on the z-axis, and the electric field has only a z component Ejz(t) given by The main value of the on-axis solution given here is to provide some idea of the behavior of the fields in the lower medium. The general off-axis solution for the source and observation point in 28 different media has not been obtained and would be a desirable ex tension. 2 TT€ 0 „ Ids n.z ( h —z)(n2 + l ) O 2 3 4 5 ct h - nz Fig. 8—The Hertz vector in the lower medium on the z-axis. 5. Dipole and field point in the lower medium When both the dipole and the observation point are located in an isotropic dielectric half-space as shown in Fig. 9, the expression for the frequency dependent scattered Hertz vector in the lower medium n jz(to) is as given by Banos[7]: Sa Jp2 + (h-z) 2 p2 + (h-z) 2 J0(Xp) Xdx 29 When the observation point is located on the z-axis (p =0), i t is possible to use the same technique which was applied to the case of the dipole and observation point in the upper medium to obtain the OjC time-dependent scattered Hertz vector n y t ) . ct n2 (t-n F T " n M (43) n j2(t) 4ire0n (h-z) The magnetic fie ld is zero on the z-axis, and the electric field has only a z component Ejz(t) which is given by 'V/c 'V-e (44) E\z{t) = nlz( t ) , 3Z - ( f J at Es (t) = = U * . 2n lz U ; . _2 4ire0n I c(h-z)‘ ct 2 L 2J ct \ 2 _J_ FT ~ n n/1"0 Ift) h-z i------az (tJ at ct h-z S The in itia l and final values of Ejz(t) can be obtained from Eq. (44), 30 -I ds (n-1)(4n+l) <«> Elz 2Tre0n2(h-z)3 n2+l The waveform is a smoothly decaying function as shown in Fig. 10. The on-axis solution given here is useful for determing the interaction of a submerged antenna or scatterer with the interface. The Fourier transform of the solution is used to calculate the change in the impedance of a submerged vertical dipole later in this Chapter. D. Horizontal Electric Dipole in Upper Medium A horizontal electric dipole above an isotropic dielectric half space along with the appropriate rectangular coordinate system is shown in Fig. 11. The determination of the fields requires both an x and a z component of the Hertz vector since the exist. However Banos[4] has determined the electric and magnetic fields directly in terms of the usual integrals so that the inter mediate calculation of the Hertz vectors is not necessary. I f the electric field is evaluated on the z-axis, only an x-component (u>) exists which can be determined from Eqs. (1) and Oa (10). 31 ////////)L/T7 Fig. 9—Vertical electric dipole within a dielectric half-space. ct n ( h -z ) 0 2 3 Iz Fig. 10—Scattered electric field in a dielectric medium. n Fig. 11—Horizontal electric dipole above an isotropic dielectric half-space. where G2j , G22, an<* ^22 were defined in Section B. The inverse transform of G2j is simply the unit step: u (t (47) r 1 [g21]| 0 = z+h The inverse transform of U22 can be evaluated by making the sub stitution x = toa, inverse transforming inside the integral, and in tegrating the result. U t U22 2 (48) z+h + Jl +(n2- l ) ( | r ) 2 In order to take to the inverse transform of the term involving V22, the following result is needed. 3 J (xp) (49) — V - X « 3p Using this result, the inverse transform of the V22 term can be taken by the same method which was used on the previous integrals. 33 1 [ 1_ a^ 22]| i i k l - (50) r LJo) 3p2 J^ =0 z+h »2 ♦ /H "2-d (lr ) Using the results of Eqs. (48), (49), and (50), i t is possible to take the inverse transform of Eq. (46) to obtain the time dependent electric field EM t). I dsM n-1 «'(* - ¥ ) , n-! *(* - (51) E"x(t) = n+1 4tte0 c (z+h) n+1 c(z+h ) 2 z+h u (t - ) c 2(z+h) L at L 1 + II + (n2- l ) ( ^ t ct ) (z+h)' The in itia l and final values of E ( t) can be obtained from Eq. (51). U A -l)(n + 2) <52> +) - ldS 3 f *SWD ------ox V c * 4 ^ z+h) 3 L n + 2-ifl-1).- ( 2n4 + 6 n3 + 5n2 + n + 2)1 , n (n+1) J rS / \ _ I ds n2 - 1 E ( ° ° ) ------k -5------4nE0 (z+h) n + 1 s A typical sketch of E ( t) is shown in Fig. 12. The main difference U /\ between i t and the waveform for the vertical dipole is the addition of the doublet term occurring at t = (z+h)/c. The magnitude of the doublet is proportional to the specular reflection coefficient (n -l)/(n + l) which is expected since the doublet is the optical term containing the (z+h)~* dependence. ox z + h Fig. 12—The scattered horizontal electric field on the z-axis as a function of time. The magnetic field on the z-axis has only a y-component, and the frequency-dependent magnetic field on the z-axis HqZ(w) given in Banos[4] can be obtained from Eqs. (1) and (10). * 1 f 3 35 The inverse transforms of the functions in Eq. (53) have been given in Eqs. (47), (48), and (49). Using these results to take the inverse * transform of H^y(u), the time-dependent magnetic field HQy(t) can be written as The delta and doublet terms in (t) and E^x( t ) are related by the free space impedance, but the remaining portions of the waveforms are quite different. The final value of H S (t) is zero, and in the perfectly conducting case (n = °») only the delta and doublet terms remain. As in the case of the vertical dipole, the on-axis solutions yield the interaction between a source and the interface. Both the electric and magnetic fields are utilized in Chapter I I to analyze the coupled horizontal electric and magnetic dipole modes of a scatterer above ground. The Fourier transform of the electric fie ld is used to determine the change in the impedance of a horizontal dipole above a dielectric ground. 36 E. Magnetic Dipole 1. On-axis solution for vertical orientation The frequency domain solutions fo r magnetic as well as electric dipoles in the presence of half-spaces are given in Banos[4], I f a vertical magnetic dipole of current K and length ds is located at a height h on the z-axis above a dielectric half-space, then the fre quency-dependent scattered magnetic Hertz vector has only a z com ponent n^(a») which in the upper medium can be obtained from Eq. ( 8). where and are given in Section A, and their inverse trans forms are given in Eqs. (47) and (48). These results can be used to take the inverse transform of Eq. (55) and obtain the time dependent o,nK magnetic Hertz vector noz(t). (56) The electric field is zero on the z-axis, and the time de- pendent magnetic field H (t) is given by From Eq. (57) i t can be shown that the final value of HQZ(t) is zero, and the in itia l value is given by | r n | nms ( z+h \ _ Kds (n-1)(n-4) (68) • As in the electric dipole case, the on-axis solution provides the interaction of a scatterer with the ground. The change in the impedanc of a vertical magnetic dipole (horizontal electric loop) could be com puted, or as in Chapter II the vertical magnetic dipole mode of a scat terer above ground could be analyzed. 2. On-axis solution for horizontal orientation The frequency domain solution for the electric and magnetic fields of a horizontal magnetic dipole over a half-space in terms of the usual integrals has been obtained by Banos[4]. If the magnetic dipole is oriented in the x direction on the z-axis at a height h, then the scattered magnetic fie ld on the z-axis has only an x com ponent H ^ (oj) which in the upper medium can be obtained from Eqs. (4) \} r\ 38 K ds [(n2+1) ( | ) 2 4iry0ja) V22 " G21 " V22 9p ] “2 [ n2( c ) V22 " G21] p=0 where G ^, V22 are defined in Section B and their inverse trans forms are given in Eqs. (47), (48), and (50). The following inverse transform is also needed: ro o> - 1 1 3^V22 1 ■-(£? “(■-¥) (60) _ j w 9P2 _ P=0 3t2 Using Eqs. (47), (48), (50), and (60), the inverse Fourier transform of H^(w) can be taken to obtain the time dependent scattered mag- UA 'Vvnc netic field on the z-axis H (t). ux s.(t . z+h) 6 f t . Zth\ K ds n-1 V c / n -1 \ c } (61) Q t ) = J Try n+1 c 2(z+h) n+1 c(z+h ) 2 z+lv U t ■( c (z+h) z+h; (z+h)‘ ms I t can be shown from Eq. (61) that the final value of H (t) is zero. WA 39 The solution for the frequency dependent scattered electric field has only a y component on the z-axis E^(o>) which in the upper medium can be obtained from Eqs. (4) and (12). 2 2 tco\ i-ms, x K ds a | n to „ r (62) Eoy( 2 2c + - e2 + 1)V 22 2 G21 3p ^W|C Using Eqs. (47) and (50), the inverse Fourier transform of EQy((o) can ^"TTIS be taken to yield the time dependent electric fie ld EQy(t). K ds (63) E ^ (t) 4'tt" n+1 c (z+h) + n+1 (z+h) 2c2t - U (z+h): (nZ+l) + 2n 1 3 . (&M z+h at ms The delta and doublet terms in EQy(t) are again related by the free space impedance, and the final value of E™y(t) is zero. These on-axis results together with the horizontal electric dipole results can be used to analyze the coupled horizontal electric and magnetic dipole modes of a scatterer above a dielectric ground. The procedure is given in Chapter II . 40 F. Approximate Expressions f or Return Fields In order to determine the interaction effects of a source element with the interface of a half-space, i t is necessary to know the fields scattered back to the source. A knowledge of these fields allows one to determine the change in the self impedance of an antenna due to the half-space (Wait[17]) or the change in the response of a scatterer from free space to the half-space environment, as w ill be shown in the following section. Consequently, the dipole solutions of interest are the on-axis solutions with z = h (or z = -h in the case of the submerged dipole). In the case of a vertical dipole above an isotropic half space, the scattered field on the z-axis Eqz(t) is given by Eq. ( 8) with 'V nv = nh = n. I f we define GvE-(t) as the time dependent electric field scattered back to the impulsive dipole source of unit current moment 6 ( t ), then <«> ■ - r e z = h. a. Although the exact expression for GvE(t) is quite complicated, the waveform as shown in Fig. 3 is quite simple. I t has been found 'Vc that E (t) can be approximated as an impulse plus a step plus a decaying exponential with a time constant (z+h)/c so that G E(t) can be written t-T T (65) GyE(t) 2l A 6 (t-T) + B + C e U(t-T), 41 where 87reQCh B 1 n2- l J 2 16ireoh n +1 C 1 2(n -l)(n 2J-2) 16TTe0h3 n(n+l) (n 2+l) and T = 2h c * The coefficient of the delta function A is exact and B and C have been chosen to yield the proper in itia l and final values as given in Eqs. O.. (23) and (24). If we define GyE(s) as the Laplace transform of Gv£( t ) , then the exact form could be obtained from the complicated expression in Eqs. (15) with w = s /j. However, the simple functions in the ap proximate expression for Gy^(t) given in Eq. (65) can be transformed to yield an approximate expression for Gv^(s). .-sT (66 ) Gv£(s) s s+l/T where A, B, and C are given in Eq. (65). In contrast to the usual high or low frequency approximations which are good at only one end of the spectrum, the expression given in Eq. ( 66 ) is good at both ends of the spectrum and fa irly accurate in between. 42 I f the dipole is in the lower medium, then the expression for "V/ the return time dependent field G ^ t ) produced by an impulsive unit current moment 6 (t) is given by (67) GvE(t) = ra? Eiz( t ) | z=-h» where Ejz( 0 is given by Eq. (44). The exact form is again quite complicated, but the exponential approximation yields the following expression: t-nT (68 ) GJE(t) 2: A£ 6 (t-nT) + B + C e U(t-nT), £ £ where -1 n-1 1 S.^cnh 2 ntl ’ -1 n2-l Bo = ? 1 ? * 16ire n h n +1 0 and -1 (n-1)(4n+l) n i l C. = £ - O n+1 16ttg n h n2+l 0 Again the coefficient of the delta function A is exact, and B and C k, iL have been chosen to yield the proper in itia l and final values given in Eq. (45). I f Gy^(s) is defined as the Laplace transform of G ^ t ) , then the exact expression can be obtained from Eq. (42). However, Eq. (68 ) can be transformed to yield the following expression: 43 Bfc C£ -nTs (69) GvE(s) = A£ + S + S+I7T where A£, B£, and C£ are given in Eq. (49). For a horizontal electric dipole in the upper medium, the return 'V electric field G^E(t) produced by a unit impulsive current moment is given by (70) GhECt) - fljg- E|x(t)!z=„ > where (t) is given by Eq. (51). The waveform is shown in Fig. 12, Ua and an exponential approximation yields the following expression (71) GhE(t) 2i Ah fi'(t-T) + Bh 6 (t-T) t-T' ch + Dh e U(t-T), where a = 1 n = l h n 2. n+1 8irr. c h 0 n-1 Bh = 2 n+1 ’ 16TTeQCh i n2- l O p * V 32ire0h n +1 44 and (n-1)(n+2) n -1 D,. = 32ite h ~ ^ 1 ) n2+1 0 + [2n4 + 6 n3 + 5n2 + n + 2] n (n+1) The doublet coefficient and the delta coefficient are exact, and and have been chosen to yield the correct in itia l and final values as given by Eq. (52). I f G ^ s ) is defined as the Laplace transform of G ^ (t), then the exact expression must be ob- tained from Eq. (46). However the approximate form of G ^ (t) in Eq. (71) can be transformed to yield ch Dh -sT (72) GhE(s) 2i V * Bh S + ?u7T where A^, B^, C^, and are given in Eq. (71). The frequency-dependent return fields given in Eqs. ( 66 ), (69), and (72) are directly related to the impedance of a short dipole in the presence of a half-space. I f the mutual impedance between a verti cal dipole and the half-space is Zmv and the free space impedance is ZQ, then the self-impedance of the dipole above the half-space Zy is given by <73> w v The mutual impedance between the dipole and the half-space is defined as minus the voltage induced in the dipole by the scattered fie ld for a unit current[18] and is given by ds (74) Z, 0Z-j —Gv^(jco) (ds)' mv where G^tjw) is given by Eq. ( 66 ). The free space space resistance of a short dipole RQ is given by <75> Ro - h i ? (kds>2- N 0 In order to normalize Zmv, Eq. (74) is divided through by RQ to give -j'kd B C mv -3 e A _ i JOL + D— (76) n J kd 1 + jkd (kd)' where d = 2h, A = n n+1 n2 - l = n2« ’ and 2(n-l)(n2+2) C = n(n+l)(n2+l) The real and imaginary parts of Zmv/RQ as a function of kd are shown in Fig. 13 for n = 2 and for a perfect conductor (n = »). For the range of kd where |Zmv| has a significant value, a ll three terms in 46 Eq. (76) are important. The various solutions for the impedance of a short vertical dipole above a conducting half-space have been sum marized by Wait[8] , and only a numerical integration of the Sommerfeld solution gives results which are valid for all kd. . 4 1— 1.0 0.8 0.6 0.4 0.2 2.5 - 0.2 0.2 0.6 0.8 2.5 3.5 - 0.2 Fig. 13—The mutual impedance of a short vertical dipole above a dielectric half-space. 47 I f the mutual impedance of a short horizontal dipole above a half-space is Zmh, then the self impedance Zh is given by (77) Zh = Zo + Zmh* The mutual impedance is given by -ELU )ds ox (78) "mh T —Gh£((ds) * where G^Uw) is given by Eq. (72). The normalized mutual impedance is obtained by dividing Eq. (78) by RQ to yield Z . ^ -jhd An(l+jkd) - j (79) ^ — kd 1+jkd Ro 2( kd) where (n-l)(n+2) 2(n -l) D = 2n4 + 6n3 + 5nZ + n + 2 - B , n(n+l) n’ n4(n+l)3 and An and Bn are given in Eq. (76). The real and imaginary parts of Zmj1/RQ are given as a function of kd in Fig. 14 for n = 2 and for a perfect conductor (n = °°). The decrease in |Zm^| is slower for in creasing kd than for the vertical dipole because of the existence of the geometrical term which varies as (kd)"* and dominates for large kd. If the impedance of a short dipole in an infinite dielectric medium is Z* and the mutual impedance between a vertical electric dipole and the upper half-space is Z*v , then the self impedance of £ the vertical dipole in the lower medium Zy is given by 48 1.4 r— kd -1 1 m 1.0 0.8 0.6 0.4 0.2 7Q 0.4 - 0.6 - 0 .4 - 1.0 - 0.8 0.2 - 0 . 4 Fig. 14—The mutual impedance of a short horizontal dipole above a dielectric half-space. 49 (80) 1Z = l l + Z£ . v ' v o mv For a dielectric half-space, Zmv is given by -E jz(w) ds (81) - GvE(ja»)(ds)< 'mv where GyE(jw) is given by Eq. (69). The resistance of a short dipole 0 in an in finite dielectric medium RQ is given by (82) R* = i r o (k„ ds) , o 6irn / e0 where k£ = m /c. Equation (81) can be normalized by dividing by Rq to yield 2Z "jk£d mv 3 e (83) i R M' where f - (n-l)(4n+l) n n n + 1 n* The real and imaginary parts of l n as a function of k d are shown mn ** in Fig. 15 for n = 2. The behavior is somewhat similar to the upper medium case in Fig. 9 except that a minus sign is introduced. 50 0.4 2.5 - 1.2 1.0 - 0.8 '- 0.6 -0 .4 - 0.2 0.2 Re(~ r) - 0.2 - 0 .4 — 0.6 - 0.8 — 1.0 - 1.2 -1 .4 - 1.6 Fig. 15—The mutual impedance of a short v ertica l dipole in a dielectric half-space. CHAPTER II EFFECT OF GROUND ON THE DIPOLE MODES OF SCATTERERS A. Introduction The problem of scattering by a target in the presence of a half space requires different techniques for the various frequency ranges. Since only the low frequency properties of the target are required in the detection scheme presented in Chapter I I I , a method which deals only with the dipole modes of the scatterer is developed in this chapter. Section B describes the low frequency representation of targets in free space by dipole modes. The alteration of uncoupled dipole modes due to the ground is treated in section C, and the most general case of coupled modes is treated in section D. Although the method for computing the backscattered fie ld due to the modi fied dipole modes is given in section E, the primary emphasis is on the shift in the s-plane poles of the target response due to the presence of the ground. This is because the detection scheme developed in Chapter I I I is based only on the location of the s- plane poles of the target in its particular environment. B. Dipole Modes of Scatterers in Free Space At sufficiently low frequencies, i t is possible to represent a scatterer by induced electric and magnetic dipoles regardless of 51 52 the type of illumination. If the incident electric field components e! . eV e! and the incident magnetic fie ld components h L H^, a jt ^ A j r £ are known at the scatterer location, then the components of the induced electric dipole moment p , p , p and the components of c a «y «*■ the induced magnetic dipole moment pmx, pmy, pmz can be obtained by a matrix multiplication involving the incident field and a polari- zability matrix a. (84) P = a F, where ' pex ' pey P = F = m m pez -* N -*•<< pmx "1 Pmy " i pmz "1 “ 11 “l2 “13 “14 “15 “ 16 a21 (*22 • • • • a - “31 ' “41 * “51 * “61 * “ 66 53 In general, the polarizability matrix a does not reduce into two 3 x 3 matrices representing electric and magnetic polarizabilities nor can i t be diagonalized by a coordinate transformation. However, for objects of sufficient symmetry, such as the sphere and prolate sphe roids considered in this chapter, the reduction occurs and the diagonalization of the two matrices is possible. This reduction also occurs in the Rayleigh lim it where the diagonalization is pos sible. The necessary conditions for Rayleigh scattering are covered in Van de Hulst[19]. The main feature of the Rayleigh lim it is that the elements of a become frequency independent, and the resultant far-zone scattered fie ld has the well-known s Rayleigh region dependence. When the polarizability matrix is allowed to be frequency de pendent, the resultant scattered field is valid into the first resonance region as well as the Rayleigh region. For example, the conducting sphere requires one scalar electric and one scalar magnetic polarizability as shown in Stratton[20]. The magnitude and phase of the far-zone backscattered field due to the electric and magnetic dipole modes are shown in Fig. 16 as a function of ka, where k is the wave number and "a" is the sphere radius. The exact solution is also shown, and the two curves are fa irly close up to ka = 1 . Although the dipole mode solution is not valid above the fir s t reso nance, the inverse Laplace transform of the dipole mode terms yields 2 a fa irly accurate ramp response since the 1/s weighting reduces the contribution of the high frequency portion of the spectrum. 54 360* DIPOLE MODES EXACT SOLUTION 180 0.5 2.0 ka Fig. 16—Backscatter from conducting sphere. The ramp response is defined as the normalized backscattered field when the incident field has a ramp time dependence, tU (t). The individual electric and magnetic dipole contributions to the ramp response and their sum are shown along with the true ramp response (Kennaugh and Moffatt[21]) in Fig. 17. The agreement between the two curves is quite good for medium and large time ( t > 2 t ) and fa irly good for smaller values of time. The main difference between the two curves is that the dipole curve has a zero slope at the 55 0.1 h . MAGNETIC r 2 - 0.1 ELECTRIC DIPOLE TERMS TRUE RESPONSE - 0.2 ELECTRIC + MAGNETIC - 0.3 Fig. 17— Ramp response of conducting sphere. origin while the slope of the true curve is non-zero. For low frequencies the electric and magnetic polarizabilities of the conducting sphere, ag(s) and am(s), are given by where Kg and Km are frequency independent. Consequently, the response of the conducting sphere has s-plane poles at s = ( - 1 /2 ± j j 3/2)c/a due to the electric dipole and at s = -c/a due to the magnetic dipole. Other poles farther from the origin resulting from higher order modes 56 have been calculated by Stratton[20], but the inverse Laplace trans form of the three dipole mode poles has been shown to yield a fa irly accurate ramp response in Fig. 17. Another target for which an analytical solution exists is the thin conducting prolate spheroid. Only a single electric dipole mode along the major axis can be supported, and the electric polari zability has the form K (8 6 ) a (s) = ------o p , e (S + y ) 2 + 3 where the complex poles are located at s = -y ± j3 . These poles have been calculated by Page and Adams[22] as a function of the prolate spheroid eccentricity. As the eccentricity goes from one (zero thick ness) to zero (sphere), the poles go from s = ( ±jtt/ 2) c/a to s = (-.5 ± jj3 /2 ) c/a, where "a" is the semi-major axis. C. Change in Dipole Modes Due to Ground When Dipole Modes Do Not Interact 1. New polarizability I f a scatterer is located above or within a half-space, then an interaction occurs between the target and the interface which alters the target response. The simplest case to consider is a thin prolate spheroid located at a height, h, above a half-space as shown in Fig. 18. The only dipole mode existing is the electric dipole along the major axis with the polarizability given in Eq. (2). I f the spheroid is oriented at an angle e to the vertical and an electric field, EQ(s), 57 Mo, *o V7~rr/-7^/'z/'r/T/ Mo » Fig. 18—Prolate spheroid above a half-space and its induced dipole moment. is applied along the major axis, then a dipole moment, Pe(s), is induced in the spheroid. This dipole moment consists of a free space term plus an in finite number of terms resulting from inter action with the half-space. The free space term of Pe(s) is given by the product of the polarizability and the incident fie ld , EQ(s) ae(s). This dipole moment produces a secondary electric fie ld at the spheroid which is given by EQ(s) ae(s) s G(s), where G(s) is the parallel component of the secondary electric field produced at the source point by a unit current moment above the half-space. This secondary electric fie ld induces an electric dipole moment term which is given by Eq(s) ct^(s) s G(s). The process can be re peated to find the succeeding interaction terms, and the total electric dipole moment is given by an in finite sum where the fir s t term (n = 0 ) is the free space term and the re maining terms represent the interaction between the spheroid and the half-space. Since Eq. (3) contains a geometric sum, i t can be summed in closed form EJs) a (s) (88) pe(s) = I - aeTsJs" gT'sT Consequently, the spheroid in the presence of the half-space can be considered equivalent to an isolated target with a new polarizability, otg(s), given by «e(s) (89) c ^ ( s ) = f T a- J sy s- G Ts")' Thus the new polarizability depends on the target location and orientation and the half-space composition, and these effects are included in G(s). Since the earlier solutions for dipoles over half-spaces were derived for either vertical or horizontal dipole orientation, i t is more convenient to express G(s) in the following manner (90) G(s) = Gy(s) cos2e + Gh(s) sin2e , where Gy(s) is the return fie ld at the source for a vertical unit current moment and G^(s) is the return field at the source for a horizontal unit current moment. The two most obvious limiting cases 59 are for the target to recede to in fin ity or the half-space param eters to approach the free space values. In either case, G(s) goes to zero and a'(s) goes to a (s), the free space value. 6 G Scatterers which can support more than one dipole mode present a more d iffic u lt problem because the possibility of coupling be tween the various dipole modes exists when the scatterer is placed in the presence of a half-space. However, there are some modes which w ill not couple and can be treated using Eq. (89). For example, consider a sphere located above a half-space as shown in Fig. 19. Both a vertical electric and a vertical magnetic dipole mode are shown because these modes do not couple with any other modes and can be treated individually using Eq. (89). The reason that the vertical electric dipole mode w ill not couple with other modes is that the secondary field of an electric dipole over a half-space contains only a vertical component of the electric field and no magnetic field at the source point. Consequently, the vertical electric mode will excite no other modes and no other modes w ill excite the vertical electric mode. The same reasoning applies to the vertical magnetic mode. Fig. 19—Sphere above a half-space with uncoupled electric and magnetic dipole modes. 60 I (s) 2 h ■ 1 | UNIT DIPOLE SOURCE Fig. 2 0 —Dipole moment induced in sphere by dipole source. An analytical check is available for the dipole moment induced in a sphere by a dipole source as shown in Fig. 20. This is the equivalent of a sphere over a perfect conductor as obtained by image theory. The source is the result of the primary dipole moment induced in the sphere and Pei-(s) is the fir s t interaction dipole moment. Harrington[5] has derived the multipole solution for the scattered field , and the following induced dipole moment is ob tained from the fir s t term of the solution. 2u, JJ(ka) Join (91) Pel-(s) = ^ H ^ k Z h ) 2n(2h) 2 k H (ka) w= s /j. However, the fir s t bracket expression is equal to the incident fie ld of the source at the center of the sphere, and the second expression is the polarizability of a sphere of radius "a". Consequently, the method of multiplying the incident field at the sphere center by the polarizability to obtain the induced dipole moment appears to be valid for a dipole source as well as a plane wave source. 61 2. New poles In the following sections, i t w ill be convenient to make an approximation on the new polarizability, c^(s), so that i t w ill have the same form as Eq. ( 8 6 ), and only the constants (y, 3 , and Kg) will change as a result of the introduction of the half-space. This approximation simplifies analysis of the target detection scheme based on the location of the target s-plane poles, y ± je. Consequently, i t is desirable to calculate the shift in the target poles due to the introduction of the half-space. The new poles of a thin vertical prolate spheroid above a half space can be determined from Eq. (89) with the appropriate G(s) function. Consider fir s t the case of vertical orientation above a perfect conductor. In this case, G(s) = Gv(s) as shown by Eq. (90) and has the following form A power series expansion of e~s^ c has been used to obtain the ap- proximate form of Gy(s) in Eq. (92). Using Eqs. ( 8 6 ), (89), and (92), the new polarizability, a^(s), can be written 62 K e (93) o'(s) = - 2Ke(c/2h)2‘ p 2 2 2 1 + S^ + 2yS + y +B 8ite0 Ch e =J 1 - (b/a)2, where b is the semi-minor axis of the prolate spheroid and y and g have been calculated as a function of b/a by Page and Adams[22], The quadratic equation can be used to find the poles of <*^(s) in Eq. (93), and Fig. 21 shows the real and imaginary parts of the poles as a function of h for three b/a ratios. Only the .013 ratio really qualifies as a thin prolate spheroid with a single dipole mode, but the other two ratios are valid for computing the shift in the uncoupled vertical electric mode poles even though other modes could be present. In fact, the .995 ratio gives essentially the pole shift in the vertical electric dipole mode of the sphere. It can be seen from Fig. 6 that the pole shift is greater for higher b/a ratios, as the spheroid approaches a sphere. This is expected since such spheroids are stronger scatterers (larger Kg) and the interaction is stronger. Figure 21 also indicates that the poles are nearly equal to their free space values for h/a > 1 0 . 63 0.013 1.6 0.6 0 .9 9 5 0.8 0 .9 9 5 0 .4 0.6 0.013 0 2 4 6 8 10 a Fig. 21—New poles for vertical prolate spheroid over perfect conductor. To repeat the analysis for a vertical prolate spheroid above a dielectric half-space, we substitute the Gy(s) function for a dielectric half-space into Eq. (89). The approximate form of Gy(s) is given in Eq. ( 6 6 ), and a low frequency expansion yields the following expression (94) G,(s) = ± < B + A + ( C - B ) f j s B-2cJ t 2-A tJ where A, B, C, and T are defined in Eq. (6 6 ). Substituting Eq. (94) into Eq. (89) and again using the quadratic formula, the new poles 64 for a vertical prolate spheroid over a dielectric half-space can be determined. The result is valid for arbitrary axial ratios, and Fig. 22 shows the new poles of the vertical electric mode of a sphere over a dielectric half-space as a function of height and refractive index. In the lim it of either n = 1 or h = », the poles go to the free space values (y = .5 c/a, 3 = .8 6 6 c/a). In the lim it of n approaching in fin ity , the results go to the con ducting half-space case of Fig. 21. The surprising feature of Fig. 22 is that the pole shift for some intermediate values of n is greater than the pole sh ift for the conducting half-space. In fact, the greatest shift in y was obtained for n approximately equal to 2. Similar results were obtained for prolate spheroids of various axial ratios. In order to find the new poles of a horizontal prolate spheroid over a dielectric or conducting ground, i t is necessary to have the G^(s) function. This function has been given for a dielectric half space in Eq. (72), and the conducting half-space solution can also be obtained by setting n equal to in fin ity . A low frequency ex pansion of Eq. (72) is substituted into Eq. (5), and the resulting new poles for a thin horizontal prolate spheroid of axial ratio .013 are given in Fig. 23. I t is not possible to use the same solution as the spheroid approaches a sphere as was done for vertical orientation because a horizontal magnetic dipole mode exists which is coupled to the horizontal electric dipole mode. 65 0.8 P i 0.6 o o y - 0.4 0.2 8 10 _h_ a Fig. 2 2—New poles of vertical electric dipole mode of sphere over dielectric half-space. Figure 23 reveals a slight increase in the magnitude of the new poles for the conducting half-space (n = °°). However, for low values of n, a small decrease is found in the magnitudes of the poles. For a general prolate spheroid orientation, the G(s) function of Eq. (90) is used. The results are applicable to thin spheroids only since coupled modes w ill result for thick spheroids at a general angle. The results for general orientation lie somewhere between the results of Fig. 22 and 23. 66 n = 1000 1.4 0.16 n = 1000 0.15 0.14 0.15 0 2 4 6 L 8 10 h a Fig. 23—New poles of a thin horizontal prolate spheroid above a dielectric half-space, axial ratio = .013. The remaining uncoupled mode which can be handled by Eq. (89) is the vertical magnetic dipole mode. It is necessary only to replace oi (s) by am(s) and to obtain the proper G(s) function. In c 171 this case, G(s) is defined as the return magnetic fie ld at the 67 source produced by a unit magnetic current moment. For a con ducting half-space, G(s) is obtained from image theory and is given by s + c_ _s 2h (95) G(s) = ------e c 8 iry QCh S In order to solve for the new pole of the vertical magnetic dipole mode of a sphere over a conductor, Eq. ( 8 6 ) and a low frequency expansion of Eq. (95) are substituted into Eq. (89) to obtain the new magnetic polarizability, “^(s) (96) a'(s) = Km nr : KT s + - + 3 16TTVi0h3 Km = - 2”a?i,oc • The new pole is simply the one real root of the denominator of Eq. (96), and i t is given by a3 (97) s = - Consequently when the sphere is touching the ground (h = a), the new pole is given by s = -.875 c/a. So far only perfectly conducting and lossless dielectric half spaces have been considered. I f the half-space is lossy, then the problem is more d iffic u lt because there is no solution for G(s) which is valid over both a large range of heights, h, and a large range of complex refractive index, N. Wait[17] has summarized the various approximations for the return fie ld of a short vertical dipole over a lossy half-space. The solution which is asymptotic in kh is not useful because we are interested in small values of h and low frequencies and consequently small values of kh. Another solution which is asymptotic in |N| is valid for all values of kh. However for large values of |N|, the solution for G(s) and the re sultant new poles d iffe r very l i t t l e from the perfectly conducting half-space case regardless of the argument. I f the magnitude of N is allowed to become small enough that an appreciable change from the perfectly conducting case results, then the asymptotic solution in |N| is questionable. Consequently, in order to handle the troubl some case of N small and complex, a numerical integration of the exact Sommerfeld solution is required as shown in Wait[6 ]. How ever, this would require a different method of finding the new poles since an analytical solution in s was used for the previous cases. A possibility would be to plot the new curve ag(jw) as a function of m and determine y and 3 from the shape and location of the resonance peak. A check of this method was attempted for the perfectly conducting half-space, but the results were not as accurate as those obtained by the low frequency approximation method. 69 D. Coupling of Dipole Modes When the dipole modes of a scatterer are coupled as a result of the presence of a half-space, then a more complicated formu lation than that of Eq. (5) is required to determine the half space effect. Two dipole modes can be considered coupled when one dipole mode produces a field which excites the other dipole mode. For instance, horizontal electric and magnetic dipole modes which are perpendicular w ill couple because each produces fields which excite the other. A sphere over a half-space supports coupled horizontal electric and magnetic dipole modes as shown in Fig. 24. The electric dipole moment, pe, is in the y direction and the magnetic dipole moment, p^, is in the x direction. Since the sphere has complete rotational symnetry the directions of the dipole modes are actually arbitrary, but the x and y axes are chosen here so that the incident field excites only the coupled pair of dipoles in Fig. 24 plus either a vertical electric or a vertical magnetic dipole. Fig. 24--Coupled horizontal electric and magnetic dipole modes of a sphere over a half-space. 70 A matrix analogy to Eq. (87) leads to the following expression for pe(s) and pjs) ■ n a S Gr asGr otE ■ p e ' CO e Ee m Em e o = I • n=0 a S G,, ot sGij a H - Pm - e He m Hm L m o J where is the return electric field produced by a unit electric current moment, is the return electric fie ld produced by a unit magnetic current moment, GHe is the return magnetic field produced by a unit electric current moment, GHm 1S the return mag netic field produced by a unit magnetic current moment, EQ is the y component of the incident electric fie ld , HQ is the x component of the incident magnetic fie ld , ag and am are the electric and mag netic polarizabilities given in Eq. (85), and all quantities are functions of s. I t can be seen from Eq. (98) that the quantities G^m and G^g provide the coupling between pg and pm* I f G^m and GHe were zero, then Eq. (98) would reduce to a pair of scalar equations of the form of Eq. (87). I t is possible to simplify Eq. (98) by using the following matrix identity for a square matrix, M: (99) I Mn = (I-M )"1, det M < 1, n=0 where I is the identity matrix. The restriction of det M < 1 is no problem since the configuration must be stable, and Eq. (98) can be rewritten -1 71 (100) Equation (100) could be used either to calculate the scattered field for particular incident fields, EQ and HQ, or to calculate the new poles resulting from the coupled electric and magnetic dipole modes. The new poles can be calculated from Eq. (100) by examining the inverse matrix. The poles of all elements of the inverse matrix are given by the zeros of the determinant: ( 101) det = 0 For the case of a perfectly conducting ground, the four G functions of Eq. (101) can be obtained from image theory and are given by e 72 I f the low frequency approximations of the functions in Eq. (102) are substituted into Eq. (101) and terms up to s are retained, then the three new poles can be obtained by solving the cubic equation in s. The resultant new poles are given in Fig. 25 as a function of n, 1.0 0.8 0.6 0 .4 S v 0.2 h a Fig. 25—New poles of coupled horizontal electric and magnetic dipole modes for a sphere over a perfect conductor. 73 and i t can be seen that as h gets large the poles go to s = -c /a , (-.5 ± j.86 6 )c/a which are the free space values. Since the sphere above the half-space can also support uncoupled vertical electric and magnetic dipole modes, there are four possible modes and six possible poles, some of which may not be excited for particular incident fields. For instance, a horizontally polarized plane wave will not excite the vertical electric dipole mode, and a vertically polarized plane wave w ill not excite the vertical magnetic dipole mode. Although the coupled dipole modes considered here were simplified because of the rotational symmetry, any object with arbitrary orientation relative to the half-space could be handled. The most general case involves 6 x 6 matrices rather than 2 x 2 as in Eqs. (98), (100), and (101), A matrix G whose elements are the components of the return fields produced by unit current moments is written G11 G12 ’13 14 G15 G16 G21 G22 (103) 31 541 351 561 66 where Gjj is the x-component of the return electric field produced by an x directed unit electric current moment, G^ i s the x-component 74 of the return electric field produced by a y directed unit electric current moment, etc. Using the same method which was used in Eqs. (98), (99), and (100), the induced dipole moments can be written (104) P = I (s a G)n a F = (I - S a G) " 1 a F, n=0 where P, a , and G are defined in Eq. (84) and I is the 6 x 6 identity matrix. The new poles of the configuration are then given by (105) det(I - s a G) = 0. E. Backscattered Field The presence of ground alters the low frequency scattering properties of a target by modifying both the polarizability of the dipole modes and the incident fie ld . The fir s t effect alone determines the shift in the s-plane poles of the target and is sufficient for difference equation calculations which involve only the poles and not the excitation. However, i f the back- scattered field as a function of frequency and aspect angle are required as in a matched f ilt e r receiver, then both effects must be included. To illustrate the method of combining the two effects, consider the case of a thin prolate spheroid vertically oriented at a height h above ground. If a vertically polarized plane wave is incident 75 at an angle 0 to the vertical as in Fig. 26, the vertical com ponent of the incident field at the center of the spheroid, Ez(s ), is given by •s cos 0 c (106) Ez(s) = Eq(s) 1 + Rv e sin 0 where Ry is the reflection coefficient of the ground for vertical polarization. This field induces a dipole moment in the z-direction, P~(s), which can be determined by Eqs. ( 8 8) and (106), and is given by 2h EQ(s) a(s) -s cos 0 (107) 1 + Rv e sin 0 . pe ^ = 1 - a(s)s G„(s) 777777777777 Fig. 26—Vertically polarized plane wave incident on a prolate spheroid over a half-space. 76 The far-zone backscattered fie ld , E (s ), in terms of the dipole moment is rtL . -s p (s) sin' •S COS 0 -s (108) E (s) = e 1 + R.. e 4ttc c“r o The complex phasor response, G (s), is defined in terms of the scattered field : r (109) Gs(s) = 2r e S c | ^ | | | Using this definition along with Eqs. (107) and (108), the complex phasor response of the spheroid over the ground can be written 2 . 2 •S — COS s sin ( g(s) c ( 110) GS(s) ! + Ry e 1 - a (s)S Gy(s) 29 tte Co 2 o The two effects of the ground on GJ(s) can be seen directly from Eq. (110). The change in polarizability is represented by the factor, [l - a(s)s G(s)]-1, and the altered excitation is represented by the -S COS factor, 1 + Ry e For objects which support more than one dipole mode, the prob lem is more complicated, but the same general procedure is used. The incident electric and magnetic fields must firs t be calculated, and the dipole moments can then be calculated in the manner indicated by Eq. ( 8 8) or Eq. (100) for coupled modes. The far-zone fie ld is the sum of the contributions radiated from each dipole moment as shown for the single electric dipole in Eq. (108). CHAPTER I I I THE DIFFERENCE EQUATION IN TARGET DETECTION AND DISCRIMINATION A. Introduction The techniques for target detection and discrimination described in this chapter require a radar system capable of obtaining ramp re sponse waveforms. Although no such operational radar now exists, a laboratory system utilizing a discrete frequency sampling technique has been used to obtain transient responses of various targets and the implementation of such a system as an operational radar appears to be technically feasible (Young, Moffatt and Kennaugh[23]). Con sequently, the purpose of this chapter is to explore the potential u tility of ramp response waveforms in detection and discrimination problems, and the implementation problems will not be considered. The detection scheme which is presented is actually based on the difference equation that the transient response of the target must satisfy rather than the actual waveform. The reason for using this type of scheme is that the difference equation which depends only on the s-plane poles is independent of aspect and polarization whereas the actual response waveform is not. A conventional matched f ilt e r receiver requires that the target waveform or frequency response be known for all aspects and as 77 78 a result the characterization of any one target is quite cumbersome. However, in the scheme presented here, the target is characterized only by its s-plane poles which are independent of the excitation. This type of characterization is analogous to a multi-port lumped circuit network where the various transfer functions may differ between ports, but the s-plane poles do not. Consequently, any scheme based only on the poles would apply equally well at any port in the same manner that the proposed target detection scheme should apply for any aspect and polarization. B. The Difference Equation for Transient Responses of Scatterers I f the Laplace transform of a target response can be written as a rational function in s, then the transient response, R(t), must satisfy the following difference equation as shown by Corrington[23] (109) R(t) = I ( - l ) k+1 Fn k R(t-kAt), k=l n*K where the order, n, of the difference equation is equal to the number of poles in the Laplace transform of R(t) and the difference coefficients, F . , are simple functions of the poles and the time n |K increment, At. Since the difference coefficients depend only on At and the poles, they are independent of aspect and polarization even though R(t) is not. The difference equation holds for all values of t > nAt, and the time derivatives of R(t) also satisfy the same difference equation. Consequently, the step and impulse responses satisfy the same difference equation as the ramp response. 79 To illustrate the functional dependence of the difference coefficients on the s-plane poles, consider the example of the backscatter from a conducting sphere in free space. The reasonably good agreement between the exact ramp response and the ramp response generated from the three dipole mode poles shown in Fig. 17 indi cates that the ramp response might satisfy a third order difference equation quite well. For a general third order difference equation, the three difference coefficients are given by the following ex pressions: S.At S?At S-At ( 110) Fg i = e + e + e (Sj+S^At (Sj+SgjAt (s2+Sg)At (sl+s2+sg)At where Sj, s2, and Sg are the three s-plane poles. In the general formulation of the difference coefficients, Fn k, the subscript n tells how many poles are to be used and the subscript k tells how many poles are to be taken at a time in forming the exponents. Consequently, F . is the sum of n!/(n-k)!k! terms. The three 11 • l\ dipole mode poles for a sphere are given by (111) where "a" is the sphere radius. When these specific values are substituted into Eq. (110) and the complex exponentials are ex pressed in terms of sines and cosines, the resultant difference coefficients for the conducting sphere are 012) F3j1 = e + 2 e COS 'h r — At + 2 e In order to determine how well the ramp response fits this d if ference equation, the difference equation along with three equally spaced values from the true ramp response was used to predict a value. By repeating this procedure for various values of time, a new curve was formed and compared to the true ramp response. The integral of the squared error was then com pared to the integral of the squared ramp response. For all values of At < a/c the error was found to be less than 10%. The same procedure was followed with a ramp response synthesized from a small number of theoretical values at odd harmonics. The integrated error was again less than 10% as long as the funda mental was sufficiently low (kQa < .4) and at least four har monics were used. An interesting result was that the f i t did not improve significantly for an increase above four in the number of 81 harmonics used. This is somewhat fortunate since the experimental results given later w ill contain only four odd harmonics. 7 7 7 7 7 7 7 777777777777 Fig. 27—Scattering from a hemispherical boss using image theory. The analytical solution for scattering from a sphere can also be employed in the problem of the backscatter from a hemi spherical boss on a perfectly conducting ground. Image theory can be used to replace the ground by an image source and to replace the hemisphere with a sphere as shown in Fig. 27. Consequently, the backscatter solution for the boss consists of the backscatter from a sphere plus the bistatic scattering term for an angle 2e. The important feature is that the s-plane poles of the hemispherical boss are the same as the poles of the free-space sphere. Using the difference coefficients for a sphere as given in Eq. (112), the synthesized ramp response of the hemispherical boss was tested for the difference equation f i t in the same way that the sphere was checked. For vertical polarization and an angle of 45°, the f i t is approximately as good as for the sphere. However, the f i t is much worse for horizontal polarization. This is probably because the electric dipole mode is shorted out and only the pole associated with the magnetic dipole mode is excited. Consequently, the errors which result from using only 3 poles become more important with the normally dominant electric dipole mode missing. The dipole modes of a sphere above a perfectly conducting half-space were analyzed previously to determine the new poles which result from the introduction of the perfectly conducting half space. Figure 28 shows the dipole modes which are excited when a vertically polarized plane wave is incident upon a sphere on a conducting ground. The poles of the vertical electric dipole mode, pev, shown in Fig. 21 are s = (-.334 ±j,624)c/a. The horizontal electric and magnetic dipole modes are actually coupled, but the pole at s = -.898 c/a is more closely associated with the magnetic dipole, pmh, and the poles at s = (-.801±j.905)c/a are more closely associated with electric dipole, peh. For low frequencies the radiation of the vertical electric dipole and its image are in phase and w ill add, and the same is true for the horizontal mag netic dipole. However, the horizontal electric dipole and its image are out of phase and w ill contribute l i t t l e to the scattered field . Consequently, instead of using all five poles to form a fifth order difference equation, i t was decided to exclude the two poles associated with the horizontal electric dipole and form 83 Fig. 28—The dipole modes of a sphere above a conducting half-space excited by vertical polarization. a third order difference equation using the other three poles, s = (-.898, -.334±j.624)c/a. The three difference coefficients were formed by substituting these values into Eq. (110). Ex perimental backscatter measurements of amplitude and phase at a 45° aspect angle were made for a sphere on a conducting ground plane at four odd harmonics with a fundamental ka of . 1732. Titus at the lowest frequency the circumference of the sphere was ap proximately l / 6 x and at the highest frequency i t was slightly greater than a wavelength. These harmonics were used to synthesize the ramp response shown in Fig. 29, and this waveform was checked for agreement with the theoretical difference equation. The in tegral of the squared error between the input ramp response of Fig. 14 and the waveform predicted by the difference equation was calculated for various values of At. A new quantity similar to a correlation is defined as 84 2 a Er Fig. 29—Ramp response for a sphere on a conducting ground for vertical polarization and a 45° aspect angle. I [ER(nAt) - ER(nAt)]2 n (H3) p« = i ------I ER(nAt) n where ER is the synthesized ramp response and ER is the predicted ramp response. The ratio of the summations is the normalized error with integrations approximated by summations, and p' is plotted vs At in Fig. 30. The result for the free-space poles is also shown, and the f i t is considerably better for the new poles. This fact tends to confirm the results obtained by the fairly simple model involving the dipole modes and their modification by a half-space which was developed previously. es i.o 0.8 0 .6 I P 2 a 0.2 0.2 0.4 0.6 0.8 - 0.2 At T Fig. 30—The correlation between the synthesized ramp response of a sphere on the ground and the predicted waveform. The fact that fa irly accurate difference equations for the ramp response can be obtained from dipole mode poles is important because the change in the dipole modes caused by the introduction of a half-space can be determined with at least some success as shown in Fig, 30, Consequently, any target detection or discrimination method involving a difference equation obtained from dipole modes can be used for targets on the ground as well as in free space. However, the difference equation for the target above the ground is formed from the modified poles of the target-ground configuration which are determined by the technique shown in Chapter I I . 86 C. Discrimination of Targets by a Difference Equation Receiver In order to test shape discrimination techniques utilizing the difference equation, the ramp response of a thin prolate spheroid was tested for f i t with the difference equation of a sphere of the same maximum dimension. The thin prolate spheroid can support only a single electric dipole mode with the polarizability as given by Eq. (8 6 ). If this single dipole mode is used, then the approxi mate ramp response is a damped sinusoid with the frequency and at tenuation constant depending on the length of the major axis and the axial ratio. For a semi-major axis "a" and an axial ratio of .013, the complex poles are located at s = (-.1495 ± jl.536)c/a, and the ramp response is shown in Fig. 31. 2 a Fig. 31—Ramp response of a thin conducting prolate spheroid. 87 PREDICTOR : SPHERE DIFFERENCE EQ. , A t. INPUT o------PROLATE SPHEROID RAMP RESPONSE PREDICTOR SPHERE DIFFERENCE EQ., A t, Fig. 32—Block diagram discriminator using two different time increments. The block diagram in Fig. 32 indicates a method for testing the discrimination properties of the difference equation utilizing two different time increments. If the prolate spheroid ramp response waveform along with the sphere difference equation is used to predict a new waveform, then this predicted waveform should d iffe r from Die input waveform and should depend on the time increment, At. Two predicted waveforms using different time increments, At^ and At,,, were formed, and their correlation as a function of Atj and At£ are shown in Fig. 33. It is desirable to make the correlation as low as possible since we are attempting to discriminate against the spheroid. Consequently, for a given value of Atj, the best choice for At^ is zero. However, the predicted waveform with At = 0 is identical to the input waveform which means that the best procedure is to com pare a predicted waveform with the input waveform. This is true because the predicted waveform changes rather gradually with At 88 = 0.4 = 0.6 0.8 z o b- < 0.2 0.4 0.6 0.8 1.0 _l UJ oc a. o o — I Fig. 33--Correlation of two predicted signals, thin prolate spheroid into sphere difference equation. as shown in Fig. 34, and the greatest difference occurs between the input waveform and the predicted waveform. Figure 34 demon strates that the in itia l interval of the waveform which can not be predicted increases with increasing At since the difference equation can be used only for t > 3At for a third order difference equation. The same procedure can be followed for discrimination against a sphere using the difference equation of a thin spheroid. I f the approximate three pole ramp response is fed into the second order difference equation for the same spheroid (.013 axial ratio), the 89 INPUT A Fig. 34—Comparison of input waveform with predicted waveforms, spheroid into sphere difference equation. change in the predicted waveform for increasing At is as shown in Fig. 35. The predicted waveform departs from the input for lower values of At than in the previous reverse case. The correlation, p, of the predicted and input waveforms was again calculated as a function of At, and quantity involving the error integration, p ', defined in Eq. (113) was also calculated. Both quantities are plotted as a function of At for both discrimination cases in Fig. 36. In both cases p' drops off more rapidly than p which means that p! is probably a more useful discrimination quantity. This w ill be shown to be particularly true in discriminating against very slowly varying waveforms with very few zero crossings. In these cases i t 90 \ Fig. 35—Comparison of input waveform with predicted waveforms, sphere into thin spheroid difference equation. ------SPHEROID INTO SPHERE EO. ------SPHERE INTO SPHEROID EQ. 1.0 0.8 0.6 0.4 0.2 0.2 0.6 0.8 At - 0.2 - 0.4 - 0.6 - 1.0 Fig. 36—Correlation of predicted and input waveforms. 91 is possible to have a very large error and s till have a high cor relation. However, p' w ill not be high and the discrimination will be provided. Another discrimination problem of interest is that of spheres of different sizes. I f the three pole ramp response of a sphere is fed into the difference equation of a sphere of radius aQ, then p' will equal 1 for a sphere of radius aQ regardless of A t, but w ill be less for sphere radii either greater or less than aQ. The behavior of p* as a function of radius is shown for several values of At in Fig. 37. I t is easily seen that discrimination against smaller = 0.4 = 0.6 t P = 0.8 0 0.5 = 0.8 0.6 Fig. 37--Discrimination against spheres of different sizes by use of the difference equation. 92 spheres is easier than discrimination against larger spheres. This is expected since rapidly varying waveforms are easier to discrimi nate against than slowly varying waveforms. In practice, the shape of the p' curve might be useful in detecting spheres within a cer tain range of size. For instance if a time increment, At = .8 t , and a threshold of p' = .8 were selected, then Fig. 37 shows that spheres ranging from approximately .88aQ to 1.23aQ in radius would be accepted while all others would be rejected. By varying the time increment and the threshold of p1, i t would be possible to obtain other ranges of acceptable sphere sizes. Although we were dealing only with pair-wise discrimination in the sphere-spheroid example, the extension to an arbitrary number of targets, N, involves identical calculations as shown in Fig. 38. The input signal is fed into the difference equation for each possible target, and N predicted signals are obtained. Each one is compared to the input waveform, and N values for p^ are obtained. A comparator picks the largest p^ and the associated target. The main advantage of this receiver is that each target is represented by a single difference equation which is independent of aspect. This is a significant simplification over a matched f ilt e r receiver where the representation of just one object at a ll aspects is generally quite complicated. 93 PREDICTOR, DIFF. EQ. I SQUARED ERROR INTEGRATOR PREDICTOR , DIFF. EQ.2 SQUARED ERROR INTEGRATOR INPUT DECISION : TARGET j PREDICTOR, DIFF. EO. N SQUARED ERROR INTEGRATOR Fig. 38—Difference equation receiver for N targets. D. Detection of Targets in the Presence of Clutter Using a Difference Equation Receiver In order to test the detection properties of the difference equation, the problem of detection of a conducting hemispherical boss on a conducting ground with experimental data was examined. Backscatter measurements of amplitude and phase at four odd harmonics were taken at an aspect angle of 45° for three cases: 1) a hemispherical boss on the ground, 2 ) coated a rtific ia l trees on the ground, and 3) coated a rtific ia l trees plus a hemispherical 94 boss on the ground. This was done for several sizes of the hemi spherical boss and for both vertical and horizontal polarization. I t was previously shown that the difference equation for the hemispherical boss is the same as the difference equation for the free-space sphere whose third order difference coefficients are given by Eq. (27). However, i t was also shown that the d if ference equation was not very accurate for horizontal polarization because the two poles of the electric dipole mode are not excited. Consequently, the detection was attempted using vertical polarization. The fir s t step was to synthesize the ramp responses from the experimental four-frequency data. The ramp responses for the three separate configurations were then fed into the difference equation for the desired size boss, and the predicted waveforms were gen erated. These waveforms were compared to the input waveforms and the p‘ quantity was calculated using the squared error integration. Since the measurements were taken using the same coated trees and different size bosses, cien the easiest boss to detect should be the largest one because the signal to clutter power ratio, S/C, should be highest. The largest boss had a fundamental ka of .75, and Fig. 39 shows a plot of p ' versus the time increment, At, for the three cases. The signal to clutter power ratio shown was computed by taking the ratio of the area under squared ramp responses of the boss alone and the trees alone. By Parseva^s theorem this is also equal to the ratio of the sum of the squares of the harmonic amplitudes with ramp weighting. This may not be 95 1.0 BOSS BOSS + TREES 0.6 I P 0.4 0.75 TREES 0.2 0.2 0.3 0.50.4 At r fitj. 39—Detection of boss with vertical polarization at a 45° aspect angle. an accurate value since the trees might partially shield the boss and lower the ratio somewhat. The important feature of Fig. 39 is the difference between the trees curve and the boss plus trees curve. The difference between the two curves is a measure of the ease of detecting the boss in the trees. For instance, if At equals .4 i, then a threshold setting on p' anywhere between ,06 and .7 would detect the boss in the trees and reject the trees alone. Detection is also possible for other values of At with different thresholds of p', but t = .4t is the optimum value 96 because the curve separation is greatest. The curve for the boss alone is also shown just to demonstrate the good f i t of the d if ference equation. I t is necessary for the target alone to f i t the difference equation well so that the curve for the target plus clutter will be well above the clutter curve. Although the difference equation f i t is important in raising the target plus clutter curve, the target to clutter power ratio is of equal importance in raising the target plus clutter curve. To illustrate this fact, Fig. 40 gives the same curves as Fig. 39, but for two smaller bosses with correspondingly smaller signal to clutter ratios. I t can be seen that the separation between the curves is less for the smaller boss even though the smaller boss s till fits the difference equation quite well. Measurements were made for even smaller bosses and the addition of the boss s till raised the value of p1. However, the change becomes quite small as the signal to clutter ratio becomes small. For instance, a smaller boss with a signal to clutter ratio of .33 provided a maxi mum shift of only .017 in p' whereas the larger boss in Fig. 39 with S/C = 3.11 provided a maximum shift of .642 in p'. However, the small separation for small targets could probably be enhanced by an integration in aspect. This integration is quite simple because the difference equation is independent of aspect. Similar measurements were also made with vertical polari zation and a 45° aspect angle for a sphere on the ground among a rtific ia l trees. The new poles were used in the difference 97 i.o BOSS 0.8 BOSS + TREES 0.6 I P 2.15 0 . 4 0.2 TREES 1.0 BOSS 0.8 BOSS + TREES 0.6 I P 1.72 0 . 4 kn 0 *= 0.675 TREES 0.2 0.2 0 .3 0 .4 0 .5 A t T Fig. 40— Detection of different size bosses in the same clutter. 98 equation since they were shown to provide a fa irly good f i t in Fig. 30. However, the sphere was quite small (kQa = .26), and the clutter level was quite high. As a result, the signal to clutter ratio was only .017 and the maximum increase in p' re sulting from the introduction of the sphere was .004. This sepa ration is much too small for reliable detection, and an aspect integration would be required to increase the separation to a reasonable value. DIFF. EO. SQUARED PREDICTOR ERROR INTEGRATOR p' > />1 t TARGET THRESHOLD COMPARATOR p' < p' : NO TARGET Fig. 41—Difference equation receiver for the detection of a target in clutter. The detection results presented so far suggest the following receiver design (shown in Fig. 41) which is quite similar to the discrimination receiver shown in Fig. 38. The input is the ramp waveform, and the difference equation of the target is used to generate a predicted waveform. The p' value is then calculated and compared to the threshold value pj. A target is assumed present when p' is greater than pj.. The two parameters which can be varied are the time increment, At, and the threshold, 99 pj., and the optimum values must in general be determined by the clutter characteristics. For the case shown in Fig. 39, the optimum value of At is ,4t and the threshold, pj., should be set somewhere between .07 and .70. In order to do a more precise analysis of determining the optimum parameters and calculating various error probabilities, specific clutter and target char acteristics would be needed. One problem existing with the receiver in Fig. 41 is that the input is not a true ramp response, but is really an approximate response synthesized by a fin ite number of harmonics. Two types of errors result from this type of synthesis: 1) lack of detail due to an insufficient number of harmonics and 2 ) running together of adjacent responses due to an insufficiently long fundamental period. The firs t type of error is not generally serious since only the dipole mode poles are being used in the difference equa tions. This means that only frequencies up through the fir s t resonance are required. The second type of error has no effect on the difference equation f i t because a superposition of identi cal shifted ramp responses w ill satisfy the same difference equa tion as the single ramp response. However, the lack of a su ffi ciently long fundamental period can be a problem because the time interval where the difference equation is valid runs from t = nAt to T/2, when only odd harmonics are used, where n is the order of the difference equation and T is the fundamental period. Con sequently i f T/2 is less than the ramp response duration, then the 100 useful Interval is shortened. This could be particularly serious when At is required to be fa irly large for clutter suppression. Consequently i t is probably best to make T/2 slightly larger than the ramp response duration. Another technique which is possible to u tilize the portion of the waveform from t = 0 to nAt is to use a negative value of At and predict the in itia l portion of the wave form, 0 < t < nAt, on the basis of future values. The major problem here is that for negative values of At, the difference coefficients are formed by combinations of growing exponentials and can be quite large. These large coefficients might cause poor receiver performance in the presence of clutter or noise, but the procedure could be worth trying in a case where lack of a sufficient time interval is already causing poor receiver performance. Although we have used only a synthesized periodic ramp wave form as the excitation signal in the examples considered here, i t is not necessary to be so restrictive. The experimental target and clutter data was available at only four discrete frequencies, and the use of only the three lowest order poles in the difference equation necessitated the use of ramp weighting to emphasize the low frequency content of the response. A step or impulse weighting of the harmonics would not be successful because the step and im pulses responses are not accurately predicted by the three pole approximation. However, in the general case where more poles are included in the difference equation other excitations might . be used successfully. For instance, if we assume that all the poles of importance are included, we could use any arbitrary excitation such as a short noise-like burst of fin ite time dura tion. Although the response waveform would not satisfy the d if ference equation during the excitation burst, i t would for times greater than nAt after the ceasing of the excitation and the de tection scheme illustrated in Fig. 25 would s t ill be used. In order to u tiliz e the time during the excitation, the poles of the response would also have to be included, but this would result in increasing the order of the difference equation which is generally undesirable. CHAPTER IV SUMMARY AND CONCLUSIONS The radiation fields of impulsive short dipoles in the presence of a dielectric half-space have been investigated. Exact time-dependent solutions have been derived for certain special cases. Although the expressions for the fie ld components appear complicated, the actual waveforms consist only of delta and doublet singularities and discontinuities occurring at the arrival time of the wavefronts to ll owed by a time function which decays monotonicly to a steady state value which can always be predicted by electrostatic theory. Consequently, the approximation of the fields for general observation points should not be too d iffic u lt since the general behavior of the waveforms is expected to remain the same. The on- axis solutions are valuable in computing the impedance change in a short dipole due to the presence of a dielectric half-space as shown in Chapter I or the interaction of the dipoles modes of scat- terers with the interface as shown in Chapter I I . Although i t is desirable to have the same solutions for a lossy half-space, the task of determining a realistic model for the complex permittivity would have to be dealt with in order to make the results physically significant. 102 103 A method has been developed to analyze the effect of the ground on the low frequency scattering of a target assuming that the free- space characteristics of the target are known. The method considers the change in the dipole modes due to the presence of the ground and particularly the resultant shift in the s-plane poles. Although the required short dipole solutions have been given only for dielectric and perfectly conducting grounds, the half-space interaction method is general and can be applied to a lossy ground i f the necessary dipole solutions are obtained. I t has shown that the s h ift in the s-plane poles can be significantly large for both dielectric and perfectly conducting grounds when the scatterer is close to the ground. The validity of the theoretical pole shift was experimentally checked for the case of a sphere on a perfect conductor using vertical polarization, and it was found that the difference equation determined front the new theoretical poles was satisfied quite well by the experimentally determined ramp response while the free space difference equation was not. However, i t was found that the results were not nearly so good for horizontal polarization, and it is be lieved that this will be generally true for conducting objects located on conducting grounds since the horizontal electric dipole modes w ill be only weakly excited. A detection method based on the uses of the difference equation of the desired target has been presented, and various examples have been given to test the detection and discrimination properties. The specific examples involving spheres vs spheroids and different size spheres indicate that for a sufficiently large time increment the discrimination can be obtained. The detection of a hemi spherical boss on a perfectly conducting ground in the presence of clutter (a rtific ia l trees) was also accomplished using vertical polarization and an aspect angle of 45° for signal to clutter ratios in the neighborhood of one. A theoretical calculation of the statistics and error probabilities would be desirable, but the required analytical clutter model is not available. The main ad vantage of the difference equation method is that the poles and resultant difference equation are independent of aspect angle. Consequently, an integration on aspect angle which might be required to build up the signal to clutter level could be accomplished using the same signal processing at a ll aspects. Although the poles for the scatterers in the examples treated here were analytically de termined, experimental data could be used to determine either the poles or the difference coefficients for more complicated targets or other target-ground configurations. 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