THE UNIVERSITY OF MICHIGAN 8579-2-Q ESD-TR-67-517, Vol. II Inverse Scattering Investigation Second Quarterly Report 3 April - 3 July 1967 by V. H. Weston and J.J. LaRue July 1967 Contract F 19628-67-C-0190 Prepared for Electronic Systems Division, ESSXK AFSC, USAF Laurence G. Hanscom Field Bedford, Massachusetts 01730

THE UNIVERSITY OF MICHIGAN 8579-2 -Q FOREWORD This report was prepared by the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan. The work was performed under Contract No. F 19628-67-C-0190, "Inverse Scattering Investigation" and covers the period 3 April - 3 July 1967. Dr. Vaughan H. Weston is the Principal Investigator and the contract is under the direction of Professor Ralph E. Hiatt, Head of the Radiation Laboratory. The contract is administered under the direction of the Electronic Systems Division, Air Force Systems Command, United States Air Force, Laurence G. Hanscom Field, Bedford, Massachusetts 01730, by Lt. H.R. Betz, ESSXS. This quarterly report was submitted by the authors on 18 July 1967. This technical report has been reviewed and is approved. Prior to release of this report to CFSTI (formerly OTS) it must be reviewed by the ESD Office of Public Information, ESTI, Laurence G. Hanscom Field, Bedford, Massachusetts 01730. Approving Air Force Authority 3ernard J. F iatreault Contracting Officer Space Defense Systems Program Office i

THE UNIVERSITY OF MICHIGAN 8579-2 -Q ABSTRACT The problem in question consists of determining means of solving the inverse scattering problem where the transmitted field is given and the received fields are measured, and this data is used to discover the nature of the target. The problem of what information can be determined about the body if the scattering matrix (phase and amplitude) is known only over an angular sector and measured in the far field, is studied further. Asymptotic analysis is used to show that in the high frequency case, portions of a piecewise smooth, convex surface can be found when knowledge of the bistatic scattered field is confined to a small cone. iii

THE UNIVERSITY OF MICHIGAN 8579-2-Q TABLE OF CONTENTS ABSTRACT iii I FURTHER COMMENTS ON THE FAR FIELD INFORMATION LIMITED TO A SOLID ANGLE 1 II FORMAL PROCEDURE FOR DETERMINING POINTS ON SURFACE 15 III PRESENT AND FUTURE PLANS 29 REFERENCES 30 DD 1473 DISTRIBUTION LIST iv

THE UNIVERSITY OF MICHIGAN 8579-2 -Q I FURTHER COMMENTS ON THE FAR FIELD INFORMATION LIMITED TO A SOLID ANGLE It was pointed out in the last quarterly that the near field could be expressed in terms of the far field quantity E (0, 0) which is related to the scattered far field electric intensity ikR E- E (0, ) (1.1) R -o by the relation 0 -ioo 2ir o 1k ik. x E() = e —E (a, J) sinadadf3, (1.2) 0 0 and a discussion was presented indicating the zone of space in which E (x) could be found when knowledge of E (0, 0) is confined to a solid angle. In practice, the scattered far field (for a fixed transmitter position), will be measured at a set of N points 0 =(0, 0 ) where n=l, 2... N, located in the solid angle 0<O<2 r and 0 < 0 < 0. With this in mind, there arises several considerations in connection with any computational procedure for determining a portion or portions of the surface of the body from the finite set of measurements, namely; the choice of near field representation, the location of the origin of the coordinate system, and the restrictions on the portions of the surface of the body which can be determined. With regard to the choice of representation, there are three which are essentially equivalent. The first is the plane wave representation given above. If the number N is very large and the points are sufficiently dense, then numerical integration of 1

THE UNIVERSITY OF MICHIGAN 8579-2-Q 0 2r E (x_) -=27~ 0 0 O o ik - x e - -E (ar,3)sinordcd/I3 -O (1.3) can be performed. This technique would have to be employed for high frequencies, in which case E (c, 3) would vary rapidly in the solid angle. Fur— O ther details on this approach will be discussed at the end of this section. For N finite (order of 20 or less) a polynomial fit may be made to E (0, 0). One practical representation would involve the spherical harmonics — O Ye (0,) = pm(cosO)COSm e mn n sin mm0 0 (1.4) in which case form the far field quantity E (0, 0) would be expressed in the following -o A kO E (, 0) = v=l r/l [as (e,0)+b T(O,0)] [-a T (0, 0) +b S (, 0)] - aUv v(O, 1/ ] b ] (., (1.5) (1.6) where the summation over v indicates summation over n=l,... oo, m=0, 1,... n, and over both even and odd components. The functions S and T are given by V Vr the relations m1T 1i a mP (cos) cos m S- = 1 a n emn sin as e mn+ sinO sinm0 o0 (1. 7) 2

THE UNIVERSITY OF MICHIGAN 8579-2 -Q e mn a P (cos0) T mn (s (1. 8) o emn do a 0ksinmv O Thus given E (0, 0) at the N points {0n,n}, one can express E (0, 0) in the form given be Eqs. (1. 5) and (1. 6) where the summation is from v=l to v= N. The unknown coefficients a, b (v = 1,..., N) are found from the set v/ U of 2N linear algebraic equations. When the half-angle 0, of the measurement cone is less than r/4, it may be best in the practical treatment, to solve for the constants a and b indirectly, by using a modified set of functions S and T in place of S and T. The functions S and T which will be linear combinav v v v tions of S and T are chosen so that they behave like 0 as 0 approaches zero. V V In this way the series k0-E = S +b T - - V VU t=l IJ c-1 r ^ ^ j k - E -a -a T +b S -o v v U v=l behaves locally as a Taylor series in the variable 0. The general expressions for S' and T will not be given here, however their explicit expressions will be given for cases m=l and n=l, 2, 3, along with the corresponding behavior for 0 - 0, S =S 1 0(1) ell ell o o 3

THE UNIVERSITY OF MICHIGAN 8579-2-Q S 1=S -3Se X 0 (2) e 12 e 12- e 11 ~ 0 0o o o o~~~~~~ S e 13 o = S - 5Se -9S e13 e12 ell O O O' 0(04) The near field is given by the relation N E(x) = v=l 7r 2ir -ioo o2 0 0 e — fa S +b T 1 v v v l +3[-a T +b S ]} sindadcdl. v v v v Integrating by parts it can be placed in the following form N 2 2/ 2 - v=l 0 0 V V a ik x ^ la e - -a - [eik x-sina Jdad N -i v=l 2r J O - 100 0 b Y a ik ) x 1 a [e3 ik +x sin x. A dd. The integral representation can be simplified on using the following results: r Lik x -Le - - [ ikx sin] isinek-x A ] A -a e - - sinma iksince - - _(x. - (x._ ik~ x = -i k Axe - _ -VAikx =-y e —X_ 4

THE UNIVERSITY 8579-2-Q OF MICHIGAN a ikx ikx. A- ik A A A aP [e - j]+, [e- sinaa ] sina me - - ik (x. +(x. )-2k i [-kAkAx+2ik] ex.k~. [-kA A + 2i - ki VA ik- x a VAVA e - x k - -L From the following relationship e mn (X o h() (kR) Ye ( 0) n e mn o 2-n 27r 2r 0 0 ik-x e -Y (c 3) sinCadad, e mn o the near field can be expressed in terms of the spherical vector wave functions -e mn - emn - o o n -e mn o - VAVA[S x] k - emno as follows N E(R,O, 0) ^ m v1xl +B n Vo uV (1. 9) 5

THE UNIVERSITY OF MICHIGAN 8579-2-Q where.n+l A =1 e mn o a e mn o (1. 10) and B = b e mn o e mn o Alternatively, if some type of polynomial representation or Taylor series was derived for E (0, 0) by curve fitting to the set of measured values /s. A E (0, 0n ) n=1, 2... N, then the partial derivatives of 0 * E and d- E with respect to 0 and 0 could be calculated. To indicate how this could be used, take the near field point to be on the z axis. Expression (1.2) becomes 2z E (0, 0, R) = / o r 0 ikR cos cy e E (a, ) sincadad3 -o (1. 11) after bending the contour from -i oo to - io. Changing the variable of inteo 2 gration a to t by the substitution cos a = 1+it the above expression becomes 2v oo k ikR / E(0, O, R) ei f 0 e -kRt EkRtE dtd -o (1. 12) which reduces to the form, 6

THE UNIVERSITY OF MICHIGAN 8579-2-Q R )n i d+ N+1 f e k~ n+1 dto n=O L (kR) at o 6o at (1. 13) upon integrating by parts. This has the form ikR e R-nE (0,) R -n n=0 00 20r expansion tis techniquivalent to Wilcoxs sources of errors; (1) the negleumberct ofints the remaining integral taken over the range of a from 0 to I- lao which is re - quired is large, and resufficiently densed (2) then numerical inteation and approximation to E ( ) in the cone 0 0 The latter source of error depends upon the -~0 0 can be performed. This technique would be best employed in the high freensity regiof the number of measurements) would vary rapidly in the cone 0 0soli 0, and an be mini—'~~~O~~0 mized by increasing this technique ther of data points. The former source of errors; () the neglect of the remaining integral taken over the range of a from 0 to 2ioo which is required in the exact results, and (2) the interpolation and approximation to E (a,/3) in the cone 0 < a < 0. The latter source of error depends upon the density of the number of measurements in the cone O < 0 < 0, and can be minimized by increasing the number of data points. The former source of error is more fundamental and will be examined in some detail here. The example of high-frequency scattering by a sphere will be treated first, where the dominant contribution in a cone 0 < 0< 0 in the back-scattered region, is the geometric optics contribution. region, is the geometric optics contribution. 7

THE UNIVERSITY OF MICHIGAN 8579-2-Q The incident field is taken in the form i ^ -ikz E =xe The geometric optics field in the far zone may be written Es (, 3) a A, 3) -2ika cos (a/2) E (a,) = -2e(a, ) e - 0 ""-; (1.15) where e is the unit vector e(a,B) =x(cosacos2 3+sin 3) -y (1-cosa) sinf cos- cos3. (1.16) It will be assumed that the far field is measured at a sufficiently dense number of points in the cone 0 < 0 4 0, such that a reasonable approximation to )1. 15) can be obtained. Employing relation (1.14) the near field is given approximately by 0 2r o S (x) = ika ( E (x) 0 0 ikf(a,'3) e e(a,/3) sinadad/3 (1. 17) where f(a, 3) = r [insina cos(0-+) + cos cosa] -2 a cos As k —oo, the dominant contribution to the integral arises from the vicinity of the stationary phase point (3= 0, a=a ) where a satisfies the equation a r sin(a -0) = asin (-) o 2 provided that 0 < a 0. By means of first order stationary phase evaluation o 0 we obtain immediately 8

THE UNIVERSITY OF MICHIGAN 8579-2 -Q Expression (1.21) thus can be reduced to the following form s in0 o ikg(0 ) - -i 1 i __ 4 4 2 2rkrsin0 e E (r, o.) Sv-^ 2 ~ ikg te iB(0)-e B(- (1.23) where ip(0 ) e 2(0 )+A (0)+A (0]) 2 o - -o o -0J B(0)) = (. 24) 0 0 kr sin(0-0 ) + ka sin (2) O 2 0 0 o o On the surface of the sphere r=a, outside the cones 00 < 0 and r —- 0 rT, 2' the dominant behavior of the amplitude is given by the factors -1 in (L -i ) + sin( - and k sin(0 + 0 -sin (-) It is seen that on this portion of the sphere expression. (1. 23) does not agree with (1. 18) the amplitude of which is unity. In addition in the shadow region of the sphere in the cone r -0 /2 < 0 < r, where the fields should vanish in the high frequency region, representation (1. 23) cannot be used as there is a stationary phase point, in which case the asymptotic evaluation of (1. 17) will give fields with amplitude the order of unity. Summing up, it is seen that the approximate representation (1. 3) will give the correct fields in the high freuency case, for a portion of the illuminated face of the scatterer. 11

THE UNIVERSITY OF MICHIGAN 8579-2-Q It is of interest to extend the analysis to scattering surfaces with edges. In this case the scatterer that will be taken will be the flat plate. For convenience the geometric optics result for the scattered field will be employed. For a plane wave of unit amplitude incident normally to a square flat plate, of length and width a, and situated on the z=0 plane of a Cartesian coordinate system, the physical optics scattered field is given by -i2 ka ka A ] E (c,)- ) sin( u) sin ( v) acosacos3 -Isin3 (1.25) -o kuv 2 2 i -2 where u = sin a cos 3 v = sin a sin Expression (1.14) will be used for the near field calculation, with measurements of the scattered field again confined to the cone 0 ~ 0 0. The near field will 0 be calculated on the x-axis in which case 0 o 2r ik x 1 e, kau.kav A ^ E (x, 0, 0) = / (cos co - n sin) sin (cosdadB 2 J UV 2 2 L 0 0 which reduces to 1 ^ ikxu ka. ka (1-u )dudv E x e sin( u) sin (v) - (1.27) 2 2 a y ^ ~ a r 2 2 A uv 1-u -v The integration in the u-v plane is over the area of the circle centered at the origin with radius sin0. The above integral can be expressed in the form O 12

THE UNIVER SITY OF 8579-2-Q MICHIGAN 1 E = xx 2T sine 0 - o -sin0 o sin u(kx+ 2 ) + sinu(-kx+ ] g) du u (1.28) where g(u) = (1-u2) V v 0 -V 0 ka sinm( v) 2 97 F dv v l-u -v with v = sinO -u 0 \ When (kav )/2 >> 1, the asymptotic behavior of g(u) for large ka is given by 0 (u) (1-u) g(u) ~ (1-u ) 2 _ r _U ka 4cos (-v ) 2 ka cos 0 v o o Since the following integral has the asymptotic behavior for I1 — > oo sin0 1 sin u/ 1 / o- s nug(u) du i g(o) sgn/, -sinO o it follows that for x< ka (i.e. [x( < k (i.e. a point on the plate) ka 2 o EN F 4cos(7 sinoo )1 - E ~ ~ r - kasin0 cosO Ic~ 0 13

THE UNIVERSITY OF MICHIGAN 8579-2-Q otherwise E will vanish for a point outside the plate. Provided that kasinO >> 1 which implies that the half-angle of the cone of observation must not be too small, the approximation expression for the near field given by Eq. (1. 14) will give the correct expression for the scattered field points on the plate which are at least a wavelength away from the edges (indicate by the condition ka that |3| = x~+ ~ 1). From the above analysis, it is seen that the approximate expression (1. 14) will give good results in the high frequency case, provided that the given data is sufficiently dense over the measurement cone (0 <O (0 ). These remarks do not include, at present, surfaces which contain cavities or protuberances. A study will have to be made of these cases. From a numerical standpoint, expression (1. 14) could be difficult to use in the high frequency case. Great care has to be taken in employing expression (1. 14) because the integrand contains a rapidly varying exponential, which could lead to serious errors in the employment of any straightforward numerical procedure. 14

THE UNIVERSITY OF MICHIGAN 8579-2 -Q II FORMAL PROCEDURE FOR DETERMINING POINTS ON SURFACE Points which may lie on the portion of a perfectly conducting body can be determined from a knowledge of the field incident upon the body and the far field scattered by the body. Specifically, if a transmitter and associated receivers are located such that they subtend a solid angle Q with respect to the scattering body then the points which can be determined will be included in the solid angle Q2. Since the criteria used for determining points on the surface is only a necessary condition, it is possible that the criteria may be satisfied by more than one point along a particular ray. However, the location of a point which lies on the surface is, of course, independent of frequency and thus the ambiguity can be removed by multifrequency measurements. Points which may determine other portions of the surface can be determined by additional transmitter siets and their associated receivers. Thus the data for the inverse scattering problem may consist of several multifrequency transmitter sites and their associated receivers. The form of the input data, the necessary coordinate rotation and the formal procedure for determining points which may lie on the surface are discussed in the paragraphs that follow. 2. 1 Input data for the Inverse Scattering Problem. The input data for the data for the Inverse Scattering problem must be given in terms of a coordinate system whose origin is in the interior of the scattering body as shown in Fig. 2-1. The orientation of the axes is arbitrary. The following data is necessary at each frequency and for each transmitter: 1. There are m transmitters located at the points T.(Rti, Oti, ti for i=l, 2 —-M where Rti is the distance in meters from the origin in the transmitter location. 0ti and 0ti are the angular displacements measured in degrees as defined by Fig. 2-1. 15

THE UNIVERSITY OF MICHIGAN 8579-2-Q z 0 Scattering r body Y x FIG. 2-1 ti 2. The field at the origin due to each transmitter is Eti== Etie e. -ti ti i where Eti is the amplitude of the transmitted field at the origin measured in volts per meter. a is the phase of the transmitted field measured in radians ti at the origin of the coordinate system. A A,4 e. eil ti+ ei is a unit vector specifying the polarization I 1 i l 12 ti of the transmitted signal. th 3. Associated with the i transmitter there are N receivers located at the points P.. (R.., 0i.,.) 1J 1J 1J 1J where R.. is the distance from the origin to the j receiver in meters. ijJ.th 0Q and 0ij are the angular displacements of the j receiver location in degrees. th th 4. The scattered field at j receiver due to the i transmitter is ia ia e 0ij +E e ij Ei Eij e 0..+Eije 1J Ji]"j ij 16

THE UNIVERSITY OF MICHIGAN 8579-2-Q where Eeij and Epij are the amplitude, in volts per meter, of the 0 and 0 components of the scattered field respectively. aij andij are the phase, in radians, of the scattered field respectively. 2.2 Coordinate Transformation for Inverse Scattering. 2.2. 1 Coordination Rotation. The data for the inverse scattering problem will be given as specified in Section 2.1. In order that the data may be used in the inverse scattering problem a rotation of the coordinate system is necessary for each transmitter location such that in the new coordinates, denoted by primes, the transmitter lies on the positive z' axis and the positive x' axis is parallel to and in the direction of the polarization vector. For the ith transmitter the position vector to the transmitter and the polarization vector are T. =T.r -1 1 ti (2. 1) e i il ti + ei2 0ti where the subscript ti denotes the ith transmitter. For convenience let ei. = cos y i1 1 (2.2) e = siny. i2 1 Using (2.1) and (2. 2) the unit vectors x!, y' and'z are 1 1 1 17

THE UNIVERSITY OF MICHIGAN 8579-2-Q A A A x = cos i 0i + sin yi t iti i ti y Z x -sin 0ti + cos ryti (2. 3) Y = ti z' =r 1 ti A/ A A In any coordinate system the unit vector r, 0, 0 at a point are related A A ^ to the unit vector x, y, z by x =sin 0cosr r+cos 0 cos 0 -sin 0 y = sine sin r + cos 0 sin 0 + cos 0 0 (2.4) A A A z = cos 0 r - sin 0 0 and A A r = sin0 cos 0 ^ + sin sin 0 y + cos 0 z AA A 0 =os 0 cos 0 x + cos 0 sin 0 y - sin 0 z (2.5) = - sin 0 x + cos 0 y The unit vector to the jth receiver is from (2. 5) r. =sin i cos ij xi +in sin ii..y.+ cos 0. z ii i]j 1 1 ij 1J 1 1i 1 =sin0' cos XI +sin' osin 0'.. + cos 0. (2.6) ij 1 1i i j 1i 1 1j 1 From Eqs. (2. 3), (2.4) and (2. 5) the following relations between the unit vectors can be obtained x.. xI = cos costi s o ti - sin s i sin 0ti A A Yi = cos i cos ti sin ti + sin yi cos ti (2. 7) X cos i sin ti tti A A z. x! = - cos -. sin O0. 1 i ti 18

THE UNIVERSITY OF MICHIGAN 8579-2-Q x. y = -siny t cos t cos 0 - cosy. sint 1 1 1 ti ti 1 ti Yi Yi A 1 i - sin yi cos 0ti sin 0ti + cos i cos 0ti 1> tiL ti 1 tiL (2. 8) = sin'y. sin ti 1 ti A A xi * Zi = sin 0i cos 0t i i ti ti A A y i z' = sin 0t sin0 i ti ti (2.9) A /A z. * z' = cos 0 i i ti From Eq. (2.6) the angle 0!. can be expressed in terms..i and 0ij as 13 13 A A ^ A z' r.. =. = sin.. cos.. z x.+ sinO.. sin0.. z! y.+cos0..z'* z 1 i i ij ij 1 i ij i js i ij o 4 o'. < Ir. (2. 10) 13 Using the result in Eqs. (2.7), (2. 8) and (2. 9) in (2.10) we obtain cos ( = sin.. sin cos ( 0 ) + cos.. cos 0 ij 13 ti ij ti 1] ti (2.11) The angle 0i can be obtained from cA A x' r sin'. cos = sin 0 cos 0 x'. x + i i j ij ij +sin O.. sin0. x' + cos 0. x' ij ij 1' z (2.12) 19

THE UNIVERSITY OF MICHIGAN 8579-2-Q y'~ r. = sino'i sin 0. = sin.. cos 0' * x + i 1j lj ii i1 +sin 0j. cos 0ij' +Cos. y (2.13) Using the result of (2. 7), (2. 8) and (2. 9) in (2. 12) and (2. 13) we obtain sin 0!. cos 0!j 13 13 - sini.. cos yi cos ti cos (0ij -ti ) + -Y O sin i sin (0ij - ij) - cos 0ij cos yi sin 0ti (2.14) and sin 0!. sin 0!. - sin, sni cos 0 cos (0ij - 0 ) i13'ij 13 ij ti ij ti + sin Oij cosi y sin (0ij- ti) + cos 0ij sin yi sin 0ti 1j 1 ti (2.15) Using the value of 0!. determined by Eq. (2. 11) Eqs. (2. 14) and (2. 15) 13 determine 0!. in the range < 01 < 2. In a similar manner the inverse transformation can be obtained as cos.. = - sin0.j sin0ti cos (0!.+ Y.) + Cos0!. os 0 0 0 < 13ij ti i1 1 ij ti i] (2. 16) and 20

THE UNIVERSITY OF MICHIGAN 8579-2-Q sin Oj cos0 = sin 0.c c os os0 cos (0' +0 ) 13n j sin ti ti ij ti -sin 0! sin 0 sin (y.+0.) ij ti 1 13 + cos 0W. sin 0. cos 0. 1j ti ti (2. 17) sin..i sin 0ij =sin 0O. coS Oti sin 0t cos (0! +.) + sin ij ti sinti I ) + sin 0. cos 0' sin (. + 0i.) 13 ti i ij + cos 0.sin sin in ti ij ti ti (2. 18) Equations (2. 17) and (2. 18) uniquely determine 0ij in the range O<0. <2 r. The components of the scattered field at the jth receiver due to the ith A A transmitter in the directions of the unit vectors 0!. and 0. must be determined. ikRij 1 1j It is convenient to remove a factor (e )/R. before developing these expressions. The expression for the scattered field at the jth receiver due to the ith transmitter at a particular frequency is i% s ij'1 E = E e O.. + E -i- 0ij i] oij ij e oij (2.19) The scattered field can be written as 21

THE UNIVERSITY OF MICHIGAN 8579-2-Q ikR.. S e E - -ij kR.. 1j ikR.. 1i e - ij kR.. 1j L 0.... 1J 1i + ij ij ij1 (2.20) The quantities t (2. 20) and are and r 1.. 1j can be determined from (2. 19) and 0.ij i(1 - kR..), J /o- ~" 1 (,f = kR. E 0.. ij 0.. iJ 1J I\,-... et \z. z 1j i(a~ -kR..) 0.. ij 0 - kR.. E e ij iij (2.22) The desired components of the scattered field are 6 0 iJ = 01 * P ij -~ i1 — 1!. 0.. j A, /A * 0..'..' 0 3 0 i j ij ij (2.23) and ti ij ij 1J - 0.. -1J1 -i 0ij A A 0' 0.. ij 1J + ^D / A' 0'ij 0. ij ij i] (2.24) The products of the unit vector can be obtained from Eqs. (2.3), (2. 4) and (2. 5). The results are 22

THE UNIVERSITY OF MICHIGAN 8579 —2-Q E',I= e1 3jcos e6 cos (i+ 0i) Lcos 0ti cos ij cos ijCos (0i- j iJ t + sin 0 sin 0ij + cos 0. sin(Y + 0.)sin(0ijti) cos Oi. tiiji ij ilij ij ti 13 - [s 6 rsin cos O.. cos (0t- ij) - costi sin 0ij} + ij - c os (i o + 0p.) cos sin (0i- 0) ijL L_ 1 1 ti t 1 + sin (Yi + 0.) cos (0 - sin 0j sin0 t sin (0ti-.i (2.25) 1 ti L 131 ij ti tilJJ Sij, J {- sin (ij+ ) os 0ti cos..j cos (0ti-0ij) +sin0i sin..ij +cos(i+0j) cos 0. sin(0 -0 + 0 I-sin(0ij+) cos 0ti sin (ti- ij) + cos (iy+ 0j) cos (0ti- i (2.26) 2.3 Points on the Surface. The scattered field due to the ith transmitter along a ray from the origin to the jth receiver at a particular frequency can be written as 23

THE UNIVERSITY 8579-2-Q OF MICHIGAN 00 Es n=l n m=0 F n+l./ \n+l (-i) +a m +(-i)n e mn -emn a m omn -omn + (-i)nb n + (-i) bm n emn-emn omn omnmn (2. 27) where - m (1) m sin m = sin h (kr) P (cos') mc. 0'j. -e, mn sin W n ij n j cos ij lj 0 1J (1) a m Cos A - h (kr'.) — Pm (cos0!.).COSmO! O! n ij O.W. n ij sin i ii nj (2.28) and _ n(n+l) -emn kr' o ij h (kr! )pm(cos 0!.)co'. r n ij n ij sinm i 1J 1 kr' ij [r h(1)(kr'.) arL. ij n 1j 1] a pm (cos 0.) O' a-.] n iij 1] + m kr'. sin0. ij 1i a r'. 1] r. 1) (kr pm (cosO'.) n ij n ij n ij (2.29) The coefficients a, a, b and b emn omn emn omn are unknown and to be determined. Assuming the receiver is in the far field of the scattering body and introducing the following notation 24

THE UNIVERSITY OF 8579-2-Q MICHIGAN - m m s =+ P (cosO'..) emn sin'.. n ij o ij sin!. m 0'! cos 1] (2.30) and a pm(cos'i)cos, emn a0'. n ij sin 1] 0 1J (2.31) the expression for the scattered field can be written approximately as ikr'. s e e E = - kr. 1J n=l n mn m Lemnen m=O + a S +b T omn omn emn emn +b T 0!. + K-a T -a T omn omnj L emn emn omn emn +b S +b S 0 e mn emn orn omn ijj From (2. 25), (2. 26) and (2. 32) we obtain 00 n = > (~a S +a S +b T +b S ) emn emn omn omn emn emn omn omn (2.32) - 1J (2.33) n=l m=O Go n 15i "ij n=l 2, (-a T -a T +b S +b S ) m — emnen onem n o mn mn emn n omn omn m=O (2.34) 25

THE UNIVERSITY OF MICHIGAN 8579-2-Q It is convenient to write Eqs. (2. 33) and (2. 34) as a single sum as follows: i. * (a S +b T ) V V2 V2 V (2.35) 0Ij ij 00 Ov Vi= (-a T +b S ) v v v v (2.36) where a -a a=1 ael a2= ao10 a3= ael1 etc. In general, the subscripts e or o, and the values of m and n can be determined for a given v as follows: 1. n is determined by (n-1) (n+2) < v < n(n+3) 2. If v-(n-l)(n+2) is even use o, odd use e 3. m is given by the integer or next highest integer to -(n-)(n+2) -1. 2 Assuming that the series in (2. 35) and (2. 36) can be approximated by a finite number of terms a system of 2N linear equations (where N is the number of receivers) can be formed to determine the av and by. There will be N equations of the form N B!. (a S +b T ), j v = 1 12=1 (2.37) 26

THE UNIVERSITY OF MICHIGAN 8579-2-Q one for each value of the index j. Similarly, there will be N equations of the form N ~,: (-a T +b S ) ij v 1 13ij = (2.38) The system of linear Eqs. (2. 37) and (2. 38) can be solved by standard methods and results used in Eq. (2. 27). The scattered field is approximately N(v) sE n=l n=1 n -n+l m +(-i) a +((-i) b n emn-emn omn-omn emn-emn m=O +b n 1 omn -omnJ (2.39) From Section 2. 1 the incident field along a ray to the jth receiver is ia -ikr'. Et = Et e e j (sin0e.cos0' r. +cos' cos'.' -sin!.'.). -t ti Jt ij j iij 1i ij Ij ij ij (2.40) From Eqs. (2. 39) and (2. 40) the total field is E =E +E -T -ti - (2.41) The criteria for determining points which may lie on the surface of the scattering body is ET x ET =0. (2.42) -T -T 27

THE UNIVERSITY OF MICHIGAN 8579-2-Q In an actual calculation Eq. (2. 42) may not be satisfied exactly and practically it may be more useful to look for minimum values of the function ET x ET 28

THE UNIVERSITY OF MICHIGAN 8579-2-Q III PRESENT AND FUTURE PLANS At present, analysis is being carried out to investigate the amount of data that is necessary in order to determine the shape and size of a dielectric body. A review of the techniques developed by Gelfand and Levitan and the technique developed by Kay for scalar scattering has been carried out. This review will be presented in detail in the next quarterly. The possibility of generalizing the results to the three dimensional vector case will be investigated. Practical techniques for determining the surface of a perfectly conducting body are being carried out on a specific example. Theoretical condition E x E 0, based upon knowledge of the total field E in the vicinity of the scattering surface, was investigated numerically, and the results indicate that this condition may be very difficult to employ in any numerical scheme where only approximate values of the total field are known. To improve its use, a proper normalization factor, yet to be found, may have to be employed. However, alternative conditions can be used, such as in the high frequency case, where the approximate condition H 2 = H 2 may be employed to find the illuminated portion of the body. This later condition appears to be much more practical in any numerical approach. Further asymptotic analysis will be carried out with the approximate near field expression Eq. (1. 14) to investigate what portions, if any, of concave surfaces can be determined when knowledge of the far field is confined to a cone 00 <0 <0 O 29

THE UNIVERSITY OF 8579-2 -Q MICHIGAN REFERENCES Weston, V.H., J.J. Bowman and Ergun Ar, (November 1966), "Inverse Scattering Investigation, " The University of Michigan Radiation Laboratory Report No. 7644-1-F. Weston, V.H. (April 1967), "Inverse Scattering Investigation-Quarterly Report No. 1," The University of Michigan Radiation Laboratory Report No. 8579-1-Q. 30

THE UNIVERSITY OF 8579-2-Q DISTRIBUTION MICHIGAN Electronic Systems Division A ttn: ESSXS L. G. Hanscom Field Bedford, Mass 01730 27 copies Electronic Systems Division ESTI L. G. Hanscom Field Bedford, Mass 01730 23 copies

UNCLASSIFIED Security Classification I N DOCUMENT CONTROL DATA - R&D (Security cloasification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1 1. ORIGINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Radiation Laboratory Unclassified Department of Electrical Engineering 2b GROUP _Ann Arbor. Michigan 48108 3. REPORT TITLE INVERSE SCATTERING INVESTIGATION 4. DESCRIPTIVE NOTES (Type of report and inclusive dats) Second Quarterly Report (3 April - 3 July 1967) 5. AUTHOR(S) (Lest name, first name, intifll) Weston, Vaughan H. and LaRue, John J. 6. REPORT QATE I 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS O July 1967 31 2 a. CONTRACT OR GRANT NO. 4. ORIGINATOR'S REPORT NUMBER(S) F 19628-67-C-0190 8579-2-Q b. pROJECT NO; c. 9b. OTHER REPORT NQ(3) (Any other numbers that may be assigned this report) d. ESD-TR-67-517, Vol. II 10. AVA ILABILITY/!IMITATION NOTICES Qualified requestors may obtain copies of this report from DDC. Prior to release to CFSTI, this report must be reviewed by ESD, ESTI, L. G. Hanscom Field, Bedford, Mass 01730. 11. SUPP..EMENTARY NOTES I. ~P9OSORING MILITARY ACTIVITY Electronic Systems Division USAF, AFSC L. G. Hanscom Field Bedford, Massachusetts 01730 J I 13. ABSTRACT The problem in question consists of determining means of solving the inverse scattering problem where the transmitted field is given and the received fields are measured, and this data is used to discover the nature of the target. The problem of what information can be determined about the body if the scattering matrix (phase and amplitude) is known only over an angular sector and measured in the far field, is studied further. Asymptotic analysis is used to show that in the high frequency case, portions of a piecewise smooth, convex surface can be found when knowledge of the bistatic scattered field is confined to a small cone.. D_ D UNCLASSIFIED __ DD I JAN 64I 1473 UNCCLASS IF IED Security Classification

UNCLASSIFIED Securitv Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE, WT Inverse Scattering Electromagnetic Theory INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 6. REPORT DATE: Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NUMIBER(S): If the report has been assigned any other repcrt numbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC." (2) "Foreign announcement and dissemination of this report by DDC is not authorized." (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through., (4) "U. S. military agencies may obtain copies of this report directly from I1DC Other qualified users shall request through,. (5) "All distribution of this report is controlled. Qualified DDC users shall request through,, If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11. SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. UNCLASSIFIED Security Classification

UN ERSITY OF MICHIGAN 9015 03627 7575