ESD-TR-67-517, Vol. 111 - -l INVERSE SCATTERING INVESTIGATION QUARTERLY REPORT NO. 3 Vaughan H. Weston Wolfgang M. Boerer Charles L. Dolph Iwov *v November 1967 8579-3-Q = RL-2185 SPACE DEFENSE SPO (4961/4741/N) DEPUTY FOR SURVEILLANCE AND CONTROL SYSTEMS ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscom Field, Bedford, Massachusetts I This document has been approved for public release and sale; its distribution is unlimited. (Prepared under Contract No. AF 19(628)-67-C-0190 by The University of Michigan, Department of Electrical Engineering, Radiation Laboratory, Ann Arbor, Michigan.)

LEGAL NOTICE When U. S. Government drawings, specifications or other data are used for any purpose other than a definitely related government procurement operation, the government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. OTHER NOTICES Do not return this copy. Retain or destroy.

8579-3-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 July to 3 October 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. L E Nyman, ESXS. This quarterly report was submitted by the authors 21 November 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 Bernard J. Filliatreault Contracting Officer Space Defense Systems Program Office i

8579-3-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. Particular aspects of this overall problem are considered, such as the effect of phase errors upon the determination of the scattering surface, polynomial interpolation of the scattered field measured at a set of discrete points, and the testing of a numerical procedure for finding the surface of a conducting body from the knowledge of the near field. In addition, a review of exact theoretical treatments for the scalar inverse problem is given. ii

8579-3-Q TABLE OF CONTENTS ABSTRACT U I THE EFFECT OF PHASE ERRORS 1 II INTERPOLATION OF FAR FIELD MEASUREMENTS 11 II ANALYSIS OF THE COMPUTATIONAL RESULTS OF DETAIL SPECIFICATION (PART ) - SUBROUTINE F6: E xE T0. 27 T -T IV MOTIVATION AND HEURISTIC DEVELOPMENT OF INVERSE SCATTERING THEORY 36 REFERENCES 54 DISTRIBUTION LIST DD 1473 iiit

8579-3-Q I THE EFFECT OF PHASE ERRORS In any practical approach to the inverse problem, the degradation of the body shape due to phase errors in the far scattered field needs to be considered. These phase errors would arise either from the measurement process or the data processing. In the latter case where the scattered field is measured at a finite set of points, the error would arise from the fact that only an approximate finite polynomial fit can be made to the scattered field. In order to investigate the effect of phase errors, the high frequency scattering case will be considered. It will be assumed that the scattering surface is perfectly conducting, and smooth except for curves or joins of discontinuity which are many wavelengths apart (i. e., the body is comprised of long smooth sections). In addition it will be assumed that in the cone of observations 0 < 9 < 6, the 0 high frequency scattered far field can be represented in the form ikR E e E (0i 0) - R -o N ikg n('., 0) E (0,0) = E (0,0) e n (1.1) n=1 where E (0, 0) are slowly varying functions of the angular coordinates. This representation corresponds to the decomposition of the far scattered field into the components that arise from the various scattering centers (characterized by the subscript n). The phase errors due to the measurement procedure or the data processing will be given by kE (0, 0) (1.2) 1

8579-3-Q in which case the scattered far field will have the form N E (e,0) = jE (0,0) expik' (0,0) + e(0,0)) (1. 3) n=1 The near-zone scattered field is derivable from the far field by the relation 0 2r E(x)= 27r e- - E (cr,,) sin adl (1.4) 0o where k = k(sin a cos 3, sin a sin 3, cos a). This is an approximate expression which is employed when the far field is known only over the cone of observation 06<40. The errors in using this expression have been discussed in the previous 0 quarterly, where it was pointed out that the expression was quite accurate in the high frequency region for determining certain illuminated portions of the scattering surface. Taking into account the phase errors in the far field, the near field will be given by E = k ikfn( E (a, 3) sin adad3 (1.5) n=1 0 0 where f (0,0) = g(0, 0) + (x sin a cos 3 + y sin a sin / + z cos ). n n Since the integral contains a set of terms, each of which has a rapidly varying phase, asymptotic analysis may be used to obtain an explicit expression for the near field. The dominant contribution of each term will arise from the stationary phase point provided that it is in the region of integration. A particular term which has this property can be easily evaluated as follows: 2

8579-3-Q ^= I S \( e ((.3)s ik n E (2 i 27 eik ( '3) E (a ) in adaOd ikf (a, S) n(f E (, =3)e n wDf(, a)]1/2 (a 11) where (a1. B1 ) is the stationary phase point given by af af n- n -0 Oaa, a (1.6) (1.7) (1.8) and a2f a2f 1 n Df(s, n) =- 1 n sin2 a aa2 af2 1 2fn 2 in- a -a3 \sin a t sei (1.9) The factor r(f) is unity unless a2f n 2 a2f > 0, 2 > 0 and Df(al, ) > 0 a2 1 1 a 3 in which case r(f) = i, and if a2f n O a1 a2f 2 <0 and Df(a, 13) >0 a1 1 l then rl(f) = -i. Expression (1. 6) may now be used to determine the effect of the phase error kE upon the near field. First the corresponding expression without the phase error has to be obtained. Set 3

8579-3-Q gn (,0) = O/n(O0,0) + (k.X /k in which case fn(0,0) = gn(0,0) + e(0,0) (1. 10) Then the corresponding integral to (1. 6), for the near-field term devoid of phase errors, is given by (x)= ik ikgn (a, 3) -n( 4 - E (a, 3) sinadadl -'1 (1. 11) If (ao, 1o) is the stationery phase point given by agn agn n o then expression (1.11) reduces to the form (g)En (a, 13) ikgr E (x)i [ ' i] 1/2 e - j) g (a, 1 / e Ia <e, 0 0 (1.12) (1.13) (a%' o) It will be assumed that the angular variation in the phase error kE is sufficiently small such that the stationary phase point (acl 13 ) is close to the point (a0, 13 ), thus enabling one to express (al,f1 ) in terms of (ao, 3 ) by expanding equations (1. 8) in terms of a Taylor series about the point (a, 13 ) as follows: as follows: af af a2f - 0 n — + (a a ) - aa1 Oa 1 0 Oa 2 0 a2f + ( ) n +" 1 o oaO (1.14) af f a 2f a2f n n n _ n 01 10 0 a/o0 132 0 4

8579-3-Q From equations (1.14) and (1. 15), together with the relations af n ae aa aa O O 0 0 af n _Oe O O a first order approximation to a1 and 1 can be obtained as follows: 2f O f a +n a3 1 0 +aaa aOa 00 0 a2f n a~o2 2 aa n O aa ap 0,/L (1.16) (1.17) 2 where A= sin a Df(a, 3 ). 0 0 0 Keeping only the first derivatives of e. /2 a2 = agn ae gn aE o1 0 aaoago aoa a 2 ap o 0 A2 2 1=^ a~, /f a~ - 2 A O O O a) 0 o 0 o the above expressions reduce to / [sina Dg(ao, '0)] [sin Dg(ao3) 0 0 0-2D~r O d (1. 18) (1.19) For the general case, only the effect of the phase error e on the phase of the near field will be considered. The phase error induced in the near field is given by kfn( a1 ) -kg,)n 5

8579-3-Q which equals 2 ag2 a2 k 1('l +1) k +k(a-a )(3-f) +( 13f3-)1 2 1 1 oaa03 0 2 2 2 2 I n aE 22 a gn a2_ a 2 gn kE(a ) +k 2 _a + - 2 g a a 1 aa0 I oa'ol 0 0I2- (1.20) r 2 E2 sina 2 Dg (a 3 In order for the near field phase error to be small, not only must 1k <~ g 1, but 2id o i u e (1. 21) 2 2 /2 )E 2 sin a Dg(a ) (2 ka 02 2side of inequalities (1. 21) Dg(. 22), a specific example will be taken, The a 0 -- 2 k 2 aa0 z-direction and polarized in thepositive x direction. In this case the far field phase is given by k/(O,) )= — 2ikacos(0/2). 6

8579 3 -Q For the point (&, 0, 8) on the surface of the sphere, the phase factor g(a, 3) is given by the relation g(a, ) = a [sin a sin0 cos ( +- p) + cos 0 cos a - 2 cos (a/2)] The stationary phase point(ao, 3 o) then satisfies the following relations cosa sin cos(3 - ) - cos sina +sin(a /2) = 0 -sina sin0 sin(3 — 0) = 0 o 0 which are derivable from equations (1. 12). The appropriate solution is a= 20, 30 = O 0 The following relations may then be derived a2g — a - — Cose, 2a O 22 C 2/cos sin a a3 o o 1 -2g _ 2 sina aa a:3~ o o o Dg(aroo) = a /4. For points on the spherical cap, inequalities (1.21) and (1. 22) can be expressed in the explicit form k <(kacos0)1/2 (1.23) aa o 7

8579-3-Q k aE-1/2 k 3c | 1/2 (1.24) sina I3 \<cos0(/ 0 0 For phase errors, that have that property that either inequalities (1. 21) and (1. 22) are not fulfilled or that ke > 0(1), then there will be a significant error in the near field, which will cause a significant degradation of the calculated scattering surface from the actual scattering surface. For perfectly conducting surfaces, the points on the surface are partially determined from the necessary condition (E + ES) x (E +Es) 0 (1.25) For the special case of high frequency specular scattering where the dominant scattered field in the vicinity of a portion of the surface can be approximated by the geometric optics result, then condition (1. 25) can be replaced by the approximate condition IEs2 IEil2 In this case large but slowly varying phase errors will not affect the result. In the numerical treatment, condition (1. 25) is replaced by the condition G F FF = 0 (1.26) where F is the real vector given by the relation F = i(E + ES) x (E + ES)A The numerical approach involves the point by point computation of the positive function G along the coordinate ray (0 = 0, 0 = 0 ) of the spherical polar coordinate system, i. e. G is computed as a function of R, at increments of AR over a prescribed range. If there are no errors in the computed total field expression, then the point R = R (0, 0 ) for which G(R, 00, 00) = 0 would 8

8579-3-Q yield a point which may be on the surface (since condition 1. 26 is a necessary but not sufficient condition). To determine the effect that small phase or amplitude far field errors have on condition (1. 26), let E be the error in the total field due to such far field errors. In this case, the function that would be computed along the ray (80,0 ) would be G = F * F (1.27) N _ i s i where F = i(E +E +e)x(E +E +E c) F + L where L = i (E + E ) xE +Ex (E +E) + ex. The computed expression then becomes G = F * (F + 2L) + L * L (1.28) and at RR = Rwhere G(Ro, 0, 00) = 0 G = L- L (1.29) The effect of the errors in the total field, upon the behavior of G, will be to produce a non zero minimum or a zero minimum at another point then (Ro, 0, 00). The computing program should search out the minimum of G rather than look for the zeros, to take into account the possibility of a non-zero minimum To obtain more qualitative results, the case where the errors produce a shift in the zero point of G will be considered. Let n be the unit normal to the surface at the point Ro, E the normal component of the total electric field, and H the tangential components of the total magnetic field at R, Expanding the total electric field E in terms of a Taylor series about the point R it can be shown that 9

8579-3-Q F(R) = n- (R-R)P(R )+ O(R-R )2. ~ -- - -X - 0 - -0 P(R) = { H+ EH] i [E nx V E -E nx VE* o n- n- n - n n- n' If AR = R - R, and y and the angle between the normal n and the ray (0, 0 ) then with retention of only linear terms in AR, R will vanish at the value AR where -(L * L) AR -o 2cosy P L The expression for L simplifies at Ro, yielding L = t E + x E xn + ExE. nC - n - - - Except for the case when the error E in the electric field has only the single component in the normal direction (yielding L = 0), the shift in the zero point of G has the following order of magnitude AR 0 o (ospY PT V En) — n1 10

8579-3-Q II INTERPOLATION OF FAR FIELD MEASUREMENTS When the far field component E (0, 0) related to the far scattered field '-O as follows Es ikr E (0 0) E= e (E(e,), R is known over the complete unit sphere, the near field at a point x in the halfspace z > z is given by the representation 2 2 r E(x) = ik e -- E (a, f3) sin adad. 0 0 The appropriate region of convergence z > z, and its relation to the scattering body is given in the final report (Weston, Bowman and Ar). In practice, measurements of E (0, 0) will be given at a finite set of points '-O (n, n0 ), n = 1... N, and in most cases these points will be confined to a measurement cone of half-angle.,.asuch that 0 O6 a. It was pointed out in previous quarterlies that in this case, it is important not only to obtain a good approximation for E (0,0) for e real, but also for complex values of 0. Thus if E (0, 0) is some polynomial approximation to E (e, 0), one would like to find the best polynomial fit which would minimize 1o (0,0) - E(0,0)| not only for 0 0 < a, but also for 6 in an extended region in the complex plane. The investigation that follows is an initial effort towards this end. The choice of best approximation will depend upon the asymptotic behavior of E (0,0) for |Im0| - oo. This can be obtained from the following representation — O 11

8579-3-Q for E 0" -o Eo@-) - 4- kIxkxJ(x)e-ikx dx' V where k = k(sine cos, sinO 0in, cos8) and V is a bounded region. If we set ie — e so that 2 cose = +, 2isine = - then each component of E (e, ) rise to a function — O f(?) =f(0,) = (x, )(- ) +B(x)(r + dx v for certain linear functions A, B, and J, which is a polynomial in { and It immediately follows that the behavior for ImO - coD is equivalent to the behavior for I C| - oo; and thus the function f(r) which is single-valued and analytic in the finite plane punctured at the origin, satisfies the growth condition X(R) <Ae where X (R) = Max f({)| 1 =IrlR. R The mathematical problem that will be considered is the following. Given functions f( r) which satisfy the above analytic and growth conditions, and which are measured at points on the arc T, given by T; |l= 1; larg rl,<a<r 12

8579-3-Q we wish to find a numerically feasible way to approximate f( ) in any bounded region, using measurements of f( C) on T. Now let us make the transformation aiw w= -(i/a) logC, = e I. The arc T is mapped onto the real segment -1 < w g 1; as I tends to zero, Im w tends to infinity, and as r tends to infinity, Im w tends to negative infinity. The ~-plane is represented on the strip - lr/a < R e w < 7r/a. If we define F by F(w) = f((w)), then F(w) is periodic, satisfying F(w+ 2)= F(w), and F(w) is entire. Furthermore, ie I R B aR F(Re.) <X(e ) <AeBe (2.1) We shall show how to approximate F(w), to within an arbitrary error c > 0, in the disc |w|<(log S) / a. This will yield an approximation, with the same tolerance, C, for f (r) in the annulus e /S I< <S/e, (2.2) for if we let [(arg r) | ~ 7r we have <( OKS - + X 2. The Legendre Coefficients of F(w) The Legendre polynomial of degree n is 13

8579-3-Q 1 dn 2 n P (x) d (x2- 1)l 2nn! dxn The Legendre polynomials are orthogonal over -1, i], with P (x) P (x) dx = -11 2j+ 1 The Legendre coefficients of a function h(x), assumed integrable over [-1, i], are a. 2 + 1 h(x) P.(x)dx -1 and the formal Legendre series of h(x) is OD h(x) - ajP.(x). We state the following theorem, which is a special case of Walsh (1935). Theorem 2.1: Let h(z) be an entire function. Then oo h(z) = E ajPj(z), (2.3) j = {aj} the Legendre coefficients of h. The series converges uniformly in any bounded set. We shall need to be able to estimate the coefficients a. in (2. 3) in terms of the growth of h(z), as one would for an ordinary power series. The proof of the following lemma is suggested by the proof in Indritz (1963, Theorem 5. 3E). We shall need the identities Ptn+ - P' (x) = (2n+) P (x), n >. (2.4) n+1 n-1 n 14

8579-3 -Q P (1) = 1., n P(.-1=( )n n (2. 5) Lemma 2. 1. Let h(x) be infinitely differentiable on 1 ill, with a. the Legendre coefficient of h(x). Then 12a1i< 1 Max I rI,<h -1 (2j - 1) (2j -3).. (2j -2k+3) h~) j+r dt (2. 6) for 2 <k <j - 1. (Here, hY is the k thderivative of h.) Proof: We have 2a i= (2j~l h(t) P(t) dt - 1 1 (t) W- P. (t J ''Lj+1 i-i1 -1 -- jh?(t) {P ~(t ) P. (t)}dt, ii -1 using (2. 4) and (2. 5), integrating by parts.- Iterating this operation again gives us 2ai= h" (t) Pj+2 _______________ -P. 2 i2j+3 2j -1 d then 1 12a.K <, Max h"l(t) P (t) dt -1 15

8579-3-Q which is (2.6) with k=2. To complete the verification of (2.6) we use induction. Assuming (2.6) with k<j-l, we have 2k 2a JI 2 (2j)... (2j-.2h3) f (2j-l)... (2j-2h+i3) Max Irl< k -1 h(k)(t) d(P i+r+l(t ) -P +r- i(t) 2 +2r1(t2j+2r+l 2'2k \< (2j-1)... (2j-2k+3) (2j-2k+1) Max I|r|k+l 1 h9k+l(t)P (t)dt J j+r -1~ and the proof is complete. Now setting k= -1 in (2. 6) we get 12a |I 2j-1 (2j-1) (2j-3)... 75 3 Max l<:r <2j-1 h(-1)(t)P (t)dt -1r 2'. (2j-1)(2j-3)...75- 3 Max h( )(x) xE [-1, 1 since IPr(t) |< 1 for -1 K t 1 (Walsh, 1935). Thus 2i (2j-2)... 42 Max 2a< (2j-)...7.5-3 (2j-2)...4.2 Max xE -1, 1 h(j-1)(x) - 2 -1)! (2j-l)! Max Ih(-1(x)I. xe [-1, 1] (2.7) Now we wish to apply (2.7) to the specific function F(w) described above. We need to estimate IJ(k)( Max xe [-1, 1] 16

8579-3-Q Now, assuming R > 2, F (k) F(w) k! 2ri ( w 2k+l R(wx)k+lw < Max F(w) (R-l)l IwI= r <2R-k Max IF(w)I IwI=R OR -k Be <2AR eB (2.8) using(2. 1) in the last step. Of course, (2. 8) remains true if we choose R to minimize the right-hand member, subject to the constraint R > 2. Let -k BeaR 0(R) 2AR e Then log (R) =log2A - k log R + BeR and (R) = +Bk e R 0(R) R We have d2k 2 aR 2 log0(R) = + Ba2 e dR R which is positive, thus 0'(R)/0(R) increases, as R increases from to oo, from negative to positive values, and there is a unique number Rk such that 0'(Rk) = 0. 17

8579-3-Q If k> 2Bae2a then Rk> 2, so that Rk satisfies the constraint which was necessary to impose on R for (2.8) to be valid. The equation '(Rk) = 0 can be put into the form aRk + log Rk = log (k/Ba). (2.9) From (2.9) we immediately get Rk < 1 log (k/Ba), (2.10) k a which is an asymptotic quality as k tends to infinity. Now let k (a) be the first integer k such that log(k/Ba) > ae+1. (2.11) Then for k > k (a) we must have Rk > e. Now by elementary calculus 0 k > e, x >e (2.12) log x since the derivative of the left hand side of (2. 12) is positive for x > e. Thus aR+logR<R Ja+1i], R>e so that for k k k(a), Rk a+ e] > Rk+lOg Rk log(k/Ba), and solving this inequality, Rk > e+ log(k/Ba). (2.13) k(2.13) 18

8579-3-Q We now use both (2. 10) and (2. 13) to calculate 0(Rk), which is an upper bound for F(k)(x) /k! on [-l,], recall, (2.8). We have, assuming of course k>k (a), (e 1k log(k/Ba) 0(Rk) < 2A ) ogk/ e e[og(k/Bak/ k < 2 (r+l)kA ek/e <2A (e /log(k/B)}. (2.14) The right-hand side of (2.14) is an upper bound for F(l (w)/k! on [-1, 1]. Combining this fact with (2.7), we have for the Legendre coefficients aj of F the upper bound lajl < j > k(a)+l. 0 (2.15) We wish to simplify (2.15) as muchas possible. We have j C(-l)!]2 (2j-1)! (2(j-1)! (2j-1)! '- J.=. 2 1 2j-1 2j-2 * j+1 < 1/2j-1 since (j-k)/(2j-k) < 1/2. Thus i 1/ t fo g '-1 J loeB 5j -1 e(j1)/= 2A og-/B 9 I B7T i **Lgi-~~ 19

8579-3-Q since a<nr, and if j-1 >(B7r) so that log(j-1)/B7r > log(j-l) we get the more convenient estimate a1 < 2A(l 1) j- (2.16) provided j>(Bir) +1; j>k (a)+1. (2.16') Equations (2.16) and (2.16') comprise the objective of this selection; bounds for the Legendre coefficients of the function F(w). 3. Error in Approximating F(w) by a Partial Sum of the Legendre Series We wish to approximate F(w) in the disc Iwl< logS. (2.17) Let SN(w) be the partial sum N SN(w) = ajP (w) j=0 of the Legendre series for F. By Theorem 2.3, oo o00 |F(w) -SN(w)= l aP. (w) <2A 2Oe1a Ip() (2. 18) j=N+l j=N+l log(j-1 20

8579-3-Q provided N >(B7) +1, N >k (a)+1, 0 (2.18') by (2.16) and (2. 16'). Now we use the well-known inequality (Indritz, 1963, p. 269) IP.(w) 1J2w3 wl > 1. (2.19) Substituting in (2. 18), we have jF(w) -SN(w) aO <4A Z j=N+1 4A < log S <4A logS ca /40el/ i log(j-1)).IA. i-1 Iw I 40o w el/j-1 < log(j-1) j 40e1i log S) c alogj (2.20) for Iwl< log S, N satisfying (2. 18'). If N is so large that 1/l 80 e80e log S log N > - -- CY then from (2. 20) we have IF(w) - SN(w) I/a 4Ae/ log S 8Ae / log S a 00 j=0 N 0 e. log S e 1log N 1 2] 1/e l og N \ flog N We have Lemma 2.2. Let N satisfy the inequalities 21

8579-3-Q (i) N >(Br)2 + 1 (ii) N>k (a)+ 1, where O o ko(a) = Min >ae + 1 (iii) logN> 80e S lo Then /alg (Oe 1/a lo N IF(W) N(S W) < 8Ae/2 lo S 10.21) 4. The Beginning Coefficients Let N = N (S,E) be the first integer N which satisfies the hypothesis of Lemma 2.17, and such that 8A el~a logS ael lg< /2 a k log N Then F(W)-SN (w) < E/2. To approximate F(w) to within an error e in the disc Iwl < (log S)/a, we need only approximate aj, for j <N, closely enough. Let F *(w) be a piecewise continuous approximation, on -1 < w 1, to F(w), and let a* be the Legendre coefficients of F*(w). Let N1 S*(w) = a Pn(w) P=0 22

8579-3-Q Then, since |P(w) < |2w for Iwj > 1, IS(w) - S (w) N1 j=o aj - a* 2 log s i a j < N(2log 1) Max j N1 Iaj-a'1 (2.22) provided 2logS >1 cy s Now Ia - aI= F(x) - F*(x)} P (x)dx 1 2 ~x) - F*(x)] dx. 1 - dx -1 -1 by Schwarz's inequality, and the norm of Pj, as mentioned in 2/(2j + ), thus Ia a < 2j+1 II F- F*I <2||F- F*|| (Here, || h|l is the inner-product norm over [-1, 1].) From (2.22) and (2.23) we see that Section 2, is (2.23) SN(w)- S(w) < E/2, IWI <,log S 1 I provided jF- F*Jj< <1 14N N1 2 log S E. (2.24) 23

8579-3-Q Now let us take interpolation points 1 x = -1+ J, m j =1-m, and define F*(x) by Then F*(x) = F(x)+ j, |F(x)-F*(x)I <nl + 1 x < Xj+ 1 m Max -1< t 1 |F'(t) 1 (2.25) To estimate the derivation in (2.25), we have |F'(t)I = - =R J:R F(w) (w-t)2 dt < R Max (R- 1)2 |w = R IF(W)I and choosing the convenient value R = 2, making use of (2.1), we get Be 2a IF'(t) <2AeBe =C Substituting in (2.26) we get (2.26) IO(x)- F-*(x) < n + C with C defedby(2.25). Therefore with C defined by (2.25). Therefore (2.27) I|F- FF*|l [ 1 n +C )2dt j m 2 = 2 [Max (7 + C 2 l+ m j = 2(r + )2 m 24

8579-3-Q where Y = Max r7 i (r) is to be interpreted as the maximal error in calculating F at the databearing points x.) From (2.24) and (2.27), we see that N(w)- S(w)| < e/2 I, w< o S provided C < 1 ( ) N1/2 m logS 5. Statement of the Result We bring together the above considerations by stating a formal theorem: Theorem 2.1: Suppose that (i) x - j, j =1-m (ii) F*(x) = F(xj)+.j, xj < x < xj+ N (iii) S*(x)= E aJP,(x), j=O 3 where 1 a* = \ F(x) P.(x)dx -1 (iv) k (a) = Min ( ogkB >ae+ 1 O lg ~ 25

8579-3-Q Then, for e >, we have I|(w)- S (w) < Iwl < g a (2.28) provided (v) N >Max (Br)2 +1, k(a) + 1 and 80e log S logN > a (vi) If r = Max i then n+ C< r1 m 8N, logS \2 logSJ N1/2 F I N the minimal N satisfying conditions (v), C defined by (2.26). 1 When (2.28) is satisfied, we have |X(z) -sN ()] < e 1/S< | | <S (2.28') where Ls() = SN(), w = -(i/a) log r \ (2.28") 26

8579-3-Q III ANALYSIS OF THE COMPUTATIONAL RESULTS OF DETAIL SPECIFICATION (PART i) - SUBROUTINE F6: ET x E* = 0. In the present CPCEI Inverse Scattering, which is restricted to the identification of perfectly conducting targets, the boundary condition ETx ET*=O was implied where ET denotes the sum of the incident and scattered electric field vectors, i.e., ET=+Es. The near field representation of E. and E is obtained from an -1 expansion into proper vector wave functions, in particular for finite convex-shaped bodies the approximation representation in vector spherical harmonics is employed. Using Stratton's notation (Stratton, 1941, Ch. 9. 25) it can be shown that the total field ET for the case of a perfectly conducting body can be given with exp (-ixt) time dependence for E A ikz ^ ikz j E = x E e = x 1 e = (R sin e cos + 0 cos 0 cos 0 0 o - 0 sin 0 ) ei(kRcos 6) (3.1) as A T T A T E R E + 0 E + 0 E (32) — r = RoER oe o0 where 2(2n+1) i 1 Cos R n= n(n + 1) einJn(p) + geinh()(p) n p oD (2n+ 1) (i)n+l (1) (cos 0) cos 0 (3. 2a) n=l P 27

8579 -3-Q Co n1l 2(2n + 1) fif j(p) + f t Ln' in oln + em -- (1 h(1) 1p Pnos9 hnp sin e P (Cos 9) p8 rCos + [g'[ i(P)] = (2n +1) (,n n=l n(n + 1) [(1) [7 P (Cos 9) sin 9 +i p (1) aPn(Cos 9) ]eCos 0 (3-2b) T _ 2(2n +1) fIf i 0111fs (1) 89) Ln= n + 12 Olo~n +f5 hnp I Pao9 I-. [ph~'~U IP (Cos 9) g~el nLP i1n(p):O + n 1 p sine0 sin 00 (2n + 1) n(n + 1) (i, (1) 0~n 8Pn (co s 9) n)0 +i p (1) n1 P n(cos 9) sine j sin 0 (3. 2c. where for a perfectly conducting sphere of ao= ka: i a-= n1 1 i, S = n +1 u(n+1 -l ai, s 2 an an= - tc (1()] I arh a (3. 2d) 28

8579-3-Q fis 5 = n n(n+ 1) bis olin 2 n i = 1 b =1 n b = - n ( 3. 2e) jn(a) h(1)(a) n (1) (p,a) = - n [n~)t(1) ' - 1(p) ca P hn (a hn (P) n( - n (a) p = or - i2 (3, 2f) (1 ) (p, aC) = J(1) (p, a) = n [(p) h( ) (a) - hn (p) ], <(f h(1) (a) n p=oa = 0 (3.2g) =0 p=c (3. 2h) It was decided to work with Stratton's representation only and to consider one test sample exlusively, namely a perfectly conducting sphere of a = ka = 2 where the associated expansion coefficients f, and g s are given in Table III-1. oln ' eln For these values the total field E,was derived from equations 2 and the boundary condition E x E = 0 subsequently applied. Since an absolute zero of this condition cannot be found a minimum searching routine was employed 29

8579-3-Q which searches for that Xmin=kRmin for which 4 ETxETTI 2 becomes a minimum along a particular aspect angle. In Table EI-2 the results for a perfectly conducting sphere of o=2 are presented for T=00(22.50)1800, V= 00(450)3150 where the Xmin2 were obtained for a searching increment of AX1=. 01 over the range 1.6 < X1 < 2.5, employing a subsequent researching subroutine over the angle Xmin- AX1 < X2 (AX2=. 001) < Xmin + AX1 yielding Xmin2 with Min -J ~ E]pxT I. The results obtained must be considered excellent, since aside from some isolated critical points, the deviation of Xmin from the exact value is less than 1 percent. Table I1-2 furthermore presents those values Xmin for which the boundary condition | | | - | E | } becomes a minimum. In Figs. 3-2a, b, c and d, the surface loci are plotted. It can be seen that the condition | E. - I E I = 0 fails to yield the exact result in the shadow region for particular aspect angles, whereas the condition ETx ET = 0 is applicable far into the shadow region. Those critical points on the sphere for which the boundary condition ET x ET = 0 fails to yield the proper results for a given incident field, will be determined from Eq. (3.2 a - h). In Weston et al (1966) it has explicitly been stated that the boundary condition ET x ET = 0 is a necessary but not sufficient condition. For example, it can be shown that for a parallel polarized plane wave Hi=x Eo/%r. exp[i(y sina-Z cosa)J as indicated in Fig. 3-1, (ETx E )/= x i sin 2 a inin (2kz cosa). Hence for normal incidence (a = 0) the behavior is identical with that of a normally polarized plane wave where ET = 0 for Z = 0 and ET x EET = 0 is identically zero and actually not applicable. If a = 0, 7/2 (ET x ET ) = 0, yields nn Z 2kos n = 1,2,..., 2k cos a >.' 30

8579-3-Q z ////// /~P/////r Y x FIG3-1: PLANE SURFACE CASE 31

8579-3-Q resulting in an infinite number of planes for which the condition (ETX ET) = 0 is satisfied, and thus the proper solution is given for n = 0 only. In the case of grazing incidence a = r/2, ET $ O at 2 = O and the nodal point of ET is off the conducting surface, however ET x EF = 0. To determine those isolated points for which the condition ET x ET = 0 may fall, the following analysis is helpful. Let p = a in (3. 2a, b, c) thus OD 2(A t 1) P (cos 0) cos ER ( =a) = (()(3. 2a') R n= aw h"() (a) E( p = a) E(p = a) = 0 For 0 = 0, c0t 2(2n + 1) P (cos ) ET + r (_) ~< (i)n R n=l e [a h( '(a)J,n nn= 2m Mm? 2ml OD i T -2(4m + 1K2m - 1)! m=l r 2 m!a ah(l)(fa)]' which is a complex nonzero constant corresponding to the case of grazing incidence of a parallel polarized plane wave onto a planar surface. For this case ET x ET = 0, although ET. InFig. 3(- 6a both IETxE I f (Xmin) and {|Ei -IE =f(Ym) are plotted versus p = kR, indicating that the first zero of (Ex ET ) = 0 is not identical with that of I I - I Et = 0, and the Ymin (O = O, 0 < 900) has a large deviation as can be seen from Figs. 3-2a, b, c, d and Figs. 3-6a and 3-7a. The condition tI Eil -1 E i } then only yields the correct result for an exceedingly large number of expansion terms. 32

8579-3-Q For 0 = 7/2, t = + 7r/2, the boundary condition for a truncated series expansion of the scattered field E. may work, since Er (p=a, n=N aco)# 0, T although ER (p=a, n=N# oo, - ~+ r/2) = 0, corresponding to the case of normal incidence of a parallel polarized or an oblique normally polarized plane wave onto a planar surface. Thus the condition ETx ET = 0 for n=Nf oo does work indeed in this case, however, the minimum searching subroutine will determine those points for which ET(p=o, n=N4 oo, =f 0, T, 0= t+ r/2) = 0. The computational results verify this point to its best, since for a searching increment of X =. 0001, Pmin= 2. 0000 for all values at + = + 7r/2. This property is also verified by the * -15 fact that the values of ETx ET = 0 are by a factor 1015 smaller. In Figs. 3-5c, 3-6c and 3-7c the corresponding I- ETxET Iand jEi | - XE4j1 values are presented, also indicating that the first minimum of ET x ET is identical with that of f IEI - Es, andpreciselyat min = 2.0000. For this particular a spect angle ( = + 90~) the condition I EJ I - | E sl holds also in the shadow region (see Figs. 3-2a, b, c and d). The distinct singular point for which E x E* = 0 fails entirely is the focal point of the shadow region or the zenith (0=0), at this particular point the number of expansion terms as well as the number of digits to which the computation is correct must be a maximum, otherwise the corresponding minimum may be at random between 0 < p < ao due to the slow convergence of the vector spherical harmonics of this singular point. 33

8579-3-Q The boundary condition E x ET = 0 is correctly applicable for 9 = 180~ and in fact for all O, yielding identical minima as shown in Fig. 3-4. In Higs 3-5b, 3-6b, and 3-7b, the values of -| ET x E eI - I are plotted versus p=kR for p = 45~, 9=135~; 90~; 67. 5~, indicating again that the condition ET-x ET = 0 is superior to - E since it is applicable farinto the shadow region. Inspecting Figs. 3-2 to 3-7 will show that the boundary condition ETX ET = 0, aside from 0 = 0, can be employed to its best. For the particular value a = ka = 2 the computed values in both the illuminated as well as the shadow region lie within one percent of the exact value. This result may not be obtained if a >> 1 in the shadow region. Since, however, in practical cases the points in the illuminated region will be considered only, this matter is of less concern. However, the deviation of pmin > s in the lluinated region or shadow region respectively needs further interpretation which will not be presented here. An entirely different question of interest must be answered; namely how to find the range for which p = a = ka, corresponding to the proper locus, for which the m - ET x E may yield the distinct point on the surface of the unknown target. To do so, Figs. 3-3a, b, c and d will be interpreted, which present the loci of successive minima over the range 1. 8 < p < 15. 5, resulting from the boundary condition ET x ET = 0. It can be seen that the locus of the first minimum describes exactly the sphere of radius cr = ka = 2. All loci associated with the minima of higher order describe concentrical, asymmetrical hyperboloids, separated from one another by almost equal spacing. In fact for 9 = 180, and p >2 a, 34

8579-3-Q A Pi = const =. 86. This particular phenomena, due to the fact that the condition ET x ET = 0 is a necessary but not sufficient condition, may be employed to find the approximate range of the scale size a = ka of the target in * question. Namely, applying the condition ET x ET = 0 at the backscatter-aspect angle at fairly large values of p over a few periods should yield the approximate range of a, since A Pmin a. In addition the properties thatlEi - E I= const for p > a and varies only near p a, with a minimum at approximately the first minimum of of ET x ET, may be used as an ultimative check. Another question of interest is related with the necessary number of expansion terms as well as the number of necessary digits to which the expansion coefficients g m f are correct. It has been shown already that proper number varies gem,n nm,n with the part cular aspect angle. However it can be shown that application of the condition ET x ET = 0 requires a smaller number of expansion terms as well as digits. This is indicated in Tables III-3 and MI-4 for the particular aspect angle = 45, 0 = 1350. It may be concluded from the presented results that the boundary condition * ET x ET = 0 although not restricted to perfectly conducting bodies, is an extremely helpful tool in the problems of inverse scattering. 35

8579-3-Q TABLE lII-1: Expansion Coefficients foin, gein for a Perfectly Conducting Sphere of a = ka = 2. n Re. f o Im oin Re {g. gei R {1 foin | I o |gein Iein 1.4884982 -.6066279 -.2764022.4472206 2.2043585.7558526 -1.341662 -.8292217 3 -2.450365.01002389.0308867 -.4293786 -4 -1 4 -.994961. 104 -. 3155869. 10 1,04377755.000191651 5.2126115 10-2 -.3013576- 10-6 -.4768353* 10-6.267442 10-2 6.3764457 - 109.8891209. 109. 1064619 - 10 -. 539721 - 10 I169I..10.592,l 36

8579-3-Q TABLE m-2: SURFACE PVIUB8 DETERMINED BY THE BOUNDARY CONDITIONS ETx E T 0 AND {|E E 0 FOR A PERFECTLY CONDUCTING PERE OF a ka 2, AND N =a EXPANSION TERMS_ 90 90 XI x {I, xT x EIS.I) eO I ~ | Xmi(-ET -_T1 T EMI | ll-IL 0 0 22.5~ 0 1.979 7.3 ~ 10'6 2.7 1.1 ~ 10-2 45 0 2.000 5.6 ~ 10'6 2.8 2.6 10'2 17.5 0 1.999 2.81 10"4 2.3 7.0 ~10' 4 90 0 2.000 7.2 10'2 1.9 7.5 * 10'3 112.5 0 2.09 5.27 106 2.02 2.3 104 135 0 2.01 2.19.10-5 2.1 4.13 10'3 157.5 0 2.012 1.09-10 4 2.05 9.8. 104 180 0 2.0000 1.38. 10'19 2.00 4.11 10'5 0 45 22.5 45 1.985 1.67 10 2.5 9.7 -108 45 45 2.001 4.49 10'7 2.6 2.1 '102 47.5 45 1.988 9.66. 10'5 2.2 8.4 - 103 90 45 2.0000 3.6 - 108 1.94 5.5.10'3 112.5 45 1.984 2.36. 10'3 2.04 5.1 103 135 45 2.009 7.32 10'3 2.04 3.5 * 10'3 157.5 45 2.001 7.27 10'4 2.0 2.1 10'3 180 45 2.000 8.25.10"20 2.00 4.11 10'5 0 90 22.5 90 1.99 1.21 10-22 2.000 9.6 ' 103 45 90 2.0000 5.1 1023 2.00 9.14. 105 67.5 90 2.004 8.03 10'23 2.00 6.39 10'3 90 90 2.0000 5.65 10 24 2.00 1.60104 112.5 90 1.989 2261022 1.99 47 112.5 90 1.989 2.26' 10 1.99 4.7 ' 10 135 90 1.987 3.31 10-21 1.76 4.3 104 157.5 90 1.997 1.15. 10'21 1.92 2.3 10'3 180 90 2 f 4.8.10-20 2.000 4.11. 10'5

85794.Q TABLE 111-2: (Continued) 00 Xmi x~ X E { xE2} y (E E- jEJ) q JEd 22.5 135 1.985 1.67 10- 2.5 9.2 103 45 13 5 2.001 4.49 10 2.6 2.1 10 67.5 135 14988 9. 66*105 2.2 8.45 10.3 90 135 2.0000 3.6.10o8 1.94 5. 5 I- 0lC 112.5 135 1.984 2.36,,10 2.04 5.1 -10 135 135 2.009 7. 32 1O3 2.05 3.5 13 157.5 135 2.01 7.2 10- 2.00 2.1 o 180 22.5 180 1.968 2. 76 10' 1' 2.7 1.1 "o2 45 180 2.001 1.26 106 2.8 2.6 o102 67.5 180 1.987 6. 62. 10'8 2.3 7.0 10-4 -7 -3 90 180 2.000 1. 12 107 1.9 7. 5. 10 8 -4 112.5 180 2.088 5.03~108 2.07 2.3 10 135 180 2.o051 2.22 10.6 2.1 4.15 10.3 157. 5 180 2.024 i.06,010-7 2.05 9.8.10'4 180 180 1.999 1.99.1020 2.00 4.1 0 225 O 3-3 22.5 225 1.985 1.67 104 2.5 9.2 ' 10-3 45 225 2.001 4.48 - 10.7 2. 6 2.1 '10'2 67.5 225 1.988 9.66' 10' 2.2 8.4 * 10 90 225 2.0000 3.60,10C8 1.94 5.5 - 1 '" 112.5 225 1.984 2.36 10' 2.04 5.1 '10 135 225 2.009 7. 32 10.3 2.04 3.5.10.3 157.5 225 2.001 7.26 104 2.00 2.1 *10-" 20.-5 180 225 2.000 8.26.10 2.00 4.1 '10 38

8579-3-Q TABLE m-2: 39

8579-3-Q TABLE mI-3: SURFACE POIUBS DETERMINED BY THE BOUNDARY CONDITIONS ET x ET = 0 AND IE. -|E = 0 FOR = ka=2 AND 6 = 1350~, = 45~ ND N EXPA NSION TERMS 40

8579-3-Q TABLE II-4: SURFACE POIUBS DETERMINED BY THE BOUNDARY CONDITIONS ET ET = 0 AND {I - E |= 0 FOR T T a = ka = 2, AND = 135, P = 45 N = 6 AND M DIGITS. 41

= 270~ = 180~ * - Xmini y - {T T} X - Xmni — {IEi - EIE.} FIG. 3-2a: RESULTING SURFACE LOCI FOR ETx E = AND E-| =0 (O = 90~ - Plane).

6 = 90~ 0 = 67.5~ 0 = 157. 5 0= 180~ 0= 157.5 W3 = 90~ 0 = 67.5~ FIG. 3-2b: RESULTING SURFACE LOCI FOR ET x ET = 9} AND |Eji - oE| = 0 (0 = 0-0 = 180 - Plane).

8 = 112.5~ = 90~ = 6750 0 = 69.0 0 = 180~ 0 = 0~ -= 112.5 \ 0 = 90~ 0 = 67.5~ FIG. 3-2c: RESULTING SURFACE LOCI FOR {ET x ET = 0 AND {El - |E|}i 0 (o = 45~ - 0= 225~ Plane). '-T 8

0 = 90 8 = 67.5~ = 112.5~, 6 = 135~ 6 = 450 = -900 0 = 157.50 | = 22.5~ =1800 0 = 00 e = 112. 25~<- 6. 7 * = 67.5~ 0 = 900 FIG. 3-2d: RESULTING SURFACE LOCI FOR T x ET = 0 AND {1E51 - |EJ}0 =. (9= 900 Plane).

15.5 o j-=900 ' 2700 Ki -Ei jpro * ( 0 o FIG. 3-3a: RESULTING SURFACE LOCI OF SUCCESSIVE MINIMA OF E x IE* (9 a - PlaE). -T -T

e90 0-0~, 8-67.5~ - 180~, 9 = 90~ 0-180~, 0 112.50' 0. 00,0. 900 0 0~0, 9 112.5~ I a 135~ K L E -1 FIG. 3-3b: RESULTING SURFACE LOCI OF SUCCESSVE MINIMA OF ET E* ( - 0~ - 0 - 180~ Plan). -T 2 E

9=45~, =67.5~ 00 = 45~, 0 = 10~ =45~, = 112.5~ - 2250, ~ 157.5~ | 45~, 0 = 157. 5~ e.180~ FIG. 3-3c: RESULTING SURFACE LOCI OF SUCCESSIVE MINIMA OF E x E * (9 = 45 - 0 = 225 Plane). -T -T

0 a 00 0s 90 ~,067.50 0 = 90.8 = 90 =90~, = 112.5~ Kt L Hi it -90~, 0 157.5~ I|.1 90~, 9 157.5~ FIG. 3-3d: RESULTING SURFACE LOCI OF SUCCES!VE MINIMA OF E x ET (0 90~ - 0 - 90~ Pla). -T X-T(09 -99 Ple)

1 f2(p) 10-14 = I-Ei - IE) = V- 1r x * 12 x Er I 10 - \ -2 - 4 10-5 10 lo-S 2 3 4 5 6 7 8 9 10 11 12 13 14 FIG. 3-4: ILLUMINATED REGION- SPECULAR POINT, 0 = 1800, ( = 0, 180). e1 = 180~ = 0, 18 0)..5 L5

I9 I i-A0 (M I 0 Uc 9A '-A CA3 0 T --- — A CA)3 CA3 U-l F-4f H ti — 01 0 cc0 i~ — 0 I' H x — 0 li11 Im I I trj OLI I —A 01n

' —4 ' —4 0 ' —4 0.1 10 IS. 0) 10 <z rz mv ' —4 lCY)I IOI C)I C r —ir —ir —lr-14 52

'Il WzI II a. '4-fq Lo ' —4 '-4 *1E-4 II ' —4 '4-4 0 0 a) '-4i Is. 0 O LO r4I II a) 0:1 -0 E10 C., 0 '-4q 5~3

LO T-4 co r —4 11 IOF4.e cq 4.4 r —4 T r —4 0 (D It CD Its_ T-A 0 O m 11 M (Z >-4 P 4 00 p 0 pq E0 p U) CD *0 Cd CD I m LO 6 -A %4 L —LUJ-L --- N r-4 I I O C) — 4 r-4 O r-4 54

Isl 10-1 10-2 fl~p): - xE c 10 10-4 10-5 i I, I I I I I, I I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 FIG. 3-6b: Shadow Boundary, 0 = 90~,, = 45~.

__,(P) i= - l i 2 1 s I 1 I 10 1 10 -2 E —3 11 _ p 10 16 I I TT cn 0, 11 1 - 1U I -4 10 I a ai I I I I I I I, I, I I 11 I I I I I I I I I I I II I I I 1 * — 2 3 4 5 6 7 8 9 10 11 12 13 14 15 FIG. 3-6c: SHADOW BOUNDARY, 0 = 90, = 90.

ry,4.0 -dwRwA cq.)IL $I E-4 OD WI WI x I E-i — T-r-, WI W I I It 11 0 It% 0 f 0 LO '-4 V CD I, C)) z 0 ' —4 0 0 so ' —4 0l 'I 0 ' —4 57

1 1o-1 (1(p) -j I.ET 01 co - 102 -I I I I I I I I I I i 1 I I I p I I I p 2 3 4 5 6 7 8 9 10 1 FIG. 3-7b: SHADOW REGION e = 67.5, 0 = 450 L1 12 13 14 15

f2( = lE I I- E 10-1 10-2 10 oCl CD 2 3 4 5 6 7 8 9 10 11 12 13 14 15 FIG. 3-7c: SHADOW REGION. 0 = 67.5, - = 90o

8579-3-Q IV MOTIVATION AND HEURISTIC DEVELOPMENT OF INVERSE SCATTERING THEORY Several different methods have been developed for the inverse scattering problem as is evident from the review article by Faddeyev (1963). The best known attack is originally due to Gelfand and Levitan (1951). For the quantum mechanical case, Levinson (1953) has motivated this attack. A very similar method was developed by Kay (1955) and Kay and Moses (1955 - 1961) for not only the quantum mechanical problem but also for the one-dimensional wave-equation. For this latter case, it is possible to give a more transparent motivation and development in a manner similar to that used by Kay (1960) in a little known paper. In addition, Moses (1956) has given an entirely different method which appears more natural to people acquainted with diffraction theory and which in addition is valid in three spatial dimensions. Both of these methods use ideas from perturbation theory as their starting point and both are limited to the case of perturbation by a re_ potential. The reality of the potential implies an analytic continuation of the reflection coefficient which appears essential for anything like a practical application. It appears that the difficulties associated with surmounting this are the chief ones preventing tbp extension of either method to the case 60

8579-3-Q of a complex potential. As is customary, we will denote the self-adjoint extension of the operator - A by H and that of the operator - A + q (x) by H so that H = H + V where V denotes multiplication by the real potential q (x). Of course for H to exist, q must satisfy certain conditions —for our purposes it will be sufficient to assume that q is locally Hoelder continuous except for a finite number of singularities and that is in L O L2over E1 or E3. With these assumptions the results of Ikebe (1960) are valid so that the interested reader can refer to this paper for detailed proofs of the arguments we are about to give in an attempt to motivate the perturbation theories underlying the two afore-mentioned approaches to scattering theory. For simplicity we will also assume that q has no bound states — for example, q could be a repulsive potential or barrier. Then if f0 (a ) satisfies the scattering integral equation iklx-yI I) 0(xk) = e - q E IX I(. yk) dy Ikebe has established the 4ollowing results: (Part of his Theorem 5) Let f(x) be an arbitrary L2 function. i) Then the generalized Fourier transform A -3/2 f f (k) = (27r) /.mj 0 (x, k) f (x) dx of f(x) exists and belongs to L2(M) where M is the 3-dimensional space formed by vectors k. if) The following expansion formula is valid: f () = (27r) "3/.i.m/M 0 (x, k) f (k) dk ill) f is in the domain of H which is equal to the domain of H if and o 61

8579-3-Q only if kl2 f (k) L2 (M) and under these circumstances we have the following representation of H: Hf(x_) = (2r) 3/2 1.i.m. X k12 ((k) dk Of course, in terms of the ordinary Fourier transform 2) f(k (2r) /1.i.m.* e — f( dx H admits the representation O 3) H f = (2r). i. |k| e f ( dk and as Ikebe shows the continuous spectra of H and H are unitarily equivalent. It is this last remark which underlies the approach of Kay and Moses (1955) and Faddeyev (1963) for it is the basis for Friedrich's (1948) method based on spectral representers. On the other hand, the Fourier transform of the scatteing integral equation (1) form the basis for the alternate approach of Moses (1956). By means of its solution, the "eigenfunctions" 0 (x, k) of H are expressed in terms of the distorted plane wave "eigenfunctions" exp (i k * x) of H. To see that this is not the only possible linear transformation relating these two let us introduce the linear operator U defined by the relatioan -3/2 iA 4) U f (x) = (2r)/.. 0 (x,k) f (k) dk 0 Then it follows that 03/ A -3/2 5) UH f = (27r)03/211im. 0(x, k) (H f) dk = (27r) 3. i. m. 0 - 0 J0(x,k) k12 f (k) dk *ince it is clear from (3) that 6) (Hi) = jk2 f ( On the other hand, it follows from (4) and (iS above that 62

8579-3-Q 7) H(Uf) = (2)3/2 1.i.m. 2 (k) f (k) dk Thus for any f in the common domain of H. H 8) HUf = UHf 0 -1 It is clear here that U exists so that (8) can be written as 9) HU = UH or as -1 10) H = U HU 0 Since 11) H f0 = Ikl2 0 it follows that 12) U-1 HU0 = Ik12 or that 13) H U = IkI2 U o o Comparison with H 0 = k 2 shows that the operator U takes o into ~ e. nd tht = -1 into 0, i.e. 0 = U 0 and that 0 = U 0. In the first approach to the inverse scattering problem the existence of such an operator U is postulated but since the eigenfunctions 0 (x, k) are not known it is sought in still a different form, namely as I + K where K is an integral operator with a kernel k(x,y) in the one-dimensional case. Fredholm operators of this kind are convenient since the existence of an inverse is necessary for the above argument. We proceed to develop this approach in detail for the one-dimensional wave equation 2 2 a2 a2 q(x) 14) - t-()- qx) U (xt)= 0 63

8579-3-Q under the assumptions on q noted above. For simplicity we will also assume q(x) = 0 for x < 0 and that initially there are no bound states. The associated time independent equation is 15) I (x,k) + k2q(X) u(x,k = 0 which, in view of the above, is to be considered as a perturbation of 16) d 2 12 u (x,k) + k u(x,k) =0 dx Thus we seek a kernel in the relation 17) 17) u(x,k) = uo(x,k) + i k(x,y) uo(y,c)dy or, equivalent in the relation 18) U(x,t) = U (x,t) + K(x, y) U (y,t) dy -00 Now if U(x, t) is a right moving transient in the sense that U(x, t) = 0 for C>t, it would seem reasonable to conjecture that it would depend on U (x, t) only through those values of x which the non-zero travelling wave would have had time to reach at t. That is that K (x, y) = 0 for y > x. The reasonableness of this is less evident for the quantum case since there is no causility but the conclusion is still true. Thus (18) can be ttively replaced by 19) U (x,t) = U (x,t) + K(x,y) U (y,t) dy -00 Now it is known that under the above hypotheses on q(x) that asymptotically ikx -ikx 20) u (x, k) ee + r(k) e x -oo ikx 21) ~ t (k) e x -D + oo These conditions correspond to a wave of unit amplitude incident from the 64

8579-3-Q left and a transnittedwave of amplitude t(k). We shall suppose that the reflection coefficient r(k) is given for all real positive values of k. In addition we shall suppose i) r(k) = r(-k) Im k=O ii) r(k) = 0(1/k) Im k> 0 iii) |r(k)I< 1, Im k = 0 iv) r(k) is analytic and its singularities lying strictly above the real axis in the k-plane consist of a finite number of simple poles on the imaginary axis having residues with positive imaginary parts and zero real parts. It will be assumed that there are no singularities on the real axis except possibly at k = 0. v) The Cauchy principal value of the Fourier transform of r(k) is coninuous with piecewise —continuous first and second derivatives for -oo < x < oo. These conditions are necessary and sufficient for the solution of the inverse scattering problem. The proof of their sufficiency is due to Kay (1955), that of their necessity to Sims (1957). A few additional comments are in order. (i) will hold automatically if q(x) is real while (ii) will hold if q(x) has a zero and first order moments. Finally, we shall initially assume that r(k) is analytic in the half-plane Im k > 0, generalizing our results to the case of (iv) subsequently. Since the differential equations (15) and (16) agree for x < 0 it is natural to set u (x, k) = u (x, k) for x<0 and hence U(x, t) = U (x, t) for O O x < 0. Now the generalized function U. = 6 (x-t) is the Fourier transinc form of the first term of (20) and 23) R(x+ t) = 2- r r(k) eik( t) dk -co is the Fourier transform of the second term. It is necessary to use (i) for the construction of this transform. In view of the (temporarily) 65

8579-3-Q assumed analytic properties of r(k), it follows from the Paley-WienerTitchmarsch theorem that r(y) = 0 for y < 0. Inserting these expressions in for U(x,t) = U (x,t) = 6 (w-t) + R(x+ t), x < 0 into (18) yields 24) U (, t) = 6(x-t) +R(x + t) +.1 K (x, y) [6(y-t) + R(y+t)] = 6 (x-t) + R(x+ t) + K (x,y) n (x-y) [6 (y-t) + + R(y+t)] dy dy = 6(x-t) + R(x + t) + K (x, t) T (x-t) + /x + 4 K(x,y) rn(x-y)R(r +t) dy where r (z) =0 if z<0 =1 if z>0 and where the lower limit follows from the fact that R(y + t) = 0 if y + t < 0. If we assume that U(x, t) is a right-moving wave that vanishes for x> t, from the above we obtain the following integral equation for K(x, t): /Ax 25) 0 = R(x+t)+ K(x,t) + 1 K(x,y) R(y+ t) dy -t To simplify this equation still further we will note that without appeal to any special properties of R(y) that it follows from (25) that if R(x) = 0 for x < -2 a then K(x, t) = 0 for x < -a. That is if t < x < -a then t + x < -2a so that R(,x+ t) = 0 by hypothesis. Moreover in the integral above y < x < -a and this coupled with the fact that t < x < -a implies that y + t < -2a so that the factor R(y + t) = 0 and thus under these conditions (25) reduces to the statement K(x, t) = 0 for t < x < -a. Applying this to R(y) = 0 for y<0 yields 66

8579-3-Q the fact that R(x + t) = 0 for x + t < 0 or x < -t and thus the fact that K(x, t) = 0 for x < -t. Thus (25) can be written as the Fredholm integral equation 26) 0 = R(x+t) + K(x,t) + J K( xy) R(y + t) dy '-x By the Fredholm alternative this will have a unique solution if the corresponding homogeneous equation X 27) (x,t)+ O x R(y + t) A (x, y) dy = 0 can be shown to possess only the trivial solution. To see that this is the case we rewrite (27) successively as 28) 0=, [R(y+t) + 6(t-y A(x, y) dy x fIo -ik (y + t) -.' r(k) e Y dk + + 2T / eik (y-t) dk}A(x y) dy Now it is readily verified that 29) r (k) e + e dk 29) J r(k) e ik (y + t) -ik(y - t)} ikt =./ (1 -r (k)|2) eik t-) dk+ J ( ik + 0 -00 -ikt iky -iky + r (k) ekt) ( ei + r (k) e k ) dk so that (28) implies that 67

8579-3-Q 30) {1 - |r(k)2} A(x, t) e- dt (x, y) iky d k + + e + r(k) e- A(x, t) dt x Y+r(k) e 'A(, y) dk O -00, -X + J x However for real k, conservation of energy demands that r(k) 2 < 1 so that each term on the left above is positive for any function -x X -ikt -X unless it is identically zero when x < 0 so that (30) implies that (x, t) = 0 if x + t = 0. Thus only the trivial solution exists and thus (26) will have a unique solution and the representation (19) will exist. In order to find the relation between K (x, y) and the potential it is merely necessary to apply the differential operator (14) to (19) and to make use of the fact that 31) dK(x,x) / dx = K (x,x) + K (x,x) Thus * f X 32) 0 = U - U -qU = -qU + K(x, y) U(y, t) + xx tt 0X -X +K(x,x) U (x,t) - K(x,-x) U (x,t) - 0x 0 x -qU U(y, t) + K (x, x) U (x, t) - 0 XX 0 X 0 -X d I(x, x) d -K (x,-x) U (x,t) + x U(x, t) + K(x,x) d Uo(,t) * For clarity, the subscript o will sometimes appear as a superscript o in the following. 68

8579-3-Q dK (x,-x) d d U(x, t) - K(x, -x) U(xt) - o x o K K U dy -YY -qU U + K(-x, x) U (x, t) -K (x,-x) U(x t) + _ xx o x o x o -x + - (x,x) U (x,t) + K(x,x) d U (x,t) - dx o dx o dK d d (x, -x) U (x, t) - K(x, -x) U (x,t) - o dx o - K(x,x) U~ (x,t) + K(x, -x) UO (x,t) + Y Y + K: (x,x) U (x,t) - K (x, -x) U (x,t) - K U dy = x (K-K - qK) U (y,t) dy + -X + [ dK(x, x - q(x) U(x, t) + 2 dK(x, -x, I dx odx That is, (14) will be satisfied if K(x, y) satisfies the partial differential equation 33) K - K - qK = 0 xx yy subject to the conditions 34) dK(x -x) 0 dK(x,x) _ dx dx 2 This Cauchy problem of the second order linear hyperbolic equation has a solution so that such a K can be found. N.B. K (x,-x) = 0 implies d K( 69

8579-3-Q Conversely suppose that q(x) is defined by (34) where K(x, -x) = 0 then applying the differential operator 2 2 '2 - - q(x) to (26) yields E2 2x 2 2 35) 0 = - -q(x)K + d2 2 2 2 X t dx dt -X dx at - q(x)K] R (y + t) dy which is the same as (27) if one now sets 2 2 dK d K A = 2 - qK dx dt It follows from the previous uniqueness argument that 2 2 a K aK dK K - 0. 2 2 d x t so that any solution of (26) will also satisfy (33). All of the above can be generalized to the case where the assumption of analyticity in the upper-half plane is replaced by the condition (iv) provided that the function R(x) is appropriately defined. Assume in accordance with this assumption that r(k) has poles at k = ir. where T.> 0. Taking for convenience r(k) in the form -2i&K r (k) = g (k) e where g(-k) = g (k) for k real and where the residue of g(k) at the above poles is r. we will now define R(x) by the expression J R() (k) -ik(x + 2a) d Z TeX (x) k) dK + 2z 3 A. i 70

8579-3-Q where the normalization constants A. will be chosen so as to make R(x) = 0 for x < -2a. Using (ii) we can close the contour in the upper half-plane by Jordan's lemma and thus obtain for x < -2a that ~1 /t g(K) _-ik(x + 2oa) T itT + T.X g(K) e dK = i x. e j 21r J -00 Thus if A. is chosen so that 1 X = -ir. e J A j J Then R(x) = 0 for x < -2a. By our previous observation this will imply that K(x, t) 0 for x < -a and hence that q(x) = 2 dK(x, x) / dx - 0 for x < -a There is actually no loss of generality in now setting a = 0 so that the potentials constructed above even in the presence of bound states will all vanish for x < 0. There is also no difficulty in extending the uniqueness argument for the homogeneous equation (27) to this case —it is carried out in detail by Kay in (1955). This completes our description of Kay's adaption of the Gelfand-Levitan approach to the one-dimensional wave equation. To actually construct q(x) it is of course necessary to solve (26) by some process such as successive iterations. The method described above does not appear to offer much promise for the solution of a large class of electromagnetic problems although as demonstrated by Moses (1967) elementary transformations will sometimes permit an electro-magnetic problem to be rephrased in such a way as to make the above applicable. In contrast the method developed also by Moses in (1956) seems more straightforward and perhaps capable of generalizations at least to the electro-magnetic problems which can be formulated in terms of vector 71

8579-3-Q integral equations analogous to (1). Cf. Dolph and Barrar (1954) and Miller (1957). Moreover there is little difference between the cases of one and three dimensions so that one may as well develop the theory for the latter. From (1) it follows that for back scattering, one has asymptotically that ik x ilkj&I 36) (-k)- / — 2 + r (2)3/2r where 1/2 e ik. x' _ ' 37) r(Q)= -(W/2)1 Je q(x') ( dx', k>O (2.)3/2 Again, although r(k) is defined only for real k > 0 in the event that q (-) is real it can be shown that 38) r(-k) = r(k) k>O or that 39) r(-k,O,0) = r(k,O,0) k> 0 It will now be shown how the scattering potential can be obtained from r (k) where k is such that it makes an angle less than r/2 radians with the positive z-axis. This restriction takes care of the over-determinancy mentioned by Faddeyev (1963). The first basic equation of this theory is just (1) written in the momentum representation. More explicitly using the known fact that tk x-y4 1 _ ikx -iky 40) J dk' 4-{ _-z[ 87r3 kt2 k' * To conform with Moses normalization, e - - in (1) has been replaced ik-x (2 -3/2 by e - - (2 72

8579-3-Q where the path in the k-plane is the one shown directly below: -k one multiplies (1) through l\y q(x) and introduces the definitions 41) T (k k) = 1/2 Je -. x q(x) i (x, k') dx (2r)3/2..... 42) V (k k') = 1 3 i(k-) x q( dx (2w) so that (1) becomes (Cf. Morse and Feshbach II p. 1077, 1954) 43) T(k-k') = V(k') +JV(k- ") T (k",k') 2 d k" 2 2 k' -k The second equation of this theory consists in essentially solving this equation for V(k k!). We first observe that if 44) (k k) = 1 ( (, d x (2,r) Then equation (1) can be written after Fourier Transform as 45) 0 (k',k) =6 (k-k') + l. i.m. 1. T (k',k) _e. 0 2 2 k - k +ie If for simplicity we again assume that there are no bound states then the completeness relation 46) (x p (xk_') dx = 5 (-_') -00 will imply the completeness relation 73

8579-3-Q -oo 47), (k"k) 0 (ktdlk" -00 ( k-k') (k', k.") - 6(k-) -00 Now by definition we have that 48) V (kk') = q ( dx (2wr) from which it follows that 49) V (k',k") 0 k ",k" = J ( i ) (k",k) dx dk" 0 (2wr) / e xq (x)dx /- '- 0 (k", ) dk" 0 -- (23/2 (2)3/2 -3 / (x( (2) 3/2 I (x _ dx =T (k',k) (2ir) in view of (47 ) and the definition of T(Q k'). Therefore we have at once that 50) V (,k') T (k' k) 0(k'k") dk Inserting the expression for o given by (45) yields the solution to (43) in the form 51) V (kk') = T (k, k')+ T T( - -- T (k',k.9 dk" k" -k' [The correspond expression in the presence of bound state is eq. 7.52a in Newton (1966), page 189]. Defining W(k) by the relation 52) W(Q = V(-) =q 8ir ei(2k x) d (x) e - I dx 74

and noting that from this it follows that 53) q (x) = V e-i2k x W(9 dk we see that (51) can be written as 54) W(k) = -() r (k) +., k2 /T ( k. —,k2 T (k, k') d k' This last equation must be extended to k < 0. We first note that the property of the potential being real implies that 55) W(-k) = W(k) k> 0 so that (54) implies that wI2 1/2 56) W(k) = W(-k) = - ( /) r(k) +fT (j- k') T (Q i_') - 2 — 2 d k' Using the definition of rl, this can be written as 57) W(k) = - 1/2 r(k_) +J T(- k') { (2) kl2 -k2 + rx/-_ T(k, k') dk' k'2 -k2 Equations (43), (52), (53) and (57) form the basis for this new theory. They must be solved simultaneously and as in the previous case the method of iterations suggests itself. One replaces r (k) by e r (k) and makes the Ansatz that 58) T (!k_') = T (k, k') W(k) =: e W (k) 00 V(kk')=:e W (-2- ) Upon substitution, one sees that T and W can be obtained from a knowledge n n 75

8579-3-Q of r (k) alone while one uses (53) to obtain q (). Alternately, one can use the expression for W(k) as given by (57) and substitute it into (53) to obtain r 2,1/2 r( 2ik_ x dk + 59) q(x) 2 ( )/ r(k) e -2 - dk + +8 Jdk q(x) 0 (') q(x") (x'", ) dx g (x + x - 2) where 60) g(. i k x(-k ') -k 3 *, -k2 + 2 2 (2r) k - k' k k' equations (59) and (1) are the basic equations to be solved. One writes 02 61) q () = e n q and ik x oo 62) 0 (x,,)= e)32 + n0 (n n (21r)j which upon substitution leads to a perturbation series. It can be shown that to any order of approximation, 0 reproduces the-reflection coefficient r(). Faddeyev (1963) expresses the view that it is quite probable that this method of Moses converges for sufficient small r(k. To support this opinion it should be noted that this procedure is so reminiscent of the Born series approach to (1) that much of the recent work giving sufficient conditions for the convergence of this series could probably be extended to equations (61) and (62), We note the crucial role played by the relations (38) and (i) in these theories respectively. Without something like them it seems difficult to see

8579-3-Q how a theory could be developed but it is perhaps worthwhile to consider sole of the work on the optical model where a complex potential is employed to give a model of nuclear scattering. As a concluding remark that Lax and Phillips (1967) give two proofs of the fact that the scattering operator for the wave equation without potential does in its time independent form in fact determine the obstacle under Dirichlet conditions. The first of these due to Schiffer proceeds along classical lines and uses the Green's representation theorem of the exterior problem and also Rellich's uniqueness theorem. An attempt to extract it in detail will not be attempted here since it involves much of the previous notation and results of previous chapters of the book. Since this was written, some recent Russian work has been noted based on A. N. Tihonov's 1943 paper on incorrectly posed problems. This should be pursued. See for example: A.N. Tihonov, 1. Dokl. Akad. Nauk, S.S.S.R. 39, 1943, p. 176. 2. Sov. Math 4, 1963, p. 1035. 3. Sov. Math 4, 1963, p. 1624. A.T. Prilepko, Sov. Math 7, 1966, p. 431 77

8579-3-Q REFERENCES Barrar, R. andC.J. Dolph(1954), Rat. Mech. and Anal. 3, p. 726. Faddeyev, L.J. (1963), Math. Physics, 4 p. 72. Friedrichs, K.O. (1948), Comm. Pure and App. Math., p. 361. Gelfand, I. and B. Levitan (1951), Am. Math Soc. Trans. 1, p. 253. Hille, E. (1962), Analytic Function Theory, Vol. II, Ginn and Co. Ikebe, T. (1960), Arch. Rat. Mech. Anal. 5, p. 1. Kay, I. (1955), "The Inverse Scattering Problem", New York University Report No. EM-74. (1960), Comm. Pure and App. Math 13 p. 371. (1962), "The Three-Dimensional Inverse Scattering Problem", New York University Report No. EM-174. and H. Moses (1955), "The Determination of the Scattering Potential from the Spectral Measure Function, Part I: Continuous Spectrum," II Nuovo Cimento, Vol. 2, p. 917. and H. Moses (1956$, "... Part II: Point Eigenvalues and Proper Eigenfun'ctions", I Nuovo Cmento Vol. 3, p. 66. and H. Moses (195X, "... Part III: Calculation of Scattering Potential from Scattering Operator for One-dimensional Schrodinger Equations," 11 Nuovo Cimento, Vol. 3, p. 276. and H. Moses (1957), "... Part IV: 'Pathological' Scattering Problems in One Dimension,", n1 Nuovo Cimento, Supp. of Vol. 5, p. 230. and H. Moses (1961a), "... Part V: The Gelfand-Levitan Equation for Three-Dimensional Scattering Problems. " II Nuovo Cimento, Vol. 22, p. 689. - and H. Moses (1961b), "A Simple Verification of the Gelfand-Levitan Equation for the Three-Dimensional Scattering Problem", Comm. Pure and Appl. Math., Vol. 14, p. 435. Lax, Peter D. and R.S. Phillips (1967) Scattering Theory, Academic Press. Levinson, N.(1953), Physics Rev. 89, p. 755. Morse, P. and H. Feshbach (1954), Methods of Theoretical Psics, McGraw-Hill. Moses, H. (1956), Physics Rev. 102, p. 559. 78

8579-3-Q Moses, H. (1963), "Properties of Dielectrics from Reflection Coefficients in One Dimension", M. I. T. Lincoln Laboratories Report No. TR-322. Mueller, C. (1957), Grundprobleme der Mathematischen Theorie Electromagnetischer Schwingunr, Springer-Verlag, Berlin. Newton, R. (1966), Scattering Theory of Waves and Particles, McGraw Hill. Sims, A. (1957), J. Soc. Indust. App. Math 5 p. 183. Walsh, J. L. (1935) "Interpolation and Approximation by Rational Functions in the Complex Domain", Amer. Math. Soc. Colloquium Publications Vol. 20. Weston, V. H. and J. J. Bowman (1966) "Inverse Scattering Investigation - Quarterly Report, 1 April - 30 June 1966", The University of Michigan Radiation Laboratory Report No. 7644-3-T. 79

8579-3-Q DISTRIBUTION Electronic Systems Division Attn: 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 _ r DOCUMENT CONTROL DATA - R & D /ca-..-rJi...fo1r.ssfiati l otitln. hndv of abstracl t Oire itdidexlnd rinnotntlon nmuNt be entered when the overall report Is claasllled) El 1. ORIGINATING ACTIVITY (Corporate author) 2o. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of Unclassified Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 N/A. 3. REPORT TITLE INVERSE SCATTERING INVESTIGATION Quarterly Report No. 3 4. DESCRIPTIVE NOTES (Type of report and Inclusive dates) Third Quarterly Report (3 July to 3 October 1967) 5. AU THOR(S) (First name, middle initial, last name) Vaughan H. Weston, Wolfgang M. Boerner, and Charles L. Dolph 6. REPORT DATE 7.~. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1967 79 25 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) F 19628-67-C-0190 ESD-TR-67-517, Vol. II b. PROJECT NO. c. (0h. OTHER REPORT NOIS) (Any other numbers that may he oas.qned this report) d. 8579-3-Q 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. 1I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Deputy for Surveillance and Control Systems Electronic Systems Division, AFSC, USAF, L.G. Hanscom Field, Bedford, Mass. 01730 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. Particular aspects of this overall problem are considered, such as the effect of phase errors upon the determination of the scattering surface, polynomial interpolation of the surfacecattered field measured at a set of discrete points, and the testing of a numerical procedure for finding the surface of a conducting body from the knowledge of the near field. In addition, a review of exact theoretical treatments for the scalar inverse problem is given......,, -........... UU D, 1NOV51473 Unclassified S t'curitv C(.'l sitic ll' iti..

Security Classification 4. KEY WORDS Inverse Scattering Electromagnetic Theory 11 1 4 i RO LI E Lm I L1NK A WT I. ROL E Lm I LINK D WT I R 0 L E I I E I I L WT I LINK C I I -rn-i mmmmm Unclassified