@~EEE T R A N S A C T I o N s o N /

ANTENNAS AND PROPAGATION MARCH 1981 VOLUME A P - 2 9 NUMBER 2

A PUBLICATION OF THE IEEE ANTENNAS AND PROPAGATION SOCIETY

PART I OF TWO PARTS

SPECIAL ISSUE ON INVERSE METHODS IN ELECTROMAGNETICS

( J S S N 00 1 8 - 9 2 6 x 3

PAPERS

Introduction to the Special Issue on Inverse Methods in Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Opening Remarks (Invited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . M . Kennaugh Transient Response Characteristics in Identification and Imaging (Invited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Spectral and Time Domain Approaches to Some Inverse Scattering Problems (Invited) . . . . . . . . . . . . . . . . . . . . . . .

Time Domain Inverse Scattering (Invited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. L. Bennett Operator-Theoretic and Computational Approaches to 111-Posed Problems with Applications to Antenna Theory (Inuited)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.Z.Nashed Newton-Kantorovich Algorithm Applied to an Electromagnetic Inverse Problem . . . . . . . . . . . . . . . . . . . .A. Roger Iterative Determination of Permittivity and Conductivity Profiles of a Dielectric Slab in the Time Domain . . . . . . .

The Applicability of an Inverse Method for Reconstruction of Electron-Density Profiles ......................

Numerical Aspects of a Dissipative Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. J . Krueger Polarization Dependence in Electromagnetic Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W-M. Boerner, M. B. El-Arini, C. Y. Chan, and P. M . Mastoris Solution of Underdetermined Electromagnetic and Seismic Problems by the Maximum Entropy Method . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.M.Bevensee Remote Sensing of Path-Averaged Raindrop Size Distributions from Microwave Scattering Measurements . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.FuruhamaandT.Ihara Determination of the Parameters of a Dipole by Measurement of its Magnetic Field . . . . . . . . J. McFee and Y. Das Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution of the Inverse Electromagnetic Scattering Problem in the Resonance Case (Invited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.C.GaunaurdandH.Uberal1

Inverse Scattering Technique Applied to Remote Sensing of Layered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W-M. Boerner, A. K. Jordan, and I. W. Kay

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. L. Moffatt, J. D. Young, A. A. Ksienski, H . C. t i n , and C. M. Rhoads

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J . C. Bolomey, D. Lesselier, C. Pichot, and W. Tabbara

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . G . T ~ h u i s

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.H.Reil1yandA.K.Jordan

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.D.RadcliffandC.A.Ba1anis

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S . Coen, K. K . Mei, and D. J . Angelakos

185 190

192

206 213

2 20 232

239

245 253

262

27 1

275 282

288

293

298

Contents continued on inside back cover

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Contents continued from front cocer

Improved Performance of a Subsurface Radar Target Identification System Through Antenna Design (Incited) . . .

Frequency Swept Tomographic Imaging of Three-Dimensional Perfectly Conducting Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . C . Chan. L . Peters. Jr., and D . L . Moffatt 301

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.K.ChanandN.H.Farhat 312 Improved Resolution in Microwave Holographic Images . . . . . . . . . . . . G . Tricoles. E . L . Rope, and R . A . Hayward 320 The K-Pulse Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . M . Kennaugh 327 Physical Optics Imaging with Limited Aperture Data . . . . . . . . . . . . . . . . . . . . . . . . . . . F . B . Gross and J . D . Young 332

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W-M . Boerner. C . M . Ho, and B . Y. Foo 336

Target Classification with Multiple Frequency Illumination . . . . . . . . . . . . . . . . . . . C . L . Bennett and J . P . Toomey 352 Time Domain Approach to Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R . H . T . Bates and R . P . Millane 359 The Inverse Electromagnetic Scattering Problem for a Perfectly Conducting Cylinder . . . . . . . . . . . . . . . . D . Colton 364 Stability Problems in Inverse Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M . Bertero and C . De Mol 368

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T . K . Sarkar. D . D . Weiner, and V . K . Jain 373 Synthetic Imaging from Coherent Backscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J . S . Yu and J . W . Williams 380 The Phase Retrieval Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . S . Taylor 386 On the Inverse Scattering Problem for Dielectric Cylindrical Scatterers . . . . . . . . . . . . . . A . K . Datta and S . C . S o m 392

Use of Radons Projection Theory in Electromagnetic Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Unified Theory of Near-Field Analysis and Measurement: Scattering and Inverse Scattering . . . . . . . . P . F .. Wacker 342

Some Mathematical Considerations in Dealing with the Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C O M M U N I C A T l O h S

Estimation of the Volume and the Size of a Scatterer from Band-Limited Time Domain Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.K.Chaudhuri

Ramp Response Radar Imagery Spectral Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . L . Moffatt On Computation of Electromagnetic Fields on Planar Surfaces from Fields Specified on Nearby Surfaces . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . E . Corey and E . B . Joy The Unique RGconstruction of a Scalar or Vectorial Field from its Values on an Arbitrary Surface to Which it has

Propagated Through an Arbitrary Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . J . Hoenders Inverse Problems and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H . P . Baltes and H . A . Ferwerda Encoding of Information in Inverse Scattering Problems . . . . . . . . M . A . Fiddy, G . Ross, and M . Nieto- Vesperinas Self-Consistent Evaluation of Complex Constitutive Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..I . S . Yu

Subsurface Radar Target Imaging Estimates .................... L . C . Chan, D . L . Moffatt, and L . Peters, Jr . Polarization Dependence of RCS-A Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . .E . M . Kennaugh

398 399

402

404 405 406 408 412 413

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-29, NO. 2, MARCH 1981 185

Introduction to the Special Issue on Inverse Methods in Electromagnetics WOLFGANGMARTIN BOERNER, SENIOR MEMBER, IEEE, ARTHUR K. JORDAN, SENIOR MEMBER, IEEE, IRVIN W. KAY

Guest Editor Associate Guest Assistant Guest Editor Editor

Abstract-Inverse methods have become a fundamental tool in the physical sciences for remotely sensing unknown objects and recon- structing their physical properties. The objective of this special issue is to present an overview of this important rapidly emerging discipline and to provide examples of the wide scope of methods used to investigate inverse problems in electromagnetics, ranging from purely theoretical considerations to some very practical problems.

I. GENERAL OVERVIEW

Inverse methods have been used to investigate a vast range of problems in the engineering sciences. They have been de- veloped in many otherwise diverse fields of physical sciences where the characteristics of a medium are estimated from ex- perimental data in a given situation. It is known that some techniques used in one field are identical, at least in principle, to those used in another entirely different area. These inter- disciplinary applications of inversion techniques are drawing increasing attention and, therefore, we consider it relevant to open this special issue with a brief overview of several references which may be useful to the general reader.

An extensive survey covering many fields in which in- verse methods may be applied is compiled in a NASA mem- orandum by Colin [20]. This has proven to be a major source for generating rapid interest in other related areas; more suc- cinct introductions t o inverse methods have been given by Newton [62], Keller [42], and Parker [63]. Theoretical developments have been summarized by Sabatier [68] and a thorough literature survey of many aspects of inverse methods in electromagnetics has been presented by Boerner [ l o ] , whereas inverse source and scattering problems in optics recently were treated in Baltes [4] , [ 51.

Mathematical Treatments

A common approach based upon the linearization of the inverse problem with the sought-after quantity represented as a small perturbation about an assumed value, leads to an underdetermined linear system. In this connection the mathe- matics of generalized inverses [ 5 11 was employed side by side with the mathematics of profile inversion and an excel- lent treatment with applications was edited by Nashed [60] in which particular emphasis is given to the Moore-Penrose inverse. Closely related with inverse problems are methods of regularization [39 ] , [73 ] which are well discussed in tutorial papers by Deschamps and Cabayan [25], as well as by Wegro- wicz [79], Twomey [75], and in recent East European publications 1741, [341, [391, 1731. In spite of the tech-

Manuscript received November 3, 1980; revised December 15, 1980. WM. Boerner is with the Department of Information Engineering,

A. K. Jordan is with the SpaceSystems,Division, Naval Research

I. W. Kay is with the Science and Technology Division, Institute

University of Illinois at Chicago Circle, Chicago, IL 60680.

Laboratory, Washington, DC.

for Defense Analyses, Washington, DC.

niques developed here, the linearized approach is highly inadequate. The complete nonlinear problem and its cor- rection to the linearized problem has been analyzed most recently by Weston [ 8 1 1, [ 821.

Of basic importance to the inverse differential geometry is the Minkowski problem [59] in which a transform proce- dure is introduced to map the topological image of a closed convex surface onto the unit sphere of directions [27] in terms of principal curvatures. In many two- and/or three- dimensional wave transmission and reflection profile recon- struction methods, the theory of reconstruction from projec- tions becomes of fundamental importance which was intro- duced by Lorentz [76] and Radon [65], who established all the basic relationships existing between transforms in image, projection, and Fourier space. The theory is well treated by John [43], whereas implementations and applica- tions of Radons projection theory were given recently in Herman [ 371.

In one way or another most inverse methods used in remote sensing deal with the inversion of the wave equation [ 201, [ 8 1 ] , which for the case of radio waves was discussed in de- tail, for example, in Budden [ 151, Chernov [ 181, and most recently in Ishimaru [ 3 8 ] , where in many solutions use is made of the Abel transform [ 771 and of the classical Wentzel- Kramer-Brillouin (WKB) approximation [401 in problems of a slowly varying profiie in (v ) . Considerable extensions of these inversion methods were obtained for the inhomogeneous wave equation in quantum mechanics known as the Gelfand- Levitan-Marchenko [ 281, [ 1 ] procedure. Implementations and applications more recently were highlighted in review arti- cles given, for example, in [201, [681, 1611, [171, [281, [ 241, [ 741. The case where the scattering data are expressed as rational functions has special interest, since closed-form expressions for the scattering potential can be obtained; see Kay [45] and Jordan and Ahn [441. It should be noted that extensions of the Gelfand-Levitan-Marchenko procedure have become of considerable interest in the solution of nonlinear problems [69], particularly solitary wave mechanics 1571 ; and in an excellent tutorial paper Lax 1521 has developed systema- tically the relationship between solitary waves with the direct, the time evolution, and the inverse problems of scattering.

Geophysical Inverse Problem

Another startling extension of the regularization method may be seen in the optimum strategy method applicable to the inversion of limited data which was introduced by Backus and Gilbert [ 21, [ 31 dealing with thegeophysicalinverse problem. This method has since found many applications [29] and was found well suited to treat a class of nonlinear inverse problems of the solitary wave type [69] . We call the readers attention to Barcilons [6] outstanding tutorial review, in which some fundamental mathematical aspects of geophysical inverse problems are discussed in a simplified presentation.

0018-926X/81/0300-0185$00.75 0 1981 IEEE

186 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-29, NO. 2, MARCH 1981 !

Today geophysical inverse methods are widely used in ex- ploration seismology [71, [SI, [191, [261, [291, [391, [411, [70], where in essence the time of arrival of seismic pulses at various locations, generated by sources at different locations, i.e., the wave equation migration [ 191 method, is used to formulate the problem in terms on nonlinear functionals. Con- siderable advances have been made in spectral analysis [ 81 and deconvolution techniques [70] which also should be very powerful methods in other fields of remote sensing where signal versus clutter suppression becomes paramount. Vertical seismic profding is an extension of the scalar methods and per- mits inverse solutions for reflection and refraction seismology in elastic media where in addition t o p , s-wave components also need to be taken into consideration [ 301.

Remote Sensing

Inverse methods are also an integral part of aeronomic remote sensing [ 2 0 ] , [ 2 3 ] , [ 3 8 ] , [ 7 2 1 , [ 3 3 ] , [561 using both passive and active wave sounding where usually the,Born and Rytov approximations for slowly varying index profiles [ 101 are relevant. Whereas in the microwave region coherent meth- ods developed in radar are more likely to be used [ 301, [421, [48] , [ 501, in the infrared and optical regions incoherent properties need to be taken into consideration as well, and we recall [ 2 2 ] , [ 161, [421. Specifically, we refer to Baltes [41, [5] in which most recent monographs on inverse problems in optics have been presented. Image formation from coherence functions is also a basic inverse problem in astronomy [ 781. We should mention here that Radons theory of image recon- struction from projections also has become a very useful tool in radio astronomy [ 3 71 as was first shown by Bracewell [ 13 1 . [ 141, and similarly various interferometric and holographic image reconstruction methods [48], [ 781 are relevant.

Medical Imaging

Inverse methods always have played, and more explicitly, are playing an increasing role in medical imaging [ 641, where in addition to X-ray radiography and its most recent exten- sions of computer-assisted tomography (CAT)-scan [ 371, [ 3 11 we need t o refer to methods of ultrasonic imaging [ 541, [ 581 as well as microwave imaging [49]. The case of inverse methods in medical diagnosis is expanding rapidly[ 121 ; and it is safe to say that diagnosis of biological systems by nonionizing electro- magnetic as well as acoustic waves is on a steady increase [ 541, [ 551 , [ 801, and applicable imaging procedures require solutions to the inverse problem of wave propagation in strongly in- homogeneous media [ 121 ,

Radar and Imaging Before introducing inverse methods directly applicable in

electromagnetics within the m-to-mm wave region, we shall here refer to some major relevant monographs and books dealing with sonar, radar, and lidar [ 4 8 ] , [ 321, [ 71 1, [ 661, since theoretical analysis of these remote ranging and sensing schemes systematically led to the rapid growth of inverse dif- fraction, inverse scattering, and imaging theories based on radar cross-section analyses [ 211, [ 671 . Specifically, we need to mention that inverse methods are implicitly used in synthe- tic aperture radar [ 361, [ 421 and other target size and shape adaptive imaging radars and scatterometers [ 321 .

11. INVERSE METHODS IN ELECTROMAGNETICS

At present there is n o generally accepted definition of the inverse problem, although several descriptions have been given,

e.g., [201, [461, [631, [91, [61. We shall attempt to present one distinction between the direct and inverse problems of wave scattering meaningful in remote sensing problems:

Whereas, on the one hand, in the direct problem of electromagnetic scattering total a priori information on the size, shape and material constitutents of an object, together with the relative geometry of the incident field vector and the object coordinate system, are given and the scattered field vector is to be determined every- where over the total frequency or time domain; on the other hand, the inverse problem is t o reconstruct the size,. shape and material characteristics of ana priori unknown scattering object from the knowledge of the incident field vector and the resulting scattered field data. This distinction [ 101 has been given in terms of vector

(electromagnetic) wave scattering; it can also be applied to the scalar (acoustic) and tensor (acoustoelastic) cases.

Considering the complexity of the boundary conditions and constitutive relations, there apparently will not exist an exact general inverse scattering theory as described above that can yield practical solutions in a finite number of computations [ 111. Rather than initially seeking exact self-consistent solu- tions for the general three-dimensional inverse problem, in- formation can be obtained by studying simplified physical models and/or using approximate analytical techniques [ 101, [121.

111. ARRANGEMENT OF PAPERS AND KEYS TO CATEGORIES

The papers which comprise this special issue address a broad spectrum of inverse methods and provide both numerical and experimental results. These papers can be somewhat arbitrarily classified according t o various criteria, as shown in Table I, where we have chosen six categories with keys indicated in parentheses.

1)NQtui-e o f model: Scattering (s) from targets [671, propagation (P) through inhomogeneous or random media [ 3 8 ] , imaging (I) of targets by processing the scattering data [30] , or the general (G) inverse problem.

2 ) Dimensionality of the physical model: The one-dimen- SiOnQl (1) scattering model is an idealized example that posses- ses a complete exact solution; physical insight into the rela- tions between the scattering data and the refractive-index pro- file can be obtained by considering this inverse scattering problem [441, [451.

Surface scattering (2) is an inverse problem which has important applications in the remote sensing of natural surfaces (terrain and oceans) and optical systems [ 141. Scattering by three-dimensional (3) convex targets has obvious importance for radar and sonar applications [43], [48]. In some cases [52] certain inverse theories are applicable to the general n-dim (n) case.

3) Frequency (F)lTime (T): Various approximations with respect to the relative wavelengths are being considered. Low- frequency, resonant-frequency, and optical-frequency approxi- mations [ 2 11 can provide simplified analyses. Related to these questions are those of transient scattering in the time domain [ 4 7 ] , [ 9 ] and the availability of broad or narrow band scat- tering data [ 531 (mixed (M)).

4 ) Polarization: While it is convenient to study the analyti- cal properties in terms of scalar theories (K), electromagnetic scattering problems will, in general, require extension to t he vector case (V). Here polarization effects must be considered [ I l l .

BOERNER et al .: INTRODUCTION 187

TABLE I

Categories Authors 1 2 3 1 5 6

H. P. Baltes and H. A. Ferwerda 1 3 F K R. H. T. Bates and R. P. Millane S 3 T V C. L. Bennett S 3 T V C. L. Bennett and J. P. Toomey S 3 T V H. Bertero and C. DeMol I 1 F V R. M. Bevensee S l F K W. M. Boerner, M. B. El-Arini,

C. Y. Chan, and P. klastories G 3 M V W . bl. Boerner, C. 11. Ho, and

B. Y. Foo I 3 M V W. M. Boerner, A. K. Jordan, and

I. W . Kay G n M V J. C. Bolomey, D. Lesselier,

C. Pichot, and W. Tabbara , S 3 T V C. K. Chan and N. H. Farhat 1 3 F V L. C. Chan, L. Peters, and

D. L. Moffatt S 3 F V L. C. Chan, L. Peters, and

D. L. Moffatt (Subsurface Radar Target Imaging Estimates) S 3 T V

S. K. Chaudhuri S 2 M K S. Coen, K. K. Mei, and

D. J. Angelakos P l F K D. Colton S 3 T V L. E. Corey and E. B. Joy S 2 T V A. K. Datta and S. C. Som S 2 F V Ivl. Fiddy, G. Ross, and

M. Nieto-Vesperinas P l F K Y. Furuhama and T. Ihara P 1 F V G. Gaunaurd and H. Uberall S 3 T V F. D. Gross and J. D. Young I 3 T K B. J. Hoenders I 3 F K E. M. Kennaugh (Opening Remarks) G 3 T V E. M. Kennaugh (Polarization

Dependence of RCS) S 3 T V E. M. Kennaugh (The K-Pulse Concept) S 3 T V R. J. Krueger P l F K J. McFee and Y. Das I 3 F V D. L. Moffatt S 3 T V D. L. Moffatt, J . D. Young,

A. A. Ksienski, H. Lin, and C. M. Rhoads S 3 T V

M . Z. Nashed G 3 M K R. D. Radcliff and C. A. Balanis S 3 T V M. H. Reilly and A. K. Jordan P l F K A. Roger P l F K T. K. Sarkar, D. D. Weiner, and

V. K. Jain S l F K L. S. Taylor P l F K A. G. Tijhuis P l T K G. Tricoles, E. L. Rope, and

R. A. Hayward S 3 F V P. F. Wacker S 3 M V J. S. Yu P 1 T K J. S. Yu and J. W. Williams I 3 F V

L R L D L D L D L R L R

L Q

L D

L Q

L D L D

L D

L D L D

U R U D U D U D

G R L D U D L D L R U D

L D L D U D L R L D

L D L D L R U D U D

U D U R U D

L D L D L D U R

Key to Categories I ) Type of Problem 4) Polarization

S Scattering K Scalar P Propagation V Vector I Imaging G General

2) Dimensionality 1 2 3 n

3) Temporal Space F Frequency T Time M Mixed Time Frequency

5) Data Format U Unlimited L Limited

6) Statistics D Deterministic R Random Q Quasi-coherent

5 ) Data l imi ta t ions: In regard to the availability of meas- urement data with respect t o aspect, frequency, and polariza- tion, we may state that narrow band (i.e.,, approaching contin- uous wave (CW)/monostatic case) techniques require multi- ple aspect data covering the total unit sphere of directions; whereas, for broad-band methods shape and size reconstruc- tion of targets becomes feasible given data for a few or only one aspect with increasing bandwidth (i.e., approaching the ideal transient case) covering the low frequency, across reson- ance, up to high-frequency regions of the target cross section under consideration [ 101 : unl imi ted da ta (U), l imi ted da ta

6 ) Stat is t ical propert ies: Scattering by volume distribu- tions of particles involve inverse problems t o determine the statistical distributions of partcicle sizes, shapes, and materials; propagation in random med ia (R) is an important application [38 ] . This is distinguished from determinis t ic (D), o r quasi- coherent ( Q ) inverse problems.

(L).

IV. EDITORS COMMENTS

The editors would like to express their thanks to the authors for their cooperation during the preparation and processing of their manuscripts and the many reviewers for their valuable comments and speedy reviews. We wish t o acknowledge sin- cerely the assistance and the advice provided by the outgoing Editor, Prof. Walter K. Kahn; and similarly we wish to thank both Prof. Vaughan H. Weston and Dr. Lucjan A. Wegrowicz for their comments during the preparation of these intro- ductory notes. A special note of thanks is extended to Dr. Ed- ward M. Kennaugh, Professor Emeritus, the Ohio State Uni- versity, for contributing the Opening Remarks. The extensive secretarial assistance of the Communications Laboratory, College of Engineering, University of Illinois at Chicago Circle as well as that of the Aerospace Systems Branch, Space Systems Division, Naval Research Laboratory are gratefully acknowledged.

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1481 M. Kock, Radar. Sonar and Holography. New York: Academic, 1973.

1491 L. E. Larson and H. H. Jacobi, Microwave scattering parameter imagery of an isolated canine kidney. Med. Phys. vol. 6 , no. 5 , Sept./Oct.. 1979.

[SO] R. W. Larson, F. Smith. R. Lawson, and M. L. Brian, Multispectral microwave imaging radar for remote sensing ap- plications. in Proc. URSI (Special Meeting on Microwave Scattering and Emission from the Earth) E. Schanda, Ed., Univ. of Bern, pp. 305-315, 1974.

15 1 1 M. M. Lavrentiev, Some Improperly Posed Problems of Mathemat- ical Physics. New York: Springer, 1967 (See also: M. M. Lavrentiev, V. G . Romanov, and S . P. Shishatsky, Ill-posed Problems of Mathematical Physics and Analysis, (in Russian), Nauka, Moscow, Oct. 1980.)

[52] P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., vol. 21, pp. 467-490, 1968.

[53] R. M. Lewis, Physical optics inverse diffraction, IEEE Trans. Antennas Propagar., vol. AP-17, pp. 308-314, 1969.

[54] J. C. Lin, Microwave Auditory Effects and Applications. Springfield, IL: C. C. Thomas, 1978.

[55] M. Linzer, Ed., Ultrasonic Imaging. New York: Academic, 1979.

1561 T . Lund, Surveillance of environmental pollution and resources by electromagnetic waves, presented at Proc. NATO Adv. Study Inst.. Spitind, Norway, Apr. 9-19, 1978.

1.571 V. G. Makhanov, Dynamics of classical solitons in non-in- tegrable systems, (Physics Reports-A review of Physics Letters (Section C)). New York: North Holland, Jan. 1978, vol. 35, no. 1. pp. 1-28.

[58] E. Marom, R. K. IMueller, R. F. Koppelmann, and G. Zilinskas, Design and preliminary test of an underwater viewing system using sound holography, in Acoustical Holography, A. F. Metherell, Ed. New York: Plenum, 1971, vol. 3 .

[59] H. Minkowski, Volumen und Oberflache, Math. Ann., vol. 57, pp. 447-495, 1903.

[60] M. Z. Nashed, Generalized Inverses and Applications. New York: Academic, 1976, also Proc. of Adv. Seminar, Univ. Wisconsin, Madison, WI, Oct. 8-10, 1973.

[61] R. G . Newton, Scattering Theory of Waves and Particles. New York: McGraw-Hill, 1966.

[62] R. G. Newton, Inverse problems in physics, SIAM Rev. vol. 6, pp. 124-134, July, 1970.

[63] R . L. Parker, Understanding inverse theory, Ann. Res. Earth Planet Sci.. vol. 5 , pp. 35-64, 1977.

1641 K. Preston, Jr., K. H. W. Taylor, S. A. Johnson, and W. R. Ayers, Eds., Medical Imaging Techniques-A Comparis0.n. New York: Plenum, 1979,

[65] J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten, Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math.-Nat. K1. vol. 69, pp. 262-271, 1917.

[66] A. W. Rihaczek, Principles of High-resolution Radar. New York: McGraw-Hill. 1969.

BOERNER e t al. : INTRODUCTION 189

G. T. Ruck, D. E. Barrick, W. D. Stuart. and c. K. Kirchbaum. Radar Cross Section Handbook, V O ~ . 1 and 11. New Plenum, 1970. p, C. Sabatier. Applied Inverse Problems, (Lecture Notes in Physics), vol. 85. New York: Springer. 1978. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin, The Soliton: a new concept in applied science, Proc. IEEE. vol. 61, no. 10, PP. 1443-1483, 1973. M. T. Silvia and E. A. Robinson, Drconvolution of geophysical rime series in rhe exploration for oil and natural gas, De- velopments in Petroleum Science, vol. IO, New York: Elsevier. 1979. M. I. Skolnik, Introduction to Radar Systems. New, York: McGraw-Hill, 1962; also: M. K. Skolnik, Radar Handbook, New York: McGraw-Hill, 1978. J. W. Strohbehn, Ed. Laser Beam Propagation in the Atmosphere. (Topics in Applied Physics, vol. 25). Berlin, Heidelberg, New, York: Springer, 1978. A. Tikhonov and V. Arsenine, Solutions of Ill-Posed Problems. New York: Wiley, 1977; also, Moscow: Nauka, 2nd ed., 1979. A. N. Tikhonov, F. Kuhnert, N. N. Kuznecov, K. Moszynski, and A. Wakulicz, Mathematical models and numerical methods. BANACH Center Publications, vol. 3 (on Inverse Problems) C. Olech and E. Fidelis, Eds. Warsaw: PWN-Polish Scientific, 1978. S. Twomey. The application of numerical filtering to the solution of integral equations encountered in indirect sensing measure- ments, J . Franklin Inst., vol. 279, no. 2, pp. 95-109, 1964. G . E. Uhlenbeck, Over een stelling van Lorentz en haar uit- breiding voor m e r dimensionale ruimten, Physica, Nederlands Tijdschrift voor Natuurkunde, vol. 5, pp. 423-428, 1925. Also, H. B. A. Bockwinkel, Versl. Kon Akad. Wis. XIV 2, p. 636, 1906. B. van der Pol and H. Bremmer, Operational Calculus. (based on the two-sided Laplace Transform), New York: Cambridge Univ.. 1955. C. van Schooneveld, Image Formation from Coherence Functions in Astronomy, Dordrecht: D. Reidel, 1979. L. A. Wegrowicz, An inverse problem for line and strip sources of EM fields above on imperfectly conducting ground plane, IEEE Trans. Antennas Propagar.. vol. AP-26, no. 2, pp. 248-257, 1978. P. N. T. Wells, Biomedical Ultrasonics. New York: Academic, 1977. V. H. Weston. Inverse problem for the reduced wave equation with fixed incident fields (Part I) J . Marh. Phys. vol. 21, no. 4, pp.

-, Inverse problem for the reduced wave equation with fixed incident field, Part 11, J . Marh. Phys. vol. 22, Oct. 1981, to be published.

758-764, 1980.

Wolfgang-Martin Boerner (S66-M67-SM75) was born on July 26, 1937 in Finschhafen, Papua- New Guinea. He received the Abitur from the August von Platen Gymnasium, Ansbach, West Germany in 1958, the Dipl.-Ing. degree from the Technische Universitiit Miinchen in 1963, and the Ph.D. degree from the Moore School of Elec- trical Engineering. University of Pennsylvania, Philadelphia in 1967.

From 1967 to 1968 he was employed as Research Assistant Engineer at the Department of

Electrical and Computer Engineering, Radiation Laboratory, University of Michigan. Ann Arbor, and during this period he was involved in inverse scattering investigations. During the summer of 1968 he joined the Electrical Engineering Department. University of Manitoba, Winnipeg.

Canada first as postdoctoral fellow and then as faculty member where he became Professor in 1976 and was continuing on research in electro- magnetic inverse problems and remote sensing. During the period of July 1975 to 1976 he spent his research year of leave in parts at the Universitat Niirnberg in Erlangen. FRG as Humboldt fellow, and the University of Canterbury in Christchurch, New Zealand, and he extended his research interest into geoelectromagnetism. In 1978 he joined the Information Engineering Department, University of Illinois at Chicago Circle. where he is employed as Professor of Electrical and Information Engineering and where he is the Director of the Communications Laboratory i n the College of Engineering.

Dr. Boerner is a member of the Canadian Association of Physicists. Sigma Xi. the Society of Photo-Optical Instrumentation Engineers, the Society of Exploration Geophysicists, the Society of Engineering Sci- ences. the American Association for the Advancement of Science. the Fulbrisht Alumni Association. He is a member of U.S. Commissions B. C, and E of the International Scientific Radio Union.

Arthur K. Jordan (M59-SM74) received the M.Sc. and Ph.D. degrees in electrical engi- neering from the University of Pennsylvania. Philadelphia, in 1971 and 1972. respectively. and

the B.Sc. degree in physics from the Pennsyl- ~ vania State University. College Park, in 1957.

He was a Research Assistant and a Research Fellow with the Moore School of Electrical Engineering, University of Pennsylvania. from 1969 to 1973. engaged i n research on the inverse scattering theory of electromagnetic waves.

Since 1973 he has been with the Space Systems Division, Naval Research Laboratory, Washington, DC. where he is conducting studies on the scattering and propagation of electromagnetic waves. the interaction of electromagnetic radiation with matter. inverse scattering theory. and radar target identification. Previously. he had industrial experience with the General Electric Company, the Radio Corporation of America, and the Philco Corporation, engaged in research and development of radar and microwave communications systems.

Dr. Jordan is a member of Sigma Pi Sigma. Sigma Xi. Commission B (Fields and Waves) of the International Scientific Radio Union (URSI), the American Physical Society. and the Optical Society of America.

Irvin W. Kay received the B.A. degree in 1948, the M.A. degree in 1949. and the Ph.D. degree in 1953. all in mathematics and all from New York University. New York.

From 1949 to 1962 he was employed at the Courant Institute of Mathematical Sciences at New York University. as a research assistant and as a research associate. During that period he became a member of the Universitys mathe- matics department faculty, serving as Assistant Professor and as Associate Professor. He spent

the summer of 1961 as Visiting Associate Professor in the Mathematics Research Center at the University of Wisconsin. From 1962 to 1970 he was employed at the Conductron Corporation i n Ann Arbor. MI where he ultimately became the Director of Applied Research. From 1970 to 1973 he was employed as Professor of Electrical Engineering at Wayne State University. Since 1973 he has been employed at the Institute For Defense Analyses in Washington, DC as a research staff member in the Science and Technology Division.

Dr. Kay is a member of the Association of University Professors. the American Mathematical Society. the American Physical Society, and U.S. Commissions B and C of the International Scientific Radio Union.

190 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-29, NO. 2, MARCH 1981

Opening Remarks

EDWARD M. KENNAUGH, MEMBER, IEEE

While the goals of the special issue can be succinctly de- fined, the contents cover a wide range of applications and spe- cial tools for inverse scattering. These remarks focus on a class of problems and several techniques that are peculiar to radar sensing of isolated targets above the earth, although not so limited in application. Considering bodies of fixed shape viewed at a single aspect in vucuo, our purpose is t o outline some approaches and concepts other than imaging that are potentially useful for inverse scattering. Several of these have a long history, and we shall attempt to identify areas of future research. Indeed many of these will be touched on in greater detail by the various papers in this issue.

One measure of electromagnetic scattering properties of an object is the radar cross section (RCS) or apparent size. In the early days of radar, it was found that rapid variation of RCS with aspect, radar polarization, and frequency complicate the relation between true and apparent sizes. As measurement capabilities improved, investigations of the variation of RCS with these parameters provided the radar analyst with a plethora of data, but few insights into this relation. In the present context, such data are essential in determining the physical features of a distant target, rather than an annoying radar anomaly. Perhaps the simplest and earliest extension of radar metrology is through polarization diversity. Here the variation of received signal with choice of transmit- receive polarization at a single aspect and radar frequency in- volves two complex parameters in addition to the RCS. One may interpret these parameters as two coordinate pairs which de- fine the target null-polarizations on the Poincare sphere. The essential features of this model were developed by 1950. While simple to obtain and analqize, target polarization properties at one aspect and frequency do not uniquely determine target shape or size. They may be used, however, t o discriminate and differentiate classes of target with special geometrical or elec- trical symmetries.

Next in order of complexity and development is the exten- sion of RCS t o a complex spectral density function C(s) or G ( j w ) for a fixed target aspect and polarization, so defined that the RCS at radian frequency o is the absolute square of G ( j o ) . By choosing a reference point in or on the target, the phase of G( jw) can be defined. Second generation chirp and short-pulse radars provided data on G ( j o ) over less than oc- tave bandwidths or the time response to modulated-carrier pulse waveforms capable of resolving in range one or more scattering centers on a target. While a discrete distribution of scattering centers is not sufficient to uniquely define a target, it is natural to speculate on the limits of information extract- able from the complete spectral density function G( jw) or its inverse, the impulse response waveform Fdt). The concept of Fdt) in radar and its approximation directly from the geome-

Manuscript receivedNovember 28,1980;revised December 15,1980. The author iswith the Ohio State University, ElectroScience Labora-

tory, Department of Electrical Engineering, Columbus, OH 43212.

try of a perfectly conducting target was first described in 1958. The significance of this relation in inverse scattering was apparent: the waveform F l ( t ) and (even more directly) its second integral with time, the ramp response F R ( t ) , should yield the approximate variation of target cross-sectional area with range over the lit side of the target. By 1962 analytical and experimental techniques for approximating F R ( ~ ) were under study.

The use of a real waveform Fl(t ) t o describe the backscat- tering from an object is preferable to the use of a complex spectral density function. The settling time for Fdt) is finite and comparable to the transit time over the target. The range history of the target features are preserved in the waveform from the arrival of the first reflection, the passage of the incident front over the target, and the subsequent variations due to creeping waves circumnavigating parts of the body or to natural resonances. Responses from subsections may in some cases be combined to estimate that for a complex target. Moment conditions satisfied for all finite target waveforms may be used t o simplify and improve estimates. The response of the target for any incident waveform can be predicted closely from Fdt) by a finite convolution integral. Separation of the target response into a forced component and those due to re-excitation of the surface by creeping waves, dif- fracted or reflected fronts is an important area of future research. By gating, portions of Fdt) more closely associated with the forced regime can be isolated. Since the forced re- sponse can be more directly related t o local surface proper- ties over the lit side of an object, this separation should be investigated to improve target shape predictions. The polari- zation properties of the target in terms of a 2 by 2 matrix of F d t ) waveforms are also of interest.

The utility of transient response in inverse scattering may be contrasted with the use of G( jw) or G(s) for investigation of complex natural resonance frequencies which characterize a scattering body. Since these frequencies are an aspect-inde- pendent property of the target, frequency-domain analysis at one or more aspects is recommended. An equivalent time- domain method can be devised, however. If the transient re- sponse is composed of N distinct complex resonance fre- quencies, any hr f 1 equally spaced samples of the transient response will satisfy a homogeneous linear difference equation, which in principle serves as an unique identification whatever the target aspect or pulsed excitation. However a forced com- ponent of response, if present, will interfere with this process. A simple correspondence between natural resonances and the physical features of a body must still be found t o establish true inverse scattering potential.

An approximate relation between Fdt) and the geometry of a perfectly conducting target is based o n a physical optics or Kirchhoff estimate, assuming only local interaction between an incident wavefront and elements of the target surface, replaced by a local tangent plane. If an impulsive planar incident wavefront arrives, impulsive surface currents are assumed t o exist along the intersection of the target with the

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KENNAUGH: OPENING REMARKS 191

incident front over the lit side, but with zero currents else- where. This crude approximation for the true surface current predicts a ramp response waveform which is a scaled replica of the projected target area with range over the illuminated por- tion of the target. Improvements t o t he Kirchhoff approxima- tion should initially focus on relating induced currents over the illuminated region t o surface features such as principal radii of curvature and composition, as well as t o t h e polariza- tion of the incident front. With the concept of impulse excita- tion, it is possible to separate effects into those which occur at the moving incident front (reasonably well predicted by the Kirchhoff estimate), and those which are due to surface rays or reflections which re-illuminate the surface after passage of the front. By use of the Radon transform, improved esti- mates of the profile function can then be combined for sev- eral aspects to yield three-dimensional target features.

The practical consequences of research findings outlined above and elsewhere in this issue will be manifest in design of future radars and sensors. From bandwidth to antenna polari-

zation and computer memory, all characteristics will be influ- enced t o a greater degree than before by inverse scattering con- cepts. Let us hope that we are on the threshold of great achievements!

Edward M. Kennaugh (A55-M60-M7 I ) was born in New York. NY, on October 3. 1922. He received the B.E.E. . M.S . (physics). and Ph.D. degrees from Ohio State University, Columbus, in 1947. 1952. and 1959. respectively.

He was i n the U.S. Army from 1943 to 1946. He has been a member of the technical staff at the Ohio State University Antenna and ElectroSci- ence Laboratories since 1948. serving as Labo- ratory Director from 1970 to 1973. A member of the faculty since 1961 and now Emeritus Pro-

fessor of Electrical Engineering at the Ohio State University, his research interests include a variety of topics in electromagnetic scattering.

Dr. Kennaush is a member of Sigma Xi and Commission B of URSI.

192 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 0 - 2 9 , NO. 2 , MARCH 1981

Transient Response Characteristics in Identification and Imaging

DAVID L. MOFFAT", JONATHAN D. YOUNG, AHARON A. KSIENSKI, FELLOW, IEEE, HENG-CHENG LIN, MEMBER, IEEE, AND CHARLES M. RHOADS, STUDEhT MEMBER, IEEE

Absrrucr-Three different solutions to problems in identification and inverse scattering using restricted far-zone scattering data are reviewed and illustrated. The data are restricted in the sense that the frequencies, aspects, and polarizations are limited. All three solutions are based in part on a fundamental time domain viewpoint whereby the far-zone scattering characteristics of any finite object are uniquely summarized by its impulse response waveforms. Insight provided by these waveforms is exploited via geometrical characteristics extracted from steady-state, forced, and free responses of the object. For the present purposes the fundamental importance of signaling waveforms whose wavelengths a re within an order of magnitude of the object dimension is demonstrated. The basic methods discussed have been previously identified in the literature as natural resonance estimation, radar imaging from ramp response signatures, and low frequency classification. Related common features, additional insight, and some new results are given. Applications of all three methods to objects ranging from simple to complex geometries are described.

I. INTRODUCTION

T HE PROBLEM of deducing the physical properties (size, shape, and composition) of finite objects using a restricted set of active far-zone scattering measurements, while it is a special case of general inversion, still represents an imposing interpretation task. For electromagnetic sensors a vector prob- lem is encountered, but for either vector or scalar fields the scattering properties of the object are perhaps best sum- marized by the impulse response concept (IRC) [ 11. With the usual linear system restrictions, the general representation used here is a multiport with each port pair relating far-zone inci- dent and scattered fields and characterized by complex fre- quency-dependent transfer functions or transform-related real time-dependent impulse response waveforms (IRWs). As- suming the number of observations to be reasonably small, the multiport is defined as a 2N-port where either 1) the separa- tion of transmitter and receiver is fixed and N orientations of the object are observed, or 2) the transmitter location is fixed and N observer locations are utilized. For vector fields there are in general four independent polarization combinations and therefore, for each port pair, four IRWs. In the above there could be N backscatter orientations in 1) and one in 2). The material in this paper is primarily concerned with the copolar-

Manuscript received April 17, 1980; revised October 28, 1980. This work was supported by the joint sponsorship of the National Science Foundation under Grant ENG76-04344, the Office of Naval Research under Contract NOOO14-78COO49, the Department of the Air Force under Contract AFOSR69-1710A, the Rome Air Development Cen- ter under Contract F30602-74-C-0170, and the Ballistic Missile De- fense Systems Command and the Ohio State University Research Foundation under Contract DASG60-77C-0133.

D. L. Moffatt, J. D. Young, A. A. Ksienski, and C. hi. Rhoads are with the Ohio State University ElectroScience Laboratory, Depart- ment of Electrical Engineering, Columbus, OH 43212.

H. C. Lin was with the Ohio State University ElectroScience Labora- tory, Department of Electrical Engineering, Columbus, OH 43212. He is now with the Harris Corporation, Melbourne, FL.

ized waveforms and exceptions from this will be carefully noted.

With the scattering cross section of the object defined as

u(w) = m 2 I G(jw) 1 2 , (1)

the normalized far-zone IRW forms the transform relation

m

G( jw) = 1 - Fl(t')e-jwt' d t ' , (2) where w is the angular frequency and c the speed of light in a vacuum. In the far zone the IRW is effectively of limited dura- tion and will include all of the object-related scattering mech- anisms which might be exploited to obtain the physical-prop- erties of the object. For example, object motion, including component motion such as turbine blades in jet engine cavi- ' ties, can in principle be included by a suitable superposition of static IRW. The proximity of other objects, the ground, and the medium in which the object is immersed can also be in- corporated into the IRW. Any postulated inversion-related scheme can be readily tested using the IRW of an object or set of objects, appropriate interrogating signals, and convolution. The response of the object to an arbitrary interrogating signal F' ( t ) is

m

F(t' ) = F,(x)F'(t' - X ) d ~ . (3) L In this paper interest is centered on the transient portion of the response to an aperiodic excitation [ 21, the response to a ramp excitation [3], and the steady-state response [4]. Gen- eral properties of the IRW and related transient response wave- forms (TRWs) have been discussed [ 11. It is emphasized again that overall size and shape of an object are obtained from an interrogating signal waveform whose highest spectral com- ponent has a wavelength greater than one-half the maximum linear target dimension. Progressively greater target detail is revealed by interrogating signals with step discontinuities and shock-type singularities. This follows from the inverse trans- form

1 "

271 -m F,(t') = - G(jw)eiw t' dw (4)

and the convolution integral. In the convolution of any two waveforms, if one of the waveforms has slow time variations, then the convolution result will also have slow time variations. Therefore, low frequency sinusoids cannot elicit rapid varia- tions in the IRW. Spatially isolated geometrical features of an object are characterized in the IRW and TRW by both approp- riate time delays and distinctive waveshapes, e.g., edge diffrac- tion [ 5 1 and rate of change of curvature discontinuities [ 61 .

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MOFFATT et a!.: TRANSIENT RESPONSE CHARACTERISTICS 193

The IRWs of an object are unique, but other TRW'S includ- ing band-limited approximations of the IRW are not Unique. The steady-state response to monochromatic excitations, as given in (6), will elicit responses from an object which are do- minated by specular contributions and geometrical com- ponents of appropriate dimension with respect to the wave- length of the excitation. Conversely, aperiodic excitations elicit first a forced response as the incident wavefront moves over the object and, subsequently, a natural or free response as the incident wavefront moves beyond the object. These three responses, i.e., steady-state, forced, and free, yield different but related geometrical information for the object.

The purpose of this paper is to review and illustrate three distinctly different solutions [ 21 -[4] to problems in identi- fication and inverse scattering. The solutions have a common background in that the initial spectral content of the interro- gating signals tested for each solution came from the IRC and from the fact, as noted above, that convolution integrals are a convenient and informative interpretation mechanism. Sec- tions I1 and 111 of the paper are concerned with the natural and forced responses of an object, respectively, while Section IV utilizes the steadystate response. A common assumption in all three sections is that the basic measurable quantity is e