Parametric Scattering Models for Bistatic Synthetic Aperture Radar

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1 Paametic Scatteing Models fo Bistatic Synthetic Apetue Rada Julie Ann Jackson Student Membe, Bian D. Rigling Membe, Randolph L. Moses Senio Membe The Ohio State Univesity, Dept. of Electical and Compute Engineeing, Columbus, Ohio, 3, USA Wight State Univesity, Dept. of Electical Engineeing, Dayton, Ohio, 535, USA Abstact Paametic scatteing cente models match to ada scene attibutes, aiding in automatic taget ecognition ATR and scene visualization. In this pape, we develop paametic models of canonical shapes fo bistatic synthetic apetue ada SAR. We genealize geometic theoy of diffaction solutions fo scatteing mechanisms in a plane to develop thee-dimensional models fo six canonical shapes: a ectangula plate, dihedal, tihedal, cylinde, top-hat, and sphee. The poposed models povide physically elevant yet compact scatteing solutions that ae easily implemented fo ada signal pocesg and ATR applications. The deived models ae shown to agee with esults obtained fom high-fequency, asymptotic scatteing simulations. Index Tems synthetic apetue ada, multistatic scatteing, ada taget ecognition I. INTRODUCTION Synthetic apetue ada SAR plays an impotant ole in combat vehicle taget identification. Attibuted scatteing models have been used to extact geometic featues fom monostatic SAR images 1 3. Bistatic SAR systems captue wide-angle views of tagets, poviding geometic infomation fo use in automatic taget ecognition ATR. In this pape, we deive paametic models which descibe canonical taget scatteing fo 3D bistatic mode SAR. Seveal geometical theoy of diffaction GTD models ae deived in Akhte fo bistatic scatteing in a plane. In this pape, we use the theoy demonstated in and genealize the models found in 1,, 5 to develop pasimonious physicsbased models of thee-dimensional bistatic scatteing. Theedimensional scatteing is appoximated by poducts of twodimensional esponses. The models pesented in this pape ae constucted to captue mainlobe scatteing in a compact paametic fom that is much less complex than exact electomagnetic scatteing solutions. To demonstate the capability of the poposed bistatic models, we compae the model esponses to asymptotic, high-fequency electomagnetic scatteing pedictions geneated by Xpatch softwae 6. II. SCENE MODEL Unde a high-fequency assumption, scatteing fom complex objects may be modeled as the sum of esponses fom in- This wok was pefomed unde fellowship funding fom the Ai Foce Reseach Lab and the Dayton Aea Gaduate Studies Institute. Appoved fo public elease #AFRL/WS-7-6. dividual scattees 7. The eceived ada signal fom bistatic scatteing of a scene may be epesented as 1 Sk = P m M Γm k, φ t,φ,θ t,θ ; Θ m e RΘm m 1 whee φ t,θ t and φ,θ, ae the azimuth, elevation coodinates of the tansmitte and eceive. In 1, k = πf c is the wavenumbe and M Γm chaacteizes the fequency esponse of the mth eflecto. RΘ m measues the distance fom the tansmitte to the mth eflecto to the eceive, elative to the distance fo a eflecto located at the scene oigin 8. M Γm and R ae scala tems that ae common acoss all polaizations. P m is a matix that chaacteizes the polaization dependence of the scattee. The vaiable Γm indicates the type of canonical scatteing cente, and theefoe indicates the paticula fom of M Γm,e.g.M plate. The vecto Θ m contains physical paametes, including object location, oientation, and size. Fo each eflecto, the tip distance RΘ m may be witten as the ange to the scatteing cente minus the ange due to object cuvatue: RΘ m = R Θ m R Θ m. To fist ode, the diffeential ange RΘ m may be appoximated unde a fa-field assumption as 8 RΘ m x x cos φ t cos θ t +cosφ cos θ +y y φ t cos θ t +φ cos θ +z z θ t +θ whee x,y,z is the object location and x,y,z is the eflection point on the object suface measued elative to x,y,z. We assume scatteing cente locations to be the local axis oigins depicted on the shapes in Figue. In Section IV, we deive the diffeential ange offset R fo the cuved canonical shapes. III. PLANAR BISTATIC SCATTERING MECHANISMS We begin the fomulation of 3D bistatic scatteing cente models by fist defining fundamental scatteing mechanisms in two dimensions. The flat plate, ight angle, and cicula mechanisms in a plane ae used in the next section to descibe the azimuthal and elevation esponses of canonical scattees in thee dimensions.

2 Fig. 1. t L ~ a Flat Plate H t H ~ b Right Angle t c Cicle t + Plana mechanisms used to build 3D canonical shape models. a Plate b Dihedal c Tihedal d Cylinde e Top-hat f Sphee Fig.. Canonical scatteing pimitives. Fist, we conside scatteing fom a flat suface, shown in Figue 1a, with length L and pose angle. We epesent the bistatic esponse of the flat suface as the sum of bistatic etuns fom two point eflectos one located at each end of the suface, witten as M flat k, t, ; L, = L c k L π t, t + π, + π. 3 Next, we model a ight angle eflecto, shown in Figue 1b. In, it is shown that the dominant bistatic esponse of a ight angle eflection is given by M ight k, t, ; H, t + = c kh t t+, t,, π + H π cos t+, t, + π, + π povided that the size of the eflecto is much geate than the tansmitted signal wavelength: H λ. Finally, we model scatteing fom a cicula suface, shown in Figue 1c. The cicula mechanism has an isotopic esponse and is constant with espect to fequency. Its bistatic amplitude esponse is a scala equal to the squae-oot of the ada coss section of the cicle: M cic k, t, ; = π 5 whee is the cicle adius. Because the suface is cuved, the location of the ange esponse migates. The ound-tip delay R fo a cicula scatteing mechanism centeed at x,y is t R cic = R cos. 6 IV. CANONICAL SCATTERING CENTER MODELS The flat, ight-angle, and cicula plana esponses given in Section III descibe the azimuthal and elevation esponses of the canonical scattees shown in Figue. Poducts of plana esponses appoximate the 3D esponses M Γm. Such models accuately descibe scatteing nea the specula angle. We scale the model intensities to the pope ada coss section of the eflecto. We also model polaization effects and vaiations due to model pose. A. Amplitude and Phase Response Fist, we conside the the plate, dihedal, and tihedal scatteing pimitives. These shapes have zeo cuvatue; thus R =. The esponses M Γm ae poducts of flat and ightangle mechanisms. We modify the azimuth esponse fo these shapes by multiplying φ t with cos θ t and by multiplying φ with cos θ to allow fo convegence of lines of longitude as elevation angles appoach ±9. Adjustments ae made to the tihedal model to account fo known phenomenology of the shape. The 3D scatteing of a ectangula plate is modeled as a poduct of flat scatteing mechanisms in both azimuth and in elevation: M plate Θ plate = M flat k, φ t,φ ; L, φ =M flat k, θ t,θ ; H, θ = = A kl c π φ t cos θ t +φ cos θ kh c θ t+θ 7 whee A = LH scales the esponse so that the peak scatte intensity is equal to the peak RCS of a plate. A hoizontally-oiented dihedal appeas as a ight angle mechanism ove vaying elevation looks and as a flat mechanism in the azimuth diection. Theefoe, we model the dihedal esponse as M dih Θ dih = M flat k, φ t,φ ; L, φ =M ight k, θ t,θ ; H, θ = = A kl c π φ t cos θ t +φ cos θ c khcos θ t cos θ { θ t+θ,θt,θ, π cos θ t+θ, θt,θ π, 8 π whee L is the length of the dihedal, H is equal to the height and depth of its sides, and A =LH. A squae-plate tihedal is initially modeled as poduct of ight angle scatteing mechanisms in azimuth and elevation. Howeve, this initial model does not fully captue the symmety of the tihedal, so we expand the azimuth tem into two ight angle tems. In addition, we know the tihedal

3 peak scatteing occus at θ t = θ = tan 1 1 = α, not at θ =5, as in the ight angle model 9. Theefoe, we model the tihedal scatteing esponse as M ti Θ ti = c kh +c kh A π c khcos θ t cos θ cos φ π cos φ + π cos θ cos φ t + π { θt+θ + π α, θ,α cos θ t+θ { cos φt+φ π cos θ cos φ t π cos θ t cos θ t } + π α, θ α, π, φ π, } φ t+φ π, φ, π whee the length of the tihedal along each edge is assumed to be equal to H and A = 3H. Next, we conside cylinde, top-hat, and sphee pimitives. These shape esponses ae composed of poducts of plana mechanisms, at least one of which is cicula. The adius of cuvatue is non-zeo; thus we also compute the ange offset tem R fo each shape. The cylinde in Figue d is centeed on the y-axis, with length L and adius. Its esponse fo φ t,φ π, π is compised of flat and cicula scatteing mechanisms and is witten as M cyl Θ cyl = cos φ cos φt M flat k, φ t,φ ; H, φ = 9 M cic k, θ t,θ ; = A cos φ cos φ t c k L φ tcos θ t +φ cos θ whee A = L and we have adjusted the amplitude esponse cos φ by cos φt to agee with the bistatic scatteing solution given in 9. As in pevious models, we account fo convegence of longitude lines in the azimuth tem. The diffeential ange tem fo a cylinde is given by R cyl θt θ = cos cos φ t +cosφ. 11 Note, in 11, we have adjusted the ange offset fo the cicula mechanism given in 6 by multiplying by 1 cos φ t +cosφ to account fo aspects outside of the plane of the cicula mechanism. A top-hat is composed of a vetical cylinde and a wide base that foms a ight angle with the cylinde. Thus, the top-hat scatteing is descibed by M top Θ top =M cic k, φ t,φ ; M ight k, θ t,θ ; H, θ = = A ckhcos θ t cos θ { θt+θ, θt,θ, π cos θ t+θ, θt,θ π, π 1 whee H is the height of the top-hat and is equal to the width of the im and is the adius of the top-hat cylinde. The amplitude scale facto is A = 8 H. The diffeential ange tem fo the top-hat is R top φt φ = cos cos θ t +cosθ. 13 Finally, we conside a sphee, which is composed of cicula sufaces in both azimuth and elevation. The sphee is invaiant to pose and is modeled as M sphee Θ sphee =M cic k, φ t,φ ; M cic k, θ t,θ ; = π 1 Due to thee-dimensional vaiation in the bistatic look angle, the ange offset fo scatteing fom a sphee of adius cannot be obtained fom the cicula mechanism ange tem, but athe is given by R sphee φt φ = cos θ t +cosθ cos θt +θ + B. Rotated Scatteing Mechanisms cos θt +θ θ t +θ 15 To obtain the esponse fo a otated scatteing mechanism, we compute the antenna aspects in the object s otated coodinate system. If a scatteing cente has oll γ, pitch θ, and yaw φ, its otation tansfomation is given by cos φ φ R φ, θ, γ = φ cos φ 1 1 cos γ γ γ cos γ cos θ θ 1 θ cos θ. 16 The tansmitte coodinates with espect to the otated object ae ˆx t cos φ t cos θ t ŷ t = R φ, θ, γ φ t cos θ t. 17 ẑ t θ t and the elative azimuth and elevation angles ae ŷt ˆφ t = actan and ˆθt = actan ˆx t ẑ t ˆx t +ŷ t. 18 The otated eceive aspect angles ˆφ, ˆθ ae found analogously. The otated scattee esponse is found by ug the otated tansmitte and eceive angles to compute M Γm k, ˆφ t, ˆφ, ˆθ t, ˆθ. C. Polaization Response We define the polaization matix P of Equation 1 as Pvv P P = vh 19 P hv P hh

4 whee the subscipts denote vetical and hoizontal eceive and tansmit polaizations. As an initial bistatic polaization model, we appoximate the bistatic polaization esponse fom monostatic scatteing matix decompositions of P into even and odd numbes of wavefom eflections. The ectangula plate, tihedal, cylinde, and sphee ae odd-bounce eflectos. The dihedal and top-hat ae even-bounce eflectos. Fo a pefect electical conducto PEC the polaization dependencies ae 1 P odd = Fig.. Bistatic esponse of a squae tihedal with side length H = 15, inteogated by a tansmitte at φ t,θ t=, 35 P even = cosζt + ζ ζ t + ζ ζ t + ζ cosζ t + ζ 1 whee ζ and ζ t ae otations of the object about the eceive and tansmitte lines of sight, espectively. The above model is valid fo small bistatic angles. Futue wok will conside bistatic polaization models fo wide bistatic angles. V. RESULTS The amplitude, phase, and polaization tems defined in Section IV ae combined via 1 to compose paametic models fo six canonical shapes. Figues 3-8 show examples of these models fo f = GHz and compae them to scatteing simulations computed with Xpatch 6. Negligible coss-polaization esponses ae excluded. The models match closely to the Xpatch pedictions nea the specula diection; Table I quantifies the specula amplitudes and eos. The models ae accuate desciptions of the complex phase histoy esponse of the 3D canonical shapes. VI. CONCLUSION In this pape, we deived six canonical shape, paametic scatteing models fo 3D bistatic SAR. The models ae poducts of D plana scatteing mechanisms. Amplitude, phase, and polaization tems wee defined. Compaisons to Xpatch pedictions show that the models accuately descibe the dominant, specula esponses of each pimitive. The poposed 3D bistatic scatteing paameteizations will enable use of attibuted, geometic taget featues in bistatic SAR ATR scene visualizations. Fig. 5. Bistatic esponse of a cylinde with dimensions L = 15, =5, inteogated by a tansmitte at φ t,θ t=, Fig. 3. Bistatic esponse of a flat plate with dimensions LH =36 1, inteogated by a tansmitte at φ t,θ t=, Fig. 6. Bistatic esponse of a top-hat with dimensions H = 18, =6, inteogated by a tansmitte at φ t,θ t=, 35

5 Plate Dihedal Tihedal Cylinde Top-hat Sphee Model Xpatch Coheent Diffeence Model VH 9.36 Xpatch VH Coheent Diffeence VH Model HV 9.36 Xpatch HV Coheent Diffeence HV Model Xpatch Coheent Diffeence TABLE I SCATTERING AMPLITUDES AND ERRORS IN DB AT SPECULAR ANGLES FOR THE EXAMPLES IN FIGURES 3-8. Fig. 7. Bistatic esponse of a = sphee inteogated by a tansmitte at θ t,φ t=, HV VH REFERENCES 1 L. C. Potte and R. L. Moses, Attibuted scatteing centes fo SAR ATR, in IEEE Tans. on Image Pocesg, Januay 1997, vol. 6, pp Y. Akyildiz, Featue Extaction fom Synthetic Apetue Rada Imagey, M.S. Thesis, The Ohio State Univesity,. 3 J. A. Jackson and R. L. Moses, Featue extaction algoithm fo 3D scene modeling and visualization ug monostatic SAR, in Algoithms fo Synthetic Apetue Rada Imagey XIII, Poc. of SPIE, E. G. Zelnio and F. D. Gabe, Eds., 6, vol N. Akhte, Fa Zone Electomagnetic Scatteing Fom Complex Shapes Ug Geometical Theoy of Diffaction, Ph.D. Dissetation, The Ohio State Univesity, B. D. Rigling and R. L. Moses, GTD-based scatteing models fo bistatic SAR, in Algoithms fo Synthetic Apetue Rada Imagey XI, Poc. of SPIE, E. G. Zelnio, Ed.,, vol D. Andesh et al., Xpatch : The next geneation in high fequency electomagnetic modeling and simulation softwae, in IEEE Intenational Rada Confeence, May, pp J. B. Kelle, Geometical theoy of diffaction, J. Opt. Soc. Ame., pp , Jan B.D.Rigling,Signal Pocesg Stategies fo Bistatic Synthetic Apetue Rada, Ph.D. Dissetation, The Ohio State Univesity, 3. 9 G. T. Ruck, Ed., Rada Coss Section Handbook, Plenum Pess, New Yok, 197. Fig. 8. Bistatic esponse of a tilted dihedal with dimensions L H = 36 1, and oll angle γ = 5, inteogated by a tansmitte at φ t,θ t=,.

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