Analysis of Optimal Panel Geometry for Self-illustration Corner Reflector

Transcription

1 Progress In Eectromagnetics Research Symposium Proceedings, Guangzhou, China, ug. 5 8, naysis of ptima Pane Geometry for Sef-iustration Corner Refector C. R. Li 1,,. S. hou 1,, and L. L. Ma 1, 1 Key Lab of Quantitative Remote Sensing Information Technoogy, Chinese cademy of Sciences, China cademy of pto-eectronics, Chinese cademy of Sciences, China bstract Sef-iuminating corner refector is the corner refector that a the rays that enter the refector s cavity experience the tripe refection on the pane and return to the radar. It has the advantage of reducing the unexpected coherent ground interaction and thus improving the accuracy of RCS and benefiting SR caibration and image quaity assessment. The optima pane geometry is the one that has the minima edge ength for a given pane area in order to aeviate edge diffraction. The pane geometry had been previousy designed to be square, pentagon and hexagon. In this paper, the genera expressions for the pane area and pane externa edge ength of arbitrariy sef-iuminating corner refectors were described by parameter equation firsty. Then the edges of refector pane were assumed as circuar arc and further eiptica arc for anaysis. Through a numerica approximation approach, the externa edge engths of those sef-iuminating trihedra corner refectors are derived and compared. The resuts showed that the sef-iuminating trihedra corner refector with circuar arc had the minima edge ength and was the optima pane geometry among the pane geometries under anaysis. 1. INTRDUCTIN SR caibration is becoming more and more important for quantitative earth observation appication [1 3]. The mosty widey used corner refector as radar bright point target is the trianguar trihedra corner refectors. It has the advantage of arge radar cross section RCS, extremey wide RCS pattern, ight weight, cheap and simpe to manufacture. The pane geometry for trihedra corner refector has been traditionay chosen as trianguar shape [4 6]. However, not a the pane area is the effective area which contributes to the nomina RCS of the refector. The additiona tip refecting area, if interacting with ground pane, wi yied an increase of RCS. This probem wi affect SR radiometric and phase caibration accuracy. The sef-iuminating corner refectors are proposed to sove this probem [7]. the rays that enter the refector s cavity experience the tripe refection on the pane and return to the radar. There are many types of sef-iuminating corner refectors, e.g., square, hexagon. Note that the nomina RCS of trihedra corner refector is cacuated by the pane area and radar waveength, however this is true ony when the eg ength is very arge compared to the radar waveength. therwise, the contribution from edge diffraction becomes an issue. Since edge diffraction is proportiona to the pane externa edge ength, the optima pane geometry is the one that has the minima externa edge ength for the same pane area. In [7], the pane geometry was assumed as poygon and optimum hexagon pane geometry was obtained. Based on the above approach, this paper anayzes more types of sef-iuminating corner refectors. For this purpose, the genera expressions for the pane area and pane externa edge ength of arbitrariy sef-iuminating corner refectors are described by parameter equation firsty. Then the edges of refector pane are assumed as straight ine, circuar arc, and eiptica arc. Through a numerica approximation approach, the externa edge engths of those sef-iuminating trihedra corner refectors are derived and compared.. TPES F SELF-ILLUSTRTIN CRNER REFLECTR ND ITS MTHEMTIC EXPRESSIN.1. Expression of Sef-iustration CR Figure 1 shows the geometry of a sef-iuminating corner refector. ssuming curve C has the foowing form: Cτ = [, fτ, gτ], τ [, t] 1 where f =, ft =, g =, gt = denote that the curve passes point and B. rbitrary point p on pane wi be mapped onto point p. ccording to the mapping reationship, the coordinate of p coud be expressed by coordinate of p as xp = z p y p = z p y p

2 146 PIERS Proceedings, Guangzhou, China, ugust 5 8, 14 C p q C C 1 B C' C' 1 q' p' ' X C' x=y Figure 1: Mapping geometry. So the curve C on X pane coud be expressed by: C = [gτ, gτ fτ, ], τ [, t]. When τ increase from to t, curve C traces from to B and curve C traces from to B. n the other hand, according to the sef-iuminating symmetry requirements, the curve mapped onto X must be identica to the curve mapped to, therefore there is point t = t, so that xq = z q y q = y q 3 Substituting 3 into, then y q zq = y = z q y q = y q q x q = y q 4 Equation 4 means that there must be one point q on ines z = y for arbitrary sef-iuminating corner refector. ccording to [7], the edge curve C 1 and C of the sef-iuminating corner refector has the foowing forms: C 1 τ = [, f 1 τ, g 1 τ], τ [, t ] 5 C τ = [, g 1 τ, g 1 τ f 1 τ], τ [, t ] 6 In Section 3, the types of edge curve C 1 i.e., straight ine, circuar, eiptica wi be discussed... RCS and Edge Length of Sef-iustration Corner Refector ccording to area integration formua, one pane area of the trihedra corner refector coud be expressed as t S = f 1 g 1 g 1f 1 dτ 7 where f denotes partia derivatives of f, and g denotes partia derivatives of g. 3. NLSIS F THE MINIML EDGE LENGTH F VRIUS TPES F SELF-ILLUSTRTIN CRNER REFLECTR ccording to curve integration formua, edge ength L of the trihedra corner refector coud be expressed as t } L = f 1 + g 1 + f 1 g 1 + g 1 dτ 8 The purpose is to choose f so that it woud minimize 8 subject to constraint Curve C 1 : Straight Line For simpicity, firsty assume the curve C 1 is a straight ine. ccording to 6, C is aso a straight ine, and thus the pane geometry is poygon. The shape of the pane geometry is dependent on the position of point q. If q ies on B, then the pane geometry is square, and edge ength of the

3 Progress In Eectromagnetics Research Symposium Proceedings, Guangzhou, China, ug. 5 8, pane geometry is. If q ies on the position that makes the three points on the same straight ine, then the pane geometry is pentagon, and edge ength of the pane geometry is.1. In order to obtain the minima edge ength, express the area and edge ength of poygon as: S = + y 9 L = y + y + y + y 1 From 9 and 1, the reation between externa edge ength L and interior eg ength coud be derived, as shown in Figure 3. The minima edge ength is L min = 441 S when =.9469 S. It can be concuded that among poygonous pane geometry, hexagon is the optima pane geometry. 3.. Curve C 1 : Circuar rc ccording to the isoperimetric theorem, the circe has the shortest perimeter among a panar shapes with the same area, so the externa edge ength of a circuar arc pane shoud be even smaer than that of hexagon pane, theoreticay. ssume that the curve C 1 is part of a circe with the center at y, x, as shown in Figure 4. From Figure 4, radius of the circe is r = y + x = x + y 11 and the ine segment x = y y y 1 q x=y y C 1 C q'' Externa edge ength L y Interior eg ength a Figure : Pane geometry of poygon. Figure 3: The reation between edge ength L with interior edge ength. C 1 y r x Externa edge ength L y Interior eg ength a Figure 4: Pane geometry of circuar arc. Figure 5: The reation between edge ength L with interior edge ength.

4 148 PIERS Proceedings, Guangzhou, China, ugust 5 8, ta C 1 y x y - Externa edge ength L t =1. t =1.1 t = Interior eg ength a Figure 6: Pane geometry of eiptica arc. Figure 7: The reation between edge ength with interior edge ength at different ratio t. Tabe 1: Comparison of the edge engths of different pane geometries. Types of CR straight ine Properties square pentagon hexagon circuar arc eipse edge Pane area S S S S S Interior eg ength S S.866 S.9469 S.93 S.93 S Minimum of externa edge ength S.1 S 441 S 8 S 8 S The parameter equation for the circuar arc is: [ y = fτ = y + r cos τ π z = gτ = x + r sin τ, τ + arctan y, π ] x Using 7, 8 and 11 13, and through a numerica approximation approach [8], the reationship between edge ength L with has been derived and shown in Figure 5. When =.93 S, the minima edge ength L min = 8 S, which is smaer than that of the hexagon pane geometry Curve C 1 : Eipse rc Now consider the more genera case of the eipse edge. ssume that the curve C 1 is part of an eipse with the center at y, x, semi-major axis b = ta t is the ratio between semi-major and short axis and semi-minor axis a as shown in Figure 6. From Figure 6, the ine segment / x = y t 4y + 4y y 14 4y The parameter equation for the eipse arc is: [ y = fτ = y + b cos τ π z = gτ = x + a sin τ, + arctan y, π ] x The reation between edge engths with interior edge ength at different ratio t are shown in Figure 7. We can see that when t = 1 i.e., the circuar arc case, the minima edge ength L min = 8 S when =.93 S. The resuts showed that the minima externa edge ength is obtained when the semi-major axis of the eipse equas to semi-minor axis, which means that the pane geometry with circuar arc has the minima externa edge ength. 4. CNCLUSIN This paper anayzed the edge engths of different types of sef-iustration corner refector. The simpest pane geometry of the sef-iustration corner refector is the poygon, but it woud not have 13 15

5 Progress In Eectromagnetics Research Symposium Proceedings, Guangzhou, China, ug. 5 8, the minima externa edge ength according to the isoperimetric theorem. This paper extended the type of pane geometry to the more genera case eiptica arc for anaysis. The resuts showed that the minima externa edge ength is obtained when the semi-major axis of the eipse equas to semi-minor axis, which means that the pane geometry with circuar arc has the minima externa edge ength. Its vaue was smaer than that of the poygon, which verified the assumption derived from the isoperimetric theorem. In the future the RCS accuracy wi be vaidated by simuation and RCS testing the prototype of the circuar arc sef-iustration corner refector. CKNWLEDGMENT This work was supported in part by the Nationa Natura Science Foundation of China Grant Nos , and Nationa High Technoogy Research and Deveopment Program of China Grant No REFERENCES 1. hou,. S., W. Hong,. P. Wang, and. R. Wu, Maxima effective baseine for poarimetric interferometric SR forest height estimation, Science China Information Sciences, Vo. 55, No. 4, , 1.. Freeman,., SR caibration: n overview, IEEE Transactions on Geoscience and Remote Sensing, Vo. 3, No. 6, , Nov Gonzaez, J. H., M. Bachmann, G. Krieger, and H. Fieder, Deveopment of the TanDEM-X caibration concept: naysis of systematic errors, IEEE Transactions on Geoscience and Remote Sensing, Vo. 48, No., , Doerry,. W., Refectors for SR performance testing, Sandia Nationa Laboratories, Döring, B. J., P. R. Looser, M. Jirousek, and M. Schwerdt, Reference target correction based on point-target SR simuation, IEEE Transactions on Geoscience and Remote Sensing, Vo. 5, No. 3, , Gray,. L., P. W. Vachon, C. E. Livingstone, and T. I. Lukowski, Synthetic aperture radar caibration using reference refectors, IEEE Transactions on Geoscience and Remote Sensing, Vo. 8, No. 3, , Sarabandi, K. and T. C. Chiu, ptimum corner refectors for caibration of imaging radars, IEEE Transactions on Geoscience and Remote Sensing, Vo. 1, No. 44, , hou,. S., C. R. Li, L. L. Ma, M.. ang, and Q. Liu, Improved trihedra corner refector for high-precision SR caibration and vaidation, Proc. 14 IEEE Internationa Geoscience and Remote Sensing Symposium IGRSS 14, Québec, Canada, Ju. 13 Ju. 17, 14.