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1 韓國電磁波學會論文誌第 17 卷第 1 號 2006 年 1 月論文 바다표면의 Bistatic Coherent Reflectivity 계산을위한 Monte-Carlo/ 모멘트법과 PO 모델비교 Comparison of Moment Method/Monte-Carlo Simulation and PO for Bistatic Coherent Reflectivity of Sea Surfaces 김상근 오이석 Sang-Keun Kim Yi-Sok Oh 요약 본논문에서는바다표면에서전파의페이딩현상을해석하는데필요한 bistatic coherent reflectivity를 Monte- Carlo/ 모멘트법과 physical optics(po) 모델을이용하여비교하고계산하였다. 모멘트법계산을위해서우선 Gaussian 형태로불규칙한바다표면을생성하였다. 풍속과유의파고등에대한태안앞바다에서의실제측정데이터를사용하여이표면을생성하였으며, 두변수간의관계식을유도함으로써측정하기쉬운변수인풍속에의해바다표면상태를예측할수있는모델을만들었다. 평면에서의결과를비교함으로써모멘트법모델을검증하였고, 거친표면의경우에비교적정확한수치해석방법인 Monte-Carlo/ 모멘트법과간단히계산이가능한이론식인 PO 모델을비교함으로써거친표면에서 PO 모델의적용가능한조건을찾아내었으며, 전파경로해석법을이용하여실제전파의페이딩현상을해석할때에사용되는 PO 모델의유효성을검증하였다. Abstract This paper proposes a method of moments(mom)/monte-carlo simulation and Physical Optics(PO) model to determine Bistatic Coherent Reflectivity of sea surfaces at various wind speeds. For the MoM simulation, a Gaussian random rough sea surface was generated based on the data of Tae-An ocean at various wind speeds and sea surface heights. The numerical results of the MoM/Monte Carlo simulations were used to verify the validity region of the PO model. It was found that the numerical result for a flat surface agrees quite well with the Fresnel reflection coefficient. The validity of the PO model on the rough sea surface is shown by using ray tracing method. Key words : Sea Surface, Coherent Reflectivity, Physical Optics, Monte-Carlo/Method of Moments Ⅰ. 서론바다위에서의무선통신의경우반사율이큰바다표면의특성상전파다중경로에의한신호세기를무시할수없다. 이러한페이딩현상을해석하는방법으로는전파경로해석법을이용하여해석하고자하는지형위에서반사점을찾아내고, 그반사점에서각도 에따른반사율을계산하여반사파에대해해석하는 two-ray 지면반사모델이널리연구되어지고있다 [1]. 바다표면의경우에는풍속에의해다양한거칠기를갖고, 거칠기의정도또한사용하고자하는주파수의파장에비해매우큰경우가많으므로다양한풍속에따른바다표면상태를고려한 bistatic coherent reflectivity 값의해석이필요하다. 본연구는한국과학재단특정기초연구 (R ) 지원으로수행되었음. 홍익대학교전자정보통신공학과 (Department of Electronic Information and Communication Engineering, Hongik University) 논문번호 : 수정완료일자 : 2006 년 1 월 10 일 39

2 韓國電磁波學會論文誌第 17 卷第 1 號 2006 年 1 月 거친표면에대한전자파산란모델연구는원격탐사를목적으로한후방산란모델에집중되어왔고, 지금까지전방산란모델에대한연구는매우적었다. 하지만최근들어 GPS와무선통신환경및다수의위성을이용한 multi-static 원격탐사등의연구가활발해지면서전방산란모델에대한중요성이높아지고있다. 거친표면에대한전자파산란을계산하는방법으로는이론적인분석법과수치해석적인방법이일반적이다. 적분방정식의이론적근사식에해당하는 PO 모델은거친유전체표면의 coherent forward scattering의경우실험적결과를통해검증된바있다 [2]. PO 모델은계산이쉽다는장점이있지만, 다중산란을포함하지않기때문에바다표면과같이거칠기가파장에비해매우큰경우에는적용할수없는범위가나타난다는단점이있다. 따라서본논문에서는빠른계산시간내에바다에서의전자파세기를예측할수있는시뮬레이터제작에적합한 PO 모델의신뢰성을확인한다. 거친표면에서비교적정확한수치해석방법인 MoM/Monte-Carlo 시뮬레이션방법을적용하여다양한거칠기를갖는바다표면에서의 forward coherent reflectivity를계산하였으며, 그결과를비교함으로써 PO 모델의신뢰성과적용가능범위를확인하였다. 태안앞바다에서측정된유의파고와풍속데이터를참조하여 [3] 풍속에따른표면거칠기예측모델을만들었으며, 이모델을이용하여실제시뮬레이션에사용되는 Gaussian 형태의불규칙하게거친표면을생성하였다. 본연구에서는시뮬레이션시간의한계를고려하여 1차원적으로거친표면을구현하여 2차원산란을계산하였다. 2차원표면을생성하여 3차원산란계산도가능하나, 이경우에는컴퓨터의계산능력상아주작은거친표면조각에대해서만계산이가능하므로정확성에문제가생기게된다 [4]. Ⅱ. 바다표면의생성전자파산란해석에사용된바다표면은그림 1과같이일정한풍속에의해거칠기가결정되는 Gaussian형태의불규칙적거친표면으로구현하였다. 이때의 rms 표면높이는태평양과같은대양에서의풍속과 sea state의관계와 sea state와파고의관계를이용 그림 1. 풍속과 rms 표면높이의관계 Fig. 1. RMS surface height with respect to wind velocity. 하여대략적으로구한실험식 [5] 을사용하지않고, 태안앞바다의경우에맞는풍속과유의파고관계데이터로부터도출된실험식으로부터결정하였다. 여기서, 유의파고는 1,024개의측정된파고데이터중에서높은쪽에서 1/3의데이터를평균한수치이다. 수심이낮은태안앞바다의특성으로인해기존의미동부태평양연안을기준으로한풍속과 rms 표면높이의관계식과의차이가그림 1에잘나타난다. 우선, 생성시킨 Gaussian random surface의 rms 표면높이 σ rms 와유의파고 h를통계적으로비교함으로써식 (1) 을유도하였다. h =4.25 σ rms 그런다음에풍속에따른유의파고의데이터를분석함으로써식 (2) 와같은풍속과 rms 표면높이의관계식을유도하였다. σ rms = ( ) U U 여기서 U(m/s) 는풍속이고, σ rms 의단위는 m 이다. 그림 2는풍속 (U) 이 6 m/s ( σ rms = 0.1 m) 인서해앞바다의표면조각을수치적으로생성한거친표면상태를보여준다. Ⅲ. Monte-Carlo/ 모멘트법시뮬레이션 (1) (2) 거친표면에서의전자파산란에대한수치해석적 40

3 바다표면의 Bistatic Coherent Reflectivity 계산을위한 Monte-Carlo/ 모멘트법과 PO 모델비교 2차원적분방정식을유도한후 MoM을이용하여미소표면전류를구하고, 그로부터산란전계를계산하였다 [7],[8]. 그림 3과같은표면에서의적분방정식은 time convention e -iωt 을이용하여다음과같다. ( 2 +k 2 ) G ( ρ, ρ')=-δ ( ρ- ρ') (3) 여기서, G( ρ, ρ')= i 4 H (1) 0 (k ρ- ρ') 는 2차원에서의그린함수이다. E i y( ρ)+ c [ E y( ρ') G ( ρ, ρ') n' 그림 2. 생성된거친바다표면 Fig. 2. Generated rough SEA surface. 인방법들중하나인 MoM(Method of Moments) 을이용하여거친바다표면에서의전파반사계산을시도하였다 [6]. 컴퓨터메모리용량의한계와계산시간을고려하여그림 3과같은 2차원적전파산란계산을수행하였으며, 이때의거친표면은 1차원적인거칠기함수를갖는다. MoM 해석은 2.2 GHz에서 λ/10 cell size 2,000 cell 의공간을설정하여약 m의범위에서이루어졌으며, 바다의유전율은 ε r =72- j 32이다. Monte- Carlo 방법으로 20회반복하여결과값을도출하였다. 1차원으로구현된표면에대한해석이므로동일편파해석에서 HH-편파의경우에는전계를이용하였고, VV-편파의경우에는자계를이용하였다. -G( ρ, ρ') E y( ρ') n' = { E y( ρ) ( ρ V 1 ) 0 ( ρ V 2 ) ] dl' 거대한바다표면의일부분을잘라서시뮬레이션하는방식이므로그림 3에서보는것과같이표면의양쪽의잘린면으로가장자리영향이생겨나게되면서입사각이큰경우에계산결과의오차가커지는현상이발생하게된다. 이를해결하기위해서본논문에서는식 (5) 와같은 Tapered 입사파를사용하였으며, 그로인해큰입사각에서의오차를줄이는결과를얻었다 [9]. E r ( r)=exp( ik(xsinθ i -zcosθ i ) (1+w( r))) exp ( - (x+z tan θ i ) 2 (x+z tan 2 θ i ) 여기서, w( r)= (2 g 2-1 ) (kg cosθ i ) 2 (4) g 2 ) (5) (6) 또한, g는 tapering parameter로서 200λ의시뮬레이션구간의경우전체구간의 1/4배가적당하다. 그리고식 (4) 에서의미지전류에해당하는값을 pulse basis function 과 point matching 방법을적용한 MoM 을이용하여구하였다. mn mn Z Z12 mn mn [ Z ] [ Z ] [ gm ] [ h ] = [ V ] 11 m m 0 (7) 그림 3. Coherent MoM 산란모델구조 Fig. 3. Geometry for coherent scattering MoM model. 여기서 E y = N g mp m, m =1 E y n' = N m =1 h mp m 이고, 41

4 韓國電磁波學會論文誌第 17 卷第 1 號 2006 年 1 月 P m 은 pulse basis 함수이며, impedance 행렬요소들은다음과같다. 11 = G ( ρ m, ρ n ') dl Cn n' n ' 12 =- G ( ρ m, ρ n ') dl n ' Cn 21 = G ( ρ m, ρ n ') dl Cn n' n ' 22 =- Cn ε r G( ρ m, ρ n ') dl n ' (8) MoM을이용하여구한표면전류값을이용하여산란전계를구하면다음과같고, E s y( ρ)= c [ E y( ρ') G( ρ, ρ') n' -G( ρ, ρ)' E y( ρ') n' ] dl', 이로부터 coherent reflectivity 값을구한다. Γ(θ) = E s y(θ) E i y(θ), ( θ=θ i =θ s ) Ⅳ. Physical Optics 모델 (9) (10) 거친표면에대한 coherent field reflection은다음과같다 [7]. Γ q (θ )=Γ qo e -2k 2 s 2 cos 2θ δ pq (11) 여기서, k는파수, s는 rms 표면높이이고, δ pq 는동일편파를보기위한 Kronecker delta 함수이며, Γ h 0,Γ v 0 는각각수평편파와수직편파에대한 Fresnel 반사계수이다. 그림 4. PO 와 MoM 의평면에대한결과비교 Fig. 4. Compared results of PO and MoM in flat surface. 뢰성을확인할수가있다. 반면에거칠기가있는표면에서는다중산란을포함하지않는 PO 모델이오차를갖게된다. 풍속이 2, 4, 6, 8 m/s인바다표면에서의전파반사를계산하였으며, 그림 5는풍속이 2 m/s인거칠기가작은표면에서의 MoM 수치해석적계산결과와 PO 근사식계산결과를비교하여보여준다. 이그림에서두결과간에생기는약간의불일치를확인할수있고, 그림 6에서는풍속이 6 m/s 인표면에서의비교결과를보인다. 그림 6에서볼수있듯이, 풍속이증가함에따라바다표면의거칠기가증가하게되면낮은입사각에서의반사도는다중산란을고려하지못하는 PO 모델의특성상 Γ ho (θ)= cos θ- ε r -sin 2 θ cos θ+ ε r -sin 2 θ (12a) Γ ho (θ)= ε rcosθε r cosθ+ ε r -sin 2 θ ε r -sin 2 θ (12b) Ⅴ. 결과및검증 그림 4에서보듯이완전평면에대한 MoM과 PO의결과는거의완벽하게일치하였다. 평면에서는 rms 높이 s=0 이므로, PO 반사계수는 Fresnel 반사계수와같으므로 PO의값이정확하고, MoM에의한결과의신 그림 5. PO 와 MoM 의결과비교 (U=2 m/s) Fig. 5. Compared results of PO and MoM(U=2 m/s). 42

5 바다표면의 Bistatic Coherent Reflectivity 계산을위한 Monte-Carlo/ 모멘트법과 PO 모델비교 그림 6. PO 와 MoM 의결과비교 (U=6 m/s) Fig. 6. Compared results of PO and MoM(U=6 m/s). 거친표면에서정확한 MoM에의한결과와맞지않게되는결과가발생한다. 즉, PO 모델의결과는높은각도에서만신뢰성을갖게되고, 바다표면이거칠어짐에따라신뢰성을갖기시작하는입사각도가점점커지는것을확인할수있다. 이결과로부터바다표면의전파산란모델을예측하는데있어서 PO 모델을적용할수있는범위를찾아내었고, 그결과는그림 7과같다. 풍속이 2 m/s 이상의경우에 PO 모델이맞지않게되는입사각에서의영역이생겨나게되고, 거칠기가매우큰풍속이 8 m/s에해당하는경우에도 73 이상의입사각에서는 PO 모델이정확한값을갖는다는것을확인할수있다. 또한 1 km 상공의일정한고도로송신기가수신기 그림 7. PO 모델의유효영역 Fig. 7. Validity region of PO model. 그림 8. 송수신기의거리에따른입사각 Fig. 8. Incident angle with respect to the distance between transmitter and receiver. 로부터멀어지는경우반사점에서송수신기간에형성되는입사각도값을광선추적법을사용하여구하면그림 8과같다. 송수신기간의거리가약 7 km 이상으로멀어지게되는경우입사각이형성되는각도가 80 이상인것을확인할수가있다. Ⅵ. 결론 본논문에서는바다표면의 coherent 전파산란을예측하기위해우선태안앞바다의실제측정자료를근거로풍속과 rms 표면높이에대한관계식을정의하였다. 이관계식으로부터 Gaussian 함수형태로불규칙한거친바다표면들을생성하였으며, 다양한풍속에따라생성된바다표면으로부터수치해석적인방법인 MoM/Monte-Calro 방법으로 coherent 전파산란을계산하였고, 이론적근사식인 PO 모델결과와비교하였다. 그비교결과로부터 PO 모델은바다표면의거칠기에따라특정입사각이하에서 MoM의결과와차이가나게되는한계영역이존재함을찾을수가있었다. 이영역은주파수가 2.2 GHz에해당하는결과이고, 높은주파수일수록상대적인거칠기가커지므로 PO의한계영역은작아질수있다. 또한실제바다표면반사에의한페이딩현상을해석하는데있어서 ray tracing 방법을이용하여찾은반사점에서의 coherent 입사각의경우일반적으로 80 이상이기때문에간단한이론식에해당하는 PO 모델만으로도바다표면반사에대한상당히정확한해석이가능하다. 43

6 韓國電磁波學會論文誌第 17 卷第 1 號 2006 年 1 月 참고문헌 [1] T. S. Rappaport, Wireless Communications - Principles and Practice, Prentice Hall PTR, [2] R. D. DeRoo, F. T. Ulaby, "Bistatic specular scattering from rough dielectric surfaces", IEEE Antennas and Propagation, vol. 42, no. 2, Feb [3] 실시간해양관측자료 ( 한국해양연구원웹페이지 ) [4] 차형준, 오이석, " 불규칙적으로거친유전체표면에서의레이더산란계수수치해석적계산 ", 대한원격탐사학회지, 16(1), pp , 2000년 3월. [5] W. Morchin, Radar Engineer's Sourcebook, Boston, Artech House, [6] R. F. Harrington, Field Computation by Moment Methods, New York: IEEE Press, [7] F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing Active and Passive, Artech House, [8] L. Tsang, J. A. Kong, Scattering of Eletromagnetic Waves Numerical Simulation, Wiley-Interscience Publication, [9] Jin A. Kong, Leung Tsang, "Tapered wave with dominant polarization state for all angles of incidence", IEEE Transactions on Antennas and Propagation, vol. 48, no. 7, Jul 김상근 2005 년 2 월 : 홍익대학교전자전기공학부 ( 공학사 ) 2005 년 2 월 ~ 현재 : 홍익대학교전자정보통신공학과석사과정 [ 주관심분야 ] 안테나, 전파산란, 이동통신 오이석 1982년 2월 : 연세대학교전기공학과 ( 공학사 ) 1988년 12월 : University of Missouri-Rolla, 전기컴퓨터공학과 ( 공학석사 ) 1993년 12월 : University of Michigan, Ann Arbor, 전기공학컴퓨터과학과 ( 공학박사 ) 1994년~현재 : 홍익대학교전자전기공학부부교수 [ 주관심분야 ] 전파산란, 마이크로파원격탐사, 안테나 44

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