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1 THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. 216 Feb.; 27(2), ISSN (Prit) ISSN X (Olie) Mathematical Modelig of Wave Propagatio Cosiderig the Atmospheric Effects ad Its Applicatio 이태승 최상혁 전주환 강성철 박동민 Taeseug Lee Saghyouk Choi Joohwa Chu Seogcheol Kag* Dogmi Park* 요약 (refractive idex), (Sell s law),,, Abstract I this paper, we propose a method that represets a mathematical form of wave propagatio by usig the fact the refractive idex determiig wave propagatio characteristic is a fuctio of altitude. Proposed method uses Sell s law that expresses relatioship betwee icidet agle ad refractio agle whe icidet wave passes medium havig a differet refractive idex. We preset the simulatio results about wave propagatio by settig the square of refractive idex i the form of the polyomial for altitude ad show that it is possible to estimate the coefficiets of the polyomial through the agle iformatio from vertical axis of multiple radar systems. Key words: Wave Propagatio, Refractive Idex, Real-Time Radiosode Iformatio, Maximum Likelihood Estimatio, Zero-Fidig Problem. 서론,,, (lie of sight wave), (refractio), (diffractio), (School of Electrical Egieerig, Korea Advaced Istitute of Sciece ad Techology) * (Hawha Thales) Mauscript received October 2, 215 ; Revised Jauary 8, 216 ; Accepted Jauary 18, 216. (ID No ) Correspodig Author: Taeseug Lee ( tslee11@kaist.ac.kr) 188 c Copyright The Korea Istitute of Electromagetic Egieerig ad Sciece. All Rights Reserved.

2 (azimuth agle), (elevatio agle), [1],[2].,,,, CRPL(Cetral Radio Propagatio Laboratory) [5] [6] 7 6, [7],,,, (Maximum Likelihood Estimatio), 2-1, , 본론 2-1 대기의고도에따른굴절률변화,, (refractive idex), c = = v e m r r (1) c, v (1) e r m r (relative permittivity) (relative permeability), (refractivity) 189

$37 1 N = Ndry + Nwet = p + e 2 T T (3) (3) T [K ], p [ mbar ], e [ mbar ] AREPS(Advaced Refractive Effects Predictio System) SPAWAR Systems Ceter PACIFIC Uiversity of Wyomig [8] AREPS Uiversity of$

3 THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb. 216., (3),, [3],[4]. RH Pw % = 1 P ( ) ws (4) 6 ( 1) 1 N = - (2) N = Ndry + Nwet = p + e 2 T T (3) (3) T [K ], p [ mbar ], e [ mbar ] AREPS(Advaced Refractive Effects Predictio System) SPAWAR Systems Ceter PACIFIC Uiversity of Wyomig [8] AREPS Uiversity of Wyomig, 1, (1) [m], (2) [hpa], (3) [ ], (4) [%] (3) e Uiversity of Wyomig, [9]. (4) P w [ mbar ], Pws [mbar ] (5) [1], e (3),,, 1,, 2 P ws [ + T - T ] exp / = 8.2 T (%) exp[ T / T ] RH e = Pw = 1 1T 2, (3),, (5) (6) 그림 1. Uiversity of Wyomig ( , ) Fig. 1. Real-time radiosode iformatio provided by the Uiversity of Wyomig. 그림 2. Uiversity of Wyomig,, ( , ) Fig. 2. The chage of temperature, pressure ad partial pressure of water vapor i accordace with the altitude of atmosphere. 19

4 그림 3. Fig. 3. The chage about square of refractivity ad refractive idex i accordace with the altitude of atmosphere(calculated through real-time radiosode iformatio - Kwagju at 28 Sep. 215). (3) 3 3, 1, m 1, m, 2-2 전파전달에대한수학적모델링 4, x-z (x R, z R ), (x t, z t) [11]., 그림 4. Fig. 4. Defiitio of the variables i order to set the mathematical model about wave propagatio. f, sif sif dx taf = cosf = 1 si = dz 2 - f (7) (Sell s law) f sif = sif Þ sif = sif (8) (7) sif dx sif = = dz si f - si f 1-2 (9) z, 1, m, 2 = az + b 1 ò dx = 2 a + x + C a + x, (8) (9) 191

5 THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb a, b y sif zt x = ò dz + x z R R 2 2 ( si f ) b - + az t é ù 1 ê ú = 2 ê sif ( b - si f + az) x 1 ú + ê a ú ë 2 û 2 sif sif 2 2 = ( b - si f ) + azt - ( b - si f ) + azr + xr a a (1) +, si, si (- f ) = -si ( f ) z zr R, (11) 2 sif sif 2 2 x = - ( b - si f ) + azt + ( b - si f ) + azr + xr a a (11) 2-3 여러개의레이다로부터측정한각도정보들을통한전파전달특성에대한변수추정방법 2-2, (1), (11) (x t, z t ), (a, b) 2 2 (Likelihood fuctio), (Maximum Likelihood Estimatio) [12] 2 ( s ) z = Φ +, ~ N, I 1 é 1 T -1 ù p( z Θ) = exp ê - ( z - Φ) R ( z - Φ) ë 2 ú û N /2 1/2 ( 2p ) det ( R ) (12) (13) (12) T Φ éf,1 f,2 ù = ë û, (White Gaussia Noise) 2-2 (13) Θ 2 Θ = [ a b] T R, T -1 ( z Φ) R ( z Φ) v = - - = T -1 T -1 T -1 T -1 z R z z R Φ Φ R z Φ R Φ (14) (13), (14),. "Zero-fidig problem", - (Newto-Raphso Algorithm) l p( z Θ T ) º f ( Θ) = Θ (15) Θ = Θ c J Θ Θ c c (16) 1 ( ) f ( ) 192

6 (16) Θ c Θ +, J ( Θc ) (Jacobia matrix) Θ $ c = éa b$ ù ë û ( ) f Θ J c ( Θ ) c T ( Θc ) ( Θ ) T f1 v v é ù é ù = ê ú = f ê a b ú êë 2 c úû ë û é f1 f1 ù ê a b ú = ê ú ê f2 f2 ú êë a b úû. 모의실험및결과 3-1 전파전달에대한수학적모델링 (17) (18) (19),,, x-z,, a b 그림 5. Uiversity of Wyomig curve fittig Fig. 5. Settig the coefficiets (a, b) of polyomial model through curve fittig. 5 Uiversity of Wyomig MATLAB "cftool" Curve fittig ( a= , y b=1.1),, + (9) z z (z R ) z (z t ) (1) 1 (1) (, ) [m] 5 m a Uiversity of Wyomig curve fittig a 표 1. Table 1. Parameters used i Sceario 1. [m] 1 a, b (xr, zr)=(, ) [m] a= 1.258e-4, 2.134e-7, 3.894e-4, 8.52e-4 b=1.1 o [deg] f = 25 = 1.3 zt zt = 5[ m] 193

THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb. 216. 그림 6.

7 THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb 그림 6. (x R, z R)=(, ) [m] 25 Fig. 6. Wave propagatio accordig the chage of coefficiets of 1 st order polyomial about square of refractive idex whe the radar is positioed o the (x R, z R)=(, ) [m] trasmits the wave havig a shootig agle of 25 [deg]. 1 7 a 1 4, a, (9) z z (z R ) 표 2. Table 2. Parameters used i Sceario 2. [m] 1 a, b (xr, zr)=(1,3, ) [m] a= 1.258e-4, 2.134e-7, 3.894e-4, 8.52e-4 b=1.1 [deg] f = -4 o = 1.3 zt z = 5[ m] t 그림 7. (x R, z R )=(1,3, ) [m] 4 Fig. 7. Wave propagatio accordig the chage of coefficiets of 1 st order polyomial about square of refractive idex whe the radar is positioed o the (x R, z R )=(1,3, ) [m] trasmits the wave havig a shootig agle of 4 [deg]. z (z t ) (11) 2 (11) 7 6, 여러개의레이다로부터측정한각도정보들을 통한전파전달특성에대한변수추정방법 2-3, 8, 3, 9 194

8 그림 8. Fig. 8. Sceario for the estimates of the two parameters (a, b) of polyomial model about altitude. 표 3. (a, b) Table 3. Parameters used i sceario for the estimates of two parameters (a, b). z = [ - ] T a = , b = 1.6 a$ = , b $ = = , 2 = az + b, 9 a iteratio 3, y b iteratio 3, b a 9 a.3,, (16) - relaxatio factor m Relaxatio factor m - relaxatio factor, "trial ad error". 결론및향후연구방향 그림 9. - / Fig. 9. Relative error about estimates of the two parameters (a, b) through Newto-Raphso algorithm., 1 (a, b),, 2 3, 195

THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb. 216. Refereces [1] R. B. Muchmore, A. D. Wheelo, "Lie-of-sight propagatio pheomea -.

$Jeh, "A study of radar elevatio-agle errors due to atmospheric refractio", Ateas ad Propagatio, IRE Trasactios o, vol. 5, o. 1, pp. 71-77, Ja. 1957. [3] D. K.$

9 THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 27, o. 2, Feb Refereces [1] R. B. Muchmore, A. D. Wheelo, "Lie-of-sight propagatio pheomea -. Ray treatmet", Proceedigs of the IRE, vol. 43, o. 1, pp , Oct [2] B. M. Fai, K. Jeh, "A study of radar elevatio-agle errors due to atmospheric refractio", Ateas ad Propagatio, IRE Trasactios o, vol. 5, o. 1, pp , Ja [3] D. K. Barto, Atmospheric Effects, Radar Equatios for Moder Radar, Norwood, MA : Artech House, 213. [4] J. A. Richards, Radio Wave Propagatio : A Itroductio for the No-specialist, Spriger Sciece & Busiess Media, 28. [5] B. R. Bea, G. D. Thayer, "CRPL expoetial referece atmosphere", Jour. Res. NBS, vol. 63, o. 3, pp , Ju [6],,,,,,, " ",, 23(7), pp , [7],,,,,, " ",, 25(1), pp , [8] Uiversity of Wyomig, "Atmospheric soudigs", weather.uwyo.edu. [Olie]. Available : upperair/soudig.html [Accessed: Sep 28, 215] [9] The Egieerig ToolBox, "Relative Humidity of Air", egieerigtoolbox.com. [Olie]. Available : [Accessed: Sep 28, 215] [1] The Egieerig ToolBox, "Water vapor ad saturatio pressure i humid air", egieerigtoolbox.com. [Olie]. Available : water-vapor-saturatio-pressure-air-d_689.html [Accessed: Sep 28, 215] [11] Codo Report, Chap 4, Sectio 3, "Basic Physical Cocepts ad Atmospheric Variables Ivolved i Light Refractio", [Olie], Available : com/shg/codo/s6chap4.html [12] S. M. Kay, Fudametals of Statistical Sigal Processig : Estimatio Theory, Pretice-Hall PTR, : & ( ) : ( ) : [ 주관심분야 ] 27 2 : ( ) 21 2 : ( ) : ( ) : [ 주관심분야 ] 196

o Vehicular Techology [ 주관심분야 ],, 21 2 : ( ) 21 4 : 214

10 1989 : Ph.D. Dept of EE, Staford Uiversity, Califoria : Member of Techical Staff, Geeral Electric, Scheectady, New York 1992 : : Editor for the IEEE Trasactio o Vehicular Techology [ 주관심분야 ],, 21 2 : ( ) 21 4 : : ( ) [ 주관심분야 ] Radar, Beamformig, RF, Sigal Processig : ( ) 21 2 : ( ) 27 3 : [ 주관심분야 ], / 197

THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. vol. 29, no. 10, Oct ,,. 0.5 %.., cm mm FR4 (ε r =4.4)

THE JOURNAL OF KOREAN INSTITUTE OF ELECTROMAGNETIC ENGINEERING AND SCIENCE. 2018 Oct.; 29(10), 799 804. http://dx.doi.org/10.5515/kjkiees.2018.29.10.799 ISSN 1226-3133 (Print) ISSN 2288-226X (Online) Method