2007 EuRad Conference: Speech on Rough Layers (ppt)
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4. Analytic ( asymptotic ) methods – State of the art 1 single rough interface Small Perturbation Method ( h << ) Reduced Rayleigh Equations ( h << ) Small Slope Approximation ( s <<| i,r | ) Full Wave Model Kirchhoff Approximation ( R c > ) Geometric Optics Approximation ( R c > + h > ) etc. : incident wavelength h : RMS surface height s : RMS surface slope R c : mean surface curvature radius Topical Review: [Elfouhaily & Guérin, WRM, 2004] For slight incidence angles i r s h R c
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8. 1 st -order Kirchhoff Approximation (KA-1) Only the first scattering is taken into account: KA-1 Multiple scattering on each interface valid for s < 0.5 (~30°) [1,2] R c > [1]: [Ishimaru, PIER, 1996] [2]: [Bourlier et al., WRM, 2004]
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13. Analytic expression of σ 2 (2D problem) Second-order Radar Cross Section 2 ~ < E 2 E 2 ’* > : depends on the Fresnel reflection and transmission coefficients at A 1 , B 1 , and A 2 probability density functions (give the specular directions) average shadowing function Є [0,1] This expression can be generalized to any order n x z B 1 A 1 A 2 i s E 2 E i - γ A1 0 + γ B1 0 γ A2 0
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15. Numerical results (2D) Bistatic RCS σ 1 & σ 1 + σ 2 : Comparison with a reference numerical method… … based on the Method of Moments [Déchamps et al., JOSAA, Feb.2006 ] V polarization Geometric optics validity domain h = 0.5 s = 0.1 (slight i ) i = {0°; -20°} s r1 =1 r2 =3 r3 =i (PC) s h H = 6 s h
16. Numerical results (2D) 1 st -order RCS σ 1 : Comparison with a reference numerical method i The shadow can be neglected Good agreement with reference method 1 i = 0°
17. Numerical results (2D) 2 nd -order contribution σ 2 : Comparison with a reference numerical method i = 0° Good agreement with reference method (model with shadow) 1 i 2
18. Numerical results (2D) 1 st -order RCS σ 1 : Comparison with a reference numerical method i = -20° The shadow can be neglected Good agreement with reference method i 1
19. Numerical results (2D) 2 nd -order contribution σ 2 : Comparison with a reference numerical method i = -20° Good agreement with reference method (model with shadow) Validation of the developed model in the high-frequency limit 1 [Pinel et al., WRCM, Aug.2007] i 2
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21. Numerical results (3D) Bistatic RCS σ 1 & σ 1 +σ 2 for a plane/rough lower interface i = 0° i = 20° In the plane of incidence s =0°: Study of the co- and cross- polarizations with respect to s i s r1 =1 r2 =3 r3 =i (PC) sx = sy = 0.1 H = 6
22. Numerical results (3D) 2 nd -order contribution σ 2 (dB) i = -20° The shadow contributes for grazing angles Contribution of the cross-polarization 1 (no shadow) o 1 ( with shadow) 1+2pl (no shadow) x 1+2pl ( with shadow)
23. Numerical results (3D) 2 nd -order contribution σ 2 (dB) i = -20° The shadow contributes for grazing angles Contribution of the cross-polarization 1+2pl (no shadow) x 1+2pl ( with shadow) 1+2r (no shadow) x 1+2r ( with shadow) Interesting means in order to detect layers (oil slicks, …)
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27. Analytic expression of σ 2 (3D problem) Second-order Radar Cross Section 2 = < E 2 E 2 ’* > : This expression can be generalized to any order n x y z
28. Results & Consequences The calculus of E 1 is simple The calculus of E 2 implies 9 variables E 2 : { x A1 ,x B1 ,x A2 , z A1 ,z B1 ,z A2 , γ A1 ,γ B1 ,γ A2 } E 1 12 23 E 2 A 1 A 2 i s s z x B 1
29. Results & Consequences Hypothesis: the lower surface S 23 is plane E 2 : { x A1 ,x B1 ,x A2 , z A1 ,z A2 , γ A1 ,γ A2 }: 7 remaining variables dependence on {z B1 ,γ B1 } suppressed E 1 12 23 E 2 A 1 A 2 B 1
30. Results & Consequences Method of stationary phase (MSP) : The main contribution comes from regions around the specular direction: γ A -> γ A 0 determined by k inc and k s1 => E 2 : { x A1 ,x A2 , z A1 ,z A2 }: 4 variables… still too much! E 1 12 23 E 2 γ A 0 θ i θ s A 1 A 2 B 1 E 1 12 γ A1 0 A 1 k inc k s1
31. Consequences Geometric optics approximation (GO) : valid if k 0 σ h >> 1 => Calculus of 2-RCS: requires only 1 numerical integration E 1 12 23 E 2 h A 1 A 2 B 1
32. Analytic expressions of σ 1 and σ 2 First-order Radar Cross Section 1 : dependent on the Fresnel reflection coefficient probability density function (gives the specular direction) shadowing function Є [0,1] E 1 A 1 i s E i γ A1 0
33. Numerical results (3D) 2 nd -order contribution σ 2 (dB) i = 0° The shadow can be neglected Contribution of the cross-polarization 1 (no shadow) x 1 ( with shadow) 1+2 (no shadow) x 1+2 ( with shadow)
34. Numerical results (3D) 2 nd -order contribution σ 2 (dB) i = -20° The shadow contributes for grazing angles Contribution of the cross-polarization Interesting means in order to detect layers (oil slicks, …) 1 (no shadow) x 1 ( with shadow) 1+2 (no shadow) x 1+2 ( with shadow)