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[object Object],Dr. Nicolas Pinel , Dr. Christophe Bourlier, Pr. Joseph Saillard IREENA Laboratory, Nantes, France E-mail:  [email_address]
Introduction: Context ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],incident wave scattered wave scattered power incident power
Introduction: Objective ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],-   extensive memory space  3   (  r3 ,  0 ) x y z 2D problem &  3D problem Scattering by  one  rough interface Scattering by  two  rough interfaces  1   (  r1 ,  0 )  2   (  r2 ,  0 )  1   (  r1 ,  0 )  2   (  r2 ,  0 )
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
Analytic ( asymptotic ) methods – State of the art ,[object Object],[object Object],[object Object],2 rough interfaces Small Perturbation Method (  h <<  )  [Fuks & Voronovich, WRM, 2000] Reduced Rayleigh Equations (  h <<  )  [Soubret et al., PRB, 2001] Full Wave Model  [Bahar et al., TAP, 1999] new:  Kirchhoff Approximation (R c >  ) Geometric Optics Approximation (R c >   +   h >  ) Presentation of the 2D case => Extension to the 3D case For slight  incidence angles
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Kirchhoff Approximation (KA) ,[object Object],R c   >   Approximation used on  both  interfaces Locally plane (infinite) interface At each surface point  A :  - the Snell-Descartes laws - the Fresnel coefficients  A  A can be used ,[object Object],E r  1   (  r1 ,  0 )  2   (  r2 ,  0 )  r  i E t E i  R c  t
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]
KA-1 improvement: the shadowing function Grazing angles (θ i , θ r , θ t ):  a part of the surface is in the shadow over-prediction of the RCS (KA-1) ,[object Object],[object Object],[object Object],= 1  if  A  is  not  in the shadow = 0  if  A  is in the shadow S 11 (θ i ,θ r )   [Wagner, JASA, 1967],    [Bourlier et al., WRM, 2002] S 12 (θ i ,θ t )   [Pinel et al., OL, 2005]  r  t  i shadow of the emitter (  i ) shadow of the reflected field (  r ) shadow of the transmitted field (  t ) A z x
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Approach of the method ,[object Object],[object Object],[object Object],[object Object], A  B A 1 z  i E i Iteration of KA-1 for each scattering inside the rough layer A 2 E 1 B 1  s z  s E 2
Approximations of the method ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Under the 1 st -order Kirchhoff Approximation (  s  < 0.5, R c  >   ): 2D problem: 2(n-1)  numerical integrations ( rough  lower interface) (n-1)   numerical integrations ( plane  lower interface)
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
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
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
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°
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
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
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
Extension of the model to the 3D case ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
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 
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)
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, …)
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object]
Conclusions & Prospects ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],~ 5s. ( approximate ) ~ 4h10mn (exact, N=50)
[object Object],Dr. Nicolas Pinel , Dr. Christophe Bourlier, Pr. Joseph Saillard IREENA Laboratory, Nantes, France E-mail:  [email_address]
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
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
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
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
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
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
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)
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)

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2007 EuRad Conference: Speech on Rough Layers (ppt)

  • 1.
  • 2.
  • 3.
  • 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
  • 5.
  • 6.
  • 7.
  • 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]
  • 9.
  • 10.
  • 11.
  • 12.
  • 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
  • 14.
  • 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
  • 20.
  • 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, …)
  • 24.
  • 25.
  • 26.
  • 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)