Bistatic radar cross section of two-dimensional random rough layers in the high-frequency limit Dr. Nicolas Pinel , Dr. Christophe Bourlier, Pr. Joseph Saillard IREENA Laboratory, Nantes, France E-mail: [email_address]

Bistatic radar cross section of two-dimensional random rough layers in the high-frequency limit

Introduction: Context EM scattering -> Scattering Coefficient (SC) SC ~ a rough single interface: relatively well-known (except from grazing incidence) two (or more) rough interfaces : research in progress Case of 2 rough interfaces: Applications: Optics (characterization of optical materials, detection of defaults, …) Remote sensing (sand on granite, ice on sea, oil slick on sea, …) … incident wave scattered wave scattered power incident power

EM scattering -> Scattering Coefficient (SC)

SC ~

a rough single interface: relatively well-known

(except from grazing incidence)

two (or more) rough interfaces :

research in progress

Case of 2 rough interfaces: Applications:

Optics (characterization of optical materials, detection of defaults, …)

Remote sensing (sand on granite, ice on sea, oil slick on sea, …)

…

Introduction: Objective Objective A fast method in order to compute the SC scattered by a homogeneous dielectric layer ( two rough layered interfaces ) Different possible approaches: rigorous asymptotic + ‘exact’ + fast - extensive computing time - restricted domain of validity - 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 )

Objective

A fast method in order to compute the SC scattered by a homogeneous dielectric layer ( two rough layered interfaces )

Different possible approaches:

rigorous asymptotic

+ ‘exact’ + fast

- extensive computing time - restricted domain of validity

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 Extension of the Kirchhoff Approximation (R c >  ) to the case of 2 rough interfaces -> a model with strongly rough interfaces (  h >  /2) 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

Extension of the Kirchhoff Approximation (R c >  )

to the case of 2 rough interfaces

-> a model with strongly rough interfaces (  h >  /2)

Outline Introduction Theoretical model: one interface Theoretical model: two interfaces Numerical results (2D & 3D) Conclusion & Prospects

Introduction

Theoretical model: one interface

Theoretical model: two interfaces

Numerical results (2D & 3D)

Conclusion & Prospects

Kirchhoff Approximation (KA) (Infinite) Tangent plane approximation: 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 directions and amplitudes of E r , E t corresponding to each surface point A (  A ), at each point of the considered medium (  1 or  2 ) E r  1 (  r1 ,  0 )  2 (  r2 ,  0 )  r  i E t E i  R c  t

(Infinite) Tangent plane approximation:

directions and amplitudes of E r , E t corresponding to each surface point A (  A ), at each point of the considered medium (  1 or  2 )

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) Illumination function Σ (A) Average shadowing function + does not increase the computing time – more precise model = 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

Illumination function Σ (A)

Average shadowing function

+ does not increase the computing time – more precise model

Outline Introduction Theoretical model: one interface Theoretical model: two interfaces Numerical results (2D & 3D) Conclusion & Prospects

Introduction

Theoretical model: one interface

Theoretical model: two interfaces

Numerical results (2D & 3D)

Conclusion & Prospects

Approach of the method Use of KA-1 on the upper interface  A (A 1 ) => reflected & transmitted fields at the point A 1 Huygens’ principle (Green function -> Weyl representation ) => E 1 & incident field on  B (B 1 ) Use of KA-1 on the lower interface  B (B 1 ) etc.  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

Use of KA-1 on the upper interface  A (A 1 ) => reflected & transmitted fields at the point A 1

Huygens’ principle (Green function -> Weyl representation ) => E 1 & incident field on  B (B 1 )

Use of KA-1 on the lower interface  B (B 1 )

etc.

Approximations of the method Calculus of the 1 st -order RCS: Simple Calculus of the n th -order RCS ( n ≥ 2 – iteration of KA-1) : Too much complicated Method of Stationary Phase : main contribution comes from regions around the specular direction Still too much complicated Geometric optics approximation (GOA) :  h >  /2 (slight  i ) : main contribution comes from highly-correlated surface points + Hypothesis: uncorrelated surfaces 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)

Calculus of the 1 st -order RCS:

Simple

Calculus of the n th -order RCS ( n ≥ 2 – iteration of KA-1) :

Too much complicated

Method of Stationary Phase : main contribution comes from regions around the specular direction

Still too much complicated

Geometric optics approximation (GOA) :  h >  /2 (slight  i ) :

main contribution comes from highly-correlated surface points

+ Hypothesis: uncorrelated surfaces

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 Introduction Theoretical model: one interface Theoretical model: two interfaces Numerical results (2D & 3D) Conclusion & Prospects

Introduction

Theoretical model: one interface

Theoretical model: two interfaces

Numerical results (2D & 3D)

Conclusion & Prospects

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 Exactly the same methodology as for the 2D case Number of numerical integrations: multiplied by 2: 4(n-1) numerical integrations ( rough lower interface) 2(n-1) numerical integrations ( plane lower interface) Increase of the numerical complexity Interests: To quantify the cross polarizations To deal with more general cases (anisotropy) N.B.: No numerical / experimental validation: Complexity of the numerical implementation

Exactly the same methodology as for the 2D case

Number of numerical integrations: multiplied by 2:

4(n-1) numerical integrations ( rough lower interface)

2(n-1) numerical integrations ( plane lower interface)

Increase of the numerical complexity

Interests: To quantify the cross polarizations

To deal with more general cases (anisotropy)

N.B.: No numerical / experimental validation:

Complexity of the numerical implementation

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 Introduction Theoretical model: one interface Theoretical model: two interfaces Numerical results (2D & 3D) Conclusion & Prospects

Introduction

Theoretical model: one interface

Theoretical model: two interfaces

Numerical results (2D & 3D)

Conclusion & Prospects

Conclusions & Prospects A fast approximate method developed (valid in the high-frequency limit): 2D problem: Office PC (1GHz proc., 256Mo RAM): Numerical validation (2D problem) [Déchamps et al., JOSAA, Feb.2006 ] Disposal of a fast method for two strongly rough interfaces (2D problem) Extension of the method to a 3D problem: To deal with more general cases (cross-polarizations) ~ 5s. ( approximate ) ~ 4h10mn (exact, N=50)

A fast approximate method developed (valid in the high-frequency limit):

2D problem: Office PC (1GHz proc., 256Mo RAM):

Numerical validation (2D problem)

[Déchamps et al., JOSAA, Feb.2006 ]

Disposal of a fast method for two strongly rough interfaces (2D problem)

Extension of the method to a 3D problem: To deal with more general cases (cross-polarizations)

Bistatic radar cross section of two-dimensional random rough layers in the high-frequency limit Dr. Nicolas Pinel , Dr. Christophe Bourlier, Pr. Joseph Saillard IREENA Laboratory, Nantes, France E-mail: [email_address]

Bistatic radar cross section of two-dimensional random rough layers in the high-frequency limit

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)