©dcns 2007 - all rights reserved / tous droits réservés scattering of electromagnetic waves from...

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Scattering of electromagnetic waves from rough surfaces at very low grazing angles.

Ph. SPIGA, North American Radio Science Meeting, URSI 200723/07/2007

M. SAILLARD

G. SORIANO

Ph. SPIGA

2 | 23/07/2007 | North American Radio Science Meeting

CONTENTS

1. Analytical results

2. Numerical model

3. Approximate models

4. The ocean surface


Scattering at low-grazing angles

q0 q

x

z

Surface S: z = f(x,y)

Plane wave

Mean plan (z = 0)

Homogeneous medium

The local perturbation of the plane model

Beam x

Source point or beam around grazing Reality


Fresnel coefficients

Brewster

Dirichlet (r=-1)

Neumann (r=+1)

The Neumann model should not be used for reflexion over finite conductivity materials around and beyond the Brewster angle


Scattering of scalar waves at LGA

Tatarskii, Charnotskii, Barrick, Fuks, Voronovich, Ishimaru

200

ikr

00r

1OE)q,q(S

r

eE,q,E,q

The scattering amplitude S(q,q0)

The backscattering cross section (q)

A

)q,q(S)q(

2

Dirichlet

00D qq)q,q(S 4D q)q(

Neumann

1)q,q(S 0N 1)q(D


Scattering of 3D electromagnetic waves at LGA

Spiga, Soriano, Saillard

rr00 H,EH,E

Perfect Electrical conductor by reflexion on the mean plane

Horizontal (TE) Vertical (TM) 0r0 qEE

1EE r0

ri EEnPi

jM

ˆ

2

10

00

1S1j:V

qSqj:Hd

0d

0

M0 and P0 are the linear integral operator of the boundary integral formalism

ri HHHn̂j

Locally perturbated plane under plane wave illumination

)q,q(S)qq(r)q,q(S 0d

0Fresnel0

Low-grazing incidence angle


Scattering of 3D electromagnetic waves at LGA

Spiga, Soriano, Saillard

Low-grazing scattering angle?

)q,q(S)q,q(S 00 Reciprocity

scattered incident polarization

Polarization

PEC HH

PEC HV

PEC VH

PEC VV

Transmission

)q,q(S 0 )q(

qq0

q

0q

1qq0

2q

4q

4q

2q

1

Conclusion

Fresnel+Reciprocity

PEC only valid in HH at LGA


CONTENTS


2. Numerical model




x

z

x

z

THE GRAZING ISSUE

i becomes very large → grazing incidence:

With present model, the number of unknowns is numerically untractable.

Backscattering energy is very weak at small grazing angles numerical accuracy is required.

i

Numerical solution of the rigorous problem


The Grazing Model

The roughness is a local perturbation of a plane.

i d

x

z


Plane wave

Mean plan (z = 0)

Length of the problem: L

i d

x

z


Plane wave

Mean plan (z = 0)

Length of the problem: L

Rough surface

Transition region

Plateaus ( z = 0 )

x

y

Rough surface

Transition region

Plateaus ( z = 0 )

x

y

ridiff EEEE

The problem is bounded.

A plane wave can be used.

The size of the problem is independent of the incidence

More precise for backscattering.


S+

S-

Not the good behavior at small grazing angles

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Scattering angle (°)

NR

CS

(dB

)

HHS+

S-S+

S-


-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5


NR

CS

(dB

)


-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5


NR

CS

(dB

)

HHS+

S-

The scattered field

ridiff EEEE

Theoretical results

x

z

S+

Extinction theorem S- = 0

Numerical results


-110 -105 -100 -95 -90 -85 -80 -75 -70

-2

0

2

4

6

8

10


u.a.

surface length = 16

surface length = 32surface length = 64

Surface waves

x

zScattered field

Surface Plasmon waves (SPW)


x

zScattered field



SPW radiate in upper and lower half-space as a sinc function.

Surface size is finite


The scattered field

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5


NR

CS

(dB

)

HHSdiff

S+

-80 -60 -40 -20 0 20 40 60 80-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5


NR

CS

(dB

)

HHSdiff

S+

The behavior of the scattered field is enforced, the truncated SPW contribution has been cancelled.

the truncated SPW contribution = S-

Sdiff = S+ - S-


CONTENTS


2. Numerical model




Comparisons with approximate methods on moderately rough surfaces

Geometrical Optics

Physical Optics

Small Perturbation Method : Perturbative expansion of the scattering amplitude Small height, rms< /20

Small Slope Approximation (by Voronovich): Based on the fundamental properties of the scattering amplitude Physical Optics integral with polarization dependence Coincide with SPM at the small height limit Small slope (rms< 0.15 ) + moderate height

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90-50-40-30-20-10

01020

HH20° MoM GO PO

NR

CS



Comparisons with SPM1 and SSA1

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

89°

80°

20°

grazing MoM SPM1 SSA1

NR

CS

(dB

)


Perfect conducting Gaussian surface, rms = /12 and lc = /2


Conclusion and future works

A rigorous model is well adapted for the grazing angle

The number of unknowns is independent of the incidence angle

Deterioration of the results of SPM1 and SSA1 at low grazing backward angles

Failure of SPM1 and SSA1 at low grazing incidence angle for Gaussian surface

Compare other approximate models, more advanced or Low grazing angle oriented (WCA, Ishimaru, …), with the Grazing boundary integral

The dielectric and impedance cases


CONTENTS


2. Numerical model




The high-frequency part of the ocean wave spectrumwave < 15 x EM

0

2

4

6

8

10

12

0 2 4 6 8 10 12fréquence GHz

err

eu

r a

bs

olu

e (

%)

u10= 5 m/s

u10= 10 m/s

u1= 15 m/s

0

2

4

6

8

10

12

0 2 4 6 8 10 12frequence GHz

err

eu

r a

bs

olu

e (

%)

u10= 5 m/s

u10= 10 m/s

u10= 15 m/s

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-90 -70 -50 -30 -10 10 30 50 70 90

MoM

SSA

-80

-70

-60

-50

-40

-30

-20

-10

0

10

-90 -70 -50 -30 -10 10 30 50 70 90

MoM

SSA

80° 89°

Error in backscattering

X band X f=10GHZ =3cm


Two-scale model for ocean scattering

Combining GO for long waves and SPM or SSA for short waves

Surface model

Tilted rough facets

Facets slopes and small-scale roughness are independent

The Shadowing issue

Local shadowing cannot enforce zero limit at grazing

Non-local shadowingshadowing functions (65 Beckmann 67 Wagner 67 Smith 02 Bourlier) gives the probability that a point on a rough surface doesn’t lie in shadow when the surface is illuminated with a parallel beam of given incidence.


The shadowing function

for grazing incident and scattering angles

Non-local shadowing functions (65 Beckmann 67 Wagner 67 Smith 02 Bourlier) gives the probability that a point on a rough surface doesn’t lie in shadow when the surface is illuminated with a parallel beam of given incidence.

The Scattering Cross Section is corrected by a multiplicative factor, the shadow-ing function F(q,q0)


The shadowing function

for grazing incident and scattering angles

The shadow function evaluation may turn very tricky when the correlation function between height and slope is taken into account.

All shadowing functions rely on ray tracing. They may be derived from the second order Kirchhoff approximation for vector waves, with a high-frequency assumption.

The Kirchhoff approximation is mainly designed for near-specular scattering. In that sense, LGA backscattering is obvious out of its validity domain. Note that on the ocean surface, shadowing starts around 80-85°, depending on the wind speed.

All the shadowing functions share the same behaviour at LGA

With as a consequence a supplementary q multiplicative factor over the backscattering cross section

)q,qmin()q,q(F 00

©dcns 2007 - all rights reserved / tous droits réservés scattering of electromagnetic waves from...

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