BISTATIC
SCATTERING
FROM
A
SEA-LIKE
ONE-DIMENSIONAL
ROUGH
SURFACE
WITH
THE
PERTURBATION
THEORY
IN
HF-VHF
BAND
Yohann
Brelet,
Christophe
Bourlier
Ecole
Polytechnique
de
l'Universite'
de
Nantes,
IREENA
La
Chantrerie,
Rue
Christian
Pauc,
BP
50609,
44036
NANTES
cedex
3,
FRANCE
ABSTRACT
We
investigate
the
scattering
of
waves
from
a
rough
surface.
From
the
analytical
theory
of
rough
surface
Green's
func-
tion
based
on
the
extension
of
the
diagram
method
of
Bass,
Fuks
and
Ito,
with
the
smoothing
approximation,
numeri-
cal
results
are
presented
for
a
sea
spectrum
and
compared
with
a
benchmark
method
by
considering
a
one-dimensional
perfectly-conducting
Gaussian
rough
surface.
The
effects
of
multiple
scattering
due
to
the
surface
roughness
are
incor-
porated
systematically
into
the
solution
through
an
effective
surface
impedance.
In
addition,
bistatic
scattering
coefficients
are
presented
with
the
first-
and
second-
orders
conventional
small
perturbation
method.
This
study
will
be
useful
for
remote
sensing
of
the
ocean
surface,
especially
when
the
transmitter
is
close
to
the
surface.
Index
Terms-
Electromagnetic
scattering,
rough
sur-
face,
sea-like
surface,
pertubation
theory
1.
INTRODUCTION
The
scattering
of
waves
from
rough
surfaces
has
been
inves-
tigated
using
a
variety
of
asymptotic
mathematical
formu-
lations
[1], [2],
[3].
The
most
popular
are
the
Small
Per-
turbation
Method
(SPM)
and
the
Kirchhoff
Approximation
(KA).
The
Small
Pertubation
Method
exploits
the
smallness
of
the
roughness
amplitude
to
generate
an
expression
for
the
scattered
field.
It
expands
the
phase
term
of
the
exponen-
tial
of
the
field
in
a
power
series
in
surface
heights,
com-
paratively
to
the
wavelength.
More
recently,
to
improve
this
scheme,
the
SPM
has
been
derived
using
Green's
formulation
[2],
[4]
based
on
the
original
published
paper
of
Feinberg
[5].
Comparatively
to
classical
SPM,
the
extended
SPM
allows
to
take
into
account,
via
the
introduction
of
an
effective
surface
impedance
due
to
surface
roughness,
the
multiple
scattering
in
the
first-order
development.
We
propose
to
present
the
ef-
fective
surface
impedances
and
the
reflection
coefficients
ob-
tained
with
the
Extended
Small
Perturbation
Method,
for
a
perfectly
conducting
(H
polarization)
rough
sea
surface
obey-
ing
to
the
Elfouhaily
et
aL
[6]
spectrum,
in
HF-VHF
band.
In
addition,
we
present
the
bistatic
scattering
coefficient
between
the
first-
and
the
second-
orders
conventional
SPM,
compared
with
the
extended
SPM
and
the
benchmark
method
(based
on
the
Method
of
Moments)
for
a
sea
spectrum.
2.
PERTUBATION
THEORY
We
consider
the
scattering
of
scalar
waves
from
a
random
rough
surface
described
by
z
=
((x).
The
geometry
is
il-
lustrated
in
Fig.
1.
The
incident
plane
wave
upon
the
rough
surface
has
a
wave
vector
ko
=
Kx
+
kz
(i)Z.
Source
ro
(xo,zo)
S
.Observation
point
r
(x,z)
c
Fig.
1.
The
rough
surface
is
described
by
z
=
((x).
The
source
point
is
at
ro,
the
observation
point
is
at
r
and
r1
is
on
the
flat
surface
at
z
=
0.
The
surface
is
assumed
to
be
infinite
in
x
direction,
and
invariant
along
y
direction
(1
D
surface).
We
study
and
present
here
the
Dirichlet
boundary
condi-
tion,
which
is
the
horizontal
polarization
for
a
perfectly
con-
ducting
rough
surface.
A
theory
was
fisrt
developed
by
Fein-
berg
[5],
based
on
the
SPM,
under
the
Green's
formulation.
We
give
briefly
a
part
of
this
theory.
We
start
with
the
Green's
function
for
a
given
point
source
located
at
r
=
ro,
satisfying
the
equation
(1)
(V2
+
k2)G(r,
ro)
=-(-
ro)
with
the
following
Dirichlet
boundary
condition
G(r,
ro)
=
°
r
=
r
Z
C
E
Then,
an
equivalent
boundary
condition
at
z
0
is
written
by
expanding
in
a
Taylor
series
the
Green's
function
in
(2)
about
the
surface
height
((x)
and
by
including
the
first
powers
in
Hence,
from
the
Green's
Theorem,
an
integral
equation
for
the
Green's
function
is
obtained
G(r
ro)
-Go0(r
r)+J
/
dri
Go0(r
ri)V(
(
i))G(ri,
ro)
JE=(
(3)
(2)
978-1-4244-2808-3/08/$25.00
C2008
IEEE
IV
-
1137
IGARSS
2008
Authorized licensed use limited to: BEIJING INSTITUTE OF TECHNOLOGY. Downloaded on September 6, 2009 at 07:59 from IEEE Xplore. Restrictions apply.