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Electromagnetic wave scattering from the sea surface in the presence
of wind wave patterns
VICTOR I. SHRIRA{, SERGEI I. BADULIN{ and
ALEXANDER G. VORONOVICH§
{Department of Mathematics, Keele University, Keele ST5 5BG, UK;
e-mail: v.i.shrira@keele.ac.uk
{Nonlinear Wave Processes Laboratory of P. P. Shirshov Institute of
Oceanology, Russian Academy of Sciences, 36 Nakhimovsky prospect,
Moscow 117218, Russia
§Environmental Technology Laboratory of the National Oceanic and
Atmospheric Administration (NOAA), Boulder, 325 Broadway, Boulder CO,
USA
(Received 26 June 2001; in final form 24 October 2002 )
Abstract. The study is concerned with electromagnetic wave (EM) scattering by
a random sea surface in the presence of coherent wave patterns. The coherent
patterns are understood in a broad sense as the existence of certain dynamical
coupling between linear Fourier components of the water wave field. We show
that the presence of weakly nonlinear wave patterns can significantly change the
EM scattering compared to the case of a completely random wave field.
Generalizing the Random Phase Approximation (RPA) we suggest a new
paradigm for EM scattering by a random sea surface.
The specific analysis carried out in the paper synthesizes the small
perturbation method for EM scattering and a weakly nonlinear approach for
wind wave dynamics. By investigating, in detail, two examples of a random sea
surface composed of either Stokes waves or horse-shoe (‘crescent-shaped’)
patterns the mechanism of the pattern effect on scattering is revealed. Each
Fourier harmonic of the scattered EM field is found to be a sum of contributions
due to different combinations of wave field harmonics. Among these ‘partial
scatterings’ there are phase-dependent ones and, therefore, the intensity of the
resulting EM harmonic is sensitive to the phase relations between the wind wave
harmonics. The effect can be interpreted as interference of partial scatterings due
to the co-existence of several phase-related periodic scattering grids. A
straightforward generalization of these results enables us to obtain, for a
given wind wave field and an incident EM field, an a priori estimate of whether
the effects due to the patterns are significant and the commonly used RPA is
inapplicable. When the RPA is inapplicable, we suggest its natural generalization
by re-defining the statistical ensemble for water surface. First, EM scattering by
an ‘elementary’ constituent pattern should be considered. Each such scattering is
affected by the interference because the harmonics comprising the pattern are
dynamically linked. Then, ensemble averaging, which takes into account the
distribution of the pattern parameters (based on the assumption that the phases
between the patterns are random), should be carried out. It is shown that,
generally, this interference does not vanish for any statistical ensemble due to
International Journal of Remote Sensing
ISSN 0143-1161 print/ISSN 1366-5901 online # 2003 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/0143116031000095907
INT
. J. REMOTE SENSING,20DECEMBER, 2003,
VOL. 24, NO. 24, 5075–5093
dynamical coupling between water wave harmonics. The suggested RPA
generalization takes into account weak non-Gaussianity of water wave field in
contrast to the traditional RPA which ignores it.
1. Introduction
Electromagnetic (EM) wave scattering from the sea surface, which is central for
sea remote sensing, has been intensively studied for more than half-a-century (e.g.
Barrick 1972). One of the main difficulties is the fact that the geometry of the
scattering sea surface is unknown a priori. The prevailing paradigm is to assume the
field statistics to be Gaussian, which, for spatially homogeneous fields, implies all
Fourier components to be completely independent and their phases totally random.
The advantages of such an approach, which for a linear wave field would be
a natural hypothesis, are quite obvious. First, it seemed to be based upon a solid
foundation, since the commonly accepted statistical description of evolution of a
wind wave field itself relies upon the same hypothesis and there is experimental
evidence in its favour (Komen et al. 1995). Second, the statistical description of a
wind wave field is enormously simplified: the field is completely characterized by its
second moment – the energy spectrum. As a rule, the spectrum is the only available
characteristic of the wave field and thus the approach is applied out of necessity as
the most plausible hypothesis.
However, there is accumulating evidence that often the Gaussian hypothesis is
inadequate, especially, for low-grazing angles. In particular, a difference in
scattering in both windward and the opposite directions was also registered (Chen
et al. 1993). Ad hoc attempts to ‘correct’ the Gaussianity hypothesis by introducing
higher-order moments of the statistical distributions are reviewed by Gilman (1997),
who simulated scattering from a few special model surfaces. The fundamental
questions of ‘why and when does the scattering differ from the Random Phase
Approximation (RPA) predictions and what is the most adequate way of describing
a wave surface?’ remain open. The present study is aimed at filling this gap by
suggesting a new paradigm enabling one to address these questions in a systematic
way.
The central idea of our study can be outlined as follows. The basic fact we start
from is that even weakly nonlinear interactions among wind waves tend to create
certain coherent patterns with the phases of the constituting Fourier components
obeying certain relations. There are both theoretical models (Shrira et al. 1996,
Annenkov and Shrira 1999) of such patterns and experimental evidence of their
existence (Su 1982, Caulliez and Collard 1999, Collard and Caulliez 1999, Badulin
et al. 2002).
In the present study we focus on identifying the mechanism of potential effect of
the presence of such patterns on EM scattering. We reveal that the mechanism is
the interference known in optics since the times of Newton and Huygens: each
Fourier harmonic of the scattered EM field is a sum of contributions due to
different combinations of wave field harmonics. The particular contributions are
phase dependent and, therefore, the intensity of the resulting EM harmonic is
sensitive to the phase relations between the wind wave harmonics. The phase
relations, in their turn, are determined by the water wave field’s intrinsic nonlinear
dynamics. The systematic weakly nonlinear approach we apply enables us to treat
5076 V. I. Shrira et al.
interactions between both EM and water waves and explicitly relates the inter-
ference of EM waves with the intrinsic water wave nonlinear dynamics.
We show that the effect of the existence of such coherent patterns can
significantly change EM scattering and analyse when and why this occurs.
Although the statistics of the wave field might be close to Gaussian, the effect due
to the presence of the patterns can be strong, since the EM scattering is very
selective – for any particular incident EM wave a relatively small number of Fourier
harmonics of water wave field is involved in the scattering.
The paper is organized as follows. In section 2 we describe the scattering model
we employ to simulate the scattering. We have chosen the widely used Small
Perturbation Method (SPM) (see, e.g. Voronovich 1994a, b) despite its known
limitations (very low-grazing angles, steep and breaking waves). Since our prime
goal in the present work is to understand the specific role played by the coherent
patterns, the use of the simplest possible model has a number of advantages. The
main advantage is that it enables us to present the results regarding the role of
coherent patterns in the most transparent way. It is straightforward to trace the
contributing Fourier components of a wind wave field for each scattered EM
harmonic and to obtain a simple formula for the EM wave intensity. The intensity
is a sum of a number of terms proportional to some correlators of Fourier
harmonics of an instantaneous realization of the water surface. The Fourier
harmonics enter into such correlators with their specific phases prescribed by the
hydrodynamic equations governing the water wave field. There are phase-
independent and phase-dependent correlators. The latter, being ensemble averaged,
are non-zero for a generic wave field. For a wave field containing coherent patterns
these correlators can be of the same order or even exceed the phase-independent
terms which are the only ones retained in the RPA. Apart from the insight into the
nature of the effect, the SPM enables us to get at a very low cost a reasonable
estimate of where the specific contribution due to the coherent patterns is expected
to be important and where a very expensive non-perturbative simulation is indeed,
necessary.
In section 3 we analyse in detail two examples of EM scattering by wave
patterns. Again, to fix the idea, we consider the simplest wave patterns: plane
Stokes wave and three-dimensional crescent-shaped patterns, the so-called horse-
shoe patterns (Su 1982, Shrira et al. 1996). The formulae for the horse-shoe patterns
we use are given in (Shrira et al. 1996, equation A7) and in the Appendix. We
compute EM scattering due to the patterns and compare it to that for the case of
the completely random field (the RPA hypothesis). The scattering proves to be
essentially different due to interference effect. We show that three-dimensionality of
the patterns is essential: it greatly increases the number of possible resonant
combinations and, thus, the number of interfering EM waves. We conclude that the
RPA based upon the linear theory of water waves is inadequate for the problems of
EM scattering, since within the framework of linear theory the existence of any
consistent phase relations between the harmonics is not possible and, thus, the
interference of EM waves cannot occur. As a more promising approach, for
preserving the interference and representing a natural generalization of the RPA,
we suggest describing the water surface as an ensemble of weakly interacting
nonlinear wave patterns.
The questions regarding the existence of such patterns in the true wind wave
EM scattering in the presence of wind wave patterns 5077
field and the possible methods of distinguishing them in experimental data are
addressed in the complementary part of the present study (Badulin et al. 2002).
The implications of the obtained results for remote sensing are briefly discussed
in the final section.
2. Scattering model
Consider an incident EM field represented by an EM plane wave of a certain
polarization:
W
in
~q
{1
=
2
0
exp ik
0
:
rziq
0
zðÞ: ð1Þ
Here r is the horizontal projection of a 3D radius-vector R and z is its vertical
component (z-axis is directed downward). Correspondingly, k
0
is the horizontal
projection of the wave vector K of the incident wave and q
0
q
0
~ K
jj
2
{ k
0
jj
2
1=2
,Imq
0
¢0 ð2Þ
is its vertical component. The scattering sea surface is assumed to be given and
characterized by elevation z~h(r). The scattered EM field beyond excursions
zvmin h(r) can be represented as a superposition of plane outgoing waves
W
sc
~
ð
S k, k
0
ðÞ
:
q
{1=2
k
exp ikr{iq
k
zðÞdk: ð3Þ
Here S(k, k
0
) is the scattering amplitude (SA) representing the amplitude of the
process of scattering of a plane wave with horizontal component of the wave vector
equal to k
0
into a plane wave with horizontal component of a wave vector equal to
k. Normalization factors q
{1=2
0
, q
{1=2
k
in equations (1) and (3) are introduced to
make the resulting formulae more symmetric.
To calculate SA we use the method of small perturbations (SPM) which is valid
for water waves of sufficiently small heights and slopes. (SPM is an asymptotic
method based on two small parameters. The first one is the steepness of the wave
surface. The use of small wave slope expansion requires smallness of the water wave
slopes compared to the grazing angle, +h
jj
% q
k
jj=
k
jj
. The second small parameter
originating from calculations of EM reflection implies that the vertical scale of an
EM wave is large compared to the surface elevations, q
k
jjh % 1; For more detail
regarding the applicability of the method see, e.g. Voronovich (1994a, b).) To the
second order the corresponding expression for SA is as follows:
S k, k
0
ðÞ~B k
0
, k
0
ðÞd k{k
0
ðÞz2i q
k
q
0
ðÞ
1
=
2
B k, k
0
ðÞh k{k
0
ðÞ
z q
k
q
0
ðÞ
1
=
2
ð
B
2
k, k
0
; jðÞh k{jðÞh j{k
0
ðÞdj:
ð4Þ
Here h(k) represents the Fourier transform of the roughness (elevation) profile
z~h(r):
h rðÞ~
ð
h kðÞexp ikrÞdk:ð5Þð
The first term in equation (4) represents a specular reflected wave with reflection
coefficient V
0
~B(k
0
, k
0
), the second term describes the Bragg scattering and the
third term corresponds to the second-order correction. The functions B (k, k
0
) and
B
2
(k, k
0
; j ) do not depend on roughness but depend on polarization of the incident
(5)
5078 V. I. Shrira et al.
and scattered waves and on the dielectric constant of water e
w
, which is generally
complex and depends on EM wavelength, salinity of the water and, possibly, some
other factors. The explicit expressions for functions B, B
2
can be found in
Voronovich (1994a). Polarization indices are omitted throughout the paper.
For the case of periodic roughness consisting of N
0
harmonics
h kðÞ~
X
N
0
n~{N
0
h
n
d k{p
n
ðÞ ð6Þ
with h
n
~h
{n
and p
n
~2p
2n
, substitution of equation (6) into equation (4) yields:
S k, k
0
ðÞ~B k
0
, k
0
ðÞd k{k
0
ðÞz
X
N
0
n~{N
0
2i q
k
q
0
ðÞ
1
=
2
B k, k
0
ðÞh
n
d k{k
0
{p
n
ðÞ
z
X
N
0
n,m~{N
0
q
k
q
0
ðÞ
1
=
2
B
2
k, k
0
; k{p
n
ðÞh
n
h
m
d k{k
0
{p
n
{p
m
ðÞ:
ð7Þ
Hence, in this case the scattered field consists of a set of discrete Fourier harmonics
corresponding to plane outgoing waves:
S k, k
0
ðÞ~
X
j
S
j
d k{k
j
, ð8Þ
with wave vector directions determined by d-functions in equation (7). Due to the
third term in equation (7) different water wave combinations (elementary scattering
processes) can contribute to the same partial scattering amplitude S
j
corresponding
to EM wavevector k
j
S
j
~
X
i
S
i
j
, ð9Þ
where S
i
j
corresponds to all possible combinations of water wave harmonics (upper
indices) that contribute to the partial reflection wave with the wavevector k
j
.
Note that the amplitudes of the harmonics h
n
are complex, and, therefore, are
specified by real amplitudes h
n
jj
and phases Q
n
:
h
n
~ h
n
jj
exp iQ
n
ðÞ, Q
n
~{Q
{n
: ð10Þ
Equation (4) can be viewed as a particular case of the standard asymptotic
procedure for wave problems, where all but the resonant terms are ignored. The
resonant conditions in our case include both space and time resonance. The
fulfillment of spatial resonances is ensured by d-functions in equations (4) and (7),
which yields
k~k
0
zp
n
; and k~k
0
zp
n
zp
m
: ð11Þ
Making use of the fact that the frequencies of the EM waves far exceed water wave
frequencies we, for simplicity, neglect the latter in the condition of temporal
(frequency) resonance, that is equivalent to the assumption of the time independent
roughness, which yields
K
jj
~ K
0
jj
: ðÞ
Although interesting Doppler effects in the scattered EM field are totally ignored
within this approximation, these effects lie beyond the scope of the present study
and will be considered elsewhere. Consider now the result of averaging the EM
EM scattering in the presence of wind wave patterns 5079
wave intensities with respect to a statistical ensemble: I
j
~S S
j
2
T. Generally, the
result of such an averaging is a sum of ‘elementary’ intensities S
i
j
2
(see
equation (9)) and all possible ‘cross’ correlators SS
i
j
S
k
j
T. The procedure of the above
averaging is literally the same as in the well-known problem of interference of light.
The interference terms in our case are expressed in terms of triple and quartic
correlators
Sh
n1
h
n2
h
n3
T~ h
n1
jj
h
n2
jj
h
n3
jj
Sexp i Q
n1
zQ
n2
zQ
n3
ðÞ½T ð12Þ
and
Sh
n1
h
n2
h
n3
h
n4
T~ h
n1
jjh
n2
jjh
n3
jjh
n4
jjSexp i Q
n1
zQ
n2
zQ
n3
zQ
n4
ðÞ½T: ð13Þ
For simplicity only, we applied a commonly used hypothesis that the wave phases
are ‘more random’ than the amplitudes. We stress that our further consideration
does not rely upon this simplification. Provided the information on the water wave
field probability distribution is complete, for example, when the phase functions Q
i
are determined by the corresponding solutions of dynamical equations, the
correlators in equations (12) and (13) can be easily calculated. Generally, these
correlators do not disappear in the nonlinear problem. However, the acceptance of
the commonly used random phase hypothesis for a wind wave field, i.e. assuming
all the phase functions to be random and statistically independent, leads to a drastic
simplification. In this approximation most of the correlators (12) and (13)
(interference terms) vanish in the expression for average SA and only the
‘elementary’ or ‘trivial’ non-zero correlators
Sh
n1
h
{n1
h
n2
h
{n2
T~ h
n1
jj
2
h
n2
jj
2
ð14Þ
remain in this expression.
Thus, EM scattering by a wave field containing coherent wave patterns and
completely random waves is essentially distinct due to the interference effects
corresponding to non-trivial combinations of water wave harmonics with non-
random phase relations.
We emphasize that the phase relations in the correlators (12) and (13), being
similar in their physical essence to the ‘optical length difference’ in classical
diffraction, are specified by the phase relations in the water wave field. This
indicates the necessity of serious consideration of the link between the scattering
characteristics and the intrinsic dynamics of a water wave field.
The ‘nontrivial’ correlators (12) and (13) can also be considered from a more
‘descriptive’ viewpoint. The correlators are obviously related to bi- and three-
spectra of the wave field (Leykin 1995) and their relations certainly merit a study of
their own. However, now we confine ourselves to the following brief comment: the
correlators we consider are due to the resonances between EM and surface waves
and, therefore, the manifold of such correlators is richer than the commonly
considered resonant combinations of surface wave harmonics.
There is a fundamental implication of the existence of the non-zero nontrivial
correlators and the way they enter into the expression for the scattering amplitude.
Due to the interference each Fourier harmonic of the scattered EM field is
a sum of contributions proportional to different correlators. However, these
partial contributions depend not only on the correlators but also on the magnitude
of the kernels B, B
2
. These kernels in their turn are specified by the specific
5080 V. I. Shrira et al.
geometry of the selected wavevectors. This implies that EM scattering from any
non-Gaussian wave field cannot be fully described in terms of higher-order
moments of the wave field, which, in particular, explains the results by Gilman
(1997).
3. Scattering by the simplest water wave patterns
In this section we analyse EM scattering by the simplest nonlinear wave
patterns.
Since we employ a perturbation method to describe the EM scattering, it is
natural to apply a similar perturbation technique to the water wave dynamics as
well. This enables us to preserve Fourier representation as the universal way of
description for both dynamical and statistical approaches. One may view the
resulting combination of two asymptotic procedures as a natural generalization of
the theory of wave resonant interactions for the case of embracing waves of
different nature. The applicability of such generalized theory is ensured
independently in water wave and EM constituent parts by the relevant independent
small parameters. Further on, we employ a simplified way of ensuring the
applicability of the method: for each value of the surface waves steepness (e)we
control the conservation of energy of scattered EM waves.
3.1. Stokes wave
3.1.1. Scattering by a Stokes wave
The Stokes wave can be viewed as the simplest example of water wave patterns.
For our purposes it is sufficient to confine our consideration to the approximate
solution for the water surface elevation g(x, t) accurate to e
3
g xðÞ~{A cos hðÞz
1
2
k
jj
A
2
cos 2hðÞ{
3
8
k
jj
2
A
3
cos 3hðÞzh A
4
, ð15Þ
where H~kx{VtzH
0
is the wave phase. The solution is a superposition
of Fourier harmonics with wavevectors and frequencies multiple of k and
V~
ffiffiffiffiffiffiffiffi
g k
jj
p
ð1z kAðÞ
2
=2Þ. In our context, their main distinctive feature is that the
amplitudes and phases are tightly related and, therefore, can be expressed in terms
of a single amplitude and phase, say, of the first harmonic. The most natural setting
of the problem would be to consider scattering of a given EM wave by an ensemble
of random Stokes waves with all their parameters, i.e. amplitude, direction of
propagation, wavelength, initial phase, being random and obeying certain a priori
given probability distributions.
First, however, we study, in detail, scattering by an elementary constituent of
such an ensemble, i.e. by a single deterministic Stokes wave. Because of specific
degeneracy with respect to the initial phase H
0
the results are the same if we
consider an ensemble of identical Stokes waves differing only in randomly
distributed initial phase H
0
. To understand the role of the interference, consider in
parallel a ‘surrogate field’ with the same Fourier amplitudes but randomized phases
H. Although such randomization of phases is, strictly speaking, incorrect, since any
combination of the same three harmonics with the phases distinct from
equation (15) is not a solution of the hydrodynamic equations, we simulate
scattering within the RPA to provide a reference point and, thus, to reveal the role
of the interference.
EM scattering in the presence of wind wave patterns 5081
The EM scattering can be naturally viewed as a superposition of the
‘elementary’ scattering processes characterized by the partial amplitudes S
j
in
equation (8). Each elementary process preserves the EM frequency in the adopted
approximation and, hence, modulo of the wavevector, while the horizontal
projection of the wavevector is determined by the resonant conditions (equa-
tion (11)). The interference phenomenon is rooted in the fact that each partial
amplitude characterizing a particular reflected EM wave results not necessarily
from one, but from several possible combinations of water wave harmonics.
The fact that several distinct combinations of water wave harmonics can contribute
to the same reflected EM wave and, thus, into the same partial scattering amplitude, is
central to our study. We analyse the corresponding effects by simulating EM scattering
from the Stokes wave. In the computations we set the wavelength of the water wave l
w
to be unity and the steepness A k
jj
~e to be e~0.1, while for the incident EM wave we
choose: the wavelength l
e
~0.3, l
w
~2p
=
K
jj
and the grazing angle 10‡.
Figure 1(a) presents the K-plane diagram of EM scattering by the Stokes wave
for the chosen set of parameters. The incident EM wave is depicted by the bold
arrow, the wavevectors of the scattered EM waves are shown by outgoing arrows.
The scattering at hand includes the first-order Bragg scattering, which occurs due to
resonant interaction with a single surface wave harmonic, and the second-order
Bragg scattering due to resonant interaction with the appropriate pair of surface
wave harmonics. Since the frequencies of EM waves far exceed the frequencies of
Figure 1. EM wave scattering by a plane Stokes wave. (a) Wavevectors of incident (bold
arrow) and reflected EM waves are given in units of the wavenumber of the plane
water wave under consideration. Reflected EM waves satisfy resonant conditions
(equation (11)) for incident EM wave and different combinations of the nonlinear
water wave harmonics. (b) Only wavevector combinations corresponding to different
resonances contributing to the same reflected EM wave are shown.
5082 V. I. Shrira et al.
the surface waves, the wavevectors of the scattered EM waves lie near the circle
K
jj
~ K
0
jj
. Each such scattered EM wave is a result of summation of several
elementary scattering processes; we refer to the result of this summation as
‘partial scattering’. For the partial scatterings labelled ‘A’, ‘B’, ‘C’ and ‘D’,
figure 1(b) sketches the possible combinations of the Stokes wave harmonics
contributing to the corresponding elementary processes. As discussed above (see
equation (4)) each elementary scattering process is first of all characterized by the
order of scattering (in our example the options are confined to the first- and the
second-order Bragg scattering). Second, the amplitudes of surface wave harmonics
causing the particular elementary processes differ considerably in magnitude
roughly characterized by the order in wave steepness e, in which they occur. In
figure 1(b) we illustrate this fact, apart from depicting vector diagrams for each
elementary process, by showing the orders e
n
above the corresponding element of
the diagram as well.
Figure 2 illustrates the role of the interference by showing the scattering
intensities resulting from several elementary processes and normalized by the
corresponding intensities of the surrogate field (computed in the RPA). The
importance of the effects for different EM waves varies enormously and, of course,
it is of interest to discuss the reasons behind such wide variations for the different
combinations of harmonics. To clarify the question we consider a few typical
examples (cases ‘A’, ‘B’, ‘C’, ‘D’) in detail. The diagrams of figure 1(b) provide us
with a very handy tool to consider the effects due to interplay of different
independent parameters and explain the results presented in figure 2.
In case ‘A’ the scattering is due to superposition of two elementary second-order
Bragg scatterings: the combination of the first and the third harmonics on the one
hand, and the combination of the two second harmonics on the other (see
figure 1(b)). The diagram also shows that from the viewpoint of the order in e, the
Figure 2. The ratio R of intensities I
St
(deterministic Stokes wave) and I
RPA
(Random Phase
Approximation for a set of harmonics representing the Stokes wave), R~I
St
/I
RPA
vs
horizontal projection of EM wavevector. Empty square denotes horizontal
polarization; inverted empty triangle indicates vertical polarization of incident
EM wave. Letters and numbers in brackets correspond to water wave harmonic
combinations shown in figure 1.
EM scattering in the presence of wind wave patterns 5083
scattering by two second harmonics of the Stokes wave yields e
2
6e
2
~e
4
, while the
scattering by the combination of the first and the third harmonics leads to the same
outcome (e
1
6e
3
~e
4
). Thus, in this case the elementary processes have the same
order in e and occur in the same order in EM scattering, which is the reason for the
strong interference effect shown in figure 2. Indeed,
I
A
~S S
A
jj
2
T~q
0
q
A
B
2ðÞ
2
S h
2
jj
4
Tz B
1ðÞ
zB
3ðÞ
2
S h
1
jj
2
h
3
jj
2
T
n
z2Re B
2ðÞ
B
1ðÞ
zB
3ðÞ
Sh
2
2
h
1
h
3
T
hio
:
ð16Þ
Here the notation for scattering coefficients corresponds to numbering of the
Stokes wave harmonics, that is, B
ð1Þ
~B
2
(k
A
, k
0
; k
A
2k), B
(2)
~B
2
(k
A
, k
0
; k
A
22k)
etc. It is easy to see that all correlators in this formula, both ‘trivial’ ones and the
‘non-trivial’ cross-term, which appears in the second line of the formula, have the
same orders in e and in the order of EM scattering. The effect of interference is
pronounced in this case because of comparable magnitude of the interfering
elementary scatterings. Note that the relative intensity of the partial scattering does
not depend on wave amplitude
R~I
St
=I
RPA
~1z
6Re B
2ðÞ
B
1ðÞ
zB
3ðÞ
hi
4 B
2ðÞ
2
z9 B
1ðÞ
zB
3ðÞ
2
: ð17Þ
In case ‘B’ the interference effect is weak although the orders of asymptotic
approximation in e are the same: combination of the first and the second Stokes’
harmonics yields the same e
3
as scattering due to the third Stokes’ harmonics.
However, because the competing elementary processes occur at different orders in
EM scattering (the second and the first Bragg, respectively), the term due to the first
Bragg dominates, thus making the interference effect insignificant.
In case ‘C’ the explanation of the weakness of the interference effect is similar to
the previous example. Two combinations of the surface harmonics (see diagram in
figure 1(b), case ‘C’, the upper and middle combinations) have the same order in e,
but they occur in the different orders in EM scattering: again we encounter a
competition between the first and the second Bragg.
Sometimes the qualitative analysis of the interference provided by the diagrams
of figure 1(b) requires additional quantitative consideration of the corresponding
kernels. The leading order terms might vanish for certain combinations of incident
and scattered waves. Case ‘D’ is aimed to illustrate this point. In this case we have
different orders both in amplitude of surface waves and in the order of EM
scattering and, normally, a single dominant process should prevail. However
figure 2 shows that there is a strong interference effect for horizontal polarization.
The peculiarity of this example lies in the fact that the first-order Bragg scattering is
anomalously small for horizontal polarization and, thus, becomes comparable to
asymptotically smaller terms. However, the apparently strong interference effect
should be considered with great caution since the SPM asymptotic expansion
breaks down in this case and a more careful analysis is needed.
3.1.2. On the Stokes wave ensemble
The analysis of EM scattering by the simplest model of wave patterns carried above
might create a false impression that we are comparing scattering from a completely
5084 V. I. Shrira et al.
random Gaussian wave field with scattering by a completely deterministic wave field.
However, we recall that the considered Stokes wave field may be viewed as an ensemble
of Stokes waves with randomly distributed initial phase H
0
– although the result would
not have been different if we had taken the deterministic limit of this ensemble
corresponding to a fixed initial phase H
0
. The main point is that the above analysis of
an elementary block of a generic random Stokes wave field enables us to draw certain
important conclusions regarding the scattering by a random Stokes wave field with
arbitrary distribution of the wave parameters. The fundamental fact we want to
emphasize is that the interference occurs within the scattering by each elementary block.
The implications are as follows. Whatever combination of the elementary blocks
is taken the interference effects cannot increase. We can expect interference effects
only for those scattered EM waves which have exhibited interference effects for
one of the elementary blocks, i.e. in our case for any of the Stokes waves constituting
the ensemble. The general recipe of handling such generic ensembles is quite
straightforward: the intensities of scattered EM waves found in our analysis for the
simplest ensemble of Stokes waves with a given set of parameters should be taken
as summands with the appropriate weight corresponding to the probability of the
particular set of the wave parameters. In fact, the suggested procedure is very similar
to the commonly used RPA approach with the principal difference lying in the
fact that our elementary blocs are nonlinear solutions of the system, while the RPA
operates with linear solutions. In this sense we can speak about a direct generalization
of the RPA.
The following observation is of independent interest. Consider an ensemble of
random unidirectional Stokes waves of the same wavelength differing only in the
initial phase and amplitude. In this context the factorization of the interfering part
of the scattered EM field, which is proportional to e
4
for the second Bragg, becomes
of prime importance. Hence, the relative importance of the interference remains the
same whatever the distribution of the amplitude. This observation holds for a much
wider class of situations as well, although an additional analysis is needed in each
particular case.
Although the above simplest model is sufficient to demonstrate the idea,
it is obvious, that for a 3D wave field the interplay of parameters is much richer
and, therefore, the interference is more important. We illustrate this point
by analysing the simplest example of the 3D wave patterns – the ‘horse-shoe’
patterns.
3.2. Scattering by the horse-shoe patterns
3.2.1. Horse-shoe patterns
The solutions describing the horse-shoe patterns were derived in Shrira et al.
(1996) as exact solutions of the so-called five-wave Zakharov equation and, thus,
they are asymptotic solutions of order e
4
to the exact hydrodynamic equations. In
our context, the following feature of these solutions is of principal importance: all
the constituent harmonics are corresponding combinations of just three indepen-
dent wave vectors. This fact ensures co-existence of different elementary scattering
processes contributing to the same partial scattering amplitude and, thus, the
interference. Since the number of harmonics in these 3D patterns far exceeds the
number of harmonics in the Stokes wave, the manifestations of the interference are
much richer in this case.
EM scattering in the presence of wind wave patterns 5085
There is a whole family of solutions describing the horse-shoe patterns which
is discussed briefly in the Appendix. In the linear approximation these solutions
are a superposition of one central windward-directed harmonic and two oblique
symmetric satellites.
Consider a one-parameter subset of such solutions having fixed amplitudes of
all harmonics and differing only in the phase between the central harmonic and the
satellites.
Figure 3 shows two examples of the nonlinear wave surfaces for two different
values of the phase difference y~p/4 and y~0 between the central harmonic and
the satellites, while the corresponding formulae derived in Shrira et al. (1996) are
given in the Appendix. The local maxima of horse-shoe patterns are spaced in a
chess-like manner. For the phase difference y~p/4 the nonlinear terms provide an
asymmetry in the vertical direction (the crests become sharper while the troughs
become shallower) and front–back asymmetry (the fronts become steeper), while for
y~0 there is no front–back asymmetry. In reality, the patterns with y close to
y~p/4 are the prevailing ones, and sometimes we refer to such patterns as the ‘true’
horse-shoes, although the occurrence of patterns with other values of y is not
completely ruled out (Annenkov & Shrira 1999).
3.2.2. Scattering by different horse-shoe patterns
An important difference between the horse-shoe patterns and the earlier
considered Stokes waves that we would like to emphasize is that in contrast to the
quite ‘rigidly’ fixed Stokes waves with only one essential parameter (the amplitude)
which can vary in a random field, the horse-shoe patterns provide freedom to vary
for a number of their parameters apart from the amplitude of the main harmonic
(amplitudes of the satellites, the phase y).
Figure 3. Wave forms for ‘horse-shoe’ pattern at y~p/4 and y~0. The amplitudes of the
central harmonic and the symmetric satellites are 0.2, and 0.02, respectively.
5086 V. I. Shrira et al.
Before proceeding to a discussion of scattering by ensembles of random patterns
it is helpful, as a first step, to investigate scattering by different horse-shoe patterns
belonging to the same family.
Figure 4 shows EM scattering from the two sample patterns of figure 3, and it is
easy to see that the manifestation of the interference, although essential in both
cases, is noticeably different. Moreover, although the pronounced interference
effect in accordance with section 3.1.2 occurs for the same scattered harmonics, now
it might be of different signs. We discuss the potential implications of this fact in
section 3.3.
In order to illustrate the importance and specific features of the interference
between the elementary scattering processes we carry out simulations of the EM
scattering by the wave surfaces described by the same asymptotic formulae but
truncated at different orders in e. Figure 5 shows the normalized intensities of EM
scattering by the ‘true’ horse-shoe patterns. Again the normalization was carried
out by using the corresponding amplitudes of the surrogate field. The grey bars
correspond to the second (quadratic in e) approximation, the black ones correspond
to the cubic approximation. The significant discrepancy is easy to explain: there
are 20 harmonics in the quadratic approximation, while the number jumps to 74
(including some identical to those from the lower approximation) in the cubic
approximation. Correspondingly, the number of elementary scattering processes
Figure 4. Scattering from two samples of the horse-shoe patterns shown in figure 3. The
ratio R of intensities I
HS
(deterministic horse-shoe pattern) and I
RPA
(Random Phase
Approximation for the set of the pattern’s harmonics), R~I
HS
/I
RPA
vs horizontal
projection of EM wavevector: (a) y~p/4, i.e. the phase shift corresponds to the ‘true’
horse-shoe patterns; (b) y~0.
EM scattering in the presence of wind wave patterns 5087
increases due to involvement of the ‘new’ harmonics of the cubic approximation.
This enhances the interference and thus explains the increase of the discrepancy
with the surrogate field.
It is worth noting that this discrepancy is most pronounced for the directions
of scattering close to the backscattering. Namely for these directions there is the
richest variety of the elementary scattering processes and, moreover, these processes
are most likely to occur in the same asymptotic order, thus ensuring the significant
effect of their interplay.
3.3. Re-definition of the wind wave statistical ensemble
The horse-shoe solutions contain a parameter specifying the phase shift y
between the central and the satellite harmonics (see equation (A1) of the Appendix).
The shape of the patterns depends dramatically on the value of this parameter. The
theoretical analysis (Shrira et al. 1996, Annenkov and Shrira 1999), supported by
the available experimental evidence (Su 1982, Caulliez and Collard 1999), predicts
the value y~p/4 as the most probable one.
The sensitivity of EM scattering to this parameter is illustrated by figure 4.
Obviously any real statistical ensemble of wave patterns is characterized by a
certain distribution of the pattern parameters such as the amplitudes of the central
harmonic and the satellites, the phase shifts y, the wavelengths, etc. The question
regarding scattering by such a field can be answered in precisely the same way as for
the case of the ensemble of Stokes’ waves. The recipe for handling ensembles of
patterns remains the same. First, we have to calculate scattering by each constituent
pattern and then to carry out ensemble averaging assuming all patterns evolving in
an incoherent way with respect to each other.
By examining scattering from two patterns with different phases we found that
the interference effect could have different signs. In view of this observation the
fundamental question is whether an arbitrary ensemble preserves the interference
effect. The question can be reformulated in a different form as follows: is it possible
Figure 5. The ratio R of intensities I
HS
and I
RPA
for different approximations of the pattern
(notation is the same as in figure 4). Grey bars correspond to the asymptotic solutions
of the second order in steepness e, black ones to the third-order solutions.
5088 V. I. Shrira et al.
to construct an ensemble made of horse-shoe patterns in which the interference
effect vanishes?
To address this point we consider the following example. Denote the master
wavevectors of horse-shoe solutions as in Appendix equation (A2): a is the central
and b
1
and b
2
are the satellite wavevectors. These master harmonics generate a
sequence of slave harmonics in accordance with the adopted weakly nonlinear
approach (Shrira et al. 1996). Consider a scattered harmonic resulting from two
different combinations of water wave harmonics (we confine our consideration to
quadratic in amplitude terms for water wave solutions).
1. The combination of the central master harmonic c
1
~a and one slave
harmonic c
2
produced by the central master harmonic and one of the two
satellites c
2
~azb
1
;
2. The combination of the satellite master c
3
~b
1
and the second central
harmonic c
4
~2a.
The corresponding partial scattering amplitude can be calculated straightfor-
wardly:
S
A
S
A
~q
A
q
0
B
1ðÞ
zB
2ðÞ
2
h
1
jj
2
h
2
jj
2
z B
3ðÞ
zB
4ðÞ
2
h
3
jj
2
h
4
jj
2
n
z2Re B
1ðÞ
zB
2ðÞ
B
1ðÞ
zB
2ðÞ
h
1
h
2
h
3
h
4
:
ð18Þ
The indices of the amplitude harmonics and of scattering coefficients correspond to
the harmonics c
i
(i~1...4) listed above. It is easy to see that in this case the partial
scattering intensity consists of the same order terms. The two first terms in brackets,
evidently, do not depend on the relative phases of the harmonics, they correspond
to the elementary scattering intensities. Moreover, the two last terms which are
related to the interference are phase independent as well! Remarkably, they do not
depend on the phase parameter y which characterizes horse-shoe pattern geometry.
This contribution due to interference can be easily found by utilizing the formulae
(A1) given in the Appendix. Quite similar to the case of the Stokes wave
(equation (16)), the relative contribution due to interference is determined
completely by scattering coefficients and by the amplitudes of water wave slave
harmonics.
The particular case of scattering we consider is rather special: the relative
contribution due to the interference does not depend on the phase parameter y of
the wave patterns and their amplitude characteristics. This results from our special
choice of elementary scattering processes and special phase relations of water wave
harmonics involved in the processes. Nevertheless, the mere existence of such
special cases implies that the effect of interference cannot be ignored for any choice
of statistical ensemble of horse-shoe patterns.
The above example illustrates a rather simple but very important point: in the
presence of patterns, any ensemble averaging should necessarily take into account
the pattern distribution, since the effect of interference never disappears.
4. Discussion
We have shown that EM wave scattering in the presence of coherent wave
patterns is indeed very different from that in the case of a completely random wave
field due to the effect of interference. This raises the question about the range of
EM scattering in the presence of wind wave patterns 5089
applicability of the commonly used RPA which assumes independence and,
therefore, complete randomness of the phases of linear modes (i.e. linear Fourier
harmonics). The approach we propose is a natural generalization of the RPA. We
view the wave field as an ensemble of nonlinear patterns, which are considered to
be independent, i.e. un-coupled in the first approximation. Similarly to traditional
RPA, the phases of these nonlinear modes are assumed to be random, however, this
does not imply the complete randomness of the phases of linear Fourier harmonics.
The use of the RPA requires the field to be strictly Gaussian, while our approach
implies a certain non-Gaussianity of the field statistics. Now we briefly discuss what
the implications are for the real wind-wave field on the basis of the analysis we have
carried out.
The wave patterns we examined were examples of steady solutions to
hydrodynamic equations. The effect of the interference we demonstrated stems
from the fact that the harmonics constituting the patterns necessarily have certain
dynamically linked phases prescribed by the hydrodynamic equations. The first step
in generalizing these results for more general random wave fields is quite
straightforward. While we considered ensembles of the nonlinear patterns of the
same kind, the ensembles made of different species of nonlinear patterns can be
treated similarly. We would like to stress that for interference to occur, the
fundamental patterns in the wave field do not need to be steady and the phases
between linear harmonics constituting the patterns be necessarily fixed, it is
sufficient if some dynamic links between the harmonics exist. The simplest example
of such kind is described by equation (A1) with time-dependent amplitudes a, b,
phase y and ‘nonlinear’ frequencies V. Then the phase relationships between the
harmonic constituents, although no longer fixed, are not random, being prescribed
by the dynamic equations (A3). Thus, the effect of interference can occur in a much
wider class of situations compared to a few strongly idealized ones we specifically
analysed. Moreover, we claim that the interference is always present even in the
absence of visible patterns. However, this point will be elaborated elsewhere. We
stress, that we are not challenging the notion that the overall wind wave field
statistics is quasi-Gaussian; however, since the EM scattering is an extremely
selective process, for a given incident EM wave a very few harmonics are involved,
and the presence/absence of deterministic phase relationships between those few
selected harmonics is the only thing which matters for the interference. In our
recent work (Badulin et al. 2002) we show that such deterministic relationships are
indeed present in real wind wave field data.
The analysis of the simplest patterns also enables us to specify the circumstances
when the effect of interference due to the presence of nonlinear wave patterns is
expected to be of importance for EM scattering. First, note the primary necessary
condition: the scales of the EM and wind waves under consideration should be
comparable. Second, since the interference contribution is due to the second and
higher-order Bragg scattering, only the situations where the second- (or higher-)
order Bragg scattering is of independent interest should be considered, e.g.
analysing Doppler spectra of HF radars where the first- and the second-order Bragg
signals are widely separated in frequency domain and the second-order Bragg
spectra are used to reconstruct wind wave spectra (Wyatt et al. 1999). For purely
geometric reasons the effect is the strongest for the backscattered EM waves (and
close directions), for pronounced three-dimensional patterns and grazing angles of
5090 V. I. Shrira et al.
incidence. The effect sharply increases with the increase of the amplitudes of the
constituent harmonics, and for the low-grazing angles. The small perturbation
method we used in the present study to fix the idea ceases to be applicable for
strongly nonlinear waves and very low-grazing angles, while the effect of the
interference has no reason to disappear. In particular, within our paradigm, wave
breaking can be viewed as an example of a strongly nonlinear spatio-temporal
pattern. However, to describe it adequately in the strongly nonlinear setting is a
problem beyond the scope of the present study.
Appendix A: Horse-shoe solutions
A rich family of approximate solutions to the exact water wave equations can be
found as exact solutions of the Zakharov equation (Zakharov 1968) following the
recipe given by Badulin et al. (1995). Solutions of the Zakharov equation in the
form of a set of d-functions in terms of the canonical Zakharov variables describe a
variety of steady and unsteady wave patterns in physical variables. In particular, a
sum of just three d-functions in terms of the canonical Zakharov’s variables
~
bb k, tðÞ
provides solutions describing the family of horse-shoe patterns
~
bbtðÞ~atðÞd k{aðÞexp {iV
b
t{yðÞ
zbtðÞd k{b
1
ðÞzd k{b
2
ðÞðÞexp {iV
b
tðÞ:
ðA1Þ
where wavevectors of the harmonics satisfy resonant conditions
3a&b
1
zb
2
3v
a
&2v
b
:
ðA2Þ
Time-dependent real amplitudes a(t), b(t) and phase y are governed by the
following ordinary differential equations (Shrira et al. 1996)
_
aa~3Wa
2
b
2
sin y
_
bb~{Wa
3
b sin y
_
yy~3V
a
{2V
b
zW 9ab
2
{2a
3
cos y:
ðA3Þ
with ‘Stokes-shifted’ (‘nonlinear’) frequencies
V
a
~v
a
zV
aaaa
a
2
z4V
abab
b
2
V
b
~v
b
z2V
abab
a
2
zV
bbbb
b
2
z2V
bcbc
b
2
:
ðA4Þ
The coefficients in the above equations are the corresponding kernels of the
Zakharov equation: W~W
ð2Þ
(k
1
, k
2
, k
3
, k
3
, k
3
)andV. Their explicit expressions
are given in Krasitskii (1994) and Shrira et al. (1996). The above canonical
presentation of nonlinear solutions eliminates all ‘slave’ harmonics which reappear
after transformation to the original in physical variables. The transformation can
be expressed in terms of power expansion in the canonical variables. In the first
order (linear approximation) one gets for surface elevation
pg xðÞ
1
~aM aðÞcos ax{V
a
t{yðÞzbM b
1
ðÞcos b
1
x{V
b
tðÞ½
z cos b
2
x{V
b
tðÞ:
ðA5Þ
EM scattering in the presence of wind wave patterns 5091
The second approximation (order e
2
) is quadratic in the component amplitudes
pg xðÞ
2
~b
2
A
1ðÞ
2b
1
, b
1
, b
1
ðÞM 2b
1
ðÞ
| cos 2 b
1
x{V
b
tðÞðÞz cos 2 b
2
x{V
b
tðÞðÞ½
za
2
M 2aðÞA
1ðÞ
2a, a, aðÞcos 2 ax{V
a
t{yðÞðÞ
z2b
2
M b
1
zb
2
ðÞA
1ðÞ
b
1
zb
2
, b
1
, b
2
ðÞcos b
1
zb
2
ðÞxz2V
b
tðÞ
z2baM b
1
zaðÞA
1ðÞ
b
1
za, b
1
, aðÞ
| cos b
1
zaðÞx{ V
b
zV
a
ðÞt{yðÞ½
z cos b
2
zaðÞx{ V
b
zV
a
ðÞtðÞ{yÞ
z2b
2
M b
2
{b
1
ðÞA
2ðÞ
b
2
{b
1
, b
1
, b
2
ðÞcos b
1
{b
2
ðÞxðÞ
zbaM a{b
1
ðÞA
2ðÞ
a{b
1
, b
1
, aðÞzA
2ðÞ
b
1
{a, b
1
, aðÞ
hi
| cos b
1
{aðÞx{ V
b
{V
a
ðÞtzyðÞ½
z cos b
2
{aðÞx{ V
b
{V
a
ðÞtzyðÞ
zb
2
M 2b
1
ðÞA
3ðÞ
{2b
1
, b
1
, b
1
ðÞ
| cos 2 b
1
x{V
b
ðÞtðÞz cos 2 b
2
x{V
b
ðÞtðÞ½
za
2
M 2aðÞA
3ðÞ
{2a, a, aðÞcos 2 ax{V
a
t{yðÞðÞ
z2b
2
M b
1
zb
2
ðÞA
3ðÞ
{b
1
{b
2
, {b
1
, b
2
ðÞcos b
1
zb
2
ðÞx{2V
b
tðÞ
z2baM b
1
zaðÞA
3ðÞ
{b
1
{a, b
1
, aðÞ
| cos ðb
1
zaðÞx{ V
b
zV
a
ðÞt{y½Þ
zcos b
2
zaðÞx{ V
b
zV
a
ðÞt{yðÞ
ðA6Þ
The third-order (cubic in amplitude) approximation can be written in the same
manner (see Shrira et al . 1996 for details).
Acknowledgments
The work was supported by ONR under grant N00014-94-1-0532, INTAS
grant 97-575, 01-234 and Russian Foundation for Basic Research N01-05-64603,
N01-05-64464, N02-05-65140.
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