This article was downloaded by: [Acadia University]On: 07 October 2013, At: 14:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Waves in Random MediaPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/twrm19

Wave diffraction by a concave statistically roughsurfaceIosif M Fuks a & Alexander G Voronovich ba CIRES/University of Colorado and NOAA/Environmental Technology Laboratory, 325Broadway, Boulder, CO 80303, USAb NOAA/Environmental Technology Laboratory, 325 Broadway, Boulder, CO 80303, USAPublished online: 19 Aug 2006.

To cite this article: Iosif M Fuks & Alexander G Voronovich (1999) Wave diffraction by a concave statistically roughsurface, Waves in Random Media, 9:4, 501-520, DOI: 10.1088/0959-7174/9/4/304

To link to this article: http://dx.doi.org/10.1088/0959-7174/9/4/304

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Waves Random Media9 (1999) 501–520. Printed in the UK PII: S0959-7174(99)03305-4

Wave diffraction by a concave statistically rough surface

Iosif M Fuks† and Alexander G Voronovich‡† CIRES/University of Colorado and NOAA/Environmental Technology Laboratory, 325Broadway, Boulder, CO 80303, USA‡ NOAA/Environmental Technology Laboratory, 325 Broadway, Boulder, CO 80303, USA

Received 9 April 1999

Abstract. We consider a statistically rough impedance surface that is concave on average incontrast to a plane. Backscattering from such a surface is considered based on the small perturbationtheory method. The diffraction problem is divided into two parts which are considered separately:the problem of scattering by small roughness (assumed to be local) and the propagation of incidentand scattered fields over a smooth large-scale concave surface. In contrast to the ‘two-scale’scattering model, the zero-order unperturbed wavefield is not assumed to be specularly reflectedfrom the local tangent plane to the smooth surface, but it is a solution of a corresponding diffractionproblem. Two particular cases of smooth surfaces are considered: first, the inner surface of aconcave cylinder with a constant radius and finite angular pattern, and second, a compound surfacethat consists of a coupled half-plane and the cylindrical surface mentioned above. In a geometricaloptics limit and with propagation at low grazing angles, the analytical results for a zero-order(unperturbed) field are obtained for these two cases in the form of a series over multiple specularreflected fields. It is shown that these non-local processes lead to the essential increase in thebackscattering cross section in comparison with the two-scale model and tangent-plane approach.

1. Introduction

Most of the existing methods of solving the problem of wave diffraction by a statisticallyrough surface are based on the assumption of a local relation between the surface field andthe surface shape at the same point (see, for example, [1–4]). This is obvious for the mostwidely used Kirchhoff approximation method, in which at every point on a rough surfacethe field is represented as a sum of the incident wave and a specularly reflected one fromthe local tangent plane to the surface. However, in the traditional small perturbation method,which is based on the boundary conditions expansion [2], the first-order scattered field atthe surface is also entirely determined by the local roughness parameters, heights and slopesat the same point of the surface. More sophisticated approaches (such as the small slopeapproximation [5], the tilt-invariant method [6], the operator expansion method [7–9] and thelocal perturbation method [10, 11]) allow the possibility of non-local interaction between thesurface and the scattered fields in the higher orders that take into account multiple-scatteringeffects [12, 13]. Some attempts have recently been made to consider non-local effects (suchas shadowing [14–16]) in the framework of geometrical optics or by iterating the appropriateintegral equation (double-scattered field calculation [17]), but up to now there is no satisfactorygeneral theory of multiple wave scattering by a statistically rough surface.

The multiple-scattering processes (in other words, non-local interaction between thewavefield and the surface) may play a significant role at a low-grazing-angle propagation. Onepossible approach to use to take into account the non-local effects is to separate the propagation

0959-7174/99/040501+20$30.00 © 1999 IOP Publishing Ltd 501

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

502 I M Fuks and A G Voronovich

problem from the scattering one. By propagation we mean diffraction of the wavefield on itsway to and from the scattering patch of the rough surface. In this instance the scattering processremains local and could be described by one of the analytical local methods mentioned above,but the wave that illuminates this scattering patch of rough surface differs from the incidentwave because of its interaction with the rough surface as it propagates above it. A separationof this problem into two parts (propagation and scattering) was made in [18, 19] to solve aproblem of calculating a scattering cross section from a rough surface at small grazing angles.The same idea is used in [20] to explain the polarization anomalies in a backscattering crosssection from a rough sea surface by considering incident- and scattered-wave diffraction by thetriangular shape of the sea crests before and after backscattering by small ripples covering thelarge-scale sea waves. In [21] the influence of the large-scale component of sea roughness inthe framework of a two-scale model was considered not only as a local slope modulation factorbut as a cause of anomalies of zero-order (unperturbed) field strength calculated numerically.

In this paper we consider an approximate solution of the problem of wave diffraction bya statistically rough impedance surface6, concave on average in contrast to a plane one. Wecan consider such a concave surface as a model of a sea trough between two adjoining crests.In this sense almost all sea surface is concave except for a small part of it that is the crests.The effect of surface curvature on the wave scattering process was considered analyticallyin [22, 23] as a correction to the Kirchhoff approximation (tangent-plane model), and in [24]in the framework of a two-scale scattering model. In these papers surface curvature plays therole of a local disturbing factor the influence of which was determined by the value of ratioλ/a, whereλ is the wavelength, anda is the surface curvature radius. In this paper we neglectthese effects, which become small in a short-wave limit, and consider only non-local effectsof multiple-scattering waves by a concave surface at low grazing angles.

2. Wave diffraction by a smooth impedance surface covered by a small roughness

2.1. Problem formulation

We consider the scalar monochromatic wavefieldU(R) (e−iωt time dependence is assumed)that is a solution of the Helmholtz equation(

1R + k2)U(R) = 0. (1)

Here,R is a radius vector in three-dimensional space,k = ω/c is the wavenumber, andc isthe wave velocity. The fieldU(R) satisfies the impedance boundary condition on the roughsurface6 (figure 1):

∂U

∂N6+ ikηU

∣∣∣∣6

= 0. (2)

Here,∂/∂N6 denotes the normal derivative at the surface6, andη is the normal surfaceimpedance. Equations (1) and (2) also describe the electromagnetic wave propagation andscattering in 2D space, if we consider separately the vertically polarized transverse-magnetic(TM) waves, in which the magnetic field vector is perpendicular to the plane of incidence, andthe horizontally polarized transverse-electric (TE) waves, in which the electric field vector isperpendicular to the plane of incidence. They are scattered by the rough boundary6 with agenerally complex dielectric permittivityε that is assumed to be sufficiently large (|ε| � 1).For these two different polarizations, impedanceη has the form:

η =

1√ε + 1

for vertical polarization (TM wave)√ε − 1 for horizontal polarization (TE wave).

(3)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 503

z

x

.(r)

RRR0

ES

N

a Curvature radius

k

Incidentwave

Tangentplane

n

Figure 1. Two-scale roughness model.

2.2. Small perturbation theory expansion

Together with the rough surface6 we consider the smooth undulating surfaceS. It is assumedthat the surface6 can be represented as small normal deviationsζ(r) (r ∈ S) from S. Wesuppose that the typical radius of curvaturea of S is large compared with the wavelength andthat the conditionka � 1 holds. Assuming that the slopesγ(r) = ∇rζ(r) of roughnessζ(r)with respect toS are also small (γ 2� 1), we can expand the boundary condition (2) in powersof ζ(r) and retain the first-order terms (∼ O(ζ )) only [2]:

∂U

∂n+ ikηU

∣∣∣∣S

= (γ(r)∇r) U − ζ(r) ∂∂n

(∂

∂n+ ikη

)U

∣∣∣∣S

(4)

where∂/∂n denotes the normal derivative with respect toS, and∇r is calculated along thelocal tangent plane toS at the pointr. Now we can represent the solution of the diffractionproblem in the formU = U0 + u, whereU0 is the unperturbed wavefield corresponding to thediffraction by the smooth surfaceS, i.e. it satisfies the Helmholtz equation (1) and the uniformboundary condition

∂U0

∂nS+ ikηU0

∣∣∣∣S

= 0 (5)

andu is a correction to the first order ofζ(r), i.e. the scattered field that satisfies the non-uniformboundary conditions atS:

∂u

∂n+ ikηu

∣∣∣∣S

= (γ(r)∇r) U0 − ζ(r) ∂∂n

(∂

∂n+ ikη

)U0

∣∣∣∣S

. (6)

Using the Green theorem and the boundary condition (6) (for details see [21]) we canrepresent the scattered fieldu at the arbitrary pointR as the following surface integral:

u(R) =∫ ∫

S

ζ(r){k2[1− η2

]G0(R, r)U0(r)−∇rG0(R, r)∇rU0(r)

}dr. (7)

Here, we assume that the inequalityka|√ε| � 1 holds, wherea is the local curvatureradius of the surfaceS, and the Green functionG0(R,R

′) of the boundary problem forS isintroduced: (

1R + k2)G0(R,R′

) = −δ (R−R′) (8)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

504 I M Fuks and A G Voronovich

∂G0(R, r)

∂n+ ikηG0(R, r)

∣∣∣∣S

= 0. (9)

Equation (7) represents the general solution of the scattering problem to the first order inζ . If we assume that at each pointr of smooth surfaceS the Green functionG0(R, r) andthe unperturbed fieldU0(r) are the same as for the plane tangent toS, we obtain the well-known two-scale scattering model that was first suggested in [25] for scalar wave scatteringand subsequently generalized for electromagnetic wave scattering in [26, 27]. Below, weinvestigate the difference between the tangent-plane approach and a more accurate method fordiffraction of the incident field by an undulating surface.

2.3. Backscattering in the far zone

We consider scattering of a plane incident waveUinc(R) = eik·R, wherek is the wavevectorwith the projections (kx = k cosψ0, ky = 0, kz = −k sinψ0). We use the cartesian coordinatesystem in which the plane{x, z} is chosen as the plane of incidence, andψ0 is the grazing angle(figure 2) relative to the planez = 0. The total unperturbed fieldU0(r) onS is represented inthe form:

U0(r)|S = Uinc(r) +Uref(r) = F(r)eik·r (10)

whereF(r) is assumed to be a slowly varying function onS (as compared with the phasefactor: |∇rF | � kF ). It follows from the reciprocity principle that in the far zone the GreenfunctionG0(R, r) can be represented as:

G0(R, r) = eikR

4πRF(r)eik·r. (11)

Substituting (10) and (11) into (7), we obtain:

u(R) = eikR

4πR

∫ ∫S

ζ(r)[k2(1− η2

)+ k2⊥]

e2ik·rF 2(r) dr (12)

wherek⊥ = k cosψ is the projection of the wavevectork on the tangent plane toS at thepoint r, andψ is a local grazing angle (see figure 1). Below, we assume that the surfaceroughnessζ(r) is a spatially homogeneous random function of spatial coordinatesr, and wedenote statistical averaging by〈. . .〉. Our goal is to estimate the average backscatter powerthat is proportional to〈|u(R)|2〉. It follows from (12) that calculation of〈|u(R)|2〉 leads to theappearance in the integrand of the autocorrelation functionW(ρ) = 〈ζ(r)ζ(r + ρ)〉 in whichthe spatial scale of variationl (correlation radius) is assumed to be small in comparison withthe spatial variation scaleL of F(r). ForL it is easy to obtain the following rough estimate:

L ' min

{√2aλ

sinψ,

λ

1ψ sinψ

}(13)

whereλ is the wavelength(λ = 2π/k), and1ψ is the characteristic width of the wavefieldU0(r) spatial angular spectrum. Based on these assumptions we obtain from (12):⟨|u(R)|2⟩ = 1

4R2

∫ ∫S

Sζ (2k⊥)∣∣k2

(1− η2

)+ k2⊥∣∣2 |F(r)|4 dr. (14)

Here, the spatial power spectrumSζ (q) of the stochastic roughnessζ(r) is introduced:

Sζ (q) = 1

(2π)2

∫ ∫ ∞−∞

W(ρ)eiq·ρ dρ. (15)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 505

k

k

k

k

R

RR

R0

R0

2R0

C

B

A

O

D

AN

ON

a

a

z

x

Figure 2. Scheme of specular reflections from a concave cylindrical surface.

According to the definition of the scattering cross sectionσ we obtain from (14):

σ = 4π〈|u(R)|2〉R2|Uinc|2 =

∫ ∫S

σ0(r) dr (16)

where the specific scattering cross sectionσ0(r) is introduced:

σ0(r) = 16πk4Sζ (2k cosψ, 0)C(ψ, r). (17)

Following [21], here, we introduce thescattering coefficientC(ψ, r):

C(ψ, r) = 1

16

∣∣1− η2 + cos2ψ∣∣2 |F(r)|4. (18)

We emphasize that the scattering coefficientC(ψ, r) depends not only on the local grazingangleψ but also on coordinater because of the non-local dependence between the fieldU0(r)

andS. In the tangent-plane approximation the termUref(r) in (10) is assumed to coincide withthe specularly reflected wave from the plane tangential toS with the local reflection coefficientV (ψ), in which case we have:

F ⇒ F0(ψ) = 1 +V (ψ) = 2 sinψ

sinψ + η(19)

C ⇒ C0(ψ) =∣∣1− η2 + cos2ψ

∣∣2 ∣∣∣∣ sinψ

sinψ + η

∣∣∣∣4 . (20)

For a perfectly conducting surfaceS the last equation leads to the simple expressions forthe scattering coefficient:

C0 (ψ) ={ (

1 + cos2ψ)2

vertical polarization (η = 0)sin4ψ horizontal polarization (η = ∞).

(21)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

506 I M Fuks and A G Voronovich

3. Scattering coefficients for a concave perfectly conducting cylindrical surface

3.1. Vertical polarization (η = 0)

In the limit |ε| → ∞ (i.e. perfect conductivity) for a vertically polarized electromagnetic fieldη = 0 (which corresponds to sound scattering from a hard surface in acoustics), the integralequation for an unperturbed surface fieldU0(r) has the following form :

U0(r) = 2Uinc(r) + 2∫ ∫

S

U0(r′) ∂G(r, r′)

∂n′dr′ (22)

whereG(r, r′) is the free-space Green function

G(r, r′

) = exp[ik|r − r′|]4π |r − r′| . (23)

Assuming thatS is a 2D concave cylindrical surface with a constant radiusa and ageneratrix directed along they-axis,Oy, we can integrate in (22) with respect to the coordinatey using the following formula:∫ ∞

−∞G(r, r′

)dy ′ = i

4H(1)0

(2ka sin

ψ − ψ ′2

). (24)

The position of every pointr at the surfaceS in (22) can be characterized by a singleparameterψ that is the angle between the incident wavevectork and the tangent plane at thispoint (figure 2). The argument of the Hankel function in (24) is the distance between twopoints (with coordinatesψ andψ ′) multiplied byk. It is convenient to count off the phase ofthe incident field from the planeCD, which is perpendicular to the wavevectork, supposingthat the phase is equal to zero at this plane. The incident fieldUinc in this notation has theform:

Uinc(ψ) = Finc(ψ)eika sinψ Finc(ψ) = 1. (25)

Let us represent the total fieldU0(ψ) in a form similar to (10):

U0(ψ) = F(ψ)eika sinψ (26)

with a slowly varying factorF(ψ). Below, we consider the case of small anglesψ , ψ0 � 1,but a large cylindrical radiusa compared with the wavelengthλ, so that the inequality

4pψ0� 1 (27)

holds, where the main large parameterp of this problem is introduced:p = ak/2� 1. Thecondition (27) allows us to replace the Hankel function in (24) by its asymptote. If we neglectthe edge wave generated at the pointO (the initial edge of the cylinder atψ = ψ0) and consideronly the forward diffracted waves, then the integral equation (22) takes the form:

F(ψ) = 2− i

2

√ip

π

∫ ψ

ψ0

dψ ′F(ψ ′)√ψ − ψ ′ exp

[ip

4

(ψ +ψ ′

)2 (ψ − ψ ′)] . (28)

The iteration procedure applied to this equation leads to multiple integrals that can be estimatedby a stationary phase method (see appendix A), provided the following condition holds:

4√pψ3

0 � 1. (29)

As a result we obtain the specularly reflected field series:

F(ψ) =N(ψ)∑n=0

Fn(ψ) N(ψ) =[ψ − ψ0

2ψ0

](30)

Fn(ψ) = 2

in√

2n + 1ei8n(ψ) 8n(ψ) = pψ3 4n(n + 1)

3(2n + 1)2. (31)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 507

Here, the [Q] denotes the integer part of the numberQ, and thenth term in the series (30)corresponds to the contribution to the total fieldU0(ψ) of then-fold specularly reflected rays atpointsψk = ψ(2k− 1)/(2n + 1), k = 1, 2, . . . , n. N(ψ) is the maximum number of specularreflections of rays that can reach the pointψ . The first termF0(ψ) = 2 corresponds to thetangent-plane approximation: the surface value of the total fieldU0 is equal to twice the valueof the incident fieldUinc becauseV (ψ) = 1 in the case under consideration. In contrast to thetangent-plane approximation, the connection betweenU0(ψ) and the incident waveUinc(ψ)

is non-local in spite of the dependence ofC(ψ) on a single parameterψ only. We considerthe value ofC(ψ) that is normalized to its tangent-plane valueC0(ψ) and refer to it as theamplification factorK(ψ), which describes the increase of the specific backscattering crosssection from the concave surface in comparison with the tangent-plane approximation:

K(ψ) = C(ψ)

C0(ψ)=∣∣∣∣ F(ψ)F0(ψ)

∣∣∣∣4 =∣∣∣∣∣1 +2(ψ − 3ψ0)

N(ψ)∑n=1

fn(ψ)

∣∣∣∣∣4

(32)

wherefn(ψ) = Fn(ψ)/2 and the Heaviside step function is introduced:

2(x) ={

0 for x < 0

1 for x > 0.(33)

3.2. Horizontal polarization(η = ∞)For the case of horizontal polarization, the wavefieldU0(R) corresponds to the tangential (tothe surfaceS) electric field which becomes equal to zero in the perfect-conductivity limit. Inthis special case (η = ∞) we can modify the general solution (7), replacing the productsηG0

andηU0 by the normal derivatives ofG0 andU0, respectively, using boundary conditions (5)and (9):

u(R) =∫ ∫

S

ζ(r)∂G0(R, r)

∂n

∂U0(r)

∂ndr. (34)

The integral equation for the normal derivative∂U0(r)/∂n at the surfaceS has a formsimilar to (22):

∂U0(r)

∂n= 2

∂Uinc(r)

∂n− 2

∫ ∫S

∂U0(r′)

∂n′∂G

(r, r′

)∂n

dr′. (35)

According to (25) the first term on the right-hand side of this equation has the form:

2∂Uinc

∂n= 2ikFinc(ψ)e

ika sinψ Finc(ψ) = − sinψ (36)

and we can seek the solution of this equation in a form analogous to (26):

∂U0

∂n= ikF (ψ)eika sinψ. (37)

After making assumptions and performing computations similar to those in the previoussubsection, we obtain the following equation forF(ψ), which is analogous to (28):

F(ψ) = F0(ψ) +i

2

√ip

π

∫ ψ

ψ0

dψ ′F(ψ ′)√ψ − ψ ′ exp

[ip

4

(ψ +ψ ′

)2 (ψ − ψ ′)] (38)

where the first termF0(ψ) = 2Finc(ψ) ' −2ψ on the right-hand side corresponds to thetangent-plane approximation. Application of the iteration procedure to this integral equationand estimation of the integrals by the stationary-phase method gives a result that can be

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

508 I M Fuks and A G Voronovich

represented in the same form (30) as for vertically polarized (TM) waves. The only differencein comparison with (31) is in the amplitude of thenth term:

Fn(ψ) = −2ψin

(2n + 1)√

2n + 1ei8n(ψ). (39)

The amplification factorK(ψ) for both polarizations can also be written in the form ofequation (32), where the termsfn(ψ) are as follows:

fn(ψ) = Anei8n(ψ) 8n(ψ) = pψ3 4n(n + 1)

3(2n + 1)2(40)

An =

1

in√

2n + 1for vertical polarization (TM wave)

in

(2n + 1)√

2n + 1for horizontal polarization (TE wave).

(41)

3.3. Configuration of cylinder and adjoining half-plane

We can generalize the results obtained for a specific composite surface,S. The compositesurface is a half-plane(z = 0, x 6 0) that is followed atx > 0 by the cylindrical surfaceconsidered above (see figure 2). Such a surface can be considered as a model of a sea-wave trough when wind blows in the negative direction of the axisOx. Although thedirect applicability of this model to the sea-surface is questionable, it allows derivation ofuseful analytical results. The presence of the additional half-plane leads to the appearanceof another plane wave (specularly reflected from that half-plane) with the same grazing angleψ0, propagating in the opposite (positive) direction of the axisOz. In the short-wave limitingcase we can neglect the penumbra zone in the vicinity of the boundaryOA between the ‘light’and ‘shadow’ (generated by the boundary rayO ′O) for the wave specularly reflected from thehalf-plane(z = 0, x 6 0). Figure 2 shows that in this assumption, pointA plays the samerole for this wave as pointO plays for the incident wave, i.e. we can use the solutionF(ψ) foran incident plane wave with grazing angleψ0, changing the argumentψ → ψ − 2ψ0. Theamplitude of this additional wave differs from the amplitude of the incident wave by a factorV (ψ0), which is the reflection coefficient from the half-plane(z = 0, x 6 0):

V (ψ0) ={

1 vertical polarization

−1 horizontal polarization.(42)

When calculating the resulting field we take into account the phase difference between theinitial incident wave and the reflected one. At pointA (figure 2) the incident wave phase isproportional to the length of the segmentA′A = a sin 3ψ0, and the phase of the additionalreflected plane wave mentioned above is proportional to the total length of the two segmentsO ′O +OA = 3a sinψ0. The phase difference is equal to the following value:

δ8 = ka (3 sinψ0 − sin 3ψ0) ' 4kaψ30 = 8pψ3

0 . (43)

We can write down the following expression for the factorF(ψ) of the zero-order fieldU0 for the surface configuration cylinder and adjoining half-plane. We denote this byFp(ψ)

as distinct from the symbolF(ψ) that denotes the field solution (30) for a single cylindricalsurface:

Fp(ψ) = F(ψ) +2(ψ − 3ψ0) V (ψ0) e8ipψ30F (ψ − 2ψ0) . (44)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 509

3.4. Numerical results

Because the local grazing angle of the incident plane wave coincides with the angular coordinateψ , for every point on the cylinder, we can substitute into (32) for the amplification factorK

the following expressions:

F(ψ)

F0(ψ)=

N(ψ)∑n=0

fn(ψ) (45)

for a cylindrical surfaceS, and

Fp(ψ)

F0(ψ)=

N(ψ)∑n=0

fn(ψ) +2(ψ − 3ψ0)F0 (ψ − 2ψ0)

F0(ψ)V (ψ0) e8ipψ3

0

N(ψ−2ψ0)∑n=0

fn (ψ − 2ψ0)

(46)

for a composite surfaceS. In these equations the functionsfn(ψ) are given by (40), (41) andN(ψ) = [ 1

2((ψ/ψ0) − 1)]. F0(ψ) = 2 for vertical polarization andF0(ψ) = −2 sinψ forhorizontal polarization. If we introduce thecurvature parameterκ

κ = pψ30 =

1

2akψ3

0 (47)

it follows from equations (45) and (46) that the amplification factorsK andKp = |Fp/F0|become functions of the argumentψ/ψ0.

In figures 3–6 the plots ofK andKp are represented as functions of the ratioψ/ψ0 fortwo values of the curvature parameter:κ = 1 andκ = 5. According to the inequality (29) wecannot consider a small value ofκ. It is clear thatK andKp are the fast oscillation functionsof the argumentd = ψ/ψ0 that increase slowly as the argumentd increases. The frequency ofthese oscillations depends on the value of the curvature parameterκ: one can see from figures 4and 6, which have a larger scale with respect to thex-axis, that the value ofκ affects only theoscillation frequency and does not influence the mean value of the amplification factors.

Let us consider the behaviour ofK andKp after averaging over these fast oscillations.This averaging, denoted below by〈〈. . .〉〉, can be performed analytically if we neglect theinterference of termsfn with different numbersn in (45) and (46) due to their fast phasevariations with respect to the argumentd (see (40)):

8n(d) = κd3 4n(n + 1)

3(2n + 1)2. (48)

Averaging (32) for the cylindrical surface, whenF(ψ)/F0(ψ) has the form of (45), leadsto the following expression for〈〈K〉〉:

〈〈K(d)〉〉 = 1 +2(d − 3)

(4I + 2I 2 −

N(d)∑n=1

|fn|4)

(49)

where

I =N(d)∑n=1

|fn|2 N(d) =[d − 1

2

](50)

|fn|2 ={

1/(2n + 1) for vertical polarization1/(2n + 1)3 for horizontal polarization.

(51)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

510 I M Fuks and A G Voronovich

2 4 6 8 10ψêψ0

0

5

10

15

20

KdB κ=5

cLVP

2 4 6 8 10ψêψ0

0

5

10

15

20

25

KpdB

κ=5

dLVP

2 4 6 8 10ψêψ0

0

5

10

15

20K

dB κ=1

aLVP

2 4 6 8 10ψêψ0

0

5

10

15

20

25

KpdB

κ=1

bLVP

Figure 3. Dependence of vertical polarization amplification factorsK andKp on the local grazingangleψ normalized by the incident angleψ0 for a perfectly conducting concave cylinder (left-handpanels) and a composite surface (right-hand panels); curvature parameterκ = 1 (upper panels) andκ = 5 (lower panels).

10 10.2 10.4 10.6 10.8 11ψêψ0

0

5

10

15

20

KdB

κ=5 cLVP

10 10.2 10.4 10.6 10.8 11ψêψ0

0

5

10

15

20

25

KpdB

κ=5 dLVP

10 10.2 10.4 10.6 10.8 11ψêψ0

0

5

10

15

20

KdB

κ=1 aLVP

10 10.2 10.4 10.6 10.8 11ψêψ0

0

5

10

15

20

25

KpdB

κ=1 bLVP

Figure 4. The same as in figure 3 but the range of the abscissa is 10–11.

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 511

2 4 6 8 10ψêψ0

−2

0

2

4

6

8

10

KdB

κ=5

cLHP

2 4 6 8 10ψêψ0

2

4

6

8

10

12

14

KpdB

κ=5

dLHP

2 4 6 8 10ψêψ0

−2

0

2

4

6

8

10K

dB

κ=1

aLHP

2 4 6 8 10ψêψ0

2

4

6

8

10

12

14

KpdB

κ=1

bLHP

Figure 5. Dependence of horizontal polarization amplification factorsK andKp on the localgrazing angleψ normalized by the incident angleψ0 for a perfectly conducting concave cylinder(left-hand panels) and a composite surface (right-hand panels); curvature parameterκ = 1 (upperpanels) andκ = 5 (lower panels).

10 10.2 10.4 10.6 10.8 11ψêψ0

−2

0

2

4

6

8

10

KdB

κ=5

cLHP

10 10.2 10.4 10.6 10.8 11ψêψ0

2

4

6

8

10

12

14

KpdB

κ=5

dLHP

10 10.2 10.4 10.6 10.8 11ψêψ0

−2

0

2

4

6

8

10

KdB

κ=1

aLHP

10 10.2 10.4 10.6 10.8 11ψêψ0

2

4

6

8

10

12

14

KpdB

κ=1

bLHP

Figure 6. The same as in figure 5 but the range of the abscissa is 10–11.

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

512 I M Fuks and A G Voronovich

0 10 20 30 40ψêψ0

0

2

4

6

8

10

12

14

K,KpdB

VP

VP

HP

K

K

Kp

Kp

HP

Figure 7. Amplification factorsK (cylindrical surface) andKp (composite surface) smoothed-overfast oscillations, for vertical (VP) and horizontal polarization (HP), as functions of the local grazingangleψ normalized by the incident angleψ0 in the perfect-conductivity limit.

For the composite surface we can neglect the interference between the incident wave andthe wave reflected from the half-plane because of the fast oscillation factor e8ipψ3

0 in the right-hand side of (46). The same averaging procedures as described above lead to the followingexpression:

〈〈Kp(d)〉〉 = 1 +2(d − 3)

{4Ip + 2I 2

p −N(d)∑n=1

|fn|4 −N(d−2)∑n=0

|gn|4}

(52)

where

Ip = I +N(d−2)∑n=0

|gn|2 (53)

gn = fn

1 for vertical polarization

1− 2

dfor horizontal polarization.

(54)

After this averaging, the functions〈〈K(d)〉〉 and〈〈Kp(d)〉〉 do not depend on the curvatureparameterκ and become universal functions of the sole argumentd (see figure 7). One cansee from this figure that the curvature of the cylindrical surfaceS mostly affects the scatteringcoefficient of a vertically polarized field, for which the amplification factor increases up to10 dB forψ/ψ0 ' 40, and has little effect on the horizontally polarized field. Conversely, fora composite surfaceS the reflection from the additional half-plane increases the amplificationfactor by' 3–4 dB for vertical polarization and by' 6–7 dB for horizontal polarization.

4. Finite surface conductivity

4.1. Specular reflection field series

For the case of an impedance surfaceS, the value of the amplitudeF(ψ) of the near-surfacezero-order fieldU0, introduced by (26), must be obtained as a solution of the corresponding

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 513

integral equation (see, for example, [4], chapter 3) that generalizes the perfect-conductivityintegral equation (28). To avoid the lengthy derivations we can easily generalize the solutionF(ψ), obtained for perfect conductivity and given by (30) and (31), to the case of finiteconductivity.

Note that the phase8n(ψ) of n times the specularly reflected fieldFn(ψ) in (31) doesnot depend on the surface conductivity and could be calculated as the difference between then+1-segment path lengthO ′O+OA+AB+. . . (see figure 2) of the multiply reflected field andthe incident one that equalsa sinψ , as follows from (25). The ray arriving at the same pointof S with the coordinateψ aftern specular reflections with grazing anglesψn = ψ/(2n + 1)passes the distance(2n + 1)a sinψn, so we have for the path length difference:

(2n + 1)a sinψn − a sinψ ' aψ3 4n(n + 1)

3(2n + 1)2(55)

which coincides with the phase8n (ψ) in (31) obtained in appendix A by applying thestationary-phase method to the subsequent terms resulting from iterations of integral equation(28). The amplitude factor 1/(in

√2n + 1) in (31) is a result of ray tube divergence and of

the caustics that are passed in a process ofmultiplereflections from the concave surface withthe constant radiusa (see appendix B for details), and it also does not depend on the surfaceconductivity. Finally, the factor of two in the amplitudeFn(ψ) in (31) is a result of a verticallypolarized field doubling at the perfectly conducting surfaceS.

We obtain the expression for then-fold specularly reflected fieldFn(ψ) from theimpedance surfaceS: we multiply the expression (31) forFn(ψ) by a factorV n(ψn), which isa result ofn-times reflection with grazing anglesψn, and replace the factor of two mentionedabove by a factor 1+V (ψn), whereV (ψn) is a Fresnel reflection coefficient from an impedanceplane surface:

V (ψ) = sinψ − ηsinψ + η

=

√ε + 1 sinψ − 1√ε + 1 sinψ + 1

for vertical polarization

sinψ −√ε − 1

sinψ +√ε − 1

for horizontal polarization.

(56)

After such modifications the equation forF(ψ) takes the form:

F(ψ) =N(ψ)∑n=0

Fn(ψ) =N(ψ)∑n=0

V n (ψn) [1 +V (ψn)]

in√

2n + 1ei8n(ψ). (57)

Taking into account thatF0(ψ) = 1 +V (ψ) (the first term in this series corresponds toa tangent-plane approximation, as mentioned above) we obtain for the amplification factorK(ψ) the same expressions (32) and (40) with the following expressions for the amplitudesAn instead of (41):

An = V n (ψn) [1 +V (ψn)]

in[1 + V (ψ)]√

2n + 1. (58)

In a perfect-conductivity limit this expression not only corresponds toAn for the verticallypolarized field that is given by the upper line in (41) (note that ifη = 0 thenV (ψ) = 1),but also coincides with the lower line in equation (41) for the horizontally polarized field: ifη = ∞ thenV (ψ) = −1 and

[1 +V (ψn)]

[1 + V (ψ)]= ψn

ψ= 1

2n + 1. (59)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

514 I M Fuks and A G Voronovich

0 5 10 15 20ψêψ0

0

1

2

3

4

5

6

7

KpdB

1

0.3

0.1

3HP

VP

2

Figure 8. Amplification factorKp (impedance composite surface with dielectric constantε = 60 + 30i) smoothed-over fast oscillations for vertical (VP) and horizontal (HP) polarizationas a function of the local grazing angleψ normalized by the incident angleψ0 in the perfect-conductivity limit. The values of the ratio ofψ0 to the Brewster angleψB are shown as numbersunder the VP plots (thick lines). The group of HP plots (thin lines) corresponds to different ratiosψ0/ψB ranging from 0.1 (upper line) to 3.0 (lower line).

4.2. Amplification factor for an impedance surface

It is shown above that the general structure of the fieldF(ψ) normalized on its tangent-plane-approximation valueF0(ψ) for the impedance surfaceS has the same form as for aperfectly conducting surface [(45) for a cylinder, and (46) for the composite surface], whereonly the amplitudesAn (58) differ from their perfectly conducting values (41), and the reflectioncoefficientV has the form (56) instead of (42). Therefore, we can calculate the amplificationfactor for the impedance surface using the same equations (45) and (46) as for a perfectlyconducting one, with the appropriate modifications offn andV . In figure 8 we display theamplification factor smoothed over fast oscillations,〈〈Kp( ψψ0

)〉〉, for vertical and horizontalpolarization and various ratios of the angle of incidenceψ0 to the Brewster angleψB, where

ψB = Re1√ε + 1

. (60)

All calculations were performed forε = 60 + 30i, which corresponds to the complexdielectric permittivity of seawater in the X-wave band, using equation (53) with the above-mentioned modifications of the factorsfn and

gn = fn F0 (ψ − 2ψ0)

F0(ψ)= fn 1 +V (ψ − 2ψ0)

1 +V (ψ). (61)

Comparison of these graphs with figure 8 shows that the finite conductivity almost does notaffect the amplification factorKp for horizontal polarization (it remains at the level of 6–7 dBfor ψ > 10ψ0), but essentially decreases the amplification factorKp for vertical polarization.The most significant difference between vertical and horizontal polarization takes place forgrazing angleψ0 close to the Brewster angleψB. It is quite clear, because of the assumed smallgrazing angles, that the Fresnel reflection coefficientV for horizontal polarization(|η| � 1)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 515

remains close to its perfect-conductivity value (−1) for all possible grazing anglesψ and|ε| � 1:

|1 +V (ψ)| =∣∣∣∣ 2 sinψ

sinψ + η

∣∣∣∣ ' ∣∣∣∣2ψη∣∣∣∣� 1. (62)

Conversely, the Fresnel reflection coefficientV for vertical polarization reduces tosufficiently small values for grazing anglesψ in the vicinity of the Brewster angleψB. In [28]this effect was investigated for reflection of the incident wave by a plane surface before itilluminates the small ripple that causes the backscattering. We see that the same phenomenonalso takes place for scattering by a concave surface. It can be seen from the plots that forgrazing anglesψ0, which exceed the Brewster angleψB or are sufficiently small in comparisonwith it, the difference between the two polarizations decreases. In the limiting caseψ0� ψB,which corresponds to the perfect-conductivity limitψB = 0, we have the opposite relation: avertical polarization amplification factor exceeds the horizontal polarization one as shown infigure 7.

5. Conclusion

It follows from the analytical and numerical results obtained above that even in the framework ofa two-scale scattering model it is possible to explain some features in radio-wave backscatteringfrom a rough sea surface. In particular, we investigate the anomalies associated with a largebackscattering cross section of horizontally polarized signals at low grazing angles. We takeinto account the multiple-scattering processes of an incident wave by a concave (on average)surface, i.e. non-local dependence between the near-surface unperturbed field and the surfaceshape. In this paper we have considered only the geometrical optics limiting case in whichthe inequalityκ = pψ3

0 � 1 holds, wherep = ak/2 is the main large parameter of theproblem. This inequality means that not only does the mean surfaceS curvature radiusaexceed the field wavelengthλ, but also the angle of incidenceψ0 and all other grazing anglesψn = ψ/(2n + 1) that appear in this problem significantly exceed the typical grazing anglesof the lowest modes of the concave structure that have the order ofp−1/3. For extremely lowgrazing anglesψ0 . p−1/3 it is necessary to use the wave-mode approach, considering thewhispering-gallery modes that can be generated by the incident wave and propagate along theconcave surface with a sufficiently small attenuation. The backscattering of these modes by asmall surface roughness can lead to the enhancement of the effect considered above.

Acknowledgments

This work was supported by the joint NOAA/DoD Advanced Sensor Application Program.The authors are grateful to Dr M Charnotskii for his very valuable comments which contributedto the final version of this paper.

Appendix A

A multiple iteration procedure applied to the integral equation (28) gives a solution in the formof the Neumann series:

F(ψ) =∞∑n=0

Fn(ψ) (A.1)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

516 I M Fuks and A G Voronovich

F0(ψ) = 2

F1(ψ) = 2

(− i

2

√ip

π

)∫ ψ

ψ0

dψ1

√ψ − ψ1 exp

[ip

4(ψ +ψ1)

2 (ψ − ψ1)]

...

Fn(ψ) = 2

(− i

2

√ip

π

)n ∫. . .

∫ ψ

ψ0

dψn dψn−1 . . . dψ2 dψ1

√ψ − ψn

×√ψn − ψn−1 . . .

√ψ3− ψ2

√ψ2 − ψ1 exp

[ip

49 (ψ,ψn, . . . ψ2, ψ1)

](A.2)

where

9 (ψ,ψn, . . . ψ2, ψ1) =n∑k=1

8(ψk+1, ψk) (ψn+1 ≡ ψ) (A.3)

8(ψk+1, ψk) = (ψk+1 +ψk)2 (ψk+1− ψk) . (A.4)

Assuming that parameterp is sufficiently large, we can estimate the multiple integral(A.2) for every ordern by the stationary-phase method. The points of stationary phase aredetermined by the set of equations:

∂9

∂ψk= ∂

∂ψk

[8(ψk+1, ψk) +8(ψk, ψk−1)

] = 0 (k > 2) (A.5)

∂9

∂ψ1= ∂

∂ψ18(ψ2, ψ1) = 0 (A.6)

that give the following relations:

ψk+1 = 2ψk − ψk−1 (k > 2) (A.7)

ψ2 = 3ψ1. (A.8)

Taking into account thatψn+1 ≡ ψ it follows from (A.7) for k = n:

ψ = 2ψn − ψn−1. (A.9)

The solution of the set of equations (A.7)–(A.9) has the form:

ψ0k =

2k − 1

2n + 1ψ (A.10)

and the trajectory of then-fold reflected ray is a broken line that connects pointsψ0k

(16 k 6 n + 1). The grazing angle at each of these points is equal toψ01 = ψ/(2n + 1). It is

evident from figure 2 that the minimum value ofψ01 corresponds to the first specular reflection

at pointO and is equal to the grazing angleψ0 of the incident wave. The inequalityψ01 6 ψ0

cuts off the infinite series (A.1) for everyψ at the maximum number

nmax≡ N =[ψ − ψ0

2ψ0

]. (A.11)

The pre-exponential factor in the integrand (A.2) can be removed from the integral at thestationary-phase pointsψ0

k (A.10):√ψ − ψn

√ψn − ψn−1 . . .

√ψ3− ψ2

√ψ2 − ψ1 '

(2ψ

2n + 1

)n/2(A.12)

and the phase9(ψ,ψn, . . . ψ2, ψ1) can be expanded in a Taylor series in their vicinity:

9 = 90 +1

2

n∑i,k=1

aik(ψi − ψ0

i

) (ψk − ψ0

k

)(A.13)

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 517

where

90 = 9 (ψ,ψ0n , . . . ψ

02 , ψ

01

) = 16n(n + 1)

3(2n + 1)2ψ3 (A.14)

aik = ∂29

∂ψi∂ψk

∣∣∣∣ψi=ψ0

i ,ψk=ψ0k

. (A.15)

Substituting (A.13) into (A.2) we obtain the standard integrand with the quadratic formin the exponent. Assuming that stationary-phase pointsψ0

k are far from the integral limits, wecan write the result of integration in the form :∫

. . .

∫ ∞−∞

exp

[ip

8

n∑i,k=1

ai,kξiξk

]dξi dξk =

(8π i

p

)n/2 1√det‖aik‖

. (A.16)

From (A.3) and (A.4) we see that the matrix‖aik‖ has only three main diagonals with thenon-zero elements:

akk = − 8ψ

2n + 1k > 2

a11 = − 12ψ

2n + 1

ak+1,k = ak,k−1 = 4ψ

2n + 1

(A.17)

and its determinant can be written in the form

det‖aik‖ = (−1)n(

4ψ

2n + 1

)nDn (A.18)

where

Dn =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

n︷ ︸︸ ︷3 −1 0 0 .

−1 2 −1 0 .

0 −1 2 −1 .

0 0 −1 2 .

. . . . .

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣= 2n + 1. (A.19)

Substituting (A.18) into (A.16) and taking into account (A.12) and (A.14) we obtain for(A.2) the asymptotic representation (31). The accuracy of this equation can be estimated bycalculating the next terms in the expansion of the integrand (A.2) in powers of(ψk −ψ0

k ) nearthe stationary phase points. This procedure leads to inequality (29).

Appendix B

In the process of multiple reflection of the incident wave at the sequence of specular pointsψ0k

(A.10), the amplitude of the reflected wave changes after each reflection because of the finitecurvature of the surfaceS. Denote byUn the strength of the field that reaches pointB withcoordinateψ ≡ ψn+1 aftern consecutive reflections at pointsψ0

1 , ψ02 , . . . , ψ

0n , and denote by

Un−1 the strength of the field that reaches the previous reflection pointA aftern− 1 specularreflections (see figure 2, which corresponds to the particular casen = 2). In the general case ofan arbitrary shape of a sufficiently smooth surfaceS and a wave phase front ofUn−1, the field

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

518 I M Fuks and A G Voronovich

Un is connected toUn−1 by the geometrical optics equation (see, for example, equation (28.6)in [2]), that for a cylindrical surface (2D case) takes the simple form:

Un = V eikρ0√1− ~ r

nρ0Un−1 (B.1)

whereV is the Fresnel specular reflection coefficient,ρ0 = 2a sinψn is the length of oneintervalAB of a ray trace,ψ is the local grazing angle,~r

n is the curvature of the reflected fieldUn phase front just after the reflection at pointψ0

n(A), which is related to the curvature~ in of the

incident fieldUn−1 phase front at the same point by the following relation (see equation (28.7b)in [2]):

~ rn = ~ i

n +2

a sinψn. (B.2)

The local grazing angleψn = ψ/(2n + 1) is a constant for the set of all specular pointsψ0k of

the same ray consisting ofn intervals.In its turn, the curvature~ i

n of the incident field at the pointψ0n(A) can be expressed in

terms of the curvature~ rn of the reflected field at the previous specular pointψ0

n−1(O) by theequation

~ in =

~rn−1

1− ~ rn−1ρ0

(B.3)

which follows from the obvious relation between the curvature radiiρ in = 1/~ i

n andρrn = 1/~r

n

for cylindrically divergent waves:

ρ in = ρr

n−1− ρ0. (B.4)

The curvatures~ in, ~

rn (and radiiρ i

n, ρrn corresponding to them) are positive if the centre of

curvature is located in front of a propagating wave and negative otherwise. If we introduce thenotationxn ≡ ~r

nρ0, then from (B.2) and (B.3) the recurrence relation follows:

xn = 4− 3xn−1

1− xn−1. (B.5)

For the plane incident wavex0 = 0, the solution of (B.5) has the form

xn = 4n

2n− 1. (B.6)

Substituting this result into the recurrent equation (B.1) with the known initial field strengthUi we obtain the solution

Un = V neinkρ0

in√

2n + 1Ui (B.7)

which coincides with (31). Thus, the factor√

2n + 1 in (31) is a result of a geometricalcylindrical divergence of the reflected fields at the sequence of specular pointsψ0

k (k =1, 2, . . . , n). The appearance of the factor in is a result of a ray passing through the causticsof the reflected fields between every subsequent pair of specular pointsψ0

k . It is enough torewrite (B.1) for the field strengthUn(ρ) at the arbitrary distanceρ from the last scatteringpoint (the previous notationUn corresponds toUn(ρ0)):

Un(ρ) = V eikρ√1− ~ r

nρUn−1. (B.8)

At distanceρcn = 1/~ rn the field strength increases to infinity as the square root of the

distance that corresponds to a simple caustic. Using the definitionxn = ~ rnρ0 and equation

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

Wave diffraction by a concave statistically rough surface 519

(B.6) we obtain the expression for the distance from thenth specular pointA to the caustic onthe intervalAB connecting the pointsψ0

n andψ0n+1:

ρcn =

ρ0

2

(1− 1

2n

). (B.9)

For arbitraryn > 1 the following inequality holds:ρ0

46 ρc

n 6ρ0

2(B.10)

which means that the multiply reflected wave at every interval crosses the caustic and everytime acquires the factor i. Because the total number of intervals forUn is equal ton, the factorin appears in (B.7).

References

[1] Beckmann P and Spizzichino A 1963The Scattering of Electromagnetic Waves from Rough Surfaces(New York:Macmillan)

[2] Bass F G and Fuks I M 1979Wave Scattering from Statistically Rough Surfaces(International Series in NaturalPhilosophy, vol 93) ed C B Vesecky and J F Vesecky (Oxford: Pergamon)

[3] Ishimaru A 1978Wave Propagation and Scattering in Random Media(New York: Academic)[4] Voronovich A G 1994Wave Scattering from Rough Surfaces(Springer Series on Wave Phenomena 17) (Berlin:

Springer)[5] Voronovich A G 1985 Small-slope approximation in wave scattering by rough surfacesSov. Phys.–JETP62

65–70[6] Charnotskii M I and Tatarskii V I 1995 Tilt-invariant theory of rough surface scatteringWaves Random Media

5 361–80[7] Milder D M 1991 An improved formalism for wave scattering from rough surfacesJ. Acoust. Soc. Am. 89529–41[8] Smith R A 1996 The operator expansion formalism for electromagnetic scattering from rough dielectric surfaces

Radio Sci. 311377–85[9] Milder D M 1998 An improved formulation of coherent forward scatter from random rough surfacesWaves

Random Media8 67–78[10] Isers A B, Puzenko A A and Fuks I M 1991 The local perturbation method for solving the problem of diffraction

from a surface with small slope irregularitiesJ. Electromagn. Wave Appl. 5 1419–35[11] Isers A B, Puzenko A A and Fuks I M 1990 Small parameters in the problem of wave scattering by a surface

with mildly sloping corrugations of arbitrary heightSov. Phys.–Acoust. 36253–5[12] Voronovich A G 1996 Non-local small-slope approximation for wave scattering from rough surfacesWaves

Random Media6 151–67[13] Anderson S J, Fuks I M and Praschifka J 1998 Multiple scattering of HF radio waves propagating across the sea

surfaceWaves Random Media8 283–302[14] Fuks I M 1979 Enhanced backscatter from rough surface with shadowingRadiotekhnika i Elektronika21633–6

(in Russian)[15] Mikhailovskii A I and Fuks I M 1994 Shadowing of sea surface at grazing radio-wave propagationRadiophys.

Quantum Electron. 37875–81[16] Mikhailovskii A I and Fuks I M 1993 Statistical characteristics of the number of specular points on a random

surface for small grazing anglesJ. Commun. Technol. Electron. 3821–32[17] Ishimaru A and Chen J S 1991 Scattering from very rough metallic and dielectric surfaces: a theory based on

the modified Kirchhoff approximationWaves Random Media1 21–34[18] Fuks I M, Tatarskii V I and Barrick D E 1999 Behavior of scattering from a rough surface at small grazing angles

Waves Random Media9 295–305[19] Fuks I M and Tatarskii V I 1998 Scattering cross section from a rough surface for small grazing anglesProc.

IEEE Int. Geoscience and Remote Sensing Symp. (Seattle, WA, USA)vol IV, pp 2279–83[20] Fuks I M and Voronovich A G 1998 Multiple radio wave scattering by sea surface at low grazing anglesProc.

IEEE Int. Geoscience and Remote Sensing Symp. (Seattle, WA, USA)vol IV, pp 2284–8[21] Voronovich A G and Zavorotny V U 1999 The effect of steep sea-waves on polarization ratio at low grazing

anglesIEEE Trans. Geosci. Rem. Sens. in press[22] Belobrov A V and Fuks I M 1985 Short-wave asymptotic analysis of the problem of acoustic wave diffraction

by a rough surfaceSov. Phys.–Acoust. 31442–5

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3

520 I M Fuks and A G Voronovich

[23] Belobrov A V and Fuks I M 1986 Depolarization of radar signal backscattered from random surfaceRadiophys.Quantum Electron. 291083–9

[24] Voronovich A G 1996 On the theory of electromagnetic waves scattering from the sea surface at low grazinganglesRadio Sci. 311519–30

[25] Kur’yanov B F 1962 The scattering of sound at a rough surface with two types of irregularitySov. Phys.–Acoust.8 252–7

[26] Fuks I M 1966 On the theory of radio wave scattering by a disturbed sea surfaceIzv. VUZ’ov Radiofiz9 876–87(in Russian)

[27] Valenzuela G R 1978 Theories for the interaction of electromagnetic and oceanic waves – a reviewBoundaryLayer Meteorol. 3161–5

[28] Hanson S G and Zavorotny V U 1995 Polarization dependency of enhanced multipath radar backscattering froman ocean-like surfaceWaves Random Media5 159–65

Dow

nloa

ded

by [

Aca

dia

Uni

vers

ity]

at 1

4:27

07

Oct

ober

201

3