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Keywords:

  • asymptotic diffraction theory;
  • backscattering cross section;
  • statistically rough surface;
  • geometrical optics;
  • polarization ratio

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[1] Diffraction corrections (up to terms ∼1/k2) to the geometric optics backscattering cross sections from a statistically rough 2-D perfectly conducting surface were derived for TE- and TM-polarized electromagnetic waves based on the high-frequency asymptotic expansions of electric and magnetic fields at the surface obtained by Fuks (2004). It was shown that at steep incident angles, where the specular reflections play the main part in scattering, diffraction results can be interpreted as scattering by a fictitious surface, the roughness of which is gentler that the real surface at HH polarization and steeper at VV polarization. The HH/VV polarization ratio (dB), being positive at steep incident angles, gradually decreases as the incident angle increases, and it becomes negative for moderate incident angles.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[2] Wave diffraction by a statistically rough surface plays an important role not only in remote sensing of natural interfaces (such as bare terrain, sea surface, snow cover, and sea ice) by different probing signals (echo sounders in hydroacoustics, radio waves in radars, lasers in optics), but in many other physical problems and engineering applications. The theoretical approaches to this problem can be divided into two main classes: the geometrical (and quasi-geometrical) optics (GO) applied to the surfaces of which roughness scales are much larger than the wavelength of the incident wave, and the small perturbation method (SPM), which deals with small-scale roughness compared to the wavelength [e.g., Beckmann and Spizzichino, 1963; Bass and Fuks, 1972; Ishimaru, 1978; Voronovich, 1998]. To interpret the experimental results obtained in the intermediate scales of roughness parameters (or for multiscale roughness, such as the sea surface and other natural interfaces), some combinations of these two basic methods are used: various two-scale models (TSM), small slope approximation (SSA), full wave approach, etc. (see the latest review on these topics in Elfouhaily and Guérin [2004]).

[3] At small and moderate grazing angles, where specular reflections can be neglected, the resonant Bragg scattering mechanism plays the main role in backscattering from a multiscale, but not extremely rough surface: only the roughness into a narrow bandwidth of a spatial power spectrum (the wavelength of which is of the order of the wavelength of the probing signal) is responsible for backscattering. The remaining part of the power spectrum describes a large-scale component of roughness and plays the role of a modulation factor, which changes the local grazing angle of the incident wave. For this case, experimental observations of the frequency and polarization dependence of the backscattering cross section can be explained in the framework of the SPM and its various modifications (such as TSM or different orders of SSA).

[4] In contrast to the small and moderate grazing angles, where the perturbation theories or their different modifications are successfully employed, the case of steep incident angles and high-frequency probing signals can be described by the Kirchhoff approximation (KA). The equations for backscattering cross sections obtained in the GO limit of the KA do not depend on either the frequency of a probing signal or on its polarization; they are determined only by the probability density function (PDF) of surface slopes, which are responsible for the backscattering specular reflections.

[5] There are only a few papers in which some diffraction corrections to these limiting GO results, following from the KA, were obtained. In McDaniel [1985], the diffraction corrections to the GO were obtained by expansion of the Huygens-Kirchhoff integral in the series of the inverse power of the Rayleigh parameter sin ψ ≫ 1 (where k = 2π/λ is the wave number of the incident field, λ is its wavelength, σ is the r.m.s. height of the surface roughness, and ψ is the grazing angle). Because the field and its derivatives at the rough surface were taken in the tangent plane approximation (TPA) (i.e., in the GO limit), the meaning of the corrections obtained to the GO is not clear. In Lynch [1970], the first-order high-frequency asymptotic corrections to the GO phase of a sound field at a pressure-release 2-D surface (with a Dirichlet boundary condition) were obtained using the variational principle. These results were extended in Rodriguez [1989, 1991] for a 3-D surface and electromagnetic wave scattering by a perfectly conducting surface. These derivations were based on the extinction theorem, and a solution of the boundary value problem was obtained as a perturbation expansion in series of a small momentum transfer, i.e., in a small angle scattering limit, but not as high-frequency asymptotic expansion in series of 1/k. In the papers cited above, only the first-order corrections ∼1/k to the GO fields at a rough surface were obtained, which were orthogonal to the zero-order GO fields; i.e., these corrections were related to the surface field phase, but not to the field amplitude.

[6] The problem of obtaining the solution beyond the TPA, i.e., taking into account the surface curvature effects, was considered by Voronovich and Zavorotny [1998] and Fuks and Voronovich [1999, 2002] in connection with the TSM improving: the zero-order field, which illuminates the small resonant scattering roughness, was corrected by diffraction (or multiple scattering at a concave surface in Fuks and Voronovich [1999]) by large-scale roughness. In Elfouhaily et al. [1999, 2001, 2003a], the field at a large-scale rough surface was considered as a solution of the surface integral equation, the zero-order iteration of which is a field obtained in the TPA. The next two iterations of the surface current integral equation, which give the higher orders of a small momentum transfer, were used to obtain the unified forms of SSA in Elfouhaily et al. [2003b, 2003c, 2004] and Elfouhaily and Johnson [2006].

[7] In this paper, we employ the equations for the surface values of the electric and magnetic field derived in Fuks [2004] by two consecutive iterations of the integral equations at a perfectly conducting 2-D (i.e., cylindrical) surface. These corrections to the GO values (the TPA solutions) were used [Fuks, 2005a, 2005b, 2006] to obtain the diffraction corrections to the scattering cross sections of the solitary specular points at a smooth surface. Here, we apply the results of Fuks [2004] to derive the diffraction corrections ∼1/k2 to the backscattering cross sections from a statistically rough surface at two linear polarizations as functions of the incident angle in a quasi-specular region, where the specular points provide the main contribution to backscattering.

2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[8] Hereinafter, we consider scattering of a monochromatic wave of frequency ω, and the phase factor exp (−iωt) is omitted. In the GO limit, backscattering (retroreflection) from a smooth surface S is caused only by the specular points Os, where the direction n of normal-to-the-surface S is opposite to the direction of the wave vector k of the incident wave ∼exp (ikR), where k = ω/c is the wave number, and c is the speed of light. For a 2-D, i.e., cylindrical surface, we can specify the equation of the surface S in the form of a single-valued function z = Z(x) (cylinder directrix) with the axis Oy directed along the surface S generatrix (Figure 1).

image

Figure 1. Geometry of backscattering.

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[9] The arbitrary polarized incident wave can be represented as a linear superposition of two eigenwaves: the TE wave with the electric field vector E0 = (0, E0, 0) directed along the surface generatrix Oy, and the TM wave with the magnetic field vector H0 = (0, H0, 0). We assume that the incident waves E0(r) and H0(r) have the unitary amplitudes, i.e., E0(r) = ey exp (ikr) for TE waves and H0(r) = ey exp (ikr) for TM waves, where ey is a unit vector along the Oy axis and ky = 0. For a perfectly conducting surface S and for these two cases of polarization, the scattered fields Esc = (0, Esc, 0) and Hsc = (0, Hsc, 0) in an arbitrary point R can be represented as surface integrals according to the Huygens-Kirchhoff principle:

  • equation image
  • equation image

where G0(R, r) is the Green function of the Helmholtz equation for a free space

  • equation image

and ∂/∂n implies the derivative along the normal n to the surface S at the point of integration rS. The total magnetic field H(r) and the normal derivative ∂E(r)/∂n of the total electrical field E(r) at surface S can be represented in the form:

  • equation image
  • equation image

where functions ue and uh are the solutions of the corresponding integral equations (see details in Fuks [2004]), which can be represented as a series in powers of 1/k:

  • equation image

where ue,h(n) ∼ 1/kn. For obtaining the diffraction corrections to the GO values ue(0) = uh(0) = 2 up to the terms of order k−2, it is sufficient to consider only the first two nonvanishing terms in representation of the surface directrix in the local frame of reference (η, Oϑ, ζ), where axis Oϑζ is directed along the normal n in each point rS and axis Oϑη is tangent to it (Figure 1):

  • equation image

[10] Here, a = −(d2ζ/2)η=0−1 is a surface curvature radius, b = −2(d3ζ/3)η = 0−1, and we take into account that / = 0 at η = 0. The following explicit expansions for the surface values of ue,h in the series of 1/k were obtained in [Fuks, 2004]:

  • equation image

where δe = 1, δh = 2, and ϑ is the local incident angle at rS (Figure 1). Note that in equation (21) in Fuks [2004], the last term proportional to sinϑ/bk2, erroneously contains the extra factor “2”. The spatial dependence of fields ue,h(r) on the coordinate rS is given by variations of surface parameters a, b and the local incidence angle ϑ, which are functions of r. Note that (8) was obtained under the assumption that corrections to the GO surface values ue(r) = uh(r) = 2 at every point rS are caused only by local surface deviations from the tangent plane; i.e., multiple scattering or shadowing are not taken into account.

3. Fields in the Far Zone

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[11] To obtain the high-frequency asymptotic expansion of backscattered fields Esc, Hsc, we substitute the surface values of H and ∂E/∂n from (4) and (5) into (1) and (2), correspondingly, and use (8) for ue,h. For the backscattering direction, in a wave zone (kRequation image 1) and at a long distance R significantly exceeding the curvature radius a0 at the specular point Os (for a concave surface the inequality Ra0 guarantees that the observation point is far away from the caustics of the reflected wave field), after integration in (1) and (2) over dy in the infinite limits (implying that the surface dimension in the direction of axis Oy essentially exceeds the linear size of the Fresnel zone equation image in this direction), we obtain:

  • equation image

[12] Here, L is the length of the surface S in the Ox direction, q = −2k is the scattering vector (qx = 2k sin θ, qz = 2k cosθ), θ is the incidence angle in the reference frame (xOz), and Z1 = dZ/dx (hereinafter we use the notation Zn = dnZ/dxn). To perform integration in (9) in an explicit form, it is necessary to have ue,h as a function of x in the frame of reference (xOz), where the surface equation has the form z = Z(x), while (8) gives ue,h in the local frame of reference (ηOϑζ). The frame of reference (η, ζ) can be obtained from the original frame of reference (x, z) by shifting the origin from point O (0, 0) to point Oϑ (x, Z) and its consequent rotation by the angle θ + ϑ. This allows us to express the local surface differential parameter a (curvature radius) introduced in (7), through the second derivative of Z(x) in the original frame of reference (see equation (24) in Fuks [2004]):

  • equation image

[13] In a high-frequency limit, only the small vicinity of the specular point Os, where equation image = 0, is essential in integrals in (9). To obtain the scattered fields in the series of 1/k with accuracy 1/k2, it is possible to neglect the factors that are proportional to sin ' in the second-order terms ∼1/k2 in (8). Taking into account that cos ϑ = −(n · k) =nz(cos equation imageZ1 sin equation image) and cos (θ + ϑ) = nz = 1/equation image, from (8) and (10) we obtain the explicit expressions for surface currents ue,h through the first and the second derivatives of the surface directrix z = Z(x):

  • equation image

[14] In Fuks [2004], equations (8) and (9) were used to obtain the explicit expressions for wave fields backscattered by a solitary specular point. Corresponding equation (41) in Fuks [2004] for backscattering cross sections contains derivatives of the surface profile at the specular point up to the sixth order. The transition from the backscattering cross section from the solitary specular point to the specific backscattering cross section from the extended rough surface is not a simple problem: even in the GO approach, when the backscattering cross section from a solitary specular point depends only on the second derivatives of the surface profile, this transition requires very sophisticated derivations [e.g., Barrick, 1968; Barrick and Bahar, 1981]. In particular, in these papers, it was shown that although the backscattering cross section from a solitary specular point depends on the surface curvature, the statistically averaged backscattering cross section from the extended rough surface is fully determined by the PDF of surface slopes and does not depend on the statistics of the surface curvature.

[15] Here, we skip the stage of derivation of the backscattering cross section from the solitary specular point and will directly perform the statistical averaging of intensities of backscattered fields. For the intensities ∣Esc2 and ∣Hsc2 of scattered fields, from (9) we obtain:

  • equation image

To obtain the representation of ∣Esc2 and ∣Hsc2 in a series of the parameter 1/k, we expand the integrand in (12) in a series of the difference ξ = x′ − x:

  • equation image
  • equation image
  • equation image

where ZnZn(x).

4. Backscattering Cross Sections

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[16] Here, we perform statistical averaging of backscattered field intensities (12) for the particular case of a Gaussian random rough surface. The integrand in (12), being expanded in accordance with (13)–(15), contains three random functions: Z1(x), Z2(x) and Z3(x). Note that for a statistically uniform (homogeneous) rough surface, Z2(x) does not correlate either with Z1(x) or Z3(x) at the same spatial point x. For Gaussian statistics, it results in the independence of Z2 from Z1 and Z3, and therefore averaging over Z2 can be easily performed. While averaging, only the two first statistical moments of Z2 appear: 〈Z2〉 = 0 and 〈Z22〉 = σ22. Then, it is possible to perform averaging of Z3 under the condition that Z1 is given:

  • equation image

where σ12 = 〈Z12〉 is the surface slope variance. The next step is integration over , which can be performed in the infinite limits if the inequality holds:

  • equation image

It results in the appearance of the δ-function and its derivatives δ(n):

  • equation image

The final averaging over the random slopes Z1 by integration of the obtained equations with a probability density function (PDF)

  • equation image

is possible to perform in the explicit form, using the δ(n) -function property:

  • equation image

The last integration over dx results simply in multiplying by factor L, because the final result of statistical averaging does not depend on the spatial coordinate x due to the statistical homogeneity of surface roughness. The specific (per unit length in the Ox direction) backscattering cross sections are introduced by the following definitions:

  • equation image

In problems related to radio wave propagation and scattering, σHH0 corresponds to a horizontal-horizontal backscattering cross section, and σVV0 – to a “vertical-vertical” one. They can be written in the form:

  • equation image
  • equation image
  • equation image

where σ0 is the backscattering cross section in the GO limit:

  • equation image

It is worth mentioning that (25) is valid for an arbitrary PDF of slopes w1, while the diffraction corrections in the forms (23) and (24) were obtained only under the assumption regarding the Gaussian statistics for the surface roughness. It this case, ΔHH and ΔVV are proportional to the second derivative variance σ22 and depend, additionally, on the surface roughness slope variance σ12. Note that diffraction corrections (of the same order ∼1/k2) to the backscattering cross sections from the solitary specular point at the 2-D smooth surface depend on the first six (!) derivatives of the directrix Z(x) [Fuks, 2004, 2005a, 2005b, 2006].

[17] In the GO limit, there is no difference between backscattering cross sections at HH and VV polarizations, so the polarization ratio RHV = σHH/σVV is equal to unity. From (22)–(26), we obtain the diffraction correction ∼1/k2 to the HH/VV polarization ratio:

  • equation image

The last equation differs from the analogous equation (63) in Fuks [2004], where the polarization ratio was introduced as averaged ratio (48) of backscattering cross sections from the solitary specular point, whereas here, by the polarization ratio we mean the ratio of statistically averaged backscattering cross sections for HH and VV polarizations. As was mentioned above, there is no direct and simple relation between the scattering cross sections from the solitary specular point and the specific backscattering cross section from the extended statistically rough surface. Besides, in Fuks [2004] the statistical averaging of polarization ratio (48) for a solitary specular point was not performed in full, because of the divergency of the corresponding integrals, and only the result of the partial averaging of (48) was produced there (equation (67) in Fuks [2004]).

5. Backscattering at Normal Incidence

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[18] The equations obtained above take a simple form in the specific case of normal incidence, θ = 0:

  • equation image
  • equation image
  • equation image

Since the GO backscattering cross section σ0 at normal incidence has the form

  • equation image

the backscattering cross sections σHH and σVV corrected by diffraction can be written in a similar form:

  • equation image

where σ1effH and σ1effV are effective rough surface slope variances of fictitious surfaces SH and SV for horizontal and vertical polarizations, respectively:

  • equation image

[19] It follows from (27) that ΔHH > 0 and, consequently, σ1effH < σ1; i.e., for horizontal polarization, the fictitious surface SH is gentler than the real surface S. In other words, diffraction smoothes the real surface in this case. As for ΔVV, from (28) it follows that ΔVV > 0 only for the gentle roughness with σ12 < (equation image − 1)/14 ≃ 0.08. For a surface with σ12 > 0.08, we have ΔVV < 0 and σ1effV > σ1; i.e., for vertical polarization, the fictitious surface SV is rougher than the real surface S in this case.

[20] Note that σ22 ≡ 〈Z22〉 is a half of the variance of the surface curvature at extrema (Z1 = 0), where specular points are located at normal incidence. We can denote it as 1/a2, where a has the meaning of a characteristic absolute value of the curvature radius. To analyze the dependence of the results obtained on parameters commonly used for describing a rough surface such as height variance σ2 = 〈Z2〉 (assuming 〈Z〉 = 0) and the correlation length l, we can employ the dimension notions to estimate σ22 and σ12 as follows:

  • equation image

Here, C1 and C2 are the dimensionless constants, the values of which depend on the specific form of the surface roughness autocorrelation function W(ξ) = 〈Z (x) Z (x + ξ) 〉. For example, for the Gaussian autocorrelation function

  • equation image

from equations

  • equation image

it is easy to obtain that C1 = 2, C2 = 12, and in this case (27)–(29) take the form:

  • equation image
  • equation image
  • equation image

Here, 1/a2 ≡ 12σ2/l4 is a half of the curvature variance at the surface extrema, where the specular points are located at normal incidence. Since here we consider only the small corrections to the GO results for backscattering cross sections and their ratio, the following inequalities have to hold: ()2, (kl)2, (kl2/σ)2 ≫ 1. In addition, to neglect multiple scattering, we have to consider only gentle surfaces with small enough slopes (σl).

[21] The first terms in (36) and (37), as well as in (27) and (28), are the same for both polarizations. They coincide with the corrections to the backscattering cross sections obtained by Lynch [1970] and Rodriguez [1989] (for Dirichlet and Neumann boundary conditions, which correspond to HH and VV polarizations in our case) and by Rodriguez [1991] for electromagnetic waves scattered by a perfectly conducting 2-D surface. In Lynch [1970] and Rodriguez [1989, 1991], these positive corrections were obtained as the first terms in expansion in a series of a small momentum transfer (which is equivalent to expansion in series of slope variance σ12 ≪ 1, given by the main terms in (27) and (28)). These terms were interpreted in the above-cited papers as a result of surface “smoothing” caused by diffraction. It is seen that at vertical polarization for steep roughness (σ12 > 0.08), diffraction results in roughness “sharpening”, but not “smoothing”. In our opinion, the positive corrections to GO backscattering cross sections proportional to ∼()−2 are caused only by the interference of waves propagating upwards from the surface, where the electric and magnetic fields take the GO values without any diffraction corrections. It is very natural that this result of interference does not depend on polarization, in contrast to the second and third terms in (36) and (37), which are caused by diffraction on the correlation length l (term ∼1/(kl)2) and on the curvature radius a (term ∼1/(ka)2). It is obvious that the limiting case kl, ka corresponds to scattering by a set of almost flat horizontally oriented facets, when there are not any diffraction effects. In this case, dependence of the scattering cross sections on the wavelength is given by the first terms in (36) and (37), originated from the propagation exponent expansion (13), rather than from diffraction corrections (14) to fields at the surface. Note that in the equations obtained above, it is impossible to make a transition to scattering by a perfectly plane surface (kl [RIGHTWARDS ARROW] ∞, σ12 [RIGHTWARDS ARROW] 0) because of inequality (17).

[22] Because the interferential terms do not depend on wave polarizations, they cancel each other out in the correction to the polarization ratio (38), and only the diffraction terms remain. Note that the diffraction corrections to HH and VV polarizations in (36) and (37) have the opposite signs: they are positive for ΔHH and negative for ΔVV.

[23] It is worth noting that initial equations (23) and (24) for high-frequency asymptotic corrections to GO backscattering cross sections depend only on two surface roughness parameters (σ12 and σ22) and do not depend on the roughness height variance σ2, and therefore (36)–(38) relate only to the specific form (34) of the roughness autocorrelation function W(ξ).

6. Diffraction at Slant Incidence

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[24] To analyze the dependence of the diffraction corrections ΔHH, ΔVV, and ΔHV on the incidence angle θ and on surface slopes σ12, we use in (23), (24) and (26) the representation of σ22 through σ2 and σ12 for the Gaussian autocorrelation function (34), which follows from (33):

  • equation image

[25] As a result, (23), (24), and (26) take the form:

  • equation image

where factors FHH and FVV are given by the terms in the square brackets in (23) and (24) multiplied by factor σ14, and FHV is equal to the term in the square brackets in (26) multiplied by factor 2σ14. Thus, these factors are functions only of the incidence angle equation image and surface slope variance σ12. In particular, at normal incidence θ = 0:

  • equation image
  • equation image
  • equation image

It is seen that FHH is always positive at θ = 0, and ΔHH monotonically increases when roughness height variance σ remains constant and surface slope variance σ12 increases. Factor FVV is equal to 1/8 at σ12 = 0, and it decreases when σ12 increases, and becomes negative for σ12 > (equation image − 1)/14 ≃ 0.08. As to the correction ΔHV to the HH/VV polarization ratio, from (40) and (43) it follows that at θ = 0 it is always positive, and it increases when roughness slope variance σ12 increases. From (39) it follows that when σ2 remains constant, increasing of the slope variance σ12 leads to increasing of the surface curvature variance σ22.

[26] In Figures 2–4, the dependence of factors FHH, FVV, and FHV on the parameter (tan θ/σ1)2 is depicted for the set of slope variances σ12. According to (40), these plots represent dependence of ΔHH, ΔHH and ΔHH on (tan θ/σ1)2 for the case σ2 = const. It is seen that FHH, being positive at θ = 0, decreases when the incident angle θ increases, and changes its sign in the vicinity of grazing angles (tan θ/σ1)2 ≃ 0.7 for the slope variance σ12 in the interval (0.1-0.5). The factor FVV has the opposite behavior: being negative at θ = 0 (for σ12 ≳ 0.08) it increases when θ increases and becomes positive for (tan θ/σ1)2 ≳ 0.8 for the same interval (0.1–0.5) of slope variance σ12. The HH/VV polarization ratio factor FHV dependence on the incident angle is very similar to FHH: it is positive for small incident angles, then changes its sign at (tan θ/σ1)2 ≃ 0.75 and remains negative for more slant angles of incidence.

image

Figure 2. Diffraction corrections FHH to backscattering cross-section σHH at horizontal polarization as functions of a normalized grazing angle θ, for different roughness slope variances σ12.

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image

Figure 3. Diffraction corrections FVV to backscattering cross-section σVV at vertical polarization as functions of a normalized grazing angle θ, for different roughness slope variances σ12.

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image

Figure 4. Diffraction corrections FHV to HH/VV polarization ratio RHV as functions of a normalized grazing angle θ, for different roughness slope variances σ12.

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[27] It is worth noting that for large incident angles tan θ ≳ σ1, it is necessary to take into account surface shadowing which is neglected here. As incident angle θ increases, the contribution of specular points to the backscattering cross section decreases exponentially, as follows from (25) for Gaussian slope PDF, and other scattering mechanisms (such as resonant Bragg scattering by small-scale roughness) start to play the main role in backscattering. For this reason, applicability equations (40), with angular dependent factors FHH, FVV and FHV depicted in Figures 2–4, is bounded by small incident angles tan θ ≲ σ1.

7. Interpretation of Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[28] It is possible to give a simple qualitative interpretation of the angular dependence of the diffraction corrections shown in Figures 2 and 3. It was shown by Fuks [2005b] (see equation (30) therein) that diffraction by a smooth curved surface results in the appearance of the phase difference Δϕeh between HH and VV specularly reflected waves:

  • equation image

where index “e” corresponds to HH polarization, index “h” corresponds to VV polarization, and a is the surface curvature radius at the specular point. Note that Δϕeh > 0 for convex specular points with a > 0, and Δϕeh < 0 for the concave ones (a < 0). This phase difference can be interpreted as a splitting of one real GO specular point into two spurious specular points (one for a HH polarized wave, and another for a VV polarized one), separated by the distance Δleh = Δϕeh/k. Thus, the specular point for HH polarization is located inside the convex surface and outside the concave one, and vice versa for VV polarization (see Figure 2 in Fuks [2005b]). For example, the spurious specular point for HH polarization at the surface of a circular cylinder is located inside the cylinder at the distance Δle = Δϕe/k = 5/(16k2a) (see equation (46) in Fuks [2005b]), and the spurious specular point for VV polarization is located outside the cylinder at the distance Δlh = Δϕh/k = 11/(16k2a) from the surface. For a parabolic type of the specular point (see equation (59) in Fuks [2005b]), these distances are equal to each other Δle = Δlh = 1/(2ak2). This divarication of the specular points results in flattening out of the real surface S to the effective “fictitious” scattering surface SH for HH polarization (it becomes more gentle, as shown in Figure 5a by the short-dashed curve), and conversely, it results in sharpening the real surface S to the steeper surface SV for VV polarization (as shown in Figure 5a by the long-dashed curve). The resulting backscattering cross sections are proportional to the corresponding slope PDF of the effective surfaces, as schematically shown in Figure 5b. It is seen that for steep incident angles, the inequalities σHH0 > σ00 > σVV0 hold (i.e., ΔHH > 0, ΔVV < 0), and for slant incidence (more grazing angles), the opposite inequalities σHH0 < σ00 < σVV0 (i.e., ΔHH < 0, ΔVV > 0) hold.

image

Figure 5. (a) A schematic sketch of a real surface S (solid curve) and fictitious scattering surfaces SH (short-dashed curve) and SV (long-dashed curve) for horizontal (HH) and vertical (VV) polarizations, respectively. (b) A schematic sketch of backscattering cross sections as functions of incident angle θ: σ0 is a GO result (solid line), σHH and σVV correspond to horizontal (short-dashed line) and vertical (long-dashed line) polarizations, respectively.

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[29] To support this qualitative interpretation, let us consider the diffraction corrections to the reflection coefficients of the coherent fields. Before applying the statistical averaging to scattered fields given by (9), we notice that at backscattering, the coherent fields are equal to zero for all incident angles θ, except for θ = 0. Then we can represent the surface fields (11) in the equivalent form (with an accuracy up to ∼1/k2):

  • equation image
  • equation image

Transition from (11) to (45) and (46) is quite similar to the “cumulant expansion” employed by Winebrenner and Ishimaru [1985a, 1985b] and Shen and Maradudin [1980], and used by Rodriguez [1989, 1991]. Substituting equations (45) and (46) into (9), we can perform statistical averaging over the two variables Z and Z2 by integration (9) multiplied by their joint PDF w2(Z, Z2):

  • equation image

where σ2 = 〈Z2〉 is the roughness height variance, and ρ = 〈ZZ2〉/σσ2 = −σ12/σσ2 is the correlation coefficient between Z and Z2. Retaining only the main two terms in the exponent and neglecting the preexponential factors, we obtain the coherent reflection coefficients for horizontal (RH) and vertical (RV) polarizations:

  • equation image

It is seen that diffraction results in changing the real surface roughness height r.m.s. σ to the “effective” r.m.s. σeffH and σeffV for H- and V-polarized waves, respectively:

  • equation image

For H-polarization, diffraction results in fictitious decreasing of the surface roughness height r.m.s. σ to the “effective” r.m.s. σeffH = σδσ, and increasing it for V-polarization σeffV = σ + δσ by the same value δσ = σ12/4k2σ. Notice that the ratio σ12/σ has the order of magnitude of the surface curvature r.m.s. at extrema where Z1 = 0 (and, correspondingly, the inverse quantity is equal to the characteristic curvature radius a). In the particular case of Gaussian autocorrelation function (34), when (33) holds with C1 = 2 and C2 = 12, the “shift” δσ of the surface height r.m.s. σ takes the form:

  • equation image

where a ≡ 1/σ2. This change of δσ of the effective surface roughness height r.m.s. σ agrees with estimations made above in this section for shifting Δle and Δlh of the actual reflection points to the spurious ones for H- and V-polarized waves.

[30] The change of the roughness height r.m.s. σ is equivalent to changing the surface slope r.m.s. σ1 by the quantity δσ1 = ∓σ13/(2)2: the fictitious scattering surface SH for HH polarization is gentler than the real one (has a smaller slope r.m.s.), and for VV polarization the fictitious scattering surface SV is steeper than the real surface S. In its turn, it results in narrowing the backscattering diagram for HH polarization and broadening it for VV polarization in comparison with the GO equation (25).

8. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References

[31] From the results obtained above, it follows that at normal incidence (θ = 0), diffraction by surface roughness results in increasing the backscattering cross section at HH polarization (ΔHH > 0) and decreasing it for steep enough roughness at VV polarization (ΔVV < 0) in comparison with the GO result. The HH/VV polarization ratio (dB), being positive at steep incident angles, gradually decreases as the incident angle increases and becomes negative for moderate incident angles. This can be interpreted as scattering in GO by the fictitious surfaces SV and SH with a slope variance σ12 dependent on the wavelength and polarization: it is greater than the real surface slope variance for VV polarization and less for HH polarization.

[32] When the incident angle increases, the absolute value of these corrections decreases, and they change their signs at some incident angles θ, where (tan θ/σ1)2 ≃ 0.7–0.8 (depending on the σ12 value). Nullification of diffraction corrections ΔHH and ΔVV at some incident angles means that diffraction corrections are there of a higher order of smallness than ∼k−2; i.e., the GO results for the backscattering cross sections at these incident angles have a wider range of applicability, and the GO approximation is valid for a longer wavelength than at other angles. Note that the quite similar dependence of the diffraction corrections ΔHH and ΔVV on the incident angle θ exists for the specular points at the surface of cylinders with a conic section directrix [Fuks, 2005a, 2005b]: ΔHH is positive and ΔVV is negative at normal incidence, when the specular point is located at the ellipse apex (the point with the maximum curvature); then, when the incident angle θ increases, the specular point is shifting down from the apex to the places with a smaller curvature and the absolute magnitudes of ΔHH and ΔVV are decreasing (see Figures 2 and 3 in Fuks [2005a]). In contrast to a solitary specular point, when ΔHH, ΔVV and ΔHV depend on the first six derivatives of the surface directrix Z(x), the equations obtained above contain only two independent parameters of surface roughness: the variance of slopes σ12 and the variance of the second derivatives σ22 (which coincides with a half of the curvature variance at the surface extrema).

[33] It is necessary to mention that the results obtained in this paper are based on the local diffraction corrections to GO fields at every point on a rough surface, which depend only on the surface derivatives at this point. In other words, none of the nonlocal effects, such as multiple scattering and shadowing, are taken into account in the framework of this theory.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Asymptotic Expansions for the Wave Fields at a Perfectly Conducting Surface
  5. 3. Fields in the Far Zone
  6. 4. Backscattering Cross Sections
  7. 5. Backscattering at Normal Incidence
  8. 6. Diffraction at Slant Incidence
  9. 7. Interpretation of Results
  10. 8. Conclusions
  11. Acknowledgments
  12. References
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