High-order high-frequency solutions of rough surface scattering problems

Authors


Abstract

[1] A new method is introduced for the solution of problems of scattering by rough surfaces in the high-frequency regime. It is shown that high-order summations of expansions in inverse powers of the wave number can be used within an integral equation framework to produce highly accurate results for surfaces and wavelengths of interest in applications. Our algorithm is based on systematic use and manipulation of certain Taylor-Fourier series representations and explicit asymptotic expansions of oscillatory integrals. Results with machine precision accuracy are presented which were obtained from computations involving expansions of order as high as 20.

1. Introduction

[2] Computations of electromagnetic scattering from rough surfaces play important roles in a wide range of applications, including remote sensing, surveillance, nondestructive testing, etc. The problem of evaluating such scattering returns is rather challenging, owing to the multiple-scale nature of rough scatterers, whose spectra may span a wide range of length scales [Valenzuela, 1978].

[3] A number of techniques have been developed to treat limiting cases of this problem. For example, the high-frequency case, in which the wavelength λ of the incident radiation is much smaller than the characteristic surface length scales, has been treated by means of low-order asymptotic expansions, such as the Kirchhoff approximation. On the other hand, resonant problems where the incident radiation wavelength is of the order of the roughness scale have been treated by perturbation methods, typically first- or second-order expansions in the height h of the surface [Rice, 1951; Shmelev, 1972; Mitzner, 1964; Voronovich, 1994]. However, when a multitude of scales is present on the surface, none of these techniques is adequate, and attempts to combine them in a so-called two-scale approach have been made [Kuryanov, 1963; McDaniel and Gorman, 1983; Voronovich, 1994; Gil'Man et al., 1996]. The results provided by these methods are not always satisfactory, owing to the limitations imposed by the low orders of approximation used in both the high-frequency and the small-perturbation methods.

[4] A new approach to multiscale scattering, based on use of expansions of very high order in both parameters λ and h, has been proposed recently [Bruno et al., 2000]. These combined methods, which are based on complex variable theory and analytic continuation, require nontrivial mathematical treatments; the resulting approaches, however, do expand substantially on the range of applicability over low-order methods and can be used in some of the most challenging cases arising in applications. Perturbation series of very high order in h have been introduced and used elsewhere to treat resonant problems, in which the wavelength of radiation is comparable to the surface length scales [Bruno and Reitich, 1993a, 1993b, 1993c; Sei et al., 1999]. In this paper we focus on our high-order perturbation series in the wavelength λ, which, as we shall show, exhibits excellent convergence in the high-frequency, small-wavelength regime. The combined (h,λ) perturbation algorithms for multiscale surfaces, which require as a main component the accurate high-frequency solvers presented in this paper, are described by Bruno et al. [2000].

[5] Our approach to the present high-frequency problem uses an integral equation formulation, whose solution ν is sought and obtained in the form of an asymptotic expansion

equation image

with p = -1 for transverse magnetic (TM) polarization and p = 0 for transverse electric (TE) polarization. This expansion is similar in form to the geometrical optics series [Lewis and Keller, 1964]

equation image

where S = S(x,y) is the unknown phase of the scattered field. Note that the phase of the density ν of equation (1) is determined directly from the geometry and the incident field and, unlike that in the geometrical optics field, it is not an unknown of the problem. In particular, the present approach does not require solution of an eikonal equation [Vidale, 1988; VanTrier and Symes, 1991; Fatemi et al., 1995; Benamou, 1999], and it bypasses the complex nature of the field of rays, caustics, etc.

[6] The validity of the expansion (2) has been extensively studied [Friedlander, 1946; Luneburg, 1944, 1949a, 1949b; Van Kampen, 1949]; in particular, it is known that equation (2) needs to be modified in the presence of singularities of the scattering surface. To treat edges and wedges, for example, an expansion containing powers of k−1/2 [Luneburg, 1949b; Van Kampen, 1949; Keller, 1958; Lewis and Boersma, 1969; Lewis and Keller, 1964] must be used; caustics and creeping waves also lead to similar modified expansions [Kravtsov, 1964; Brown, 1966; Ludwig, 1966; Lewis et al., 1967; Ahluwalia et al., 1968]. Proofs of the asymptotic nature of expansion (2) were given in cases where no such singularities occur [Miranker, 1957; Bloom and Kazarinoff, 1976]. In practice, only expansions (2) of very low orders (one or, at most, two) have been used, owing in part to the substantial algebraic complexity required by high-order expansions [Bouche et al., 1997]. First-order versions of expansion (1), on the other hand, were treated by Lee [1975], Chaloupka and Meckelburg [1985], and Ansorge [1986, 1987].

[7] The region of validity of our asymptotic expression (1), on the other hand, corresponds to configurations where no shadowing occurs. At shadowing, the wave vector of the incident plane wave is tangent to the surface at some point, which causes certain integrals to diverge; see section 4. Thus a different kind of expansion, in fractional powers of 1/k, should be used to treat shadowing configurations: A first-order version of such an expansion was discussed by Hong [1967]; see Friedlander and Keller [1955], Lewis and Keller [1964], Brown [1966], and Duistermaat [1992] for the ray-tracing counterpart.

[8] In this paper we show that high-order summations of expansion (1) can indeed be used to produce highly accurate results for surfaces and wavelengths of interest in applications for both TE and TM polarizations; in section 7, for example, we present results with machine precision accuracy, which were obtained from computations involving expansions of order as high as 20. Our algorithm is based on systematic use and manipulation of certain Taylor-Fourier series representations, which we discuss in section 5. Operations such as product, composition, and inversion of Taylor-Fourier series lie at the core of our algebraic treatment; as shown in section 5, certain numerical subtleties associated with these operations require a careful treatment for error control.

[9] In order to streamline our discussion we first treat, in sections 25, the complete formalism in the TE case; the changes necessary for the TM case are then described in section 6. In detail, in section 2 we present our basic recursive formula for the evaluation of the coefficients νn(x) of equation (1) for the TE case. These coefficients depend on certain explicit asymptotic expansions of integrals, which we present in sections 3 and 4. A discussion of the Taylor-Fourier algebra then ensues in section 5. As we said, the modifications necessary for the TM case are discussed in section 6. A variety of numerical results for both TE and TM polarizations, finally, are presented in section 7.

2. High-Frequency Integral Equations: TE Case

[10] The scattered field u = u(x,y) induced by an incident plane wave impinging on the rough surface y = f(x) under TE polarization is the solution of the Helmholtz equation with a Dirichlet boundary condition. As is known [Voronovich, 1994], the field u(x,y) can be computed as an integral involving a surface density ν(x,k) and the Green's function G(x,y,x′,y′) for the Helmholtz equation

equation image

where ν satisfies the boundary integral equation

equation image

In what follows we will use the relations

equation image

where H11 is the Hankel function, θ is the incidence angle measured counterclockwise from the vertical axis, and k = 2π/λ is the wave number.

[11] A useful form of the integral equation (4) results as we factor out the rapidly oscillating phase function eiαxiβf(x)

equation image

which cancels the fast oscillations in all nonintegrated terms and thus suggests use of an asymptotic expression of the form

equation image

Substitution of the asymptotic expression (6) into equation (5) then yields

equation image

where

equation image

To solve equation (7), we use asymptotic expansions for the integrals In(x,k), collect coefficients of each power of 1/k, and then determine, recursively, the coefficients νn(x). In detail, we obtain in section 3 an expansion which gives In(x,k) in terms of derivatives of νn(x)

equation image

where the functions Bq−l(x) are determined from the profile and incidence angle only. (We point out, however, that our algebraic treatment yields an expression for Iqn(x) which, although equivalent to that of equation (8), is different in form; see section 3 and equation (25).) From equations (7) and (8) we then find a recursion which gives νn(x) as a linear combination of derivatives of the previous coefficients νn−1−q(x)

equation image

Use of the Taylor-Fourier algebra of section 5 allows us to perform accurately the high-order differentiations required by our high-order expansions; the needed expansion (8) of the integral In, in turn, is the subject of section 3.

3. Asymptotic Expansion of In(x,k)

[12] We first split the integral In as a sum In = In + I+n where

equation image

We evaluate in detail the asymptotic expansion for I+n (x,k); the corresponding expansion for In then follows analogously.

[13] Using t = x′ − x, we obtain

equation image

where

equation image

For the treatment presented here, f(x) is assumed to satisfy the condition

equation image

so that the map t ↦ϕ+(x,t) is invertible. (This condition is generally satisfied by rough surfaces considered in practice: For a sinusoidal profile f(x) = acos (x), for example, the inequality (11) holds as long as a < 1.) Then setting

equation image

equation (10) becomes

equation image

Calling

equation image

I+n(x,k) reduces to

equation image

The unknown νn(x) is contained as a factor in the function F+n(x,u). Notation (12) is useful in that it helps present the integrand as a product of two distinct factors: a nonoscillatory component F+n(x,u) and an oscillatory component equation image.

[14] In addition to equation (11) we assume that the profile y = f(x) is an analytic function, so that the map uF+n(x,u) is analytic as well. Using the Taylor series

equation image

the integral I+n(x,k) takes the form

equation image

Thus the 1/k expansion of I+n results from the corresponding expansions of the integrals

equation image

These nonconvergent integrals must be reinterpreted by means of analytic continuation, in a manner similar to that used in the definition and manipulation of Mellin transforms [Bleistein and Handelsman, 1986]. An explicit expansion of A+(k,m,x) is given in section 3.1.

3.1. Expansion of the Integrals I+n and In

[15] Using the Taylor expansion of

equation image

in the variable u together with the identity ψ+(x,0) = 0, expansion of the function exp {−i [kψ+(x, {v/k) − ψ1+(x)v]} leads to the expression

equation image

for certain (function) coefficients al,n+(x). Then, defining

equation image
equation image

we obtain

equation image

and the series

equation image

for I+n results. We see that this expansion is given in terms of the integrals (17), which, like those of equation (15), are nonconvergent and require analytic continuation. An explicit expression for this integral as a function of p and x is given in section 4.

[16] The case of In(x,k) can be treated similarly: We use t = xx′ and the definitions

equation image

Then, letting al,n(x) be the coefficients in the expansion of equation image

equation image

calling

equation image

and defining the functions An(m,x) by

equation image

we obtain the expansion for the integral In:

equation image

3.2. A Simplified Expression for the Integral In(x,k)

[17] The expansions for I+n(x,k) and In(x,k) can be combined into an expression which depends only on the functions pn,l+(x), al,n+(x), and

equation image

Indeed, using the identity ϕ(x, t) = ϕ+(x, − t), we find

equation image

The 1/k-expansion of In(x, k) now follows from equations (20), (22), and (24)

equation image

Using equations (18), (21), and (24), we thus obtain our key formula

equation image

[18] As we have seen, the function S(m,x) is defined by divergent integrals; an explicit expression for this function is given in section 4. The coefficients aj,m+ and pn,m+, in turn, are defined as products, quotients, compositions, and inverses of certain power series expansions; accurate methods for such manipulations of power series are given in section 5. Note that aj,l+(x) and S(q + j,x) depend on the scattering profile and incidence angle only; the coefficient pn,m+ depends on the geometry and the derivatives of the coefficient νn of order ≤ m.

4. Computation of S(q,x)

[19] Interestingly, a closed-form expression can be given for the function S(q, x). Indeed, since ψ1(x) = ψ1+(x), we may write

equation image

or

equation image

Using formulas (11.4.19) and (11.4.16) of Abramowitz and Stegun [1964] together with the Taylor expansion of sin (x) and cos (x), we then obtain the closed-form expression

equation image

Thus S(q, x) is a series in powers of ψ1+(x), whose coefficients can be evaluated explicitly in terms of the Γ function.

[20] It is easy to see that the radius of convergence of the power series in equation (26) is 1 and that the series actually diverges for ψ1+ = 1. This condition has an interesting physical interpretation; since

equation image

the condition ψ1+(x) = 1 is equivalent to

equation image

or, equivalently, some rays in the incident plane wave are tangent to the scattering surface. Alternatively, using the asymptotic expansion

equation image

we see that the oscillatory term in A0+(p,x) is equation image, which becomes nonoscillatory for ψ1+(x) = 1 and thus causes the integral to diverge. Therefore, as mentioned in the introduction, the present algorithm applies only to configurations for which no shadowing occurs. Extensions of these methods to configurations including shadowing are forthcoming.

[21] In addition to the infinite series (26), the function S admits a finite closed-form representation, namely,

equation image

where Pq(x) is a polynomial of degree equal to the integer part of q/2. These polynomials can be computed easily and efficiently through a Taylor expansion of the product S(q,ψ)(1−ψ2)(q + 3/2). For example, for the first few values of q we have

equation image

5. Computation of pn,q-l+ and aj,l+: Taylor-Fourier Algebra

[22] As indicated previously in section 3, the functions pn,q−l+ and aj,l+ in equations (12), (13), and (16) can be obtained through manipulations of Taylor−Fourier series, which we define, quite simply, as Taylor series whose coefficients are Fourier series. Thus a Taylor-Fourier series f(x, t) is given by an expression of the form

equation image

The manipulations required by our methods include sum, products, and composition, as well as algebraic and functional inverses. These operations need to be implemented with care, as we show in what follows.

[23] Compositions and inverses of Taylor-Fourier series require consideration of multiplication and addition, so we discuss the latter two operations first. Additions do not pose difficulties: Naturally, they result from addition of coefficients. Multiplications and divisions of Taylor-Fourier series, on the other hand, could, in principle, be obtained by means of fast Fourier transforms (FFTs) [Press et al., 1992]. Unfortunately, such procedures are not appropriate in our context. Indeed, as we show below, the very rapid decay of the Fourier and Taylor coefficients arising in our calculations is not well captured through convolutions obtained from FFTs. Since an accurate representation of this decay is essential in our method, which, based on high-order differentiations of Fourier-Taylor series, greatly magnifies high-frequency components, an alternate approach needs to be used.

[24] Before describing our accurate algorithms for manipulation of Taylor-Fourier series, we present an example illustrating the difficulties associated with use of FFTs in this context. We thus consider the problem of evaluating the subsequent derivatives of the function

equation image

through multiplication and differentiation of Fourier series. For comparison purposes we note that S actually admits the closed form

equation image

the value a = 10 is used in the following tests.

[25] In Table 1 we present the errors resulting in the evaluation of a sequence of derivatives of the function S at x = 0 through two different methods: FFT and direct summation of the convolution expression. (Here errors were evaluated by comparison with the corresponding values obtained from direct differentiation of expression (28) by means of an algebraic manipulator.) We see that, as mentioned above, use of Fourier series obtained from FFTs leads to substantial accuracy losses. Indeed, FFTs evaluate the small high-order Fourier coefficients of a product through sums and differences of “large” function values, and thus they give rise to large relative errors in the high-frequency components. These relative errors are then magnified by the differentiation process, and all accuracy is lost in high-order differentiations: Note the increasing loss of accuracy that results from use of a larger number of Fourier modes in the FFT procedure. The direct convolution, on the other hand, does not suffer from this difficulty. Indeed, direct convolutions evaluate a particular Fourier coefficient an of a product of series through sums of terms of the same order of magnitude as an. The result is a series whose coefficients are fully accurate in relative terms, so that subsequent differentiations do not lead to accuracy losses. We point out that full double precision accuracy can be obtained for derivatives of orders 20 and higher provided that sufficiently many modes are used in the method based on direct convolutions.

Table 1. Values of the Derivatives of the Function S(x) at x = 0 for Various Orders of Differentiationa
OrderExact Value at 020 Modes30 Modes40 Modes
FFTConv.FFTConv.FFTConv.
  • a

    The columns marked “20 modes,” “30 modes,” and “40 modes” list the relative errors of the derivatives computed by summing differentiated Fourier series truncated at 20, 30, and 40 modes, respectively. Columns FFT and Conv. resulted from use of Fourier coefficients obtained through FFTs and direct convolution, respectively. Read −7.376924249352232e-01 as −7.376924249352232 × 10−1.

2−7.376924249352232e-011.3e-125.4e-161.7e-112.7e-161.0e-112.7e-16
42.377008924791275e+006.8e-113.0e-144.2e-092.9e-159.8e-102.9e-15
6−1.702966504696142e+015.5e-101.8e-124.6e-070.0e-001.7e-070.0e-00
82.184589621949499e+022.7e-085.8e-112.9e-052.3e-154.1e-052.3e-15
10−4.361708943655447e+038.8e-071.2e-091.1e-036.9e-163.8e-036.9e-16
121.248619506829422e+051.4e-051.7e-083.1e-020.0e-002.2e-018.0e-16
14−4.844671808314213e+061.4e-041.8e-076.5e-016.0e-158.7e+001.0e-15
162.445839768254340e+081.1e-031.5e-061.0e+011.3e-132.7e+021.6e-15
18−1.557531553377787e+106.5e-039.4e-061.3e+021.9e-126.6e+030.0e-00
201.220898732494702e+123.1e-025.0e-051.4e+032.2e-111.3e+058.2e-16

[26] In addition to sums and multiplications our approach requires use of algorithms for composition as well as algebraic and functional inverses of Taylor-Fourier series. In view of the previous considerations a few comments will suffice to provide a complete prescription. Compositions result from iterated products and sums of Fourier series, and thus they do not present difficulties. As is known from the theory of formal power series [Cartan, 1963], functional inverses of a Taylor-Fourier series (27) with f0 = 0 results quite directly once the algebraic inverse of the Fourier series f1(x) ≠ 0 is known. We may thus restrict our discussion to evaluation of algebraic inverses of Fourier series.

[27] As in the case of the product of Fourier series, two alternatives can be considered for the evaluation of algebraic inverses. One of them involves point evaluations and FFTs; in view of our previous comments it is clear that such an approach would not lead to accurate numerics. An alternative approach, akin to use of a direct convolution in evaluation of products, requires solution of a linear system of equations for the Fourier coefficients of the algebraic inverse. In view of the decay of the Fourier coefficients of smooth functions, such linear systems can be truncated and solved to produce the coefficients of inverses with high accuracy.

[28] In sum, manipulations of Taylor-Fourier series should not use point-value discretizations if accurate values of functions and their derivatives are to be obtained. The approach described in this section calls, instead, for operations performed fully in Fourier space. In practice we have found that the procedures described here produce full double precision accuracies for all operations between Taylor-Fourier series and their subsequent high-order derivatives in very short computing times.

6. High-Frequency Integral Equations: TM Case

[29] In the transverse magnetic (TM) polarization the scattered field u = u(x, y) induced by an incident plane wave impinging on the rough surface y = f(x) is the solution of the Helmholtz equation with a Neumann boundary condition. The field u(x, y) can be computed [Voronovich, 1994] as an integral involving a surface density ν(x,k) and the Green's function G(x, y, x′, y′) for the Helmholtz equation

equation image

where ν satisfies the boundary integral equation

equation image

In what follows we will use the relations

equation image

where H11 is the Hankel function, θ is the incidence angle measured counterclockwise from the vertical axis, and k = 2π/λ is the wave number. Since

equation image

calling equation image, we can rewrite equation (30) as follows:

equation image

As in section 2, a useful form of the integral equation (31) results as we factor out the rapidly oscillating phase function eiαxiβf(x)

equation image

which cancels the fast oscillations in all nonintegrated terms. Using an expansion for the function equation image(x,k) similar to equation (6)

equation image

in equation (32) then yields

equation image

where

equation image

The solution of equation (34) requires asymptotic expansions for the integrals In(x, k). These expansions are obtained, as in the TE case, by the methods of section 3. In particular, we obtain the following expressions for the coefficients equation imagen(x):

equation image

7. Numerical Results

[30] Our numerical method proceeds to obtain the integral densities ν(x, k) through equation (6) in TE polarization and equation (33) in TM polarization, with coefficients νn and equation imagen obtained from equations (9) and (35), respectively, and with Iqn given by equation (25). The Taylor-Fourier expansions required in equation (12) for the functions ϕ+−1 and g/ϕ′+ are precomputed, as they depend only on the profile f and they are independent of wave numbers, incidence angles, etc. The precomputation time was 0.5 s for the profile of Figure 1a and Figure 1b and 0.7 s for the profile of Figure 1c. (This and all subsequent calculations were performed in a DEC Alpha workstation (600 MHz).) Once the density has been obtained, all field-related quantities can be evaluated easily from equation (3) in the TE case and equation (29) in the TM case.

Figure 1.

Various profiles considered in the text: (a) f(x) = (h/2) cos (2πx), (b) f(x) = (h/2)[cos (2πx) + cos (4πx)], and (c) f(x) = (h/2)[−cos (2πx) + 0.35cos (4πx) − 0.035 cos (6πx)].

[31] In this section we present the results produced by our algorithm for the energy radiated in the various scattering directions. To do this, we use the periodic Green's function equation image of period d [Petit, 1980]

equation image

to obtain from equation (3) (TE) or from equation (29) (TM) the Rayleigh series for the scattered field

equation image

Here, the coefficients Bn are “Rayleigh amplitudes,” which are given in TE polarization by

equation image

and in TM polarization by

equation image

The required integrals were computed by means of the trapezoidal rule, which for the periodic functions under consideration is spectrally accurate and can be computed very efficiently by means of the FFT.

[32] Our numerical results show values and errors corresponding to the “scattering efficiencies” en [see Petit, 1980], which are defined by

equation image

and which give the fraction of the energy which is scattered in each one of the (finitely many) scattering directions. To test the accuracy of our numerical procedures, we compare our high-frequency (HF) results to those of the method of variation boundaries (MVB) [Bruno and Reitich, 1993a, 1993b, 1993c] in an “overlap” wavelength region, in which both algorithms are very accurate; additional results, in regimes beyond those that can be resolved by the boundary variation method, are also presented. Note that the HF and MVB methods are substantially different in nature: One is a high-order expansion in 1/k whereas the other is a high-order expansion in the height h of the profile. In Tables 210 we list relative errors for the computed values of scattered energies in the various scattering directions. The figures given in the columns denoted by orders 0–19 are the relative errors for the values of the scattered energy calculated from the high-frequency code to orders 0–19 in the corresponding scattering direction. In all cases, errors were evaluated through comparison with a highly accurate reference solution; in Tables 28 the reference solution was produced by means of the boundary variations code mentioned above; in Tables 9 and 10 the reference solution was obtained through a higher-order application of our high-frequency algorithm (order 15). The first term in the high-frequency expansion happens to coincide with the classical Kirchhoff approximation. Note that the Kirchhoff approximation can also be obtained as the zeroth-order term of the Neumann series for equation (4). We emphasize, however, that the high-frequency method used in this paper is of a completely different nature compared to that arising from use of Neumann series.

Table 2. Results for the Profile of Figure 1a in TE Polarization With h = 0.025, λ = 0.025, and Incidence Angle θ = 30°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 11Order 17
  • a

    The run time was 10 s for the calculation of order 17.

07.538669511479800e-043.8e-026.3e-051.1e-085.0e-122.4e-13
11.194293110668300e-012.0e-043.1e-064.9e-092.8e-132.2e-14
24.713900020760300e-031.5e-021.7e-052.7e-086.5e-133.3e-14
39.472951023686101e-022.4e-035.8e-051.7e-091.2e-134.0e-15
41.606247510782500e-011.2e-048.9e-052.8e-107.3e-148.6e-15
58.121747375826800e-021.5e-031.3e-042.3e-083.5e-147.9e-15
62.068175899532900e-022.7e-031.9e-041.2e-083.1e-134.4e-15
73.171379802403400e-033.9e-032.5e-043.5e-081.5e-135.3e-15
Table 3. Results for the Profile of Figure 1a in TM Polarization With h = 0.025, λ = 0.0251, and Incidence Angle θ = 30°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 11Order 17
  • a

    The run time was 10 s for the calculation of order 17.

01.148002904781718e-033.1e-023.4e-052.6e-097.2e-134.0e-13
11.196487185464779e-011.5e-041.1e-052.2e-111.8e-143.3e-14
23.930863630633380e-031.6e-022.9e-052.8e-101.5e-131.1e-13
39.729981709870275e-022.4e-032.3e-052.6e-104.3e-143.7e-14
41.603959300090739e-011.4e-044.0e-053.6e-101.9e-142.4e-14
57.991729015156721e-021.5e-036.5e-052.8e-101.3e-146.9e-15
62.011674295356339e-022.7e-049.8e-053.4e-104.0e-141.3e-14
73.052813099869383e-033.9e-031.4e-042.5e-094.4e-149.6e-14
Table 4. Results for the Profile of Figure 1a in TM Polarization With h = 0.025, λ = 0.025, and Incidence Angle θ = 30°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 11Order 17
  • a

    The run time was 10 s for the calculation of order 17.

06.978718873398379e-0044.0e-024.8e-053.2e-094.0e-131.6e-15
11.193803726254851e-0012.1e-041.1e-051.9e-111.3e-149.3e-16
24.854671479355886e-0031.5e-022.5e-052.6e-103.9e-145.4e-16
39.427330239288337e-0022.4e-032.3e-052.5e-106.8e-152.9e-16
41.606619051666006e-0011.1e-043.9e-053.5e-104.3e-155.2e-16
58.146471443830940e-0021.5e-036.4e-052.8e-104.3e-150.0e-16
62.079411505463193e-0022.7e-049.6e-053.1e-102.2e-141.0e-15
73.195973191313253e-0033.9e-031.4e-042.3e-091.4e-131.9e-15
Table 5. Results for the Profile of Figure 1b in TE Polarization With h = 0.01, λ = 0.025, and θ = 0°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 9Order 11
  • a

    The run time was 5 s for the order 11 calculation.

01.983702874853860e-011.4e-036.7e-066.3e-101.8e-139.8e-16
12.125625186015414e-024.3e-037.3e-068.6e-104.6e-141.5e-14
25.109656298137152e-024.2e-035.3e-061.3e-093.5e-141.1e-14
31.350594564861170e-011.1e-033.0e-061.8e-105.1e-146.6e-15
41.670755436364386e-024.3e-03928e-061.5e-091.8e-132.7e-15
51.041839113172000e-018.7e-041.1e-054.7e-103.8e-142.9e-15
63.029977474761340e-027.6e-041.1e-054.8e-101.0e-159.0e-15
72.828409217693459e-022.5e-033.2e-051.8e-093.5e-149.4e-15
Table 6. Results for the Profile of Figure 1b in TM Polarization With h = 0.01, λ = 0.025, and θ = 0°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 9Order 11
  • a

    The run time was 4 s for the order 11 calculation.

01.985778821348800e-011.3e-031.1e-055.6e-119.1e-155.0e-15
12.203189065423864e-024.3e-036.1e-068.5e-122.0e-142.2e-14
24.989624245086630e-024.2e-031.2e-051.3e-106.8e-158.3e-15
31.363942224141270e-011.1e-033.0e-066.6e-111.1e-146.7e-15
41.685456723960805e-024.3e-031.7e-051.5e-102.7e-141.5e-14
51.040033802018770e-018.9e-049.9e-078.4e-112.9e-152.0e-15
62.994981016528542e-027.8e-041.5e-061.9e-101.4e-151.0e-14
72.795532080716518e-022.5e-031.5e-051.2e-113.2e-151.1e-15
Table 7. Results for the Profile of Figure 1c in TE Polarization With h = 0.02, λ = 0.04, and θ = 0°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 11Order 19
  • a

    The run time was 27 s for the calculation of order 19.

02.762105662320035e-011.9e-31.6e-57.4e-96.5e-142.4e-15
15.735818584364873e-023.1e-32.0e-51.9e-81.3e-126.0e-16
29.154897389472935e-023.7e-35.1e-68.2e-91.4e-126.7e-15
31.051875097051952e-014.6e-45.1e-61.6e-91.0e-129.2e-16
46.713521833646909e-021.7e-32.7e-51.4e-113.6e-122.1e-16
52.830374622545111e-023.9e-37.3e-51.4e-82.2e-136.7e-15
69.270117932865375e-035.9e-31.3e-43.2e-81.1e-113.0e-15
72.435385416440963e-037.9e-32.2e-48.4e-84.1e-121.8e-16
Table 8. Results for the Profile of Figure 1c in TM Polarization With h = 0.02, λ = 0.041, and θ = 0°a
EfficiencyScattered EnergyOrder 0Order 1Order 5Order 11Order 19
  • a

    The run time was 23 s for the calculation of order 19.

01.291589358261453e-012.2e-031.6e-052.7e-102.1e-129.5e-14
11.662663939465338e-011.7e-035.1e-052.8e-101.6e-124.1e-14
23.920583564991646e-038.5e-035.3e-067.0e-091.3e-116.2e-13
31.878743087516774e-021.5e-022.7e-062.6e-096.9e-121.5e-13
47.416580102755271e-023.7e-032.8e-054.6e-096.7e-129.4e-15
57.840714643630796e-022.4e-042.0e-056.5e-096.6e-132.4e-14
65.345392880491968e-022.6e-031.6e-056.5e-099.6e-124.9e-14
72.611186407051911e-025.1e-036.8e-054.3e-098.6e-135.1e-14
Table 9. Results for the Profile of Figure 1a in TE Polarization With h = 0.025, λ = 0.001, and θ = 70°a
EfficiencyScattered EnergyOrder 0Order 1Order 3Order 7Order 9
  • a

    The run time was 12 s for the order 9 calculation.

09.405896918172547e-039.1e-062.3e-071.2e-101.7e-146.1e-16
14.691450406852652e-031.2e-051.1e-078.4e-112.3e-162.2e-16
24.977619850889408e-031.2e-051.3e-077.3e-117.5e-159.1e-17
38.970028199252082e-031.1e-052.1e-071.8e-105.5e-153.6e-16
41.591650541515982e-038.4e-064.2e-082.5e-115.2e-173.9e-16
51.151449525094811e-026.0e-062.7e-072.7e-103.8e-155.7e-17
61.509705192413105e-042.8e-064.9e-098.3e-131.2e-152.0e-16
71.232860573339191e-026.3e-072.9e-073.5e-104.2e-157.6e-16
Table 10. Results for the Profile of Figure 1a in TM Polarization With h = 0.025, λ = 0.0011, and θ = 70°a
EfficiencyScattered EnergyOrder 0Order 1Order 3Order 7Order 9
  • a

    The run time was 19 s for the order 9 calculation.

04.663322085313096e-032.9e-031.8e-051.4e-088.5e-149.9e-15
15.732453099268077e-032.4e-032.0e-051.5e-081.2e-141.0e-14
29.742422239435774e-031.4e-031.9e-051.5e-085.0e-142.7e-15
31.692802029774497e-035.8e-032.2e-051.7e-085.1e-162.9e-15
41.271613899298100e-025.4e-041.9e-051.7e-084.9e-142.5e-15
59.550848574956618e-052.7e-023.1e-052.7e-081.7e-138.7e-15
61.351161517190391e-022.2e-052.0e-051.9e-086.8e-145.3e-15
71.898341280339258e-042.0e-021.1e-059.9e-092.9e-137.6e-14

[33] Our first example, presented in Table 2, corresponds to the profile in Figure 1a illuminated by a TE-polarized plane wave with h = 0.025, λ = 0.025, and an incidence angle θ = 30°. The run time was 10 s for the calculation of order 17; we see that, as claimed, the present approach produces results with full double precision accuracy in short computing times.

[34] The results for the profile in Figure 1a under TM polarization are given in Table 3. Here we take h = 0.025 and λ = 0.0251, with an incidence angle θ = 30°. Our choice of wavelength in the present TM case, which is slightly different from the value λ = 0.025 we used in the TE case, was made to avoid the Wood anomaly [Hutley, 1982] that occurs at the latter value, for which the test boundary variation code fails. Our high-frequency method, however, does not suffer from that drawback, and results for the profile in Figure 1a with h = 0.025, λ = 0.025, and an incidence angle θ = 30° are given in Table 4. The reference solution in this case is the high-order high-frequency solution of order 21. The convergence of our expansion in this case is similar to the convergence observed in the previous case where λ/d = 0.0251. Again, full double precision accuracies are reached in a 10 s calculation.

[35] Our method is not restricted to sinusoidal surfaces, of course. In Table 5, for instance, we present results corresponding to the profile of Figure 1b with h = 0.01, λ = 0.025, and θ = 0° in TE polarization. The run time was 5 s for the order 11 calculation. Table 6 shows results for the same profile under TM polarization with h = 0.01, λ = 0.0251, and θ = 0°. The run time was 4 s for the order 11 calculation.

[36] We next consider the third-order “Stokes” wave [Kinsman, 1965] shown in Figure 1c. The results presented in Table 7 assumed the parameter values h = 0.02, λ = 0.04, and θ = 0° and TE polarization. The run time in this case was 27 s for the calculation of order 19. Table 8 shows the results for TM polarization and h = 0.02, λ = 0.041, and θ = 0°. The run time was 23 s for the order 19 calculation.

[37] Table 9 presents results for a low grazing angle example for the profile of Figure 1a with h = 0.025, λ = 0.001, and θ = 70° (20° grazing) in TE polarization. As mentioned above, for reference in this case we used the high-frequency solution of order 15. The run time for the order 9 calculation was 12 s. The results for the profile of Figure 1a under TM polarization with h = 0.025, λ = 0.0011, and θ = 70° are given in Table 10. The run time was 19 s for the order 9 calculation.

[38] A final remark concerning the order of the Fourier series used in the examples above is now in order. For the examples given in Tables 28, no more than 30 Fourier modes were used. The number of Fourier modes needed depends on the incidence angle, the height of the profile, the order of the high-frequency expansion used, and the accuracy required; in the cases considered in Tables 9 and 10, for example, it was necessary to use 45 Fourier coefficients to achieve the accuracies reported.

8. Conclusions

[39] We have shown that high-order summations of expansions of the type of equation (1) can be used to produce highly accurate results for problems of scattering by rough surfaces in the high-frequency regime in TE and TM polarizations. Our algorithm is based on analytic continuation of divergent integrals and careful algebraic manipulation of Taylor-Fourier series representations. Our results show accuracies which improve substantially over those given by classical methods such as the Kirchhoff approximation. As shown recently [Sei et al., 1999], such accuracies are needed to capture important aspects of rough surface scattering involving very low scattering returns and occurrences of unusual polarization ratios. Further, the results of Bruno and Reitich [1993a, 1993b, 1993c] clearly suggest that a multiscale perturbation algorithm of the type proposed by Bruno et al. [2000] should yield the required accuracies for multiscale surfaces provided that an accurate high-frequency solver, such as the one presented in this paper, is used. This paper thus extends the range of applicability of classical asymptotic methods, producing a versatile, highly accurate, and efficient high-frequency numerical solver.

Acknowledgments

[40] This effort was sponsored by DARPA/AFOSR under contract F49620-99-C-0014, by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grants F49620-99-1-0010 and F49620-99-1-0193, and through NSF contracts DMS-9523292 and DMS-9816802. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.

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