## 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

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]

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 2–5, 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 ν_{n}(*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.