506 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 7, NO. 3, JULY 2010 Low-Frequency Limit of Unified Models for Backscattering From Oceanlike Surfaces Christophe Bourlier, Member, IEEE, and Nicolas Pinel, Member, IEEE Abstract—In the context of electromagnetic-wave backscatter- ing from oceanlike surfaces, by using the first two orders of unified models, like the small slope approximation and the local curvature approximation, we recently proposed an original technique to reduce the number of numerical integrations to two for easier nu- merical implementation. In this letter, this technique is simplified in the low-frequency limit, allowing us to bring a correction to the first-order small perturbation method. Index Terms—Radar cross sections, remote sensing, sea surface electromagnetic scattering. I. I NTRODUCTION F ROM the 1960s, the derivation of the microwave backscat- tering normalized radar cross section (BNRCS) from oceanlike surfaces is a topic of investigation which is making progress and remains a challenging task. The first developed model is the Two-Scale Model [1], [2]. In the last two decades, another group of scattering models was proposed, namely, the local unified models [3]. One of the most popular is the small slope approximation (SSA) [4]; more recently, models based on the same decomposition of the scattering matrix as SSA were developed, like the local curvature approximation (LCA) [5], [6]. It is well known that such backscattering models are ex- tremely difficult to implement in the full 3-D case, because of the fourfold integral that is involved (with two space variables and two frequency variables) and because of the strongly oscil- lating behavior of the integrand. That is why, recently, Bourlier and Pinel presented an original technique to reduce this com- putation to a twofold integral (with one space variable and one frequency variable) by resorting to the azimuthal harmonic expansion of the BNRCS and by using Bessel functions [7]. In this letter, this technique is tested in the low-frequency (LF) limit, allowing us to bring a correction to the first-order small perturbation method. This approximation allows us to sig- nificantly simplify the numerical implementation and to reduce the computing time of the BNRCS. In Section II, the technique developed by Bourlier and Pinel is briefly summarized, and in Section III, it is simplified in the LF limit. In Section IV, numerical comparisons are presented. Manuscript received October 22, 2009; revised December 3, 2009. Date of publication February 22, 2010; date of current version April 29, 2010. The authors are with the Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique (IREENA) Laboratory, Université Nantes Angers Le Mans, Polytech’Nantes, 44306 Nantes Cedex 3, France (e-mail: christophe.bourlier@polytech.univ-nantes.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2010.2040366 II. BNRCS OF UNIFIED MODELS In the literature, from microwave experimental data (for instance, see [9] and the references therein), it was established that the BNRCS can be expressed for pq = {VV,HH} copo- larizations in the form σ pq (θ,φ; u)= σ pq 0 (θ; u)+ σ pq 2 (θ; u) cos(2φ) (1) where φ is the observation azimuthal angle with respect to the wind direction, θ is the observation elevation angle, and u is the wind speed. In addition, σ pq 1 (θ; u)=0 (coefficient along cos φ) because the surface is assumed to be Gaussian. By considering the first two orders of the kernels of unified models, ˆ N pq (θ,φ; ξ) ≈N pq 1 (θ,φ)+ ˆ N pq 2 (θ,φ; ξ), which depend on the chosen model, the BNRCS is equal to the sum of two terms, σ pq (θ,φ; u)= σ pq 11 (θ,φ; u)+ σ pq 12 (θ,φ; u). The subscript “11” results from the autocorrelation of the first-order scattered field, whereas the subscript “12” results from the cross correlation between the first- and second-order scattered fields. Bourlier and Pinel recently showed that the BNRCS even harmonics {σ pq 11,0 (θ; u),σ pq 11,2 (θ; u)} related to “11” can be expressed as [7]  σ pq 11,0 (θ; u)= A 1  ∞ 0 J 0 (a)[e β I 0 (b) − 1]rdr σ pq 11,2 (θ; u)=2A 1  ∞ 0 e β J 2 (a)I 1 (b)rdr (2) where ⎧ ⎨ ⎩ a(r)=2Kr sin θ, b(r)= Q 2 z W 2 (r) β(r)= Q 2 z W 0 (r), A 1 =2πA |N pq 1 (θ)| 2 e −Q 2 z σ 2 η A = 1 πQ 2 z , Q z =2K cos θ. (3) In addition ⎧ ⎨ ⎩ W 2 (r, φ r )= W 0 (r) − cos(2φ r )W 2 (r) W 0 (r)=  ∞ 0 ˆ W 0 (ξ )J 0 (rξ )dξ W 2 (r)=  ∞ 0 ˆ W 2 (ξ )J 2 (rξ )dξ (4) where ˆ W 0 and ˆ W 2 stand for the isotropic and anisotropic parts of the sea spectrum, respectively. W 0 and W 2 are their respective associated correlation functions, and σ 2 η = W 0 (0) is the height variance. In polar coordinates (ξ,φ ξ ), the sea spectrum is assumed to be ˆ W (ξ,φ ξ )=[ ˆ W 0 (ξ )+ ˆ W 0 (ξ ) cos(2φ ξ )]/(2π) (5) which is consistent with the sea spectrum of Elfouhaily et al. [8]. J m and I m are the Bessel functions of the first and second kinds, respectively, and of order m. Equation (2) shows that the BNRCS is obtained from a single numerical integration over the radial distance r. 1545-598X/$26.00 © 2010 IEEE Authorized licensed use limited to: University of Nantes. Downloaded on July 23,2010 at 08:37:28 UTC from IEEE Xplore. Restrictions apply.