ILLUMINATED HEIGHT PDF OF A RANDOM ROUGH SURFACE AND ITS IMPACT ON THE FORWARD PROPAGATION ABOVE OCEANS AT GRAZING ANGLES C. Bourlier 1 , N. Pinel 1 , and V. Fabbro 2 1 IREENA, Universit´ e de Nantes, Polytech’Nantes, Rue C. Pauc, La Chantrerie, BP 50609, 44306 Nantes Cedex 3, France 2 ONERA-DEMR, 2 Avenue Edouard Belin, 31055 Toulouse Cedex 4, France ABSTRACT When solving electromagnetic rough-surface scattering problems, the effect of shadowing by the surface rough- ness often needs to be considered, especially as the illu- mination angle θ approaches grazing incidence. Indeed, due to the surface roughness, only a part of the surface is illuminated. This phenomenon is characterized by the statistical illumination function which gives the probabil- ity that a point on a rough surface is illuminated. In this paper, we propose to calculate the bistatic statistical illu- mination function for any one-dimensional random rough surface and to analyse its impact on the forward propaga- tion above rough sea surfaces by considering Gaussian statistics and for grazing angles φ of the order of one de- gree. 1. INTRODUCTION When solving electromagnetic rough-surface scattering problems, the effect of shadowing by the surface rough- ness often needs to be considered, especially as the illu- mination angle approaches grazing incidence. In this case only the highest elevations are likely to be illuminated. In this paper, for one-dimensional random rough surfaces of any statistics and for a bistatic configuration (transmitter and receiver are distinct), we propose to study this prob- lem by calculating the height PDF (Probability Density Function) of the illuminated points. Usually, for forward (specular direction) propagation above oceans at graz- ing angles, the calculation of the reflection coefficient is based on the simple model of Ament [1] or Miller-Brown [2], in which the shadowing phenomenon is neglected. In addition, in this paper we propose to include the shadow- ing effect in this model and to present comparisons with a benchmark method [3]. In Section 2, the shadowing effect is presented for any statistics, and the height PDF of the illuminated points is derived analytically in the forward direction and for Gaussian statistics. This study leads to a new height PDF of the illuminated points, in which the mean value and the standard deviation of the illuminated surface heights are calculated. In addition, this analytical approach is compared with a Monte-Carlo method. In Section 3, a new reflection coefficient, based on the simple model of Ament, is derived from the height PDF presented in Sec- tion 2. For the horizontal polarisation, the interest of the proposed approach is proven and discussed in Section 4 by comparison with a statistical Monte Carlo benchmark method based on the MoM (Method of Moments) com- bined with an accelerated spectral method and a multigrid iterative approach (MGIA) [3]. This model will be used in the paper of V. Fabbro et al. [4]. 2. ILLUMINATED HEIGHT PDF The problem of shadowing from rough surfaces was con- sidered in the textbook of Bass and Fuks [5] by means of the theory of random function overshoots. The sta- tistical illumination function was then expressed from an infinite Rice series of multiple integrals. The shadow- ing effect was also studied by Wagner [6] and Smith [7], who retained only the first term of the series. Moreover, Smith used Wagner’s approach by introducing a normal- ization function. For monostatic and bistatic configura- tions, these authors assumed a one-dimensional surface with an uncorrelated Gaussian process of surface heights and slopes. This means that the statistical illumination function is independent of the surface height correlation function. Recently, for one- and two-dimensional sur- faces with Gaussian statistics, Bourlier et al. [8], [9] ex- tended the Wagner and the Smith formulations by taking into account the correlation. More recently, the statisti- cal illumination function in reflection was extended to the transmission case [10] corresponding to the case where the transmitter is located below the surface. In this sec- tion, by considering Gaussian statistics and the forward direction, the illuminated height PDF is derived by using the Wagner and the Smith approaches with and without correlation. In order to keep the best formulation, these approaches will be compared with a Monte-Carlo method for small grazing angles. When the shadowing effect is included, the bistatic (sub- script b) illuminated height PDF is written as ˘ p b (φ 1 ,φ 2 ; ξ )= (1)