## 1. Introduction

[2] Radar probing of the Earth's surface has now been in use for a long time. Several Earth observation missions have already proven radar potentialities, with numerous scientific and operational products being distributed. Microwave and millimeter wave instruments present the major advantage of being able to sense the Earth's surface through cloud cover. Their main drawback compared to optical or infrared sensors is their weak spatial resolution due to the longer electromagnetic wavelength they use. This problem can be overcome using synthetic aperture techniques, leading, however, to particular disturbing effects such as speckle, target displacements, etc…

[3] Radar echoes are directly related to the scattering properties of the observed scenes. Therefore a number of methods have been developed for scattering simulations and for the retrieval of geophysical parameters from microwave measurements. Among these models, empirical and theoretical ones should be distinguished. Though the first ones are usually very efficient considering their computation time, they do not allow a full understanding of the physical processes implied. Physically based theoretical models, on the other hand, allow the analysis of the sensitivity of the scattering characteristics to both target parameters and radar technical specifications. Moreover, being tabulated for particular instrumental configurations or implemented on high-performing recent computers, they can also be used as the basis for inversion algorithms for operational systems. Finally, when proven efficient, theoretical models could also be used as references for the intercalibration of multichannel active and passive instruments.

[4] Since the basic work of *Rice* [1951], an abundant literature has covered the subject of random rough surface scattering, both for the scalar case (acoustics) and for the vector case (electromagnetics), under either extreme boundary conditions (soft and hard surfaces, or perfectly conducting surfaces) or more general conditions. Extensive lists of references are to be found in some recent textbooks, e.g., those by *Fung* [1994] or *Voronovitch* [1994]. Among all proposed methods, two-scale methods work rather well for the case of the ocean surface, because the separation between the large-scale component and the small-scale component is somehow related to gravity and capillary waves. We have developed a two-scale approach [*Guissard and Sobieski*, 1987; *Guissard et al.*, 1992], based on the boundary perturbation method (BPM) [*Burrows*, 1973]. It leads to numerical results that compare rather well with experimental data [*Lemaire et al.*, 1999]. Another method that has encountered a large audience in recent years is the integral equation method (IEM) proposed by *Fung et al.* [1992] and *Fung* [1994], which can be applied for both ocean and soil surfaces.

[5] We present in this paper a comparison between these two theoretical models for rough-surface-scattering computation. Since both methods lead to the same asymptotic solution in the cases of either very small vertical displacements of the surface with small slopes or very large surface displacements with large curvature radii, we shall focus on the case of a composite surface where different scales of roughness may be present. In section 2, the two models are presented following a scheme of increasing complexity, so that it remains as clear as possible and facilitates the following discussions. The presentation is restricted to the main steps that highlight the differences between the methods, and the reader is referred to the literature whenever the mathematical developments are not relevant to the purpose of this paper. Next, the results obtained with both methods are compared in the case of the ocean surface and for various radar configurations. Since the solution appears as a series development for both methods, we analyze the results starting with the zeroth-order term and then continuing with higher-order terms.