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Remote sensing of terrain and ocean surfaces is circumscribed in the physical domain of electromagnetic scattering by rough surfaces. The development of accurate models has gathered a great deal of efforts since the 80’s. Until that moment there were two classical approaches to be applied to two different asymptotic cases: the surfaces with small roughness and those having long correlation length. The first situation was dealt successfully via the small perturbation method (SPM)) whereas the second one was the target of the Kirchhoff approximation (KA). In effect, the abundance of models in the last two decades has made it very difficult for the Earth Observation practitioner to properly classify them and choose between them. The most important effort to that purpose was made by Tanos Elfouhaily in Elfouhaily & Guerin (2004), and we refer to his work for those interested in having a comprehensive account of the available methods for the problem. We focus here on the model that has arguably awakened the largest share of interest within the remote sensing community, that is, the Integral Equation Model (IEM) presented by Fung and Pan in Fung & Pan (1986) and later corrected in a long series of amendments by the same authors Fung (1994); Hsieh et al. (1997); Chen et al. (2000); Fung et al. (2002); Chen et al. (2003); Fung & Chen (2004); Wu & Chen (2004); Wu et al. (2008). In effect, there has been a number of issues that made the model theoretically inconsistent, even if each amendment was accompanied by properly suiting numerically simulated results. In 2001 the author of this chapter carried out a complete revision of Fung’s work and proposed a corrected IEM that successfully achieved one of the objectives of the rough surface scattering models developed so far: to unify in a single equation both the SPM and the KA in the most general case of bistatic scattering. This corrected IEM was named IEM with proper inclusion of multiple scattering at second order or IEM2M. This chapter aim is twofold: on the one had a quick summary of the IEM2M is given and on the other an extension of it is proposed to include those surfaces comprising both a zero-mean height, random component and a deterministic component that we call here “topographical”.

The Extended Integral Equation Model IEM2M for Topographically Modulated Rough Surfaces