# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 1 Introduction  
The scattering of waves from rough surfaces is an important phenomenon in diverse areas of science, such as optics, geophysics, communications, acoustics, hydraulics [1], elastodynamics for non-destructive testing, etc. For instance, in the particular case of ultrasonic waves, scattering from rough surfaces leads to many applications in non-destructive testing [2], in underwater acoustics for high resolution ultrasonic sonars [3] and for medical sciences [4, 5]. The modeling of such phenomena is a century-old and still active research topic. Scattering models of acoustic waves from rough surfaces could be classified into analytical models, numerical ones and combinations of both analytical and numerical methods. However, such a classification is rather ambiguous, since some analytical models (usually called semi-analytical) employ numerical calculations and since some numerical methods start from analytical approximations. The current review focuses on analytical or semi-analytical methods and the validity of each method will be emphasized as well as some applications examples. Some analytical methods are presented or used in the following in more detail: the Kirchhoff approximation (KA), the perturbation theory (PT), the integral equation (IE) method, the parabolic equation (PE) method and the small slope approximation (SSA). The Kirchhoff and PT methods represent early classical approaches but are still very often employed in the literature, notably the Kirchhoff one. The other methods represent more modern approaches which have larger domains of validity but are more complex and therefore more difficult to implement or can lead to a higher calculation time. All the previously cited methods havebeen found to be amongst the most common in the literature and many of the other methods of the literature are based on or have much in common with these approaches. This review focuses thus only on the scattering of acoustic waves and does not deal with the transmission through a rough surface of waves within a solid material. Indeed the models used in acoustics are often also employed in elastodynamics. For instance, the elastic Kirchhoff model has been devised to deal with rough defects [6; 7; 8] and to take into account mode-conversions between compressional and shear waves. A variety of analytical models involving reflection and transmission waves (for instance through a rough fluid-solid interface), such as the phase screen approximation [9; 10; 11; 12] and Kirchhoff theory, have been developed to understand the elastic fields in solids or scattered from different embedded scatterers. Several reviews of scattering from rough surfaces exist in the literature, notably in general physics (diverse areas of sciences) for random rough surfaces [13; 14; 15] or in electromagnetism [16; 17]. The main review describing works in acoustic scattering from rough surfaces has been written by [PERSON] in the 80s [18], although its review does not only deal with acoustics. This review has notably summarized the two classical approximations ([PERSON] and the perturbation theory) but it is now out of date: since this review, important models have been devised for grazing incidence or observation, such as the small slope approximation, and new criteria have notably been established for defining the validity of the Kirchhoff approximation, which is probably the most used analytical model for roughness scattering. Some other textbooks also deal with acoustic scattering from rough surfaces notably in the view of ocean applications [19; 20]. At [PERSON]'s time, the accuracy of [PERSON] approximation came from [PERSON]' work [21] and was determined by the ratio of the height correlation length \(L\) of the random rough surface to the wavelength. It is also the case in [PERSON]'s more recent textbook (see Figure 1) [20], where approximations KA, PT and SSA are all said to require this ratio less than a constant in order to be valid; another different criterion has even been proposed for radar applications [22].