# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 3 Historical Approximations
### The Kirchhoff Approximation  
#### 3.2.1 Theory and History  
The main hypothesis of the Kirchhoff approximation (also referred to as the physical optics approximation) is to assume that each point of the surface behaves as if it belonged to the infinite tangent plane to this surface. Then it is practical to employ geometrical acoustics to approximate the exact fields on the local surface: the field on the illuminated part of surface is the sum of the incident field and the field reflected by the local tangential plane. This supposed surface field is then propagated far from the surface using a propagator (called Green function). Indeed the scattered far field is expressed by the way of an integral over the flaw surface (the Kirchhoff-Rayleigh integral) whose integrand is--for the example of Dirichlet conditions--the product of the Green function by the derivative of the geometrical field on the flaw surface. For complex incident fields or scatterer shapes, this integral is evaluated numerically. The Kirchhoff approximation has been extensively used to model scattering of waves in optics (early introduced by [PERSON] [54]), electromagnetism [55; 56], acoustics, etc. In both acoustics [57] and elastodynamics [58; 59], the Kirchhoff approximation also has the advantage in dealing with anisotropic materials and impedant surfaces [57]. Note that when doing an asymptotic evaluation of the KA integral on a finite surface, the KA resulting leading terms describe the reflected waves, and the other terms describe diffracted waves by the surface edges which have the same nature (cylindrical or conical wave fronts) but not the same amplitude as those predicted by the geometrical theory of diffraction (GTD) [55; 60]. The main advantage of the Kirchhoff approximation is to provide finite results for the scattered field in the whole space, contrary to GTD [61; 62; 63; 64; 65]. One of the main inconveniences of the Kirchhoff approximation (KA) is its incorrect quantitative prediction of edges diffraction, since the approximation of the surface field by the geometrical field is invalid near the edges. To handle this drawback, the physical theory of diffraction (PTD) has been proposed in electromagnetism [66], acoustics [59; 67] and elastodynamics [68; 69; 70] and compared to other uniform scattering models [67; 70; 71]. In the literature, the Kirchhoff approximation has been studied by different authors. A review has been carried out by [PERSON] [18]. The first one to introduce the Kirchhoff approximation is [PERSON] [72]. [PERSON] [27] has well described the theory for 1D rough profile (Figure 2) and Dirichlet conditions (soft interface) extending [PERSON]'s works. [PERSON] et al. proposed a Kirchhoff-based theory extended to shadowing effects for the backscattering of acoustic waves by randomly rough surfaces [37]. [PERSON] et al. employed in fact the potential-method formulation developed by [PERSON] which showed by iterating the scattering integral equation that the scattered pressure can be expressed as a series: the first term corresponds to the singly and the others to the multiply scattered field components. [PERSON] et al. considered only the first term. Their experimental validations of their model using Gaussian rough (liquid-solid interfaces) surfaces have shown that the experimental measurements of the intensity backscattered by randomly rough interfaces (liquid-solid) are in good agreement with the theory until \(40^{\circ}\) backscattering angles (with respect to the normal to the mean plane). Nevertheless, an anomaly has been observed in [PERSON]'s predictions for the Rayleigh's angle of incidence.