# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 4 More Modern Approaches with Larger Domains of Validity
### Small Slope Approximation (SSA)  
The SSA technique was described first by [PERSON] [107; 108; 109]. An alternative small slope approximation has originally been developed by [PERSON] and then studied by [PERSON] (1994) [110]. All these methods attempt to exploit the assumed smallness of the surface slope without restricting the RMS surface height to wavelength ratio. The SSA model from [PERSON] has been investigated by [PERSON] and [PERSON] [111; 112]. SSA consists in introducing a roughness correction to the Helmholtz integral. This correction is expanded into a functional Taylor series in slope whose terms depend on the integral of the wave number spectrum of the surface height function. The approximation of each order of SSA is based on the approximations of previous orders. Owing to its principles SSA takes into account shadowing and diffraction. [PERSON] and [PERSON] showed that, at lowest order the small slope approximation of Urusovskii/McDaniel is limited to forward scattering [110] but that the lowest order term in [PERSON]'s SSA was incorrect in [PERSON]'s SSA. SSA tends to the perturbation approximation for small k \(\sigma\) (adimensional wave numbers in terms of RMS surface height \(\sigma\)). The second order SSA (SSA (2)) reduces to the Kirchhoff approximation (KA) at high frequencies, hence does not take into consideration the shadowing effects. The predictions of SSA(2), SSA(3), and SSA(4) are compared [112] with Monte Carlo integral equation (IE) calculations devised earlier by [PERSON] [21], and with other approximations: the perturbation approximation, the Kirchhoff approximation (KA), and geometrical optics (GO). The scattering geometry is described in Figure 8.  
Figure 8: Figure 1. [PERSON] Figure 1 [112]. Scattering geometry. Reproduced with permission from [112], AIP Publishing, 1997. SSA and IE scattering predictions are presented in Figure 9 for respective k \(\sigma\) and kL 0.5 and 4.0, for 45\({}^{\circ}\) incident angle and RMS slope s = 0.177 (\(s=\sqrt{2}_{L}^{\sigma}\) for Gaussian surface), and the RMS slope angle \(\gamma=\tan^{-1}s\) is 10\({}^{\circ}\). The contribution of specular reflection--leading to a peak in the IE prediction--has not been integrated in the SSA models. In Figure 10, k \(\sigma\) and kL have both been increased by a factor of three to 1.5 and 12.0, respectively. The increase in the surface roughness (resp. correlation length) causes disappearance of the specular peak (resp. decrease of the backscattered strength). A higher SSA order improves the prediction in backscattering. Figure 10: [PERSON] Figure 3 [112]. Comparison of SSA(2), SSA(3), SSA(4), Kirchhoff approximation (KA) and IE scattering strengths. Here k \(\sigma\) = 1.5, kL = 12.0, \(\gamma\) is 10\({}^{\circ}\) and \(\theta_{i}\) is 45\({}^{\circ}\). Reproduced with permission from [112], AIP Publishing, 1997. Figure 9: [PERSON] Figure 2 [112]. Comparison of the second-order SSA (2), third-order SSA(3), and fourth-order SSA(4) small slope approximation and Monte Carlo integral (IE) scattering strengths (=10 log of the scattering cross section) for a modest rms surface slope angle. Here, k \(\sigma\) = 0.5, kL = 4.0, the RMS slope angle \(\gamma\) is 10\({}^{\circ}\), and the incident angle \(\theta_{i}\) is 45\({}^{\circ}\). Reproduced with permission from [112], AIP Publishing, 1997. According to [PERSON] [21], the [PERSON] approximation KA is accurate away from low grazing angles for rough surfaces smooth on the scale of a wavelength, assuming small surface slopes. For a Gaussian roughness spectrum, the KA is said to be generally accurate for kL \(>\)6, that is when \(\frac{L}{\lambda}>1\). We have seen in Section 3.2.2 the recent definition of a more precise criterion for the KA validity. In the following of this section the SSA and KA are thus considered for several examples with kL \(>\) 6. In [PERSON] previous Figure 3[112], KA is valid away from grazing angles, whereas PT (not shown) is not.