# 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(2) result reduces to the KA result in the specular region; the third and fourth-order results do as well. KA behaves better than SSA for backscattering, but it is the contrary in the forward direction at grazing angles. Finally, several cases have been examined beyond the region of validity of the fourth-order Perturbation Theory PT(4) and of KA. For example, in their Figure 6[112] since k \(\sigma=1\), PT(4) breaks down whereas KA is globally correct and SSA(4) the more realistic (Figure 11). Other validations showed that the fourth-order SSA (SSA(4)) mimics some shadowing phenomena and the SSA results are globally accurate for all observation angles for incident angles up to 80\({}^{\circ}\), but the validity decreases with the incident angle. Another study [113] of the Bragg scattering from surface waves by a rough interface recently provided comparisons between the Kirchhoff approximation, the perturbation theory and the integral equation method; [PERSON] leads to results relatively close to the integral equation method in the studied configurations (in forward and backscattering). [PERSON] (see Figure 1.1 in [20]) argued that SSA is valid for \(\sigma\)/\(L\) less than a constant value. According to him, KA and PT also require this condition but PT needs in addition a small k\(\sigma\) and KA a great kL. SSA has also been developed for rough solid elastic interfaces [114]. [PERSON] [115] showed that it provides an effective solution compared to Monte-Carlo integral equations simulations for grazing incident angles, whereas Kirchhoff leads to an overall important underestimation. SSA and first-order PT give rather good predictions and exhibit minima or peaks at the Rayleigh angle that can be exaggerated or discontinuities at critical angles which are not physical. The authors noted that Kirchhoff is rather smooth or does not have the valid behaviour for these particular angles: it is logical because the associated phenomena (Rayleigh or head waves) are not accounted on the simplified Kirchhoff theory (see also the perspectives section). We nevertheless notice that Kirchhoff yields an  
Figure 11: [PERSON] Figure 6[112]. Comparison of the SSA(4), KA, PT(4) and IE scattering strengths for a case when neither PT(4) nor the KA is accurate. Here, k \(\sigma=1.0\), kL = 4.3, \(\gamma=18.2^{\circ}\), and \(\theta_{i}=45^{\circ}\). Reproduced with permission from [112], AIP Publishing, 1997.  
overall rather good prediction versus scattering angle despite some quantitative discrepancies. We also emphasise that one comparison shows an overestimation by first-order PT of the specular amplitude, which is a strong inconvenient for numerous applications as NDE. SSA for layered fluid seafoors has only been studied recently. A study published this year [116] reviewed three different approximations. [PERSON] [117] has proposed an acoustic SSA method for rough water-sediment interfaces with a fluid layer of finite thickness overlying a semi-infinite fluid layer. A second method consists in using in acoustics the SSA developed for the general layered case in electromagnetics [118; 119; 120]. A third approach is developed in [116] by translating usual small-slope ansatz from k-space to coordinate space. It is then shown in [116] using different examples that the different approximations give different results in the case of strong internal reflections and also differ from Kirchhoff in specular direction. In an example with a mud layer overlying a harder basement, the first approximation differs strongly from the two others and consequently seems erroneous. Unfortunately, no numerical validation is provided to define the best ansatz.