# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 3 Historical Approximations
### The Kirchhoff Approximation
#### 3.2.2 Validity  
Historically KA has been considered--in the point of view of its local tangential plane assumption--as accurate when the rough surface can be seen planar by the incident wave. In [PERSON]'s review [7], KA is thus said valid for \(\mathrm{kr_{c}\cos^{3}\theta i>>1}\), with \(\mathrm{r_{c}}\) the surface local radius of curvature and \(\theta\)i the incident angle. This historical criterion forgets that multiple scattering and shadowing effects are not taken into account in KA leading to large errors for grazing incidences or higher roughness. KA prediction has been compared to Integral Equation (IE) results by [PERSON] [21] for 1D rough profiles. [PERSON] proposed to use the surface correlation length \(L\) as main parameter for defining the Kirchhoff validity away from the low grazing angle region: KA is said to be accurate for a correlation length greater than the wavelength or even for lower correlation lengths if shadowing effects are not important. We notice this first criterion proposed by [PERSON] at this step is too simple and not involve the rms height. In the low grazing angle region some examples of validation with comparison to another methods are given in next Section 4; at low grazing angles, additional validity dependence enters through the RMS slope which is for Gaussian rough surfaces: \(s=\tan\gamma=\sqrt{2}\frac{L}{L}\) i.e., the rms height is now involved. However, at shorter correlation lengths, the needed shadowing correction proposed by [PERSON] becomes greater to improve Kirchhoff prediction at low grazing angles in the case of backscattering. Note that in order to improve its prediction for low grazing angles, [PERSON] [76] provided an improved Kirchhoff formula (using a smoothing special function) notably by including the correlation length of the scattering surface into the reflection coefficient. The modification impact is not noticeable since shadowing and multiple scattering will dominate for low grazing angles. [PERSON] showed that the PT and KA results differ except in the limit of infinite correlation length (smooth surface). [PERSON] et al. [77] recently proposed for elastic materials (also for shear waves [78; 79]) and for compressional waves a different surface parameter than the surface correlation length \(L\) used by [PERSON], a function \(\sigma^{2}/L\) of both \(L\) and \(\sigma\), to define a criterion when the KA use is valid. They found that, with a normal incidence angle \(\theta\)i = 0\({}^{\circ}\), KA is valid with a scattering angle \(-70^{\circ}<\theta\)s \(<70^{\circ}\) and when \(\sigma^{2}/L\leq c\), the constant upper bound \(c\) being in two dimensions 0.20 \(\lambda\)p and in three dimensions 0.14 \(\lambda\)p (\(\lambda\)p being the compressional wavelength). Figure 5 shows a comparison between KA and a finite element (FE) method for a 2D case when \(\sigma^{2}/L\)-\(c\). A small incidence angle of 30\({}^{\circ}\) can improve the KA accuracy when \(\sigma^{2}/L\) exceeds \(c\) as in the case shown in Figure 6. We can however regret that the numerical evaluation of this proposed criterion was limited to small incidences angles: however, this criterion is likely to be suitable for many NDE configurations for which incidence is close to normal. A study of this criterion near grazing incidence should also be useful. As said in the introduction, another different KA validity criterion has been proposed for radar applications [22] and is also based on numerical calculations: it is given by Equation (1) in [22] (see also their Figure 3, Kirchhoff being called the scalar approximation) and ensures a KA validity for \(L^{2}/\sigma\) more than a constant times the wavelength and a condition \(kL>6\) is also associated. We find the condition on \(L^{2}/\sigma\) very disbelieving and in contradiction with the previously cited studies and also that given by [PERSON] (\(\sigma/L<C\) and large kL; see Figure 1.1 in [20]). The condition \(kL>\)6 is also found in other references as [52].