# Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances
## 3 Historical Approximations
### The Kirchhoff Approximation
#### 3.2.3 Elastic Targets and Applications  
The Kirchhoff approximation models bulk waves, and does not take into account surface acoustics waves (SAW) as Rayleigh waves neither head waves. Head waves led to recent researches [70; 86; 87; 88] and a modification [89] of the Kirchhoff approximation is henceforth proposed for head waves account. Head waves on irregular surfaces can also be modelled [90].  
The Kirchhoff approximation for elastic rough targets is proposed by [PERSON] [13] and then revisited by [PERSON] [91] and [PERSON] [92]. The Kirchhoff approximation validity have also been specifically considered in the particular cases of rigid and soft targets in a modeling study [93] of the reflected field including multiple scattering. One example of KA application to impedance surfaces [57] for non-destructive evaluation (NDE) is the modeling of ultrasonic telemetry of immersed targets [94; 59].  
We report here an application of KA simulation to impedance rough surfaces for NDE. Ultrasonic inspection of a planar steel component immersed in water (water path 25 mm) at frequency = 2 MHz (wavelength \(\lambda\) = 0.75 mm) is simulated with the Kirchhoff approximation for elastic rough entry surfaces of varying roughness using CIVA software inspection simulations tools [95; 57; 96]. A phased-array (represented in yellow in Figure 7a,b) is used to emit an incident beam directed at 117\({}^{\circ}\) with respect to the surface thanks an emission delay law (applied in red in Figure 7a to only some left elements) and all elements receives the scattered waves without delay law in the angular domain (118\({}^{\circ}\), 58\({}^{\circ}\)).  
In Figure 7a,b, the true B-scans (true-to-geometry imaging) are surperimposed to the specimen in color codes; they represent the echographic signal versus time for all elements in reception (corresponding to scattering angles in the angular domain (118\({}^{\circ}\), 58\({}^{\circ}\))). Figure 7c shows that increasing the roughness deteriorates the specular reflection echo and causes appearance of echoes at non-specular angles. Figure 7d shows that the scattering directivity patterns obtained for three different generated rough surfaces of the same statistical properties are slightly different.  
A recent study [97] has compared the accuracy of KA and PT using a finite element (FE) model for fluid and elastic ocean bottoms. For a fluid bottom, KA is close to FE with maximal gap of 2 dB except for small grazing incident angles for which PT is more effective. PT behaves globally well except near specular observation direction notably for small grazing incident angles. For the elastic case, the disagreement between KA and FE increases to 5 dB and KA is inaccurate for incident angles less than the longitudinal critical angle: the interferences between transversal, longitudinal and surface waves are not taken into account in KA. The PT prediction is generally good, but another limitation appears in the elastic case for incident angles near the longitudinal critical one. However, we regret the provided validations are established only for a small roughness k \(\sigma\)= 0.6 favorable to PT.