1. Diffusion par des surfaces rugueuses: approximations faibles pentes Marc Saillard LSEET UMR 6133 CNRS-Université du Sud Toulon-Var BP 132, 83957 La Garde cedex, France marc.saillard@lseet.univ-tln.fr

2. Outline Boundary integral formalism Approximate scattering theories Kirchhoff approximation Small perturbation theory Two-scale model Small Slope Approximation IEM Small Slope Integral Equation Conclusion

3. Integral representation of fields Perfect conductor

4. Scattering matrix

5. Approximate scattering theories Kirchhoff approximation

6. Small perturbation theory Small height limit of KA

7. Two-scale model (KA + SPM)

8. Small slope approximation

9. Comparison of MoM with Kirchhoff approx. (KA), Small Perturbation Method (SPM) 1st order Small Slope Approx. (SSA) Comparison of MoM (solid line) with 1st order SSA (dashed line) n = 1.6 r.m.s height 0.17 - correlation length Numerical examples: Gaussian spectrum

10. Band limited [k/30,4k] power-law spectrum (K -4 ); h = /5; s = 0.1; khs = 1/8 Numerical examples: power-law spectrum

11. IEM

12. Horizontal distance d distance R Slope s Height h = sd Validity : khs<<1. Matrices associated to integral operators are 2D Toeplitz Storage : 2N instead of N 2 Product : 2Nlog 2 N instead of N 2 Small slope integral equation (Meecham – Lysanov) 1 st order

13. Band limited [k/30,4k] power-law spectrum (K -4 ); khs = 1/8 Gaussian spectrum ; khs = 1/4

14. Conclusion Domain of validity that covers both that of SSA1 and of the tangent plane approximation (as SSA2 or OEM) No assumption on the surface statistics The accuracy can be estimated Very low computational cost It provides an estimation of the cross-polarized component in the plane of incidence It is an alternative to statistical approximate methods but requires Monte-Carlo process