A non-periodic two scale asymptotic method to take account of rough topographies for 2D elastic wave propagation

SUMMARY We propose a two scale asymptotic method to compute the effective effect of a free surface topography varying much faster than the minimum wavelength for 2-D P-SV elastic wave propagation. The topography variation is assumed to be non-periodic but with a de-terministic description and, in the present paper, the elastic body below the topography is assumed to be homogeneous. Two asymptotic expansions are used, one in the boundary layer close to the free surface and one in the volume. Both expansions are matched appropriately up to the order 1 to provide an effective topography and an effective boundary condition. We show that the effective topography is not the averaged topography but it is a smooth free surface lying below the fast variations of the real topography. Moreover, the free boundary condition has to be modified to take account of the inertial effects of the fast variations of the topography above the effective topography. In other words, the wave is not propagating in the fast topography but below it and is slowed down by the weight of the fast topography. We present an iterative scheme allowing to find this effective topography for a given minimum wavelength. We do not attempt any mathematical proof of the proposed scheme, nevertheless, numerical tests show good results.