Second order homogenization of the elastic wave equation for non-periodic layered media

In many cases, in the seismic wave propagation modelling context, scales much smaller than the minimum wavelength are present in the earth model in which we wish to compute seismograms. For many numerical methods these small scales are a challenge leading to high numerical cost. The purpose of this paper is to understand and to build the effective medium and equations allowing to average the small scales of the original medium without losing the accuracy of the wavefield computation. In this paper, only the simple layered medium case is studied, leaving the general 3-D medium case for future work. To obtain such an effective medium and equations, we use high order two scale homogenization applied to the wave equation for layered media with rapid variation of elastic properties and density compared to the smallest wavelength of the wavefield. We show that the order 0 homogenization gives the result that was obtained by Backus in 1962. Order 0 homogenized models are transversely isotropic even though the original model is isotropic. It appears that order 0 is not enough to obtain surface waves with correct group and phase velocities and higher order homogenization terms up to two are often required. In many cases, the order one and two simply require to correct the boundary conditions of the wave equation to obtain an accurate solution, even for surface waves. We show how to extend the theory from the periodic case to the non-periodic case. Examples in periodic and non-periodic media are shown. The accuracy of the results obtained by homogenization is checked against the normal mode solution computed in the original medium and shows good agreement.