2-D non-periodic homogenization to upscale elastic media for P–SV waves

SUMMARY The purpose of this paper is to give an upscaling tool valid for the wave equation in general elastic media. This paper is focused on P–SV wave propagation in 2-D, but the methodology can be extended without any theoretical difficulty to the general 3-D case. No assumption on the heterogeneity spectrum is made and the medium can show rapid variations of its elastic properties in all spatial directions. The method used is based on the two-scale homogenization expansion, but extended to the non-periodic case. The scale separation is made using a spatial low-pass filter. The ratio of the filter wavelength cut-off and the minimum wavelength of the propagating wavefield defines a parameter e0 with which the wavefield propagating in the homogenized medium converges to the reference wavefield. In the general case, this nonperiodic extension of the homogenization technique is only valid up to the leading order and for the so-called first-order corrector. We apply this non-periodic homogenization procedure to two kinds of heterogeneous media: a randomly generated, highly heterogeneous medium and the Marmousi2 geological model. The method is tested with the Spectral Element Method as a solver to the wave equation. Comparing computations in the homogenized media with those obtained in the original ones shows that convergence with e0 is even better than expected. The effects of the leading order correction to the source and first correction at the receivers’ location are shown.