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Non-periodic homogenization of the elastic wave equation for wave propagations in complex media

When considering numerical acoustic or elastic wave propagation in media containing small heterogeneities with respect to the minimum wavelength of the wavefield, being able to upscale physical properties (or homogenize them) is valuable, for mainly two reasons: first, replacing the original discontinuous and very heterogeneous media by a smooth and more simple one, is a judicious alternative to the necessary fine and difficult meshing of the original media required by many wave equation solvers; second, it helps to understand what properties of a medium are really “seen” by the wavefield propagating through it, which is an important aspect in an inverse problem approach. We present here a solution to solve this up-scaling problem for non-periodic complex media with rapid variations in all directions based an a non-periodic homogenization procedure. We first present a pedagogical introduction to non-periodic homogenization in 1-D, allowing to find the effective wave equation and effective physical properties of the wave equation in a highly heterogeneous medium. It can be extended from 1D to a higher space dimension and a special care of boundary conditions is required. This development can be seen as an extension of the classical two-scale periodic homogenization theory applied to the wave equation for non-periodic media. To validate this development, we then present two examples of wave propagation in 2D complex elastic models: a geometrically square model with random heterogeneities, and the Marmousi2 model. A reference solution is computed with the Spectral Element Method with meshes honoring all interfaces. Furthermore, we compare the results obtained in the homogenized model and in a low-pass filtered model with respect to the reference solution.