Non-periodic homogenization of 3-D elastic media for the seismic wave equation

International audienceBecause seismic waves have a limited frequency spectrum, the velocity structure of the Earth that can be extracted from seismic records has a limited resolution. As a consequence, one obtains smooth images from waveform inversion, although the Earth holds discontinuities and small scales of various natures. Within the last decade, the non-periodic homogenization method shed light on how seismic waves interact with small geological heterogeneities and 'see' upscaled properties. This theory enables us to compute long-wave equivalent density and elastic coefficients of any media, with no constraint on the size, the shape and the contrast of the heterogeneities. In particular, the homogenization leads to the apparent, structure-induced anisotropy. In this paper, we implement this method in 3-D and show 3-D tests for the very first time. The non-periodic homogenization relies on an asymptotic expansion of the displacement and the stress involved in the elastic wave equation. Limiting ourselves to the order 0, we show that the practical computation of an upscaled elastic tensor basically requires (i) to solve an elastostatic problem and (ii) to low-pass filter the strain and the stress associated with the obtained solution. The elastostatic problem consists in finding the displacements due to local unit strains acting in all directions within the medium to upscale. This is solved using a parallel, highly optimized finite-element code. As for the filtering, we rely on the finite-element quadrature to perform the convolution in the space domain. We end up with an efficient numerical tool that we apply on various 3-D models to test the accuracy and the benefit of the homogenization. In the case of a finely layered model, our method agrees with results derived from Backus. In a more challenging model composed by a million of small cubes, waveforms computed in the homogenized medium fit reference waveforms very well. Both direct phases and complex diffracted waves are accurately retrieved in the upscaled model, although it is smooth. Finally, our upscaling method is applied to a realistic geological model. The obtained homogenized medium holds structure-induced anisotropy. Moreover, full seismic wavefields in this medium can be simulated with a coarse mesh (no matter what the numerical solver is), which significantly reduces computation costs usually associated with discontinuities and small heterogeneities. These three tests show that the non-periodic homogenization is both accurate and tractable in large 3-D cases, which opens the path to the correct account of the effect of small-scale features on seismic wave propagation for various applications and to a deeper understanding of the apparent anisotropy