Residual homogenization for seismic forward and inverse problems in layered media

An elastic wavefield propagating in an inhomogeneous elastic medium is only sensitive in an effective way to inhomogeneities much smaller than its minimum wavelength. The corresponding effective medium, or homogenized medium, can be computed thanks to the non-periodic homogenization technique. In the seismic imaging context, limiting ourselves to layered media, we numerically show that a tomographic elastic model which results of the inversion of limited frequency band seismic data is an homogenized model. Moreover, we show that this homogenized model is the same as the model that can be computed with the non-periodic homogenization technique. We first introduce the notion of residual homogenization, which is computing the effective properties of the difference between a reference model and a 'real' model. This is necessary because most imaging technique parametrizations use a reference model that often contains small scales, such as elastic discontinuities. We then use a full-waveform inversion method to numerically show that the result of the inversion is indeed the homogenized residual model. The full-waveform inversion method used here has been specifically developed for that purpose. It is based on the iterative Gauss-Newton least-square non-linear optimization technique, using full normal mode coupling to compute the partial Hessian and gradient. The parametrization has been designed according to the residual homogenized parameters allowing to build a real multiscale inversion with progressive frequency band enrichment along the Gauss-Newton iterations.