Homogenization for wave propagation in deterministic media with no scale separation, such as geological media, has been recently developed. With such an asymptotic theory, it is possible to compute an effective medium valid for a given frequency band, such that effective waveforms and true waveforms are the same up to a controlled error. So far, this method has been mainly used as a preprocessing step to simplify complex media to reduce the forward modeling numerical cost. Nevertheless, for the inverse problem, it raises an interesting point: for a given dataset and a given frequency band, two models are now solution of the full waveform inverse problem (FWI): the true model and the homogenized model. Because homogenized models belong to a finite dimensional space (which is no the case of potential true models), we claim it is easier (and probably the only option) to search for an homogenized model than for a true model. In this work we investigate this idea of a FWI constrained by homogenization.
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