To compute the effective matrix from the stack of the layers we use the Baker-Campbell-Hausdorff (BCH) series. From the truncated BCH series we derive the velocity dispersion equation that correctly describes the wave propagation at low frequencies. The explicit equations derived for an acoustic medium, periodically layered medium, medium with monoclinic anisotropy and the vertical propagation case. The derived equations are tested on the two-layer periodically layered medium and on the real well-log data. The first-order correction term in the velocity dispersion equation results in more accurate phase velocity at low frequencies. This correction is an extension of the Backus averaging method and can be used for upscaling of the well-log data and seismic modeling.
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