Low-Frequency Extension of The Backus Averaging Method

The standard Backus (1962) averaging method is widely used for upscaling of the well-log data to seismic frequency range. In many cases, with strong heterogeneity within the upscaling unit, the Backus averaging is not accurate enough. We propose to extend the Backus averaging method for the low-frequency case and introduce the dispersive Backus model using the Baker-Campbell-Hausdorff (BCH) series (Serre, 1965). We derive the first- and second-order terms of this series, and extend this technique to the medium with arbitrary number of layers in a period. That results in the correction term for velocity dispersion at low frequencies. We show that the phase velocity in such media is the even function of frequency.<br>The accurate description of velocity dispersion for effective medium is very important in seismic modelling and inversion of seismic data into effective reservoir properties.