Low-frequency layer-induced anisotropy

It is often useful to compare velocities estimated from seismic experiments with those measured by well logs at much smaller scales and higher frequencies. In the presence of fine layering, seismic velocities and anisotropy are scale- and frequency-dependent and upscaling of well logs is necessary to allow comparison. When upscaling well-log data measured at sonic frequencies, we assumed a layered medium with layer thickness given by logging step. Standard upscaling gives the exact solution for effective properties of a layered medium assuming that all constituents of the medium are linearly elastic and there are no anelastic energy losses. This method is static, meaning that the solution is obtained in the zero frequency limit. We expand upscaling to low frequencies and propose to use a seismic wavelet estimate for weighted averaging of effective properties. By expanding the logarithm of the propagator matrix computed from the stack of horizontal transversely isotropic layers with a vertical symmetry axis (VTI) in series with frequency and keeping the first three terms in this series, we obtain the low-frequency extension of Backus upscaling. The effective medium properties computed for individual nonzero frequencies correspond to a nonphysical medium with a vertical symmetry axis. To preserve the VTI symmetry, the effective slowness surface obtained for each individual frequency is approximated by a VTI medium by fitting the coefficients of the Taylor series derived from corresponding eigenvalues of the effective system matrix and similar series obtained for the vertical slownesses of different wave modes. The frequency-dependent anisotropy parameters are obtained from Taylor series coefficients for individual wave modes. Afterwards, these coefficients are averaged with frequencies weighted by the spectrum of a seismic wavelet extracted at the corresponding depth interval. The proposed averaging technique is data-driven and takes into account the low-frequency behavior of seismic waves with near-vertical propagation.