Long‐wave anisotropy in stratified media: A numerical test

Abstract

When a seismic signal propagates in a stratified earth, there is anisotropy if the dominant wavelength is long enough compared to the layer thickness. In this situation, the layered medium can be replaced by an equivalent nondispersive transversely isotropic medium. Theoretical and experimental analyses of the required minimum ratio of seismic wavelength to layer spacing based on kinematic considerations yield different results, with a much higher value in the experimental test. The present work investigates the effects of layering by wave simulation and attempts to establish quantitatively the minimum ratio for which the long‐wave approximation starts to be valid. We consider two‐constituent periodically layered media and analyze the long‐wave approximation for different material compositions and different material proportions in 1-D and 2-D media. The evaluation of the minimum ratio compares snapshots and synthetic seismograms visually and through a measure of coherence. Layering induces scattering with wave dispersion or anisotropy depending upon the wavelength‐to‐layer thickness ratio. The modeling confirms the dispersive characteristics of the wave field, the scattering effects in the form of coda waves at short wavelengths, and the smoothed transversely isotropic behavior at long wavelengths. 1-D numerical tests for different media indicate that the minimum ratio is highest for the midrange of compositions, i.e., equal amount of each material, and for stronger reflection coefficients between the constituents. For epoxy‐glass, the value is around R = 8, while for sandstone‐limestone, it is between R = 5 and R = 6. Recent wave‐propagation experiments done in epoxy‐glass also imply a highest minimum ratio for midrange of composition; however, the 1-D numerical tests confirm the long‐wave approximation at shorter wavelengths than experimentally. The 2-D case shows that the more anisotropic the equivalent medium, the higher the minimum ratio, and that the approximation depends upon the propagation angle with longer wavelengths required in the direction of the layering.