Velocity shift in random media

Seismic waves in a random medium (with standard deviation ∈ and correlation distance a of the relative slowness fluctuations) prefer fast paths, and therefore the apparent velocity of wave propagation is larger than the velocity which corresponds to the volume average of slowness. This velocity shift can be determined by ray perturbation theory (Snieder & Sambridge 1992), by the Huygens method (Podvin & Lecomte 1991) and by wave theory (Müller, Roth & Korn 1992). We apply all three methods to plane-wave propagation through a 2-D acoustic medium with Gaussian or exponential autocorrelation function of the slowness fluctuations. Ray perturbation theory gives numerical and analytical results, but has path-length (L) limitations. The Huygens method, which also gives the ray-theoretical velocity shift, can be used for L/a ratios of seismological interest. Wave theory shows that the velocity shift also depends on the wavelength λ and that for λ/a less than about 0.1 the velocity shift agrees with the result of the Huygens method. For λ/a = 1 the wave-theoretical (i.e. true) shift is lower than the Huygens-method shift by a factor of 0.25 to 0.5. Simple formulae for the e dependence of the Huygens-method shift at long path lengths (L/a≥ 80) are given, and a correction factor is derived which approximately transforms plane-wave 2-D into spherical-wave 3-D velocity shifts; the latter correspond to 3-D two-point ray tracing.

For short-period seismic waves, propagating to teleseismic distances, mantle heterogeneity with ∈=1 per cent and a = 100 km produces a velocity shift of about 0.2 per cent. Shifts of this order can explain the difference in earth models, derived from free oscillations on the one hand and from short-period body waves on the other. A velocity shift (or velocity dispersion) due to anelasticity would be additional.