Traveltime biases in random media and the S-wave discrepancy

Abstract

In the first part of this paper, we develop expressions for the frequency-dependent traveltime bias in Gaussian and exponentially correlated random media. This effect, which is the finite-frequency manifestation of geometrical rays bending to find the fastest path between the source and receiver, necessitates that we first derive a finite-frequency expression for the second-order traveltime perturbation, because the linear perturbation provides no contribution upon ensemble averaging to derive the bias. The expression for the mean traveltime-shift in Gaussian and exponential media is of the form ɛ2σL2/a multiplied by a dimensionless function of a/(λL)1/2, where ɛ is the RMS strength of the relative slowness perturbations, σ is the background slowness, L is the source-receiver distance, a is the scale length of the heterogeneity and λ is the characteristic wavelength of the probing wave. The dependence of the mean traveltime, 〈δT〉, upon the wavelength, λ, has the implication that a slightly heterogeneous medium is apparently dispersive in the frequency band where the wavelengths are comparable to the scale lengths of heterogeneity. Observational evidence for such an intrinsic dispersion can be found in the baseline shift, required by Gilbert & Dziewonski to reconcile their normal-mode based earth models with observed S-wave traveltimes at higher frequencies. Though most of this so-called S-wave discrepancy can be accounted for by anelastic dispersion, any leftover S-wave discrepancy provides a constraint upon the elastic heterogeneity structure of the mantle. For both Gaussian and exponentially correlated random heterogeneity, we derive the maximum allowable RMS heterogeneity, ɛ, as a function of the scale length, a, for the upper and lower mantle. For Gaussian heterogeneity, minima in these curves occur at a≈ 50 km, corresponding to ɛ≈ 0.5–1 per cent in the upper mantle and ɛ≈ 0 –0.5 per cent in the lower mantle. Stronger heterogeneity is admissible if we assume that the mantle is exponentially correlated: in this case, the curves indicate that RMS heterogeneity of approximately 1 to 2 per cent may be present in the upper mantle for scale lengths, a, greater than 100 km. In contrast, exponential structure in the lower mantle is weaker for these scale lengths, with ɛ values admissible up to only 1 per cent. The data are consistent with the upper mantle being substantially more heterogeneous than the lower mantle, by at least a factor of 2.