# Frequency dependence of long-period *t**

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## Abstract

Multi-phase long-period *t** measurements are among the key evidences for the frequency-dependent mantle attenuation factor, *Q*. However, similarly to *Q*, poorly constrained variations of Earth’s structure may cause spurious frequency-dependent effects in the observed *t**. By using an attenuation-coefficient approach which incorporates measurements of geometric spreading (GS), such effects can be isolated and removed. The results show that the well-known increase of body *P*-wave *t** from ~0.2 s at short periods to ~1–2 s at long periods may be caused by a small and positive bias in the underlying GS, which is measured by a dimensionless parameter *γ** ≈ 0.06. Similarly to the nearly constant *t** at teleseismic distances, this GS bias is practically range-independent and interpreted as caused by velocity heterogeneity within the crust and uppermost mantle. This bias is accumulated within a relatively thin upper part of the lithosphere and may be closely related to the crustal body-wave GS parameter *γ* ~ 4–60 mHz reported earlier. After a correction for *γ*, P-wave *t* _{ P }* becomes equal ~0.18 s at all frequencies. By using conventional dispersion relations, this value also accounts for ~40 % of the dispersion-related delay in long-period travel times. For inner-core attenuation, the attenuation coefficient shows a distinctly different increase with frequency, which is remarkably similar to that of fluid-saturated porous rock. As a general conclusion, after the GS is accounted for, no absorption-band type or frequency-dependent upper-mantle *Q* is required for explaining the available *t** and velocity dispersion observations. The meaning of this *Q* is also clarified as the frequency-dependent part of the attenuation coefficient. At the same time, physically justified theories of elastic-wave attenuation within the Earth are still needed. These conclusions agree with recent re-interpretations of several surface, body and coda-wave attenuation datasets within a broad range of frequencies.

## Keywords

Attenuation Body waves Frequency dependence Geometric spreading Dispersion Inner core Mantle Structure*Q*

*t**

## Notes

### Acknowledgments

This research was supported by NSERC Discovery Grant RGPIN261610-03. This work was conducted in part during the author’s 2010 sabbatical visit at the Air Force Research Laboratory, Hanscom AFB, sponsored by the US National Research Council. I thank Dr. Anton Dainty for hosting this visit. I also thank two anonymous reviewers for constructive comments.

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