As they propagate through the Earth, seismic waves experience energy loss, which is summarized by , where is the internal energy lost by a seismic wave in one cycle. Attenuation is high and nearly constant within a certain frequency band and rapidly falls off with frequency away from this band. The frequency dependence of can be described using a power law, , with a model-dependent that is usually thought to be smaller than 0.5 within the absorption band.

Though seismological efforts at constraining globally-averaged
within the absorption band have benefited from numerous
measurements of surface wave or normal mode attenuation (see
`http://mahi.ucsd.edu/Gabi/rem.html`), the determination of has
been confounded by the fact that oscillations at different frequencies
can have very different depth sensitivities to elastic and anelastic
properties of the earth. As a result of this tradeoff between
frequency and depth effects, radial variations of attenuation can
obscure the signal. The only studies attempting to obtain
within the absorption band have found ranging from
0.1 to 0.3 while emphasizing the lack of resolution on the deduced
values. More recent studies have relied upon analysis of body waves
to argue for values of in the 0.1-0.4 range (see *Romanowicz and Mitchell*, 2007). However, these
studies were restricted to frequencies higher than 40mHz and were of
regional extent, leaving unanswered the question of the average mantle
.

In light of these difficulties, seismic studies routinely assume that,
within the seismic band, cannot be resolved and thus
implicitly rely on the frequency-independent attenuation model of
*Kanamori and Anderson* (1977). We re-examine the
model's applicability to the mantle using a new method based on the
standard analysis of *Backus and Gilbert* (1970) that
allows us to separate the effects of the radial profile from those
due to frequency dependence of as described by .

We can relate a mode attenuation measurement to material
properties within the Earth via sensitivity (Fréchet) kernels
and :

The sensitivity kernels of fundamental modes with similar frequencies
are very similar, implying that the datasets are highly
redundant. We seek to exploit this redundancy and divide modes into a
low and high frequency bin, denoted by superscript and ,
respectively. Each linear combination of Fréchet kernels of modes
in each bin defines a new ``hyperkernel'':

(2.12) |

Each particular choice of and will yield hyperkernels with different depth sensitivities. Therefore, by requiring that and yield hyperkernels with identical sensitivities to the radial attenuation profile, it is possible to remove the trade-off between depth and frequency dependence of attenuation measurements. Since we focus on the effective in the mantle, we require the hyperkernels to be zero outside the mantle while providing maximally uniform sensitivity in the mantle. In order to eliminate the contribution from poorly-constrained mantle bulk attenuation, we seek hyperkernels that are insensitive to .

To each hyperkernel corresponds a value, which is a weighted
average of the measurements of its constituent normal modes:

(2.13) |

(2.14) |

After validating our method on a synthetic dataset, we apply it to
existing attenuation measurements of free oscillations and surface
waves spanning the period range 3200s-50s. We observe that effective
is likely to be frequency dependent. Specifically,
is negative at periods longer than 1000s and positive and increasing
at shorter periods (see Figure 2.46). This conclusion runs against both the
assumption of frequency-independent attenuation often used in
seismology, and the constant, positive model suggested by
laboratory studies (*Jackson et al.*, 2005). A frequency-dependent effective is nevertheless physically plausible. This is because the effective that one would obtain by analyzing the normal-mode and surface-wave
attenuation measurements is the result of an interplay of the *actual* associated with the material at a given depth and the
position of the absorption band. This interplay can give rise to a
negative effective as long as the actual is negative
at long periods at some depths. Furthermore, our preferred model of frequency dependence of attenuation is consistent with earlier studies that have relied upon body waves and have focused on higher frequencies (see Figure 2.46).

A non-zero value of carries important implications for the construction of radial profiles of attenuation. Efforts at determining the radial profile of attenuation in the Earth have routinely assumed that attenuation is frequency independent. The resulting models have, therefore, mapped the signal of frequency-dependence of into its depth profile.

Relating lateral variations of attenuation in terms of temperature
requires knowing , since when when is zero,
is a constant, whereas when is positive,
is exponentially dependent on temperature
(*Minster and Anderson*, 1981). Recent studies of lateral attenuation variations
rely on data with periods shorter than 300s (e.g.
*Gung and Romanowicz*, 2004), at which periods our preferred model suggests that
0.3. This value implies an exponential temperature
dependence of attenuation, and justifies the interpretation of
lateral attenuation variations in terms of temperature variations.

Intrinsic attenuation causes dispersion of seismic velocities, which must be corrected when datasets with different frequency content are used to simultaneously constrain Earth structure. Both the values of and of q significantly affect the magnitude of the dispersion correction. For an value of 0.3, the assumption of frequency-independent attenuation will result in 25% error for a frequency ratio of 10 and a 50% error for a frequency ratio of 100.

Finally, the precise knowledge of seismic velocity, its dispersion and associated attenuation is important for meaningful comparisons with other geophysical observables, such as the geoid. Future work should thus be aimed at improving the precision of measurements and the development of radial profiles that properly account for the frequency dependence of .

Backus, G.E. and J.F. Gilbert, Uniqueness in the inversion of inaccurate gross earth data, *Phil. Trans. R. Soc. London, 266*, 123-192, 1970.

Gung, Y. and B. Romanowicz, Q tomography of the upper mantle using three-component waveforms, *Geophys. J. Int.,157*, 813-830, 2004.

Jackson, I., S. Webb, L. Weston, and D. Boness, Frequency dependence of elastic wave speeds at high temperature: a direct experimental demonstration, *Phys. Earth Planet. Inter., 148*, 85-96, 2005.

Kanamori, H. and D.L. Anderson, Importance of physical dispersion in surface wave and free oscillation problems: Review, *Rev. Geophys. Space Phys., 15*, 105-112, 1977.

Minster, B. and D.L. Anderson, A model of dislocation-controlled rheology for the mantle, *Phil. Trans. R. Soc. Lond., 299*, 319-356, 1981.

Romanowicz, B. and B. Mitchell, Deep Earth structure: Q of the Earth from crust to core. In: Schubert, G. (Ed.), *Treatise on Geophysics, 1*, Elsevier, 731-774, 2007.

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