Global Imaging of the Earth’s Deep Interior: Seismic Constraints on (An)isotropy, Density and Attenuation - Physics and Chemistry of the Deep Earth - page 335

attenuation tomographies are based on variants of
Equations (11.17) and (11.18) because they allow
us to eliminate the contribution of the potentially
uncertain seismic moment by measuring ampli-
tude ratios of different seismic phases (e.g. Reid
et al. , 2001; Cheng & Kennett, 2002; Kennett &
Abdullah, 2011).
Early studies of Q in the Earth were primar-
ily based on the decay of free oscillation peaks
and long-period surface wave amplitudes (e.g. An-
derson et al. , 1965; Canas & Mitchell, 1978;
Anderson & Hart, 1978; Sailor & Dziewonski,
1978). They established that attenuation in shear
largely dominates over attenuation associated
to bulk deformation, meaning that Q μ << Q κ
almost anywhere in the Earth. Attenuation mea-
surements of the Chandler wobble at sub-seismic
frequencies on the one hand, and surface waves on
the other hand furthermore indicated that Q μ is
mildly frequency-dependent, with Q μ ∝ ω α and α
in the range of 0.1-0.5 (e.g. Anderson & Minster,
1979). These results were confirmed by numer-
ous body wave analyses (e.g. Sipkin & Jordan,
1979; Flanagan &Wiens, 1998; Cheng & Kennett,
2002) and laboratory experiments (e.g. Jackson,
2000, 2007; Karato, 2008). In a recent study,

Lekic et al. (2009) developed a method that sep-
arates the depth- and frequency-dependencies of
normal-mode and surface-wave attenuation mea-
surements. Their results suggest that the effective
α in the mantle is negative (

0.4) at periods
longer than 1000 s, transitioning to positive val-
ues around 0.3 for periods shorter than 500 s.
Despite the unequivocal evidence for a power-
law frequency dependence, tomographic inver-
sions mostly assume Q to be constant across the
seismic frequency band, i.e. from

≈−

10 − 3
Hz. This simplification is motivated by the dif-
ficulty to robustly constrain Q variations in the
Earth even within a narrow frequency band. The
constant- Q model is closely related to the notion
of a continuous absorption band, i.e. a continu-
ous distribution of relaxation mechanisms that
lead to nearly frequency-independent absorption
(Liu et al. , 1976), as shown in Figure 11.6. This
is in contrast to the appearance of isolated ab-
sorption peaks that are, for instance, commonly
found in metals (e.g. Zener, 1948. Outside the
absorption band, Q is predicted to be proportional
to ω − 1 at the low-frequency end, and to ω at the
high-frequency end.
A direct consequence of attenuation in gen-
eral is dispersion, i.e. the frequency-dependence

≈ ≈

1to

5.08

Fig. 11.6 Phase velocity (dashed)
and Q − 1 (solid) as a function of
frequency in an absorption band
model with a nearly constant Q of
400 in the frequency band from
≈

1/400

1/500

10 − 5 Hzto1Hz.A
frequency-dependent Q within
the absorption band is shown
schematically in the form of a
gray line. A power-law frequency
dependence, Q

5.04

1/1000

∝ ω α with a
slightly positive α around 0.3 is
consistently found in seismic and
laboratory studies, but commonly
ignored in tomographic
inversions.

Q − 1
Phase velocity

5.00

− 8 − 6 − 4 − 2 0 2 2

Log (frequency)