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of elastic velocity and attenuation models will be
incompatible, because the deduced 3D variations
in
Q
were not accounted for in the interpretation
of travel times used to derive the velocity model
that was initially needed to remove the elastic
effect from amplitudes. This discrepancy under-
lines the intrinsic nonseparability of elastic and
anelastic effects that precludes any consistent
removal of focusing effects from seismic wave
amplitudes.
Both the multi-observable/multi-parameter na-
ture of the inverse problem for
Q
structure and
the inconsistency of many simplifications explain
why our progress in attenuation tomography has
been rather slow, compared to tomography for
elastic structure.
deviations were found to range between 10% and
20%, depending on depth.
The distribution of bulk attenuation, required
by the decay of radial free oscillations, is the
subject of a long-lasting debate, in the course
of which a large variety of models have been pre-
sented. Some of these are displayed in Figure 11.7.
Dziewonski & Anderson (1981) confined a com-
paratively large bulk attenuation in PREM to
the inner core, where
Q
κ
=
1327.7 compared to
Q
κ
=
57823 anywhere else in the Earth. They
noted, however, that ''this should be only under-
stood as a way to lower
Q
of radial modes in
order to make them more compatible with obser-
vations. The problem is highly non-unique and its
early resolution is not likely.'' Similar to PREM,
Anderson & Hart (1978) and Okal & Jo 1990 lo-
cate high bulk attenuation (small
Q
κ
) in the inner
core. In contrast, more recent models based on
higher-quality data clearly prefer a distribution of
low
Q
κ
in the upper mantle and the transition
zone (Durek & Ekstr om, 1996), or in the whole
mantle and the outer core (Widmer
et al.
, 1991;
Resovsky
et al.
, 2005). The variety of
Q
κ
models
is proportional to the ill-posedness of the inverse
problem. As in the case of density tomography,
the observational constraints on
Q
κ
are weak, so
that the solution of a deterministic inverse prob-
lem is dominated by the unavoidable and largely
subjective regularization. Any particular value of
Q
κ
should be interpreted with caution, since plau-
sible values of bulk attenuation may span nearly
one order of magnitude (Resovsky
et al.
, 2005).
Alloftheradial
Q
μ
models shown in Figure
11.7 are based on the assumption of frequency-
independent attenuation, despite the well-known
power law relation
Q
μ
∝
ω
α
with
α
mostly be-
tween 0.1 and 0.5. Additional free parameters to
describe frequency dependence are unlikely to be
resolvable. As pointed out by Lekic
et al.
(2009),
this simplification may lead to incorrect
Q
μ
pro-
files, because higher-frequency waves with higher
Q
μ
constrain shallower structure, whereas lower-
frequency waves with lower
Q
μ
constrain deeper
structure. Thus, attenuation estimates may be bi-
ased towards low
Q
μ
values in the deep mantle
and towards high
Q
μ
values in the upper mantle.
11.6.3 Constraints on the radial attenuation
structure
Constraints on the spherically averaged attenu-
ation in the Earth are mostly derived from the
amplitude decay of body waves, long-period sur-
face waves and free oscillation peaks.
Estimates of the radial shear attenuation by
various authors, summarized in Figure 11.7, are
relatively consistent. The only notable exception
is the lithosphere where values range between
Q
μ
≈
200 (Durek & Ekstr om, 1996) and
Q
μ
≈
600
(Dziewonski & Anderson, 1981; Okal & Jo, 1990).
This variability partly reflects the presence of
strong heterogeneity in the lithosphere that pre-
vents the determination of a physically meaning-
ful average. Within a narrow asthenospheric layer,
Q
μ
is generally found to be low (60-90). This drop
is followed by a rapid increase through the transi-
tion zone to
Q
μ
≈
300 near 660 km depth. Within
the lower mantle some authors favour a slight
decrease (Okal & Jo, 1990; Widmer
et al.
, 1991),
while others prefer a constant
Q
μ
≈
300 down
to the core-mantle boundary (Dziewonski & An-
derson, 1981; Durek & Ekstr om, 1996). Reliable
uncertainty estimates for the radial
Q
μ
profile
were provided by Resovsky
et al.
(2005), who ap-
plied a probabilistic inversion to normal-mode
and surface-wave data. Typically, the standard
