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of phase velocity (Figure 11.6). In the constant-
Q
model (Liu
et al.
, 1976), the phase velocity
dispersion relation is given by
infers that 15-20%of the observed phase delays in
long-period surface wavesmay in fact be the result
of 3D heterogeneity in
Q
μ
. It follows, in conclu-
sion, that any inference on the spatial distribution
of
Q
requires the solution of an intrinsically cou-
pled multi-observable/multi-parameter problem.
The inverse problem is complicated by the dif-
ficult nature of seismic wave amplitudes that are
frequently used as the only measurement to con-
strain
Q
. In addition to attenuation, amplitudes
are strongly affected by focusing. Waves trav-
eling through a low-velocity region are focused
and amplified, whereas defocusing and amplitude
reduction occur in high-velocity zones. Further-
more, amplitude measurements are affected by
miscalibrated instruments, scattering and seismic
source characteristics. Small-scale heterogeneity
in the immediate vicinity of the receiver, such
as layers of nearly saturated sediments, may give
rise to nonlinear amplitude effects. These fac-
tors add frequency-dependent source and receiver
corrections to the list of unknowns.
The daunting complexity of the inverse prob-
lem for
Q
explains why substantial simplifica-
tions are common. Instead of jointly inverting
for all the necessary free parameters, both elastic
focusing effects and source/receiver corrections
are frequently ignored. While convenient, this ap-
proach was shown to produce incorrect estimates
of attenuation structure in body and surface wave
studies (Dalton
et al.
, 2008; Tian
et al.
, 2009). To
isolate the signature of attenuation, several stud-
ies proposed to remove the focusing effect from
observed amplitudes with the help of 3D elastic
velocity models derived from travel time tomog-
raphy. While being a significant step forward,
this strategy suffers from the subjective choice
of an elastic model, and its smoothness proper-
ties in particular. Overly rough elastic models
may lead unrealistically strong focusing, while
overly smooth models will compensate the un-
derestimated elastic effect by excessive
Q
vari-
ations. Interestingly, this scale-length problem
is similar to the one encountered in anisotropic
tomography where the roughness of isotropic ve-
locity variations trades off with the strength of
anisotropy. In any case, the resulting combination
π Q
ln
ω
,
v
phase
(
ω
)
v
phase
(
ω
ref
)
1
=
1
+
(11.19)
ω
ref
where
ω
ref
is a reference frequency. Equation
(11.19) holds, provided that
Q >>
1, and that
ω
lies within the absorption band. Equation (11.19)
suggests that phase velocity increases monoton-
ically with increasing frequency; and this effect
is most pronounced in highly attenuative media
where
Q
is small.
It is important to make a clear distinction be-
tween velocity as a property of a traveling wave,
and velocity as a material property. The phase
velocity in Equation (11.19) is strictly speaking
the propagation speed of a monochromatic plane
wave in an isotropic and homogeneous medium.
This is not generally identical with a velocity as
material property, given by the square root of an
elastic modulus divided by density.
11.6.2 The nature of the inverse problem
The dependence of the phase velocity on both
Q
and
ω
has profound implications on our ability
to infer the absorption properties of the Earth.
On the one hand, there is no seismological ob-
servable that is sensitive to variations in
Q
only,
while being practically insensitive to variations
in any other parameter. On the other hand, there
is also no seismological observable that is unaf-
fected by
Q
. The amplitudes of both long-period
(50-200 s) surface waves and intermediate-period
(5-30 s) body waves are found to be dominated
by focusing induced by 3D variations of purely
elastic structure (Sigloch, 2008; Zhou, 2009; Tian
et al.
, 2009). This does not mean that Equation
(11.17) is incorrect, but it is a reminder that a
proportionality derived from plane wave analysis
in homogeneous media should not be mistaken
for an equality that holds in the real Earth. The
impact of attenuation on travel times should also
not be underestimated. Based on the analysis of
finite-frequency sensitivity kernels, Zhou (2009)
