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most in the vicinity of mid-ocean ridges where
characteristic low-velocity regions can lead to a
strong focusing of wave energy that must be ac-
counted for correctly. The strength of the lateral
variations in global attenuation models varies by
a factor of up to 4, which is significantly above the
error estimates of around 10%, given, for instance,
by Romanowicz (1995). The various models are
therefore, technically speaking, inconsistent. The
large discrepancies between
Q
μ
models and the
overly optimistic error estimates are again related
to the properties of an ill-posed inverse problem
where weak constraints and strong trade-offs lead
to a dominance of subjective and often implicit
regularization.
Despite the difficulties involved in attenuation
tomography, some large-scale robust features can
be deduced from the combined analysis of all
available global attenuation models. Above 250
kmdepth, most studies find a correlation between
Q
μ
and surface tectonics. The most notable con-
sistency is the appearance of high
Q
μ
values in
stable continental interiors, including Australia's
Pilbara and Yilgarn cratons, the East European
Platform and the Canadian Shield. Below 250
km depth the attenuation pattern changes, and
low
Q
μ
can roughly be associated with the posi-
tion of hotspots as imaged in isotropic velocity
tomographies.
The variability of
Q
μ
models is due to a strong
methodology-dependence that reflects the diffi-
culty of extracting a robust signal from seismic
wave amplitudes that can be attributed to attenu-
ationwith sufficient confidence. From the current
perspective, much progress remains to be made
in order to arrive at a stage where attenuation
models can be interpreted quantitatively in terms
of the Earth's thermo-chemical structure.
as anisotropy, density and attenuation. The
consensus amongst seismologists decreases from
isotropic to anisotropic, density and attenuation
models. The reason for this is the nonsymmetric
nature of the resolution operator (Equation
11.13). The Fr echet derivatives are such that
trade-offs to isotropic parameters are smallest.
This explains the considerable overlap between
different studies using different approximations
and data. Good data exist to image the elastic
part of the problem, but density is weakly
constrained. The Fr echet derivatives to anelastic
parameters are much smaller with common
parametrizations and misfit functionals. We
see mainly three general future directions: (1)
progressing towards full waveform inversion
with appropriate resolution analysis, (2) principal
component analysis and (3) finding new observ-
ables which are sensitive to a limited range of
parameters.
Full waveform inversion uses gradient-type im-
plementations of Equation (11.10) with exact
Fr echet derivatives computed with purely numer-
ical solutions of the forward problem (Bamberger
et al.
, 1982; Tarantola, 1988; Igel
et al.
, 1996).
This approach accounts for the nonlinear rela-
tion between structure, phase and amplitudes in
the construction of tomographic models, thus
progressing beyond linear approximations com-
monly used. Since the update of the model is just
a scaled version of themisfit kernel, there is no ex-
plicit regularization required, beyond smoothing
the kernels and terminating the inversion after
a finite number of iterations. The first regional-
to continental-scale seismological applications of
full waveform inversion indicate that resolution
indeed improves (e.g. Chen
et al.
, 2007; Ficht-
ner
et al.
, 2009; 2010; Tape
et al.
, 2010). In
particular, the amplitudes of the heterogeneities
increase, and more small-scale features are ro-
bustly imaged. We anticipate that full waveform
inversion techniques will make particularly valu-
able contributions to attenuation tomography.
The replacement of source/receiver correction
factors by more elaborate forward and inverse
modeling techniques that correctly account for
11.7 Promising Future Directions
From the above discussions it is clear that global
tomography is, in general, a multi-observable/
multi-parameter problem. The complicated
nature of this problem has to be addressed very
carefully, especially when inverting for weakly-
constrained and scale-dependent properties such
