Geoscience Reference
In-Depth Information
able to recover reliably from long-period normal-
mode data.
It is becoming increasingly evident that new
inferences on the details of Earth's 3D density
structure will force us to progress beyond both
traditional data analysis and modeling. The accu-
rate measurement of splitting functions on data
from recent megathrust earthquakes, combined
with proper mode coupling, has the potential
to yield more robust information, also about
odd-degree density structure (Deuss
et al.
, 2011).
Several nonseismological developments may
also help to improve our knowledge concerning
density. Lateral variations of the Earth's body
tides have recently been shown to be above the
observational error, and may thus yield additional
constraints on global density structure (Latychev
et al.
, 2009). Also on a global scale, neutrino
tomography
et al
., originally conceived in the
early 1980s (e.g. De Rujula
et al.
, 1983) - may
provide very direct insight into density structure,
that is free from the trade-offs that plague
seismological approaches (e.g. Gonzalez-Garcia
et al.
, 2008). On small scales (up to a few kilome-
tres) muon transmission tomography has been
used successfully to image low-density magma
conduits inside active volcanoes (e.g. Tanaka
et al.
, 2003, 2010). Both tidal and elementary
particle tomographies are, however,
where
E
and
R
denote an activation energy and
the gas constant, respectively. Before delving into
the details of the Earth's
Q
structure, we give
a brief summary of the description of seismic
wave attenuation, complemented by the most
fundamental observations.
11.6.1 Description of seismic wave
attenuation, basic observables
and observations
Assuming that all seismologically relevant re-
laxation mechanisms can be modeled by lin-
ear rheologies, shear attenuation is commonly
described by the frequency-domain visco-elastic
stress-strain relation
σ
(
ω
)
=
μ
(
ω
)
ε
(
ω
),
(11.15)
where
μ
denotes the complex shear modulus in
the frequency domain. The ratio between the real
and imaginary parts of
μ
defines the shear quality
factor
Q
μ
:
μ
(
ω
)
Im
Q
μ
(
ω
)
=
μ
(
ω
)
.
(11.16)
The bulk quality factor
Q
κ
is defined similarly
on the basis of the complex bulk modulus
κ
.
The observability of
Q
in the Earth is closely
related to the amplitude decay of seismic waves.
For instance, when
Q
μ
>>
1, the amplitude
A
of a
plane S wave propagating through a homogeneous
and isotropic medium behaves as
˜
in their
infancy.
11.6 Attenuation Tomography
exp
,
−
ω x
2
v
s
Q
μ
Seismic waves propagating through the Earth are
attenuated due to various relaxation mechanisms
that lead to the transformation of elastic energy
into heat. The large interest in the absorption
properties of the Earth, parametrized in terms of
the quality factor
Q
, is mostly related to their
temperature dependence. While seismic veloci-
ties are quasi-linearly related to temperature,
T
,
the temperature-dependence of
Q
is well approxi-
mated by an exponential Arrhenius-type law (e.g.
Jackson, 2000)
∝
A
(11.17)
where
x
denotes the distance traveled. Relation
(11.17) is frequently adapted in ray-theoretical
studies that replace
x/Q
μ
v
s
by
ds
Q
μ
(
x
)
v
s
(
x
)
t
∗
=
,
(11.18)
C
with
C
and
ds
denoting the ray path and a path
segment, respectively. Analogous relations can
be derived for plane P waves. Most body wave
RT
,
e
−
Q
−
1
∝
(11.14)
