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interpretations concerning the upper mantle
only. Significant azimuthal anisotropy has been
observed for Pn waves beneath oceans (Hess,
1964) and continents (Smith & Ekstr om, 1999) as
well as for travel time residuals of teleseismic P
waves (e.g., Babuska
et al.
, 1998). There are clear
observations of polarization anomalies of long
period P-waves which have been used to infer
upper mantle anisotropy (Schulte-Pelkum
et al.
,
2001), but the biggest wealth of observations
comes from SKS-splitting measurements (for a
review see e.g., Long & Silver, 2009). Surface
waves also exhibit a clear anisotropic behaviour.
Both Love and Rayleigh waves show azimuthal
dependencies for fundamental modes (e.g., Tram-
pert & Woodhouse, 2003; Ekstr om, 2011) as well
as overtones (Visser
et al.
, 2008). The azimuthally
averaged phase velocities of Love and Rayleigh
waves see a different vertically averaged elastic
structure. This is known as the Love-Rayleigh
discrepancy and was first interpreted by (Ander-
son 1961) in terms of anisotropy. A transverse
isotropic medium with vertical symmetry is
sufficient the reconcile Love and Rayleigh wave
propagation. Most joint inversions of Love and
Rayleigh wave phase and/or group velocities for
three-dimensional structure therefore employ a
transverse isotropic description of the uppermost
mantle (e.g., Montagner & Tanimoto, 1991; Gung
et al.
, 2003; Kustowski
et al.
, 2008). There are
few clear inferences of anisotropy in the mantle
transition zone, but surface wave overtones
suggest azimuthal (Trampert & van Heijst, 2002)
and radial (Visser
et al.
, 2008) anisotropy. There is
a large consensus that the lower mantle is devoid
of anisotropy with the exception of D'' (e.g.,
Maupin
et al.
, 2005; Panning & Romanowicz,
2006). Wave propagation through the inner core
is best explained using anisotropy although the
details are not fully mapped yet (e.g., Morelli
et al.
, 1986; Woodhouse
et al.
, 1986; Ishii &
Dziewonski, 2002; Beghein & Trampert, 2003;
Deuss
et al.
, 2010).
In mathematical terms, seismic anisotropy is
easily understood. From continuum mechan-
ics we know that if the local elastic tensor
has more than two independent parameters,
wave propagation depends on the direction of
propagation. With seismic observations we can-
not estimate the elastic tensor at a specific point
in space, but only spatial averages from tens to
thousands of kilometres depending on the waves.
This makes the seismic inferences difficult to
interpret. From geodynamics and mineral physics
modeling we know that anisotropic minerals
or isotropic inclusions can align themselves
in preferred strain orientations. The former is
known as lattice preferred orientation (LPO) and
the latter as shape preferred orientation (SPO)
(e.g., Karato, 2008). Because the orientation of
these crystals and inclusions is not instanta-
neous, the observation of seismic anisotropy
potentially gives us valuable constraints on
the geological history of the strain field (e.g.,
Wenk
et al.
, 2011). As mentioned above, seismic
observations cannot image individual crystal or
inclusion, but only spatial averages of them.
The (an)isotropic description therefore is scale-
dependent. Backus (1962) showed that a stack
of thin (sub-wavelength) isotropic layers is seen
by seismic waves as a large-scale anisotropic
medium. This concept can be generalized, and
in fact any large scale description of small scale
isotropic heterogeneity has to include anisotropy
(e.g., Capdeville
et al.
, 2010). While this is
a fundamental property of the mathematical
description at a limited wavelength, another
ambiguity is often forgotten. To infer seismic
anisotropy from travel time or polarization
anomalies, an inverse problem is usually solved.
Starting from Equation (11.1), let us explicitly
partition the model parameters into isotropic (
i
)
and anisotropic (
a
) parts. Neglecting data errors
and the starting model for notational simplicity,
we find
(
G
i
G
a
)(
m
i
m
a
)
T
.
d
=
|
|
(11.12)
From Equation (11.8), we see that the inverse
operator will also partition and hence
m
i
˜
m
a
S
i
G
i
S
a
G
a
m
i
m
a
S
i
G
a
=
(11.13)
S
a
G
i
The ill-posedness of the inverse problem requires
regularization which implies that the resolution
