Geoscience Reference
In-Depth Information
is needed to assess the quality of
m
, unless one of
P-velocity models are almost exclusively
constructed from large collections of travel
time residuals. The uneven distribution of
earthquakes and seismic stations implies that
mainly tectonically active regions are sampled
by these data, and the main features to be imaged
are descending lithospheric plates (e.g., van der
Hilst
et al.
, 1997). The depth extent of the
imaged slabs varies, some reaching the lower-
most mantle, others stagnant in the transition
zone (Fukao
et al.
, 2009). If S-wave travel time
residuals are used, remarkably similar images
are retrieved (Grand
et al.
, 1997). Travel time
models are often referred to as high-resolution
models. This is misleading since the fine block
or pixel parametrization, usually employed in
the construction of these models, only indicates
a potential maximum resolution. The actual
achieved resolution is frequently unknown,
despite the many proposed synthetic tests.
S-velocity models are more often constructed
from long-period waveforms and/or a combi-
nation of long-period body wave travel time
residuals, surface wave dispersion measurements
and normal-mode splitting functions (e.g. Megnin
& Romanowicz, 2000; Kustowski
et al.
, 2008;
Ritsema
et al.
, 2011). The main robust features
appearing are low-velocity mid-oceanic ridges
recognizable down to a depth of 100-150 km
(Figure 11.1, left). There is a clear ocean-continent
difference disappearing at around 250 km depth.
The transition zone and the lower-most mantle
are dominated by large low-velocity zones
beneath Africa and the Pacific Ocean and higher
velocities in a circum-Pacific belt (Figure 11.1,
middle and right). These features have been
known since the pioneering studies of (Masters
et al.
1982), (Woodhouse &Dziewonski 1984) and
(Dziewonski 1984). They have since been con-
firmed by virtually all successive studies. These
S-velocity models employ data with a more even
coverage than high-frequency P-wave travel time
residuals, and the parametrization emphasizes
the long-wavelength structure by employing low-
order spherical harmonic expansions. Overall,
the resolution is more even over the globe, but
that is not to say that it is known precisely.
them is dominant.
The only way to avoid possibly unrealistic
Gaussian distributions and the effects that reg-
ularization has on our inferences, is to directly
sample the probability distribution (11.5), using,
for instance, Monte Carlo techniques (e.g. Sam-
bridge & Mosegaard, 2002) or neural networks
(e.g. Meier
et al.
, 2007). These are, however, com-
putationally much more involved than solving
the regularized analytical expressions based on
Gaussian statistics.
Finally, we note that seismic tomography is
in reality not a linear inverse problem because
our data generally depend nonlinearly on the
properties of the Earth. Within a probabilistic
framework, non-linearity can naturally be ac-
commodated in the model space sampling. In
deterministic problems, the nonlinearity is most
often addressed using a perturbation approach.
This involves updating the model iteratively:
m
n
=
m
n
−
1
+
P
n
−
1
(
d
−
G
n
−
1
m
n
−
1
)
+
Q
(
m
n
−
1
−
m
0
) ,
(11.10)
where
n
indicates the iteration step and
G
n
−
1
the Fr echet derivatives of the nonlinear forward
functional with respect to
m
n
−
1
. Commonly used
algorithms are, for instance, conjugate gradient
or Newton schemes which each define separate
operators
P
n
−
1
and
Q
. It is worth noting that
solution (11.5) holds for linear as well as nonlinear
problems. In the following, we will refer to these
different ways of solving the inverse problem in
the context of (an)isotropic and (an)elastic seismic
tomography.
11.3 Isotropic Velocity Tomography
Over the last three decades seismic tomography
has produced a large number of models with a
high degree of overlap as documented in many
review articles (e.g., Woodhouse & Dziewonski,
1989; Ritzwoller & Lavely, 1995; Masters
et al.
,
2000; Romanowicz, 2003; Trampert & van der
Hilst, 2005; Rawlinson
et al.
, 2010).
