Geoscience Reference
In-Depth Information
the help of large computers, state-of-the-art
techniques allow us to model quasi-exact wave
propagation in quite complex Earth models. Seis-
mic tomography is the corresponding ill-posed
inverse problem and still quite a challenge. The
main limiting factors are the uneven illumination
of the Earth's interior by seismic waves and the
computational resources. The former is addressed
by using more of the available data, although
the nonuniform distribution of earthquakes and
seismic stations cannot entirely be undone. The
computational burden is reduced by implement-
ing various theoretical approximations during
imaging.
The most common approach to seismic tomog-
raphy is based on travel times of identified body
and surface waves, which are inverted for ve-
locities using approximations of elastic isotropy
and high frequency wave propagation (ray theory).
Recent reviews provide extensive reference lists
going back to the beginning of this research field
in the 1970s (e.g., Romanowicz, 2003; Trampert &
van der Hilst, 2005; Nolet, 2008; Rawlinson
et al.
,
2010). Sometimes, a collection of normal-mode
frequencies is added to provide a global long-
wavelength constraint on the structure (Ritsema
et al.
, 2011). While these studies have shaped a
clear picture of the Earth's internal structure, they
have not provided unambiguous information on
their origin and evolution. Trampert & van der
Hilst 2005 argued that estimates of uncertainty
and information on the density structure are es-
sential to distinguish between temperature and
chemical origins. A convincing case can further
be made that information on intrinsic attenuation
and anisotropic parameters significantly helps our
quest to understand the thermo-chemical evo-
lution of the Earth (e.g., Karato, 2008). Mantle
discontinuities also provide interesting informa-
tion and is discussed in chapter 10 by Deuss
et al
.
The purpose of this chapter is to review on
which class of models there is agreement and
where further investigation is required. The main
emphasis will be on the imaging itself and not
so much its thermo-chemical interpretation,
although we will provide some indications. In
turn we will treat the case of isotropic velocity
tomography, anisotropy, density and attenuation
imaging in decreasing order of consensus. Finally,
we will make some suggestions for future work.
11.2 An Introduction to Linearised
Inverse Theory
Assume that we have a collection of measure-
ments gathered in a data vector
d
.First,wehave
to decide on a mathematical description of the pa-
rameter field we want to image. Often the model
parameters are expanded onto a finite number of
global or local basis functions, e.g. spherical har-
monics, blocks and many others. The coefficients
for these basis functions are called the model
parameters and are collected in a vector
m
.Math-
ematically,
d
and
m
belong to vector spaces, the
data space
D
and the model space
M
, respectively.
The mapping from
M
to
D
is called the forward
operator. It is, in the case of a linear problem,
represented by the matrix
G
. For instance, if we
use ray theory, the data could be travel time resid-
uals, the model parameters constant slownesses
in blocks, and a row of
G
contains the lengths of
a particular ray in each block. Formally, we may
write
d
=
Gm
+
e
.
(11.1)
The data are not perfect and therefore we added
an error vector
e
which is implicitly recorded
together with the data
d
. The inverse problem
consists of finding a linear mapping
S
from
D
to
M
. Since
G
is usually not invertible,
S
is not
the exact inverse of
G
, and thus we only find an
estimator
m
of
m
. This is formally written as
˜
m
=
m
0
+
S
(
d
−
Gm
0
),
(11.2)
where we have introduced an optional starting
model
m
0
. The interesting question is how our
estimated model
˜
m
is related to the true model
m
that we intended to find. We therefore insert
Equation (11.1) into Equation (11.2) to get
m
˜
−
m
0
=
R
(
m
−
m
0
)
+
Se
,
(11.3)
