Global Imaging of the Earth’s Deep Interior: Seismic Constraints on (An)isotropy, Density and Attenuation - Physics and Chemistry of the Deep Earth - page 332

Along similar lines, Simmons et al. , 2010 as-
similated seismic travel times, the global free air
gravity field, the divergence of tectonic plates, dy-
namic surface topography and topography of the
core-mantle boundary into a joint velocity and
density model. Their method relies on the radial
viscosity profile of the Earth (Mitrovica & Forte,
2004), and on the assumption that lateral hetero-
geneities are most likely of thermal origin. While
being a significant step forward, this approach is
limited by the bias towards thermally induced
density variations, the large uncertainties in the
viscosity profile, and the absence of a formal error
analysis that accounts for plausible variations of
the forward modeling assumptions.
Recognizing that enforced velocity-density cor-
relations may prevent important inferences on
thermo-chemical structure from the outset, Ishii
& Tromp (1999, 2001, 2004) inverted normal-
mode splitting and free air gravity for a global
degree-6 model of velocities and density that were
not related a priori. Their results suggest a decor-
relation of velocity and density variations, and
in particular the presence of high-density piles in
the deep mantle beneath the African and Pacific
super-plumes.
The robustness of Ishii and Tromp's density
models was at the centre of a long debate that
reflects the difficulty of solving a deterministic
multi-parameter inverse problem that is char-
acterized by weak constraints, strong trade-offs
and the nearly complete absence of true physical
prior information. Resovsky & Ritzwoller (1999)
pointed out that the resolution analysis in a de-
terministic inverse problem depends on the prior
knowledge, i.e. the choice of the prior model co-
variance σ m in Equations (11.8) and (11.9). Since
we have little prior knowledge on density vari-
ations in the Earth, σ m corresponding to density
should be very large. The weakness of our con-
straints furthermore implies that the entries of
the forward modeling or sensitivity matrix G are
small. It follows that the matrix G T G

the inversion in a subjective way, and not to
represent an objective state of knowledge. When
the data constraints are weak, this subjectivity
dominates the perceived resolution, expressed, for
instance, in terms of the posterior covariance C

m
(Equation 11.9). Based on a series of test inversions
with different choices of the prior information,
Resovsky & Ritzwoller 1999 concluded that a
decorrelation of S velocity and density could not
be detected reliably. A similar conclusion was
reached by Romanowicz (2001), who found that
density in the lower mantle trades off strongly
with the topography of the core-mantle bound-
ary, and that gravity data hardly discriminate
between different density models. However, Ro-
manowicz (2001) was able to infer rough bounds
for the depth-dependent relationship between the
degree-2 structure of density and S velocity. The
robustness of 3D density variations in determinis-
tic inversions was furthermore questioned by Kuo
& Romanowicz (2002) on the basis of synthetic
experiments.
To circumvent the subjectivity of determinis-
tic tomographies, Resovsky & Trampert (2003)
worked with a probabilistic formulation of the in-
verse problem, that does not require explicit regu-
larization to stabilize and reduce trade-offs. Using
a combined set of normal-mode and surface-wave
data they explored the model space with the help
of a neighbourhood algorithm (Sambridge, 1999a;
b) to produce probability distributions for velocity
and density heterogeneities. These probability
densities provide a complete description of our
state of knowledge, including uncertainties. The
results of Resovsky & Trampert (2003) suggest
that long-wavelength variations in v s and ρ are
unlikely to be correlated as much as variations
in v p and v s anywhere in the mantle. Especially
within the transition zone the data favour a
d ln v s - d ln ρ anti-correlation. Using mineral
physics relations between seismic properties,
temperature and composition, Trampert et al.
2004 were able to discriminate between thermal
and compositional contributions to observed
density variations, suggesting that high-density
anomalies in the deep mantle (2000-2891 km
depth) are most likely of compositional origin.

˜

+ σ d σ − m I
in Equation (11.8) may practically not be invert-
ible, unless σ m is chosen smaller than it actually
is on the basis of our prior information. The
prior model covariance is then used to regularize