# The Earth in three dimensions

```The Earth in three dimensions
From seismology to composition and
temperature
Is this the only model
compatible with the data?
If we project some chosen
model onto the null space
of the forward operator
Then add it to original model.
The same data fit is
guaranteed!
Deal et al. JGR 1999
Seismic tomography (Masters et al., 2000)
Radial variation of dlnVs/dlnVp most popular diagnostic
Robertson and Woodhouse, 1996
‘’The observations of RW96 and SD97 reinforce the
conclusion that we should seek a compositional
interpretation of tomographically observed velocity
anomalies in the lower mantle’’ Stacey, 1998.
Interpretation: ratios
Vp=[(K+4/3G)/r] Vs=[G/r]
The ratio dlnVs/dlnVp=2.25 if the only
variation is in G for a Poisson solid.
This is also observed in laboratory
experiments for thermal variations.
Average ratios of dlnVs/dlnVp are not useful if
more than one cause is at play. It can best give a
qualitative indication.
The full histogram gives statistical evidence of
compositional anomalies, but they can not be
located in the Earth
Direct inversion
Example of a descending slab
Using lnVp/T=-1.9x10-5, lnVs/T=-3.0x10-5
lnVp/X =0.056 and lnVs/X=-0.025, X is the perovskite
content
Taking dlnVp=0.003 and dlnVs=0.006, we find dT=-190K and
dX –0.011 which is -1.1% perovskite variation, as expected
Probabilistic tomography
Averaged and gravity
filtered models over 15x15
degree equal area caps
Likelihoods in each cap are
nearly Gaussian
Most likely model (above
one standard deviation)
Uniform uncertainties
dlnVs=0.12 %
dlnVΦ=0.26 %
dlnρ=0.48 %
Likelihood of
correlation
between
seismic
parameters
Evidence for
chemical heterogeneity
Vs-VF
Incompressibility:
KS=-V(P/V)S
Thermal expansivity a=-1/r(r/T)P
Gr&uuml;neisen parameter (P/T)V=gCV/V
Anderson-Gr&uuml;neisen : d=-1/aKs(Ks/T)P
Anderson-Gr&uuml;neisen : e=-1/aKs(G/T)P
d = Ks’-1-g+q
Ks’~4, q~1, g~1, e&gt;6
dlnr/dlnVs~0.2
Likelihood of
rms-amplitude
of seismic
parameters
Evidence for
chemical heterogeneity
RMS density
Likelihood of
correlation between
seismic and thermochemical parameters
in the lowermost
mantle
oRicard et al. JGR 1993
oLithgow-Bertelloni and Richards
Reviews of Geophys. 1998
oSuperplumes
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