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concept derives from the CLT and contrasts with
the idea of a convectively homogenized MORB
mantle reservoir, and different reservoirs for OIB
where homogenization of the source is achieved
by convective stirring and mixing. In contrast,
in the SUMA model, the isotopic compositions
of MORB and OIB are the outcome of homog-
enization during sampling, by partial melting
and magma mixing. The primary homogeniza-
tion process
sampling upon melting and averaging
,
SUMA, does not require the participation of dis-
tinct (e.g. lower mantle) reservoirs to explain OIB
compositions. Statistical distributions of litho-
logic components and sampling theory replace
the concept of distinct, isolated geochemical
reservoirs, and extensive solid-state convective
stirring prior to sampling. In sampling theory
terms, SUMA is the heterogenous population to
be sampled -- the probability density function.
MORB represents a large scale sample, or average,
from this population; near-ridge seamounts are
a smaller-scale sample; grain boundaries, fluid-
inclusions and melt-inclusions are very small-
scale samples. MORB is uniform and does not
have the extremes of composition because it is
large-scale average of the sampled mantle.
Math-
ematical and statistical treatments of
isotopic heterogeneity of basalts and
upper mantle assemblages
arestartingto
replace the static reservoir and convective stir-
ring concepts.
from the 'wrong' reservoir, and then statistics is
applied. In seismology, a
prior
or
reference
model
is adopted and perturbations are made to this to
satisfy new datasets.
Which kind of statistical approach is prefer-
able in these situations?
Bayesian meth-
ods have a long and controversial his-
tory
. Bayesian reasoning has emerged from an
intuitive to a formal level in many fields of
science.
Subjective probability
was developed to
quantify the plausibility of events under circum-
stances of uncertainty. Bayes' theorem is a natu-
ral way of reasoning, e.g.
www.ipac.caltech.edu/
level5/March01/Dagostini/Dagostini2.html.
There is an apparent contradiction between
rigorous normal statistics and the intuition of
geologists. Geologists' intuition resembles the
less familiar bayesian approach; conclusions
should not deviate too much from prior beliefs.
On the other hand, a common objection to
bayesian statistics is that science must be objec-
tive -- there is no room for belief or prejudice.
Science is not a matter of religion. But scien-
tific beliefs and assumptions are always there,
but often well hidden.
There are good reasons for applying bayesian
methods to geological problems and geochemi-
cal datasets. First of all, geochemical data are
often ratios, they cannot be negative. Conclu-
sions should not depend on whether a ratio or
its inverse is analysed. In the absence of infor-
mation to the contrary it can be assumed that
all values of 0 to infinity are equally probable
in the underlying distribution (i.e. the magma
source -- or mantle -- prior to sampling and averag-
ing). Ruling out negative values already is a prior
constraint. Sampling of a heterogenous source,
according to the central limit theorem (CLT) will
yield a peaked distribution that, in the limit of
a large sample volume, is
normal
or
log-normal
.
Many geochemical samples can be considered to
have sampled fairly large volumes. These consid-
erations are more critical for
3
He/
4
He than for
heavier isotopes since the spread of values about
the mean is larger, and median values are not
far from 0. Similarly, histograms of seismic wave
travel-time residuals, which can be negative, or
heat flow values, depend on the distance the rays
travel or the number of samples averaged used to
Bayesian statistics
The use of prior probabilities and subjective
constraints external to the dataset is known
as bayesian statistics [
Harold Jeffreys baye-
sian statistics
]. Bayesian reasoning is com-
mon in geophysical inversion problems and in
the statistical treatments of isotope data. For
example, it is often assumed that there are two
populations in isotopic data -- the MORB reser-
voir, corresponding to the 'convecting degassed
upper mantle', and the OIB reservoir, assumed
to be an isolated, more primitive, less-degassed,
more variable reservoir in the lower mantle. Data
are corrected or filtered to remove the influ-
ence of contamination or pollution by materials
