Geoscience Reference
In-Depth Information
n
is the
sample size
for each mean and is not the
number of samples. The
sample size
is the number
of measurements that goes into the computation
of each mean. In the case of sampling by a vol-
cano of a heterogenous mantle, the volume of
the sample space is equivalent to the number of
discrete samples that are averaged; the volcano
does the averaging in this case.
Suppose the mantle has a variety of discrete
components, or blobs, with distinctive isotope
ratios. The probability density function has, say,
five peaks; call them DM, EM1, EM2, HIMU and
Q. The CLT states that a sufficiently large sam-
ple, or average, from this population, will have
a narrow Gaussian distribution. MORB has this
property; it has nothing to do with the homo-
geneity of the mantle or convective stirring.
MORB is best viewed as an
average
, not a reservoir
or component.
If one has several large datasets there are sta-
tistical ways to see if they are drawn from the
same population. If two datasets have different
means and different standard deviations they
may still be drawn from the same population.
But if the datasets have been
filtered
,
trimmed
or
corrected
, then statistics cannot be applied. Often
these corrections involve some hypothesis about
what the data should look like. This is a very com-
mon error in seismology and geochemistry; data
can be selected, filtered and discarded for vari-
ous reasons but there are formal and statistically
valid ways of doing this.
that might be found in a rock collection. Magmas
average over a large volume of the mantle, just as
river sediments can average over a large area of a
continent. Similarly, seismic waves average over
tens to hundreds of km and see many different
kinds of lithologies; they do not see the extremes
in properties of the material that they average.
Anisotropy of the mantle may be due to
anisotropy of individual mantle minerals or due
to large-scale organized heterogeneity. A 1-second
P-wave has a wavelength of about 10 km and
therefore does not usually care whether it is
microphysics or macrophysics that gives the
fabric. Similar considerations apply to seismic
anelasticity. Geophysics is as much a matter of
composites, and averaging, as it is a science of
crystallography and mineral physics. The prob-
lem of averaging occurs throughout the sciences
of petrology, geochemistry and geophysics. Much
of deep Earth science is about unravelling aver-
ages. In seismology this is called
tomography
.
The scale becomes important if it becomes
big enough so that gravity takes over from diffu-
sion. A large blob behaves differently, over time,
than a small impurity, because of buoyancy. If
the heterogeneities are large, a
fertile blob
can be
confused with a
hot plume
.
Scrabble statistics
The central limit theorem can be illustrated
with the game of Scrabble, which has a well-
defined distribution function of letters or scores
(Figure 12.1). Put all the letter tiles into a can
and draw them out repeatedly. Plot the number
of times each letter is pulled out on a cumu-
lative histogram. The histogram is very ragged;
this is what the population -- the world of let-
ters -- looks like. Now play a game with only
two-letter combinations -- words -- allowed, and
plot the average scores. Continue this process
with three- and four-letter words (but do not fil-
ter out the offensive words or words that do not
make sense). The histograms get smoother and
smoother and narrower and narrower. By the
time one gets to three-letter words one already
has nearly a Gaussian distribution, with very few
average scores of 4 or 5, or 20 and higher. If this
The problems of scale
Volcanoes sample and average large volumes of
the mantle. It is not necessary that the source
region correspond to a rock type familiar in
hand-specimen size. The 'grains' of the source
may be kilometres in extent, and the melt
from one source region may rise and interact
with melts from a different lithology, or smaller
degree melts from the same lithology. The prop-
erties of individual crystals are greatly washed
out in the average. The hypothetical rock types --
pyrolite and piclogite -- and hypothetical reser-
voirs -- MORB, FOZO -- may only exist as averages,
rather than as distinct 10 cm by 10 cm 'rocks'
