Geoscience Reference
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can be construed to be an a priori model that can be compared with data without
fudging.
Another approach would be to use the function 1
þ
B tanh
ð
K
½
x
þ
y
Þ
in place
of (1
B). Note that this function goes smoothly to (1
þ
B) when (x
þ
y)is
large-positive and to (1
B) when (x
þ
y) is large-negative. It also provides a
smooth continuous equation to replace the abrupt discontinuous Imbrie model.
As K
)1
, the tanh function approaches the step function used by the Imbries.
A reasonable value for K might perhaps be about 3.
Pisias and Shackleton (1984) embellished the Imbrie model by adding a term
to represent the effect of greenhouse forcing due to variable CO
2
content in the
atmosphere during glacial-interglacial cycles. However, they did not reveal how
they did this, so it is dicult to reproduce their results. Furthermore, in the
process of adding a term for CO
2
forcing, they seem to have fitted the model to
ocean sediment data, thus producing a kind of self-confirming curve that fits the
data. Nevertheless, adding a term to the Imbries' differential equation to provide
for CO
2
forcing may be a good thing to do if done judiciously and independently
of sediment data.
9.6.3 Memory model
Berger (1999) developed a model that dealt with the change in ice volume as a
function of solar variations, a model that seems to be highly contrived. His
function was:
dx
dt
¼
K
xy
A
m
C
in which, as before, x is ice volume, y is insolation (0 to 1 scale), and m is a so-
called memory function representing the average value of x for the previous 57
years. The powers A and C are about 4. While he was able to find good agreement
with SPECMAP data, data tuning and model artifices leave this writer with the
impression that this was little more than imaginative curve fitting.
9.7 MODELS BASED ON ECCENTRICITY OR OBLIQUITY
9.7.1 A model based on eccentricity
Bol'shakov (2008) raised a number of objections to the conventional astronomical
theory of ice ages. One objection was based on his claim that glaciations appear to
occur in both hemispheres in synchrony, whereas the peaks and valleys in solar
input are completely out of phase in the north and the south. Furthermore, he
was critical of the conventional version of the astronomical theory that is based
on the belief:
a glaciation can only develop if the summer high northern
latitudes are cold enough to prevent
''
...
the winter snow from melting,
thereby
allowing a positive annual balance of snow and ice.'' He asserted that this:
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