The results concerning the signs of the Thomsen parameters obtained in the preceding sections are summarized in Table 1.
1.2
The nonnegativity of and
for layered
models is well-known. The fact that
can be either
positive or negative, and the circumstances leading to negative values
have been little appreciated before. A quick glance at the Table
seems to indicate that
is either zero or negative and this is
perhaps why there has been so much confusion about the possibility
of
for layered models. We have shown that the correct
inference about positive
follows rather simply from the
result expressed in the last column of the Table. Since
when Poisson's ratio is constant or when
, it is clear that we must have (for example)
and
if
is to be positive. The second expression can be rearranged
to
. Now we see that there are
two possible cases, either
or
. The first case results in positive
, while the second case does not when
.This also shows that in perfectly random layered media one expects
to account for 50% of the models. In our simulation
we found
accounted for only about 25% of the models
produced, but this apparent discrepancy has been traced to a bias in the
particular algorithm we used to generate the models. In any case,
the earth does not have to obey perfectly random statistics and there
is no reason to suppose that real layered earth will conform to these
statistical considerations. Our main point is merely that
is entirely possible and quite understandable for layered earth models, and
so it is not at all surprising that
is often observed in
real data examples.