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An important anisotropy parameter for quasi-*SV*-waves (which is our
main interest in this paper) is Thomsen's parameter
, defined in equation (2).
Formula (12) for *a* may be rewritten as

| |
(14) |

which can be rearranged into the convenient and illuminating form
| |
(15) |

This formula is very instructive because the term in square brackets
is in Cauchy-Schwartz form
[], so this factor is nonnegative. Furthermore,
the magnitude of this term depends
mainly on the fluctuations in the Lamé
parameter, and is largely independent of , since
appears only in the weighting factor .Clearly, if , then this bracketed factor
vanishes identically, regardless of the behavior of
. Large fluctuations in will tend to make this term large.
If in addition we consider Thomsen's parameter written in a
similar fashion as
| |
(16) |

we find that the term enclosed in the first bracket on the right hand
side is again in Cauchy-Schwartz form showing that it always makes a positive
contribution unless , in which case it vanishes.
Similarly, the term enclosed in the second set of brackets is always
non-negative, but the minus preceding the second bracket causes
this contribution to make a negative contribution to unless , in which case it vanishes.
So, in general the sign of is indeterminate.
The Thomsen parameter may have either a positive or a
negative sign for a TI medium composed of arbitrary thin isotropic layers.
Thomsen (2002) states that if *K* and are
positively correlated. But (16) shows that such correlations
only produce with certainty if they are also
supplemented by the stronger condition that [in fact, implies that there is a positive correlation
between *K* and , but the reverse does not necessarily hold
unless we also assume that the fluctuations in are quite small --
an assumption that we do not make here].

Fluctuations of in the earth have important implications
for oil and gas exploration. As we
recalled in our earlier discussion, Gassmann's well-known results
(Gassmann, 1951) show that, when isotropic porous elastic media are saturated
with any fluid, the fluid has no mechanical effect on the shear
modulus , but -- when these results apply --
it can have a significant effect on the bulk modulus
, and therefore on . Thus, observed
(high spatial frequency) variations in layer shear modulus
should have no direct information about fluid content,
while such variations observed in layer Lamé parameter
, especially if they are large variations,
may contain important clues about variations in fluid content.
So the observed structure of in (16)
strongly suggests that small positive and all negative values of
may be important indicators of significant fluctuations
in fluid content (Berryman *et al.*, 1999).

** Next:** Thomsen's
** Up:** THOMSEN PARAMETERS AND
** Previous:** THOMSEN PARAMETERS AND
Stanford Exploration Project

5/23/2004