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An important anisotropy parameter for quasi-*SV*-waves is Thomsen's parameter
, defined in equation (2).
Formula (18) for *a* may be rewritten as

| |
(20) |

which can be rearranged into the convenient and illuminating form
| |
(21) |

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, 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
| |
(22) |

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, 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.

Helbig and Schoenberg (1987) discuss an interesting case where large
fluctuations in combined with large fluctuations in ,including for one component, lead to wavefronts with
anomalous polarizations in layered TI media. Schoenberg (1994) also
discusses several shale examples having large fluctuations in both
and .

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 variations
in layer should have no direct information about fluid content, while
observed variations in layer , especially if they are large variations,
may contain important clues about variations in fluid content.
So the observed structure of in (22)
strongly suggests that small positive and all negative values of
may be important indicators of significant fluctuations
in fluid content.

From (21), we can infer in general that

| |
(23) |

so we have an upper bound on in finely layered media
stating that
| |
(24) |

where is the density.

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

10/16/2003