Thomsen's parameter is defined by

= (f+l)^2 - (c-l)^22c(c-l). This parameter is considerably more difficult to analyze than either or for various reasons, some of which we will enumerate shortly.

Because of the controversy surrounding the sign of for
finely layered media, we have performed a series of Monte Carlo
simulations with the purpose of establishing the existence or
nonexistence of layered models having positive .The simulations to be presented here were performed on
models limited to two types of layers. Another set of
simulations was performed having three types of layers, since
it is known (Backus, 1962) that this is the most general case that
needs to be considered, but we will not present that work here.
The set of constant density models was chosen randomly by specifying
the possible values of *v*_{p} to lie in some range
. The values chosen for the limiting
wave speeds in the example shown in Figures 1 and 2 were
*v*_{p}^{min} = 2.5 km/sec and *v*_{p}^{max} = 5.5 km/sec.
The range of the shear wave velocity was similarly specified
by constraining the ratio *v*_{s}/*v*_{p} to lie within a range
, where for the case being presented
*r*_{min} = 0.35 and *r*_{max} = 0.65. The density was chosen
to be kg/m^{3}, but any choice of constant density would have
resulted in the same dimensionless Thomsen parameters.
For comparison, note that the density of quartz is about 2650 kg/m^{3}
and the density of calcite is about 2720 kg/m^{3} (*e.g.*,
Wilkens *et al.*, 1984).

Figure 1 shows the Monte Carlo results for plotted versus from 500 random models, where

= -1 + 2. The parameter has been shown to be useful in seismic processing by Alkhalifah and Tsvankin (1995). It is particularly useful in the present context because it is known that for layered media this parameter will always be positive because will always be positive (Postma, 1955; Backus, 1962; Berryman, 1979). We see that most models are clustered around the origin , but there is no question that a large fraction of the models have positive .For this particular method of generating the models, we find that about 25% of the two-layer models have positive .

Figure 1

Encouraged by the Monte Carlo simulation results, we have analyzed in light of the Backus formulas in order to determine whether it was possible to understand those circumstances in which are most likely to occur. Using the formulas (avea)-(avem) derived by Backus (1962), it is not hard to show that one natural form of the expression for is

= 2<++2> <+(+2)>^-1 [<1> <++2> -<+(+2)>]. It is easy to show that the prefactors are all always positive. So the sign of is determined by the expression in brackets. Using simple algebraic manipulations, this expression can be rewritten in a number of different but useful forms, including

<1> <++2> -<+(+2)> = <1+2> <+2> - <(+2)> <+2> = <1+2> <++2> - <+(+2)> <+2> = <1+2> - <1> <+2>. The manipulations required to arrive at these results will be described in more detail elsewhere. The important point about this sequence of equalities is that each successive one can be used to prove something general about the sign of for layered media. In each case, we note that if some multiplicative factor is constant in all layers of the finely layered medium, then that factor may be removed from the averages. Using this technique, the first transformation shows that if , then the expression and therefore is nonpositive. Similarly, the second equality in the sequence shows that if , then is also nonpositive. The third equality shows that if ,then again is nonpositive. The very first expression may also be used in a slightly different way to arrive at a general result, if Poisson's ratio is constant (Thomsen, 1986), for in that case the ratio is also constant and so is constant.

It is straightforward to show for layered media having only two constituents that the last expression in (variousforms) implies that will be positive if

_1_2 < _1_2 < 1, showing that the fluctuations in the Lamé constant must be greater than those in the shear modulus to observe positive .

As discussed earlier, *v*_{s}/*v*_{p} ratios are
decreased by the presence of pores, flat cracks,
or the addition of clay, dolomite, feldspar, or
calcite to silicic rocks. Any of these factors can
cause fluctuations in without greatly changing .

For layered media with constant density, the final expression in (variousforms) shows that

sign() = sign(<v_p^-2> - <v_s^-2> <v_s^2/v_p^2>).

Because our analysis establishes a similarity in the circumstances
between the occurrence of positive and the occurrence of
small positive or negative (*i.e.*, both
occur when Lamé is fluctuating greatly from layer to
layer), we have plotted versus in Figure 2.
The Monte Carlo results
used to produce Figure 2 are exactly the same as those used to produce
Figure 1. We see that, as expected from the analysis, the positive
values of are in fact most highly correlated with the
small but positive values of . We should also keep
in mind the fact that is always true
for layered models and this fact also plays a role in Figure 2,
determining the unoccupied upper left hand corner.

Figure 2

11/11/1997