THOMSEN PARAMETER $\delta$

Thomsen's parameter $\delta$ is defined by

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

Because of the controversy surrounding the sign of $\delta$ 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 $\delta$.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 $v_p^{min} \le v_p \le v_p^{max}$. 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 $r_{min} \le v_s/v_p \le r_{max}$, where for the case being presented r_min = 0.35 and r_max = 0.65. The density was chosen to be $\rho = 2670$ kg/m³, 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/m³ and the density of calcite is about 2720 kg/m³ (e.g., Wilkens et al., 1984).

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

= -1 + 2.   The parameter $\eta$ 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 $\epsilon - \delta$ will always be positive (Postma, 1955; Backus, 1962; Berryman, 1979). We see that most models are clustered around the origin $(\eta,\delta) = (0,0)$, but there is no question that a large fraction of the models have positive $\delta$.For this particular method of generating the models, we find that about 25% of the two-layer models have positive $\delta$.

 

eta.del.500
Figure 1 Crossplot of $\delta$ versus $\eta$ for 500 random examples of layered media according to the algorithm in the text.

Encouraged by the Monte Carlo simulation results, we have analyzed $\delta$ in light of the Backus formulas in order to determine whether it was possible to understand those circumstances in which $\delta \gt 0$ 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 $\delta$ is

= 2<++2> <+(+2)>^-1 [<1> <++2> -<+(+2)>].   It is easy to show that the prefactors are all always positive. So the sign of $\delta$ 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 $\delta$ 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 $\lambda= constant$, then the expression and therefore $\delta$ is nonpositive. Similarly, the second equality in the sequence shows that if $\lambda+ \mu= constant$, then $\delta$ is also nonpositive. The third equality shows that if $\lambda+ 2\mu= constant$,then again $\delta$ is nonpositive. The very first expression may also be used in a slightly different way to arrive at a general result, $\delta = 0$ if Poisson's ratio is constant (Thomsen, 1986), for in that case the ratio $\lambda/\mu$ is also constant and so $(\lambda+\mu)/(\lambda+2\mu)$ is constant.

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

_1_2 > _1_2 > 1,   or if

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

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 $\lambda$ without greatly changing $\mu$.

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 $\delta$ and the occurrence of small positive or negative $\epsilon$ (i.e., both occur when Lamé $\lambda$ is fluctuating greatly from layer to layer), we have plotted $\delta$ versus $\epsilon$ 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 $\delta$ are in fact most highly correlated with the small but positive values of $\epsilon$. We should also keep in mind the fact that $\epsilon - \delta \ge 0$ is always true for layered models and this fact also plays a role in Figure 2, determining the unoccupied upper left hand corner.

 

eps.del.500
Figure 2 Crossplot of $\delta$ versus $\epsilon$ for 500 random examples of layered media according to the algorithm in the text.