On uniqueness of $\lambda$-diagrams

Since the possible linear combinations of the elastic bulk and shear moduli (K and $\mu$) are infinite, it is natural to ask why (or if) the choice $\lambda = K - {2\over3}\mu$ is special? Is there perhaps some other combination of these constants that works as well or even better than the choice made here? There are some rather esoteric reasons based on recent work (Berryman et al., 1999) in the analysis of layered anisotropic elastic media that lead us to believe that the choice $\lambda$ is indeed special, but we will not try to describe these reasons here. Instead we will point out some general features of the two types of plots that make it clear that this choice is generally good, even though others might be equally good or even better in special circumstances. First, in the diagram using the ($\rho/\mu$, $\lambda/\mu$)-plane, it is easy to see that any plot of data using linear combinations of the form ($\rho/\mu$, $(\lambda + c\mu)/\mu$), where c is any real constant, will have precisely the same information and the display will be identical except for a translation of the values along the ordinate by the constant value c. Thus, for example taking $c = {2\over3}$, plots of ($\rho/\mu$, $K/\mu$)will have exactly the same interpretational value as those presented here. But, if we now reconsider the data-sorting plot (e.g., Figure 2) for each of these choices, we need to analyze plots of the form ($\rho/(\lambda + c\mu)$,$\mu/(\lambda + c\mu)$). Is there an optimum choice of the parameter c that makes the plots as straight as possible whenever the only variable is the fluid saturation? It is not hard to see that the class of best choices always lies in the middle of the range of values of $\lambda/\mu$ taken by the data. So setting $-c = {1\over2}(\min{(\lambda/\mu)} + \max{(\lambda/\mu}))$will always guarantee that there are very large positive and negative values of $\mu/(\lambda + c\mu)$, and therefore that these data fall reliably (if somewhat approximately) along a straight line. But the minimum value of $\lambda/\mu$ has an absolute minimum of $-{2\over3}$, based on the physical requirement of positivity of K. So $c < {2\over3}$ is a physical requirement, and since $\max{\lambda/\mu} \simeq + {2\over3}$ is a fairly typical value for porous rocks, it is expected that an optimum value of $c \le 0$ will generally be obtained using this criterion. Thus, plots based on bulk modulus K instead of $\lambda$ will not be as effective in producing the quasi-orthogonality of porosity and saturation that we have obtained in the data-sorting style of plotting. We conclude that the choice $\lambda$ is not unique (some other choices might be as good for special data sets), but it is nevertheless an especially simple choice and is also expected to be quite good for most real data.