We have shown an example where S&M interpolation is considerably better than nearest-neighbor interpolation. As we have stated, there is an apparent drawback in that interpolated grid points near lithologic boundaries will usually require increased anisotropic complexity. In particular, an average of two differing isotropic materials will not be isotropic. The long-term solution is clear. There is no longer much justification for the idea that the Earth can be assumed isotropic at seismic scales. We should develop and preferentially use modeling schemes that allow for a general Hooke's Law relation with a complete set of elastic constants. In the short term there is an intermediate solution. S&M theory is based on conservation arguments, and we can validly ask which isotropic material best represents the elastic properties of the anisotropic composite.
In any case, it is clear the elastic constants should be interpolated as a unified tensor quantity. In our S&M example in Figure 1, for example, is zero in both of the isotropic homogeneous media, but is distinctly nonzero in the tilted-axis transversely-isotropic medium necessary to represent the border grid cells. No interpolation method that considers each elastic constant independently of the others could interpolate a nonzero between two sets of zeroes. While we have not proven that our S&M-based method is the optimal tensor interpolation scheme for this application, it does have the advantage that it is simple to understand and easy to implement, and does seem to work well.
How might we do better? There are two areas we have not discussed. First, we have taken the non-overlapping tile as being a good representation of the region of influence of a point. By analogy with the sort of thing we do in geophone array design, we might consider using a tile of twice the linear dimension and tapered from the center out to the corners. These thick tiles would still uniformly cover the region, and might offer a better compromise between smooth transition and compact size. Second, our scheme is based on static arguments. As frequencies rise, so do the results deteriorate. Introducing viscous damping into the scheme may help, but the theory is not yet developed, and it would add double the number of parameters required to describe the elastic properties. Again, this may be a worthwhile complication to be building into our modeling, since heterogeneity lowers Q factor as well as increasing anisotropy.