Backus (1962) presents an elegant method of producing the effective constants for a thinly layered medium composed of either isotropic or anisotropic elastic layers. This method applies either to spatially periodic layering or to random layering, by which we mean either that the material constants change in a nonperiodic (unpredictable) manner from layer to layer or that the layer thicknesses may also be random. For simplicity, we will assume that the layers are isotropic, in which case equation (sscij) becomes
_11_22 _33 _23_31 _12 = +2& & & & & & +2& & & & & & +2& & & & & & & 2& & & & & & & 2& & & & & & & 2 e_11 e_22 e_33 e_23 e_31 e_12 . Although the materials forming the thin layers that make up the VTI medium do not need to be isotropic materials, any VTI medium that is anisotropic due to thin layering can be represented using a model with layers composed of a minimum of three isotropic materials (Backus, 1962). The key idea presented by Backus is that these equations can be rearranged into a form where rapidly varying coefficients multiply slowly varying stresses or strains. This analysis is well-known so we will not repeat it here.
The results are
a = c_11= <+2>^2 <1+2>^-1 + 4<(+)+2>,
b = c_12 = <+2>^2 <1+2>^-1 + 2 <+ 2>,
c = c_33 = <1+2>^-1
f = c_13 = <+2> <1+2>^-1,
l = c_44 = <1>^-1 ,
m = c_66 = <>. One very important fact that is known about these equations is that they reduce to isotropic results with a=c, b=f, and l=m, if the shear modulus is a constant, regardless of the behavior of .