Backus (1962) presented 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 might also be random.
For simplicity, we will assume that the physical properties of the
individual layers are constant and isotropic.
[For applications to porous earth materials, we implicitly make the
typical assumptions of spatial stationarity within these layers as
well as scale separation -- *i.e.*, the sizes of the pores are
much smaller than the thickness of the individual layers in which
they reside.]
The key idea presented by Backus is that these equations
can be rearranged into a form where rapidly varying (in depth)
coefficients multiply slowly varying stresses or strains.

The derivation has been given many places including Schoenberg and
Muir (1989) and Berryman (1999a). Another illuminating derivation
has been given recently by Milton (2002).
We will not repeat any of the derivations here. The final results
will be expressed in terms of averages , where
the brackets surrounding a variable *Q*(*z*)
indicate the volume average (or, equivalently, the linear average with depth
in the vertically layered medium under consideration) of the quantity *Q*.
It follows that
the anisotropy coefficients in equation (1) are then
related to the layer parameters by the following well-known expressions:

(8) |

(9) |

(10) |

(11) |

(12) |

When the layering is fully periodic, these results may be attributed to Bruggeman (1937) and Postma (1955), while for more general layered media including random media they should be attributed to Backus (1962). The constraints on the Lamé parameters and for each individual layer are and . Although, for physically stable materials, shear modulus and bulk modulus must both be nonnegative, these relations mean that (and also Poisson's ratio ) may be negative (but nevertheless bounded below, since , and ). Large fluctuations in for different layers are therefore entirely possible, in principle, but may or may not be an issue for any given region of the earth.

Large fluctuations in are also possible, and the Backus
averaging technique is fully capable of handling all such fluctuations
properly. But, if these fluctuations are too large,
then the weak anisotropy assumption of Thomsen's original work
(Thomsen, 1986) will be violated and some care must be taken
when writing approximate equations. We do *not* at any point
assume weak anisotropy in this paper [except equations (5) and
(28)], since the shear behavior
we are trying to study will be shown to depend on the presence of
strong anisotropy in this sense. We will also find it
useful to develop alternatives to some of Thomsen's formulas
in order to deal with the strong anisotropy that arises in our analysis.

One very important fact known about the Backus averaging equations
(Backus, 1962) 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 .This fact is also very important for applications involving partial
and/or patchy saturation (Mavko *et al.*, 1998; Johnson, 2001).
Furthermore, this fact is closely related to the well-known bulk modulus
formula of Hill (1963) for isotropic composites having uniform shear
modulus, and also to the Hashin-Shtrikman bounds
(Hashin and Shtrikman, 1961), which can be used to provide an
elementary proof of Hill's equation. Nevertheless, this limit will
not be of much interest to us here except as a boundary condition on the
results obtained. Furthermore, one of the main purposes of the paper
is to show how deviations from these limiting and rather restictive
results affect the predictions of the referenced work
on partial and patchy saturation.

5/23/2004