J Elast (2017) 127:179–196
On Backus Average for Generally Anisotropic Layers
Len Bos1·David R. Dalton2·Michael A. Slawinski2·
Received: 22 June 2016 / Published online: 7 November 2016
© Springer Science+Business Media Dordrecht 2016
Abstract In this paper, following the Backus (in J. Geophys. Res. 67(11):4427–4440, 1962)
approach, we examine expressions for elasticity parameters of a homogeneous generally
anisotropic medium that is long-wave-equivalent to a stack of thin generally anisotropic
layers. These expressions reduce to the results of Backus (1962) for the case of isotropic
and transversely isotropic layers.
In the over half-a-century since the publications of Backus (1962) there have been numer-
ous publications applying and extending that formulation. However, neither George Backus
nor the authors of the present paper are aware of further examinations of the mathematical
underpinnings of the original formulation; hence this paper.
We prove that—within the long-wave approximation—if the thin layers obey stability
conditions, then so does the equivalent medium. We examine—within the Backus-average
context—the approximation of the average of a product as the product of averages, which
underlies the averaging process.
In the presented examination we use the expression of Hooke’s law as a tensor equation;
in other words, we use Kelvin’s—as opposed to Voigt’s—notation. In general, the tensorial
notation allows us to conveniently examine effects due to rotations of coordinate systems.
Keywords Backus averaging ·General anisotropy ·Seismology ·Approximation ·Upper
1Dipartimento di Informatica, Università di Verona, Verona, Italy
2Department of Earth Sciences, Memorial University of Newfoundland, St. John’s, Newfoundland,