On problematic case of product approximation in Backus average

Abstract

Seismic anisotropy is often a combined effect of the intrinsic anisotropy and the anisotropy induced by thin-layering. The Backus average, a useful mathematical tool, allows us to describe the latter one quantitatively. The results are meaningful only if the underlying physical assumptions are obeyed, such as the low frequency of the propagating wave. In this paper, however, we focus on the only mathematical assumption of the Backus average, namely, product approximation. It states that the average of the product of a varying function with nearly-constant function is approximately equal to the product of the averages of those functions. We discuss particular, problematic case for which the aforementioned assumption is inaccurate. Further, we examine numerically if this inaccuracy affects the wave propagation in a homogenous medium---obtained using the Backus average---equivalent to thin layers. We take into consideration various material symmetries, including orthotropic, cubic, and others. We show that the problematic case of product approximation is strictly related to the negative Poisson's ratio of constituent layers. Therefore, we discuss the laboratory and well-log cases in which such a ratio has been noticed. Upon thorough literature review, it occurs that examples of so-called auxetic rocks (rocks that have negative Poisson's ratio) are not extremely rare exceptions as thought previously. The investigation and derivation of Poisson's ratio for materials exhibiting symmetry classes up to monoclinic become a significant part of this paper. Except for the main objectives of the paper, we additionally show that the averaging of cubic layers results in an equivalent medium having tetragonal (not cubic) symmetry. Also, we present concise formulations of stability conditions for low symmetry classes, such as trigonal, orthotropic, and monoclinic.