# Statistical and numerical considerations of Backus-average product approximation

## TL;DR

This paper analyzes the accuracy of Backus-average product approximation in layered solids, providing statistical insights and identifying conditions where the approximation remains reliable or fails, especially in physical versus material science contexts.

## Contribution

It offers a statistical analysis of the Backus-average product approximation, extending previous bounds and identifying scenarios where the approximation is effective or may produce spurious results.

## Key findings

- The approximation is generally accurate in physical scenarios modeled by Backus averaging.
- Certain cases can lead to deterioration or spurious values in the approximation.
- The analysis extends the understanding of the approximation's applicability beyond previous bounds.

## Abstract

In this paper, we examine the applicability of the approximation, $\overline{f\,g}\approx \overline f\,\overline g\,$, within Backus (1962) averaging. This approximation is a crucial step in the method proposed by Backus (1962), which is widely used in studying wave propagation in layered Hookean solids. According to this approximation, the average of the product of a rapidly varying function and a slowly varying function is approximately equal to the product of the averages of those two functions. Considering that the rapidly varying function represents the mechanical properties of layers, we express it as a step function. The slowly varying function is continuous, since it represents the components of the stress or strain tensors. In this paper, beyond the upper bound of the error for that approximation, which is formulated by Bos et al. (2017), we provide a statistical analysis of the approximation by allowing the function values to be sampled from general distributions. Even though, according to the upper bound, Backus (1962) averaging might not appear as a viable approach, we show that$-$for cases representative of physical scenarios modelled by such an averaging$-$the approximation is typically quite good. We identify the cases for which there can be a deterioration in its efficacy. In particular, we examine a special case for which the approximation results in spurious values. However, such a case$-$though physically realizable$-$is not likely to appear in seismology, where Backus (1962) averaging is commonly used. Yet, such values might occur in material sciences, in general, for which Backus (1962) averaging is also considered.