I'm working through the paper (1) which uses the elastic wave equation to model an anisotropic medium.
When the authors move to the "$\bf{Weakly}$ $\textbf{Anisotropic}$ $\textbf{Regime}$" (section 4) I'm confused about their scaling choice. They state:
"We assume from now on that anisotropy is weak and that the correlation length of the fluctuations of the symmetry axis and the typical wavelength of the incoming wave are much smaller than the propagation distance $L$. We introduce the small dimensionless parameter $\epsilon$ to characterize this scaling regime."
$$\kappa=\varepsilon \tilde{\kappa}, \qquad \psi(z)=\tilde{\psi}\left(\frac{z}{\varepsilon^{2}}\right), \qquad f(t)=\tilde{f}\left(\frac{t}{\varepsilon^{2}}\right)$$
with the scaled Fourier transforms
$$\hat{f}^{\varepsilon}(\omega)=\frac{1}{\varepsilon^{2}} \int f(t) e^{i \frac{\omega}{\varepsilon^{2}} t} d t, \qquad f(t)=\frac{1}{2 \pi} \int \hat{f}^{\varepsilon}(\omega) e^{-i \frac{\omega}{\varepsilon^{2}} t} d \omega.$$
My first thought was to non-dimensionalise the space and time variables by
$$\tilde{z} = \frac{z}{L}, \quad \tilde{t} = \frac{\bar{c t}}{L}$$
where $\bar{c}$ is the effective speed. I don't understand where the small parameter comes from, and what it is a ratio of as they don't explicitly state it.
I would appreciate if someone could explain what is happening in this step of their analysis.
(1) Garnier, J., & Sølna, K. (2016). Apparent attenuation of shear waves propagating through a randomly stratified anisotropic medium. Stochastics and Dynamics, 16(04), 1650009. doi:10.1142/S021949371650009X