Multiscale Window Transform

Abstract

A new analysis apparatus, the multiscale window transform (MWT), is developed to generalize the classical mean-eddy decomposition (MED) in fluid mechanics to include three or more ranges of scales, and to ensure a faithful representation of localized energy processes. The development begins with the introduction of a sequence of finite-dimensional subspaces of $L_2[0,1]$, $\{V_{\varrho,j}\}_{0\le j\le j_2}$, based on a multiresolution analysis of $L_2(\mathbb{R})$. All the $V_{\varrho,j}$ are sampling spaces, i.e., spaces spanned by some translation invariant basis. The upper bound of the index of scale level, $j_2$, is prescribed in accordance with the signal of concern. Within $V_{\varrho,j_2}$, the concepts of large-scale, mesoscale, and submesoscale windows are introduced, each being a subspace of $V_{\varrho,j_2}$ and containing exclusively a range of scales. A transform-reconstruction pair is constructed on each window, representing the features for the corresponding range of scales. The resulting MWT has several useful properties. Of particular importance is a property of marginalization, which allows for a simple representation of energy on specified windows and localizations. The MWT has been connected to other existing analysis techniques: The classical MED turns out to be a particular case, with averaging acting as both a transform and a reconstruction in the MWT framework; it has also been compared with wavelet analysis and the Hilbert–Huang transform. As a realization, we show how an MWT is built with an orthonormalized cubic spline system. Computational techniques leading to a fast transform scheme are also presented. The MWT is validated with two examples, and its applications in the study of hydrodynamic stability analysis (fully nonlinear) are discussed and exemplified.