Multiscale Window Transform

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 L₂[0,1], ,j\0≤ j≤ j₂, based on a multiresolution analysis of L₂(R). All the V,j are sampling spaces, i.e., spaces spanned by some translation invariant basis. The upper bound of the index of scale level, j₂, is prescribed in accordance with the signal of concern. Within V,j₂, the concepts of large-scale, mesoscale, and submesoscale windows are introduced, each being a subspace of V,j₂ 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.