Introduction to Orthonormal Wavelet Analysis with Shift Invariance: Application to Observed Atmospheric Blocking Spatial Structure

Abstract

Orthonormal wavelet analysis (OWA) is a special form of wavelet analysis, especially suitable for analyzing spatial structures, such as atmospheric fields. For this purpose, OWA is much more efficient and accurate than the nonorthogonal wavelet transform (WT), which was introduced to the meteorological community recently and which is more suitable for time series analysis. Whereas the continuous WT is strictly correct only for infinite domains, OWA is derived from periodizing and discretizing the infinite-domain case and so is correct for periodic boundary conditions. Unlike Fourier spectra, OWA is not shift invariant. Nor is it equivariant like the WT; that is, the OWA output does not shift as its input shifts. Two remedies are to combine all possible shifts, known as the overcomplete, nonorthogonal shift equivariant WT, or else to use a ?best shift,? known as best shift wavelet analysis. Although shift invariant and orthonormal w.r.t. arbitrary inputs, the latter?s optimization generally depends nonlinearly on a reference structure. OWA is a generalization of Fourier series, and the associated multiresolution analysis (MRA) generalizes Reynolds averaging. Like these, OWA/MRA on discrete and continuous domains satisfy analogous identities arithmetically exactly, unlike the WT. OWA/MRA is much more efficient than Fourier series for analyzing several examples, including nearly singular synthetic structures, and a 90-day Hovmöller diagram of observed geopotential height, containing five atmospheric blocking events. OWA?s variance compression is comparable to that of EOF analysis, but is much faster and less data-dependent. Implementing the basic OWA operations is documented with WaveLab, a freely available package requiring MATLAB.