The leakage problem of orthonormal wavelet transforms when applied to atmospheric turbulence

Orthonormal wavelet transforms are becoming common in the study of turbulence phenomena. Although they are powerful tools in representing a signal, their use as tools to study the characteristics of turbulent structures can create appreciable errors in interpretation. It is shown here that although the orthonormal wavelet transform is computationally economical by taking advantage of multiresolution analysis, it has insufficient resolution in both scale and location to resolve detailed information of turbulence structures. Lacking in resolution, the energy at a particular frequency (or wavelength) may leak into neighboring frequencies and may pass down to smaller scales to produce an artificial “cascade” of energy (with a slope close to −2/3). The choice of wavelet basis function is important to the wavelet spectrum, especially in the study of turbulence flows dominated by coherent structures, since the method most accurately senses energy contained in pulses that have a similar pattern to the wavelet function. To use the method as a filter can be problematic owing to the low resolution of the orthonormal wavelet transform; nonorthonormal wavelet analysis should be employed when high resolution is important. When orthonormal wavelet transforms have to be used for signal analysis, segmented averaging should be employed.