Anisotropic multi-resolution analysis in 2D, application to long-range correlations in cloud mm-radar fields

摘要

Because of Earth's gravitational field, its atmosphere is strongly anisotropic with respect to the vertical; the effect of the Earth's rotation on synoptic wind patterns also causes a more subtle form of anisotropy in the horizontal plane. The authors survey various approaches to statistically robust anisotropy from a wavelet perspective and present a new one adapted to strongly non-isotropic fields that are sampled on a rectangular grid with a large aspect ratio. This novel technique uses an anisotropic version of Multi-Resolution Analysis (MRA) in image analysis; the authors form a tensor product of the standard dyadic Haar basis, where the dividing ratio is (lambda)(sub z) = 2, and a nonstandard triadic counterpart, where the dividing ratio is (lambda)(sub x) = 3. The natural support of the field is therefore 2(sup n) pixels (vertically) by 3(sup n) pixels (horizontally) where n is the number of levels in the MRA. The natural triadic basis includes the French top-hat wavelet which resonates with bumps in the field whereas the Haar wavelet responds to ramps or steps. The complete 2D basis has one scaling function and five wavelets. The resulting anisotropic MRA is designed for application to the liquid water content (LWC) field in boundary-layer clouds, as the prevailing wind advects them by a vertically pointing mm-radar system. Spatial correlations are notoriously long-range in cloud structure and the authors use the wavelet coefficients from the new MRA to characterize these correlations in a multifractal analysis scheme. In the present study, the MRA is used in realistically although only a few parameters are used to control the randomness of the LWC's wavelet coefficients.