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

Taking a wavelet standpoint, we survey on the one hand various approaches to multifractal analysis, as a means of characterizing long-range correlations in data, and on the other hand various ways of statistically measuring anisotropy in 2D fields. In both instances, we present new and related techniques: (i) a simple multifractal analysis methodology based on Discrete Wavelet Transforms (DWTs), and (ii) a specific DWT adapted to strongly anisotropic fields sampled on rectangular grids with large aspect ratios. This DWT uses a tensor product of the standard dyadic Haar basis (dividing ratio 2) and a non-standard triadic counterpart (dividing ratio 3) which includes the famous 'French top-hat' wavelet. The new DWT is amenable to an anisotropic version of Multi-Resolution Analysis (MRA) in image processing where the natural support of the field is 2ⁿ pixels (vertically) by 3ⁿ pixels (horizontally), n being the number of levels in the MRA. The complete 2D basis has one scaling function and five wavelets. The new MRA is used in synthesis mode to generate random multifractal fields that mimic quite realistically the structure and distribution of boundary-layer clouds even though only a few parameters are used to control statistically the wavelet coefficients of the liquid water density field.