Semi-discrete wavelet transforms are discrete in scale, as in Stéphane Mallat’s multi-resolution analysis and Ingrid Daubechies’ binary sampling, but continuous in position. The number of coefficients and algorithmic complexity then grows only as NlogN where N is the number of points (pixels) in the time-series (image). The redundancy of this representation at each scale has been exploited in denoising and data compression applications but we see it here as a safeguard when cumulating spatial statistics. The wavelets are scaled so that the exponents of the statistical moments of the coefficients are the same as for structure functions at all orders, at least for nonstationary signals with stationary increments. By now, we have effectively relaxed the habitual constraints of orthogonality and normalization in wavelet theory, and even done away the standard decimation by powers-of-2.

We then apply 1D and 2D semi-discrete transforms to remote sensing data on cloud structure from a variety of sources: NASA’s MODerate Imaging Spectroradiometer (MODIS) on Terra and Thematic Mapper (TM) on LandSat; high-resolution cloud scenes from DOE’s Multispectral Thermal Imager (MTI); and an upward-looking mm-radar at DOE’s climate observation sites supporting the Atmospheric Radiation Measurement (ARM) Program. We show that the scale-dependence of the variance of the wavelet coefficients is always a better discriminator of transition from stationary to nonstationary behavior than conventional methods based on auto-correlation analysis, 2nd-order structure function (a.k.a. the semi-variogram), or spectral analysis. Examples of stationary behavior are residual (delta-correlated) instrumental noise at very small scales and large-scale decorrelation of cloudiness; here, wavelet coefficients decrease with increasing scale. Examples of nonstationary behavior are the horizontal structure of cloud layers as well as instrumental or physical smoothing in the data; here, wavelet coefficients increase with scale. In all of these regimes, we have theoretical predictions for and/or empirical evidence of power-law relations for wavelet statistics with respect to scale as expected in physical (finite-scaling) fractal phenomena. In particular, this implies the presence of long-range correlations in cloud structure coming from the important nonstationary regime.

Finally, we discuss the implications of our findings for cloud-radiation interaction and dynamical cloud modeling, two intensely researched sub-problems in global and regional climate modeling.