Multifractals, cloud radiances and rain

The extreme variability of rainfall over huge ranges of space-time scales makes direct rain gauge measurements of areal rainfall impossible; assumptions about the rainfall scaling-whether trivial (homogeneous), or multifractal (heterogeneous)are required even for interpolation. The alternative is to use rain surrogates such as radar reflectivities or those based on visible-infra red radiances. In this paper, we argue that cloud radiances should be studied to obtain basic information about the range and type of scaling in the atmosphere. Since, rain and clouds are strongly non-linearly coupled-and since the scaling of the fields, the scale invariance of the generators/exponents is a symmetry principle-a break in the scaling in one of the fields would cause a break in the other. Using 909 images from three satellites and six sensors (visible and infra red) collectively spanning the range of scales 5000-1 km, we demonstrate that power law scaling is respected to within an error of +/- 0.3-0.5%; that an upper bound on the deviations from the theoretical universal multifractal scaling is 1-2% per octave in scale. We also show that the outer scale of the cascade is very close to 20,000 km, the largest great circle distance on the earth. Allowing for (one-parameter) subpower law (logarithmic) scaling corrections we show that universal multifractal cascades starting at this scale explain the isotropic moments (order <= 1.6) to within an error of +/- 0.8%. We argue that the scaling of these isotropic statistics shows that the diversity of cloud morphologies reflects differences in anisotropies which are effectively washed out by the isotropic statistical methods used. We compare and contrast existing multifractal models showing which can be used as realistic cloud and rain models. We go on to use continuous in scale, anisotropic, space-time multifractal rain and cloud simulations (including radiative transfer) to show how diverse cloud, rain and radiance morphologies can be compatible with the observed isotropic scaling statistics. Finally, we argue that these will be necessary for solving measurement problems including the use of rain gauge, radar and visible/infra red surrogate fields. (c) 2005 Elsevier B.V. All rights reserved.