Radiative smoothing in fractal clouds

Spectral and structure function analyses are used to study the smoothness properties of the radiation fields for stratiform clouds whose horizontally fluctuating extinction fields are modeled with multiplicative cascades. Models of this type are “scale invariant,” meaning that their two-point statistics obey power laws in the scale parameter. The independent pixel approximation (IPA) treats each pixel as a plane-parallel layer and yields scale-invariant albedo and radiance fields with the same exponents as the associated optical depth field. This is not the case with exact Monte Carlo (MC) results for which we confirm the existence of a characteristic “radiative smoothing” scale q. For scales larger than 7, IPA and MC reflectance fields fluctuate together, and the IPA can be invoked to infer optical depths from measured radiances. We use a multifractal characterization of structure functions to assess the performance of such retrievals. For scales smaller than TJ, MC fields are much smoother than their IPA counterparts, and IPA-based retrievals of the tihderlying optical depth field are unreliable. The scale break location TJ has been found to be closely related to the characteristic size (p) of the “spot” of multiply scattered light excited by illumination with a narrow beam, the random variable p being the horizontal distance between photon entry and exit points. New analytical arguments are presented for thick homogeneous media showing that (p) Lh[(l-g)z]-‘I*, given the cloud’s optical (2) and geometrical (h) thicknesses (g is the asymmetry factor); this result is shown to hold numerically for fractal cloud models too. An improved “nonlocal” IPA is defined as the convolution product of the IPA field with a gamma-type smoothing kernel dependent on (p).