Radiative Transfer in Multifractal Atmospheres: Fractional Integration, Multifractal Phase Transitions and Inversion Problems

Abstract

This paper is devoted to studying the inhomogeneity of the radiation field resulting from propagation through a multifractal cloud field by relating the orders of singularities and codimensions of both fields. This direct relationship is of fundamental importance for climate studies, whereas the inverse problem is fundamental for remote sensing. We point out similarities between smoothing by scattering and fractional integration, showing they are exactly analogous for certain cases: 1-D medium and plane parallel atmospheres (with a few extra hypotheses). We therefore deduce that there is a limited range of singularities susceptible to exhibiting identical multifractal characteristics before any inversion. The lower bound (γD′) is defined by a first order multifractal phase transition which occurs when the dimension D′ of the fractional integration is insufficient to smooth out the singularities of the cloud field, whereas the upper bound (γs) is defined by a second-order phase transition and corresponds to the limitations induced by the finite size of the samples. These two critical singularities drastically reduce the range of relevant singularities and justify some essentially ad hoc procedures used in multifractal estimation.