Scalar multifractal radar observer's problem

The classical radar observer's problem in rain is to interpret the fluctuating radar echo from precipitation. Contrary to the usual homogeneity assumption involving Poisson statistics and incoherent scattering, we make a (scaling) heterogeneity assumption involving multifractal statistics and (some) coherent scattering. We consider the simplest problem, which is to relate the liquid water (sigma) statistics to the (measured) effective radar reflectivity statistics (Z(e)) and to the (theoretical) radar reflectivity factor (Z; Z(e)=Z for incoherent scattering). We ignore polarization effects (that is, we use the scalar wave approximation), and denote the pulse length I, wavelength lambda(w), the inner (homogeneity) scale of the rain field (eta), and the outer (largest) scale of rain (L). For the simplest (conservative) multifractal sigma the two main effects are 1) as in the standard theory, Z approximate to sigma(2); however, because of the strong subpulse volume gradients, there is a bias of (l/lambda(w))(K sigma(2)); (K-sigma(2) is the scaling exponent of sigma(2)); 2) because of partial coherence, there is an enhancement: Z(e)/Z approximate to(lambda(w)/eta)(D-K sigma(2)), where D is the (effective) dimension of space. For nonconservative multifractals (parametrized by H we obtain the overall bias in the means: <Z(e)>/<Z>approximate to(lambda(w)/eta)D-K sigma(2))(L/lambda(w))(-2H)). Using available data, we estimated this as typically approximate to 10(-3) which is <<1; Z should therefore not be used as a proxy for Z(e). New theories relating radar measurements to rain must therefore be developed. Finally, we show that radar ''speckle'' (the drop ''rearrangement'' problem) is a general consequence of multifractal liquid water/drop correlations.