SCALAR MULTIFRACTAL RADAR OBSERVERS PROBLEM

The classical radar observer's problem in rain is to interpret the flu ctuating radar echo from precipitation. Contrary to the usual homogene ity assumption involving Poisson statistics and incoherent scattering, we make a (scaling) heterogeneity assumption involving multifractal s tatistics and (some) coherent scattering. We consider the simplest pro blem, which is to relate the liquid water (sigma) statistics to the (m easured) effective radar reflectivity statistics (Z(e)) and to the (th eoretical) radar reflectivity factor (Z; Z(e)=Z for incoherent scatter ing). We ignore polarization effects (that is, we use the scalar wave approximation), and denote the pulse length I, wavelength lambda(w), t he inner (homogeneity) scale of the rain field (eta), and the outer (l argest) scale of rain (L). For the simplest (conservative) multifracta l sigma the two main effects are 1) as in the standard theory, Z appro ximate to sigma(2); however, because of the strong subpulse volume gra dients, there is a bias of (l/lambda(w))(K sigma(2)); (K-sigma(2) is t he scaling exponent of sigma(2)); 2) because of partial coherence, the re 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 me ans: <Z(e)>/<Z>approximate to(lambda(w)/eta)D-K sigma(2))(L/lambda(w)) (-2H)). Using available data, we estimated this as typically approxima te to 10(-3) which is <<1; Z should therefore not be used as a proxy f or Z(e). New theories relating radar measurements to rain must therefo re be developed. Finally, we show that radar ''speckle'' (the drop ''r earrangement'' problem) is a general consequence of multifractal liqui d water/drop correlations.