A fast analytical radiative transfer model to account for propagation of unpolarized monochromatic radiation in random media with a plane-parallel geometry is presented. The model employs an Eddington-like approach combined with the delta phase-function transformation technique. The Eddington approximation is extended in a form that allows us to unfold the azimuthal dependence of the radiance field. A first-order scattering correction to the azimuth-dependent Eddington radiative model solution is also performed to improve the model accuracy for low-scattering media and flexibility with respect to use of explicit arbitrary phase functions. The first-order scattering-corrected solution, called the generalized Eddington radiative model (GERM), is systematically tested against a numerical multistream discrete ordinate model for backscattered radiance at the top of the medium. The typical mean accuracy of the GERM solution is generally better than 10% with a standard deviation of 20% for radiance calculations over a wide range of independent input optical parameters and observation angles. GERM errors are shown to be comparable with the errors due to an input parameter uncertainty of precise numerical models. The proposed model can be applied in a quite arbitrary random medium, and the results are appealing in all cases where speed, accuracy, and/or closed-form solutions are requested. Its potentials, limitations, and further extensions are discussed.

Abstract. Aim of this work is to provide a new insight into the physical basis of the meteorological-radar theory in attenuating media. Starting form the general integral form of the weather radar equation, a modified form of the classical weather radar equation at attenuating wavelength is derived. This modified radar equation includes a new parameter, called the range-bin extinction factor, taking into account the rainfall path attenuation within each range bin. It is shown that, only in the case of low-to-moderate attenuating media, the classical radar equation at attenuating wavelength can be used. These theoretical results are corroborated by using the radiative transfer theory where multiple scattering phenomena can be quantified. From a new definition of the radar reflectivity, in terms of backscattered specific intensity, a generalised radar equation is deduced. Within the assumption of first-order backscattering, the generalised radar equation is reduced to the modified radar equation, previously obtained. This analysis supports the conclusion that the description of radar observations at attenuating wavelength should include, in principle, first-order scattering effects. Numerical simulations are performed by using statistical relationships among the radar reflectivity, rain rate and specific attenuation, derived from literature. Results confirm that the effect of the range-bin extinction factor, depending on the considered frequency and range resolution, can be significant at X band for intense rain, while at Ka band and above it can become appreciable even for moderate rain. A discussion on the impact of these theoretical and numerical results is finally included.

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