The aR<sup>b</sup>relation in the calculation of rain attenuation

Because of its simplicity, the empirical relation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A = aR^{b}</tex> between the specific attenuation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</tex> and the rainrate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</tex> is often used in the calculation of rain attenuation statistics. Values for the frequency-dependent parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> are available, however, for only a limited number of frequencies. Some of these values, furthermore, were obtained experimentally, and may contain errors due to limitations in the experimental techniques employed. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">aR^{b}</tex> relation is shown to be an approximation to a more general relation, except in the low-frequency and optical limits. Because the approximation is a good one, however, a comprehensive and self-consistent set of values for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> is presented in both tabular and graphical form for the frequency range <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f = 1-1000</tex> GHz. These values were computed by applying logarithmic regression to Mie scattering calculations. The dropsize distributions of Laws and Parsons, Marshall and Palmer, and Joss et al., were employed to provide calculations applicable to "widespread" and "convective" rain. Empirical equations for some of the curves of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(f)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(f)</tex> are presented for use in systems studies requiring calculations at many frequencies. Some comparison is also made with experimental results, and suggestions are given regarding application of the various calculations.