Theoretical study of the VHF radar echo statistics

In this article the statistical characteristics of the VHF radar returns, which are assumed to comprise components generated from the atmospheric isotropic turbulences plus anisotropic irregularities are theoretically studied. By employing a theory of the random variable, the analytic form of the amplitude and phase probability density functions of the VHF radar echoes are derived. After a somewhat tedious and complicated calculation it shows that the conventional Rayleigh distribution, the Rice distribution, and the Hoyt distribution are closely related to the derived generalized probability density function. The Nakagami m parameter corresponding to this generalized probability density function of the VHF radar signal amplitude is derived as well. It indicates that the magnitude of Nakagami m parameter is governed by the radar echo parameters, that is, 2S², σ², and μ, where 2S² is the power of the radar echo component scattered from the isotropic turbulences, μ and σ² are the mean and variance, respectively, of the radar signal component generated from the anisotropic irregularities. After examining the general behavior of the derived Nakagami m parameter in more detail, it is found that no matter what the S value is, the magnitude of Nakagami m parameter, m, is always in one of the following three categories, depending on the relative changes between μ and σ. Namely, m = 1 if σ² = (2^½ ‐ 1)μ²; 0.5 < m < 1 if σ² > (2^½ ‐ 1)μ²; and m > 1 if σ² < (2^½ ‐ 1)μ². These results are quite different from those expected with the conventional theories. Therefore great care should be taken when the probability density function of signal amplitude and phase and the Nakagami m parameter are employed to distinguish the echo mechanism of VHF radar.