Statistical Reliability of Neighboring Range Bin Estimates of Coherent Fractional Contributions to Radar Backscattered Power

Abstract

It was recently demonstrated that magnitudes of the power-normalized cross-correlation functions of complex amplitudes in neighboring range bins are identical to the fractional contributions made by radar coherent backscatter in the direction of propagation to the total backscattered power in rain and snow. Here, a theoretical framework is presented for calculating the noise associated with estimates of these normalized cross correlations. This noise is identical to the statistical uncertainties in . Radar signals consist of two components: the usual incoherent backscatter often modeled by a Gaussian process and a coherent component modeled for the purposes of these calculations by a phasor C of fixed magnitude that rotates at a constant angular velocity ?C. Using the representation of the cross-correlation function as the average over the real part of the phasor dot products, it is found that the noise in this function comes from the dot products of C with the incoherent-scatter phasors in each range bin as well as the dot product between the two incoherent phasors. Furthermore, as long as ?C ? 0 and the number of statistically independent realizations (samples) k is sufficiently large, the noise is represented well by a normal distribution with mean 0 and with a variance that goes as 1/(2k). It is then shown that as the magnitude of C increases it acts to suppress the variance of . A formula is derived that gives the standard deviation of as a function of the number of statistically independent samples in the observation and the observed value of . Two examples, one in rain and the other in snow, are also presented.