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In statistically inhomogeneous rain, however, the coefficient relating Z and R varies in an unknown fashion so that one must resort to statistical fits, often power laws, in order to relate the two quantities empirically over widely varying conditions. The justification' for non-linear power law Z-R relations is not physical, but rather statistical in that they provide convenient parametric fits for estimating mean R from measured mean Z in statistically inhomogeneous rain. Accordingly, the interpretations of drop size distributions in statistically inhomogeneous rain now depend upon the measurement process. Furthermore, in statistically inhomogeneous rain, different remote sensing instruments, even if pointed at the same target, will see different total drop size distributions simply because the beam-widths are not the same. This will affect all algorithms which assume that different instruments are viewing the same set of drops.

Finally, examples suggest that such generalized relations between two variables defined by such sums are potentially useful over a wide range of remote sensing problems and over a wide range of scales. Application of such relations, however, requires the extraction from the non-Rayleigh signal statistics, common to almost all radar and radiometer observations, of the component of the variances due to the distribution of mean values.