Implications of a λ - µ relationship for Z-R relationships

            Reports of a systematic relationship between the shape (μ) and size (λ) parameters of gamma size distribution functions fitted to observed raindrop spectra have been the subject of some debate.  Problems have been noted with some of the data selection and statistical analysis procedures underlying those findings.  Nevertheless, similar findings continue to emerge from more careful analyses of observed raindrop size spectra.  For purposes of the present discussion we take those findings as a point of departure, albeit with some reservations.

            Relationships between the radar reflectivity Z (sixth moment M₆) associated with a given gamma drop size distribution and other moments M_i of the distribution can be expressed in terms of the gamma parameters λ and μ.  Using a λ-μ relationship reduces those relationships to M₆-M_i relationships depending only upon μ.  If μ were constant, M₆ would be directly proportional to M_i.  Though μ probably varies, taking the typical quadratic form of the λ-μ relationship and reported values of its parameters together with plausible values of μ suggests that the M₆-M_i relationships depend only weakly upon the value of μ – and are therefore nearly linear.  Since the M₆-M₃ and M₆-M₄ relationships bracket the Z-R relationship, this suggests that the Z-R relationship must also be nearly linear – a behavior postulated earlier by Jameson and Kostinski (2001).

            Fitted values of μ (based on substantial sample sizes and reasonably accurate fitting procedures) could be plotted against the sample values of M₃, M₄, and R to investigate this matter further.  An example in Chu and Su (2008) suggests no strong dependence of μ upon R.