The Shape–Slope Relation in Observed Gamma Raindrop Size Distributions: Statistical Error or Useful Information?

Abstract

The three-parameter gamma distribution n(D) ϭ N 0 D exp(Ϫ⌳D) is often used to characterize a raindrop size distribution (DSD). The parameters and ⌳ correspond to the shape and slope of the DSD. If and ⌳ are related to one another, as recent disdrometer measurements suggest, the gamma DSD model is simplified, which facilitates retrieval of rain parameters from remote measurements. It is important to determine whether the-⌳ relation arises from errors in estimated DSD moments, or from natural rain processes, or from a combination of both statistical error and rain physics. In this paper, the error propagation from moment estimators to rain DSD parameter estimators is studied. The standard errors and correlation coefficient are derived through systematic error analysis. Using numerical simulations, errors in estimated DSD parameters are quantified. The analysis shows that errors in moment estimators do cause correlations among the estimated DSD parameters and cause a linear relation between estimators and. However, the slope and intercept of the error-induced relation depend on the expected values and ⌳, ⌳ and it differs from the-⌳ relation derived from disdrometer measurements. Further, the mean values of the DSD parameter estimators are unbiased. Consequently, the derived-⌳ relation is believed to contain useful information in that it describes the mean behavior of the DSD parameters and reflects a characteristic of actual raindrop size distributions. The-⌳ relation improves retrievals of rain parameters from a pair of remote measurements such as reflectivity and differential reflectivity or attenuation, and it reduces the bias and standard error in retrieved rain parameters.

Shape Slope Parameter Distribution Modelling of Electromagnetic Scattering by Rain Drops

“…3, the two fits from DG have very low µ values compared to all other fits, indicating very weak events in Scotland. This agrees with Zhang's statement [6] that µ-Λ relations vary with the location since each location has different types of rain. Another reason for their fit to have a low µ value is the use of different distrometers and their use of a new procedure to estimate the gamma parameters.…”

“…The spread of scattered µ and Λ points in Fig. 2, are also due to measurement errors, which are relatively larger for low rain rates than high rain rates [6]. Tokay et al [21] compared the video distrometer and Joss distrometer drop size measurements.…”

“…They filtered the database to exclude rain rates smaller than 5 mm/hr, and drop counts smaller than 1000 prior to curve fitting. Zhang et al [6] have used the µ-Λ relation proposed by [5] and shown that the relation contains useful information and characterizes the natural rain DSD variations well. He noted that the coefficients of the µ-Λ relation might change with location and season.…”

Remote Sensing of Precipitation from Airborne and Spaceborne Radar

“…In principle, then, it is possible to retrieve Dm from the DFR (after attenuation correction has been performed) and an assumption regarding μ, such as a constant value (Liao et al, 2014), reflectivity-dependent value (Munchak and Tokay 2008), μ-Λ relationship (Zhang et al, 2003), or σm-Dm relationship (Williams et al, 2014). At some frequency pairs, such as Ku-Ka, there is an ambiguity in that two values of Dm can be associated with the same DFR for a given μ (Figure 9).…”

“…Analysis of disdrometer observations has shown that neither the three free parameters in (9) nor the physical quantities in (10)-(12) are statistically independent in rain (e.g., Haddad et al, 1996;Zhang et al, 2003;Munchak and Tokay, 2008;Williams et al, 2014. ) and relationships between the parameters can be formulated to reduce the degrees of freedom in (9).…”