Describing the Shape of Raindrop Size Distributions Using Uncorrelated Raindrop Mass Spectrum Parameters

Abstract

Rainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parameters N w , D m , and m. If only two independent measurements are available, as with the dualfrequency precipitation radar on the Global Precipitation Measurement (GPM) mission core satellite, then retrieval algorithms are underconstrained and require assumptions about DSD parameters. To reduce the number of free parameters, algorithms can assume that m is either a constant or a function of D m . Previous studies have suggested m-L constraints [where L 5 (4 1 m)/D m ], but controversies exist over whether m-L constraints result from physical processes or mathematical artifacts due to high correlations between gamma DSD parameters. This study avoids mathematical artifacts by developing joint probability distribution functions (joint PDFs) of statistically independent DSD attributes derived from the raindrop mass spectrum. These joint PDFs are then mapped into gamma-shaped DSD parameter joint PDFs that can be used in probabilistic rainfall retrieval algorithms as proposed for the GPM satellite program. Surface disdrometer data show a high correlation coefficient between the mass spectrum mean diameter D m and mass spectrum standard deviation s m . To remove correlations between DSD attributes, a normalized mass spectrum standard deviation s 0 m is constructed to be statistically independent of D m , with s 0 m representing the most likely value and std(s 0 m ) representing its dispersion. Joint PDFs of D m and m are created from D m and s 0 m . A simple algorithm shows that rain-rate estimates had smaller biases when assuming the DSD breadth of s 0 m than when assuming a constant m.

Atmospheric polarimetric effects on GNSS radio occultations: the ROHP-PAZ field campaign

“…12), liquid or ice water content (LWC, IWC), effective particle diameter (D eff ), mean weighted diameter (D m ), and rain rate (R). Further details of the relation between these magnitudes and N (D) can be found in the literature, for example in Williams et al (2014).…”

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).…”

“…These names describe the mathematical form of the distribution moreso than physical quantities, and it is difficult to impart any physical meaning to a value for anyone one of the parameters given in isolation. However, it is possible to re-cast these parameters in terms of physical quantities through the following relationships (Testud et al, 2001;Williams et al, 2014):…”

“…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).…”

“…To demonstrate the relationship between PSD parameters and radar reflectivity Z, Figures 4 and 5 show reflectivity and extinction coefficients integrated over PSDs with Dm ranging from 0.1 to 3 mm and μ ranging from -1 to 3 for snow and following the σm-Dm relationships given by Williams et al (2014) for rain. The integrated water content of all PSDs was normalized to W=1 g m -3 to emphasize the importance of the shape of the PSD in terms of its mean and dispersion on the reflectivity and extinction.…”