### Hydrometeor Droplet Size Distributions  
Although theoretical droplet size distributions (DSD) are often formulated in terms of the number of droplets above some threshold radius (or diameter), observed droplet size distributions measure hydrometeor counts in diameter classes and thus are processed to generate number concentrations (dN) within some finite size interval (dD). They are expressed as the number concentrations of raindrops per cubic meter as a function of their diameter normalized for a fixed size interval (_dN/MD_ or _N_(_D_), #m\({}^{-3}\) mm\({}^{-1}\)). It has long been recognized that increasing precipitation intensity (or rain rate, RR) shifts the hydrometeor DSD towards an increase in the relative abundance of larger droplets [57]. Accordingly, theoretical DSD are described as a function of prevailing RR and/or liquid water content (_LWC_) of air. Frequently, the number DSD for rain droplets is assumed to follow a two- or three-parameter gamma distribution [58], although there is some evidence to suggest a three-parameter lognormal may be optimal under some circumstances [59]. The normalized gamma distribution as a function of _LWC_ (in gm\({}^{-3}\)) is given by:  
\[N(D)\ =\ N_{w}f(u)\bigg{(}\frac{D}{D_{m}}\bigg{)}^{\mu}exp\bigg{[}-(4+\mu) \frac{D}{D_{m}}\bigg{]} \tag{6}\]  
where \(N_{w}\) is:  
\[N_{w}\ =\ \frac{3.67^{4}10^{3}LWC}{\pi\rho_{w}D_{0}^{4}} \tag{7}\]  
and \(D_{m}\) is:  
\[D_{m}\ =\ \frac{4+\mu}{3.67+\mu}D_{0} \tag{8}\]  
where \(D_{0}\) = median drop diameter, \(\rho_{w}\) = density of water and \(\mu\) = shape parameter of the distribution that can be derived by fitting to observed DSD [45]. Details on estimation of \(N_{w}\) and \(f(\mu)\) are given in [PERSON] et al. [60]. The Marshall-Palmer approximation for DSD for rain was experimentally derived based on data collected in Montreal, Canada in 1946 [61]. It assumes that rain droplets follow an exponential distribution and can be easily used to predict the size distribution of liquid hydrometeors (rain droplets) based on the prevailing rainfall rate (RR, in mmhr\({}^{-1}\)) [61]. It is most frequently expressed as the number of droplets above radius, \(R\), per cubic meter of air (\(N\), m\({}^{-3}\)):  
\[N\ =\ \frac{N_{0}}{\Lambda}e^{-\Lambda R} \tag{9}\]  
where \(\Lambda\ =\ 8200(RR)^{-0.21}\) (m\({}^{-1}\)) and \(N_{0}\ =\ 1.6\times 10^{7}\) m\({}^{-4}\). The Det Norske Veritas (DNV) Recommended Practice for LEE testing issued in December 2020 and amended in October 2021 [46] proposes use of the so-called Best rain DSD to characterize the droplet size spectrum. This distributional form was published in 1950 and is based on measurements made in the United Kingdom using filter paper [41]. It has the form:  
\[N(D)\ =\ \frac{W}{V}\bigg{(}\frac{k_{B}\cdot D^{k_{B}-1}}{a^{k_{B}}}\bigg{)}e^{ -\lfloor D/a\rfloor^{k_{B}}} \tag{10}\]  
where \(V\) = spherical volume of the droplet \(\Big{(}\frac{1}{6}\tau D^{3}\Big{)}\) (mm\({}^{-3}\)), \(D\) = droplet diameter (mm), \(W\) = total water volume which is given as 67 \(RR^{0.846}\) (mm\({}^{3}\)) \(RR\) = rainfall rate in mmhr\({}^{-1}\), \(k_{B}\) = 2.25, \(a\) = 1.3 \(RR^{0.232}\). The DNV Recommended Practice also suggests 6 mm as an upper bound for droplet diameter asserting that above this diameter \"aerodynamic forces break up the droplet\" [46]. As shown herein, observations derived using ground-based disdrometers suggest the existence of droplets at diameters considerably above 6 mm. Fewer studies have examined the size distribution of hail stones [62; 63]. The simplest approximations use a single fitting parameter (\(\lambda\)), e.g.,:  
\[N(D)\ =\ 115\lambda^{3.63}e^{-\lambda D} \tag{11}\]  
where \(D\) = hail stone diameter [64]. Values of \(\lambda\) of 0.3 to 0.6 mm\({}^{-1}\) were proposed based on measurements in Canada [64; 65]. It should be noted that rain droplets are generally not spherical and are increasingly oblate as \(D\) increases.