_{0}, λ, and μ). One problem with these parameters is that they are mathematical parameters such that they only have physical meaning when the DSD has an exponential shape (i.e., when μ = 0). Also, due to computational limitations, the shape parameter is often held constant in two-moment microphysical model parameterization schemes.

With the aim of using observations to improve model parameterizations and understanding how aerosols influence deep convective clouds, this study develops a framework to describe raindrop size distributions using physical attributes of observed DSDs. Specifically, the DSD shape is characterized using the mean and variance of the mass spectrum and the DSD scale is described using the total number concentration. All three of these physical attributes can be estimated from surface disdrometers without assuming a Gamma or lognormal distribution. Disdrometer observations show that the mass spectrum mean and variance are highly correlated, such that the effective variance (i.e., variance normalized by the squared mean) provides a physical interpretation of when distributions become narrower or broader.

In summary, similar to describing cloud droplet and aerosol distribution breadth with the effective radius and effective variance, this study uses mass spectrum mean diameter and mass spectrum effective variance to describe the shape of raindrop size distributions. The developed mathematical model can be mapped to previous works using Gamma-function and lognormal DSD models. And potentially, the new framework can be used to identify when DSD shapes are influenced by aerosol - cloud interactions.