How estimates of uncertainty in measured particle size distributions can be used to characterize the variability of gamma/exponential functions used in numerical models

Gamma and exponential functions are used to represent particle size distributions (SDs) in mesoscale and cloud resolving models that predict 1, 2 or 3 moments of various hydrometeor species. The functions are characterized by intercept (N_0/), slope (λ) or shape (μ) parameters that are either derived from prognosed model fields or diagnosed based on fits to SDs measured in-situ. Currently, all such fits derive the most likely value of N₀, λ, and μ as functions of other parameters, such as mass concentration or temperature. Here, a new approach for deriving fit parameters is introduced, where uncertainties in measured SDs due to statistical variability and unknown numbers of artificially generated small ice crystals from the shattering of large ice crystals on the tips/inlets of in-situ probes are used to define the allowed tolerance in fit parameters. Thereafter, that tolerance is used to determine a range of plausible fit parameters, which is represented as a three-dimensional volume in N₀-λ-μ phase space that accounts for both the permitted variability of individual parameters and their co-dependence. Such volumes can be derived for either individual SDs or for families of SDs obtained in similar conditions. Because the eigenvalues and eigenvectors of the Hessian matrix obtained from the fits to measured SDs can be used to characterize the orientation and axes length of the ellipsoid characterizing the volume, Monte Carlo model simulations can be conducted to determine how uncertainties in measured SDs cascade up to predictions of microphysical processes rates and modeled fields. The application of this technique is demonstrated using data acquired in arctic cirrus during the Indirect and Semi-Direct Aerosol Campaign and in hurricanes during the NASA African Monsoon Multidisciplinary Analysis campaign.