A General N-Moment Normalization Method for Deriving Raindrop Size Distribution Scaling Relationships

Abstract

A general drop size distribution (DSD) normalization method is formulated in terms of generalized power series relating any DSD moment to any number and combination of reference moments. This provides a consistent framework for comparing the variability of normalized DSD moments using different sets of reference moments, with no explicit assumptions about the DSD functional form (e.g., gamma). It also provides a method to derive any unknown moment plus an estimate of its uncertainty from one or more known moments, which is relevant to remote sensing retrievals and bulk microphysics schemes in weather and climate models. The approach is applied to a large dataset of disdrometer-observed and bin microphysics-modeled DSDs. As expected, the spread of normalized moments decreases as the number of reference moments is increased, quantified by the logarithmic standard deviation of the normalized moments, σ. Averaging σ for all combinations of reference moments and normalized moments of integer order 0–10, 42.9%, 81.3%, 93.7%, and 96.9% of spread are accounted for applying one-, two-, three-, and four-moment normalizations, respectively. Thus, DSDs can be well characterized overall using three reference moments, whereas adding a fourth reference moment contributes little independent information. The spread of disdrometer-observed DSD moments from uncertainty associated with drop count statistics generally lies between values of σ using two- and three-moment normalizations. However, this uncertainty has little impact on the derived DSD scaling relationships or σ when considered.

Using an Arbitrary Moment Predictor to Investigate the Optimal Choice of Prognostic Moments in Bulk Cloud Microphysics Schemes

“…For the other two processes, the optimal configuration includes both high‐ and low‐order moments. This result agrees with Morrison et al () as discussed in section .…”

“…Predicting the 3 rd and fourth moments or 3 rd and 6 th moments seem optimal. Morrison et al () speculated that this may be the case based on their analysis of the relationships between moments of rain drop size distributions.…”

“…From a statistical standpoint, Morrison et al () find that knowledge of just the 0 th and 3 rd moments gives little constraint on higher‐order moments. They suggest that predicting a combination of high and low moments such as is done by triple‐moment schemes may be best for reducing uncertainty in the simulations of all moments.…”

Radar-derived precipitation climatology for wind turbine blade leading edge erosion

“…Most observational studies indicate an exponential form (Uijlenhoet, 2001), and the Marshall-Palmer distribution is the most widely applied. However, a range of different forms have been proposed to describe the size spectrum of rain droplets (dN/dR) including gamma (Ulbrich, 1983) and lognormal (Feingold and Levin, 1986), an alternative exponential form (Best, 1950), and more complex non-parametric forms (Morrison et al, 2019). There is also evidence that droplet size distributions may exhibit a functional dependence on near-surface wind speed (Testik and Pei, 2017).…”

Measurements and Modeling of the Full Rain Drop Size Distribution

“…The scaling normalization of the DSD as proposed in [12] and the generalization given in [13] to include double-moment normalization [14,15] using two reference moments has proven to be useful to arrive at the intrinsic or generic shape of the distribution (referred to as h(x) defined later in Section 2, Equation 2). The hypothesis is that a large part of the DSD variability (around 80%; [16]) can be accounted for by the two reference moments, with variations in h(x) playing a secondary role to the extent that the shape of h(x) may be considered as "stable" or "invariant" [17]. In other words, after normalization, the scatter in h(x) is greatly reduced, enabling a good fit to its shape relative to more commonly used models such as exponential, standard gamma, and log-normal [15,[18][19][20].…”