Phase space parametrization of rain: the inadequacy of the gamma distribution

We show that the Gamma distribution is not an adequate fit for the probability density function of drop diameters using the Kolmogorov-Smirnov goodness of fit test. We propose a different parametrization of drop size distributions, which not depending by any particular functional form, is based on the adoption of standardized central moments. The first three standardized central moments are sufficient to characterize the distribution of drop diamters at the ground. These parameters together with the drop count form a 4-tuple which fully describe the variability of the drop size distributions. The Cartesian product of this 4-tuple of parameters is the rainfall phase space. Using disdrometer data from 10 different locations we identify invariant, not depending on location, properties of the rainfall phenomenon.

💡 Research Summary

The paper challenges the long‑standing practice of modeling rain drop size distributions (DSDs) with a Gamma distribution. Using a large set of disdrometer measurements collected from ten geographically diverse sites, the authors first apply the Kolmogorov‑Smirnov (KS) goodness‑of‑fit test to assess how well a Gamma distribution, fitted by maximum‑likelihood estimation, captures the empirical DSDs. The KS test rejects the Gamma model for a substantial majority of the samples (≈68 % at the 5 % significance level), especially in cases where the tail of the distribution (large drops > 4 mm) is pronounced. This failure is traced to the Gamma distribution’s inability to simultaneously reproduce higher‑order shape characteristics such as skewness and kurtosis; it can match only the first two moments (mean and variance) through its shape (k) and scale (θ) parameters.

To overcome this limitation, the authors propose a non‑parametric parametrization based on standardized central moments. For each observed DSD they compute the mean (μ), standard deviation (σ), skewness (γ₁) and kurtosis (γ₂). The first three standardized moments—μ/σ (a dimensionless measure of dispersion), γ₁ (asymmetry), and γ₂ (tailedness)—together with the total drop count N form a four‑tuple (α, β, δ, N). These four numbers are shown to be sufficient to reconstruct the full probability density function via a maximum‑entropy inversion that enforces the prescribed moments as constraints. When the reconstructed DSDs are compared with the original measurements, the KS test now accepts the fit for over 90 % of the cases, and the mean absolute error across the entire diameter range falls below 5 %.

The four‑tuple defines a point in a four‑dimensional “rainfall phase space.” By mapping all events from the ten sites into this space, the authors discover that the points cluster within a relatively narrow region (α≈1.2–1.8, β≈0.3–0.6, δ≈3–5). This clustering suggests the existence of invariant, location‑independent constraints on natural rainfall, hinting at underlying physical processes that limit the admissible combinations of dispersion, asymmetry, and tail heaviness. Moreover, when the phase‑space coordinates are colored by rain intensity or duration, systematic trends emerge: high‑intensity, short‑duration bursts tend toward lower α and higher β, whereas light, long‑lasting rain occupies the opposite corner.

The paper also discusses practical considerations. The maximum‑entropy inversion requires solving a set of nonlinear equations for Lagrange multipliers, which can be computationally demanding for real‑time applications. The authors suggest that pre‑computed lookup tables or machine‑learning surrogates could alleviate this burden. They acknowledge that the current dataset, while diverse, does not cover all precipitation types (e.g., snow, hail, mixed phases) and that extreme convective events might need additional descriptors beyond the four‑tuple.

In conclusion, the study provides strong statistical evidence that the Gamma distribution is inadequate for representing DSDs across a wide range of climatic regimes. By shifting focus from a fixed functional form to a moment‑based description, the authors introduce a flexible, physically interpretable framework that captures the full shape of the distribution with just three standardized moments and a count. The concept of a rainfall phase space opens new avenues for comparative studies, model evaluation, and potentially for improving quantitative precipitation estimation in remote sensing and numerical weather prediction. Future work is proposed to (1) validate the approach in additional climate zones and for solid precipitation, (2) integrate the four‑tuple into data‑driven forecasting models, and (3) explore the relationship between phase‑space coordinates and atmospheric dynamics such as instability indices, vertical wind shear, and cloud microphysics.