Particle size distributions#

A radar observable is an integral over the number concentration of scatterers per unit volume per unit diameter, \(N(D)\). rustmatrix.psd provides the standard analytical forms plus a generic binned distribution, all compatible with PSDIntegrator’s cached tabulation.

Exponential (Marshall–Palmer)#

[Marshall and Palmer, 1948] observed that stratiform rain drop-size distributions are well approximated by

\[ N(D) = N_0 \exp(-\Lambda D), \]

with \(N_0 \approx 8000\) mm⁻¹ m⁻³ and \(\Lambda\) set by the rain rate. The two-parameter exponential is the simplest non-trivial PSD and is wired up as psd.ExponentialPSD(N0=..., Lambda=...).

Use it when you want a tractable stress test — it has no free shape parameter, so convergence and band dependence are easy to isolate. The radar-band sweep tutorial uses it for exactly that reason.

Gamma (Ulbrich)#

Real rain spectra show a roll-off at small diameters that the exponential cannot capture. [Ulbrich, 1983] proposed

\[ N(D) = N_0 D^\mu \exp(-\Lambda D), \]

adding a shape parameter \(\mu\) — positive for convective rain, negative for drizzle-dominated distributions. Implemented as psd.UnnormalizedGammaPSD(N0, mu, Lambda).

The three Ulbrich parameters are strongly correlated across rain events, so rustmatrix also provides the normalised form below which disentangles concentration from shape.

Normalised gamma (Testud / Bringi–Chandrasekar)#

The normalised-gamma PSD [Bringi and Chandrasekar, 2001, Testud et al., 2001] is parameterised by quantities that are roughly independent across rain events:

\(D_0\) — median volume diameter (mm),
\(N_w\) — normalised intercept, constant for a Marshall–Palmer distribution,
\(\mu\) — dimensionless shape parameter.

\[ N(D) = N_w \, f(\mu) \, \left(\frac{D}{D_0}\right)^\mu \exp\!\left[-(3.67 + \mu)\,\frac{D}{D_0}\right], \]

where \(f(\mu)\) is a normalisation that keeps \(N_w\) fixed across \(\mu\). This is the form used in the gamma-PSD rain tutorial and throughout the operational-radar literature. Constructed as psd.GammaPSD(D0=..., Nw=..., mu=...).

Binned / empirical#

For disdrometer observations, reanalysis output, or any case where you have \(N\) in arbitrary diameter bins, use psd.BinnedPSD(bin_edges, N) — it interpolates linearly within bins and evaluates to zero outside. The standard PSDIntegrator machinery works unchanged.

Numerical integration#

psd.PSDIntegrator:

On init_scatter_table(s), evaluates \(\mathbf{S}(D)\) and \(\mathbf{Z}(D)\) at num_points diameters between \(D_\min\) and \(D_\max\). This is the parallel Rust kernel — the only expensive step.
For each PSD assigned to s.psd, integrates the cached tables against \(N(D)\) by trapezoidal rule.
Any number of different PSD shapes can be evaluated from the same cached table — swap s.psd = psd.GammaPSD(...) and re-read the observables at a cost near zero.

Tips:

D_max: set a few times the expected largest drop. Too small truncates the tail and under-estimates \(Z_h\) at C-band and below; too large wastes quadrature points on near-zero contributions.
num_points: 64 is almost always enough for rain; oriented ice at the Mie-resonance bands may want 128.
geometries: pass both back-scatter and forward-scatter in the same tuple so the table is built once for all observables you need.