Beam pattern × scene integration#

The closed-form beam-broadening width \(\sigma_\mathrm{beam} = |\mathbf{u}_h|\,\theta_b / (2\sqrt{2\ln 2})\) [Doviak and Zrnić, 1993] assumes every scatterer inside the beam sees the same reflectivity and the same wind — a uniform resolution volume. When the scene is not uniform (deep convection with sub-beam-scale reflectivity cells, strong shear across the beam, a scanning antenna over a horizontally-structured storm), the closed form breaks and you have to integrate the beam pattern against the scene explicitly. That is what rustmatrix.spectra.beam does.

What goes wrong with closed-form \(\sigma_\mathrm{beam}\)#

The Doviak–Zrnić result assumes

a Gaussian beam pattern \(G(\theta)\) of angular full-width-half- maximum \(\theta_b\),
a spatially uniform radar reflectivity and PSD inside the beam,
a single horizontal wind \(\mathbf{u}_h\) across the beam,

and then derives the Doppler-width contribution from the in-beam distribution of line-of-sight velocities as a closed-form Gaussian.

If any of the three assumptions fails, the measured spectrum is not \(\mathcal{N}(w, \sigma_t^2 + \sigma_\mathrm{beam}^2)\); it’s a \(G\)-weighted mixture of different reflectivities and velocities. The wind × turbulence sensitivity tutorial shows closed- form \(\sigma_\mathrm{beam}\) agreeing with the explicit integration under uniform conditions; the beam × scene tutorial shows the case where it fails.

The integration#

For a beam pointing in direction \(\hat{\mathbf{n}}\) with angular pattern \(G(\theta)\), and a scene with reflectivity field \(Z(\mathbf{x})\) and wind field \(\mathbf{u}(\mathbf{x})\), the received spectrum at Doppler velocity \(v\) is

\[ P(v) = \int G(\theta)^2 \, Z(\mathbf{x}(\theta)) \, \mathcal{K}\!\left[v - \hat{\mathbf{n}}\cdot\mathbf{u}(\mathbf{x}(\theta))\right] \, d\Omega, \]

with \(\mathcal{K}\) the turbulence + fall-speed kernel at each beam angle. rustmatrix.spectra.beam discretises the beam angles on a Gauss–Legendre grid, evaluates the scene at each angle, runs the per-angle spectrum through SpectralIntegrator, and sums weighted by \(G^2\).

Beam patterns#

Two patterns are built in:

GaussianBeam(theta_b, ...) — the classic Gaussian analytic pattern, matched in FWHM to theta_b. Sensible default.
AiryBeam(theta_b, ...) — the diffraction pattern from a circular aperture; has non-zero side-lobes.
TabulatedBeam(theta, gain) — wrap a measured antenna pattern as a 1-D (angle, gain) table.

Scene shortcuts#

Building a full 3-D reflectivity + wind scene is tedious. For the common case of vertically pointing radars over a 2-D horizontal reflectivity map, spectra.beam ships helpers:

marshall_palmer_scene(Z_map, N0=8000) — turn a dBZ map into a per-pixel Marshall–Palmer PSD.
convective_cell_scene(...) — parameterised Gaussian cells for sensitivity studies; the NB14 tutorial drives an interactive slider over cell spacing.

When to reach for this#

Use spectra.beam when any of:

your beam covers horizontally-structured reflectivity (convective cells, stratiform-convective transition, the edge of a storm);
you have strong shear across the beam (upper-level jets, gust fronts, scanning geometries with large azimuth steps);
you need to reproduce a measured antenna side-lobe contamination signature.

For uniform stratiform rain with a narrow beam, the closed-form \(\sigma_\mathrm{beam}\) in SpectralIntegrator is faster and sufficient.