Polarimetric observables#

Dual-polarization weather radars transmit and receive orthogonal linear polarizations (conventionally horizontal h and vertical v). A handful of combinations of the resulting amplitudes and phases form the standard polarimetric observables implemented in rustmatrix.radar. Definitions, sign conventions, and units below follow [Bringi and Chandrasekar, 2001] and [Doviak and Zrnić, 1993].

The scattering matrix#

For a single particle at a fixed geometry, the far-field scattered electric field is related to the incident field by the \(2 \times 2\) amplitude scattering matrix \(\mathbf{S}\):

\[\begin{split} \begin{pmatrix} E_h^s \\ E_v^s \end{pmatrix} = \frac{e^{ikr}}{r} \begin{pmatrix} S_{hh} & S_{hv} \\ S_{vh} & S_{vv} \end{pmatrix} \begin{pmatrix} E_h^i \\ E_v^i \end{pmatrix}. \end{split}\]

The \(S_{ij}\) are complex — they carry both amplitude and phase. rustmatrix returns them from the T-matrix solver at any specified incident / scattered geometry via rustmatrix.scatter.amplitude_matrix.

Polarimetric observables come in two flavours:

Back-scatter quantities (\(Z_h\), \(Z_{dr}\), \(\rho_{hv}\), \(\delta_{hv}\), LDR) use \(\mathbf{S}\) at the 180° back-scatter geometry and integrate \(|S|^2\)-type quantities over the PSD.
Forward-scatter quantities (\(K_{dp}\), \(A_h\), \(A_v\)) use \(\mathbf{S}\) at the 0° forward-scatter geometry and integrate \(\Re\) / \(\Im\) linear in \(S\).

Switch geometries with s.set_geometry(geom_horiz_back) / s.set_geometry(geom_horiz_forw). The T-matrix itself is cached on the Scatterer, so the switch is cheap.

Back-scatter observables#

Reflectivity factor \(Z_h\)#

\[ Z_h = \frac{\lambda^4}{\pi^5 |K_w|^2} \int |S_{hh}^{b}(D)|^2 \, N(D)\, dD \]

Units: mm⁶ m⁻³ (linear); dBZ after \(10 \log_{10}\). The dielectric factor \(|K_w|^2\) for water at the radar band is tabulated in rustmatrix.tmatrix_aux.K_w_sqr. Use radar.refl(s, h_pol=True).

Differential reflectivity \(Z_{dr}\)#

\[ Z_{dr} = 10 \log_{10} \frac{Z_h}{Z_v}. \]

Positive \(Z_{dr}\) means oblate scatterers aligned with their long axis horizontal (the equilibrium-drop configuration). Rain at C-band gives \(Z_{dr} \approx 0.3\)–3 dB; pristine ice columns give slightly positive values; randomly-tumbling aggregates give ~0 dB.

Co-polar correlation \(\rho_{hv}\)#

\[ \rho_{hv} = \frac{ \left| \int S_{vv}^b S_{hh}^{b*} \, N(D)\, dD \right| }{ \sqrt{\int |S_{hh}^b|^2 \, N\, dD \cdot \int |S_{vv}^b|^2 \, N\, dD} }. \]

Bounded by \([0, 1]\). Values near 1 mean a uniform population; drops below 0.97 indicate mixed-phase or irregular scatterers. A key discriminator between meteorological and non-meteorological echo.

Differential backscatter phase \(\delta_{hv}\)#

\[ \delta_{hv} = \arg \int S_{hh}^b S_{vv}^{b*} \, N(D)\, dD. \]

Non-zero only when scatterers are large enough that the Rayleigh approximation fails — so a resonance fingerprint at C-band for \(D \gtrsim 5\) mm. The HydroMix tutorial shows it for mixed rain + ice.

Linear depolarization ratio LDR#

\[ \mathrm{LDR} = 10 \log_{10} \frac{\int |S_{hv}^b|^2 \, N\, dD}{\int |S_{hh}^b|^2 \, N\, dD}. \]

Requires non-zero cross-polar response, i.e. non-trivial canting. LDR is small (< −25 dB) for rain and rises sharply in the melting layer and for oriented ice [Kumjian, 2013, Ryzhkov et al., 2005].

Forward-scatter observables#

Specific differential phase \(K_{dp}\)#

\[ K_{dp} = \frac{180}{\pi}\, \lambda \, \Re \!\int \! \left[ S_{vv}^f(D) - S_{hh}^f(D) \right]\, N(D)\, dD. \]

Units: ° km⁻¹. Positive for horizontally-oriented oblate particles (rain), near-zero for spheres and tumbling ice, slightly negative for vertically-oriented crystals. \(K_{dp}\) is immune to attenuation and calibration bias — the workhorse for rain-rate retrieval.

Specific attenuation \(A_h\), \(A_v\)#

\[ A_i = 8.686 \cdot 10^{-3} \, \lambda \, \Im \! \int \! S_{ii}^f(D) \, N(D)\, dD. \]

Units: dB km⁻¹. Rises sharply at the higher radar bands (Ka, W) — see the radar-band sweep tutorial.

Sign and geometry conventions#

rustmatrix follows Bringi & Chandrasekar [Bringi and Chandrasekar, 2001]:

Horizontal polarization is h, vertical is v.
Equilibrium drop axis ratio is reported as \(h/v\) ≥ 1; the Scatterer(axis_ratio=...) argument expects \(v/h\) (the value the Mishchenko code wants), so rain scripts use axis_ratio = 1.0 / dsr_thurai_2007(D).
Back-scatter geometry is geom_horiz_back from rustmatrix.tmatrix_aux (the BSA convention — incident antenna frame). Forward-scatter is geom_horiz_forw.
Doppler velocity is positive downward (toward the radar for a vertically-pointing profiler) throughout rustmatrix.spectra — matching the convention in [Kollias et al., 2002, Zhu et al., 2023].

Relationship to the scattering cross-sections#

Back-scatter cross-section \(\sigma_b\) and total scattering \(\sigma_{sca}\) are also available via rustmatrix.scatter.radar_xsect and scatter.sca_xsect — useful for Mie parity checks and for feeding spectra.SpectralIntegrator a custom \(\sigma_b(D)\).