Multi-Doppler Wind Synthesis (Stage 1)

Stage 1 is a gridpoint-independent, overdetermined, weighted least-squares dual-Doppler horizontal wind retrieval (u, v), streamed through an accumulator that mirrors the ScalarGridAccumulator model and carries a preserved per-component uncertainty. It follows CEDRIC appendix F (Miller & Anderson, 1991), §1, "Cartesian components of motion from radial velocities".

It is not vertical velocity, mass continuity, the variational solution, fallspeed/advection correction, dealiasing, or spectral-grid integration — those are stage 2 / upstream QC / follow-ons.

Observation equation

For each contributing gate m (from any radar or sweep) reaching a grid point, with the vertical term dropped (W neglected — the two-equation dual-Doppler approximation):

\[v_{r,m} \approx u\, a_m + v\, b_m, \qquad a_m = \sin(\mathrm{az}_m)\cos(\mathrm{el}_m), \qquad b_m = \cos(\mathrm{az}_m)\cos(\mathrm{el}_m)\]

where $\mathrm{az}_m$ is the azimuth clockwise from +y (true north) and $\mathrm{el}_m$ the refraction-corrected elevation. Both are the effective line-of-sight angles from the gate's radar origin to the grid point — exactly the gridpt_az/gridpt_el the shared gate-geometry kernel already computes for the scalar gridding path. (The deferred three-unknown extension restores the third column $c_m = \sin(\mathrm{el}_m)$, the coefficient of W.)

Weighted least squares → streamed normal system

Minimizing $\sum_m w_m (u\,a_m + v\,b_m - v_{r,m})^2$ with the existing gridding weight $w_m = \text{range\_weight}\cdot\text{angle\_weight}$, each gate contributes a rank-1 update to a per-gridpoint normal system. Writing $g_m = [a_m, b_m]$:

\[A^\top W A \mathrel{+}= w_m\, g_m g_m^\top, \qquad A^\top W b \mathrel{+}= w_m\, v_{r,m}\, g_m, \qquad M_2 \mathrel{+}= w_m^2\, g_m g_m^\top\]

No per-radar pre-averaging and no retention of individual gate rows: multi-radar synthesis is simply continuing to grid more sweeps into the same accumulator. All updates are in the Cartesian basis. The symmetric $A^\top W A$ and $M_2$ are stored packed ([S_aa, S_ab, S_bb]).

Per-gridpoint solve and covariance

At finalize, each grid point is solved independently:

\[D = S_{aa}S_{bb} - S_{ab}^2, \qquad C_i = (A^\top W A)^{-1} = \frac{1}{D}\begin{bmatrix} S_{bb} & -S_{ab} \\ -S_{ab} & S_{aa}\end{bmatrix}, \qquad \begin{bmatrix}u\\v\end{bmatrix} = C_i \begin{bmatrix}S_{av}\\S_{bv}\end{bmatrix}\]

The covariance is the equal-variance "sandwich", with $M_2$ the second accumulated matrix:

\[\mathrm{cov} = \sigma_{vr}^2\, C_i\, M_2\, C_i\]

Each gate's radial velocity is modeled with the same variance $\sigma_{vr}^2$ (diagonal, geometry-independent — the standard "1 m/s, diagonal" assumption); the beam-power×range weights are interpolation/representativeness weights, not a noise model. $\sigma_{vr}^2$ (velocity_variance) is a decoupled scalar that only rescales to physical units; setting it to 1 gives normalized (CEDRIC USTD/VSTD) σ. At equal gate weights $M_2 = A^\top W A$ and $\mathrm{cov} = \sigma_{vr}^2 C_i$, recovering the unweighted CEDRIC Eq-13 $\sum g^2$ form. As the grid point approaches the baseline $D \to 0$ and the covariance blows up — the direct baseline-instability detector.

Output frame

The Cartesian solution is rotated into an active orthonormal output frame $R =$ rotation_at(frame, x, y, z):

\[\begin{bmatrix}c_1\\c_2\end{bmatrix} = R\begin{bmatrix}u\\v\end{bmatrix}, \qquad \mathrm{cov}_{\text{frame}} = R\,\mathrm{cov}\,R^\top, \qquad \sigma_i = \sqrt{(\mathrm{cov}_{\text{frame}})_{ii}}\]

Because R depends only on the grid point (not the gate), it factors out of every accumulated sum, so accumulation stays Cartesian and the frame is a pure finalize-time view — one accumulator can be finalized in several frames without re-gridding. Nothing hardcodes U=zonal/V=meridional. Stage 1 ships only the identity CartesianFrame ((c_1,c_2)=(u,v)); plane-aligned (RHI), polar/cylindrical (hurricane), and 3D coplane frames are pure additions later.

Quality control (non-destructive)

Quality is judged solely by the per-component normalized uncertainty (CEDRIC DTEST style), evaluated in the output frame, against the independent thresholds max_sigma[1]/max_sigma[2] — one component can be well-determined while the other is not. There is no radar-count and no gate-count threshold: the adequacy of the geometry (how many radars, crossing angle, baseline proximity) is already expressed in σ.

Masking is non-destructive: where the system is solvable, comp1, comp2, sigma1, sigma2 are always written, and a separate quality_flag records which σ threshold(s) failed — so stage 2 can recover those points using global information. Two intrinsic (non-tunable) states blank the components to fill_value:

The singular guard is scale-relative ($D \le \varepsilon\, S_{aa}S_{bb}$): a numerically rank-1 single-look system is caught (rather than producing a falsely-confident $(0,0)$ with $\sigma = 0$), while a physically near-baseline pair keeps a positive $D$ and survives, its instability surfacing as a large σ caught by the thresholds.

Stage-2 handoff

The WindGridAccumulator is the stage-2 handoff artifact: its AtWA/AtWb/M2/weight_total/counts are exactly what a variational stage needs to recover masked baseline points. finalize_wind produces the single-stage product; the accumulator is the richer input for stage 2. Because accumulation is linear, merge_wind_accumulators! across files and radars is an exact elementwise sum.