= Two-track thermal-wind covariance
For a steady density field sampled by two ships at angles $\theta$ and zero, write $\mathbf y=C\bigl(\begin{smallmatrix}\sin\theta&-\cos\theta\\0&-1\end{smallmatrix}\bigr)(u,v)^{\mathsf T}$. Independent measurement errors of variance $\sigma_y^2$ give
$$
\mathsf S_{uv}=\frac{\sigma_y^2}{C^2}\begin{pmatrix}(1+\cos^2\theta)/\sin^2\theta&\cot\theta\\\cot\theta&1\end{pmatrix}.
$$
Nearly parallel tracks poorly constrain $u$ and induce positive component covariance for $0<\theta<\pi/2$. Orthogonal tracks remove this covariance and give equal component precision under the stated independent-noise model.
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