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Balances and density gradients. The first two relations describe horizontal geostrophic balance between pressure gradient and Coriolis force accelerations. The vertical relation is hydrostatic balance between the vertical pressure force and gravity. They describe a slowly varying large-scale interior circulation, excluding the frictional surface/bottom layers and large ageostrophic accelerations.

Differentiate the horizontal balances vertically and use $p_z=-\rho g$, treating $f,{\rho }_0$ as vertically constant:These are the oceanic thermal wind relations. With no motion at $z=-H$, the exact surface relations involve the depth-integrated density gradients. If the measured density is representative of the vertical mean, or if those gradients are approximately uniform with depth,Equivalently these hold exactly for the horizontal gradients of $\bar{\rho }=H^{-1}{\int }_{-H}^0\rho dz$ in this fixed-depth reference model. A single arbitrary surface-density gradient does not otherwise determine a vertical integral. In the Northern Hemisphere the inferred surface current runs along horizontal isopycnals with denser water on its left when the no-motion reference and this vertical-gradient approximation apply.

For a steady density pattern and a ship whose actual track velocity is $v_s(\cos \theta ,\sin \theta )$,If the field changes during the survey an Eulerian time derivative must also be included. Here $H=\Delta z>0$ is the upward separation from reference depth to surface, so its sign must not be reversed.

Inversion and velocity estimate. Let $C=f{\rho }_0{v}_s/(gH)$ and denote the two ship measurements by $y_{\theta },y_0$, with the east-going ship assigned the subscript zero. The observation system and its inverse areIt is invertible for $\sin \theta \ne 0$. For two measurements and two unknowns this inverse is also the stated least-squares estimate; the nearly parallel-track geometry explains its poor conditioning at small angle.

The supplied parameters give $C=5.654\times 10^{-5}$ in units converting velocity to the measured density tendency. With $y_{\theta }=1.9\times 10^{-5}$ and $y_0=1.8\times 10^{-5}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$,The current therefore has eastward and southward components. Signs follow from the positive-upward, east–north coordinate convention and the specified no-motion reference.

Uncertainty and track design. The two-track thermal-wind covariance exposes the geometrical limitation. The stated error already applies to the difference of successive density observations. Dividing by the sampling interval, rather than adding an extra factor of $\sqrt{2}$, gives ${\sigma }_y=7.2\times 10^{-4}/3600=2.0\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. Assume independent errors of that size for the two ships, as in the supplied covariance model. ThenAt $5^{\circ }$,The correlation is positive,Both estimates depend on the east-going ship; its positive measurement error lowers $v_g$ and also lowers $u_g$, explaining the sign of the covariance. If the other ship goes north instead, $\theta =90^{\circ }$ gives orthogonal observations: $u_g=y_{90}/C$ and $v_g=-y_0/C$. Both standard errors become $0.00354\mathrm{m}{\mathrm{s}}^{-1}$ and their covariance vanishes. The north-going track improves the poorly constrained zonal component by about a factor of sixteen, while leaving the meridional precision unchanged under the same noise assumptions.