= Correlated position errors of pressure taps
\b[Correlated position errors of pressure taps] occur when uncertainty in one tap coordinate is statistically related to uncertainty in another. A shared origin, common scale or common image transformation can couple all inferred positions. Treating every tap as independent can either overestimate or underestimate the uncertainty in a reconstructed <pressure> load, depending on the sensitivities and their signs.
For fixed reconstruction weights $w_i$ and a differentiable <pressure coefficient> profile, the linearized position contribution is
$$
\delta Q\simeq\sum_i b_i\varepsilon_i,\qquad b_i=w_iC_p'(\xi_i),\qquad
u_Q^2=\mathbf b^T\Sigma_\xi\mathbf b.
$$
Here $\varepsilon_i$ is dimensionless coordinate error and $\Sigma_\xi$ its <covariance matrix>. The general covariance propagation law is the same one used for other correlated inputs in a <measurement model>.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3, covariance terms in uncertainty propagation]{ref}
An original two-tap example takes equal coordinate standard uncertainties $s=0.002$ and <correlation coefficient> $r=0.9$. With sensitivities $\mathbf b=(1,-1)$,
$$
u_Q^2=2s^2(1-r),\qquad u_Q\simeq0.0008944.
$$
Discarding correlation gives $u_Q=\sqrt2s\simeq0.002828$, which is too large here. For $\mathbf b=(1,1)$ instead, $u_Q=\sqrt{2s^2(1+r)}\simeq0.003899$; the same independence assumption is now too small. Positive correlation alone does not determine the direction of the error in the output uncertainty.
A <common-origin pressure-tap position error> can be written $\varepsilon_i=\delta+\eta_i$, with an independent shared offset of variance $s_0^2$ and independent residual errors of variances $s_i^2$. Then
$$
u_Q^2=s_0^2\left(\sum_i b_i\right)^2+\sum_i b_i^2s_i^2.
$$
The common contribution cancels to first order when the sensitivities sum to zero. Repeatedly reading the same misplaced coordinate origin does not remove its uncertainty. <Pressure-tap position calibration> must therefore retain shared reference information, not just a list of per-hole scatters. If the coordinates also determine the <quadrature> weights or surface geometry, their changes belong in the full sensitivity calculation; the fixed-weight formula alone is incomplete.
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