= Rank-one update of pressure-map information
A <rank-one update of pressure-map information> describes the extra constraint supplied by one independent scalar <pressure> measurement. Suppose the existing <pressure-map information matrix> is $\mathcal I=H^{\mathsf T}R^{-1}H$, and a new <sensor> reports $z=a^{\mathsf T}p+\eta_a$ with known <variance> $\sigma_a^2>0$, independent of the existing errors. Adding its squared residual to the <weighted least squares> objective gives
$$
\boxed{\mathcal I_+=\mathcal I+\frac{aa^{\mathsf T}}{\sigma_a^2}.}
$$
The added <matrix> has <rank> one when $a\ne0$. It is positive semidefinite because its <quadratic form> is $(a^{\mathsf T}v)^2/\sigma_a^2$. Thus the new observation cannot reduce information in any direction under this fixed <linear map> and noise model. The noise normalization is the same <whitening transformation> used to express a correlated residual metric as an ordinary squared norm.<SciPy 1.16.2 documentation>[Covariance.whiten, description and Examples]{ref}
The update can remove a missing direction only if the new channel actually sees it. More precisely,
$$
\ker\mathcal I_+=\ker\mathcal I\cap\{v:a^{\mathsf T}v=0\}.
$$
This follows because a sum of two nonnegative quadratic forms is zero exactly when both are zero. Repeating an existing mixture improves its precision but cannot identify a <pressure> contrast that all mixtures annihilate.
When $\mathcal I$ is positive definite, write $C=\mathcal I^{-1}$. The <Sherman–Morrison formula> yields the updated estimator <covariance matrix>
$$
\boxed{C_+=C-\frac{Caa^{\mathsf T}C}{\sigma_a^2+a^{\mathsf T}Ca}.}
$$
Multiplication by $\mathcal I_+$ verifies the identity directly. For a particular quantity $v^{\mathsf T}p$, its <variance> falls by $(v^{\mathsf T}Ca)^2/(\sigma_a^2+a^{\mathsf T}Ca)$. A small noise level is valuable only together with a useful sensitivity direction.
For an original calculation in kilopascal coordinates, let $C=\operatorname{diag}(4,1)$, $a=(1,1)^{\mathsf T}$ and $\sigma_a^2=1$. Then
$$
C_+=\begin{pmatrix}4&0\\0&1\end{pmatrix}-\frac16\begin{pmatrix}16&4\\4&1\end{pmatrix}
=\begin{pmatrix}4/3&-2/3\\-2/3&5/6\end{pmatrix}.
$$
The <variance> of the sum drops from $5$ to $5/6\,\mathrm{kPa^2}$. The negative off-diagonal entry means the two inferred coefficients now have negatively correlated errors; it does not mean either individual <variance> is negative.
A <correlated-channel update of pressure-map information> needs the new channel's residual after predicting it from old channel noise. Simply adding $aa^{\mathsf T}/\sigma_a^2$ double-counts information when a common amplifier or reference contributes correlated errors. A proposed array experiment uses repeated known loads to estimate sensitivity and noise separately, adds one channel, and compares withheld-load errors with $C_+$. Agreement tests the full assumed model, including independence and unchanged <sensor> response, rather than the <matrix> identity alone.
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