= Correlated-channel update of pressure-map information
A <correlated-channel update of pressure-map information> measures what a new <pressure> channel adds after accounting for noise that it shares with the old channels. The <rank-one update of pressure-map information> for independent noise is a special case. An extra channel may be a precise-looking duplicate of an existing noisy signal and contribute almost no new information.
Write the old and new observations as $y=Hp+e$ and $z=a^{\mathsf T}p+\eta$. Suppose their known joint error <covariance matrix> is positive definite,
$$
\operatorname{Cov}\begin{pmatrix}e\\\eta\end{pmatrix}
=\begin{pmatrix}R&r\\r^{\mathsf T}&s\end{pmatrix},\qquad R>0.
$$
Subtract the part predicted from old noise by defining
$$
z_*=z-r^{\mathsf T}R^{-1}y=b^{\mathsf T}p+\nu,
\quad b=a-H^{\mathsf T}R^{-1}r,
\quad \nu=\eta-r^{\mathsf T}R^{-1}e.
$$
Direct multiplication gives $\operatorname{Cov}(e,\nu)=0$ and $\tau^2=\operatorname{Var}\nu=s-r^{\mathsf T}R^{-1}r>0$, a <Schur complement>. Thus the invertible residual transformation separates the weighted residual objective into its old term and one scalar innovation term:
$$
\boxed{\mathcal I_+=H^{\mathsf T}R^{-1}H+\frac{bb^{\mathsf T}}{\tau^2}.}
$$
For jointly <normal random variables>, uncorrelatedness here also gives independence, and the matrix is the parameter <Fisher information>. For more general zero-mean errors with this fixed covariance, the displayed algebra still describes the <generalized least squares> information matrix. This is a partial <whitening transformation>; the primary covariance interface documents whitening and its identity-covariance check.<SciPy 1.16.2 documentation>[Covariance.whiten, description and Examples]{ref}
An original scalar example observes the same pressure twice, with unit error variances and error correlation $\rho\in(-1,1)$. Then $H=a=R=s=1$, $r=\rho$, so
$$
\mathcal I_+=1+\frac{(1-\rho)^2}{1-\rho^2}=\frac2{1+\rho},
\qquad \operatorname{Var}\widehat p=\frac{1+\rho}{2}.
$$
At $\rho=0.8$, the true estimator variance is $0.9$, whereas an independent-noise calculation would claim $0.5$. At $\rho=-0.8$, the variance is $0.1$: opposite noise contributions can cancel. Correlation can reduce or increase the benefit relative to the independent case; it does not make the correctly computed added information negative.
The effective sensitivity is $b$, not merely the original $a$. If $b=0$ and $\tau^2>0$, the new residual carries no additional pressure information. A <zero-innovation-variance pressure channel> requires separate treatment: division by $\tau^2=0$ is invalid, and the limiting relation may be a redundant duplicate or an exact constraint, depending on $b$.
A proposed array characterization measures simultaneous repeated readings under fixed known loads, estimates the complete cross-channel covariance, and tests reconstruction on withheld repetitions. A common reference offset can survive indefinitely despite more channels. The derivation assumes known, unchanged sensitivities and covariance; estimating them from few samples adds uncertainty not included by this update.
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