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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,Subtract the part predicted from old noise by definingDirect multiplication gives $\mathrm{Cov}(e,\nu )=0$ and ${\tau }^2=\mathrm{Var}\nu =s-r^{\mathsf{T}}{R}^{-1}r>0$, a Schur complement. Thus the invertible residual transformation separates the weighted residual objective into its old term and one scalar innovation term: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.^[ref]

An original scalar example observes the same pressure twice, with unit error variances and error correlation $\rho \in (-1,1)$. Then $H=a=R=s=1$, $r=\rho$, soAt $\rho =0.8$, 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.