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 givesThe added matrix has rank one when $a\ne 0$. 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.^[ref]

The update can remove a missing direction only if the new channel actually sees it. More precisely,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 matrixMultiplication 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 $C=\mathrm{diag}(4,1)$, $a=(1,1{)}^{\mathsf{T}}$ and ${\sigma }_a^2=1$. ThenThe variance of the sum drops from $5$ to $5/6\mathrm{kP}{\mathrm{a}}^2$. 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 $a{a}^{\mathsf{T}}/{\sigma }_a^2$ 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.

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.