 Symmetric-overlap normalization drift

Symmetric-overlap normalization drift is a time-dependent change in the multiplicative scale of two overlap readings. The symmetric log-overlap ratio cancels one shared scale, but sequential settings need not experience that same scale.

Let the stable optical contributions at the two radius settings be $F_{+}$ and $F_{-}$, and letwhere $a(t)=\log A(t)$. The ideal log-ratio is $D_0=F_{+}-F_{-}$. A positive-setting reading at time 0 followed by a negative-setting reading at time $\Delta$ instead gives $D_{\mathrm{obs}}=D_0+a(0)-a(\Delta )$. A drift in normalization can therefore mimic a signed radius mismatch.

An original four-reading design uses settings $+,-,-,+$ at equally spaced times $0,\Delta ,2\Delta ,3\Delta$. FormThe scale contribution vanishes exactly when $a(t)$ is linear in time: both settings have the same mean observation time. This cancels linear drift in logarithmic scale, not arbitrary drift in every measured quantity.

If $a(t)=a_0+bt+c{t}^2$, the remaining drift contribution is $2c{\Delta }^2$. For stipulated $D_0=0.05$, $b=0.02$ per time unit and $\Delta =1$, a simple two-reading comparison gives 0.03 under purely linear log-drift, while the four-reading design gives 0.05. Adding $c=0.003$ per squared time unit changes the latter to 0.056. These are analytical timing examples, not measurements from a beam experiment.

With independent equal log-readout variance ${\sigma }^2$, the four-reading estimator has variance ${\sigma }^2$, because it combines four coefficients of magnitude one half. A single two-reading difference has variance $2{\sigma }^2$ at half the acquisition count. Correlated noise and longer acquisition time can change the practical tradeoff.

State-dependent gain in symmetric overlap measurements does not cancel through this ordering. If the scale changes systematically with the radius-setting state, it remains confounded with $F_{+}-F_{-}$. The optical mismatch must also remain stable over the sequence; a changing beam is not merely an electronic scale drift.

NIST controlled mode and polarization matching while calibrating the receiver's electrical response.^[ref] The ordering above is an original drift-control calculation, not an attributed NIST procedure. Test its assumptions with timing records, repeated sequences and independent monitoring before feeding the resulting ratio into the radius inversion.