Run arithmetic: co-add and co-subtract
Two runs of the same measurement add to one run with twice the statistics; a laser-on run minus its laser-off reference isolates the photo-excited signal. Asymmetry does both at the raw-count level — it sums (or subtracts) the detector histograms and reduces afterwards, never averaging the finished asymmetry curves. The combined row that appears in the Data Browser is a first-class run: it carries real histograms, so it can be regrouped, deadtime-corrected, count-fitted, and Fourier/MaxEnt-transformed exactly like a loaded run.
Both actions live in the Data Browser context menu. Select two or more runs and choose Co-add Selected; or select one run and choose Subtract Reference Run… to pick the reference to remove. “Separate Combined” restores the constituents.
Why count-level, not curve-level
The asymmetry and its error must see the total statistics. Summing counts first and reducing once gives the correct pooled Poisson error; averaging two reduced curves does not, unless the two runs happen to have identical statistics. For forward/backward totals \((F_1, B_1)\) and \((F_2, B_2)\) the count-sum route reduces \((F_1+F_2,\, B_1+B_2)\) to a single Poisson error, while the curve-mean route forms \(\tfrac12\sqrt{\sigma_1^2 + \sigma_2^2}\) — equal to the pooled error only when the runs match.
The gap grows with the imbalance. Co-adding a low-statistics run with one carrying ten times the events, the old curve-mean error bar over-estimates the combined error by 53 % relative to the correct pooled value; for two equal runs the two routes agree to better than 0.1 %. The asymmetry value is unchanged at \(\alpha = 1\) (both routes are linear in the counts there), but any nonlinear correction — deadtime, an \(\alpha \neq 1\) balance, a background subtraction — only comes out right when it acts on the summed counts. This is why the combined row keeps its histograms rather than a finished curve.
Co-add
Counts add bin-by-bin per detector; detectors whose time-zero differs (PSI
multi-\(t_0\) data) are aligned to a common bin before summing, as for a
single run’s grouping. Good frames accumulate, so the deadtime normaliser sees
the combined exposure. Temperature and field are reported as the
event-weighted mean over the constituents (weighted by good frames, the
defensible summary of an inhomogeneous group), and the spread is recorded under
temperature_spread / field_spread so a group spanning, say, 4.0(1)–10.0(1) K
is visible rather than hidden behind a single averaged number. Period-mode
(red/green) runs sum per period, preserving the two-period structure for the
G∓R reduction.
Co-subtract (reference run)
A reference subtraction forms
per detector bin \(i\), where \(s\) is the good-frame ratio sample/reference — the same exposure scale the background-run correction uses, so two runs of unequal beam time are matched before subtraction. The two count spectra are independent, so the errors add in quadrature,
and the reduced asymmetry of the difference is formed from these propagated errors rather than from a Poisson assumption that no longer holds. Bins that go negative after subtraction are unphysical as expected counts; their count is recorded in the combined run’s provenance so an over-subtraction is not silent.
WiMDA performs the same count-level subtraction (cosign = -1) but leaves the
errors untouched; Asymmetry propagates them (see the porting study). WiMDA also
keeps the master run’s temperature and field on a co-add, where Asymmetry
event-weights them and records the spread.
When to use which.
Co-add — the same physical measurement repeated for more statistics. The result is one higher-statistics run; reduce, fit, or transform it normally.
Co-subtract (reference run) — remove a signal carried by a separate exposure: laser-OFF from laser-ON in photo-μSR, or a known reference run. Frame-scaled, errors add in quadrature.
Background-run correction (a reduction step, not a combined row; see backgrounds) — subtract a scaled background as part of reducing one run, when the background is a steady detector floor rather than a co-measured signal. It shares the same count-level subtraction arithmetic with co-subtract but does not create a new dataset.
Provenance and projects
The combined run records its constituents (combined_from), the operation and
per-constituent weights/scales, the time-zero alignment, and any negative-count
tally under metadata["combination"] — mirroring the simulation provenance
block. Projects store only the definition (the source runs and the operation),
so reloading a .asymp recomputes the combined row from its sources through
the same count-level path; combined curves saved before this correction will
recompute to the statistically correct values on load.
References
S. J. Blundell, R. De Renzi, T. Lancaster, and F. L. Pratt, Muon Spectroscopy: An Introduction (Oxford University Press, Oxford, 2022).