Calibration and the measurement pipeline

A recorded number is the beginning of a measurement

A detector file records indications: digital numbers, channel assignments, timestamps and other instrument outputs. The measurand is the quantity intended to be measured, such as a source count rate in a specified aperture and band. Defining it requires more than naming the object. State the observing interval, region, wavelength or energy range, and how background and instrumental effects enter.

Calibration establishes relationships between reference quantity values and indications, with their uncertainties, and uses those relationships to obtain a measurement result. This follows the International Vocabulary of Metrology distinction between calibration, adjustment of an instrument, and verification against a requirement. Running a correction program does not by itself certify that the resulting scientific interpretation is valid.

A pipeline is an ordered set of processing steps. Its science values, uncertainty information, quality flags and reference-file identities together describe what happened. A display image is only one representation of that record. A flagged pixel can retain a numerical value while being unsuitable for a particular extraction.

This lesson uses two original synthetic instruments. Their values are supplied teaching inputs, not Hubble or Chandra data. Real pipelines require instrument-specific documentation, references and checks.

The order follows the measurement model

Consider one linear, unsaturated pixel whose raw indication D is measured in analogue-to-digital units, abbreviated ADU. Let B be its additive electronic bias in ADU, Q its accumulated dark contribution in ADU, f a dimensionless relative response, and t the exposure duration in seconds. Define the reference-response rate r by D = B + Q + ftr. Then r = (D − B − Q)/(ft).

Here r has units of reference-response ADU s⁻¹. It is not yet a photon rate or physical energy flux. Those conversions need further information such as gain, collecting area, throughput, spectral response and the extraction definition. Q is already the dark contribution accumulated during this exposure; a dark rate would first require multiplication by the appropriate duration.

Supply D = 20 ADU, B = 4 ADU, Q = 2 ADU, f = 0.5 and t = 10 s. The signal remaining after additive corrections is 14 ADU, so r = 14/(0.5 × 10) = 2.8 reference-response ADU s⁻¹. Dividing D by f and then subtracting the unscaled B and Q gives (40 − 4 − 2)/10 = 3.4 in the same nominal output units. It implements a different model.

Operations can sometimes be rearranged, but their terms must transform too. Writing D/f − B/f − Q/f preserves the original numerator divided by f. Writing D/f − B − Q does not. Correct order is therefore a consequence of what each term means, rather than a universal list that can be copied between instruments.

A corrected pixel still has an observing context

The WFC3 UVIS handbook provides a concrete instrument example: its documented processing includes bias and dark corrections before flat-field division, alongside error and data-quality arrays. A flat field characterizes relative sensitivity; it does not identify all compromised pixels or supply every correction needed for every astronomical measurand.

The handbook also explains a geometric distinction. Its flat-fielded images are made uniform for uniform illumination, while point-source aperture photometry on those products requires the appropriate pixel-area treatment. A correction suitable for one product and measurand should not be silently reused for another. These are documented instrument conditions, not extra numerical assumptions in our one-pixel toy.

Reference selection is part of the result. The Calibration References Data System uses versioned rules and observing configuration to select reference files. Retaining the actual context and references makes the calibration choices inspectable; asking for whatever is considered best at a later date can change the processing.

Useful intermediate records retain units, reference identities, processing decisions, masks and uncertainty information. They let an analyst ask whether an anomalous source value arose in the recorded signal, a correction, a geometry conversion or the subsequent model.

A response is sometimes a matrix rather than a scalar

Suppose x = (x1, x2) contains expected incoming photon numbers in two true bands during one fixed exposure. Let y contain the two recorded channel counts and let b = (5, 7) be known background count means. Define E[y] = Rx + b, with output channels as rows and input bands as columns. Use the dimensionless response R = [[0.7, 0.1], [0.2, 0.6]].

Each column gives the probabilities that an incoming photon in that band is registered in either output channel. The column sums are 0.9 and 0.7; the missing probabilities represent losses. A photon from one true band can appear in the other recorded channel. Independent division by two diagonal efficiencies would discard that redistribution.

For the supplied incoming means x = (100, 40), the source channel means are (74, 44), and the total means are (79, 51). The 140 incoming photons in expectation produce 118 registered source photons in expectation. These are model means, not a declaration that every realization has fractional-free or identical counts.

The Chandra spectral-fitting reference describes the corresponding forward-fitting idea: predict detector-channel counts from a physical source model through instrumental response, then compare predictions with observed counts under a chosen statistic. In real spectroscopy the effective area and redistribution have distinct roles and energy dependence. Our small matrix illustrates mixing without claiming to reproduce that instrument.

Algebraic recoverability and stability are separate questions

For this first matrix, R⁻¹ = [[1.5, −0.25], [−0.5, 1.75]]. Applying it to the exact background-subtracted means (74, 44) recovers (100, 40). Dividing by diagonal entries alone would instead give about (105.7143, 73.3333). An exact inverse addresses mixing; it does not remove random counting variation or uncertainty in the response.

Now consider the different response Rpoor = [[0.505, 0.495], [0.495, 0.505]]. It predicts source means (70.3, 69.7) for the same x. Change those recorded means by only (+0.1, −0.1), to (70.4, 69.6). Exact inversion now returns (110, 30): a small change in the difference between channels becomes a much larger change in the inferred band balance.

For a fixed invertible matrix, its condition number in a specified norm bounds relative solution changes from relative right-hand-side changes. In the maximum-row-sum, or infinity, norm, κ∞ is 1.8 for the first matrix and 100 for Rpoor. This is a worst-case sensitivity measure, not a prediction that every perturbation reaches the bound. It describes the problem as well as the arithmetic used to solve it.

Real observed counts are random, backgrounds and responses may be uncertain, and an unconstrained inverse can produce negative estimates. Keeping the detector data supports an appropriate forward likelihood and physical constraints instead of pretending that inverse processing creates noiseless true-band observations. The next lesson follows uncertainty and covariance through such a transformation.

Worked examples

Try the reasoning

1. A dark reference is supplied in ADU s⁻¹. Can it be subtracted directly from a raw ADU value?

   Show the answer and reasoning

   No; obtain the accumulated contribution for the relevant duration first.

   Additive terms must represent the same quantity in compatible units.

2. Does applying a flat field make a quality mask unnecessary?

   Show the answer and reasoning

   No.

   Relative response and pixel suitability describe different aspects of the measurement.

3. What does a response-column sum of 0.7 mean in the supplied exclusive-output toy?

   Show the answer and reasoning

   The registration probability is 0.7 and the loss probability is 0.3 for that input band.

   Columns refer to input bands; their entries cover the two possible recorded channels.

4. Does an invertible response guarantee a stable reconstruction?

   Show the answer and reasoning

   No.

   A poorly conditioned inverse can amplify small data changes even with exact algebra.

What to carry forward

• Define the measurand, units and correction model before processing an indication.
• Retain science values together with quality, uncertainty and calibration provenance.
• Channel mixing and conditioning determine what an inverse can recover reliably.