Calibration Equation#

This page presents the final calibration formula that produces antenna temperature \(T_A^*\) from raw counts.

The Calibration Equation#

For each spectral channel \(\nu\), dump \(d\), receiver \(r\), array \(a\), and subscan \(s\):

\[T_A^*(\nu, d, r, a, s) = \frac{\big(C_{ON}(\nu,d,r,a,s) - C_{REF}(\nu,r,a,s)\big) \cdot \gamma(\nu,r,a)} {\big(C_{hot}(\nu,r,a) - C_{cold}(\nu,r,a)\big) \cdot t_r^{sig}(\nu)}\]

where:

\(C_{ON}\) — raw counts from the ON-source subscan
\(C_{REF}\) — reference counts (mode-dependent, see Observation Modes)
\(\gamma\) — gain calibration factor (see Calibration Loads)
\(C_{hot}, C_{cold}\) — mean HOT/COLD load counts
\(t_r^{sig}\) — signal sideband atmospheric transmission (see Atmosphere)

Atmospheric Correction#

Only the signal sideband transmission is applied in the denominator. The image sideband contribution is already accounted for in \(\gamma\) through the sideband gain weighting \((g_s x_s + g_i x_i)\). This is the standard single-sideband (SSB) calibration convention.

Implementation: cal-core/src/calibrate.rs → calibrate_full()

Vectorized Implementation#

The inner loop precomputes a combined factor to minimise per-element operations:

\[F(\nu, r, a) = \frac{\gamma(\nu, r, a)} {\big(C_{hot}(\nu, r, a) - C_{cold}(\nu, r, a)\big) \cdot t_r^{sig}(\nu)}\]

Then the calibration reduces to:

\[T_A^*(\nu, d, r, a, s) = \big(C_{ON}(\nu,d,r,a,s) - C_{REF}(\nu,r,a,s)\big) \cdot F(\nu, r, a)\]

Loop order is C → R → A → D → S for row-major cache-friendly access, with the innermost loop (S) achieving stride-1 memory access on the subscan axis.

System Temperature#

The system temperature includes receiver noise, load coupling, and atmospheric contributions:

\[T_{sys} = T'_{rec} + T'_{hot} + (1 - f_{amb}) \Big[ g_s x_s \cdot T_{hot,eff}^{sig} \cdot \big(a_s (1 - t_r^{sig}) - 1\big) + g_i x_i \cdot T_{hot,eff}^{img} \cdot \big(a_i (1 - t_r^{img}) - 1\big) \Big]\]

\(T_{sys}\) is computed per channel, per pixel, per subscan and written to the L1 output.

Implementation: cal-core/src/math/temperature.rs → tsys_formula()

Flag Propagation#

The calibration stage propagates flags from the bad channel mask (bit 0 = BAD_CHANNEL) to the output flags array. All dumps and subscans for a flagged channel receive the flag. The corresponding spectral values are set to NaN.

See /source/developer/flag-system for the complete flag architecture.

Output: CalibrationResult#

The calibration produces a CalibrationResult containing:

Field	Shape	Description
`spectra`	[C, D, R, A, S]	Calibrated \(T_A^*\) (K)
`t_sys`	[C, R, A, S]	System temperature (K)
`t_int`	[S]	Integration time (s)
`t_rec_ssb`	[C, R, A]	Receiver temperature SSB (K)
`gamma`	[C, R, A]	Gain factor
`tau_signal`	[C]	Zenith opacity, signal (Np)
`tau_image`	[C]	Zenith opacity, image (Np)
`t_sky`	[C, R, A]	Sky contribution (K)
`flags`	[C, D, R, A, S]	u16 quality bitmask
`signal_freqs`	[C]	Signal frequencies (Hz)
`image_freqs`	[C]	Image frequencies (Hz)
`pwv`	scalar	Fitted PWV (mm)

Plus scan metadata (source, mode, arrays, pixel names, strategies).

QA Metrics#

Quality metrics are computed per scan and stored as L1 attributes:

Mean and median \(T_{sys}\) across channels
Fraction of flagged channels
PWV convergence information

Implementation: cal-core/src/qa.rs → QaMetrics