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