Calibration

Calibration of direct imagers

The signal chain of each antenna applies a complex, frequency-dependent gain to the incident voltage:

E_{a} = g_{a} E_{a}^{true} + n_{a} .

In an FX correlator the visibilities take the form $V_{ab} = g_{a} g_{b}^{*} V_{ab}^{true}$ . Because the visibilities are recorded, the gains may be determined and removed after correlation.

In a direct imager, the antenna signals are combined on the grid before any time averaging, and visibilities are not formed. The gain estimates must therefore be applied to the voltages, $E_{a}^{'} = E_{a} / \overset{g}{^}_{a}$ , before gridding, and updated in real time. Malfunctioning antennas cannot be excised after imaging and must be flagged beforehand (Memo 002).

EPICal

Beardsley et al. (2017) developed EPICal, following a proposal by Morales (2011). EPICal correlates the voltage of each antenna with the image pixel at the position $\hat{s}_{0}$ of a calibrator source:

K_{a, \hat{s}_{0}} = ⟨ E_{a} (t) \tilde{E}^{' *} (\hat{s}_{0}, t) ⟩_{t} .

The image pixel is a weighted sum over all antennas, so $K_{a, \hat{s}_{0}}$ contains the sum $\sum_{b} V_{ab}$ , weighted by the current gain estimates. The self term is subtracted to give $C_{a, \hat{s}_{0}}$ . The updated gain is then the ratio of $C_{a, \hat{s}_{0}}$ to its prediction from model visibilities $V_{ab}^{M}$ :

g_{a}^{(n + 1)} = \frac{N _{a} C _{a, \hat{s}_{0}}}{\sum _{b \neq = a} W _{b}^{*} ( s ^ _{0} ) e ^{- 2 π i f \hat{s}_{0} \cdot r_{b} / c} V _{ab}^{M}} .

The iteration is damped to suppress oscillation:

g_{a}^{(n + 1)} \leftarrow (1 - γ) g_{a}^{(n + 1)} + γ g_{a}^{(n)}, 0 \leq γ < 1.

The damping corresponds to an effective integration time $t_{eff} = t_{cal} (1 + γ) / (1 - γ)$ .

The algorithm has the following properties (Beardsley et al. 2017):

Cost. The cost per time sample is $O (N_{a})$ per calibrator pixel, so it scales in the same way as EPIC itself.
Accuracy with a full sky model. The gain noise exceeds that of visibility-based calibration by approximately 23%.
Accuracy with a single-source model. The noise floor is lower than that of visibility-based calibration, because the correlation effectively beamforms on the calibrator.
Convergence. In simulation, the gains converged within approximately 12 iterations (0.12 s).
LWA data. On 2 s of LWA data, calibrating on Cygnus A and Cassiopeia A, image dynamic range increased from 80.4 to 133.1.
Extension. Correlation against multiple pixels permits estimation of direction-dependent gains and of the antenna beam.

Corrections applied at LWA-SV

EPICal is not implemented in the real-time LWA-SV imager (Kent et al. 2019; Krishnan et al. 2023), and epic++ does not perform antenna-based gain calibration.

The imager applies a per-stand, per-polarization, per-channel complex weight. The weight is derived from the station model in the LWA Software Library (LSL) and recomputed whenever the station is retuned:

ϕ_{a} (f) = \frac{exp [ 2 π i f ( τ _{a}^{cable} ( f ) - z _{a} / c ) ]}{G _{a}^{cable} ( f )} .

Its components are:

Cable delay. $τ_{a}^{cable} (f)$ is the frequency-dependent cable delay.
Height delay. $z_{a} / c$ is the geometric delay corresponding to the height of the stand. Together with the cable delay, it phases the station to zenith.
Cable gain. $G_{a}^{cable} (f)$ is the cable attenuation model.

Two classes of stand are excluded by setting their weight to zero:

stands with a combined status code below 33 in either polarization;
the outlying stands 122–128, 185, 186 and 256, which keeps the aperture compact.

The F-engine samples are multiplied by $ϕ_{a}$ during gridding.

Flux scale. No flux calibration is applied, and image values are in arbitrary units. The FITS keyword BUNIT is set to UNCALIB. Where a flux scale is required, Krishnan et al. (2023) normalize to the brightest A-team source, using the spectral models of Perley & Butler (2017) and the directive gain of the dipole.

Bandpass. No bandpass correction is applied. Pixel spectrograms consequently show a spectral gradient and slow gain variations (Memo 013). For time-domain analyses, normalization of each channel by a running median is sufficient for most purposes.

Astrometry. A comparison of EPIC images with images from the LWA-SV FX imager (Memo 012) found two discrepancies:

a bulk offset of one to two pixels, which was traced to the image orientation and corrected;
a rotation of approximately 1°–3°, which increases toward the horizon and whose origin was not identified.

Source positions at low elevation should therefore be interpreted with caution.

The EPIC Algorithm

Derivation of the EPIC imaging equation, its equivalence to visibility-based imaging, gridding, autocorrelation removal, and computational cost.

The LWA-SV Station

The Long Wavelength Array station at Sevilleta, its digital signal processing, and the commensal interface through which EPIC receives data.