Calibration of ALMA as a Phased Array. ALMA Observations During the 2017 VLBI Campaign

The APS performs phase adjustments to the individual ALMA antenna signals to create a phased array from a designated subset of the full observing array. The phasing corrections are computed relative to a specified reference antenna. The phased signals are then summed within the ALMA BL correlator cards to create a virtual antenna (the sum antenna) whose signals are fed back into the ALMA BL correlator and correlated as if it were a real, much larger antenna co-located with the reference antenna. This allows the sum antenna to be cross-correlated with other non-phased antennas (within the comparison array) to provide feedback on the efficiency of the phasing process. The phasing adjustments are calculated within the ALMA Telescope Calibration (TelCal; Broguière et al. 2011) subsystem of the software and then applied by the BL correlator. The information is therefore processed in a closed "phasing loop" between the BL correlator (where measurements are made and applied) and TelCal (where corrections are calculated). We discuss some important details on the spectral specifications and the timing of the phasing loop in the next two subsections.

2.1.1. Spectral Specifications of the Phasing Loop

2.1.2. Timing of the Phasing Loop

Each "VLBI scan" is partitioned into "subscans" for correlation and for processing in TelCal. To choose a suitable integration period to calculate the phasing corrections in the channel averages, one should consider that longer integrations result in better (less noisy) phase corrections in stable atmospheric conditions, whereas shorter integrations are required for acceptable efficiency during sub-optimal observing conditions. The operational compromise adopted for Cycle-4 was to program the correlator for four 4-s integrations (4.032 s) per subscan with 2 s (2.016 s) channel averages across four adjacent TFBs.¹⁵ These 16 s (16.128 s) correlator subscans require a setup and dump gap between them of 2.064 s. Thus the total loop time (between phasing updates) is 18 s (18.192 s). The phasing corrections are calculated at the end of each 16-s subscan and applied at the beginning of the next.

Note that the full 16-s subscan cannot be used to determine the phasing solution. In practice, the transfer of data to TelCal is not instantaneous, so a few seconds are needed to obtain the data to process (Telcal does the data processing in <1 seconds); the arrival of the phasing adjustment from TelCal to the correlator is usually within the first 4-s integration. Therefore, 12 or 14 s of channel average data are typically considered valid for the next TelCal computation. Note also that the correlator continues to process data through the gaps for the VLBI recordings, but that these 2 s intervals are not represented in the ALMA ASDM files (the raw data in the ALMA archive). An operational consequence of this is that the ALMA data only cover 90% (16.128/18.192) of the VLBI data and some interpolation through the gap is required. A timing diagram sketching the timings associated with the different components of the system is shown in Figure 2.

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A final consideration is that the phasing system is imperfect: there are ∼10 deg (rms) phasing errors under typical conditions. This is captured within the ASDM data sets and calculations may be made after the observations to calculate the phasing efficiency and correct the amplitude of the summed VLBI signal for decorrelation losses (see Section 6.2.3 for details).

Conventionally VLBI is performed using circularly polarized feeds (with quarter wave plates) to avoid parallactic angle issues in the correlation. The ALMA antennas however have linearly polarized feeds, which provide a high polarization purity, i.e., a low polarization leakage between polarizers (e.g., Rudolf et al. 2007; see also Section 5.1). Several options were possible for the adaptation of the ALMA linear polarizations into a circular basis for VLBI. These included applying the conversion to the raw data streams either at the antenna frontends or computing it at the correlation stage at the VLBI backend. This option however, has two major drawbacks: first, the additional hardware required can potentially increase the instrumental polarization effects, and, second, it is an irreversible process. Therefore, the final chosen strategy was to apply a post-correlation conversion for the ALMA signal polarization. The VLBI correlation is executed with the DiFX software (Deller et al. 2011), which correlates data streams and has no intrinsic understanding of polarization other than as labels. Because ALMA provides X and Y polarization recordings, while the rest of the VLBI stations provide R and L circular polarization signals, DiFX reports XL, XR, YL and YR correlation products in its native binary (so-called SWIN) output. The VLBI fringes in this mixed-polarization basis, can then be converted into a pure circular polarization basis using an algorithm based on hybrid matrices in the frame of the Radio Interferometer Measurement Equation (Hamaker et al. 1996; Sault et al. 1996).

The process of polarization conversion can be divided into two main parts. In the first part, the visibilities among the ALMA antennas (computed by the ALMA correlator, simultaneous with the VLBI observations) are calibrated using ALMA-specific algorithms for full-polarization data reduction (see Sections 4 and 5). Within the ALMA organization, this process is known as "Level 2 Quality Assurance" (QA2). In the second part, the calibration tables derived in the QA2 stage are sent to the VLBI correlators, where a software tool known as PolConvert (Martí-Vidal et al. 2016) (run at the correlator computers) applies these tables directly to the VLBI visibilities produced by the DiFX software. It is the PolConvert program that transforms the linear-polarization ALMA data streams into a circular basis for VLBI, and generates the calibration information for phased-ALMA. One of the main advantages of this "off-line" conversion is that it is "reversible", in the sense that one can perform the full QA2 analysis of the ALMA data multiple times, to find the best estimates of the pre-conversion correction gains prior to the polarization conversion. Details about this process are given in Section 6 (see also Martí-Vidal et al. 2016, for a full description of the PolConvert algorithm).

In summary, VLBI observations with phased-ALMA is a four-part process: (i) observe at ALMA using the APS, (ii) correlate the VLBI data from ALMA and the other participating stations with DiFX in a mixed-polarization basis, (iii) calibrate the ALMA data for polarization and other calibration products, and finally (iv) apply these products to the DiFX output using PolConvert in a postprocessing step prior to VLBI data calibration.

The GMVA observed the three approved programs on three consecutive nights spanning 2017 April 2–4 (Table 1). The spectral setup includes a total of four SPWs, centered at 86.26, 88.2, 98.26, and 100.26 GHz (Table 3).

Note that only the lowest frequency (∼86 GHz) SPW (0) was recorded on VLBI disks due to the limited recording rates available at the other GMVA stations. The list of observed sources and their calibration intent is given in Table 4.

The EHT observed the six approved projects in five nightly schedules or tracks, labeled A through E (see Table 2), during an observing window of ten days (where the observations are triggered based on the weather). As a consequence, subsets of different projects shared common tracks. While this strategy was adopted to optimize the efficiency of the VLBI campaign, the arrangement of different projects within the same observing block is not a standard scheduling mode for ALMA.

The EHT spectral setup includes a total of four SPWs, centered at 213.1, 215.1, 227.1, and 229.1 GHz (Table 5). Note that only the SPWs in the upper sideband (SPW = 2, 3) were recorded on VLBI disks as done by the other EHT stations.

The list of observed sources and their calibration intent is given in Table 6.

The VLBI schedule is governed by the VLBI EXperiment (VEX) file. ALMA observes the VLBI targets specified in the VEX file with the APS actively phasing the array. To enable calibration of the ALMA array, a block of 15-min duration before the start of the VLBI schedule is devoted to observations of flux density, bandpass, and polarization calibrators in ordinary interferometric mode (i.e., with the APS off). Scans on the phase and polarization calibrators (also in ordinary interferometric mode) are then cycled through the schedule in the gaps between VLBI scans. Therefore, the ALMA scheduling blocks (SBs) include scans when the phasing is activated (APSscans) and scans during ordinary ALMA observations (ALMAscans). These two modes of operation are usually referred to as ALMA-mode and APS-mode. This operational scheme enables full calibration of the ALMA visibilities.

In principle, the ALMA calibrations within each project on any given track would be sufficient to properly calibrate the project. However, in practice, some calibration scans were not completed, and it became necessary to extend the calibration across the full observing night (track). This is not normally done with ALMA observations. Instead, ALMA would normally re-observe such SBs. For VLBI, however, this is not an option given the participation of the other global sites. As a result, ad-hoc calibration procedures were developed to handle the QA2 of VLBI experiments (see Sections 4 and 5).

3.3.1. Choice of Reference Antenna

4.1.1. Data Import

4.1.2. Data Flagging

The bandpass calibration tables are derived from observations of the bandpass calibrators in ALMA mode. The same targets are usually also observed in the VLBI schedule for the VLBI bandpass calibration. Figure 5 shows the amplitudes and phases of the bandpass solutions for one of the phased antennas in both Band 3 and Band 6. Two different bandpass tables are used in the calibration. One is obtained from an ordinary calibration using the ALMAscans and is applied to the ALMAscans. The other table is a copy of the first one, but with all phases zeroed; this is applied to the APSscans. This scheme is necessary because of the intrinsic difference between ALMAscans and APSscans. In ALMA-mode, any difference between the X and Y bandpass phases reflects residual cross-delays from the ALMA correlator model, which is not used in APS-mode. In APS-mode, the TelCal phase adjustments introduce additional X-Y cross-phases (in frequency chunks) which are, by construction, zero for the reference antenna. The phases of the chunks must be solved for using the polarization calibrator, but with no bandpass phases applied to the data.

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Because APP observations may be done with a "flexible" array²² in which different antennas can be present at different times during the execution of a given project or observing track, there could be cases where some antennas are not in the array during observations of the bandpass calibrator (but are either added later or dropped earlier). Such antennas would miss bandpass calibration and would be flagged following standard procedures. Two solutions are implemented in the APP calibration scheme to address this issue: a) more than one source can be listed as bandpass calibrator (in ALMAscans); b) in cases where some (well-functioning) antennas did not observe any suitable bandpass calibrator during an entire track, a flat bandpass is assumed by setting a unity-gain for the bandpass amplitude solutions (this avoids these good antennas to be flagged). The latter option was never used in the Cycle-4 data processing.

The phase gains are also intrinsically different between the ALMAscans and APSscans (the bandpass tables to be pre-applied are different), and are therefore found separately for each set of scans. The gaincal task derives phase gains tables with a solution interval equal to the integration time of the ALMA correlator (solint = 'int'). Figure 6 shows the time evolution of phase gains for a typical phased antenna both in APSscans and ALMAscans. For the APSscans (Figure 6, left), the phases are around zero with no offset between the polarizations. During ordinary ALMA observations (Figure 6, right), there are clear phase offsets between polarizations (X/Y cross-phase) and the phases are offset from zero.

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Figure 7 shows the time evolution of phase gains for a "comparison" antenna (i.e., an antenna participating in the observations, but not being phased). Also in the case of a comparison antenna, there is a difference (in the phases of each polarization channel) between APSscans and ALMAscans, owing to the fact that the corrections from the ALMA correlator model are not applied in the APS-mode.

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The amplitude calibration and the absolute flux-density scaling are mostly performed as in standard ALMA observations. This process uses a primary flux-density calibrator and is based on self-calibration of the observed sources, and consists of three steps:

• 1.  
  The CASA task setjy is used to scale the model of the flux-density calibrator to its correct value. The models of all the other sources are set to a flux density of 1 Jy.
• 2.  
  The CASA task gaincal is then used to calibrate the amplitudes (with a solution interval equal to the scan length) of the antenna gains for all sources, after applying on the fly the bandpass corrections (Section 4.2) and the phase gains corrections (Section 4.3).
• 3.  
  The CASA task fluxscale uses the amplitude gains (generated in the previous step) to bootstrap the flux-density from the primary flux-density calibrator into all the observed sources which will also be scaled to Jy units.

We note that the changes in atmospheric opacity (for each source) are encoded in the gain tables generated in step #2. For instance, a higher opacity for a given VLBI scan will automatically result in a higher amplitude correction for all the affected antennas in that scan.

Figure 8 shows the amplitude gains for the phased antenna DA41 in Band 6, including APSscans and ALMAscans. The left panel shows the gains assuming a normalized flux density for all sources (calculated in step #2). The right panel shows the amplitude gains after bootstrapping the flux density from the primary calibrator to all sources (calculated in step #3). Note that most of the spread observed in the gains is removed after the absolute flux-density scaling²³ (although some dependence on the elevation remains, especially near the end of the track when sources are typically setting).

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At this stage there are a few subtle differences with respect to standard ALMA amplitude calibration procedures. First, because each mode needs a different bandpass and phase gain tables, the amplitude gains must be found separately for APSscans and ALMAscans. second, the gain calibration is performed in "T" mode (i.e., one common gain for the two polarizations), to avoid altering the X/Y amplitude ratios for the polarization calibrator (this would affect the estimate of the QU Stokes parameters from the XX and YY visibilities versus parallactic angle; see Section 5.2.3). Third, the T_sys measurements at the individual ALMA antennas, normally used to track the atmospheric opacity, are not used in the calibration. Instead, in the APS data calibration, any effect from the time-variable atmospheric opacity during the observation of a given source is tracked by the amplitude self-calibration. While this scheme effectively removes the bulk of the opacity effect, it leaves a global scaling factor in the amplitude gains that is related to the difference between the opacity correction in the observation of the primary flux calibrator and the (average) opacity in the observation of a given source. Such a difference should be of the order of a few %, for high antenna elevations, but it could be much higher if the air mass difference between the primary flux calibrator and the target sources is higher (i.e., at low elevations). In summary, not accounting for the T_sys of the individual ALMA antennas introduces an amplitude offset which is source-dependent and constant during the observing epoch. Appendix A.1 provides an estimate of this amplitude scaling factor for each target (see values in Tables 9 and 10).

4.4.1. Primary Flux-density Calibrators

In interferometers having antennas with linearly polarized feeds (like ALMA) both orthogonal linear polarizations (X and Y) are received simultaneously and the data are correlated to obtain XX, YY, XY, and YX correlations. The polarization response can be described assuming that each feed is perfectly coupled to the polarization state to which it is sensitive, with the addition of a complex factor times the orthogonal polarization; this is called the "leakage" or "D-term" model. In the limit of negligible higher order terms in the instrumental polarization and zero circular polarization,²⁴ the cross-correlations for linear feeds on a baseline between antenna i and antenna j is given by (e.g., Nagai et al. 2016)

$$\begin{eqnarray}&&{X}_{i}{X}_{j}^{* }=(I+{Q}_{\psi })+{U}_{\psi }({D}_{{X}_{j}}^{* }+{D}_{{X}_{i}}),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{X}_{i}{Y}_{j}^{* }={U}_{\psi }+I({D}_{{Y}_{j}}^{* }+{D}_{{X}_{i}})+{Q}_{\psi }({D}_{{Y}_{j}}^{* }-{D}_{{X}_{i}}),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{Y}_{i}{X}_{j}^{* }={U}_{\psi }+I({D}_{{Y}_{i}}+{D}_{{X}_{j}}^{* })+{Q}_{\psi }({D}_{{Y}_{i}}-{D}_{{X}_{j}}^{* }),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{Y}_{i}{Y}_{j}^{* }=(I-{Q}_{\psi })+{U}_{\psi }({D}_{{X}_{j}}^{* }+{D}_{{X}_{i}}),\end{eqnarray} \tag{ 4 }$$

where ${Q}_{\psi }=Q\cos 2\psi +U\sin 2\psi$, ${U}_{\psi }=-Q\cos 2\psi +U\sin 2\psi$, ψ is the parallactic angle, and D_X and D_Y are the instrumental polarization D-terms. Therefore, a contribution from Stokes Q, U, and parallactic angle ψ appears in the real part of all correlations. Each Stokes parameter can be obtained from these four equations, and thus the calibration residuals of ${D}_{{X}_{i}}$, ${D}_{{X}_{j}}$, ${D}_{{Y}_{i}}$, ${D}_{{Y}_{j}}$ affect the Q and U visibilities. The instrumental contribution to the cross-hands visibilities (i.e., the effect due to leakage) is independent of parallactic angle (and thus is constant with time), whereas the contribution of linear polarization from the source rotates with parallactic angle for alt-az mount antennas and it is therefore time-dependent. This makes it possible to uniquely separate the source and instrumental contributions to the polarized interferometer response. To that end, observations with an array using linear feeds need to include frequent measurements of an unresolved calibrator over a wide range of parallactic angle.

5.2.1. Gains for the Polarization Calibrator

To examine the polarization calibrator, the first gains are determined with gaincal using gaintype = 'G' (i.e., independent solutions for the XX and YY correlations), providing in output the calibration table '<label>.Gpol1' : the gain corrections in this table absorb all of the polarization contributions. The (linear) source polarization can be displayed by plotting the (antenna-based) amplitude polarization ratio versus time (using poln = '/' in plotcal), which reveals a clear variation as the linear polarization rotates with parallactic angle as a function of time (Figure 9, left panel). Once a full-polarization source model is obtained for the polarization calibrator (Section 5.2.3), one can revise the gain calibration using such a model, yielding in output the calibration table '<label>.Gpol2', where any signature of the source polarization is removed from the gains (Figure 9, right panel).

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5.2.2. X-Y Cross-phase

The bandpass and gain tables computed in Section 4 are adequate for the parallel hands. Because the phase of the reference antenna is set to zero in both polarizations, yielding relative phases for all other antennas,²⁵ a single residual phase bandpass relating the phase of the two hands of polarization in the reference antenna remains in the cross-hands of all baselines. The APS has already removed a bulk cross-delay (linear phase slope) in this phase bandpass,²⁶ but any residual non-linear phase bandpass in the XY-phase needs also to be solved for, so that both cross- and parallel-hands can be combined to extract correct Stokes parameters. This can be computed running gaincal in mode XYf+QU.²⁷ Figure 10 shows the X-Y cross-phases of the reference antenna obtained for EHT track D in APSscans.

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There a number of considerations that apply specifically to handling the X-Y cross-phases for APS observations.

• 1.  
  Differences are expected between ALMAscans and APSscans due to the different corrections applied by the APS software. For instance, the cross-phases in individual SPWs of the APSscans show small jumps among the frequency chunks used by TelCal (Section 2.1.1).
• 2.  
  The X-Y phase offset determined for the APSscans is independent of the antenna used as the reference in the QA2, as such an offset was applied to all the antennas prior to the cross-correlation, while keeping all phases close to zero. For the ALMAscans, using a different reference antenna in the QA2 calibration changes the derived X-Y offset.
• 3.  
  It is imperative for the success of the polarization conversion (Section 6.1), to flag "noisy" solutions in the X-Y phase difference in the calibration tables obtained during ordinary calibration (see Section 4.3), to minimize leakage-like noise in the VLBI visibilities (see Appendix C). To this end, a cross-polarization running average of the phase gains is applied before polarization calibration.
• 4.  
  It is necessary to check and fix possible X/Y cross-phase jumps of 180 degrees within each SPW.
• 5.  
  Only the X-Y phases are solved for, while the X-Y cross-delay is not computed: the bulk of the cross-delay is already removed by the APS.

5.2.3. Estimating QU for the Polarization Calibrator

5.2.4. Polarization Leakage (D-terms)

Once the polarization model is obtained for the polarization calibrator and the X-Y phase offsets have been calibrated, one can solve for the instrumental polarization: the leakage terms or D-terms are estimated using the CASA task polcal. This task produces an absolute instrumental polarization solution on top of the source polarization and ordinary calibrations. Note that no reference antenna is used here because the polarized source provides sufficient constraints to solve for all instrumental polarization parameters on all antennas, relative to the specified source polarization. This is not possible in the case of an unpolarized calibrator (or in the circular basis, even if the calibrator is polarized), where only relative instrumental polarization factors among the antennas may be determined with respect to the reference antenna.

The values fitted are of the order of a few percent (and generally <10%; see Figures 11, 12). The D-terms are then computed into the sky frame and arranged in a Jones matrix that can apply corrections (up to second order) to all four correlation products (XX, XY, YX and YY).

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We explicitly note that the problem of the leakage calibration of each individual ALMA antenna is not critical, as the effect of the D-terms on the final VLBI calibration is relatively small (below the thermal limit of the VLBI baselines).

5.2.5. Amplitude Gain Ratios between X and Y

Based on a comparative study among all potential polarization calibrators observed in the VLBI campaign, the source 3C 279 was established as the best calibrator. It is a strong mm source (∼13 Jy in Band 3 and ∼9 Jy in Band 6) with a high fractional polarization (12%–15%) and was observed with a large parallactic-angle coverage. 3C 279 was also observed on multiple (consecutive) days, allowing a check of the stability of the source (and/or of the array) polarimetry across the whole campaign. During three nights (Apr 2, 3, 7) 3C279 was not included in the VLBI schedules, and alternative polarization calibrators had to be employed. Despite the fact that their fractional polarizations are an order of magnitude lower than 3C 279, they allowed us to achieve a satisfactory calibration of the X-Y phases at the reference antenna. Tables 7 and 8 show the polarization properties of 3C 279 and the "alternative" polarization calibrators, in Bands 3 and 6, respectively.

Tables 7 and 8 show an appreciable Faraday Rotation of the polarization calibrators used as alternatives to 3C279. A rotation of the electric vector position angle (EVPA) of several degrees across the SPWs (as seen in Band 3), yields a rotation measure (RM) of order of 3 × 10⁴ rad/m². Such a high rotation can potentially bias the calibration of the leakage terms (D-terms) if one single (i.e., frequency-averaged) polarization model is used in the polcal CASA task, as per the official QA2 procedure. This effect is shown in Figures 11 and 12, which compare the D-terms in one of the Band 3 experiments as estimated with a standard (QA2) procedure (i.e., one single model for source polarization in all SPWs), versus independent models for each SPW. The systematics clearly seen in the former case are minimized in the latter. Based on this evidence, each SPW was separately calibrated in polarization, by accounting for the different EVPA of the polarization calibrator in each SPW (i.e., accounting for the RM in the calibrator), as well as for the spectral index. This is especially important for the GMVA (3 mm) observations.

A couple of final remarks are in order. The polarization calibration can be done using either ALMAscans or APSscans or both. During the April 2017 VLBI campaign (good) polarization calibrators were not observed often enough in ALMA mode, therefore only APSscans were used (which ensured a good coverage of the parallactic angle). Therefore, the ALMAscans were not calibrated in polarization. Finally, only data from antennas that were phased during the entire observing track are used for the polarization calibration.²⁹

Details of this process are described in Martí-Vidal et al. (2016), and here we summarize the main concepts for completeness.

As described in Section 2.2, the DiFX software is "blind" to polarization and provides XL, XR, YL and YR correlation products. The visibilities in mixed-polarization basis can be arranged in a matrix form as

$$\begin{eqnarray}{V}_{+\odot }=\left(\begin{array}{cc}{V}_{{XR}} & {V}_{{XL}}\\ {V}_{{YR}} & {V}_{{YL}}\end{array}\right),\end{eqnarray} \tag{ 5 }$$

where ALMA is assumed to be the first antenna in the baseline (linear polarization basis, denoted as +) and all the other stations observe in circular polarization basis (denoted as ⊙). The matrix in pure circular basis (i.e., ALMA converted to circular) can be arranged as

$$\begin{eqnarray}{V}_{\odot \odot }=\left(\begin{array}{cc}{V}_{{RR}} & {V}_{{RL}}\\ {V}_{{LR}} & {V}_{{LL}}\end{array}\right).\end{eqnarray} \tag{ 6 }$$

The visibility matrix in circular-circular polarization can be recovered directly from the matrix in mixed-polarization by applying a simple matrix product:

$$\begin{eqnarray}&&{V}_{\odot \odot }={C}_{\odot +}{V}_{+\odot },\end{eqnarray} \tag{ 7 }$$

where (see Equation (5) of Martí-Vidal et al. 2016)

$$\begin{eqnarray*}{C}_{\odot +}=\displaystyle \frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 & -i\\ 1 & i\end{array}\right)\end{eqnarray*}$$

is the matrix that converts polarizations from linear to circular (C stands for Conversion).

Equation (7) assumes that the visibilities are free from instrumental effects (absence of noise and perfectly calibrated signals). The observed visibility matrix is then related to the perfectly-calibrated visibility matrix by the equation (as computed by PolConvert)

$$\begin{eqnarray}&&{V}_{\odot \odot }^{{obs}}={C}_{\odot +}{\left({J}^{A}\right)}^{-1}{V}_{+\odot },\end{eqnarray} \tag{ 8 }$$

where J^A is the calibration Jones matrix of the phased array

$$\begin{eqnarray}&&\begin{array}{l}{J}^{A}\,=\\ \ \left(\begin{array}{cc}\left\langle {\left({B}_{0}^{i}\right)}_{X}\,{\left({G}_{p}^{i}\right)}_{X}\,{G}_{a}^{i}\right\rangle & \left\langle {\left({D}^{i}\right)}_{X}\,{\left({B}_{0}^{i}\right)}_{X}\,{\left({G}_{p}^{i}\right)}_{X}\,{G}_{a}^{i}\right\rangle \\ \left\langle {\left({D}^{i}\right)}_{Y}\,{\left({B}_{0}^{i}\right)}_{Y}\,{\left({G}_{p}^{i}\right)}_{Y}\,{G}_{a}^{i}\right\rangle & \left\langle {\left({B}_{0}^{i}\right)}_{Y}\,{\left({G}_{p}^{i}\right)}_{Y}\,{G}_{a}^{i}({{XY}}_{a}^{i})\right\rangle \,({{XY}}_{p}^{r})\end{array}\right).\end{array}\end{eqnarray} \tag{ 9 }$$

J^A includes all the calibration matrices (i.e., gain, bandpass, D-terms) of all the antennas in the phased array. In particular, ${G}_{a}^{i}$ and ${G}_{p}^{i}$ are the amplitude and phase gains, ${B}_{0}^{i}$ is the bandpass, and Dⁱ are the D-terms for the ith ALMA antenna; ${{XY}}_{p}^{r}$ is the phase offset between the X and Y signals of the reference antenna (indicated with index r); $\left\langle ...\right\rangle$ means averaging over phased antennas at each integration time. Note that the G_a gains do not distinguish between X and Y (solutions are forced to be the same for both polarizations using the "T" mode—Section 4.4). PolConvert interpolates individual gains (linearly, in amplitude-phase space) in both frequency and time directions to the values at the VLBI correlator (see Appendix D; see also Appendix C for a discussion on possible issues related to this interpolation).

For the case where phased ALMA is the second station in the baseline :

$$\begin{eqnarray}&&{V}_{\odot \odot }^{{obs}}={V}_{\odot +}{\left[{\left({J}^{A}\right)}^{H}\right]}^{-1}{C}_{\odot +}^{H},\end{eqnarray} \tag{ 10 }$$

where H is the Hermitian operator. Note that the application of Equations (8) and/or 10 automatically calibrates for the post-converted R-L delay and phase at ALMA. Thus, using phased ALMA as the reference antenna in the VLBI fringe-fitting will account for the absolute EVPA calibration and, furthermore, no fringe-fitting of the R-L delays will be needed.

6.1.1. Post-conversion Effects of Residual Cross-polarization Gains

In the estimate of the ${{XY}}_{p}^{r}$ solutions, CASA assumes that the amount of circular polarization in the calibrator is negligible. If this is not the case, Stokes V appears as an imaginary component in the XY and YX correlation products, given in the sky frame. This component might be partially absorbed in the estimate of ${{XY}}_{p}^{r}$, hence introducing a phase offset between X and Y that cannot be corrected following standard QA2 procedures. Nevertheless, if a phase offset is introduced in ${{XY}}_{p}^{r}$, it will appear as a leakage-like effect in the polconverted VLBI visibilities, which can then be calibrated downstream in the VLBI data processing. The proof of this statement is straightforward.

Given any gain Jones matrix in linear basis, J₊, it can be converted into circular basis as ${J}_{\odot }={C}_{\odot +}{J}_{+}{C}_{+\odot }$. If J₊ has the form

$$\begin{eqnarray}{J}_{+}=\left(\begin{array}{cc}1 & 0\\ 0 & \rho \end{array}\right),\end{eqnarray} \tag{ 11 }$$

where ρ is a residual complex gain between the phased sums in X and Y, then

$$\begin{eqnarray}{J}_{\odot }\propto \left(\begin{array}{cc}1 & D\\ D & 1\end{array}\right),\end{eqnarray} \tag{ 12 }$$

which has the shape of a leakage matrix with equal D-terms for R and L. In this equation,

$$\begin{eqnarray}&&D=\displaystyle \frac{1-\rho }{1+\rho },\end{eqnarray} \tag{ 13 }$$

which means that ρ can be derived from D, so that it is possible to use the VLBI calibration to improve the alignment of the X and Y phases at ALMA, thus allowing us to detect Stokes V with a higher accuracy.

According to the specifications devised by the APP, the D factor in Equation (13) should be lower than 3% in absolute value, which translates into a ρ of less than ∼5% in amplitude and ∼5 degrees in phase. These figures fall within the requirements of ordinary ALMA full-polarization observations (i.e., a 0.1% sensitivity in polarization or better, as per the Call for Proposals³⁰ ). Therefore, the requirement for a post-conversion polarization leakage below 3% is easily met after the QA2 calibration is performed.

PolConvert applies a modified version of Equations (8) (or 10) as following

$$\begin{eqnarray}&&{V}_{\odot \odot }=\displaystyle \frac{K}{\sqrt{{K}_{00}{K}_{11}}}{V}_{+\odot }.\end{eqnarray} \tag{ 14 }$$

where phased-ALMA is assumed to be the first antenna in the baseline, $K={C}_{\odot +}{\left({J}^{A}\right)}^{-1}$ is the total (calibration plus conversion) matrix applied by PolConvert (Equation (8)), and K₀₀ and K₁₁ are the diagonal elements of K (the ratio ensures that the amplitude correction applied in Equation (14) is normalized). For compatibility with other VLBI stations, PolConvert stores the VLBI amplitude corrections A for phased-ALMA as a combination of T_sys (one value per intermediate frequency and integration time) and an instrumental gain given in degrees per flux unit or DPFU (assumed to be stable over time and frequency)

$$\begin{eqnarray}&&A=\sqrt{{T}_{{sys}}/\mathrm{DPFU}}=\sqrt{\displaystyle \frac{\left|{K}_{00}{K}_{11}\right|}{N}}\end{eqnarray} \tag{ 15 }$$

or equivalently

$$\begin{eqnarray}&&{T}_{{sys}}=\displaystyle \frac{\left|{K}_{00}{K}_{11}\right|}{N}\mathrm{DPFU},\end{eqnarray} \tag{ 16 }$$

where N is the number of phased antennas at the given integration time. The $\sqrt{N}$ factor in Equation (15) comes from the scaling between the amplitude calibration from the intra-ALMA cross-correlations and that of the summed signal, as we demonstrate in Appendix B.

6.2.1. DPFU, T_sys, and SEFD

In principle, the DPFU of the phased array should scale with the number of phased antennas. Because the number of phased antennas changes during the observations, the DPFU would also change. In order to keep the DPFU of phased-ALMA constant over a given epoch (for calibration purposes), we set the DPFU of phased-ALMA to the antenna-wise average of DPFUs (instead of the antenna-wise sum). As a consequence, there is an N factor that must be absorbed by the T_sys in order to keep the same amplitude correction A, according to Equation (15). In fact, we explicitly notice that the only meaningful calibration information for the VLBI fringes with ALMA is given by the combination of T_sys and DPFU (and not by their independent values), via the factor $\sqrt{{K}_{00}{K}_{11}/N}$ in Equation (16). Using a different DPFU for phased-ALMA will thus result in a different set of T_sys values computed by PolConvert, in such a way that the VLBI amplitude corrections remain unchanged.

To compute the antenna-wise DPFU average, DPFU_i values for the individual antennas can be estimated using the following equation:

$$\begin{eqnarray}&&{\mathrm{DPFU}}_{i}=\displaystyle \frac{\langle {A}_{i,k}^{-1}{\rangle }^{2}}{\left\langle {\left({T}_{{sys}}\right)}_{i,k}^{-1}\right\rangle },\end{eqnarray} \tag{ 17 }$$

where the sub-index k runs over all scans where a T_sys is measured at the antenna i, A_i,k is the amplitude correction of the antenna i (see Equation (23) in Appendix B) for scan k, and $\langle$...$\rangle$ is the median operator (the median is more insensitive to outliers than the average). In Cycle 4, there was a slightly different number of phased antennas on each day, therefore the average $\langle$ DPFU $\rangle$ is also slightly different. We therefore set the average DPFU = 0.031 K/Jy as representative of the phased array during the whole campaign.

One could also define a system-equivalent flux density (SEFD) as the total system noise represented in units of equivalent incident flux density, which can be written as

$$\begin{eqnarray}&&\mathrm{SEFD}=\langle {T}_{{sys}}\rangle /\mathrm{DPFU}.\end{eqnarray} \tag{ 18 }$$

Using Equation (16) and DPFU = 0.031 K/Jy, one can derive an effective phased-array system temperature for each scan (these are shown in Figure 13 for the Band 6 tracks). By taking the median of all T_sys values (∼2.3 K), one derives SEFD = 74 Jy in Band 6.

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One could also compare this estimate with the theoretically expected SEFD as provided by the ALMA observatory:

$$\begin{eqnarray}&&\mathrm{SEFD}=\langle {T}_{{sys}}\rangle \ 2{k}_{B}/{\eta }_{A}{A}_{\mathrm{geom}},\end{eqnarray} \tag{ 19 }$$

where T_sys is the opacity corrected system temperature, k_B is the BoltzmannÕs constant, η_A is the aperture efficiency, and A_geom is the geometric collecting area of the telescope. For a diameter of 73 m for the area (equivalent to 37 12 m antennas), η_A = 0.68 (Matthews et al. 2018), and taking a representative value of 76 K for the opacity corrected T_sys (calculated for median elevations of the sources and typical atmospheric opacities in the 2017 observations), one derives SEFD = 74 Jy in Band 6, which matches perfectly the estimate from the QA2 gain calibration. We therefore take SEFD = 74 Jy as the representative sensitivity of phased-ALMA during the 2017 campaign in Band 6. A similar analysis provides SEFD = 65 Jy in Band 3.

6.2.2. ANTAB Files

6.2.3. Phasing Efficiency

An ideal phased array of N elements is equivalent to a single aperture with N times the effective area of one of the individual antennas. However, a real phased array will suffer from efficiency losses which translate in a decrease of effective collecting area. These losses can be characterized in terms of a "phasing efficiency", η_p, where η_p = 1 corresponds to perfect efficiency. Following Matthews et al. (2018), the phasing efficiency can be written as a function of the cross-correlation between the summed signal and that of a comparison antenna, divided by the averaged cross-correlations between the comparison antenna and the individual phased elements:

$$\begin{eqnarray}&&{\eta }_{p}=\displaystyle \frac{\left\langle {V}_{{sum}}{V}_{c}\right\rangle }{\sqrt{N}\left\langle {V}_{i}{V}_{c}\right\rangle },\end{eqnarray} \tag{ 20 }$$

where N is the number of phased antennas.³²

The main source of efficiency losses in the APS is due to imperfections in the phasing solutions. These imperfections are the unavoidable consequence of residual delay errors, troposphere fluctuations, and cosmic source structure, but also the necessarily finite time lag in the application of the phasing corrections (see Section 2.1 and Figure 2). All of these factors increase the rms fluctuations in the phases of the antennas used to form the phased sum, σ_ϕ, which in turn induce a decrement in the correlated amplitude by a factor $\epsilon \approx {e}^{\tfrac{-{\sigma }_{\phi }^{2}}{2}}$ (e.g., Carilli & Holdaway 1999; Matthews et al. 2018). In general, the atmosphere dominates , hence efficiency losses will be larger during poor atmospheric conditions and at higher frequencies. In this respect, the phasing efficiency provides a metric for comparing performance of the system under various conditions (e.g., weather conditions, different observing frequencies, array configurations). In the following, by "phasing efficiency" we refer to the efficiency of the phase solver as reported by TelCal, which does not include other losses (see Matthews et al. 2018, for a list of all possible losses.).

The phases calculated by the APS and applied to the data are stored in the ASDM metadata (in the CalAppPhase.xml table). These phases can be used to understand the phasing efficiency. Figure 14 displays per-subscan averages of the phasing corrections applied to the Band 6 observations of the 2017 VLBI campaign. The average is over all 4 SPWs (and all spectral channels)³³ and all the phased-antennas for every correlator subscan. When the efficiency is good, only small phase corrections (≲20°) are needed (>80% of the time). When the atmosphere is generating phase fluctuations faster than the APS can correct them, larger phase corrections are generated and do not necessarily improve the result (i.e., the APS cannot keep up with the atmosphere).

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Figure 15 shows the corresponding phasing efficiency calculated by TelCal and extracted from the same ASDM file metadata used to produce Figure 14. This efficiency is calculated by comparing the visibilities of the sum and reference antennas on baselines to one comparison antenna. The plot shows that in Cycle 4, for about 80% of the time, the APS has met 90% phasing efficiency, which was the goal specified in the original operational requirements (Matthews et al. 2018). When the phasing efficiency is less than 90% the problem is ascribable to atmosphere and/or low target elevation. In fact, the VLBI tracks were scheduled to cover each science target from elevation limit (15°) to elevation limit.³⁴ Both Figures 14 and 15 show a clear systematic degradation of phasing efficiency associated with low elevations.

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It is worth noting that while the metadata contained in the CalAppPhase.xml table of the ASDM files are useful to understand, quantify, or display the phasing efficiency, they are not actually used to calibrate the data (e.g., to make an appropriate correction for the absolute VLBI correlation amplitudes). In fact, the effects of phasing efficiency are automatically taken into account via the Jones matrix for the phased array (Equation (9)). The latter is computed at each integration time and for each frequency channel of the intra-ALMA visibilities, and provides the full calibration of the summed VLBI signal (see also Equations 13 and 14 of Martí-Vidal et al. 2016), including the scaling factor that calibrates the VLBI amplitudes to Jy units (Equation (14)). In other words, the effects of the phasing efficiency are already included in the amplitude- and phase-gains computed for the indvidual ALMA antennas and are absorbed in the T_sys calculated by PolConvert (Equation (16)). As a complementary source of information for data quality assessment, the phasing efficiency can also be estimated from the gain calibration of the intra-ALMA visibilities as

$$\begin{eqnarray}&&{\eta }_{p}=\displaystyle \frac{\left|\,\left\langle {G}_{p}^{i}{G}_{a}^{i}\right\rangle ,\,\right|}{\left|\,\left\langle {G}_{a}^{i}\right\rangle ,\,\right|}.\end{eqnarray} \tag{ 21 }$$

where ${G}_{p}^{i}$ and ${G}_{a}^{i}$ are the phase and amplitude gains for the ith phased antenna.

ALMA normally tracks the atmospheric opacity by measuring system temperatures at each antenna. The system temperatures are however not used in the phased-array data calibration. The reason for this choice is that the phased signal for VLBI is an unweighted sum of the signals from all the phased antennas, therefore all phased antennas should be treated equally in the gain calibration for PolConvert. In particular, in APS observations, antennas with bad or failed T_sys measurements cannot be removed from the analysis (since they have already been added to the phased sum); if T_sys were used, opacity corrections would be applied only to a subset of phased antennas, biasing the calibration by PolConvert. As explained in Section 4.4, the bulk of the opacity effects is removed with self-calibration, whereas T_sys (usually measured a few times per scheduling block) would correct for second-order opacity effects, related to the difference between the opacity correction in the observation of the primary flux calibrator and the (average) opacity in the observation of any given source.

To obtain a more accurate absolute flux-density calibration by accounting for these second-order opacity effects encoded in the T_sys measurements, the original ASDM files (containing the T_sys information) can be recovered and calibrated (in bandpass, flux, and phase) following ordinary QA2 procedures, but using the same primary flux-density calibrator used for the ALMA-VLBI QA2. The resulting visibilities (which will have the T_sys effects applied) may then be imaged with the same weighting and gridding as the ones used in the ALMA-VLBI QA2.

Here, we offer an alternative method to estimate the opacity correction which uses the antenna-wise average of valid T_sys measurements (the same atmospheric opacity for all ALMA antennas is assumed) to estimate post-QA2 a global scaling factor, which is source-dependent and constant during the observing epoch, related to the difference between the opacity of a given source and the primary flux calibrator. In particular, let ${S}_{{QA}2}^{a}$ be the flux-density estimate of source a, obtained from the <label>.flux_inf QA2 table; g^a(t) be the antenna-wise average of the amplitude gains; and ${T}_{{sys}}^{a}$(t) be the antenna-wise average of valid system temperatures (linearly interpolated in time for each source). Then, the opacity corrected flux density for source a, ${S}_{\tau }^{a}$, will be given by

$$\begin{eqnarray}&&{S}_{\tau }^{a}={\left(\displaystyle \frac{\left\langle {g}^{a}(t){T}_{{sys}}^{a}(t)\right\rangle }{\left\langle {g}^{P}(t){T}_{{sys}}^{P}(t)\right\rangle }\right)}^{2}{S}_{{QA}2}^{a},\end{eqnarray} \tag{ 22 }$$

where the superindex P stands for the primary flux-density calibrator and $\left\langle ...\right\rangle$ is the time average.

Besides providing an opacity correction to the flux densities, Equation (22) can also be used to derive a global absolute flux-density calibration for the whole VLBI campaign. If we assume the same SEFD for the ALMA system during the VLBI campaign (as assessed in Section 6.2.1), we can use Equation (22) to estimate the flux densities of all sources in all the VLBI tracks, by just using the gains g^P(t) from the primary calibrator of one of the tracks. This is particularly useful to improve the flux-density scale in tracks with no SSOs, as pointed out in Section 4.4.1. In Tables 9 and 10, we show the opacity corrected flux densities for all the observed sources, using Callisto from project 2016.1.00413.V (in Band 3) and Ganymede from Track B (in Band 6) as the primary flux calibrators, respectively. This is numerically equivalent to run the task fluxscale on all experiments combined, using the SSO in one of the experiments as primary calibrator (and correcting for the T_sys for each source). The significance of this correction is given by the ratio ${S}_{\tau }^{a}/{S}_{{QA}2}^{a}$ (columns 6 and 7 in Tables 9 and 10). Although in most cases this ratio is just a few percent, in some cases it can be as high as 10%–20%. These large corrections can be explained by a combination of significant air mass difference between the primary flux calibrator and sources observed at low elevations (typical in VLBI observations) and the flux variability of QSOs used as primary flux calibrator.

To assess the accuracy of the flux density calibration, the measured flux densities may be compared with values derived from independent flux monitoring done at ALMA. Specifically, every ∼10 days, the ACA observes bright reference sources (called Grid Sources or GSs) along with SSOs, with the goal of monitoring the flux evolution of the GSs, anchored to the modeled fluxes from the observed SSOs. The use of GSs as flux calibrators in regular observations is expected to provide a 5% to 10% absolute flux calibration uncertainty. Some GSs were also observed during the ALMA-VLBI campaign either as ALMA calibrators or VLBI targets. As a reference, the list of observed sources considered as GSs at the time of writing are: 3C273 (J1229+0203); 3C279 (J1256-0547); 3C84 (J0319+4130); 4C01.28 (J1058+0133); 4C09.57 (J1751+0939); J0510+1800; J0750+1231; J1229+0203; OJ287 (J0854+2006); QSO B0003-066 (J0006-0623); QSO B1730-130 (J1733-1304); and QSO B1921-293 (J1924-2914). This allows a direct comparison between fluxes measured during the ALMA-VLBI campaign and archival GS fluxes measured close in time. Figure 16 shows such a comparison, which can be used to identify possible systematic trends (e.g., variability of the flux calibrators). The expected flux density of GSs at a given time and frequency can be retrieved from the ALMA archive via the getALMAflux() function implemented in the CASA analysis utils. The flux values of the sources observed either in ALMA-mode or APS-mode are estimated in the uv-plane using the CASA task fluxscale, which adopts a point-source model. This assumption is correct since GSs are unresolved on ALMA baselines.³⁵ Tables 9 and 10 report the measured flux values (per SPW) for all sources observed in Band 3 and Band 6 (respectively) during the 2017 ALMA-VLBI campaign and, when available, the archival flux values for GSs. The flux values per SPW are opacity corrected using Equation (22), whereas flux values at the representative frequency are reported both with and without correction, respectively. In the following analysis, we consider only the opacity corrected flux density values. The ratio between the measured and the archival flux values can then be used to identify possible systematics, and eventually improve the flux calibration. The tables also report the time difference between the VLBI observations and the archival entry. In the following, we consider only ACA measurements within 10 days from the VLBI observations for a meaningful comparison.

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In Band 3, experiments 2016.1.00413.V and 2016.1.01216.V used a SSO, Callisto, as the absolute flux density calibrator. While for 2016.1.01216.V the flux measurements are generally consistent with the archival values (within the expected 5% uncertainty), sources in 2016.1.00413.V have flux values apparently lower (9%–14%), which may point to a ∼10% systematic error. Experiment 2016.1.01116.V used J0510+1800 as the absolute flux density calibrator, and shows the most significant discrepancy seen in Band 3, where OJ287 appears to have a flux 15% lower than expected.

In Band 6, tracks B and C used a SSO, Ganymede, as a flux calibrator, and the flux measurements are generally within 10% of the archival values. Exceptions are QSO B0003-006 and 3C84 in Track C and J1058+0133 (4C01.28) in Track B, which appear to be 18%, 12%, and 17% weaker than the archival predicted values. While we cannot exclude intrinsic variability in these sources, we note that they have only 1–2 T_sys measurements per track, making the T_sys correction to the flux scale less reliable. Another exception is 3C273 in Track B, which appear 15% higher than the archival predicted value, possibly pointing to variability. In tracks D, A, and E, 3C279 was used as flux calibrator. To provide a self-consistent calibration among the different tracks, the flux-density estimate for 3C279 in track B (which used Ganymede) was used in tracks D, A, and E (under the assumption that it remained constant across that week). ACA observed 3C279 in Band 3 on Apr 1 and in Bands 3 and 7 on Apr 13. These measurements can be used to predict a Band 6 flux value at the representative frequency of 221 GHz, which appears to be lower than the measured value in Track B (8.91 versus 8.21 Jy, corresponding to a <10% difference). After using Equation (22), the opacity corrected flux-density values for 3C279 (bootstrapped from Track B) are 8.96, 9.39, 8.51, and 8.19 in Tracks D, B, A, and E, respectively. This indicates that the flux-density of 3C279 has apparently varied by 13% during the EHT campaign (note the consistency between the ACA measurement and the bootstrapped value towards the end of the campaign). As additional confirmation of the goodness of the flux calibration, GS sources in Tracks D, A, and E do not show discrepancies larger than about 10% with respect to the archived measurements (within 10 days from the VLBI campaign), with the exception of J0750+1231 (ALMA target) in track E, which is 14% weaker than the archival value (note that also this source had only 1 T_sys measurement in the full track).

Besides the GS sources, Figure 17 shows also the time evolution of the fluxes measured for the non-GS sources observed in the EHT campaign, which appear to have constant flux density values over the course of the observing week (only Sgr A* and J0837+2454 show an appreciable variability).

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It is worth noting that the flux-density values obtained during QA2 from the fluxscale task (Col. 7 in Tables 9 and 10) show larger (often >10%–20%) discrepancies with the archival predicted values (Col. 8 in Tables 9 and 10), with respect to the corrected values using Equation (22). In addition, there are also some apparent systematic trends, where Tracks B, C, and D exhibit a tendency toward lower fluxes compared to those measured in tracks A and E (between 10 and 20%), Track C has generally lower values than B (of the order of 10%), and Track E shows an apparent systematic increase in flux values with respect to Track A (also of the order of 10%). These apparent systematic trends are reconducible to both the secondary opacity effects described in Appendix A.1 and the flux variability of 3C279, and are fixed after correcting the flux estimates with Equation (22) and Ganymede in Track B as primary calibrator. The final flux-density values for SPW 0 to 3 are reported in columns 2 to 5 of Tables 9 and 10, respectively.