ALMA High-frequency Long-baseline Campaign in 2019: Band 9 and 10 In-band and Band-to-band Observations Using ALMA's Longest Baselines

Our experiments were made during the winter months 2019 June and July in Chile, where phase stability conditions are generally improved compared to the summer months (Maud et al. 2023), and to directly coincide with the longest baseline array configuration offering baselines up to 16 km. Table 1 provides the overview of the 13 experiments analyzed as part of this work. The in-band observations were set up like standard ALMA observations at band 9 or band 10, while the corresponding B2B modes were arranged to use the low frequencies of band 4 or 6 for band 9 calibration (denoted B9-B4, B9-B6) and band 3 or 7 for band 10 calibration (denoted B10-B3, B10-B7). In total, four experiments were made at band 9 and one at band 10 for in-band mode. The band pairs for the B2B observations are indicated in Table 1.

These observations were the final experiments planned as part of the HF-LBC-2019 and were run on the telescope using scheduling blocks (SBs) produced with the ALMA Observing Tool (OT; Biggs & Warmels 2018) in the exact same manner as PI observations. ⁹ Each observation is recorded as an execution block (EB) with a unique identification ID. Our observations included all required calibrators to ensure the full calibration of the science target, including bandpass, flux, phase, and check source calibrators for the in-band observations, and the inclusion of the DGC source for the B2B mode (see Asaki et al. 2020a). It is worth remembering that check sources are included in all ALMA high-frequency user observations as to provide a means to "check" the phase transfer. These should be located equidistant from the phase calibrator as the science target is. In our experiments, due to the difficultly of finding high-frequency QSOs that are bright, we do not adhere to this rule for check sources and simply include them in the process of forming correct SBs. We highlight that they are farther away from the phase calibrator than the target sources are.

The B2B observations use the so-called harmonic frequency switching, which facilities the change in frequency within ∼2−3 s and avoids a ∼20 s delay per switching event (nonharmonic mode). The DGC source was included at the start of the observation, after ∼50 minutes (roughly marking the middle of the observation), and once again at the end for observation. In some cases, observations were terminated early but still have two DGC source visits. These so-called DGC source blocks are ∼10 minutes long and consist of 10 low-frequency 18 s duration scans interleaved with nine high-frequency scans of 32 s duration (see Asaki et al. 2020a; Maud et al. 2020). For the B2B mode, an independent bandpass source was not observed for the low-frequency band as the DGC source low-frequency scans are suitable for this purpose.

We use phase-referencing switching cycle times shorter than the typical ALMA long-baseline setup of ∼76 s (54 s on target, 18 s on the phase calibrator, ∼4 s slew times). Irrespective of in-band or B2B mode, the cycle times were the same, ∼56 s for band 9 and ∼51 s for band 10, where the changes were made to shorten the target scan durations to 30 s and 24 s for bands 9 and 10, respectively.

The SBs were arranged to cover a range of local sidereal times (LST) in order to maximize the scheduling feasibility of these tests in between Cycle 6 PI observations. The main requirements for an observation to be conducted were that the precipitable water vapor (PWV) content was below the ALMA requirements (0.658 and 0.472 mm for bands 9 and 10, respectively), that the observation was started at least ∼1 hr before the LST where the target sources would drop below ∼30° elevation, and that the short-term phase rms was <1 rad as measured by the so-called ALMA go/no-go test ¹⁰ (see Section 3.1.1). Band 10 observations were arranged only in the LST range between 21:0 and 01:00 hr where strong enough QSOs to act as the target and calibrators for our in-band tests could be found. The band 9 tests covered three ranges, 13:00–16:00, 16:00–20:00, and 20:00–02:00, as it was easier to find suitable QSO combinations. Paired in-band and B2B observations were arranged to run one after the other to ensure the best continuation of weather and phase stability conditions such that a fair comparison of the two modes could be made, although not all of our experiments were continuous due to time constraints (see Maud et al. 2020). The QSOs selected as targets and calibrators were known suitable sources based on our HF-LBC-2017 studies and allowed us to use phase-calibrator-to-target separations angles between 07 and 38. Given the testing time allocated to the HF-LBC-2019 experiments, we observed the same source groups numerous times, but on different days and hence in different conditions. Of the 13 observations, the in-band data used phase calibrators with separation angles of either 37 or 38 to the target, while the B2B observations predominantly used separation angles of either 07 or 08, with one observation using 3.7°.

For both the in-band and B2B modes, there was one observation setup (EBs ending X70fe, Xdac, and X7682) that used the same QSO as both the science target and the phase calibrator ¹¹ , i.e., the phase calibrator and target are identical. This provided a case to investigate a 00 phase-calibrator-to-target separation angle such that only temporal phase referencing was conduced without any position switching. The list of sources observed for each of our experiments and any specific data reduction notes are given in Table 2. As highlighted in Table 2, the above mentioned EBs observed the check source J1720−3552, which, at the time, the coordinates were found to be in error by ∼57 mas considering our observations were the most up to date and highest spatial resolutions ones made.

2.2.1. Spectral Configuration Issues

Data reduction was made using slightly modified scripts that were produced by the ALMA Quality Assurance script generator (Petry et al. 2020; Petry 2021) using casa (CASA Team et al. 2022) version 6.2.1. The in-band data were processed almost exactly as per standard ALMA observations including applying system temperature correction, water vapor radiometer (WVR) solutions, updated antenna positions, and standard bandpass and phase corrections. Due to the weaker nature of high-frequency phase calibrators, the signals for all eight SPWs were combined in order to boost the signal-to-noise ratio (S/N).

The main deviation from standard PI calibration for the in-band data was that we used a single flux calibrator for amplitude correction because of the difficulty in finding a secondary amplitude gain calibrator at high-frequency bands (Maud et al. 2020). Typically, for standard data reduction, the phase calibrator is used as a secondary gain calibrator where amplitude gains are solved temporally, but bootstrapped from the main flux calibrator. We use the single flux calibration method for both in-band and B2B modes as to provide a consistent amplitude calibration process.

For all of our observations, the WVR solutions are applied by default as these generally help to improve phase stability on the shortest integration timescales. For higher-frequency observations (bands 9 and 10), the typical improvement factor is between 1.1 and 1.3 when comparing historical data (see Maud et al. 2023). All phase rms values subsequently reported are after WVR corrections have been applied. The phase rms remaining in all observations after WVR application are composed of uncorrected tropospheric fluctuations that, at high frequencies, are likely caused by variations of the refractive index, i.e., dry air component, which is totally unmeasured by the WVR system (see also Nikolic et al. 2013; Maud et al. 2017).

B2B data reduction follows essentially the same method as in-band calibration with the addition of the DGC step. The DGC source is used to solve the phase offsets between the low-frequency calibrator and the high-frequency target data. Furthermore, we use the interpolation option "linearPD" in the applycal task within casa to correctly scale the low-frequency phase calibrator solutions to the high-frequency scans of the target. To solve the DGC offset, the DGC source low-frequency phase solutions are applied to the high-frequency scans using "linearPD" interpolation as to correct the temporal phase variations. Subsequently, the now-corrected high-frequency DGC source scans are grouped from each DGC block (start, middle, and end) and then solved. The result is the DGC phase offset solutions per high-frequency SPW and per polarization at three time intervals ¹³ . The solutions for each DGC block are applied to the target data with a linear interpolation in time. The DGC offsets are typically stable with time, although slight changes of the order of 10°–20° can be expected due to long-term phase drifts between the high- and low-frequency bands (Asaki et al. 2020a). Despite the aforementioned issues with the low-frequency spectral setup, the low-frequency basebands from the same signal path were matched with the corresponding high-frequency basebands in order to transfer the low-frequency calibrator solutions to the high-frequency data of the target. For further information about B2B mode data reduction, please see Asaki et al. (2020a) and Maud et al. (2020). As noted in Table 2, in some experiments we used the bandpass or DGC source as the primary flux calibrator rather than the independent flux calibrator because the latter was not always matched between the observations at different epochs, and we require the use of the same flux calibrator across our different observations to facilitate flux consistency checks (see Section 3.5).

After the phase calibration, as described above, we also self-calibrate the target sources as to compare with what we achieve from phase referencing. We note that the self-calibration does not use the lowest solution interval of the integration time (2 s for these data) due to the low S/N on some targets (see Cornwell & Wilkinson 1981; Brogan et al. 2018), but instead we generate a solution per target scan time (30 s and 24 s for bands 9 and 10, respectively). As such, there may be very minor short-term phase decoherence remaining (see also Sections 3.1.2 and 4).

2.3.1. Imaging

Below we separate the assessment of the phase stability conditions and the achieved image coherence in addressing goal 1.

3.1.1. Assessment of Phase Stability Conditions

Before observing, go/no-go stability checks were run in order to assess the phase stability conditions. On June 4, the astronomer on duty instead assessed the stability using the bandpass observations from a previously executed testing observation (which are not available for public release) rather than using a go/no-go. On July 5, two go/no-go tests were run, as the conditions appeared marginal but improved thereafter. We specified a "go" to observe when the phase rms was <1 rad (∼573) measured on baselines >80th percentile (P80, typically >7 km) for the 2 minute stare at a QSO during the go/no-go test. Based upon Maud et al. (2022), in the long-baseline array configurations, the phase rms for baselines >1500 m follows $\sim \sqrt{\mathrm{time}}$ if measured over <2 minute periods. Our 2 minute stability limit corresponds to <35°–40° phase rms for the phase-referencing cycle times of 51 and 59 s used in the band 10 and band 9 experiments. The expected image coherence should be >0.7 when adhering to this strict phase rms criterion (see Equation (1) in Maud et al. 2022). Table 4 lists the go/no-go checks and the phase rms over the 2 minute observation. We note that any elevation differences are not accounted for (see Butler 1997; Holdaway 1997; Maud et al. 2020), but that our targets are generally within 10° elevation of the QSOs used in the stability tests.

Table 4. Details of the Go/No-go Checks Run before the Observations

Execution Block ID	Date	Source	Elevation	Phase rms (over 2 minutes)
			(deg)	(um)	at Band 9 (deg)	at Band 10 (deg)
uid://A002/Xdd0d37/Xa9ff	2019-06-03	J1229 + 0203	65.0	34.9	28.5	35.9
... a	2019-06-04	J1256−0547	67.5	49.4	39.7	50.8
uid://A002/Xdd4cf3/X5e6f	2019-06-07	J1229 + 0203	61.7	40.7	32.6	41.9
uid://A002/Xde63ab/X5c49	2019-07-05	J2253 + 1608	50.4	72.6	58.4	74.6
uid://A002/Xde63ab/X5d67	2019-07-05	J2253 + 1608	50.8	65.7	52.8	67.5

Notes. Phase rms is post-WVR application and can be scaled to different timescales using $\sqrt{\mathrm{time}({\rm{s}})/120\,{\rm{s}}}$. Given the prescription of assessing the median phase rms of the longer baselines, a spread of ± 20% is not unlikely for the phase rms uncertainty. Elevation is the median value over the 2 minutes stare.

^a Internal testing data were used that are not for public release and cannot be identified.

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To provide a direct measure of the phase stability at the start of all of our experiments, we use the bandpass source scan. Following Maud et al. (2022), we calculate the phase rms extracted over the time period equivalent to the phase-referencing cycle time, which is a good proxy for understanding what the remaining phase residuals would be after ideal phase referencing. Figure 1 shows the SSF as produced from the bandpass source of the band 10 in-band EB ending X105d2. The gray points show the SSF (baseline length versus phase rms) for the total observing time of the bandpass source, ∼300 s, while the red points indicate the SSF produced by selecting chunks of time equal to the cycle time for the phase rms calculation (51 s). The black bar represents the median phase rms as established on baselines >P80 (∼7 km).

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The last two columns in Table 1 show the phase rms in units of microns and in degrees with respect to the observing high frequency as measured over the cycle time. The values are corrected for the target source elevation. We scale the phase rms by sin(EL_bp)/sin(EL_tar), where EL is the elevation in degrees, to account for the geometrical effect of longer or shorter lines of sight through the atmosphere (see Butler 1997; Holdaway 1997; Maud et al. 2020). Considering the elevation-corrected phase rms as the proxy for the phase residuals that would remain in our targets after phase referencing, and based only on the underlying stability of the atmosphere, we expect our target images to have coherence values, i.e., an expected coherence, of >0.8 (using Equation (1) from Maud et al. 2022) after successful phase transfer.

3.1.2. Achieved Coherence

In Figure 2 we plot the comparison between the achieved image coherence, of the target sources in the left panel and the check sources in the right panel, against the expected coherence as calculated from the phase rms. Image coherence values should be considered as upper limits as the comparison self-calibration images use the scan timescale, which averages the target visibilities over 30 and 24 s for bands 9 and 10, respectively. This is because short-term phase variations are not fully corrected in the scan-based self-calibration, such that when considering the phase rms (Section 3.1.1), we would expect our reported self-calibrated fluxes to be ∼8% lower on average (also see Section 4) as compared to integration timescale self-calibrated images. The yellow and green colors indicate band 9 and band 10 data, while the circles and squares represent in-band and B2B modes, respectively. The dashed line in both panels is that of equal value. In the left panel, the three larger symbols are those where the target and phase calibrator are identical (EBs X70fe, Xda5c, and X7682), while the two outlined symbols are the band 10 observations with the same setup using a target-to-phase-calibrator separation angle of 37 (EB X105d2 and Xf3ab).

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All but one of the five in-band observations have image coherence values <0.6, although >0.8 is expected. The only in-band observation to meet the expected value is EB X70fe, where the target and phase calibrator are identical (temporal referencing only, larger symbol). All but two of the eight B2B mode observations achieve an image coherence of >0.69, close to our pragmatic limit of 0.7. Of these, two EBs are where the target and phase calibrator are identical (larger symbols), while the other four observations use closer calibrators, <1°, when compared to the in-band observations. The remaining two B2B EBs with low coherence comprise the band 10 observation that purposefully used a more distant calibrator (Xf3ab—37) and X13757, which was observed down to very low elevation (see Appendix A.1).

In the right panel, we see that all check sources have an image coherence much lower than would be expected considering the phase rms. However, these check sources are farther away from the phase calibrator (22–65) when compared with the target sources per observation.

Table 5 reports the image parameters for the target sources, while Table 6 gives them for the check sources. The fitted flux is measured by fitting 2D Gaussian to the source, while the area flux is measured by integrating over a circular aperture with a 15 pixel radius centered on the source. In a number of cases, the larger area fluxes, when compared to the fitted fluxes, correspond to the spreading of source emission away from a pointlike structure as a result of calibration (phase) defects. For long-baseline observations, the aim is to resolve and correctly image small-scale structures around complex scientific targets, and hence it is imperative that the phase correction is accurate and that emission is not spread around the map (see also Section 3.4).

Table 5. Target Source Image Parameters from the HF-LBC-2019 Experiments

Exec. Block	Target	Phase Referenced					Coherence	Peak/Flux	Self-calibrated
Suffix		Beam	Peak	Fitted Flux a	Area Flux b	Noise			Peak a	Fitted Flux a	Noise
		(mas)	(mJy bm−1)	(mJy)	(mJy)	(mJy bm−1)			(mJy bm−1)	(mJy)	(mJy bm−1)
In-band—Band 9
Xb0b8	J1259−2310	8 × 7	64.29	83.36	151.52	0.59	0.34	0.42	186.72	191.17	0.30
X62b6	J1259−2310	9 × 7	105.73	131.66	148.89	0.50	0.56	0.71	188.99	192.49	0.31
X70fe c	J1713−3418	9 × 6	160.42	164.58	167.53	0.29	0.91	0.96	176.47	178.19	0.26
X64c3	J2229−0832	9 × 7	48.36	172.74	189.60	0.88	0.14	0.26	341.27	345.68	0.37
In-band—Band 10
X105d2	J2229−0832	8 × 5	67.34	88.72	135.39	1.05	0.34	0.50	193.71	198.79	0.83
B2B—Band 9-4
Xda5c c	J1713−3418	9 × 8	151.18	149.10	144.71	0.49	0.85	1.04	177.53	178.12	0.31
X13757	J1259−2310	12 × 8	75.49	86.52	99.09	1.07	0.46	0.76	161.95	163.41	1.09
X6945	J1259−2310	12 × 7	127.90	133.49	139.71	0.61	0.69	0.92	185.37	182.68	0.50
B2B—Band 9-6
Xc5f1	J1259−2310	10 × 8	118.52	119.09	122.74	0.52	0.69	0.97	171.73	168.99	0.35
X7682 c	J1713−3418	14 × 7	139.62	144.80	151.13	0.53	0.81	0.92	171.71	174.60	0.45
X5dcd	J2229−0832	7 × 7	242.64	245.91	236.42	0.78	0.72	1.03	337.69	324.54	0.44
B2B—Band 10-3
X7fe3	J2229−0832	7 × 6	189.28	196.95	197.09	1.02	0.79	0.96	238.87	238.86	0.73
B2B—Band 10-7
Xf3ab	J2229−0832	7 × 5	108.56	114.73	183.40	1.22	0.44	0.59	246.31	244.91	0.78

Notes.

^a Fitted flux from a 2D Gaussian fit to the central target source. ^b Integrated flux within a 15 pixel radius circle located at the center of the image—used for peak/flux ratio that is representative of how pointlike the image is, under the assumption that all of the flux is recovered within the circle. ^c Recall that the target source and phase calibrator are identical (0° separation angle, only temporal phase referencing). Beam reported to the nearest whole milliarcsecond.

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Table 6. Check Source Image Parameters from the HF-LBC-2019 Experiments

Exec. Block	Target	Phase Referenced					Coherence	Peak/Flux	Self-calibrated
Suffix		Beam	Peak	Fitted Flux a	Area Flux b	Noise			Peak	Flux	Noise
		(mas)	(mJy bm−1)	(mJy)	(mJy)	(mJy bm−1)			(mJy bm−1)	(mJy)	(mJy bm−1)
In-band—Band 9
Xb0b8	J1259−2310	8 × 7	40.79	54.41	113.89	0.94	0.27	0.36	152.82	165.12	0.67
X62b6	J1259−2310	9 × 7	78.42	103.38	115.58	0.86	0.52	0.68	152.03	154.76	0.72
X70fe	J1713−3418	9 × 6	115.76	121.91	124.82	0.79	0.76	0.93	152.95	154.81	0.68
X64c3	J2229−0832	9 × 7	4.61	... c	29.73	0.84	0.04	0.15	117.34	115.90	0.68
In-band—Band 10
X105d2	J2229−0832	8 × 5	14.82	42.84	53.97	1.63	0.13	0.27	116.86	134.67	2.10
B2B—Band 9-4
Xda5c	J1713−3418	9 × 8	102.12	109.95	124.68	0.94	0.68	0.82	150.68	159.50	0.76
X13757	J1259−2310	12 × 8	45.18	213.39	199.17	4.40	0.08	0.23	568.78	598.96	2.47
X6945	J1259−2310	12 × 7	138.34	223.17	307.44	3.05	0.21	0.45	646.54	609.74	1.33
B2B—Band 9-6
Xc5f1	J1259−2310	10 × 8	104.18	451.37	456.89	3.40	0.16	0.23	670.66	687.83	0.99
X7682	J1713−3418	14 × 7	50.12	66.77	75.78	1.24	0.31	0.66	161.60	162.19	1.07
X5dcd	J2229−0832	7 × 7	85.15	97.23	121.71	0.94	0.46	0.70	186.57	175.11	0.65
B2B—Band 10-3
X7fe3	J2229−0832	7 × 6	95.57	116.51	130.79	1.76	0.54	0.73	176.17	176.02	1.59
B2B—Band 10-7
Xf3ab	J2229−0832	7 × 5	38.78	54.24	76.92	1.81	0.29	0.50	132.25	135.17	1.60

Notes.

^a Fitted flux from a 2D Gaussian fit to the central Check source. ^b Integrated flux within a 15 pixel radius circle located at the center of the source—used for peak/flux ratio that is representative of how pointlike the image is, under the assumption that all of the flux is recovered within the circle. ^c The fitting did not converge due to a poor image. Beam reported to the nearest whole milliarcsecond.

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Based upon the metric of the phase rms, we expected high image coherence for all of our observations (>0.8). The fact that we do not see direct one-to-one correlations in expected and image coherence means that we need to carefully assess the degradation imparted due to the calibrator separation angle (Section 3.4), which is the only free parameter of these observations. Maud et al. (2020) expected a coherence loss of ∼0.2 for long-baseline observations made in band 7 (∼345 GHz) when using calibrators at ∼4° separation angle from the target source. In extrapolating their results to band 9 and 10, they reported that achieving >0.7 image coherence may not be possible. To address goal 1 outlined in Section 1, we report that even when the phase stability conditions are correct, and optimal, it is not a given fact that we will achieve the expected image coherence, and that a more careful consideration of the target-to-phase-calibrator separation angles is required.

Figure 3 shows the pairs of in-band and B2B data sets that targeted the same source but did not necessarily use the same phase calibrators. The circles represent the observations taken immediately after one another on the same night, while the squares show those taken on different nights. Yellow and green colors again represent band 9 and band 10 data, respectively, while the larger symbols highlight the observations where the target and phase calibrator are identical, and the circled symbol highlights the band 10 experiments using the phase calibrator 37 from the target.

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The B2B data sets typically have a higher image coherence. Except the unique data sets mentioned above, all in-band observations use phase calibrators farther from the target source than the paired B2B observations (e.g., Maud et al. 2020). We note that the B2B EB X13757 has much lower-than-expected coherence, which is due to the low source elevation during the latter half of the experiment (see Appendix A.1).

For the experiments where the phase calibrator and target are identical (larger symbols), the image coherence is high, and the in-band and B2B values are within a 0.1 of each other. This supports the findings of Maud et al. (2020, 2022), who reported that the in-band image coherence will be marginally higher than the B2B image coherence, as the DGC step can cause a slight degradation for the B2B mode.

In our band 10 experiments that use the phase calibrator at 37 (in-band X105d2 and B2B Xf3ab, circled symbol) the B2B image coherence is slightly higher, but still within 0.1 of the in-band value. Despite selecting optimal calibrators, we did use SPW combination to achieve a sufficient S/N for the in-band data, while flagging of each data set was also slightly different. Hence, we do not over-interpret this as evidence that B2B is superior at Band 10, simply that the data reduction can cause slight, few percent, differences.

Figure 4 shows the target source images from six observations and compares the in-band (left) and B2B (right) calibration modes. In all cases except where the target and phase calibrator are identical (top—X70fe and Xda5c), the B2B data used a closer calibrator and indicate a higher image coherence and more pointlike image structure.

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In addressing goal 2 as outlined in Section 1, when comparing in-band and B2B modes, we advocate that in stable atmospheric conditions any observation using a phase calibrator at a given distance from a target source will provide target images of equal quality regardless of the mode used, in-band or B2B. It is apparent that the observations using the closest phase calibrators will always provide the most optimal phase correction, and the closer calibrators will be provided by the B2B mode.

In order to understand the quality of the phase calibration, other than as measured from image parameters, we examine the average phase offset and phase rms, per antenna, of the target sources after phase referencing. These parameters from a subset of six observations are shown in Figure 5. The left panels are in-band observations while the right panels are the paired B2B observations of the same target source. The gain tables from which the values are extracted were produced from the self-calibration solutions using the scan time for the target sources after phase referencing. For plotting clarity, we extract only the solutions of all antennas with the reference antenna, although all baselines follow the same trend. Ideally, after phase calibration the targets should be point sources at the pointing center such that the gain solutions should have a zero degree average phase offset and where the phase rms was reduced down to the expected level when considering the atmospheric variations (Section 3.1.1). In each plot the upper section indicates the phase rms while the lower section shows the phase offset, both as a function of baseline length from the reference antenna. The gray bars show the median values, while the dotted gray line for the phase offset is the standard deviation when considering the averages of each antenna. The dashed black lines are the expected values that (for the phase rms) are taken from Table 1, while for the phase offset, the expected value is 0°. Antenna-averaged phase offsets and residual phase rms values from all experiments are listed in Table 7.

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The ideal case is illustrated by the in-band EB X70fe and the B2B EB Xda5c in the top panels of Figure 5 where the target and phase calibrator are identical. Both plots show a near 0° phase offset and a median phase rms slightly better than expected. We acknowledge that the apparent improved phase rms in these cases is likely due to the fact we use scan-based solutions rather than the integration time (as due to S/N requirements). Thus, we likely miss some of the real (i.e., not S/N related) short-term, seconds-long timescale variability. The B2B mode has a larger scatter in the phase offset values that is likely imparted from the DGC step (also see Maud et al. 2020). The phase offset scatter is also larger in the B2B EB X7682, which is not shown but where the target and phase calibrator were also identical.

For the band 9 data, we show the in-band observation Xb0b8 and the B2B observations Xc5f1 in the middle panels of Figure 5. These both have phase offsets with an average close to zero degrees with a similar spread. However, the phase rms is almost 90° for the in-band observations, significantly higher than the expected ∼27°, while it is ∼46° for the B2B observation. The B2B data do not achieve the expected value of ∼28°, but the phase rms is over a factor of 2 better than the paired in-band observation. Given the finite calibrator-to-target separation, it is not unreasonable to consider that the larger difference in sky position from the target to phase calibrator hinders the phase-referencing effectiveness. Regardless of using the scan-based solutions, the phase rms in these cases is likely dominated by the suboptimal calibration creating considerable scatter rather than any underlying small timescale variability. Note that the B2B observation may also have a higher-than-expected phase rms because the source elevation in the latter half of the observation is low (<40°), such that the expected phase rms is likely underestimated when scaled from the bandpass scan in comparison to the median target elevation. The two-lowest elevation B2B observations X13757 and X6945 (not shown) both have average phase offsets of ±20°, which could also be a consequence of the low elevation impacting the DGC solutions.

At band 10 we compare the in-band EB X105d2 and the B2B mode using a closer calibrator in EB X7fe3 in the bottom panels of Figure 5. Both have a similar average offset with a similar spread. Most noticeable is that the in-band phase rms is >70° after phase referencing and points to a coherence loss of ∼0.5. By comparison, the B2B observation phase rms is less than half the in-band value (∼36°), although it is not quite as low as expected (∼21°). The target source elevation is >45° during this observation, and thus the higher-than-expected phase rms can only be attributed to the finite separation of the target and phase calibrator, despite it being <1°. The coherence calculated from the phase rms remaining in the target data after B2B phase referencing is 0.81 and is comparable with the image coherence achieved of 0.79. We note that the band 10 B2B EB Xf3ab (not shown) that uses the same calibrator setup as X105d2 (separation from target of 37) also has a similarly poor phase rms (∼70°). This suggests such a separation angle is already too distant to provide any useful reduction in phase rms when employing phase referencing for band 10 long-baseline observations. The phase offset for Xf3ab is also nonzero, but is ∼20° better than X105d2. Nonzero phase offsets for the in-band observations could point to underlying antenna position uncertainties and cannot be excused as caused by the DGC step as for B2B data. That said, DV01 was used for the reference antenna in all data, and the phase offsets do not appear to be biased across the different observations. We acknowledge the fact that at the highest frequencies the antenna position uncertainties will be more acute and hence act to create more defects in the image. We can only state that careful and accurate measurements of the antenna positions are important when conducting such high-frequency observations.

To address goal 3 outlined in Section 1, we find that only the observation with closest calibrators, those using B2B mode in this case (excluding our experiments where the target and phase calibrator are identical), can provide a reduction in phase rms to almost the expected level. When the phase calibrators are too distant, as has been shown before by Asaki et al. (2016) and Maud et al. (2020, 2022), phase referencing is ineffective at notably reducing the phase rms and combating decoherence. The phase offsets averaged over all antennas are generally within 10° of zero for all of our observations, supporting that the phase referencing accurately positions the source, on average, to the pointing center during imaging.

Figure 6 shows the image coherence of both the target and check sources as a function of the target-to-phase-calibrator (and check-to-phase-calibrator) separation angle. Yellow and green represent band 9 and 10 observations, while the circles and squares indicate in-band and B2B modes, respectively. The smaller symbols are those for the check sources. The lower limit bars illustrate the coherence we might expect if we had compared with the integration self-calibrated images (we increased the self-calibrated image peak flux to account for the expected reduction when considering the phase rms). The observation where the target and phase calibrator are identical provides an anchor point at 0.0° (X70fe, Xda5c, X7682). For these observations, we see a loss of coherence from only the temporal phase-referencing process before we even considering moving to a different sky location. For these three data sets, we see that the in-band experiment has the best image coherence and corresponds to that expected when calculated from the phase rms (see Table 1). The two B2B observations are slightly worse, and we attribute the extra image coherence reduction to the DGC step (see also Maud et al. 2020). When moving to larger separation angles, it is clear that the image coherence degrades rapidly. Within the parameter space of these experiments, the degradation in image coherence can be approximated to a linear fit (intercept = 0.8, slope =−0.109 ± 0.014). We do not separate the band 9 and 10 data, nor the in-band or B2B modes given the low number statistics.

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To address goal 4 outlined above, considering that all of our observations were taken in stable atmospheric conditions (Section 3.1.1), we can already quantify that with a coherence requirement of >0.7 (Asaki et al. 2020a, 2020b; Maud et al. 2020), 16 km baseline high-frequency observations (>660 GHz) with ALMA need to use phase calibrators within ∼1°. Compared with the extrapolated results from Maud et al. (2020), who predicted band 9 observations up to maximal baselines of 8.5 km require a 1° separation angle, we find that reality is more optimistic. If a reduced image coherence is acceptable when considering the operational difficulties of such high-frequency long-baseline observations (see Section 4.4), then image coherence levels of ∼0.6 and ∼0.5 are expected for separation angles at ∼2° and ∼3°, respectively. Further investigation on separation angle effects are detailed in Section 4.

The absolute flux accuracy of ALMA at the higher-frequency bands is quoted as 20% in the documentation ¹⁴ . Considering the fluxes after self-calibration, regardless of the initial observing mode, at band 9: J1259−2310 ranges from 163.41−192.49 mJy, J1713−3418 from 174.60−178.19 mJy, and J2229−0832 from 324.54−345.68 mJy, while at band 10 the only target J2229−0832 ranges from 198.79−244.91 mJy. The percentage differences when considering these minimum and maximum values are 16.0% for J1259−2310, 2% for J1713−3418, and 6% for J2229−0832 at band 9, which are all within the accuracy reported by ALMA. The difference is 21% for J2229−0832 at band 10. We did ensure the use of the same calibrators for the amplitude scaling, but note that the fluxes extracted from the ALMA catalog did vary depending on observing date. It is also important to note that our self-calibration process used the scan time for these targets, not the integration time where the S/N was significantly reduced, and hence the fluxes of the targets after self-calibration should be regarded as lower limits. We do see some minor residual extended flux in the band 10 image of J2229−0832 from the in-band observation X105d2, which, if accounted for, boosts the flux to ∼220 mJy and thus would be within ALMA's prescribed uncertainty. Using the estimated phase rms and scaling between the integration and the scan times for the target sources, we expect an 8% flux reduction on average compared to ideal integration time self-calibration, which could be considered as an added uncertainty in our fluxes.

We cannot comment on longer-term high-frequency flux accuracy greater than a few days, but in reality for a given ALMA array configuration and such high frequencies, any user observations would likely be completed within a short period of time while all observing parameters are suitable. Of course, a main caveat is that not all targets can be self-calibrated, and thus it is absolutely imperative that ALMA users fully understand the effects of decoherence for these challenging observations, as the target source flux without self-calibration can be >20% discrepant regardless of the absolute flux accuracy adhering to the ALMA quoted uncertainties. Only in the observations where the target and phase calibrator were identical did we achieve a target coherence >0.8, while even using close calibrators in the B2B mode and with good phase stability conditions all target sources lost more than 20% of the peak flux density.

We follow exactly the procedure outlined in Maud et al. (2020) in using the trend in antenna position uncertainties as measured by Hunter et al. (2016) to calculate phase corruptions. For completeness, as per Maud et al. (2020), the path length uncertainties ℓ(μm) are calculated as Δρ · Δθ where Δρ is the baseline position uncertainty and Δθ is the calibrator-to-target separation angle (in radians). As a function of baseline length, b, the uncertainties are: east (0.140 mm + b km · 0.071 mm km⁻¹); north (0.110 mm + b km · 0.054 mm km⁻¹); and vertical (0.220 mm + b km · 0.198 mm km⁻¹). A value is randomly attributed to each antenna, as separated from the reference antenna, from a Gaussian distribution centered on the absolute uncertainty with standard deviations in the east, north, and vertical directions of 0.086, 0.076, and 0.129 mm and 0.121, 0.118, and 0.296 mm for baselines shorter and longer than 2.5 km, respectively. Corrupting gain tables are created using gencal in casa for calibrator separation angles ranging from 1.0−70 (in 10 steps). The errors manifest as almost constant phase offsets per antenna.

The top panel in Figure 7 shows the residual phase rms and phase offset remaining for the target source after corrupting the ideal target data with the antenna position uncertainties assuming a phase-calibrator-to-target separation angle of 40. As there was no phase noise added, the phase rms is 0°; however, the phase offsets can reach > ± 20° for some antennas. The average offset considering all antennas still remains close to 0°. In contrast, the largest angular separation case observed (38; bottom panel in Figure 5) has an overall phase offset and suggests there is a more systematic antenna position uncertainty for those observations. We surmise that it is possible for the antenna position uncertainties to be more systematic across the array in contrast to random values for our corruption process.

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The solid black lines in Figure 6 indicate the reduction in image coherence with separation angle. As Maud et al. (2020) also found, the antenna position uncertainties alone cannot explain the degradation of the target images, and in fact the image coherence does not begin to fall below 0.9 and 0.8 until separations angles of ∼4° and ∼6°, respectively.

Maud et al. (2020) extensively discussed the unaccounted for degradation of their target images as caused by suboptimal phase referencing. In Figure 5 we also identified that even after phase referencing the target residual phase rms can be significantly higher than the expected phase rms. Simply, calibrating a target using solutions from a vastly different patch of sky is unlikely to provide a reasonable phase correction. To examine the effect more closely, we use a frozen-flow turbulent phase screen model (see the Appendix) to create mock observations using calibrators separated from the target by 00–70 (in 10 steps). The 00 case is that where the target and phase calibrator are identical and only temporal phase referencing occurs.

In short, each antenna is assigned absolute path length values from the screen (i.e., a phase value) for the target and phase calibrator as a function of time and position. The screen is moved at a representative speed of 5.0 ms⁻¹ (Table 1), while a position offset is also included for the phase calibrator scans as a function of separation angle from the target. The direction of the position offset can be parallel or perpendicular to the screen motion. The phase rms level is normalized to 30° over a 60 s period when observing a single source through the phase screen as to match the conditions of our experiments. The model phase screen corruptions for the target and phase calibrator are applied to the self-calibrated (ideal) data and essentially create mock raw observational data. Phase calibration is then undertaken, as would be conducted for real observations, and phase transfer is made to the target before imaging. Further details are provided in the Appendix.

In Figure 6 the yellow lines indicate the image coherence after phase referencing through a mock phase screen as a function of separation angle (dashed and solid lines represent position offsets for the phase calibrator parallel and perpendicular to the screen motion, respectively). For consistency with our observations, we calculate the image coherence compared with the scan time self-calibrated image of J2229−0832 that has a flux of 345.68 mJy bm⁻¹ at band 9 (EB X64c3, which is lower than the integrated self-calibrated value; see above). At 00 the image coherence is 0.8, as expected given the remaining phase rms over the cycle time. However, as a function of increasing separation angle, the image coherence degradation from the phase screen model is not as extreme as the observations themselves. At worst, the coherence is ∼0.6 at 7° in the case where the phase calibrator offset position is perpendicular to the screen motion. We understand that the worse image coherence, in this case, is caused by the lack of any possibility to align the target phase variations with those of the phase calibrator. For a given baseline, such a case may occur when the phase calibrator position offset direction through the screen is parallel to the screen motion because the target will see the same phase variations as the phase calibrator when they arrive some time later due to the screen motion (e.g., Asaki et al. 1998).

In the middle panel of Figure 7 the residual phase rms and phase offset remaining for the target source after corruption from the phase screen model using phase referencing with a 40 calibrator is shown. The phase rms remaining even after phase referencing is ∼50° despite the phase rms conditions setting the model at 30°. It is clear that the different lines of sight through the model screen mean that the target is not optimally calibrated, although the remaining phase rms is notably lower than in our observations (Figure 5). There is also little change in the phase offsets indicating that the screen model acts only to spread the phase offset due to underlying uncorrected long-term phase variations between the target source and phase calibrator. We do not see systematic offsets, as is also apparent in the observational data (Figure 5), suggesting that the change in phase at the positions of the target and calibrator is not significantly different.

The phase screen model itself is not the focus of this paper; however, it is worth noting caveats due to our simplifications and possible modifications for the future. The phase screen is a single-layer static frozen flow where it simply moves over the array with fixed Kolmogorov turbulent power scales (Dravskikh & Finkelstein 1979), where the line of sight of each antenna is through a single pixel in the screen, and where we assume the screen height and wind speed are representative. For certain baselines, considering the time between the target and calibrator scans and the screen motion, the target will be well corrected as the frozen in-phase variations match up for the calibrator as they were recorded for the target (see Asaki et al. 1998). In reality, a finite thickness phase screen with constant variability in all directions along with a general screen motion would likely be a more realistic representation. The turbulent power scaling is only one representation of the variations and does not always necessarily match those at the ALMA site, which do vary (Matsushita et al. 2017). We also model an idealized case of a zenith observation without any additional scaling for a different line-of-sight path between the calibrator and target source, and so we expect that elevation scaling would become more important for larger separation angles and low elevation <40°–50°. Most notably, the phase differences between the target and phase calibrator are insufficient in corrupting the model target. An increase in screen speed and/or the turbulent layer height would increase the distance, on the screen, for the lines of sight to the target and calibrator and should result in a worse calibration of the target. Ishizaki & Sakamoto (2005) used two short (300 m) test baselines at the ALMA site to measure the screen speed. Their best fit indicates that the screen speed is 1.2 times greater than a ground-based wind measurement, and as such, the small increase for our screen model is unlikely to cause sufficient additional corruption. Ishizaki & Sakamoto (2005) also reported that the screen height should generally be <1000 m, counter to what we require in the models. One possibility, as introduced by Asaki et al. (2016) to explain a notable worsening of the phase rms on the long baselines (as compared to short baselines) of their targets after phase referencing with distant calibrators, is that a second high-altitude layer (∼15–20 km) with less turbulence but a faster screen speed could provide an additional phase difference between a target and phase calibrator. We recommend a detailed parameter space investigation of such screen models for future studies.

The final case to examine is that of both the phase screen and antenna position uncertainties. The green lines in Figure 6 indicate the image coherence as caused by phase corruptions from the screen models with parallel (dashed) and perpendicular (solid) phase calibrator offsets applied in concert with the corruption in antenna positions.

For both cases with parallel and perpendicular phase calibrator position offsets, the coherence losses appear dominated by the antenna position uncertainties. Only beyond a ∼3° separation angle do we see additional coherence loss compared to what the antenna uncertainties cause alone. The perpendicular case provides a further 2%–3% coherence loss. However, neither case provides enough degradation for the image coherence to match that seen in the experiments. The overall coherence loss is not simply the sum of the corruptions caused by the antenna position uncertainties and phase screen applied alone. The bottom plot in Figure 7 shows the residual phase rms and phase offsets remaining for the target source. Comparing with the top and middle panels, we see that the bottom panel is the combination of the antenna positions dominating the phase offset, with the phase rms provided by the phase screen model. However, in comparing with the observations presented in Figure 5 for the 38 separated calibrator, we can see that the residual phase rms is lower from our models, i.e., there is not enough phase rms corruption.

In light of the above discussion, with a detailed screen model investigation, we would expect that the combined effect of the screen and antenna positions would be broadly representative of the experiments overall. Here we summarize that both components, sky position between the target and calibrator and antenna position uncertainties, must be acknowledged for high-frequency long-baseline observations despite the fact that the phase rms as parameterized by a measurement of a strong source would have suggested very stable conditions (phase rms ∼30°). At present, our current corruption models are too optimistic.

Using the Phase rms Database (Maud et al. 2023), we take all unique target sources stored (6757) and compare each with the calibrators from the ALMA calibrator catalog ¹⁶ in a target–calibrator pair matching exercise. We use a best-case scenario to calculate the required calibrator fluxes, assuming bandwidths of 15.0 and 7.5 GHz for in-band and B2B modes, respectively, 12 s scan times, and an S/N requirement of 15. For B2B we use an S/N of 45 as the scaling of the calibrator solutions also scales the noise (we use a factor of 3 for ease, but this should strictly be the ratio of the high-frequency to the low-frequency band pair). At Bands 9 and 10, we would need calibrators of 247 mJy and 638 mJy, and at band 3, 4, 6, and 7 (for the B2B modes related to bands 9 and 10 used in this work) calibrators would need to be 34, 36, 45, and 66 mJy, respectively. Table 8 shows the median separation angles and also the percentage chance that a calibrator is within 1°, 2°, and 3° (see also Asaki et al. 2020a). We highlight the effectively zero chance for finding in-band calibrators, while B2B, although still difficult, will allow a good portion of observations to be conducted.

In the short (immediate) term, we recommend that the empirical trend from the limited data we have available is the best parameterization of expected image corruption when observing in the correct conditions for PI observations. We follow the same recommendations as Maud et al. (2020) suggested from their lower-frequency data, and state that the phase rms stability should be <30° over the intended phase-referencing cycle time while the phase calibrator should be within the strict limit of 1° if we are to expect a target image coherence >0.7 for bands 9 and 10, after phase referencing (and standard WVR corrections). This should ensure that any high-frequency observation, irrespective of the target source strength, can be calibrated to a sufficiently high quality. As we have illustrated, in operations the use of the so-called go/no-go stability checks are suitable, and should continue to be used, for measuring the stability. We caution that with our limited number of test observations we do not separate in-band and B2B data and cannot comment directly on the minor negative effect the DGC step imparts. For consideration, we can assume that the fast frequency switching employed at bands 9 and 10 for the DGC scans would yield a phase rms residual <20°–25°, and when following the parameterization in Maud et al. (2020), a 0.05 difference in image coherence between in-band and B2B observations is likely.

We highlight that self-calibration can also be considered for the brightest targets, but caution that it should not be depended on a priori. ALMA undertakes quality assurance (see Petry et al. 2020) as to confirm the user requirements are met, and in some cases observations are re-queued if these are not. For high-frequency long baselines, the calibrator separation angle could be relaxed (>2°–3°) if self-calibration was guaranteed, such that the target source properties drive the final phase correction accuracy. However, if after the observation self-calibration was not possible, due to unknown changes in the target source, then the final products would be degraded or in the worst case useless due to using too distant calibrators. Indeed, a high priority for ALMA in 2023 is to include self-calibration into pipeline calibration (Hunter et al. 2023 submitted) as to achieve the most optimal products if it can be conducted. Considering the flux requirements outlined in Table 8 for pointlike calibrators, science targets at bands 9 and 10 do need to be sufficiently bright, and likely more than shown in Table 8 if they are extended (see also, Asaki et al. 2023 submitted) and dependent on the solution S/N and the self-calibration timescale. Although it is easier to find more distant calibrators, we reaffirm that the strict criteria above (i.e., ∼1° calibrator-to-target separation angle) must be adhered to, in particular for weaker targets where self-calibration is not possible.

In the long term, it may be worth considering whether a more realistic phase screen model could be built at runtime from measurable observing conditions and used to dynamically adjust the observing scenario. Maud et al. (2022) outlined how the phase rms measured as a function of time can be used to adjust the observing cycle time to meet a given requirement, i.e., using faster phase referencing to reduce the phase rms (see also Holdaway 2001) or slower referencing in stable conditions as to improve efficiency. If, in aligning measurable parameters from, for example, a bandpass-like scan, one could build a more realistic phase screen model, it could be used to assess whether with the suggested cycle times were feasible in achieving a give coherence level for the available calibrators or if another observation would be better suited to the conditions. A phase-referencing experiment could also be conducted before observations take place, similar to the go/no-go test. However, it is likely to be time consuming in requiring numerous visits to two or more QSOs on the sky in order to reasonably establish the image coherence after conducting phase referencing. Investigations into improvement of antenna position measurements may also alleviate the strict limitation on the calibrator-to-target separation angle, which could be tested in future long-baseline campaigns, while elaborate paired antenna calibration scenarios could also be envisaged.

The observation of EB X64c3 has the lowest coherence of all of the experiments. These data are in-band using one of the most distant calibrators of the sample and where the target and phase calibrator fall to lower elevations <40° for the last half of the run. During the observations it was also noted that the stability conditions were degrading, which is further supported by the fact that the phase rms is degraded as compared to the B2B EB X5dcd, which was run prior to X64c3. Excluding the latter half of the observations during imaging, the target image coherence does increase slightly to 0.18. We conclude that the combination of overall lower elevation, worsening conditions and large target-to-phase-calibrator separation cumulates to a poor final image. There is no evidence of instrumental issues.

The B2B experiment EB X13757 is reported to have the second lowest elevation (246) of all of our experiments; only X6945 is reported to have been observed down to lower values (200; see Table 1). Despite this, target images using the X13757 observation are notably worse (0.23 lower coherence) than X6945, even when considering the same measured phase stability. Upon inspection, it is apparent that X13757 is a shorter observation and that the final DGC block was not taken due to the projected elevation dropping below 20°. In comparison with X6945, the target, phase calibrator, and check source are all at a few degrees lower elevation throughout the entire observation for X13757 while the last enclosing DGC is missing. If we exclude the latter half of the observations in X13757, the image coherence increases to 0.57, and the image peak-to-flux ratio is 0.92. Thus, we conclude the poorer-than-expected parameters we report in the main text are due to suboptimal observations where the sources are at very low elevation and the final DGC block was not taken.