Selective Dynamical Imaging of Interferometric Data

Interferometric astronomical observations offer much larger resolving power than do single telescopes, with the interferometric resolution depending on the distance between the elements rather than the diameters of the individual apertures. Since an interferometer probes the Fourier transform of an on-sky source (and not the source image itself), the placement, selection, and availability of baselines to maximize coverage of the (u, v)-plane is an important and open optimization problem inherent to the interferometric image synthesis. For very long baseline interferometry (VLBI), which represents some of the most extreme interferometric observations, the arrays are generally very sparse, and optimized placement or selection of baselines becomes critical to recovering reliable information about the source.

For any ground-based interferometric observations, including VLBI, Earth rotation causes each individual baseline to trace an elliptical path in the (u, v)-plane. Such "Earth rotation aperture synthesis" can be used to improve the (u, v)-coverage of sparse arrays. Standard VLBI imaging uses this additional coverage under the assumption that the target source structure remains unchanged for the duration of the observation. For the cases of sources that vary on shorter timescales (e.g., the Galactic center, X-ray binaries, microquasars), "snapshot images" can be produced using only time intervals over which the source can be considered static (e.g., Martí-Vidal et al. 2011; Massi et al. 2012; Miller-Jones et al. 2019). In the most extreme scenarios, only instantaneous coverage may be suitable for use in static image reconstructions, severely limiting the (u, v)-coverage and resulting image quality. Such a case is the main concern of this paper. Moreover, this limitation will generally result in portions of an observation that are better than others for imaging because the snapshot coverage is time dependent.

In particular, directionally biased (u, v)-coverage results in a nonisotropic resolution that complicates interpretation of the reconstructed source geometry. The (u, v)-coverage of a given array varies with the decl. of the source, due to its dependence on the projected baseline lengths and orientations. The impact of source decl. on the (u, v)-coverage varies throughout a night of observation as the target rises and sets. A network of baselines that produces an evenly distributed (u, v)-coverage of an equatorial source will be east–west biased when targeting a northern source. Observations where the source morphology and orientation of the source are unknown cannot exploit a priori knowledge to determine whether a directionally biased (u, v) configuration will be able to reproduce source structure reliably in an image. Therefore, for cases where the source morphology is unknown, the optimal (u, v)-coverage for producing high-quality image reconstructions will be those that are approximately isotropic in angular distribution.

Periods of sustained isotropic (u, v)-coverage may allow multiple high-quality image reconstructions on short timescales to be strung together, resulting in "dynamical" reconstructions. For instance, the most basic dynamical reconstruction can be produced by joining a series of snapshot images. The quality of such a reconstruction will be time dependent owing to the evolving snapshot coverage, and it may be necessary to exclude periods with poor (u, v)-coverage. In periods of time where the (u, v)-coverage is optimal, the coverage used to produce each snapshot of a dynamical reconstruction will be sufficiently isotropic to reproduce source features on snapshot timescales, allowing for the production of reconstructions that adequately recover the evolution of highly time-variable sources. By contrast, periods of suboptimal (u, v)-coverage may be unable to provide high-quality reconstructions on snapshot timescales, making the algorithmic detection and subsequent flagging of these periods important for the production of meaningful dynamical reconstructions.

The challenges of observing a variable source with a sparse interferometer can be exacerbated by poor coverage geometry. In particular, given the shortage of available facilities supporting high-frequency radioastronomical observations, corresponding VLBI arrays exhibit substantiantial variance in coverage quality in time.

The Event Horizon Telescope (EHT) is a unique VLBI network of telescopes that exploits the full diameter of Earth and the performance at the challenging 1.3 mm (230 GHz) wavelength to achieve the required angular resolution for horizon-scale tests of general relativity for the largest black holes on the sky (see, e.g., Johannsen & Psaltis 2010; Psaltis et al. 2015; Psaltis 2019). The EHT operates in millimeter wavelengths, the optimal range that enables the resolution of the Sgr A* black hole shadow and reduces the impact of the interstellar medium scattering effects dominant for longer wavelengths, while still being observable and manageable in the radio interferometric framework. Operating at 230 GHz, the EHT achieved a resolution of ∼20–25 μみゅーas, which the EHT Collaboration (EHTC) used to produce the first images of a supermassive black hole (Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f). These data and the resulting images were used to estimate a mass of M ≈ 6.5 × 10⁹ M_⊙ for the supermassive black hole in M87 (Event Horizon Telescope Collaboration et al. 2019f), based on an observed angular shadow diameter of ∼ 42 μみゅーas (Event Horizon Telescope Collaboration et al. 2019a, 2019d, 2019f).

During the 2017 campaign, the EHT also observed the radio source Sgr A* in the Galactic center (Event Horizon Telescope Collaboration et al. 2022a, 2022b, 2022c, 2022d, 2022e, 2022f), associated with a supermassive black hole with M ≈ 4.1 × 10⁶ M_⊙ (GRAVITY Collaboration et al. 2018a; Do et al. 2019; Event Horizon Telescope Collaboration et al. 2022f). The expected mass-to-distance ratio (M/D) of Sgr A* yields a predicted angular shadow diameter of ∼50 μみゅーas (GRAVITY Collaboration et al. 2018b) and a minimum variability timescale (light-crossing time) of GM/c³ ≈ 20 s. The corresponding timescale for M87 is ∼ 1600 times longer owing to the larger mass. Indeed, structural variability of the M87 shadow has been reported on timescales from ∼1 week (Event Horizon Telescope Collaboration et al. 2019d) to several years (Wielgus et al. 2020b). The rapid minimum variability timescale of Sgr A*, combined with the extreme sparsity of the EHT, presents an urgent and unique need to characterize the effects of time-dependent instantaneous (u, v)-coverage.

In this paper, we develop a procedure for selective imaging of highly variable sources. In Sections 2 and 3, we summarize the synthetic data generation and imaging methods used herein. In Section 4, we show the limitations of imaging in sparse and uneven coverage. In Section 5, we survey several metrics capable of ranking (u, v)-coverage quality. Additionally, we derive a novel isotropy-based metric that addresses the limitations described in Section 4. In Section 6, we apply these metrics to the 2017 EHT coverage of Sgr A*, validate their ability to predict reconstruction quality from (u, v)-coverage geometry, and make recommendations for selective dynamical imaging of the 2017 EHT Sgr A* data set. In Section 7, we briefly discuss the utility of coverage metrics in ranking and selecting between different available observing periods. Finally, in Section 8, we summarize our results.

In order to test the ability of various EHT array configurations to recover source variability in different observation periods, we designed and generated synthetic data for three different models. The models are chosen owing to their structural similarity (in image and visibility domain) to expected images of Sgr A*. Similarity between the model and data was characterized as either displaying time variability or producing a Bessel function Fourier representation with nulls between 2 and 4 Gλらむだ and between 6 and 9 Gλらむだ. The models are described in Section 2.1. The synthetic data generation is expanded on in Section 2.2.

2.1. Models

Here we describe the suite of models used to test the effects of coverage on reconstructions. Examples of each model with CLEAN beam convolution can be seen in the first row of Figure 1.

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2.1.1. Rotating Elliptical Gaussian

The rotating elliptical Gaussian model is generated using a bivariate exponential with major-axis FWHM Γがんま_a , minor-axis FWHM Γがんま_b , and overall flux density A:

$$\begin{eqnarray}&&I(x,y)=\displaystyle \frac{4A\mathrm{ln}2}{\pi {{\rm{\Gamma }}}_{a}{{\rm{\Gamma }}}_{b}}\exp \left(-4\mathrm{ln}2\left[{\left(\displaystyle \frac{x}{{{\rm{\Gamma }}}_{a}}\right)}^{2}+{\left(\displaystyle \frac{y}{{{\rm{\Gamma }}}_{b}}\right)}^{2}\right]\right).\end{eqnarray} \tag{ 1 }$$

The image is padded and rotated by polar angle φふぁい after the model is generated. By default, the overall flux density A was set to 1.0 Jy. Periods for the rotation ranged between 30 and 1000 minutes (longer than one night of observation).

2.1.2. Ring and Orbiting Hot Spot

The ring model is generated by the subtraction of two concentric uniform-brightness disks, equivalent to the crescent model described in Kamruddin & Dexter (2013), with the parameters a = b = 0. The positive disk has a radius of 25 μみゅーas, and the subtracted disk has a radius of 18 μみゅーas. All ring models in this paper use these parameters with the exception of the ring model in Figure 4, which has a subtracted disk radius of 20 μみゅーas. The ring model is used with two synthetic data tests: the static ring (with no hot spot) and the dynamic ring (a static ring plus an orbiting hot spot, referred to as ring+hs). The underlying ring has a diameter of 50 μみゅーas and a flux density of 1.0 Jy. A hot spot total flux density of 0.25 Jy and an FWHM of 10 μみゅーas are added to the image, centered on the ring. After construction, the total flux density of each image is normalized to 1.0 Jy. In the dynamic ring model, the orbiting hot spot is centered on the ring and circularly orbiting at a distance of 21.5 μみゅーas with periods of 30 and 270 minutes. The static ring model is a special case of the ring+hot spot model with the flux density of the hot spot set to zero.

2.2. Synthetic Data Generation

Since the interferometric measurements are often incomplete in the Fourier domain, the inverse problem of reconstructing an image from the observed data set is usually underdetermined. Consequently, the image reconstruction requires prior information, assumptions, or constraints to derive a reasonable image from the infinite number of possibilities that can explain the measurements.

The two most popular categories of imaging methodologies are inverse modeling (e.g., CLEAN) and forward modeling (e.g., regularized maximum likelihood). See Event Horizon Telescope Collaboration et al. (2019d) for a general overview of the two methods. For time-variable sources, both approaches may allow for more effective reconstructions of dynamic structures than snapshot imaging by including assumptions or constraints on temporal variations of the source structure in addition to the spatial properties regularized in static imaging (e.g., Event Horizon Telescope Collaboration et al. 2019d). One such way imposes a temporal similarity constraint between images at different times and between each time snapshot and the time-averaged structure. In the following subsections, we briefly describe each dynamical reconstruction method used in this paper. See Event Horizon Telescope Collaboration et al. (2022c) for more details.

3.1. Inverse-modeling Approaches

3.2. Forward-modeling Approaches

In this section, we explore limitations of imaging with limited and directionally biased (u, v)-coverage. In particular, we will show that sparse (u, v)-coverage results in predictable limitations (i.e., deviations from the true source morphology), which are determined by the geometry of the (u, v)-coverage.

Figure 2 shows how imaging using directionally biased (u, v)-coverage (column (a)) fails to properly recover the orientation of an intrinsically noncircularly symmetric source when the (u, v)-coverage does not sufficiently sample the source structure in the relevant direction. By contrast, the same baselines—oriented in a more isotropic way (column B)—are capable of recovering the source profile in all directions. In addition, reconstructions of circularly symmetric sources by directionally biased (u, v)-coverage (column (a) of Figure 3) can introduce a lack of circular symmetry that is not present in the underlying source or in reconstructions performed on nondirectionally biased coverage (column (b) of Figure 3). Due to the minimization of the artifacts introduced into a reconstruction via incomplete (u, v)-coverage, imaging algorithms work better when applied to isotropic (u, v)-coverages.

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In addition to angular inhomogeneity, radially inhomogeneous coverage also leads to ambiguous image reconstruction. The Fourier transforms of various source types (e.g., rings, crescents, Gaussians) are approximately degenerate when observed using an interferometer that only marginally resolves the image (e.g., Thompson et al. 2017; Issaoun et al. 2019). More complex approximate degeneracies exist for radially inhomogeneous interferometers that only probe short and long but not intermediate baselines (e.g., Doeleman et al. 2008). For the EHT observing a ∼ 50 μみゅーas diameter ring, as expected for Sgr A*, short baselines correspond to those with length <2 Gλらむだ and long baselines correspond to those with length > 6 Gλらむだ. Figure 4 demonstrates that a Gaussian model describes the simulated EHT observation of a 50 μみゅーas static ring nearly as well as the static ring model itself if only particular subsets of the data are fit. Even an infinitesimally thin ring model, when only fit to medium and long baselines, can provide a high-quality fit while misrepresenting the total intensity of the source. Without sufficient radial homogeneity (i.e., coverage of short, medium, and long baselines, as seen in column (c) of Figure 2), fitting and interpreting a model confidently can be difficult.

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Periods of (u, v)-coverage where these limitations are more likely to occur can be identified by constructing a metric that scores directional bias and radial homogeneity (i.e., coverage of short, medium, and long baselines). The prevalence and severity of reconstruction artifacts that result from coverage limitations form a continuum that can be used to rank different (u, v) configurations. A metric based on these limitations could be applied to a full observation to distinguish different observing periods (composed of many evolving (u, v) configurations) by their ability to produce high-quality reconstructions.

Multiple (u, v)-coverage metrics with different underlying considerations exist in the literature. Here we summarize several metrics and compare the way they score a given observation. In addition, we develop a novel "isotropy metric" that has been tailored to the specific vulnerabilities detailed in Section 4.

Our "selective dynamical imaging" approach uses such a metric to identify intervals during an observation where the coverage is optimally configured for imaging. Importantly, these intervals are chosen before image reconstruction is attempted. In contrast, methods such as lucky imaging (e.g., Fried 1978) identify particularly useful images (e.g., with minimal distortion) after and on the basis of the reconstruction. A single score computed directly from (u, v)-coverage known a priori is preferable to an empirical approach (e.g., performing a simulated observation of a synthetic model and comparing reconstructions with the model), as it provides source-structure-agnostic assessments with a substantially lower performance cost.

Any metric capable of scoring different periods of (u, v)-coverage would have demonstrable limitations. Within a single observation, a comparison of (u, v)-coverage at two different points in time is a reliable way of determining which time region of the observation will produce superior image reconstructions. However, certain reconstruction-impacting data quantities can vary independently of the (u, v)-coverage. Sensitivity, calibration, and systematic uncertainty can also be important factors but are not probed by coverage metrics.

5.1. Normalized Cross-correlation

The normalized cross-correlation between two images is a measure of their similarity. By performing a dynamical reconstruction on a simulated observation and comparing each image of the dynamical reconstruction to the model, we can heuristically identify which portions of the observation produced the best reconstruction (i.e., the portions of the reconstruction with the greatest similarity to the model). We define the normalized cross-correlation ρろー_NX(X, Y) of two images X and Y in an identical fashion to Event Horizon Telescope Collaboration et al. (2019d),

$$\begin{eqnarray}&&{\rho }_{{\rm{NX}}}(X,Y)=\displaystyle \frac{1}{N}\sum _{i}\displaystyle \frac{({X}_{i}-\langle X\rangle )({Y}_{i}-\langle Y\rangle )}{{\sigma }_{X}{\sigma }_{Y}}.\end{eqnarray} \tag{ 2 }$$

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A normalized cross-correlation between a model and the associated reconstruction is the most straightforward way to identify trustworthy periods of an observation, assuming that the reconstruction on synthetic data will behave in a similar fashion to a reconstruction on real data. This assumption is only upheld if care is taken to ensure that the qualities of the synthetic data match those of the real data. In addition, heuristic tests such as the normalized cross-correlation can be biased depending on the structure and inherent variability of the model chosen. If the source randomly aligns with collimated (u, v)-coverage at some point in the observation, it can result in a misleadingly high normalized cross-correlation that cannot be replicated for a different source model.

5.2. (u, v) Filling Fraction

5.3. Largest Coverage Gap

An alternative metric probing the coverage isotropy is based on identifying the largest gap in the (u, v)-coverage; hence, we refer to it as the largest coverage gap (LCG) metric (Wielgus et al. 2020a). In this approach we consider the coverage as a set of sampled (u, v)-plane locations, to find the largest circle that can be drawn within the limits of the coverage that does not contain a coverage point in itself. Such a largest circular gap can be efficiently calculated with Delaunay triangulation of the coverage set (Barber et al. 1996). Then, the diameter of the gap ${d}_{\max }$ can be turned into a metric coefficient with

$$\begin{eqnarray}&&{m}_{\mathrm{LCG}}=1-{d}_{\max }/{\rho }_{\max },\end{eqnarray} \tag{ 3 }$$

where ${\rho }_{\max }$ is the longest projected baseline length. If we demand that the (u,v) distance corresponding to the center of the circle is less than ${\rho }_{\max }$, then we have 0 ≤ m_LCG ≤ 1, with m_LCG = 1 corresponding to the limit of a complete continuous coverage. A coverage consisting of a single detection would correspond to m_LCG = 0. Unlike the filling fraction metric, the LCG metric is independent of the assumed field of view.

5.4. Isotropy and Radial Homogeneity

We propose a novel metric of (u, v)-coverage isotropy and radial homogeneity (hereafter referred to as the "isotropy metric") based on the limitations described in Section 4. Similarly to the LCG metric, the isotropy metric penalizes anisotropy of the coverage, although the two approaches differ appreciably. In this approach, we treat the distribution of baselines in the Fourier plane as a mass distribution and quantify the radial and angular homogeneity using the second moments of inertia. We define the isotropy metric coverage parameter ${ \mathcal C }$ for a given snapshot as

$$\begin{eqnarray}&&{ \mathcal C }=I\left(1-\displaystyle \frac{{ \mathcal K }}{{{ \mathcal K }}_{{\rm{\max }}}}\right),\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&I=1-\displaystyle \frac{\sqrt{{\left(\langle {u}^{2}\rangle -\langle {v}^{2}\rangle \right)}^{2}+4\langle {uv}{\rangle }^{2}}}{\langle {u}^{2}\rangle +\langle {v}^{2}\rangle },\end{eqnarray} \tag{ 5 }$$

where 〈u²〉, 〈v²〉, and 〈uv〉² are the second moments of the baseline distribution, ${ \mathcal K }$ is the Kolmogorov–Smirnov (K-S) distance of the radial distribution of baseline lengths from uniform, and ${{ \mathcal K }}_{{\rm{\max }}}$ is the maximum value of ${ \mathcal K }$ at any point in time during the observation (or an arbitrary value for the purpose of cross-observation comparisons). The isotropy metric has the benefit of being fully analytic and automatically normalized between 0 and 1. A full derivation of the isotropy metric is presented in Appendix B.

5.5. Discussion of Metrics

We apply the coverage metrics discussed in Section 5 to the EHT (u, v)-coverage of Sgr A* corresponding to 2017 April 7. The instantaneous metric values for each snapshot of the observation are shown in Figure 6. This method of scoring the observation clearly identifies distinct periods of varying coverage quality. The time region from ∼01:30 GMST to ∼03:10 GMST (denoted as "Region II" in Figure 6) has the highest overall isotropy and baseline density of the observation. We select this period as a candidate time region for high-quality imaging (a "good" time region, i.e., one where the typical metric score is high). In contrast, the time region from ∼19:45 GMST to ∼21:00 GMST (denoted as "Region I" in Figure 6), while relatively stable, displays substantially lower coverage quality. We select this time region to examine the behavior of reconstructions in periods of ambiguous coverage quality. Exact time stamps for these time regions are given in Table 1.

By performing reconstructions in these time periods, we can validate the capability of the (u, v)-coverage metrics to predict reconstruction quality based on coverage alone. We reconstruct four configurations of the ring+hs toy model detailed in Section 2: a 270-minute orbital period clockwise and counterclockwise, and a 30-minute orbital period clockwise and counterclockwise. The reconstructions in each time region are produced according to the RML and CLEAN imaging methods in Section 3, and we perform feature extraction on each image using the REX module of eht-imaging (Event Horizon Telescope Collaboration et al. 2019d) to recover the position angle of the hot spot at each moment in time. The extracted hot spot angles and the model orbits for Regions I and II are shown in Figures 7 and 8, respectively. Images sampled from dynamical reconstructions generated by each method are displayed in Section C. The "success" of a reconstruction is determined by the successful extraction of the hot spot position angle.

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We find that Region II produces reconstructions that facilitate accurate recovery of dynamical variability. The reconstructions show a ring of approximately 50 μみゅーas in diameter with a distinct hot spot. The recovered hot spot orientations are shown in Figure 8, along with a comparison to the model values. To compare the N recovered position angles ${\varphi }_{\gamma }^{r}$ with the model angle ${\varphi }_{\gamma }^{m}$, we use a phase-adjusted rms ${ \mathcal R }$, defined as

$$\begin{eqnarray*}{ \mathcal R }={\left[\displaystyle \frac{1}{N}\sum _{\gamma }\left\{\begin{array}{ll}{\left({\varphi }_{\gamma }^{r}-{\varphi }_{\gamma }^{m}\right)}^{2}, & \mathrm{if}\ | {\varphi }_{\gamma }^{r}-{\varphi }_{\gamma }^{m}| \leqslant \pi \\ {\left(2\pi -{\varphi }_{\gamma }^{r}+{\varphi }_{\gamma }^{m}\right)}^{2}, & \mathrm{if}\ | {\varphi }_{\gamma }^{r}-{\varphi }_{\gamma }^{m}| \gt \pi \end{array}\right.\right]}^{1/2}.\end{eqnarray*}$$

Overall, the Region II reconstructions successfully recover the dynamical variability in the model, with rms of the recovered angles and model varying between 0.16 and 0.20 rad. Decreases in coverage quality as measured by the metrics (indicated pointwise in Figure 8 by increases in transparency, using the isotropy metric as an example) correlate with lower-quality recovery of the hot spot position angle. These low-quality angle recoveries are most obvious in the RML reconstructions of the counterclockwise T = 30-minute case. The CLEAN algorithm appears to be more resistant to the sudden loss of coverage and maintains reconstruction quality even through drops in metric score. Excluding these lapses in coverage quality, Region II clearly facilitates the recovery of source structure and at least one kind of dynamical variability, covering a wide range of periods and directions.

By contrast, a comparatively low scoring time region (Region I) does not produce reconstructions capable of accurately recovering dynamical variability. Dynamically reconstructed images show a ring-like feature, but the time variation of the brightness asymmetry does not match the model. Overall, the Region I reconstructions fail to recover the dynamical variability in the model, with rms of the recovered angles and model varying between 1.21 and 1.73 rad. The rms on the recovery in Region I is between 7 and 10 times higher than in Region II. For all tests, the scatter around the model is substantially larger than the scatter in Region II, rendering the accurate extraction of a period difficult or impossible. Reconstructions in periods outside of Regions I and II cannot recover even basic source structure without significant a priori information.

Based on these results, we expect that reconstructions on real data will produce the most accurate and robust recoveries of orbital dynamical variability in Region II of April 7 of the 2017 EHT observations of Sgr A*. Reasonable imaging and feature extraction procedures failed to produce meaningful results in Region I. This ranking is consistent with the predictions of the coverage metrics described in Section 5 and Appendix B. A wide array of factors impact reconstructions, and the tests presented here do not compose a realistic synthetic data suite for assessing whether or not Region II can accurately recover dynamical variability for Sgr A*. The tests provided here solely demonstrate that Region II is the best time region for performing dynamical reconstructions based on coverage considerations. Additional testing on more complex source types with realistic data corruptions can be found in Event Horizon Telescope Collaboration et al. (2022c).

Go/no-go decisions about whether to proceed with an observing run are often made with limited information about the readiness or weather conditions at particular sites. Simulating (and scoring) array configurations with different dropouts—and characterizing changes in the size and quality of the identified candidate temporal regions—can facilitate those go/no-go choices while incorporating uncertainties about station status. One observation configuration can be considered "better" than another if it provides a candidate time region with a higher metric score, or a similar metric score for a longer duration. Figure 9 shows an example of this kind of analysis performed with a hypothetical ngEHT array. The top panel shows the isotropy metric (see Section 5.4) score per snapshot for the full array during a night of observation in which every station is observing, which represents the ideal scenario. The middle and bottom panels reproduce the metric score per snapshot assuming that four sites are unable to observe. By identifying a candidate time region and characterizing its quality (based on, e.g., average metric score in the time region) and duration (denoted as Δでるたτたう in Figure 9), we can track these characterizations through different combinations of site dropouts and estimate how critical the sites are to the array. Based on the computations in Figure 9, ALMA, JCMT, LMT, and SMA dropping out would be catastrophic to the array performance, as the optimal time region for dynamical imaging reduces in duration by ≈80% and reduces in quality by ≈50%. By comparison, the combined dropout of PV, PDB, CARMA, and LMT does not substantially change the duration or quality of the identified candidate time region.

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A metric-based comparison additionally provides a natural and quantitative ranking for identifying which day of an observation campaign produced the most optimal coverage for dynamical imaging. We can rank observations on separate days in a similar fashion to the dropout scenarios visualized in Figure 9 by comparing the duration and quality of the identified candidate time regions for each day. An example of such a comparison between the April 7 and April 10 runs of the 2017 EHT observation campaign is shown in Figure 10. Instantaneous metric scores are computed for each scan of both days, and candidate time regions for dynamical imaging (green) are identified. While both candidate time regions are of approximately the same duration, the candidate time region associated with April 7 displays substantially better coverage, making April 7 a better choice for dynamical imaging than April 10.

We have demonstrated that limitations of imaging associated with sparse baseline distribution can be inferred from the specific geometric properties of the (u, v)-coverage. Highly anisotropic coverage produces artifacts in reconstructions that distort the image in a direction consistent with stripes in the dirty beam. Additionally, an uneven radial distribution of the Fourier plane (u, v)-coverage tends to result in ambiguous image reconstruction. These limitations can be partially avoided by imaging in radially and angularly homogeneous coverage. The reconstruction issues associated with sparse coverage are exacerbated by rapid short-timescale variability, as seen in a wide array of astrophysical sources, including Sgr A* and the precessing jets of X-ray binaries.

Next, we surveyed and compared existing geometric measures of (u, v)-coverage, in addition to deriving a novel isotropy-based metric that addressed the specific limitations demonstrated in Section 4. The examined metrics included the normalized cross-correlation (2), the (u, v) filling fraction (see Palumbo et al. 2019), and the LCG metric (Equation (3); see Wielgus et al. 2020a). The isotropy metric treats the distribution of baselines as a mass distribution and examines the second moment to rank coverages by homogeneity in the Fourier plane. The isotropy metric gives similar results to other (u, v)-coverage-based metrics while being more computationally efficient.

These metrics were applied to the April 7 data of the 2017 EHT coverage of Sgr A* (Event Horizon Telescope Collaboration et al. 2022a, 2022b, 2022c, 2022d, 2022e, 2022f) and used to select candidate time regions for high-quality dynamical imaging. All metrics identify a period from ∼01:30 GMST to ∼03:10 GMST ("Region II") that minimizes the coverage limitations each metric addresses. We also select a period from ∼19:45 GMST to ∼21:00 GMST ("Region I") with reconstruction capability. Reconstructions of time-variable sources allowed successful recovery of the characteristic source variability in Region II. In contrast, reconstructions in Region I were unable to recover the characteristic motion. The ranking determined by the suite of reconstructions performed on synthetic data verifies the predictions made by the examined coverage metrics. We expect that attempts to recover variability in real EHT observations of the Galactic center will produce the most robust and accurate recoveries in Region II of the 2017 April 7 data set, and therefore we recommend performing dynamical imaging procedures in that time region.

Coverage metrics have additional utility for ranking interobservation comparisons based on their ability to recover dynamical variability, which has a variety of applications to the broader field of interferometery. These metrics provide the ability to make select observation time slots based on the capability of available antennas to recover particular dynamical evolution in the target. Such a scored assessment of coverage could well prove of use to other VLBI arrays both as they make go/no-go decisions about whether to observe on a particular night and then when identifying the periods of best coverage to perform static and dynamic imaging.

We thank the National Science Foundation (awards OISE-1743747, AST-1816420, AST-1716536, AST-1440254, AST-1935980) and the Gordon and Betty Moore Foundation (GBMF-5278) for financial support of this work. This work was supported in part by the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation to Harvard University. Support for this work was also provided by the NASA Hubble Fellowship grant HST-HF2-51431.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.

The Event Horizon Telescope Collaboration thanks the following organizations and programs: the Academy of Finland (projects 274477, 284495, 312496, 315721); the Agencia Nacional de Investigacin y Desarrollo (ANID), Chile via NCN19₀₅₈ (TITANs) and Fondecyt 3190878, the Alexander von Humboldt Stiftung; an Alfred P. Sloan Research Fellowship; Allegro, the European ALMA Regional Centre node in the Netherlands, the NL astronomy research network NOVA and the astronomy institutes of the University of Amsterdam, Leiden University, and Radboud University; the Black Hole Initiative at Harvard University, through a grant (60477) from the John Templeton Foundation; the China Scholarship Council; Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico, projects U0004-246083, U0004-259839, F0003-272050, M0037-279006, F0003-281692, 104497, 275201, 263356); the Delaney Family via the Delaney Family John A. Wheeler Chair at Perimeter Institute; Direccin General de Asuntos del Personal Acadmico-Universidad Nacional Autnoma de Mxico (DGAPA-UNAM, projects IN112417 and IN112820); the European Research Council Synergy Grant "BlackHoleCam: Imaging the Event Horizon of Black Holes" (grant 610058); the Generalitat Valenciana postdoctoral grant APOSTD/2018/177 and GenT Program (project CIDEGENT/2018/021); MICINN Research Project PID2019-108995GB-C22; the European Research Council for advanced grant "JETSET: Launching, propagation and emission of relativistic jets from binary mergers and across mass scales" (grant No. 884631); the Gordon and Betty Moore Foundation (grant GBMF-3561); the Istituto Nazionale di Fisica Nucleare (INFN) sezione di Napoli, iniziative specifiche TEONGRAV; the two Dutch National Supercomputers, Cartesius and Snellius (NWO grant 2021.013); the International Max Planck Research School for Astronomy and Astrophysics at the Universities of Bonn and Cologne; Joint Princeton/Flatiron and Joint Columbia/Flatiron Postdoctoral Fellowships, with research at the Flatiron Institute being supported by the Simons Foundation; the Japanese Government (Monbukagakusho: MEXT) Scholarship; the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellowship (JP17J08829); the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJ-SSW-SYS008, ZDBS-LY-SLH011); the Leverhulme Trust Early Career Research Fellowship; the Max-Planck-Gesellschaft (MPG); the Max Planck Partner Group of the MPG and the CAS; the MEXT/JSPS KAKENHI (grants 18KK0090, JP21H01137, JP18H03721, 18K03709, 18H01245, 25120007); the Malaysian Fundamental Research Grant Scheme (FRGS) FRGS/1/2019/STG02/UM/02/6; the MIT International Science and Technology Initiatives (MISTI) Funds; the Ministry of Science and Technology (MOST) of Taiwan (103-2119-M-001-010-MY2, 105-2112-M-001-025-MY3, 105-2119-M-001-042, 106-2112-M-001-011, 106-2119-M-001-013, 106-2119-M-001-027, 106-2923-M-001-005, 107-2119-M-001-017, 107-2119-M-001-020, 107-2119-M-001-041, 107-2119-M-110-005, 107-2923-M-001-009, 108-2112-M-001-048, 108-2112-M-001-051, 108-2923-M-001-002, 109-2112-M-001-025, 109-2124-M-001-005, 109-2923-M-001-001, 110-2112-M-003-007-MY2, 110-2112-M-001-033, 110-2124-M-001-007, and 110-2923-M-001-001); the Ministry of Education (MoE) of Taiwan Yushan Young Scholar Program; the Physics Division, National Center for Theoretical Sciences of Taiwan; the National Aeronautics and Space Administration (NASA, Fermi Guest Investigator grant 80NSSC20K1567, NASA Astrophysics Theory Program grant 80NSSC20K0527, NASA NuSTAR award 80NSSC20K0645); the National Institute of Natural Sciences (NINS) of Japan; the National Key Research and Development Program of China (grant 2016YFA0400704, 2016YFA0400702); the National Science Foundation (NSF, grants AST-0096454, AST-0352953, AST-0521233, AST-0705062, AST-0905844, AST-0922984, AST-1126433, AST-1140030, DGE-1144085, AST-1207704, AST-1207730, AST-1207752, MRI-1228509, OPP-1248097, AST-1310896, AST-1555365, AST-1615796, AST-1715061, AST-1716327, AST-1903847, AST-2034306); the Natural Science Foundation of China (grants 11650110427, 10625314, 11721303, 11725312, 11933007, 11991052, 11991053); NWO grant No. OCENW.KLEIN.113; a fellowship of China Postdoctoral Science Foundation (2020M671266); the Natural Sciences and Engineering Research Council of Canada (NSERC, including a Discovery Grant and the NSERC Alexander Graham Bell Canada Graduate Scholarships-Doctoral Program); the National Youth Thousand Talents Program of China; the National Research Foundation of Korea (the Global PhD Fellowship Grant: grants NRF-2015H1A2A1033752, 2015-R1D1A1A01056807; the Korea Research Fellowship Program: NRF-2015H1D3A1066561, Basic Research Support Grant 2019R1F1A1059721); the Netherlands Organization for Scientific Research (NWO) VICI award (grant 639.043.513) and Spinoza Prize SPI 78-409; the New Scientific Frontiers with Precision Radio Interferometry Fellowship awarded by the South African Radio Astronomy Observatory (SARAO), which is a facility of the National Research Foundation (NRF), an agency of the Department of Science and Technology (DST) of South Africa; the Onsala Space Observatory (OSO) national infrastructure, for the provisioning of its facilities/observational support (OSO receives funding through the Swedish Research Council under grant 2017-00648); the Perimeter Institute for Theoretical Physics (research at the Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science); the Spanish Ministerio de Ciencia e Innovacin (grants PGC2018-098915-B-C21, AYA2016-80889-P, PID2019-108995GB-C21, PID2020-117404GB-C21); the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award for the Instituto de Astrofsica de Andaluca (SEV-2017-0709); the Toray Science Foundation; the Consejera de Economa, Conocimiento, Empresas y Universidad of the Junta de Andaluca (grant P18-FR-1769), the Consejo Superior de Investigaciones Cientficas (grant 2019AEP112); the M2FINDERS project, which has received funding by the European Research Council (ERC) under the European Unions Horizon 2020 Research and Innovation Programme (grant agreement No. 101018682); the US Department of Energy (USDOE) through the Los Alamos National Laboratory (operated by Triad National Security, LLC, for the National Nuclear Security Administration of the USDOE (contract 89233218CNA000001); the European Unions Horizon 2020 research and innovation program under grant agreement No. 730562 RadioNet; Shanghai Pilot Program for Basic Research, Chinese Academy of Science, Shanghai Branch (JCYJ-SHFY-2021-013); ALMA North America Development Fund; the Academia Sinica; Chandra DD7-18089X and TM6-17006X; and the GenT Program (Generalitat Valenciana) Project CIDEGENT/2018/021. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF grant ACI-1548562, and CyVerse, supported by NSF grants DBI-0735191, DBI-1265383, and DBI-1743442. The XSEDE Stampede2 resource at TACC was allocated through TG-AST170024 and TG-AST080026N. The XSEDE JetStream resource at PTI and TACC was allocated through AST170028. The simulations were performed in part on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, and on the HazelHen cluster at the HLRS in Stuttgart. This research was enabled in part by support provided by Compute Ontario (http://computeontario.ca), Calcul Quebec (http://www.calculquebec.ca), and Compute Canada (http://www.computecanada.ca). C.C. acknowledges support from the Swedish Research Council (VR). We thank the staff at the participating observatories, correlation centers, and institutions for their enthusiastic support. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2016.1.01154.V. ALMA is a partnership of the European Southern Observatory (ESO; Europe, representing its member states), NSF, and National Institutes of Natural Sciences of Japan, together with National Research Council (Canada), Ministry of Science and Technology (MOST; Taiwan), Academia Sinica Institute of Astronomy and Astrophysics (ASIAA; Taiwan), and Korea Astronomy and Space Science Institute (KASI; Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, Associated Universities, Inc. (AUえーゆーI)/NRAO, and the National Astronomical Observatory of Japan (NAOJ). The NRAO is a facility of the NSF operated under cooperative agreement by AUI. Hector Olivares and Gibwa Musoke were supported by Virtual Institute of Accretion (VIA) postdoctoral fellowships from the Netherlands Research School for Astronomy (NOVA). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden). The SMA is a joint project between the SAO and ASIAA and is funded by the Smithsonian Institution and the Academia Sinica. The JCMT is operated by the East Asian Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the Ministry of Finance of China, Chinese Academy of Sciences, and the National Key R&D Program (No. 2017YFA0402700) of China. Additional funding support for the JCMT is provided by the Science and Technologies Facility Council (UK) and participating universities in the UK and Canada. Simulations were performed in part on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen cluster at the HLRS in Stuttgart, and on the Pi2.0 and Siyuan Mark-I at Shanghai Jiao Tong University. The computer resources of the Finnish IT Center for Science (CSC) and the Finnish Computing Competence Infrastructure (FCCI) project are acknowledged. Junghwan Oh was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1A6A3A01086420). We thank Martin Shepherd for the addition of extra features in the Difmap software that were used for the CLEAN imaging results presented in this paper. The computing cluster of Shanghai VLBI correlator supported by the Special Fund for Astronomy from the Ministry of Finance in China is acknowledged. The LMT is a project operated by the Instituto Nacional de Astrofisica, Optica, y Electronica (Mexico) and the University of Massachusetts at Amherst (USA). The IRAM 30 m telescope on Pico Veleta, Spain, is operated by IRAM and supported by CNRS (Centre National de la Recherche Scientifique, France), MPG (Max-Planck- Gesellschaft, Germany), and IGN (Instituto Geogrfico Nacional, Spain). The SMT is operated by the Arizona Radio Observatory, a part of the Steward Observatory of the University of Arizona, with financial support of operations from the State of Arizona and financial support for instrumentation development from the NSF. Support for SPT participation in the EHT is provided by the National Science Foundation through award OPP-1852617 to the University of Chicago. Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago. The SPT hydrogen maser was provided on loan from the GLT, courtesy of ASIAA. Support for this work was provided by NASA through the NASA Hubble Fellowship grant No. HST-HF2-51494.001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5–26555. The EHTC has received generous donations of FPGA chips from Xilinx Inc., under the Xilinx University Program. The EHTC has benefited from technology shared under open-source license by the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER). The EHT project is grateful to T4Science and Microsemi for their assistance with Hydrogen Masers. This research has made use of NASA's Astrophysics Data System. We gratefully acknowledge the support provided by the extended staff of the ALMA, both from the inception of the ALMA Phasing Project through the observational campaigns of 2017 and 2018. We would like to thank A. Deller and W. Brisken for EHT-specific support with the use of DiFX. We thank J. Delgado for helpful discussions and feedback. We acknowledge the significance that Maunakea, where the SMA and JCMT EHT stations are located, has for the indigenous Hawaiian people.

Here we present representative samples of the visibility amplitudes and closure phases that are generated from the synthetic data described in Section 2.1. Sample amplitudes are shown in Figure 11, and sample closure phases are shown in Figure 12.

Section 4 demonstrated that quantifying the isotropy of a (u, v)-coverage configuration can indicate whether it is suitable for producing accurate reconstructions of a dynamic source. We adopt a coverage metric of the form

$$\begin{eqnarray}&&{ \mathcal C }=I(\{{\vec{u}}_{i}\})R(\{{\vec{u}}_{i}\}),\end{eqnarray} \tag{ B1 }$$

where $\{{\vec{u}}_{i}\}=\{({u}_{i},{v}_{i})\}$ is the set of 2N baselines (including their Hermitian conjugates), I is a measure of the isotropy of the coverage, and R is a measure of the radial homogeneity of the coverage.

To estimate the radial homogeneity, we compare the cumulative distribution function (CDF) of the distribution of baseline lengths against the uniform CDF via the K-S test. The uniform distribution examined for this test ranges from 0 Gλらむだ to the maximum baseline length achieved in the observation. This test returns a "distance" ${ \mathcal K }$ between the distributions, which increases as the distribution becomes less radially homogeneous, making use of the test in this context a measure of radial inhomogeneity. To convert the result of the K-S test into a measure of radial homogeneity, we select an upper bound ${{ \mathcal K }}_{{\rm{\max }}}$ corresponding to the maximum distance from uniform any individual baseline distribution obtains throughout the observation. We then subtract the result of the K-S test from this maximum, i.e.,

$$\begin{eqnarray}&&{ \mathcal K }^{\prime} =\displaystyle \frac{{{ \mathcal K }}_{{\rm{\max }}}-{ \mathcal K }}{{{ \mathcal K }}_{{\rm{\max }}}}=1-\displaystyle \frac{{ \mathcal K }}{{{ \mathcal K }}_{{\rm{\max }}}},\end{eqnarray} \tag{ B2 }$$

where $R(\{{\vec{u}}_{i}\})={ \mathcal K }^{\prime}$ is our new metric of homogeneity. This metric is conveniently bounded between 0 and 1. To make this metric absolute, a fixed value of ${{ \mathcal K }}_{{\rm{\max }}}$ can be chosen arbitrarily and applied to multiple observations. For the absolute comparisons in this paper (see, e.g., Section 7) a value of ${{ \mathcal K }}_{{\rm{\max }}}=0.513338437261774\approx 0.513$ is adopted. This value of ${{ \mathcal K }}_{{\rm{\max }}}$ is chosen to be the maximum value of ${ \mathcal K }$ achieved during the April 7 observation.

In order to measure the isotropy of the coverage, we examine the second moment (moment of inertia) of the distribution of baselines. As a spatial configuration of points with uniform weighting, the (u, v)-coverage can be treated as a mass distribution. For a two-dimensional mass distribution, a disk is considered isotropic, a rod is considered anisotropic, and the spectrum between the two cases can be probed using the moment of inertia tensor. Given 2N baselines and conjugate baselines with coordinates $\{{\vec{u}}_{i}\}=\{({u}_{i},{v}_{i})\}$, we can compute the second moments of the distribution as

$$\begin{eqnarray}\begin{array}{rcl}\langle {u}^{2}\rangle & = & \displaystyle \frac{1}{2N}\sum _{i}^{2N}{u}_{i}^{2},\\ \langle {v}^{2}\rangle & = & \displaystyle \frac{1}{2N}\sum _{i}^{2N}{v}_{i}^{2},\\ \langle {uv}\rangle & = & \displaystyle \frac{1}{2N}\sum _{i}^{2N}{u}_{i}{v}_{i}.\end{array}\end{eqnarray} \tag{ B3 }$$

The moment of inertia tensor ${ \mathcal M }$ is

$$\begin{eqnarray}{ \mathcal M }=\left[\begin{array}{cc}\langle {u}^{2}\rangle & \langle {uv}\rangle \\ \langle {uv}\rangle & \langle {v}^{2}\rangle \end{array}\right].\end{eqnarray} \tag{ B4 }$$

The principal moments of inertia can be computed from the eigenvalues λらむだ₁ and λらむだ₂ of ${ \mathcal M }$. From these, we can derive the following orientation-independent measure of isotropy:

$$\begin{eqnarray}\begin{array}{rcl}\tilde{I}(\{{\vec{u}}_{i}\}) & = & 1-\displaystyle \frac{| {\lambda }_{1}-{\lambda }_{2}| }{{\lambda }_{1}+{\lambda }_{2}}\\ & = & 1-\displaystyle \frac{\sqrt{{\left(\langle {u}^{2}\rangle -\langle {v}^{2}\rangle \right)}^{2}+4\langle {uv}{\rangle }^{2}}}{\langle {u}^{2}\rangle +\langle {v}^{2}\rangle }.\end{array}\end{eqnarray} \tag{ B5 }$$

Note that $\tilde{I}(\{{\vec{u}}_{i}\})$ can also be written in terms of the FWHM of the dirty beam,

$$\begin{eqnarray}&&\tilde{I}(\{{\vec{u}}_{i}\})=1-\displaystyle \frac{1/{\theta }_{\min }^{2}-1/{\theta }_{\mathrm{maj}}^{2}}{1/{\theta }_{\min }^{2}+1/{\theta }_{\mathrm{maj}}^{2}}\end{eqnarray} \tag{ B6 }$$

$$\begin{eqnarray}&&=\displaystyle \frac{2{\theta }_{\min }^{2}}{{\theta }_{\mathrm{maj}}^{2}+{\theta }_{\min }^{2}}.\end{eqnarray} \tag{ B7 }$$

This measure of isotropy is naturally normalized between 0 and 1. Substituting this expression and Equation (B2) into Equation (B1) gives the following expression for a coverage quality metric:

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal C } & = & I\left(\{{\vec{u}}_{i}\}\right)R\left(\{{\vec{u}}_{i}\}\right)\\ & = & \left(1-\displaystyle \frac{\sqrt{{\left(\langle {u}^{2}\rangle -\langle {v}^{2}\rangle \right)}^{2}+4\langle {uv}{\rangle }^{2}}}{\langle {u}^{2}\rangle +\langle {v}^{2}\rangle }\right)\left(1-\displaystyle \frac{{ \mathcal K }}{{{ \mathcal K }}_{{\rm{\max }}}}\right).\end{array}\end{eqnarray} \tag{ B8 }$$

This measure of coverage quality can be applied to partition an arbitrary VLBI observation into time regions ranked by their ability to accurately reconstruct dynamical sources.

In Section 6, we test the coverage metrics examined in Section 5 by performing reconstructions of synthetic observations in selected time regions of the 2017 EHT observational coverage of Sgr A*. Here we provide representative snapshot images from each of the time regions. Sampled snapshot images from Region I are shown in Figure 13, and sampled snapshot images from Region II are shown in Figure 14.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Reconstructions in both Region I and Region II demonstrate clear recovery of a ring-like feature of approximately 50 μみゅーas in diameter, as both sufficiently probe the radial distribution of the source Fourier transform to constrain the overall size of the source. However, the directionally biased coverage of Region I produces incorrect reconstructions of hot spot location. By contrast, reconstructions in Region II repeatedly recovery the correct hot spot location across all periods, directions, and imaging algorithms.

Though both time regions recover a ring-like feature of the approximately correct size, the Region II reconstructions provide a more accurate ring-to-central-depression flux density ratio. The increased accuracy is present in both the CLEAN and RML reconstructions. By contrast, reconstructions in Region I fail to consistently provide a visually distinctive depression and misrepresent the angular brightness profile of the source.