electron temperature ratio $R_{\mathrm{high}}=160$, and viewing inclination
$i=50^{\circ}$. Although failed in other constraints (see Paper V appendix A),
this model satisfies the same criteria used for selection of the geometric
models as seen by the resulting visibility amplitudes (Section 5.1.1) and
temporal variability (see Section 5.1.2), as shown in Figure 9. Figure 7 shows
a single snapshot frame of the GRMHD movie along with a time-averaged
structure. The GRMHD movie frames contain a sharp photon ring with a faint
emission broadly extended over $\sim 100\,\mu$as. Some of the frames contain
notable spiral arm features surrounding the photon ring and extending beyond
the compact structure (refer to the SANE frames shown in Figure 25). This
spiral arm feature is smoothed out by averaging over the observational time.

In addition, Appendix H shows the result of applying the same imaging
procedure on a strongly magnetized “magnetically arrested disk” (MADs) model
with positive spin, which passes more observational constraints and is in the
“best bet region” considered by Paper V. By using these two GRMHD models,
generated with different physical parameters, we demonstrate that our imaging
procedure and the resulting performance is robust against the details of GRMHD
models.

##  6 Imaging Surveys with Synthetic Data

We conducted surveys over a wide range of imaging assumptions with four
scripted imaging pipelines using RML, CLEAN and a Bayesian posterior sampling
method. The surveys were performed on the synthetic data sets presented in
Section 5 as well as on the real Sgr A∗ data. Reconstructing synthetic data
with exactly the same procedure used on Sgr A∗ allows us to assess our ability
to identify the true underlying time-averaged morphology. Both synthetic and
real Sgr A∗ data sets were pre-processed with a common pipeline described in
Section 6.1. We describe the RML and CLEAN imaging parameter surveys in
Section 6.2 and imaging with a Bayesian posterior sampling method in Section
6.3. Images of synthetic data from imaging surveys are described in Section
6.4. We present images of the actual Sgr A∗ data in Section 7.

###  6.1 Common Pre-imaging Processing

To reconstruct a time-averaged image of Sgr A∗, each pipeline used the
original calibrated data sets described in Section 2 and/or data sets further
normalized by the time-dependent total flux density of Sgr A∗. Within each
imaging pipeline, data sets were first time-averaged to enhance the S/N of
visibilities; each pipeline adopted a single integration time or explored
multiple choices of the integration time (see Tables 3-5). After time
averaging, fractional errors of 0%, 2 % or 5 % were added to the visibility
error budget in quadrature to account for the non-closing systematic errors
(refer to Paper II).

As described in Sections 3.1 and 3.2, we employ additional strategies to
mitigate extrinsic scattering and intra-day variations. To assess these
proposed strategies, and our ability to account for these two effects in the
imaging process, we incorporate parameterized error budgets and systematically
explore the various assumptions on these two effects in the RML and CLEAN
surveys discussed in Section 6.2. In total, we potentially include up to three
additional noise budgets that account for 1) systematic error, 2) interstellar
scattering, and 3) time variability.

The second budget accounts for the substructure arising from refractive
scattering. We added the anticipated refractive noise level for the observed
(i.e. interstellar-scattered) visibilities using two base models described in
Section 3.1. The first model we explore is the Const model, which adds a
constant noise budget across all visibilities prior to deblurring; we examined
two noise levels: 0.4 % and 0.9 % of the total flux density at each time
segment (i.e., Const and 2$\times$Const). The second model explored is the
J18model1 (J18) model, which is no longer constant in the $(u,v)$-space; we
adopt two scaling factors of 1.0 and 2.0 for this noise floor (i.e, J18model1
(J18) and 2$\times$J18model1 (2$\times$J18). The two noise levels adopted in
each model reasonably cover differences in the noise levels caused by
different potential intrinsic structures (see Section 3.1). After including
one of the above budgets for the refractive noise, visibilities were divided
by the diffractive scattering kernel based on the J18 model to mitigate
diffractive scattering. In addition to the above four sets of the scattering
mitigation schemes, we attempted imaging without any form of scattering
mitigation to probe the interstellar-scattered source structure of Sgr A∗
(referred to as on-sky images).

The third noise budget explored accounts for the structural deviations from
the time-averaged morphology due to the intra-day variations. We further
inflated the visibility error budget using the variability noise model
described by Equation 2 in Section 3.2. This budget was added in quadrature to
the visibility noise budget, after being normalized by the time-dependent
total flux density. We systematically explored various sets of parameters in
Equation 2 including the variability rms level at 4 G$\lambda$ ($a$), the
break location ($u_{0}$), and the variability power-law spectra index at long
baselines ($b$). Similar to scattering, we also attempted reconstructions
without this error budget (i.e., assuming no intra-day variation in data).

###  6.2 RML and CLEAN Imaging Parameter Surveys

In a manner similar to previous EHT imaging of M87∗ (; ), we explore how
recovered images are influenced by different imaging and optimization choices.
In particular, we objectively evaluate each set of imaging parameters in
scripted RML and CLEAN imaging pipelines using synthetic data with known
ground truth images. Each parameter survey leads to a Top Set of parameters:
parameter combinations that each produce acceptable images on our entire suite
of synthetic data. The distribution of Sgr A∗ images recovered with the Top
Set parameter combinations reflects our uncertainty due to modeling and
optimization choices made in imaging; thus, it is different from a Bayesian
posterior and instead attempts to characterize what is sometimes referred to
as epistemic uncertainty.

Table 3: Parameters in the DIFMAP Pipeline Top Set

April 7 (8400 Param. Combinations; 1626 in Top Set)
---
Systematic | 0 | 0.02 | 0.05
error | 25.6% | 36.8% | 37.5%
$\bf Ref\ Type$ | No | Const | 2$\times$Const | J18 | 2$\times$J18
| 14.9% | 20.7% | 21.3% | 22.1% | 20.9%
$\bf a_{\bf\rm psd}$ | No | 0.015 | 0.02 | 0.025
| 5.1% | 28.5% | 32.1% | 34.3%
$\bf b_{\bf\rm psd}$ | No | 1 | 3 | 5
| 5.1% | 20.2% | 35.5% | 39.2%
$|u|_{0}$ | No | 2
| 5.1% | 94.9%
Time average | 10 | 60
(sec) | 45.0% | 55.0%
ALMA weight | 0.1 | 0.5
| 41.1% | 58.9%
UV weight | 0 | 2
| 54.7% | 45.3%
Mask Diameter | 80 | 85 | 90 | 95 | 100 | 105 | 110
($\mathrm{\mu as}$) | 0.2% | 2.5% | 22.3% | 25.0% | 21.3% | 20.0% | 8.7%

Note.  In each row, the upper line with bold text shows the surveyed parameter
value corresponding to the parameter of left column, while the lower line
shows the number fraction of each value in Top Set. The total number of
surveyed parameter combinations and Top Set are shown in the first row.

Table 4:  Parameters in the eht-imaging Pipeline Top Set

April 7 (112320 Param. Combinations; 5594 in Top Set)
---
Systematic | 0 | 0.02 | 0.05
error | 21.4% | 36.7% | 41.8%
$\bf Ref\ Type$ | No | Const | 2$\times$Const | J18 | 2$\times$J18
| 27.0% | 23.9% | 20.6% | 16.4% | 12.1%
$\bf a_{\bf\rm psd}$ | No | 0.015 | 0.02 | 0.025
| 11.4% | 40.6% | 26.6% | 21.4%
$\bf b_{\bf\rm psd}$ | No | 1 | 2 | 3 | 5
| 11.4% | 24.8% | 20.4% | 21.5% | 21.8%
$|u|_{0}$ | No | 2
| 11.4% | 88.6%
TV | $\bf 0$ | 0.01 | 0.1 | 1
| 13.2% | 16.0% | 36.1% | 34.7%
TSV | $\bf 0$ | 0.01 | 0.1 | 1
| 29.5% | 32.3% | 26.6% | 11.5%
Prior size | 70 | 80 | 90
($\mathrm{\mu as}$) | 33.9% | 34.7% | 31.4%
MEM | $\bf 0$ | 0.01 | 0.1 | 1
| 7.8% | 18.4% | 54.3% | 19.6%
Amplitude | 0 | 0.1 | 1
weight | 0.9% | 23.3% | 75.8%

Note.  Same as Table 3

Table 5: Parameters in the SMILI Pipeline Top Set

April 7 (54000 Param. Combinations; 2763 in Top Set)
---
Systematic | 0 | 0.02 | 0.05
error | 33.9% | 33.3% | 32.8%
$\bf Ref\ Type$ | No | Const | 2$\times$Const | J18 | 2$\times$J18
| 15.7% | 22.1% | 18.5% | 21.4% | 22.3%
$\bf a_{\bf\rm psd}$ | No | 0.015 | 0.02 | 0.025
| 7.5% | 37.2% | 26.7% | 28.6%
$\bf b_{\bf\rm psd}$ | No | 1 | 2 | 3 | 5
| 7.5% | 16.8% | 27.7% | 31.9% | 16.1%
$|u|_{0}$ | No | 1 | 2
| 7.5% | 40.7% | 51.8%
TV | $\bf 10^{2}$ | $\bf 10^{3}$ | $\bf 10^{4}$ | $\bf 10^{5}$
| 8.6% | 46.7% | 44.4% | 0.3%
TSV | $\bf 10^{2}$ | $\bf 10^{3}$ | $\bf 10^{4}$ | $\bf 10^{5}$
| 38.7% | 53.3% | 8.0% | 0.0%
Prior size | 140 | 160 | 180
($\mathrm{\mu as}$) | 33.0% | 33.6% | 33.3%
$\bf\ell_{1}$ | 0.1 | 1 | $\bf 10$
| 44.8% | 54.8% | 0.3%

Note.  Same as Table 3

####  6.2.1 Imaging Pipelines

Similar to previous EHT work (; ), we designed three scripted imaging
pipelines utilizing the DIFMAP, eht-imaging and SMILI software packages. After
completing the common pre-imaging processing of data (Section 6.1), each
pipeline reconstructs images using a broad parameter space (e.g., weights for
the regularization functions, mask sizes, station gain constraints,
variability noise budget parameters, etc.). We describe each pipeline in
detail in Section D.1, Section D.2, and Section D.3.

Each pipeline explored on the order of $10^{3-5}$ parameter combinations, as
summarised in Table 3, 4, and 5 for DIFMAP, eht-imaging, and SMILI,
respectively. Each pipeline has some unique choices that are fixed (e.g. the
pixel size, or the convergence criterion) and surveyed (e.g., the regularizer
weights), while some parameters are commonly explored (e.g. parameters for the
scattering and intra-day variations in Section 6.1).

While all imaging pipelines adopt the common pre-processing of data described
in Section 6.1, there are some differences in data processing. For instance,
the noise budgets for refractive scattering and intra-day variability are
updated during self-calibration rounds in SMILI. RML imaging pipelines (eht-
imaging and SMILI) adopt the same prior and initial images across all
synthetic and real data sets. The DIFMAP pipeline uniformly explores a library
of initial models for a first phase self-calibration, selecting the one that
provides the best fit to the closure phases after a first run of cleaning (see
Section D.1). All three pipelines use combined low- and high-band data for
imaging without any data flagging (including the intra-site baselines).

####  6.2.2 Top Sets of Imaging Parameters via Surveys on Synthetic Data

Large imaging surveys on synthetic data facilitate the evaluation of different
potential parameter combinations. Following M87∗ Paper IV, the principal
output from each parameter survey is a Top Set: a set of parameter
combinations that produce acceptable images on the suite of synthetic data
presented in Section 5.

Figure 10: Normalized cross correlation, $\rho_{\rm NX}$, between time-
averaged ground truth images and their corresponding blurred images, as a
function of the blurring kernel size. These curves are shown for both the
intrinsic images (top panels) and the scattered (i.e., on-sky) images (bottom
panels). The dashed black line indicates the angular resolution equivalent to
the maximum fringe spacing of Sgr A∗ observations, 24 $\mathrm{\mu as}$; the
corresponding value of $\rho_{\rm NX}$ at 24 $\mathrm{\mu as}$ is used to
define a threshold that selects which DIFMAP, eht-imaging and SMILI imaging
parameter combination are applied to Sgr A∗ data.

The fidelity of synthetic image reproduction is measured using the normalized
cross-correlation between the reconstructed images and ground truth images. We
define the normalized cross-correlation of two images $X$ and $Y$ made of $N$
pixels as

$\rho_{\rm
NX}=\frac{1}{N}\sum_{i}\frac{(X_{i}-\braket{X})(Y_{i}-\braket{Y})}{\sigma_{X}\sigma_{Y}}.$
(10)

Here $X_{i}$ and $Y_{i}$ denote the image intensity at $i$-th pixel,
$\braket{X}$ and $\braket{Y}$ denote the mean pixel values of the images, and
$\sigma_{X}$ and $\sigma_{Y}$ are the standard deviations of pixel values. The
position offset between the frames is corrected by shifting one frame relative
to the other along R.A. and decl. and identifying the shift coordinate that
corresponds with the largest $\rho_{\rm NX}$.

In order to recover a Top Set, a threshold for $\rho_{\rm NX}$ is defined for
each synthetic data set. Imaging parameter combinations that recover images
that score above that thresholds for all synthetic data set are selected as
Top Sets. The threshold values of $\rho_{\rm NX}$ are determined in a manner
similar to M87∗ Paper IV. For each “ground truth" image (obtained by time
averaging the synthetic movie), we evaluate $\rho_{\rm NX}$ between the ground
truth image and the same image blurred with a Gaussian beam of FWHM equivalent
to the maximum fringe spacing of the Sgr A∗ observations, 24 $\mathrm{\mu
as}$. This value of $\rho_{\rm NX}$ quantifies the potential loss of the image
fidelity due to the limited angular resolution. Figure 10 shows examples of
the $\rho_{\rm NX}$ curves between unblurred and blurred ground truth images
as a function of the blurring size; the critical value corresponds to those at
$\alpha=24$ $\mathrm{\mu as}$. Note that this value depends on the true source
structure. Unlike in M87∗ Paper IV, we find that a relaxation of the
$\rho_{\rm NX}$ threshold is required to account for the fact that we
reconstruct static images from time variable data sets. Hence the critical
$\rho_{\rm NX}$ values for all training data sets are multiplied by the
relaxation factor of 0.95. In other words, the threshold for each synthetic
data set is set to $0.95\times\rho_{\rm NX}$ for $\rho_{\rm NX}$ evaluated at
$\alpha=24\mathrm{\mu as}$. This relaxed threshold allows for a large enough
number of Top Set parameters to be identified for all epochs and imaging
pipelines. We ensure that the relaxed threshold still reconstructs all
representative ground truth morphologies; in Appendix E, the worst $\rho_{\rm
NX}$ images are shown for Top Set reconstructions of each model with each
imaging pipeline to demonstrate that the representative features are recovered
even in the worst-fidelity Top Set images.

In Tables 3, 4 and 5, we summarize the parameters and surveyed values in each
pipeline’s survey. These tables indicate the fraction of images corresponding
to that parameter in each pipeline’s Top Set for April 7 data. The results on
April 6 data are summarized in Appendix E. The tables also provide the total
number of surveyed parameter combinations as well as the number of
combinations selected for each Top Set. As seen in each table, there are more
than 1000 parameter combinations in each pipeline’s Top Set for April 7, which
we find is sizable enough for downstream analysis. In contrast, we find that
the April 6 Top Set sizes are much smaller, likely due to the poorer
$(u,v)$-coverage.

###  6.3 Image Posteriors with Themis

Table 6: Bayesian Imaging Priors  Parameter | Units | Prior††Linear priors from $a$ to $b$ are represented by $\mathcal{U}(a,b)$, logarithmic priors from $a$ to $b$ are represented by $\mathcal{L}(a,b)$, normal priors with mean $\mu$ and standard deviation $\sigma$ are represented by $\mathcal{N}(\mu,\sigma)$.
---|---|---
Control points | Jy $\mu{\rm as}^{-2}$ | $\mathcal{L}(10^{-3},0.1)$
FoVx | $\mu{\rm as}$ | $\mathcal{U}(50,500)$
FoVy | $\mu{\rm as}$ | $\mathcal{U}(50,500)$
Raster rotation | rad | $\mathcal{U}(-0.25\pi,0.25\pi)$
Shift in $x$ | $\mu{\rm as}$ | $\mathcal{U}(-100,100)$
Shift in $y$ | $\mu{\rm as}$ | $\mathcal{U}(-100,100)$
$\ln(\sigma_{\rm ref})$ | – | $\mathcal{N}(-5.5,1.0)$
$\ln(f)$ | – | $\mathcal{N}(-4.6,1.0)$
$a$ | – |  $\mathcal{L}(a_{25\%},a_{75\%})$‡‡$x_{25\%}$ and $x_{75\%}$ are the bottom and top quartile values of the quantity $x$ given in Table 2.
$u_{0}$ | – |  $\mathcal{L}(u_{0,25\%},u_{0,75\%})$‡‡$x_{25\%}$ and $x_{75\%}$ are the bottom and top quartile values of the quantity $x$ given in Table 2.
$b$ | – |  $\mathcal{L}(b_{25\%},b_{75\%})$‡‡$x_{25\%}$ and $x_{75\%}$ are the bottom and top quartile values of the quantity $x$ given in Table 2.
$c$ | – | $\mathcal{L}(1.5,2.5)$

The Bayesian imaging method employed in Themis differs from those described
above in a number of respects. Most significantly, apart from sampler tuning –
which affects only the efficiency with which the posterior is explored – the
splined raster model has only two free parameters. Image resolution, raster
orientation, the brightness at each control point, and noise model parameters
are determined self-consistently by the fitting process (see Appendix A). This
is achieved by replacing the hyperparameters associated with field of view,
scattering threshold in the Const prescription, systematic error budget, and
those that define the variability mitigation noise with fit parameters,
precluding the need to survey over them. Priors for each quantity are listed
in Table 6, and make use of the pre-imaging constraints described in Section 3
and listed explicitly in Table 2.

Importantly, eliminating the hyperparameters associated with the additional
contributions to the visibility uncertainties eliminates the noise-related
data preparation steps described in Section 6.1; we do not add any additional
uncertainty prior to the Themis analysis. However, to reduce the data volume
(and thus computational expense of the posterior sampling) we incoherently
average the flux-normalized data over scans. To prevent significant coherence
losses, prior to averaging we calibrate the phase gains of the JCMT using the
intra-site baseline, JCMT-SMA, and assume that the source is unresolved at the
corresponding spatial scales probed by that baseline.

The remaining two unspecified hyperparameters are the raster dimensions,
$N_{x}$ and $N_{y}$. Initial guesses for these are provided by the diffraction
limit; for a typical source size of $80~{}\mu{\rm as}$, $N_{x}=N_{y}=5$ is
sufficient to marginally super-resolve the source. This may then be refined
via a modest survey over potential values, with the final values selected by
maximizing the Bayesian evidence, computed in Themis via thermodynamic
integration (Lartillot & Philippe, 2006). In practice, due to the expense of
such a survey, we restrict ourselves to $N_{x}=N_{y}=5$ for the validation
with synthetic data sets with a sole exception. We performed a raster
dimension survey for the GRMHD data set presented in Section 6.4.2, finding
that $N_{x}=N_{y}=6$ is preferred.

For application to Sgr A∗, we perform raster dimension surveys independently
for April 6 and 7 as described in Section D.4, finding preferred dimensions of
$N_{x}=N_{y}=6$ and $N_{x}=N_{y}=7$, respectively.

###  6.4 Synthetic Data Images

We first present Top Set and posterior images recovered from geometric model
datasets in Section 6.4.1, followed by images recovered from the GRMHD dataset
in Section 6.4.2.

####  6.4.1 Geometric Model Images

(a) Descattered Reconstructions

(b) On-sky Reconstructions

Figure 11:  Reconstructed images of synthetic data sets on April 7 (a) with
and (b) without scattering mitigation for seven geometric models and the GRMHD
model. Reconstructions of each geometric model by DIFMAP, eht-imaging and
SMILI pipelines are made using a parameter combination identified via cross-
validation: the imaging parameters that result in the the best average
$\rho_{\rm NX}$ across all other geometric models. These cross-validation
results demonstrate the ability of the selected parameters to correctly
reproduce novel source morphologies. GRMHD reconstructions for DIFMAP, eht-
imaging, and SMILI are produced from an imaging parameter combination that
performs best on all geometric models. In contrast, for Themis reconstructions
the average posterior image is shown for each model; the average posterior
image appears to correctly identify the true source morphology in all
synthetic data sets tested.

In Figure 11, we show the time-averaged ground truth images and corresponding
image reconstructions for each of the synthetic data sets with April 7
$(u,v)$-coverage. Images from all pipelines are obtained with and without
scattering mitigation (henceforth, descattered and on-sky reconstructions).
These descattered and on-sky reconstructions are compared to the time-averaged
ground truth images of the intrinsic and scattered structure, respectively.

DIFMAP, eht-imaging and SMILI images for each geometric model in Figure 11 are
obtained using cross-validation: the parameter combination that provides the
best mean $\rho_{\rm NX}$ across other geometric models (i.e., except for the
model being tested) are selected. The cross-validation images in Figure 11
(which are contained in each pipeline’s Top Set) successfully recover the
representative morphology of each geometric model, demonstrating the
capability of a single imaging parameter combination to identify various
source structures. The manifestation of structure significantly different from
the ground truth morphology is only seen in a small fraction of the cross-
validated Top Set parameters. For instance, using the ring classification
method described in Appendix F, only 2%, $<$1%, and $<$1% of the cross-
validated Top Set reconstructions for the Ring model are identified as not
having a ring feature for DIFMAP, eht-imaging and SMILI, respectively; the
reconstruction of a non-ring morphology is also found to be limited to small
fraction of 3%, 5%, and 1% for the Crescent model. Similarly, only a small
fraction of the reconstructions from non-ring models are reconstructed with a
ring morphology — in particular, for the DIFMAP, eht-imaging, and SMILI
pipelines, 7%, 11%, and 3% and $<$1%, $<$1%, and $<$1% of the cross-validated
Top Sets reconstructed a ring feature from the Point and Double source models,
respectively.

For Themis reconstructions, the mean posterior image is shown for each model.
Figure 11 shows that the posterior images from Themis reconstructions identify
the general morphology of each geometric model. Most of Themis posterior
images satisfy the criteria based on a minimum threshold of $\rho_{\rm NX}$
used in Top Set selections for DIFMAP, eht-imaging and SMILI pipelines (see
Section 6.2.2), with the exception of a few models discussed below. For April
7 images, all posterior images show higher $\rho_{\rm NX}$ than the threshold
except for 1% and 7% of descattered reconstructions for Double source and
Ring+hs models, respectfully. For April 6, the images below the threshold are
limited to 1% and 22% of descattered reconstructions for the Point and Double
source models, respectively. However, 95% of on-sky reconstructions of Point
source model are above the $\rho_{\rm NX}$ threshold on April 6 for Themis.
The high fractions of images beyond the $\rho_{\rm NX}$ threshold for all
models on April 7 demonstrate the capability of Themis pipeline to recover
various representative morphologies at an acceptable fidelity.

Figure 11 also shows the resiliency of the reconstructed morphology to the
scattering prescriptions. While the on-sky reconstructions without scattering
mitigation tend to be slightly more blurry than those with descattering, there
are not many other notable differences in their appearance. In particular, the
refractive substructure, which adds spatial distortion of images on scales
finer than the angular resolution of the EHT, is not well constrained in our
EHT data and therefore does not strongly appear in any reconstructions. We
further discuss the effects of scattering prescriptions for Sgr A∗ images in
Section 7.5.2.

####  6.4.2 GRMHD Reconstructions

Figure 12:  The distribution of images obtained from synthetic data provided
using a GRMHD movie with April 7 $(u,v)$-coverage. From left to right
(separated by vertical lines), we show the distributions of Top Set images
from the DIFMAP, eht-imaging and SMILI pipelines and posterior samples from
the Themis pipeline; each vertical panel is further subdivided into clusters
identifying common morphologies recovered by each pipeline. The figure is
comprised of three horizontal panels separated by horizontal lines. The top
panel shows individual images randomly sampled from different clusters. The
middle and bottom panels visualize the distributions of reconstructed
descattered and on-sky images for each cluster, respectively. In each panel,
from top to bottom, we show the average of each cluster, the distributions of
the radial profiles, and the distributions of azimuthal intensity profiles. In
the radial profiles, each horizontal slice corresponds to the azimuthally-
averaged intensity profile of an image, normalized by its peak value.
Similarly, the azimuthal profiles show the azimuthal distribution of the
radial peak intensity within a radius of 10 to 40 $\mathrm{\mu as}$, also
normalized by its peak value. The dotted lines in the profiles indicate the
peak radius or position angle of the ground truth GRMHD movie. The vertical
order of images in both profiles is independently sorted by the peak radius or
position angles; therefore the images are ordered differently in each profile
distribution image. These results on synthetic data show that the Top Set and
posterior samples from the imaging pipelines are able to recover ring images
that resemble the true time-averaged structure of a GRMHD movie (see Figure
7). However, the imaging methods sometimes produce non-ring images that still
fit the data well.

We show example GRMHD reconstructions on April 7 in the rightmost panels in
Figure 11. This GRMHD simulation contains a ring with a diameter of $\sim 51$
$\mathrm{\mu as}$. GRMHD images from DIFMAP, eht-imaging, and SMILI pipelines
are reconstructed with the Top Set parameter combinations that corresponds
with the largest average $\rho_{\rm NX}$ value across all seven geometric
models. For Themis reconstructions, on the other hand, the expected (i.e.,
mean) posterior image is shown444Unlike for Sgr A∗, for this particular GRMHD
data set the Themis pipeline used a large scale Gaussian to account for
extended emission in the underlying source model..

Unlike with the simple geometric synthetic data sets, the distribution of
GRMHD reconstructions show wide variations in the image appearance555Note that
this resembles the varied Sgr A∗ reconstructions in Section 7 and Appendix C.
Although GRMHD reconstructions in Figure 11 commonly show a ring-like
morphology with a diameter of $\sim 50$ $\mathrm{\mu as}$, the azimuthal
intensity distribution is not consistent across the Top Sets. In the top
panels of Figure 12, we show images from GRMHD April 7 data from each
pipeline, which are randomly selected from DIFMAP, eht-imaging and SMILI Top
Sets and posterior images from Themis. Most of the images have clear
asymmetric ring features, but a few images show non-ring structures. The
diameters of ring features are broadly consistent across different
reconstructions, comparable to the ground truth image. On the other hand, the
azimuthal distributions are not uniquely constrained by the Top Set images —
different position angles are seen in the randomly sampled images from each
Top Set.

To visualize the distribution of images with different morphology, we
categorize all images into three major groups: ring images peaked at PAs
within (a) the range of $-180^{\circ}\leq{\rm PA}<0^{\circ}$ where the ground
truth value of $-124^{\circ}$ is located, (b) the range of $0^{\circ}\leq{\rm
PA}<180^{\circ}$, and (c) the remaining images comprising non-ring or other
ring-like images with much less consistency. The definition of a ring used in
this paper to classify ring versus non-ring images is described in Appendix F.
Note that the particular definition of a ring will influence the quoted
percentages of ring and non-ring images recovered. We find that the particular
definition chosen in this paper results in classification that largely aligns
with human perception. However, the classification of images that are
borderline between ring and non-ring classification are sensitive to the exact
criteria used. Therefore ring definitions that make use of different criteria
can lead to classification that still largely aligns with human perception but
varies somewhat in the ring classification percentages quoted in this paper.

In the middle and bottom panels of Figure 12, we summarize the distribution of
images from each pipeline with and without scattering mitigation,
respectively. The GRMHD images within each pipeline’s Top Set are clustered
into image modes. The upper sub-panel shows the mean image of each cluster,
indicating a common or representative morphology recovered. The middle sub-
panel shows the distributions of the azimuthally-averaged radial intensity
profiles, where the vertical order of images is sorted by the peak radii of
the profiles. The lower sub-panel shows the azimuthal distribution of the
radial peak intensity within a radius of 10 to 40 $\mathrm{\mu as}$.

Figure 12 demonstrates that most of the Top Set or posterior images
reconstruct ring images from the GRMHD data set. In particular, the radial
profiles of the first two ring modes show a broad consistency of the peak
radius around $\sim$25 $\mathrm{\mu as}$ which implies a diameter of $\sim$50
$\mathrm{\mu as}$ consistent with that of $\sim$50 $\mathrm{\mu as}$ for the
ground truth model. The capability of identifying a ring with the consistent
diameter does not appear to depend on the scattering mitigation. Similarly to
geometric synthetic data (see Section 6.4.1), the scattering mitigation does
not significantly affect the resulting source morphology in the reconstructed
images, except that the images without scattering mitigation tend to be
slightly broader. Refractive substructure does not appear in the reconstructed
on-sky images, again indicating that the appearance of the refractive
substructure is not strongly constrained with EHT 2017 data.

As seen in the azimuthal profiles, each pipeline provides at least one
asymmetric ring mode with the peak position angle roughly consistent with that
of the mean ground truth image of $-124^{\circ}$. For this particular GRMHD
example, this mode is found to be the most popular mode in most pipelines.
However, Figure 12 indicates that the ring mode with the correct orientation
is not always identified as the most popular mode among image samples — for
instance, the correct orientation is not identified as the most popular ring
mode for the eht-imaging on-sky pipeline. Therefore, caution should be taken,
as the popularity of a mode in an RML or CLEAN pipeline’s Top Set does not
necessarily always correspond with the true underlying structure.

We find that key takeaways from the GRMHD example are consistent with the
“best-bet” GRMHD models presented in Paper V, identified based on various
criteria using both EHT and non-EHT data. In Appendix H, we show example
reconstructions of a “best-bet” GRMHD model. We find the identification of a
ring morphology for the vast majority of the Top Set reconstructions, however
multiple ring modes and non-ring images are still reconstructed. Note also
that the peak position angle of the ground truth “best-bet” GRMHD model is not
necessarily identified as the most popular mode reconstructed in each Top Set.
These results indicate that the same trends are seen across multiple GRMHD
models that are broadly consistent with EHT data.

##  7 Horizon-scale Images of Sgr A∗

Figure 13:  In the top main panel we show the representative image of Sgr A∗
obtained with the EHT from observations on 2017 April 7. This top image is
obtained by averaging the bottom four images. On the bottom from left to
right, we show the average images of three prominent ring clusters with
different azimuthal structures and a non-ring cluster. The height of the
colored bar (bottom left in each panel) represents the relative fraction of
images in the Top Sets for each cluster. We note that Themis posterior sample
only includes ring images. In each cluster, the image is computed through a
weighted average over the descattered reconstructions, including all Top Set
images from the three imaging methods (DIFMAP, eht-imaging, and SMILI) and
1024 images randomly selected from descattered posteriors from Themis. Images
are weighted by the inverse of the total number of Top Set or posterior images
used from each pipeline, so that pipelines are represented equally in each
image. Note that DIFMAP model images are convolved with 20 $\mathrm{\mu as}$
beam (represented by the inset circle), while no blurring is applied to the
rest of images.

Having determined Top Set imaging parameters for RML and CLEAN, and validated
posterior estimation for Themis via tests on synthetic data (Section 6), we
now show the result of these methods applied to Sgr A∗ data from the 2017 EHT
observations. Unlike in the previous EHT imaging of M87∗ (; ), Sgr A∗’s
recovered structure depends somewhat on the imaging strategy and parameter
choices. Thus, this section presents our main imaging results and analyzes how
the image structure is affected by different imaging choices.

We begin in Section 7.1 by giving an overview of the results, followed by a
more detailed discussion of the image structures recovered in each pipeline’s
Top Set or posterior in Section 7.2. Average images across pipelines,
calibrated data sets, and observing days are discussed in Section 7.3. In
Section 7.4 we present Sgr A∗ imaging obtained by combining the data sets for
April 6 and 7. In Section 7.5 we address the question of whether Sgr A∗ is a
ring, and explore how the recovered images are affected by the scattering and
temporal variation mitigation strategy.

###  7.1 Overview of Recovered Sgr A∗ Structure

Figure 13 summarizes the common morphologies recovered from Sgr A∗ data by the
four imaging pipelines for April 7. We find that the vast majority are rings
that can be classified into three different clusters with varying azimuthal
structures (Section 7.2), shown in the three bottom left panels of Figure 13,
and a small fraction of non-ring images that also fit the Sgr A∗ data well
(bottom right panel in Figure 13). Since these non-ring images are not as
consistent in structure they largely blur out when averaged together and
primarily emphasize a double-structure that is sometimes present. The
representative Sgr A∗ image obtained by averaging reconstructions from all
four clusters is shown in the large top panel of Figure 13, corresponding to
that of a ring with a diameter of $\sim$50 $\mu$as (Section 8).

Figure 14:  The distribution of reconstructed Sgr A∗ images on April 7. We
show the distribution of images from each pipeline for each cluster with the
same convention as Figure 12. From left to right (separated by vertical
lines), we show the distributions of Top Set images from the DIFMAP, eht-
imaging, and SMILI pipelines and posterior samples from the Themis pipeline;
each vertical panel is further subdivided into clusters identifying common
morphologies recovered by each pipeline. The figure is comprised of three
horizontal panels separated by horizontal lines. The top panel shows
individual images randomly sampled from different clusters. The middle and
bottom panels visualize the distributions of reconstructed descattered and on-
sky Sgr A∗ images for each cluster, respectively. In each panel, from top to
bottom, we show the average of each cluster, the distributions of the radial
profiles, and the distributions of azimuthal intensity profiles. Note that
Themis images have only three clusters for each of descattered and on-sky
reconstructions — their descattered posterior does not contain a non-ring
cluster and their on-sky posteriors do not contain an east PA ring cluster.
Figure 15:  Comparison of image characteristics between each cluster. The
radial and azimuthal values are the fractional central brightness and
azimuthal peak brightness PA, respectively. The left panel shows the
distribution of all Top Set and posterior images with all imaging pipelines
(DIFMAP, eht-imaging, SMILI, and Themis) and scattering mitigation
(descattered and on-sky), including the three ring clusters with different
peak azimuthal brightness [red ($-180^{\circ}\leq{\rm PA}<-70^{\circ}$), blue
($-70^{\circ}\leq{\rm PA}<40^{\circ}$), and green ($40^{\circ}\leq{\rm
PA}<180^{\circ}$)], and a non-ring cluster (white). The right eight panels
show distributions within a single imaging pipeline and scattering mitigation.
The majority of azimuthal peak brightness among each ring cluster is shown in
the outer gray histogram in each panel.  Figure 16:  Closure phases plotted as
function of GMST on four selected triangles from April 7 observations. Each
line indicates the corresponding closure phase curves from a single Top Set
and posterior image randomly selected from each cluster. The error bars of Sgr
A∗ data include the fractional 1 % noise budget for systematic error and a
representative budget for scattering and temporal variability – in particular,
the J18model1 refractive noise model and a variability model with parameters
$a=0.02$, $u_{0}=2$, $b=2.5$, and $c=2$. These additional noise budgets are
all added to the 60 second complex visibility noise budget prior to forming
closure quantities. All images show reasonable fits to Sgr A∗ data as they all
are within two standard deviations of the observed data.

###  7.2 Clustering of Recovered Sgr A∗ Images

To effectively visualize the distributions of Sgr A∗ images, we cluster all
reconstructed Top Set and posterior images using a similar criteria to that
used in Section 6.4.2 and Figure 12 (see Appendix F for details). Images with
a ring feature are grouped by the peak PAs in the southwest
($-180^{\circ}\leq{\rm PA}<-70^{\circ}$), northwest ($-70^{\circ}\leq{\rm
PA}<40^{\circ}$), and east ($40^{\circ}\leq{\rm PA}<180^{\circ}$) directions.
The Sgr A∗ image clustering results for each pipeline are summarized in Figure
14; here, we focus on the observed date of April 7 with the HOPS data
reduction pipeline (see Section 7.3 for April 6 and CASA-based
reconstructions). The results of the DIFMAP, eht-imaging, SMILI, and Themis
pipelines are displayed from left to right. Images are separately clustered
within each pipeline.

The middle and bottom panels of Figure 14 show each cluster’s average image
for the descattered and on-sky images, respectively. To better visualize the
properties of individual Sgr A∗ images, we present three randomly selected
descattered images from each cluster in the columns of Figure 14’s top panel;
within each cluster images appear to have largely consistent morphologies.

The percentages in each panel of Figure 14 show the fraction of Top Set or
posterior images contained in that particular cluster. These percentages
indicate that most of the Top Set and posterior images have ring structures.
For instance, the fraction of non-ring cluster images is $\leq 5$ % of the Top
Set descattered images from DIFMAP, eht-imaging, and SMILI imaging pipelines.
Although this is significant, we note that in the case of the Top Sets this
does not constitute a likelihood and therefore the reported fractions should
not be considered as an exact measure of our degree of certainty. In addition,
only ring images appear in Themis posterior estimation of descattered images.

The bottom row of both the middle and bottom panel of Figure 14 shows the
azimuthal distribution of the radial peak intensity within a radius of 10 to
40 $\mathrm{\mu as}$. These profiles are sorted within each cluster by the
location of peak brightness to best accentuate variations within a cluster. By
inspecting the profiles in each cluster it can be seen that three primary
brightness distributions, with a peak brightness at $\sim-140^{\circ}$,
$-40^{\circ}$, or $70^{\circ}$, appear across the pipelines. Thus, even when
we restrict our attention to ring-like morphology reconstructions, it is
difficult to constrain the azimuthal profile around the ring via imaging. This
azimuthal uncertainty could be due to data properties (e.g., sparsity or low
S/N data) or variation in the intrinsic azimuthal structure of Sgr A∗ due to
intra-day evolution.

To visualize the image-domain differences among each cluster, we compare in
Figure 15 the relationship between the fractional central brightness ($f_{c}$,
radial values) and the azimuthal peak brightness (${\rm PA}$, azimuthal
values). The polar distribution of Top Sets and posterior images among all
imaging pipelines and with and without scattering mitigation (left panel in
Figure 15) shows that most of the ring images within each different cluster
(indicated by the red, blue, and green points) have a smaller fractional
central brightness ($f_{c}\lesssim 0.5$) than that for the images in the non-
ring cluster ($f_{c}\gtrsim 0.5$).

The histogram of the azimuthal peak brightness distributions shown in left
panel in Figure 15 provides a clear visualization of the clustering of ring
images around position angles of $\sim-140^{\circ}$, $-40^{\circ}$, or
$70^{\circ}$, which correspond to the locations of the three knots that
commonly appear in ring images. Changes in position angle mostly reflect
variations in the relative brightness of these knots. For the case of non-ring
images the position angles are, as expected, more randomly distributed. Figure
15 also confirms that the fractional central brightness are systematically
larger for the on-sky images due to the angular broadening produced by the
interstellar scattering.

Both the ring and non-ring morphologies identified in Top Set and posterior
images show reasonable fits to Sgr A∗ data. In Appendix E, we show $\chi^{2}$
distributions of Top Set images to Sgr A∗ data. After adding the budgets of
non-closing systematic errors, representative refractive noise, and time
variability, all Top Set images result in a $\chi^{2}<2$ for both closure
phases and log closure amplitudes – we refer the reader to (Paper IV) and
(M87∗ Paper IV) for a discussion of these data products and $\chi^{2}$
distributions. In Figure 16 we compare closure phases of Sgr A∗ to those of
individual images from each cluster and pipeline for four selected closure
triangles. As indicated by the $\chi^{2}$ metrics, multiple ring and non-ring
images fit the observed closure phases within the range of deviations
anticipated due to temporal variability and refractive scattering effects.
Therefore, none of the identified clusters can be excluded from possible Sgr
A∗ morphologies in terms of the goodness of fit to Sgr A∗ data (or through
synthetic data tests presented in Section 6).

###  7.3 Average Sgr A∗ Images across Pipelines

(a) April 6 |  (b) April 7
---|---
Figure 17:  The dominant recovered morphology in Sgr A∗ descattered and on-sky
reconstructions identified from two VLBI data products (CASA and HOPS data)
with all four imaging pipelines (DIFMAP, eht-imaging, SMILI and Themis) for
two observing days (April 6 and 7). Each panel shows the average image of the
corresponding Top Set images for DIFMAP, eht-imaging and SMILI pipelines, and
the average posterior image for Themis pipeline. Only the HOPS data has been
imaged using the Themis pipeline.

Figure 17 shows the average of Sgr A∗ images reconstructed by each of the four
imaging pipelines (DIFMAP, eht-imaging, SMILI, and Themis) from each
calibrated data set (CASA and HOPS) on each observing day (April 6 and 7).
Only images from the HOPS data product have been reconstructed using the
Themis pipeline. The images from DIFMAP, eht-imaging and SMILI are obtained by
averaging their respective Top Set images; these average images show the
dominant features identified across different combinations of the selected
imaging parameters in the Top Sets (refer to Section 6). For the Themis
reconstructions we instead show the mean of each posterior obtained by
averaging all the posterior samples. Figure 17 shows that the majority of
images contain a ring-like structure. This ring morphology is common among all
imaging pipelines, and is resilient to the scattering mitigation strategy
employed (as discussed in Section 7.5.2). Additionally, we recover largely
consistent images between data calibrated by the HOPS and CASA calibration
pipelines. Although we recover images with a ring morphology in the majority
for all pipelines on April 7, these average images also highlight that the
azimuthal brightness distribution is sensitive to small changes in the data
and imaging strategy.

###  7.4 Imaging Combining April 6 and 7 Data Sets

Figure 18:  The effect of removing the Chile-LMT baseline from April 6 data
reconstructions. Each panel shows the average image of the Top Set images for
the DIFMAP, eht-imaging and SMILI pipelines from April 6 and the HOPS data
product for the descattered and on-sky reconstructions. For comparison we show
the average images obtained from full data sets as well as image obtained from
data without the Chile-LMT baselines. This Chile-LMT baseline appears near the
visibility null, and appears to exhibit significant intra-day variations on
April 6 that is likely not captured by the variability noise model presented
in Section 3.2.  Figure 19: Mitigation of the apparent strong intra-day
variations seen in April 6 through Themis imaging of the combined April 6 and
7 data. For comparison we show the average posterior images from April 6 and 7
independently and that obtained by combining the data sets for April 6 and 7,
exhibiting a clear ring for the descattered and on-sky reconstructions.

In Figure 17 we show results from April 6 data, reconstructed using the same
imaging procedure as April 7. We note that although a ring feature appears in
most of these reconstructions, it is less prominent. The images also contain a
diagonal rail-like feature going from northeast to southwest that corrupts the
ring. This feature is especially prominent in the Themis April 6 descattered
reconstruction. This corrupted or non-ring mode is likely emphasized in Themis
imaging compared to RML and CLEAN pipelines due to the goal of Themis image
samples to characterize the probability of an image rather than represent the
variety of possible images that can fit the data.

Through an in-depth inspection of the April 6 data, documented in Paper IV,
the Chile-LMT baselines (i.e., ALMA-LMT and APEX-LMT baselines) were
identified as having large coherent visibility swings that are not effectively
mitigated by the noise model on a single day, causing this particular feature
to arise. Figure 18 shows the reconstructions using the DIFMAP, eht-imaging
and SMILI imaging pipelines when these particular baselines have been flagged,
resulting in a cleaner ring structure.

The large visibility swings on the Chile-LMT baselines could be the cause of
variability around Sgr A∗ that exceeds expectations set by the incorporated
stationary noise model presented in Section 3.2. In particular, we believe
that the variability noise model should capture Sgr A∗’s stochastic evolution
in expectation, but a single night may contain non-stochastic short-lived
variability that can bias a single day’s reconstruction. Correlated
variability may be mitigated via multi-day fits, which combine statistically
independent structural fluctuations; this better matches the assumption within
our variability mitigation scheme of an underlying stochastic process, though
does carry with it the additional assumption that Sgr A∗ is statistically
stationary over the multiple days combined. Therefore, in addition to the
single day analyses performed by all imaging methods, the Themis image model
was also fit to the combined April 6 and 7 scan-averaged data from high and
low bands for the HOPS data product. The resulting static image shown in
Figure 19 represents an image recovered from the combined data sets. All
images within the multi-day Themis posterior exhibit a clear ring-like
structure.

###  7.5 Is Sgr A∗ a Ring?

Our primary imaging goal is to answer the question: “Is Sgr A∗ a ring?”.
Although our reconstructions are overwhelmingly dominated by ring images,
there is a small number of non-ring images that fit the data well and cannot
easily be excluded through additional tests. There are at least three possible
reasons for the recovery of non-ring reconstructions of Sgr A∗: 1) Sgr A∗’s
intrinsic structure is not ring shaped, 2) scattering causes a distortion of a
ring morphology, resulting in a non-ring image, and 3) imaging algorithms
recover an incorrect source structure, aggravated by challenges of sparse
$(u,v)$-coverage and Sgr A∗’s intra-day variation. In this section we will
explore these three possible origins of non-ring structure. Based on this
exploration we conclude that there is evidence that the non-ring
reconstructions are caused by our imaging algorithms resulting from the
limited $(u,v)$-coverage and rapid variability, rather than being intrinsic to
Sgr A∗.

####  7.5.1 Manifestation of Rings from Intrinsic Non-rings

The first possible explanation for the small percentage of non-ring
reconstructions of Sgr A∗ is simply that Sgr A∗ does not possess a ring
morphology. Although this possibility cannot be ruled out completely, we
explore the possibility of a potential bias in our imaging approach towards
recovering ring images from underlying non-ring sources.

There is always a possibility that the parameters initially explored by the
RML and CLEAN parameter surveys were inadvertently biased to produce mostly
ring images. This would result in an artificially high percentage of ring
reconstructions that could give overconfidence in an incorrect ring solution.
To explore this hypothesis we inspect the percentage of ring and non-ring
images of Sgr A∗ that are in the initial parameter survey versus the
restricted Top Set on April 7. We find that although ring images make up a
large percentage of the initial parameter surveys, 91%, 84%, 60% for DIFMAP,
eht-imaging, and SMILI respectively for the descattered reconstructions, this
number becomes significantly larger after Top Set selection. In particular,
the percentage of ring images raises to values of 95%, 97%, 98%, respectively.
Therefore, when we restrict to those parameters that can best disambiguate
between the different ring and non-ring source morphologies contained in our
synthetic data sets (Section 5.1), the number of ring images increases. We
also note that Themis posterior samples for April 7 only contains ring images.

To further test the possibility that we are biased to recover primarily rings
from underlying non-ring sources, we have explored the performance of our
imaging methods on non-ring synthetic data sets to see how they compare to Sgr
A∗ results. In particular, we performed a Top Set analysis on the Point and
Double source models. Two “cross-validation" Top Sets were identified by
excluding the performance of either the Point or Double source model and only
using the remaining six geometric models in selection of the imaging
parameters. These two cross-validation Top Set parameter sets were then
applied to the Point and Double synthetic data sets. Although each cross-
validation Top Set does incorrectly reconstruct some ring images, each image
set primarily contains reconstructions that look similar to the ground truth
Point ad Double source structure. In particular for descattered
reconstructions, only 9% and $<$1% percentage of the cross-validation Top Set
reconstructions possess a ring morphology, compared to 97% in Sgr A∗
reconstructions of April 7. The Themis pipeline produces no ring images in the
posterior samples for both the Point and Double data sets. Thus, in the non-
ring synthetic data sets we have explored we find that the number of incorrect
ring reconstructions is much less than the number that we recover for Sgr A∗.
In summary, our imaging pipelines do not appear prone to artificially create a
majority of ring structures in sources that do not possess an intrinsic ring
morphology.

####  7.5.2 Scattering’s Effect on Image Reconstruction

The second possible explanation for the non-ring reconstructions of Sgr A∗ is
that interstellar scattering causes a non-ring morphology. This could take
either two forms: a) Sgr A∗’s ring morphology has been distorted to a non-ring
morphology by an interstellar scattering screen, or b) our imaging algorithms
have incorporated an incorrect scattering model that reconstructs a corrupted
non-ring morphology.

To address the possibility that interstellar scattering is causing Sgr A∗’s
intrinsic structure to be distorted to a non-ring, we inspect differences in
the descattered versus on-sky reconstructions. As outlined in Section 3.1,
descattered reconstructions attempt to mitigate two primary effects of
interstellar scattering — diffractive scattering that causes angular
broadening and refractive scattering that introduces small-scale structure to
the on-sky image. Sgr A∗ reconstructions in Figure 14 show that on-sky images
are systematically blurrier than descattered images, as expected due to
deblurring that is performed before recovering a descattered image (refer to
Section 3.1.1). This systematic difference between on-sky and descattered
images is also seen in synthetic data reconstructions presented in Section
6.4. This angular broadening causes the central dip of a ring to be less
prominent. Nonetheless, we note that the vast majority of the on-sky (i.e.,
without any scattering mitigation prescription) images are still rings, with
percentages of 93%, 98%, 90%, and 98% for CLEAN, eht-imaging, SMILI, and
Themis, respectively. We note that it is highly unlikely that a non-ring
morphology was distorted into an on-sky ring morphology by interstellar
scattering.

To address the possibility that our assumed model of the interstellar
scattering is reconstructing a corrupted image of the descattered source, we
explore the use of multiple refractive noise models to mitigate effects of
refractive substructure before deblurring (Section 6.2). The contribution of
the included refractive noise budget used to mitigate scattering (for
descattered reconstructions) is non-negligible for long baseline data (see
Section 3.1.2 and Figure 5). Nontheless, the RML and CLEAN imaging surveys
show that the choice of the refractive noise model does not significantly
affect the resulting distributions of image structures. Figure 20 shows the
average descattered Sgr A∗ images (clustered into the same four morphologies
as is presented in Figure 13) for each refractive noise prescription explored.
The comparable fractions of images in each cluster across the different
descattering strategies indicates the resiliency of the recovered image
structures to different refractive noise models. Another piece of evidence
suggesting that the scattering prescription does not strongly affect results
can be seen in the Themis results; in Themis a constant refractive noise floor
is able vary in posterior sampling of descattered images. Although an
essentially unlimited refractive noise component is permitted by the Themis
model, the posterior estimation instead typically chooses a noise level of
$<25$ mJy, only 1% of the total flux of the source.

Figure 20: The range of descattered Sgr A∗ images recovered using different
refractive scattering noise models. From left to right, images are shown
clustered in the same order as Figure 15. Each panel shows an average of all
descattered Top Set images of Sgr A∗ on April 7 from the DIFMAP, eht-imaging,
and SMILI pipelines that were generated using the specified refractive noise
model (from top to bottom: Const, 2$\times$Const, J18model1, and
2$\times$J18model1). The number on each panel shows the percentage of
descattered Top Set images that were reconstructed using the specified
refractive noise model. These percentages indicate that there is not a clear
preference for a particular refractive noise model in Top Set selection.
Additionally, the recovered image structure does not appear to correlate with
the refractive noise model used.

In summary, results indicate that for EHT measurements of Sgr A∗, interstellar
scattering does not significantly affect the recovered morphology. Both on-sky
and descattered images contain a majority of ring morphologies. In addition,
we find the particular choice of scattering mitigation strategy only minimally
affects the recovered image structures.

####  7.5.3 Variability’s Effect on Image Reconstruction

The third possible origin for non-ring Sgr A∗ images is poor reconstruction
quality of the imaging methods. Imaging is solving an ill-posed inverse
problem due to sparse $(u,v)$-coverage, which always will have the possibility
of recovering an incorrect image. This is further exasperated in imaging Sgr
A∗ by the challenges that come with recovering an evolving source. In this
section we explore this possibility and find that our imaging methods often
reconstruct non-ring sources for variable ring sources with a comparable small
percentage as found for Sgr A∗.

The first natural question is how our imaging methods perform on
reconstructing ring sources with similar data properties as Sgr A∗. We find
that our methods produce non-ring images, even when the underlying source
structure is ring-like. As an example, recall that for the variable GRMHD
ring-like sources analyzed in Section 6 and Appendix H, our imaging methods
produced mostly rings, but also a very small fraction of non-ring images.
Therefore, we expect that for a variable ring source we would recover some
non-rings that fit the data well. However, note that the number of non-rings
was still fairly small in the case of the GRMHDs: 5% for the GRMHD presented
in Figure 12 and 4% for the “best-bet” GRMHD presented in Figure 37. These
values are comparable to the 3% of non-rings reconstructed for Sgr A∗
descattered images across all pipelines. We also find that cross-validated Top
Set images reconstructed of the Crescent and Ring geometric sources produce a
small fraction of non-ring images. Note that these non-ring percentages for
the GRMHD, Crescent, and Ring sources are much less than the percentage of
non-rings reconstructed for the variable Point and Double source, as discussed
in Section 7.5.1.

Due to Sgr A∗’s intra-day evolution, mitigation of temporal variability in the
data is important for reconstructing a single static image of Sgr A∗. We next
explore how the variability mitigation approach affects the proportion of ring
versus non-ring images reconstructed. Section 3.2 introduced an approach to
model the temporal variability as an additional noise budget that could be
added in quadrature to the visibility thermal noise budget. The dependence of
the temporal variability noise model parameters on resulting Sgr A∗
reconstructions for the RML and CLEAN pipelines was investigated. Although a
variability noise budget generally helps with imaging (as evidenced by a
higher percentage of Top Set parameters selected – 89-95% of the Top Set
parameter combinations include a variability noise model on April 7), similar
to scattering, we find that there are no significant differences between the
images recovered under different variability noise parameters. We suspect that
this is partly caused by the complex interplay between regularizers and
different noise model parameters (e.g., variability, scattering, and
systematic), although no significant trends were identified.

In Appendix G, we demonstrate how our results are insensitive to a different
method of variability mitigation. In particular, we show time-averaged
morphologies identified by a large survey over full-track RML dynamical
imaging parameters (refer to Section 4.4.1) that does not rely on the
variability noise model presented in Section 3.2.2. We again find the ring
modes to be dominantly reconstructed, while a small fraction of non-ring
structures are also identified. The broad consistency indicates that our
results are resilient to at least two different methods to recover time-
averaged morphology.

In summary, we find that our imaging methods do reconstruct a small percentage
of non-ring images from ring sources with comparable variability as that seen
in Sgr A∗ data. Similar behavior is observed across a variety of different
imaging approaches to mitigate variability. Therefore, we conclude that we
would expect our methods to produce a small fraction of non-ring images from
an underlying variable ring morphology.

##  8 Image Analysis

###  8.1 Ring Parameter Fitting

To analyze the Sgr A∗ images we use two tools, REx (Chael, 2019) and VIDA
666https://github.com/ptiede/VIDA.jl (Tiede et al., 2022), both of which are
able to extract quantitative and pertinent information from the Top Set
images. Here we briefly review the two image extraction techniques.

REx attempts to extract ring-like features by directly characterizing the
features of the Top Set images. This is the same tool used in 2017 M87∗
analysis (M87∗ Paper IV). The detailed definition of REx ring parameters
follow those of the M87∗ analysis (M87∗ Paper IV).

In REx, the ring center ($x_{0}$, $y_{0}$) is determined so that the
dispersion of intensity peak radii is minimized. Around the center, a polar
intensity map $I(r,\theta|x_{0},y_{0})$ is constructed. The ring radius
$r_{0}$ (or diameter $d=2r_{0}$) is defined as the radius where azimuthally-
averaged radial intensity peaks. The ring width $w$ is $\left<{\rm
FWHM}[I(r,\theta|x_{0},y_{0})-I_{\rm floor}]\right>_{\theta}$, where
$I_{\rm{floor}}=\left<I\left(r_{\rm max}=60\ {\rm\mu
as},\theta|x_{0},y_{0}\right)\right>_{\theta}$777We set $r_{\rm max}=60\
\rm\mu as$ instead of the $50\ \rm\mu as$ used in M87∗ Paper IV due to the
larger ring size.. To characterize the azimuthal structure, we define the
normalized first circular moment at radius $r$ as

$m_{1}(r)=\frac{\int^{2\pi}_{0}I(r,\theta|x_{0},y_{0})\cos(\theta)\mathrm{d}\theta}{\int^{2\pi}_{0}I(r,\theta|x_{0},y_{0})\mathrm{d}\theta}.$
(11)

The ring position angle $\eta$ and asymmetry $A$ are given by the radially
averaged (from $r_{0}-w/2$ to $r_{0}+w/2$) argument and amplitude of
$m_{1}(r)$ respectively. Finally, a central fractional brightness $f_{c}$ is
defined as a ratio of the mean brightness within 5 $\mu$as from the center to
the azimuthally-averaged brightness along the ring ($r=r_{0}$).

VIDA takes a forward modeling or template matching approach for image analysis
(Tiede et al., 2022). That is, we approximate the images with parametric
families or templates $f_{\Theta}$, such as as rings, crescents, or Gaussians.
The template used depends on both the observed image structure and the
features of interest. VIDA’s approach is therefore similar to geometric
modeling presented in (Paper IV), except it is applied to the image
reconstructions rather than in the visibility domain. The image features, such
as diameter, are then given by the parameters of the optimal template. This
differs from REx which defines its quantities directly on the image.

To find the optimal template we first re-normalize the Top Set image to form a
unit flux image $\hat{I}(x,y)$. We then take the L2 norm as our objective
function:

$J(\Theta)=\mathrm{LS}(f_{\Theta}||\hat{I})=\int|f_{\Theta}(x,y)-\hat{I}(x,y)|^{2}\mathrm{d}x\mathrm{d}y,$
(12)

where $\Theta$ denotes the template parameters.

For analyzing Sgr A∗ Top Set images, we use VIDA’s SymCosineRingwFloor
template. This template is characterized by a ring center ($x_{0}$, $y_{0}$),
diameter $d$, FHWM width $w$, and a cosine expansion to describe azimuthal
brightness distribution $S$:

$\displaystyle
S(\theta)=1-2\sum_{m=1}^{M}A_{m}\cos\left[m(\theta-\eta_{m})\right].$ (13)

For this paper we take $m=4$. Note we also restrict $A_{m}<0.5$ to restrict
negative intensity in the template888It is still possible for negative
intensity with this restriction. To prevent negative intensity we further take
${\rm max}(0,S(\theta))$ when computing the template.. The position angle of
the image is taken as the phase of the first order cosine expansion, i.e.,
$\eta_{1}=\eta$. Similarly we define the asymmetry $A=A_{1}$ to match REx’s
definition above. Note, this azimuthal structure is very similar to the
$m$-ring model described in Section 9.

In addition, we add a central disk to constrain the central brightness
depression of the ring. This disk is forced to have the same radius as the
ring, and a Gaussian taper is included that matches the width of the ring. We
then compute the central fractional brightness, $f_{c}$, of the optimal
template, following the definition above.

In the next subsection, we provide the results of both REx and VIDA for five
ring parameters, namely ring diameter $d$, position angle $\eta$, ring width
$w$, fractional central brightness $f_{c}$, and asymmetry $A$. Note that while
each methods parameter definition are similar to each other REx produces
estimates on the image directly, while VIDA parameter values are defined from
the optimal template. If the template provides a “good” approximation to the
on-sky image these estimates should be similar. Note, the negative pixels of
Themis images are treated as zero in the REx and VIDA analyses.

###  8.2 Ring Fitting Results

Figure 21: The ring diameters measured from April 7 images, shown separately
for each cluster with a ring morphology, each pipeline, and descattered or on-
sky reconstructions. Circle and triangle points and associated error bars
indicate the means and standard deviations of diameters measured with REx and
VIDA, respectively. Note that Themis error bars appear significantly smaller
than the error bars on DIFMAP, eht-imaging, and SMILI. This is partly due to
the fact that Themis is primarily quantifying aleatoric (e.g., statistical)
uncertainty whereas the goal of the other imaging surveys is to also
characterize epistemic (e.g., systematic) uncertainty.

In this subsection, we present ring parameter results only for images from
ring morphology clusters (Figure 13). In Figure 21, we show the diameters
measured for on-sky and descattered ring images for each pipeline. We find
that the ring images are consistent with a diameter of $\sim$50 $\mu$as, as
shown for each ring morphology cluster separately in Section 7.2 and Figure
14.

Figure 22: Ring fit results for Sgr A* Top Set descattered images in the ring
clusters reconstructed from April 7 data. Each panel shows the distribution of
ring parameters corresponding to the images resulting from a single pipeline.
In each panel, the scatter plot on the left shows a ring parameter extracted
from each image using REx (orange) and VIDA (green); the vertical histogram on
the right shows the resulting kernel density estimation from the collection of
extracted image parameters. Images are ordered by clusters described in
Section 7.2, of which boundaries are shown with vertical solid lines. From
left to right, the panels show the results for DIFMAP, eht-imaging, SMILI and
Themis. For the radius and position angle, we also show the radial/azimuthal
brightness distribution of each image in the background.  Figure 23: Ring fit
results for Sgr A* Top Set on-sky images reconstructed from April 7 data.
Refer to the caption of Figure 22 for more detail.

For more detailed distributions of ring parameters, in Figure 22 we show the
ring parameter fitting results of Sgr A∗ descattered Top Set or posterior
images for April 7. From top to bottom, distributions of ring radius, position
angle, ring width, fractional central brightness, and asymmetry are presented
for four pipelines. Results from both REx and VIDA are shown, and for
comparison azimuthally or radially averaged brightness distributions are shown
in the background of diameter $d$ and position angle $\eta$ plots.

Radius distributions are clearly peaked at $\sim$ 25 $\mu$as and mostly
concentrated between 25 and 30 $\mu$as, being consistent across the three
pipelines. Note that in some occasions, REx and VIDA show discrepancy in the
radius (and some other parameters), which is mostly due to the difference of
the responses to an image with a salient feature. However, as seen in the
kernel density distributions of the radius, contributions of such outliers are
negligible for determinations of the mean ring radius. Meanwhile, the PA value
is by far less consistent across the pipelines or even within a single
pipeline. Multiple modes clearly appear in the Top Set images with various PA
values, as already seen in Section 7. Note that the scatter of PAs tends to be
larger when the ring has multiple bright spots that affect the resulting PA,
or when the azimuthal profile is close to uniform without a clear peak. Ring
width values are $\sim 30\ \rm{\mu as}$, and this may come from the angular
resolution of the observation. Fractional central brightness is mostly $\sim
0-0.3$ for RML and Themis while minor non-ring modes (right-most clusters in
each panel) tend to show somewhat higher values. These values confirm that
majority of the images show ring-like structures with clear central
depression. CLEAN images tend to have a slightly larger value of width and
central brightness than RML and Themis, as expected due to the beam-
convolution effect for CLEAN imaging. Asymmetry values are $\sim 0.1$,
indicating that most of Sgr A∗ images have a nearly symmetric azimuthal
intensity distribution on ring. Apart from time variability, Sgr A∗’s apparent
symmetry could be one of possible reason for difficulty in constraining the
PA.

Figure 23 shows the ring parameter fitting results of the on-sky Top Set or
posterior images. Comparing the ring parameters with and without scattering
mitigation, PA values are slightly different and FWHM and fractional central
brightness values are larger for the on-sky images. These differences are
reasonable when one considers the angular broadening effect due to scattering
that remains in on-sky images. On the other hand, radius and asymmetry show
similar distributions between descattered and on-sky images – we obtain a ring
diameter of $\sim$50 $\mu$as for on-sky images, indicating that the ring size
is robust with little dependence on the scattering correction.

In Table 7, we summarize the mean diameters and their standard deviations over
the ring images reconstructed by each pipeline for both descattered and on-sky
images and on both April 6 and 7. As seen in the table, the mean ring radii
for on-sky images are slightly smaller than those of the descattered cases by
a few $\mu$as. This small reduction of ring radii is mainly due to the
difference of effective resolutions with/without scattering mitigation.
Nevertheless, the ring radii are within their standard deviations and thus
broadly consistent regardless of scattering mitigation.

Comparing the results of April 6 and 7 in Table 7, diameters derived from ring
reconstructions are consistent within the computed standard deviations,
regardless of the pipelines. The other parameters (except for PA) also give
consistent values for both days. Again, PA values have a large scatter in
April 6 images; the existence of large scatter in PA indicates that it is
difficult to constrain the azimuthal brightness distributions along the ring.
In general, most of the Top Set images show a ring morphology with a
consistent diameter around $\sim$50 $\mu$as.

In Appendix I we list the fitted diameter, width, position angle, asymmetry,
and fractional central brightness measured for each one of the identified
imaging clusters and different pipelines shown in Figure 22 and Figure 23.

Table 7: Mean and standard deviation of diameter $d$ and width $w$ measured from Top Set or posterior Sgr A∗ images for each pipeline. | Descattered |  |  |
---|---|---|---
|  | $d\ \rm{(\mu as)}$ | $w\ \rm{(\mu as)}$
DIFMAP |  |  |
April 6 Ring | REx | $46\pm 4.1$ | $33\pm 3.5$
| VIDA | $51\pm 3.1$ | $33\pm 3.1$
April 7 Ring |  | $49\pm 2.1$ | $32\pm 2.9$
|  | $51\pm 1.5$ | $33\pm 1.7$
eht-imaging |  |  |
April 6 Ring |  | $56\pm 4.5$ | $24\pm 2.4$
|  | $59\pm 11.3$ | $30\pm 10.4$
April 7 Ring |  | $54\pm 2.0$ | $26\pm 2.6$
|  | $54\pm 4.6$ | $27\pm 3.5$
SMILI |  |  |
April 6 Ring |  | $57\pm 3.4$ | $24\pm 1.9$
|  | $46\pm 12.0$ | $50\pm 16.6$
April 7 Ring |  | $52\pm 4.7$ | $26\pm 2.1$
|  | $52\pm 4.0$ | $27\pm 4.7$
Themis |  |  |
April 6 Ring |  | $51\pm 3.9$ | $25\pm 1.2$
|  | $54\pm 0.9$ | $24\pm 0.9$
April 7 Ring |  | $53\pm 0.7$ | $22\pm 0.5$
|  | $56\pm 1.2$ | $27\pm 0.7$
| On-sky |  |  |
---|---|---|---
|  | $d\ \rm{(\mu as)}$ | $w\ \rm{(\mu as)}$
DIFMAP |  |  |
April 6 Ring | REx | $46\pm 3.0$ | $34\pm 4.3$
| VIDA | $47\pm 1.9$ | $39\pm 3.4$
April 7 Ring |  | $48\pm 2.7$ | $38\pm 2.7$
|  | $49\pm 2.1$ | $37\pm 2.0$
eht-imaging |  |  |
April 6 Ring |  | $49\pm 3.9$ | $28\pm 3.6$
|  | $50\pm 5.6$ | $41\pm 6.7$
April 7 Ring |  | $50\pm 2.5$ | $32\pm 2.7$
|  | $50\pm 4.1$ | $35\pm 3.8$
SMILI |  |  |
April 6 Ring |  | $43\pm 0.4$ | $28\pm 3.1$
|  | $39\pm 3.9$ | $46\pm 8.8$
April 7 Ring |  | $47\pm 4.7$ | $33\pm 2.7$
|  | $47\pm 3.8$ | $35\pm 5.2$
Themis |  |  |
April 6 Ring |  | $46\pm 2.1$ | $30\pm 1.6$
|  | $47\pm 3.1$ | $33\pm 2.0$
April 7 Ring |  | $50\pm 1.5$ | $26\pm 1.1$
|  | $56\pm 2.6$ | $32\pm 0.9$

##  9 Short-timescale Dynamic Properties on Select Observation Window

The dynamical timescale at the location of the innermost stable circular orbit
for Sgr A∗, $t_{\rm g}=12\pi\sqrt{6}GM/c^{3}$ for zero spin, is approximately
30 min and can be smaller by a factor of $\sim 10$ if the black hole is
spinning rapidly. Variability at these timescales across the electromagnetic
spectrum, including at 230 GHz, is one of Sgr A* salient features – see
Wielgus et al. (2022) and references therein. As discussed in Section 3.2 and
Paper II, a few EHT closure phase triangles show measurable variability across
the 2017 observing campaign that can be attributed to intrinsic source
variability. In this section, we explore the level and characteristics of
structural changes in the Sgr A∗ image that are consistent with the observed
variability.

Recovering time-resolved structures on these short timescales is especially
challenging due to the sparse snapshot $(u,v)$-coverage for the EHT array.
Indeed, without additional constraints, any observed change in the visibility
domain can be interpreted as caused either by intrinsic variability or simply
by the rotation of the baselines with the Earth and their probing of different
spatial structures – though fitting fast fluctuations in the visibilities with
static emission requires larger fields-of-view. This is especially true for
baselines that probe regions of the $(u,v)$-space in which the visibility
amplitudes show deep minima (or nulls), across which the complex visibilities
change by order unity over infinitesimal changes in baseline length.

In attempts to describe the EHT observations with a static image, we assign
any observed variation to spatial structures and mitigate potential effects of
variability by inflating the error budget. In this section, we instead attempt
to fit the time-evolving data directly to produce spatially- and temporally-
resolved images of Sgr A∗ on minute timescales. Our analysis of dynamic
properties presuppose that the 230 GHz emission from Sgr A∗ is compact (see
Section 2.3) and ring-like such that the short-timescale variability we see
can be attributed to changes in the image with time. We combine two
independent analysis methods – dynamic imaging with temporal regularization
between frames and snapshot geometric modeling – to identify trends in the
spatial evolution of Sgr A∗.

Our analysis shows that significant uncertainty exists in any attempt to
characterize the spatially-resolved dynamics of Sgr A∗ using EHT 2017 data. We
expect that future observations with an expanded EHT array will yield
significantly improved time- and spatially-resolved movies of Sgr A∗.

###  9.1 Selecting an Observation Window

The rotation of the Earth causes the EHT’s snapshot $(u,v)$-coverage to change
over time. Static imaging and modeling approaches assume the source is
unchanging in time, which allows these approaches to combine data from a full
night of observations. However, recovery of short-timescale evolution requires
that we only consider coverage synthesized on the variability timescale. This
“snapshot" coverage is extremely sparse and introduces artifacts into image
reconstructions. To minimize these artifacts, we constructed and evaluated
metrics to assess the performance of the snapshot coverage and identify the
most promising time windows for dynamic analysis. These metrics rely purely on
the $(u,v)$-coverage rather than the properties of the underlying Sgr A∗
visibilities. The construction and validation of a suite of these metrics are
reviewed in Farah et al. (2021).

We consider metrics that assess several attributes of the $(u,v)$-coverage,
including the largest gap in coverage (Wielgus et al. 2020), the fraction of
the $(u,v)$ plane covered (Palumbo et al. 2019), and the geometric properties
of the coverage (Farah et al. 2021). We summarize the application of these
metrics to the 2017 April 7 EHT Sgr A∗ data set in Appendix J. These three
metrics identify a period from approximately 1.5-3.2 GMST on April 6 and 7
that maximally mitigates the EHT’s snapshot coverage limitations. During this
time window, all sites participate in observing Sgr A∗ except PV, though there
is a notable dropout of the LMT between approximately 2.4-2.9 GMST on both
days. All dynamic analyses discussed in the remainder of Section 9 are
performed _only in this selected time window_.

Figure 24 shows the $(u,v)$-coverage for April 7 during the roughly 100 min
observation window selected for dynamic analysis, along with the coverage for
a single 60 second “snapshot" integration. Closure phases from two informative
triangles are overlaid for April 6 and 7 during this time window. These
closure phases show distinct evolution in the resolved structure of Sgr A∗
during the same 100 min window on April 6 and 7.

###  9.2 Dynamic Imaging and Modeling Methods

Figure 24: (Left) $(u,v)$-coverage for the selected time window for dynamic
imaging and modeling. The light blue points show the coverage of the full
night of observation, while the dark blue points and the red points represent
the coverage for the selected dynamic imaging region and a single 60s snapshot
from that region, respectively. (Right) Closure phases for Sgr A∗ (green and
yellow) on two representative triangles during the selected time region.
Figure 25:  Position angle (PA) recovered from synthetic data from three
different GRMHD simulations on April 7 EHT coverage during the dynamic
analysis window using both StarWarps dynamic imaging and DPI snapshot
geometric modeling techniques. (Top row) Ground truth GRMHD movie snapshots
(including interstellar scattering) from each of the three simulations at 1.7,
2.3, and 3.2 GMST. (Middle row) Plots of PA vs time for the reconstructions
compared with the simulation ground truth (in red). The shaded red region
indicates the circular standard deviation of the ground truth PA computed
using REx (refer to Section 8 and Chael (2019)). Modeling histograms (blue)
correspond to actual marginal posterior distributions, whereas for StarWarps
imaging the histograms represent the distribution of PAs and their associated
uncertainties for a collection of movies reconstructed under different
parameter settings. The gray band at roughly 2.6 GMST indicates the time
period where the LMT dropped out of the observation. (Bottom row) Visibility
variance in the $(u,v)$ plane over the selected time window for the ground
truth simulation movies (left, red) and the reconstruction (right, green). In
Simulations 1 & 2, both dynamic imaging and snapshot geometric modeling
methods are often able to correctly identify the PA of evolving GRMHD movies
during this time window, but they show significant offsets from the correct PA
in Simulation 3. From left to right, the maximum variance of the ground-truth
(reconstructed) movie is 0.85 (2.62), 4.45 (12.07), and 3.94 (7.79) $\times
10^{-2}$ Jy2. Contours start at 90% of the peak variance and decrease by
successive factors of 2 until they reach 0.7%.  Figure 26:  (Left) Mean
azimuthal brightness profiles from the StarWarps movie reconstructions,
unwrapped around the ring as a function of time, and PA distributions obtained
from dynamic imaging and snapshot geometric modeling reconstructions of EHT
Sgr A∗ data on April 6 (top) and 7 (bottom) in the selected time window.
Geometric modeling distributions are marginal PA posteriors from DPI. Imaging
histograms represent the distribution of PAs and their associated
uncertainties from a collection of StarWarps movies reconstructed under
different parameter settings with the spatial prior mean $\mu$ and temporal
regularization $\beta_{Q}^{-1}$ held fixed (refer to Equation 7). Blank spaces
indicate time regions without any data. The gray band at roughly 2.6 GMST
indicates the region where the LMT dropped out and data coverage is poor. Both
dynamic imaging and modeling appear to identify a nearly constant PA on April
6 but a variable PA over the same time window on April 7. In the
reconstructions in this figure, both dynamic imaging and modeling make a prior
assumption that the source morphology is ring-like; StarWarps imaging uses a
prior/initialization image of a uniform ring, while geometric modeling uses a
second order $m$-ring ($m=2$) model. Both dynamic imaging and modeling recover
“descattered" movies using the J18model1 refractive noise model.
(Right) Focus on the two modes reconstructed by StarWarps on April 7. For each
mode, the top panels show three reconstructed snapshots at different times,
and bottom panels compare the fitting of the corresponding reconstructed movie
(magenta or yellow) and a representative static reconstruction from the eht-
imaging static imaging pipeline (white) to the closure phase data measured by
three key triangles. The dynamic reconstruction on the top (magenta) shows an
evolving PA shift over the observation window. In contrast, the reconstruction
on the bottom has a nearly constant PA of $\sim 100^{\circ}$ (yellow). In the
selected closure phase plots (bottom rows), the measured data is averaged in
60 s snapshots and the error bars do not include a variability noise model.
The static image visibilities (white) capture the general trend of the data,
but they do not well fit variability in the closure phases. In contrast, both
selected StarWarps dynamic reconstructions better fit the data on minute
timescales. We find that the fit’s behavior on the SMT-LMT-SMA triangle has a
large influence on the resulting PA of the movie on April 7. A positive SMT-
LMT-SMA closure value tends to result in a southeast PA ($\sim 100^{\circ}$)
whereas a negative value results in a more northwest PA ($\sim-80$∘).

To analyze Sgr A∗’s spatially-resolved dynamics during the 100 min selected
region of time on April 6 and 7, we use two methods: dynamic imaging and
snapshot geometric modeling (also simply referred to as dynamic modeling).
Both methods fit EHT 2017 data on 60s snapshot integrations within the
selected observation window but make different prior assumptions about the
structure of the source in space and time. Note that unlike in static imaging,
we do not flux normalize the data before dynamic analysis. Because the
$(u,v)$-coverage is sparse even in the best available time window, both
methods need to make strong prior assumptions about the spatial and/or
temporal structure of the source to constrain the fits to the data. Note that
when performing dynamic imaging/modeling fits we do not include a variability
noise budget as is done in static imaging (Section 3.2.2).

##### Dynamic Imaging

Dynamic imaging methods reconstruct a time-evolving image that best fits the
observed evolution of Sgr A∗. Our dynamic imaging approach is based on
StarWarps (Bouman et al., 2018), which enforces continuity across an image and
in time by means of spatial and temporal regularization (Section 4.4.2).
Temporal regularization is set by a parameter $\beta_{Q}^{-1}$ which
corresponds to the allowed variance between pixels in snapshots that are
typically 60s apart999Frames are sometimes separated by more than 60s due to
the interval between scans.; smaller values of $\beta_{Q}^{-1}$ correspond to
stronger continuity in time. Spatial regularization is imposed by a
multivariate Gaussian prior on snapshots with a mean $\mu$ and covariance
$\Lambda$ that encourages spatial smoothness (see Equation 7 and Bouman et al.
2018). We examine the sensitivity of time-variable image features (e.g.,
position angle) to different settings in the StarWarps imaging algorithm by
running surveys over different values of the spatial regularization covariance
$\Lambda$ and data weights of the visibility amplitude and log closure
amplitude; we typically keep $\beta_{Q}^{-1}$ and the mean image $\mu$ fixed
and examine the sensitivity of our results to these parameters separately
across different surveys. Unless specified otherwise, in this paper we set
$\beta_{Q}^{-1}$ to $5\times 10^{-6}$ (Jy/pixel)2 and the prior mean $\mu$ and
initialization image to an image of a uniform ring blurred by a 25 $\mu$as
beam. These surveys result in distributions of the image features at each
snapshot in time. For each of the measurements obtained from 54 parameter
combinations, we draw 100 random samples from a normal distribution
characterized by the image feature measurement and its associated error. These
survey results are _not_ posterior probability distributions, but they do
provide a sense of the robustness of the sensitivity of movie reconstructions
to changes in the algorithm parameters.

##### Geometric modeling

In our geometric modeling approach, a simple geometric model is fit to each 60
s snapshot _independently_ , with no enforced correlations in time.101010In
the language of the temporal regularization parameter defined above, for
geometric modeling $\beta_{Q}^{-1}\rightarrow\infty$. We consider several
different $m$-ring models (Johnson et al., 2020), described by infinitesimally
thin rings with azimuthal brightness variations decomposed into Fourier modes,
which are subsequently blurred with a circular Gaussian kernel. The complexity
of an $m$-ring model depends on the maximum number of Fourier modes, $m$, that
are added (e.g. $m=1$ corresponds to a simple crescent). To model a central
floor we include a Gaussian that is located at the center of the ring; the
size and brightness of the Gaussian are additional model parameters.

For each $m$-ring model considered, we produce a multi-dimensional posterior
using two modeling approaches. First, we consider a variational inference
based approach, DPI, that fits to the log closure amplitudes and closure
phases (Sun & Bouman, 2021; Sun et al., 2022). Second, we consider a sampling
based method, Comrade (Tiede, 2022)111111https://github.com/ptiede/Comrade.jl,
which fits to visibility amplitudes and closure phases. Comrade uses the
nested sampling package dynesty (Speagle, 2020) and the probabilistic
programming language Soss (Scherrer & Zhao, 2020). The different data products
used by DPI and Comrade imply different assumptions made about the telescope
amplitude gains – they are unconstrained in DPI, while in Comrade the gain
amplitudes are more constrained and are included as model parameters during
fitting (see Paper IV). As a result of these different data products, DPI and
Comrade produce slightly different posteriors. Details of the geometric
modeling approaches are further explained in Paper IV121212Note that model
fits in Paper IV uses 120 second snapshots, while in this section, we use 60
second snapshots..

##### Comparing dynamic imaging & modeling

Both dynamic imaging and modeling share a common goal of extracting time- and
spatially-resolved structure in Sgr A∗ but there are key differences between
the methods in how prior assumptions about the spatial and temporal
variability are incorporated. StarWarps imaging allows for more freedom in the
recovered spatial structure but assumes strong temporal regularization between
frames. In contrast, snapshot geometric modeling is restricted to a
parameterized set of spatial structures, but makes no assumptions on image
correlations in time. Although snapshot geometric modeling cannot recover
spatial structures outside of the $m$-ring model specifications, it allows for
quantifying the uncertainty in $m$-ring model features (and their temporal
variability) as it estimates full posterior distributions for the particular
geometric model used.

##### Diagnostics

To characterize our dynamic reconstructions, we mostly investigate trends of
the brightness position angle (PA; see Eq. 21 in M87∗ Paper IV) with time. The
PA is a simple and easily characterizable feature of the brightness
distribution around an asymmetric ring. For Starwarps reconstructed movies, we
extract the ring PA on the different snapshots using REx; in $m-$ring model
fitting results, the PA is obtained directly from the fitted model as the
argument of the first azimuthal Fourier mode.

##### Ring assumption

Many of the results in this section apply strong prior assumptions that Sgr
A∗’s underlying structure is ring-like, motivated by the ring morphology
recovered in static image reconstructions using the full $(u,v)$-coverage
(Section 7). StarWarps reconstructions enforce a ring constraint by setting
the mean prior image $\mu$ to a ring with $\approx 50\,\mu$as diameter and a
width set by a circular Gaussian blurring kernel. In geometric modeling, the
ring assumption is intrinsically imposed by the structure of the $m$-ring
model. In Section 9.4.1 and in Appendix J we explore the sensitivity of our
results to the choice of mean image $\mu$ in StarWarps.

###  9.3 Synthetic Data Tests

Sparse snapshot $(u,v)$-coverage can lead to artifacts in both imaging and
geometric modeling results. These artifacts appear in static imaging, but are
further amplified in dynamic imaging due to the far-sparser coverage (Farah et
al., 2021). Thus, it is important to assess how the sparse $(u,v)$-coverage
during the selected time window may affect the recovered results and whether
it may introduce biases in recovered image features, particularly the position
angle (PA) of ring-like images.

####  9.3.1 Static Crescents

In Section J.3 we present synthetic data tests conducted to characterize the
effect of the sparse snapshot EHT2017 $(u,v)$-coverage on PA recovery from
static crescent images. These tests show that there are significant biases in
the recovered PA from 60s snapshots when the brightness asymmetry of the
ground truth ring image is low. When there is a strong asymmetry in the
brightness distribution around the ring, however, the PA is accurately
recovered even with 60s snapshot $(u,v)$-coverage.

####  9.3.2 GRMHD Simulations

We explored how our methods perform in recovering time-varying position angles
from three selected GRMHD simulation movies. We used three representative
GRMHD movies from the GRMHD library presented in Paper V 131313 Simulation 1 &
2 is using a MAD GRMHD model with parameters $a_{*}=0$, $i=10$, $R_{\rm
high}=10$. Simulation 3 is using a SANE GRMHD model with $a_{*}=-0.94$,
$i=50$, $R_{\rm high}=160$. (see Section 5.2). We generated visibility data
from the three movies over the 100 min dynamic analysis window on April 7
using the same procedure described in Section 5, including atmospheric noise,
telescope gain errors, and polarimetric leakage.

Figure 25 presents results obtained from both dynamic imaging and modeling
reconstructions of these three synthetic data sets. The ground truth
simulation PA evolution is recovered (with $\sim$30∘) for the first two models
(Simulation 1 & 2). However, there are several localized deviations in the
recovered PA distributions from the ground truth in these models, especially
when the the instantaneous $(u,v)$-coverage worsens (e.g. during the LMT
dropout time region). For the third model (Simulation 3) both the dynamic
imaging and modeling methods recover significant offset from the true PA. One
potential cause of this offset is the prominent extended jet structure to the
northwest of the ring in the SANE simulation. This extended structure cannot
be captured in either the dynamic imaging or modeling methods due to their
strong prior assumptions of a ring-like morphology.

In the bottom row of Figure 25, we investigate the complex variance of the
Fourier transform of the reconstructed images across the selected time window
for both the ground truth (scattered) movies and the reconstructions. We find
that in all three cases the reconstructions tend to introduce more variability
than is present in the ground truth. In Simulation 1 and 2 this excess
variability does not prevent the reconstructions from qualitatively recovering
the correct PA trend, but the PA results are not correctly recovered in
Simulation 3.

These results indicate that although we often recover the PA accurately from
some realistic synthetic GRMHD data sets, we should remain cautious when
interpreting Sgr A∗ dynamic results. Our methods sometimes incorrectly recover
the PA, especially if extended structure is present that is not captured by
the prior assumptions on the source structure made by the dynamic imaging and
modeling methods. In addition to effects from extended structure, there may be
additional uncharacterized systematic uncertainty from prior assumptions in
the reconstructions in these results derived from extremely sparse snapshot
Sgr A∗ coverage.

###  9.4 Sgr A∗ Spatiotemporal Characterization and Uncertainty

Here we present results of our analysis on Sgr A∗’s spatially resolved
temporal variability on minute timescales on April 6 and 7, using both dynamic
imaging and snapshot geometric modeling methods. In Figure 26, we show
detailed results for the Sgr A∗ PA evolution and data fits reconstructed using
a restricted range of dynamic imaging and modeling parameters. In Figure 27 we
show PA results obtained under a broader range of parameter settings.

Figure 27: The PA for the 2017 Sgr A∗ data recovered using dynamic imaging and
geometric modeling techniques under different assumptions. StarWarps imaging
results were obtained using a spatial prior image set to either a uniform ring
convolved with a circular Gaussian kernel with FWHM of 20 or 25 $\mu$as (see
Figure 7), or a uniform disk blurred with a kernel with FWHM of 15 $\mu$as.
Descattered modeling results were obtained from geometric models with
increasing complexity ($m$-ring 2 vs $m$-ring 3) and different fitting
software packages (DPI vs Comrade). All results were obtained from low-band
data on April 6 and 7 that has been descattered with a J18model1 refractive
noise floor. The gray band at roughly 2.6 GMST indicates the region where the
LMT has dropped out and data coverage is poor.

Figure 28: Comparing the effect of temporal regularization on the
reconstructed StarWarps movies for April 6 and 7. The temporal regularization
strength is increased from left to right ($\beta_{Q}^{-1}=5\times 10^{-4}$,
$5\times 10^{-6}$, and $5\times 10^{-8}$ (Jy/pixel)2). For each value of
$\beta_{Q}^{-1}$, we show the mean unwrapped movie around the ring (top), the
mean closure phase values on the triangle SMT-LMT-SMA (middle), and the
variance of the complex visibilities across the $(u,v)$ plane (bottom). As
temporal regularization is increased the recovered movies become more static,
and the degree of $(u,v)$ plane variability decreases. From weak to strong
regularization, the maximum variance of the reconstructed movie is 5.37, 1.11,
and 0.73 $\times 10^{-2}$ Jy2 on April 6, and 33.81, 13.24, and 0.50 $\times
10^{-2}$ Jy2 on April 7. Contours start at 90% of the peak variance and
decrease by successive factors of 2 until they reach 0.7%. For comparison, the
variance in the light curve over this interval is ${\sim}0.5\times 10^{-2}{\rm
Jy}^{2}$. Thus, the leftmost reconstruction with the weakest temporal
regularization produces a movie with visibility variance substantially
exceeding the light curve variance due to overfitting to the thermal noise.

In general, we find that snapshot geometric modeling results performed under
different $m$-ring orders, scattering mitigation strategies, and modeling
codes produce fairly consistent results. The modeling results show broad
posteriors of PA at each 60 s snapshot but still indicate significant
differences between April 6 and 7 and between the first and second halves of
the 100 min window on April 7. In rough terms, the PA on April 6 is centered
around $\sim-50^{\circ}$, with some scatter around this value over the time
window. In contrast, on April 7 the modeling results show PA posteriors that
are initially centered around $\sim 90^{\circ}$ and then shift to values
centered around $\sim-50^{\circ}$ in the second half of the time window.

Compared to snapshot geometric modeling, dynamic imaging allows for more
freedom in spatial and temporal regularization, and as a result is more
sensitive to parameter choices. Dynamic imaging results can produce movies
that reproduce the PA trends on April 6 and 7 recovered by snapshot modeling.
These PA trends – a stable PA on April 6 and a shifting PA on April 7 – are
predominantly seen in dynamic imaging reconstructions with low temporal
regularization and ring-like spatial priors. Geometric modeling makes similar
assumptions, imposing no temporal regularization and enforcing ring structure
in the form of the model. In Figure 26, we directly compare dynamic modeling
and imaging results under these strong assumptions.

Even in the limited space of dynamic imaging reconstructions conducted with
weak temporal regularization and ring-like mean prior images, the imaging
results are sensitive to other hyperparameters. In particular Figure 26
indicates that we recover two modes of position angle evolution on April 7
even with the mean prior image and temporal regularization level fixed. We
present representative snapshots and fit to the closure phase data from these
two modes in the right panels of Figure 26.

When the ring-like mean image prior is changed or the weak temporal
regularization is increased in StarWarps dynamic imaging, significantly
different PA variations can be recovered from the same data. In Figure 27, we
show that when the ring assumption is relaxed and a disk prior is used in
reconstruction, StarWarps results show drastically different PA trend over
time. In particular, in reconstructions initialized with a disk prior, the PA
curves on April 6 and 7 appear to be flipped by $180^{\circ}$ (i.e., on April
7 the PA transitions from $\sim 90^{\circ}$ to $\sim-50^{\circ}$). We further
show in Figure 28 that when using stronger temporal regularization in the
StarWarps dynamic imaging, the PA stays constant, eliminating the shift from
positive to negative PA trend on April 7. We discuss these tests further in
the Section 9.4.1 and Section J.5.

Note that April 6 and 7 have nearly identical $(u,v)$-coverage during the
selected 100 min region of time. We can thus compare the results obtained over
these two days to help disentangle effects of $(u,v)$-coverage from any
effects due to intrinsic evolution. If the PA trends that we recover are due
primarily to biases from the $(u,v)$-coverage, we would expect to recover the
same PA trends on both days. However, we consistently see different PA trends
with time using the same parameter settings in both dynamic imaging and
modeling methods. This implies that differences in the visibilities, not the
$(u,v)$-coverage, drive differences in the PA evolution we see on the two
nights in some reconstructions, but it does not help select among any of the
different reconstruction modes on either day.

####  9.4.1 Effect of Model and Imaging Choices

The PA evolution recovered with dynamic imaging and modeling methods is
sensitive to choices made in the imaging and modeling procedures that enforce
constraints on the spatial and temporal structure of the reconstructions.
Enforcing strong spatial or temporal priors will suppress any structural
variability while adding too much flexibility in a model with sparse data
constraints will lead to overfitting or uninformative posteriors. In Section
J.5 we present in detail several tests of these choices for both imaging and
modeling methods; here, we highlight the most important results.

##### Spatial Priors

Constraints on the spatial structure are enforced via the $m$-ring order in
geometric model fitting and via the choice of mean image prior in dynamic
imaging. To test the effects of different mean prior images in StarWarps, we
produced reconstructions using uniform ring priors with increasing widths
(from convolution of the ring described in Section 5 with circular beams of
11, 15, 20, and 25 $\mu$as FWHM; henceforth
ring$\textasteriskcentered$11$\mu$as, ring$\textasteriskcentered$15$\mu$as,
ring$\textasteriskcentered$20$\mu$as, and ring$\textasteriskcentered$25$\mu$as
priors, respectively). We also used a tapered disk with diameter $\sim$
74$\mu$as (see Figure 7) as a prior that does not feature any central dip. The
disk is tapered via convolution with a circular beam of 15 $\mu$as
(henceforth, disk$\textasteriskcentered$15$\mu$as prior). We discuss the
details of these prior choices in Section J.5.1. For geometric modeling, we
tested ring-like models of increasing complexity in their azimuthal brightness
distribution, from crescent models ($m=1$) to higher order $m$-rings. When
fitting $m$-rings to Sgr A∗ snapshot data, we explored $m$ from 1 to 4 and
selected the order to use based on the Bayesian evidence across all data sets
– settling on $m=2$ (see Section J.5.2).

Figure 27 shows histograms of the dynamic imaging PA results made using
different mean prior images and PA posterior distributions from geometric
modeling results from different $m-$ring orders. We also compare modeling
results from two different modeling codes in Figure 27. The reconstructed PA
trends are fairly consistent on both days among the different $m$-ring orders
in geometric model fitting. In StarWarps imaging, reconstructions using ring-
like mean prior images of several different thicknesses produce similar
trends, with a stable PA on April 6 and a PA transition from positive to
negative values on April 7. However, when a disk prior is used in StarWarps
the PA trends of both April 6 and 7 change drastically and appear to be
flipped by $180^{\circ}$. Figure 44 in Appendix J shows image snapshots and
data fits for StarWarps reconstructions with both disk and ring mean prior
images. Note that although the position angle evolution is different than the
movie reconstructed using a ring prior, the movie reconstructed with a disk
prior still results in a ring-like structure, though with a less prominent
central brightness depression.

##### Temporal Regularization

Geometric modeling enforces no correlations in between temporal snapshots,
while dynamic imaging can enforce correlations via temporal regularization.
Figure 28 shows that when using stronger temporal regularization in StarWarps
(lower values of $\beta^{-1}_{Q}$) the PA becomes constant in time on both
April 6 and 7 – a result of the method enforcing strong continuity between
frames. Note that in the case of high temporal regularization, the SMT-LMT-SMA
closure triangle fits in the second half of the time window on April 7 appear
offset with respect to the data, although still within two standard deviations
of most data points.

We also show that reconstructions with low levels of temporal regularization
produce prominent variance in the model visibilities plane at $(u,v)$ points
that are not sampled by our coverage during this time window141414As shown in
the GRMHD synthetic data tests, it is not necessary that the peaks in the
$(u,v)$-plane variability map align with measured data points to correctly
identify PA evolution.. In contrast, reconstructions with more temporal
regularization lower the overall variance of model visibilities everywhere in
the Fourier plane and place the peaks in the variance maps at $(u,v)$ points
sampled by the EHT. We discuss the interpretation of the different temporal
regularization values $\beta^{-1}_{Q}$ used here in Section J.2 and further
tests of the StarWarps temporal regularization further in Section J.5.1.

##### Scattering

Another choice made in both dynamic imaging and modeling procedures common to
both static and dynamic reconstruction methods is the strategy for mitigating
the effects of interstellar scattering in Sgr A∗ data. We investigate the
effects of the same five prescriptions for scattering mitigation we use in
static imaging on the dynamic reconstructions in Section J.4. In general, we
find that choices made in the scattering mitigation procedure contribute less
to our overall uncertainty than choices related to the spatial prior or
temporal regularization.

###  9.5 Sgr A∗ Dynamic Property Conclusions

Our aim in this section has been to use the 2017 EHT data to explore
spatially-resolved dynamics of Sgr A∗ on minute timescales. First, we
identified the time windows with the best $(u,v)$-coverage during the
observation run – a roughly 100 min window on April 6 and 7. We identify a
significant difference between the closure phases on April 6 and 7, signifying
that the underlying structure is different on the two days. We reconstruct
movies from this small slice of the EHT data using dynamic imaging and
geometric snapshot modeling methods. We track the average position angle (PA)
in our dynamic imaging and modeling results as a way of following a specific,
dynamic, and measurable aspect of the source over time. We find that we are
able to recover the PA in synthetic EHT data from some GRMHD simulation
movies; however, there are prominent cases when this is not the case and both
geometric modeling and dynamic imaging methods recover biased results.

On April 6, most dynamic imaging and modeling results show a stable PA in the
Sgr A∗ images over the selected window. In contrast, the recovered PA
evolution on April 7 is more dependent on prior assumptions on the spatial
structure and temporal regularization. On April 7, when using a ring image as
a spatial prior and weak temporal regularization, dynamic imaging results
largely align with geometric modeling results and show an evolution in the PA
of $\sim 140^{\circ}$ over the $\sim 100$ min window. However, we also see
several other PA trends in the dynamic imaging results, including a PA
evolution in the opposite direction and modes where the PA is static on both
days.

These results, along with our synthetic data tests, show that while the 230
GHz image of Sgr A∗ may exhibit interesting and measurable dynamics, our
current methods cannot conclusively determine the PA evolution of Sgr A∗.
Dynamic reconstructions of Sgr A∗ with EHT2017 coverage should thus be
interpreted with caution. This analysis provides a promising starting point
for further studies of future evolution seen in Sgr A∗ EHT observations with
denser $(u,v)$-coverage.

##  10 Summary and Conclusions

We present Sgr A∗ static and dynamic imaging results for data taken with the
EHT in April 2017. Sgr A∗ was observed with eight EHT stations at six
geographic sites over five observing days, out of which the highly sensitive
phased ALMA array participated in April 6, 7, and 11. April 7 is the only day
in which the easternmost station PV participated, providing the best
$(u,v)$-coverage that probes a null at $\sim$3.0 G$\lambda$. On April 11 Sgr
A∗ exhibits the highest variability in the light curve during the 2017
campaign (Wielgus et al., 2022), possibly related to an X-ray flare observed
shortly before the start of the EHT observations (Paper II), rendering static
imaging for this day particularly challenging. For these reasons, April 7 has
been used as the primary dataset, while April 6 is considered as a secondary
data set. Results from the remaining 2017 EHT observations of Sgr A∗ will be
presented in future publications.

Similar to the 2017 EHT observations of M87∗ (; ; ; ; ; ), the data sets
analyzed in this paper are the first with sufficient sensitivity and
$(u,v)$-coverage to reconstruct images of Sgr A∗ on event-horizon scales with
an angular resolution of $\sim$20 $\mathrm{\mu as}$. Imaging Sgr A∗ with the
EHT is however significantly more challenging than M87∗ due to the
interstellar scattering towards the Galactic Center, and most importantly the
rapid intra-day variability that characterizes Sgr A∗, with timescales much
shorter than the duration of our typical EHT observing runs.

To mitigate the scattering effects in our Sgr A∗ reconstructions we have
developed a strategy based on Fish et al. (2014), Psaltis et al. (2018) and
Johnson et al. (2018) to account for the angular broadening and substructure
induced by diffractive (Section 3.1.1) and refractive (Section 3.1.2)
scattering, respectively. Images of Sgr A∗ have been obtained with and without
these scattering mitigation prescriptions to asses their impact in our
reconstructions.

It is however the rapid intra-day intrinsic variability of Sgr A∗, coupled
with the sparse $(u,v)$-coverage as compared with regular VLBI observations,
that poses the strongest challenge for reconstructing horizon-scale images of
Sgr A∗ with the EHT. With a typical variability timescale of a few minutes,
the horizon-scale brightness distribution can change significantly during our
typical multi-hour observing runs, which violates the fundamental assumption
for Earth rotation VLBI aperture synthesis. To overcome this extraordinary
challenge we have included a “variability noise budget" in the observed
visibilities (Section 3.2.2) that facilitates the reconstruction of static
full-track images capturing the time-averaged structure.

Static full-track imaging of Sgr A∗ has been conducted through surveys over a
wide range of imaging assumptions using the classical CLEAN algorithm
(implemented in DIFMAP), regularized maximum likelihood methods (eht-imaging
and SMILI), and a Bayesian posterior sampling method (Themis), as described in
Section 4. Imaging surveys, exploring $\sim$103-5 parameter combinations, were
first performed on synthetic data sets designed to be qualitatively consistent
with Sgr A∗ measurements, including its characteristic temporal variability
(Section 5). The use of synthetic data sets allows us to asses the capability
to accurately reproduce different morphologies with our imaging methods, and
to select the “Top Sets": imaging parameter combinations for RML and CLEAN
methods that successfully reproduce the known ground truth movies (Section
6.2.2).

Unlike for M87∗, where a persistent ring structure is observed across imaging
pipelines (M87∗ Paper IV), our static reconstructions of Sgr A∗ show
structural changes within and across the different imaging methods used. The
variety of images can be classified into four main clusters of images
corresponding to ring images with three different azimuthal brightness
distributions, and a cluster that contains a small number of non-ring images
with multiple morphologies. This classification is resilient to the scattering
mitigation prescription used, including no mitigation.

Although the relative fraction of non-ring images in the Top Set
reconstructions from RML and CLEAN is very small ($\leq$5% in the April 7
descattered images), we note that the Top Sets do not sample from the Bayesian
posterior likelihood and are instead meant to characterize the range of
possible images due to epistemic uncertainty. On the other hand, the full
Bayesian imaging approach implemented in the Themis pipeline does provide a
Bayesian posterior exploration that characterizes aleatoric uncertainty.
Scattering mitigated Sgr A∗ reconstructions from Themis for April 7 only
contain ring images with a very similar structure as that found for RML and
CLEAN methods, although it also identifies a small number (2%) of on-sky non-
ring images (Section 7).

The April 6 data set suffers from poorer $(u,v)$-coverage and likely more
unusual intrinsic variability. This results in April 6 reconstructions that
contain a less prominent ring structure in the RML and CLEAN pipelines.
Nonetheless, the diameter of the ring structures recovered for April 6 are
consistent with that of the April 7 ring images. In addition, although
descattered Themis samples of April 6 primarily show a corrupted ring or non-
ring structure, Themis posterior samples for the combined April 6 and 7 data
sets exhibits only clear ring-like images for both, the on-sky and scattered
mitigated reconstructions.

Imaging of a synthetic GRMHD data set with data properties similar to those of
Sgr A∗ and characterized by a ring image of $\sim$50 $\mu$as (Section 6.4.2)
shows similar imaging results to those found in Sgr A∗: RML and CLEAN methods
recover ring images with different azimuthal orientations and a small ($\leq
5\%$ ) Top Set fraction of non-ring images, while Themis posterior sample only
recovers ring images.

We conclude that the Event Horizon Telescope Sgr A∗ data show compelling
evidence for an image that is dominated by a bright ring of emission. This
conclusion is based on our extensive analysis of the effects of sparse
$(u,v)$-coverage, source variability and interstellar scattering, as well as
studies of simulated visibility data, which find that non-ring images are
recovered in the minority by our imaging pipelines for variable sources with
an intrinsic ring morphology (Section 7.5). Representative first event
horizon-scale images of Sgr A∗ are shown in Figure 13, obtained from the April
7 data set by averaging similar Top Set and posterior images together. Despite
the different azimuthal brightness distributions observed, all ring images
have a ring diameter of $\sim$50 $\mu$as on both April 6 and 7, consistent
with the expected “shadow" of a $4\times 10^{6}M_{\odot}$ black hole in the
Galactic Center located at a distance of 8 kpc.

We also present preliminary dynamic imaging and modeling analysis of Sgr A∗ on
horizon scales in an attempt to characterize the azimuthal variations on
minute timescales and their uncertainties. We applied dynamical analysis
methods to a 100-minute interval of the 2017 Sgr A∗ data with the best
coverage on April 6 and 7. On April 6, most dynamic imaging and modeling
methods recover a stable position angle. In contrast, on April 7 the recovered
PA evolution is more dependent on spatial and temporal regularization; April 7
frequently shows an evolving position angle over the 100 minute window when we
impose strong priors on the spatial structure but weak temporal
regularization. However, when expanding the parameter space available to the
imaging and modeling methods, we recover disparate modes in the position angle
behavior, including some reconstructions that are nearly static. Our initial
results show that significant uncertainty exists in any attempt to
characterize the spatially-resolved dynamics of Sgr A∗ using sparse EHT 2017
data, but this analysis provides a promising starting point for further
dynamical studies of Sgr A∗ with future EHT observations.

EHT.

We thank the anonymous referee for helpful suggestions that improved this
paper.

The Event Horizon Telescope Collaboration thanks the following organizations
and programs: the Academia Sinica; the Academy of Finland (projects 274477,
284495, 312496, 315721); the Agencia Nacional de Investigación y Desarrollo
(ANID), Chile via NCN$19\\_058$ (TITANs) and Fondecyt 1221421, 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 ALMA North America
Development Fund; the Black Hole Initiative, which is funded by grants from
the John Templeton Foundation and the Gordon and Betty Moore Foundation
(although the opinions expressed in this work are those of the author(s) and
do not necessarily reflect the views of these Foundations); Chandra DD7-18089X
and TM6-17006X; the China Scholarship Council; China Postdoctoral Science
Foundation fellowship (2020M671266); Consejo Nacional de Ciencia y Tecnología
(CONACYT, Mexico, projects U0004-246083, U0004-259839, F0003-272050,
M0037-279006, F0003-281692, 104497, 275201, 263356); the Consejería de
Economía, Conocimiento, Empresas y Universidad of the Junta de Andalucía
(grant P18-FR-1769), the Consejo Superior de Investigaciones Científicas
(grant 2019AEP112); the Delaney Family via the Delaney Family John A. Wheeler
Chair at Perimeter Institute; Dirección General de Asuntos del Personal
Académico-Universidad Nacional Autónoma de México (DGAPA-UNAM, projects
IN112417 and IN112820); the Dutch Organization for Scientific Research (NWO)
VICI award (grant 639.043.513) and grant OCENW.KLEIN.113; the Dutch National
Supercomputers, Cartesius and Snellius (NWO Grant 2021.013); the EACOA
Fellowship awarded by the East Asia Core Observatories Association, which
consists of the Academia Sinica Institute of Astronomy and Astrophysics, the
National Astronomical Observatory of Japan, Center for Astronomical Mega-
Science, Chinese Academy of Sciences, and the Korea Astronomy and Space
Science Institute; the European Research Council (ERC) Synergy Grant
“BlackHoleCam: Imaging the Event Horizon of Black Holes" (grant 610058); the
European Union Horizon 2020 research and innovation programme under grant
agreements RadioNet (No 730562) and M2FINDERS (No 101018682); 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 Institute for Advanced Study; the Istituto Nazionale
di Fisica Nucleare (INFN) sezione di Napoli, iniziative specifiche TEONGRAV;
the International Max Planck Research School for Astronomy and Astrophysics at
the Universities of Bonn and Cologne; DFG research grant “Jet physics on
horizon scales and beyond” (Grant No. FR 4069/2-1); Joint Princeton/Flatiron
and Joint Columbia/Flatiron Postdoctoral Fellowships, research at the Flatiron
Institute is supported by the Simons Foundation; the Japan Ministry of
Education, Culture, Sports, Science and Technology (MEXT; grant
JPMXP1020200109); the Japanese Government (Monbukagakusho: MEXT) Scholarship;
the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS
Research Fellowship (JP17J08829); the Joint Institute for Computational
Fundamental Science, Japan; the Key Research Program of Frontier Sciences,
Chinese Academy of Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJSSW-SYS008,
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JP18K13594, 18K03709, JP19K14761, 18H01245, 25120007); the Malaysian
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of Education (MoE) of Taiwan Yushan Young Scholar Program; the Physics
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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 National Institute of Natural Sciences (NINS) of
Japan; the National Key Research and Development Program of China (grant
2016YFA0400704, 2017YFA0402703, 2016YFA0400702); the National Science
Foundation (NSF, grants AST-0096454, AST-0352953, AST-0521233, AST-0705062,
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Foundation of China (grants 11650110427, 10625314, 11721303, 11725312,
11873028, 11933007, 11991052, 11991053, 12192220, 12192223); 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, the Korea Research Fellowship Program:
NRF-2015H1D3A1066561, Brain Pool Program: 2019H1D3A1A01102564, Basic Research
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Perimeter Institute is supported by the Government of Canada through the
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Spanish Ministerio de Ciencia e Innovación (grants PGC2018-098915-B-C21,
AYA2016-80889-P, PID2019-108995GB-C21, PID2020-117404GB-C21); the University
of Pretoria for financial aid in the provision of the new Cluster Server nodes
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the Shanghai Pilot Program for Basic Research, Chinese Academy of Science,
Shanghai Branch (JCYJ-SHFY-2021-013); the State Agency for Research of the
Spanish MCIU through the “Center of Excellence Severo Ochoa” award for the
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through the Los Alamos National Laboratory (operated by Triad National
Security, LLC, for the National Nuclear Security Administration of the USDOE
(Contract 89233218CNA000001); and the YCAA Prize Postdoctoral Fellowship.

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. (AUI)/NRAO, and the National
Astronomical Observatory of Japan (NAOJ). The NRAO is a facility of the NSF
operated under cooperative agreement by AUI. This research used resources of
the Oak Ridge Leadership Computing Facility at the Oak Ridge National
Laboratory, which is supported by the Office of Science of the U.S. Department
of Energy under Contract No. DE-AC05-00OR22725. We also thank the Center for
Computational Astrophysics, National Astronomical Observatory of Japan. The
computing cluster of Shanghai VLBI correlator supported by the Special Fund
for Astronomy from the Ministry of Finance in China is acknowledged.

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 Research and
Development Program (No. 2017YFA0402700) of China and Natural Science
Foundation of China grant 11873028. Additional funding support for the JCMT is
provided by the Science and Technologies Facility Council (UK) and
participating universities in the UK and Canada. The LMT is a project operated
by the Instituto Nacional de Astrófisica, Óptica, y Electrónica (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 Geográfico 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.

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. XSEDE Stampede2 resource at
TACC was allocated through TG-AST170024 and TG-AST080026N. XSEDE JetStream
resource at PTI and TACC was allocated through AST170028. This research is
part of the Frontera computing project at the Texas Advanced Computing Center
through the Frontera Large-Scale Community Partnerships allocation AST20023.