Improved Image Quality over 10' Fields with the ʻImaka Ground-layer Adaptive Optics Experiment

The fields' positions and guide star coordinates for each run are given in Table 2. The two fields used are shown in Figure 2. For the 2017 January and 2017 February runs, we observed the Pleiades (Field 1). This field has four guide stars, each positioned around the periphery of the science field. For the 2017 May run, we switched to a second field in the galactic plane (Field 2) in order to observe a higher density of stars. This field differed slightly in that it had its fourth guide star in the center of the field, though this star was not always used in the correction (refer to Table 3). For Field 2, three different guide star and wavefront sensor (WFS) configurations were tested: all four (4WFS) and three outer WFS 0, 1, and 2 (3WFS). For further technical details, refer to Chun et al. (2016).

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Table 2.  Field and Guide Star Positions

Label	Name	R.A. (J2000)	Decl. (J2000)	R mag
Field 1 (January, February)
Field center	⋯	3h48m580	24°10'001	⋯
WFS 0	HD 23873	3h49m217	24°22'544	6.6
WFS 1	HD 23763	3h48m296	24°20'489	6.9
WFS 2	HD 23886	3h49m259	24°14'517	7.9
WFS 3	HD 2377	3h48m348	24°10'523	8.7
Field 2 (May)
Field center	⋯	19h33m448	7°49'458	⋯
WFS 0	HD 184362	19h33m465	7°55'050	7.5
WFS 1	HD 184451	19h34m111	7°47'088	6.9
WFS 2	HD 184244	19h33m121	7°46'220	7.7
WFS 3	HD 184336	19h33m448	7°49'457	8.0

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In order to quantify the gains of ʻImaka, we made observations both with GLAO correction (AO-on) and with no correction (AO-off). To capture both modes in a range of seeing conditions, we cycled through different modes while taking data (i.e., one image with no correction, one image with GLAO correction, and then back to no correction). Rather than either freezing or resetting when the mode switched from AO-on to AO-off, the DM was set to an average of the last 5–10 s at the beginning of each AO-off exposure. On average, each night produced 36 frames in each mode.

Science images were taken with exposure times of 30–60 s; these times were chosen in order to average over the seeing. We used I-band and R-band Johnson filters (centered at 806 and 658 nm, respectively) and a "long-pass" filter with an effective wavelength of approximately 1 μm. Image position and filter information are listed for each night in Table 3. In addition to our science images, we took dithered sky images throughout each night, as well as twilights at the beginning of each run for use as flat fields.

In order to evaluate ʻImaka performance, we also recorded the AO system wave-front sensor measurements and DM commands. This data set (the "telemetry") provides a means to independently estimate the uncorrected aberrations, the level of correction, and the atmospheric seeing. The telemetry data are taken concurrently with our science images; however, the total recording time for the telemetry stream is limited to ∼10 s, while the science exposures are 3–6 times longer. The WFS telemetry data are taken without a filter and have an effective wavelength similar to a V+R-band filter.

The telemetry is used to reconstruct the seeing at a range of altitudes using SloDAR techniques (Wilson 2002), although turbulence profiles are not used in this paper. Instead, the telemetry is used to simply derive the integrated, ground-layer and free-atmosphere seeing independently from MASS/DIMM. Telemetry data were acquired with no AO correction and are used to compute the slope covariance matrices and maps. These are global in the sense that the covariances are computed across different wave-front sensors (the cross-covariances), as well as with themselves (the auto-covariances). A full description of the telemetry data and its analysis is described in Lai et al. (2018). The result of the telemetry processing is the estimated seeing for the ground layer and the free atmosphere for both AO-off and AO-on images.

In addition to our own measurements, we use the seeing reported by the Maunakea Weather Center⁷ (MKWC). The MKWC reports seeing measured by the Maunakea seeing monitor (MKAM) installed on a 7 m tall tower between the Canada–France–Hawaii telescope and the Gemini-north telescope. MKAM is approximately 150 m north of the UH88'' telescope. The seeing monitor consists of a Multi-aperture Scintillation Sensor (MASS) and a Differential Image Motion Monitor (DIMM; Tokovinin 2002). While the DIMM measures integrated seeing of the whole atmosphere, the MASS generates a profile of seeing at a range of altitudes (0.5, 1, 2, 4, 8, and 16 km). Note that the MKWC MASS data are processed with version v2.047 of the MASS software. However, there are improvements to the MASS data analysis (Lombardi & Sarazin 2015), and we have reprocessed the data using the most recent version of the software (Kornilov 2017). Note that the data reprocessing still uses the MKAM MASS calibrations from 2009. Throughout this work, we will refer to the MASS seeing as the integrated seeing in the entire MASS profile (e.g., from 500 m to 16 km).

Both instruments report seeing at 500 nm throughout the night. Every night of focal plane data has corresponding MASS/DIMM measurements with the exception of 2017 January 13 UT, when the MASS was not working. We exclude this night from the rest of our analysis.

It should be noted that the MASS/DIMM instruments do not necessarily observe the same atmospheric profile as ʻImaka. Ideally, the telemetry's integrated seeing should be comparable to the DIMM, while the free-atmosphere seeing should match the MASS. However, the MASS/DIMM points close to zenith, while we observe at a range of altitudes. The differing paths through the atmosphere may result in different turbulence profiles. Additionally, the instruments are located at different places on the summit of MaunaKea and therefore experience different local seeing (e.g., seeing very close to the ground at the seeing monitor and dome seeing at the UH 2.2 m telescope).

Before establishing a parametric model for the image plane PSFs, we took a numerical approach to their characterization using three different metrics, described below. Each metric quantifies the PSF shape and size slightly differently; thus, the values vary for the same image and can be seen in Figure 3.

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Empirical FWHM: Pixels were counted in the vicinity of a star's centroid with ADU counts above half of the maximum pixel value as a brute-force method of calculating a star's FWHM. Assuming that this area is circular, we calculated the corresponding diameter: the empirical FWHM.

Encircled Energy (EE) Diameter: To obtain a more detailed profile of the sources' PSFs, we also calculated EE diameters for our sources. To do this, we created brightness profiles of each source, counting flux contained within a series of concentric annuli centered on the source's centroid. From this profile, we measured the diameter within which 25%, 50%, and 80% of the total flux was enclosed.

Noise Equivalent Area (NEA): An additional way we measure the PSF is an "NEA." A full derivation of the NEA of an arbitrary PSF can be found in King (1983), but simply put, it is the area of a region centered on a star's centroid in which the S/N is unity. This is calculated with a similar method to the EE radius, in which concentric circular regions of increasing size around a star are isolated until reaching an S/N of 1. The NEA of a Gaussian profile, for example, is 4πσ². For the sake of comparison to PSF measurements that are widths rather than areas, we define the "NEA width" to be the diameter corresponding to a circle with area NEA.

Though the above metrics provide a good sense of image quality, they are not a complete description of stars' PSFs. These models ignore the shape of the source, which is problematic, as our sources are not radially symmetric. This is particularly apparent in AO-off images, where stars show significant elongation. As such, we parametrically modeled the sources' shape.

We tried a variety of models, including bivariate Gaussian, Lorentz, Moffat, and sinc distributions. Additionally, we tried two-component models, including combinations (e.g., Gaussian + Moffat). For each model, the goodness of fit was quantified with the fraction of variance unexplained:

$$\begin{eqnarray}&&{\rm{FVU}}=\displaystyle \frac{{\sum }_{i}^{{N}_{\mathrm{pix}}}{[{\mathrm{PSF}}_{\mathrm{obs},i}-{\mathrm{PSF}}_{\mathrm{mod},i}]}^{2}}{{\sum }_{i}^{{N}_{\mathrm{pix}}}{[{\mathrm{PSF}}_{\mathrm{obs},i}-\overline{{\mathrm{PSF}}_{\mathrm{obs}}}]}^{2}}.\end{eqnarray} \tag{ 1 }$$

This value was calculated in a box centered on the star with a side length of about 4 times the FWHM. Using this quantity to compare different models, we found that for AO-off images the best model was a single-component elliptical Moffat profile. We had predicted that for the AO-on images a two-component elliptical Moffat would be ideal: a narrower, brighter component representing the corrected portion of the final image, and a dimmer component with a size similar to that of the AO-off image, representing the uncorrected portion of the PSF. However, as even a carefully constrained double-component model showed no improvement in the FVU, we ultimately chose a single-component Moffat for AO-on images as well.

The model for a 2D elliptical Moffat with rotation is defined as

$$\begin{eqnarray}f(x,y) & = & a[1+A{(x-{x}_{o})}^{2}+B{(y-{y}_{o})}^{2}\\ & & +\ {C(x-{x}_{o})(y-{y}_{o})]}^{-\beta }\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&A={\left(\displaystyle \frac{\cos \phi }{{\alpha }_{x}}\right)}^{2}+{\left(\displaystyle \frac{\sin \phi }{{\alpha }_{y}}\right)}^{2},\ B={\left(\displaystyle \frac{\sin \phi }{{\alpha }_{x}}\right)}^{2}+{\left(\displaystyle \frac{\cos \phi }{{\alpha }_{y}}\right)}^{2}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&C=2\sin \phi \cos \phi \left(\displaystyle \frac{1}{{\alpha }_{x}}-\displaystyle \frac{1}{{\alpha }_{y}}\right),\end{eqnarray} \tag{ 4 }$$

where the star's centroid is given by x_o and y_o, the amplitude is a, β describes the slope of the PSF, α_x and α_y describe its width, and ϕ is the rotation of the major axis from horizontal. The data were fit with a Levenberg–Marquardt algorithm and least-squares statistic, with 3σ outliers rejected iteratively over two passes. The corresponding FWHM θ is calculated as

$$\begin{eqnarray}&&\theta =2\alpha \sqrt{{2}^{1/\beta }-1}.\end{eqnarray} \tag{ 5 }$$

Note that unless otherwise indicated, the Moffat minor FWHM will be used to describe PSF size in further analyses in order to remove the contribution of telescope jitter apparent in the AO-off images.

The quality of the model fit is illustrated in Figure 4, which shows a radial profile of a AO-off and AO-on PSF and the best-fit Moffat profile along the minor axis. There are small discrepancies between the observed and model PSF, particularly in the 2D residuals (Figure 5). The distinct high-order structure evident in the residuals is likely due to the Moffat fitting with a single exponent in the presence of jitter, since the PSF profile follows a different power law along the narrow axis than the elongated one. Static aberrations are calibrated out by optimizing the image quality on artificial sources and recording the centroid offsets on the wave-front sensors, so they are likely not the cause of this residual structure in the PSF. However, the telescope environment may introduce differential aberrations on the various wave-front sensors, or some poorly controlled mirror modes could introduce such a pattern once the loop is closed and noise is injected into the system. Table 4 compares the result of the Moffat fit, specifically, the rms of the minor and major FWHM to the previously described metrics.

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The seeing-limited AO-off images are well described by a Moffat profile with a median β ≃ 4.8, which is slightly above the β of roughly 3–4 typically measured in astronomical images (Saglia et al. 1993). Additionally, they show relatively little dependence on conditions or FWHM, as expected (Figure 6). For the AO-on images, there is a correlation between β and FWHM.

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As previously stated in Section 3, the data comprising our observations were taken through three different filters (details in Table 3). For ease of comparison, the MASS/DIMM data (which are observed at 500 nm) are converted to the wavelength of the corresponding ʻImaka science images. The telemetry data have been similarly converted. The data are converted in the following way.

For a given wavelength λ, the Fried parameter describing the amount of wave-front distortion due to atmospheric turbulence scales as

$$\begin{eqnarray}&&{r}_{o}\propto {\lambda }^{6/5}.\end{eqnarray} \tag{ 6 }$$

To scale a given seeing measurement θ_obs observed at a wavelength, λ_obs, we assume an infinite outer scale and that the seeing measurement is proportional to the ratio of the imaging wavelength and the Fried parameter, which gives

$$\begin{eqnarray}&&\theta \propto \displaystyle \frac{\lambda }{{r}_{o}}\propto {\lambda }^{-1/5}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\theta }_{\mathrm{conv}}={\theta }_{\mathrm{obs}}\cdot {\left(\displaystyle \frac{{\lambda }_{\mathrm{obs}}}{{\lambda }_{\mathrm{conv}}}\right)}^{1/5},\end{eqnarray} \tag{ 8 }$$

where θ_conv is the seeing scaled to a wavelength λ_conv.

The simplest analysis that can be done to characterize the GLAO performance is to compare the FWHM in AO-off images to that in AO-on images, as shown in Figure 7. The FWHM along the minor axis decreases from 076 in the AO-off case to 045 with AO correction in I band, a change of 51%. Changes in all other PSF size metrics at all observed wavelengths are given in Table 4 and show roughly consistent improvements.

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Though already showing a fair amount of correction, this comparison does not capture the full impact of the instrument. As was visible in Figure 5, a change occurs not only in the PSF size but in its shape as well. Static and dynamic aberrations within the telescope and instrument are also corrected by the GLAO system. As noted above, the static aberrations are removed in the AO-off images by applying average voltages to the DM during AO-off images. The correction of the PSF elongation, caused predominately by telescope jitter, is readily seen in the images in Figure 5. AO-off images show elongation in one direction by a factor of ∼1.3. We attribute this elongation to telescope jitter, as it is generally oriented along the east–west direction (i.e., R.A. drive of the telescope). The elongation could be caused by a combination of astigmatism and focus if the DM voltages used for AO-off images were not averaged for a sufficiently long period. However, this effect would give rise to a random direction and amplitude to the image elongation. Since the direction of the PSF elongation is constant over many hours, its unlikely that a static aberration from the DM shape during AO-off images is not the dominant source of elongation. Regardless, elongation is well corrected by the AO system and is nearly nonexistent in the AO-on images. This stark difference can be seen in Figure 8, which shows the elongation parameter:

$$\begin{eqnarray}&&E=\displaystyle \frac{{\theta }_{\mathrm{maj}}}{{\theta }_{\min }}.\end{eqnarray} \tag{ 9 }$$

The median value of the elongation decreases from 1.34 to 1.08. This residual elongation of 8% is consistent with the plate-scale variation in the ʻImaka design, which is roughly 8% in the east–west direction. Even more, the AO-on elongation values demonstrate substantially less spread than the AO-off values, with standard deviations of 0.28 and 0.35, respectively.

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In addition to changes in FWHM, we can use our other PSF metrics to quantify ʻImaka's gains. As seen in Table 4, the AO-on/AO-off change in the 50% EE diameter is greater than the same change in the 80% diameter (e.g., in R band, the improvements in 50% EE and 80% EE are 24% and 16%, respectively). The relatively low change in the 80% diameter is indicative of the less substantial correction the system makes to the extended halo. The radius in which any correction of the PSF is possible for a given AO system is set by the number of actuators in the DM. ʻImaka's current DM has 36 actuators; increasing this number could potentially increase the radius of its AO correction, and subsequently increase this amount of correction for the higher encircled energies.

The improvement in our final metric, NEA, is shown in Figure 9. Note that unlike other metrics and the representation of NEA previously, this figure reports NEA as an area; this representation is useful in addition to the NEA width because photometric precision scales with NEA, while astrometric precision scales with NEA^1/2. The NEA decreases from 1.96 square arcseconds to 0.89 in I band, a change of over a factor of two.

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Besides improvements in individual PSFs, AO-on images demonstrate higher stability throughout each night of observing. The spread of minor FWHM for each night in both AO-off and AO-on images can be seen in Figure 10. Regardless of seeing conditions for a given night, there is noticeably more variation in the size of AO-off PSFs than those of the AO-on images. This effect is particularly apparent when the free atmosphere is weak. This is well illustrated in the May run, where the night-averaged standard deviation of FWHM decreases from 059 to 005, a change of more than an order of magnitude. In contrast, when the total seeing is dominated by poor free-atmosphere seeing (e.g., Run 1), the PSF variability even in GLAO mode is more pronounced. This trend can also be seen with more detail in Figure 16, where the same data are shown with MASS/DIMM data over time for each night. Generally, even when the ground-layer seeing has more variation over the course of the night than the free atmosphere, the AO-on FWHM remains more stable than in the AO-off case.

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The comparison between AO-off and AO-on images already begins to characterize the gains from GLAO. However, a more objective indicator of success would be to compare the GLAO PSFs to the seeing above the ground layer using MASS/DIMM measurements described in Section 3.3. First, we compare the FWHM of the focal plane images to the contemporaneous MASS/DIMM seeing in Figure 11, where the top and bottom panels show the correlations between AO-off images and DIMM and between AO-on images and MASS, respectively.

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Assuming accurate measurements by the MASS/DIMM instruments and no error in our correction, the DIMM should show some correspondence with the AO-off image FWHM, and similarly the MASS should correspond to the AO-on data. To examine this relationship, we looked at the correlation between the focal plane data, telemetry, and MASS/DIMM seeing. Across all nights, we calculated the correlation coefficient for six combinations: three in the total atmosphere regime (AO-off vs. free-atmosphere telemetry, AO-off vs. DIMM, integrated seeing telemetry vs. DIMM) and three in the free-atmosphere regime (AO-on vs. integrated seeing telemetry, AO-on vs. MASS, free-atmosphere seeing telemetry vs. MASS). In the case of AO-on images, we see good correlation with the MASS, with a correlation coefficient of 0.46 in I band and 0.52 in R band. These correlations are shown in Figure 11. The image data are similarly well correlated with ʻImaka's telemetry in I and R band, though some discrepancies occur at 1000 nm. We note that the amount of data taken in this filter is small, so the lack of correlations here may not be significant. A full list of correlation coefficients is presented in Table 5.

Although the AO-on images are correlated with the MASS seeing, there is a distinct offset between their cumulative distributions, with the MASS seeing being smaller than the AO-on FWHM. One limiting factor in the agreement is the apparent floor in the AO-on images at approximately 03. Across all analyses, our GLAO PSF sizes generally do not go below this minimum even when the seeing predicted by the MASS does drop below 03. This limitation is likely due to systematic effects (e.g., static aberrations, internal seeing), which the low-order system (due to a relative lack of DM actuators and the low bandwidth of our wave-front sensor cameras) cannot correct.

As mentioned in Section 3.3, comparisons to MASS/DIMM are also limited by the fact that they are not observing along the same lines of sight through the atmosphere. To compensate for this, we compare the image quality to our estimates of the integrated and free-atmosphere seeing derived from our telemetry measurements. These have the advantage that these data are synchronized with the images. Figure 12 compares the science image PSF FWHM to both the telemetry measured by ʻImaka and the seeing measured by the MASS/DIMM. Median values and standard deviations are given for all these data sets in Table 6.

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Figure 12 shows the cumulative distributions for the image FWHM and the seeing from the MASS/DIMM and the ʻImaka telemetry. There is a mix of agreement and disagreement between the distributions of seeing and image FWHM. During the nights when we observed in I band, both the AO-on and AO-off images' FWHM distributions were shifted to values larger than the I-band-converted MASS/DIMM seeing and telemetry-derived seeing. In these cases the distribution of MASS seeing values is significantly lower than either the telemetry-derived free-atmosphere seeing or the AO-on image FWHM. On the other hand, on nights when we observed in R band, the AO-off image FWHMs are smaller than the telemetry seeing but worse than the MASS/DIMM seeing, but the AO-on image FWHMs are comparable or better than the MASS or telemetry-derived free-atmosphere seeing.

Some of this difference could come from the fact that the I-band and R-band nights had quite different seeing conditions. During most of the nights when we observed in I band, the free-atmosphere seeing was weak, and both the MASS seeing and DIMM seeing are better than their corresponding image FWHMs. During the R-band nights, the total seeing was dominated by large free-atmosphere seeing. Here, the MASS seeing cumulative distribution function (CDF) and AO-on image CDF are similar for the better free-atmosphere seeing but skewed to values larger than the AO-on image FWHM CDF in the poorer free-atmosphere seeing conditions. Together, the I-band and R-band AO-on data suggest that we have an instrumental floor that prevents GLAO images of better than about 03. Alternatively, the data may be indicating that the MASS underestimates the free-atmosphere seeing under good seeing conditions, as seen in Lombardi & Sarazin (2015).

For the telemetry data the shapes of the distribution functions are similar in all cases to the distributions for the image FWHM, but with fixed offsets between them. One possibility is that the effective wavelength of the wave-front sensors changes between the observed fields. All of the R-band data were taken on the Pleiades field, where the guide stars are generally bluer than the guide stars in the other fields. Since we assume a wavelength of 0.7 μm for the WFSs, we may be overestimating our seeing estimates in the Pleiades field (e.g., R-band data). Alternative explanations could be an effect that happens on longer timescales, such that it is seen in the 30–45 s science camera exposures but not in the 15 s telemetry exposures. Additionally, if the dome seeing consists of mostly high frequencies (which the AO system cannot correct for), these residual wave-front aberrations could contribute to a broadening of the wings of the PSF, in turn slightly enlarging the FWHM.

We can examine the wavelength dependence of GLAO correction by plotting the raw FWHM values at different wavelengths, as shown in Figure 13. Ideally, we would simply look at how FWHM varies with wavelength. However, because the observations in the different filters were done over different nights and over a wide range of observing conditions, direct comparisons cannot be made from night to night. For this reason, we present each data set with the MASS/DIMM seeing measurements for the corresponding times, converted from 500 nm to the relevant wavelength. As in previous figures, we see an offset between AO-off and DIMM, as well as between AO-on and MASS. Despite this discrepancy, at each wavelength there is a clear improvement between AO-off and AO-on, sometimes larger than the difference between DIMM and MASS. Though we do not yet have enough data in consistent seeing conditions to conclude anything about the specific dependence of GLAO correction on wavelength, we can already see that ʻImaka achieves substantial correction at all currently observed wavelengths.

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From simulations the GLAO PSF is predicted to be quite uniform across the field. However, as GLAO is deployed over wider and wider fields, it is important to ensure that the instrumental PSF quality does not suffer with increasing field size. The left panel of Figure 14 shows the FWHM of each star as a function of position in the stacked AO-on image from 2017 May 19 UT (chosen because its seeing conditions were typical of the run), with the location of guide stars marked in green. To better view any global trends across the field, these data had outliers removed through an iterative sigma-clipping routine, rejecting points more discrepant than 3σ from the median, with five iterations.

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The dominant field variation is a gradient in FWHM, with PSFs increasing in size toward the east side of the field. In single guide star AO, the expectation is for the PSF quality to improve closer to the guide star. The trend visible here is not well correlated to the distance to the field center or to any single guide star. The trend is repeated over multiple nights and appears fixed to the sky, as we describe in more detail below.

We used several methods of quantifying the global variation. First, the data shown in the left panel of Figure 14 were fit in three dimensions (x, y, and FWHM) to a plane. The best-fit plane is defined by two quantities. The first, ϕ, is the position angle of the plane's normal vector projected onto the focal plane and measured east of north in degrees. The second is the gradient, which measures the spatial change of FWHM in arcseconds per arcminute. The right panel of Figure 14 shows the residual FWHM after subtraction of the best-fit plane for the stacked image on 2017 May 19 UT, with an offset of 044 added for the purpose of comparison. In addition to these parameters, we calculated the range of FWHM values before and after the plane removal (${\rm{\Delta }}{\mathrm{FWHM}}_{i}$ and ${\rm{\Delta }}{\mathrm{FWHM}}_{f}$).

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As seen in Figure 14, though some faint, high-order structure appears to remain, the variation decreases from the left to the right panels substantially. Averaged over the five nights of the May run, the range decreases from 0145 to 0082. Finally, the average variation of FWHM across the best-fit planes in the direction of greatest change is 00086 per arcminute of the field.

To justify the removal of the fitted plane, the variability would have to be attributed to an instrumental effect, such as a tilt in the CCD relative to the incoming beam. To check whether the planar variability could be due to an atmospheric effect, similar plane fits were generated for each individual frame on the same night. On a frame-by-frame basis for all images taken on 2017 May 18 UT, neither the direction nor the gradient showed significant correlation to time, ground-layer seeing, wind speed or direction, or zenith angle of the telescope (a potential cause of flexure in the instrument). Based on the lack of any trends in these analyses, we conclude that the field variability of the FWHM is dependent on an instrumental effect such as misalignment, which will be addressed in the future.

Further evidence for an instrumental origin of the PSF field variability comes from the stability of the plane over multiple nights. The stacked AO-on images from each night of the 2017 May run were fit with a plane in the same manner as above, and results are listed in Table 7. The plane parameters ϕ and gradient are given, along with ΔFWHM_i and ΔFWHM_f. The uncertainties in angles were generated using a full sample bootstrap with replacement routine in the fit. In order to generalize this effect, we report 4.5% as the typical variation in FWHM across the field, calculated as the mean of each night's percent change from the median to the minimum point. As such, we can tentatively apply a correction of 4.5% to any focal plane PSF size measurements owing to field variability.

Table 7.  Nightly Variability Plane Fit for Run 3

Date	${\rm{\Delta }}{\mathrm{FWHM}}_{i}$	${\rm{\Delta }}{\mathrm{FWHM}}_{f}$	Direction ϕ	Gradient
(UT)	(arcsec)	(arcsec)	(deg)	(arcsec/arcmin)
2017 May 17	0.128	0.032	153.4 ± 6.0	(16.6 ± 0.2) × 10−3
2017 May 18	0.201	0.110	97.0 ± 9.7	(8.8 ± 0.1) × 10−3
2017 May 19	0.104	0.091	95.8 ± 9.7	(8.2 ± 0.1) × 10−3
2017 May 20	0.115	0.086	86.0 ± 14.5	(5.0 ± 0.1) × 10−3
2017 May 21	0.178	0.092	92.3 ± 31.3	(4.2 ± 0.1) × 10−3

Note. For each night of the May run, the mean range of FWHM values of all stars in AO-on images is given for both before (${\rm{\Delta }}{\mathrm{FWHM}}_{i}$) and after (ΔFWHM_f) the removal of a best-fit plane. The value ϕ represents the direction of the best-fit plane (in degrees, east of north with respect to the detector), while the gradient describes the variation in arcseconds per arcminute of the plane.

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The variation in the data of Table 7 may be explained by adjustments made to the instrument between nights. For example, the science camera was removed and rotated by 45° between 2017 May 17 UT and 2017 May 18 UT, corresponding to the large change in the direction of variation. Also, between 2017 May 19 and 20, the camera was taken off and returned to the same position. The constant direction with different inclination suggests that after remounting the camera, we modified the camera tilt slightly. Though not all the changes between nights can be explained with camera changes, these data do present us with potential experiments we can conduct in the future in order to understand the source of this aberration.

It should be noted that the entirety of this variability analysis comprises only AO-on images. The gradient apparent in these images is not seen in AO-off frames, as the image quality is too poor to detect the small change in the FWHM over the FOV. Any attempts at fitting the variability had uncertainties too large to provide valuable insight, so they were omitted from this analysis.