# Study of Mean Whisker¶

This page describes study of the mean whisker in ~8000 community exposures from 1 March - 19 April 2013 using the data assembled in Klaus's file on IQCatalog. Work done by Gary Bernstein through 23 May 2013.

All the work is done using the mean across the focal plane of the asymmetric second moments of the PSF, which have units of arcsec^2. In terms of the quantities in the file that come from Image Health, these are m1 = *covxx* - *covyy*, m2 - 2*_covxy_. The optics as designed should yield zero for both components of this quantity. There is a requirement that the RMS whisker amplitude be <0.2 arcsec in riz bands, which means |m| < 0.04 arcsec^2 is needed. In fact the FOV-mean moment should be below this because variation across the field will increase the RMS.

This is a summary of conclusions:

**Top level:** In *griz* there is a mean (*m1,m2*)=(-0.008, 0.011) arcsec^2 that must be due to *static misalignment or error in the optics.* There is an RMS of ~0.020 arcsec^2 about this mean. Roughly speaking: 1/3 of this variation comes from expected sources: *atmospheric dispersion, guiding errors* (with contribution from wind shake at >15 km/h winds), and *focus variations about the reference wavefront* (not the design). Another 1/3 of the moment variation comes from short-term fluctuations with little correlation from one exposure to the next, perhaps due to *rapid image motion* (seeing) or maybe measurement noise. The last 1/3 has an hour-scale coherence time, changes on slews, but is not a consistent function of orientation. This could be some kind of *drift in primary mirror figure or other alignment.*

- The expected PSF elongation from Atmospheric Dispersion is detected in g and r bands. It is expected to be undetectable in izY. I subtract the predicted elongation from further PSF analyses.
- After dispersion correction, the mean and RMS of PSF moments are similar in
*griz*bands: in*riz*the mean is (-0.005, 0.011) and the RMS is (0.020, 0.018).**The**This could be a detector effect.*Y*filter has mean m1 that is -0.020 arcsec^2 away from the other filters. - Wind shake: There is no detectable difference in the average or the RMS of the mean moments between high and low wind conditions, so wind shake is not the dominant contributor to the mean moments of an exposure. But:
- Poorer tracking in RA is seen in the TCS telemetry at wind speeds >15 km/h.
- The mean Moment 1 of the TCS, guider, and PSF all increase by about 0.003 as wind speed rises. This suggests that wind shake is mostly manifested as low-frequency HA tracking error.

- Guider motion: There is a strong correlation of PSF moments with
**guider motion**moments. Regressing PSF against guider yields slope of ~0.7. A slope of 1 would be expected if image motion outside the guide camera bandwidth are uncorrelated with the slower image motions. This slope may be <1 due to noise in the guider position readouts. - Focus Dependence: PSF moments correlate with
**focus error**reported by the donut system (*dodz*). This is expected if there is astigmatism from optical misalignments/errors. There is no focus position that makes the mean moment go to zero. This agrees with Chris Davis's analysis of PSFs expected from defocusing the observed reference wavefront. - Time dependence: After removing the predicted atmospheric dispersion and the signals correlated with dodz and ge1, the mean moments still show:
- A static offset of (m1,m2)=(-0.008,+0.011) (different in Y). The RMS about this mean in
*riz*is (0.017,0.016) arcsec^2, so the effects removed above have a significant but not dominant contribution to the variation.**This static avg moment probably comes from some fixed optical error/misalignment.** - A slow variation (~hour timescales) with 0.012 arcsec^2 RMS per moment that changes values on large slews, and does not repeat on returning to the same telescope orientation.
**Could be primary-mirror shape variation or other alignment issue.**This slow component shows no correlation with any of the donut alignment outputs beyond focus. The slow variations do seem larger farther from the zenith. - A rapid variation, nearly uncorrelated from one exposure to the next. The RMS variation of this fast component rises from 0.008 arcsec^2 in good seeing (0.35<r50<0.45 arcsec) to 0.016 arcsec^2 in poor seeing (r50~0.85 arcsec).
**This could be image motion due to seeing that is too fast for the guider to see.**

- A static offset of (m1,m2)=(-0.008,+0.011) (different in Y). The RMS about this mean in
- The RMS variation of PSF moments, guider moments, and dodz all rise continuously as the PSF radius R50 rises.

*Note added 31 May: Steve Kent says the perfect optical model develops mean moment of 0.007 arcsec^2 for a 1 mm hexapod shift. Near zero for tilts about focal plane.*

## Atmospheric dispersion and mean moments¶

The symbols in this first plot shows the average moments for all exposures divided into filters and split into the upper and lower halves of the*windspd*distribution. The ellipses are the RMS variation of the mean moments from exposure to exposure (with sigma clipping to remove outlier exposures). Solid ellipses are for low wind speeds, dashed for high wind speeds [all of the Y-band images were taken on a small number of low-wind-speed nights.]

We see that:

- For a given filter, the averages and the variation of the moments are the same for high/low winds.
*There is no evidence that wind shake dominates mean whisker or its variations.* - The griz filters all have the same, non-zero average whisker. This suggests
*a common static misalignment or figure error*in the optical system. - The Y filter has a more negative average m1. This could be a fluke as we have just 1 or 2 nights' Y data here. Or could also be extra charge-spreading along one of the CCD axes in Y band, where the photons penetrate the thick devices. Not necessarily an optical difference to chase down. This shift of Y band persists after other corrections are made.
**I will leave Y band out of most of the further analysis.** - The g filter has larger variations in m1 than the others.

The moment variations in g band could be signatures of atmospheric dispersion (wavelength dependence of the refraction). This will cause PSF elongation toward the zenith, with an amplitude that scales as tan(zenith angle)^2. This plot shows the g-band m1 vs this tan^2 factor (including some trig factors giving the proper projection from zenith elongation onto m1). The red line is the prediction from Andres Plazas for the moment shift expected for a G star observed in the g filter. Agreement is pretty good!

The dispersion is expected to be ~5x weaker in r band, but here we can see it (in the m2 component; red dots are binned averages, red line is again the a priori prediction for a G star). Atmospheric dispersion is several times weaker yet in izY bands so can be ignored. *All further analyses have the expected atmospheric dispersion signature subtracted from the observed moments.*

With the atmospheric dispersion removed, the variation of g-band moments is now comparable to the other filters:

**Conclusions: Wind shake does not dominate either the avg or variation of the mean PSF moments. Dispersion is present as predicted. After dispersion correction, there are no large differences in whisker characteristics between filters except a possible shift in Y band.**

## Wind shake¶

First I look for any change in TCS telemetry at higher wind speed. In this plot we see that the RMS HA error seen by the TCS increases above 15 km/h. Dec RMS does not change, so the *tcs1* moment increases above 15 km/h.

Next I look for this extra RA motion feeding into the (xx-yy) moments of the guider motion and the PSF. These are noisier than the TCS but show a similar pattern and amplitude.

**The extra HA motion seen in TCS at high wind speed is slow enough to be captured by the guider.**So there is not evidence of wind shake causing faster motions, at least none that are coherent along HA.- Since the TCS Dec RMS is not changing at high wind speed, there is
**no evidence of wind shake inducing motions outside the HA tracking problems.**

## Guider motion¶

We expect the guider centroid motion to result in equivalent PSF motion. Plotting PSF*m1*(

*m2*) against the guider moment

*ge1*(

*ge2*) shows a clear correlation. Regressing the PSF against the guider motion shows a slope of ~0.7. There are some reasons why this could depart from unity:

- There is noise in the measurement of guider motion, which will tend to suppress the slope.
- If there are high-frequency motions in the same direction as the low-frequency ones caught by the guider, the slope could be >1. High-frequency image motion due to turbulent seeing would be damped or invisible in the guider motion record.

Henceforth *I will subtract 0.7 x (guider moment) from the _m1* and *m2* quantities.

## Focus dependence¶

Here are the regressions of *m1* and *m2* (after removal of the dispersion and the guider motion signals) against the donut focus measure *dodz* for the *riz* bands. In this plot the red dots are individual exposures, the blue (green) dots are the means in bins of *dodz* (*m1* or *m2*). The red lines are slopes d(m1)/d(dodz)=-0.00025, d(m2)/d(dodz)=+0.00005. (The moment and dodz data have been passed through a high-pass filter to remove some other conflicting signals from the moments - see next section).

A linear dependence of moments on defocus is a very good fit. Here is another view of this (without any high-pass filtering now), showing that the mean PSF moment traverses the (*m1*,_m2_) plane linearly as the focus changes, but **no focus position results in the prescribed zero moments.**

Note that there is measurement noise both on the PSF moments and on *dodz.* The latter will tend to suppress the slope of the moments-focus measurement.

Although the mean moment should be zero even for a defocused perfect telescope, we do find that the dependence of moments on defocus agrees pretty well with a prediction by Chris Davis of the effect of defocus on the moments of a PSF with the *observed* reference wavefront. Below I have overplotted the solid line, the observed mean moment vs defocus behavior, on top of Chris's model prediction from his talk at the April 2013 DES Collaboration Meeting. The agreement is pretty good!

**Conclusions:**

**the mean whisker dependence on donut***dodz*is observed with roughly the behavior expected given the observed wavefront.**there must be aberrations besides defocus in the optical system that break the azimuthal symmetry of the optics and cause nonzero mean moment at any dz.**

*I will henceforth subtract the mean behavior with focus from the PSF moments:*

- m1 -= -0.00025*dodz - 0.008
- m2 -= +0.00008*dodz + 0.011

Note that this will couple noise in the *dodz* meaurement into the PSF moments.

## Time dependence¶

Now we can examine the behavior of the PSF moments after I have applied corrections for the expected and confirmed sources of anisotropy: atmospheric refraction, guider motion, and defocus. After these corrections the RMS of the moments in *riz* bands is reduced from its original (0.020, 0.018) to (0.017,0.016) arcsec^2. [Keep in mind that my corrections may have introduced additional measurement noise.] *The identified effects account for a significant, but not dominant, portion of the PSF moment variation about the static mean value.*

Here is the covariance of the corrected *m1* and *m2* signals vs the lag in number of exposures. [For these plots I only consider exposure pairs within 1 hr of HA of each other, or with (0.1 hrs)*lag of time, so I'm not correlating exposures with slews or long breaks between them.] We can see that *the variations in m1 are dominated by some component with a very long coherence time, and both components have a variance component that does not correlate between exposures at all since the 0-lag point is high.* The latter could of course be measurement noise.

I will now try to examine these two components - the fast and slow variations in moments - by running a smoothing window of length +- 1/2 hour across the data. (The window is truncated whenever the telescope slews by more than 1 hr in HA.) The moving average is the "low pass" moment variation and the difference between the original and the moving average values is the "high pass" moment sample.

### High-pass moment behavior¶

The high-pass filter suppresses, as expected, covariance on scales longer than the 10-20 exposure (1 hr) width of the smoothing window. We note that now the zero-lag RMS is reduced to 0.012 arcsec^2 - **about half of the corrected-moment variation is on short time scales.** We can also see some much weaker covariance at 1 or 2 exposure lags.

- measurement noise, either in the moments or in the guider moments and dodz that have been used to correct the moments
- Image motion from seeing, which is too fast to be captured by the guider exposures.
- Some other physical effect that is changing between every exposure, but is not manifested in any of the other telemetry I've checked, such as TCS errors.

As a further diagnostic, here is the mean and RMS of the high-passed *m1* signal as a function of the size of the PSF (in R50 units). (*m2* has similar behavior and is not shown.) There is no tendency for *dodz*, guider moments, or PSF moments to have bias with seeing (they have all been high-passed anyway). But **the RMS of the guider motion, PSF moments, and focus adjustments all grow roughly linearly with the seeing.** This increase is compatible with an explanation that either the RMS is due mostly to image motion (since worse seeing has larger image motion and larger moments) or noise (since both the guider positions and the moment measurement have higher noise for larger stars). I don't know why the *dodz* variance would depend upon seeing. **(Aaron, any ideas???)**

Note that in good seeing (0.35<r50<0.45") the RMS moment variation is only 0.008 arcsec^2, and it doubles for bad seeing (r50~0.85").

**Conclusion: there is about 0.012 arcsec^2 of variation in each of m1 and m2 that is not correlated with the telemetry I've checked and has only weak, short-term temporal correlation. This could be explained perhaps as turbulent image motion as it scales strongly with the PSF size R50. Or it could be measurement noise, with some correlations induced by noise in our dodz corrections.**

### Low-pass moment behavior¶

The low-pass filter leaves behind a moment signal with RMS of 0.012 arcsec^2 per component. This is very unlikely to be noise since we have averaged many exposures, nor can it be associated with turbulent seeing.

The first plot shows the low-pass-filtered moments as whiskers vs the HA and Dec position at which they were taken. (The mean moments have been subtracted, and the moments have been corrected for defocus, dispersion, and guider motion.) Color coding is by date of observation. The second plot shows the whiskers as a function of time, with the color coding being airmass - a sudden change of color indicates a slew. This second plot reads like a book - time advances left to right, then to the next row for the following night. Whiskers are drawn every 15 min. Remember the smoothing filter is +-30 minutes.

- There is a strong, slowly-varying component.
- It can change abruptly on a slew, so it's not just a time-dependent thing.
- At the same RA/Dec, the slowly-varying component can take different values on different nights, so it is not purely a flexure response.
- The slowly-varying component tends to be higher at higher airmass. At X<1.2, the slow component has RMS = 0.0095 arcsec^2. At X>1.2, it's 0.0142 arcsec^2.
- (not plotted): there is some tendency for higher slow moments in worse seeing too. Hard though to disentangle this from the airmass dependence.
- (not plotted): the slowly-varying moments don't show any correlation with the donut outputs
*dod[xy], do[xy]t, z*- except perhaps a correlation between^{56}delta, z^{56}theta[xy]*m2*and*doxt*.

**Conclusion: I don't know what this is. It could be changes in the primary mirror figure or some other alignment that are driven by gravity but also have some time variation or hysteresis. This would be consistent with the above behaviors, e.g. worse farther from the zenith. It would be nice to find some other signs of aberration in the donuts that correlate with this.**