[Aavso-photometry] Re: Dark Sky Annulus

[Michael Newberry]

That's some great advice from Arne. I would just add a couple of small 
points to what he says about the sky annulus.

About choosing the inner radius:

First, I wholeheartedly agree with Arne's rule of thumb about setting the 
diameter of the inner sky annulus to 5 times the FWHM of the star. Another 
way to think of this rule is to make the *radius* of the inner sky aperture 
be 2.5 times the FWHM of the star profile. Here is a graph that gives a 
visual representation of these numbers:
    http://www.mirametrics.com/pub/SkyAnn1.png

This shows a radial profile plot of the pixel values around a star in the 
field of BL Lac. The square points show the pixel values sampled in all 
azimuths around the centroid position of the star. The solid curve shows a 
Gaussian + Constant model fit to the data. The FWHM, obtained from the fit, 
is 6.5 pixels (radius = 3.25 pixels). The plot radius extends to 2.5 times 
the FWHM, or 16 pixels. The right edge of the graph is where the sky 
sampling would begin according to the rule of thumb Arne mentions. You can 
see that the star profile is pretty flat aat that point nd has essentially 
merged into the background. So radius = 2.5 x FWHM is a pretty good rule of 
thumb.

About choosing the outer radius:

I also like Arne's choice of ~10 pixels between the inner and outer sky 
radius. We know that making the annulus larger adds pixels, which reduces 
the uncertainty in the sky value becasue the error of the mean equals the 
mean error divided by the number of samples (pixels). Reducing the error of 
the mean sky value is important because the sky uncertainty is folded into 
the calculation of the star's magnitude error. However, you can get problems 
by making the outer radius of the sky annulus too large. There are a number 
of issues that can come into play: First, if the sky is not a "perfectly 
flat plane (either constant or inclined), then the sky estimate can be 
biased positive or negative, depending upon whether the sky is humped or 
bowl-shaped. This can result from imperfect flat fielding, scattered light 
from something bright that is inside or outside the field, and spherical 
aberration which enlarges the wings of the star profile relative to the 
core. In addition, the larger the radius the more likely it is to include 
things that are not underneath the star, like faint fuzzy galaxies. All 
these effects can bias the background toward a higher value, making the star 
appear systematically too faint.

I have published a couple of papers that go into some detail about the sky 
annulus issue. See the proceedings of the American Astronomical Society 
Workshop on Precision Photometry in 1999 (ASP Conferences Series, Vol 189, 
p. 74). In particular, see section 3, starting on page 79. Also see my paper 
on Signal To Noise considerations for CCD photometry, in Publications of the 
Astronomical Society of the Pacific, 1991 January, Vol 103, p. 122.

Here's a numeric example: Let's say our goal is to beat down the background 
noise of 1 pixel (i.e., the mean error) by a factor of 100. That requires 
100^2, or 10,000 pixels. The number of pixels in the sky annulus is simply 
the difference in area between the two bounding apertures, whose radii are 
r1 and r2. The area, or number of pixels, n, in the annulus is given by n = 
pi * (r2^2 - r1^2). Then we have 10,000 / 3.14 = 3200 = (r2^2 - r1^2). Arne 
mentioned a good rule of thumb as being r1 = 2.5 * FWHM. In my example, r1 = 
16 pixels. So, r2 = sqrt(3200 + 16^2) or about 59 pixels for the outer 
radius. I have uploaded another picture showing this:
    http://www.mirametrics.com/pub/SkyAnn2.png

This shows an aperture set using 8, 16, and 59 pixels for the object, inner 
sky, and outer sky radii. As you can see, the sky annulus is pretty large. 
Before using an aperture that large I would want to be sure that it was not 
picking up the tail of other star profiles or a faint galaxy which would 
bias the sky to a higher value. In other words, trying to beat down the 
noise by a factor of 100 may be giving an impression of increasing the 
precision but at the expense of losing accuracy. Here is a picture showing 
the 10 pixel radius that Arne suggests:
    http://www.mirametrics.com/pub/SkyAnn3.png

This picture shows radii of 8, 16, and 26. As you can see, this is more 
conservative sky annulus that doesn't reach too far from the star. It is a 
good trade-off between reducing the sky noise by a huge amount versus 
increasing the risk of catching light from something else. In this case, the 
area of the annulus works out to be ~400 pixels, which beats down the sky 
noise by a factor of 20.

Some other ways to beat down the sky noise are to 1) observe in a darker 
sky, 2) use a camera with lower readout noise, and 3) use software 
processing techniques that minimize the addition of noise through the image 
calibration process (bias, dark, flat corrections). Since these factors lead 
to images with less noise in the first place, you do not need to reduce it 
so large a factor. If you can't control your sky brightness, then you can 
try to choose a camera with lower readout noise. If your sky is fairly dark, 
the lower readout noise can be noticeable. But a large factor often 
overlooked is to use better image calibration techniques that minimize the 
amount of noise added at each step of the calibration process. The paper I 
mentioned in PASP 1991 discusses this point in great detail.

Michael Newberry


----- Original Message ----- 
From: "arne" <arne at aavso.org>
To: "Aavso-Photometry" <aavso-photometry at mira.aavso.org>
Sent: Wednesday, January 04, 2006 11:16 AM
Subject: [Aavso-photometry] Re: Dark Sky Annulus


> Richard Huziak wrote:
>> Can you give me a brief summary of what the 'optimal size' of the 
>> dark-sky annulus should be in aperture photometry (in a 'normal' field)? 
>> Is there some preferred ratio of dark annulus to aperture that is 'best'? 
>> I see a very small effect when resizing the annulus, except in a really 
>> poor sky, where making the annulus smaller seems to smooth the data. 
>> This is contrary to what I thought - thinking a very large dark sky 
>> annulus would give a nice quiet, average background.  I don't see 
>> variations of more than 1 percent of so except on the worst nigths when 
>> playing with the annulus.
>>
> The "dark-sky annulus" (also just called the sky annulus) needs to be:
>   (a) concentric with the target object, so that any gradient in the sky
> background is handled;
>   (b) large enough so that the noise in the sky measurement is reduced
> to the point that it has little effect on the noise of the target 
> measurement
> (remember, you subtract sky from the target aperture);
>   (c) inner radius needs to be large enough so that little of the target
> object profile is included in the annulus
>
> Setting the size of the annulus is a "black art" (so to speak).  I usually
> set the inner radius to be about 5x fwhm and width about 10 pixels, but 
> the
> main idea is to have more pixels in the annulus than in the aperture. 
> Where
> you run into potential problems are for faint stars (near the sky limit) 
> or
> crowded fields (where a bright contaminating star might be in the sky 
> annulus).
> The sky annulus size is not critical; you can use a large one on one 
> object and
> a small one on another.   You may not see any difference between different
> annuli because the sky noise may be dominated by the Poisson noise of the
> target object.
>
>> Part 2 then becomes, if annulus size has little effect on the 
>> measurement, then can you use zero diameter annulus for a measurement of 
>> a supernova in a galactic arm, and a tight aperture to eliminate 
>> background as much as possible (assuring, of course, you minimize 
>> clipping of the image)?
>>
> By zero diameter, I assume you mean a 1-pixel-wide annulus?  Not wise
> because the noise in the sky measurement will be large.  Supernovae are
> really hard to do right because the underlying galaxy is nonuniform, and
> how you set the sky annulus will affect the final result.  When the SN is
> bright, you can be pretty crude about the process, but as it fades, the
> best solution is to remove the underlying galaxy before starting the 
> measure.
> This can be done with an image without the SN present (either precursor
> or far post-cursor), by removing an analytical model, etc.  See
> Richmond et al., 1995 AJ 109, 2121 for more details.  In general, though,
> measurement of a supernova requires the smallest measurement aperture with
> which you feel comfortable, and playing around with the sky measurement
> until you get consistent results.  Watch out for the wierd colors when
> the SN gets faint; transformations become very difficult.  I prefer to
> leave SNe alone or only work on them when they are bright or isolated
> from the galaxy.  SN2002ap was an exception; there I followed it for a
> long time since it is a GRB SNe prototype.  Image subtraction is an
> interesting technique that ought to work well, as long as you have
> a good non-SN galaxy template.
> Arne
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