1. Eliminating noise and other sources of error
CCD Calibrations
Eliminating noise and other sources of error

2. Types of CCD calibration data
Additive systematic noise (bais, dark)
Multiplicative systematic noise (flat)
 fig 9.1
1.Bias or zero image : obtain by resetting the charges on a CCD and then immediately reading it out with the shutter closed
Readout noise : simply by measuring the standard deviation of the pixel values in a bias image.
Bias offset voltage fluctuates over a small range during a night and from night to night.
 taking frequent bias images.
2. bias offset  use the overscan region of a CCD image (overscan line contains information on the value of the bias offset voltage at the time it was read out – no information on bias variations across a line).
 prepare a “master bias” from many bias images to correct the bias pattern & the overscan region gives the instantaneous value of the offset for each line

5. Dark image
Dark image obtained with the CCD shutter closed. – charges are collected for exp times : thermal agitation of the silicon crystal lattice (unless cooled less than -80 C)
Temperature stable ; many dark images master dark images
Night sky emission lines produce fringes on a CCD image; also additive noise ; usually in near IR I band :observe the sparse field with the same exp of the targets. (fig 9.3)

7. Pixel to pixel QE variations
Multiplicative(or scaling) noise : due to pixel to pixel QE variations & non-uniform illumination
QE curve of CCD : average from all the pixels on the chip
Mirrors, lenses, filters & the CCD window
 non-uniform illumination
 flat correction

8. Observation of Flats
1. twilight sky ; most accessible source of uniform illumination, While large brightness gradient, but have a point with very small gradient
 after sunset about 20 deg from the zenith, opposite the Sun : reverse is the case just before sun rise.
 careful preparation : rapidly changing sky brightness & different transmissions of the filters
S/N of flat greater than that of the best science image
Require more than 1 flat image per filter
Sometimes stars in flat image when the twilight brightness is low
 1. tele sidereal drive is turned on ;star image remain fixed on the same pixels. Move tele for next flat image. Then normalizing and combining the flat images with a median or mode
2. turn off drive. Cause trails of stars on each image. The combine the same way, most of the trails removed (some trails overlapped, so not so effective as the first one)
2. Dome flat : can obtain any time, no stars, but not quite uniform , photometry not better than 1% with a dome screen
3. night sky flat : combine with mode ; remove stars. But low S/N so rarely used in optical, but used in the Infrared.

9. Calibrating the Science Data
Correct all data for the bias offset ; subtracting the master bias from the dark, flat, target and fringe image
Correct dark (scaled) : flat, target, & fringe images
Correct fringe for target, flat images
Divide target images by the Normalized flat image
Data on CCD : two-D array of numbers
B[x,y], D[x,y], Fl[x,y], Frl[x,y] & Sl[x,y]
Bias, dark, flat, fringe, science data

10. Science Data Neglecting fringing Normalized flat Fln[x,y] :
Fln[x,y] =( Fl[x,y] –B[x,y])/MODE(Fl[x,y] –B[x,y])
Reduced science image
Sl*[x,y] = (Sl[x,y]-(ts/td)D[x,y] – B[x,y])/Fln[x,y]
ts, td ;exp time of science and dark

11. Characteriszing CCD Random noise : read noise & poisson or shot noise
Gain of CCD:
array of counts : ADUs(analog-to-digital Unit) or DNs(Data numbers)
Gain = the numbers of detected photons(=number of photoelectrons or electrons) corresponding to one count : read noise into data stream
A/D converter ; discreet steps
 digitization ; a type of noise
The greater number of bits in an A/D converter, the less important this noise source.
Professional ; 16 bits ( levels) cost speed & cost,
many sys : 12 bits(4096) or 14bits( level)
Digitization noise is related to the value of the gain
In units of electron sdig = [(g2-1)/12]1/2
Manufacture sets the gain such that the read noise properly sampled.
Total read noise = quadrature sum of the amplifier read noise and the digitization noise

12. constraint of Gain : Full Well Capacity
Gain be set such that the dynamic range of each pixel matches the count range of the A/D converter :
if FWC = , A/D =16bits (65 536)
 gain = 3 electrons per count
(but, it may not optimal to sample the read noise)
For very faint target ;gain should set to a low value
For high count ;gain should be set to a high value
Digitization noise scales with the gain
Smaller A/D converter for given full well, require higher gain  greater digitization noise
Use highest resolution(16 bits) A/D converter

13. Poisson noise Variance corresponding to N counts spoiss2 = gN
Total random noise = poisson + read noise
s2 = gN + sR2 (read noise variance term not multiplied by g2 since given in units of electrons)
Determine Gain & read noise :
method 1 : Two bias B1, B2, and two flat F1, F2 :
obtained close in time = eliminate the sys noise
Form two difference images from the pair of biases and flats
 variance on each difference image obtained with standard statistical methods, (ie width of the histogram of the counts of an image = variance)
Variance of the flat difference image = sum of the poisson and read noise variances
g2sF1-F22 = g2spoiss2 + g2sB1-B2 2
g2spoiss2 = g [F2 +F1] – (B1 + B2)] :average over all pixels on each images
g2sB1-B2 2 = g2(sB12 +sB22) = 2g2sB12 = 2g2sB22=2sR2
 variance of the bias difference image equals twice the variance of a single bias image

14. Variance plot or photon transfer curve
a plot of variance against mean counts
The variance of a image with low counts is dominated by the read noise =read noise regime,
at high counts, dominated by the Poisson niose= photon and shot noise regime
s2count = Ncount /g + sR2/g2
Log scount = ½ log (Ncount /g + sR2/g2)

15. Variance plot
Y intercept : sR2/g2,
slop in photon noise regime = ½

16. From real data
1. Obtain several images of a uniformly illuminated CCD within a range of expsures
 images with low count sample, read noise regime, images with high count sample, the photon noise regime
2. from a single image if contains both bright and dark areas. Devide into many small regions, each treating independently , Or each of several images binned into many samll regions for statistics and plot on the same variance plot

17. Variance plot from 5 bias and flat images
50,000 bins each 10*10 pixels in area –fig
This method : Systematic noise present
1. non-uniform illumination
2. Pixel –to-pixel QE variance as noise = called scalenoise (systematic noise) : greater for higher counts
Nonlinearity regime :
Very high counts

18. Change CCD characteristics
Gain :
faint source = low value, bright source = high value
Chip clocking frequency ;
high = fast read out time, high read noise, often change gain as well
Binnig ;
on-chip binning = high potential well & larger pixel
 reducing readout noise per physical pixel & readout time, but reduce spatial resolution

19. Point Sources

20. Signal and Noise
The Poisson noise in a signal (the standard deviation s of the individual measurements from the true signal) can be estimated as the square root of the signal
The total noise from the four independent noise sources
sB is the background noise, ss is the object noise, sT is the thermal signal, and sR is the readout noise.
SNR in one pixel

21. Growth of Signal, noise with t
the growth of Signal, Noise, and Signal to Noise Ratio (SNR) with increasing exposure time t.
Note that, in this example, the background signal B is stronger than the signal S from the object under observation. The background noise is sB, the object noise is sS. The readout noise is independent of the exposure time and it is
therefore not drawn.
The signal S grows linear with increasing exposure time, as do the background signal B and the thermal signal T. Fortunately, the background noise (sB = ÖB) and the thermal noise (sT = ÖT) grow slower. Doubling the exposure time will increase all signals (S,B,T) by a factor of 2, but the noise levels (ss, sB, sT, s) by a factor of only Ö2, so the SNR increases by 2 ÷ Ö2 = Ö2.
With increasing exposure, the faint object will eventually emerge from the noise, even though the background signal is always stronger than the signal from the object.

22. S/N at peak intensity
0.6m f/3.3 reflector, with a central obstruction of 0.2m.
a CCD with 24μm square pixels ( 2.5” at the focal length of 1.98m),
a dark current of one electron per second per pixel,
a readout noise of ten electrons per pixel,
a mean quantum efficiency of 70% over (400nm to 800nm).
the target : a stellar object of 20 mag
sky background to be 18mag per square arc second, and the FWHM of the stellar image to be 4”.
In that spectral range, receive about 4´1010 photons per second per square meter from a star of 0mag. A difference of 1mag corresponds to a factor of 2.5 in the brightness, so there will be only 4´1010 ÷ 2.520, or about 440 photons per second per square meter from our target.

23. S/N
The light collecting area of the 0.6m telescope is 0.25m², so it will accumulate 11’000 photons in a 100 second exposure. With a quantum efficiency of 0.7, this will generate about 7’700 electrons in the CCD.
 Assuming that the PSF of the object can be described with
, and that the peak brightness is located exactly at the centre of one pixel, this pixel collects about 29% of the total light, or about 3’190 photons, which will generate 2’233 electrons in that pixel. The Poisson noise of this signal is Ö2’233 ~ 47.

24. S/N
the flux from the sky background (18mag per square arc second) to be 4´1010 ÷ 2.518, or about 2’748 photons per second per square meter. The telescope collects about 68’700 photons from each square arc second during the exposure. Each pixel covers 6.25 square arc seconds, and therefore, about 429’375 photons from the sky background will be collected during the exposure in each pixel. This will generate about 300’563 electrons, with a Poisson noise of ~ 548 electrons.
During the exposure, the dark current will generate 100 electrons in each pixel, and the dark noise is therefore Ö100 = 10. The readout adds further 10 noise electrons
Ö(47² + 548² + 10² + 10²) ~ 550
 The Peak Signal Noise Ratio is now found to be 2’233 ÷ 550 ~ 4.1.

25. Exposure time
To increase the limiting magnitude by one full magnitude, the exposure time would have to be extended by a factor of 6.25

26. S/N

27. Telescope Aperture
1.5m telescope : 6.25 times more light collecting area than the 0.6m instrument, the Peak SNR is now only 3.4. Compared to the Peak SNR of 4.1 that was found for the 100 second integration with the 0.6m telescope, this is a loss of ~0.2mag in limiting magnitude.
Due to the long focal length of the telescope, each pixel now covers only 0.71” ´ 0.71”. (cf pixel size 2.5” ´ 2.5” for 0.6m),
 8% of the area.
By combining the increased light collecting power, and the smaller pixel scale,  each pixel receives only about 6.25 ´0.08 » 0.5 times the light collected in one pixel of the CCD by the smaller telescope. As both the light from the object and from the sky background (the dominant source of noise) drop by the factor of 0.5, the SNR should decrease approximately by a
factor of 0.5 ÷ Ö0.5 ~ 0.7.( true factor about 0.8 since PSF is a non-linear function

28. Pixel Size and Sampling
the relative size of the pixel to the FWHM of the stellar images is an important factor in obtaining the highest possible SNR.
the highest Peak SNR is obtained when the pixels are
about 1.2 ´ FWHM in size

29. Sampling
To retain the information of the objects on the CCD image, the scale must be chosen so that the FWHM of stellar
sources spans at least 1.5 to 2 pixels . This scale is called “critical sampling”,
Critically sampled images will give the highest SNR and deepest limiting magnitude possible with a given equipment in a certain exposure time, without loosing important information contained in the image.

30. Error Estimates
Fitting a PSF profile to a faint, noisy detection is naturally less precise than for bright stellar images with a high SNR
The fractional uncertainty of the total flux is simply the reciprocal value of the Signal to Noise Ratio, 1 ÷ SNR
By converting this uncertainty to magnitudes :
SNR is the total SNR of all pixels involved :
S is the total integrated signal from the object in the measurement (i.e., within the aperture), and n is the number of pixels within the aperture

31. Errors
sAST is the estimated one-sigma error of the position of the object, sPSF the Gaussian sigma of the PSF