The signal-to-noise ratio S / N (signal to noise)

The S / N ratio of the continuum of a spectrum is an important „quality criterion“, which describes the statistical significance of the information contained in the spectrum.
Basic knowledge see here (german).

The information is available in the measured CCD images in the single pixels. In addition to the desired flux measurements are present in the photographs yet (statistical) effects caused by the photon noise, electronics and optics. These are eliminated via the biases, darks, and flats up to unavoidable statistical fluctuations.

Below you see a excerpt of a single pixel row with an ST-7 produced bias (2×2 binning, 0.5 s exposure, as abscissa the pixel numbers are plotted). The mean signal (S) is 90 ADU, the fluctuation noise (N) is approximately 10% (9 ADU) of S. The estimated S / N of the bias is therefore approximately 90 / 9 = 10.bias.png

The same CCD excerpt from a 10-minute dark under the same conditions. Most pixels show with about 100 ADU an increased level compared with the bias, caused by the dark current. But many pixels are several times higher in intensity. That are the hot pixels, producing higher dark current.dark.png

The same excerpt of a 10 minutes flat, recorded by the complete equipment (telescope + Lhires III). The pixels are saturated to about 30% (16 Bit, max. ADU = 2 ^ 16 = 65536). There exists a clear gradient . The hot pixels are visible, of course, they also contribute their increased rate of dark-current. The normal „noise“ is about 150 ADU (excluding hot pixels). The S / N in the flat can be roughly estimated at 20,000 ADU to 20000/150 = 133.

Subtracting from the flat the dark, is in the resulting masterflat even the 10 ADC noise of the dark introduced.


The text below shows the printout of a command in MIDAS. There are calculated the image statistics for the images analyzed above. Important are the average (mean) and the standard deviation of pixel intensities.

The mean (S) of the bias is 89.1 ADU with a standard deviation of 8.23 (N). The median S / N in the bias is therefore 89.2 / 8.23 = 10.8. That corresponds fairly closely to the expectation for a random distribution. In such a case, the standard deviation (= noise) ~ squareroot (signal).

In the dark, of course, is the mean value increased on 135.8 caused by the warm pixels and the dark current, compared with the above <1s exposed bias. The standard deviation is much higher (336.8). However, during the actual data reduction, the effects of dark current and hot pixels is eliminated by the dark subtraction from the object images.

It is important to look at the flats. The exposure time is set such that the pixels are saturated to 30 to 50%. In the flat but in general, structures are found: the shadow of dust grains on the glass front of the CCD chip, horizontal dark streaks caused by dust in the slit, vignettings… Therefore, the standard deviation over all pixels of the flat is very high. This standard is not synonymous with noise but is the result of the depicted structures. The noise has only a small proportion in the standard deviation, roughly in the magnitude of the darks.

As shown above, is the noise in the investigated section of the flats pixel rows around 150. Theoretically, expected to be root (20000) = 141, ie practically the same value.

The Midas command ’statistics / ima‘ yields for the flat: mean = 11,606, standard deviation 11065 ADU.

Midas 001> STATIS/IMAG
frame: (data = UI2, format = FITS), complete area of frame
minimum, maximum: 0.000000e+00 1.830000e+02 at pixel (1,1),(185,81)
mean, standard_deviation: 8.912154e+01 8.233002e+00
3rd + 4th moment: -5.60833 51.7889
total intensity: 8.68133e+06
median, 1. mode, mode: 8.914917e+01 3.588235e-01 9.006470e+01
total no. of bins, binsize: 256 7.176471e-01
# of pixels used = 97410 from 1,1 to 382,255 (in pixels)

Midas 002> STATIS/IMAG
frame: (data = UI2, format = FITS), complete area of frame
minimum, maximum: 0.000000e+00 3.908300e+04 at pixel (286,17),(325,184)
mean, standard_deviation: 1.357749e+02 3.368284e+02
3rd + 4th moment: 54.0146 4278.68
total intensity: 1.32258e+07
median, 1. mode, mode: 8.455499e+01 7.663333e+01 7.663333e+01
total no. of bins, binsize: 256 1.532667e+02
# of pixels used = 97410 from 1,1 to 382,255 (in pixels)

Midas 003> STATIS/IMAG
frame: (data = UI2, format = FITS)complete area of frame
minimum, maximum: 5.190000e+02 5.573700e+04 at pixel (369,1),(74,238)
mean, standard_deviation: 1.160649e+04 1.106488e+04
3rd + 4th moment: 0.432029 1.74744
total intensity: 1.13059e+09
median, 1. mode, mode: 1.236926e+04 8.438118e+02 8.438118e+02
total no. of bins, binsize: 256 2.165412e+02
# of pixels used = 97410 from 1,1 to 382,255 (in pixels)

IMPORTANT: By any correction using biases or darks or flats additional statistical errors are introduced into the resulting spectrum. If Ni is the noise in image i, and N is the noise in the sum spectrum, the mean square of all is (in the sum of n involved single shots + darks + flats):

N = squareroot ( N1 ^2 + N2 ^2 +…Nn ^2)

A sum spectrum, which is composed of one or a few long exposed pictures, provided a better (higher) S / N as a sum spectrum, which is integrated from many short exposed images (same total exposure time). Each recording smuggles its own noise in the sum spectrum. This noise in each individual recording is also generated by the electronic noise (readout noise, etc.), which is the photon noise superimposed. Scientifically valuable are spectra whose S / N is over 100. For analysis of time series even S / N up to 1000 is desirable. I target, whenever possible – in general, an S / N of > 200.