the astroscopic labs - article: Math & Myth - Signal To Noise Ratio and Image Combining

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article: Math & Myth - Signal To Noise Ratio and Image Combining

There have been a lot of discussions in the amateur scene about improving the Signal To Noise Ratio (SNR) by image combining (frame stacking). There is the myth that the SNR in image combining is increasing by the square root of the combined images while a single exposure of the equal exposure time is somehow much better than the combination. But nobody ever explained why it should be like that. In fact that is not the case and this article will explain what is really going on.

Let's first have a look at a formula which is accepted by professionel astronomers to give a good estimation of the SNR under certain observation circumstances:

The CCD Equation:

	The signal from the object of interest, i.e. a star. The photons entering one pixel of the CCD must be multiplied by the QE of the CCD camera to calculate the detected electrons. Because the flux of photons is given per second of time we can also write: S(star) = flux * exposure time
	The signal from the object again, but now as the Photon Noise. It describes the uncertainty of the incoming light from the object because this light is generated randomly.
	This term describes the noise introduced by incoming photons of the sky background. Again the number of photons must be multiplied by the QE of the CCD camera to calculate the detected electrons.
	The noise by the Dark Current. A Dark Frame includes the electrons generated by the dark current plus the Bias Noise and Offset Level (positive offset to prevent negative values during readout) plus the Readout Noise. It can simply be recorded by covering the telescope so that no light can enter the CCD. It is strongly depending on the temperature of the CCD chip.
	This term describes the readout noise introduced by the camera electronics. Unlike the other noises this noise is not Poisson distributed but it behaves like shot noise. Hence it must be squared.
With this formula the SNR of a CCD image (in one pixel) can be estimated quite well. Note, that the Spatial Noise is not regarded here and that the Bias Noise is included in the noise of the Dark Frame. For more detail see Howell, 2000

The signal form the object (i.e. a star) is obviously inceasing linear with time. Two times the length of an exposure is giving two times the number of counted photons. Because this signal contains photon noise (and neglecting all other noises) we can write:

Now if we expand this formula to devide the given exposure time by a number of frames each of this frame is containing it's own readout noise which is decreasing the SNR. When the noise from readout goes to zero then both formulas will be identical. Meaning a 1-shot exposure cannot be distinguished from a combined exposure.

Example calculations

I estimated the SNR for 4 different kinds of CCD cameras. They varied from the best readout noise to the worst currently found in the amateur scene. Exposure times up to 600 seconds (10 minutes) are regarded. Each 1 shot exposure is compared to 60 exposures giving the same total exposure time. The estimation is for a bright source under a very dark sky with typical dark noises for that class of camera.

The conclusion is obvious: the owners of low readout cameras can safely make their images from combined frames while camera owners with bad readout noise should avoid combining whenever possible! Fortunately most of the new models of CCD and CMOS cameras have medium to low readout noise.

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