Signal-to-Noise Calculations for IR Imaging

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Important Reminder

It is possible to use the new NOAO Exposure Time Calculator to estimate the anticipated time to carry out a particular observation. However, blind reliance on such a "calculator" without an understanding of the limitations and assumptions can yield misleading results. This is a brief discussion of what is involved in this exercise.

Calculating Signal and Noise

Using the information in the User's Manual, one can estimate the expected signal for an object of given brightness per unit time, at a given telescope and filter , as well as the statistical noise contribution from the object signal, sky background, dark current, and detector read noise. One may thus estimate the anticipated signal-to-noise (S/N) of a particular observation or, alternatively, the observation time necessary to achieve a desired S/N, which is often a prerequisite for a successful observing proposal. These calculations may be carried out quickly using the exposure time calculator available on the Web, which contains the relevant parameters for both optical and IR imagers at KPNO and CTIO.

This calculation makes two significant assumptions, which must be kept in mind throughout:

Note that it is necessary to always use detected electrons in these calculations. Values given in units such as "ADUs", "counts", "data number", etc. must be converted to electrons using the system gain.

Depending on the brightness of the source, one is generally limited by the photon noise of the source itself or that of the sky background within the detection aperture. The sky background in the IR is sufficiently high (except for narrowband filters at short wavelengths) that detector read noise is rarely a limiting factor in imaging.

If we define the following:

St = signal level from a star, integrated over an aperture of n pixels, per unit time

Sk = signal level from sky, per pixel, per unit time

Rd = read noise per pixel in rms electrons

Nd = dark current per pixel per unit time

n = number of pixels in detection aperture

T = integration time

Then the S/N is:

S/N = St / [Sk*n*T + St*T + Nd*n*T + Rd2*n]0.5

This formula is that used in the Exposure Time Calculator. Keep in mind the following caveats:

In the typical case in which the background sky level is the dominant noise source, note that S/N is proportional to St*T0.5. If the source brightness doubles, the S/N will double as well, or the same S/N may be obtained in 1/4 the time.

Finally, when using this formula to estimate the amount of telescope time needed, remember to add system overheads for source acquisition, small telescope motions and array readout. The last is the dominant time factor for standard star observations, for example. You may wish to consult with the instrument support scientists for estimates on these factors.
16 September 1998