Estimating Signal-To-Noise Ratio of a Point Source Measurement for IRAC
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Note that a tool to calculate the Signal-To-Noise Ratio and
other quantities mentioned in this memo is available at

http://ssc.spitzer.caltech.edu/tools/snirac.pro

When planning observations that need to measure the flux density or
variations in the flux density of a point source to a certain precision,
it is important to include the Poisson noise of the source in the
estimation of the signal-to-noise ratio. The online sensitivity
estimator, SENS-PET, that determines the detection signal-to-noise ratio
(how significant a source is compared to the background) only includes
the read noise of the instrument and the Poisson noise of the background
in the calculation of the uncertainty. This memo discusses in detail how
to estimate the uncertainty in a measurement of the source flux density
for the benefit of observers who are interested in variations in source
brightness (as in observations of extrasolar planet transits, etc).

The uncertainty in the measured flux density of a point source in a
single exposure is 

    sigma^2 = sigma_rn^2 + sigma_s^2 + ap_fac * sigma_bg^2 + 
              sigma_sys^2

where sigma_rn is the read noise of the detector, sigma_s is the Poisson
noise of the source + background, sigma_bg is the Poisson noise of the
subtracted background alone, ap_fac is a scaling factor accounting for
the noise in the annulus used to estimate the background to subtract, and
sigma_sys is any systematic/confusion error. In this memo, sigma is
calculated for a single exposure and in the limit where sigma_sys = 0;
therefore, sigma decreases as the square-root of the number of
exposures.

For an optimal source extraction, the number of pixels contributing to
the source is the noise pixels (npix) for that particular array (see
table 6.1 in the IRAC chapter of the Spitzer Observer's Manual, SOM). 
Then the read noise for the source is 

    sigma_rn^2 = npix * sigma_readnoise^2, 
     
where sigma_readnoise is the read noise in electrons for the desired
frametime (see Table 6.3 of the SOM). If a smaller aperture is used, then
npix is the number of pixels in the aperture, but the flux density of
the source should be scaled by the fraction of the source flux density
enclosed by the aperture.

The term sigma_s^2 is the Poisson noise of the source plus the background
in the aperture used. Then, sigma_s^2 = e_s + npix * e_bg, where e_s is
the signal of the source in electrons and e_bg is the signal of the
background per pixel in electrons.

Typically, the source flux density (fd) is given in micro-Jy. To convert
to electrons detected, use

     e_s = fd(micro-Jy) / scale * GAIN * FRAMETIME / FLUXCONV

where scale (34.98) converts from micro-Jy to equivalent MJy/sr, so that
the the correct number of electrons are determined when using the
FLUXCONV and GAIN values supplied; FLUXCONV converts from MJy/sr to DN/s;
GAIN converts from DN to electrons; and, FRAMETIME is the frame time
used. Note that GAIN, FRAMETIME and FLUXCONV can all be found in the
headers of a BCD image. GAIN and FLUXCONV are constants for a given IRAC
channel. FLUXCONV can also be found in Table 5.1 of the IRAC Data
Handbook and GAIN is tabulated in Table 6.5 of the SOM.

Simplifying the expression, 

     e_s = fd * FRAMETIME * fd_to_e

where fd_to_e = 0.867, 0.764, 0.182, 0.537 electrons / (uJy * s) for
channels 1-4, respectively.

Likewise, the number of electrons contributed by the background is given 
by

     e_bg = background_sb * FRAMETIME * sb_to_e 

where the background_sb is an estimate of the surface brightness of the
background in MJy/sr (such as the estimate given by Spot) and sb_to_e is
the scaling between surface brightness and electrons. Note that for
channels 3 and 4, the Spot estimates of the background surface
brightness should be scaled up by ~1.5, to account for extra background
signal due to internal scattering in the arrays.

The term sb_to_e = 30.331, 26.729, 6.384, 18.802 electrons / (MJy/sr * s)
for channels 1-4, respectively.

To be conservative, assume that the background subtraction used in either
aperture photometry or PSF fitting contributes noise from the background
summed over the number of noise pixels for the source. Then, ap_fac =
npix and 

 sigma_bg^2 = e_bg  / nback = background_sb * FRAMETIME * sb_to_e / nback

where nback is the number of pixels used in a reasonably-sized background
annulus.

A reasonable estimate of the noise (in electron units) in a single
exposure is then 

   sigma^2 = sigma_rn^2 + FRAMETIME * (fd * fd_to_e + 
           npix * background_sb * sb_to_e + 
           npix * background_sb * sb_to_e / nback) .

To be conservative, scale the noise upward by 30%-50% to account for
systematics, etc.

The signal-to-noise ratio for one exposure is simply

fd * fd_to_e * FRAMETIME / sigma.