In the bright target limit, Poisson noise sets the SNR and
where S is the number of detected photons, and Robject is given by the above Equations 6.2 through 6.4, and t is the exposure time.
In the background limited case (e.g. read noise, dark current, or sky noise limited) the SNR is a function not only of the expected number of detected photons S from the source but also of the average effective background count rate B in each pixel, the point spread function 
, and the weights used to average the signal in the pixels affected by the source. It is easy to show that the signal-to-noise ratio for optimal weights (which are proportional to the point spread function) is given by:
where sharpness is effectively the reciprocal of the number of pixels contributing background noise. The summation is tabulated for a few representative cases in Table 6.5. To estimate the signal-to-noise, multiply the signal-to-noise obtained, assuming all the flux is in one pixel, by the square root of the value in the table.
We note that PSF fitting is equivalent to convolving the image with the PSF, and then measuring the peak counts for stellar objects. Also, the location of the star on the pixel grid will be impossible to know in advance of the observation (i.e. pixel center vs. pixel corner in Table 6.5). In general, the lower "pixel corner" values should be used, so as to insure adequate SNR.
The average effective background counts per exposure and per pixel can be expanded to include various sources:
where terms include the read out noise of the CCD (readnoise), the dark current (Pdark), sky background count rate (Psky), and the count rate of any diffuse background light from astrophysical sources (Pbackground). Herein we will use "P" to represent count rates per pixel, and "R" to represent the total counts for an object. The exposure time is represented by t.
For example, Table 2.2 lists the faintest V magnitude star, V=28.19, measurable with a signal-to-noise ratio of 3 in a 3000s integration in F569W in the Wide Field Cameras. The calculation to check this goes as follows. The efficiency of the filter is 0.02343 from Table 6.1. The sky background in each pixel is 23.3+5=28.3, assuming an ecliptic latitude of 90° from Table 6.3, and the pixel area correction for the WFC given in that section. The total sky background collected per pixel in 3000 seconds is given by Equation 6.1 as 84.1 electrons. Note that the AB
color correction required for the sky in the wavelength range of the filter is 0.0 from Table 6.2. From Table 4.4, the read noise for WF3 is 5.2 electrons. From Table 4.2, the median dark current at -88 °C is 0.0045. Therefore the total dark current (on which there will be shot noise) is only 13.5 electrons. The equivalent background per pixel is then given as B=84.1+5.22+13.5=124.5. The total number of detected electrons from a star with V=28.19 is S=93 electrons, again using Equation 6.1. (We note that AB is approximately zero at this wavelength, so the spectral class is unimportant.) The expected peak count is 28 detected electrons using Table 5.4 (peak near pixel center), which is much less than B, requiring the use of Equation 6.5 for the background limited case. The sharpness for the WF camera in the best case, when the star is centered on a pixel, is given in Table 6.5 as 0.128. Then Equation 6.5 above gives the signal-to-noise as 3.0:
If, instead, the peak count rate comes out much greater than the background, the observation is photon noise limited, and the signal-to-noise should be computed as the square root of the signal S in electrons.
In principle, one should also include contributions in the signal-to-noise for flat fielding uncertainties, noise in the bias and dark calibration files, and quantization noise. Flat fielding errors will be of order 1%, and will limit SNR in the large-signal limit. Noise in the bias and dark calibration files will be unimportant in most pixels, although these could become important if many (>10) non-dithered frames of the same field are combined.
Quantization noise can be estimated as 
(i.e., 
in the 7 e- DN-1 channel, and 
in the 14 e- DN-1 channel). In nearly all situations it can be ignored. In the weak signal case, the quantization noise is effectively included in the read noise values given throughout this Handbook; in the strong signal case it is very small compared to the Poisson noise and can be ignored.
A generalized equation for estimating point source signal-to-noise ratio per exposure is given below (Equation 6.6). It is exact in both the bright and faint object limits, and is a reasonable approximation to the intermediate case. Pbackground represents any generalized source of diffuse background light (e.g. galaxy on which target is superposed). Table 6.6 gives rough values for some of the parameters, along with references for more accurate values.
Note that in this formulation, sharpness-1 is the equivalent number of pixels the weighted signal is integrated over. In the event that multiple exposures are taken (e.g. to remove cosmic rays), the signal-to-noise ratio for the final averaged image is approximately given by:
where N is the number of images averaged.
| Parameter |
Description |
Units |
Approx. Value |
Better Value |
|---|---|---|---|---|
| Robject |
object count rate |
e- s-1 |
|
Equation 6.1, 6.2, or 6.3 |
| Pdark |
dark count rate |
e- s-1 pixel-1 |
0.004 |
Table 4.2; Eqn 4.1 on page 90 |
| Psky |
sky count rate |
e- s-1 pixel-1 |
||
| Pbackground |
count rate from background light (if any) |
e- s-1 pixel-1 |
|
|
| read noise |
|
e- |
ATD-GAIN=7 use 5.31 ATD-GAIN=15 use 7.5 |
|
| sharpness |
|
|
WFC use 0.11 PC1 use 0.06 |
|
| t |
exposure time |
s |
|
|
| 1ATD-GAIN defaults to 7 unless otherwise specified on Phase II proposal. |
|
Space Telescope Science Institute http://www.stsci.edu Voice: (410) 338-1082 help@stsci.edu |