System Throughput and SNR / Exposure Time Estimation

Below we give further examples of SNR calculations. The Appendix also gives tables of SNR values for a wide range of representative cases.

First we estimate the count rate for our target. Consulting Equation 6.2, Table 6.1, and Table 6.2 we have:

in units of e^{-} s^{-1}. Next we fill out Equation 6.6. To keep things simple we just use values from Table 6.6, and get the sky count rate from Table 6.4. There is no background light (i.e. no superposed galaxy), so P_{background}=0. The exposure time t=600 for each exposure of the CR-SPLIT:

The SNR for the total 1200s exposure, i.e. both halves of the CR-SPLIT, would simply be:

At these high SNR levels, it is likely that flat fielding would limit the photometric accuracy, rather than the noise. If we have a look at the terms in the SNR equation, we can see that the Poisson noise dominates; the term containing the sharpness and background noise sources is unimportant.

Just for fun, let us see what happens if we keep everything the same, but give the target V=25. Now we have R_{object}=0.66 e^{-} s^{-1}, and:

We see that now the term with the background noise (in particular, the read noise) limits the SNR. For the full 1200s exposure the SNRtotal=17.6.

Apparently using aperture photometry with a 0.5" radius aperture reduces the SNR by a factor ~4 as compared to PSF fitting, for this background limited case.

program, which is available on the WFPC2 WWW pages at:

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http://www.stsci.edu/ftp/instrument_news/WFPC2/wfpc2_top.html
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To use this program, access the above address with NCSA Mosaic, Netscape, or a similar program. Once in the WFPC2 area, select the "Software Tools" page, and then the "ETC" page. For the first example above, choose the "Point Source" form and complete it as shown in Figure 6.2 for the 600s sub-exposure. Then clicks the "calculate" button and after a few seconds the result is displayed (Figure 6.3). The answer, SNR=198, is comparable to that obtained by the manual calculation above for the 600s sub-exposure (SNR=197). Alternatively, one can input the total exposure time (1200s), and then use the result farther down the output page for "No. Sub-Exposures = 2" (see Figure 6.4), thereby obtaining SNR=277 for the total 1200s CR-SPLIT exposure.

We begin by computing the total count rate for the target. Using Table 6.2 we see that this target will have V=25.31. From Table 6.1 we obtain the filter efficiency and mean wavelength. Interpolating by mean wavelength in Table 6.2 we obtain AB_{nu}=-0.83 for the B0 star. Using Equation 6.2 we have:

in units of e^{-} s^{-1}. Next we consider the background light from the superposed galaxy. We set sigma(V)=22 mag arcsecond^{-2} in Equation 6.11, and AB_{nu}=3.63 for a gE galaxy at lambda=3000Å (filter F300W) from Table 6.2. Hence the count rate per pixel due to the background light is:

For the sky background, we note that Table 6.4 has no entry for F300W, so that the sky must be unimportant. If we wanted to calculate it anyway, as a check, we would use Table 6.3 for the sky brightness, Table 6.2 for the sky's AB_{nu}, and again Equation 6.11. We will assume the target is near the ecliptic pole.

For the sharpness function we will use "pixel corner" values (least optimistic choice) from Table 6.5. Using read noise and dark current from Table 6.6, and Equation 6.6 for point source SNR:

for this single exposure. The SNR for multiple 40 min. exposures would be simply 16.8(N^{1/2}), where N is the number of exposures.

and then click on "calculate." Figure 6.6 shows some of the results.

If the scale of features in the target is larger than one pixel, the signal-to-noise can sometimes be improved by smoothing the observed image or - if read noise is a significant contributor - by reading the image out in **AREA mode** (see section 2.8, "CCD Orientation and Readout", on page 31).

The signal-to-noise ratio calculation for point-like or extended emission-line sources is similar to that for continuum sources. However, the details of the calculation are different, because of the units used for the line flux, and because the flux is in a narrow line. The integrated filter efficiency is not relevant for the signal calculation; what matters is the total system throughput

If the source is extended, the expected signal per arcsecond must be multiplied by the effective pixel area: 0.0099 arcsec^{2} for the WF, 0.0021 for the PC. For a line flux of, say, F = 10^{-15} erg s^{-1} cm^{-2} arcsec^{-2}, this corresponds to 15 electrons in 1000 seconds for a WFC pixel. The noise is now dominated by the background, and the single-pixel signal-to-noise ratio is 15/(33 + 15)^{1/2} ~ 2.1.

First we must estimate the flux per square arcsecond. Using the nebula diameter, the average brightness is *I*_{nu}* *= 2.0x10^{-14} erg s^{-1} cm^{-2} arcsec^{-2}. From the plots in section A.2, "Passbands including the System Response", on page 211, we see that QT=0.053. Using Equation 6.10 for the target count rate per pixel:

Next we estimate the SNR for each 900s sub-exposure using Equation 6.14 and Table 6.9. For this narrow filter the sky background can be ignored. We presume there is no background light from astrophysical sources:

Hence SNR=3.1 per pixel for each 900s sub-exposure. The SNR per pixel for the total 1800s is

The SNR for the entire nebula is this SNR per pixel times the square root of the number of pixels in the image, or ~420. In actuality, uncertainties in the photometric calibration and flat fields, would limit the SNR to ~100.

We have selected "[OIII] 5007" on the emission line menu, and have left the redshift (z) set to zero. The PC and F502N filter are selected. Note we have entered the exposure time as 1800s. Scrolling down through the output page we find a table of SNR for various CR-SPLITings of the exposure (See Figure 6.8). "No. Sub-Exposures = 2" gives the answer we want, SNR=4.5 per pixel.

Since the redshift is significant, we cannot observe with the F656N filter. Instead we will use the Linear Ramp Filter (LRF). The observed wavelength will be 8007Å. From Table 3.7 on page 48 we see that this will be observed using the FR868N filter on CCD WF3. Combining the LRF transmission from Figure 3.2 and the "WFPC2 + OTA System Throughput" from Figure 2.4 we estimate QT=0.054. We compute the count rate using Equation 6.4:

To estimate the SNR we use Equation 6.6, which assumes that PSF fitting will be used to analyze the image. Since the filter is narrow, we will ignore the sky emission. We use Table 6.6 for the WFC sharpness and also the read noise.

which is for an un-split 2400s exposure. The Poisson noise and background noises contribute nearly equally. For three such exposures over three orbits

The result is SNR=13.1 for the un-split 2400s exposure (Figure 6.10), which is comparable to the manual calculation of SNR=14.

**6.7.1**- Point Sources- Simple Star, Manual Calculation, PSF Fitting
- Simple Star, Manual Calculation, Aperture Photometry
- Simple Star, SNR Tables, PSF Fitting
- Simple Star, On-Line Calculator, PSF Fitting
- Star Superposed on Galaxy, Manual Calculation
- Star Superposed on Galaxy, On-Line Calculator
**6.7.2**- Extended Sources**6.7.3**- Emission Line Sources- Extended Line Emission Source, Manual Calculation
- Extended Line Emission Source, On-Line Calculator
- Line Emission Point Source w/ LRF, Manual Calculation
- In this example we consider an unresolved source of Ha emission in a galaxy at redshift z=0.22 with flux F=1.5x10
^{-16}erg s^{-1}cm^{-2}. We want the SNR for a 2400s exposure without CR-SPLITing. - which is for an un-split 2400s exposure. The Poisson noise and background noises contribute nearly equally. For three such exposures over three orbits
- Line Emission Point Source w/ LRF, On-Line Calculator