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Signal to Noise Ratio and the Subexposure Duration
Copyright Steve Cannistra
Methods to determine subexposure durations for CCD imaging have received a great deal of attention.  John Smith has developed a subexposure calculator based upon the need to reduce the camera's read noise component by overwhelming it with sky noise.  I have approached this problem from a different angle, asking what subexposure duration is needed to maximize the signal to noise ratio of the final cumulative exposure.   Even though the final equations are different, the results are similar to John's, but conceptually I find this approach more intuitive.  The math is straightforward and graphs are provided to illustrate the points (I suggest right-clicking on the graph links to open them in a separate window).  At the end, I provide a link to an Excel Spreadsheet that you may use to calculate subexposure duration at your own site.

The signal to noise ratio per pixel, for a single sub, is expressed as follows (ignoring the contribution from dark noise):                                                            
1. SNR (for single sub) = (Obj)*tsub / sqrt[(Sky+Obj)*tsub + R2]

where SNR = signal to noise ratio per pixel; Obj = object flux in electrons/pixel/minute; Sky = sky flux in electrons/pixel/minute; tsub = subexposure time in minutes; R = read noise in e RMS.

2. By combining N exposures, the noise per pixel becomes:
 Noise (Tot) = sqrt[N*{(Sky+Obj)*tsub + R2}] /  N                                                         

Therefore, by combining N exposures, the SNR (Tot) per pixel becomes:
SNR (Tot) = N*(Obj)*tsub / sqrt[N*{(Sky+Obj)*tsub + R2}]

where SNR(Tot) refers to SNR per pixel in the total exposure (assumes an average, median, or sigma reject combine for the signal component); N = number of subexposures.

3. Simplifying, this becomes SNR (Tot) = sqrt[N]*(Obj)*tsub / sqrt[(Sky+Obj)*tsub + R2]

4. Let K = total exposure time, which equals N*tsub.  So N = K / tsub
 
5. Substituting K / tsub for N in equation 3:  SNR (Tot) = sqrt[K / tsub]*(Obj)*tsub / sqrt[(Sky+Obj)*tsub + R2]

6. SNR (Tot) = sqrt[K*tsub]*(Obj) / sqrt[(Sky+Obj)*tsub + R2]

7. This yields the following representative graph of SNR versus subexposure duration for a fixed total exposure K (4 hours in this example).  The values are chosen to represent a faint Ha emitting object at my imaging site, with the Maxcam CM10 (R= 8.24 e RMS).  Note that the curve rises rapidly over a short span of subexposure times, and then flattens out.  In this example, the gains in SNR for subexposure times beyond 10 minutes are minimal.  As indicated below, it can be shown that the curve starts to plateau when the photon noise contribution dominates the read noise.

8.  The curve predicted from equation 6 has an asymptote that can be easily calculated as subexposure time approaches infinity.  Specifically, as subexposure time approaches infinity, the contribution of R becomes negligible, and this factor can be ignored in determining the asymptote.  The equation becomes:  SNR asymptote = sqrt[K]*(Obj) / sqrt[(Sky+Obj)].  Thus, in photon noise limited conditions, the SNR is minimally affected by the read noise or subexposure duration (i.e., assuming that the subexposure duration is sufficiently long).

9.  Here is a graph that shows the asymptote for the previous example, along with a new metric that I refer to as "F".  F is the fraction of the maximum SNR (at asymptote) achieved at a given subexposure time.  In this example, a value of 0.9 is chosen, along with its corresponding subexposure time (x-axis).  The choice of 0.9 is arbitrary- as long as the F value is a sizable fraction of the maximum achievable SNR (e.g., F greater than or equal to 0.9, for instance), the subexposure time should be fine.

10.  It is easy to calculate the subexposure time for a desired value of F.  F = SNR(Tot) / SNR (asymptote), which is derived further in equation 11 below.

11.  F = (sqrt[K*tsub]*(Obj) / sqrt[(Sky+Obj)*tsub + R2]) / (sqrt[K]*(Obj) / sqrt[(Sky+Obj)])

12.  Solving for tsub:  tsub =  F2*R2 / [(Sky+Obj)*(1-F2)].  For Sky >> Obj, the equation becomes tsub = F2*R2 / [Sky*(1-F2)], where F is the desired fraction of maximum SNR you wish to achieve, R is the read noise (e RMS), and Sky is the sky flux (e/minute).  This equation has similarities to the one derived by John, but it is not the same.

13.  Note that the subexposure time is related to the square of the read noise, which makes having a low read noise camera ideal.  Here is a representative graph showing this relationship.  Even a small increase in read noise can significantly increase subexposure time.

14.  As mentioned in point #7 above, the SNR curve rises quickly, and there is little to be gained beyond a certain subexposure duration.  However, the shape of the curve (specifically how fast it flattens out) is dependent upon the sky noise level, as shown in the following graph, which demonstrates the relationship between SNR and subexposure time as a function of sky flux.  Notice that the curve rapidly approaches the asymptote in a relatively light polluted site (sky flux through an Ha 6nm filter of 43 e/min), and but takes longer to do so at a dark site (sky flux 2 e/min, used for illustrative purposes only).  Because of this effect, the dark site requires a longer subexposure to achieve an F value of 0.9 (in this example).

15.  Feel free to download my Excel spreadsheet that will allow you to calculate your subexposure time based upon this analysis.  I developed this spreadsheet, and then Neil Fleming added some nice frills, including a camera drop down button and a blue background (thanks Neil!).  A few things to note when using the spreadsheet:

 A.  Measuring the Sky ADU is the most important user-defined parameter.  On a moonless night, choose a region of sky near the zenith that is free of large nebulae or galaxies that might obscure background sky.  Take an image for 3 minutes, and then calibrate it (dark frame and flats).   Use this calibrated image to measure the Sky ADU value.  For instance, in Maxim you could use the annulus tool to measure the average ADU in an area of the image free of nebulae or stars.  Get a few readings from different parts of the background and average them together to get a representative value.

B.  The manufacturer-determined camera gain and read noise values are automatically chosen for you in this spreadsheet.  This will get you very close, but the spreadsheet will also allow you to insert other values if the measured gain and read noise for your camera differs.  John Smith describes the procedure for determining your camera's gain and read noise on his webpage, and AIP4WIN software will also do this for you.

C.  This spreadsheet will work for the conditions of your set up at any given time.  It takes into account your binning value or choice of filter, in the sense that the measured Sky ADU value will be self-correcting for this.  For instance, if you bin 2x2, your measured Sky ADU will be higher, and your calculated subexposure duration will be lower.  Likewise, if you are using an Ha filter, your measured Sky ADU will be lower, and your calculated subexposure duration will be higher.  So don't worry about any of this, except to note that your calculated subexposure duration needs to be determined for a given set of conditions (binning, filter, f ratio, camera, imaging site, etc.).  If these change, you need to remeasure the Sky ADU and obtain a more relevant subexposure duration for your new conditions.

D.  Many software packages, including Maxim, will add a value of 100 ADU to the calibrated image (i.e., the image that you will use to measure the Sky ADU).  This means that a value of 100 must be subtracted from the measured ADU in order to obtain an accurate determination of Sky Flux.  Depending upon the software that you choose in the drop down menu, my spreadsheet will automatically subtract this out, so you don't have to worry about it. 


Please note that you use this at your own risk, and I take no responsibility for problems that this may cause with your computer, etc.  If you find this analysis interesting, please feel free to refer others to this link.

Steve

 Copyright Steve Cannistra
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