Ideal exposure time using SNR

Signal-to-Noise Ratio Explanation

1.     Final images is to reduce these noise sources as much as possible. 

2.     CCDs are designed to generate low dark current, and cameras are cooled to remove as much thermal noise as possible.  

3.     Dark frames tries to eliminate dark noise it from the final images. 

4.     Cameras are designed to have low readout noise, dark sites to help eliminate light pollution and reduce this noise.

Combining Exposures to Reduce Noise

·        The usual method for reducing noise in astronomical images is to stack multiple exposures.

·        In an image, there is signal and there is noise. 

·        While the signal (the light from the object being imaged) stays the same from image

·        to image, the noise changes. 

·        The signal-to-noise ratio (SNR) in an image measures how much signal there is relative to the noise levels. 

·        Decreasing the amount of noise in an image increases the signal-to-noise ratio and results in a better picture.

For Example

An image with a 300 signal to a 150 noise =2:1

Noise decreases to 100 = 3:1

SNR  200

SNR 300 (noise reduced)

·        Since signal stays constant, does not change from image to image

Taking two image and  averaging two exposures together doubles the signal.  But noise, averaging two exposures increases the noise by only √2, about 1.4 times.

 

Two images = 2x2 = 4 signal

Two images = square root 4 =1.4 noise

All these are part of why it  increase or reduce  SNR

·        Exposure Time

·        Number of Exposures

·        Object Flux

·        Sky Background Flux

·        Binning

·        Resolution

·        Focal Ratio

·        Dark Current

·        Readout Noise

Exposure Time

This is probably the most important factor. The most obvious way to increase ratio (SNR-increasing the ratio increases the quality of the image) is simply to increase exposure time. 

If we increase the exposure time (say from 300 sec to 600 sec)

For most deep-sky images, doubling the exposure time increases the SNR by √2 = 1.4 times. 

Sky glow, from light pollution sources, prevents us from taking indefinitely long exposures so SNR must be increased through other means. 

Sky glow limitations also imply that there may be an optimal exposure time for a given imaging system and location, which we will see is true.

Number of Exposures

We saw earlier how stacking multiple exposures increased SNR (better image)

Perhaps stacking multiple exposures taken at the optimal exposure time would be preferable to a single longer exposure. 

We will see that this is true, and for a variety of reasons. 

There are myriad ways to combine image files, and they are discussed in more detail below.  The basic method is to average exposures, taking the mean value of each common pixel to produce a result with less noise. 

Combining N (36) exposures this way leads to a Sign / Noise Ratio,  increase of √N (36) = 6.

As seen in the examples above, averaging 2 exposures yields a √2 = 1.4 increase in SNR,

 and averaging 10 exposures gives an increase of √10 = 3.16. 

averaging 20 = 4.47

From 1.4 to 3.16 (3,16/1,4x100) is 225% increase in SNR

From 1.4 to 4,47 =319% , its only 94% increase, while it goes from 10 to 20 (100% increase). It can also be seen that there is a point of diminishing returns,

As will be seen below there are other reasons to use a larger number of subframes;  for example, it might be preferable to take ten 5-minute exposures rather than five 10-minute exposures.

Sky Background Flux

This is the flux of the sky glow, determined primarily by light pollution factors. 

The sky (whether lit by city lights or the moon or natural airglow) produces photons that are captured by the CCD and turned into the background of the image.  The image becomes contaminated with noise.

Note that this background in an image is not perfectly black but has some value. 

This value is a function of the sky background flux and the exposure time.  For example, the sky from a dark site might have a flux of 2e-/sec (this is also a function of focal ratio since it is measured in terms of what the CCD counts rather than what the sky itself is producing). 

 In a 5 minute exposure, the background will reach a value of 300sec x 2e-/sec = 600e-.

10 mins 600 x2e= 1200e-

·        An image taken from a suburban location and has a background ADU count of 2500.

·        Converting using the equations above gives a value of 3120e-. (Exposure time was again 600 seconds and the flux is 5.2e-/sec),

·        indicating that the sky is much brighter from this location. Compare to the 600e- of the dark site.

Binning

·        Binning affects the SNR by effectively increasing the sensitivity of the CCD chip. 

The effect is similar to making the focal ratio faster. 

·        Binning a CCD 2x2 combines each 2x2 group of pixels into one "super pixel" which can gather 4 times as much light as a single smaller pixel during a given exposure. 

Thus the system becomes 4 times faster when binned 2x2, equivalent to a 2-stop reduction in focal ratio.  However, as described below, resolution determines SNR as well, and since binning decreases resolution, it can also decrease SNR, and if the SNR decreases the noise will increase.

Resolution

Resolution is a major factor in determining fine detail SNR such as that of stars. 

·        Increased resolution gives increased SNR. 

·        However, sampling is an important factor as well. 

·        (See the section on Nyquist Theorem for more details on sampling.) 

·        Undersampled images (such as those taken with short focal length scopes and/or binned CCD chips) will have worse SNR than properly sampled images.

Focal Ratio

Focal ratio is the primary determinant of imaging speed. 

·        Deep-sky imagers all know the importance of having a fast scope for reducing exposure times. If the exposure time is reduced and we increase the number of subframes, this will reduce noise.

·        Focal ratio also affects resolution, assuming a constant aperture (in other words, using a focal reducer on a given telescope) and thus affects SNR in the same way. 

More importantly, focal ratio determines exposure time necessary to achieve a given sky background flux.



Determining Optimal Exposure Times

So, in the end, what is the best exposure time to use for subframes? 

1.     Shorter exposures allow better combination methods for greater noise removal. 

Shorter exposures are also an advantage in that, if something goes wrong during the exposure (wind, tracking errors, etc.), less data is lost. 

2.     The trade off between taking more short exposures versus fewer long exposures in terms of SNR loss is very slight, as John Smith recommends the following exposure recommendation:

This calculator allows you to determine the ideal exposure time for subframes that will be stacked. It requires that you take a test exposure using your CCD imaging setup and measure the background sky value. Please refer to the CCD Imaging Theory page on Optimum Exposures for the details behind these calculations.

How to Use the Ideal Exposure Calculator

• 1. Take a test exposure using your standard CCD setup.  Factors that will influence the calculated ideal exposure time include light pollution, focal ratio, filters, CCD camera, binning, and object elevation (altitude).  It is recommended that you image an object near zenith, unfiltered, and on a moonless night.  If you change cameras, telescopes, or observing locations, you will need to adjust accordingly, or take test exposures for each setup.
• 2. Take an equivalent length dark frame. Do this even if you have a low-noise camera that may not normally use darks. This will remove the bias level from the image.
• 3. Calibrate your light frame by subtracting the dark frame.
• 4. Measure the background ADU count (background value) using, for example, the Information tool in MaxIm DL, or similar tool in the program of your choice.  Be sure to avoid measuring the value of any faint nebulosity.  Take a few measurements around the image and take an average to be sure of getting an accurate value.
• 5. Select your CCD camera from the pull-down menu below.
• 6. Enter your test exposure time in minutes.
• 7. Enter the measured sky background value from the test exposure.
• 8. Select one of the standard choices for the percent contribution of readout noise.  5% is the usual figure used but a higher noise tolerance will give a shorter exposure time.
• 9. Press the Calculate button to determine the ideal exposure time.

 https://starizona.com/tutorial/ideal-exposure-calculator/