Next Stargazing April 22nd
Next Stargazing April 22nd

# Optimum Exposures

Capturing faint detail in deep-sky astronomical images is all about overcoming noise. Noise comes from a variety of sources:
• Heat (dark current)
• Light pollution/sky background
• Cosmic rays
• Pixel defects
Optimal images will reduce these noise sources as much as possible. CCDs are designed to generate low dark current, and cameras are cooled to remove as much thermal noise as possible. Dark frames that capture the remaining dark noise are subtracted to eliminate it from the final images. Cameras are designed to have low readout noise. And observers travel to dark sites to help eliminate light pollution.   Combining Exposures to Reduce Noise The usual method for reducing noise in astronomical images is to stack multiple exposures. This works due to the random nature of noise. 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) 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. Above: Single exposure. Faint star circled has SNR of 5. Above: 10 stacked exposures. Faint star now has SNR of 15, an increase of 3 times. Noise is noticeably reduced. Since signal stays constant, averaging two exposures together doubles the signal. But due to the random nature of noise, averaging two exposures increases the noise by only √2, about 1.4 times.   Signal to Noise Ratio An exact equation describing signal to noise ratio is not easily determined due to the large number of factors and inexact nature of those parameters. However, for most purposes, it is sufficient to describe SNR as a function of the following factors (each described more thoroughly below):
• Exposure Time
• Number of Exposures
• Object Flux
• Sky Background Flux
• Binning
• Resolution
• Focal Ratio
• Dark Current
We can consider each of these factors in turn to see what their effect is and the quantity or significance of their effects. Also, this will lead us to a method of determining optimal exposure times and other methods for reducing noise and increasing SNR. Note: The above factors determine SNR for random noise sources. There still exist non-random sources which will be considered later. Exposure Time This is probably the most important factor. The most obvious way to increase SNR is simply to increase exposure time. 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. 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 exposures this way leads to a SNR increase of √N. 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. It can also be seen that there is a point of diminishing returns, with an increase in exposures N yielding only a slight increase in SNR. Going from 2 to 10 exposures gives a 225% increase in SNR, while increasing to 20 exposures gains only another 140%. 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. Object Flux The flux is simply the rate at which light from the target reaches the CCD chip. Flux is often measured in photons per second, but in the case of a CCD camera, which is converting photons to electrons, a more practical value is electrons per second, abbreviated e-/sec. The brighter the object, the greater the flux. With astronomical subjects the flux is typically low, often not much higher than the sky flux described below. 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. Note that the 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-. Using image processing software, the sky background can be easily measured, although there is a slight calculation necessary to convert to the background sky flux in e-/sec. The above image has an average background sky pixel value of 950. Most software packages add a pedestal value of 100 to the pixel values in order to prevent negative numbers. This means the true value is 850. This is called the Background ADU Count. To convert to a value in electrons, this number must be multiplied by the camera gain. In this example, an ST-10XME camera was used, which has a gain of 1.3e-/ADU. Therefore the actual sky background value is 1105e-. The exposure was 600 seconds, implying a sky background flux at this observing location of 1.8e-/sec, indicating a dark site. Again this is a function of the focal ratio, in this case f/7. This image was 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. (Note that the focal ratio of the second picture was f/5.4, which means the equalized sky background flux is 3.1e-/sec, indicating the sky was 72% brighter from this location than where the top picture was taken.)
 Measuring Sky Background In MaxIm DL, use the Information command to measure the background ADU count. Set the Aperture Radius to a moderate value such as 6 or 8 to average the values out over a larger area. Be sure to move the cursor to a few different background spots and take an average. Estimating to the nearest 50 or 100 should be more than adequate, given all the variables that will affect an exposure. Be sure to avoid faint areas of nebulosity and keep away from the corners of the image where there might be vignetting that will lower the value.
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. 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. 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, the importance of which will be seen in the next section. Dark Current Dark noise is generated within the camera by heat sources. This is the reason CCD cameras are cooled; colder cameras have less dark current. Most cameras, even after cooling, still have residual dark noise which is then removed using dark frames. Assuming a sufficient number of dark frames have been taken and combined to make a master dark frame, dark noise can essentially be ignored for SNR calculations. Readout Noise This is an important determinant when it comes to selecting optimal subframe exposures. Take a look at a simplified equation for SNR: This equation assumes that dark current is not a significant factor (because it has been removed) and that the sky flux is large relative to the object flux (as is the case for faint deep-sky objects). The remaining factors are:
• N = Number of Exposures
• Eobj = Object Flux
• Esky = Background Sky Flux
• t = Exposure Time
Readout noise is generated by the CCD when the data from the chip is transferred to the computer. This is a measurable quantity and is quoted by CCD manufacturers in the spec sheets for their products. The SBIG ST-10XME used in the above examples has a typical readout noise of 7e-. In terms of choosing an ideal exposure time, there is little we can do about the object flux, and the background flux will be more or less constant for a given location. What we can do, however, is minimize the contribution of the CCD readout noise to the overall image noise. How to do this is discussed in the next section, but it can be seen from the equation above that making the sky background flux, Esky, large compared to the readout noise, Ron, will minimize the contribution from readout noise.   Sky Limited Exposures Note: Equations are given in the discussion below but a JavaScript calculator is provided at the end of this section for easy calculations. A very important idea is that of sky limit. An exposure in which the primary limiting factor is the background sky flux is called sky limited. In such an exposure the sky background flux is the primary factor in determining SNR. Astroimager Stan Moore states, "sky limit is the zone where the sky noise overpowers the readout noise". Just where this point occurs is somewhat arbitrary but there is a generally accepted guideline:
• Stan Moore recommends keeping the readout noise contribution to just 5% of the total noise, the point where the sky noise is 3 times greater than the readout noise.
• John Smith's excellent article on the subject also adopts this 5% readout noise rule. The equations which follow are from that article.