6800 7000 7200 7400
Fig. 4.1. Histogram of a typical flat field image. Note the fairly Gaussian shape of the histrogram and the slight tail extending to lower values. For this R-band image, the filter and dewar window were extremely dusty leading to numerous out of focus "doughnuts" (see Figure 4.4), each producing lower than average data values.
is F = 6950 ADU and its width (assuming it is perfectly Gaussian (Massey & Jacoby, 1992)) will be given by
We have made the assumption in this formulation that the Poisson noise of the flat field photons themselves is much greater than the read noise. This is not unreasonable at all given the low values of read noise in present day CCDs.
Let us now look at how bias frames and flat field images can be used to determine the important CCD properties of read noise and gain. Using two bias frames and two equal flat field images, designated 1 and 2, we can proceed as follows. Determine the mean pixel value within each image.1 We will call the mean values of the two bias frames BB 1 and BB2 and likewise F and F2 will be the corresponding values for the two flats. Next, create two difference images (B1 - B2 and F1 - F2) and measure the standard deviation
1 Be careful here not to use edge rows or columns, which might have very large or small values due to CCD readout properties such as amplifier turn on/off (which can cause spikes). Also, do not include overscan regions in the determination of the mean values.
V F ■ Gain ^ADU = Gain of these image differences: (^B,— b? and f?- Having done that, the gain of your CCD can be determined from the following:
and the read noise can be obtained from
4.4 Signal-to-noise ratio
Finally we come to one of the most important sections in this book, the calculation of the signal-to-noise (S/N) ratio for observations made with a CCD.
Almost every article written that contains data obtained with a CCD and essentially every observatory user manual about CCDs contains some version of an equation used for calculation of the S/N of a measurement. S/N values quoted in research papers, for example, do indeed give the reader a feel for the level of goodness of the observation (i.e., a S/N of 100 is probably good while a S/N of 3 is not), but rarely do the authors discuss how they performed such a calculation.
The equation for the S/N of a measurement made with a CCD is given by
unofficially named the "CCD Equation" (Mortara & Fowler, 1981). Various formulations of this equation have been produced (e.g., Newberry (1991) and Gullixson (1992)), all of which yield the same answers of course, if used properly. The "signal" term in the above equation, N*, is the total number of photons1 (signal) collected from the object of interest. N* may be from one pixel (if determining the S/N of a single pixel as sometimes is done for a background measurement), or N* may be from several pixels, such as all of those contained within a stellar profile (if determining the S/N for the
1 Throughout this book, we have and will continue to use the terms photons and electrons interchangeably when considering the charge collected by a CCD. In optical observations, every photon that is collected within a pixel produces a photoelectron; thus they are indeed equivalent. When talking about observations, it seems logical to talk about star or sky photons, but for dark current or read noise discussions, the number of electrons measured seems more useful.
measurement of a star), or N* may even be from say a rectangular area of X by Y pixels (if determining the S/N in a portion of the continuum of a spectrum).
The "noise" terms in the above equation are the square roots of N*, plus npix (the number of pixels under consideration for the S/N calculation) times the contributions from NS (the total number of photons per pixel from the background or sky), ND (the total number of dark current electrons per pixel), and NR (the total number of electrons per pixel resulting from the read noise.1
For those interested in more details of each of these noise terms, how they are derived, and why each appears in the CCD Equation, see Merline & Howell (1995). In our short treatise, we will remark on some of the highlights of that paper and present an improved version of the CCD Equation. However, let's first make sense out of the equation just presented.
For sources of noise that behave under the auspices of Poisson statistics (which includes photon noise from the source itself), we know that for a signal level of N, the associated 1 sigma error (1o-) is given by VN. The above equation for the S/N of a given CCD measurement of a source can thus be seen to be simply the signal (N*) divided by the summation of a number of Poisson noise terms. The npix term is used to apply each noise term on a per pixel basis to all of the pixels involved in the S/N measurement and the NR term is squared since this noise source behaves as shot noise, rather than being Poisson-like (Mortara & Fowler, 1981). We can also see from the above equation that if the total noise for a given measurement
^N* + npix(NS + Nd + NR) is dominated by the first noise term, N* (i.e., the noise contribution from the source itself), then the CCD Equation becomes
N =&= , yielding the expected result for a measurement of a single Poisson behaved value.
This last result is useful as a method of defining what is meant by a "bright" source and a "faint" source. As a working definition, we will use the term bright source to mean a case for which the S/N errors are dominated by the source itself (i.e., S/N ~ vN), and we will take a faint source to be the case in which the other error terms are of equal or greater significance compared with N*, and therefore the complete error equation (i.e., the CCD Equation) is needed.
1 Note that this noise source is not a Poisson noise source but a shot noise; therefore it enters into the noise calculation as the value itself, not the square root of the value as Poisson noise sources do.
The CCD Equation above provides the formulation for a S/N calculation given typical conditions and a well-behaved CCD. For some CCD observations, particularly those that have high background levels, faint sources of interest, poor spatial sampling, or large gain values, a more complete version of the error analysis is required. We can write the complete CCD Equation (Merline & Howell, 1995) as
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