Data for Cookbook 245, 378-wide External Binning, Low-Dark-Current Mode

lines (relatively low frequency). If a pattern is present, it may vary randomly from line to line, or it may be fixed from one frame to the next. Patterns may originate inside the camera or may be interference from a nearby computer or monitor. Note whether the pattern is the same from one frame to the next (and therefore generated in the CCD electronics), or whether it changes from frame to frame (and therefore comes from a source that is not synchronized with the CCD and its associated electronics). Power supplies that exhibit ripple or switching transients are prime suspects. If the pattern repeats exactly, it can be averaged and subtracted out. If the pattern is different in each bias frame, it is important to identify and eliminate the cause.

• Burst noise will most likely appear as an isolated band across one or more bias frames. It may be caused by electrical equipment such as a motor operating in the house, in the observing area, or elsewhere on the same electrical circuit. The source of burst noise should be found, if possible.

• Popcorn noise (also called 1//noise) usually appears as a semirandom variation in the pixel values from line to line. It may originate in the camera electronics. Since popcorn noise does not repeat from one bias frame to the next, it cannot be averaged and subtracted.

• Slopes are systematic variations across the bias frame from top to bottom or left to right. They may result from the bias injection part of the CCD electronics. Slopes are typically small in amplitude and highly repeatable, so they are subtracted during image calibration.

• Hot regions are caused by the CCD's on-chip amplifier, as is the case in the TC211 chip, or by charge leakage during readout. Hot regions that repeat precisely are routinely subtracted out during image calibration.

• Dark pixels are defects in the chip manufacturing process. The total number should be less than the total of "point defects" specified by the quality grade of the CCD.

Scrutinize the median bias in the same way you did each of the individual frames. Features of the bias frame that remain the same from frame to frame will be present in the median bias, but features that change should be absent. This is one of the best ways to determine whether a noise source is coherent (related to the camera's frame-taking process) or incoherent and therefore probably external to the camera system.

The outstanding characteristic of a good bias frame is blandness. The pixel value is nearly constant across the frame, and the random variation of the pixel values should itself be very bland. Repeatable noise features less than one ADU in amplitude are seldom cause for concern since they are subtracted during routine image calibration. However, random noise and burst noise sources should be identified and removed.

Charge skimming occurs when an abnormal photosite retains a few hundred electrons during readout. The characteristic signature of charge skimming is a dark pixel at the location of the charge "trap" and a dark tail of pixels following the trap.

Because the signal levels are so low in this check, it is necessary to shoot multiple frames and combine them to reduce the noise level. Furthermore, because of the hot pixels present even in short-integration flat fields, it is necessary to make and subtract a master dark frame to remove the hot pixels.

To check for charge skimming, make a median of the nine skim dark frames and save it as SKIMDMED.FTS. Make another median of the skim frames and save this frame as SKIMLMED.FTS. Add 1,000 to SKIMLMED and then subtract SKIMDMED from it. Examine this image carefully for dark pixels and their associated tails.

You can obtain some feel for the number of electrons caught in the trap by measuring how much lower the dark pixel is than the surrounding pixels, or by

Mean Pixel Value (ADUs)

Figure 8.6 The CCD transfer curve graphs the relationship between mean pixel value of a uniformly illuminated region and the variance from the mean pixel value in the region. The inverse of the slope of the line is the conversion coefficient between ADUs and electrons, about 55 electrons per ADU in this example.

Mean Pixel Value (ADUs)

Figure 8.6 The CCD transfer curve graphs the relationship between mean pixel value of a uniformly illuminated region and the variance from the mean pixel value in the region. The inverse of the slope of the line is the conversion coefficient between ADUs and electrons, about 55 electrons per ADU in this example.

summing the excess pixel values in the tail. To convert the measured pixel value from ADUs to electrons, multiply by g, the conversion factor.

8.3.4.3 Plotting the Transfer Curve

The transfer curve and the test for linearity use the same set of measurements from the flat-field data, but the two tests use it in different ways.

The transfer curve is a graph of the variance, opv , as a function of the average pixel value, 5pv , for a set of flat-field images. The variance at different levels of pixel value is found by subtracting the pairs of flat-field frames. From the transfer curve, you can determine the conversion factor of your CCD in electrons per pixel value and the readout noise of the CCD. The theory behind this determination is really quite elegant.

Deducing the transfer factor relies on the counting statistics of events such as the generation of photoelectrons having a Poissonian distribution; that is, the standard deviation in the number of photoelectrons, Gpe, from each sampling interval equals the square root of the average number of photoelectrons, Se. For example, if on the average, the light falling on a given photosite generates 10,000 photoelectrons per integration period, then the standard deviation will be

J\0, 000 photoelectrons, and the number of photoelectrons expected to arrive during a given sampling period will be 10, 000 ± VlO, 000.

In a healthy CCD, the signal consists of a combination of the number of electrons plus the random noise from Poisson counting statistics plus the readout noise of the CCD. Since standard deviations add in quadrature, we expect the standard deviation of the combined signal, oe, in units of equivalent electrons at the detection node of the CCD, to be:

where Gpe is the standard deviation in the number of photoelectrons and Gron is the readout noise expressed in electrons.

For each of the flat-field images, we can directly measure the average pixel value, 5pv , in ADUs. The mean number of photoelectrons at the detection node, Se, equals Spv times g, the conversion factor for the number of electrons per pixel value unit:

For the photoelectron component of the electrons reaching the charge detection node, the standard deviation, ope, equals , so the variance, Gpe, equals Se. Thus, by substitution:

Another quantity we can measure is the standard deviation of pixel values in a flat-field frame, cpv . Even though flat-fields are never perfectly "flat," by subtracting one flat-field image from another made under identical conditions, we can eliminate almost all nonuniformities, and thus determine the quadratic sum of the noise. The noise measured in electrons is scaled by the conversion factor into pixel value units:

Note that the measured variance using the difference-of-two-flats technique is twice the variance of a single flat-field frame. You must divide each measured variance by 2 to obtain the variance of a single flat frame.

Recall that noise adds in quadrature, so that the total noise measured is the square root of the sum of the standard deviations of the photoelectron noise and the readout noise:

Substituting the equalities developed earlier into the noise equation:

Multiplying through by 1 /g , we obtain:

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