We have seen that the relationship between signal variance and signal is not linear. This may be worrysome for us, but it is not really the non-linearity we originally feared. We would be in great trouble if signal level was not a linear function of incident intensity. In the body of data generated to show the variance-vs-level nonlinearity we also have the material to test whether level rises proportionally with exposure time. Of course, we do not really trust the shutter – it sticks and is variable at times – and we have no way to vary intensity of our lamps, but we start by inspecting the signal vs exposure time relationship in the V filter:From top left to bottom: Box 1: same as shown before – variance vs signal strength. Box 2: residuals after linear and 2nd order (red) fits have been subtracted. Box 3: signal level against nominal exposure time with 2nd order polynomial fit shown in red. Box 4: residuals from Box 3 when linear and 2nd order fits are subtracted. Box 4: same as shown before – nominal uncertainty on the parabola fit seen in Box 1.Inspecting the above plot in Box 3 we see that the signal level rises remarkably linearly as a function of nominal exposure time. Now, it could be that the exposure time is all wrong (we only have the time we ask for, not the time we actually get) but it seems unlikely that the linear relationship should arise by chance. The 2nd order coefficient shown in Box 3 is very small compared to the one in Box 1 – the signal-exposure time relationship is more linear than is the variance-signal strength relationship. It thus seems that electrons are not lost in whatever process causes the flattening out of the variance at high signal levels and it has been suggested in the literature that electrons migrate to neighbouring less-filled pixels. Thereby total signal is conserved but bright areas tend to blur into the darker neighbouring areas.If this is what happens we also have to think about the consequences of this – we must think of using DS/BS analysis methods that avoid using small bright regions for reference. This was at one point the core in the EFM method – the model of the observed image was constrained to pass through a region on the BS selected by the user – and often we chose the brightest parts near crater Tycho. This is now no longer used in the EFM – we now use a flux-conservation constraint to scale the model image. This would seem to be rather safe in the current situation.Eventually the DS/BS ratios need to be converted to albedo and this step is influenced by the choice of reference areas – something BBSO also must worry about in their reductions, but have not discussed.
Ahmad has tested the linearity of the CCD in Lund, and Henriette presents work on tests based on data taken at MLO. Since even small deviations from linearity in the CCD may be important for us we are revisiting the issue, and present here some results based on the variance-vs-level method which is nicely explained in Ahmad’s report on the Andor Camera.In the upper panel is plotted the variance against the level for all filters. Plots for the individual filters are available but the scatter is not much less. The solid line is the fitted least squares linear regression. As red is plotted the best-fitting parabola, and the dashed line is the diagonal. Beside the panel are written the coefficients of the best fitting 2nd-order function. In the middle panel is the difference between the data and the linear fit (black symbols), and between data and 2nd-order fit (red symbols).Lower panel shows the magnitude of the formally estimated uncertainty of the fitted parabola, in percent of the signal level. Dashed line shows the fiducial 0.1% error level we strive for. Variance and level was calculated from a central area of 190×190 pixels in the pairs of images. Images were bias subtracted using the ‘superbias’ we have, without scaling to surrounding dark field levels.The nonlinearity is clearly quite large since at 25000 counts a 0.1% difference is just 25 counts – small compared to the differences seen above. At 50.000 the observed difference of 1.500 corresponds to 3%. At count levels above 55.000 the relationship rapidly breaks down and saturation occurs. For observations below 55.000 we can probably correct all observed images we already have. In the future it may we best to observe only up to 52.000.Acceptable linearity may only be available up to count levels of 10- or 15.000. We have to correct our observations – but at least it can be done at any time in the future. We should now and then gather and archive a huge set of lamp images as were used above.
This image of the Moon was obtained on JD 2455849.
Notice the strong saturation at the lower left corner – on the sky! Other parts of the Moon are safely exposed in the 10000-30000 counts range.
Here is a supposed ‘dark’ frame showing something similar:
Most of the dark frame is at the bias level of 397 – the corner shoots into 65000!
This was during our ‘everything looks like a bias frame we think something is getting stuck’-period.
Both frames above are available in the BADEXAMPLES/ directory as FITS files.