In this section, we will consider what is measured by the mean counts
in a set of exposures of different exposure times when the CCD is
illuminated by a stable light source such as a 
light.
If F is the number of signal electrons collected per second by the CCD
(which equals the incident flux times the quantum efficiency in the range of
interest), g is the CCD conversion factor in electrons/ADU, and 
is
the actual exposure time (ie. not necessarily the measured exposure
time), then the true mean count level, 
, is:
However, no CCD is strictly linear and nonlinearities may be modelled
by a polynomial dependence of g on 
:
There may also be an error in the bias subtraction such that the observed
mean counts, 
, are in error by an amount b, such that:
Finally, the observed exposure time, 
, is the true exposure
time, 
, less the shutter delay s (which is assumed to be constant
for all exposure times):
Therefore, the mean counts measured by the experiment are:
Where F, 
, s and b are unknowns. For now, we
shall assume that F is a constant (ie. the light source is constant)
and that b is small enough that we can disregard it.
Ideally, we should like to fit Equation 8
to real data and solve for the unknowns. In practise, this is a
difficult task, if not impossible. For example, if we assume that the
CCD exhibits only first-order nonlinearities (
), then
Equation 8 becomes:
Therefore a fit of a quadratic to a plot of the observed mean counts
in an image versus the reported integration time (the ``linearity'' curve,
by its normal definition) will provide
measurements of 3 coefficients. However, the coefficients of 
in equation 9 include 4 unknowns (
, 
,
s and F) and so a complete solution is not possible. Nevertheless,
it is possible to extract a measurement of s, since at 
,
.
can be estimated as follows - if we fit the equation
to the observed linearity curve, then, after some tortuous algebra, the following relation is found to hold:
Since 
is measured by the transfer curve (to first order only,
since for completeness, we should apply a similar analysis of
nonlinearities in the transfer curve), we can therefore estimate the
value of 
and thus the magnitude of a first-order nonlinearity
(and, if desired, F).
This measurement of 
rests on a number of untested
assumptions - that s and F are constant, b is small and there
are no higher nonlinearities present in the response of the CCD. It
is true that the residuals of the fit might provide some clues as to
the presence of higher-order nonlinearities and possible variations in
the light source (or F) during the test, but even if these are
small, then the precision of the measurement of 
is still
difficult to estimate.
In practise, we are most interested in determining the shutter error as precisely as possible, particularly at short exposure times (where the relative error is large and because the observer normally takes calibration exposures at short exposure times), an estimate of the total amplitude of any nonlinearities in the typical dynamic range used for observations, and some idea of the stability of the light source over the test.
We have converged on a practical strategy which exposes these three factors in a clear, qualitative fashion and, in the right circumstances, produces reliable quantitative measures for the shutter delay and nonlinearity of the CCD system.
The count rate may be derived by dividing equation 8
by the exposure time. Ideally, we should like to divide
by the true exposure time, 
, but in practise we must divide by
the reported exposure time, 
. If the shutter error, s, is
non-zero, then the measured count rate will tend to anomolously large
values with shorter exposure times (or small values if s < 0, which
is unusual). Therefore, in order to obtain an estimate of s, we
compute the count rates, C, with the formula (ignoring b):
where 
is a test value of the shutter delay. When
, the count rate is given by:
which, in the linear case (n = 0), is a constant (
), or,
in the case with first-order nonlinearity (n = 1), a straight line
(with respect to 
). As a further refinement, divide by the
mean count rate, 
, over the test to yield:
Thus a plot of this count rate computed with observed exposure times corrected for the shutter delay versus the observed exposure time (or the mean counts in the image) provides a curve whose amplitude of deviation from unity is a measure of the total nonlinearities present in the system over the dynamic range explored by the test, expressed as a fraction of the linear term (and may therefore described as a ``fractional'' nonlinearity).
For this to be valid, 
must be as close to s as
possible. A suitable estimate can be obtained by assuming that any
nonlinearities of second order or higher are negligible. In that
case, the count rate curve should be a straight line. Therefore, the
optimum value of 
may be selected by the straight line fit
to 
versus 
which has the smallest standard deviation of
fit. Note that this computation implicitly gives greater weight to
those images of shorter exposure time as the count rates for these are
most affected by errors in the reported exposure time.
Typically, any variations in the brightness of the light source used
to illuminate the CCD occur on a timescale comparable to the time
taken to collect the data, or longer. Such variations may also be
comparable to the amplitudes of the nonlinearities in the system. For
example, a 
light shows a variation in generated flux of 
% per degree centigrade change in temperature. As a result, we
must require that the temperature of the housing of the 
light
be stable to within 
C in order to confirm the
presence of 
nonlinearities. This kind of temperature
stability is very hard to achieve in an open instrument mounted at
Cass. or even in a Coudé room. Therefore, until we can obtain a
light source with the required stability, we are forced to express the
nonlinearities present in the system as an upper limit based on what
we know of the variability of the light source.
If the images are collected in a simple sequence of increasing (or decreasing) exposure time, any variation in illumination will normally (unless the temperature changes quickly) take a form very similar in form to a smooth nonlinearity in the response of the CCD. Instead, if the data are collected in two sequences, say one of increasing exposure time and the other of decreasing exposure times (with different, interleaved exposure times in each case), then a smooth change in illumination level will appear as two distinct bands of data. An example is shown in figure 10 where the light level has decreased over the course of the test.
If, on the other hand, the two sequences follow the same curve, then we may be reasonably sure that the illumination level did not change during the test (unless it followed the exact same change in reverse through the second sequence as through the first, which is unlikely). In these circumstances, we can obtain a precise measure of the form of any nonlinearities present and use the resulting relation to correct science data for them.
One further point remains to be addressed - how precise does the bias
subtraction need to be to avoid contaminating the results? If we assume
a linear response from the CCD, 
becomes:
Therefore, we may require:
or
Typically, 
sec and 
ADU/pixel/sec and
thus we require b << 50, which is reasonable under almost any
circumstances.
