Midterm summary of FMS calibration and pieces we need to move on

    It is clear that following the position of LED peak in order to translate the set of gains calibrated for a specific dataset to the rest has been suffering from the problem

of pulser instability. Therefore, time-dependent intensity corrections of the LED pulsers are needed. The current strategy is to break up the PbGl cells array into

groups that are fed by the same LED channel respectively, and extract the common behavior of LED events for those cells within the group on a run-by-run level.  The

common mode of LED variations will be described in terms of the mean ADC for the run in question relative to that of the baserun, then averaged over the cells in

the current group. Detailed explanation can be found here.

    The key point for this common mode extraction to be effective is including the right candidates (stable cells) into the group. And there have been some efforts to

identify cells that have unstable gains, i.e. large ADC variations in LED events within a run, see Chris. It's likely that this uncertainty will result in a fluctuating mean

ADC on a run-by-run level which is not correlated with the LED intensity. Therefore these cells should be eliminated in trying to extract the common mode. This part

of the work has been taken care of by Chris.

    Further refinement of the list of good cells to be included into the group is currently split into the following options:

    1. Only include the ones that show an apparent threshold behavior in their ADC spectrum ( of physics events ). The reason of doing this is to get a handle on the

actual change of gain for those candidate cells over a certain time period, so that we can remove the part that comes from a gain shift when relating the change of

observed mean ADC to the change of pulser intensity. It seems we have made a loop here in going from the actual gain variation to the actual gain variation. The

merit of this option will be explained later. For now the seeming loop is:

                                             actual gain variation (from adc spectrum) -(1)-> remove it from mean ADC variation -(2)-> change of pulser intensity (common mode)

                                                         II                                                                                                                                                               I

                                             actual gain variation (event-by-event)      <-(4)- remove it from mean ADC variation<-------(3)----------------------

    It should be noted that the actual gain variation at the end is estimated on a different level compared to its counterpart in the beginning. From the start we need

several sets of data (e.g. each set contains all runs from a single day, and we take 5 different days to get 5 sets) in order to have a valid slope or number over

threshold for each set, and compare with each other to get the relative gain shift.  Then the same gain shift will be substracted out from all the runs within the set.

This step completes the raw correction to mean ADC variation by eliminating set-by-set (or day-by-day) gain drift.

    Then we proceed to extract the common mode of mean ADC variations on a run-by-run level, and remove it from the ADC variations of individual cells. What left is

an event-by-event LED variation that expands all of our data sets, and we consider it as the change of actual gain that can be corrected on an event-by-event basis.

    If we going to following this option, we need to build a list of cells which passed the pre-selection and have a gain that is high enough to show a threshold behavior

within the set of data as mentioned above.

    2. In the 2nd option, we note that without a raw correction accounting for the gain drift (step (1), as shown above in the loop), only the low frequency component

of gain drift that is common to all the cells within the group might enter the common mode of mean ADC variation and be misidentified as the change of pulser intensity

(see examples below). If we adopt the assumption that the actual gain shift usually occurs at higher frequency, and tends to get averaged out in a long time period,

we can negelect this common mode of low frequency drift. By doing this, we will expand the collection of good cells in our list compared to the above scheme, therefore

improve the precision of estimating intensity variations, if the aforementioned misidentification turns out to be small. And the differential mode of slow gain drift might

as well be account for, if there is any. In this way, slopes of adc spectra will instead become an auxiliary tool to examine the long term stability of the tube gain.

    Here is an simple example to explain the difference of these two options. Say we have two sets of data come from different days: SetA (10 runs from day80),

SetB ( 10 runs from day95). For each data set we have measured the slopes of adc distributions for a group of cells, and they indicate that the gains of all of

these cells have dropped down by 10%. And we observe an overall 30% decrease of run-by-run mean ADC of LED events in going from SetA to SetB. Since

ADC*gain = Energy, option 1 will give the correct 40% decrease of LED intensity whereas option 2 will only give 30% which is from the common mode of ADC

variation. In this case the common mode of gain drift from day80 to day95 (10%) has been ignored by option 2. On other hand, if 50% of the cells have gains

that went up by 10% and the other 50% have gains went down by 10%, after extract common mode of mean ADC variations option 1 will give 30% decrease

of LED intensity ( [ (30+10)+(30-10) ] /2 ), and option 2 will give the same result. In real practice, the common mode of gain drift over such a long time period

is expected to be small and the distinction between these two method in this aspect will be less than the extreme case above.

    If we are going to follow option 2, it will be necessary to examine the size of slow drift in the tube gain, especially to see if cells from the same group have

any common parttern.

    In either case, we should analyze the change of actual gain in a long time period by looking at adc spectra. Then according to the outcome we can make the

chioce between a simpler scheme (option 2, ignoring the common mode of slow gain drift) or a slightly complicated one (option 1, correct for gain drift). And of

course in option1 we can apply the same correction to the cells that didnt have a threshold behavior but otherwise stable, so that we can include them into the group to

estimate the pulser intensity variations.