This memo will cover the following topics:

                    A.   Filter wheel calibration
                    B.   Tot-to-PE conversion using the occupancy method
                    C.   Comparison of occupancy pes and adc pes
                    D.   PMT saturation
                    E.    A proposed explanation for the difference between occupancy and adc pes

Filter wheel calibration

A few weeks ago, a memo by Andy described a problem that we had with the filter wheel, namely that the slope of the optical density (OD) vs. filter angle did not seem to be the same for different parts of the filter wheel:  The darker part of the filter wheel seemed to have a slope of ~0.02 OD/degree, while the lighter part seemed to have a slope of ~0.013 OD/degree.  Because of this break in the slope, it is not possible to calculate the number of PEs at high light levels (where the occupancy method is not useful anymore)  by merely extrapolating from the plot of average PEs vs. filter angle at low light levels as calculated from the occupancy method.  Indeed, that I tried to do this in the first place, before the problem was discovered, was what led to the big difference between the adc pes and occupancy pes when I compared them.  To get over this problem, I calibrated the filter wheel as follows:

I disconnected from the optical switch the optical fiber which was collecting light coming from the filter wheel and which was supplying light to the optical switch.  I shined the (laser) light coming from this fiber onto a 2-inch PMT.  Triggering on laser light detected by a photodiode which is located before the filter wheel, the areas of the PMT pulses at a particular filter angle were measured using the digital scope.  The digital scope made a histogram of the areas, and I measured the peak of the histogram.  This was done for different filter angles.  When the light level going into the PMT was too high, the light level was attenuated by using makeshift filters.  Though the absolute value of the PEs detected by the PMT was not known, it was still possible to get a calibration curve by making several sets of measurements which overlapped in filter angle.  The final calibration curve was obtained by minimizing the difference in log(area) where the different data sets overlapped in filter angle.  The resulting filter wheel calibration curve is a plot of log(area) -- which is proportional to optical density -- vs. filter angle.  The slope of log(area) vs. filter angle for 20 to 110 degrees (the dark part of the filter wheel) measured through this method agrees with the slope given by the occupancy method to within 5%.

Tot-to-PE conversion using the occupancy method

To calibrate a grid, I fit  a y-intercept to the plot of log(mu) vs. filter angle using a slope of ~0.0205 OD/degree for filter angles up to 110 degrees, where mu is the average number of PEs as determined using the occupancy method.  Once the PEs for the low light levels are known, the PEs for higher light levels and larger filter angles are calculated by using the calibration curve shown above.  The sample plot of occupancy PEs vs. adc counts shows that for this grid, saturation starts at ~75 PEs (lower plot).  The upper plot shows when the high tot starts being used -- in this case, ~10 PEs.  This sample plot was produced using shower data.

Comparison of occupancy PEs and adc PEs

The next step in determining if I got it right this time is, of course, to compare the occupancy PEs with the adc PEs.
I have made a profile plot of occupancy PEs and adc PEs as a function of  low tot (upper plot).  All grids which have been calibrated contribute to this profile plot.  Larger values of low tot were not included because I did not want to worry about the effects of low tot saturation.  As you can see, the occupancy PEs are consistently larger than the adc PEs.  The lower plot gives occupancy PEs divided by adc PEs, as calculated from the upper plot.  The ratio as a function of low tot is flat, and the horizontal line drawn in the plot corresponds to a value of ~1.17 occupancy PEs per adc PE.

There's more.  I also have a profile plot of occupancy PEs vs. adc PEs where the PEs were calculated using high tot.  The slope of the line drawn is 1.17, consistent with the ratio obtained using the low tot.

PMT saturation

From the profile plot of occupancy PEs vs. adc PEs given above, it can be seen that on the average, PMT saturation sets in at ~70 occupancy PEs (that is,  if you believe the plot :) ).  However, I have seen a few grids which begin to saturate at ~40 occupancy PEs.

A proposed explanation for the difference between occupancy and adc PEs

Now what?  What's causing the ~20% difference between the occupancy and adc PEs?  I've been thinking about it for days and I still don't know why.  Just kidding, I just wanted to give Todd a heart attack!

I guess you are all familiar with the adc method of finding the PEs.  Using shower data, an adc distribution is produced for each grid.  The adc method makes a fit to the single-PE bump and defines this fit as 1.000... PE.  However, the occupancy method will never define this bump as exactly one PE.  The occupancy method will give a value slightly larger than one to correspond to this "single-PE" bump.  What this value is depends on the effective occupancy that a PMT sees from the shower data.  And if the two methods don't agree at one PE, I don't expect them to agree at any other PE.

One thing that Todd pointed out is that the adc distributions were obtained using a 100-tube trigger, which probably means that a tube is hit ~50 percent of the time.  It is possible that this 100-tube trigger requirement can cause the effective occupancy seen by a tube to be large enough to shift the adc single-PE peak to slightly larger values.  In other words, at a 100-tube trigger, the two-PE contamination of the single-PE bump might be significant enough to shift the fit to the adc peak slightly to the high side.

To check if the number of tubes in the trigger requirement affects the adc distribution, I compared adc distributions
produced with a 100-tube trigger to adc distributions produced with a 25-tube trigger.  Check out the sample adc distributions for grid 1, grid 2, and grid 150 (a baffled tube -- no I don't mean that the tube is confused, just that it's wearing a baffle :) ).  The upper plot for each grid was taken using a 100-tube trigger, while the lower plot was taken using a 25-tube trigger (Feb. 15, 1998, by Morgan and Andy).  The two data sets were taken on the same day, about ~5 hours apart.  (Note the shift in the adc distributions by ~5 counts.  I have no idea what caused this.)  From the plots, it can be seen that for  a 25-tube trigger, the single-PE bump is not as well-defined as that for a 100-tube trigger (with about the same number of events for the two distributions).  It can also be seen that the slope of the 25-tube trigger distribution on the high side of the single-PE peak is steeper than the corresponding slope for the 100-tube trigger distribution, indicating that there might be  a significant contribution from two-PEs in the 100-tube single-PE bump.

At Todd's suggestion, I also investigated the effect of two-PE contamination on the single-PE peak of the laser calibration data.    Look at the plots of the adc distributions for grid 1, for filter positions 20 to 40 degrees, filter positions 50 to 70 degrees, and filter positions 80 to 100 degrees.  The plots give the occupancy of each distribution and the corresponding mean of the gaussian fitted to the peak of each distribution.  Just so you can orient yourselves, for an occupancy of 0.85 (histogram 1090), there is ~24% probability for one-PE events, and ~27% probability for two-PE events.  I have plotted adc counts minus pedestal (357.5 for this grid) as a function of occupancy.  There are similar plots for grid 116, grid 6, and grid 134.  The plots show that starting at an occupancy of about 20 to 30 percent, the fits to the peak shift as a function of occupancy, indicating that two-PE contamination of the single-PE peak causes shifts in the fits.

(The adc distributions for grid 116 are also available:  filter 10 to 30, filter 40 to 60, filter 70 to 90, filter 100).

That's it.  That's how I propose to explain the difference between the occupancy and adc pes.  I think it all boils down to how the two methods define the adc "single-PE" bump.  The adc method calls it 1.0000... PE; the occupancy method calls it, on the average, 1.17 PEs because of the occupancy of the shower data at a 100-tube trigger.  What do you think?

I hope you're not as baffled as I am -- and I don't mean that I'm wearing a baffle!