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   Progress on the calibration of the EEMC using a MIP analysis for run 9 has been documented previously in blogs by Zach Nault and Alice Bridgeman.  This note will comment on problems observed in obtaining reasonable values for the gains and the most probable values (MPV) from Landau and Gaussian fits to the ADC spectra for the EEMC towers.  The calibration is currently in the first pass of obtaining gains for all elements of the EEMC; relative gains have been made for the SMDs, first and second pre-shower detectors, and the post-shower detector.  The next step in the calibration is to use these elements to identify a mip passing through a given tower and fit its ADC spectrum with a combination of a Gaussian and Landau curve.  This is shown in plots given in the 1 July 2010 posting in the znault blog.  At this time, it is believed that the only cut on the tower ADC spectra consist of hits in one or two adjacent, isolated SMD strips in both U and V planes. 

 

Problem

   Problems were identified with the tower fits, as noted in the 7 July 2010 posting in the znault blog.  These problems can be briefly summarized:

 

(1) the fit to a Landau-shaped ADC spectrum included a portion of the pedestal causing a low gain value, and

(2) the error in the gain was calculated to have an extremely large value.

 

   After some investigation of the code, fitTower.C, the first problem may be traced to a search for the peak in the ADC spectrum with the largest number of counts for a given ADC channel.  When the pedestal peak is larger than the peak in the Landau region, an incorrect fitting range, lower than the correct range, for the spectrum is used.  This explains the low gain values obtained for about 90 towers.  The second problem is probably due to the selection of the upper limit of the fit range.  The code chooses a lower limit as 10 channels below the ADC channel with the largest number of counts (limited to positive ADC channels) and an upper limit as twice the ADC channel with the largest number of counts.  If the ADC channel with the largest number of counts is close to zero, as in the case of picking up the pedestal, the upper limit of the fit range will be small as well.  This selection then makes the number of ADC channels in the fitting range too small, which limits the number of degrees of freedom in the fit.  Consequently, the error will be large. 

 

Solution

   An attempt to solve these problems associated with the fit range was made.  The fit range was changed so that the lower limit was set to ADC channel 2, which is just above the pedestal peak, and the upper limit was set to ADC channel 35, which includes a number of channels in the tail above the ADC channel with the largest number of counts (larger channel number than the pedestal).  This choice removes the possibility of including the pedestal in the fit and the selection also allows for a fixed number of ADC channels within the fit range.  These changes in the fit range should then fix the two problems above. 

   

   A check of this method was made by making individual fits to the EEMC tower ADC spectra.  The fit was made manually in ROOT to a function that added a Gaussian function with a Landau function.  This is similar to what was done in fitTower.C, except no coefficients to these functions were added.  The fits to three separate towers in sector 9, subsector E (eta bins 1, 4, and 12) are shown below:

 

 

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Note that in 09TE01 and 09TE12, the number of counts in the pedestal peak at ADC channel 0 is larger than those from the fit (MPV) peak.  Also, the fit range shown here is from ADC channel 2 to channel 35.  The MPV for each of the fits, given by the mean value in the statistics, visually looks like a good fit and will give a good value.  The Gaussian+Landau function fits the data well, as shown by the good values of chi²/df.  Fits to the data using only a Gaussian function and only a Landau function were tried, but gave worse results, in both locating the MPV and the chi²/df than the sum of the two functions. 

 

   The table below lists the data from the first pass of fits to the tower ADC spectra, showing the MPV and its uncertainty.  The table also provides data from the fit of the Gaussian+Landau function, showing the MPV and its uncertainty, the chi², the number of degrees of freedom, and calculated chi²/df for the to the ADC spectra of towers in sector 9, subsector E, for the 12 eta bins:

v\:* {behavior:url(#default#VML);}
o\:* {behavior:url(#default#VML);}
w\:* {behavior:url(#default#VML);}
.shape {behavior:url(#default#VML);}

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First Pass Fit   Shifted Range Gauss+Landau Fit                           Tower   MPV Error in   MPV Error in   C2 / d.f.   C2/df Name   (ADC ch) MPV   (ADC ch) MPV                                       09TE01   0.5 80.1   6.60 0.09   23.3 / 23   1.01 09TE02   0.5 99.2   7.28 0.09   36.6 / 24   1.53 09TE03   0.5 32.6   7.94 0.16   33.4 / 28   1.19 09TE04   8.3 0.08   8.26 0.08   33.4 / 29   1.15 09TE05   9.3 0.09   9.09 0.01   31.7 / 29   1.09 09TE06   10.6 0.12   10.46 0.11   39.9 / 29   1.38 09TE07   11.0 0.13   10.78 0.12   46.6 / 30   1.55 09TE08   0.5 140.6   11.58 0.14   37.7 / 28   1.35 09TE09   0.5 173.5   12.40 0.13   22.2 / 29   0.77 09TE10   0.5 168.8   13.42 0.16   39.7 / 29   1.37 09TE11   0.5 163.4   14.98 0.15   17.3 / 27   0.64 09TE12   0.5 198.0   15.01 0.16   34.3 / 29   1.18

It can be seen from this table, that the MPV values listed as bad in the first pass have been corrected by shifting the fit range to exclude the pedestal.  The values of the chi²/df also indicate that the fits using the Gaussian+Landau function are very good.  The systematic increase in the values of MPV as a function of eta bin (1-12) shows the geometrical effect of an increase in detector material that a particle traverses.  

 

   An alternate fit method using the convolution of the Gaussian and Landau functions was reported in the 20 July 2010 blog by aliceb.  Fits to the tower ADC spectra were made using this method and the plots showing the fit can be found at the above entry.  There are several plots that show that the fits need improvement and do not give a good value for the MPV.  Also, the number of degrees of freedom can be seen to vary more than the shifted-range method, along with the chi²/df.  On a visual inspection of the fits to the ADC spectra by the convolution method, the tails in the Landau distribution are not included as completely as the other method.  The results of this convolution method are tabulated below:

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   A plot directly comparing the MPV for the three methods is presented below for the same sector and subsector:

 

 

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The error bars are included in the above plot for the convolution and shifted-range methods, but not for the first pass, since these would dominate the graph.  The convolution and shifted-range methods give reasonable MPV values that fix the problem of a pedestal included in the fit.  A difference of one or two channels per eta bin for the MPV can be observed between the two methods. 

 

   Another comparison between the convolution and shifted-range methods can be seen in the plot of the chi²/df for the same towers in the previous plot. 

 

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In the plot, the shifted-range method shows good values for the chi²/df over the entire eta range of the subsector.  The large excursions in the chi²/df from the convolution method can be seen visually as poor fits to the ADC spectra.  Specifically, in tower 09TE06, the tail of the Landau distribution is not fit well; in 09TE09, the fit is too broad; in 09TE11, the fit is both broad and poor in the tail; and in 09TE12, the fit is shifted to higher ADC channels. 

 

Conclusion

   The problems listed above with bad gains observed in the tower ADC spectra can be solved by changing the fitting routine.  Two methods were used to fix these problems: one used a convolution of Gaussian and Landau functions to fit the ADC spectra and the other used a shifted fit region to exclude the pedestal from the fit.  A comparison of the two methods can be seen in plots of the MPV and in the chi²/df for a given subsector of towers.  There is a difference of about 1-2 ADC channels in the MPVs found for each method and the chi²/df shows that the shifted-range method has a more consistent fit over the entire eta range.  Other considerations in implementing a new method include the ease of incorporating a fix to the analysis code and the time needed to run it with the patch over the entire data sample.

 

 

NOTE:  A change was made in the convolution method fits to the data since this analysis was performed.  The plots and results have also been changed, and a further comparison of methods similar to that presented above has not been done at this time.