Milagrito Angular Reconstruction

Gus Sinnis

September 3, 1997

In this memo, I explore the utility of the standard CYGNUS angle fitter for Milagrito data. Unlike CYGNUS, the timing distributions in Milagrito have exceptionally long late tails at small pulse heights. Thus, one would expect a fitter that is based on a Guassian model for the timing residuals to perform poorly. The point of this memo is to determine a baseline: how well does the Chi squared method work for our data and is there a subset of the data for which this technique yields acceptable results?

Sampling Correction
Weights as a Function of PEs
Fit Results (Deleo, etc.)
Fit Results (Deleo, etc.): RELAXED
Comparison to Previous Fitter
Conclusions

The Data Set:

For the following analysis I have used the slewing corrections as derived from the laser calibration data by Isabel Leonor, the timing pedestals (also from the laser data) as derived by Peter Nemethy, Roman and Lazar Fleysher, and the TOT to PE conversions derived from the ADC by Kelin Wang.

The problem is then to measure the sampling and curvature corrections from the data itself. To do this one requires an angle fitter. However, without the proper sampling corrections the results may be biased. To minimize such problems the following cuts where made on the events:

  1. Zenith angle < 20 degrees
  2. Core < 20 meters from the center of the pond.
  3. NFIT > 20 counters
  4. PES > 2 for a PMT to be included in the fit.

A slightly modified version of the online fitter was used in this preliminary pass.

Technique:

After the angle fit is performed the residuals from all of the hit PMTs are calculated and stored in a 2-D histogram: Tchi(PE,Rcore). For PE bin the average Rcore is found and for each Rcore interval the average PE value within the PE bin is found. These true averages are used in deriving the sampling and curvature corrections. The table below gives the results (in ns) of this preliminary pass at the data. As "usual" a negative time corresponds to a late arrival time.

PEs Rcore Median Peak Average RMS Nentries
0.267095 7.118501 -22.5 -8.5 -16.23145 14.83136 160732
0.264923 15.15495 -26.5 -4.5 -13.98170 15.38400 437384
0.296054 24.46268 -25.5 -7.5 -12.79270 14.94166 338626
0.345033 33.91194 -24.5 -6.5 -12.57292 14.91664 125898
0.371727 42.40936 -22 -2.5 -12.03710 14.93752 17087
0.723976 6.980463 -17 -6.5 -15.36188 13.74021 126717
0.716533 15.17315 -19.5 -5 -13.47943 14.57037 314504
0.712671 24.70510 -19 -5.5 -12.23524 14.1936 291509
0.711906 34.19790 -19.5 -5.5 -12.21947 14.43905 156349
0.720642 42.53039 -19.5 -7.5 -11.71681 14.57728 29292
1.4653 6.817902 -11 -4 -12.88333 11.96277 154906
1.454009 15.04027 -11 -3.5 -12.12933 12.96005 316758
1.450059 24.61347 -10.5 -4 -10.96694 12.54672 258771
1.444222 34.16461 -10.5 -4.5 -10.82554 12.72309 126540
1.428921 42.48268 -11 -2 -10.55157 13.15150 22211
2.477205 6.64449 -7.5 -4 -9.740786 10.25593 110235
2.469505 14.96655 -7 -2.5 -9.551287 11.39532 200931
2.467353 24.60181 -6.5 -3 -8.77859 11.17141 157974
2.465624 34.14288 -6.5 -2.5 -8.516298 11.30792 75221
2.464132 42.46207 -6.5 -3 -8.107101 11.46713 12871
3.940044 6.494463 -5 -2.5 -6.850049 8.632158 159359
3.920908 14.93237 -4.5 -1.5 -6.738891 9.638292 259999
3.91376 24.59291 -4 -2 -6.331322 9.770061 200826
3.910582 34.12962 -4 -1.5 -6.165928 9.950109 94508
3.911752 42.43478 -4 -2.5 -6.100883 10.56923 15627
5.973475 6.30899 -2.5 -1 -3.93609 6.679157 124143
5.947416 14.87571 -2 -0.5 -3.855116 7.593038 176658
5.935386 24.57009 -2 -0.5 -3.75843 7.967135 130968
5.929262 34.09703 -2.5 -0.5 -3.865756 8.462518 59373
5.930631 42.42386 -2 0.5 -3.737195 8.848928 9606
8.473679 6.263705 -2 -1.5 -2.909206 5.635937 37376
8.428466 14.86537 -2 -1 -2.794891 6.27739 51263
8.405581 24.56638 -1.5 -0.5 -2.755658 6.887471 38090
8.398643 34.09077 -1.5 -0.5 -2.789349 7.332652 17688
8.417135 42.49469 -1.5 -1 -2.571507 7.58237 2741
12.38283 5.928576 -1 -0.5 -1.336146 4.401685 54820
12.36335 14.85055 -0.5 0 -1.091778 4.620873 68159
12.34546 24.57109 -0.5 0 -0.898836 5.115211 48387
12.33415 34.10803 -0.5 0 -0.796106 5.590997 21597
12.29493 42.35923 -0.5 0 -0.901933 6.082977 3518
17.37747 5.691648 -0.5 0 -0.628129 3.999388 40186
17.35511 14.82281 0 0 -0.50893 4.179274 48322
17.34096 24.48245 0 0.5 -0.414766 4.779402 33103
17.33351 34.10130 0 0 -0.373912 5.167998 14712
17.33278 42.34863 0 1 -0.386445 5.567493 2228
24.65351 5.387527 0 0 -0.089354 3.730998 56668
24.53464 14.76977 0 0.5 -0.075215 3.943362 61896
24.49157 24.48757 0 0.5 -0.013354 4.38043 41487
24.50113 34.09296 0 0 -0.050458 4.883304 17886
24.54165 42.39799 0 0.5 -0.080075 5.687932 2941
38.91114 4.866641 0.5 0.5 0.568916 3.375527 67372
38.57323 14.73106 0.5 0.5 0.455819 3.402687 61509
38.44298 24.49291 0.5 0.5 0.531074 4.031707 39518
38.33975 34.11068 0.5 0.5 0.477479 4.738723 16829
38.47187 42.40413 0.5 0.5 0.366934 5.213709 2867
69.10617 4.330937 1 1 1.176867 2.993901 63460
67.83467 14.60966 1 1 1.003142 3.086671 44561
67.04486 24.38399 1 1 1.064233 3.563003 25937
67.37074 34.17498 1 1.5 1.019543 4.064615 10848
67.06462 42.30343 1 1 0.773429 4.98562 1878
134.9008 3.767374 2 2 2.040394 2.91173 20139
129.1826 14.39104 1.5 1.5 1.722651 3.247738 9068
127.0093 24.29037 2 2 1.951822 3.891673 4556
129.9621 34.42427 2 2.5 1.882984 4.288491 1910
131.0391 42.25512 1.5 1 1.88427 4.567912 445

One can see that there is no evidence in the data for a "curvature" correction. This may be due to contamination from events where the core locator incorrectly determined the shower core.

I used the above table to determine the sampling correction (independent of core distance) by averaging over all core distances for each pulse height bin. For this analysis, I chose to correct the peaks (mode) of the Tc distributions to 0.

The black line is the fit and the blue diamonds are the data. The smallest PE bin (0-0.25) was not used in the fit. The fit function is:

The next step is fitting the sigmas and the RMSís of the residuals as a function of Pes. The blue diamonds are the data and the black line is the fit:

The first equation is used if Pes < 12.3, the second otherwise.

Finally, I fit the RMS's of the TCHI distributions as a function of Pes.

The blue diamonds are the data and the black line is the fit:

 

The Final Fitter:

Once I have the above functional forms for the sampling correction, the sigmas, and the RMS's of the TCHI distributions as a function of pulse height I can begin to perform the angle fitting. There is still a wide selection of options available.

  1. How to select the PMTs to be used in the fit
  2. What weights (errors) to assign to each arrival time.

Using DELEO as my guide I arrived at the following choices:

  1. Use 1/(RMS*RMS) as the weight of each PMT in the fit.
  2. Only PMTs with 2 Pes or more are included in the fit.

Below is a plot of DELEO, NFIT, DELEO vs. Zenith angle, DELEO vs. NFIT, DELEO for NFIT > 40, and DELEO vs. Zenith angle for NFIT > 40 for the above fitter.

 

 

 

Limitations of Technique:

One can see that for NFIT>40 DELEO looks quite good. However DELEO has only half the counters to work with. Therefore one might expect the fitter itself to do well if NFIT>20. Below I show a plot of the RMS of TCHI distribution (for all PMTs with 10<PES<20) as a function of NFIT. Superimposed on the plot is the function 1/sqrt(NFIT), normalized to large values of NFIT. We see that indeed at an NFIT value of 20, the RMS has a similar behavior to DELEO for NFIT of 40.

Fraction of Showers Fit:

Roughly 90% of all triggers (100 PMT multiplicity requirement) pass the fitting algorithm. It is believed that the failed 10% are not air showers but triggers due to large angle muons. If the additional requirement of NFIT>20 is made, 70% of the fit showers pass. In an attempt to recover some of these events I have modified the above fitting algorithm in the follwoing manner.

Relax:

There are 4 passes through the fitter, at each pass PMTs are excluded/included in the fit based on the DISTANCE = (PMT time - fit time)/(sigma of PMT hit). At each pass the number of PMTs used in the fit is checked. If NFIT < 20 the PE requirement is relaxed. After the first pass if less than 20 PMTs are in the fit the PE requirement is relaxed to 1.5 PEs. After the second pass if there are still less than 20 PMTs in the fit the PE requirement is relaxed to 1 PE. However, at the same time the DISTANCE criteria is made stricter so that only those low PE hits that are consistent with the already fitted plane are used in the next pass. Below I show the resulting NFIT distribution and DELEO distribution.

With this version of the fitter 91% of the triggered events are reconstructed and of these 87% have 20 or PMTs used in the fit. The Deleo distribution (for NFIT > 40) indicates that the angular resolution on this set of events is ~0.5 degrees.

Systematic Offset of Previous Fitter:

The previous online fitter was using timing pedestals derived from the data itself. Peter Nemethy showed atour last collaboration meeting that this should have given rise to a systematic pointing error of 2 degrees due west. The following figures show the angle difference between this version of the fitter and the previous version running online and the projections of that difference along the east-west axis and the north-south axis.

Here are the reconstructed theta and phi distributions:

Conclusions:

I have implemented the laser calibration data and the adc calibration data into the angular reconstruction code. Using a sampling correction only (no curvature correction) that is derived by aligning the peaks in the distributions of the timing residuals, and "optimizing" the PMT selection criteria we can attain an angular resolution of ~0.5 degrees for ~80% of the triggered events.