Time Pedestals II
(Enhancements)

Lazar Fleysher, Roman Fleysher, and Peter Nemethy.

November 1, 1997

In our previous note the formulation and results of time-pedestal determination problem were stated. In a continuing effort to define and improve the quality of knowledge of the pedestals the following ideas were considered.

General

1. Most important, pedestals should not depend on calibration procedure. In particular, should not depend on laser ball used to determine them. Thus, pedestal mismatch distribution Delta(j)= PED(j, ball_A)-PED(j, ball_B) becomes a useful diagnostic tool.
2. Mean mismatch in pedestals measured from two laser-balls gives fiber delay difference between them.
3. The width of pedestal mismatch distribution is a measure of pedestal quality. Ideally should be zero.
4. Introduction of a cut against low energy (~ small Low-ToT ) eliminates worry about asymmetry of slewing-corrected start time distributions. We introduced the extreme cut of Hi ToT > 0. The RMS of difference between pedestals with and without the cut is 0.19 counts; the pedestals are robust.
5. Water level (height of laser balls above the PMT plane) is an input to the multiparameter fit program at fixed value of 90 cm. Changing this value to 75 cm produced 0.12 counts RMS difference in pedestals. They are not sensitive to exact water level, as expected.

Speed of Light

Our multiparameter fit of the laser ball positions had yielded measured values of the speed of light, but was designed to reduce, rather than enhance the sensitivity to the speed (we used nearby balls and PMT's) and suffered from the correlation problems intrinsic to a multiparameter fit.

We have now used the RMS sigma of the pedestal mismatch for a new, sensitive determination of the speed.

The principle is that using the wrong speed of light will symmetrically widen the mismatch distributions from two laser balls. PMT's near ball A will have nearly correct times using signal from A, but incorrect timing from a distant B and vice versa. (See laser ball locations in the pond on the left.)

With this in mind, to be most sensitive, two furthest away laser balls in pairs (0 - 9) and (5 - 9) were used to judge pedestal quality. Repeating the entire pedestal determination procedure using different fixed values of the speed of light we watched how the mismatch changes. Here are the plots of RMS of pedestal mismatch versus speed of light.

The two curves have clear minima at the speed of light in water of 11.06 cm/count ( = 22.12 cm/ns). Compare with the predicted value of 22.04 cm/ns and previously measured 21.92 cm/ns. The new value is on the "good side of Todd":

"One question though: Why is the speed of light that you measure
consistently below that expected from the calculation?...
I'd expect the OPPOSITE result to that which you find.
This puzzles me..."

Todd H.(7/97)

Table of RMS pedestal mismatch for all laser ball pairs

Pair    Sigma at OLD speed       Sigma at NEW speed
	  10.96 cm/count          11.06 cm/count

0-2           0.84                      0.77
0-5           0.42                      0.42
0-7           0.97                      0.66
0-9           1.55                      0.83

2-5           0.87                      0.65
2-7           0.86                      0.64
2-9           1.09                      0.77

5-7           0.98                      0.53
5-9           1.62                      0.79


7-9           0.97                      0.75

The new sigmas are significantly smaller and much closer to each other.

At this speed average distances of laser balls from nominal positions are:

   BALL	       X, cm       Y, cm

    0          44.5         32.1
    2        -221.6        -14.1
    5           9.7         56.9
    7         -49.0        -69.8
    9         -52.8        -40.0

Mean values of pedestal mismatch distributions yield the following fiber optics delays:

    Ball      Delay, counts

     0          0 (definition)
     2         +0.1
     5         -0.4
     7          0.0
     9         -1.2

Pedestals

Final pedestals, which were obtained using the best speed of light, the above fiber delays, laser ball coordinates and high ToT cut are found in peds_new.dat file, e-mailed to all by PN on 10/17/97.

Fitting a plane to the differences ped(new)-ped(old), we find, in pond oriented coordinates (y points to pond north', x to pond east'):

DELTA(THETA_X)=0.01 deg and DELTA(THETA_Y)=-0.23 deg

We can estimate the quality of the pedestals by comparing them with natural (Tony's, after High-Voltage correction) pedestals and ones obtained from LED calibration. The dispersion of a sum of independent random variables (pedestals) is a sum of their dispersions. The measured sigma of differences can therefore be solved for estimates on individual sigmas from the true values.

sigma^2(Laser) + sigma^2(LED)=sigma^2(Laser-LED)=7.6
sigma^2(Laser) + sigma^2(N)=sigma^2(Laser-N)=2.8
sigma^2(N) + sigma^2(LED)=sigma^2(N-LED)=7.9

sigma(Laser) = 1.1
sigma(N)     = 1.2
sigma(LED)   = 2.5

Conclusion

A new, more sensitive method of determination of speed of light in water and fiber delays was proposed and used to recalculate the time pedestals of PMTs. The new pedestals undo 0.23 degree tilt towards Pond South'. The tilt change is entirely dominated by the fiber delay change of -1.2 counts for laser ball 9, used to calibrate North end of the pond. That is why it is straight pond North'-South' effect. We do not see any room for a futher tilt of ~1 deg (~3 counts across pond) from remaining pedestal systematics. The random error of individual pedestals is about one count.