Twalk correction compensates for the fact that signals with small amplitude cross discriminator threshold later than signals with large amplitudes. So even though small and large signal would originate at the same time, TDC would record different arrival times for them.
This effect should be distinguished from anther effect: signals that originate close to the RS module end have less time to propogate till they reach phototube. They also tend to have larger amplitude, because there is less attenuation. So it leads to the same phenomena: signals with larger amplitudes arrive earlier.
In general for module j TDC arrival time for the track that crossed it in the middle at t=0 and generated signal of amplitude aj is given by:
t = t0j + twalk(aj),
where
t0j is the calibration constant. Twalk correction
twalk is assumed to be the same
for all channels j
Different channels j have different t0j and different distribution of amplitudes aj. Because amplitude distribution aj is different averaging over different channels j to extract twalk is not correct. Thus I use different technique: calculating derivative of arrival times dt/da. Indeed:
dtj/da = d twalk(a) /da ,
so averaging of
derivatives dt/da over large sample of events and over
different modules j will give necessary derivative of the
function. After that it can be integrated to extract twalk.
As each event is measured with certain precision statistical methods should be applied. Calculations show that:
nj * ({ta}j-{t}j{a}j)
dtwalk/da = Sumj -----------------------
nj * ({a2}j-{a}2j)
,
where nj is the number of events in module
j. And {ta}j is the average of
t*a over all events in module j. Similarly
{t}j, {a}j,
{a2}j are defined. As
dtwalk/da is non-liniear this derivative should be computed in
several ranges of amplitudes. In this particular study 20 different
amplitude ranges were used.
The cosmic trigger data was used, as that was the only data we had at the moment. To select tracks that cross the middle of RS module amplitude from 2 ends were required to be equal to each other within 10% error. Illumination plot for different secotrs and layers and amplitude distribution for events that satisfied the cuts are shown.
Cosmic tracks do not cross the module at t=0, but we assumed that this offset is constant, anyway it cancels out when we calculate the derivative. Though there may be a catch... see systematic error discussion.
The following code dplot.F was used to compute averages {t}j etc for each module j and store them in the ntuple. Following runs were processed twalk.sh . there were 20 runs with 360K events in total. All the necessary files are sitting in source directory, including Makefile and log file.
Once individual averages like {t}j were
computed and stored in ntuple they have to be summed up according to formula
and derivative dtwalk/da can be calculated. Following COMIS
function twalk.for and twalk.kumac were used to process ntuple.
Derivative dtwalk/da was calculated for each run
independently and after that RMS for 20 different runs was used as an error
estimate.
Following function was used for twalk and it's derivative:
twalk(a) = p(1)/(p(2)+a)
d twalk(a) / da = - p(1)/(p(2)+a)**2,
where p(1) and p(2) are fitting parameters.
As a sanity check twalk was calculated as an average of all modules j . As we discussed it is not quite right as different modules have different amplitude distribution aj, but as they are close it should give about the right answer. The results of averaging and fit are give here PS . Note that errors on the plot were done using nt/pl ... option=prof and thus have nothing to do with reality.
Results of fit of dtwalk/da averaged over all modules and averaged over 20 runs are given here PS . Errors computed in derivative fit were used to compute errors on twalk . Usual error propogation was used, we also used correlation coeefficient 0.755 between p(1) and p(2) which I got from fit of twalk derivative. On the second plot middle red line represent expectation value for twalk function, surrounding dashed lines represent statistical error. Statistical error as a function of amplitude is shown on the third plot
Twalk function can be computed for any subset of modules. To study systematics I computed twalk function for different sectors and compared them to each other. 4 twalk function were computed based on data from sectors 1..6, 7..12, 13..18, 19..24. Similarly 5 twalk functions were computed for different layers: 1+19, 2..6, 7..10, 11..14, 15..18. Results for twalk derivative fit for each subset are shown here PS Results for different twalk functions are shown here PS . Red line represent overall twalk function computed for all modules in the detector.
Clearly difference in twalk for different subsets of data is 2-3 times bigger than the statistical error. So there is clearly some systematic bias involved. Part of that can come from data itself, because cosmic runs do not illuminate all the sectors uniformly. Twalk calculated for different layers are quite similar and layers are illuminated more uniformly PS. One should note that all layers are illuminated uniformly but selection of events which have closer amplitudes benefits layers 2..5 which are made from more transparent plastic. Illumination is a suspect though it is not clear exacly how non-uniform illumination of sectors can translate in large deviations between twalk functions for different sectors. I did not push on systematic studies more because anyway we will repeat this study on more relevant sets of data.
Based on 360K cosmic triggers twalk function was studied. The suggested function is:
twalk(a) = 87 / (7.4 + a),
where twalk is in nanoseconds and a is amplitude in ADC counts. The
statistical error is of order 0.1nsec. Systematic error is 2-3 times as big.
Further studies should reduce systematic error once we start to analyze more
appropriate sets of data.