15/7/2025

THE TIME AND SIZE OF MUV1 PULSES: COMPARISON OF DIRECT AND FIT MEASUREMENTS.
----------------------------------------------------------------------------

"All models are wrong, but some are useful" George Box

INTRODUCTION
------------

The distribution of the digitised signals from MUV1 as a function of time are
shown here together with the mean time and rms of the data
and the results of a split-normal fit.
For these low intensity data, all data pulses are single-peak, are asymmetric, and have a low background.

A simple description of a pulse is in terms of its size, ie the sum of digitising counts, and the
the mean time of the pulse; this will be referred to as the direct measurement. The baseline
is taken to be the minimum count.
It should be noted that the direct measurement is pulse-shape independent.
This pulse-shape independence is important because both the pulse-shape and the statistical error of the digitisings
vary with pulse-amplitude and measurement channel.

A more detailed description of the pulse can be made using a split-normal function fitted to the data;
this allows the shape of the pulse to be determined.
For the fit the baseline is the minimum count; the fitted parameters are amplitude, mode, sigma1 and sigma2.
From these parameters the mean time and rms of the distribution can be calculated. The size of the
pulse is calculated from the fitted split-normal function evaluated at each digitising time.

The objective of this note is to compare the pulse parameters , time and size, from the direct measurement,
with those from the split-normal model.

split-normal reference
All MUV data is from Dan.

COMPARISON PLOTS
----------------

Compare1 LOW BEAM INTENSITY COMPARISON PLOTS (pulses selected by C++ program)

Compare2 LOW BEAM INTENSITY COMPARISON PLOTS - low amplitude included
plus additional plots of mean-time, pulse-size and rms.
0.6% of fits are out-of-time; 20% more pulses are found than with the C++ code..

Compare3 MUV1 Comparison plots fits vs direct. test code. muvfit1m1test.pdf
Pages 5 -7 show plots demonstrating the dependence of the pulse shape on the amplitude of the MUV1 pulse,
Both the mean of the RMS of the pulse and variance of the RMS reduce with inceasing amplitude.
The variance of the trailing edge is larger than that of the leading edge.

Description of Compare2 plots
-----------------------------

Page 1
------

a) Amplitude comparison.

The black curve shows the distribution of max - min digitisings for all data.
The red curve shows the distribution of single peak events selected by the
peak finder; 47843 events are selected for max - min greater than 20.
The green curve shows the distribution of the amplitude determined by the split-normal fit;
this agrees well with the max - min of the selected events showing that the
split-normal model fits the data well.

b) Pulse size comparison

Here the pulse-size from the fit is compared with that from the direct measurement.
The agreement is good.

Page 2
------

These plots show the distributions of pulse-size and amplitude.
There is a 2-peak structure, possibly corresponding to muons
and pions. The pulse-size distribution shows good agreement
between raw data (direct measurement) and fit.

Page 3
------

The LHS plot shows(black) the distribution of the time of the mean of
the pulse with the corresponding distribution from the fit(red)) overlayed.
The agreement is good. A similar plot (RHS) is shown for the rms.
Again the agreement between the two mwasurements is good.

Conclusion
----------

In all cases, there is good agreement between the parameters derived
directly from the data and those determined by the fit. Consequently a fit is unnecessary.

Allthough this conclusion follows from a low intensity run, it
is applicable also to single peak pulses from high-intensity runs, provided
early out-of-time data are removed from the pulse.

For two-peak pulses, a fit is required to select the component pulses, see the high-intensity plots here.

ADDITIONAL LOW INTENSITY PLOTS:
-------------------------------

muvfit1devbli.pdf LOW INTENSITY split-normal fits

muvfit1ttll3.pdf LOW INTENSITY comparison plots with chi**2 vs amplitude
The chi**2 plot shows that error = 2*sqrt(amplitude/50) is correct to better than 40%.

muvfit1devbbli.pdf LOW INTENSITY fit parameters vs channel number.
Shows a dependency of pulse size and shape on channel number.
See the corresponding HIGH INTENSITY plots below for comments.

TWO-PEAK FITS
-------------

muvfit2.pdf HIGH INTENSITY 2-peak fits to two split-normals.
baseline = minimum count.

-----------------------------------------------------------------------------------------------------------
-----------------------------------------------------------------------------------------------------------
-----------------------------------------------------------------------------------------------------------
-----------------------------------------------------------------------------------------------------------

MISC. HIGH INTENSITY SINGLE PEAK PLOTS
--------------------------------------

Compare3hi HIGH INTENSITY MUV1 Comparison plots fits vs direct. muvfit1m1testhi.pdf.

51% (1.3%) of pulses are selected as single (double) peak for pulse-amplitudes greater than 20
from 75267 pulses input of minimum amplitude ~ 10.

muvfit1tthh2.pdf HIGH INTENSITY comparison plots fit vs direct. Limited range
Comment: removal of early-out-of-time pulses not yet optimised.

muvfit1tthh3.pdf HIGH INTENSITY comparison plots with chi**2 vs amplitude.

The chi**2 plot shows that error = 2*sqrt(amplitude/50) is correct to better than 40%.

muvfit1devb.pdf HIGH INTENSITY fits. Limited range.

muvfit1devbb.pdf HIGH INTENSITY fit parameters vs channel number

(1) The sigma2 plots show that the channels 1 - 8 and 40 - 44 are anomalously high in comparison
with 9 - 40 , see sides 0 and 2 in particular.
(2) The minimum count is channel dependent.
(3) The mean pulse amplitude shows the channel dependence that results from
the shower energy distribution.

MUV2 plots - high intensity 05/08/25
---------------------------------------

muvfit1m2.pdf MUV2 HIGH INTENSITY comparison plots fit vs direct.

Compare with MUV1 comparison plots fit vs direct:
muvfit1m1.pdf MUV1 HIGH INTENSITY full-range version comparison plots (05/08/25)

Conclude: MUV2 pulses similar to MUV1

muvfit1m2bb.pdf MUV2 HIGH INTENSITY fit parameters vs channel number
Shows less channel variability than MUV1.

muvfit1m2plot.pdf MUV2 HIGH INTENSITY fits

Development and test versions MUV1 HIGH INTENSITY full range + cuts.

muvfit1m1dev.pdf Comparison plots fits vs direct. select time data 160-220 ns.
(scatter plot axes corrected in code, selection update 08/08/25)

muvfit1m1test.pdf MUV1 Comparison plots fits vs direct. test code.
Find 6% of fits out of time range (0 - 300 ns) for high intensity data.

Set no OoT cuts + T GT 120 ns cut for data and fit.
Optimum OoT selection? More study needed.
Remove HSTATI rounding error effects by addition 0.001 to pulse digi
for plot 2000 ( 25/08/25). No no significant changes otherwise.
Problem with skewness under study - could it be defined relative to mode
instead of mean? Not normalised in split normal wiki.
Skewness understood: problem of lack of high-time tail of distribution
for data; the fit calculation avoids this problem . Skewness from from fit
formula is relative to mean.

muvfit1m1testli.pdf LOW INTENSITY MUV1 Comparison plots fits vs direct. test code.
Set with no OoT cuts. Compare2 has 'standard' cuts.

muvfit1m1testli2.pdf LOW INTENSITY MUV1 Comparison plots fits vs direct. test code.
Low amplitude check.

muvfit1.f muvfit1.kumac xfun used.
Fits made using HBOOK routine HFITH.

MUV2.html (old muv1.html MUV1.html.190725)