Signal-to-noise and counting time optimization
for E00102
KF 010829 

The following is a writeup describing what I did to
optimize the left-right counting times for e00102 via
anticipated counting rates and signal-to-noise ratios.
In order to make sense of this writeup, you will need
the attached figures ep.ps, tc_minus.ps, and tc_plus.ps.
You will also need the attached tables best_rates.html,
sn_ratios.html, and counting_times.html.  If you prefer,
there are .ps.gz files corresponding to each of the
previously mentioned .html files.

The bottom line: 

I present the ultimate results of all my signal-to-noise
simulation work below: 

kin      <lhs time>          ratio     suggested
         1p12 1p32      1p12/1p32      lhs time 

i-+       0.60 0.58           1.04         0.590
h-+       0.56 0.48           1.16         0.520
g-+       0.43 0.38           1.15         0.405
f-+       0.41 0.39           1.04         0.400
e-+       0.43 0.42           1.02         0.425
d-+       0.42 0.41           1.03         0.415
c-+       0.45 0.44           1.03         0.445
b-+       0.47 0.45           1.03         0.460
a-+       0.48 0.47           1.02         0.475 

Happily, they are quite consistent, both as a function of
kinematics, and as a function of state.  Values averaged
over the two states should suffice, and are presented in
the column "suggested lhs time".  However, particularly
at the large pmiss values corresponding to kinematics
i-+, h-+, and g-+, it is a very good idea if we spend
the time doing the exploratory signal-to-noise measurement
(24 hours) we currently have in place in our draft runplan.
That way we have the data we need to make any necessary
adjustments should reality and simulation disagree
substantially. 

Overview of the method of calculation: 

The calculation of the experiment signal-to-noise ratios
for the purpose of optimizing the counting times on either
side of qvec for the various kinematics was in principle
straightforward, but in practice, it took fair amount of
extra thought.  The first step was to generate the most
realistic signal-to-noise ratios I could.  This took
shape in the following manner: 

o the phase-space matched foreground rates for the p-shell
 states of 16O for the various kinematics were determined. 

o for the exact same kinematics and input parameters, the
 (e,e'), (e,p), (e,pi+), and (e,pi-) rates were estimated
 using the QFS and EPC codes as incorporated into MCEEP.
 It should be noted right from the start that these codes
 are suspect at these energies.  I proceed from here with
 the assumption that the overall shape may be OK. 

o indeed, a discontinuous step in the (e,p) rates as the
 hadron spectrometer was rotated through qvec was noticed
 and independently verified (Sarty).   An algorithm was
 determined to "patch" the transition so that a more physical
 distribution could be obtained.  See the appendix below
 entitled "Investigation of (e,p) singles from MCEEP"
 for details.  You will also need the plot ep.ps.
 The rates were evaluated per MeV for three scenarios:
 full coincidence HRS^2 Emiss acceptance, a 2 MeV Emiss
 bin centered on the state in question, and finally a
 2 MeV Emiss bin centered on the state in question
 with the phase space match used to determine the foreground
 rates tacked on. 

o the background rates were combined as per the prescription
 of Knoll - "Radiation Detection and Measurement" assuming a
 2 ns basewidth to the CRT peak to determine the accidental
 coincidence rates per MeV of Emiss.  Given the anticipated
 pi+/- suppression ratio of 1000, only the accidental
 (e,e'p) rate was of significance. 

o at this stage I realized the signal-to-noise ratios I
 quoted in p00102 were incorrect, and I re-evaluated them
 to illustrate their consistency with the new numbers
 (the kinematics changed only slightly). 

A complete summary of the work presented above may be found
in the file best_rates.html.  A complete description of the
table may be found in the appendix below entitled "The rate
table presented in best_rates.html". 

At this stage I realized that the signal-to-noise ratios
I was coming up with were more-or-less unreasonable, so
I set out to improve my estimates.  Step two, determine
some sort of overall normalization factor to tie the
simulations to reality. 

o I reanalyzed the e89003 theta_pq = +/- 20 deg data for
 signal-to-noise ratio (plots from Liyanage).  I also
 simulated these kinematics using the exact same simulation
 engine I have been using for e00102.  This allowed me
 to determine hard normalization coefficients to tie
 e89003 simulations to e89003 data. 

o when I compared these coefficients to those that I
 would extract for theta_pq = +/- 20 deg for e00102
 (supposing the e89003 data), they were remarkable similar.
 I thus decided to apply them uniformly to the e00102
 simulation points to extract what I consider to be the
 best possible estimate of the signal-to-noise ratio
 we can make. 

The e89003 data consists of two files:  tc_minus_20.ps
and tc_plus_20.ps.  A complete description of the analysis
may be found in the appendix below entitled "Measured
signal-to-noise ratios from e89003".  A complete summary of
the work presented above may be found in the file
sn_ratios.html.  A complete description of the table may be
found in the appendix below entitled "The signal-to-noise
table presented in sn_ratios.html". 

The last step was to combine the signal-to-noise ratios
and rate estimates according to a modified version of Larry's
presciption to determine what ratio of time we should
spend counting on which side of qvec to optimize our
measurement.  I also looked to see if these parameters
depended upon the p-state in question.  A complete
summary of the work presented above may be found in the
file counting_times.html.  A complete description of the
table may be found in the appendix below entitled "The
estimated counting times presented in counting_times.html" 

Appendix A:
Investigation of (e,p) singles from MCEEP 

Consider the plot ep.ps.  I generated this plot from the
rates predicted by MCEEP for (e,p).  The solid circles are
the data points, one for each kinematic point we intend to
measure. 

The striking feature in this plot is the marked
discontinuity in (e,p) rate as one crosses the momentum
transfer.  This discontinuity leads to wild signal-to-noise
ratios.  I cross-checked it against the output of several
versions of EPC (thanks Sarty) and concluded the problem
is not MCEEP, but rather EPC.  I don't think it is physical,
and have thus tried to make a patch. 

First, I fit exponentials to both the -'ve pmiss and the
+'ve pmiss points separately.  The open squares represent
the extension of the fits into the respective complementary
domain.  Based on my knowledge of the real signal-to-noise
ratios from e89003, I chose the upper set of points to
form my "patched" (e,p) singles distribution for the e00102
kinematics.   I extended the patch to the other various
simulated kinematics using the scale factor determined for
the e00102 points.  These numbers all appear as "patched
(e,p)" in the file "best_rates.html". 

Appendix B:
The rate table presented in best_rates.html 

This table is the output of the spreadsheet work I
did to estimate the signal-to-noise ratios.  If you would
prefer to have the spreadsheet (note:  gnumeric!), let
me know. 

I begin by breaking down the table in broad terms.  I start
with the three general table sections. 

The first general section of the table (labeled "Proposal")
is the kinematics originally put forward in the proposal
p00102 to illustrate the signal-to-noise ratios I worked out
there were flawed.  The proper projected values for the
signal-to-noise ratios for the p00102 kinematics are presented
here.  Note that just because they are the "proper projected
values" does not mean they have any basis in reality.
Subsections are for the 1p12 and 1p32 states. 

The second general section of the table shows the results of
a simulation of the e89003 theta_pq = +/-20 deg data points.
I did this because I have actual, measured signal-to-noise
ratios to compare to here to use to benchmark what MCEEP is
telling me.  "Full E89003" labels rate projections for the
full HRS2 coincidence acceptance, "dEmiss E89003" labels rate
projections for the HRS2 coincidence acceptance cut around
a 2 MeV Emiss bite centered at the 1pXX state in question,
and "dEmiss, phase E89003" labels rate projections for both
the Emiss cut above AND an optimal (omega, q, Emiss, pmiss)
overlap.  Again, subsections are for the 1p12 and 1p32 states. 

The third general section of the table shows the results of an
extensive simulation for the agreed-upon e00102 kinematics.
These signal-to-noise ratios are of interest for determining
the allocated beamtimes.  As previously, "Full E00102" labels
rate projections for the full HRS2 coincidence acceptance,
"dEmiss E00102" labels rate projections for the HRS2 coincidence
acceptance cut around a 2 MeV Emiss bite centered at the 1pXX
state in question, and "dEmiss, phase E00102" labels rate
projections for both the Emiss cut above AND an optimal
(omega, q, Emiss, pmiss) overlap.  Again, subsections are for
the 1p12 and 1p32 states. 

Now I will address each column.  "kin" is the kinematics label
(if appropriate).  "Theta_pq" labels the opening angle between
the HRSr central angle and the momentum transfer.  "Pm" labels
the corresponding missing momentum.  "D_Em" refers to the width
of the missing energy distribution in MeV that I averaged over
to get rates which come in the next few columns.  "(e,e')",
"(e,pi-)", "(e,p)", and "(e,pi+)" are the hourly rates output
by MCEEP.  "patched (e,p)" is due to the makeshift patch I apply
to the MCEEP (e,p) rates to make the distribution look more
physical - see the section titled "Investigation of (e,p) singles
from MCEEP".  "dt" is the coincidence timing peak base width I
use - 2 ns.  For each of "(e,e'p) / 2 MeV",
"patched (e,e'p) / 2 MeV", "(e,pi-p) / 2 MeV",
"patched (e,pi-p) / 2 MeV", "(e,e'pi+) / 2 MeV", and
"(e,pi-pi+) / 2 MeV", I calculate the accidental rate for a 2 MeV 
missing energy bin, and then calculate twice the product as per
Knoll, "Radiation Detection and Measurement".  "rate" is the
hourly matched signal rate that I extracted from MCEEP, also for
a 2 MeV missing energy bin.  And finally, "S/N" and "patched S/N" 
are the signal-to-noise ratios.  It is the right-most column that 
is of ultimate interest. 

Appendix C:
Measured signal-to-noise ratios from e89003 

Consider the two plots tc_minus_20.ps and tc_plus_20.ps that
Nilanga sent me.  First, I evaluated the signal-to-noise
ratios as follows: 

tc_plus_20.ps:  S + N :  11237
                    N :    354
                S     :  10883 <- 

tc_minus_20.ps:  S + N :  10340
                    N :   3600
                S     :   6740 <- 

However, there are both p12 and p32 protons in these peaks. 

Thus, I completely reran the MCEEP simulations for e89003
kinematics for theta_pq = +/- 20 deg using the best calcs of
Udias e t c.  I ran both phase space decks to get the
optimal overlap and physics decks to get the rates.  I
determined the phase space overlap in exactly the same way
I did for the e00102 rates and obtained cut hourly rates for
the previous experiment. 

theta_pq    1p12     1p32  1p12/1p32 ratio 

  -20.0  171.33   476.45            0.356
  +20.0  560.95  1498.80            0.374 

I then decoupled the yields above 

tc_minus_20.ps:  S + N :  10340
                     N :   3600
                 S     :   6740 = 2399 1p12 + 4341 1p32 

tc_plus_20.ps:   S + N :  11237
                     N :    354
                 S     :  10883 = 4070 1p12 + 6813 1p32 

This left me with measured signal-to-noise ratios from
e89003 given by 

theta_pq             1p12 S/N             1p32 S/N 

   -20.0  2399 / 1800 =  1.33  4341 / 1800 =  2.41
   +20.0  4070 /  177 = 22.99  6813 /  177 = 38.49

Note that I assume 50% of the noise affects the 1p12
measurement and 50% of the noise affects the 1p32
measurment.

I present these S/N ratios in the column labeled 'data'
in the table sn_ratios.html. 

Appendix D:
The signal-to-noise table presented in sn_ratios.html: 

This table is the output of the spreadsheet work I
did to summarize the signal-to-noise ratios.  If you would
prefer to have the spreadsheet (note:  gnumeric!), let
me know. 

I begin by breaking down the table in broad terms.  I start
with the two general table sections. 

The first general section of the table shows the results for
the e89003 theta_pq = +/-20 deg data points.  Subsections are
for the 1p12 and 1p32 states. 

The second general section of the table shows the results for
the agreed-upon e00102 kinematics.  Again, subsections are for
the 1p12 and 1p32 states. 

Now I will address each column.  "kin" is the kinematics label
(if appropriate).  "Theta_pq" labels the opening angle between
the HRSr central angle and the momentum transfer.  "Pm" labels
the corresponding missing momentum.  "full patched S/N" labels
the signal-to-noise ratios extracted for the full acceptance
(e,p)-patched background rates.  "dEmiss patched S/N" labels
the signal-to-noise ratios extracted from the 2-MeV wide bin
straddling the 1pXX bound state for the (e,p)-patched background
rates.  "phase dEmiss patched S/N" labels the signal-to-noise
ratios extracted from the above Emiss cut together with an optimal
(omega, q, Emiss, pmiss) overlap for the (e,p)-patched background
rates.  "data" is just the signal-to-noise ratios I extracted
from the e89003 data - see the section entitled "Measured
signal-to-noise ratios from e89003".  "fp S/N / data = Kfp"
labels the ratio of "full patched S/N" to "data",
"dEp S/N / data = KdEp" labels the ratio of "dEmiss patched S/N"
to "data".  Since these previous two columns employ essentially
the same technique varying only the Emiss bin, the average is
presented in "av(fp,dEp) S/N / data = Kav".
"phdEp S/N / data = KphdEp" contains the ratio of
"phase dEmiss patched S/N" to "data".  Note the similarities
between all the ratios for the two experiments!  Finally,
"pdEp S/N / Kav" represents the simulated signal-to-noise
ratios normalized to the scaling factor "Kav" to tie them to the
e89003 data, while "pdEp S/N / KphdEp" represents the simulated
signal-to-noise ratios normalized to the scaling factor "KphdEp"
to tie them to the e89003 data.  They are rather similar.
Finally, "av(Kav,KphdEp)" is just the average value. 

Appendix E:
The estimated counting times presented in counting_times.html: 

This table is the output of the spreadsheet work I
did to estimate the how we should divide the counting times
based on Larry's derivation and the signal-to-noise ratios
presented in sn_ratios.html.  If you would prefer to have the
spreadsheet (note:  gnumeric!), let me know. 

There is only one section to this table.  The subsections are
for the 1p12 and 1p32 states. 

Now I will address each column.  "kin" is the kinematics label.
"Theta_pq" labels the opening angle between the HRSr central
angle and the momentum transfer.  "Pm" labels the corresponding
missing momentum.  "pdEp S/N / Kav" represents the simulated
signal-to-noise ratios normalized to the scaling factor "Kav" to
tie them to the e89003 data, while "pdEp S/N / KphdEp" represents
the simulated signal-to-noise ratios normalized to the scaling
factor "KphdEp" to tie them to the e89003 data.  "av(Kav,KphdEp)"
is just the average of these two values.  These three columns
were taken from sn_ratios.html.  "rate" represents my best estimate
of the phase-space matched rate for e00102 taken from best_rates.html.
Columns "(4)", "(5)", and "(6)" are intermediate steps in the
time optimization based on "pdEp S/N / Kav", "pdEp S/N / KphdEp",
and "av(Kav,KphdEp)" respectively.  The left-right kinematics
are then combined in "kinLR".  "Tl (4)", "Tl (5)", and "Tl (6)"
are the projected proportion of time to be spent on the left-hand
side of qvec for "pdEp S/N / Kav", "pdEp S/N / KphdEp",
and "av(Kav,KphdEp)" respectively.  "av" presents the average
of these three values.  "Tl p12 / p32 (4)", "Tl p12 / p32 (5)",
and "Tl p12 / p32 (6)" presents the ratio between the states of
the p-shell for time-to-be-spent for "pdEp S/N / Kav",
"pdEp S/N / KphdEp", and "av(Kav,KphdEp)" respectively.  Note
that the ratio is essentially 1 for all kinematics.  And
finally, "Tl p12 / p32 (av)" is the average of these three values.