[gate-users] Energy Spectrum Woes

[Angela M K Foudray]

A few people have written back to the list (and to me) about the energy
spectrums from photon one and photon two that are generated from the
Coincidence file in a sphericalPET system model and their ideas for why
these spectra are different.  I feel my question comes down to the
specifics of how the two photons are modelled, i.e.:

- How are energies assigned for each event?
- How is time assigned for each event?


Specifically, to use some nomenclature already used in this list, let's
call photon one S0 and photon two S1.  I am assuming the following:

1 - Since we are dealing with a simulation that presumably models the
two annihilation photons independently, the first photon should have a
random chance of being initially directed anywhere in the 4pi solid
angle.  

2 - We arbitrarily chose this first photon to model first - I assume we
can't and shouldn't automatically label this photon S0, we assign it
some time stamp which we have "jittered" based on a gaussian
distribution whose standard deviation is the time resolution.

3 - Every event depositing energy in the detectors is written in the
singles list.  A time window and energy window, specified by the user,
is used to comb the singles data, looking for pairs of events fitting
this criteria.  S0 is assigned to the event with the earlier time of the
pair (not necessarily the first photon modelled in a event pair of
trues), and S1 assigned to the one with the later time stamp.

If 1, 2 and 3 are true, (particularly the time jittering on the time
stamp), S0 and S1 have an equally likely chance of being the first
photon modelled.  And since we generally have scattering media in the
system, they obviously need not be even of the same annihilation event
(they could both be the first photon modelled or both be the second).  

Therefore,

S0 and S1 cannot have different energy spectra (qualitatively with low
statistics, or "at all" with enough statistics).  A) S0's event location
is randomly distributed about the detection system, and B) it can be
either the first or second photon modelled (this shouldn't make a
difference anyway, right?).  S0 and S1 for all intents and purposes, if
1, 2 and 3 are correct, are the "same" photon for true events, or have
statistically averaged in differences for scatter and random events. 
Even with backscatter, this should add just as often to our slightly
earlier time jittered event or the later (and really this shouldn't
happen much at all, right, because this requires that the photon
traverses on average half of the phantom without other-than-forward
scatter, backscatters at pretty much 180 degrees, traverses the entirety
of the phantom without other-than-forward scatter as well as the second
photon traversing half the phantom without other-than-forward scatter -
seems like we shouldn't notice this effect much).

So, how could these photons have different physics (i.e. different
energy spectra)?   It can't even be that the first photon modelled is
automatically labelled S0 - how do we get randoms then?  And how would
the first photon have difference physics than the second anyway?! (on
average)


Ange