CKD will only work for *N* > 2 since there are no independent scattering
angle measurements for a single Compton scatter followed by a
photoabsorption. It turns out that the ordering of two-site photopeak events can
still be determined with a high probability; however, the ability to reject
background events is lost. As is discussed further in Sect. 6,
the loss of background rejection, coupled with
low peak-to-Compton ratios, and a larger fraction of backscatter events
mean that the inclusion of two-site events will likely hurt the
sensitivity of a GCT; however, discussion of event ordering is still
included for completeness.

Given a two-site event, the first test one can perform is to determine whether both possible orderings of the interaction sites are energetically compatible with a single Compton scatter, i.e. are compatible with the requirement that (Eq. 2). In Fig. 5, the fraction of spatially-resolved, photopeak events with unique orderings are plotted versus energy. Also plotted are the fraction with ambiguous orderings. At energies below 0.4 MeV the majority of resolved photopeak events have a unique ordering, while at higher energies most events are ambiguous.

As an empirical test of the ambiguous events, the relative magnitude of
the energy lost in the initial scatter (*E*_{1}) compared to the
photoabsorption (*E*_{2}) can be
compared. The fraction of resolved two-site photopeak events which have
ambiguous orderings with
*E*_{1} > *E*_{2} is plotted in Fig. 5. At higher energies,
nearly all of the resolved photopeak events with ambiguous interaction orders
have
*E*_{1} > *E*_{2}, which can be used to determine the most likely interaction order.
This empirical result can be easily understood, in hindsight, by the fact that
photons which deposit most of their energy in the initial Compton scatter are
much more likely to be photoabsorbed in the second interaction.

Therefore, a simple Single Scatter Discrimination (SSD) technique to determine the most likely interaction ordering of two-site events follows. First one determines whether a physically unique ordering exists; if not, the larger energy deposit is assumed to be the initial Compton scatter. Only at the lowest energies are two-site events possible in which neither ordering is acceptable (unresolved events, Compton continuum), and some background rejection is possible.

The fraction of two-site photopeak events which are spatially resolved is shown in Fig. 6 as a function of energy, for the instrument model in Appendix A. Roughly of all events from 0.2-20 MeV are resolved. This number is about lower than the 3+ site events, due to the smaller path lengths of the lower energy scattered photons in two-site photopeak events. Also shown in Fig. 6 is the fraction of events which SSD has properly reconstructed. SSD allows proper reconstruction (hence imaging) of of the photopeak events, while improperly imaging the remaining into the off-source background. Only a relatively small number of low energy events can be rejected outright. SSD is least effective around 0.5 MeV, where the unique/ambiguous ordering signatures are not as clear. For comparison, if the order of the interaction sites were randomly chosen would be properly imaged, while the remaining would be improperly imaged into the background.

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