## 5 Data Model  
Clarifying the impact of the challenges described in the previous section on RIID algorithm performance requires a model for the data. Under standard assumptions, the counts \(C\) in a given energy channel that does not drift in calibration over the repeats will be distributed as Poisson (\(\mu\)). Repeat measurements vary because of this Poisson variation and also because the detected counts will be less than the true counts in a given time period because the efficiency \(\varepsilon\) of any detector will be less than 1. The efficiency \(\varepsilon\) at a given channel is the probability that a \(\gamma\)-ray incident on the detector will be detected.  
Figure 3: A principal coordinates plot of the three spectral pairs from Figure 2.  
Given \(C\), the simplest effective model for the detected counts \(D\) in time \(t\) is therefore D\(|\)C \(\sim\) Binomial(C, \(\varepsilon\)) and C \(\sim\) Poisson(\(\mu\)), where D\(|\)C denotes the conditional distribution of \(D\) given \(C\). Although it appears to not be well known in the literature, it is not difficult to then show that D\(|\mu\) \(\sim\) Poisson(\(\mu\)e). To simplify notation, we will write \(C\) for the detected counts and include the efficiency term \(\varepsilon\) in the term \(\mu\), so \(C\) \(\sim\) Poisson(\(\mu\)).  
The simple model \(C\) \(\sim\) Poisson(\(\mu\)) has often been found to adequately describe detected counts at a given energy channel in NaI spectral data, even after effects such as Gaussian broadening [12]. If \(C\) \(\sim\) Poisson(\(\mu\)), then variance(C) \(=\)\(\mu\), so a simple check for the Poisson assumption is to compare the sample variance across measurement repeats to the sample mean at each energy channel. Overdispersion, defined as var(\(C\)) \(>\)\(\mu\), typically occurs when the mean \(\mu\) is not constant across repeat measurements. Underdispersion, defined as var(\(C\)) \(<\)\(\mu\), can occur, for example, at high values of \(\mu\) due to detector dead time effects [12].  
Following the notation in [13] for an unshielded source, the mean count rate \(\mu_{i}\) at channel \(i\) can be expressed as \(\mu_{i}\) \(=\)\(\sum\limits_{j}R_{y}a_{j}\) for isotopic composition vector \(a_{j}\) describing the activity in decays per second of each isotope present and response matrix \(R_{y}\). The response matrix \(R_{y}\) either is or can be thought of as a representative reference spectra for each unshielded isotope in the library. When attenuating absorbers are present or could be present, the response matrix \(R_{y}\) with matrix elements \(\mu_{i}(j,\mathrm{ref})\) is modified to \(R_{y}(\rho)\) with matrix elements \(\mu_{i}(j,\rho)\) \(=\)\(\mu_{i}(j,\mathrm{ref})\) \(\times\) exp \(\{-\)\(\sum\limits_{z}\rho_{z}\
u_{i}(j,z)\}\), where \(\rho_{z}\) is the density of absorber z (the relative amount of the \(z\)th absorber), \(\
u_{i}(j,z)\) \(=\) \(-\) ln\(\left(\dfrac{\mu_{i}(j,z)}{\mu_{i}(j,\mathrm{ref})}\right)\), and the sum is over absorbers \(z\) in a library of absorbers. The multiple-isotope material basis set approach described in Section 9 typically allows only up to three to five representative absorbers such as Pu, Steel, and Fe to fit the spectral effects of arbitrary absorber mixtures. Without such a reduction in the absorber library, the number of unknowns could be quite large, with approximately 200 possible source isotopes and approximately 100 absorbers.  
The RIID algorithms described in Section 9 each assume that all relevant isotopes, including all possible NORM isotopes of interest are in the spectral library. One practical challenge in constructing such a comprehensive spectral library is that each measurement scenario and detector could require its own library, particularly if effects such as secondary interactions leading to minor peaks and backscatter from nearby structures are nonnegligible and dependent on the measurement scenario.