# Application of the Approximate Bayesian Computation Algorithm to Gamma-Ray Spectroscopy
## 2 Background  
Figure 1 plots real (smoothed using the iterative bias reduction method [23; 24]) and simulated spectra (from the GADRAS 18.8.10 code [21]) from a relatively low-energy resolution and widely deployed NaI detector of the isotope I-131 without shielding effects and with background effects subtracted. I-131 is a medical-use isotope that is sometimes detected at border crossings, is a radionuclide that vendor's field instruments are required to recognize and is a significant fission product that can be released into the atmosphere. Shielding attenuates lower energy counts more than higher energy counts, and can reduce or sometimes eliminate evidence of low energy peaks. The vertical lines indicate the birth energies of the key source of gamma energies from the decay of I-131; note that some peaks are not likely to be detected. The approach and an example function to simulate such spectra are given later in the section. Figure 2 plots real and simulated spectra of I-131 for two energy ranges in Figure 2a,b, and the corresponding relative model bias calculated as (simulated/-measured)/simulated in Figure 2c,d. Other radioisotopes have different gamma birth energies with different relative intensities so in theory, the potential signature of most radioisotopes is at least moderately distinct from that of other isotopes. In practice, for relatively low resolution detector types such as NaI detectors (commonly deployed detector and the main type of interest here, although higher resolution is also discussed), RIID performance can be surprisingly poor [15; 16; 17; 18; 19; 20; 1; 15; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47;  
The ABC-based RIID reported in this paper is shown to be _well calibrated_, and it can rely (depending on the user choice of summary statistics) on locating peaks and estimating peak areas. Section 5 demonstrates that (1) the extracted peak areas are impacted by modeling bias and (2) that there can be some advantage in using non-parametric methods to estimate peak areas. Both parametric and nonparametric options are evaluated in real and simulated data. The parametric method must first select an analytical model such as the one used in the Gamma Detector Response and Analysis software (GADRAS [21]) or the one used in the Fixed Energy Response Function Analysis with Multiple Efficiencies software (FRAM [28]) and then estimate the model parameters. The DRF is a mapping from the gamma birth energy to the distribution of possible detected energies. The DRF must account for the main interactions of gamma rays in the detector, including photoelectric absorption, Compton scattering, and pair production, as well as the processes in the conversion process from deposited energy in the detector to a measured voltage pulse that leads to increased variance in the measured  
Figure 1: Simulated and real (smoothed) I-131 spectral data. Figure 2: Similar to Figure 1, but for 2 energy regions: simulated and real I-131 spectra in (**a**,**b**). The real data are background adjusted, but model bias remains, as seen in (**c**,**d**). energy. Figure 1 is an illustration with the vertical lines depicting the gamma birth energies with the observed count rates in each energy bin arising from the probability of detecting a gamma with birth energy E\({}_{1}\) at another energy E\({}_{2}\). An example DRF (for a range of detector resolutions) is:  
\[f(E) = h_{1}e^{-\frac{(E-\mu_{1})^{2}}{\sigma_{1}^{2}}}+h_{2}e^{\mathrm{i}(P_{ 1}+P_{2}\mu_{1})+(P_{3}+P_{4}\mu_{1})(E-\mu_{1})\mid\times(1-e^{-\frac{(E-\mu_{ 2})^{2}}{\sigma_{2}^{2}}})} \tag{1}\]  
which defines a simplified version of a DRF in FRAM with the Gaussian broadening of each peak, plus a tail and background contribution from each peak (FRAM sets \(h_{1}=h_{2}\)\(\mu_{1}=\mu_{2}\) and sets \(\sigma_{2}^{2}=2.5\sigma_{1}^{2}\)). Previous versions of FRAM set \(P_{2}=P_{4}=0\) to get what is referred to as model 2 in Section 5.