# Application and Evaluation of Surrogate Models for Radiation Source Search
## 6 Radiation Source Localization  
To illustrate the use of these surrogate models for Bayesian inference, we consider the problem of locating a source within the domain depicted in Figure 1 and outlined in Section 2.2. We begin by simulating a \(3.22\times 10^{8}\) Bq source at the location \((x,\,y)=(158,\,98)\) as in Ref. [16]. We then employ the ray-tracing model from Equation (3) to simulate detector responses and DRAM to perform Bayesian inference. A complete description of DRAM and the methodology used for source localization can be found in Ref. [6]. We initialize the MCMC chains in the center of the uniform parameter prior distributions--i.e., an approximately \(3.7\times 10^{7}\) Bq source in the center of the domain--and we draw \(10^{5}\) samples, with the first half discarded as burn-in. Visual inspection of the sample histories indicate that this number of samples is a very conservative choice. We plot the posterior obtained from DRAM in Figure 8 and note that the mode of the posterior is within approximately \(1.5\) m of the true source location. Additionally, we have captured the true source location within the posterior distribution, which covers approximately a \(10\) m by \(10\) m portion of the domain.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Surrogate** & **rRMSE** & **Surrogate Computation** & **Surrogate Evaluation** \\  & & **Time (s)** & **Time (s)** \\ \hline Legendre & \(2.13\times 10^{-1}\) & 33.42 & 3.86 \\ Legendre Reg. & \(3.05\times 10^{-1}\) & 50.21 & 3.95 \\ MARS & \(4.18\times 10^{-1}\) & 66.43 & 0.02 \\ RBF & \(2.08\times 10^{-1}\) & 134.45 & 0.29 \\ GP & \(1.98\times 10^{-1}\) & 425.64 & 2.32 \\ NN & \(2.62\times 10^{-1}\) & 124.11 & 0.13 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of surrogate models using Gauss–Legendre training points where Legendre refers to the model constructed via Equation (21) and Legendre Reg. refers to the model constructed via Equation (26).  
Figure 7: Root mean squared relative error for the 10 surrogate models, corresponding to the 10 detectors, constructed using these methods.  
Next, we use the same simulated source and employ the surrogate models to generate detector responses to solve the source localization problem with DRAM. We employ the same DRAM setup as with the ray-tracing model and we plot the posteriors in Figure 9. We note that the mode of the posterior is within 2 m of the true source location and that the posteriors are more diffuse than before. The wider posteriors are caused by the model discrepancy errors \(\epsilon_{i}^{m}\) that arise from our smoothing of the ray-tracing model solution. These discrepancies lead to less precise localization results, however we have still accurately localized the source to within a 35 m by 35 m portion of the domain and this amount of precision in the posterior results is sufficient for most applications.  
Figure 8: Posterior density for source location employing the ray-tracing model for Bayesian inference with DRAM. We plot (**a**) the full domain and (**b**) the portion of the domain where the posterior is located.  
Figure 9: Posterior density for source location employing the Gaussian processes surrogate model for Bayesian inference with DRAM. We plot (**a**) the full domain and (**b**) the portion of the domain where the posterior is located.