# Application and Evaluation of Surrogate Models for Radiation Source Search
## 1 Introduction  
Significant attention has been focused on accurate and efficient determination of the location of radioactive materials. This is especially important in urban environments, which are particularly susceptible to threats due to high population density. One strategy is to deploy a network of detectors, which count ionizing gamma rays that reach their location. The resulting problem is to locate a radioactive source in a complicated domain using this noisy detector data. The difficulties are compounded when the signal-to-background radiation ratios are low, which is often the case. One avenue of current research has focused on fusing detector data to determine the source location while attempting to decrease the number of detectors, increase the accuracy of the location estimate, and increase the efficiency of the algorithm [1; 2; 3]. This problem requires modeling the paths of the gamma particles emitted by the source by employing the Boltzmann transport equation to determine the origination location. Two standard methods for numerically solving the Boltzmann transport equation are to solve for the explicit, deterministic solution and the implicit, stochastic solution using Monte Carlo simulations of many particle histories. However, both are computationally expensive. In prior work [2], we employed a 2D ray-tracing algorithm to model detector responses to a radiation source in a simulated city block. We used the open source code gefry to solve for the detectorresponses within this test environment [4]. Whereas this algorithm is based on Boltzmann transport theory, scattering is neglected to significantly improve the efficiency. It is shown in Ref. [5] that, despite these simplifying assumptions, the code can resolve source locations to within meters for the considered geometry and background levels. To solve the source localization problem, we employ Bayesian inference by using a Delayed Rejection Adaptive Metropolis (DRAM) algorithm [6], which requires thousands to millions of model evaluations to infer the 2D position \((x,y)\) and intensity \(I\), which are the design variables used throughout this paper. Whereas the use of DRAM is tractable for this model, two issues motivate the construction of surrogate models: (1) extension to 3D models employing Monte Carlo N-Particle (MCNP) [7] simulations that incorporate more comprehensive physics; and (2) addressing the non-differentiable likelihood due to the presence of buildings in the geometry. We extend the analysis presented in Ref. [2] by exploring the choice of surrogate modeling technique and training point selection in this paper. We performed preliminary research regarding the use of Monte Carlo simulations [7] to generate measurement data for nine point detectors in three dimensions with scattering in an extremely simplified geometry detailed in Ref. [8]. A Monte Carlo simulation was run once to generate synthetic measurements, and calibration was then performed against the calculated detector responses using the ray-tracing model. In this geometry, four rectangular prisms in a 2 \(\times\) 2 layout were considered in a 100 m \(\times\) 100 m domain, each adjusted to have wall thicknesses of 0.5 m of concrete. Employing the ray-tracing model for the source localization problem for a 662 keV source placed 1 m above the ground, we were able to localize the source to within 5 m. This level of accuracy is sufficient in this large-space source search problem. However, it required several CPU hours of computation to generate one set of detector measurements employing MCNP code [7; 8]. For a realistic 3D geometry, such as a 3D version of the geometry depicted in Figure 1, thousands of CPU hours would be required to solve the source localization problem using only Monte Carlo simulations. Furthermore, the model solution to the 2D problem exhibits discontinuous derivatives due to the buildings in the geometry, which block radiation paths to detectors. Hence, gradient-based sensitivity analysis and optimization are not directly applicable.