# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 2 Radiation Transport Model and Surrogate Formulations  
Gamma transport phenomena, as derived from [PERSON] transport theory, can be modeled by the partial differential equation (PDE)  
\[ \begin{split}\mathbf{\hat{\Omega}}\cdot\
abla I&( \mathbf{r},E,\mathbf{\hat{\Omega}})+\Sigma_{t}(\mathbf{r},E)I(\mathbf{r},E, \mathbf{\hat{\Omega}})\\ &=S(\mathbf{r},E)+\int_{0}^{\infty}dE^{\prime}\int_{\mathcal{A }\pi}d\mathbf{\hat{\Omega}}^{\prime}\Sigma_{s}(\mathbf{r},E^{\prime}\to E, \mathbf{\hat{\Omega}}^{\prime}\rightarrow\mathbf{\hat{\Omega}})I(\mathbf{r},E^{\prime},\mathbf{\hat{\Omega}}^{\prime}).\end{split}  \tag{1}\]Here \(I\) and \(S\) respectively denote the gamma intensity per unit area and external gamma source in the medium characterized by the position vector \(\mathbf{r}\), energy \(E\), and unit vector in the direction of the gamma \(\hat{\mathbf{\Omega}}\). The parameters include the total macroscopic cross-section for gamma interactions \(\Sigma_{t}\), and the double-differential macroscopic scattering cross-section \(\Sigma_{s}\), which depends on the change in gamma energy from incident energy \(E^{\prime}\) to emergent energy \(E\) (i.e., \(E^{\prime}\to E\)) and the change in gamma direction from incident direction \(\Omega^{\prime}\) to \(\Omega\) (i.e., \(\Omega^{\prime}\rightarrow\Omega\)). We refer readers to [PERSON] and [PERSON] (2018) for a more detailed treatment of transport theory. The problem of inferring the radiation source location and intensity from sensor measurements requires the evaluation of the Boltzmann radiation transport model (1) at various points in the admissible parameter space. Numerically solving the PDE (1) is computationally expensive even on HPC systems. Solving an inverse problem constrained by Equation (1) or forward propagating uncertainties using Monte Carlo simulations are not computationally feasible since require solving Equation (1) for many times. Instead, we employ a model that only considers gamma rays that travel directly from source to detectors, without taking into account photons that incur collisions. This approach relies on the assumption that photons undergoing interactions in the medium have a very small probability of ever arriving at a detector. We also assume that the physical scale of our problem is sufficiently large so that both the source and detectors can be localized to points inside the domain. We will denote the location of the source as \(\mathbf{r}_{s}\) and associated intensity by \(S_{0}\). \(S_{0}\) can be treated as time-independent. Most radionuclides of interest for source search have half-lives on the order of several years to tens-of-thousands of years. Consequently, radioactive decay of the source is insignificant during the measurement. Under these assumptions, Equation (1) can be simplified to  
\[\hat{\mathbf{\Omega}}\cdot\
abla I(\mathbf{r},E,\mathbf{\Omega})+\Sigma_{t}( \mathbf{r},E,\hat{\mathbf{\Omega}})I(\mathbf{r},E,\hat{\mathbf{\Omega}})= \frac{S_{0}}{4\pi}\delta(E-E_{0})\delta(\|\mathbf{r}-\mathbf{r}_{s}\|_{2}), \tag{2}\]  
where \(\hat{\mathbf{\Omega}}\) is a unit vector pointing in the traveling direction of the gamma rays and \(E_{0}\) is the source emission energy and delta denotes the Dirac delta function; see [PERSON] and [PERSON] (2018) for more details. Equation (2) can be solved to determine the intensity of photons arriving at any point \(\mathbf{r}\) inside domain. This enables the computation of the count rate measured by the \(i\)-th detector \(D_{i}\) assuming that detectors are point detectors with face area \(A_{i}\) and dwell time \(\Delta t_{i}\). The detector intrinsic efficiency \(\epsilon_{i}\in[0,1]\) is usually known in practice.