## 3 Gaussian Process Regression Model for Radiation Contamination Mapping  
GPR is a non-parametric regression technique, where the data is modeled as a Gaussian process (\(GP\)). A \(GP\) is defined as a collection of random variables where any finite number of the random variables have a joint Gaussian distribution [23; 24]. The output of a \(GP\)  
Figure 1: Simulated radiation contamination site showing gamma-ray dose rate distribution. Global average of terrestrial background dose rate is approximately \(5.5\times 10^{-3}\) mrem/h.  
model, \(\mathbf{y}=f(\mathbf{x})\) is fully described in terms of the input, \(\mathbf{x}\), with the mean function \(m(\mathbf{x})\) and the covariance function \(k(\mathbf{x},\mathbf{x}^{\prime})\) defined as:  
\[m(\mathbf{x})=\mathbb{E}[f(\mathbf{x})], \tag{1}\]  
\[k\big{(}\mathbf{x},\,\mathbf{x}^{\prime}\big{)}=\mathbb{E}[f(\mathbf{x})-m(\mathbf{x}))\big{(}f \big{(}\mathbf{x}^{\prime}\big{)}-m\big{(}\mathbf{x}^{\prime}\big{)})]. \tag{2}\]  
The \(GP\) can empirically be written as:  
\[\mathbf{y}=f(\mathbf{x})\ \sim\ GP\big{(}m(\mathbf{x}),\ k\big{(}\mathbf{x},\,\mathbf{x}^{\prime} \big{)}\big{)}. \tag{3}\]  
To apply the GPR technique for predicting dose rates from the detected count rates, and in turn, the contamination at various locations in the area of interest, we define the collection of previously observed data as (\(\mathbf{P}=(\mathbf{p_{1}},\,...,\,\mathbf{p_{n}}^{\top})\), \(\mathbf{c}=(c_{1},\,...,\,c_{n})^{\top}\)) where \(\mathbf{c}\) is an \(n\times 1\) vector containing the radiation count rate, and \(\mathbf{P}\) is an \(n\times 2\) matrix that contains two features describing the location (i.e., the corresponding latitudinal and longitudinal coordinates) for \(n\) observations [24]. This model assumes that the observed count rate \(c\) behaves as a normal distribution about the corresponding mean count rate \(m(\mathbf{p})\) for observation location \(\mathbf{p}\), with a covariance directly related to the relationship between all observation locations (e.g., \(\mathbf{p}\) and \((\mathbf{p_{1}},\,\ldots,\,\mathbf{p_{n}})\). Using the aforementioned definitions and employing the properties of \(GP\), we define the GPR model as  
\[\begin{pmatrix}c_{*}\\ c_{1}\\ \vdots\\ c_{n}\end{pmatrix}\sim\mathcal{N}\left(\mu=\begin{pmatrix}m(\mathbf{p_{*}})\\ m(\mathbf{p_{1}})\\ \vdots\\ m(\mathbf{p_{n}})\end{pmatrix},\ \Sigma=\begin{pmatrix}k_{**}&k_{*}\\ k_{*}&k_{.}\end{pmatrix}\right), \tag{4}\]  
with the components of the covariance matrix \(\Sigma\) defined as  
\[k_{**}=k(\mathbf{p_{*}},\,\mathbf{p_{*}})\, \tag{5}\]  
\[k_{*}=(k(\mathbf{p_{*}},\,\mathbf{p_{1}}),\cdots,\ k(\mathbf{p_{*}},\,\mathbf{p_{n}})), \tag{6}\]  
\[k_{*}=(k(\mathbf{p_{1}},\,\mathbf{p_{*}}),\,\cdots,\ k(\mathbf{p_{n^{\prime}}}\,\,\mathbf{p_{ n}}))^{\top}\, \tag{7}\]  
\[k_{-}=\begin{pmatrix}k(\mathbf{p_{1}},\,\mathbf{p_{1}})&\cdots&k(\mathbf{pconsiders uncertainties independent of the predictions made. There are many commonly used kernels for describing the covariance function and several different types were tested in this work by using a five-fold cross-validation test [23]. The cross-validation testing involves taking a fixed number of random samples from the data to be used as the training set, whereas the remaining partition of the data is used as the test set to evaluate the fidelity of the kernel used detailed in Section 3.2. Using the GPR model described in (4), the observed count rates and the respective latitudinal and longitudinal coordinates are used as input data and the remaining unvisited locations \(\mathbf{P}_{*}\) are used to predict the respective count rates \(\mathbf{c}\). Once all the count rates at the unvisited locations are predicted, it can be projected as a two-dimensional grid corresponding to their respective locations. The predicted GPR full-map is defined as  
\[\phi(g),\ \ \ for\ g\in G,\ \ \ G\subset\mathbb{R}^{2}. \tag{10}\]