## 5 Results and Discussion  
To quantify the grid-by-gird accuracy of the predicted map, we compare each of the estimated grid locations in the entire area of interest to the true simulated distribution using  
\[E=1-\frac{\sum_{g\in G}\left|\phi_{true}(g)-\phi_{pred}(g)\right|}{\sum_{g\in G} \phi_{true}(g)}\, \tag{18}\]  
where \(\phi_{true}(g)\) is the simulated flux distribution provided by MCNP6.2, and \(\phi_{pred}(g)\) is the convergent predicted contamination map given by the GPR model. The results for the five- and seven-sensor survey using the PDM-based motion planning procedure are presented in Figures 3 and 4; the true distribution is shown in Figure 1. The velocity was set to 5 m/s with a dwell time of 5 s per measurement location, where we assume a rotocopter type UAV for this study [2], but the presented algorithm is applicable to any type of UAV or UGV that is able to turn in any direction from its current position. The initial starting positions of the UAV sensors were distributed in the southern region of the one-square-kilometer predefined survey area to demonstrate a case where the UAV sensors are deployed from a common destination. The predicted contamination map in Figures 3 and 4 are for an \(E\) value of 0.90 (i.e., 90% mean accuracy), the choice of 0.90 is used here for visual purposes. Figure 5 shows the optimized motion-path trajectories from the PDM-based mapping routine overlaid on the predicted distributions. The GPR-predicted full-maps also provide prediction uncertainties for unvisited locations within the area of interest, which is a unique feature unavailable to the traditional uniform survey methods that can only provide statistical uncertainties and in turn confidence intervals at the visited locations. Figure 6 shows the GPR predicted values for dose rates through \(y=0\), where the true distribution is primarily contained within the 95% confidence interval of the predicted values. It is important to note here that there is no prediction uncertainty in the observed locations because no prediction is made but rather an actual observation. Therefore, the uncertainty in the visited locations would in principle reflect the counting statistics obtained based on the type of detector used, source strength, and the dwell time. As the number of visited locations increases (i.e., longer survey times), the prediction uncertainty of the overall full-map prediction would decrease as there would be fewer unvisited locations and in turn fewer predictions made. Figure 3: Five-sensor survey results showing (**a**) the predicted distribution for \(E=0.90\) and (**b**) the grid-by-grid fractional difference to the true distribution shown in Figure 1.  
An important distinction from typical uniform survey routines, the prediction uncertainty provided by the GPR maps is independent of knowing the true magnitude of the radiation contamination at a location but rather reflects the uncertainty based on the vicinity of the unvisited locations to the visited locations. In practice, the prediction uncertainty can be used to establish a quantitative metric for deciding a stopping criterion for the mapping routine in addition to the statistical uncertainties at visited locations that arise from the instruments used as well as the established dwell time. For example, the amount of additional survey time required to reach a desired accuracy could be inferred from the prediction uncertainties. Figure 4: Seven-sensor survey results showing (**a**) the predicted distribution for \(E=0.90\) and (**b**) the grid-by-grid fractional difference to the true distribution shown in Figure 1.  
Figure 5: Optimized motion path trajectories dictated by the PDM-informed motion planning procedure for (**a**) the five-sensor and (**b**) seven-sensor surveys. The white markers represent locations where a measurement has been taken.