# Kernel Density Derivative Estimation of Euler Solutions
## 3 Results
### Case Study in British Columbia, Canada  
The survey area was located in the Quesnellia Region in south-central British Columbia, Canada, a region famous for copper and gold porphyr deposits. Several gold and silver (plus copper, lead, and zinc) deposits have been mined in this area since 1884 (Figure 12). The region is covered by a thick layer of sand and gravel left behind by glaciers. Airborne gravity surveys were used to identify potential targets and help understand the geology of this area [91; 92]. In our study, the survey block consisted of two connected parallelograms, where the larger one was about 386 km in length and 120 km in width, and the smaller one was 120 km in length and 60 km in width. Figure 13 shows the construction of gravity data for irregular gravity anomaly data, with a grid size of 250 \(\times\) 250 m. Figure 13a illustrates the topographic relief contours of the study area, and the measured heights ranged from 717 to 2609 m above the mean sea level. Potential data continuation from an uneven surface to the horizontal planes \(-\)2700 m and \(-\)5700 m was necessary to obtain the new gravity and background field in Figure 13b. The continuation height corresponding to the background field was determined by the radial average spectrum using Oasis montaj software [93]. The background field was subtracted from the new gravity anomaly to obtain the local gravity anomaly shown in Figure 13c. The FTG data were derived from the local gravity anomaly using the FFT, but only the gzz component of the FTG is illustrated in Figure 13d. The entire survey gravity map was covered by a series of moving windows with sizes ranging from 4 \(\times\) 4 to 12 \(\times\) 12 data points to form tight clusters of solutions. A total of 2,592,878 solutions were generated using tensor Euler deconvolution before filtering. The Euler solutions are drawn in the plan and perspective views in Figure 14. Difficulties were encountered in separating adjacent clusters of solutions (Figure 14a) and determining the depths of the causative bodies (Figure 14b). Figure 12: Locations map. Figure 14: Scatter plots of the Euler solutions: (**a**) Plan view (from top to bottom); (**b**) Perspective view. Figure 13: Construction of FTG data for irregular gravitational anomaly data. (**a**) The study area’s topographic relief is shown with contours. (**b**) The regional gravity field. (**c**) The local gravity field. (**d**) The \(g_{zz}\) component of the FTG. To show the details of the 3-D KODE results, we selected a small survey grid (red line in Figure 15) with a similar process to that in Figure 14, to yield 155,327 Euler solutions, as shown in Figure 16.  
The traditional Euler deconvolution tried to determine the locations of geological targets using a series of tentative \(N\). We used different minimum \(N\) and \(z\) to filter Euler solutions to simulate this process, as shown in Figures 17 and 18. The scatter plots of the Euler solutions filtered by minimum \(N\) (Figure 17) show that the large clusters, especially those in Figure 17d-i, corresponded to the gravity high of \(\mathrm{g_{z}}\) in the survey grid in Figure 15. However, it was difficult to determine which cluster corresponds to weak anomalous sources. Figure 16: Scatter plots for Euler solutions in (a) the plan view (from bottom to top) and (b) the perspective view, respectively. Figure 15: The square survey grid is colored in red over the images of \(g_{z}\). The size of the survey grid was \(250\times 250\) m.  
The scatter plots of the Euler solutions filtered by different depths (Figure 18) show that multiple-ridge clusters correspond to the boundaries of the gravity high of g\({}_{z}\) in the survey grid in Figure 15. However, this correspondence became worse as the depth increases. Therefore, it was difficult to determine which cluster indicated an actual geological body.