# Kernel Density Derivative Estimation of Euler Solutions
## 4 Discussion  
Spurious solutions may be because of noise, either natural or due to the FFT [32], which was employed to calculate the spatial gradients. This easily produces spurious solutions and yields poor results, especially for potential profile data [24; 69]. According to our numerical results, satisfactory probability density distribution results were obtained using 3-D KDDE combined with tensor Euler deconvolution compared to the traditional Euler deconvolution, Euler deconvolution with truncated singular value decomposition [17], and joint Euler deconvolution of gravity gradiometry data [97]. The \(N\) is a function of the moving window size and the source-to-observation distance. Consequently, for an arbitrarily shaped source, Euler deconvolution easily yields spurious solutions because no single \(N\) is correct at all source-to-observation distances [22]. In addition, using a series of predefined SIs according to a particular step size, the procedure of Euler deconvolution will yield a large number of solutions [25]. To overcome these problems, the \(N\) was not predicted in this paper. Figure 21: Probability density images obtained using 3-D KDDE to analyze the subset \(\{x_{0},y_{0},z_{0}\}\) in Figure 16, at different depths, \(z\): (**a**) \(-600\); (**b**) 350; (**c**) 1350; (**d**) 2300; (**e**) 3300; (**f**) 4250; (**g**) 5250; (**h**) 6200; and (**i**) 7200. Figure 22: Probability density isosurfaces obtained using 3-D KDDE to analyze the subset \(\{x_{0},y_{0},z_{0}\}\) (Figure 16) in (**a**) the plan view (from bottom to top) and (**b**) the perspective view. Finally, following the suggestions of [PERSON] et al. [69], we proposed a multivariate KDDE approach based on nonparametric estimation to overcome the problem that traditional discrimination techniques are challenging to analyze Euler solutions for visualizing Euler solutions on modern graphics workstations. For a given Euler solution dataset, the grid size is inversely proportionate to the bandwidth. The bandwidth features a trade-off between the estimator's bias and its variance. When the bandwidth decreases, the estimated point density has abrupt changes; when it increases, the estimated point density change becomes smoother. The quality of the final probability density distribution of multivariate KDDE is determined by the bandwidth parameters/the grid size. However, if the grid size increases, the computational efficiency of multivariate KDDE with nested for-loops will dramatically decrease. Moreover, when the sample data are combined with high-dimensional sparse data, determining the \"best\" bandwidth/grid size is time-consuming [62]. The multivariate KDDE, in conjunction with the fast linear binning approximation and the FFT, was proposed with high computational efficiency to overcome this difficulty. In applying the multivariate KDDE to field data, while the computational efficiency of the multivariate KDDE is essential, we were also concerned with the ability for noise suppression and distinguishing spurious solutions. Hence, based on adjacent anomalous sources with various separations, the results of employing FCM, DBSCAN, and 3-D KDDE were compared and analyzed. Of them, using the probability density contours/isosurfaces of the 3-D KDDE allowed efficient separation of adjacent anomalous sources. Furthermore, the results of the 3-D KDDE for different Gaussian noises indicated that it can effectively separate adjacent anomalous sources and perform consistently, which provides a basis for the application of the multivariate KDDE to complex data. Because of the high-computational efficiency of the multivariate KDDE method, we could try different-sized grids or/and select various \(l\)th layer contours to find interesting geological structures more effectively. For example, the 3-D KDDE, in conjunction with the subset \(\{x_{o},y_{o},z_{o}\}\) in Figure 16, was carried out with different-sized cubic grids, whose length along one direction was 16, 32, 64, 128, 256, and 512.