# Kernel Density Derivative Estimation of Euler Solutions
## 5 Conclusions  
The Geoscience BC project provided the field data used in the study. The residual Bouguer gravity anomaly was the difference between the gravity data continuing upward to \(-2700\) and \(-5700\) m to remove the contributions induced by near-surface. By comparing the results of two clustering methods in Figures 8 and 19 and traditional discriminate techniques in Figures 17 and 18, the results of the multivariate KDDE in conjunction with the subset \(\{x_{o},y_{o},z_{o}\}\) obtained using tensor Euler deconvolution were more intuitive and can effectively characterize the geological targets in Figures 9, 21 and 22. In addition, compared to the Occam's -style algorithm failing to recover the depth resolution from the surface gravity data (Figure 20), our results effectively obtained the depth resolution of the anomalous sources. In illustrating Euler solutions, because many Euler solutions need to be drawn one by one, the top/newly drawn ones will obscure the bottom/previously drawn ones, whether using a three-dimensional or two-dimensional scatter plot. Therefore, the multivariate KDDE method of Euler solutions for characterizing geological bodies proposed in this paper contains the following processes: selecting a series of different-sized or fixed-size moving windows, traversing the gridded data using Equation (4) to obtain Euler solutions, constructing an estimation grid with predefined grid size \(M\) or bandwidth \(H\), and binning the subset \(\{x_{o},y_{o},z_{o}\}\) into the estimation grid via the fast linear binning approximation to yield grid counts \(c\), calculating kernel function evaluations \(k\) based on Equation (13), constructing the zero-padded versions of the grid count \(C\) and the kernel evaluation \(K\), and carrying out the convolution between \(C\) and \(K\) quickly by FFT to get the probability density distribution of the subset \(\{x_{o},y_{o},z_{o}\}\). Tests on the field data showed that the multivariate KDDE, which was used in conjunction with the subset \(\{x_{o},y_{o},z_{o}\}\) obtained using tensor Euler deconvolution, proposed in this paper, does not rely on a priori geological information. Therefore, it is suitable for analyzing and interpreting large-area potential field data. In addition, the depth resolution information extracted from the surface gravity data using 3-D KDDE can be used as a constraint to analyze geological targets. Consequently, the ideal geological targets would be compact three-dimensional bodies with sharp boundaries, such as salt domes, ore bodies, and igneous intrusions. The estimated grid size is proportional to the bandwidth size for a given input data instance. Therefore, only one parameter, \(H\) or \(M\), was required in the manuscript regarding the multivariate KDDE to process the sample data. Regarding the analysis of the multivariate KDDE of the field data, the required data and parameters were \(g_{z}\), \(w_{x}\), \(w_{y}\), \(H\), or \(M\). Conceptualization, methodology, software, writing, review and editing, S.C.; software, validation, Y.D. ; data curation, investigation, writing original draft, B.Y. ; resources, supervision, writing, review and editing, G.L. ; formal analysis, resources, X.H. ; validation, visualization, Y.M. ; software, S.H. ; conceptualization, supervision, Z.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China under Grant 41704138, Grant 41974148, and in part by the Hunan Provincial Science & Technology Department of China under Grant 2017 JJ3069, and in part by the Project of Doctoral Foundation of Hunan University of Science and Technology under Grant E51651, and in part by the Hunan Provincial Key Laboratory of Share Gas Resource Exploitation under Grant E21722. Not applicable. Data Availability Statement: Data are unavailable. We acknowledge Geoscience BC for making the data available. Valuable comments by two anonymous reviewers are gratefully acknowledged.