# Kernel Density Derivative Estimation of Euler Solutions
## 1 Introduction
Such as, Rayleigh wave inversion [54], electromagnetic inversion [55], gravity inversion [56], and joint inversion of gravity and magnetic data [57; 58; 59; 60]. Unlike the histogram, Kernel density estimation (KDE) produces smooth probability density distributions, indicating multimodality more convincingly. The deviation of the kernel estimator is one order of magnitude greater than that of the histogram estimator [61]. The KDE is a nonparametric method for estimating the probability density function of a random variable [62]. It can reject sparsely located solutions and retain dense clusters formed by compact Euler solutions. Moreover, KDE lays a solid foundation for density-based clustering algorithms, which are based on the idea that objects forming dense regions should be grouped into a cluster [63; 64]. [PERSON] and [PERSON] [65] applied KDE to filter spurious solutions; then, a fuzzy c-means clustering algorithm (FCM) was employed in the filtered data set to locate anomalous sources. [PERSON] [66] used KDE to estimate the relationship between data and predictions from geophysical 1D inversions. The former can be applied when the relationship deviates from linear and Gaussian assumptions. [PERSON] and [PERSON] [67] employed bivariate KDE for generating the joint probability distribution of significant wave height and energy period. Traditional discrimination techniques use markers with different shapes, colors, or/and sizes to indicate depth and SI values for Euler solutions for locating anomalous sources [68; 33; 69]. However, these methods can hardly analyze complex Euler solution datasets. Furthermore, traditional discrimination techniques cannot evaluate the overall quality of the solutions or determine how dense/compact the clusters are [34; 65], such as dense distributions (for example, the centroids of geological bodies) and sparse distributions, for example, the gaps among geological bodies, of Euler solutions. The multivariate Kernel Density Derivative Estimation (KDDE), extended from KDE, can be used to determine density clusters and distributions for given sample data [70]. Non-parametricstatistical techniques, such as KDE and KDDE, have higher computational requirements because they need to evaluate joint probability density distributions on a multi-dimensional domain. Both KDDE and KDE can be considered a convolution, which can be quickly implemented using a FFT [71;72]. The tensor Euler deconvolution converges faster [33;73] and produces tighter clusters than traditional Euler deconvolution [74]. Therefore, we obtained Euler solutions from tensor Euler deconvolution of FTG data for coordinates \(x_{o}\), \(y_{o}\), \(z_{o}\), and \(N\). Then, we used multivariate KDDE with a comparative analysis process to obtain probability density curves, images, and isosurfaces for various combinations of elements of Euler solutions \(x_{o}\), \(y_{o}\), \(z_{o}\), and \(N\), such as \(\{x_{o}\}\), \(\{x_{o},z_{o}\}\), \(\{x_{o},z_{o},N\}\), and \(\{x_{o},y_{o},z_{o},N\}\). The results of synthetic models and field data show that probability density isosurfaces effectively discriminate spurious solutions and isolate complex geological sources.