# Kernel Density Derivative Estimation of Euler Solutions
## 1 Introduction  
The studies on the edges of geological bodies are significant for potential field data interpretation. Furthermore, when studying lateral inhomogeneity, especially the position of the edges of anomalous sources, gravity and magnetic surveys have advantages for detecting geological structures like faults and tectonic boundaries [1]. With the development of gravity gradient measurement methods and corresponding interpretation techniques, many edge-enhancement techniques have been proposed in recent years, such as edge detection [2], total variation [3], edge-preserved inversion [4], and focusing inversion [5]. Gravity gradient measurements are a multicomponent gradiometric surveying technology that measures different components of Full Tensor Gravity (FTG) data and have been widely carried out and have yielded large amounts of high-precision FTG data. However, FTG data impose higher requirements on their corresponding interpretation techniques because they often generate more complex anomalies for given sources [6]. Compared with other interpretation techniques of FTG data, Euler deconvolution is suitable for analyzing and interpreting large-area potential field data because it does not rely on prior geological information [7, 8]. Euler deconvolution is based on the Euler homogeneity relationship and uses a moving window to traverse potential field data todetermine locations and structural indices (SIs) of geological targets [9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. The SIs are associated with the source geometry [19; 20; 21] and need to be determined in advance for performing Euler deconvolution [22]. Nevertheless, the use of a single structural index (\(N\)) makes it difficult to characterize multiple anomaly sources [22; 23; 24; 25; 26]. Many extended Euler deconvolution methods have been presented to reduce the influence of the SIs [27; 28; 29; 30; 31]. However, traditional Euler deconvolution produces spurious solutions possibly because of noise, either natural or due to the fast Fourier transform (FFT) [32; 33; 34; 35; 36]. To overcome this difficulty, various methods have been proposed to optimize the determination of SIs [37; 38; 39; 40; 41], many extended Euler deconvolution methods are proposed to eliminate the dependence on predicted \(N\)[23; 42; 43; 44; 45; 46; 47], and several methods have been proposed to filter out spurious solutions [17; 30; 34; 35]. For example, [PERSON] and [PERSON] [14] eliminated spurious solutions using the differential similarity transformation based on the error estimations of singular point coordinates and SIs. [PERSON] et al. [48] proposed a horizontal gradient filtering criterion for removing solutions with a small horizontal gradient modulus, a distance constraint criterion for rejecting solutions far from their corresponding moving windows, and a constraint criterion for eliminating solutions with low convergence. [PERSON] et al. [30] filtered spurious solutions according to their clustering, misfit, and the depth and distance from the moving window. Euler deconvolution with the correct predicted \(N\) yields a tight cluster, which is formed by Euler solutions [19; 25; 49]. [PERSON] et al. [50] developed a clustering technique based on artificial intelligence to select reliable Euler solutions. However, clustering methods cannot discriminate spurious solutions efficiently in shallow sources and neighboring bodies [50; 51]. [PERSON] et al. [34] reported spurious solution rejection using skewness and kurtosis obtained by histogram; however, it was difficult to correct for multiple anomalous sources. The density histogram was employed to visualize each formation to check the coherency of the inversion process [52]; however, it was difficult to efficiently divide a multi-dimensional space into non-coherent histogram bins to keep the error rates small [53]. A probability density distribution is a continuous version of a histogram with density and is widely used in geophysical inversions.