# Kernel Density Derivative Estimation of Euler Solutions
## 3 Results  
### Model Studies  
#### 3.1.1 Verification of the Validity of the Multivariate KDDE Algorithm  
Two simple models (cube and cylinder) were established to validate the tensor Euler deconvolution algorithm and the multivariate KDDE algorithm. Both models have a density contrast of 0.36 g/cm\({}^{3}\). The geometric parameters and theoretical \(N\) of the two models are shown in Table 1. Then, a rectangular survey grid, with 200 points, with 100 m grid interval in both horizontal directions, was employed for forward processing. The study used the noise-free FTG data obtained by the analytical solutions and noise-corrupted data, which contaminated the FTG data by Gaussian noise with zero-mean and variance \(\sigma(p)\), which is a function of the percentage \(p\) of input data:  
\[\sigma(p)=\frac{p}{100}\sqrt{\frac{\sum_{i=1}^{n_{obs}}obs_{i}^{2}}{n_{obs}}} \tag{17}\]  
where \(obs_{i}\) is the \(i\)th data and \(n_{obs}\) is the number of the given data sample. Equations (1)-(3) were applied to the noise-free and the noise-corrupted FTG data to estimate \(x_{\rm o}\), \(y_{\rm o}\), \(z_{\rm o}\), and \(N\). The size of the square moving window was \(w_{x}=w_{y}=15\). As shown in Figure 4, the solutions were filtered by \(N>0\). \begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **Centroid** & **Radii** & **Lengths** & **Theoretical \(N\)** \\ \hline Cube & (\(-1000\), \(-2000\), \(1500\)) & / & \(1000\times 1000\times 1000\) & 2 \\ Cylinder & (\(0,0,2500\)) & 1000 & 4000 & 1\(\sim\)2 \\ \hline \end{tabular}
\end{table} 
Table 1: The geometric parameters and structural indices for two simple models. Figure 4: Scatter plots of the Euler solutions. (**a,b**) Contours of noise-free gravity. (**c,d**) Euler solutions derived from noise-free FTG. (**e,f**) Euler solutions derived from noise-corrupted FTG. (**g,h**) Euler solutions for noise-corrupted FTG in the plan view, for the cube and the cylinder, whose frames are solid black. As Figure 4b shows, the Euler solutions were around the center of the cylinder and their SIs (\(N\approx 2\)) were consistent with the theoretical values. In Figure 4c,d, the SIs of the Euler solutions in red were consistent with their theoretical values (Table 1). Compared to the cylinder's Euler solutions, the cube's ones formed a tighter cluster in Figure 4c. There were no spurious solutions in the Euler solution dataset in Figure 4c,d. Figure 4e-h show that the magnitudes of the SIs for the Euler solutions around the anomalous sources were consistent with their theoretical values. However, there were many more spurious solutions. The histogram was used to discriminate spurious Euler solutions [34]. When there was no noise in the input data, the tensor Euler deconvolution yielded very pure solutions, indicating that many similar or duplicate solutions were generated and may be clustered tightly in the vicinity of the real sources (Figures 4c and 5a). Conversely, the tensor Euler deconvolution produced many spurious solutions in Figures 4e-h and 5b,d. This phenomenon shows that the tensor Euler deconvolution was susceptible to noise. Moreover, as for the cylinder model, the SIs derived from the noiseless data fluctuated between 0.41 and 2.25; in contrast, the solutions derived from noise-corrupted data tended to diverge because their maximum \(N\) value was smaller than the former. For comparison, using the 7 th, 8 th, and 9 th strategies proposed by [PERSON] et al. [34] to select the most reliable Euler solutions, we could effectively remove spurious solutions (such as \(N>3\) or \(N<0\)) to quickly determine the anomaly source's location in Figure 5c,d. Compared to the Euler solutions drawn from the noise-free data, the magnitudes of all SIs decreased for the ones derived from the noise-corrupted data in the cylinder model. [PERSON] [22] pointed out that the \(N\) changes discontinuously as the source-to-observation distance changes. Therefore, only using the SIs made it difficult to determine optimal solutions.