# Kernel Density Derivative Estimation of Euler Solutions
## 2 Materials and Methods  
### Tensor Euler Deconvolution  
Euler deconvolution is a semi-automatic/automatic interpretation method based on Euler's homogeneity equation, and it has been widely used to interpret profile, grid, and ungridded potential field data. The higher-order differentiation of gravitational potential is the homogenous function in [PERSON]'s sense. The tensor Euler deconvolution equation is composed of two similar Equations (1) and (2) and the traditional Euler Equation (3):  
\[x_{o}\frac{\partial g_{x}}{\partial x}+y_{o}\frac{\partial g_{x}}{\partial y} +z_{o}\frac{\partial g_{x}}{\partial z}-N(B_{x}-g_{x})=x\frac{\partial g_{x}}{ \partial x}+y\frac{\partial g_{x}}{\partial y}+z\frac{\partial g_{x}}{ \partial z}+\gamma_{x} \tag{1}\]  
\[x_{o}\frac{\partial g_{y}}{\partial x}+y_{o}\frac{\partial g_{y}}{\partial y} +z_{o}\frac{\partial g_{y}}{\partial z}-N\big{(}B_{y}-g_{y}\big{)}=x\frac{ \partial g_{y}}{\partial x}+y\frac{\partial g_{y}}{\partial y}+z\frac{ \partial g_{y}}{\partial z}+\gamma_{y} \tag{2}\]  
\[x_{o}\frac{\partial g_{z}}{\partial x}+y_{o}\frac{\partial g_{z}}{\partial y} +z_{o}\frac{\partial g_{z}}{\partial z}-N(B_{z}-g_{z})=x\frac{\partial g_{z}} {\partial x}+y\frac{\partial g_{z}}{\partial y}+z\frac{\partial g_{z}}{ \partial z}+\gamma_{z} \tag{3}\]  
where (\(x\), \(y\), \(z\)) are the coordinates of the observation points; (\(x_{o}\), \(y_{o}\), \(z_{o}\)) is the location of the source; \(g_{x}\), \(g_{y}\), \(g_{z}\) are the components of the gravity vector; for all pairs (\(\alpha,\beta\)) in \(\{x,y,z\}\), \(g_{\alpha\beta}\) represents a component of FTG, which is obtained by analytical solutions, or a FFT; the structural index \(N\) depends on the source geometry; the parameter \(B_{\alpha}\) is simply the background, or regional field [33]; \(\gamma_{\alpha}\) is a constant that usually vanishes except for \(N=0\)[65]. The \(N\) defining the geometry of the anomalous source needs to be determined in advance [22;38]. However, the use of a single \(N\) makes it difficult to characterize multiple anomaly sources [22;23;24;25;26]. Traditional Euler deconvolution uses a series of predefined SIs and yields many spurious solutions. Alternatively, using tentative SIs makes Euler deconvolution subjective and time-consuming because the interpreter/user must evaluate the quality of clusters of Euler solutions as the SIs change [19]. To overcome these problems, \(N\) will not be predicted [14;34;35].  
Tensor Euler deconvolution uses a moving window (length \(\times\) width = \(w_{x}\times w_{y}\)) with \(w_{n}\) observation points to traverse gridded potential field data. Then, each window location generates a solution consisting of the causative source's location (\(x_{o}\), \(y_{o}\), \(z_{o}\)) and \(N\). For the \(i\)th moving window, Equations (1)-(3) are rewritten in matrix form as follows:  
\[\mathbf{Am}=\mathbf{b} \tag{4}\]  
where \(\mathbf{A}=[\mathbf{A}_{1},\ \ldots,\mathbf{A}_{l},\ \ldots,\mathbf{A}_{w_{n}}]^{T}\), \(\mathbf{A}_{l}=[(\partial g_{\alpha}/\partial x)_{l},(\partial g_{\alpha}/\partial y )_{l},(\partial g_{\alpha}/\partial z)_{l},\ (g_{\alpha}-B_{\alpha})_{l}]\), \(\mathbf{b}=[\mathbf{b}_{1},\ \ldots,\mathbf{b}_{l},\ \ldots,\mathbf{b}_{w_{n}}]^{T}\), and \(\mathbf{b}_{l}=(x\partial g_{\alpha}/\partial x)_{l}+(y\partial g_{\alpha}/ \partial y)_{l}+(z\partial g_{\alpha}/\partial z)_{l}\). The subscript \(l\) represents the sequence number of the moving sliding window, \(m\) represents the Euler solution datasets \(\{x_{o},y_{o},z_{o},N\}\), and \(\mathbf{m}_{i}\) is the \(i\)th Euler solution written as \([x_{o}^{i},y_{o}^{i},z_{o}^{i},N^{i}]\).