## 2 Methodology
\begin{array}{c}x_{0}(\frac{\partial T_{i}}{\partial x}-\frac{\partial T_{ c}}{\partial x})+y_{0}(\frac{\partial T_{i}}{\partial y}-\frac{\partial T_{ c}}{\partial y})+z_{0}(\frac{\partial T_{i}}{\partial z}-\frac{ \partial T_{c}}{\partial x})+a(N+1)(x_{i}-x_{c})\\ +b(N+1)(y_{i}-y_{c})+c(N+1)(z_{i}-z_{c})+N(T_{c}-T_{i})\\ =x_{i}\frac{\partial T_{i}}{\partial x}+y_{i}\frac{\partial T_{i}}{\partial y }+z_{c}\frac{\partial T_{i}}{\partial z}-x_{c}\frac{\partial T_{c}}{\partial x }-y_{c}\frac{\partial T_{c}}{\partial y}-z_{c}\frac{\partial T_{c}}{\partial z} \end{array}  \tag{6}\]  
The obtained equation, Equation (6), derived from [PERSON]'s homogeneity equation for a case when the measured field contains a homogeneous anomalous field and a linear background, \(B\) is linear to the coordinates of the field source. Therefore, the source parameters and the coefficient of the background field can be estimated simultaneously. If \(N\) is given, there are only six unknown parameters, which can be obtained by solving overdetermined equations. If we make the following abbreviations,  
\[ \begin{array}{c}\delta T_{xi}=\frac{\partial T_{i}}{\partial x}-\frac{\partial T_ {c}}{\partial x},\delta T_{yi}=\frac{\partial T_{i}}{\partial y}-\frac{ \partial T_{c}}{\partial y},\delta T_{zi}=\frac{\partial T_{i}}{\partial x}- \frac{\partial T_{c}}{\partial z}\\ \delta T_{i}=T_{i}-T_{c},\delta x=x_{i}-x_{c},\delta y=y_{i}-y_{c},\delta z=z_ {i}-z_{c},\\ \delta xyT=_{i}\frac{\partial T_{i}}{\partial x}+y_{i}\frac{\partial T_{i}}{ \partial y}+z_{c}\frac{\partial T_{i}}{\partial z}-x_{c}\frac{\partial T_{c}} {\partial x}-y_{c}\frac{\partial T_{c}}{\partial y}-z_{c}\frac{\partial T_{c} }{\partial z}\end{array}  \tag{7}\]  
from Equation (6) for every grid point of a sliding window, we obtain  
\[ \begin{array}{c}x_{0}\delta T_{xi}+y_{0}\delta T_{yi}+z_{0}\delta T_{zi}+a( N+1)\delta x+\\ b(N+1)\delta y+c(N+1)\delta z+N\delta T_{i}=\delta xyT\end{array}  \tag{8}\]  
or, in a matrix form,  
\[\text{Gp}=\text{y} \tag{9}\]  
where p represents the unknown parameters, G is a coefficient matrix, composed of the field anomaly, the gradient of the anomaly and the coordinate points within the window, and y is a vector of the sums of the product differences of the coordinates and the respective gradients of the field. The least squares value of p is given by  
\[\text{p}=(\text{G}^{\text{T}}\text{G})^{-1}\text{G}^{\text{T}}\text{y} \tag{10}\]  
Theoretically, the background function can be regarded as a higher-order functional form (e.g., 2-order or 3-order) and the finite-difference method can still be applied to the Euler equation. Owing to the complexity and unknown nature of the background function, the variation form of the function is unknown. When the selected window size is large, treating the background function as a high-order case is reasonable. However, a large window size may lead to missing small-scale field sources. Therefore, in practical applications, the window size should not be too large. Windows containing between \(5\times 5\) and \(10\times 10\) grid points may be the best choice [23]. If the selected order of the background field is high (e.g., 2, 3, etc. ), it will lead to overfitting and more valuable anomalies being separated, and this will eventually lead to inaccurate inversion results. Hence, accounting for a linear background in the anomalous field, the Euler deconvolution may perform well in the presence of interfering fields. In the next section, we will compare the inversion accuracy of our algorithm with that of the Euler inversion algorithm based on DST and the finite-difference Euler deconvolution algorithm.