## 2 Methodology  
The classical formulation of Euler deconvolution can be written as  
\[x_{0}\frac{\partial T}{\partial x}+y_{0}\frac{\partial T}{\partial y}+z_{0}\frac{ \partial T}{\partial z}-N(T-B)=x\frac{\partial T}{\partial x}+y\frac{\partial T }{\partial y}+z\frac{\partial T}{\partial z} \tag{1}\]  
where \(x\), \(y\) and \(z\) are the observation positions. \(T\), \(\frac{\partial T}{\partial x}\), \(\frac{\partial T}{\partial y}\) and \(\frac{\partial T}{\partial z}\) are the potential field data and their gradients with respect to the variables \(x\), \(y\), and \(z\). \(x_{0}\), \(y_{0}\) and \(z_{0}\) are the source positions. \(B\) is the background value, and \(N\) is the structural index (SI). In Euler deconvolution, the SI is generally assumed to be constant in each sliding-window. Assuming that for all the points in a window the background \(B\) is constant, the product of \(N\) and \(B\) are constant and we can write  
\[NB=(x_{c}-x_{0})(\frac{\partial T}{\partial x})_{c}+(y_{c}-y_{0})(\frac{ \partial T}{\partial y})_{c}+(z_{c}-z_{0})(\frac{\partial T}{\partial z})_{c }+NT_{c} \tag{2}\]  
where \(c\) is the central window point. Substituting Equation (2) into Equation (1), we can obtain  
\[ \begin{array}{c}x_{0}(\frac{\partial T_{i}}{\partial x}-\frac{\partial T_{ c}}{\partial x})+y_{0}(\frac{\partial T_{i}}{\partial y}-\frac{\partial T_{ c}}{\partial y})+z_{0}(\frac{\partial T_{i}}{\partial z}-\frac{\partial T_{ c}}{\partial z})-N(T_{i}-T_{c})\\ =x_{i}\frac{\partial T_{i}}{\partial x}+y_{i}\frac{\partial T_{i}}{\partial y }+z_{i}\frac{\partial T_{i}}{\partial z}-x_{c}\frac{\partial T_{c}}{\partial x }-y_{c}\frac{\partial T_{c}}{\partial y}-z_{c}\frac{\partial T_{c}}{\partial z }\end{array}  \tag{3}\]  
where \(i\) is the window point. Equation (3) is linear with an unknown source position and SI, and the least squares method can be applied to solve the unknown parameters. It is the basic equation of the finite-difference Euler deconvolution algorithm proposed by [PERSON] [16]. However, the method limits itself to accounting for only a constant background. To some extent, the finite-difference Euler deconvolution method improves the accuracy of Euler solutions. However, the background field is not constant and always varies with the position. In Euler deconvolution, 5-10 times the grid interval may be a good choice for the window size [23] and we consider approximating the background as a linear function in a window more realistic, especially in the case of closely located anomalies that interfere with each other. In this paper, we assume that \(B=ax+by+cz+d\), and that \(NB\) varies with \(x\), \(y\), and \(z\). Substituting \(B\) into Equation (1),  
\[ \begin{array}{c}x_{0}(\frac{\partial T}{\partial x}-a)+y_{0}(\frac{\partial T }{\partial x}-b)+z_{0}(\frac{\partial T}{\partial z}-c)-N(T-ax-by-cz-d)\\ =x(\frac{\partial T}{\partial x}-a)+y(\frac{\partial T}{\partial y}-b)+z( \frac{\partial T}{\partial z}-c)\end{array}  \tag{4}\]  
In Equation (4), \(d\) is constant and hence, \(N^{\ast}d\) can be treated as a constant. Based on the finite-difference Euler deconvolution algorithm, for all the window locations, we can write  
\[ \begin{array}{c}Nd=x_{c}\frac{\partial T_{c}}{\partial x}+y_{c}\frac{ \partial T_{c}}{\partial y}+z_{c}\frac{\partial T_{c}}{\partial z}-x_{0}\frac{ \partial T_{c}}{\partial x}-y_{0}\frac{\partial T_{c}}{\partial y}-z_{0}\frac{ \partial T_{c}}{\partial z}+ax_{0}+by_{0}+cz_{0}\\ +NT_{c}-Nax_{c}-Nby_{c}-Ncz_{c}-ax_{c}-by_{c}-cz_{c}\end{array}  \tag{5}\]  
\(c\) is the central window point. Substituting \(Nd\) into Equation (4), we can obtain a new formula:  
\[