# A Stable and High-Precision Downward Continuation Method of Magnetic Data
## 2 Methodology of the Proposed Method  
The proposed method is based on the equivalent source method; it is different from the conventional downward continuation technique, and there is no need to modify the continuation operator to amplify the signal, thus avoiding the influence of noise. It focuses on inverting a set of equivalent sources corresponding to the observed data, and then the magnetic data of the target depth can be obtained by forward calculation. Figure 1 shows the space relationship of the observation surface, equivalent source cells, and measured data. However, the coordinates for measured data at the lower interface are hard to match with the observation network appropriately all the time. Therefore, its position needs to be interpolated to the observation network. We suppose that the coordinate of a constrained point is \((x_{ct},y_{ct},z_{ct})\) and the value is \(d_{ct}\); its position at the grid is shown in Figure 2, and the nearest joint point is \((x_{dj},y_{dj},0)\). Figure 1: Sketch map of the space relationship of the observation surface, downward continuation surface, equivalent source, and constrained data. Correspondingly, the forward operator takes the form of Equation (4) after adding constraint data:  
\[ \begin{split}& g_{ij}=\frac{H\cdot K}{4\pi}\big{\{}d_{1}\cdot\ln[r+(x_{mj }-x_{ci})]+d_{2}\cdot\ln[r+(y_{mj}-y_{ci})]+d_{3}\cdot\ln[r+(z_{mj}-z_{ci})]\\ &\quad+d_{4}\cdot\arctan(\frac{(x_{mj}-x_{ci})(y_{mj}-y_{ci})}{(x _{mj}-x_{ci})^{2}+r(z_{mj}-z_{ci})+(z_{mj}-z_{ci})^{2}}\\ &\quad+d_{5}\cdot\arctan\frac{(x_{mj}-x_{ci})(y_{mj}-y_{ci})}{(y _{mj}-y_{ci})^{2}+r(z_{mj}-z_{ci})+(z_{mj}-z_{ci})^{2}}\\ &\quad+d_{6}\cdot\arctan\frac{(x_{mj}-x_{ci})(y_{mj}-y_{ci})}{r( z_{mj}-z_{ci})}\big{\}}\big{[}c_{x_{jm}}^{(x_{jm})(y_{mx})(z_{jm})}\]  
\[\left\{\begin{array}{l}d_{1}=\cos\mathrm{l}_{0}\cdot\sin\mathrm{A}_{0}^{ \prime}\cdot\sin\mathrm{l}+\sin\mathrm{I}_{0}\cdot\cos\mathrm{I}\cdot\sin \mathrm{A}^{\prime}\\ d_{2}=\cos\mathrm{l}_{0}\cdot\cos\mathrm{A}_{0}^{\prime}\cdot\sin\mathrm{l}+ \sin\mathrm{I}_{0}\cdot\cos\mathrm{I}\cdot\cos\mathrm{A}^{\prime}\\ d_{3}=\cos\mathrm{l}_{0}\cdot\sin\mathrm{A}_{0}^{\prime}\cdot\cos\mathrm{l}+ \cos\mathrm{I}_{0}\cdot\sin\mathrm{A}_{0}^{\prime}\cdot\cos\mathrm{I}\cdot \cos\mathrm{A}^{\prime}\\ d_{4}=\cos\mathrm{l}_{0}\cdot\cos\mathrm{A}_{0}^{\prime}\cdot\cos\mathrm{I} \cdot\cos\mathrm{A}^{\prime}\\ d_{5}=\cos\mathrm{l}_{0}\cdot\sin\mathrm{A}_{0}^{\prime}\cdot\cos\mathrm{I} \cdot\cos\mathrm{A}^{\prime}\\ d_{6}=-\sin\mathrm{I}_{0}\cdot\sin\mathrm{I}\end{array}\right.\end{split}  \tag{4}\]  
In Equation (4), \(H\) is the magnetic field intensity, and \(K\) represents the permeability of cells. \(x_{ci}\), \(y_{ci}\) and \(z_{ci}\) represent the coordinates of \(t\)th constraint point, \(j=1\cdots M\), \(i=1\cdots N+t\). \((cx_{i}\mathrm{max},cx_{i}\mathrm{max})\), \((cy_{i}\mathrm{max},cy_{i}\mathrm{max})\) and \((cz_{i}\mathrm{max},cz_{i}\mathrm{max})\) are integral limits of \(x_{mi}\), \(y_{mi}\), and \(z_{mi}\), respectively, which represent the minimum and maximum coordinate for the \(j\)th equivalent cell corner point. \(r=[(x_{mj}-x_{ci})^{2}+(y_{mj}-y_{ci})^{2}+(z_{mj}-z_{ci})^{2}]^{1/2}\), the \(d_{1\cdots 6}\) is as Equation (5) shows, where \(\mathrm{I}_{0}\), \(\mathrm{A}_{0}^{\prime}\), and \(\mathrm{I}\), \(\mathrm{A}^{\prime}\) are the inclination angle of the geomagnetic field to the direction of the total magnetization and the declination angle relative to the \(x\)-axis. The objective equation of the model is as follows [20]:  
\[ \begin{split}\Phi_{k}&=\alpha_{s}\int_{y}\zeta(r)[k( r)-k_{0}]\big{\}}^{2}dv+\alpha_{x}\int_{y}\bigg{\{}\frac{2\zeta(r)[k(r)-k_{0}]}{ \partial x}\bigg{\}}^{2}dv\\ &\quad+\alpha_{y}\int_{y}\bigg{\{}\frac{\partial\zeta(r)[k(r)-k_{0 }]}{\partial y}\bigg{\}}^{2}dv+\alpha_{z}\int_{y}\bigg{\{}\frac{\partial\zeta(r )[k(r)-k_{0}]}{\partial z}\bigg{\}}^{2}dv\end{split}