## 2 Forward and Inversion Theory  
### Three Dimensional Forward Modelling of the Magnetic Method  
The forward model studied in this paper does not consider remanence and demagnetization, so forward inference can therefore be regarded as an analytic solution problem. We divided the subsurface into cubic units of the same size and different magnetic susceptibilities and then calculated the magnetic anomalies caused by the unit at a certain observation point. The total magnetic anomaly at the measuring point was obtained by means of the summation of the magnetic anomaly caused by all the cube elements at the measuring point position.  
The parameters of a single cube unit are shown in Figure 1. Its central coordinates are set to (x\({}_{0}\),y\({}_{0}\),z\({}_{0}\)); the length, width and height of the cube unit are a, b and c, respectively; and the observation point is s (x, y, z).  
The magnetic anomaly \(\Delta\)T caused by the anomalous body element at the observation point s (\(x,y,z\)) is [18]:  
\[\begin{array}{rl}\Delta\text{T}(x,y,z)&=G^{\text{mag}}{}_{K}\\ &=\frac{\mu_{0}}{4\pi}\kappa\{k_{1}ln[r+(\xi-x)]+k_{2}ln[r+(\eta-y)]+k_{3}ln[r+( \xi-z)]\\ &+k_{4}arctan\frac{(\xi-x)(\eta-y)}{(\xi-x)^{2}+r(\xi-z)+(\xi-z)^{2}}\\ &+k_{5}arctan\frac{(\xi-x)(\eta-y)}{(\xi-y)^{2}+r(\xi-z)^{2}+(\xi-z)^{2}}\\ &+k_{6}arctan\frac{(\xi-x)(\eta-y)}{r(\xi-z)}\}|_{x_{0}-\frac{\xi}{2}}^{x_{0} +\frac{\xi}{2}y_{0}+\frac{\xi}{2}z_{0}+\frac{\xi}{2}},\\ \end{array} \tag{1}\]  
where x is the magnetic susceptibility matrix of the rectangular element, \(\mu_{0}\) is the vacuum permeability, \(G^{\text{mag}}\) is the Jacobian matrix, and a, b, and c are the lengths of the cuboid in the xyz direction.  
\[\begin{array}{c}r=\sqrt{(\xi-x)^{2}+(\xi-y)^{2}+(\xi-z)^{2}},\\ k_{1}=hN+nM,k_{2}=lN+nL,k_{3}=lM+hL,k_{4}=lL,k_{5}=hM,k_{6}=-nN;\\ l=cosI_{0}cosA_{0},h=cosI_{0}sinA_{0},n=sinI_{0};\\ L=cosIcosA,M=cosIsinA,N=sinI;\end{array} \tag{2}\]  
where \(l\), \(h\), \(n\), \(L\), \(M\) and \(N\) are the direction cosines of the geomagnetic field and the total magnetization intensity, respectively; \(I_{0}\), \(A_{0}\), \(I\) and \(A\) are the inclination angles of the geomagnetic field and the total magnetization and the deviation angles relative to the \(X\)-axis, respectively; and \(G^{\text{mag}}\) is the kernel function of the magnetic anomaly.  
Because the magnetic outlier at measuring point s is the superposition of the magnetic outlier generated by all underground rectangular units at the point, the magnetic outlier at measuring point s is calculated using the following formula:  
\[\Delta T_{i}=\sum_{j=1}^{j=M}G_{ij}^{mag}\kappa_{j}, \tag{3}\]  
where \(\Delta T_{i}\) is the magnetic anomaly at observation point \(i\), \(\kappa_{j}\) is the magnetic susceptibility of the Jth model element, and \(G_{ij}^{mag}\) is the kernel function corresponding to rectangular element \(j\) at observation point \(i\).  
The three-dimensional forward modeling of magnetic methods can be regarded as the problem of obtaining the abnormal data \(\Delta T\) from magnetic methods by using the kernel G  
Figure 1: Cube unit diagram.  
matrix and the magnetic susceptibility of the model vector. The forward mapping to the global inversion formula is a large formula. It can be simplified as follows:  
\[d=Gm, \tag{4}\]  
representing the process of obtaining the data result d by using the G matrix and model vector m.