## 2 Materials and Methods  
### Forward Model  
When the distance between a magnetic dipole source and the measurement sensor exceeds 2.5 times the size of the magnetic source, the anomaly generated by the target magnetic source can be considered characteristic of a magnetic dipole. This assumption is consistent with most \"far-field\" detection scenarios. In measurements involving objects such as UXO, while modeling techniques based on ellipsoidal magnetic dipoles that account for their self-demagnetization effects have significantly advanced, practical limitations remain. Specifically, the UXO's dimensions, burial depth, and the distance between the sensor and the target cause the response from self-demagnetization effects induced by the target's shape characteristics to attenuate with distance, complicating detection in practical applications. Therefore, the point dipole forward modeling formula sufficiently addresses most detection scenarios [40; 41; 42]. The response of the magnetic dipole source is expressed by the following relation [43]. \[\mathbf{B_{a}}=\frac{\mu_{0}}{4\pi}\bigg{(}\frac{3(\mathbf{m}\cdot\mathbf{R})\mathbf{R}}{R^{5}} -\frac{\mathbf{m}}{R^{3}}\bigg{)} \tag{1}\]  
where \(\mu_{0}\) is the permeability in free space; \(\mathbf{m}\) is the magnetic moment vector of the dipole, \(\mathbf{m}=m\cdot[m_{x},m_{y},m_{z}]^{T}\); and \(\mathbf{R}\) is the distance between the magnetic sensor and the target. When \(R=((x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}+(z_{1}-z_{0})^{2})^{1/2}\), the coordinates \(x_{1}\), \(y_{1}\), and \(z_{1}\) represent the location of the observation point and \(x_{0}\), \(y_{0}\), and \(z_{0}\) correspond to the position of the magnetic dipole source. The magnetic anomalies of the target body for magnetic anomaly detection are generally much smaller than the geomagnetic field values. So the magnetic anomaly field \(\Delta\)B can be regarded as the projection of \(\textbf{B}_{a}\) in the direction of the geomagnetic field \(\textbf{B}_{e}\) when the scalar magnetic sensor is used for measurement. The formula is as follows:  
\[\Delta\textbf{B}=\frac{\mathbf{B}_{e}\cdot\textbf{B}_{a}}{|\mathbf{B}_{e}|}=\textbf{u }\cdot\textbf{B}_{a} \tag{2}\]  
where \(\textbf{u}=(\text{cos}(I)\text{cos}(D))\textbf{e}_{x}+(\text{cos}(I)\text{sin }(D))\textbf{e}_{y}+(\text{sin}(I))\textbf{e}_{z}\), \(D\) is the declination, and \(I\) is the inclination. In scalar magnetic measurements, the total field detected by a magnetometer generally results from the combination of the geomagnetic field and anomalous fields produced by nearby sources. These observations are typically performed within a Cartesian spatial framework where the \(z\)-axis is oriented vertically downward. The induced anomaly from a magnetic dipole target can be mathematically modeled using an established dipolar field equation [44]:  
\[\Delta T=\frac{\mu_{0}m}{4\pi}\sum_{n=1}^{6}\,b_{n}\varphi_{n} \tag{3}\]  
where  
\[\begin{array}{l}\varphi_{1}=\frac{-x^{2}-y^{2}+2z^{2}}{[x^{2}+y^{2}+z^{2}]^{ \frac{1}{2}}},\varphi_{2}=\frac{2x^{2}-y^{2}-z^{2}}{[x^{2}+y^{2}+z^{2}]^{\frac {1}{2}}}\\ \varphi_{3}=\frac{-x^{2}+2y^{2}-z^{2}}{[x^{2}+y\[ \begin{array}{l}b_{1}=\sin{I_{0}}\sin{I},\ b_{2}=\cos{I_{0}}\cos{A_{0}}\cos{I} \cos{A}\\ b_{3}=\cos{I_{0}}\sin{A_{0}}\cos{I}\sin{A}\\ b_{4}=\sin{I_{0}}\cos{I}\cos{A}+\cos{I_{0}}\cos{A_{0}}\sin{I}\\ b_{5}=\cos{I_{0}}\sin{A_{0}}\cos{I}\cos{A}+\cos{I_{0}}\cos{A_{0}}\cos{I}\sin{A} \\ b_{6}=\sin{I_{0}}\cos{I}\sin{A}+\cos{I_{0}}\sin{A_{0}}\sin{I}\end{array}  \tag{5}\]  
where \(I\) and \(A\) are the inclination and declination of the Earth's magnetic field, respectively, \(I_{0}\) and \(A_{0}\) are the inclination and azimuth of the magnetic dipole source, as illustrated in Figure 1, and \(\mathbf{m}_{hc}\) is the projection of the magnetic dipole moment vector onto the x-y plane, with \(I_{0}\) and \(A_{0}\) as shown in Figure 1. \(x=(x_{1}-x_{0})\), \(y=(y_{1}-y_{0}),z=(z_{1}-z_{0})\).