## 3 Simulation Setups and Results
### Results  
To examine the relative effects of the shield, the gain G is expressed by \(G=20*log\left(\frac{|H_{x}\hat{z}|}{|H_{0}\hat{z}|}\right)\) dB, which is the ratio of the scattered field in the absence of the hidden object to the total field in the presence of the hidden object. Figure 5 shows the gain G for different sizes of the hidden box as a function of spatial coordinates. For this set of simulations, the hidden object is assumed to be a perfect electrical conductor (PEC) in the shape of a cube. The conductivity of the outside shell is \(10^{6}\) S/m, and the distortion magnetic fields are measured at the negative z-direction. Figure 5a shows the gain fall-off with receiver distance for different widths and lengths of the hidden PEC box with the same height. Specifically, the gain is measured at \(x=y=0.6\) m (central axis of the system). The gain decreases with the receiver stand-off distance with an \(\frac{1}{r^{2}}\) profile as expected from that of a magnetic dipole; however, notable differences are observed for the different sized objects. For instance, at \(z=-0.005\) m, when the length (\(y\)) increases from 0.2 to 0.4 m, the G value increases by 2 dB (29.9 to 31.9 dB). When the area of the inside box is the same (\(x\) 0.2 m, \(y\) 0.4 m vs. \(x\) 0.4 m, \(y\) 0.2 m), the gain \(G\) values are almost the same. When the width and length increase from 0.2 to 0.4 m, the \(G\) value increases 1.52 times (29.9 to 45.4 dB). At  
Figure 4: Illustration of simulation setup of an inside box and outside box shield in the near-field of an infinitesimal point magnetic dipole. \(T_{x}\) and \(R_{x}\) represent a transmitter of the dipole and receiver. The receivers are located on the light blue plane and negative z-direction. The thickness of the outside shell is 2.7 mm. \(z=-1\) m, the G value for the 0.04 m\({}^{2}\) area is 4.6 times higher (0.0327 to 0.151 dB) than in the case of the 0.16 m\({}^{2}\). As the stand-off distance of the receiver is increased (negative z-direction), the difference in gain between the different conditions is accentuated. This is consistent with the theoretical expectations from Section 2 where the induced magnetic dipole moment is proportional to the volume of the hidden object and the corresponding fields fall off as the cubic distance from the object's location. Figure 5b shows the spatial gain profile for three different heights of the hidden box (from 0.2 to 0.8 m). Note that the spatial profile is shown in the region everywhere below the external shield (negative z). The cross-sectional areas of the three boxes at \(z=-0.005\) m have the same highest G values in all conditions, as shown in Figure 5b. However, the gain increases and spreads out at the greater height of the object. This is an important feature since it demonstrates that the field distortion is sensitive to the height of a hidden object and not simply its cross-section, which allows investigating features of hidden objects that are \"deep\" inside the enclosure. This is also consistent with the theoretical expectations that induced magnetic dipole moment on the object is proportional to the volume of the object. The results shown in Figure 5 suggest that many of the basic analytical features described in Section 2 still hold well even when the geometry of the hidden objects and shield deviate from that of a sphere. This provides support for developing simplified and computationally feasible inversion algorithms for non-destructive evaluation using ELF/VLF magnetic fields. The results shown in Figure 5 demonstrate that the field distortions (or gain) are sensitive to the size of the hidden object inside an enclosure. However, different passive electrical properties of the shield can also induce different magnetic distortion fields. Figure 6 shows the G value for different conductivities of inside and outside boxes for 50 Hz magnetic fields. The relative permeability of all cases is fixed at 1.