## 2 Formulation of the Problem and Proposed Solution
### Proposed Solution
To this purpose, it must be emphasized that the commercial software was not designed to deal with scattering problems involving layered media, thus  
Figure 2: Normalized truncation error for a zero-thickness PEC disk of radius \(a=5,\;10,\;15,\;20\) cm buried in a lossy half-space with \(\varepsilon_{r2}=3.5-\text{j}0.3\) and \(\mu_{r2}=1\) at a depth \(d=3\) cm, and illuminated by a Hertzian dipole directed along the \(y\) axis with \(\left|I_{0}\right|=10^{-3}\text{Am}\), operating at 800 MHz and located in air, \(\varepsilon_{r1}=\mu_{r1}=1\), at \(x_{0}=-2\) m, \(y_{0}=2\) m, \(z_{0}=2\) m, as a function of (**a**) \(M\) for \(N=7\) and as a function of (**b**) \(N\) for \(M=8\). forcing an approximation of the infinite layered medium to a finite one. Unfortunately, a fairly accurate solution can be obtained only when a very large, with respect to the wavelength, layered region is involved, with an exorbitant number of mesh-cells and an endless computation time to perform the simulations. For this reason, comparisons are provided only in homogeneous medium, in particular, even when medium 2 in Figure 1 is filled with air, i.e., \(\varepsilon_{r1}=\mu_{r1}=\varepsilon_{r2}=\mu_{r2}=1\). In Figure 3, the amplitude of the components of the surface current density on a zero-thickness PEC disk of radius \(a=10\) cm, located at \(d=3\) cm, and illuminated by a Hertzian dipole directed along the \(y\) axis with \(\left|\underline{I}_{0}\right|=10^{-3}\)Am, operating at 800 MHz and located at \(x_{0}=-2\) m, \(y_{0}=2\) m, \(z_{0}=2\) m, is plotted along the \(x\) and \(y\) axes, i.e., for \(\left|x/a\right|\leq 1\), \(y=0\), and for \(x=0\), \(\left|y/a\right|\leq 1\), and compared with the ones reconstructed by means of CST-MWS. The agreement is excellent. However, the proposed method achieved a normalized truncation error less than \(10^{-3}\) for \(M=4\) and \(N=5\) with a computation time of 15 s, whereas the time-domain solver of CST-MWS required about 30 million mesh-cells with a computation time of about 3 h to accurately reconstruct the solution. As an example of the application of the proposed method, a configuration was considered that resembled a measurement system of the field scattered from a landmine based on a two drones, one equipped with a transmitting antenna and the other equipped with a receiving antenna. As with the cases examined before, a zero-thickness PEC disk, of radius \(a=10\) cm buried in a lossy half-space with \(\varepsilon_{r2}=3.5-\)j0.3 and \(\mu_{r2}=1\) at the depth \(d=3\) cm, was considered. Now, the transmitting Hertzian dipole, directed  
Figure 3: Amplitude of the components of the surface current density along the \(x\) and \(y\) axes on a zero-thickness PEC disk of radius \(a=10\) cm, located at \(d=3\) cm, and illuminated by a Hertzian dipole directed along the \(y\) axis with \(\left|\underline{I}_{0}\right|=10^{-3}\)Am, operating at 800 MHz and located at \(x_{0}=-2\) m, \(y_{0}=2\) m, \(z_{0}=2\) m, for \(\varepsilon_{r1}=\mu_{r1}=\varepsilon_{r2}=\mu_{r2}=1\). (**a**) \(\left|\underline{I}_{\rho}\right|\) along the \(x\) axis, (**b**) \(\left|\underline{I}_{\rho}\right|\) along the \(y\) axis, (**c**) \(\left|\underline{I}_{\rho}\right|\) along the \(x\) axis, and (**d**) \(\left|\underline{I}_{\rho}\right|\) along the \(y\) axis. Blue solid line: this method; red dashed line: CST-MWS. along the \(y\) axis with \(\left|\dot{I}_{0}\right|=10^{-3}\)Am and operating at 800 MHz, was located in the upper half-space filled by air, \(e_{r1}=\mu_{r1}=1\), at \(z_{0}=2\) m and four different positions in the \(xy\) plane: (1) \(x_{0}=-2\) m, \(y_{0}=2\) m, (2) \(x_{0}=0\) m, \(y_{0}=2\) m, (3) \(x_{0}=-2\) m, \(y_{0}=0\) m, and (4) \(x_{0}=0\) m, \(y_{0}=0\) m. The scattered field was evaluated in the horizontal area \((x,y)\in[-2\) m, \(2\) m\(]\times[-2\) m, \(2\) m\(]\) for \(z=1\) m. In Figure 4, two separate plots for the amplitudes of the incidence and the scattered electric fields in the area of interest are provided for the four cases examined.