*2.3. Numerical Simulations of Electomagnetic Wave Propagation*

In order to better interpret the results obtained from the GPR studies, several numerical models with inclusions expected under the tested floor were prepared. The calculations with the models were performed to give information about the behavior of electromagnetic waves under the influence of anomalies, which can be used to analyze the GPR B-scans of the floor. Numerical modeling of electromagnetic wave propagation was carried out by the finite-difference time-domain (FDTD) method using the gprMax open source software (release 3.1.5) [44]. Two groups of 2D models were created. The first one (models #1) corresponded to the floor, while the second one (models #2) represented the concrete slab. The models were discretized using a 1 mm × 1 mm grid. The time step was selected automatically based on the Courant–Friedrichs–Lewy (CFL) condition. The outer space of the models was restricted by perfectly match layer (PML) absorbing boundary conditions. The excitation signal emitted by the transmitting antenna was the Ricker function with a central frequency of 2 GHz, and the distance between the transmitting and receiving antenna was set as 6 cm, according to the actual distance in the IDS antenna.

The FDTD models of the floor are shown in Figure 9. The models with external dimensions of 2.96 m × 1.12 m were prepared in four variants. The aim of the simulations made on models #1.1–1.4 was to enhance the interpretation of the GPR surveys by analyzing how different a prior known underfloor inclusion influenced the registered echograms. Each model included four stone tiles with dimensions of 43 cm × 4 cm and a tombstone with the dimensions of 120 cm × 15 cm. An air gap with a thickness of 1 cm was inserted under two stone tiles (on the left side of the tombstone). The other two stones were laid directly on the ground (on the right side of the tombstone). Model #1.1 (Figure 9a) included a plain layer of sand under the stone tiles and tombstone. In model #1.2, three walls were inserted to represent underfloor crypts. The walls were made of bricks with dimensions of 6.5 cm × 12 cm and a 1 cm thick mortar. Additional two models (#1.3 and #1.4) comprised of concentrated inclusions in the form of brick rubble. The following values of the electric permittivity were adopted: ε*<sup>r</sup>* = 9 (tiles), ε*<sup>r</sup>* = 3 (sand), ε*<sup>r</sup>* = 6 (brick) and ε*<sup>r</sup>* = 4 (mortar). The conductivity for all materials was set as σ = 0.01 S/m. A-scans were registered at 280 nodes, starting from 0.08 m and giving the scan length of 2.79 m.

**Figure 9.** Finite-difference time-domain (FDTD) model of the floor: (**a**) model #1.1; (**b**) model #1.2; (**c**) model #1.3; (**d**) model #1.4.

Figure 10 illustrates the FDTD models of two concrete slabs separated with an air gap. The 2D models with external dimensions of 0.7 m × 0.45 m consisted of two concrete sections with dimensions of 0.49 m × 0.1 m separated by an air gap with a thickness of 1 (model #2.1), 4 (model #2.2), 20 (model #2.3) and 50 mm (model #2.4). In the upper slab, a circular air inclusion with a diameter of 40 mm was inserted. The electric permittivity of concrete ε*<sup>r</sup>* = 4, corresponding to the velocity of electromagnetic waves equal to 15 cm/ns, was determined using the "depth to known reflector" method [45]. The conductivity of concrete was adopted as σ = 0.01 S/m. During the FDTD simulations, 41 A-scans were registered, giving the scan length of 0.4 m.

**Figure 10.** FDTD model of the concrete slabs with an air gap of the thicknesses (**a**) 1 mm (model #2.1); (**b**) 4 mm (model #2.2); (**c**) 20 mm (model #2.3); and (**d**) 50 mm (model #2.4).
