The finite difference time domain method is employed to calculate the radar signatures of an obliquely penetrating tunnel in various background media. The relatively faster propagation of electromagnetic signal and stronger distortion in the first peak were discovered due to the existence of an empty tunnel as its penetration angle becomes more oblique. It renders the arrival time of the first peak to be increased relatively slower than that of the first received signal. To analyze the different behaviors of two arrival times, those data are compared with the straight line model and the ray-tracing model. Finally, the relation between the penetration angle of an empty tunnel and the arrival time of received pulse signatures is evaluated with the curve fitting model in three different background media. These results are compared with the measured data in a well suited tunnel test site.
 Since the 1980s, borehole radar systems have been used in Korea to detect deeply located empty tunnels with an approximate 2 m by 2 m cross-section. Underground rocks are highly weathered, jointed, and fractured. Hence, the detection of such an empty tunnel is very difficult due to the strong noise level generated from geological irregularities such as a fault, joint, lode, or underground water. Many researchers have struggled to find a proper way to extract tunnel signatures from received signal patterns significantly contaminated by such noise [Ellis and Peden, 1997; Kim and Kim, 2010; Schneider and Peden, 1993; Takahashi and Sato, 2006; Zhou and Sato, 2004]. Until now, two types of cross-borehole radar systems have been operated in Korea. One type is cross-borehole continuous wave (CW) radar where the most important feature is the double dip pattern in the received signal [Lytle et al., 1979]. Strong attenuation of specific frequencies has been observed at the positions corresponding to the top and bottom boundaries of an empty tunnel [Lee et al., 1989]. A double dip pattern was constructed by superposing an incident wave and its out-of-phase scattered wave [Peden et al., 1992]. This kind of pattern played a key role in detecting the 4th tunnel in Korea [Kim and Ra, 1993].
 The other type is cross-borehole pulse radar. The relatively fast signal propagation through an empty tunnel has been used to detect the position of an empty dormant tunnel. In addition, the tunnel's boundaries cause severe attenuation and multiple reflections that distort the waveform of the received pulse signal. Olhoeft  categorized these characteristic features into three parameters—velocity, attenuation, and dispersion—by extracting the first and second peaks of a received signal in the time domain. These parameters have mainly been used in the analysis of actual measured radar data because of simplicity and accuracy. In particular, the propagation velocity is estimated by the time-of-peak (TOP) obtained from the arrival time of the first peak. Once the tunnel signature is discovered in radar signatures, several holes are bored to determine the direction of tunnel in real measurement. If the penetration angle can be estimated from these measured parameters, the number of these additional boring could be minimized.
 In spite of its excellent capability for detecting a perpendicularly penetrating tunnel, the double dip pattern in a CW radar system suffers from severe degradation as the penetrating angle of an empty tunnel becomes oblique [Lee et al., 1989]. For the pulse radar system, some propagation models have been suggested to explain the physical mechanism on the variation of the Olhoeft's  velocity parameter according to the penetrating angle of an empty tunnel. Moran and Greenfield  and Alleman et al.  investigated the effect of the angle variation of tunnel axis using 2.5D analytic solution and the radar data measured in a well suited tunnel test site, respectively. Their results commonly showed that a pulse signal was severely distorted when an air-filled tunnel penetrated at an extremely oblique angle. To explain this phenomenon, the critical angle on the tunnel boundary and the waveguide model were adopted but those accuracies were not assured.
 Recently, we estimated the oblique angle of an empty man-made tunnel using the arrival time data measured in a well suited test site by employing our cross-borehole pulse radar [Kim et al., 2007]. According to the above results, the TOA with 1% amplitude level of the peak value was more effective for estimating the oblique angle of the empty tunnel. However, the proposed scheme cannot provide accurate estimation of the oblique angle of tunnels in different rocks because the received tunnel signatures may be significantly affected by the electric properties of its background rocks. Also, the number of the available tunnel test sites is limited.
 As an alternative, we used the self-made borehole radar simulator [Kim and Kim, 2000] in order to generate the radar signatures from an obliquely penetrating tunnel in various background media. The simulator was implemented by adopting the 3-dimensional finite difference time domain (FDTD) method [Yee, 1966; Taflove and Hagness, 2005] for a lossy and dispersive medium defined by Debye formula [Luebbers et al., 1990; Uno et al., 1997]. To generate more realistic radar data, dipole antenna of our cross-borehole pulse radar and an actual boring situation were considered here. The simulations were repeated in three different cases of low, medium, and high dielectric rocks.
 The simulation result in the medium rock condition concurred with the measured data in a well suited tunnel test site, as shown in the previous paper [Kim et al., 2010]. The time-of-arrival (TOA) extracted from the time of the first arriving signal became drastically faster than the time-of-peak arrival (TOP) determined by the arrival time of the first peak as the penetrating angle became more oblique. The variation profiles of the TOA and TOP according to the penetrating angle of an empty tunnel are compared with data extracted from the straight line and the raypath models. The TOA variation profile, calculated from simulation results in 3 different background media at the depth of the tunnel center, may be useful for estimating the penetration angles of empty tunnels in various rock conditions.
2.1. Modeling of a Cross-Borehole Radar
 The geometrical configuration of our simulation is shown in Figure 1. The background medium is considered Granite with lossy properties. Its complex permittivity is expressed as a function of angular frequency as
where ɛr and σ are relative permittivity and conductivity in S/m respectively [Schneider and Peden, 1993]. The values of ɛr and σ of underground rocks in a wide range of mountains in Korea were measured by employing an open-ended coaxial probe [Jung and Kim, 2007]. According to our measured data, the relative permittivity (ɛr) varied from 5 to 12 and the conductivity (σ) changed from 0.001 to 0.007 S/m. In order to simulate various rock conditions, we considered three cases of the constitutive parameters, as shown in Table 1. In case 1, the relative permittivity and conductivity were 9 and 0.0031 S/m respectively as the average medium of granite in Korea. Case 2 was the low dielectric medium with the relative permittivity of 5.8 and conductivity of 0.001 S/m. In case 3, the high dielectric medium had the relative permittivity of 12 and conductivity of 0.005 S/m. Strong attenuation through the background medium renders only frequency components below 100MHz to be detectable in the receiver of a cross-borehole pulse radar [Duff and Zook, 1993]. Hence the input voltage at the transmitting (Tx) antenna can be closely approximated by a Gaussian pulse with a maximum frequency of 100 MHz, where
Table 1. Constitutive Parameters of the Background Media
 The locations of the Tx and receiving (Rx) antennas are determined by the corresponding borehole profiles. The deviation profiles of the Tx and Rx boreholes are measured by a logging device during the real boring process. In this paper, we assumed the Tx borehole as a straight line without any deviation. The relative deviation of the Rx borehole was adjusted to fit the deviation data measured in the actual borehole site. Hence, the separation distance between the Tx and Rx antennas may be changed according to the depth. When the radar data was calculated at depths (D) between 63 m and 83 m, the separation distance (S) was linearly changed between 15.9 and 16.8 m, as shown in Figure 1a. An empty tunnel with a cross section of 2 m by 2 m was assumed. The cylindrical tunnel along the y axis was located at a depth of 73 m and in the middle of the horizontal trace path between the Tx and Rx antennas. The oblique angle (θ) was defined as the angle between the trace path and the normal line of the tunnel axis, as shown in Figure 1b, where θ = 0°, the trace path penetrated the tunnel perpendicularly. The trace path between Tx and Rx was rotated at the intersection point between the tunnel axis and the trace path at the depth of D = 73 m in order to analyze the radar signature at various angles. The oblique angle was changed from 0° to 75° with an angle interval of 15°.
2.2. FDTD Geometry
 We used the borehole radar simulator [Kim and Kim, 2000] by adopting the 3-dimentional lossy and dispersive FDTD method [Luebbers et al., 1990]. The permittivity of the background media was not changed according to the frequency, as listed in Table 1. Considering the 10 cells per the minimum wavelength determined by the relative permittivity in Table 1 and the maximum frequency of source waveform in (2), we adjusted the spatial sampling width of the FDTD calculation to 0.1 m. To satisfy the Courant stability condition [Taflove and Hagness, 2005], the temporal sampling width was taken by 0.1732 ns. As shown in Figure 2a, the length and radius of the dipole antenna used in this simulation were 3 m and 0.03 m respectively. Because the radius of the antenna was smaller than the FDTD cell size, the dipole antenna was simplified to a thin wire [Watanabe and Taki, 1998]. The upper boundary of the tunnel was assumed to be round. To minimize numerical error due to the staircase model of the tunnel, the contour-path method [Jurgens et al., 1992] was implemented here [Kim and Kim, 2000].
 To keep the depth position of the Tx antenna equal to that of the Rx antenna, the FDTD computation was repeated at every 20 cm in the depth between 63 m and 83 m. The outer boundary of the computational region was filled with 4 cells of the perfectly matched layer (PML) [Berenger, 1994; Uno et al., 1997]. To reduce the computational resource and time, the computational region was redefined into 100 different simulation cases according to the locations of the tunnel and both antennas. As some typical examples, the redefined FDTD computational regions for 4 cases are shown in Figures 2a–2d. In Figures 2a and 2b, the tunnel penetrated at the oblique angle of 0° and 75° respectively, while both antennas were located at a depth of 73 m. By considering the PML thickness and the computational region including the tunnel and both antennas at 73 m, the total numbers of FDTD cells at the oblique angle of 0° and 75° became 191 × 31 × 59 cells and 68 × 186 × 59 cells respectively. When both antennas moved to 83 m in depth, the FDTD cells at the oblique angle of 0° and 75° were extended to 196 × 30 × 153 cells and 73 × 191 × 153 cells respectively, as shown in Figures 2c and 2d. When both antennas were rotated simultaneously, the position of both antennas could be deviated from the FDTD cells within half of one cell size. In this simulation, the positions of both antennas were adjusted to fit the nearest FDTD cells in view of z-component of the electric field. In order to achieve the pulse response up to 500 ns, each FDTD simulation was performed for 3000 time steps.
2.3. Arrival Time
 The pulse signal excited at a transmitter and the pulse response captured by a receiver are depicted together in Figure 3. The amplitudes of both signals were normalized to 1 because the received signal was attenuated exponentially during its propagation through underground rock. To extract the arrival time of the received signal, the TOP is generally used in a conventional radar system [Olhoeft, 1988]. It is useful when the pulse signal, excited by the transmitting antenna, travels in such a homogeneous and non-dispersive background medium as air without any distortion of its shape. Unfortunately, the ground is usually composed of inhomogeneous, lossy and dispersive media with many scatterers. Hence the pulse response captured by a receiver is usually severely distorted, dissipated and dispersed. Therefore, if the pulse travel time is extracted by the TOP, the exact velocity cannot be achieved.
 To overcome this problem, Alleman et al.  used the first anomaly time to find the first arrival time. This arrival time may be a similar concept to the TOA measurement used in other fields [Kelly et al., 1993]. The TOA is simply the earliest time when the amplitude of the received signal exceeds a specific threshold. Any earlier signal except the pulse response did not exist in our simulation because of our homogeneous background condition. In a real situation, however, some unwanted signals due to noise and background inhomogeneity may occur in the received signal. Hence, the amplitude criterion of the TOA was also defined by 1 percent of its maximum under the same condition as used in the previous paper [Kim et al., 2010].
3.1. TOP and TOA
Figure 4 illustrates our FDTD simulation results on the propagation of the pulse signal for a perpendicularly penetrating empty tunnel, as shown in the case 1 of Table 1. The amplitude distribution of the z-directional electric fields on the cross-sectional plane including the Tx and Rx antennas is represented in dB scale. The fields were captured at 107.4 ns, 142.0 ns, 147.2 ns, and 162.8 ns. In Figure 3a, the pulse signal radiated at the Tx antenna travels through underground rock. It reached the left boundary of the tunnel at 142.0 ns, as shown in Figure 4b. Figure 4c shows that the air inside the tunnel rendered the pulse velocity to be quicker. The distorted wavefront, due to the empty tunnel, continued to propagate toward the Rx antenna, as shown in Figure 4d. From this relatively fast propagation through the empty tunnel, we easily found the basic criterion for how to single out a tunnel signature in the radar profile measured by a cross-borehole pulse radar.
Figure 5 shows the calculated radar profiles when the tunnel oblique angle θ was changed from 0° to 75°. This result shows that the fast arrivals were discovered in the first peak of the radar signals near a tunnel depth of D = 73 m. In comparison with Figure 5a, where θ = 0°, Figure 5f, where θ = 75°, illustrates the fast arrival of the transmitted pulse signal. Figure 6 shows the received radar signals that were extracted at the tunnel center depth of D = 73 m from all radar profiles shown in Figure 5. The positions of the TOP and TOA are marked by inverse triangles and triangles, respectively. As the trace path approached the tunnel axis, the TOA varied more significantly than the TOP. This variation implies that the earlier part of the first peak became wider and its amplitude became lower due to the strong distortion as an empty tunnel penetrates more obliquely. Hence the TOA occurred much earlier than the TOP as the tunnel oblique angle was increased.
 Both the TOP and TOA were extracted from all scan data as indicated in Figure 7. If a tunnel does not exist, both parameters should be linear because the separation distance varies from 15.9 m at 63 m depth to 16.8 m at 83 m depth and the velocities of all trace paths are identical. However, the TOP profile had a narrow variation in the region between 72 m and 74 m due to an air-filled tunnel, as shown in Figure 7a. By considering a linear interpolation of TOP at a depth between 63 and 83 m, the TOP at 73 m in case of no tunnel would be about 270 ns. Due to the existence of an empty tunnel, the TOP may appear about 10 ns to 25 ns early as the tunnel oblique angle increases. In contrast, Figure 6b illustrates that the TOA was significantly varying in the wider depth region between 69 m and 79 m. Compared to the TOP, the TOA at the tunnel depth of 73 m advanced drastically about 10 ns to 70 ns according to variation of the tunnel oblique angle.
3.2. Velocity Profile
 In order to eliminate the effect of the separation distance due to deviation in the receiving borehole and analyze the effect of an air-filled tunnel on the propagation path between Tx and Rx, the TOP and TOA velocity profiles were calculated, as shown in Figure 8. According to the rule illustrated in Figure 3, the TOP and TOA in the transmitted pulse signal were determined by 100 ns and 83.6 ns, respectively. By considering the separation distance and the TOP and TOA offsets in the source waveform, the TOP and TOA velocity profiles can be calculated, as shown in Figures 8a and 8b. At first, the accuracy of the two estimated velocities was verified in the case that no tunnel exists in the trace path between both antennas, as shown in Figures 2c or 2d. According to Figures 8a and 8b, the TOP and TOA velocities at 83 m depth were calculated by 0.098 m/ns and 0.1 m/ns respectively. At the depth of 83 m, the pulse may have been propagated along the line-of-sight path between the Tx and Rx antennas because the tunnel was located far away from this propagation path. Hence, the ideal velocity between two antennas at 83 m is 0.1 m/ns since the relative permittivity of the background rock is 9. This result shows the accuracy of the conversion of TOA and TOP velocities. The minute decrease of the TOP velocity may have come from the small pulse dispersion due to the lossy property of the background medium.
 Since the tunnel was filled with air, the transmitted pulse propagated faster inside the tunnel, as shown in Figure 4. It rendered both of TOP and TOA velocities to be increased monotonically as the propagation path between two antennas approached the depth of the tunnel center. So, we focus on the variation of the propagation velocity according to the penetrating angle of an empty tunnel at the tunnel center depth of 73 m. When the penetration angle of the empty tunnel increased from 0° to 75°, both velocities increased steeply. As shown in Figure 8a, the TOP velocity at 73 m changed from 0.106 m/ns at θ = 0° to 0.1164 m/ns at θ = 75°. In contrast, Figure 8b illustrates that the TOA velocity increased from 0.1092 m/ns at θ = 0° to 0.1748 m/ns at θ = 75°. From these results, as illustrated in Figure 8, one may find that the TOA and TOP velocities exhibit different behaviors as the tunnel oblique angle is increased.
4.1. Propagation Models
 In a cross-borehole pulse radar system, there are many propagation paths connecting the Tx to the Rx antennas. The received signal may be composed of several delayed pulses along the lower attenuated paths. The peak is formed in the case where the main components are in phase. Then, the TOP velocity is considered as the average velocity of the major components among several possible paths. On the other hand, the TOA velocity may be determined as the fastest way among all possible paths although the corresponding amplitude is relatively small.
 From this point on, we considered two propagation models for explaining the different behaviors of the TOP and the TOA velocities. At first, we use the straight line model to describe the TOP velocity. Because the TOP velocity reflects the average velocity, its path is assumed as a straight line between the Tx and Rx antennas, as shown in Figure 9a. When the length of the trace path (S) and the width of the tunnels (W) are defined, as shown in Figure 9a, the average value of the relative permittivity distribution along the trace path (ɛavg) is given by
where dA = W · cos θ is the length of the air portion. ɛG and ɛA are the relative permittivities of background rock and air, respectively. The average velocity of the straight line model (VS) can be determined by
where c denotes the velocity of light.
 In contrast, to describe the TOA velocity, we considered the raypath method. The shortest time path through the empty tunnel is determined by Snell's law as
where θG and θA denote the incidence and the refraction angles respectively, as shown in Figure 8b. The travel time (TR) and the velocity (VR) of the raypath model are given below
 In comparison with the TOP and TOA velocities, the velocities corresponding to the two propagation models are listed in Table 2 and described in Figure 10. One may find that the TOP velocity was slower than the velocity estimated from the straight line model. In comparison, the TOA velocity existed between the velocities estimated from two different propagation models. It means that the real propagation path is very complex. In the real operation of a cross-borehole pulse radar system, the TOA velocity may be effective for estimating the penetrating angle of a deeply located man-made tunnel.
Table 2. The Variation of Four Different Kinds of Velocities According to the Penetrating Angle of an Empty Tunnel With Depth of Its Center (D = 73 m)
Straight Line Model
4.2. Estimation of Tunnel Penetration Angle
 To analyze the variation of radar profiles more effectively, some normalization procedures are applied. First, the amplitude of the pulse signal measured at each depth is normalized by its maximum amplitude to prevent the strong attenuation of the tunnel signature from being concealed by other signals in the wiggle trace graph. Next, the gradual variation in the arrival time due to the deviation of the involved boreholes is compensated by adopting normalization process like the normal move out (NMO) process [Sato and Feng, 2005]. Although the NMO process is normally used to compensate the variation of the distance between the Tx and Rx antenna in the reflection mode by using time shift and rescaling time axis, we applied only time shift to extract the advanced time caused by an air-filled tunnel. This process converts the TOP and TOA acquired at different separation distances into the TOP and TOA at the same distance. Using the velocity profiles in Figures 8a and 8b, one may construct the TOP and TOA at the same distance. Finally, the normalized arrival time can be obtained by calculating the difference between the arrival times converted by normalization process at 73 m (center of the tunnel) and at 83 m. It means the difference between the normalized arrival times in cases of tunnel and no tunnel.
Figure 11 depicts the normalized arrival times of TOP and TOA at the depth of the tunnel center (D = 73 m). They are compared with the field data measured at the tunnel test site in Korea [Kim et al., 2010]. As shown in Figure 11a, the TOP was gradually increased as the tunnel oblique angle was increased. But the TOP variation for 3 different rocks according to the tunnel oblique angle seemed to be nearly constant. Since the cross-section of an empty tunnel is about 2 m by 2 m, the lower frequency components of the electromagnetic pulse cannot consider the empty tunnel as a scatterer. But the higher frequency components suffer from strong attenuation due to medium conductivity. Especially, the main portion of the received pulse consists of the lower frequency components after propagating along the line-of-sight between the Tx and Rx antennas in case of extremely oblique penetration angle. This result suggests that TOP propagates through empty tunnel like the straight line model in the previous section.
 In comparison, the TOA was greatly increased according to the tunnel penetration angle, as shown in Figure 11b. As the complex permittivity was increased, the variation width also became wider. The constitutive parameter of underground rock in the previous paper [Kim et al., 2010] was close to that of the case 2 (low dielectric) in this study. Hence, the TOA extracted from the measured data was very similar to the simulation result in the case 2. Based on three relation curves between the tunnel oblique angle and the TOA in Figure 11b, the TOA in ns was expressed by a quadratic equation to the tunnel oblique angle (θ) in degree as
where ɛr denotes the relative permittivity of background rock. Three coefficients in (8) were also expressed by quadratic equations to ɛr as
 It should be mentioned that the conductivity (σ) of the background rock with a fixed value ɛr is automatically given by the proportional relation from the data in Table 1. Hence the medium with high ɛr is automatically highly lossy. The bold line in Figure 11b denotes the TOA curve constructed by applying our estimation equation (8) in the case of the background medium corresponding to ɛr = 7. One may easily find that the simulated quadratic curve in Figure 11b agrees well with the measured TOA data. The relation curves in Figure 11b and the quadratic equation in (8) will help us to estimate the penetration angle of an empty tunnel with the width of 2 m in real sites from the TOA data measured by a cross-borehole pulse radar.
 FDTD simulations were performed to provide effective guidance on estimating the penetrating angle of a deeply located empty tunnel when operating a cross-borehole pulse radar in real sites. Our simulation results illustrated that the TOA velocity at the depth of the tunnel increased more significantly than the TOP velocity as the empty tunnel penetrated more obliquely. These results imply that the TOA velocity may be effective for detecting an empty tunnel with an arbitrary penetrating angle. In addition, we used the straight line and raypath models to estimate the TOP and TOA velocities. The TOP velocity was slower than that in the straight line model. The TOA velocity was located between two velocities estimated from the straight line and the raypath models. In particular, the arrival time of a transmitted pulse varied significantly according to the electric property of background rocks. Hence, the simulations were repeated in three different media of low, medium, and high conductivity. Compared to the TOP, the simulated TOA agreed well with the data measured in a well suited tunnel test site. The TOA variation profiles according to the tunnel penetration angle in three different background media may be useful for estimating the penetration angle of an empty tunnel in real sites.
 This work was supported in part by the KIST Internal Project and in part by Ministry of Culture, Sports and Tourism (MCST) and Korea Creative Content Agency (KOCCA) in the Culture Technology (CT) Research and Development Program 2011.