3D GPR Polarization Analysis for Imaging Complex Objects. Jeffrey J. Daniels1 ..... This point is emphasized by the two slice views that are shown in Figure 6.
3D GPR Polarization Analysis for Imaging Complex Objects Jeffrey J. Daniels1, Lucian Wielopolski2, Stan Radzevicius3, and Jess Bookshar1 1
Dept. Geological Sciences, The Ohio State University, Columbus, OH 43210 2 Brookhaven National Laboratory, Upton, NY 11973 3 APA Division, ENSCO Inc., 5400 Port Royal Rd., Springfield, VA 22181-2312
Abstract Ground Penetrating Radar (GPR) polarization is an important consideration when designing a GPR survey and is useful to constrain the size, shape, orientation, and electrical properties of buried objects. The polarization of the signal measured by the receive antenna is a function of the polarization of the transmit antenna and scattering properties of subsurface targets. These polarization dependent scattering characteristics can be used to discriminate between different classes of targets for optimal target detection. The polarization of an electromagnetic wave is a fundamental property of propagation that provides the GPR technqique with a unique opportunity for producing improved images of objects in the subsurface. Conversely, ignoring the polarization aspects of GPR can lead to false interpretations of the shape and orientation of objects in the field, and in the extreme case can cause buried objects to be totally missed in an interpretation. The effects of polarization become very critical in the case of 3D displays. The direction of measurement (e.g., line direction) can strongly influence the resulting image. Even simple features, like dipping layers, can mask the individual objects that are the target of a survey.
Introduction Investigations by Roberts (1994) and Roberts and Daniels (1996, 1997) have demonstrated the potential of using the polarization characteristics of GPR for defining the size, shape, orientation, and material properties of buried objects. A more recent study by Radzevicius (2001), and Radzevicius, et al. (2003), show the polarization properties of antennas in the near field. The general characteristics of GPR polarization are shown in Figure 1. In a practical sense the polarization properties of electromagnetic waves are exploited by utilizing a particular orientation of the transmit and receive antennas. The common arrangements are shown in Figure 1c. However, it should be emphasized that polarization is a combined function of the relative orientation of the transmit-receiver antenna pair and the orientation of the objects in the subsurface relative to the orientation of the antennas.
FIGURE 1. EM polarization: a) Polarization components of an electromagnetic wave b) primary types of polarization that can be measured in the field, and c) antenna polarization components.
The electromagnetic field at a given point in space, at a given time, has both a magnitude and a direction, and thus is described by a vector. As the electromagnetic wave propagates, the orientation and magnitude of these vectors change as a function of time. Polarization describes the magnitude and direction of the electromagnetic field as a function of time and space. When the time varying EM fields vary sinusoidally (time harmonic), polarization may be classified as linear, circular, or elliptical. If the vector that describes the electric field at a point in space as a function of time is always directed along a straight line, the field is said to be linearly polarized. If the vector sweeps out a circle, then it is referred to as circular polarization. Linear and circular polarization are special cases of elliptical polarization, in which the electric field traces out an ellipse. An arbitrary electromagnetic vector field can be described by three orthogonal basis vectors. Since the electric and magnetic fields are orthogonal to the direction of propagation, if we choose one of the basis vectors in the direction of propagation, the electric field can be decomposed into two orthogonal basis vectors. The electric field of a wave traveling in the z direction can be described by two orthogonal components as given in Balanis 1989: Ex(z,t) = Exoe-αzcos(ωt-βz-φx) and E y(z,t) = Eyoe-αzcos(ωt-βz-φy)
(1)
where α represents the attenuation constant, β represents the phase constant, ω the angular frequency, φ the phase, and Exo and Eyo are the maximum amplitudes of the Ex and Ey components respectively. Linear PolarizationFor a wave to have linear polarization, the time-phase difference between the two components must be ∆φ = φy - φx = nπ n=0,1,2,3,... (2) Circular PolarizationCircular polarization is achieved only when the magnitudes of the components are the same and the time-phase difference are odd multiples of π/2. 1 Ex = Ey and ∆φ = φy - φx = ± --- + 2nπ , where + & - refer to clockwise (CW) or counter2 clockwise (CCW) rotation. n=0,1,2,3,...
(3)
Elliptical PolarizationElliptical polarization is achieved only when the time-phase difference between the two components are odd multiples of π/2 and their magnitudes are not the same or when the time-phase difference between the components are not equal to multiples of π/2, regardless of their magnitudes, including the following two cases. Case1 1 E x ≠ E y and ∆φ = φy - φx = ± --2- + 2nπ
n=0,1,2,3,...
(4)
where + & - refer to Clockwise or Counter Clockwise rotation. Case2 nπ ∆φ = φy - φ x ≠ ± ------ n=0,1,2,3,... 2 ∆φ > 0 for Clockwise rotation ∆φ < 0 for Counter Clockwise rotation. When EM waves are obliquely incident on an plane interface between two media, it is necessary to consider two cases: the first case is when the electric field is polarized perpendicular to the plane of incidence, while the second case is when the electric field is polarized parallel to the plane of incidence (see Figure 1a). Electric field polarization orthogonal to the plane of incidence is achieved with antennas in broadside (bistatic) mode (see Figure 1c) and polarization parallel to the plane of incidence is achieved with antennas in end-for-end (called endfire) mode (figure 4b). The reflection coefficient for each case is different, and is a function of the incidence angle (antenna offset) and the electrical properties of the media above and below the interface. For example, consider the partitioning of an electromagnetic wave at the boundary of a planar, dielectric, nonmagnetic interface. The ratio of reflected to incident wave amplitudes, known as the Fresnel reflection coefficients, are given below. 2
R⊥
cos ϑ i – ( ε 2 ⁄ ε 1 ⋅ 1 – ( ε 1 ⁄ ε 2 ⋅ sin ϑi ) ) = --------------------------------------------------------------------------------------------------- (19) 2 cos ϑ i + ( ε 2 ⁄ ε 1 ⋅ 1 – ( ε 1 ⁄ ε 2 ⋅ sin ϑi ) ) 2
– cos ϑ i + ( ε 1 ⁄ ε 2 ⋅ 1 – ( ε 1 ⁄ ε 2 ⋅ sin ϑi ) ) R ll = -------------------------------------------------------------------------------------------------------(20) 2 cos ϑ i + ( ε 1 ⁄ ε 2 ⋅ 1 – ( ε 1 ⁄ ε 2 ⋅ sin ϑi ) ) where R ⊥ is the Fresnel reflection coefficient for electric field component which is polarized perpendicular to the plane of incidence, R ll is the Fresnel reflection coefficient for electric field component which is polarized parallel to the plane of incidence, ϑ i represents the incident angle, ε represents permittivity. In general the reflection and transmission coefficients are complex quantities, and their amplitudes and phases are a function of the angle of incidence (i.e. antenna separation). For parallel polarization, the reflection coefficient is equal to zero for a specific angle of incidence that is known as the Brewster angle ϑ B = atan ( ε 2 ⁄ ε 1 ) (21). The magnitude of the reflection coefficient is also unity for a specific incidence angle that is called the critical angle. The critical angle ϑ C = asin ( ε 2 ⁄ ε 1 ) (22) is independent of polarization and can only occur when the underlying layer velocity is greater than the overlying layer velocity. Tables 1 through 4 summarize the relative amplitude and phases for different angles of incidence and different relative permittivity contrasts across the boundaries. The point should be made here that the different relative angle of incidence
can arise either from situations related to the orientation of the antennas on the surface, or from different dip angles of the boundaries. Table 1: slow layer over a fast layer for R ll incidence angle
complex/ real
amplitude
phase
ϑ i < ϑ B < ϑ C real
variable
0
ϑ B < ϑ i < ϑ C real
variable
180
ϑi = ϑB
real
zero
0 to 180
ϑi ≥ ϑC
complex
unity
variable
Table 2: fast layer over a slow layer for R ll incidence angle
complex/ real
amplitude
phase
ϑi < ϑB
real
variable
180
ϑi = ϑB
real
zero
180 to 0
ϑi > ϑB
real
variable
0
Table 3: slow layer over a fast layer for R ⊥ incidence angle
complex/ real
amplitude
phase
ϑi < ϑC
real
variable
0
ϑi ≥ ϑC
complex
unity
variable
Table 4: fast layer over a slow layer for R ⊥ incidence angle all ϑ i
complex/ real real
amplitude variable
phase 180
The effect of polarization can be seen in a practical sense by observing the GPR resonse over dipping dielectric and conducting planes. Examples of different combinations of antenna polarization orientation for dielectric planes buried in a sand-filled pit are shown in Figure 2. These data were measured using a special antenna system that was constructed for this purpose. The amplitude of the reflections for the cross-pole mode are significantly diminished. In fact, the reflection for the cross-pole mode for a flat plane should be zero for a perfectly polarized linear dipole antenna system in the far-field. The far-field distance for a dipole antenna in this medium (sand with a dielectric constant of between 6 and 7) is approximately 0.5 m. This is approximately the depth (a 2-way travel time of around 10 ns) that would be computed from the strong refection on all four crosspole and co-pole orientation-dip combinations that are shown in Figure 2.
FIGURE 2. Polarization components over dipping planes in the pit. Conducting plane. (a) and (b) High dip angle 45 degrees. (c) and (d) Low dip angle 16 degrees. Parallel polarization (a) and (c). Perpendicular polarization (b) and (d).
The primary point of Figure 2 is that for a dielectric material in the far-field, the cross-pole component has greatly diminished amplitude. Therefore, a cross-pole can be used in 3D imaging to distinguish planer objects from objects with sharp or rounded boundaries. Figure 3 shows more aspects of the effect of the orientation of the antenna with respect to the dip angle of the plane. A conductive plane with twenty degrees of dip (Figures 3a-3d) is shown for comparison with the conductive plane with forty degrees of (Figure 3e-3h). The original orientation of the co-pole bistatic antennas is perpendicular to the line direction, while the rotation angle of 60 degrees for the line causes the antenna to be
much closer to the surface of the plane, which presents a much shallower dip. This rotation effectively changes the refection angle from the plane for both the co-pole and the cross-pole orientations, as can be seen in each of the cross-sections in Figure 3. It also has the effect of changing the scattering direction of some of the energy.
FIGURE 3. Co-pole bistatic (a, c, e, and g) and cross-pole bistatic (b, d, f, and h) cross sections over a plane that is dipping at 20 and 40 degrees. Two different travel dirctions are shown. The dipping plane was contained in a pit containing sand with a dielectric constant of approximately 7. The antennas had an a center-band freqency of approximately 500 Mhz.
A cylinder is the other idealized model that is often used to obtain a basic understanding of polarization over a two dimensional object. The response of bistatic GPR antennas over dielectric and conductive cylinders is shown (unmigrated) in Figure 4. There are two significant observations from this experiment: 1) the response over the conductive pipe is nearly zero when the antennas are perpendicular to the axis of the pipe, and
2) the response for the dielectric pipe is nearly independent of the orientation of the antenna. The back scattered fields from a large, smooth plane are not depolarized, but the scattered fields from cylinders may be strongly depolarized depending on the orientation of the cylinder relative to the antennas and the radius of the cylinder compared to the incident wavelength. As the radius to wavelength ratio of metal and plastic pipes decreases, the scattering properties become increasingly more polarization dependent. When using linearly polarized dipole antennas, metallic pipes and low impedance dielectric pipes, are best imaged with the long axis of the dipole antennas oriented parallel to the long axis of the pipes. Thin, high impedance, dielectric pipes, are best imaged with the long axis of the dipoles oriented orthogonal to the long axis of the pipe. When the radius of high impedance pipes becomes greater than approximately one tenth the wavelength, they are best imaged with the long axis of the dipoles oriented parallel to the long axis of the pipe. The pipes shown in Figure 4 are very thin compared to the wavelength of the 500 Mhz (centerband frequency) used for the measurements that are shown. Polarization of Complex objects in the Field The plane and the cylinder are so heavily studied and modeled because they provide a foundation for understanding the effects of polarization on objects in the field. However, the effects of polarization can be compounded even in the simplest layered environment. Figure 4 shows the 3D models of two field surveys run with conventional antennas (bistatic parallel, 300 Mhz center-band antennas). The lines in Figure 5a were run in the east-to-west, while the lines in the lower block in Figure 5b were run north-to-south. The subtle differences in these two surveys can make significant difference between the way that data are interpreted. This point is emphasized by the two slice views that are shown in Figure 6. Our final visual example is from detailed surveys of tree roots. These data are shown in Figure 7. The surveys were run along parallel lines spaced approximately 2 cm apart, providing a high density of samples for constructing the 3D block views shown in Figure 7. The surveys were run with parallel bistatic antennas in two orthogonal directions in a manner that was similar to the surveys run in Figure 6 over the larger area. These data clearly show the effect of polarization/depolarization on linear objects that was stated previously in the discussion on cylinders.
FIGURE 4. Polarization over pipes in a test pit showing the effect of the relative orientation of the antennas with respect to the orientation of the cylinder. The transmitter and receiver transducers consisted of was a 500 Mhz center-band antenna system. The objects were buried in a sand pit with a dielectric constant of approximately 6.
FIGURE 5. Two polarization directions of BNL 3D GPR data from the field, showing the block views. Note that the perspective rotations are slightly different, but the blocks are showing the same relative data volume.
FIGURE 6. Two polarization directions of BNL 3D GPR data from the field, showing the blockslice views. Note that the perspective rotations are slightly different, but the blocks are showing the same relative data volume.
FIGURE 7. Two views of a detailed survey over tree roots. These data were run along parallel lines that were spaced approximately 2 cm apart. A 500 Mhz (center-band) antenna was used.
Conclusions The effect of polarization of GPR signals in even the simplest cases can be important factors in both 2D and 3D imaging of the subsurface. The direction of a line will strongly influence the success, or failure of a survey to detect subsurface features. The same argument can used in reverse, since polarization properties of the transmitter and receiver can be utilized to emphasize features with a particular shape, size and orientation. The polarization factor becomes particularly critical when it comes to using GPR in a 3D mode. Even simple dipping layers can mask the ability to detect individual objects, and the direction of the survey lines with respect to the dip of beds is important to any display of 3D data. In the case of very complex objects (e.g., tree root systems), the
effect of polarization is even more critical to ultimately obtaining a good 3D view of individual objects within a group of complex features.
References Beckmann, P., 1968, The Depolarization of Electromagnetic Waves: The Golem Press, Boulder, CO, 214 p. Roberts, R.R., 1994, Analysis and theoretical Modeling of GPR Polarization Data: Ph.D. Dissertation, The Ohio State University, 429 p. Roberts, R.L., and J.J. Daniels, 1996, Analysis of GPR Polarization Phenomena: Journal of Environmental and Engineering Geophysics, vol. 1, No.2, p. 139-157. Roberts, R.L., and J.J. Daniels, 1997, Modeling Near Field GPR in three dimensions using the FDTD Method: Geophysics, Vol. 62, No. 4, p. 1114-1126. Radzevicius, S.J.(2001) Dipole antenna properties and their effects on ground penetrating radar data: Ph.D. Dissertation, The Ohio State University. Radzevicius, S.H., C.-C. Chen, L. Peters, and J.J. Daniels (2003 ), Near-field radiation dynamics through FDTD modeling: Jour. Applied Geoph., v. 52, p. 75-91.
