Ground penetrating radar polarization and scattering from cylinders

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1 Journal of Applied Geophysics Ground penetrating radar polarization and scattering from cylinders Stanley J. Radzevicius ), Jeffrey J. Daniels Department of Geological Sciences, The Ohio State UniÕersity, 125 South OÕal Mall, Columbus, OH USA Received 18 October 1999; accepted 10 July 2000 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. Circular cylinders represent important environmental and engineering targets such as buried pipes, wires, and rebar. The backscattered fields from cylinders may be strongly depolarized depending on the orientation of the cylinder relative to the antennas, the electrical properties of the cylinders, and the radius of the cylinder compared to the incident wavelength. These polarization dependent scattering properties have important implications for target detection, survey design, and data interpretation. As the radius-to-wavelength ratio of metal and plastic pipes decreases, the backscattering properties become 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. High impedance, dielectric pipes, are best imaged with the long axis of the dipoles oriented orthogonal to the long axis of the pipes. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Ground penetrating radar; Polarization; Cylinders 1. Introduction Ground penetrating radar Ž GPR. is a common geophysical technique for investigating the shallow subsurface ŽAnnan and Davis, 1989; Olhoeft, 1992; Peters et al., 1994; Daniels et al., The vector nature of the GPR electro- ) Corresponding author. address: radzev@geology.ohio-state.edu Ž S.J. Radzevicius.. magnetic field, commonly referred to as polarization, is described ŽBeckman, 1968; Born and Wolf, 1980; Mott, 1986; Balanis, and is largely ignored by interpreters of GPR data. Investigations by Roberts Ž and Roberts and Daniels Ž 1996, have demonstrated the potential of using the polarization characteristics of GPR for defining the size, shape, orientation, and material properties of buried objects. This paper describes polarization and cylinder scattering theory and concepts relevant for the GPR practitioner. Analytic solutions and r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S

2 112 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics GPR data examples over buried pipes of varying radii are used to illustrate polarization concepts and verify the applicability of theory to commercial dipole antennas and physical models. This manuscript also describes which dipole antenna configurations will result in optimal survey design, depending on whether the targets of interest are metallic or plastic pipes. 2. Polarization 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 vectors. 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 Žtime harmonic., polarization may be classified as linear, circular, or elliptical. If the vector that describes the electric field 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, it is referred to as circular polarization. Both are special cases of elliptical polarization, in which the electric field traces out an ellipse. An arbitrary electromagnetic 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 compo- nents as given by Balanis 1989 : E z,t se e ya z x x0 cosž v tyb zyfx. and E Ž z,t. se e ya z cos v tyb zyf Ž 1. y y0 y where a represents the attenuation constant, b the phase constant, v the angular frequency, f the phase, and Ex0 and Ey0 are the maximum amplitudes of the Ex and Ey components, re- spectively Linear polarization For a wave to have linear polarization, the time-phase difference between the two components must be Dfsf yf snp ns0,1,2,3,... Ž 2. y 2.2. Circular polarization x Circular polarization is achieved only when the magnitudes of the components are the same and the time-phase differences are multiples of pr2. ExsEy and 1 Dfsfyyfxs" ž q2n p Ž 3. 2 / where q and y refer to clockwise Ž CW. or counterclockwise Ž CCW. rotation. n s 0, 1, 2, 3, Elliptical polarization Elliptical polarization is achieved only when the time-phase difference between the two components are odd multiples of pr2 and their magnitudes are not the same or when the timephase difference between the components are not equal to multiples of pr2, regardless of their magnitudes. Case 1: E x/ey and 1 Dfsfyyfxs" ž q2n/ p Ž 4. 2 ns0, 1, 2, 3,...,where q and y refer to CW or CCW rotation.

3 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Case 2: np Dfsfyyf x/" 2 ns0,1,2,3,... Ž 5. Df)0 for CW Df-0 for CCW. Pipes and other objects scatter energy preferentially, depending on the incident polarization. The polarization and orientation of the transmit antenna is thus important to ensure sufficient energy is scattered from subsurface targets to allow measurement by the receive antenna. Preferential scattering may result in depolarization of the incident field. Depolarization occurs when the amplitude or phase of the incident field components ŽEq. Ž 1.. are modified such that the scattered field results in a different polarization. The ability of the receive antenna to measure these scattered fields is determined not only by the power of the scattered fields, but also by the polarization match between the scattered fields and the receive antenna. The polarization of the field incident on the receive antenna is determined by the polarization of the field radiated by the transmit antenna and the degree of depolarization experienced by scattering from subsurface objects. It is thus important to understand the polarization properties of GPR antennas and scattering from subsurface objects. Most commercial GPR antennas are dipole or bow-tie antennas that radiate linearly polarized energy with the majority of the radiated electric field oriented along the long axis of the dipole or bow-tie. For a description of dipole fields over a half-space, consult Annan Ž 1973., Annan et al. Ž 1975., Arcone Ž 1995., Engheta et al. Ž 1982., Smith Ž and Radzevicius et al. Ž 2000b.. A complete polarization mismatch using dipole antennas results when the scattered field and polarization of the receive antenna are both linearly polarized ŽEq. Ž 2.. and oriented at right angles to each other. For example, rotating ideal dipole antennas orthogonal to each other Ž crossed-dipoles. results in a complete polarization mismatch. Spiral, or other circularly polarized antennas, are also used for pipe detection. A complete polarization mismatch using circularly polarized antennas results when the scattered field and receive antenna are both circularly polarized ŽEq. Ž 3.., but have electric fields with opposite rotation directions Žleft and right circular.. 3. Normal incidence plane wave scattering by circular cylinders ( fundamental theory and conclusions from analytical solutions) Cylinders represent an important class of objects for GPR since they represent important environmental and engineering targets and also because their scattering properties are strongly polarization dependent. A brief discussion of the importance of polarization on the scattering of plane waves normally incident on both dielectric and conductive circular cylinders is now described. The reader is referred to Balanis Ž and Ruck et al. Ž for a more detailed explanation of equations and for oblique incidence. Two linearly independent basis vectors Ž polarizations. ŽEq. Ž 1.. are necessary to describe scattering from both dielectric and perfectly conducting cylinders. It is convenient to choose these polarization vectors such that one vector is oriented along the long axis of the cylinder ŽE parallel or transverse-magnetic Ž TM.. and the other vector oriented orthogonal to the long axis of the cylinder ŽE perpendicular or transverse electric Ž TE.. Ž Fig. 1.. TM polarization is achieved when the long axis of the transmit and receive dipole antennas are oriented parallel to the long axis of the cylinder and the survey direction is orthogonal to the long axis of the cylinder Ž Fig. 2a.. TE polarization is achieved when both antenna axes are oriented orthogonal to the long axis of the cylinder and the survey direction is orthogonal to the long axis of the cylinder Ž Fig. 2b.. The scattered field is a function of the electrical properties of the cylinder and surrounding

4 114 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics The radar cross-section Ž RCS. represents a convenient way to describe the strength of scattered fields observed in the far-field. The RCS is defined as the the area intercepting the amount of power that when scattered isotropically, produces at the receiver a density that is equal to the density scattered by the actual target ŽBal- anis, < E s < 2 2 RCSs lim 4p r 2 Ž 6. r ` < i E < For 2D objects such as an infinite cylinder, the RCS becomes the scattering width Ž SW. or Fig. 1. Definitions of E parallel Ž TM. and E perpendicular Ž TE. polarizations relative to a cylinder, as defined in Balanis Ž TM polarization occurs when the electric field is parallel to the long axis of the cylinder. TE polarization occurs when the electric field is perpendicular to the long axis of the cylinder. E and H represent the electric and magnetic fields respectively, while a represents the cylinder radius, b represents the propagation vector, f represents the scattering angle, r represents the radial distance from the cylinder, and x, y, z represent the coordinates of a cartesian coordinate system. material, the distance from the cylinder, and the scattering angle f, as defined in Fig. 1. A scattering angle of 1808 represents backscattering. Since circular cylinders have a cylindrical shape, their scattering properties are conveniently described using Hankel and Bessel functions because they represent cylindrical waves. Incident and scattered fields for conductive and dielectric cylinders are described in terms of cylindrical coordinates in Appendix A. Fig. 2. Examples of GPR surveys using linearly polarized coincident Ž common-offset with small separation. antennas to image cylinders. The GPR survey direction is into the page for all the cases. Ž. a Dipoles oriented parallel to the long axis of the cylinder are best for imaging conductive and small diameter, low impedance, dielectric cylinders. Ž. b Dipoles oriented orthogonal to the long axis of the cylinder are best for imaging high impedance dielectric cylinders. Ž. c Crossed-dipoles at 458 are good for imaging both conductive and dielectric cylinders.

5 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics alternatively the RCS per unit length ŽBalanis, < E s < 2 SWs lim 2p r 2 Ž 7. r ` < i E < All scattering widths of cylinders in this manuscript are normalized with respect to the wavelength of the incident field and Figs. 3 5 are plots of normalized scattering widths as a function of scattering angle for different radius to incident wavelength ratios. and s will be used to represent dielectric permittivity and conductivity. Several universal features are observed in the scattering widths as a function of scattering angle for both conductive and dielectric cylinders. As the radius of the cylinder becomes small compared to wavelength: Ž 1. scattering width amplitudes for both polarizations oscillate less, Ž 2. TM polarization scattering widths become nearly constant as a function of scattering angle, and Ž 3. TE polarization scattering widths form a single, low amplitude null, at a scattering angle of 908. Most GPR surveys used to image the subsurface are conducted in common-offset mode using closely spaced antennas. This results in a scattering angle of approximately 1808 Žback- scattered., depending on antenna separation and target depth. It is thus important to further investigate backscattered scattering widths as a function of pipe radius. Fig. 6a represents backscattering from metallic pipes as a function of pipe radius. The TM polarization backscattering width is greater than TE backscattering width for most radius-to-wavelength ratios. The TE component oscillates about the TM component and TE converges toward TM as the radius becomes large compared to wavelength. TM polarization is the preferred polarization for the detection of metallic pipes, as illustrated by Figs. 2a and 6a. High impedance dielectric cylinders Žcylin- ders with a permittivity less than the surround- ing soil. represent such targets as PVC pipes filled with air or hydrocarbons and surrounded by higher permittivity soil. Fig. 6b is a plot of the backscattering widths for dielectric pipes embedded in a medium having a permittivity seven times greater than the pipe. This represents air filled PVC pipes surrounded by a typical moist sand. The TE polarization backscattering width is greater than the TM backscattering width for small diameter, high impedance, dielectric pipes, and Fig. 2b represents the best survey geometry. Fig. 6c is a plot of backscattering widths for dielectric pipes embedded in a medium having a permittivity seven times less than the pipe. The TM polarization backscattering width is greater than the TE backscattering width for small diameter, low impedance, dielectric cylinders, and Fig. 2a represents the best survey geometry. It is often informative to describe scattering by a scattering matrix defined as: s yikr E e Sxx Sxy E i x x s s Ž 8 i. E r S S E y yx yy y nates for thin metal pipes oriented with their long axis of symmetry along the x axis, whereas thin dielectric pipes with an impedance greater than the surrounding soil have S yy)s xx. Cross-pole antennas Ž Fig. 2c. are useful for improving antenna isolation and can be used to reduce clutter under appropriate field conditions and when stratigraphy is not the objective of the GPR survey Ž Radzevicius et al., 2000a.. Objects are visible using linear, crossed-dipole antennas where subscripts x and y denote an orthogonal set of coordinates and superscripts i and s denote the incident and scattered components of the electric E fields, k is the wavenumber, r is the distance from the target to observation point. The scattering matrix is useful in describing scattering from thin metal pipes Žradius small compared to wavelength.. The S term domixx

6 116 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Fig. 3. Scattering widths for metallic cylinders normalized by the wavelength Ž l. of the incident field. As the radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TM polarization backscattering widths are greater than the TE polarization scattering widths for most cylinders. when they scatter electric field components orthogonal to the field components radiated by the transmit antenna. Scattered cross-components are produced by scattering from rough planes or most small objects. Cross-components Žcompo- nents not present in incident field. are not introduced by scattering of plane waves normally incident on infinite circular cylinders, as seen in

7 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Fig. 4. Scattering widths for high impedance dielectric cylinders, normalized by the wavelength l of the incident field. As the radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TE polarization backscattering widths are greater than the TM polarization scattering widths for small diameter cylinders. Eqs Balanis 1989 describes the case of a plane wave obliquely incident on both dielectric and conducting cylinders. Balanis observed that scattering from a perfectly conducting infinite cylinder does not introduce additional components in the scattered field that are not present in the incident field. This is not the case for dielectric cylinders, which introduce orthogonally polarized components under oblique wave incidences. Cylinders having a

8 118 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Fig. 5. Scattering widths for low impedance dielectric cylinders, normalized by the wavelength l of the incident field. As the radius of the cylinder becomes small compared to wavelength, TM scattering widths become nearly constant as a function of scattering angle and TE scattering widths form a single, low amplitude null, at a scattering angle of 908. The TM polarization backscattering widths are greater than the TE polarization scattering widths for small diameter cylinders. finite length also produce depolarization from edge scattering. It is not necessary for a target to produce a scattered cross-component to be visible with cross-pole antennas. A long metallic pipe, while a strong depolarizer Ž Fig. 3. does not produce field components that were not originally present in the incident field. The linear geometry of

9 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics pipes results in field components that are oriented both parallel and orthogonal to the long axis of the pipe. The reflected and transmitted fields for thin pipes are related by the following relationships Ž Daniels et al., 1988.: yikr e Sxx Sxy EssE ysinu cosu tž / S S r yx yy = cosu Ž 9. sinu ž / E e yikr s s ž S yy ys E 2r t xx =sin2uys sin 2 uqs cos 2 u Ž 10. xy where Es and and Et are the scattered and transmitted fields, respectively, and u is the angle between the long axis of the transmit dipole and the long axis of the cylinder. For linear targets Sxy and Syx are small compared to other components and thus E e yikr s s Ž S yy ys xx. sin2u Ž 11. E 2r t ž / The ratio EsrEt is maximized when us458 for both dielectric and conductive pipes and thus crossed-dipole antennas at 458 with respect to cylinders represent the best antenna geometry to image cylinders Ž Fig. 2c.. yx / 4. Linear polarization and scattering from pipes physical model examples To test the applicability of the analytical Ž. solutions Eqs and Fig. 6, data were Fig. 6. Backscattering widths as a function of radius Ž a., normalized by the wavelength Ž l. of the incident field. Solid lines represent TM polarization and dashed lines represent TE polarization. Ž. a TM backscattering widths are greater than TE backscattering widths for most metallic cylinders. Ž. b TE backscattering widths are greater than TM backscattering widths for small diameter, high impedance, dielectric cylinders. Ž. c TM backscattering widths are greater than TE backscattering widths for small diameter, low impedance, dielectric cylinders.

10 120 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Fig. 7. Geometry of bow-tie antenna elements for 500 MHz Ž air. multi-component antenna. The antenna consists of four transmitting elements Ž T1, T2, T3, T4. and two receiving elements Ž R1, R2.. The figure is drawn to scale. recorded in a sand test pit using a Geophysical Survey Systems GSSI bow-tie, multi-component antenna having a center frequency of 500 MHz Ž in air. Ž Fig. 7.. The antenna consisted of two receiving elements and four transmitting elements Žeight different transmitting receiving combinations. that were used to record the data. Fig. 8 is a plot of the amplitude spectrum obtained by taking the Fourier transform of both co-pole and cross-pole traces with no pipes present. The soil interface influences current distribution and impedance of GPR antennas by an amount determined by the antenna design and the electromagnetic properties of the ground. A soil interface causes the antenna to radiate at lower frequencies Ž 270 MHz peak on interface. than in an air whole-space Ž 500 MHz peak.. The amplitude spectrum varies smoothly as a function of frequency Žno nulls at a given frequency. and is approximately Gaussian in shape. A large polarization mismatch with cross-pole antennas is observed in Fig. 8 Žco-pole larger amplitude than cross-pole. for all frequencies because each frequency is linearly polarized. Constant polarization for all frequencies is a desirable feature of dipole antennas Žde Jongh et al., While time-domain impulse radar Fig. 8. Amplitude spectrum for co-pole solid and cross-pole dashed traces for an antenna located on the soil interface with no buried pipes. The amplitude spectrum varies smoothly as a function of frequency and is approximately Gaussian in shape with a peak frequency of 270 MHz. The strong polarization mismatch for cross-pole antennas results in larger co-pole amplitudes for all frequencies because each frequency is linearly polarized.

11 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics radiates a pulse composed of many frequencies Ž Fig. 8., the frequencies centered about 270 MHz have the largest amplitudes and are the most significant for determining the response observed in field data. To illustrate the above concepts, data were recorded over plastic and metal pipes with survey directions normal to the long axis of the pipes and at 158, 308, and 458 angles to the long axis of the pipes. Long pipes were used to avoid resonance and edge effects. Pipes of varying radii were used to demonstrate the most significant features observed in the analytic solutions. Fig. 9 shows the results of data recorded over a m radius copper pipe buried at a depth of 0.46 m and having a length of 3.05 m. A soil permittivity probe, described by Caldecott et al. Ž and operating at 40 and 60 MHz frequencies gave relative permittivity values of 4 near the surface and graded to a permittivity of 7 at a depth of 0.6 m. The peak 270 MHz frequency has a nominal wavelength of 0.42 m and thus the copper pipe with a radius of m yields a radius-to-wavelength ratio of While this radius-to-wavelength ratio is only for a single frequency, most of the energy radiated from the antenna is composed of frequencies that have TM polarization scattering widths greater than TE polarization scattering widths Ž Figs. 3 and 6a.. Fig. 9 demonstrates that this is the case with the TM component Ž T2R1. much greater than the TE component Ž T1R2.. Crosspole components Ž T2R2 and T3R1. yield maximum values when the dipole antennas are oriented at 458 to the long axis of the cylinder, as described by Eqs. Ž 9. Ž 11.. No difference between cross-pole configurations would be expected with ideal antennas over homogeneous soils or at a 458 survey angle for co-pole configurations. Coupling between the four transmitting elements and two receiving elements, in addition to small construction and alignment differences, results in a slightly different antenna response between the two cross-pole Ž T2R2 and T3R1. and co-pole Ž T2R1 and T1R2. configurations. Heterogeneities in the soil also Fig. 9. GPR survey normal and at 158, 308, 458 to the long axis of a copper pipe buried at a depth of 0.46 m, having a radius of m, and a length of 3.05 m. A 270 MHz center frequency antenna on soil with a relative permittivity of 7 has a nominal wavelength of 0.42 m and thus the copper pipe yields a radius-to-wavelength of Figs. 3 and 6a suggest that this small radius-to-wavelength ratio results in larger TM scattering widths Ž T2R1 Fig. 7. than TE scattering widths Ž T1R2 Fig. 7.. Cross-pole components Ž T2R2 and T3R1. yield maximum values when the dipole antennas are oriented at 458 to the long axis of the pipe as described by Eqs. Ž. 9 Ž 11.. produce differences due to the antenna element offsets. Data were also recorded over metal and plastic pipes, having a radius of m, a length of 3.66 m, and buried at a depth of 0.61 m. A 270 MHz antenna yields a nominal radius-towavelength ratio of approximately Figs. 3 and 6a suggest similar backscattering widths for TM and TE polarizations for metallic pipes

12 122 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics having this radius-to-wavelength ratio. In contrast to the thin metal pipes, Fig. 10 demonstrates that the TM polarization Ž T2R1. is only slightly larger than the TE polarization Ž T1R2., as predicted from analytic solutions. Cross-pole components Ž T2R2 and T3R1. still yield maximum values when the dipole antennas are oriented at 458 to the long axis of the metal pipe, as in the thin metallic pipe case. In contrast to Fig. 10. GPR survey normal and at 158, 308, 458 to the long axis of a steel pipe buried at a depth of 0.61 m, having a radius of m, and a length of 3.66 m. A 270 MHz center frequency antenna yields a nominal radius to wavelength ratio of approximately Unlike the thin pipe, Figs. 3 and 6a suggest similar TM and TE scattering widths for metallic pipes having this radius-to-wavelength ratio. In this figure, the TM Ž T2R1. polarization is only slightly brighter than the TE polarization Ž T1R2., as predicted. Cross-pole components Ž T2R2 and T3R1. still yield maximum values when the dipole antennas are oriented at 458 to the long axis of the pipe, as in the thin pipe case. Fig. 11. GPR survey normal and at 158, 308, 458 to the long axis of a PVC pipe buried at a depth of 0.61 m, having a radius of m, and a length of 3.66 m. A 270 MHz center frequency antenna yields a nominal radius to wavelength ratio of approximately Figs. 4 and 6b suggest larger backscattering widths for TE compared to TM, given this radius-to-wavelength ratio. This figure verifies that the TE polarization is greater than the TM polarization. Cross-pole components Ž T2R2 and T3R1. still yield maximum values, as in the metallic pipe case, when the crossed-dipole antennas are oriented at 458 to the long axis of the pipe. metal pipes, Fig. 11 demonstrates that the TE polarization Ž T1R2. is larger than the TM polarization Ž T2R1., as predicted from analytic solutions. Cross-pole components Ž T2R2 and T3R1. still yield maximum values when the dipole antennas are oriented at 458 to the long axis of the plastic pipe, as in the metallic pipe case.

13 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics Conclusions The scattering properties of cylinders are strongly polarization dependent. It is thus important to understand the scattering properties of cylinders and the polarization properties of antennas used in the GPR survey. Knowledge of the electrical properties of the buried pipe and surrounding soil allows one to constrain the diameter of the pipe. The analytical solutions and field data demonstrate the importance of conducting GPR surveys, with two axially rotated measurements to avoid nulls over thin metallic pipes, when using linearly polarized co-pole or cross-pole antennas. An alternative is to use circularly polarized antennas that automatically rotate the polarized vector in space and thus removes the direction of signal nulls. As the radius of the metallic pipes increased, the TM and TE polarization backscattering widths become more similar and the need for axially rotated measurments diminished. When using linearly polarized dipole antennas, metallic pipes are best imaged with the long axes of the dipoles oriented parallel to the long axis of the pipe. Small diameter, high impedance, dielectric pipes are best imaged with the dipole axes oriented orthogonal to the long axis of the pipe. Crossed-dipole antennas can be used to reduce clutter and improve antenna isolation when stratigraphy is considered clutter and only pipes or other depolarizing targets are of interest. Maximum amplitudes are observed over pipes when the crossed-dipoles are oriented at 458 to the pipe. The best choice of antennas and polarizations for a particular survey depends on the targets of interest and the field conditions. Appendix A. Cylinder scattering equations Below are equations, in cylindrical coordinates, that describe normally incident plane wave scattering from infinitely long conductive and dielectric circular cylinders. The scattered field is a function of the electrical properties of the cylinder and surrounding material, the distance from the cylinder, and the scattering angle. Hankel and Bessel functions represent cylindrical waves and are useful for describing scattering from cylindrical cylinders. Eqs. Ž 12. Ž 23. demonstrate that cross-components Žcomponents not present in the incident field. are not introduced by scattering of plane waves normally incident on infinite cylinders. The reader is referred to Balanis Ž and Ruck et al. Ž for a more detailed explaination of equations. TM polarization normally incident on a perfectly conducting cylinder case: E i se e yib 0 x z 0 Ž 12. ` JnŽ b0 a. s yn Ž2. inf Ez sye0 Ý i Hn Ž b0 r. e Ž2. nsy` Hn Ž b0 a. Ž 13. E s se s s0 Ž 14. r f TE polarization normally incident on a perfectly conducting cylinder case: H i sh e yib 0 x z 0 Ž 15. H syh i H b r e ` X Jn b0 a s yn Ž2. inf X z 0 Ý Ž2. n Ž 0. nsy` Hn Ž b0 a. Ž 16. H s sh s s0 Ž 17. r f TM polarization normally incident on a dielectric cylinder case: E i se e yib 0 x z 0 Ž 18. ` s z 0 Ý yn i nsy` E se = ' X X Jn Ž b0 a. JnŽ b1a. y errmr JnŽ b0 a. Jn Ž b1a. X Ž2. Ž2. ' X r r n 1 n 0 n 1 n 0 e rm J Ž b a. H Ž b a. yj Ž b a. H Ž b a. = H Ž2. Ž b r. e inf 19 n 0 E s se s s0 Ž 20. r f

14 124 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics TE polarization normally incident on a dielectric cylinder case: H i sh e yib 0 x z 0 Ž 21. ` s z 0 Ý yn i nsy` H sh = ' X X Jn Ž b0 a. JnŽ b1a. y mrrer JnŽ b0 a. Jn Ž b1a. X Ž2. Ž2. ' X r r n 1 n 0 n 1 n 0 m re J Ž b a. H Ž b a. yj Ž b a. H Ž b a. = Hn Ž2. Ž b0 r. e inf Ž 22. H s sh s s0 Ž 23. r f The TE polarization case is expressed in terms of magnetic fields for convenience and one may convert between electric and magnetic fields using Ampere s and Faraday s laws. E i is the incident field, E s is the scattered electric field, H i is the incident magnetic field, H s is the scattered magnetic field, r, f, z are the standard coordinates in a cylindrical coordinate system, r is the radial distance of the observation point, a is the cylinder radius, xsrcosf Ž 24. J Ž b r. n is the Bessel function of the first kind or order n, J X Ž b r. n is the derivative of the Bessel function of the first kind with respect to the entire argument of the Bessel function, Ž2. H Ž b r. n is the Hankel function of the second Ž2. kind of order n, H X Ž b r. n is the derivative of the Hankel function of the second kind with respect to the entire argument of the Bessel function, b0 is the phase constant of the material surrounding the cylinder, b1 is the phase constant of the cylinder, r is the ratio of the permittivity of the cylinder to the surrounding material, and mr is the ratio of the permeability of the cylinder to the surrounding material. The e iwt time convention is used in this manuscript. References Annan, A.P., Radio interferometry depth sounding: Part 1. Theoretical discussion. Geophysics 38 Ž. 3, Annan, A.P., Davis, J.L., Ground-penetrating radar for high-resolution mapping of soil and rock stratigraphy. Geophys. Prospecting 37, Annan, A.P., Waller, W.M., Strangway, D.W., Rossiter, J.R., Redman, J.D., Watts, R.D., The electromagnetic response of a low-loss, 2-layer, dielectric earth for horizontal electric dipole excitation. Geophysics 40 Ž. 2, Arcone, S.A., Numerical studies of the radiation patterns of resistively loaded dipoles. Appl. Geophys. 33, Balanis, C.A., Advanced Engineering Electromagnetics. Wiley, New York, NY. Beckman, P., The Depolarization of Electromagnetic Waves. The Golem Press, Boulder, CO. Born, M., Wolf, E., Principles of Optics. Pergamon, New York, NY. Caldecott, R., Poirier, M., and Svoboda, D.E., A radio frequency probe to measure soil electrical properties. Final Report , US Army Engineer Waterways Experiment Station Corps of Engineers, P.O. Box 631, Vicksburg, MS Daniels, D.J., Gunton, D.J., Scott, H.F., Introduction to subsurface radar. IEE Proc. F 135 Ž. 4, Daniels, J.J., Brower, J., Baumgartner, F., High resolution GPR at Brookhaven National Laboratory to delineate complex subsurface targets. J. Environ. Eng. Geophys. 3 Ž. 1, 1 5. de Jongh, R.V., Yarovoy, A.G., Ligthart, L.P., Kaploun, I.V., Schukin, A.D., Design and analysis of new GPR antenna concepts. Proceedings of the Seventh International Conference on Ground Penetrating Radar, May 27 30, Lawrence, KS. pp Engheta, N., Papas, C.H., Elachi, C., Radiation patterns of interfacial dipole antennas. Radio Sci. 17, Mott, H., Polarization in Antennas and Radar. Wiley, New York, NY. Olhoeft, G.R., Geophysical detection of hydrocarbon and organic chemical contamination. Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems: 2, April 26 29, Oak Brook, IL. Society of Engineering and Mining Exploratino Geophysics, Golden, CO, pp Peters, L. Jr., Daniels, J.J., Young, J.D., Ground penetrating radar as a subsurface sensing tool. Proc. IEEE 82 Ž 12., , December. Radzevicius, S.J., Daniels, J.J., Guy, E.D., Vendl, M.A., 2000a. Significance of crossed-dipole antennas for high noise environments. Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems, February 20 24, Arlington, VA. pp Radzevicius, S.J., Daniels, J.J., Chen, C., 2000b. GPR H-plane antenna patterns for a horizontal dipole on a

15 S.J. RadzeÕicius, J.J. DanielsrJournal of Applied Geophysics half-space interface. Proceedings of the Eighth International Conference on Ground Penetrating Radar, May 23 26, Gold Coast, Australia. pp Roberts, R.L., Analysis and theoretical modeling of GPR polarization data. PhD Dissertation, The Ohio State University, Columbus, OH. Roberts, R.L., Daniels, J.J., Analysis of GPR polarization phenomena. J. Environ. Eng. Geophys. 1 Ž. 2, Roberts, R.L., Daniels, J.J., Modeling Near Field GPR in 3D Using the FDTD Method, 62 Ž Ruck, G.T., Barrick, D.E., Stuart, W.D., Krichbaum, C.K., Radar Cross Section Handbook. Plenum, New York, NY. Smith, G.S., Directive properties of antennas for transmission into a material half-space. IEEE Trans. Antennas Propagation AP-32 Ž. 3,