Transmitted field in the lossy ground from ground penetrating radar (GPR) dipole antenna

Transcription

1 Computatonal Methods and Expermental Measurements XVII 3 Transmtted feld n the lossy ground from ground penetratng radar (GPR) dpole antenna D. Poljak & V. Dorć Unversty of Splt, Croata Abstract The paper deals wth the evaluaton of transmtted electrc feld n the ground due to the GPR dpole antenna. The frequency doman formulaton s based on the ntegro-dfferental equaton of the Pocklngton type. The nfluence of the earth ar nterface s taken nto account va the smplfed reflecton/transmsson coeffcent arsng from the Modfed Image Theory (MIT). The space-frequency Pocklngton equaton s solved va the Galerkn Bubnov varant of the Indrect Boundary Element Method (GB-IBEM) and the correspondng transmtted feld s obtaned by numercally computng feld ntegrals. Some prelmnary results for the electrc feld transmtted nto materal meda are presented. Keywords: ground penetratng radar, dpole antenna, lossy half-space, transmtted feld, numercal soluton procedures. 1 Introducton Ground-penetratng radar (GPR) fnds numerous applcatons n dfferent areas of geophyscs and underground engneerng, such as cvl engneerng [1]. An mportant, one of the most crtcal, component regardng GPR system performance s an antenna whose type and sze strongly depend on partcular applcaton []. Namely, the knowledge of the energy transmtted nto the dsspatve half-space enables better antenna desgn and more realstc nterpretaton of the target reflected wave. The use of physcally larger, low frequency antenna provdes deeper penetraton nto the ground, whle hgh frequency antennas are convenent for the tasks requrng less penetraton depth and better resoluton. do:10.495/cmem150011

2 4 Computatonal Methods and Expermental Measurements XVII Consequently, a deeper nsght nto the behavor of the feld transmtted wthn the lossy ground s of contnuous nterest n GPR research. In general, the analyss can be carred out n the frequency or tme doman, respectvely, [3 5]. Many effcent GPR antenna models based on the Fnte Dfference Tme Doman (FDTD) method of soluton have been reported, e.g. [6, 7]. Contrary to the wdely used FDTD approach the present paper deals wth the assessment of transmtted electrc feld n the ground due to the GPR dpole antenna by means of the Boundary Element Method (BEM). The formulaton s based on the space-frequency ntegro-dfferental equaton of the Pocklngton type and correspondng fled formulas. The presence of the earth ar nterface s taken nto account va the smplfed reflecton/transmsson coeffcent arsng from the Modfed Image Theory (MIT). The space-frequency Pocklngton equaton s numercally solved va the Galerkn Bubnov varant of the Indrect Boundary Element Method (GB-IBEM) and the correspondng transmtted feld s obtaned by numercally computng the related feld ntegrals. Some llustratve results for the transmtted electrc feld are gven n the paper. Formulaton Geometry of nterest s related to the dpole radatng above a lossy medum, as t s shown n Fg 1. ~ GPR dpole antenna ncdent wave reflected wave ar ( 0, 0 ) transmtted wave lossy ground (,, ) Fgure 1: GPR dpole antenna above a lossy half-space. Generally, a drectve transmsson of sgnal nto a materal half-space can be analyzed by means of the rgorous Sommerfeld ntegral formulaton, or the approxmate Fresnel reflecton/transmsson coeffcent capproach [3]. The valdty of each approach has been dscussed elsewhere and some general remarks can be, for example, found n [8] or [9]. Ths work features the use of the modfed mage theory (MIT) [9] n the transmtted feld formula. The ntegro-dfferental equaton for the current nduced along the dpole can be derved by enforcng the nterface condtons for the tangental components of the electrc feld at the wre surface: exc sct e E E (1) x 0

3 Computatonal Methods and Expermental Measurements XVII 5 nc where the exctaton feld s composed from the ncdent feld E ref reflected from the lossy ground E : exc nc ref E E E and the feld The scattered feld s expressed n terms of magnetc vector potental A and electrc scalar potental : sct E j A (3) Accordng to the thn wre approxmaton [3] expresson (3) becomes: sct Ex jax (4) x wth: L/ Ax I( x') g( x, x') dx' (5) 4 L/ L/ 1 I( x') ( x) gxx (, ') dx' j4 0 x' L/ () (6) where I(x ) s the nduced current along the antenna, whle g(x,x ) denotes the total Green functon: gxx (, ') g( xx, ') R g( xx, ') (7) where g 0 (x, x ) s the free space-green functon: g o 0 x, x' whle g (x, x ) arses from the mage theory: g x, x' e R TM jkoro o (8) e R jkor (9) where R o and R denote the correspondng dstance from the source to the observaton pont, respectvely. The reflecton coeffcent for the transverse magnetc polarzaton R TM, whch accounts for the presence of a lower lossy medum, s gven by: ncos nsn RTM (10) ncos nsn The refracton ndex n and angle θ are gven by: n r j x x ', arctg (11) 0 h Insertng (4) (6) nto () yelds the Pocklngton s ntegro-dfferental equaton for the unknown current dstrbuton nduced along the dpole:

4 6 Computatonal Methods and Expermental Measurements XVII L/ L/ exc 1 I x' Ex j Ix' gx, x' dx' gx, x' dx' 4 j4 x x' (1) L/ 0 L/ The transmtted electrc feld components n the XZ plane are gven, as follows: L/ L/ 1 I( x') G( x, x', z) Ex dx' G( x, x', z) I( x') dx' j4 eff x' x' (13) L/ L/ where G(x,x ): E z L/ 1 I( x') G( x, x', z) dx' j4 (14) x' z eff L/ MIT Gxx (, ') g( xx, ', z) (15) tr MIT And the transmsson coeffcent tr arsng from the modfed mage theory (MIT) s gven by: MIT n tr (16) n 1 The soluton of ntegro-dfferental equaton (1) can be obtaned by applyng the Galerkn Bubnov scheme of the Indrect Boundary Element Method (GB- IBEM). Once calculatng the nduced current the related transmtted electrc feld s obtaned by numercally computng feld ntegrals (13) and (14). 3 Numercal procedures The Galerkn Bubnov scheme of the Indrect Boundary Element Method (GB- IBEM) for the soluton of ntegro-dfferental equaton (1) s documented n detal elsewhere, e.g. n [3]. The components of the electrc feld transmtted nto the materal medum due to the dpole radaton are evaluated usng the BEM formalsm. The procedure s outlned n ths secton, for the sake of completeness. The current and ts frst dervatve at the -th boundary element are gven by: x x' x' x1 I( x') I1 I (17) x x I( x') I I1 (18) x ' x where I 1 and I are the values of current at the local nodes of the -th boundary element, wth coordnates x 1 and x, Δx= x -x 1 denotes the element length. Dscretzng (13) and (14) and substtutng (17) and (18) nto (13) and (14) results n the followng expressons: N x j 1 x j I I1 G( x, x', z) x x' x' x1 (19) Ex dx ' I1 I G( x, x ', z) I( x ') dx ' j4 eff 1 xj x' x x x1j x 1j E z M N x j j 1 Ij I1 j Gxx (, ', z) E dx' j4 x z (0) eff j 1 1 j x1j where N j denotes the total number of boundary elements along the wre.

5 Computatonal Methods and Expermental Measurements XVII 7 Integrals n (19) and (0) are numercally evaluated usng the Gaussan quadrature. The quas-sngularty of the Green functon s avoded by approxmatng the frst-order dfferental operator wth fnte dfferences [3]. 4 Numercal results The computatonal example s related to the dpole antenna of length L=1m and radus a=mm horzontally placed at heght h=0.5m above a real ground wth permttvty ε rg =10 and conductvty σ=10ms/m. Termnal voltage s V T =1V. The operatng frequency s vared from 1MHz to 300MHz. Fgs to 4 show the related felds components for f=1mhz, f=10mhz and f=100mhz. (a) E x component. (b) E z component. Fgure : Transmtted feld (V/m) nto the ground at f = 1MHz.

6 8 Computatonal Methods and Expermental Measurements XVII (a) E x component. (b) E z component. Fgure 3: Transmtted feld (V/m) nto the ground at f = 10MHz.

7 Computatonal Methods and Expermental Measurements XVII 9 (a) E x component. (b) E z component. Fgure 4: Transmtted feld (V/m) nto the ground at f = 100MHz. Analyzng the numercal results presented n Fgs to 4 t can be observed that the feld dstrbuton remans relatvely stable over the consdered frequences. Fg. 5 shows the E x component of the transmtted feld versus depth n the broadsde drecton for dfferent operatng frequences.

8 10 Computatonal Methods and Expermental Measurements XVII Fgure 5: Broadsde transmtted feld (V/m) nto the ground for dfferent frequences. Note that, due to the symmetry of the problem, E z component n the broadsde drecton s zero. Ths s easly vsble n Fgs b, 3b and 4b. 5 Concluson The paper deals wth the analyss of the electrc feld transmtted nto the materal half-space due to the GPR dpole antenna radaton. The frequency doman formulaton s based on the Pocklngton ntegro-dfferental equaton and related feld formulas. The nfluence of the earth ar nterface s taken nto account va the smplfed reflecton/transmsson coeffcent arsng from the Modfed Image Theory (MIT). The Pocklngton equaton s numercally solved va the Galerkn Bubnov varant of the Indrect Boundary Element Method (GB-IBEM) and the correspondng transmtted feld s determned by usng BEM formalsm, as well. Some computatonal examples for the electrc feld transmtted nto the materal medum are presented. Ths work should be consdered as an opener to the subject and the future work wll deal wth coupled dpole arrays above a lossy ground for GPR applcatons n both frequency and tme doman. Moreover, wthn future actvtes t s planned to carry out a stochastc collocaton analyss of the transent current nduced along the wres radatng over a lossy medum. Acknowledgement Ths work benefted from networkng actvtes carred out wthn the EU funded COST Acton TU108 Cvl Engneerng Applcatons of Ground Penetratng Radar.

9 Computatonal Methods and Expermental Measurements XVII 11 References [1] L. Pajewsk et al., Applcatons of Ground Penetratng Radar n Cvl Engneerng COST Acton TU108, 013. [] C. Warren et al., Radaton Characterstcs of a Hgh-Frequency Antenna n Dfferent Delectrc Envronments, 15 th Internatonal Conference on Ground Penetratng Radar GPR 014, pp , Brussels, Belgum, 014. [3] D. Poljak, Advanced Modelng n Computatonal electromagnetc Compatblty, John Wley and Sons, New York 007. [4] D. Poljak et al., Transent Analyss of Two Coupled Horzontal wres over a Real Ground, IEE Proc. Mcrowave Antennas Propagat., 147, 87 94, 000. [5] M. Fernandez Pantoja et al., Tme Doman Analyss of Thn Wre Antennas over Lossy Ground usng the Reflecton-Coeffcent Approxmaton, Rado Scence, Vol. 44, 009. [6] J. M. Bourgeos, G. Smth, A Fully Three- Dmensonal Smulaton of a Ground-Penetratng Radar: FDTD Theory Compared wth Experment, IEEE Transactons on Geoscence and Remote Sensng, Vol. 34, No 1, Jan., pp [7] L. Gürel, U. Oguz, Three-Dmensonal FDTD Modelng of a Groundpenetratng Radar, IEEE Trans. On Geoscence and remote Sensng, Vol. 38, No 4, July 000. [8] E. K. Mller, A. J. Poggo, G. J. Burke, E. S. Selden, Analyss of wre antennas n the presence of a conductng half-space. Part II. The horzontal antenna n free space, Canadan Journal of Physcs, 50, 197, pp [9] D. Poljak, Poljak, K. El Khamlch Drss, K. Kerroum, S. Sesnc, Comparson of analytcal and boundary element modelng of electromagnetc feld couplng to overhead and bured wres, Engneerng analyss wth boundary elements. 35, 3; , 011.