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Analysis of Horn Antenna Transfer Functions and Phase-Center Position for Modeling Off-Ground GPR

Analysis of Horn Antenna Transfer Functions and Phase-Center Position for Modeling Off-Ground GPR
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  IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 5, MAY 2011 1649 Analysis of Horn Antenna Transfer Functions andPhase-Center Position for ModelingOff-Ground GPR Khan Zaib Jadoon, Sébastien Lambot, Evert C. Slob, and Harry Vereecken  Abstract —The antenna of a zero-offset off-ground ground-penetrating radar can be accurately modeled using a linear systemof frequency-dependent complex scalar transfer functions underthe assumption that the electric field measured by the antennalocally tends to a plane wave. First, we analyze to which extentthis hypothesis holds as a function of the antenna height abovea multilayered medium. Second, we compare different methodsto estimate the antenna phase center, namely, 1) extrapolationof peak-to-peak reflection values in the time domain and 2) fre-quency-domainfull-waveforminversionassumingbothfrequency-independent and -dependent phase centers. For that purpose, weperformed radar measurements at different heights above a per-fect electrical conductor. Two different horn antennas operating,respectively, in the frequency ranges 0.2–2.0 and 0.8–2.6 GHzwere used and compared. In the limits of the antenna geometry,we observed that antenna modeling results were not significantlyaffected by the position of the phase center. This implies that thetransfer function model inherently accounts for the phase-centerpositions. The results also showed that the antenna transfer func-tion model is valid only when the antenna is not too close to thereflector, namely, the threshold above which it holds correspondsto the antenna size. The effect of the frequency dependence of the phase-center position was further tested for a two-layeredsandy soil subject to different water contents. The results showedthat the proposed antenna model avoids the need for phase-centerdetermination for proximal soil characterization.  Index Terms —Antenna modeling, antenna phase center,frequency dependence, ground-penetrating radar (GPR). Manuscript received April 29, 2009; revised May 28, 2010; acceptedSeptember 19, 2010. Date of publication January 6, 2011; date of currentversion April 22, 2011. This work was supported by the ForschungszentrumJülich GmbH (Germany), the Université Catholique de Louvain and Fondsde la Recherche Scientifique (Belgium), Delft University of Technology (TheNetherlands), and DIGISOIL project financed by the European Commissionunder the Seventh Framework Program for Research and Technological Devel-opment, Area “Environment,” Activity 6.3 “Environmental Technologies.”K. Z. Jadoon and H. Vereecken are with the Institute of Bio- and Geo-sciences, Agrosphere (IBG-3), Forschungszentrum Jülich GmbH, 52425 Jülich,Germany (e-mail:; Lambot is with the Institute of Bio- and Geosciences, Agrosphere(IBG-3), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany, and alsowith the Earth and Life Institute- Environmental Sciences (ELI-e), Uni-versité Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium (; C. Slob is with the Department of Geotechnology, Delft University of Technology, 2628 CN Delft, The Netherlands (e-mail: versions of one or more of the figures in this paper are available onlineat Object Identifier 10.1109/TGRS.2010.2089691 I. I NTRODUCTION P RECISE location of the phase-center position of an an-tenna is becoming more important because of new de-velopments and requirements in navigation, landing, tracking,aircraft,aerospace, and environmental researchand engineering[1]–[5]. The phase center is a virtual source point, whichrepresents the srcin of the radiated field from where thesphericaldivergenceappearstobeinitiated[6].Forthepracticalantennas such as arrays, reflectors, horns, and others, there isno single ideal phase center [7]. Estimating the position of theantenna phase center by using analytical formulations exists fora limited number of configurations.Experimentalmethodshavebeenproposedtoestimateappar-ent phase-center positions for different antennas [5], [8]–[11].For instance, Bares  et al.  [9] used an empirical model tocompute the frequency-dependent phase-center position of alog-periodic dipole (LPD) antenna and validated their approachby measurements. Doppler effects generated by the phase-center displacement are computed for frequency-modulatedcontinuous-wave radar applications. McKinney and Weiner[10] demonstrated photonic-synthesis techniques for arbitraryelectromagnetic waveforms, which enables the conjugate an-tenna phase response to be applied directly to the transmittedwaveform in an ultrawideband (UWB) system. Menudier  et al. [11] showed that the electromagnetic bandgap antenna can beused to feed reflector and quantify the influence of phase-center variation with frequency on the aperture efficiency of the reflector antenna. Wang  et al.  [5] reported a new numericalmethod to calculate the phase center for any kind of antennasas long as its far-field radiation expression is determined, forexample, the horn and LPD antennas. Liu  et al.  [12] proposeda 3-D coherent radar backscatter model for forest canopies toimprove the interpretation of interferometric synthetic apertureradar data, and they observed that the height of the scatteringphase center depends on canopy height, attenuation of canopy,and the gaps within the canopy.The phase center of a double-ridged horn antenna existsmostly between its imaginary apex point and its aperture. Atthe phase center and in the far-field region, the phase responseof the radiation pattern in the vicinity of the main beam willbe reasonably constant [13]. Double-ridged horn antennas havebeen used as an off-ground ground-penetrating radar (GPR)for accurate characterization of subsurface electrical properties[14]. Lambot  et al.  [15] estimated a single effective phase 0196-2892/$26.00 © 2011 IEEE  1650 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 5, MAY 2011 center by performing measurements with the antenna at dif-ferent heights above a perfect electrical conductor (PEC) andby extrapolating the peak-to-peak   ( PtP  )  reflection values inthe time domain [16]. In their approach, the medium propertiesare estimated by full-waveform inversion of zero-offset UWBfrequency-domain radar data, thereby maximizing informationretrieval from a single GPR measurement. Phase (travel time)and amplitude information is inherently exploited. The tech-nique relies on an accurate and computationally effective radarforward model. This includes internal antenna and antenna–soilinteraction propagation effects through frequency-dependentcomplex scalar transfer functions and solves exactly the 3-DMaxwell’s equations for wave propagation in multilayeredmedia. The proposed method has been validated in laboratoryconditions for identifying both the electric permittivity and theelectric conductivity of a sandy soil subject to a range of watercontents [17], [18] to monitor the dynamics of water in a labo- ratory sand column and subsequently derive the soil hydraulicproperties using hydrodynamic inverse modeling [19], to mapthe soil surface water content in field conditions [20]–[22], tocombine with passive radar to better estimate the field scale soilmoisture [23], and to estimate soil hydraulic properties fromintegrated hydrogeophysical inversion of time-lapse off-groundGPR data [24], [25].The exact location of the phase center of double-ridged hornsdepends on the dimensions of the horn, particularly on itsflare angle and tapper, and on the frequencies of operation.Significant variation can be observed in the phase center withfrequency when the pattern angles extend across the main beam[13]. For large flare angles and/or high frequencies, the phasecenter is closer to the apex, and for small flare angles and/or lowfrequencies, the phase center moves toward the aperture of thehorn [7]. Historically, the work of Walton and Sundberg [26] on the double-ridged horn was important, and it is still usedtoday for the design of these horns. Such horn antennas haverecently been analyzed numerically [27]–[30], but the effect of frequency on the phase-center position has not been quantifiedin these studies. Antenna analysis is, nevertheless, necessaryfor precise measurements of the source point above the groundand to correctly parameterize the forward electromagnetic radarmodel.In this paper, we compare different methods for estimatingthe phase center of two GPR double-ridged horn antennas,namely, by extrapolation of peak-to-peak reflection values inthe time domain and by frequency-domain full-waveform in-version assuming both frequency-independent and -dependentphase centers. In addition, we investigate to which extent theantenna transfer function model developed by Lambot  et al. [15] holds as a function of the antenna height above a mul-tilayered medium. For these purposes, we performed radarmeasurements at different heights above a PEC. Two differenthorn antennas operating in different frequency ranges, but withthe same bandwidth (0.2–2.0 and 0.8–2.6 GHz, respectively),were used, and their results were compared. A new inversionscheme is introduced to determine the phase-center position forboth a frequency-dependent phase center  ( ψ c ( f  ))  and a singleoptimal effective frequency-independent phase center  ( ψ c ) . Aconceptual requirement of the forward model is to operatewith the antenna sufficiently far from the ground. However, inorder to minimize the footprint, minimize losses by sphericaldivergence in wave propagation, keep a high signal-to-noiseratio, and ensure a high spatial resolution, it is essential tominimize the distance between the antenna and the ground. Forthat purpose, we determined the minimum antenna height forwhich the antenna forward model remains accurate. Finally, wetested the different phase-center approaches for characterizingthe electromagnetic properties of a two-layered sand subject todifferent water contents.II. M ATERIALS AND  M ETHODS  A. Radar System We used UWB stepped-frequency continuous-wave (SFCW)radar combined with an off-ground monostatic (zero-offset)horn antenna. The SFCW radar was set up using a vectornetwork analyzer (VNA, ZVRE, Rohde & Schwarz, Munich,Germany). Two antenna systems were tested, and they con-sisted of linear polarized double-ridged broadband horn an-tennas: models BBHA 9120-A and BBHA 9120-F fromSchwarzbeck Mess-Elektronik,Schönau, Germany.Thedimen-sions of the BBHA 9120-A are 22-cm length and  14  ×  24  cm 2 aperture area, its nominal frequency range is 0.8–5 GHz, andits isotropic gain ranges from 6 to 18 dBi. The relatively small3-dB beamwidth of the antenna (45 ◦ in the E-plane and 30 ◦ in the H-plane at a frequency  f   = 1  GHz, and 27 ◦ in theE-plane and 22 ◦ in the H-plane at  f   = 2  GHz) makes it suitablefor using off-ground. The dimensions of the BBHA 9120-F are96-cm length and  68  ×  95  cm 2 aperture area, its nominal fre-quency range is 0.2–2 GHz, and its isotropic gain ranges from6 to 18 dBi. The 3-dB beamwidth is 45 ◦ in both the E- andH-planes at 1 GHz.The antennas were connected to the reflection port of theVNA via a high-quality N-type  50 - Ω  impedance coaxial ca-ble of 2.5-m length (Sucoflex 104PEA, Huber + Suhner AG,Herisau, Switzerland). We calibrated the VNA at the connec-tion between the antenna feed point and the cable using anOpen-Short-Match reference calibration kit. The frequency-dependent complex ratio  S  11 ( ω )  between the returned signaland the emitted signal was measured sequentially for both an-tennas, with  ω  = 2 πf   being the angular frequency. For antennaBBHA 9120-A,  S  11 ( ω )  was measured at 301 evenly steppedoperating frequencies over the range 0.8–2.6 GHz using a6-MHz frequency step, and for antenna BBHA 9120-F, thefrequency range was set to 0.2–2 GHz, with 901 evenly steppedoperating frequencies with a step of 2 MHz.  B. Modeling of the Radar Signal1) Electromagnetic Model:  The constitutive parametersgoverning electromagnetic wave propagation are electric per-mittivity  ε  ( Fm − 1 ) , electric conductivity  σ  ( Sm − 1 ) , and mag-netic permeability  µ  ( Hm − 1 ) . In this paper, we assume that  µ  isequal to the permeability of free space,  µ 0  = 4 π  ×  10 − 7 Hm − 1 ,which is valid for nonmagnetic soil materials as prevalent inmostsubsurfaceenvironments.Therelativeelectricpermittivity  JADOON  et al. : HORN ANTENNA TRANSFER FUNCTIONS AND PHASE-CENTER POSITION FOR OFF-GROUND GPR 1651 is defined as  ε r  =  ε/ε 0 , where  ε 0  = 1 / ( µ 0 c 20 )  is the electricpermittivity of free space, with  c 0  = 299792458  ms − 1 beingthe speed of light in free space.The air–subsurface system is modeled as a 3-D multilayeredmedium consisting of   N   horizontal layers separated by  N   −  1 interfaces. The medium of the  n th layer is homogeneous andcharacterized by  ε n ,  σ n , and thickness  h n . The solution of Maxwell’s equations for electromagnetic waves propagating inmultilayered media is well known. As an assumption to modelthe antenna, we define Green’s function as the backscattered(upwardcomponentdenotedbytheuparrowin G ↑ xx ) x -directedelectric field (first subscript  x  in  G ↑ xx ) at the antenna phasecenter for a unit-strength  x -directed electric source (secondsubscript  x  in  G ↑ xx ) situated at the same position above themultilayered medium. The antenna is, therefore, modeled as apoint source and receiver, and hence, the radiation pattern isemulated by that of a point dipole. Following the approach of Lambot  et al.  [15], the analytic expression for the zero-offsetGreen’s function in the spectral domain (2-D spatial Fourierdomain) is found to be ˜ G ( k ρ ) = 18 π  Γ 1 R TM  1 η 1 −  ζ  1 R TE  1 Γ 1  exp( − 2Γ 1 h 1 ) .  (1)In this expression, the subscripts denote layer indexes,  R TM  and  R TE  are, respectively, the transverse magnetic (TM) andtransverse electric (TE) global reflection coefficients account-ing for all reflections and multiples from inferior interfaces[31],  Γ  is the vertical wavenumber defined as  Γ =   k 2 ρ  −  k 2 ,while  k 2 =  ω 2 µ ( ε  −  (  σ/ω ))  with  ω  being the angular fre-quency. For the free-space layer 1, we have  k 21  = ( ω/c ) 2 , with c  being the free-space wave velocity.The transformation of (1) from the spectral domain to thespatial domain is carried out by employing the 2-D Fourierinverse transformation, i.e., G  = + ∞   0 ˜ G ↑ xx ( k ρ ) k ρ dk ρ  (2)which reduces to a single integral in view of the invarianceof the electromagnetic properties along the  x  and  y  coordi-nates. We developed an optimal procedure to properly evaluatethat integral, which contains singularities [32]. To avoid thesingularities (branch points and poles), the integration path isdeformed in the complex  k ρ  plane. In addition, oscillationsare minimized by defining an optimal path, which makes theintegration faster. Defining k ρ  as the complex number  ( x  +  jy ) ,the following relationship was found for the constant phaseintegration path: y ( x ) =  x   xcω  2 + 1 (3)where  c  is the free-space electromagnetic wave velocity. 2) Radar-Antenna Model:  The radar-antenna–subsurfacesystemismodeledusingtheblockdiagramrepresentedinFig.1[15]. This model of complex linear transfer functions assumes Fig. 1. Block diagram representing the radar-antenna–multilayered mediumsystem modeled as linear systems in series and parallel, where  a  and  b  are,respectively, the emitted and received waves at the radar calibration plane,  ω is the angular frequency,  H  i  is the return loss transfer function,  H  t  and  H  r are, respectively, the transmitting and receiving transfer functions,  H  f   is thefeedback loss, and  G ↑ xx  is the transfer Green’s function of the air–subsurfacemultilayered medium [15]. that the spatial distribution of the backscattered electric fieldmeasured by the antenna does not depend on the air andsubsurface layers, i.e., only the amplitude and phase of the fieldchange (local plane wave approximation over the antenna aper-ture). This simplification holds when the antenna is sufficientlyfar above a multilayered medium. The corresponding transferfunction model, expressed in the frequency domain, is given by S  11 ( ω ) =  b ( ω ) a ( ω ) =  H  i ( ω ) +  H  ( ω ) G ↑ xx ( ω )1  −  H  f  ( ω ) G ↑ xx ( ω ) (4)where  S  11  is the quantity measured by the VNA,  b  and  a  are,respectively, the backscattered and incident waves at the VNAreference calibration plane,  H  i  is the return loss accountingfor the multiple reflections occurring in the antenna indepen-dently of the target (global reflectance),  H   =  H  t H  r  is thetransmitting–receiving transfer function accounting for antennagain and propagation time (global transmittance),  H  f   is thefeedback loss accounting for the multiple reflections occurringbetween the antenna and the ground (global reflectance), and G ↑ xx  is the transfer Green’s function of the air–subsurfacesystem modeled as a 3-D multilayered medium. C. Antenna Transfer Function Determination The characteristic antenna transfer functions [ H  i ( ω ) ,  H  ( ω ) ,and  H  f  ( ω ) ] can be determined by solving the system of equations (4) for different model configurations (denoted  k ,ranging from 1 to  n ). We use well-defined model configurationswith the antenna situated at different heights above a metalsheet playing the role of an infinite PEC. Green’s functions G ↑ xx,k ( ω )  can, therefore, be computed, while the functions S  11 ,k ( ω )  can be readily measured. It is worth noting thatthe return loss transfer function  H  i ( ω )  can also be measured  1652 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 5, MAY 2011 Fig. 2. Flowchart representing inversion of the GPR data collected over a perfect electric conductor (e.g., copper sheet) to optimize the antenna phase center.(Shaded boxes) Operators. (White boxes) Variables. directly by performing measurements in free-space conditions,i.e., for which  G ↑ xx ( ω ) = 0 . We recommend generating anoverdetermined system of equations, i.e., to take  n >  3 , for anaccurate characterization of the antenna transfer functions [33]becauseusingonlythreeequationsmaynotleadtoacompletelyindependent system of equations in the whole frequency range.To solve this system, (4) can be rewritten as S  11 ,k  =  H  i  +  S  11 ,k G ↑ xx,k H  f   +  G ↑ xx,k ( H   −  H  i H  f  ) .  (5)The linear system of (5) can be written in matrix form as b  =  Ax  (6)where b  =  S  11 , 1 ... S  11 ,k ... S  11 ,n  (7) A  =  1  S  11 , 1 G ↑ xx, 1  G ↑ xx, 1 ......... 1  S  ↑ 11 ,k G ↑ xx,k  G ↑ xx,k ......... 1  S  ↑ 11 ,n G ↑ xx,n  G ↑ xx,n  (8) x  =  H  i H  f  H   −  H  i H  f   .  (9)Using the least squares approach, we can easily calculate thevector of unknowns as x  = ( A H  A ) − 1 A H  b  (10)where the superscript  H   denotes the Hermitian.  D. Phase-Center Determination The phase center is the point from which the electromagneticradiation emanates spherically outward. To estimate the phase-center position,  S  11 ( ω )  was measured with the antenna at dif-ferent heights over a PEC. Two different approaches were usedto determine the phase-center position, i.e., 1) a single phasecenter  ψ  ptp  obtained by  PtP   magnitude of the backscatteredfields in the time domain [16] and 2) by inversion of the GPRdata assuming both a frequency-independent phase center  ψ c and a frequency-dependent phase center  ψ c ( f  ) .In the far-field region of the antenna, the virtual sourcepoint represents the srcin of the radiated field from whichthe  1 /R  spherical divergence is initiated, with  R  being thepath distance from the observation point to the source point.Conceptually, at the phase center,  PtP   tends, therefore, toinfinity, and then, the inverse of   PtP   tends to zero. As a result, ψ  ptp  can be determined by linear extrapolation of the measured 1 /PtP   values as a function of the travel path, i.e., two timesthe height of the antenna aperture (taken as reference) fromthe PEC. Such methodology can be used only for far-fieldmeasurements, when the antenna is at some minimal distancefrom the PEC.Fig. 2 illustrates the inversion scheme to optimize either thesingle frequency-independent phase center or two unknown pa-rametersof thefrequency-dependent phase-center function. As-suming a phase center (parameterized by  p , i.e., the unknownparameter vector), the electromagnetic forward model was usedto obtain G ↑ xx ( f,h, p ) , the modeled Green’s function, where  h correspondstotheantennaapertureheightplusthephase-centerposition from the antenna aperture. For the different model cal-ibration heights, the raw radar data  S  11 ( f,h )  and G ↑ xx ( f,h, p ) were used in (4) to estimate the corresponding antenna transferfunctions  H  i ( ω ) ,  H  ( ω ) , and  H  f  ( ω ) . These transfer functionsand  S  11 ( f,h )  were used in the forward radar-antenna model tocalculate themeasured Green’sfunction  ( G ↑∗ xx ( f,h )) . Then, themodeledandmeasuredGreen’sfunctionsarecomparedthroughan objective function, defined as φ ( p ) =  G ↑∗ xx  − G ↑ xx  T    G ↑∗ xx  − G ↑ xx   (11)  JADOON  et al. : HORN ANTENNA TRANSFER FUNCTIONS AND PHASE-CENTER POSITION FOR OFF-GROUND GPR 1653 where Green’s function matrices  G ↑ xx  =  G ↑ xx ( f,h, p )  and G ↑∗ xx  =  G ↑∗ xx ( f,h )  are arranged versus frequency and antennaheight plus frequency-dependent phase-center position. Sincethese response functions are complex functions, the differencebetween observed and modeled data is expressed by the ampli-tude of the errors in the complex plane. This objective functionis minimized to retrieve the optimal phase-center parameters ( p ) . We used the global multilevel coordinate search algorithm[34] combined sequentially with the classical Nelder–Meadsimplex algorithm [35], [36] to minimize (11).For the frequency-dependent phase center, the distance of the phase center  ψ c ( f  )  from the aperture of the antenna iscomputed using ψ c ( f  ) =  cd 2 πf   +  e  (12)where  c  is the velocity of light,  f   is the frequency, and  d and  e  are two variables defining the frequency dependence ( p  = [ d,e ]) . Bares  et al.  [9] used a similar empirical model toestimate the frequency-dependent phase center of the LPD.  E. Laboratory Experiment  WeusedthelaboratorydatasetofLambot etal. [15],inwhichradar measurements were carried out in a controlled laboratorysetup on a wooden sandbox filled with a two-layered sandysoil [see Fig. 12(a)]. The antenna system consisted of a linearpolarized double-ridged broadband transverse electromagnetichorn (BBHA 9120 D, Schwarzbeck Mess-Elektronik). Thefrequency-dependent complex ratio  S  11  between the returnedsignal and the emitted signal was measured sequentially at 126stepped operating frequencies over the range 1–2 GHz witha frequency step of 8 MHz. The volumetric water content of the the top layer was subject to nine different water contentlevels, ranging from 0 to 0.26 m 3 m − 3 , whereas the bottomlayer was fixed at about 0.10 m 3 m − 3 . The size of the sandboxwas  1 . 45  ×  1 . 30  m 2 area, and the sand was packed horizontally.The thickness of the bottom layer was equal to 0.13 m, whereasthe thickness of the top sand layer varied from about 0.01 to0.14 m, as a function of the imposed water content level. Belowthe sand layer, a horizontal metal sheet was installed to controlthe bottom boundary condition in the electromagnetic model.Materials underneath this metal sheet have no influence on themeasured backscattered signal.We considered the electrical conductivity to be frequencydependent  [ σ ( f  )] , with the frequency dependence describedby a linear model in the limited frequency range 1–2 GHz[15], i.e., σ ( f  ) =  σ 1GHz  +  a ( f   −  10 9 )  (13)where  σ 1GHz  is the reference apparent electrical conductivityat 1 GHz, and  a  is the linear variation rate of   σ ( f  ) . Differentwater contents were imposed by the addition of water to thesandandbymixingmanuallytogetahomogeneous distributionof water within the whole sand layer. After each radar mea-surement, three time-domain reflectory (TDR) measurementswere performed in the footprint of the radar antenna, and then,three cylindrical soil samples of 100 cm 3 were collected at thesame locations to determine the actual water content by usingthe standard oven-drying method at 105  ◦ C for at least 24 h.Three electromagnetic parameters were optimized for eachsand layer, namely,  ǫ r ,  σ 1GHz , and  a , by considering frequency-independent and -dependent phase centers. The bottom layerwas characterized independently of the second layer by per-forming measurements before setting up the second layer. Forthe two-layered cases, the bottom layer was then assumed asknown. The layer thickness was directly measured and fixedduring the inversions. For all the nine scenarios, large parame-ter space ( 2 . 5  < ǫ r  <  15 ;  1  ×  10 3 < σ 1GHz  <  1  ×  10 − 1 Sm − 1 ; 1  ×  10 12 < a <  1  × − 10 Ssm − 1 ) was considered during inver-sion. The reader is referred to Lambot  et al.  [15] for additionaldetails about that experiment.III. R ESULTS AND  D ISCUSSION  A. Phase-Center Estimation We performed radar measurements with the antenna situatedat 30 increasing heights above a PEC, i.e., copper sheets  2  × 2 m 2 and 4  ×  4 m 2 areaforthehigh-andlow-frequencydouble-ridged horn antennas, 0.8–2.6 and 0.2–2.0 GHz, respectively.The height of the 0.8- to 2.6-GHz antenna aperture varied fromabout 1 to 25 cm above the ground and from 8 to 125 cm forthe 0.2- to 2.0-GHz antenna. To infer the phase center of theantenna, i.e., the virtual source and receiver point, we used twodifferent approaches as explained in Section II-D.Fig. 3 illustrates the frequency-independent single phasecenter estimated by plotting  1 /PtP   as a function of twotimes the height  h  of the antenna aperture above the PEC.The  PtP   value is the difference between the maximum andminimum magnitudes of the measured signal, correspondingto the PEC reflection, expressed in the time domain. At thephase center, the  PtP   value of the signal tends to infinity,and hence, the inverse of   PtP   tends to zero. The phase centercan, therefore, be determined by linear extrapolation of themeasured  1 /PtP   values as a function of   2 h  (corresponding tothe two-way travel time). The best linear fit was observed forthe measurements of the low-frequency antenna as comparedwith the high-frequency antenna. From the fitted linear curve, 1 /PtP   is zero when  2 h  is equal to  − 5.66 cm and  − 144.20 cm(see Fig. 3), so the phase center is at 2.83 and 72.10 cm from theantenna aperture, respectively, for the high- and low-frequencyantennas. The negative values of   2 h  at  1 /PtP   = 0  indicate thatthe phase center is inside the antenna and from the antennaaperture toward the antenna feed point. The misfit between thedata and the linear regression is to be partly attributed to theinternal antenna reflections that are not removed at this stage. Inaddition, slight oscillations may also arise from the proximityof the near-field region of the antenna, for which the linearbehavior does not hold.Fig. 4 shows the objective function calculated for a rangeof frequency-independent phase centers by using the inversemodeling procedure depicted in Fig. 2. A range of   ψ c  wasselected between  − 0.1 to 2.0 m and 0 to 1.5 m, with an evenstep of 0.01 and 0.03 m for high- and low-frequency antennas,
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