IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 49, NO. 5, MAY 2011 1649
Analysis of Horn Antenna Transfer Functions andPhaseCenter Position for ModelingOffGround GPR
Khan Zaib Jadoon, Sébastien Lambot, Evert C. Slob, and Harry Vereecken
Abstract
—The antenna of a zerooffset offground groundpenetrating radar can be accurately modeled using a linear systemof frequencydependent 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 peaktopeak reﬂection values in the time domain and 2) frequencydomainfullwaveforminversionassumingbothfrequencyindependent and dependent phase centers. For that purpose, weperformed radar measurements at different heights above a perfect 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 phasecenterpositions. The results also showed that the antenna transfer function 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 phasecenter position was further tested for a twolayeredsandy soil subject to different water contents. The results showedthat the proposed antenna model avoids the need for phasecenterdetermination for proximal soil characterization.
Index Terms
—Antenna modeling, antenna phase center,frequency dependence, groundpenetrating 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 Development, Area “Environment,” Activity 6.3 “Environmental Technologies.”K. Z. Jadoon and H. Vereecken are with the Institute of Bio and Geosciences, Agrosphere (IBG3), Forschungszentrum Jülich GmbH, 52425 Jülich,Germany (email: k.z.jadoon@fzjuelich.de; h.vereecken@fzjuelich.de).S. Lambot is with the Institute of Bio and Geosciences, Agrosphere(IBG3), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany, and alsowith the Earth and Life Institute Environmental Sciences (ELIe), Université Catholique de Louvain, 1348 LouvainlaNeuve, Belgium (email:s.lambot@fzjuelich.de; sebastien.lambot@uclouvain.be).E. C. Slob is with the Department of Geotechnology, Delft University of Technology, 2628 CN Delft, The Netherlands (email: e.c.slob@tudelft.nl).Color versions of one or more of the ﬁgures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identiﬁer 10.1109/TGRS.2010.2089691
I. I
NTRODUCTION
P
RECISE location of the phasecenter position of an antenna is becoming more important because of new developments 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.Experimentalmethodshavebeenproposedtoestimateapparent phasecenter positions for different antennas [5], [8]–[11].For instance, Bares
et al.
[9] used an empirical model tocompute the frequencydependent phasecenter position of alogperiodic dipole (LPD) antenna and validated their approachby measurements. Doppler effects generated by the phasecenter displacement are computed for frequencymodulatedcontinuouswave radar applications. McKinney and Weiner[10] demonstrated photonicsynthesis techniques for arbitraryelectromagnetic waveforms, which enables the conjugate antenna 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 phasecenter 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 farﬁeld radiation expression is determined, forexample, the horn and LPD antennas. Liu
et al.
[12] proposeda 3D 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 doubleridged horn antenna existsmostly between its imaginary apex point and its aperture. Atthe phase center and in the farﬁeld region, the phase responseof the radiation pattern in the vicinity of the main beam willbe reasonably constant [13]. Doubleridged horn antennas havebeen used as an offground groundpenetrating radar (GPR)for accurate characterization of subsurface electrical properties[14]. Lambot
et al.
[15] estimated a single effective phase
01962892/$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 different heights above a perfect electrical conductor (PEC) andby extrapolating the peaktopeak
(
PtP
)
reﬂection values inthe time domain [16]. In their approach, the medium propertiesare estimated by fullwaveform inversion of zerooffset UWBfrequencydomain radar data, thereby maximizing informationretrieval from a single GPR measurement. Phase (travel time)and amplitude information is inherently exploited. The technique relies on an accurate and computationally effective radarforward model. This includes internal antenna and antenna–soilinteraction propagation effects through frequencydependentcomplex scalar transfer functions and solves exactly the 3DMaxwell’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 timelapse offgroundGPR data [24], [25].The exact location of the phase center of doubleridged 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 doubleridged 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 phasecenter 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 doubleridged horn antennas,namely, by extrapolation of peaktopeak reﬂection values inthe time domain and by frequencydomain fullwaveform inversion assuming both frequencyindependent 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 multilayered 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 phasecenter position forboth a frequencydependent phase center
(
ψ
c
(
f
))
and a singleoptimal effective frequencyindependent 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 signaltonoiseratio, 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 phasecenter approaches for characterizingthe electromagnetic properties of a twolayered sand subject todifferent water contents.II. M
ATERIALS AND
M
ETHODS
A. Radar System
We used UWB steppedfrequency continuouswave (SFCW)radar combined with an offground monostatic (zerooffset)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 consisted of linear polarized doubleridged broadband horn antennas: models BBHA 9120A and BBHA 9120F fromSchwarzbeck MessElektronik,Schönau, Germany.Thedimensions of the BBHA 9120A are 22cm 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 small3dB beamwidth of the antenna (45
◦
in the Eplane and 30
◦
in the Hplane at a frequency
f
= 1
GHz, and 27
◦
in theEplane and 22
◦
in the Hplane at
f
= 2
GHz) makes it suitablefor using offground. The dimensions of the BBHA 9120F are96cm length and
68
×
95
cm
2
aperture area, its nominal frequency range is 0.2–2 GHz, and its isotropic gain ranges from6 to 18 dBi. The 3dB beamwidth is 45
◦
in both the E andHplanes at 1 GHz.The antennas were connected to the reﬂection port of theVNA via a highquality Ntype
50

Ω
impedance coaxial cable of 2.5m length (Sucoﬂex 104PEA, Huber + Suhner AG,Herisau, Switzerland). We calibrated the VNA at the connection between the antenna feed point and the cable using anOpenShortMatch reference calibration kit. The frequencydependent complex ratio
S
11
(
ω
)
between the returned signaland the emitted signal was measured sequentially for both antennas, with
ω
= 2
πf
being the angular frequency. For antennaBBHA 9120A,
S
11
(
ω
)
was measured at 301 evenly steppedoperating frequencies over the range 0.8–2.6 GHz using a6MHz frequency step, and for antenna BBHA 9120F, 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 permittivity
ε
(
Fm
−
1
)
, electric conductivity
σ
(
Sm
−
1
)
, and magnetic 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 PHASECENTER POSITION FOR OFFGROUND 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 3D 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 unitstrength
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 zerooffsetGreen’s function in the spectral domain (2D 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 accounting 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 frequency. For the freespace layer 1, we have
k
21
= (
ω/c
)
2
, with
c
being the freespace wave velocity.The transformation of (1) from the spectral domain to thespatial domain is carried out by employing the 2D 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
coordinates. 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 freespace electromagnetic wave velocity.
2) RadarAntenna Model:
The radarantenna–subsurfacesystemismodeledusingtheblockdiagramrepresentedinFig.1[15]. This model of complex linear transfer functions assumes
Fig. 1. Block diagram representing the radarantenna–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 aperture). 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 independently 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 3D 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 welldeﬁned 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 freespace 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. PhaseCenter Determination
The phase center is the point from which the electromagneticradiation emanates spherically outward. To estimate the phasecenter position,
S
11
(
ω
)
was measured with the antenna at different heights over a PEC. Two different approaches were usedto determine the phasecenter 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 frequencyindependent phase center
ψ
c
and a frequencydependent phase center
ψ
c
(
f
)
.In the farﬁeld 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 farﬁeldmeasurements, when the antenna is at some minimal distancefrom the PEC.Fig. 2 illustrates the inversion scheme to optimize either thesingle frequencyindependent phase center or two unknown parametersof thefrequencydependent phasecenter function. Assuming 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
correspondstotheantennaapertureheightplusthephasecenterposition from the antenna aperture. For the different model calibration 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 radarantenna 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 PHASECENTER POSITION FOR OFFGROUND 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 frequencydependent phasecenter position. Sincethese response functions are complex functions, the differencebetween observed and modeled data is expressed by the amplitude of the errors in the complex plane. This objective functionis minimized to retrieve the optimal phasecenter 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 frequencydependent 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 frequencydependent 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 twolayered sandysoil [see Fig. 12(a)]. The antenna system consisted of a linearpolarized doubleridged broadband transverse electromagnetichorn (BBHA 9120 D, Schwarzbeck MessElektronik). Thefrequencydependent 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 measurement, three timedomain 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 ovendrying 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 frequencyindependent and dependent phase centers. The bottom layerwas characterized independently of the second layer by performing measurements before setting up the second layer. Forthe twolayered cases, the bottom layer was then assumed asknown. The layer thickness was directly measured and ﬁxedduring the inversions. For all the nine scenarios, large parameter 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 inversion. The reader is referred to Lambot
et al.
[15] for additionaldetails about that experiment.III. R
ESULTS AND
D
ISCUSSION
A. PhaseCenter 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
areaforthehighandlowfrequencydoubleridged horn antennas, 0.8–2.6 and 0.2–2.0 GHz, respectively.The height of the 0.8 to 2.6GHz antenna aperture varied fromabout 1 to 25 cm above the ground and from 8 to 125 cm forthe 0.2 to 2.0GHz antenna. To infer the phase center of theantenna, i.e., the virtual source and receiver point, we used twodifferent approaches as explained in Section IID.Fig. 3 illustrates the frequencyindependent 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 twoway travel time). The best linear ﬁt was observed forthe measurements of the lowfrequency antenna as comparedwith the highfrequency 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 lowfrequencyantennas. 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 nearﬁeld region of the antenna, for which the linearbehavior does not hold.Fig. 4 shows the objective function calculated for a rangeof frequencyindependent 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 lowfrequency antennas,