Satoshi Ebihara

Takashi Yamamoto

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RESONANCE ANALYSIS OF CIRCULAR DIPOLE ARRAY ANTENNA  
IN CYLINDRICALLY LAYERED MEDIA FOR DIRECTIONAL BOREHOLE RADAR 
 
 
Satoshi EBIHARA and Takashi YAMAMOTO 
Osaka Electro-Communication University  
18-8 Hatsucho, Neyagawa-shi, Osaka, 572-8530 Japan 
s-ebihara@m.ieice.org
 
 
 
1. Introduction 
Borehole radar is one of the ground penetrating radar (GPR) techniques [1]-[3]. In the borehole 
radar, radar operates in a borehole, and they can detect some targets such as fractures and geological 
layers. Most conventional borehole radar system uses dipole antennas, which are omnidirectional. 
Recently, 3-D estimation of target positions becomes important. For the measurement, we proposed 
the circular dipole array antenna with an optical modulator [3], as directional borehole radar. The basic 
idea to estimate direction of arrival wave (DOA) with dipole antenna array is to make use of difference 
in phase and amplitude of received signals by the antennas. 
It is well known that mutual coupling among circular array antenna elements causes 
phase-sequence resonance [4]-[8]. This resonance is sometimes made use of in some antennas such as 
a supergain antenna having a large number of elements and electrically large size [8]. On the other 
hand, the element number and the antenna size is small in the proposed dipole array antenna for the 
borehole radar in [3]. However, even if in this condition, the phase-sequence resonance may occur. 
This resonance might influence estimation of DOA with the proposed method in [3]. Especially, if the 
MoM model did not agree to experimental data, we would fail in direction finding. In this paper, we 
investigate influence of the phase-sequence resonance of the array antenna on direction finding. 
 
2. Formulation 
A. The MoM solution 
MoM analysis for the dipole array antenna in a borehole was given in [3], and we briefly review 
that here. In the analysis, a borehole filled with some fluid was approximated by infinitely long 
multiple dielectric cylinders. Consider the circular dipole antenna array in the layers shown in Fig. 1 
(or see Fig. 7 in [3]). The M vertical dipole antennas are arranged uniformly on a circle at  and bρ =
2( 1)
k
k
M
πφ φ −= =  . We utilize Richmond’s MoM, which use Galerkin’s method and 
piece-wise sinusoidal functions. We should notice that we need to calculate the modified Green’s 
function in [2]-[3], since the function should include scattered waves from the cylindrical boundaries. 
The diameter of all the dipole antennas is 2a, and is small enough for the thin wire approximation to 
be applied. In [3], we assumed that a plane wave is incident on the array antenna with the azimuth 
angle 
( 1, 2, ,k = )M
φ ϕ= , and the electromagnetic field can be represented as (17) in [3], or as 
 
( ) ( )
1 1
( , ; , )
( )
( , ; , )
z
inc
z jk z jn
ninc
nz
E z
e e J k
H z
φ ϕ
ρ
φ θ ϕ
ρ
φ θ ϕ
∞
− − −
=−∞
⎡ ⎤
=⎢ ⎥
⎢ ⎥⎣ ⎦
∑ a n .    (1) 
 
It should be noted that the incident wave can be broken down to into cylindrical harmonic with 
different jne φ−  dependence. 
 
B. Circular array resonance 
When the resonance having order m (0 / 2)m M≤ ≤  occurs, the voltage  at the feeding point 
of the k-th circular array antenna element in MoM has the form [4]-[8] like 
kV
 
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PROCEEDINGS OF ISAP2005, SEOUL, KOREAPOS-A-11 
ISBN: 89-86522-78-0   94460    KEES
0
kjm
kV e V
φ−= ,           (2) 
 
where  is a complex value, which is independent of 0V kφ . In order to satisfy (2), electric field and 
magnetic field should be the following form instead of (1). 
 
( ) ( )
1
( ) ( ) ( ) ( )
1 1
( ) ( 0),
( , ; , )
( , ; , )
( ) ( ) ( 0
z
z
jk z jn n
n iinc
lz n m lN
inc
z jk z jn n jn n
n i n m lN n m lN
l
e e J k m
E z
H z
e J k e e m
φ ϕ
ρ
φ ϕ φ ϕ
ρ
ρ
φ θ ϕ
φ θ ϕ
ρ
+∞
− − −
=−∞ = +
+∞
− − − − −
= + =− +
=−∞
⎧
=⎪
⎡ ⎤ ⎪=⎢ ⎥ ⎨
⎢ ⎥ ⎪⎣ ⎦ )+ ≠⎪
⎩
∑
∑
a
a a
 (3) 
 
It should be noted that (3) is the m-th azimuth dependency component extracted from the total plane 
wave incidence field in (1). If the antenna was excited by the incident field in (3) at the resonance 
frequencies corresponding to the order m, the current at feeding point of the antennas would be large. 
Using parameters: layer number N = 2, 1rε =1, 1σ = 0, 2rε = 10, 2σ = 0.001 S/m, 2a1 = 0.1 m, 
antenna element diameter 2a = 1 mm, circular array diameter 2b = 0.07 m, antenna full length l = 1.5 
m, element number M = 7, terminated impedance , we calculated voltage at the feeding point 
of the antenna located at 
1pFtjZ =
1 0φ φ= =  with the TM wave incidence. The calculated voltages are broken 
or dotted lines in figure 2. The red solid line is response by the total field shown in (1), and the four 
dotted lines correspond to response by the fields in (3). According to the peak positions of the three 
dotted lines for m = 0, 1 and 2 in the figure, we find that the circular array in a borehole is resonated at 
70 MHz, 120 MHz and 135 MHz, which correspond to phase-sequence factor m = 0, 1 and 2, 
respectively. Around the three frequencies, we can see that the received signal excited by the total field 
has maximum values. These imply that the phase-sequence resonance occur around the three 
frequencies. 
 
φ
zεr1, μ0, σ1
εr2, μ0, σ2
2a1
2a2
εrN, μ0, σN
ρ
z
(ρ, φ, z)
Dipole antenna
x
 
20 40 60 80 100 120 140 160 180 200
-60
-50
-40
-30
-20
-10
0
10
Frequency(MHz)
A
m
pl
itu
de
(d
B)
m = 0
m = 1
m = 2
m = 3
total
 
 
Fig. 1. MoM model. Fig. 2. The calculated voltage at a feeding point of the dipole 
antenna 1.  
 
3. Influence of phase-sequence resonances on direction finding 
A. Phase-sequence resonance analysis of the circular array 
Form of the incident field in (1) implies that the received signals  can be represented by the 
linear combination of the M phase-sequence voltage as 
kV
 
2( 1)1 (
0
kM jn
M
k
n
V e
π ϕ−− − −
=
= ∑
)
nA ,       (4) 
 
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where  is a complex value. Inverse digital Fourier transform of the received signals may extract 
the m-th phase-sequence component from the received array signals . Since both the order m and 
M-m represent the same m-th phase sequence resonance, we will evaluate the resonance with the 
following value 
nA
kV
mς : 
 
2 ( )21 1
2 2
0 0
1 ( ) ( )
k M mkmM Mj j
M M
m k k m
k k
v e v e A A
M
ππ
ς
−− −
−
= =
= + =∑ ∑ 2 2M m+ , (0 / 2)m M≤ ≤  (5) 
 
Note that the value mς  is the function of a frequency. We can calculate the power mς  corresponding 
to the m-th phase-sequence with the equation (5). Fig. 3 (a) shows the calculated value mς  with the 
theoretical signals generated by the MoM model in Fig. 1. A color scale in the figure corresponds to 
the value mς . At around 120 MHz and 135 MHz, the power have the high value at m = 1 and m = 2, 
respectively. These frequencies correspond to the resonant frequencies of m = 1 and 2, which were 
evaluated in Fig. 2. We should notice that the diameter 2b = 0.07 m is much smaller than wave length, 
which is 0.63 m 8(3 10 m/s)/( 10 150MHz)= × ×  at 150 MHz. This leads to little phase difference 
among the antenna elements. If there was no mutual coupling among the antennas, the value mς  
should be highest at m = 0. That is the reason why the power is highest at m = 0 below 60 MHz. 
 
B. Analysis of experimental data 
We arranged the dipole array antenna in the borehole, BR2, at the field test site of O.E.C.U., 
Shijonawate, Japan, as shown in Fig. 4. The antenna consists of seven dipole antennas uniformly 
arranged on a circle, i.e. M =7. The parameters of the antennas such as a diameter of this circle and 
length of the antennas are the same as ones used in the section 2. Fig. 3 (b) shows the value mς , which 
were calculated with the experimental data. We can see that the power mς  is highest at m = 1 and 115 
MHz. Also, the value mς  is high at m = 2 and 130 MHz. We can see some similarity between Fig. 3 (a) 
and Fig. 3 (b), and this implies that the phase-sequence resonance corresponding to m = 2 and 3 occurs 
in the experiment actually. 
Fig. 3 (c) shows the averaged coherence among the array signals, and Fig. 3 (d) shows the 
estimated direction of arrival. We formed a steering vector with the MoM, and estimated the direction 
with a Fourier-based method using the steering vector. Below 60 MHz in the figure, the estimated 
direction is correct. However, we can see some spurious solutions between 110 MHz and 120 MHz, or 
around 135 MHz. These might be caused by the phase-sequence resonance having order m = 2 and 3. 
Also, the estimated direction is biased around 80 MHz, and this frequency is near the frequency of the 
phase-sequence having m = 0 in Fig. 2. It should be noted that there is some error in direction finding, 
when the phase-sequence resonance happens. In higher frequencies than 180 MHz, inhomogeneous 
medium around the borehole might cause the error.  
 
4. Conclusions 
Using the MoM model including influence of the borehole, we investigated frequency of the 
phase-sequence resonance in a circular dipole array antenna in a borehole. We applied the inverse 
digital Fourier transform to both the experimental and theoretical data, and we found that the 
phase-sequence resonance corresponding order m = 1 and 2 occur in the experiment data. Comparison 
between the resonance analysis and estimation of direction of arrival showed that the phase-sequence 
resonance increases error in direction finding with a dipole array in a borehole. 
 
Acknowledgements 
Thanks are due to Akira Kusama, Kazushi Kondo and Kouhei Minamoto for their assistance in 
doing experiments in Shijonawate, Japan. This work was supported by Grant-in-Aid for Young 
Scientists (A), MEXT. KAKENHI 15686037, Japanese government. 
 
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1
0
0.5
0
1
2
3
0
1
2
3
20020 10040 60 80 120 140 160 180
Frequency (MHz)
(a) Resonance analysis (MoM)
(b) Resonance analysis (Experiment)
(d) Estimation of direction of arrival (Experiment)
O
rd
er
 m
O
rd
er
 m
0
90
180
270
360
A
zi
m
ut
h 
φ
(d
eg
re
e)
20020 10040 60 80 120 140 160 180
Frequency (MHz)
20020 10040 60 80 120 140 160 180
Frequency (MHz)
(c) Coherence (Experiment)
 
Fig. 3. Analysis results. In (a), (b) and (d), the values are normalized by the maximum one at each 
frequency. In (d), the white broken line represents the true direction of arrival, which is 197φ = ° . 
Surface
Tx
2a2 = 0.1 m
Granite (εr2 = 10, μ0, σ2 = 0.001 S/m )
Rx
(dipole 
antenna array)
Compass
BR3
6 m
Optical fibers
FRP vesselAir (εr1 = 1, μ0, σ3 = 0 )
BR2
2a1 = 0.1 m
1.344 m1.5 m
2b = 0.07 m
 
Fig. 4. Experiment to transmit a wave to the dipole array in granite 
 from outside the borehole in O.E.C.U., Shijonawate, Japan. 
 
References 
[1] S. Ebihara, M. Sato, and H. Niitsuma, “Super-resolution of Coherent Targets by a Directional Borehole 
Radar,” IEEE Trans. on Geoscience and Remote Sensing, Vol.38, No.4, pp. 1725 -1732, July 2000. 
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[3] S. Ebihara, “Directional borehole radar with dipole antenna array using optical modulators ,” IEEE Trans. on 
Geoscience and Remote Sensing, Vol. 42, No. 1, pp. 45 -58, January 2004. 
[4] R. W. P. King et al., Cylindrical Antennas and Arrays, Cambridge University Press, pp. 112-152, 2002. 
[5] R. W. P. King, “Supergain antennas and the Yagi and circular arrays,” IEEE Trans. Antennas. Propagat, Vol. 
37, No. 2, pp. 178 -186, Feb. 1989. 
[6] G. Fikioris, R. W. P. King and T. T. Wu, “The resonant circular of electrically short elements,” J. Appl. Phys., 
vol. 68 (2), pp. 431-439, July 1990. 
[7] D. K. Freeman and T. T. Wu, “Variational-principle formulation of the two-term theory for arrays of 
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[8] R. W. P. King, M. Owens and T. T. Wu, “The resonant circular of electrically short elements,” IEEE Trans. 
Antennas. Propagat, vol. 51, no. 1, pp. 103-109, Jan. 2003. 
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Resonance analysis of circular dipole array antenna in cylindrically layered media for directional borehole radar

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Resonance analysis of circular dipole array antenna in cylindrically layered media for directional borehole radar

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