Resonance Analysis of a Circular Dipole Array Antenna ... - IEEE Xplore
22
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
Resonance Analysis of a Circular Dipole Array Antenna in Cylindrically Layered Media for Directional Borehole Radar Satoshi Ebihara, Member, IEEE, and Takashi Yamamoto
Abstract—In this paper, we discuss the influence of a resonance on estimating direction of arrival (DOA) with a circular dipole array in a borehole (CAB). The resonance is caused by the phasesequence of currents on the dipole antennas. Making use of method of moments (MoM) analysis, we predict resonant frequencies of the CAB and describe a mechanism for the resonance theoretically, making use of the fact that a plane wave can be broken down into cylindrical harmonics. In order to examine the agreement between the MoM and experimental data, we propose a signal processing method to extract phase-sequence components from the actual received array signal. This method is useful for confirming excitation of the phase-sequence resonance in experimental data, as predicted by the MoM. Using the MoM, we examine the influence of the resonances on DOA estimation, and we conclude that the resonances give some spurious solutions. In order to confirm these results, we conduct experiments at a field test site. We arrange the CAB, which consisted of seven receiving dipole antennas. In this experiment, a plane wave is incident on the CAB from a transmitter in another borehole. We apply the proposed signal method, and the measured results demonstrate the existence of the phase-sequence resonance in a physically real CAB. The spurious solutions found to occur in the DOA estimation at the resonant frequencies are also found experimentally. Index Terms—Circular array, cylindrical layers, dipole antenna, directional borehole radar, method of moments (MoM), mutual coupling, phase-sequence resonance.
I. INTRODUCTION
B
OREHOLE radar is one form of ground-penetrating radar [1]–[8]. Most conventional borehole radars use dipole antennas that are omnidirectional. However, in many engineering applications, the three-dimensional position of targets must be accurately determined. Such measurements can be achieved using one borehole, if a directional borehole radar can be used [9]–[16]. To realize such a radar, control of the radiation pattern in a borehole has been attempted using a dielectric medium [9], and a cavity-backed antenna was used with mechanical rotation in a borehole [10]–[12]. Also, a cross-loop antenna, which consists of loops having a figure-eight radiation pattern in air, was tried [13]. These previous studies made use of antenna
Manuscript received May 30, 2005; revised August 22, 2005. This work was supported by a Grant-in-Aid for Young Scientists (A), MEXT. KAKENHI 15686037, Japanese Government. S. Ebihara is with the Department of Electronic Engineering and Computer Science, Faculty of Engineering, Osaka Electro-Communication University, Osaka 572-8530, Japan (e-mail: [email protected]). T. Yamamoto is with the Tsuruyayoshinobu Inc., 602-8434 Kyoto, Japan. Digital Object Identifier 10.1109/TGRS.2005.860486
radiation patterns to achieve directional borehole radar. This means that only the amplitude information of the antennas was exploited to obtain a direction-of-arrival (DOA) estimation. Directional borehole radar could be realized by arranging some array antenna elements in circle in a borehole [14]. We have proposed an array antenna using optical modulators [16]. In this system, a circular dipole array in a borehole (CAB) consisting of vertical dipole antennas was arranged. The voltages received at antenna elements of the CAB are converted to optical signals with the optical modulators near driving points of the antennas. In this radar, in order to estimate DOA, we make use of differences in both phase and amplitude among the received array signals. Steering vectors to analyze the array signals for DOA estimation were formed using the method of moments (MoM) analysis [17]–[22]. In this theoretical analysis, the borehole effect [23]–[31], which is the influence of a borehole on antenna characteristics, was considered. of the antenna In the CAB, parameters such as number of the dipole antenna element elements and the half length of array elements inare optional. Generally, if the number creases, the number of signals which can be estimated would also increase [32]. As far as the dipole antenna length is concerned, we should in general use relatively long antennas rather than electrically short antennas in order to obtain higher sensitivity [33]. However, in the CAB, arranging many long dipole antennas in the narrow space may lead to strong mutual coupling among the elements. This may cause resonances of the CAB, and these phenomena might influence the DOA estimation. For many years, array antennas making use of resonance among array elements, such as the Yagi–Uda array [34], have been studied. It has also been realized that another useful antenna array can be obtained by bending the Yagi–Uda array in a circle. According to previous studies [35]–[44], it is known that a phase-sequence resonance occurs in a circular array. This resonance is sometimes useful for supergain [36] or superdirectivity [43]. However, according to [35], [38], and [43], the radiation pattern of the circular array antenna has multiple sharp peaks at the resonant frequencies. These peaks might make DOA estimation difficult with the CAB. Furthermore, resonance of antennas generally makes frequency bandwidth narrower and worsens estimations of time delay resolution in impulse radar measurement. Predicting and knowing the resonant frequencies may be important before array antennas can be useful. The two-term theory [35] has been developed to predict frequencies of the phase-sequence resonance of a circular dipole array. This theory uses an approximation solution to the coupled
0196-2892/$20.00 © 2006 IEEE
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
integral equations for the currents along the dipoles. The computational cost of the solution is relatively low, even if the number of elements is large. Furthermore, agreement between experimental results and the theory is good [41]. However, there are some assumptions in this theory. The two-term theory is adequate, when the close-coupling effects [35], [38] are avoided. This is the case when interelement spacing of the circular array of the dipole, i.e., . In is larger than the half-length addition, the theory requires that the array elements should be , where is the electrically short dipole antennas, i.e., spatial wavelength. Since electromagnetic waves at higher frequencies are attenuated in subsurface media [1], operating frequencies should be lower than about 200 MHz. Therefore, in order in to obtain sufficient detectable range, the wavelength should be longer than about 0.47 m media having m/s MHz . Even if the high frequencies of the borehole radar are used, the distance between the antenna . This elements is much smaller than the wavelength, i.e., is because the diameter of a borehole is usually about 0.1 m, m . Furthermore, if the and the distance should be half-length of the dipole antenna is longer than the wavelength, , the mutual coupling among dipole antennas i.e., influences the antenna array characteristics significantly. These , imply that we cannot ignore conditions, i.e., the close-coupling effects among the antenna elements at the high frequency of the borehole radar. This differs from the assumptions in the two-term theory. Another important difference between previous work [35]–[44] and the CAB case is the fact that media around the antenna is not homogeneous in the CAB. Since a borehole wall always exists around the antenna, we need to assume that the antennas are in cylindrically layered media rather than in homogeneous media [23]–[31]. The main difference between [16] and this paper is that in the previous work there was no discussion of the existence of phase-sequence CAB. In [16], the number of antenna elements in the circle was small or the electrically small dipole antenna elements were used. In [16], we did not consider the influence of the resonance on DOA, since we used the dipole array antenna only in situations of no resonance in the dipole array antenna. Therefore, we could estimate DOA without consideration of the resonance. In this paper, we will investigate the influence of the phase-sequence resonance on DOA estimation with the CAB. In Section II, we will propose a theoretical method to predict frequencies of the phase-sequence resonance for the CAB. We will show numerical results and predict the resonant frequencies. This is followed by a proposed method to estimate the resonant frequencies from the received signals of the CAB. In Section III, we discuss the influence of resonance on DOA estimation with the MoM analysis. In Section IV, we describe the associated experimental data, and we will discuss the influence of the resonance using experimental data.
Fig. 1.
23
MoM model for a CAB.
A. Review of the MoM Solution for the CAB Consider the CAB in a cylindrical coordinate system as shown in Fig. 1. This antenna consists of an array of identical, parallel, center-driven coplanar cylindrical vertical dipoles that are equally spaced about the circumference of a circle. The th vertical dipole antenna is arranged at and (1) Here we apply Richmond’s MoM, which uses Galerkin’s method and piecewise sinusoidal functions [18]. The th segments, each of length dipole antenna is divided into , and the complex constants denote the current function sampled at the junctions . The diameter of all the dipole antennas is , and is small enough for the thin wire approximation to be applied. The delta-gap source modeling is applied for a driving point. The basis function, which is same as the test function, is (2) where is the wavenumber in the innermost layer. We can obtain a set of linear equations as
(3) II. FORMULATION First, briefly, we review the MoM solution, which was previously formulated for the CAB [16]. This is followed by consideration of a method to predict resonance frequency and its resonant mechanism.
and
(4)
24
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
where (5)
(6)
(7) The driving point of all the dipole antenna element is located at , and is the index of the segment junction, where the driving point at the center of each dipole antenna exists. The termiat all the nated impedance at the driving point is the constant is the component of the impressed driving points. . The function is electric field at the Green’s function, when the -directed Hertzian dipole as the and the comsource exists at . ponent of the field at the observation point exists at infinThese source point and observation points are inside itely long cylindrical layers, where boundaries exist at . The function is the same as equation (10) in [16]. It should be noted that the Green’s function includes scattered fields from the cylindrical boundaries, and this is an important difference between the borehole radar case and homogenious media case. We should notice that this Green’s function involves the evaluation of Sommerfeld integrals. The numerical consideration and treatment of the Sommerfeld integration have been presented by various authors such as [45] and [46]. B. Driving Point Impedance of the CAB Fikioris et al. have recently reported difficulties in MoM analysis of a resonant circular array [44]. In this work, they assumed antenna elements in air, and a circular array having they found that the driving-point conductance has a large negative value near the resonant frequencies. Here we will confirm that there is no meaningless result in our calculation with the MoM. We consider the CAB having several parameters, which can be used in real borehole radar measurements, and will confirm that the MoM stated here is capable for dealing with actual borehole radar measurements. Parameters used in the calculation are , number of cylindrical the number of antenna elements: , , , , S/m, layers: m, antenna element diameter mm, cirm, antenna full length cular array diameter m, and terminated impedance pF at the center of each dipole antenna. This impedance corresponds to an input impedance of the optical modulators used in the experiments described in Section IV. Now we will consider input impedance of one element of the CAB. For the CAB with one element driven, the driving point admittance is proportional to the value of the current on the driving point at the center of the driven element. Fig. 2(a) and (b) shows the driving-point conductance and susceptance. The MoM considering the borehole effect predicts three distinct
Fig. 2. Driving point impedance. The computer simulation results. (a) Conductance. (b) Susceptance. The three arrows are at 116, 132, and 138 MHz.
resonances, where the conductance has maximum value and the susceptance is near zero. It should be noted that there is no large negative conductance values in the figure unlike in [44]. A significant difference between the present calculation and that of the elements. The increased may of [44] is the number cause the large negative conductance seen in [44]. As stated in the latter paper, decoupling the integral equations would be needed, since the increased elements lead to high phase-sequence among the currents at the center of the antennas. In the CAB, we cannot increase the number of the antenna elements, since the space available for the array is limited. Based on the evidence, we believe that the MoM reviewed in the last subsection works well for the relatively small number of elements. C. Resonance Analysis With the MoM Once the impedance matrix is calculated, the current induced by any impressed electric field can be calculated using the MoM. Using the current calculated with frequency changed, all the antenna characteristics such as resonant frequency are also obtained. However we cannot know the mechanism of the resonance only with the MoM calculation. According to [35] and [38], the circular array resonances can be associated with a parwith even, ticular phase-sequence where resonances with larger occur at higher frequencies. It is worthwhile knowing the relationship between the resonant frequency and the corresponding phase-sequence for designing optimal array antenna parameters. In [35]–[39], the multiple integral equations were reduced to a single equation for each phase-sequence , and a resonant
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
frequency corresponding to the was found using the assumptions stated in Section I. However we cannot directly employ such an approach here. This is because the two assumptions in [35]–[39] are not valid for the CAB as we stated in Section I. One assumption is that the close-coupling effects among the antenna elements can be avoided. The other is that media around the antennas is homogeneous. These do not satisfy the CAB case at high frequencies in borehole radar measurement. Here we will try to combine the resonant frequency and the resonance’s phase-sequence , exploiting and modifying the MoM analysis. at of the th dipole antenna can be The currents phase-sequence currents represented by a superposition of as follows [35]–[39]:
(8) It should be noted that the above equation is valid even for a circular array in inhomogenous media, although the circular array is in homogeneous media in [35]–[39]. In cylindrically symmetrical media as the CAB, the phase-sequence currents can be associated with the th phase-sequence resonance. The th phase-sequence currents are excited by the following elements: phase-sequence voltages applied to the
25
wave is defined as a plane wave, the magnetic field of which is transverse to the direction. The (11) implies that the incident plane wave can be broken down into cylindrical harmonics with dependence. The component having dedifferent in (11), contributes to exciting the th pendence, i.e., phase-sequence electric field in (7). We should notice that the phase-sequence is decided by the relationships among phases of the electric fields, not at spatially continuous points, but at some discrete points in space. According to this fact, the comhaving other than may contribute to ponent the excitation of the th phase sequence electric field. Considering an analogy to aliasing in sampling continuous signals, the following contributes to exciting the electric field having the th phase-sequence in (11): for
(15)
for
(16)
and
and According to the above equation, both phase-sequence give the same . This implies that it is enough with even and only to consider with odd. Consequently we may decompose the plane wave in (11) to components contributing to only the th phase-sequence as
(9) where
is the th phase-sequence voltages at . In order to excite the above voltage , the impressed electric field to the array antenna should be the following th phase-sequence: (17) (10) Now, let us assume that a plane wave with the elevation angle and the azimuth angle traveling parallel to the plane is incident on the CAB in multiple cylindrical layers shown in Fig. 1. According to [16] and [45], the impressed electric field in (7) should be given by (11) where (12) (13) (14) , and is the wavenumber of a plane wave in the th layer. The reflection matrixes such as and , and the transmission matrix are given in the appendix of [16] or [45, ch. 3]. If a TM plane is equal to zero, wave is incident on the cylindrical layers, is the magnitude of the electric field strength. The TM and
with
even or
with
odd
When a plane wave is incident on the CAB at the resonant frequency corresponding to the phase-sequence , the impressed in the electric field may be approximated by the field with above (17), instead of the exact field in (11). This implies that, at the th resonant frequency, usage of either (11) or (17) with will give almost the same result of the current calculation. We calculated voltages received at the driving point of the with the TM wave incidence. antenna located at This plane wave propagates from direction of the azimuth angle and the elevation angle . All the parameters of the CAB are the same as those used in the previous subsection except for usage of the CAB as a receiver. Note that all the pF. The caldriving points are terminated with the culated voltages are shown as solid, broken, or dotted lines in Fig. 3. The solid line is the response impressed by the total TM wave field shown in (11), and the other four lines correspond to responses impressed by the fields in (17). According to the solid line in the figure, we find three distinct peaks, and we find that the CAB resonates at these three frequencies: 116, 132, and 138 MHz. These frequencies correspond to the peak positions of
26
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
This processing corresponds to a spatial digital Fourier transform (DFT) of the array signals. In this paper, we will utilize the spatial DFT to obtain information on the phase sequence as the array voltage resonance. We define signals received by the th antenna element of the CAB at a certain frequency. Considering a Fourier transform of the periodic signals, the signals can be represented by the linear phase-sequence voltages as combination of the
(18)
Fig. 3. Calculated voltage at a feeding point of the dipole antenna 1. The three arrows are at 116, 132, and 138 MHz.
the curves calculated with the phase-sequence respectively. Using this correspondence, we reveal the relation between the resonant frequencies and the corresponding phase-sequences. It should be noted that the wavelength at very low m of frequencies is much larger than the diameter the circular array. If there was no mutual coupling among the antenna elements, the plane wave field might be approximated phase-sequence component in (17). This is the by the reason why the curves with the total field and the phase-senearly coincide at frequencies below 60 MHz. quence We should notice that the resonant frequencies would get higher, if the length of the dipole antenna got shorter. This is because the close coupling among the antenna elements would be reduced by shortening the antenna length, as we stated in Section I. Therefore, we do not need to give careful consideration to the resonances in use of the CAB consisting of very short dipole antenna elements. However, if we used long dipole antennas as we considered in this subsection, it is important to investigate influence of the resonance on DOA estimation. D. Resonance Analysis of the Circular Array Signals In the last subsection, we predicted the resonant frequency and the corresponding phase-sequence with the MoM. Even if we modeled the CAB with the MoM, there would be no guarantee that the MoM could represent the experimental data. This is because there might be inhomogeneity such as stones and voids around the antenna. This makes agreement between the MoM and the experimental data worse. If we had a signal processing method to confirm that the resonances actually occur in experiments as the MoM predicted, it would be useful for showing the validity of MoM predictions. In this subsection, processing received array signals actually acquired by the CAB, we will try to obtain information on the phase-sequence. This information could be associated with the phase-sequence resonance, and we would know the mechanism of the resonances from the array signals. In [47], [48], and [52], the inner product of the array signal vector and the weight vector was taken in order to extract the th phase sequence component from circular array signals.
where is an arbitrary complex value. If we compare the form and can be of (8) and (18), we find that the values component. This implies connected to the phase-sequence may have information that the values on the phase sequence resonance. In this paper, we propose evaluating the resonance with the following value :
(19) with
even or
with
odd
is a function of frequency. Note that the value We assume that a wavelength of a plane wave in the outermost layer at the operating frequencies is much larger than the diameter of the circular array. Under this assumption, if there was no mutual coupling among the array elements and no resonance, the value would be the largest of all possible at each frequency. This is because a values small phase and amplitude difference among the array signals will make large. However, as the operating frequency is increased from zero, the phase sequence resonance corresponding will be excited and the will become the largest value to . After this resonance, the next resonance of will be appear at higher frequency than the resowith , and , will become the largest. nance corresponding to This process will repeat until the phase-sequence m becomes or . It should be noted that the resonances with larger phase-sequence occur at higher frequencies [35], [38]. calculated with the theoretical sigFig. 4 shows the value nals of the MoM. All the parameters are the same as those used in the last subsection. A color scale in the figure corresponds to below the value . In the figure, the power is highest at 60 MHz. In these frequencies, the wavelength is much larger than the antenna, and there is no mutual coupling among the antennas. However, we should notice that the phase-sequence to number of the relative maximum power shifts from just below 116 MHz. After this, the relative maximum at around 132 and 138 MHz, respecvalue shifts to tively. These frequencies are evaluated as the resonant frequencies in the last subsection. We confirmed that we can determine the resonant frequency from the array signals and its phase-sequence with plots of the value .
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
27
in the above, and this can be where we used shown from symmetry. Generally, if the azimuth angle of the other than , we approximated the incident wave is as signal (24) where the constant is related to both the terminated impedance and phase sequence current. At the frequency of the th phase-sequence resonance, using the above approximation, the mode vector can be written as
(25) Fig. 4. Resonance analysis. Computer simulation results. Note that the value is normalized at each frequency, and the minimum value and the maximum value are 0 and 1, respectively. The three arrows are at 116, 132, and 138 MHz.
and the measurement vector is approximated as
III. INFLUENCE OF RESONANCES ON DOA ESTIMATION The CAB is used as a directional receiver in directional borehole radar [16]. Our main interest is the influence of the phasesequence resonance on estimation of target positions in 3-D space in the radar measurement. This can be estimated by the time delay of arriving waves as well as DOA consisting of an azimuth angle and an elevation angle . However, in this paper we will focus on the influence of the resonance on estimation of DOA, especially the azimuth angle only. This is because the limitation of the diameter of a borehole is strict, and estimation of the azimuth angle is the most difficult of the three parameters. We assume that the TM wave is incident with the elevation . The azimuth angle can be estimated by the angle following customary procedure [49], [50]: (20) where (21) (22) Note that the mode vector was derived with the MoM analysis at certain frequency in s Section II-A, and the measurement vector is based on data actually acquired by the CAB at the same frequency. In previous works [15], [16], [53], the superresolution technique was applied to estimate the parameters. It should be noted that the above estimation procedure works under conditions of high signal to noise ratio and is limited to a single arriving wave. Although this condition is rare in borehole radar measurement, we assume these conditions here to further our investigation of the influence of the resonance. To simplify our investigation, we approximate the vectors as , follows. If the azimuth angle of the incident wave is of received at the th antenna at the approximated signal the th phase-sequence resonant frequency is
(23)
(26) is a true DOA. Substituting (25) where the azimuth angle and (26) into (20), at the resonant frequency, the approximation of the estimator in (20) is (27) We should notice that the above function has peaks of the estimator at some other angles as well as the true angle. This spurious solutions in the DOA esimplies that we obtain timation at the th phase-sequence resonance frequency, when . Note that these facts correspond to that the fact that sharp peaks of the th phase-sequence resonance produces radiation pattern when a circular array is used as a transmitter [35], [43]. Computer simulations were performed to confirm the existence of the spurious solutions. In order to avoid the approximation, we formed the measurement vector as well as the mode vector, not with (25) and (26), but with the MoM results in Section II-a. We assumed that the TM wave with azimuth and elevation angle was incident on the CAB. Other parameters are the same as those used in Section II-C. Fig. 5(a) shows a color scale plot of the estimator value (20) between 5 and 200 MHz. At all the frequencies, we can see that the estimator value has the relative maximum value , which is a correct solution. However there are at multiple relative maximum values around the frequencies 116, 132, and 138 MHz. At these frequencies, the phase-sequence resonances are excited as we stated in the last section, and we can see multiple peaks at each frequency. Note that the number . For example, there are two of the peaks corresponds to and around relative maximum values at 116 MHz, where the phase-sequence resonance having occurs. Fig. 5(b) shows a plot of the estimator value (20) only at 138 MHz. Note that there are six relative maximum values at and as well as , at 138 MHz. This is because there should be six peaks at the phase sequence resonance. According to the figure, we
28
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
Fig. 6. Experiment to transmit a wave to the dipole array in granite from outside the borehole in O.E.C.U., Shijonawate, Japan.
Fig. 5. DOA estimation. Computer simulation results. Note that the value is normalized at each frequency, and the minimum value and the maximum value are 0 and 1, respectively. (a) The estimator values between 5 and 200 MHz. The white broken line represents the true DOA, which is ' = 70 . The three arrows are at 116, 132, and 138 MHz. (b) The estimator values at 138 MHz. Note that this 2-D plot corresponds to a section of (a) at 138 MHz.
confirm that there are spurious solutions. It should be noted that the phase-sequence resonance leads to ambiguity in DOA estimation with the CAB, although we can estimate DOA correctly at other than the resonant frequencies.
IV. VERIFICATION WITH EXPERIMENTAL DATA Field experiments were conducted at a site composed largely of granite. We arranged the CAB at a depth of 6 m in the borehole, BR2, at the field test site of Osaka Electro-communication University, Shijonawate, Japan, as shown in Fig. 6. We set a compass around the CAB, and measured the positions of the antenna elements in the borehole. All the parameters of the antennas, such as circular diameter and length, were the same as those used in Section II-C. It should be noted that modified versions of the optical modulators of the OEFS1 system (NEC TOKIN Corporation) were used as in [16] in order to avoid metallic feeding cables. Another dipole antenna serving as a transmitter was set at the same depth as the CAB, in the BR3 borehole, which is 6 m away from BR2. The measurement system is based on a network analyzer (Agilent technology
Fig. 7. Resonance analysis: experimental results. The value is normalized at each frequency. The three arrows are at 116, 132, and 138 MHz.
E5061A) like [16]. In the following analysis, all the received signals were windowed in the time domain, and we reduced scattered wave power from scatterers such as fractures. Since we used this windowing, the measurement is treated as if there is only a direct wave from the transmitter in the analyzed data. Fig. 7 shows the values , which were calculated from the is largest between zero and experimental data. The value 60 MHz, since there is little phase difference at very low frequency. We can see that the relative maximum value shifts from to at around 100 MHz. Also, the value is and MHz, and at and MHz. We large at should notice that an interval between resonant frequencies of and is small, according to Figs. 2 and 3. This may make worse in Fig. 7. It resolution of the two resonances of should be noted that we can see some similarity between Figs. 4 and 7, and this implies that the phase-sequence resonance correand actually occurs in the experiment. sponding to Fig. 8 shows the estimated DOA with the experimental data. According to the compass data near the CAB and the geometrical arrangement of the boreholes, the correct azimuth of the . We formed a steering vector with direct wave is the MoM, and used the estimator in (20). Below 60 MHz in the
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
Fig. 8. DOA estimation. Experimental results. The white broken line represents the true DOA, which is ' = 72 . The value is normalized at each frequency. The three arrows are at 116, 132, and 138 MHz.
Fig. 9. Averaged coherence. Experimental results. The three arrows are at 116, 132, and 138 MHz.
figure, the estimated direction is almost correct. However, we can see other spurious solutions than the correct solution around 11, 132, and around 138 MHz. These frequencies are those of respecthe expected phase-sequence resonance tively. In particular, we can clearly see two relative maximum values at 110 MHz. This might be caused by the phase-sequence as we expected in resonance having phase-sequence the last section. At 132 MHz, there are three relative maximum , , and , although we cannot see values at . These multiple a relative maximum value clearly at solutions may be generated by the phase-sequence resonance . At 138 MHz, which is predicted as the resonant having , the estimated angle is not accurate. At frequency of this frequency, we cannot see the six relative maximum values clearly unlike the prediction in the last section. Since the resois very narrow as shown in nance having phase-sequence Figs. 2 and 3, even the small difference between the predicted resonant frequency and the actual one in the experiment may make much difference between Figs. 5(a) and 8. At frequencies higher than 160 MHz, low signal-to-noise ratio caused instability of the color plot and errors in the estimation. Fig. 9 shows coherence [49], [50] among the array signals. This was calculated by applying cross spectrum analysis be-
29
tween all the combinations of the CAB signals. In the figure, we find that the coherence was low around the expected resonant frequencies, especially between 130 and 140 MHz. In this frequency band, we expected two resonances having different phase-sequence. These resonances have very narrow frequency bandwidth, and large variation of phase with frequency might make the coherence low. When we changed the depth of both the transmitter and the receiver, the difference between the MoM results and the experwas increased at some depths. imental results in the plot of This may be caused by inhomogeneity in subsurface or location error of the antennas in the borehole. What is important is that we could see influence of the phase-sequence resonance approximately between 100 and 150 MHz on DOA estimation even at these depths. Even when there is small inhomogeneity, which is not considered in the MoM model, the prediction with the MoM shown in this paper might be useful to design the optimal CAB. pF as the In the MoM analysis, we adopted terminated impedance, and this impedance is near a typical value as an input impedance of an optical modulator [51]. We found that the resonant frequencies predicted by the MoM are in the MoM slightly shifted, if the terminated impedance model is changed. Also, the shape of the actual driving point, which is shown in [16, Fig. 2], may be difficult to model with the delta gap source exactly. Both the terminated impedance and the shape of the driving point may influence the resonant frequency. However, we should notice that our objective in this paper is not to predict the resonant frequency very accurately, but to investigate influence of the resonance on the DOA estimation. According to Fig. 7, we found that the phase-sequence resonances are actually excited in the experiment, and the resonant frequencies in the experimental data were not far from the MoM predictions. Since we knew that the phase-sequence resonance makes the spurious solutions in the computer simulation in Section III, we should design the CAB optimally so as to avoid the phase-sequence resonance.
V. CONCLUSION Using the MoM model including the influence of the borehole, we investigated the phase-sequence resonance in a circular dipole array antenna in a borehole. In order to combine resonant frequencies and the resonant phase-sequence, we modified the impressed field and artificially excited the phase-sequence resonance in MoM. Using this method, we can theoretically analyze the resonance, even if the antenna is in a borehole. Furthermore, using the values generated from the spatial digital Fourier transform of the CAB signals, we showed that we can confirm the excitation of the phase-sequence resonance. According to the proposed MoM analysis, we concluded the phase-sequence resonance gives some spurious solutions in DOA estimation with the conventional estimator. In order to verify the theoretical results, we have done experiments with the CAB consisting of seven dipole antennas in granite. We found that the phase-seand occur in quence resonance corresponding to the experiment data. Comparison between the resonance analysis and estimation of DOA showed that the phase-sequence resonance makes spurious solutions in direction finding with a circular dipole array in a borehole.
30
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
ACKNOWLEDGMENT Thanks are due to Y. Fujita, A. Kusama, K. Kondo, and K. Minamoto for their assistance in doing experiments in Shijonawate, Japan. The authors thank the three anonymous reviewers for the detailed and helpful comments. REFERENCES [1] D. J. Daniels, Ground Penetrating Radar, 2nd ed. London, U.K.: IEE, 2004. [2] H. Nickel, F. Sender, R. Thierbach, and H. Weichart, “Exploring the interior of salt domes from boreholes,” Geophys. Prospect., no. 31, pp. 131–148, 1983. [3] D. L. Wright, R. D. Watts, and E. Bramsoe, “A short-pulse electromagnetic transponder for hole-to-hole use,” IEEE Trans. Geosci. Remote Sens., vol. GE-22, no. 6, pp. 720–725, Nov. 1984. [4] J. A. Bradley and D. L. Wright, “Microprocessor-based data-acquisition system for a borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. GE-25, no. 4, pp. 441–447, Jul. 1987. [5] O. Olsson, L. Falk, O. Forslund, L. Lundmark, and E. Sandberg, “Borehole radar applied to the characterization of hydraulically conductive fracture zones in crystalline rock,” Geophys. Prospect., vol. 40, no. 2, pp. 109–142, 1992. [6] T. Miwa, M. Sato, and H. Niitsuma, “Subsurface fracture measurement with polarimetric borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 2, pp. 828–837, Mar. 1999. [7] T. Miwa, M. Sato, and H. Niitsuma, “Enhancement of reflected waves in single-hole polarimetric borehole radar measurement,” IEEE Trans. Antenna Propagat., vol. 48, no. 9, pp. 1430–1437, Sep. 2000. [8] H. Zhou and M. Sato, “Subsurface cavity imaging by crosshole borehole radar measurements,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 2, pp. 335–341, Feb. 2004. [9] R. J. Lytle and E. F. Laine, “Design of a miniature directional antenna for geophysical probing from boreholes,” IEEE Trans. Geosci. Remote Sens., vol. GE-16, no. 4, pp. 304–307, Oct. 1978. [10] C. W. Moulton, D. L. Wright, S. R. Hutton, D. von G. Smith, and J. D. Abraham, “Basalt-flow imaging using a high-resolution directional borehole radar,” in Proc. 9th Int. Conf. Ground Penetrating Radar, Santa Barbara, CA, Apr. 2002, pp. 13–18. [11] K. W. A. van Dongen, A Directional Borehole Radar System for Subsurface Imaging. Delft, The Netherlands: Delft Univ. Press, 2002. [12] K. W. A. van Dongen, R. van Waard, S. van der Baan, P. M. van den Berg, and J. T. Fokkema, “A directional borehole radar system,” Subsurf.Sens. Technol. Appl., vol. 3, no. 4, pp. 327–346, Oct. 2002. [13] E. Mundary, R. Thierbach, F. Sende, and H. Weichart, “Borehole radar probing in salt deposits,” presented at the 6th Int. Symp. Salt, vol. 1, Toronto, ON, Canada, 1983. [14] M. Sato and T. Tanimoto, “A shielded loop array antenna for a directional borehole radar,” in Proc. 4th Int. Conf. GPR Geological Survey of Finland, Rovaniemi, Finland, Jun. 1992, Special Paper 16, pp. 323–327. [15] S. Ebihara, M. Sato, and H. Niitsuma, “Super-resolution of coherent targets by a directional borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1725–1732, Jul. 2000. [16] S. Ebihara, “Directional borehole radar with dipole antenna array using optical modulators,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 1, pp. 45–58, Jan. 2004. [17] R. F. Harrington, Field Computation by Moment Method. New York: Macmillian, 1968. [18] J. H. Richmond and N. H. Gery, “Mutual impedance of nonplanar-skew sinusoidal dipoles,” IEEE Trans. Antennas Propagat., vol. AP-23, no. 3, pp. 412–414, May 1975. [19] K. Sawaya, “Numerical techniques for analysis of electromagnetic problems,” IEICE Trans. Commun., vol. E83-B, no. 3, Mar. 2000. [20] H. Nakano, S. R. Kerner, and N. G. Alexopoulos, “The moment method solution for printed wire antennas of arbitrary configuration,” IEEE Trans. Antennas Propagat., vol. 36, no. 12, pp. 1667–1674, Dec. 1988. [21] R. Li and H. Nakano, “Numerical analysis of arbitrary shaped probe-excited single-arm printed wire antennas,” IEEE Trans. Antennas Propagat., vol. 46, no. 9, pp. 1307–1317, Sep. 1998. [22] T. J. Cui and W. C. Chew, “Modeling of arbitrary wire antennas above ground,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 1, pp. 357–365, Jan. 2000. [23] M. Sato and R. Thierbach, “Analysis of a borehole radar in cross-hole mode,” IEEE Trans. Geosci. Remote Sens., vol. 29, no. 6, pp. 899–904, Nov. 1991.
[24] S. Ebihara, M. Sato, and H. Niitsuma, “Analysis of a guided wave along a conducting structure in a borehole,” Geophys. Prospect., vol. 46, pp. 489–505, 1998. [25] T. B. Hansen, “The far field of a borehole radar and its reflection at a planar interface,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 4, pp. 1940–1950, Jul. 1999. [26] F. L. Teixeira and W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1530–1543, Jul. 2000. [27] S. Liu and M. Sato, “Electromagnetic logging technique based on borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 9, pp. 2083–2092, Sep. 2002. [28] K. Holliger and T. Bergmann, “Numerical modeling of borehole georadar data,” Geophysics, vol. 67, no. 4, pp. 1249–1257, Jul./Aug. 2002. [29] Y. Chen and M. L. Oristaglio, “A modeling study of borehole radar for oil-field applications,” Geophysics, vol. 67, no. 5, pp. 1486–1494, Sep./Oct. 2002. [30] K. J. Ellefsen and D. L. Wright, “Radiation pattern of a borehole radar antenna,” Geophysics, vol. 70, no. 1, pp. K1–K11, Jan.–Feb. 2005. [31] J. Tronicke and K. Holliger, “Effects of gas- and water-filled borehole on the amplitudes of crosshole georadar data as inferred from experimental evidence,” Geophysics, vol. 69, no. 5, pp. 1255–1260, Sep.–Oct. 2004. [32] R. O. Schmidt, “Multiple emitter location and signal parameters estimation,” IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [33] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [34] Y. Mushiake, Antenna and Radio Propagation (in Japanese). Tokyo, Japan: Corona, 1961. [35] R. W. P. King, G. J. Fikioris, and R. B. Mack, Cylindrical Antennas and Arrays. Cambridge, U.K.: Cambridge Univ. Press, 2002. [36] 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. [37] , “The large circular array with one element driven,” IEEE Trans. Antennas Propagat., vol. 38, no. 9, pp. 1462–1472, Sep. 1990. [38] G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys., vol. 68, no. 2, pp. 431–439, Jul. 1990. [39] D. K. Freeman and T. T. Wu, “Variational-principle formulation of the two-term theory for arrays of cylindrical dipoles,” IEEE Trans. Antennas Propagat., vol. 43, no. 4, pp. 340–349, Apr. 1995. [40] G. Fikioris, R. W. P. King, and T. T. Wu, “Novel surface-wave antenna,” Proc. Inst. Elect. Eng., Microw. Antennas Propagat., vol. 143, no. 1, pp. 1–6, Feb. 1996. [41] G. Fikioris, “Experimental study of novel resonant circular arrays,” Proc. Inst. Elect. Eng., Microw. Antennas Propagat., vol. 145, no. 1, pp. 92–98, Feb. 1998. [42] D. Margetis, G. Fikioris, J. M. Myers, and T. T. Wu, “Highly directive current distributions: General theory,” Phys. Rev. E, vol. 58, no. 2, pp. 2531–2547, Aug. 1998. [43] R. W. P. King, M. Owens, and T. T. Wu, “Properties and applications of the large circular resonant dipole array,” IEEE Trans. Antennas. Propagat, vol. 51, no. 1, pp. 103–109, Jan. 2003. [44] G. Fikioris, P. J. Papakanellos, J. D. Koundouros, and A. K. Patsiotis, “Difficulties in MoM analysis of resonant circular arrays of cylindrical dipoles,” Electro. Lett., vol. 41, no. 2, Jan. 2005. [45] W. C. Chew, Waves and Field in Inhomogeneous Media. New York: IEEE Press, 1995. [46] S. Ebihara and W. C. Chew, “Calculation of Sommerfeld integrals for modeling vertical dipole array antenna for borehole radar,” IEICE Trans. Electron., vol. E86-C, no. 10, pp. 2085–2096, Oct. 2003. [47] C. P. Mathews and M. D. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2395–2407, Sep. 1994. [48] K. M. Reddy and V. U. Reddy, “Analysis of spatial smoothing with uniform circular arrays,” IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1726–1730, Jun. 1999. [49] S. M. Kay, Modern Spectral Estimation. Upper Saddle River, NJ: Prentice-Hall, 1988. [50] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1992. [51] N. Kuwabara, K. Tajima, R. Kobayashi, and F. Amemiya, “Development and analysis of electric field sensor using LiNbO3 optical modulator,” IEEE Trans. Electromagn. Compat., vol. 34, no. 4, pp. 391–396, Nov. 1992.
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
[52] D. E. N. Davies, The Handbook of Antenna Design. London, U.K.: Peter Peregrinus, 1986, ch. 12. [53] S. Ebihara, K. Nagoya, N. Abe, and M. Toida, “Experimental studies for monitoring water level by dipole antenna array radar fixed in subsurface,” Near Surf. Geophys., 2006, to be published.
Satoshi Ebihara (M’97) was born in Chiba, Japan, on December 10, 1968. He received the B.S., M.S., and Dr.Eng. degrees in resources engineering from Tohoku University, Sendai, Japan, in 1993, 1995, and 1997, respectively. From 1995 to 1997, he was a Research Fellow with the Japan Society for the Promotion of Science, Tokyo, Japan. From 1997 to 2003, he was a Research Associate with the Center for Northeast Asian Studies, Tohoku University. He is currently an Assistant Professor with the Department of Electronic Engineering and Computer Science, Faculty of Engineering, Osaka Electro-Communication University, Osaka, Japan. His current research interests include superresolution techniques, array signal processing, and borehole radar measurement. Dr. Ebihara was awarded “Best Paper” of the Second Well-Logging Symposium of Japan, Chiba, Japan, in September 1996 and was awarded the Miyagi Foundation Young Researchers Award of the Miyagi Foundation for Promotion of Industrial Science, Sendai, Japan, in March 2001.
31
Takashi Yamamoto was born in Kyoto, Japan, on September 30, 1981. He graduated from Osaka Electro-Communication University, Osaka, Japan, in 2005. He is currently with Tsuruyayoshinobu, Inc., Kyoto.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
Resonance Analysis of a Circular Dipole Array Antenna in Cylindrically Layered Media for Directional Borehole Radar Satoshi Ebihara, Member, IEEE, and Takashi Yamamoto
Abstract—In this paper, we discuss the influence of a resonance on estimating direction of arrival (DOA) with a circular dipole array in a borehole (CAB). The resonance is caused by the phasesequence of currents on the dipole antennas. Making use of method of moments (MoM) analysis, we predict resonant frequencies of the CAB and describe a mechanism for the resonance theoretically, making use of the fact that a plane wave can be broken down into cylindrical harmonics. In order to examine the agreement between the MoM and experimental data, we propose a signal processing method to extract phase-sequence components from the actual received array signal. This method is useful for confirming excitation of the phase-sequence resonance in experimental data, as predicted by the MoM. Using the MoM, we examine the influence of the resonances on DOA estimation, and we conclude that the resonances give some spurious solutions. In order to confirm these results, we conduct experiments at a field test site. We arrange the CAB, which consisted of seven receiving dipole antennas. In this experiment, a plane wave is incident on the CAB from a transmitter in another borehole. We apply the proposed signal method, and the measured results demonstrate the existence of the phase-sequence resonance in a physically real CAB. The spurious solutions found to occur in the DOA estimation at the resonant frequencies are also found experimentally. Index Terms—Circular array, cylindrical layers, dipole antenna, directional borehole radar, method of moments (MoM), mutual coupling, phase-sequence resonance.
I. INTRODUCTION
B
OREHOLE radar is one form of ground-penetrating radar [1]–[8]. Most conventional borehole radars use dipole antennas that are omnidirectional. However, in many engineering applications, the three-dimensional position of targets must be accurately determined. Such measurements can be achieved using one borehole, if a directional borehole radar can be used [9]–[16]. To realize such a radar, control of the radiation pattern in a borehole has been attempted using a dielectric medium [9], and a cavity-backed antenna was used with mechanical rotation in a borehole [10]–[12]. Also, a cross-loop antenna, which consists of loops having a figure-eight radiation pattern in air, was tried [13]. These previous studies made use of antenna
Manuscript received May 30, 2005; revised August 22, 2005. This work was supported by a Grant-in-Aid for Young Scientists (A), MEXT. KAKENHI 15686037, Japanese Government. S. Ebihara is with the Department of Electronic Engineering and Computer Science, Faculty of Engineering, Osaka Electro-Communication University, Osaka 572-8530, Japan (e-mail: [email protected]). T. Yamamoto is with the Tsuruyayoshinobu Inc., 602-8434 Kyoto, Japan. Digital Object Identifier 10.1109/TGRS.2005.860486
radiation patterns to achieve directional borehole radar. This means that only the amplitude information of the antennas was exploited to obtain a direction-of-arrival (DOA) estimation. Directional borehole radar could be realized by arranging some array antenna elements in circle in a borehole [14]. We have proposed an array antenna using optical modulators [16]. In this system, a circular dipole array in a borehole (CAB) consisting of vertical dipole antennas was arranged. The voltages received at antenna elements of the CAB are converted to optical signals with the optical modulators near driving points of the antennas. In this radar, in order to estimate DOA, we make use of differences in both phase and amplitude among the received array signals. Steering vectors to analyze the array signals for DOA estimation were formed using the method of moments (MoM) analysis [17]–[22]. In this theoretical analysis, the borehole effect [23]–[31], which is the influence of a borehole on antenna characteristics, was considered. of the antenna In the CAB, parameters such as number of the dipole antenna element elements and the half length of array elements inare optional. Generally, if the number creases, the number of signals which can be estimated would also increase [32]. As far as the dipole antenna length is concerned, we should in general use relatively long antennas rather than electrically short antennas in order to obtain higher sensitivity [33]. However, in the CAB, arranging many long dipole antennas in the narrow space may lead to strong mutual coupling among the elements. This may cause resonances of the CAB, and these phenomena might influence the DOA estimation. For many years, array antennas making use of resonance among array elements, such as the Yagi–Uda array [34], have been studied. It has also been realized that another useful antenna array can be obtained by bending the Yagi–Uda array in a circle. According to previous studies [35]–[44], it is known that a phase-sequence resonance occurs in a circular array. This resonance is sometimes useful for supergain [36] or superdirectivity [43]. However, according to [35], [38], and [43], the radiation pattern of the circular array antenna has multiple sharp peaks at the resonant frequencies. These peaks might make DOA estimation difficult with the CAB. Furthermore, resonance of antennas generally makes frequency bandwidth narrower and worsens estimations of time delay resolution in impulse radar measurement. Predicting and knowing the resonant frequencies may be important before array antennas can be useful. The two-term theory [35] has been developed to predict frequencies of the phase-sequence resonance of a circular dipole array. This theory uses an approximation solution to the coupled
0196-2892/$20.00 © 2006 IEEE
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
integral equations for the currents along the dipoles. The computational cost of the solution is relatively low, even if the number of elements is large. Furthermore, agreement between experimental results and the theory is good [41]. However, there are some assumptions in this theory. The two-term theory is adequate, when the close-coupling effects [35], [38] are avoided. This is the case when interelement spacing of the circular array of the dipole, i.e., . In is larger than the half-length addition, the theory requires that the array elements should be , where is the electrically short dipole antennas, i.e., spatial wavelength. Since electromagnetic waves at higher frequencies are attenuated in subsurface media [1], operating frequencies should be lower than about 200 MHz. Therefore, in order in to obtain sufficient detectable range, the wavelength should be longer than about 0.47 m media having m/s MHz . Even if the high frequencies of the borehole radar are used, the distance between the antenna . This elements is much smaller than the wavelength, i.e., is because the diameter of a borehole is usually about 0.1 m, m . Furthermore, if the and the distance should be half-length of the dipole antenna is longer than the wavelength, , the mutual coupling among dipole antennas i.e., influences the antenna array characteristics significantly. These , imply that we cannot ignore conditions, i.e., the close-coupling effects among the antenna elements at the high frequency of the borehole radar. This differs from the assumptions in the two-term theory. Another important difference between previous work [35]–[44] and the CAB case is the fact that media around the antenna is not homogeneous in the CAB. Since a borehole wall always exists around the antenna, we need to assume that the antennas are in cylindrically layered media rather than in homogeneous media [23]–[31]. The main difference between [16] and this paper is that in the previous work there was no discussion of the existence of phase-sequence CAB. In [16], the number of antenna elements in the circle was small or the electrically small dipole antenna elements were used. In [16], we did not consider the influence of the resonance on DOA, since we used the dipole array antenna only in situations of no resonance in the dipole array antenna. Therefore, we could estimate DOA without consideration of the resonance. In this paper, we will investigate the influence of the phase-sequence resonance on DOA estimation with the CAB. In Section II, we will propose a theoretical method to predict frequencies of the phase-sequence resonance for the CAB. We will show numerical results and predict the resonant frequencies. This is followed by a proposed method to estimate the resonant frequencies from the received signals of the CAB. In Section III, we discuss the influence of resonance on DOA estimation with the MoM analysis. In Section IV, we describe the associated experimental data, and we will discuss the influence of the resonance using experimental data.
Fig. 1.
23
MoM model for a CAB.
A. Review of the MoM Solution for the CAB Consider the CAB in a cylindrical coordinate system as shown in Fig. 1. This antenna consists of an array of identical, parallel, center-driven coplanar cylindrical vertical dipoles that are equally spaced about the circumference of a circle. The th vertical dipole antenna is arranged at and (1) Here we apply Richmond’s MoM, which uses Galerkin’s method and piecewise sinusoidal functions [18]. The th segments, each of length dipole antenna is divided into , and the complex constants denote the current function sampled at the junctions . The diameter of all the dipole antennas is , and is small enough for the thin wire approximation to be applied. The delta-gap source modeling is applied for a driving point. The basis function, which is same as the test function, is (2) where is the wavenumber in the innermost layer. We can obtain a set of linear equations as
(3) II. FORMULATION First, briefly, we review the MoM solution, which was previously formulated for the CAB [16]. This is followed by consideration of a method to predict resonance frequency and its resonant mechanism.
and
(4)
24
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
where (5)
(6)
(7) The driving point of all the dipole antenna element is located at , and is the index of the segment junction, where the driving point at the center of each dipole antenna exists. The termiat all the nated impedance at the driving point is the constant is the component of the impressed driving points. . The function is electric field at the Green’s function, when the -directed Hertzian dipole as the and the comsource exists at . ponent of the field at the observation point exists at infinThese source point and observation points are inside itely long cylindrical layers, where boundaries exist at . The function is the same as equation (10) in [16]. It should be noted that the Green’s function includes scattered fields from the cylindrical boundaries, and this is an important difference between the borehole radar case and homogenious media case. We should notice that this Green’s function involves the evaluation of Sommerfeld integrals. The numerical consideration and treatment of the Sommerfeld integration have been presented by various authors such as [45] and [46]. B. Driving Point Impedance of the CAB Fikioris et al. have recently reported difficulties in MoM analysis of a resonant circular array [44]. In this work, they assumed antenna elements in air, and a circular array having they found that the driving-point conductance has a large negative value near the resonant frequencies. Here we will confirm that there is no meaningless result in our calculation with the MoM. We consider the CAB having several parameters, which can be used in real borehole radar measurements, and will confirm that the MoM stated here is capable for dealing with actual borehole radar measurements. Parameters used in the calculation are , number of cylindrical the number of antenna elements: , , , , S/m, layers: m, antenna element diameter mm, cirm, antenna full length cular array diameter m, and terminated impedance pF at the center of each dipole antenna. This impedance corresponds to an input impedance of the optical modulators used in the experiments described in Section IV. Now we will consider input impedance of one element of the CAB. For the CAB with one element driven, the driving point admittance is proportional to the value of the current on the driving point at the center of the driven element. Fig. 2(a) and (b) shows the driving-point conductance and susceptance. The MoM considering the borehole effect predicts three distinct
Fig. 2. Driving point impedance. The computer simulation results. (a) Conductance. (b) Susceptance. The three arrows are at 116, 132, and 138 MHz.
resonances, where the conductance has maximum value and the susceptance is near zero. It should be noted that there is no large negative conductance values in the figure unlike in [44]. A significant difference between the present calculation and that of the elements. The increased may of [44] is the number cause the large negative conductance seen in [44]. As stated in the latter paper, decoupling the integral equations would be needed, since the increased elements lead to high phase-sequence among the currents at the center of the antennas. In the CAB, we cannot increase the number of the antenna elements, since the space available for the array is limited. Based on the evidence, we believe that the MoM reviewed in the last subsection works well for the relatively small number of elements. C. Resonance Analysis With the MoM Once the impedance matrix is calculated, the current induced by any impressed electric field can be calculated using the MoM. Using the current calculated with frequency changed, all the antenna characteristics such as resonant frequency are also obtained. However we cannot know the mechanism of the resonance only with the MoM calculation. According to [35] and [38], the circular array resonances can be associated with a parwith even, ticular phase-sequence where resonances with larger occur at higher frequencies. It is worthwhile knowing the relationship between the resonant frequency and the corresponding phase-sequence for designing optimal array antenna parameters. In [35]–[39], the multiple integral equations were reduced to a single equation for each phase-sequence , and a resonant
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
frequency corresponding to the was found using the assumptions stated in Section I. However we cannot directly employ such an approach here. This is because the two assumptions in [35]–[39] are not valid for the CAB as we stated in Section I. One assumption is that the close-coupling effects among the antenna elements can be avoided. The other is that media around the antennas is homogeneous. These do not satisfy the CAB case at high frequencies in borehole radar measurement. Here we will try to combine the resonant frequency and the resonance’s phase-sequence , exploiting and modifying the MoM analysis. at of the th dipole antenna can be The currents phase-sequence currents represented by a superposition of as follows [35]–[39]:
(8) It should be noted that the above equation is valid even for a circular array in inhomogenous media, although the circular array is in homogeneous media in [35]–[39]. In cylindrically symmetrical media as the CAB, the phase-sequence currents can be associated with the th phase-sequence resonance. The th phase-sequence currents are excited by the following elements: phase-sequence voltages applied to the
25
wave is defined as a plane wave, the magnetic field of which is transverse to the direction. The (11) implies that the incident plane wave can be broken down into cylindrical harmonics with dependence. The component having dedifferent in (11), contributes to exciting the th pendence, i.e., phase-sequence electric field in (7). We should notice that the phase-sequence is decided by the relationships among phases of the electric fields, not at spatially continuous points, but at some discrete points in space. According to this fact, the comhaving other than may contribute to ponent the excitation of the th phase sequence electric field. Considering an analogy to aliasing in sampling continuous signals, the following contributes to exciting the electric field having the th phase-sequence in (11): for
(15)
for
(16)
and
and According to the above equation, both phase-sequence give the same . This implies that it is enough with even and only to consider with odd. Consequently we may decompose the plane wave in (11) to components contributing to only the th phase-sequence as
(9) where
is the th phase-sequence voltages at . In order to excite the above voltage , the impressed electric field to the array antenna should be the following th phase-sequence: (17) (10) Now, let us assume that a plane wave with the elevation angle and the azimuth angle traveling parallel to the plane is incident on the CAB in multiple cylindrical layers shown in Fig. 1. According to [16] and [45], the impressed electric field in (7) should be given by (11) where (12) (13) (14) , and is the wavenumber of a plane wave in the th layer. The reflection matrixes such as and , and the transmission matrix are given in the appendix of [16] or [45, ch. 3]. If a TM plane is equal to zero, wave is incident on the cylindrical layers, is the magnitude of the electric field strength. The TM and
with
even or
with
odd
When a plane wave is incident on the CAB at the resonant frequency corresponding to the phase-sequence , the impressed in the electric field may be approximated by the field with above (17), instead of the exact field in (11). This implies that, at the th resonant frequency, usage of either (11) or (17) with will give almost the same result of the current calculation. We calculated voltages received at the driving point of the with the TM wave incidence. antenna located at This plane wave propagates from direction of the azimuth angle and the elevation angle . All the parameters of the CAB are the same as those used in the previous subsection except for usage of the CAB as a receiver. Note that all the pF. The caldriving points are terminated with the culated voltages are shown as solid, broken, or dotted lines in Fig. 3. The solid line is the response impressed by the total TM wave field shown in (11), and the other four lines correspond to responses impressed by the fields in (17). According to the solid line in the figure, we find three distinct peaks, and we find that the CAB resonates at these three frequencies: 116, 132, and 138 MHz. These frequencies correspond to the peak positions of
26
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
This processing corresponds to a spatial digital Fourier transform (DFT) of the array signals. In this paper, we will utilize the spatial DFT to obtain information on the phase sequence as the array voltage resonance. We define signals received by the th antenna element of the CAB at a certain frequency. Considering a Fourier transform of the periodic signals, the signals can be represented by the linear phase-sequence voltages as combination of the
(18)
Fig. 3. Calculated voltage at a feeding point of the dipole antenna 1. The three arrows are at 116, 132, and 138 MHz.
the curves calculated with the phase-sequence respectively. Using this correspondence, we reveal the relation between the resonant frequencies and the corresponding phase-sequences. It should be noted that the wavelength at very low m of frequencies is much larger than the diameter the circular array. If there was no mutual coupling among the antenna elements, the plane wave field might be approximated phase-sequence component in (17). This is the by the reason why the curves with the total field and the phase-senearly coincide at frequencies below 60 MHz. quence We should notice that the resonant frequencies would get higher, if the length of the dipole antenna got shorter. This is because the close coupling among the antenna elements would be reduced by shortening the antenna length, as we stated in Section I. Therefore, we do not need to give careful consideration to the resonances in use of the CAB consisting of very short dipole antenna elements. However, if we used long dipole antennas as we considered in this subsection, it is important to investigate influence of the resonance on DOA estimation. D. Resonance Analysis of the Circular Array Signals In the last subsection, we predicted the resonant frequency and the corresponding phase-sequence with the MoM. Even if we modeled the CAB with the MoM, there would be no guarantee that the MoM could represent the experimental data. This is because there might be inhomogeneity such as stones and voids around the antenna. This makes agreement between the MoM and the experimental data worse. If we had a signal processing method to confirm that the resonances actually occur in experiments as the MoM predicted, it would be useful for showing the validity of MoM predictions. In this subsection, processing received array signals actually acquired by the CAB, we will try to obtain information on the phase-sequence. This information could be associated with the phase-sequence resonance, and we would know the mechanism of the resonances from the array signals. In [47], [48], and [52], the inner product of the array signal vector and the weight vector was taken in order to extract the th phase sequence component from circular array signals.
where is an arbitrary complex value. If we compare the form and can be of (8) and (18), we find that the values component. This implies connected to the phase-sequence may have information that the values on the phase sequence resonance. In this paper, we propose evaluating the resonance with the following value :
(19) with
even or
with
odd
is a function of frequency. Note that the value We assume that a wavelength of a plane wave in the outermost layer at the operating frequencies is much larger than the diameter of the circular array. Under this assumption, if there was no mutual coupling among the array elements and no resonance, the value would be the largest of all possible at each frequency. This is because a values small phase and amplitude difference among the array signals will make large. However, as the operating frequency is increased from zero, the phase sequence resonance corresponding will be excited and the will become the largest value to . After this resonance, the next resonance of will be appear at higher frequency than the resowith , and , will become the largest. nance corresponding to This process will repeat until the phase-sequence m becomes or . It should be noted that the resonances with larger phase-sequence occur at higher frequencies [35], [38]. calculated with the theoretical sigFig. 4 shows the value nals of the MoM. All the parameters are the same as those used in the last subsection. A color scale in the figure corresponds to below the value . In the figure, the power is highest at 60 MHz. In these frequencies, the wavelength is much larger than the antenna, and there is no mutual coupling among the antennas. However, we should notice that the phase-sequence to number of the relative maximum power shifts from just below 116 MHz. After this, the relative maximum at around 132 and 138 MHz, respecvalue shifts to tively. These frequencies are evaluated as the resonant frequencies in the last subsection. We confirmed that we can determine the resonant frequency from the array signals and its phase-sequence with plots of the value .
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
27
in the above, and this can be where we used shown from symmetry. Generally, if the azimuth angle of the other than , we approximated the incident wave is as signal (24) where the constant is related to both the terminated impedance and phase sequence current. At the frequency of the th phase-sequence resonance, using the above approximation, the mode vector can be written as
(25) Fig. 4. Resonance analysis. Computer simulation results. Note that the value is normalized at each frequency, and the minimum value and the maximum value are 0 and 1, respectively. The three arrows are at 116, 132, and 138 MHz.
and the measurement vector is approximated as
III. INFLUENCE OF RESONANCES ON DOA ESTIMATION The CAB is used as a directional receiver in directional borehole radar [16]. Our main interest is the influence of the phasesequence resonance on estimation of target positions in 3-D space in the radar measurement. This can be estimated by the time delay of arriving waves as well as DOA consisting of an azimuth angle and an elevation angle . However, in this paper we will focus on the influence of the resonance on estimation of DOA, especially the azimuth angle only. This is because the limitation of the diameter of a borehole is strict, and estimation of the azimuth angle is the most difficult of the three parameters. We assume that the TM wave is incident with the elevation . The azimuth angle can be estimated by the angle following customary procedure [49], [50]: (20) where (21) (22) Note that the mode vector was derived with the MoM analysis at certain frequency in s Section II-A, and the measurement vector is based on data actually acquired by the CAB at the same frequency. In previous works [15], [16], [53], the superresolution technique was applied to estimate the parameters. It should be noted that the above estimation procedure works under conditions of high signal to noise ratio and is limited to a single arriving wave. Although this condition is rare in borehole radar measurement, we assume these conditions here to further our investigation of the influence of the resonance. To simplify our investigation, we approximate the vectors as , follows. If the azimuth angle of the incident wave is of received at the th antenna at the approximated signal the th phase-sequence resonant frequency is
(23)
(26) is a true DOA. Substituting (25) where the azimuth angle and (26) into (20), at the resonant frequency, the approximation of the estimator in (20) is (27) We should notice that the above function has peaks of the estimator at some other angles as well as the true angle. This spurious solutions in the DOA esimplies that we obtain timation at the th phase-sequence resonance frequency, when . Note that these facts correspond to that the fact that sharp peaks of the th phase-sequence resonance produces radiation pattern when a circular array is used as a transmitter [35], [43]. Computer simulations were performed to confirm the existence of the spurious solutions. In order to avoid the approximation, we formed the measurement vector as well as the mode vector, not with (25) and (26), but with the MoM results in Section II-a. We assumed that the TM wave with azimuth and elevation angle was incident on the CAB. Other parameters are the same as those used in Section II-C. Fig. 5(a) shows a color scale plot of the estimator value (20) between 5 and 200 MHz. At all the frequencies, we can see that the estimator value has the relative maximum value , which is a correct solution. However there are at multiple relative maximum values around the frequencies 116, 132, and 138 MHz. At these frequencies, the phase-sequence resonances are excited as we stated in the last section, and we can see multiple peaks at each frequency. Note that the number . For example, there are two of the peaks corresponds to and around relative maximum values at 116 MHz, where the phase-sequence resonance having occurs. Fig. 5(b) shows a plot of the estimator value (20) only at 138 MHz. Note that there are six relative maximum values at and as well as , at 138 MHz. This is because there should be six peaks at the phase sequence resonance. According to the figure, we
28
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
Fig. 6. Experiment to transmit a wave to the dipole array in granite from outside the borehole in O.E.C.U., Shijonawate, Japan.
Fig. 5. DOA estimation. Computer simulation results. Note that the value is normalized at each frequency, and the minimum value and the maximum value are 0 and 1, respectively. (a) The estimator values between 5 and 200 MHz. The white broken line represents the true DOA, which is ' = 70 . The three arrows are at 116, 132, and 138 MHz. (b) The estimator values at 138 MHz. Note that this 2-D plot corresponds to a section of (a) at 138 MHz.
confirm that there are spurious solutions. It should be noted that the phase-sequence resonance leads to ambiguity in DOA estimation with the CAB, although we can estimate DOA correctly at other than the resonant frequencies.
IV. VERIFICATION WITH EXPERIMENTAL DATA Field experiments were conducted at a site composed largely of granite. We arranged the CAB at a depth of 6 m in the borehole, BR2, at the field test site of Osaka Electro-communication University, Shijonawate, Japan, as shown in Fig. 6. We set a compass around the CAB, and measured the positions of the antenna elements in the borehole. All the parameters of the antennas, such as circular diameter and length, were the same as those used in Section II-C. It should be noted that modified versions of the optical modulators of the OEFS1 system (NEC TOKIN Corporation) were used as in [16] in order to avoid metallic feeding cables. Another dipole antenna serving as a transmitter was set at the same depth as the CAB, in the BR3 borehole, which is 6 m away from BR2. The measurement system is based on a network analyzer (Agilent technology
Fig. 7. Resonance analysis: experimental results. The value is normalized at each frequency. The three arrows are at 116, 132, and 138 MHz.
E5061A) like [16]. In the following analysis, all the received signals were windowed in the time domain, and we reduced scattered wave power from scatterers such as fractures. Since we used this windowing, the measurement is treated as if there is only a direct wave from the transmitter in the analyzed data. Fig. 7 shows the values , which were calculated from the is largest between zero and experimental data. The value 60 MHz, since there is little phase difference at very low frequency. We can see that the relative maximum value shifts from to at around 100 MHz. Also, the value is and MHz, and at and MHz. We large at should notice that an interval between resonant frequencies of and is small, according to Figs. 2 and 3. This may make worse in Fig. 7. It resolution of the two resonances of should be noted that we can see some similarity between Figs. 4 and 7, and this implies that the phase-sequence resonance correand actually occurs in the experiment. sponding to Fig. 8 shows the estimated DOA with the experimental data. According to the compass data near the CAB and the geometrical arrangement of the boreholes, the correct azimuth of the . We formed a steering vector with direct wave is the MoM, and used the estimator in (20). Below 60 MHz in the
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
Fig. 8. DOA estimation. Experimental results. The white broken line represents the true DOA, which is ' = 72 . The value is normalized at each frequency. The three arrows are at 116, 132, and 138 MHz.
Fig. 9. Averaged coherence. Experimental results. The three arrows are at 116, 132, and 138 MHz.
figure, the estimated direction is almost correct. However, we can see other spurious solutions than the correct solution around 11, 132, and around 138 MHz. These frequencies are those of respecthe expected phase-sequence resonance tively. In particular, we can clearly see two relative maximum values at 110 MHz. This might be caused by the phase-sequence as we expected in resonance having phase-sequence the last section. At 132 MHz, there are three relative maximum , , and , although we cannot see values at . These multiple a relative maximum value clearly at solutions may be generated by the phase-sequence resonance . At 138 MHz, which is predicted as the resonant having , the estimated angle is not accurate. At frequency of this frequency, we cannot see the six relative maximum values clearly unlike the prediction in the last section. Since the resois very narrow as shown in nance having phase-sequence Figs. 2 and 3, even the small difference between the predicted resonant frequency and the actual one in the experiment may make much difference between Figs. 5(a) and 8. At frequencies higher than 160 MHz, low signal-to-noise ratio caused instability of the color plot and errors in the estimation. Fig. 9 shows coherence [49], [50] among the array signals. This was calculated by applying cross spectrum analysis be-
29
tween all the combinations of the CAB signals. In the figure, we find that the coherence was low around the expected resonant frequencies, especially between 130 and 140 MHz. In this frequency band, we expected two resonances having different phase-sequence. These resonances have very narrow frequency bandwidth, and large variation of phase with frequency might make the coherence low. When we changed the depth of both the transmitter and the receiver, the difference between the MoM results and the experwas increased at some depths. imental results in the plot of This may be caused by inhomogeneity in subsurface or location error of the antennas in the borehole. What is important is that we could see influence of the phase-sequence resonance approximately between 100 and 150 MHz on DOA estimation even at these depths. Even when there is small inhomogeneity, which is not considered in the MoM model, the prediction with the MoM shown in this paper might be useful to design the optimal CAB. pF as the In the MoM analysis, we adopted terminated impedance, and this impedance is near a typical value as an input impedance of an optical modulator [51]. We found that the resonant frequencies predicted by the MoM are in the MoM slightly shifted, if the terminated impedance model is changed. Also, the shape of the actual driving point, which is shown in [16, Fig. 2], may be difficult to model with the delta gap source exactly. Both the terminated impedance and the shape of the driving point may influence the resonant frequency. However, we should notice that our objective in this paper is not to predict the resonant frequency very accurately, but to investigate influence of the resonance on the DOA estimation. According to Fig. 7, we found that the phase-sequence resonances are actually excited in the experiment, and the resonant frequencies in the experimental data were not far from the MoM predictions. Since we knew that the phase-sequence resonance makes the spurious solutions in the computer simulation in Section III, we should design the CAB optimally so as to avoid the phase-sequence resonance.
V. CONCLUSION Using the MoM model including the influence of the borehole, we investigated the phase-sequence resonance in a circular dipole array antenna in a borehole. In order to combine resonant frequencies and the resonant phase-sequence, we modified the impressed field and artificially excited the phase-sequence resonance in MoM. Using this method, we can theoretically analyze the resonance, even if the antenna is in a borehole. Furthermore, using the values generated from the spatial digital Fourier transform of the CAB signals, we showed that we can confirm the excitation of the phase-sequence resonance. According to the proposed MoM analysis, we concluded the phase-sequence resonance gives some spurious solutions in DOA estimation with the conventional estimator. In order to verify the theoretical results, we have done experiments with the CAB consisting of seven dipole antennas in granite. We found that the phase-seand occur in quence resonance corresponding to the experiment data. Comparison between the resonance analysis and estimation of DOA showed that the phase-sequence resonance makes spurious solutions in direction finding with a circular dipole array in a borehole.
30
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 44, NO. 1, JANUARY 2006
ACKNOWLEDGMENT Thanks are due to Y. Fujita, A. Kusama, K. Kondo, and K. Minamoto for their assistance in doing experiments in Shijonawate, Japan. The authors thank the three anonymous reviewers for the detailed and helpful comments. REFERENCES [1] D. J. Daniels, Ground Penetrating Radar, 2nd ed. London, U.K.: IEE, 2004. [2] H. Nickel, F. Sender, R. Thierbach, and H. Weichart, “Exploring the interior of salt domes from boreholes,” Geophys. Prospect., no. 31, pp. 131–148, 1983. [3] D. L. Wright, R. D. Watts, and E. Bramsoe, “A short-pulse electromagnetic transponder for hole-to-hole use,” IEEE Trans. Geosci. Remote Sens., vol. GE-22, no. 6, pp. 720–725, Nov. 1984. [4] J. A. Bradley and D. L. Wright, “Microprocessor-based data-acquisition system for a borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. GE-25, no. 4, pp. 441–447, Jul. 1987. [5] O. Olsson, L. Falk, O. Forslund, L. Lundmark, and E. Sandberg, “Borehole radar applied to the characterization of hydraulically conductive fracture zones in crystalline rock,” Geophys. Prospect., vol. 40, no. 2, pp. 109–142, 1992. [6] T. Miwa, M. Sato, and H. Niitsuma, “Subsurface fracture measurement with polarimetric borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 2, pp. 828–837, Mar. 1999. [7] T. Miwa, M. Sato, and H. Niitsuma, “Enhancement of reflected waves in single-hole polarimetric borehole radar measurement,” IEEE Trans. Antenna Propagat., vol. 48, no. 9, pp. 1430–1437, Sep. 2000. [8] H. Zhou and M. Sato, “Subsurface cavity imaging by crosshole borehole radar measurements,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 2, pp. 335–341, Feb. 2004. [9] R. J. Lytle and E. F. Laine, “Design of a miniature directional antenna for geophysical probing from boreholes,” IEEE Trans. Geosci. Remote Sens., vol. GE-16, no. 4, pp. 304–307, Oct. 1978. [10] C. W. Moulton, D. L. Wright, S. R. Hutton, D. von G. Smith, and J. D. Abraham, “Basalt-flow imaging using a high-resolution directional borehole radar,” in Proc. 9th Int. Conf. Ground Penetrating Radar, Santa Barbara, CA, Apr. 2002, pp. 13–18. [11] K. W. A. van Dongen, A Directional Borehole Radar System for Subsurface Imaging. Delft, The Netherlands: Delft Univ. Press, 2002. [12] K. W. A. van Dongen, R. van Waard, S. van der Baan, P. M. van den Berg, and J. T. Fokkema, “A directional borehole radar system,” Subsurf.Sens. Technol. Appl., vol. 3, no. 4, pp. 327–346, Oct. 2002. [13] E. Mundary, R. Thierbach, F. Sende, and H. Weichart, “Borehole radar probing in salt deposits,” presented at the 6th Int. Symp. Salt, vol. 1, Toronto, ON, Canada, 1983. [14] M. Sato and T. Tanimoto, “A shielded loop array antenna for a directional borehole radar,” in Proc. 4th Int. Conf. GPR Geological Survey of Finland, Rovaniemi, Finland, Jun. 1992, Special Paper 16, pp. 323–327. [15] S. Ebihara, M. Sato, and H. Niitsuma, “Super-resolution of coherent targets by a directional borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1725–1732, Jul. 2000. [16] S. Ebihara, “Directional borehole radar with dipole antenna array using optical modulators,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 1, pp. 45–58, Jan. 2004. [17] R. F. Harrington, Field Computation by Moment Method. New York: Macmillian, 1968. [18] J. H. Richmond and N. H. Gery, “Mutual impedance of nonplanar-skew sinusoidal dipoles,” IEEE Trans. Antennas Propagat., vol. AP-23, no. 3, pp. 412–414, May 1975. [19] K. Sawaya, “Numerical techniques for analysis of electromagnetic problems,” IEICE Trans. Commun., vol. E83-B, no. 3, Mar. 2000. [20] H. Nakano, S. R. Kerner, and N. G. Alexopoulos, “The moment method solution for printed wire antennas of arbitrary configuration,” IEEE Trans. Antennas Propagat., vol. 36, no. 12, pp. 1667–1674, Dec. 1988. [21] R. Li and H. Nakano, “Numerical analysis of arbitrary shaped probe-excited single-arm printed wire antennas,” IEEE Trans. Antennas Propagat., vol. 46, no. 9, pp. 1307–1317, Sep. 1998. [22] T. J. Cui and W. C. Chew, “Modeling of arbitrary wire antennas above ground,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 1, pp. 357–365, Jan. 2000. [23] M. Sato and R. Thierbach, “Analysis of a borehole radar in cross-hole mode,” IEEE Trans. Geosci. Remote Sens., vol. 29, no. 6, pp. 899–904, Nov. 1991.
[24] S. Ebihara, M. Sato, and H. Niitsuma, “Analysis of a guided wave along a conducting structure in a borehole,” Geophys. Prospect., vol. 46, pp. 489–505, 1998. [25] T. B. Hansen, “The far field of a borehole radar and its reflection at a planar interface,” IEEE Trans. Geosci. Remote Sens., vol. 37, no. 4, pp. 1940–1950, Jul. 1999. [26] F. L. Teixeira and W. C. Chew, “Finite-difference computation of transient electromagnetic waves for cylindrical geometries in complex media,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1530–1543, Jul. 2000. [27] S. Liu and M. Sato, “Electromagnetic logging technique based on borehole radar,” IEEE Trans. Geosci. Remote Sens., vol. 40, no. 9, pp. 2083–2092, Sep. 2002. [28] K. Holliger and T. Bergmann, “Numerical modeling of borehole georadar data,” Geophysics, vol. 67, no. 4, pp. 1249–1257, Jul./Aug. 2002. [29] Y. Chen and M. L. Oristaglio, “A modeling study of borehole radar for oil-field applications,” Geophysics, vol. 67, no. 5, pp. 1486–1494, Sep./Oct. 2002. [30] K. J. Ellefsen and D. L. Wright, “Radiation pattern of a borehole radar antenna,” Geophysics, vol. 70, no. 1, pp. K1–K11, Jan.–Feb. 2005. [31] J. Tronicke and K. Holliger, “Effects of gas- and water-filled borehole on the amplitudes of crosshole georadar data as inferred from experimental evidence,” Geophysics, vol. 69, no. 5, pp. 1255–1260, Sep.–Oct. 2004. [32] R. O. Schmidt, “Multiple emitter location and signal parameters estimation,” IEEE Trans. Antennas Propagat., vol. AP-34, no. 3, pp. 276–280, Mar. 1986. [33] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [34] Y. Mushiake, Antenna and Radio Propagation (in Japanese). Tokyo, Japan: Corona, 1961. [35] R. W. P. King, G. J. Fikioris, and R. B. Mack, Cylindrical Antennas and Arrays. Cambridge, U.K.: Cambridge Univ. Press, 2002. [36] 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. [37] , “The large circular array with one element driven,” IEEE Trans. Antennas Propagat., vol. 38, no. 9, pp. 1462–1472, Sep. 1990. [38] G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys., vol. 68, no. 2, pp. 431–439, Jul. 1990. [39] D. K. Freeman and T. T. Wu, “Variational-principle formulation of the two-term theory for arrays of cylindrical dipoles,” IEEE Trans. Antennas Propagat., vol. 43, no. 4, pp. 340–349, Apr. 1995. [40] G. Fikioris, R. W. P. King, and T. T. Wu, “Novel surface-wave antenna,” Proc. Inst. Elect. Eng., Microw. Antennas Propagat., vol. 143, no. 1, pp. 1–6, Feb. 1996. [41] G. Fikioris, “Experimental study of novel resonant circular arrays,” Proc. Inst. Elect. Eng., Microw. Antennas Propagat., vol. 145, no. 1, pp. 92–98, Feb. 1998. [42] D. Margetis, G. Fikioris, J. M. Myers, and T. T. Wu, “Highly directive current distributions: General theory,” Phys. Rev. E, vol. 58, no. 2, pp. 2531–2547, Aug. 1998. [43] R. W. P. King, M. Owens, and T. T. Wu, “Properties and applications of the large circular resonant dipole array,” IEEE Trans. Antennas. Propagat, vol. 51, no. 1, pp. 103–109, Jan. 2003. [44] G. Fikioris, P. J. Papakanellos, J. D. Koundouros, and A. K. Patsiotis, “Difficulties in MoM analysis of resonant circular arrays of cylindrical dipoles,” Electro. Lett., vol. 41, no. 2, Jan. 2005. [45] W. C. Chew, Waves and Field in Inhomogeneous Media. New York: IEEE Press, 1995. [46] S. Ebihara and W. C. Chew, “Calculation of Sommerfeld integrals for modeling vertical dipole array antenna for borehole radar,” IEICE Trans. Electron., vol. E86-C, no. 10, pp. 2085–2096, Oct. 2003. [47] C. P. Mathews and M. D. Zoltowski, “Eigenstructure techniques for 2-D angle estimation with uniform circular arrays,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2395–2407, Sep. 1994. [48] K. M. Reddy and V. U. Reddy, “Analysis of spatial smoothing with uniform circular arrays,” IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1726–1730, Jun. 1999. [49] S. M. Kay, Modern Spectral Estimation. Upper Saddle River, NJ: Prentice-Hall, 1988. [50] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1992. [51] N. Kuwabara, K. Tajima, R. Kobayashi, and F. Amemiya, “Development and analysis of electric field sensor using LiNbO3 optical modulator,” IEEE Trans. Electromagn. Compat., vol. 34, no. 4, pp. 391–396, Nov. 1992.
EBIHARA AND YAMAMOTO: RESONANCE ANALYSIS OF A CIRCULAR DIPOLE ARRAY ANTENNA
[52] D. E. N. Davies, The Handbook of Antenna Design. London, U.K.: Peter Peregrinus, 1986, ch. 12. [53] S. Ebihara, K. Nagoya, N. Abe, and M. Toida, “Experimental studies for monitoring water level by dipole antenna array radar fixed in subsurface,” Near Surf. Geophys., 2006, to be published.
Satoshi Ebihara (M’97) was born in Chiba, Japan, on December 10, 1968. He received the B.S., M.S., and Dr.Eng. degrees in resources engineering from Tohoku University, Sendai, Japan, in 1993, 1995, and 1997, respectively. From 1995 to 1997, he was a Research Fellow with the Japan Society for the Promotion of Science, Tokyo, Japan. From 1997 to 2003, he was a Research Associate with the Center for Northeast Asian Studies, Tohoku University. He is currently an Assistant Professor with the Department of Electronic Engineering and Computer Science, Faculty of Engineering, Osaka Electro-Communication University, Osaka, Japan. His current research interests include superresolution techniques, array signal processing, and borehole radar measurement. Dr. Ebihara was awarded “Best Paper” of the Second Well-Logging Symposium of Japan, Chiba, Japan, in September 1996 and was awarded the Miyagi Foundation Young Researchers Award of the Miyagi Foundation for Promotion of Industrial Science, Sendai, Japan, in March 2001.
31
Takashi Yamamoto was born in Kyoto, Japan, on September 30, 1981. He graduated from Osaka Electro-Communication University, Osaka, Japan, in 2005. He is currently with Tsuruyayoshinobu, Inc., Kyoto.
