Abstract

The 1-D range profiles are suitable features for target identification and target discrimination because they provide discriminative information on the geometry of the target. To resolve features of the buried target, the contribution from individual scattering centers of the buried target in the range profiles need to be identified. Thus, the study of complex scattering mechanisms from which the range profiles are produced is of great importance. In order to clearly establish the relationship between the range profile characteristics and the complicated electromagnetic (EM) scattering mechanisms, such as reflections and diffractions, a buried cuboid possessing straight edges is chosen as the buried target in this paper. By performing an inverse discrete Fourier transform (IDFT) on the wideband backscattered field data computed with an accurate and fast EM method, the 1-D range profiles of the buried cuboid is successfully simulated. The simulated range profiles provide information about the position and scattering strength of the cuboid’s scattering centers along the range direction. Meanwhile, a predicted distribution of the scattering centers is quantitatively calculated for the buried cuboid based on the ray path computation. Good agreement has been found between simulated and predicted locations of the range profiles. Validation for amplitudes of the range profiles is further provided in the research. Both the peak amplitudes and locations of the range profiles could be understood and analyzed based on the knowledge of the scattering mechanisms. The formation of the 1-D range profiles has been revealed clearly from the full analysis of the scattering mechanisms and contributions. The problem has been solved for both near and far field regions. Finally, the buried depth and the characteristic size of the object are reasonably deduced from the simulated range profiles.

© 2011 OSA

1. Introduction

The model of dielectric objects buried in a half-space has numerous applications in geophysical exploration and subsurface sensing of landmines, target identification and target discrimination [16]. At sufficiently high frequencies, the scattering response of an object can be well approximated as a sum of responses from individual scattering centers, which occurs at geometrical discontinuities on the object and provides a physically relevant, yet concise description of the object. Then, identification and discrimination of a buried target can be achieved by using 1-D range profiles, which provide information about the position and scattering strength of the target’s scattering centers along the range direction [7,8]. To resolve features of the buried target, the contribution from the individual scattering centers of the buried target in the range profiles need to be identified. Thus, the study of complex scattering mechanisms which give rise to range profiles is very important in the area of target identification and discrimination. However, very little investigations have been done for the complicated scattering mechanisms from buried target scattering, and even less attention has been paid for the edge diffractions of the buried target. In the past, the modeling of ultra-wideband short-pulse radar echoes from the buried target has been studied [16]. The wide-band scattering from buried wires [1] and from bodies of revolution [24] buried in a half space has been investigated by means of MOM algorithm. A fast and accurate algorithm for the short-pulse scattering form a buried conducting circular plate is presented in [5]. The short-pulse scattering from a finite set of buried perfectly conducting cylinders is addressed in [6] with an analytical-numerical technique. However the electrical size of the buried target in the mentioned studies is relatively small, so the condition for discontinuities on the buried target to be scattering centers is not satisfied. In view of this, an electrically large buried cuboid possessing straight edges is chosen as the buried target in this paper, where the contribution of the edge diffractions could be observed clearly. The investigation in this paper aims to explore the relationship between the range profile characteristics and the complicated scattering mechanisms of the buried target.

Recently, a fast and accurate algorithm for scattering from electrically large objects buried in layered media has been developed in our previous research [9,10]. The proposed method offers conditions for large-scale computation of wideband backscattered field from electrically large buried target scattering. Hence, by performing an inverse discrete Fourier transform (IDFT) on the wideband backscattered field data, the 1-D range profiles of 3-D electrically large buried target could be achieved and studied in this paper based on the accurate electromagnetic (EM) simulation.

A lot of studies have been done for scattering from target buried in a lossy half-space [16]. The impact caused by the lossy media mainly reflected in the wave attenuation. In order to focus on the formation of the range profiles and the study of the high frequency scattering mechanisms, the ground is chosen as lossless dry sand ground in this research.

It should be noted that the coupling interaction between the buried target and the air-ground interface is exist and inevitable. Consequently, the scattering mechanisms of the buried target are complicated by the multiple reverberations between buried target and the air-ground interface and it causes great difficulty for the analysis of the radar returns. In our previous research [8], the range profiles of a target above a rough surface have been studied and the study of coupling between target and rough surface could provide great help for the coupling analysis of the buried target in a half space.

With the accurate and fast algorithm proposed for buried target scattering in our previous research [9,10], the simulated results of the range profiles (locations and amplitudes) are reliable in our study. To establish the relationship between the range profile characteristics and the complicated scattering mechanisms, the ray path prediction is an effective and useful tool for the analysis of scattering features of the buried target. Based on the ray path computation, explicit formulas could be derived for the prediction of the range profile peak locations in the coordinate system. Then a predicted distribution of the scattering centers along the range direction is quantitatively calculated for the buried target. Good agreement has been found between the simulated and the predicted locations of the range profiles. Validation for the amplitudes of the range profiles is further provided in the latter. Both the peak amplitudes and locations of the range profiles could be understood and analyzed based on the knowledge of the scattering mechanisms. The formation of the 1-D range profiles has been revealed clearly from the full analysis of the scattering mechanisms and contributions. In addition to the far-field results, the near-field results could also be understood and explained by the scattering mechanisms. Finally, the buried depth and the characteristic size of the object could be reasonably deduced from the simulated range profiles.

Since antennas for the buried target detection usually work in near earth condition, the EM wave is non-uniformly incident on the buried object in the near field condition, which makes the analysis of the EM scattering mechanisms more complicated than the case that the far field condition is satisfied. From the angle of simplicity, firstly the plane wave incidence is chosen and the far region range profiles (by Fourier transforming the wide band field data received in the far field) of the buried target are studied. Afterwards, the near region range profiles are investigated under the excitation of the dipole located in near field. To summarize, the investigation of the range profiles will be carried out according to the different location set of the transmitters and receivers.

The remainder of the paper is organized as follows. In Section 2 we briefly describe the numerical model of a buried 3-D dielectric object and the problem in this paper. In Section 3, the 1-D range profiles of a buried 3-D dielectric cuboid are presented and their relationship with complicated scattering mechanisms is fully analyzed. Finally, conclusions are addressed in Section 4.

2. Problem statement

Figure 1 shows the geometry of our problem, a 3-D dielectric object is buried in the lower region of a half-space characterized by relative permittivities ε1 and ε2. Suppose the buried object is a dielectric cuboid with complex permittivity εr(r) and located parallel to the interface with the size of Lx×Ly×Lz. The cuboid top is separated from the interface by h. The transmitters are set on the survey line parallel to x axis. Then the wideband scattered response from the buried cuboid is collected by the receivers located on the survey line at a height of Habove the interface. The survey line of the transmitters or the receivers is set in the far or near field. The coordinate of the receiver could be described as (L,0,H).

 

Fig. 1 Coordinate system of a 3-D dielectric object buried in a half-space.

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During the following numerical simulations, the upper region is assumed to be free space ε2=1 and the lower region is assumed to be dry sand ε1=4. For analysis convenience, the first example is chosen as a dielectric cuboid with size of 3m×2m×0.05m. Then for further validation and real application, a cuboid of size 3m×0.5m×0.8mwith increased height is considered. The frequency response of the scattering system is simulated by stepped frequency waveform (SFW). The backscattered field is sampled from 100MHzto 600MHz. Thus, the bandwidth is B=500MHzand the high-resolution is ΔR=c1/2B=0.15m. A frequency step Δf=5MHz is considered to obtain the sufficient unambiguous range Ru=c1/2Δf=15m.

By performing an IDFT on the wideband backscattered field data, the 1-D range profiles of the buried 3-D dielectric object are achieved. The accomplishment of the whole EM simulation for multi-frequency and multi-angle case desires an efficient EM algorithm for electromagnetic scattering from the electrically large object in a half-space. In this paper, the improved BCGS-FFT method proposed in our previous research [9,10] is used to save the CPU time and reduce the computational complexity.

Based on the ray path computation, explicit formulas could be derived for the prediction of the range profile peak locations. Then a reference for the simulated locations is provided to help the identification of the contribution from individual scattering centers of the buried target. To illustrate this, we take one certain scattering center of the buried target (contributed by the edge point A') as an example. The ray path of the radar signal from the transmitter to the receiver contributed by this scattering center could be marked as abcd in Fig. 1. According to the Snell’s law, θ and θ' are given by ε2sinθ=ε1sinθ. The time delay of the radar signal between the transmitter and the receiver is calculated as

τ=(b+c)/c1+(a+d)/c2
where c1=c0/ε1,c2=c0/ε2 are the wave velocity in the lower region and the upper region, respectively; c0is the wave speed in free space. Apparently, the range profile location determined by the time delay of the radar signal is related to the specific travel distance and travel speed in the upper and lower space.

SupposeRT andRR is the distance between the transmitter/receiver and the reference point O, respectively. Then to calculate the path difference of path abcd to the reference pointO, the travel time that the wave needs to travel along RT and RR is calculated firstly,

τo=(RT+RR)/c2

The path difference to the reference point O in the upper region could be achieved as,

ΔR2=((ττo)c2)/2

The corresponding path difference in the lower region ΔR1is related toΔR2 by the following formula,

ΔR1/ΔR2=c1/c2

So we get,

ΔR1=ΔR2/(ε1/ε2)=((ε1/ε2)(b+c)+a+dRTRR)/(2ε1/ε2)

The location of the range profile contributed by any other scattering centers of the buried target could be quantitatively calculated based on the ray path computation. Then a predicted distribution of the individual scattering centers along the range direction is provided for the buried target. A detailed comparison between the numerical simulation and the ray path prediction will be made in the next section to help the analysis of the complicated scattering mechanisms.

Note that the contribution of the planar surface to the radar return only exists when the incident direction is normal to the air-ground interface. In order to focus on the independent scattering of the target, the direct scattering contribution from the ground surface is subtracted in the radar echoes. In Section 3.1 the transmitters and the receivers are set in the far field region while in Section 3.2 the near field condition is satisfied and considered.

3. Numerical results

3.1 Far region range profiles under plane wave incidence

In this case, both the transmitter and the receiver are set at the same location in the far field region for the monostatic case as shown in Fig. 1. A plane wave with parallel polarization and normalized electric field is incident from the upper space. As shown in Fig. 1, the backscattered fields are collected at N=99 receivers on the survey line at a height of H=200m, where thex coordinate ranges from 200m to 200m.

By performing an IDFT on the wideband backscattered field at multi receivers along the survey line, the lateral distribution of 1-D range profiles along xaxis is obtained, as shown in Fig. 2 . The reference point O in Fig. 1 corresponds to a range location of zero in Fig. 2. The lateral distribution is symmetrical in the cross range along x axis due to the symmetry of the scattering model.

 

Fig. 2 The lateral distribution of far region range profiles along x axis.

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It is illustrated that when the incident direction is normal to the air-ground interface, the peak amplitude of the range profile is the largest. That’s because the strong specular scattering from the top surface of the cuboid AA'BB' leads to the strongest peak amplitude at the incident angleθ=ϕ=0. Specifically, the range profiles at θ=ϕ=0 are separated from the lateral distribution of 1-D range profile and the peaks are marked as 1 and 3 in Fig. 3 . The simulated locations of the peaks in Fig. 3 are listed in Table 1 and will be validated by the predictions based on the ray path computation. According to the ray theory, peak 1 is caused by the echo reflected directly from the cuboid top surface and peak 3 is caused due to a reverberation (viz. the 1st-order interaction) between the cuboid top and the air-ground interface. The path difference of peak 1 to the reference point O equals the buried depth h=1.5m. And path difference of peak 3 to the reference point O is twice the depth. In Table 1, the simulated results and the predicted locations agree well with each other.

 

Fig. 3 Range profiles of the buried object along θ=ϕ=0.

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Tables Icon

Table 1. Simulated and Expected Range Locations of a Buried Object (Unit: m)

Furthermore, the peak amplitudes of the range profiles could be evaluated according to the Fresnel theory. Suppose that the buried object is one layer of the stratified structure, the simplified four planarly layers with relative permittivities ε1=4,ε2=8,ε3=4,ε4=1 are illustrated in Fig. 4 .

 

Fig. 4 The equivalent multilayered media.

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Based on the Fresnel theory, the expected amplitude ratio RP of peak 3 to peak 1 could be estimated as follows,

RP=|Г43TMR˜32TMR34TMR˜32TMГ34TMej2k3hГ43TMR˜32TMГ34TMej2k3h|    =|R34TMR32TM+R21TMej2k2Lz1+R32TMR21TMej2k2Lz|
where k2,k3 are the wave numbers in layer 2 and 3. R,Г andR˜are the Fresnel reflection coefficients, Fresnel transmission coefficients and generalized reflection coefficients separately [11]. Considering the center frequency 300MHz, the corresponding amplitude ratio is calculated as 0.0883, which is very close to the simulated results 0.0984. The above analysis about the locations and amplitudes provides validation for the simulated range profiles. Apparently, the buried depth is related to the location of peak 1 in Fig. 3 when the target is incident vertically and could be deduced as h=1.575m.

In Fig. 5 , the incident direction is set asθ=π/4,ϕ=0 and the range profiles along the direction obliquely to the air-ground are demonstrated and indicated by three peaks marked as 1, 2 and 4.

 

Fig. 5 Range profiles of the buried object along θ=π/4,ϕ=0.

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Figure 6 shows the ray paths of the buried target when the oblique incident direction is considered. The scattering mechanisms of the range profiles strongly depend on the aspect angle between the radar and target. Instead of the specular reflections from the top surface in Fig. 3, the contribution to the range profiles in Fig. 5 corresponds to the diffractions of the two edges AA' and BB', which lead to path 1 abba and path 2 abbaas illustrated in Fig. 6. Due to the 1st-order interaction between the edges and the air-ground interface, the 1st-order ray paths are formulated, viz. path 3 abccba and path 4 abccba .

 

Fig. 6 Ray paths of the buried object.

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The range locations of the range profiles caused by path 1 and path 3 could be estimated by calculating the paths difference to the reference pointO similar to formula (5):

ΔR(path1)=(a+ε1bR)/ε1
ΔR(path3)=(a+ε1(b+c)R)/ε1
Where RT=RR=R. The path length of a,b,c could be calculated by the geometrical relations and Snell's law:

a=R+(Lx/2htanθ)sinθb=h/cosθ,c=h,sinθ=ε1sinθ

The same prediction could be done for path 2 and path 4.

It is found in Table 1 that the simulated locations of peak 1, 2, 4 in Fig. 5 match with the predicated range locations well. However the expected peak by path 3 at the position of 2.373m is hardly observed in the simulated results. This may be caused because the contribution to the anticipated peak by path 3 is too small to be reflected. Furthermore, a huge peak (peak 2 at the location of 1.933m) close to the location of the expected peak 3 makes it even difficult for the small peak to be distinguished.

To further validate the simulated results in Fig. 5, a detailed study of the range profile amplitudes for peak 1 and peak 2 is accomplished in the following. As shown in Fig. 7 , the 1-D range profiles of the same dielectric object in the spaceε1=4 are simulated for comparison. The incident angle is chosen the same as the refraction angle in the half-space problem, viz, θ=sin1(sin(π/4)/2),ϕ=0 . Compared with marked peaks 1 and 2 in Fig. 5, the range profiles marked as 1 and 2 in Fig. 7 are also caused by the diffractions from the edges of AA' and BB'under the same incident angle and in the lower space. The amplitude ratios of peak 2 to peak 1 are supposed to be the same in the two cases. In fact, the amplitude ratio of peak 2 to peak 1 in Fig. 7 is calculated as 0.938, which provides a reference for the amplitude ratio of peak 2 to peak 1calculated in Fig. 5 as 0.946.

 

Fig. 7 Range profiles of the same object in free space along θ=sin1(sin(π/4)/2),ϕ=0.

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According to the distribution of the individual scattering centers (peak 1 and peak 3) in Fig. 5 and the given geometrical relations in Fig. 6 with incident direction θ=π/4, the length of the buried cuboid in xdimension could be reconstructed as L'x=2.9698m, which matches well with the real size Lx=3m.

3.2 Near region range profiles under the excitation of the dipole in near field

The electric dipole α^t(IΔl)=e^x as the excitation incidence is set in the near field region as shown in Fig. 1. Both the transmitter and the receiver are set at the same location in the near filed for the monostatic case. The other parameters keep same as the previous case. Using the half-space dyadic Green’s functions [12], we easily obtain the incident electric field

E1inc(r,r)=η2k2G¯ee12(r,r)e^x

Here, k2,η2are the wavenumber and wave impedance in the upper space; G¯ee12is the spatial-domain electric field dyadic Green’s function when the observation point in lower space and the source point is in upper space.

Figure 8 , Fig. 9 and Fig. 10 show the 1-D range profiles indicated by marked peaks when the dipole locates at (0m,0m,2m), (4m,0m,2m)and (0m,4m,2m)respectively.

 

Fig. 8 Range profiles of the buried object when the dipole locates at (0m,0m,2m).

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Fig. 9 Range profiles of the buried object when the dipole locates at (4m,0m,2m).

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Fig. 10 Range profiles of the buried object when the dipole locates at (0m,4m,2m).

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Figure 11 illustrates the ray paths of the buried target when the dipole is located at a certain point on the survey line in the near field.

 

Fig. 11 Ray paths of the buried object.

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When the condition thatLLx/2satisfies, the ray paths contributed by the direct reflection of the cuboid top surface exist, viz. path abba and path abbbba. The marked peaks 1 and 3 in Fig. 8 correspond to these two paths respectively.

For the simulated results in Fig. 9, the condition thatLLx/2doesn’t satisfy, so the range profiles are expected to correspond to the four paths contributed by the edge diffractions, viz. path 1 abba and path 2 abba, path 3 abccba and path 4 abccba . As shown in Table 2 , it is observed that the marked peaks 1, 2 and 3 in Fig. 9 match well with the predicted locations of the ray paths. However, the expected peak by path 4 is too small to be distinguished.

Tables Icon

Table 2. Simulated and Expected Range Locations of Fig. 8, Fig. 9 and Fig. 10 (Unit: m)

For the simulated results in Fig. 10, the condition thatLLy/2doesn’t satisfy either, so the four ray paths could be derived similarly. All the expected peaks are observed at the corresponding range locations marked as 1, 2, 3 and 4 in Fig. 10.

Specifically, the range locations of the range profiles in Fig. 9 and Fig. 10 caused by the path 1 and path 3 could be estimated according to the following explicit formulas,

ΔR(path1)=(a+ε1bR)/ε1
ΔR(path3)=(a+ε1(b+c)R)/ε1

And the path length of a,b,c is calculated based on the geometrical relations and Snell’s law, where θ1 is determined by a numerical root searching method such as Muller’s method,

asinθ1+bsinθ1=Lx/2+Lb=h/cosθ1,a=H/cosθ1,c=h,sinθ=ε1sinθ

According to the location of peak 1 in Fig. 8, the buried depth is reconstructed as h=1.575m. Based on the distribution of the individual scattering centers (peak 1 and peak 2) in Fig. 9 and the geometrical relations in Fig. 11 with given incident direction, it is deduced thatOA(A')=1 .631mand OB(B')=1 .491m. Then the length of the buried cuboid in x dimension could be reconstructed as L'x=OA+OB=3 .122m, which matches well with the real size Lx=3m. Similarly, the length of the buried cuboid in y dimension could be reconstructed as L'y=2 .134m, which matches well with the real size Ly=2m.

Suppose the survey line is parallel to xaxis which is the diagonal of xaxis and yaxis. And the observation range is from 6.4m to 6.4malong xaxis. Then the lateral distribution of 1-D range profiles along xaxis is demonstrated in Fig. 12 .

 

Fig. 12 The lateral distribution of near region range profiles along x'axis.

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Figure 13 shows the 1-D range profiles when the dipole locates at (4.5255m,4.5255m,2m) on the diagonal survey line.

 

Fig. 13 Range profiles of the buried object when the dipole locates at (4.5255m,4.5255m,2m).

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Instead of the edge diffractions in Fig. 9 and Fig. 10, the individual scattering centers of the buried object correspond to the diffractions of four cornersA,A',B,B'in Fig. 13. Take the diffraction point B as an example, Fig. 14 shows the ray paths contributed by B. E' is the intersection point between the ray path and the air-ground interface. The angles satisfy the relationshipsinθ=ε1sinθ. Axis u is determined by the plane of incidence. The direct ray path is abba and the 1st order path is abccba. So in all, eight paths are contributed by diffractions of the four corners and the corresponding reverberations. The range locations caused by the paths could be estimated and illustrated in Table 3 based on the ray path prediction. It is observed that the expected peak 5 and peak 7 contributed by the 1st order interaction of the diffraction corners B'andA'seem too small to be discovered in the simulated results. While the other simulated range profile locations match well with the expected results.

 

Fig. 14 Ray paths of the buried object.

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Tables Icon

Table 3. Simulated and Expected Range Locations of Fig. 13 (Unit: m)

For real application, the height of the cubic is increased in the following case and a 3-D cuboid with size of 3m×0.5m×0.8mis considered. The other parameters are chosen the same as εr=8 and h=1.5m. Figure 15 shows the 1-D range profiles when the dipole locates at (4m,0m,2m).

 

Fig. 15 Range profiles of the buried object when the dipole locates at (4m,0m,2m).

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The range locations caused by the scattering centers of the buried object could be estimated according to Fig. 11 based on the ray path prediction and the expected results match well with the simulated locations in Table 4 . Only the expected peak 5 is too close to peak 3 to be distinguished.

Tables Icon

Table 4. Simulated and expected range locations of Fig. 15 (Unit: m)

Based on the distribution of the individual scattering centers in Fig. 14 and the geometrical relations in Fig. 11 with given incident direction, it is deduced thatOA(A')=1 .418mand OB(B')=1 .418m. Then the length of the buried cuboid in x dimension could be reconstructed as L'x=OA+OB=2 .899m, which matches well with the real size Lx=3m. Similarly, the length of the buried cuboid in y dimension could be reconstructed successfully.

In all, the buried depth of the object can be estimated from the locations of the range profiles when the incident direction is normal to the air-ground interface. Then, given the oblique incident direction, the characteristic size of the buried object can be deduced from the simulated range profiles.

4. Conclusion

The detail contributions from individual scattering centers of the buried target in the range profiles have been studied and identified in this paper. The 1-D range profiles of the buried target are simulated based on the accurate and fast EM numerical method. A cuboid possessing straight edges is chosen as the buried target to focus on the important scattering mechanisms, such as reflections and diffractions. The ray path prediction is employed to analyze the scattering features of the buried target. Good agreement has been found between the simulated and the predicted locations of the range profiles. The relationship between the range profile characteristics and the complicated scattering mechanisms is clearly established. Finally, the buried depth and the characteristic size of the object are reconstructed from the simulated results, which is helpful for target identification and target discrimination.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61001059), the China Postdoctoral Science Foundation, and the Fundamental Research Funds for the Central Universities. The authors would like to thank the reviewers for their helpful and constructive suggestions.

References and links

1. S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

2. S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996). [CrossRef]  

3. S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997). [CrossRef]  

4. N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999). [CrossRef]  

5. V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003). [CrossRef]  

6. F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007). [CrossRef]  

7. K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001). [CrossRef]  

8. S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009). [CrossRef]  

9. Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006). [CrossRef]  

10. L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010). [CrossRef]  

11. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

12. T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003). [CrossRef]  

References

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  1. S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).
  2. S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
    [CrossRef]
  3. S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
    [CrossRef]
  4. N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
    [CrossRef]
  5. V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
    [CrossRef]
  6. F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
    [CrossRef]
  7. K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
    [CrossRef]
  8. S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
    [CrossRef]
  9. Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
    [CrossRef]
  10. L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
    [CrossRef]
  11. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  12. T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
    [CrossRef]

2010 (1)

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

2009 (1)

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

2007 (1)

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

2006 (1)

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

2003 (2)

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

2001 (1)

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

1999 (1)

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

1997 (1)

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

1996 (1)

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

1995 (1)

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Aydiner, A. A.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Boix, R. R.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Carin, L.

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Chen, H.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Chew, W. C.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Cui, T. J.

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Deng, F.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Frezza, F.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Geng, N.

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

He, S.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

He, S. Y.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Hu, W.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Hu, W. D.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Kim, H. T.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Kim, K. T.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Le, F. H.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Losada, V.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Martinelli, P.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Medina, F.

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

Pajewski, L.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Ressler, M. A.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Schettini, G.

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

Seo, D. K.

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

Sturgess, K.

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

Vitebskiy, S.

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

Xiao, B. X.

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

Ye, X. B.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Yu, W.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Yu, W. X.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Zhang, Y. H.

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

Zhu, G.

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

Zhu, G. Q.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

Zhuang, L.

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

IEE Proc., Radar Sonar Navig. (1)

K. T. Kim, D. K. Seo, and H. T. Kim, “Radar target identification using one-dimensional scattering cernters,” IEE Proc., Radar Sonar Navig. 148(5), 285–296 (2001).
[CrossRef]

IEEE Geosci. Remote Sens. Lett. (1)

F. Frezza, P. Martinelli, L. Pajewski, and G. Schettini, “Short-pulse electromagnetic scattering by buried perfectly conducting cylinders,” IEEE Geosci. Remote Sens. Lett. 4(4), 611–615 (2007).
[CrossRef]

IEEE Trans. Antenn. Propag. (3)

S. He, F. Deng, H. Chen, W. Yu, W. Hu, and G. Zhu, “Range profile analysis of the 2-D target above a rough surface based on the electromagnetic numerical simulation,” IEEE Trans. Antenn. Propag. 57(10), 3258–3263 (2009).
[CrossRef]

S. Vitebskiy, K. Sturgess, and L. Carin, “Short-pulse plane-wave scattering from buried perfectly conducting bodies of revolution,” IEEE Trans. Antenn. Propag. 44(2), 143–151 (1996).
[CrossRef]

N. Geng and L. Carin, “Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium,” IEEE Trans. Antenn. Propag. 47(4), 610–619 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

S. Vitebskiy and L. Carin, “Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space,” IEEE Trans. Antennas Propag. 43(11), 1303–1312 (1995).

IEEE Trans. Geosci. Rem. Sens. (5)

T. J. Cui, W. C. Chew, A. A. Aydiner, and Y. H. Zhang, “Fast-forward solvers for the low-frequency detection of buried dielectric objects,” IEEE Trans. Geosci. Rem. Sens. 41(9), 2026–2036 (2003).
[CrossRef]

V. Losada, R. R. Boix, and F. Medina, “Fast and accurate algorithm for the short-pulse electromagnetic scattering from conducting circular plates buried inside a lossy dispersive half-space,” IEEE Trans. Geosci. Rem. Sens. 41(5), 988–997 (2003).
[CrossRef]

S. Vitebskiy, L. Carin, M. A. Ressler, and F. H. Le, “Ultra-wideband, short-pulse ground-penetrating radar: simulation and measurement,” IEEE Trans. Geosci. Rem. Sens. 35(3), 762–772 (1997).
[CrossRef]

Y. H. Zhang, B. X. Xiao, and G. Q. Zhu, “An improved weak-form BCGS-FFT combined with DCIM for analyzing electromagnetic scattering by 3-D objects in planarly layered media,” IEEE Trans. Geosci. Rem. Sens. 44(12), 3540–3546 (2006).
[CrossRef]

L. Zhuang, S. Y. He, X. B. Ye, W. D. Hu, W. X. Yu, and G. Q. Zhu, “The BCGS-FFT method combined with an improved discrete complex image method for EM scattering from electrically large objects in multilayered media,” IEEE Trans. Geosci. Rem. Sens. 48(3), 1180–1185 (2010).
[CrossRef]

Other (1)

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).

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Figures (15)

Fig. 1
Fig. 1

Coordinate system of a 3-D dielectric object buried in a half-space.

Fig. 2
Fig. 2

The lateral distribution of far region range profiles along x axis.

Fig. 3
Fig. 3

Range profiles of the buried object along θ = ϕ = 0.

Fig. 4
Fig. 4

The equivalent multilayered media.

Fig. 5
Fig. 5

Range profiles of the buried object along θ = π /4 , ϕ = 0.

Fig. 6
Fig. 6

Ray paths of the buried object.

Fig. 7
Fig. 7

Range profiles of the same object in free space along θ = sin 1 ( sin ( π / 4 ) / 2 ) , ϕ = 0.

Fig. 8
Fig. 8

Range profiles of the buried object when the dipole locates at ( 0 m , 0 m , 2 m ) .

Fig. 9
Fig. 9

Range profiles of the buried object when the dipole locates at ( 4 m , 0 m , 2 m ) .

Fig. 10
Fig. 10

Range profiles of the buried object when the dipole locates at ( 0 m , 4 m , 2 m ) .

Fig. 11
Fig. 11

Ray paths of the buried object.

Fig. 12
Fig. 12

The lateral distribution of near region range profiles along x ' axis.

Fig. 13
Fig. 13

Range profiles of the buried object when the dipole locates at ( 4.5255 m , 4.5255 m , 2 m ) .

Fig. 14
Fig. 14

Ray paths of the buried object.

Fig. 15
Fig. 15

Range profiles of the buried object when the dipole locates at ( 4 m , 0 m , 2 m ) .

Tables (4)

Tables Icon

Table 1 Simulated and Expected Range Locations of a Buried Object (Unit: m)

Tables Icon

Table 2 Simulated and Expected Range Locations of Fig. 8, Fig. 9 and Fig. 10 (Unit: m)

Tables Icon

Table 3 Simulated and Expected Range Locations of Fig. 13 (Unit: m)

Tables Icon

Table 4 Simulated and expected range locations of Fig. 15 (Unit: m)

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

τ = ( b + c ) / c 1 + ( a + d ) / c 2
τ o = ( R T + R R ) / c 2
Δ R 2 = ( ( τ τ o ) c 2 ) / 2
Δ R 1 / Δ R 2 = c 1 / c 2
Δ R 1 = Δ R 2 / ( ε 1 / ε 2 ) = ( ( ε 1 / ε 2 ) ( b + c ) + a + d R T R R ) / ( 2 ε 1 / ε 2 )
R P = | Г 43 T M R ˜ 32 T M R 34 T M R ˜ 32 T M Г 34 T M e j 2 k 3 h Г 43 T M R ˜ 32 T M Г 34 T M e j 2 k 3 h |      = | R 34 T M R 32 T M + R 21 T M e j 2 k 2 L z 1 + R 32 T M R 21 T M e j 2 k 2 L z |
Δ R ( path 1 ) = ( a + ε 1 b R ) / ε 1
Δ R ( path 3 ) = ( a + ε 1 ( b + c ) R ) / ε 1
a = R + ( L x / 2 h tan θ ) sin θ b = h / cos θ , c = h , sin θ = ε 1 sin θ
E 1 i n c ( r , r ) = η 2 k 2 G ¯ e e 12 ( r , r ) e ^ x
Δ R ( path 1 ) = ( a + ε 1 b R ) / ε 1
Δ R ( path 3 ) = ( a + ε 1 ( b + c ) R ) / ε 1
a sin θ 1 + b sin θ 1 = L x / 2 + L b = h / cos θ 1 , a = H / cos θ 1 , c = h , sin θ = ε 1 sin θ

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