JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA, VOL. 5, NO. 3, SEPTEMBER 2007 252

Abstract⎯A fast SAR imaging algorithm for near– field subsurface forward–looking ground penetrating radar (FLGPR) is presented. By using nonstationary convolution filter, the refocused image spectrum can be reconstructed directly from the backscattered signal spectrum of target area. The experimental results show the proposed method can fast achieve image refocusing. Also it has higher computational efficiency than the phase–shift migration approach and the delay–and–sum (DAS) approach.

Index Terms⎯Delay-and-sum (DAS) beamforming,

forward–looking ground penetrating (FLGPR), nonstationary filter, phase–shift migration.

1. Introduction The synthetic aperture radar (SAR) imaging technology

is a very significant topic for forward-looking ground penetrating radar (FLGPR) investigation. SAR is useful for improving the spatial resolution, increasing the signal–to–noise ratio of the survey, and improving the landmine detection capability. Due to these merits, SAR has become an active topic in FLGPR investigation systems in recent years[1]-[5]. In FLGPR, refraction occurs when the transmitted pulse encounters the ground surface and the non–homogeneity of the medium is reflected in the variation of the velocity. That is, the velocity is time–variant. Here, the classical SAR algorithms with constant velocity are not suitable. Current several SAR imaging techniques for FLGPR have been discussed in the literature, such as space–variant matched filter algorithm[1], delay-and-sum (DAS) approach[2], phase–shift migration approach[2],[3], spectral estimation approach[2],[4], and modified wavefront reconstruction algorithm[5]. Among these techniques, space–variant matched filter and DAS need to determine refraction points on the ground and the process must be

Manuscript received May 22, 2007; revised June 14, 2007. This work was

supported by the National Nature Science Foundation of China under Grant No. 60472014.

Y. Fan and Z.-O. Zhou are with School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, China (e-mail: fanyong704@163.com and zzo@uestc.edu.cn).

J.-L. Xu is with Department of Electronic and Information Engineering, Chengdu University, Chengdu, 610106, China (e-mail: xujl@163.com ).

J.-S. Lin is with Department of Electronic and Information Engineering, Putian University, Putian, 351100, Fujian, China (e-mail: linjs@163.com ).

repeated for every focal point. Modified wavefront reconstruction needs to modify the wavefront for every focal point. Therefore they are time consuming. Furthermore, the computational complexity will increase drastically when multilayer ground structure is involved. Phase–shift migration can reconstruct image but which is recursive rather than direct.

In this paper, a fast near–field subsurface SAR imaging algorithm is presented for FLGPR image formation. This algorithm is based on nonstationary filter. The fundamental difference between stationary and nonstationary linear filter is that the impulse response of the latter must be allowed to vary arbitrarily with time. The complete description of a general nonstationary filter requires that its impulse response is known for any and all times. The nonstationary filter theory was first introduced by Margrave[6], and it has many possible applications in seismic data[6], including: the one–way propagation of waves through complex media, time migration, normal moveout removal, time variant filtering, and forward and inverse Q filtering. Here the SAR imaging algorithm directly reconstructs the spectrum of the refocused image from the spectrum of backscattered signal from the target area. It avoids the time–consuming processing needed by space–variant matched filter, DAS, and modified wavefront reconstruction. Experimental results are used to demonstrate the excellent performance of the new imaging algorithm as compared with the DAS and the phase–shift migration.

2. SAR Imaging Algorithm For FLGPR, the backscattered echoes from the surface or

subsurface buried target are shaped as hyperbolas in image plane. When the electromagnetic wave propagates in two different media, the propagation of the resultant spherical wave in the dielectric half–space (air) can be conceptually decomposed into an infinite number of plane waves, each with a different incidence angle at the air–ground interface. Here, we use the stratified earth assumption which is used in much of seismic data processing currently [7]. That is, there is no lateral velocity variation. So the plane wave velocity v is a function of r and incidence angle, where r is the slant–range of the focal point to x–axis.

Here, we discuss only 2–D imaging in image plane, whose incidence angle equals the target incidence angle. Then the wave velocity v is a function of r. Considering any

Fast SAR Imaging Algorithm for FLGPR Yong Fan, Zheng-Ou Zhou, Jia-Li Xu, and Jin-Shan Lin

FAN et al.: Fast SAR Imaging Algorithm for FLGPR 253

scalar field component u(x, r, t) resulting from an exploding source, the field must satisfy the scalar wave equation[3],[7]:

2 2 2

2 2 2 2

1 ( , , )( , , ) u x r tu x r tx r v t

⎡ ⎤∂ ∂ ∂+ =⎢ ⎥∂ ∂ ∂⎣ ⎦

(1)

where x is the cross–track dimension, t is the real time of the observer.

Taking a 2–D Fourier transform of (1) over x and t, because v varies with r, the 2–D Fourier transform of (1) is a second–order ordinary differential equation with a variable coefficient. Using the coefficient approximate solution[7], the upgoing wave equation can be expressed by the following under the stratified earth assumption:

2 2

0( , , ) ( , 0, ) exp[ ( ) ]

r

x x xU k r U k r j v k drω ω ω= = −∫ (2)

where U(kx, r=0, ω) is the Fourier transform of the recorded data over x and t; kx is the horizontal wavenumber and ω is the frequency.

Taking the inverse Fourier transform of (2) over ω, setting t=0, and defining a new variable τ viaτ=r/v (migration time), we have

1( , , 0) ( , 0, ) ( , , )exp( )2x x xU k t U k r k j dτ ω α τ ω ωτ ωπ

= = =∫ (3)

where 2 2

0( , , ) exp[ ( ( ) )]x xk j vk d

τα τ ω ω τ ωτ= − −∫ (4)

is the transfer function of the nonstationary migration filter. According to nonstationary filter theory[6], (3) is a

nonstationary combination filter expressed in the mixed domain and it is a nonstationary convolution filter expressed in the Fourier domain by the forward Fourier transform of the τ dependence of α (kx, τ, ω). Taking the forward Fourier transform of (3) over τ , and reversing the order of integration, the reconstructed spectrum, U(kx, η), of U(kx, τ, t=0) is computed as follows:

1( , ) ( , 0, ) ( , , )2x x xU k U k r A k dη ω η ω ω ωπ

= = −∫ (5)

where ( , , ) ( , , ) exp( )x xA k k j dη ω α τ ω ητ τ= −∫ (6)

If U(kx, r=0, ω) represents the input signal spectrum, and A(kx, η, ω) is an nonstationary filter, then the reconstructed spectrum U(kx, η) is the product of the nonstationary filter output and a coefficient 1/2π. So once the velocity is evaluated[8], the transfer function of the nonstationary migration filter can be reconstructed directly by (4) and (6). The refocused image spectrum can be reconstructed directly from (5). We see that the reconstructed spectrum can be achieved directly by one migration filter process.

Taking the inverse Fourier transform of (5) over η and kx, the refocused image can be expressed as follows:

2

1( , , 0) ( , ) exp[ ( )]4 x x xu x t U k j k x dk dτ η ητ ηπ

= = −∫∫ (7)

In summary, the SAR imaging algorithm has four steps. (1) Take Fourier transform of u(x, r = 0, t) with respect to

x, t. (2) Reconstruct the nonstationary filter by (4) and (6). (3) Reconstruct spectrum by (5). (4) Take inverse Fourier transform to obtain u(x, τ, t=0)

by (7).

3. Experimental Analysis The geometry of the measurement is shown in Fig. 1,

where x, y and z denote the cross-track, along-track, and depth coordinate, respectively. The experiments were performed in a sand container, which is 3m×5m×2m. The bistatic antenna with incidence angle (θ) of 42° was oriented at height (H) of 0.66 m over the ground plane in a forward- looking manner. A plastic anti-tank mine (M) with radius 0.14 m was located at (0.91, 0.63, 0.13) m. The impulse radar system transmitted wide-bandwidth pulse signal with a duration of 2 ns and a pulse repetition frequency (PRF) of 100 kHz. Experimental data were collected every 0.093 m in the cross-track dimension.

M z

x

y

H

N

Fig. 1. Measurement and imaging geometry.

After pre-processing by time gating and subtracting from each A-scan an averaged value of an ensemble of A-scans[9], Fig. 2 shows the B-scan radar image of the plastic anti-tank mine. Fig. 3(a) shows the image refocused in the x-y plane by the proposed SAR imaging algorithm. Note that the SNR increases significantly after the SAR imaging. Here we compare the proposed SAR imaging algorithm with the phase-shift migration approach and the DAS approach. Fig. 3(b) and Fig. 3(c) show images refocused by the phase-shift migration approach and the DAS approach, respectively. All imaging algorithms were coded using Matlab and performed on a Pentium4–1.4 GHz PC with 256 M RAM. Table 1 summarizes the processing time, SNR, and 3D coordinate estimation. It shows that they can accurately locate the buried target. But the proposed SAR imaging algorithm has very high computational efficiency.

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T im e (n s) Fig. 2. B-scan radar image after pre-processing.

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Fig. 3. Imaging results in x–y plane: (a) The proposed SAR imaging result, (b) Phase-shift migration imaging result, (c) DAS imaging result.

Table 1 Numerical comparisons of the proposed SAR imaging,

Phase-shift migration and DAS

SAR imaging techniques Processing time (s)

SNR (dB)

Coordinate estimation (x,y,z) (m)

Proposed SAR imaging 32 15.37 (0.93, 0.62, 0.11) Phase-shift migration 246 14.07 (0.94, 0.62, 0.11)

DAS 1178 15.10 (0.94, 0.62, 0.11)

4. Conclusions An efficient SAR imaging algorithm for FLGPR based

on nonstationary filter is proposed. It shows that the proposed SAR imaging algorithm can be used to refocus the SAR imagery for vertical velocity variant. Also the SAR imaging algorithm can locate accurately for the buried target using experimental data and has very high computational efficiency due to direct migration filter process. For multi–layer medium, when the propagation velocity of each medium is known, the SAR imaging algorithm can be used to refocus

the SAR imagery in multi-layer medium under the stratified earth assumption.

References [1] F. G. Joaquim, “A novel 3-D subsurface radar imaging

technique,” IEEE Transactions on Geoscience and Remote Sensing, vol. 40, no. 2, pp. 443-452, Feb. 2002.

[2] G.-Q. Liu, Y.-W. Wang, J. Li, and M. Bradley, “SAR imaging for a forward-looking GPR system,” in Proc. Detection and Remediation Technologies for Mines and Minelike Targets VIII, Proceedings of SPIE, Orlando, Florida, USA, 2003, pp. 322-333.

[3] K. Gu, G. Wang, and J. Li, “Migration based SAR imaging for ground penetrating radar systems,” IEE Proc. Radar Sonar Navig, vol. 151, no. 5, pp. 317-325, Oct. 2004.

[4] Y.-W. Wang, X. Li, Y.-J. Sun, J. Li, and P. Stoica, “Adaptive imaging for forward-looking ground penetrating radar,” IEEE Trans. on Aerospace and Electronic Systems, vol. 41, no. 3, pp. 922-936, Jul. 2005.

[5] T. Jin, Z. Zhou, and W. Chang, “Modified wavefront reconstruction imaging formation for stand-off GPEN SAR,” Electronics Letters, vol. 41, no. 10, pp. 63-64, May 2005.

[6] G. F. Margrave, “Theory of nonstationary filtering in the Fourier domain with application to time-variant filtering,” Geophysics, vol. 63, pp. 244-259, 1998.

[7] E. A. Robinson, “Migration of seismic data by the WKBJ method,” Proceedings of the IEEE, vol. 74, no. 3, pp. 428-439, 1986.

[8] L. van Kempen, H. Sahli, J. Brooks, and J. Cornelis, “New results on clutter reduction and parameter estimation for landmine detection using GPR,” in Proc. Eighth International Conference on Ground Penetrating Radar, SPIE, Gold Coast, Australia, 2000, pp. 872-879.

[9] D. J. Daniels, Ground Penetrating Radar, 2nd ed. London: The Institution of Electrical Engineers, 2004, ch. 7.

Yong Fan was born in Sichuan, China, in 1971. He received the M.S. degree from the School of Electronic Engineering, University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 2003. He is currently pursuing the Ph.D. degree with UESTC. His research interests include subsurface object detection and image processing in remote sensing.

Zheng-Ou Zhou was born in Yongzhou, Hunan, China, in 1940. He now is a professor and doctoral supervisor with UESTC. His current research interests include in radar signal processing and digital communications.

Jia-Li Xu was born in Sichuan, China, in 1971. She received the M.S. degree from the School of Computer Science and Engineering, UESTC, in 2004. She is now a lecturer with College of Electronic and Information Engineering, Chengdu University. Her research interests include data mining, mobile agent, mobile communication, and signal processing.

Jin-Shan Lin was born in Fujian, China, in 1971. He is an associate professor with Department of Electronic and Information Engineering, Putian University. His current research interests include wireless communication, computer intelligence, and artificial intellective network.

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