as illustrated in Figure 8(a). During the B-scan process, the backscattering parameter (S 11 ) measurements were taken at 2 cm spacing along a straight path of 134 cm. At each spatial point, the frequency of the VNA was stepped between 0.8 and 5 GHz with 8.4 MHz increments. Since the relative permittivity of the soil is unknown, the space-time image [Fig. 8(b)] was obtained by simply taking the 1D IFT of the spatial-frequency data along the frequency axis. The dominant scattering mechanism from air-ground surface is easily detected around at t ϭ 18 ns and seen throughout the whole synthetic aperture. As expected, the unprocessed GPR image in Figure 8(b) is unfocused around the buried objects. After applying our algorithm, well focused images of the three buried objects are successfully acquired as depicted in Figure 8(c). The same interpolation technique just as in the previous cases was employed in this study as well. The dominant scattering mechanisms from buried objects are clearly identified at their true locations with resolutions in either direction. CONCLUSIONIn this work, we presented a frequency domain version of f Ϫ k based SAR focusing technique for B-scan GPR imagery. The formulation of the algorithm was given in detail. The algorithm is first tested with a set of simulated data obtained by an EM simulation code. Almost perfect focusing was achieved after applying the proposed technique to the simulated GPR data. The performance of the method was also checked by laboratory sandpool and outdoor soil experiments. Measured B-scan GPR images of metallic and nonmetallic objects after applying the algorithm demonstrate the real performance of the method. The algorithm successfully focuses any hyperbolic behavior with good fidelity without distorting any flat behavior such as air-ground interface in the image. Thanks to sharp focusing feature of the method, resolutions in the new 2D GPR images are so good that even very close scattering mechanisms can be easily distinguished. This can be seen from Figures 5(c) and 6(c), where the scattering from the top of the pipes can be easily discerned from the creeping wave scattering mechanisms. The necessary interpolation step in the algorithm is the main source of the numerical noise in the resultant focused GPR images. ACKNOWLEDGMENTS

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Many commonly used velocity estimation procedures assume that the reflectors are horizontal. Because of this, their performance tends to degrade as the reflectors become curved or discontinuous. Much of this degradation can be traced to the fact that data recorded over nonhorizontal reflectors need not resemble in detail the subsurface in the area where they were recorded. Diffraction and scattering are the major complicating factors. Beginning with the scalar wave equation and using a small dip assumption, approximate wave equations which quite accurately model both near‐ and wide‐angle reflections generated by one or more sources can be found. Finite difference formulations of these equations can be used to demonstrate that surface recorded seismic reflections which have been downward continued to the depth of their source reflectors must resemble those reflectors in detail. This property of downward continuation can be exploited to improve velocity estimates by using downward continuation as a preprocessor for velocity estimation techniques. Both synthetic and field data examples show that estimates based on downward continued data do not exhibit diffraction effects and are not dependent upon reflector dip. Synthetic data examples also illustrate that the use of downward continuation allows accurate velocity estimates to be made from no record data recorded over an earth in which the reflectors are random functions of the horizontal and vertical coordinates. For reasonable data parameters, theoretical considerations indicate that the coherence of properly downward continued random reflector data measured along the true velocity hyperbolic should be greater than a similar measure on the corresponding surface data. This coherence increase should make velocity estimates based on downward continued random reflector data less susceptible to noise than estimates based on surface recorded data.

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