Because of the space limit, we will not detail the formulation. A full description of the method will be reported elsewhere [Song et al., 2004]. For a description of the contrast source inversion method for objects in a homogeneous background medium, the reader is referred to Abubakar et al. [2002a, 2002b] and the references therein. Below we will only summarize the key steps in the inverse method. If the target is embedded in the ith layer, the contrast function χ(r) is defined as χ(r) = − 1. According to Xu and Liu , the scattered electric field measured in the first layer (air) can be written as
where the contrast source w is defined as w(r) = χ(r)E(r) for r ∈ D, ki is the wave number in the ith layer, Gmi is the dyadic Green's function for the magnetic vector potential in the layered medium [Michalski and Mosig, 1997]. Furthermore, from the integral equation, the contrast source can be written as
 Similar to the contrast source inversion in a homogeneous background [van den Berg and Kleinman, 1997], a cost functional is defined as
where the first term in the right-hand side of equation (12) measures the misfit in the data equation (9) and the second term measures the misfit in the object equation in (10). The latter term plays a role of regularizing the ill-posed problem. Hence the method has an advantage that there is no need to choose a artificial regularization parameter. For the detailed updating steps to minimize the above functional, the reader is referred to van den Berg et al.  and Abubakar and van den Berg  for a homogeneous background. Note that the major difference between this work and the previous works for 3-D inversion in a homogeneous background [Abubakar and van den Berg, 2000; Zhang and Liu, 2004] are the operations GS,D. They are now involving the dyadic Green's function for the magnetic vector potential in a layered medium, rather than a scalar Green's function in a homogeneous background. The fast method for these operations is described by Xu and Liu  and Millard and Liu [2003, 2004].
 The main computational burden of the CSI method for layered media lies on the operations of GS,D and their adjoint operations. GS and its adjoint operator require O(MN) arithmetic operations, where M is the number of data points and N the number of voxels inside D. GD and its adjoint operator require O(MTN log N) arithmetic operations by using the fast Fourier transform (FFT) algorithm, where MT is the number of transmitters. Since the CSI method is performed in a functional formulation instead of matrix formulation, the memory is not a big issue. In the CSI procedure, one only needs to store contrast source vectors w and contrast χ and several auxiliary vectors of the same size; thus the memory requirement is O(MTN).
3.2. Three-Dimensional Inverse Scattering Results
 We now examine some numerical results of the 3-D CSI method for reconstructing buried objects in a layered medium. The “measured” scattered fields are obtained by synthetic results using the forward modeling method presented by Millard and Liu . Since the inverse method does not use the forward solver during inversion, there is no question of the so-called “inverse crime” being committed here.
 Figure 6 shows the configuration for a two- or three-layer background characterized by εr1 = 1.0, σ1 = 0 S/m; εrm = 1.5, σm = 0.01 S/m; εr2 = 2.0, σ2 = 0.02 S/m. In this configuration, the positive z direction points upward and the positive x and y directions are defined with the right-hand system. The origin of the coordinates is referred to the center of the D domain. We use this configuration to simulate two scenarios: (1) for a two-layer medium, z1 = z2 = 2.15 m so that the middle layer (index m) is absent and (2) for a three-layer medium, z1 = 2.15 m and z2 = 2.11 m. The center of test domain D is at (1.0, 1.0, 2.04) m and its dimension is 9.2 × 9.2 × 9.2 cm3. The distance from the top side of the D domain to the interface at z1 is d = 6.4 cm. Domain D is divided into a number of cubes, assuming the contrast function to be piecewise constant. In D domain, there are two buried objects to be reconstructed, with the electrical parameters εr = 4.0 and σ = 0.16 S/m. Note that the permeability for background and the objects is all equal to μ0 in free space. Under the operating frequency of 2 GHz, the dielectric wavelength in lower half-space is around 10.6 cm.
Figure 6. The geometry for reconstructing 3-D objects buried in a two- or three-layer medium characterized by εr1 = 1.0, σ1 = 0; εrm = 1.5, σm = 0.01 S/m; εr2 = 2.0, σ2 = 0.02 S/m. For the two-layer case, z1 = z2 = 2.15; for the three-layer case, z1 = 2.15 m, z2 = 2.11 m.
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 In the numerical simulation for the synthetic data, the incident electric field is generated by a z-directed electrical dipole J = δ(r − r0), where r0 is the source location. We assume a 2-D planar array with 64 sources and 64 receivers uniformly distributed, with an aperture dimension of 60 × 60 cm2 and 3 cm high above z1. Thus we have complex data of 4096 considering one component Ez of the scattered data in the reconstruction.
 In the following simulations, the initial guess is derived from a single frequency migration. Following Claerbout , such migration is performed by applying the adjoint or migration operator GS† of the forward modeling operator GS in equation (9) to the data to derive the migrated image of the contrast source w0 = αGS†f, where α may be used to scale the migration operation in a least squares sense [van den Berg et al., 1999]. Once w0 is derived, the initial contrast χ0 can be calculated by the constitutive relation w = χE. In the migration, the dyadic kernel and the wave number terms in the integral equations are all changed to their complex conjugates for back-propagating wave fields from receiver positions. Thus such migration operation is also known as back-propagation [van den Berg et al., 1999].
 Note that in the first part of this report the 2-D scalar migration is implemented in frequency-wave number domain for a wide-band data under a far field approximation of a free space Green's function. In this part, a migration operation is adapted for easily producing an initial model directly usable in the inverse scattering procedure and performed in space-frequency domain for a single frequency data considering the exact dyadic Green's function in a layered medium.
 First, we consider the two-layer case where domain D is divided into 15 × 15 × 15 voxels, thus 3375 complex unknowns to be reconstructed. The problem is the so-called mixed-determined (it is not overdetermined because of the nonuniqueness even though the number of data points is larger than the number of unknowns). Figures 7a and 7b show the ground truth of dielectric constant and conductivity profiles of two buried objects, on three orthogonal cross sections. The simultaneously reconstructed profiles of dielectric constant and conductivity at iteration 50, 100, 300, 500 are displayed in Figures 8 and 9, respectively. By way of comparison, we observe that the dielectric constant profile in Figure 8a at early iteration 50 has revealed the shapes and locations of the objects, although the inverted dielectric constant values are still small compared to the true ones. After 50 more iterations, as illustrated in Figure 8b, the inverted profile is further improved in the level of dielectric constant. At iteration number to 300 and 500, the inverted profiles Figures 8c and 8d become closer and closer to the original profile in the spatial distribution and dielectric constant level. The inverted conductivity profiles of Figure 9 also exhibit a fashion similar to the case of dielectric constant. It is visually demonstrated that the convergence of our algorithm is fairly stable for this problem. To examine the effect of an initial guess, we perform an inversion starting from the migration with multifrequency data (0.50–3 GHz at the interval of 0.50 GHz). After 500 iterations, the corresponding inverted result of dielectric constant and conductivity in Figures 10a and 10b are almost identical to Figures 8d and 9d. In Figure 11, it is observed that the initial guess from the multifrequency data yields the smaller errors at the beginning of the iteration. If the iteration was to be terminated early, say at around 100 iterations, for cases with high noise, this improved initial guess would have accelerated the inversion process. However, under the present lower noise case, the fitting errors for the data and object equations would eventually overlap with the case where the initial guess was provided by a single-frequency back-propagation. In summary, it is seen that the errors are reduced significantly around after 100 iterations and then continue to decrease monotonically with iterations. For this example, the CPU time of 500 iterations is around 5 hours on an IBM 690 and the memory requirement is 80 MB.
Figure 10. With the initial guess from migration in multifrequency data, the inverted profiles of (a) dielectric constant and (b) conductivity at iteration 500 for the case in Figure 7.
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Figure 11. The fitting errors of (a) the data and (b) the object as the function of iteration number for the two-layer case in Figure 7, corresponding to the single-frequency (SF) and multifrequency (MF) initial guesses.
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 Now using the same model parameters as the above, we study the performance of the 3-D imaging technique in the form of the actual 1-D array configuration. We simulate the two kinds of 1-D arrays. The first configuration is a multistatic array having 8 sources and 8 receivers uniformly distributed along cross-track (x) direction; this 1-D array is then moved 8 times at constant interval of 8.6 cm along down-track (y) direction to collect data. The total number of data is 8 × 8 × 8 = 512. The second configuration is a monostatic array with 8 sensors, each being a source and receiver to perform a monostatic measurement. In this case, the number of data is only 8 × 8 = 64. After 500 iterations, Figures 12a and 12b illustrate the reconstruction of dielectric constant and conductivity using the multistatic array and Figures 12c and 12d the dielectric constant and the conductivity using the monostatic array. Compared to the results in Figures 8d and 9d using a full 2-D array, it can be observed that the inversion of the locations and shapes of the objects are generally good under such sparse data, although the level of the inverted electric parameters are lower than those of the actual ones, especially in the case of the monostatic array.
Figure 12. The inverted profiles of (a) dielectric constant and (b) conductivity with 1-D multistatic arrays and (c and d) with 1-D monostatic arrays for the case in Figure 7.
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 Next we continue the simulation with the 2-D array. In order to examine the effects of the topsoil on the inversion, we perform the imaging for a three-layer case. The configuration geometry is referred to Figure 6. The true and reconstructed profiles (at iteration 500) of dielectric constant and conductivity are shown in Figures 13 and 14, respectively. It appears that the spatial distribution of the two buried objects is generally well reconstructed, although the maximum values of the dielectric and conductivity are somewhat lower than the previous case.
 Finally, we present an example to reconstruct one buried object in the two-layer case using the same configuration geometry as in the first example. The size of the object is 5 × 5 × 5 cm3. In this case, we discretize domain D into 25 × 25 × 25 voxels, thus there are 15625 complex unknowns, but the number of data points remains to be 4096. The purpose of this test is concerned with the performance of the reconstruction algorithm for more unknowns. Figures 15 and 16 depict the true and the reconstructed profiles (at iteration 600) of dielectric constant and conductivity, respectively. It is evident that the buried object is reconstructed but its position appears to shift a little downward in Figure 15b and a little upward in Figure 16b. This may be attributed the fact that the one-side limited view inherently has the potential to introduce various distortions in the inversion, probably leading to the shift in the locations of the reconstructed objects. This phenomenon is somewhat observed in the other examples. Such ambiguity may be reduced by imposing additional a priori information if available or increasing view aperture from the other side. However, we restrain from doing that in this work. In light of the stable convergence behavior in the CSI, it may have possible improvement by continuing more iterations. Overall, it is observed that the reconstruction captures the features of the true object distribution even when the number of unknowns is much larger than the number of independent measurements. For this example, the CPU time of 600 iterations is around 20 hours on an IBM 690 and the memory requirement is 368 MB.
Figure 15. For a two-layer case with 25 × 25 × 25 voxels, (a) the original and (b) inverted profiles of dielectric constant.
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