Analysis of Approximations and Aperture Distortion for 3D Migration of Bistatic Radar Data with the Two-Step Approach
© L. Zanzi and M. Lualdi. 2010
Received: 31 December 2009
Accepted: 7 June 2010
Published: 29 June 2010
The two-step approach is a fast algorithm for 3D migration originally introduced to process zero-offset seismic data. Its application to monostatic GPR (Ground Penetrating Radar) data is straightforward. A direct extension of the algorithm for the application to bistatic radar data is possible provided that the TX-RX azimuth is constant. As for the zero-offset case, the two-step operator is exactly equivalent to the one-step 3D operator for a constant velocity medium and is an approximation of the one-step 3D operator for a medium where the velocity varies vertically. Two methods are explored for handling a heterogeneous medium; both are suitable for the application of the two-step approach, and they are compared in terms of accuracy of the final 3D operator. The aperture of the two-step operator is discussed, and a solution is proposed to optimize its shape. The analysis is of interest for any NDT application where the medium is expected to be heterogeneous, or where the antenna is not in direct contact with the medium (e.g., NDT of artworks, humanitarian demining, radar with air-launched antennas).
In 1983, Gibson et al.  introduced the fast two-step migration technique for 3D poststack seismic data. In a companion paper, Jakubowicz and Levin  showed that in a constant velocity medium the method is equivalent to the classical one-step 3D migration. In their paper, Gibson et al. performed a detailed analysis of the differences between the two-step approach and the one-step approach when the velocity varies within the medium. They showed that in normal conditions these differences are negligible so that the method was suggested as a quite attractive solution for fast 3D migration of poststack seismic data. The extension of the two-step approach to prestack 3D migration is not straightforward although achievable as shown by Canning and Gardner . They proposed a scheme composed of 3D DMO, cross-line 2D PSI, inline 2D DMO−1, velocity analysis, and 2D inline depth migration. A variation of this scheme was proposed by Meinardus et al. .
Here, it is shown that under a very restrictive condition, that is, when the source-receiver azimuth is constant, the two-step approach can be directly extended to non-zero-offset data. In seismics, this would be the case of 3D marine data collected by a single ship, equipped with a single cable, and shooting along parallel lines. Of course this is not very interesting for the seismic community where 3D acquisitions are designed aiming at a balanced azimuth distribution to get a good picture of 3D structures. Instead, the result is interesting for GPR applications where 3D experiments are normally executed by maintaining a constant orientation of the antenna box, that is, a constant TX-RX azimuth. With the present hardware technology, this approach is what is needed by GPR users to achieve the goal of real-time visualization of 3D migrated volumes. Thus, the following sections discuss the non-zero-offset extension of the two-step method, the approximations resulting from the application to vertically variable velocity fields, and finally the distortion effects on the aperture of the migration operators. The quantitative results are derived assuming the usage of an ultra high-frequency radar with air-launched antennas. This type of hardware is normally preferred to ground-coupled antennas to speed up the NDT acquisitions on highways and bridges. It is also preferred for humanitarian demining to prevent mine activation and for diagnostic inspections on cultural heritage and artworks to prevent damages to delicate decorations, paintings, precious materials, and so forth. The air gap that separates the antenna from the medium generates a situation where the migration velocity field varies vertically even if the medium is homogeneous. Nevertheless, the discussion that follows is also of interest for radars with ground-coupled antennas when they are used to investigate a medium that is vertically heterogeneous. This is a frequent situation when a GPR is applied to NDT inspections of layered structures such as walls, floors, and pavements.
2. The Two-Step Approach
The two-step approach consists of performing a 2D migration in the -direction according to (2) followed by a 2D migration in the direction according to (3). Note that in the monostatic case any summation order is valid.
The extension of the two-step approach consists of performing a 2D non-zero-offset migration in the -direction according to (5) followed by a 2D zero-offset migration in the direction according to (6). Note that in the bistatic case the summation order is relevant, that is, the first step must be in the azimuth direction. The conclusion is that the extension of the two-step approach to a homogeneous medium investigated with a bistatic radar is possible, and the algorithm is totally equivalent to an exact one-step 3D migration.
3. Approximations for the Vertically Heterogeneous Medium
The standard method to migrate the diffractions observed in a medium where the velocity varies vertically consists of using the rms velocity function to extend the use of the equations derived for the constant velocity medium. Another approach was successfully experimented at the Lawrence Livermore National Laboratories by Johansson and Mast . This approach is applicable when the radar measurements are performed with air-launched horn antennas as those often used for nondestructive testing of highways and bridges, for humanitarian demining, for diagnostic investigations on art-works, and so forth. The air gap that separates the antenna from the medium generates a situation where the migration velocity field varies vertically even if the medium is homogeneous. Johansson and Mast proposed a method based on an approximate estimation of the inflection point, that is, the point on the medium surface where the antenna-target raypath is bent according to Snell's law. Both methods preserve the property discussed above, that is, the possibility to split the 3D migration operation into a sequence of two bidimensional migrations provided that a proper order is followed when the system is bistatic. Let us examine both approximations, shortly indicated in the following as rms and LLNL solutions, and let us perform a kinematical analysis of the expected errors with respect to the exact 3D migration. We will see that the final errors are the combination of the errors induced by the approximation adopted to estimate the diffraction surface plus the additional errors induced by the application of the two-step approach. Thus, let us discuss first the errors for the monostatic case and the bistatic case when the migration is performed in one step, and then let us consider the additional errors introduced by the two-step approach.
Finally, let us extend the error analysis to the two-step approach. With respect to the one-step approach, we have to include a further error, that is, the difference between the 3D approximation of the diffraction surface and the actual diffraction surface over which the contributions are taken when the two-step approach is applied. For the rms approximation, the difference comes from the velocities that are applied to perform step 1 with (5) and step 2 with (6): in principle, both of these velocities should be equal to the rms velocity observed at the zero-offset time , whereas in practice the velocity applied for the first step is the rms velocity observed at a higher zero-offset time given by . In other words, the problem is due to the fact that when the velocity varies vertically, for a given , the intersection of the one-step 3D diffraction surface with a vertical plane parallel to the -axis, (5) is not only a function of but also depends on the zero-offset time (see (6)). A similar comment is applicable to the LLNL approach. Again the problem comes from the fact that the first step should collapse in contributions that belong to different intersection curves depending on the final where the diffraction is going to be focused.
Finally, let us remind the reader that the migration aperture that we are discussing here is the aperture that we would like to select to perform a constructive interference of the summed contributions. As we are going to see in the next section, the actual shape of the migration aperture that we can obtain with the two-step approach might be very different. Besides, we want to stress the point that the aperture discussed in this section has nothing to do with the real footprint of the radar system, that is, with the area actually illuminated by the antenna that depends on many other factors related with the antenna distance from the medium, the medium absorption, the medium permittivity, and so forth. Nevertheless, the results of the analysis are encouraging because the real footprint measured on experimental data with physical parameters similar to those assumed in the above examples is seldom wider than the conservative aperture of the two-step operator suggested by Figure 4 to prevent destructive interference of unfocused data.
4. Design of the Operator Aperture
5. Application Examples
A few examples are shortly presented to illustrate situations where the vertical heterogeneity of the medium is successfully handled by using the two-step migration approach with the rms velocity approximation to focus the data collected with high-frequency bistatic radar systems.
A direct extension of the two-step approach for fast 3D migration of bistatic GPR data is possible provided that the source-receiver azimuth is constant. As for the zero-offset case, the two-step operator is exactly equivalent to the one-step 3D operator for a constant velocity medium and is an approximation of the one-step 3D operator for a medium where the velocity varies vertically.
Two methods have been considered (rms and LLNL) for migrating data collected with air-launched antennas where the air gap that separates the antennas from the medium generates a situation that requires a vertically variable migration velocity even if the medium is homogeneous. Both methods are suitable for the application of the two-step approach, and they have been compared in terms of accuracy of the final 3D operator. The result of the analysis is that both the rms and the LLNL methods can be applied with the two-step approach producing a negligible degradation of the migration accuracy. A solution has been also proposed for an optimal shaping of the two-step operator aperture.
The impact of the two-step algorithm on the CPU cost of the 3D migration is quite interesting, as 3D images such as the mine reconstruction of Figure 6 can be produced in a few seconds, that is, in real time, with a standard personal computer. The advantage is of great interest if we consider that currently the GPR suppliers are producing multi-channel GPR equipment with more and more antennas mounted in a cart to increase the productivity. These systems generate huge amounts of data that cannot be migrated in real time by a single computer unless a very effective algorithm is used.
Finally, the rms method and the accuracy discussion are also of interest when radars with ground-coupled antennas are used to investigate a medium that is vertically heterogeneous. This is a frequent situation when the GPR is applied to NDT inspections of layered structures such as walls, floors, and pavements.
The authors are grateful to RST GmbH that developed the stepped frequency radar prototype for humanitarian demining, to the Joint Research Center in Ispra that gave free access to the mine test field, to Dr. G. Lenzi of ISMES S.p.A. who performed the acquisitions on the marble monument in Rome, and to IDS S.p.A. that supplied the 2 GHz system for the experiments in the Venetian Palace.
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