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Figure 2. The domain of object function in the spectral domain: (a) full-view, (b) receiving mode, (c) transmission mode, (d) angle mode. Accordingly, when the incident and observation angles change from 0 to 27z, the full object information in the spectral domain is achieved since, as shown in Figure 2a, the radius of the blue circle is 26, its area will be A = 7(2B)" ; thus, the NDF is equal to Aghr(2B)2 /(9n)* = 16qb7 /)2. For the sake of illustration, we recall three modalities from [30] to combine the im. ringing angles of the plane waves (transmitter) and observation directions (receiver) for he aspect-limited data case. First, it is supposed that the transmitters and receivers are ocated on the same side in the receiving mode, where the back-scattered fields are only vailable; it may be useful in GPR imaging and oil and gas exploration. Second, on the ‘ther hand, the transmission mode is specified when the receivers are on the opposite ide of the transmitters, and the forward scattered fields are only available, for instance, n seismic prospection. Third, the angle mode is defined when the transmitters and the eceivers are located on adjacent sides, which may occur in TWI applications. In particular

Figure 2 The domain of object function in the spectral domain: (a) full-view, (b) receiving mode, (c) transmission mode, (d) angle mode. Accordingly, when the incident and observation angles change from 0 to 27z, the full object information in the spectral domain is achieved since, as shown in Figure 2a, the radius of the blue circle is 26, its area will be A = 7(2B)" ; thus, the NDF is equal to Aghr(2B)2 /(9n)* = 16qb7 /)2. For the sake of illustration, we recall three modalities from [30] to combine the im. ringing angles of the plane waves (transmitter) and observation directions (receiver) for he aspect-limited data case. First, it is supposed that the transmitters and receivers are ocated on the same side in the receiving mode, where the back-scattered fields are only vailable; it may be useful in GPR imaging and oil and gas exploration. Second, on the ‘ther hand, the transmission mode is specified when the receivers are on the opposite ide of the transmitters, and the forward scattered fields are only available, for instance, n seismic prospection. Third, the angle mode is defined when the transmitters and the eceivers are located on adjacent sides, which may occur in TWI applications. In particular

Related Figures (11)

Figure 1. The geometry of a 2D rectangular. This section addresses a two-dimensional (2D) rectangular investigation, as shown in Figure 1. Its length along the x and y is indicated by a and D, respectively. An unknown scatterer is located within this finite domain, i.e., the investigation domain (ID). We denote the angle providing the direction of propagation of the incident plane wave E; by 6; and E,; indicates the far-field response of the scatterer in the direction @;. The plane waves and observations angular domains are supposed to be limited so that 6; € [a1,a2 | and 6; € [1,2 |, respectively. The inverse scattering problem is stated as to determine the characteristics of the unknown scatterer from EF; and E,. It should be mentioned that the scattering sensing is investigated by using multiple plane waves with different incident angles to increase the NDF and to improve the reconstruction performance (multi-view configuration).
Figure 1. The geometry of a 2D rectangular. This section addresses a two-dimensional (2D) rectangular investigation, as shown in Figure 1. Its length along the x and y is indicated by a and D, respectively. An unknown scatterer is located within this finite domain, i.e., the investigation domain (ID). We denote the angle providing the direction of propagation of the incident plane wave E; by 6; and E,; indicates the far-field response of the scatterer in the direction @;. The plane waves and observations angular domains are supposed to be limited so that 6; € [a1,a2 | and 6; € [1,2 |, respectively. The inverse scattering problem is stated as to determine the characteristics of the unknown scatterer from EF; and E,. It should be mentioned that the scattering sensing is investigated by using multiple plane waves with different incident angles to increase the NDF and to improve the reconstruction performance (multi-view configuration).
Figure 3. The behavior of the normalized singular values of the linearized inverse scattering: (a) full view, (b) three sensing modes. The dashed lack black line provides the analytically predicted NDF. The aim of the first numerical simulation is to confirm the estimated NDFs for the full view case and the three modes at hand. Therefore, Figure 3 shows the pertinent behavior of normalized singular values of the relevant operator. It is worth noting that the singular values have a distinct step-like behavior from which the NDF can be evaluated. As predicted, the results confirm that the NDF is different for each mode in the aspect-limited case. In particular, Figure 3b confirms that the upper bound NDF of receiving, transmission, and angle modes is equal to 57, 114, and 200, respectively. It should be noted that the NDF of the transmission mode is twice the NDF of receiving mode; Figure 3a verifies that the NDF of the full view case is equal to 1257.
Figure 3. The behavior of the normalized singular values of the linearized inverse scattering: (a) full view, (b) three sensing modes. The dashed lack black line provides the analytically predicted NDF. The aim of the first numerical simulation is to confirm the estimated NDFs for the full view case and the three modes at hand. Therefore, Figure 3 shows the pertinent behavior of normalized singular values of the relevant operator. It is worth noting that the singular values have a distinct step-like behavior from which the NDF can be evaluated. As predicted, the results confirm that the NDF is different for each mode in the aspect-limited case. In particular, Figure 3b confirms that the upper bound NDF of receiving, transmission, and angle modes is equal to 57, 114, and 200, respectively. It should be noted that the NDF of the transmission mode is twice the NDF of receiving mode; Figure 3a verifies that the NDF of the full view case is equal to 1257.
Figure 4. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yo = 2A (green lines) when x = —A in receiving mode. Figure 4. Comparison of the normalized amplitude of the actual (solid lines) and approximated of both PSFs for three points is the same along with the cut in the three modes, which means the resolution is space invariant along the vertical lines. The results confirm that the approximated PSF overlaps quite well with the actual PSF for each mode, which means that the approximation works satisfactorily in the main lobe in all three modes to predict the achievable resolution for every point scatterer. approximated PSF overlaps quite well with the actual PSF for each mode, which means
Figure 4. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yo = 2A (green lines) when x = —A in receiving mode. Figure 4. Comparison of the normalized amplitude of the actual (solid lines) and approximated of both PSFs for three points is the same along with the cut in the three modes, which means the resolution is space invariant along the vertical lines. The results confirm that the approximated PSF overlaps quite well with the actual PSF for each mode, which means that the approximation works satisfactorily in the main lobe in all three modes to predict the achievable resolution for every point scatterer. approximated PSF overlaps quite well with the actual PSF for each mode, which means
Figure 5. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yp = 2A (green lines) when x = —A in transmission mode.
Figure 5. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yp = 2A (green lines) when x = —A in transmission mode.
Figure 6. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yo = 2A (green lines) when x = —A in angle mode. Figure 6. Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 6. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for yo = —3A (red lines), yo = 0 (blue lines) and yo = 2A (green lines) when x = —A in angle mode. Figure 6. Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 7. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in receiving mode. Figure | 7. - Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 7. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in receiving mode. Figure | 7. - Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 8. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in transmission mode. Figure 8. Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 8. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in transmission mode. Figure 8. Comparison of the normalized amplitude of the actual (solid lines) and approximated
Figure 9. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in angle mode.
Figure 9. Comparison of the normalized amplitude of the actual (solid lines) and approximated (dashed lines) PSFs for x9 = —A (red lines), x9 = 0 (blue lines) and x9 = A2 (green lines) when y = 0 in angle mode.
Figure 10. The reconstructed image of two point-like scatterers in the receiving mode when their spac- ing equals (a) and is less (b) the resolution. The red dots indicate the position of the scatterer points.
Figure 10. The reconstructed image of two point-like scatterers in the receiving mode when their spac- ing equals (a) and is less (b) the resolution. The red dots indicate the position of the scatterer points.
Figure 11. The reconstructed image of four point-like scatterers in the angle mode when their spacing equals (a) and is less (b) the resolution. The red dots indicate the position of the scatterer points. The reconstruction of four scatterer points in the angle mode is considered in the second example. Now, the resolution is the same for horizontal and vertical lines, i.e., Rha = Roa = 0.66A. Figure 11a illustrates the reconstructed images when the distance between two points is equal to the resolution; it is observed that the considered points can be entirely resolved. Then, four points appear as a single scatterer when the distance is equivalent to half of the resolution, as observed in Figure 11b.
Figure 11. The reconstructed image of four point-like scatterers in the angle mode when their spacing equals (a) and is less (b) the resolution. The red dots indicate the position of the scatterer points. The reconstruction of four scatterer points in the angle mode is considered in the second example. Now, the resolution is the same for horizontal and vertical lines, i.e., Rha = Roa = 0.66A. Figure 11a illustrates the reconstructed images when the distance between two points is equal to the resolution; it is observed that the considered points can be entirely resolved. Then, four points appear as a single scatterer when the distance is equivalent to half of the resolution, as observed in Figure 11b.
Figure 12. The normalized reconstructed image of ten point-like scatterers: (a) angle mode, (b) re- ceiving mode. The red dots indicate the position of the scatterer points.
Figure 12. The normalized reconstructed image of ten point-like scatterers: (a) angle mode, (b) re- ceiving mode. The red dots indicate the position of the scatterer points.
Related topics:
MathematicsInverse ProblemsInverse Scattering Problems
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