Tomographic inversion of simulated data

Figure 5.1 shows single point backscattering curves for the difference between the simulated Detail Model (DM) (A)–(E) (Table 4.2) and the Homogeneous Model (HM) which contains the constant background permittivity distribution withϵ^′_r =4 and lacks any interior details. The time axis has been scaled from the unitless computa-tional system to the SI-units used in a real measurement by using the spatial scaling factor s = 1911 and the time scaling formula in Table 2.1. The measurement fre-quency corresponded to a 2 MHz bandwidth signal, and the measurement distance was to 4.59 kilometres in SI-scale (Table 4.5). The curves show the measured nor-malised antenna voltages for the signal travelling inside of the target asteroid domain D and scattering to the receiver. The time axis indicates the two-way traveltime of the signal, starting from the surface of the target. The first peak after 29µscan be in-terpreted to originate from the mantle-interior interface of each of the DMs, as that is the first structural difference between the HM and the DM. The echo from the shallow crack shown in the dashed black line is clearly at approximately 30µs. The effect it has on the echo is also visible with respect to the other models, as the crack was placed very close to the surface affecting the wave propagation at the mantle-interior interface. The ellipsoidal internal details of the void, highly porous mate-rial, and high density boulder are located close to 30.5 µs. The ellipsoidal detail causes the signal to fluctuate, and the effect of the detail permittivity value affects the direction of the signal peak. The deep crack signal is clear at 32µs, which is the location where it is expected to be seen based on two-way traveltime of the signal in a medium with an average relative permittivity of 4.

Similar curves can be obtained for each of the 64 measurement points in the nu-merical experiment point cloud (Figure 4.3). The total variation inversion procedure was used to obtain the tomographic inversion of the data. The inversion

regularisa-Figure 5.1 The normalised backscattering difference signals between the DM and HM for the five aster-oid interior structures shows how the interior details can be interpreted from simple antenna voltage curves. The time axis has been scaled to real size measurement and presents the two-way traveltime from the transmitter to the receiver. The solid black curve indicates the void model (A). The shallow crack (Model (E), dashed black) is easily detectable from the signal alone. The deep crack (Model (D), dotted black) is clearly detectable at 32µs. The porous ellipsoid (solid grey) gives a signal which is only a slightly different from the void signal, whereas the signal for the high density ellipsoid (dashed grey) clearly is in opposite phase to that of the void.

tion parameters for the models presented here wereα=0.01, andβ=0.005. The parameters were adjusted to these levels based on preliminary experiments in which the goal was to maximise the distinguishability of the interior details and to obtain a range of permittivity values with respect to the exact model.

Figure 5.2 shows the exact model structure and the resulting 3D reconstructions for the models (A), (D), and (E), the void, deep crack, and shallow crack models, respectively. For the void model (A) reconstruction, it also shows the difference between the monostatic and bistatic measurement, where the bistatic measurement appears to give a better-defined and more robust reconstruction of the void detail in comparison to the monostatic one.

The shallow crack (Model (E)) reconstruction was able to localise the crack mod-erately well, although the shape of the crack could not be reconstructed exactly. This could be expected based on the signal curves in Figure 5.8, because there was such a well-defined echo in the temporal region when the signal was expected to reach the location. The reconstruction for the deep crack (Model (D)) was poor, which is sur-prising considering how well the crack could be seen in the signal curve. However, as the reconstruction is built from all measured signals, the deep interior details may have such high uncertainty that the exact location and shape of the detail cannot be

Exact void model Bistatic reconstruction Monostatic reconstruction

Exact shallow crack model Shallow crack reconstruction

Exact deep crack model Deep crack reconstruction

Figure 5.2 3D cut-views of reconstructions of the void and crack models showing the localisation of the details. The colour bar indicates the relative permittivity values in the images.

reconstructed reliably.

The results from the reconstructions were also quantified by calculating the mean square error (MSE), mean absolute error (MAE), and the relative mean absolute er-ror (MAE-R) for the reconstructions (Table 5.1). The MSE and MAE values were computed separately for the interior detail (ellipsoidal anomaly or complex-shaped cracks), the mantle, and the global reconstruction containing the whole structure.

The MAE-R was computed for the details only to compare the goodness of detail reconstruction between the models. The errors were noticed to be the greatest in the detail areas and the smallest in the surface layer. This is also evident by visual inspection of the reconstructions, because the majority of the volumes in the void and cracks are not captured in the reconstruction, causing an evident source of error.

Figure 5.2 also shows a comparison between the void detection by a monostatic and bistatic measurement. In the monostatic case, the signal is transmitted and re-ceived at the same location, whereas in the bistatic case the signal is transmitted at one point and received at two, giving both the backscattering data and the bistatic data. The reconstruction shows that the bistatic measurement approach provides a slightly more robust reconstruction, as it contains more information on the target during the inversion process.

The effect of noise on the reconstruction was investigated by adding Gaussian noise in the system during inversion. The results are shown with respect to the

Table 5.1 The mean square errors (MSE), mean absolute errors (MAE), and relative mean absolute errors (MAE-R) of the reconstructions for Models (A), (D), and (E) (Table 4.2). The reconstruc-tions for the Models (A), (D), and (E) are shown in Figure 5.2 and reconstrucreconstruc-tions and these error values for all the models are available in the Publication I.

Model Error Detail Mantle Global

(A) MSE 8.21 0.99 1.19

MAE 2.80 0.89 0.93

MAE-R 0.93 . . . .

(D) MSE 7.10 0.93 1.05

MAE 2.49 0.89 0.90

MAE-R 0.83 . . . .

(E) MSE 6.86 0.90 1.04

MAE 2.38 0.87 0.89

MAE-R 0.79 . . . .

-25 dB -22 dB -15 dB

-8 dB -4 dB 0 dB

Figure 5.3 The effect of noise on the reconstructions of the void model. The colour scale is as in Figure 5.2.

bistatic measurement of the Model (A) containing the ellipsoidal void (Figure 5.3).

The threshold at which noise appears to start affecting the reconstruction is between -15 and -8 dB, between which the localisation and shape detection of the void is lost and the amount of artefacts due to noise increases clearly. This result supports the earlier findings in[74]that the critical noise threshold is around -10 dB.

5.2 The effect of the higher-order Born approximation on