44
2 Voxel-Based Inversion Via Set-Theoretic Estimation
coil
anomalous
region
top layer
bottom layer
8 mm
Fig. 2.17 Fastener with multisite damage as a surface-breaking anomaly
will report only the results of the S-estimator, which we call ‘stest’ in the captions
of several figures.
Problem No. 1: Fastener with Multisite Damage in a Slab In the first problem,
the anomalous region breaks the surface of a slab, as shown in Fig. 2.17. The data
are gathered at 200 Hz.
The results for this problem are shown in Fig. 2.18.
Problem No. 2: Layer-Stripping Using Multifrequencies In an attempt to
improve the results of Problem No. 1, we redo that problem by adopting a strategy,
that we call layer-stripping using multifrequencies. The idea is to isolate the top
layer of the anomalous region of Fig. 2.17 from the bottom layer. We do this by
gathering data at a very high frequency, such that the incident field produced by
the sensors does not penetrate to the bottom layer. This means that the anomalous
currents in the cells of the bottom layer will not contribute to the measured
impedances. Hence, we simply constrain these currents to be zero when we do
the reconstruction starting with the QR-decomposition. We used VIC-3D® to
determine suitable frequencies, trying 85, 170, and 340 kHz; we will report the
results for 340 kHz.
The results of the inversion are shown in Fig. 2.19. The important point to note
here is that the ‘zeroes’ in the upper layer are very sensitive to the systematic
and random errors, and are therefore not well reconstructed, whereas the flaw and
fastener cells are well reconstructed. This numerical experiment (and others which
we will not report here) is the basis for determining a heuristic rule, which will be
discussed now.
The stest-estimator generates a curve of scale, S, versus the normalized anomalous conductivity. The conductivity-estimate for each cell is chosen to be that
which produces the (unique) minimum of this curve. In Fig. 2.20 we plot S versus
conductivity for cells 86 and 87 of Fig. 2.19. Cell no. 86 contains flawed material,
whereas no. 87 contains host material.
The minimum for cell no. 87 would, ideally, be sharply peaked at zero, but is
quite broad, and lies too far to the left. Furthermore, it is a shallow null; i.e., the
2 Voxel-Based Inversion Via Set-Theoretic Estimation
coil
anomalous
region
top layer
bottom layer
8 mm
Fig. 2.17 Fastener with multisite damage as a surface-breaking anomaly
will report only the results of the S-estimator, which we call ‘stest’ in the captions
of several figures.
Problem No. 1: Fastener with Multisite Damage in a Slab In the first problem,
the anomalous region breaks the surface of a slab, as shown in Fig. 2.17. The data
are gathered at 200 Hz.
The results for this problem are shown in Fig. 2.18.
Problem No. 2: Layer-Stripping Using Multifrequencies In an attempt to
improve the results of Problem No. 1, we redo that problem by adopting a strategy,
that we call layer-stripping using multifrequencies. The idea is to isolate the top
layer of the anomalous region of Fig. 2.17 from the bottom layer. We do this by
gathering data at a very high frequency, such that the incident field produced by
the sensors does not penetrate to the bottom layer. This means that the anomalous
currents in the cells of the bottom layer will not contribute to the measured
impedances. Hence, we simply constrain these currents to be zero when we do
the reconstruction starting with the QR-decomposition. We used VIC-3D® to
determine suitable frequencies, trying 85, 170, and 340 kHz; we will report the
results for 340 kHz.
The results of the inversion are shown in Fig. 2.19. The important point to note
here is that the ‘zeroes’ in the upper layer are very sensitive to the systematic
and random errors, and are therefore not well reconstructed, whereas the flaw and
fastener cells are well reconstructed. This numerical experiment (and others which
we will not report here) is the basis for determining a heuristic rule, which will be
discussed now.
The stest-estimator generates a curve of scale, S, versus the normalized anomalous conductivity. The conductivity-estimate for each cell is chosen to be that
which produces the (unique) minimum of this curve. In Fig. 2.20 we plot S versus
conductivity for cells 86 and 87 of Fig. 2.19. Cell no. 86 contains flawed material,
whereas no. 87 contains host material.
The minimum for cell no. 87 would, ideally, be sharply peaked at zero, but is
quite broad, and lies too far to the left. Furthermore, it is a shallow null; i.e., the