12.8 Inverse Problems
335
the measured data in Fig. 12.12 was performed at T = 300 ◦ K, then we can solve
(12.32) for E g
E g = kT ln[2σ 0 /σ (E g ) − 1]
= 0.02586 ln
10000
σ (E g )
− 1
= 0.0157eV .
(12.33)
Now, assuming that all other parameters in (12.28) are known, we can solve for
by inserting ΔE gap = −0.0157 into the left-hand side.
This example illustrates the fact that we can apply the methods of eddy-current
NDE to carbon nanotubes once we have rigorous models for computing the electromagnetic properties of these tubes. These models require quantum-mechanical
calculations. Further examples of more complicated model-based inverse problems,
in which a number of variables are jointly determined, are given in [111].
We can scan the system of Fig. 12.17 and apply voxel-based inverse methods to
map out E g throughout the region using the models in Chap. 2.
12.8.4 Voxel-Based Inversion: A Surface-Breaking
Checkerboard at 50 MHz
In this experiment we consider the complex checkerboard flaw that is shown in
Fig. 12.14, and perform a 16×16 raster scan with the transmitter and receiver probes
at 50 MHz. The number in each voxel is the anomalous conductivity, which is the
actual conductivity minus the host conductivity, divided by the host conductivity.
Thus, a zero indicates that that voxel contains host material, whereas −1 indicates
that the actual conductivity is zero.
The checkerboard is a difficult anomaly to reconstruct, because the ‘scene’
changes so rapidly; i.e., it contains high spatial frequencies and the receiver scan
must be fine enough to reconstruct these frequencies. The reconstruction using the
LMS-estimator is quite good (corresponding results were obtained using the Sestimator and classical estimator). We show the reconstruction of the middle row
and column in Fig. 12.15.
12.8.5 Voxel-Based Inversion: A Buried Checkerboard
When we bury the checkerboard below a layer of host material we get the added
complication of increasing the number of unknowns that are to be determined during
the reconstruction. This, together with the fact that the checkerboard scene contains
335
the measured data in Fig. 12.12 was performed at T = 300 ◦ K, then we can solve
(12.32) for E g
E g = kT ln[2σ 0 /σ (E g ) − 1]
= 0.02586 ln
10000
σ (E g )
− 1
= 0.0157eV .
(12.33)
Now, assuming that all other parameters in (12.28) are known, we can solve for
by inserting ΔE gap = −0.0157 into the left-hand side.
This example illustrates the fact that we can apply the methods of eddy-current
NDE to carbon nanotubes once we have rigorous models for computing the electromagnetic properties of these tubes. These models require quantum-mechanical
calculations. Further examples of more complicated model-based inverse problems,
in which a number of variables are jointly determined, are given in [111].
We can scan the system of Fig. 12.17 and apply voxel-based inverse methods to
map out E g throughout the region using the models in Chap. 2.
12.8.4 Voxel-Based Inversion: A Surface-Breaking
Checkerboard at 50 MHz
In this experiment we consider the complex checkerboard flaw that is shown in
Fig. 12.14, and perform a 16×16 raster scan with the transmitter and receiver probes
at 50 MHz. The number in each voxel is the anomalous conductivity, which is the
actual conductivity minus the host conductivity, divided by the host conductivity.
Thus, a zero indicates that that voxel contains host material, whereas −1 indicates
that the actual conductivity is zero.
The checkerboard is a difficult anomaly to reconstruct, because the ‘scene’
changes so rapidly; i.e., it contains high spatial frequencies and the receiver scan
must be fine enough to reconstruct these frequencies. The reconstruction using the
LMS-estimator is quite good (corresponding results were obtained using the Sestimator and classical estimator). We show the reconstruction of the middle row
and column in Fig. 12.15.
12.8.5 Voxel-Based Inversion: A Buried Checkerboard
When we bury the checkerboard below a layer of host material we get the added
complication of increasing the number of unknowns that are to be determined during
the reconstruction. This, together with the fact that the checkerboard scene contains