2.5 Some Examples of the Inversion Algorithm
31
necessary as we add more rows to the constraint equation; this will happen as we
satisfactorily reconstruct more and more cells.
The second method uses direct elimination by premultiplication using both
orthogonal and nonorthogonal transformation matrices, and the third solves the
problem by using weighted least-squares. We have tested the second and third
methods, and have found them to produce good results. We used the second method
in the examples reported later in this chapter.
Once the minimum-norm solution of the constrained least-squares problem has
been computed, the algorithm proceeds as before. The currents are substituted into
(2.41), thereby determining the total field in each cell (we can exclude cell klm,
unless we believe that it could be improved). Then the feasibility set is fitted with a
straight line by means of robust estimation; the slope of this line is accepted as the
normalized conductivity of the cell.
If we are still not satisfied with the results of one or more cells, we will perform
a new experiment. The experiment could be simply a new frequency of excitation,
which would make the process a multifrequency algorithm.
2.5 Some Examples of the Inversion Algorithm
Introduction The model examples in this section include surface-breaking and
buried flaws in a half-space. We simulate a transmit-receive (T/R) probe configuration, which includes a single transmitting and receiving probe. Each probe
is scanned independently of the other, but the receiving probe is assumed to be
connected to an infinite-impedance amplifier. This means that it carries no current;
hence, the excitation of the flaw is accomplished solely by means of the transmitter
probe. In all cases the transmitting and receiving coils are identical.
In all the problems considered in this chapter, the host region is a half-space,
whose conductivity is 10 5 S/m. The anomalous conductivity is normalized to have
a value σ a = σ f /σ h − 1, where σ f is the conductivity of the flaw, and σ h is the
conductivity of the host. Hence, for a void, in which the conductivity of the flaw,
σ f = 0, we have σ a = −1, and this is the smallest value that the anomalous
conductivity can take, because σ f ≥ 0. For an unflawed cell, σ f = σ h , which
implies that σ a = 0.
In principle, there is no upper limit to the value that σ a can assume, but in practice
the conductivity of the flaw is not likely to exceed that of copper (= 5.8 × 10 7 ),
which means that for typical host materials, whose conductivities are in the range
10 6 to 10 7 , σ a is likely to be bounded by 9 or so. In the numerical experiments
presented here, we used this as our upper bound in searching for the best fit for the
conductivity of each cell.
A Surface-Breaking Slot at 50 kHz Consider a cubic region intersecting the
surface of the half-space and extending 3 mm vertically (the z-direction) into the
half-space. The (x, y) dimensions are also 3 mm, centered at the origin in (x, y)-
31
necessary as we add more rows to the constraint equation; this will happen as we
satisfactorily reconstruct more and more cells.
The second method uses direct elimination by premultiplication using both
orthogonal and nonorthogonal transformation matrices, and the third solves the
problem by using weighted least-squares. We have tested the second and third
methods, and have found them to produce good results. We used the second method
in the examples reported later in this chapter.
Once the minimum-norm solution of the constrained least-squares problem has
been computed, the algorithm proceeds as before. The currents are substituted into
(2.41), thereby determining the total field in each cell (we can exclude cell klm,
unless we believe that it could be improved). Then the feasibility set is fitted with a
straight line by means of robust estimation; the slope of this line is accepted as the
normalized conductivity of the cell.
If we are still not satisfied with the results of one or more cells, we will perform
a new experiment. The experiment could be simply a new frequency of excitation,
which would make the process a multifrequency algorithm.
2.5 Some Examples of the Inversion Algorithm
Introduction The model examples in this section include surface-breaking and
buried flaws in a half-space. We simulate a transmit-receive (T/R) probe configuration, which includes a single transmitting and receiving probe. Each probe
is scanned independently of the other, but the receiving probe is assumed to be
connected to an infinite-impedance amplifier. This means that it carries no current;
hence, the excitation of the flaw is accomplished solely by means of the transmitter
probe. In all cases the transmitting and receiving coils are identical.
In all the problems considered in this chapter, the host region is a half-space,
whose conductivity is 10 5 S/m. The anomalous conductivity is normalized to have
a value σ a = σ f /σ h − 1, where σ f is the conductivity of the flaw, and σ h is the
conductivity of the host. Hence, for a void, in which the conductivity of the flaw,
σ f = 0, we have σ a = −1, and this is the smallest value that the anomalous
conductivity can take, because σ f ≥ 0. For an unflawed cell, σ f = σ h , which
implies that σ a = 0.
In principle, there is no upper limit to the value that σ a can assume, but in practice
the conductivity of the flaw is not likely to exceed that of copper (= 5.8 × 10 7 ),
which means that for typical host materials, whose conductivities are in the range
10 6 to 10 7 , σ a is likely to be bounded by 9 or so. In the numerical experiments
presented here, we used this as our upper bound in searching for the best fit for the
conductivity of each cell.
A Surface-Breaking Slot at 50 kHz Consider a cubic region intersecting the
surface of the half-space and extending 3 mm vertically (the z-direction) into the
half-space. The (x, y) dimensions are also 3 mm, centered at the origin in (x, y)-