2.5 Some Examples of the Inversion Algorithm
39
Fig. 2.12 Reconstruction of
the middle row and column of
the buried checkerboard
y
x
-0.97
-0.97
-0.96
-0.96
0.00
0.00
0.00
0.00
-1.00
0.00
0.00
0.00
0.00
When we attempted to reconstruct the buried checkerboard using the previous
16 × 16 raster scan with the receiver, we obtained a reasonable reconstruction– one
that was good enough to indicate the presence of a checkerboard flaw beneath a
layer of host material, but not as good as Fig. 2.12. See Fig. 2.13 for these results.
A Double Checkerboard When we stack two checkerboards with the opposite
polarity on top of each other, and reconstruct at 50 MHz using a 31 × 31 receiver
raster scan, we have difficulty obtaining an accurate reconstruction, except for the
zeros of the top layer.
We have developed an empirical rule for determining the quality of a reconstructed cell, when using either the LMS or S-estimators. We plot the median of the
squares of the residuals, in the case of the LMS-estimator, or the scale factor, in the
case of the S-estimator, versus the sought-for parameter, θ , for −1 ≤ θ ≤ 9 (say, or
some other upper bound, perhaps 0). If the minimum value of the minimum median
or scale factor is two or more orders of magnitude smaller than the maximum value
of median or scale, then the resulting answer is reliable; otherwise, it is suspect.
For example, in Fig. 2.14 we plot the curve of the logarithm of the median versus
θ , for −1 ≤ θ ≤ 0, for cell 1 (corner cell, top layer, conductivity = 0) and cell 2
(conductivity = −1). The curve for cell 1 clearly satisfies our empirical rule for a
reliable reconstruction, whereas that for cell 2 does not. The reconstructed value for
cell 1 is zero, whereas it is not well defined for cell 2.
The source of this problem in our inversion algorithm is the quantity and quality
of data that are presented to the QR-decomposer in the first stage of the algorithm.
Typically, the QR-method in least-squares analysis produces reliable results if the
system of equations is quite over-determined. Hence, there are three possible ways
39
Fig. 2.12 Reconstruction of
the middle row and column of
the buried checkerboard
y
x
-0.97
-0.97
-0.96
-0.96
0.00
0.00
0.00
0.00
-1.00
0.00
0.00
0.00
0.00
When we attempted to reconstruct the buried checkerboard using the previous
16 × 16 raster scan with the receiver, we obtained a reasonable reconstruction– one
that was good enough to indicate the presence of a checkerboard flaw beneath a
layer of host material, but not as good as Fig. 2.12. See Fig. 2.13 for these results.
A Double Checkerboard When we stack two checkerboards with the opposite
polarity on top of each other, and reconstruct at 50 MHz using a 31 × 31 receiver
raster scan, we have difficulty obtaining an accurate reconstruction, except for the
zeros of the top layer.
We have developed an empirical rule for determining the quality of a reconstructed cell, when using either the LMS or S-estimators. We plot the median of the
squares of the residuals, in the case of the LMS-estimator, or the scale factor, in the
case of the S-estimator, versus the sought-for parameter, θ , for −1 ≤ θ ≤ 9 (say, or
some other upper bound, perhaps 0). If the minimum value of the minimum median
or scale factor is two or more orders of magnitude smaller than the maximum value
of median or scale, then the resulting answer is reliable; otherwise, it is suspect.
For example, in Fig. 2.14 we plot the curve of the logarithm of the median versus
θ , for −1 ≤ θ ≤ 0, for cell 1 (corner cell, top layer, conductivity = 0) and cell 2
(conductivity = −1). The curve for cell 1 clearly satisfies our empirical rule for a
reliable reconstruction, whereas that for cell 2 does not. The reconstructed value for
cell 1 is zero, whereas it is not well defined for cell 2.
The source of this problem in our inversion algorithm is the quantity and quality
of data that are presented to the QR-decomposer in the first stage of the algorithm.
Typically, the QR-method in least-squares analysis produces reliable results if the
system of equations is quite over-determined. Hence, there are three possible ways