28
2 Voxel-Based Inversion Via Set-Theoretic Estimation
only with the transmitter, because the transmitter is the sole exciter of the system
(the receiver is assumed to carry no current). Hence, (2.40) is replaced by
Z 1 (v) = −
LMJ
E
(1)
LMJ (v) · J LMJ (v)
Z 2 (v) = −
LMJ
E
(2)
LMJ (v) · J LMJ (v)
. . .
Z N s (v) = −
LMJ
E
(N s )
LMJ (v) · J LMJ (v) ,
(2.42)
where E
(n)
LMJ (v) is the “incident” field produced by the sensor when it is in its nth
scan position, during the vth view. This field is known a priori, because we know
the location and geometry of the receiving sensor during the vth view.
The minimum-norm solution of (2.42) is given by
J LMJ (v) = M
† (v)
⎡
⎢
⎣
Z 1 (v)
. . .
Z N s (v)
⎤
⎥
⎦ ,
(2.43)
where M † (v) is the pseudoinverse of the matrix in (2.42). In the examples of
this chapter, we used the QR-decomposition [62] to compute the minimum-norm
solution. The algorithm that is presented in [62] to solve for the minimum-norm
solution allows the user to define a tolerance level, from which an effective pseudorank is obtained for the system matrix. The tolerance is chosen to produce a stable
solution in the presence of noisy data.
The electric field produced by this current is gotten by substituting (2.43) into
(2.41):
E lmj (v) = E
(i)
lmj (v) +
LMJ
G
(ee)
jJ (l − L, m − M; ω) · J LMJ (v) .
(2.44)
Note that this is the correct electric field-moment corresponding to the current J, and
VIC-3D® computes this field very quickly and accurately. This gives VIC-3D®
the advantage over other field-solving methods, such as finite-elements or finitedifferences. More important, however, is that (2.44) is the unique field-moment
associated with the current, and is calculated to the same precision as the current,
which will ensure that the final step in the algorithm, namely the statistical decision
step that is described below, will be meaningful.
Given the electric field moments, we can then calculate an expansion for the
electric field within the flaw. The resulting electric field expansion coefficients will
2 Voxel-Based Inversion Via Set-Theoretic Estimation
only with the transmitter, because the transmitter is the sole exciter of the system
(the receiver is assumed to carry no current). Hence, (2.40) is replaced by
Z 1 (v) = −
LMJ
E
(1)
LMJ (v) · J LMJ (v)
Z 2 (v) = −
LMJ
E
(2)
LMJ (v) · J LMJ (v)
. . .
Z N s (v) = −
LMJ
E
(N s )
LMJ (v) · J LMJ (v) ,
(2.42)
where E
(n)
LMJ (v) is the “incident” field produced by the sensor when it is in its nth
scan position, during the vth view. This field is known a priori, because we know
the location and geometry of the receiving sensor during the vth view.
The minimum-norm solution of (2.42) is given by
J LMJ (v) = M
† (v)
⎡
⎢
⎣
Z 1 (v)
. . .
Z N s (v)
⎤
⎥
⎦ ,
(2.43)
where M † (v) is the pseudoinverse of the matrix in (2.42). In the examples of
this chapter, we used the QR-decomposition [62] to compute the minimum-norm
solution. The algorithm that is presented in [62] to solve for the minimum-norm
solution allows the user to define a tolerance level, from which an effective pseudorank is obtained for the system matrix. The tolerance is chosen to produce a stable
solution in the presence of noisy data.
The electric field produced by this current is gotten by substituting (2.43) into
(2.41):
E lmj (v) = E
(i)
lmj (v) +
LMJ
G
(ee)
jJ (l − L, m − M; ω) · J LMJ (v) .
(2.44)
Note that this is the correct electric field-moment corresponding to the current J, and
VIC-3D® computes this field very quickly and accurately. This gives VIC-3D®
the advantage over other field-solving methods, such as finite-elements or finitedifferences. More important, however, is that (2.44) is the unique field-moment
associated with the current, and is calculated to the same precision as the current,
which will ensure that the final step in the algorithm, namely the statistical decision
step that is described below, will be meaningful.
Given the electric field moments, we can then calculate an expansion for the
electric field within the flaw. The resulting electric field expansion coefficients will