24
2 Voxel-Based Inversion Via Set-Theoretic Estimation
E
y
klm =
1
3
J
y
klm
σ klm
+
1
6
J
y
kl+1m
σ kl+1m
+
1
6
J
y
kl−1m
σ klm
+
1
3
J
y
klm
σ kl+1m
δxδyδz . (2.25)
By adding the following equations for E x
kl+1m and E
y
k+1lm ,
E
x
kl+1m =
1
3
J x
kl+1m
σ kl+1m
+
1
6
J x
k+1l+1m
σ k+1l+1m
+
1
6
J x
k−1l+1m
σ kl+1m
+
1
3
J x
kl+1m
σ k+1l+1m
δxδyδz (2.26)
E
y
k+1lm =
1
3
J
y
k+1lm
σ k+1lm
+
1
6
J
y
k+1l+1m
σ k+1l+1m
+
1
6
J
y
k+1l−1m
σ k+1lm
+
1
3
J
y
k+1lm
σ k+1l+1m
δxδyδz , (2.27)
we obtain four equations in the four unknowns, σ klm , σ k+1lm , σ kl+1m , σ k+1l+1m .
These equations can be written in matrix form as:
⎡
⎢
⎢
⎣
x 00 x 10 0 0
y 00 0 y 01 0
0 0 x 01 x 11
0 y 10 0 y 11
⎤
⎥
⎥
⎦
⎡
⎢
⎢
⎣
ρ klm
ρ k+1lm
ρ kl+1m
ρ k+1l+1m
⎤
⎥
⎥
⎦ δxδyδz =
⎡
⎢
⎢
⎣
E x
klm
E
y
klm
E x
kl+1m
E
y
k+1lm
⎤
⎥
⎥
⎦ ,
(2.28)
where ρ klm = 1/σ klm and
x 00 =
J x
klm
3
+
J x
k−1lm
6
, y 00 =
J
y
klm
3
+
J
y
kl−1m
6
(2.29)
x 10 =
J x
klm
3
+
J x
k+1lm
6
, y 01 =
J
y
klm
3
+
J
y
kl+1m
6
(2.30)
x 01 =
J x
kl+1m
3
+
J x
k−1l+1m
6
, y 10 =
J
y
k+1lm
3
+
J
y
k+1l−1m
6
(2.31)
x 11 =
J x
kl+1m
3
+
J x
k+1l+1m
6
, y 11 =
J
y
k+1lm
3
+
J
y
k+1l+1m
6
.
(2.32)
Note that on the borders of the flaw, some of the expansion coefficients for the
anomalous currents vanish. In particular
J
x
klm =
nonzero 0 ≤ k ≤ N x − 2, 0 ≤ l ≤ N y − 1, 0 ≤ m ≤ N z − 1
zero
otherwise
(2.33)
J
y
klm =
nonzero 0 ≤ k ≤ N x − 1, 0 ≤ l ≤ N y − 2, 0 ≤ m ≤ N z − 1
zero
otherwise
(2.34)