Chapter 2
Voxel-Based Inversion Via Set-Theoretic
Estimation
2.1 The Electromagnetic Model Equations
The starting point for any electromagnetic inverse problem is Maxwell’s equations:
∇ × E(r) = −jωμ 0 H(r)
∇ × H(r) = jωω 0 E + σ (r)E(r)
= jωω 0 E(r) + (σ f (r) − σ h )E(r) + σ h E(r)
= jωω 0 E(r) + σ
(a) (r)E(r) + σ h E(r) ,
(2.1)
where σ f (r) is the flaw conductivity, σ h is the uniform host conductivity, and σ (a) (r)
is the anomalous conductivity. The product, σ (a) (r)E(r) defines the anomalous
electric current density, J(r).
The formal solution of Maxwell’s equations can be obtained by equating the total
electric field, E(r) = J(r)/σ a (r), to the sum of the incident field, that is produced
by the current in the exciter coil, and the scattered field, that is due to the anomalous
electric current:
E
(i)
x (r) =
J x (r)
σ a (r)
− E
(s)
x (r) [J]
E
(i)
y (r) =
J y (r)
σ a (r)
− E
(s)
y (r) [J]
E
(i)
z (r) =
J z (r)
σ a (r)
− E
(s)
z (r) [J] .
(2.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_2
19