P. Schmoldt, PhD - MTNet - DIAS

3.4. Boundary conditions
materials and their condition, it is important to examine the effects of such conductivity
interfaces on the electric current and EM fields.
From Maxwell’s Equations (Eqs. 3.1 - 3.4), with the aid of Stokes’ and Gauss’ theorems
(Eqs. 3.6 and 3.7), it becomes immediately evident that the relationship for EM
fields and electric currents on conductivity interfaces are such as given in Table 3.2. In
Continuity on conductivity interface
Component
Normal Transverse
B
H
E
D
Field
El. current Normal Transverse
J
Tab. 3.2.: Behaviour of electromagnetic fields and electric current on conductivity interfaces with their components in regards to the
orientation of the interface.
summary, the magnetic fields B and hence the magnetic field strength H are constant
on conductivity interfaces (presuming no magnetic permeability variation along with the
conductivity change). The electric current J exhibits a continuous normal and a discontinuous
transverse component, and since the electric field E is related to the current via
the local conductivity as described by Ohm’s Law (Eq. 3.5) its behaviour is exactly opposite
(and so is the electric displacement fields D, presuming constant permittivity). The
relation between the electric components of two areas can be given in a form similar to
Snell’s Law, i.e.
and
J1T
J2T
E1N
E2N
= σ1ET
σ2ET
= σ2JN
σ1JN
= σ1
σ2
= σ2
σ1
= ρ2
ρ1
= ρ1
ρ2
= tan(α1)
, (3.57)
tan(α2)
= tan(α2)
, (3.58)
tan(α1)
with αi representing the angles between the flow lines of electric current and a vector
normal to the conductivity interface.
3.4.1. Horizontal interfaces
To illustrate the effect of horizontal interfaces on MT observations consider a layered
subsurface with Q layers, hence Q interfaces (including the air-surface interface at depth
z = 0). In each of the first Q-1 layers the general form of the EM fields as described in
Equation 3.20 is valid, including the upward term F1, whereas in the bottom half-space
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