P. Schmoldt, PhD - MTNet - DIAS

3. Mathematical description of electromagnetic relations
The general solution for Equation 3.19 is
F(z) = F0e −kz + F1e +kz · e ıωt
(3.20)
with the (positive) wave number
k = √
ωµσ
ıωµσ = (1 + ı) . (3.21)
2
However, since F1e +kz is unreasonable as it becomes infinitive for large values of z, i.e.
lim
z→∞ F1e +kz = ∞, the fields to be considered here are of the form
F(z) = F0e −kz−ıωt . (3.22)
Applying the time dependence of magnetic field described by Equation 3.16 to Faraday’s
Law (Eq. 3.2) yields
∇ × E = −∂t B = −ıωB. (3.23)
From this, two decoupled relationships between the horizontal electric fields and the respective
orthogonal horizontal magnetic field can be derived as shown in the following
paragraphs. In MT, the two cases are commonly referred to as transverse electric (TE)
and transverse magnetic (TM) modes, corresponding to the field’s orientation relative to
a lateral conductivity interface (described in more detail in Chapter 4).
For a setting with a complex 3D subsurface, both modes can be correlated, resulting in
further complication of the fundamental MT relationships. For the purpose of illustration,
only the case of a layered subsurface (or a situation where the only horizontal conductivity
interface is aligned with the coordinate system) is considered here, yielding simple
relationships for the EM fields. First, the case of an electric field in x-direction and a
magnetic field in y-direction is examined; therein no assumptions are made which one is
aligned with the conductivity interface. Considering only the terms of Equation 3.23 that
have an êy component, and, again presuming that the waves are uniform, yields
∂zEx = −ıωBy (3.24)
Eq.3.22
⇐⇒ kEx0✘✘✘✘ e −kz−ıωt = ıωBy0✘✘✘✘ e −kz−ıωt e −ıφxy , (3.25)
with the index of the phase indicating the relation to the phase difference between Ex
and Hy fields. The relationship between the electric and magnetic field can therefore be
written as
34
Ex0
By0
= ıωe−ıφxy
√ ıωµσxy
=
ıω
µσxy
· e −ıφxy , (3.26)