Specifications

University of Pretoria etd – Combrinck, M (2006)
In these equations the dielectric permittivity, magnetic permeability and electric
conductivity should all be regarded as tensor functions of angular frequency, position, time,
temperature, pressure and magnetic/electric field strength. However, in order to derive
analytical solutions some assumptions regarding these parameters are necessary.
ASSUMPTIONS:
• All media are linear, isotropic and homogeneous and possess physical properties,
which are independent of time, temperature and pressure (implying scalar
presentation of physical properties rather than tensors).
• The magnetic permeability of all media is assumed to be equal to that of free space,
i.e. µ = µ
0
.
The TDEM solutions are derived by calculating the frequency domain electromagnetic
(FDEM) responses for equivalent models and then applying Fourier transforms to these
results. In order to solve the FDEM equations analytically some further assumptions are
necessary.
ADDITIONAL ASSUMPTIONS:
• There are no free electric charges or current in the medium, i.e.
.0;0
=
⋅
∇
=
⋅
∇
= ED
ρ
• We assume a harmonic time varying field, i.e.
and
ti
e
ϖ −
=
0
EE .
ti
e
ϖ −
=
0
BB
• Displacement currents are much smaller than induction currents, i.e.
j
D
<<
∂
∂
t
and EH
σ
=
×
∇
implying a wavenumber of
in the solution of the wave (Helmholz) equation. ωµσ ik =
2
The receiver and transmitter are in the same plane of observation (z=0), also indicating the
air-surface boundary in the case of a half space or layered earth.
Incorporating these assumptions into calculations the analytical solutions for a number of
different geological models can be found. For the purposes of this study there are four
cases deserving special attention, namely:
• Conductive half space
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