P. Schmoldt, PhD - MTNet - DIAS

3. Mathematical description of electromagnetic relations
tween electric and magnetic fields. However, it is worthwhile to examine under what
circumstances this assumption is valid.
The relationship between electric and magnetic fields under consideration of the permittivity
is
êx : Hx = 1
ıεω + σyx Ey
k
(3.61)
and
êy : Hy = − 1
ıεω + σxy Ex
k
(3.62)
for the two orthogonal horizontal directions. Impedance and apparent resistivity are derived
from
Zi j = Ei
H j
(3.63)
and
ρai j
1
=
µω
Zi j
2 , (3.64)
respectively, with i, j ∈ [x, y] (cf. Sec. 3.2). To investigate the effect of permittivity,
absolute values of the impedance derived with (ρwith) and without (ρwithout) the first term
in Equations 3.61 and 3.62 are compared
ρwithout
ρwith
= |Zwithout| 2
= 2
|Zwith| 1
k (iεω + σ) 2
1
kσ = 1 +
2
εω
2 ; (3.65)
σ
for the sake of convenience indices of resistivity and impedance are dropped in here.
Using ω = 2π f and σ = ρ −1 this can be rewritten as
ρwithout
ρwith
= 1 + (2περ f ) 2 , (3.66)
showing that the effect of neglecting the permittivity term is given by the second term in
the right hand side of Equation 3.66, i.e.
Dperm := 2περ f. (3.67)
Thus, the deviation is positive proportional to permittivity, resistivity, and frequency, suggesting
an assessment in respect to the range of contributing parameters. Permittivity is
the product of the permittivity of free air ε0 = 8.89 · 10 −12 As/Vm and the permittivity of
the material εr with typical values of εr for Earth’s material lying between 3 (ice) and
81 As/Vm (water), and an average value of 20 As/Vm for dry rocks [Telford et al., 1990].
Common resistivity values for Earth’s materials derived by laboratory studies range from
10 −7 for graphite to 10 7 Ωm for dry igneous rocks (cf. Figs. 3.5 and 3.6, Chap. 5).
44
Since ε and ρ are both positive, deviation increases with increasing frequencies as