P. Schmoldt, PhD - MTNet - DIAS

3. Mathematical description of electromagnetic relations
Fig. 3.3.: Illustration of apparent resistivity ρa and phase φ behaviour in the complex plane using the relevant period range for the case
of (a) a resistive medium underlying a more conductive layer, and (b) a conductive medium underlying a more resistive layer. ρa(m1)
and ρa(m2) refer to the values of the upper and lower layer respectively, observed for periods that are unaffected by the resistivity
interface.
only the downward term F0 exists. The impedance Z in every layer m can be derived from
wave number k, and thickness t of the layer, and impedance of the layer m+1 below using
a modified version of Wait’s recursion formula [Wait, 1954], i.e.
Zm = Γm (kmZm+1 + Γm+1 tanh(kmtm))
, (3.59)
Γm+1 + kmZm+1 tanh(kmtm)
with Γm = ıωµm. The resistivity measured at the surface for a certain frequency is therefore
not related to the resistivity of a single layer but can be pictured as the weighted
integral over the electric parameters of all layers, usually referred to as apparent resistivity
ρa(T) = 1
µω |Zz=0(T)| 2 . (3.60)
From Equations 3.32, 3.59, and 3.60 it can be deduced that for the case of a resistive
medium underlying a more conductive layer, for the period range sensitive to the transition
zone, the apparent resistivity increases with period, whereas the phase decreases. For
longer periods, the phase will then re-establish its original value of 45 degrees, presuming
that no other effects exert influence on the present EM fields. For the opposite case of
a conductive medium underlying a more resistive layer, the effect is reversed, exhibiting
a decrease of apparent resistivity and an increase of phase. In both cases, the phase
effect is usually observable at shorter periods than the resistivity change (cf. Sec. 4.2.1);
however, this is simply due to the greater relative response of the phase at shorter periods.
The different behaviour of apparent resistivity and phase at an interface becomes obvious
when displayed in terms of real and imaginary values of resistivity; respective curves in
the complex rho-plane for two layered 1D models are shown in Figure 3.3.
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