P. Schmoldt, PhD - MTNet - DIAS

Depth
d s
d 1
d 2
d n-2
d n-1
t 1
t 2
t n-1
t' n
F n=F 0/e
F n-1
Magnitude of penetrating wave
F 2
3.3. Magnetotelluric induction area
F 1
F 0
layer 1: r 1, m 1
layer 2: r 2, m 2
layer n-1: r n-1, m n-1
layer n: r n, m n
Fig. 3.2.: Change of wave magnitude with depth for the case of a 1D subsurface with n-layers; with Fi Magnitude of the wave at the
bottom of the i-th layer, di depth to the bottom of the i-th layer, ti thickness of the i-th layer, ρi electric resistivity of the i-th layer, µi
magnetic permeability of the i-th layer, and δs the skin depth.
3.42 for the layer n:
F(δs) = 1
e F0 = F0e −1 = Fn−1e −Re(kn)t ′ n (3.48)
where Fn−1 is the amplitude of the penetrating wave at the bottom of the layer n-1. The
only unknown in Equation 3.48, i.e. Fn−1, can be represented using the wave amplitude
at the bottom of the layer n-2 (Fn−2) as well as absorption kn−1 and thickness tn−1 of the
layer n-1:
F0e −1 = Fn−2e −Re(kn−1)tn−1
e −Re(kn)t ′ n. (3.49)
Fn−1
In turn, Fn−2 can be represented using the wave amplitude at the bottom of the layer n-3
(Fn−3) and absorption kn−2 and thickness tn−2 of the layer n-2:
F0e −1 = Fn−2 e −Re(kn−1)tn−1 e −Re(kn−1)tn−1 −Re(kn)t
e ′ n. (3.50)
Fn−2
This process can be repeated up to the Earth surface where the initial amplitude of the
wave F0 is used, yielding the modified form of Equation 3.48:
F0e −1 = F0e −Re(kn)t ′ n · Π n−1
i=1 e−Re(ki)ti
= F0e −Re(kn)t ′ n− n−1
i=1 Re(ki)ti .
(3.51)
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