IEEE Geoscience and Remote Sensing Magazine - March 2019 - 77

Propagation Matrix
As an alternative to the reflection coefficient rTE, one can formulate the propagation of the EM wave through the subsurface in terms of a propagation
matrix P (S1) [2], [20]. Hence, (1)-(9) and (11) can be used by replacing rTE
with the ratio P 21 /P 11:
P =c

N
P 11 P 12
m = M1 % Mn
P 21 P 22
n =2

(S1)

Function propagation matrix (sensor, model, lambda mn)

with
J1
n0 u1
K 2 a 1+ n 1 u 0 k
M 1 =K
K 1 a 1- n 0 u 1 k
n1 u0
L2

n0 u1 N
1a
k
2 1- n 1 u 0 OO
n0 u1 O
1a
k
2 1+ n 1 u 0 P

(S2)

and
J
N
n n -1 u n
n n -1 u n
1a
1a
k
k
K
O
2 1+ n n u n -1
2 1- n n u n -1
O,
M n =K
K 1 a 1- n n -1 u n ke -2u n -1 t n -1 1 a 1+ n n -1 u n ke -2u n -1 t n -1O
n n u n -1
n n u n -1
2
L2
P

M
H PZZ = 4r
H SZZ = 4M
r

∞

(10)

^ rTE e u 0^ z -hhhm 2 J 0(mr) dm.

(11)

0
∞

#
0

(S3)

^ e -u 0 ^ z +hhh m 2 J 0(mr) dm,

#

Expressions of the present free-space magnetic fields
H 0 (A/m) used in the normalization are given for an X-directed magnetic dipole source located at (0, 0, -h):
∞

M 1 2x 2
H 0X =- 4r c r - 3 m # ^e -u0 ^ z +hhh m J 1(mr) dm
r
M x2
- 4r 2
r

∞

#
0

(12)

a Y -directed magnetic dipole source located at (0, 0, -h):
∞

M 1 2y
H 0Y =- 4r d r - 3 n # ^e -u0 ^ z +hhh m J 1(mr) dm
r
2

2
M y
- 4r 2
r

∞

#
0

(13)

and a Z-directed magnetic dipole source located at (0, 0, -h):
M
H 0Z = 4r

∞

#
0

^e -u 0 ^ z +hhh m 2 J 0(mr) dm.

(14)

The normalized response H [or coupling ratio; in parts
per million (ppm)] is then given by
H =10 6 $

HT - HP
HS
=10 6 $ 0 .
0
H
H

(15)

Alternatively, normalization is often described as follows (in ppm):
H =10 6 $
march 2019

HT - HP
HS
=10 6 $ P .
HP
H

calculate the magnetic permeability of each layer n n (28)
calculate the wavenumber of each layer u n (25)
calculate for the first layer M 1 (S2)
FoR every layer starting at the second layer
toward basement layer
5
calculate for each layer M n (S3)
6 EnD
7 calculate P (S1)
8 REtuRn the ratio P21/P11
EnD

Note that H 0 equals H P if full coupling takes place [(15),
(16)], e.g., with a ZZ [or horizontal coplanar (HCP)] coil
orientation. One should pay attention to the normalization method because, e.g., H SZX is normalized [according
to (15)] using H 0Z = H PZZ . Ultimately, FDEM instruments
use a phase-sensitive measurement between the (complex) coupling ratio, i.e., an IP (or real) and QP (or imaginary) measurement (in ppm):

QP = Im ^H h.

(16)

ieee Geoscience and remote sensing magazine

(17)
(18)

REFLECTION COEFFICIENT
The reflection coefficient rTE is defined by [25] as
rTE =

0

^e -u 0 ^ z +hhh m 2 J 0(mr) dm;

1
2
3
4

IP = Re ^H h,

0

^e -u 0 ^ z +hhh m 2 J 0(mr) dm;

where n 0 and n n are, respectively, the magnetic permeability of a free
space (equal to 4r10 -7 H/m) and of the nth layer [H/m; see (28)], n n is the
wavenumber of the nth layer [1/m; see (25)] and t n is the thickness of the nth
layer (m). Analogously, the quasi-static assumption can be applied.
The pseudocode of the propagation matrix function, which replaces the
reflection coefficient function, is given below.

Y0 - Yt 1
,
Y0 + Yt 1

(19)

where Y0 is the intrinsic admittance [see (22)] of the air halfspace, and Yt 1 is the surface admittance (at z = 0), which
can be determined recursively by starting at the basement
half-space (n = N) toward the surface (n =1) . The recursive
formula is outlined by Ward and Hohmann [25] for an Nlayered half-space (n 1 N):
Yt n +1 + Yn tanh (u n t n)
Yt n = Yn
,
Yn + Yt n +1 tanh (u n t n)

(20)

where Yn is the admittance of the nth layer [see (23)], u n is
its wavenumber [1/m; see (25)], and t n is its thickness (m).
It can be desirable to formulate tanh (u n t n) in terms of negative exponentials for greater numerical stability during numerical calculations [38]; however, this approach was not
followed herein. For the basement half-space (n = N), the
intrinsic admittance is
77
IEEE Geoscience and Remote Sensing Magazine - March 2019

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