IEEE Geoscience and Remote Sensing Magazine - March 2019 - 79

where p d[m, m + Tm n], Err(Tm n) is the error made on
the sensitivity due to the first-order approach, Tm n is the
perturbation (defined relative or absolute), and e is the
numerical error made (e.g., roundoff errors). The forward
model must be resolved N times, with N representing the
number of layers (hence brute force, due to the number of
computations)-each time with a perturbation Tm n of the
governing nth layer model parameter m n, where m can represent one of three physical properties: 1) EC v n , 2) magnetic susceptibility l n, or 3) dielectric permittivity e n .
PSEUDOCODE
To provide a more practical approach in terms of modeling implementation, a general pseudocode (the schematic
overview in Figure 3) is passed along such that a smooth
integration into one's preferred software package can be
provided. The pseudocode was based on a combination of
diverse literature [2], [14], [17], [19], [20], [24], [25].
The quasi-static assumption can be used by adapting the
theoretical approach as discussed previously or by setting
the dielectric permittivity e n of each layer equal to zero
and the free-space dielectric permittivity e 0 equal to zero,
such that one can, on a regular basis, switch between the
assumptions in the algorithm.
INPUTS
To obtain a specific FDEM instrument's forward response
or sensitivity profile above a given layered half-space, the

INPUT Sensor

INPUT Model

characteristics of both the considered FDEM instrument
(sensor characteristics) and layered half-space (model characteristics) need to be defined. Additional input specifications are given in Table S1 in "Code Input Structure." Although we made use of structures in MATLAB and objects
in Python for the sensor and model input, the choice is entirely up to the user.
FUNCTIONS
The preceding theoretical description of forward modeling
is subdivided into different general functions (Figure 3) to
allow a more user-friendly implementation, i.e., 1) forward
modeling (or, alternatively, normalization), 2) magnetic

theoretical Sensitivity (Reflection coefficient)
Calculating the theoretical sensitivity requires for each layer n [14], [41]
2Y0
2rTE
2Yt1
2v n =- ^Y0 +Yt1h2 2v n ,

(S4)

2Yt1 2Yt1 2Yt2
2Ytn -1 2Ytn
2v n = 2Yt2 2Yt3 g 2Ytn 2v n ,

(S5)

Y 2n -1(1- tanh 2 (u n h n))
2Ytn -1
=
t
2Y n
(Yn -1 +Ytn tanh (u n h n))2

(S6)

2Ytn
2v n =g.

(S7)

with [18]

with

and

It should be noted that (S7) is not presented in full following the straightforwardness of its derivative and the length of its full expression in the linear domain.
Derivations for magnetic susceptibility and dielectric permittivity are analogous.

FUNCTION Sensitivity Modeling

QP and IP
Sensitivity

FUNCTION Forward Modeling

QP and IP
Response

FUNCTION Magnetic Fields

FUNCTION Digital Filter

FUNCTION Primary Sampled Function
FUNCTION Secondary Sampled Function
FUNCTION Reflection Coefficient
or
FUNCTION Propagation Matrix
(See "Propagation Matrix")
FIGURE 3. A schematic overview of the pseudocode inputs and

inPut sensor
x coordinate of receiver x (m; Figure 1)
y coordinate of receiver y (m; Figure 1)
z coordinate of receiver z (m; note: positive axis pointed down; Figure 1)
Height of transmitter h (m; Figure 1)
Transmitter moment M (A/m2)
Operating frequency f (Hz)
Coil orientation (any two-letter combination of X, Y, and Z)
Angular frequency ~ [rad/s; (24)]
9 Coil spacing r (m; Figure 1)

1
2
3
4
5
6
7
8

inPut model
1
2
3
4

EC of layer(s) v n (S/m)
Magnetic susceptibility of layer(s) l n (-)
Dielectric permittivity of layer(s) e n (F/m)
Thickness of layer(s) t n (m)

functions for both the forward and sensitivity modeling.
march 2019

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IEEE Geoscience and Remote Sensing Magazine - March 2019

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2019
