Content uploaded by Joost Hase
Author content
All content in this area was uploaded by Joost Hase on Jul 18, 2023
Content may be subject to copyright.
Citation: Hase, J.; Gurin, G.; Titov, K.;
Kemna, A. Conversion of Induced
Polarization Data and Their
Uncertainty from Time Domain to
Frequency Domain Using Debye
Decomposition. Minerals 2023,13,
955. https://doi.org/10.3390/
min13070955
Academic Editors: André Revil,
Damien Jougnot and Jacques Deparis
Received: 25 May 2023
Revised: 14 July 2023
Accepted: 14 July 2023
Published: 17 July 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
minerals
Article
Conversion of Induced Polarization Data and Their Uncertainty
from Time Domain to Frequency Domain Using
Debye Decomposition
Joost Hase 1,* , Grigory Gurin 2,3, Konstantin Titov 2and Andreas Kemna 1
1Geophysics Section, Institute of Geosciences, University of Bonn, 53115 Bonn, Germany;
kemna@geo.uni-bonn.de
2Institute of Earth Sciences, St. Petersburg State University, 199034 St. Petersburg, Russia;
k.titov@spbu.ru (K.T.)
3JSC ”VIRG-Rudgeofizika”, Office 244, Building 8A, Aerodromnaya Str., 197348 St. Petersburg, Russia
*Correspondence: hase@geo.uni-bonn.de
Abstract:
The time-domain (TD) induced polarization (IP) method is used as an extension of direct
current (DC) resistivity measurements to capture information on the ability of the subsurface to
develop electrical polarization. In the TD, the transient voltage decay is measured after the termi-
nation of the current injection. To invert tomographic TD IP data sets into frequency-domain (FD)
models of complex electrical resistivity, a suitable approach for converting TD IP transients and
their corresponding uncertainties into the FD is essential. To apply existing FD inversion algorithms
to TD IP measurements, a conversion scheme must transform the measured decay curves into FD
impedances and also propagate the corresponding measurement uncertainty from the TD to the FD.
Here, we present such an approach based on a Debye decomposition (DD) of the decay curve into a
relaxation-time distribution and the calculation of the equivalent spectrum. The corresponding FD
data error can be obtained by applying error propagation through all of these steps. To accomplish
the DD we implement a non-linear Gauss–Newton inversion scheme. We test the conversion scheme
in a synthetic study and demonstrate its application to field data on a tomographic TD IP data set
measured on the Maletoyvaemskoie ore field (Kamchatka, Russia). The proposed conversion scheme
yields accurate impedance data for relaxation processes, which are resolved by the TD measurements.
The error propagation scheme provides a reasonable FD uncertainty estimate, as confirmed by a
Monte Carlo analysis of the underlying parameter distributions.
Keywords: induced polarization; Debye decomposition; complex resistivity tomography
1. Introduction
In geoelectrics, the time-domain induced polarization (TD IP) method is used as an
extension of direct current (DC) resistivity measurements to capture information on the
ability of the subsurface to develop electrical polarization. Theoretical concepts fundamen-
tal to the IP method have been studied over the past few decades, primarily motivated by
the method’s application to mineral and reservoir characterization (e.g., [
1
–
7
]). Although
many field-scale IP measurements are conducted in the time domain (TD), the lithological,
textural, and hydraulic properties of the targeted rock have been found to relate especially
to the spectral characteristics of the IP phenomenon in numerous frequency-domain (FD)
laboratory studies (e.g., [
8
–
14
]). In the context of increasing the measurement accuracy
of laboratory and field instruments and emerging FD IP analysis approaches in terms of
complex resistivity, the petrophysical and hydrogeophysical communities have started to
establish the diagnostic potential of the IP method in their research fields (e.g., [
9
,
15
–
19
]).
Independent of the specific application at hand, accurate conversion between TD IP and
FD IP data is essential to exploit the FD information contained in TD IP measurements in a
Minerals 2023,13, 955. https://doi.org/10.3390/min13070955 https://www.mdpi.com/journal/minerals
Minerals 2023,13, 955 2 of 20
quantitative manner. An accurate conversion provides the possibility to draw quantitative
conclusions from the FD characteristic of the IP phenomenon with regard to the petrophys-
ical parameters of interest. Therefore, taking extra effort to ensure an accurate and stable
conversion of the data set is sensible.
Analytical representations of TD transients can be converted to the FD via the Fourier
and Laplace transforms. Specifically, in the conversion of TD IP measurements using
standard instrumentation, one faces the problem of sparsely sampled signals, which are
typically discretized at 20 time steps or less. To ensure a stable conversion, even with noisy
transients, adequate assumptions on their expected shape must be made and included
in the conversion. Care must be taken in formulating these assumptions since the use of
unsuitable or under-parameterized models can lead to a misrepresentation of the transients
and subsequent errors in the conversion. During tomographic inversions of electrical data
in the TD and FD, data points are often weighted with their respective errors (e.g., [
20
,
21
]).
Given individual error estimates for the transients, adequate error propagation through the
conversion scheme to the corresponding FD error estimates is therefore essential.
For the conversion of TD IP data sets to the FD, previous studies (e.g., [
3
,
22
,
23
]) have
used the assumption of a frequency-independent, constant phase angle (CPA), finding it
to be a valid approximation for many laboratory and field applications. As quantitative
applications of TD IP measurements call for increasingly high precision, and measurement
systems continue to improve, the limitations of the CPA approximation become relevant
and must be overcome to fully utilize the quantitative diagnostic potential of the TD IP
method. Using a parameterization of the FD complex resistivity in terms of the empirical
Cole–Cole model, other studies [
24
–
27
] have been able to invert TD IP data for the spatially
distributed spectral behavior of electrical properties in the subsurface. Given scenarios
for which the assumption of a Cole–Cole-like behavior is valid, this inversion approach
yields easy-to-interpret tomographic results and provides the possibility of relating imaged
Cole–Cole parameters to petrophysical properties using relations established in laboratory
experiments (e.g., [
28
]). A drawback of the approach is the Cole–Cole model’s limitation
with respect to the possible complexity of the FD characteristics it can represent such as
the superposition of multiple polarization processes or spectral characteristics that appear
asymmetric in log-log plots in general.
In this work, we describe a general-purpose approach for converting tomographic TD
IP data sets to the FD. Importantly, the approach also provides the possibility to accurately
propagate a TD data error estimate to a corresponding FD error estimate. The TD to FD
conversion scheme presented in this paper is based on the concept of Debye decomposition
(DD) (e.g., [
29
–
31
]). The DD can fit various types of TD transients while including very few
prior assumptions on their shape. A measured TD transient is decomposed into a number
of exponential decays. The decomposition is performed on a grid of predefined relaxation
times. Using an inversion approach, the appropriate contribution of each exponential decay
to the superposition can be determined, yielding a relaxation-time distribution (RTD).
Related by the Laplace transform, equivalent formulations of the Debye decomposition
exist in the TD and FD. The FD response can be calculated from the RTD. Martin et al.
[
32
] demonstrated this equivalence by performing the Debye decomposition on TD and
FD IP data sets obtained for the same samples. They found that the estimated RTDs were
mostly in agreement with each other, attributing deviations to limitations in TD IP data
quality. After converting each transient to the FD, we use a tomographic FD inversion code
to compute subsurface images of the complex electrical resistivity at single frequencies.
This work is structured as follows. After a short recap of the theoretical background
underlying the Debye decomposition, we formulate the inverse problem inherent in TD to
FD conversion. The inverse problem is solved using parameter optimization. We describe
the optimization algorithm and explain our choices for the hyperparameters. In a synthetic
validation study, we investigate the accuracy and limitations of the TD to FD conversion
scheme, complemented by a validation of the error propagation scheme. We conclude with
a demonstration of the overall approach on a tomographic TD IP field data set.
Minerals 2023,13, 955 3 of 20
2. Time-Domain to Frequency-Domain Conversion of Induced Polarization Data
2.1. Induced Polarization in the Time Domain
During a TD IP measurement, the transient voltage decay
V(t)
between the potential
electrodes is measured as a function of time
t
after the termination of the current injection.
After an abrupt decrease in the voltage from its primary value
V0
, associated with the DC
resistivity measurement to a secondary value V1, the voltage decreases continuously:
lim
t−→∞V(t) = 0. (1)
Normalization with respect to the primary voltage gives the normalized IP transient
η(t) = V(t)
V0
. The ratio of the initial voltage drop at the end of the current injection
t=
0 is
known as the chargeability [
2
]. The occurrence of this electrical relaxation phenomenon is
an expression of the subsurface’s ability to develop electrical polarization under an applied
electric field.
2.2. Debye Decomposition
The Debye decomposition (DD) is a semi-phenomenological model with the ability
to represent various types of electrical relaxation responses based on the superposition of
individual Debye responses. We adopted and modified the FD forward operator of the DD
after Nordsiek and Weller [31]:
ˆ
Z(ω) = R0−
M
∑
k=1
γk1−1
1+iωτk, (2)
where
M
is the number of Debye terms,
ω
is the angular frequency, and
i2=−
1 is the
imaginary unit. These parameters were used to calculate the spectrum of the complex elec-
trical impedance
ˆ
Z(ω)
, which is equivalent to the measured TD transient. The relaxation
time
τk
is the characteristic time constant of the
k
-th Debye term. The DC resistance
R0
is
the magnitude of the electrical impedance at the low-frequency limit of the spectrum:
R0=lim
ω−→0|ˆ
Z(ω)|. (3)
The parameters
γk
have the unit of resistance and scale the contributions of the
different Debye terms to the superposition. By choosing
γk
as scaling parameters and thus
deviating from the formulation of Nordsiek and Weller [
31
], we avoid correlated errors
between the different Debye terms in the superposition. Plotting the values of
γk
against
the corresponding relaxation times τkyields the RTD.
We estimate the values of
γk
by inverting the measured TD transient on a grid of
predefined relaxation times
τk
using the TD forward operator, which is equivalent to
Equation (2) (e.g., [30,33,34]):
η(t) = 1
R0
M
∑
k=1
γkexp −t
τk. (4)
A closer investigation of Equation (2) shows that it is reasonable to restrict the interpre-
tation of the derived spectrum to the angular frequencies
1
τmax <ω<1
τmin
, which assumes
that there is a smallest relaxation time
τmin
and a largest relaxation time
τmax
that can be
resolved with a given measurement. In this work, we chose
τmin
and
τmax
to be equal to
the first and last time values used for the discretization of the transient.
2.3. Formulation of the Inverse Problem
The goal of the DD is to decompose a measured TD transient into an RTD. We do
not use the discrete values of the normalized TD transient as data since their errors are
correlated due to division by the measured quantity V0. Instead, we choose
Minerals 2023,13, 955 4 of 20
d=R0(η1, . . . , ηi, . . . , ηN)T, (5)
at discrete time steps
t=(t1, . . . , ti, . . . , tN)T, (6)
which provides us with uncorrelated measurements to invert. In the model domain, the
inverse problem is discretized on a grid of
M
predefined,
log10
-spaced, relaxation times
τk
.
It is critical that the model domain discretization covers a wide enough parameter range and
provides enough degrees of freedom. With the first and last time steps of the discretized
transient being
t1
and
tN
, we choose
τk∈[
10
log10(t1)−1.5
, 10
log10(tN)+1.5 ]
, extending the
discretization of the model domain by 1.5 decades to the left and right of the sampled time
frame. The number of relaxation times
M
is based on the number of decades covered by
the model-domain discretization so that the density of the model-domain discretization is
consistent for differently sampled transients. For each decade in
τk
, we use 25 relaxation
times for the discretization of the model domain, which is 5 relaxation times more than the
20 per decade recommended by Weigand and Kemna [
35
], ensuring enough degrees of
freedom. To account for the wide range of values within
γk
and to restrict the inversion
from yielding results associated with
γk<
0, the natural logarithms of
γk
are used as the
model parameters:
m=ln γ1
γ0, . . . , ln γk
γ0, . . . , ln γM
γ0T
. (7)
Note that the division by
γ0=
1
Ω
is necessary to ensure that the argument of
ln(·)
is
dimensionless. For simplicity, this is implied in the notation
ln γk
γ0=ln (γk)
from here
on. The TD forward operator of the DD is modified as
ˆ
Z(t) =
M
∑
k=1
exp ln (γk)−t
τk, (8)
leading to the discrete TD forward operator (e.g., [36])
fi(m,ti) =
M
∑
k=1
Gik exp(mk), (9)
with
Gik =exp−ti
τk. (10)
Setting up the forward operator as a matrix-vector multiplication is favorable in terms
of computational performance due to the possibility of parallel computing. Note that we
formulate
f(·)
as a function of
m
and
t
. In most cases,
t=const
. The only exception is in
Section 2.4, where we discuss the stacking of subsequent injection pulses for a given model
realization; hence.
m=const
. From here on we only explicitly pass the argument to the
forward operator that is not constant.
The inverse problem that results from Equation (9) is non-linear. The corresponding
cost function
Ψ(m) = 1
2(d−f(m))TC−1
D(d−f(m))+1
2λmTRm, (11)
is minimized iteratively using a pseudo-Newton model update, as described in Taran-
tola [37]:
mq+1=mq+α∆m=mq−αJT
TD C−1
DJTD +λR−1JT
TD C−1
Df(mq)−d+λRmq. (12)
The Jacobian and the step length are represented by
JTD
and
α
, respectively. Since
the inverse problem is underdetermined, regularization λ>0 has to be applied to ensure
Minerals 2023,13, 955 5 of 20
that
JT
TD C−1
DJTD +λR
can be inverted. We use a roughness operator
R
that minimizes
the squared difference between the model parameters of the neighboring Debye terms.
The strength of the applied regularization is controlled by the parameter λ. Measurement
uncertainties are accounted for in the data-precision matrix
C−1
D
. The root-mean-square
error (RMSE)
ε=s(d−f(m))TC−1
D(d−f(m))
N(13)
is used to estimate the goodness of the data fit achieved by a given model realization in the
context of measurement uncertainties. For each iteration, the step length
α
is optimized
automatically using a line-search approach described in Kemna [
38
]. During the line search,
three trial-model updates are calculated for
αtrial ∈ {
0,
α/
2,
α}
and their corresponding
data fits
ε
are evaluated. The updated step length is then chosen as the minimum of a
parabola fitted through
ε(αtrial)
. The inversion is terminated if the norm of the model
update
|∆m|
vanishes, indicating that the maximum a posteriori (MAP) solution has
been found.
2.4. Adaptation to Common Data-Acquisition Strategies
Most TD IP measurement systems use gate integration during the measurement in
order to reduce high-frequency noise. This means that the transient returned by the instru-
ment is actually a sequence of mean values, each of which is calculated for a corresponding
time gate. Different choices can be made for the widths of the time gates. Given the case
where the widths of the time gates increase in a
log10
manner, Fiandaca et al. [
25
] suggested
using the center of the time gates, calculated using the geometric mean, as the time points to
represent the averaged values in the transient. Since this resolves the problem of adjusting
the forward operator to the gate integration, we use transients sampled at
log10
-spaced
time steps.
During a TD IP measurement, a sequence of opposing current pulses is injected into
the subsurface. The measured TD transient is calculated as the mean of the transients
induced by these pulses to increase the signal-to-noise ratio. For a given model realization
m=const, the forward operator has to be extended as (e.g., [25,34])
fStackj(t) =
j
∑
m=1
2
∑
k=1
(−1)m+kf(t+1(k−1)TOn +1(j−m)(TOn +TOff))(14)
and
fA(t) = 1
NStacks
NStacks
∑
j=1
(−1)j+1fStackj(t), (15)
where
f(·)
represents the forward calculation given in Equation (9),
fStackj
represents the
j
-th transient with the superposition of the transients from prior injection pulses taken into
account, and
fA
is the averaged transient after
NStacks
stacks. We use the formulation in
Equation (15) to adjust the forward calculation
f(·)←fA(·)
appropriately when dealing
with stacked transient data.
3. Uncertainty Approximation and Propagation
Choosing a suitable regularization strength is essential for the conversion, as well
as the error propagation from the TD to the FD. In our work, we base the choice of an
appropriate regularization strength on
ε
, assuming that we have a good estimate of the data
uncertainty. We choose the regularization strength in such a way that the MAP solution
achieves
ε=
1, fitting the measurements appropriately in the context of their respective
uncertainties (e.g., [
20
,
38
,
39
]). Details of our approach for finding the optimal regularization
strength
λf inal
can be found in Appendix A. The analytical propagation of measurement
Minerals 2023,13, 955 6 of 20
errors from the TD to the FD is based on an adequate uncertainty characterization in
the model domain. Here,
λR
can be identified as the precision matrix of the prior term.
Assuming normally distributed model-parameter uncertainties near the MAP solution, the
posterior covariance matrix [37]
CM=JT
TD C−1
DJTD +λR−1, (16)
is approximated from the result of the inversion. It can be used as an uncertainty measure
for the estimated model parameters, considering both the measurement uncertainty and
prior uncertainty. Using error propagation, the following formulation can be used for the
mapping of the data errors to the model domain [40]:
CE=CMJT
TD C−1
DJTD CM. (17)
Equation (17) isolates the propagated uncertainty of the inverted measurements and
is, therefore, the approximation used for the error propagation from the TD to the FD. The
Jacobian of the FD response can be calculated from the partial derivatives with respect to
the model parameters ln(γk). For the k-th model parameter, we find
JFD,k=∂Z0(ω)
∂ln(γk)
∂Z00(ω)
∂ln(γk)T
k=−γk(ωτk)2
1+(ωτk)2−γkωτk
1+(ωτk)2T
k
, (18)
for the formulation in terms of the real and imaginary parts. The covariance matrix for
the real and imaginary parts is calculated from
JFD
and
CE
while also accounting for the
uncertainty of R0:
cov(Z0,Z00) = var(R0)1 0
0 0 +JFDCEJT
FD . (19)
From the covariance matrix of the real and imaginary parts, the covariance matrix of
the logarithmic magnitude and phase can be obtained according to:
cov(ln |Z|,φ) = ∂ln |Z|
∂Z0∂ln |Z|
∂Z00
∂φ
∂Z0∂φ
∂Z00 !cov(Z0,Z00) ∂ln |Z|
∂Z0∂ln |Z|
∂Z00
∂φ
∂Z0∂φ
∂Z00 !T
. (20)
4. Synthetic Validation Study
4.1. Accuracy of the TD to FD Conversion
To investigate the accuracy of the approach described above when extracting in-
formation at a frequency of 1 Hz, we tested it on 30 synthetic transients with varying
relaxation times
d(ti) = R0V1
V0
exp−ti
τs, (21)
with
R0=
1
Ω
,
V1
V0=
0.1, and
τs∈[
10
−2
, 10
]
s. The synthetic transients were discretized
over 20 time gates, with geometric means
ti
that were
log10
-spaced between 0.1 s and
1 s. Noise with a relative error of 1% and an absolute error of 10
−6Ω
was added to the
synthetic measurements. The pseudo-random number generator was initialized with the
same seed for all synthetic transients so that the added noise realization was always the
same. During the Debye decomposition, the standard deviation of the synthetic noise was
accounted for by modification of the data-precision matrix
C−1
D
. During the inversion of
all synthetic transients, the regularization strength was adapted to the data uncertainty to
achieve a data fit of
ε=
1. We evaluated the accuracy of the conversion by investigating its
ability to estimate the phase of the complex electrical impedance, the results of which are
shown in Figure 1. The expected phase value for a given synthetic transient is calculated
analytically. Although the misfit between the expected phase and estimated phase is high
for the conversion of very fast relaxation processes, the estimation is exact if the relaxation
Minerals 2023,13, 955 7 of 20
time
τs
of the synthetic transient lies within the sampled time frame from 0.1 s to 1 s. For
slower relaxation processes, the accuracy of the conversion does not drop as drastically
as it does for faster relaxation processes. Up to a relaxation time of
τs=
10 s, a good
estimation of the phase is possible through the extrapolation of the information contained
in the synthetic measurements. The achieved data fits
ε
were all close to the target value
of
ε=
1, as shown in Figure 1. The optimized regularization strength
λf inal
was minimal
for synthetic transients with relaxation times within the sampled time frame. Here, the
inversion was dominated by the measurements. For values of
τs
outside the sampled
time frame, the inversion was dominated by the prior information, which corresponded to
higher estimates of λf ina l.
10 210 1100101
s
[s]
0
10
20
30
40
50
[mrad] ( and -)
(a)
expected
estimated
10 210 1100101
s
[s]
101
102
103
104
105
final
(-)
(b)
0.0
0.5
1.0
1.5
2.0
(--)
final
Figure 1.
Results of the synthetic validation study to investigate the accuracy of the conversion
scheme. The conversion was performed on input signals of varying relaxation times. (
a
) Phase
estimates obtained from the conversion in comparison to their expected, analytically calculated
counterparts. The dotted red vertical lines mark the lower and upper bounds of the sampled time
frame. For signals resolved by the measurement, the conversion scheme achieved a good fit between
the expected phase and the estimated phase. (
b
) Achieved data fit
ε
and optimized estimate for the
regularization strength λf inal plotted against the relaxation time of the respective input signal.
4.2. Validation of Error Propagation
The error propagation from the TD to the FD was based on the existence of a suitable
covariance matrix for describing the estimated RTDs’ uncertainty. Equation (17) provides
such an estimate for the mapping of the data errors to the model domain, assuming normal
data error and model-parameter distributions. To validate the error propagation from the
TD to the model domain, we created 10
4
noise realizations of Equation (21) with
V1
V0=
0.1,
τs=
0.5 s, and
R0=
1
Ω
, as exemplarily shown in Figure 2. The error on
R0
was simulated
by adding a synthetic noise realization with a 10% relative error and a 5
×
10
−3Ω
absolute
error to
R0
. The synthetic transient was discretized over 20
log10
-spaced time gates with
geometric means
ti
between 0.1 s and 1 s to create the synthetic measurements. The noise
added to the synthetic transient had a relative error of 1% and an absolute error of 10
−6Ω
.
Inversions were performed with a fixed regularization strength, meaning that variations
in the cost function (11) only occurred in the data misfit term and not in the prior term,
isolating the scatter in the estimated model parameters that was caused by the noise on
the synthetic measurements. Because the discretization at 20 time steps only provided a
limited sample of the noise, and the regularization strength was not adapted to the specific
noise realization, the achieved values for
ε
showed a scatter. The regularization strength
was chosen to be λ=1, yielding scattering εvalues with a mean ¯
ε≈1.
The synthetic validation study showed that although the distributions of the model
parameters deviated from the normal distribution, they were otherwise well-behaved (see
Figure 2for an example of a model parameter). The distributions of the model parameters
obtained during the synthetic study were used to calculate the reference covariance matrix
and to validate the choice of the uncertainty estimator in the model domain, which was
used for the error propagation and calculated according to Equation (17). Figure 3shows
a comparison of the reference matrix calculated from the scatter of the model-parameter
estimates and the approximations of
CM
and
CE
using Equations (16) and (17), which
Minerals 2023,13, 955 8 of 20
were calculated from the inversion result of the noise-free decay. As expected,
CM
showed
strong deviations from the reference matrix since it included the uncertainty of the prior
information and did not isolate the mapping of the data errors to the model domain. The
matrix
CE
was a much better approximation of the reference matrix. However, although it
captured the overall shape,
CE
also showed deviations from the reference matrix. These
can be traced back to the assumption of normally distributed model-parameter estimates
underlying Equation (17), which was violated in some cases, as shown in Figure 2. We,
therefore, assumed
CE
to be a valid but not completely exact approximation of the un-
certainty in the model domain. To investigate the distributions of the FD estimates, the
FD responses of all 10
4
RTDs were calculated at 1Hz, and are shown in Figure 2. The
distributions of ln |Z|and φwere not normal but were well-behaved.
10 1100
t
[s]
0.02
0.04
0.06
0.08
R
0(
t
)
(a)
Scatter TD
11 10 9
m
40
0
100
200
300
400
500
counts
(b)
Scatter model domain
0.6 0.4 0.2 0.0 0.2
ln|
Z
|
0
100
200
300
400
500
counts
(c)
Scatter FD
100 80 60 40 20
[mrad]
ln|
Z
|
Figure 2.
(
a
) Noise realizations of the input transient. (
b
) Scatter of the sample model parameter
m40
.
(c) Scatter of the obtained FD estimates.
0 20 40 60 80
0
20
40
60
80
(a)
C
M
0 20 40 60 80
0
20
40
60
80
(b)
covariance of scattered
model parameters
0 20 40 60 80
0
20
40
60
80
(c)
C
E
0
10
20
30
40
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.25
0.50
0.75
1.00
1.25
1.50
Figure 3.
Comparison of the different approximations for the covariance matrix of the estimated
RTD. (
a
) Covariance matrix calculated according to Equation (16). (
b
) Reference covariance matrix
obtained from the Monte Carlo analysis. (
c
) Covariance matrix calculated according to Equation
(17). It can be seen that the covariance matrix calculated using Equation (17) represents the superior
approximation in this comparison.
The scatter of the FD responses calculated using the inversion results with
λ=
1 was
used to estimate the reference standard deviation for the individual components of the
FD estimates. To validate the accuracy achieved by the error propagation scheme, we
performed a conversion for 10
3
noise realizations, for which we adapted
λ
, calculated the
FD error estimate for each FD estimate, and compared the mean of the error estimates to
the reference. Figure 4shows the distributions of the estimated standard deviations for the
different components of the FD estimates. The standard deviations of both
ln |Z|
and
φ
were well estimated by the proposed error propagation scheme.
Minerals 2023,13, 955 9 of 20
0.10 0.12 0.14 0.16 0.18
std(ln|
Z
|)
0
20
40
60
80
100
counts
(a)
mean of analytical
error estimates
reference
analytical error
estimates
3456789
std( ) [mrad]
0
20
40
60
80
100
120
counts
(b)
Figure 4.
Results of the synthetic validation study to test the error propagation scheme for (
a
) the
magnitude, and (
b
) the phase. The dashed red vertical lines indicate the values of the reference
standard deviations obtained from the distributions of the FD estimates displayed in Figure 2. Shown
in blue are the distributions of the error estimates propagated analytically using the proposed scheme
for multiple noise realizations of the same TD transient. It can be seen that the mean values of
the analytically propagated error estimates, indicated by the black vertical lines, approximated the
reference values very well, indicating the validity of the proposed error propagation scheme.
5. Demonstration of the Application to Tomographic Field Measurements
We demonstrated the application of the new conversion approach to tomographic
field measurements on a TD IP data set that was measured in the Maletoyvaemskoie ore
field (MOF), located in the central part of the Maketoyvaemskoie ore cluster in the North
Kamchatka region (Russia, compare Figure 5). Geologically, the field site is located in the
central part of a volcanic-tectonic structure that represents a stratovolcano. The strato-
volcano is located within the Koryak-Central Kamchatsky belt of the Neogen-Quaternary
age and is composed of Early Miocene stratified volcanic and sedimentary rocks (effusive,
pyroclastic, and tuff-sedimentary rocks predominantly of Andesitic composition) with
different degrees of hydrothermal and metasomatic alteration. The MOF includes three
gold objects considered small HS (high sulfidation)-type gold deposits (Yugo-Zapadnoye,
Gaching, and Yubileinoye), several gold anomalies superimposed with copper-arsenic
sulfosalt mineralization, and copper anomalies [
41
,
42
]. Most of the discovered ore bodies
are of high chargeability (e.g., [
43
]). The MOF features massifs of secondary quartzites
with pronounced horizontal zonality, which are typical for HS-type deposits. Epithermal
gold deposits of HS type are the new types of deposits found in the far east of Russia. To
date, only a few economically important HS deposits have been discovered within the
region. They are settled within young volcano-sedimentary belts, where hydrothermally
altered (advanced argillic) rocks have formed due to the aside alteration of initial rocks
by hydrothermal processes within active hydrothermal systems, surrounding the gold HS
deposits. Numerous fields of altered rocks have been discovered within the Kamchatka
peninsula, being potential sources of high-grade and large-tonnage epithermal gold-silver
and copper-porphyry deposits. Starting in the year 2000, gold prospecting was carried out
in the MOF.
There have been significant efforts to characterize the field site using geophysical and
petrophysical surveys. Geoelectric measurements have been complemented by information
from other geophysical methods, such as geomagnetic measurements, and geological
surveys. Gurin [
43
] presented the results and interpretations of multiple geophysical and
petrophysical surveys carried out between 2016 and 2017 in the central part of the MOF. The
IP characteristics of the field site have been previously analyzed (e.g., [44]). The extensive
studies that have been conducted in the area provide the basis on which the validity of our
results, obtained from the conversion and subsequent tomographic inversion of the TD IP
data, can be assessed. The TD IP data set was obtained using pole-dipole measurements,
which were carried out along a 2400 m long profile, utilizing an injection and off-time
of 1 s and a total of 15 stacks. A potential dipole with an electrode separation of 20 m
was used while the current electrode was moved on a 100 m grid. The second current
electrode, assumed to be located at an infinite distance during the tomographic inversion,
Minerals 2023,13, 955 10 of 20
was placed 3500 m away from the nearest point on the profile. For data acquisition, an
AIE-2 instrument was used, featuring a VP-1000 (1 kW power) transmitter and a VP-MPP
receiver. No low-pass filter was used during the data acquisition, eliminating the need to
correct the early times of the transients for the corresponding effects. The workflow we
followed during the processing of the tomographic TD IP field data set is presented in
Figure 6.
Figure 5.
Kamchatka, with the location of the Maletoyvaemskoie ore field indicated in red. Map by
OpenStreetMap (OSM) [
45
]. Coordinate System: WGS 84. A detailed geological map of the field site
is provided in Gurin [43].
TDIP transient data
Relaxation time distribution
FDIP spectrum
Transient error model
Covariance matrix
FDIP errors
Resistance-error model
Fitting of reference decays
Bin analysis on residuals
Fitting of error model to
standard deviation estimates
Filter
Debye decomposition
Resistance data
Complex resistivity tomography
Filter Filter
Figure 6. Illustration of the workflow followed during the TD IP field-data processing.
Minerals 2023,13, 955 11 of 20
To assess the general data quality, filter transients that showed highly erratic behavior,
and later quantify errors, we followed Flores Orozco et al. [
46
] and started the data
processing by fitting a power-law model to the measured TD transients. Note that the
power-law model
d(ti) = atib(22)
approximates the TD equivalent of the CPA spectrum of the electrical impedance [
3
] given by:
ˆ
Z(ω) = R0K(iω)−b. (23)
By basing the filtering on the Pearson correlation coefficient between the measured
decay and the response of the fitted power-law model, we defined erratic behavior as a
strong deviation from the CPA behavior. We excluded transients with
r<
0.9 from the data
processing, interpreting them as non-physical. This filter reduced the total size of our data
set by 9%. Examples of transients and their corresponding Pearson correlation coefficients
are shown in Figure 7.
10 1100
t
[s]
10 2
10 1
(
t
)
r = 0.4483
r = 0.8338
r = 0.9998
Figure 7.
Examples of transients from the TDIP field data set, for which the fitted power-law model
achieved specific Pearson correlation coefficients. Transients with
r<
0.9, representing 9% of the data
set, were rejected by our filter and excluded from further processing and inversion.
Since the error quantification aimed to characterize the statistical uncertainty of the
data, performing it after filtering non-physical transients from the data set was reasonable.
Based on the scatter of the measurements around the responses of the fitted power-law
models, we estimated the standard deviations of
di
by performing a bin analysis [
47
]. We
calculated the residuals of all
di
readings to the power-law model response and plotted
them against the
di
readings themselves. After subdividing the data set into
log10
-spaced
bins, we estimated the standard deviations corresponding to the different bins from the
scatter of the residuals within them. Finally, we fitted a linear error model through the
estimated standard deviations, as shown in Figure 8. To account for the varying number
of residuals within the different bins, we weighted the estimated standard deviations by
1
√NBin
during the fit. The fitted error model was used to estimate the standard deviations
of all
di
readings during the TD to FD conversion. To account for the simplicity of the
fitted power-law model and allow for more complex RTDs, we slightly adjusted the fitted
error model by reducing the absolute error (see Figure 8). For the DC resistances, we used
a linear error model with relative and absolute errors of 9% and 5
·
10
−3Ω
, respectively.
Lacking reciprocal measurements, these parameters were chosen conservatively and in
such a way that the results of the TD to FD conversion, as well as those of the tomographic
inversion, were robust with regard to small changes in the values chosen for the DC error
model.
Minerals 2023,13, 955 12 of 20
10 610 410 2100
R
0 [ ]
10 8
10 6
10 4
10 2
residuals [ ]
(a)
residuals
bins
10 410 2100
R
0 [ ]
10 6
10 5
10 4
10 3
10 2
std(
R
0) [ ]
(b)
error model
error model adjusted
estimated std for bin
0.0
0.5
1.0
1.5
log10
NBin
Figure 8.
Results of the error analysis performed on the TD IP data set to obtain an error model for
the TD transients. (
a
) Bin analysis. The scatter of the residuals within each bin was used to calculate a
corresponding standard deviation. (
b
) Linear error model fitted to the estimated standard deviations.
The error model used for the inversion of the TD transients was adjusted manually and is indicated
in red. The estimated standard deviations for the bins are indicated in shades of gray, representing
the number of data points in the corresponding bin.
The inversions of all the measured TD transients into RTDs were performed with
the adaptation of the regularization strength to the data uncertainty. Figure 9shows
the development of
ε
and
λ
during the inversion process for the sample transients. All
inversions started with a high
ε
of the starting model, which decreased significantly during
the first 10 iterations. Note that the development of
ε
showed discontinuities in the
iterations, at which time the regularization strength λwas updated.
10 2100
[s]
10 4
6 × 10 5
2 × 10 4
3 × 10 4
k
[ ] (-)
(a)
Transient: 1
final
= 39.64
10 2100
[s]
10 5
(b)
Transient: 10
final
= 3.74
10 2100
[s]
10 4
6 × 10 5
7 × 10 5
8 × 10 5
9 × 10 5
(c)
Transient: 400
final
= 410.36
10 1100
t [s]
0.002
0.004
0.006
0.008
R
0 [ ]
(d)
data
response
10 1100
t [s]
0.0001
0.0002
0.0003
0.0004
(e)
data
response
10 1100
t [s]
0.001
0.002
(f)
data
response
0 20 40
iteration
100
101
102
103
(-)
(g)
0 200 400 600 800
iteration
101
102
103
104
(h)
0 10 20 30
iteration
100
101
102
103
(i)
102
103
104
105
evolution
evolution
101
103
105
evolution
evolution
102
103
104
105
(..)
evolution
evolution
Figure 9.
Results obtained from the Debye decomposition of transients 1, 10, and 400. (
a
–
c
) Esti-
mated RTDs, which were of varying complexity. (
d
–
f
) Achieved data fits. (
g
–
i
) Evolution of the
regularization strength λand data fit εover the course of the optimization.
The estimated RTDs were of varying complexity, as displayed in Figure 9. Although
some showed no or only one peak, others featured multiple peaks and structures that indi-
cated the presence of relaxation phenomena, which could not be quantitatively described
by the CPA assumption or single Debye or Cole–Cole terms. Generally, the estimated RTDs
achieved good data fits, as shown in Figure 9.
Minerals 2023,13, 955 13 of 20
For every converted transient, we extracted electrical impedance data from the FD
responses at 1 Hz and 20 Hz and used the magnitude and phase values as input for the
tomographic inversions. Prior to the tomographic inversions, we filtered the data set
based on the histograms of the apparent resistivity magnitude and phase, as shown in
Figure 10, and excluded all measurements associated with geometric factors larger than
50,000. Figure 11 shows the FD error estimates at 1 Hz. We fitted the error models through
the data set to demonstrate the systematic behavior of the error estimates. For the phase-
error model, we used the inverse power-law relation proposed by Flores Orozco et al. [
48
]
std(φ) = a|Z|−b+c, (24)
whereas for the magnitude-error model, we followed the standard linear assumption, fea-
turing relative and absolute errors. For the tomographic inversion, we used the individual
error estimates rather than the ones associated with the fitted-error models. We were un-
able to obtain stable tomographic inversion results when including data points with phase
errors
std(φ)<
1 mrad, so we excluded them. As can be seen from the scatter plot of the
phase errors in Figure 11, this mainly corresponded to data points with a high impedance
magnitude
|Z|
, which mainly occurred in our data set for the very shallow region between
meters 2000 and 2500 of the profile. Due to the resulting local sparsity of the data points,
the tomographic inversion result was less data-driven in that region.
2 4
log
10(|
a
| [ m])
0
50
100
150
200
250
300
counts
(a)
raw
3000 2000 1000 0
a
[mrad]
0
200
400
600
800
counts
(b)
raw
123
log
10(|
a
| [ m])
0
50
100
150
counts
(c)
filtered
80 60 40 20
a
[mrad]
0
20
40
60
80
100
counts
(d)
filtered
Figure 10.
Distributions of the FD estimates obtained from the conversion, before and after the
filtering of outliers. (
a
,
b
) Estimates of the apparent resistivity and phase before the application of the
filter. (c,d) Estimates of the apparent resistivity and phase after the application of the filter.
10 310 210 1100
|
Z
| [ ]
10 2
10 1
std(|
Z
|) [ ]
(a)
propagated standard deviations
a
|
Z
| +
b
10 310 210 1100
|
Z
| [ ]
100
101
102
std( ) [mrad]
(b)
propagated standard deviations
a
|
Z
|
b
+
c
Figure 11.
Error estimates for the individual data points at a frequency of 1 Hz obtained from the
error propagation scheme. Commonly used error models, plotted in orange, were able to successfully
capture the systematic behavior of the estimated standard deviations. The error models were fitted
only for demonstration purposes. During the tomographic inversion, the individual error estimates
were used. (
a
) Standard deviations of the estimated magnitudes, which followed a linear trend. (
b
)
Standard deviations of the estimated phases, which followed a trend that could be fitted using a
power law.
The tomographic inversions were performed using a finite element-based, smoothness-
constrained complex resistivity inversion code developed by Kemna [
38
]. Figure 12 displays
the results of the tomographic inversion at 1 Hz, alongside a plot of the estimated coverage,
Minerals 2023,13, 955 14 of 20
which shows the regions of the model that were less constrained by the measurements.
The obtained images of the complex resistivity were realistic, given the expected geological
context. The estimated phases achieved values from 0 mrad to
−
60 mrad. Large absolute
phase values between
−
40 mrad and
−
60 mrad were primarily located at the start (left) of
the profile and extended up to 1700 m. These large absolute phase values also coincided
with larger values of
σ00
, as shown in Figure 13, which is approximately proportional to
the metal factor (e.g., [
49
,
50
]) for the phase angles we considered. In contrast to the phase,
large absolute values of
σ00
can be seen in the region between 2500 m and 3000 m. In the
shallow region around 1500 m, a reduction in
σ00
occurred. We can attribute this reduction
in
σ00
near the surface to the presence of quartzites and interpret the large
σ00
values at
depths from 600 to 1800 m to be caused by copper-porphyry deposits, located within
moderately to strongly sulfidized rocks. With respect to the possible exploration of deposits
in the MOF, the described region can be classified as a promising area for potential copper
mining. For further information on prospective areas in the MOF deduced from geophysical
measurements at the study site, we refer the reader to Gurin [
43
]. Although the frequency
dependence of the magnitude was negligible, the frequency dependence of the phase can
be clearly seen from the comparison of the inversion results at 1 Hz and 20 Hz, as displayed
in Figure 14. This spectral behavior can be interpreted further, for example, in terms of
the associated relaxation times. Assuming a single dominant relaxation process
τpeak
, the
difference plot at the bottom in Figure 14 would indicate
τpeak <1
2π
s in the red regions and
τpeak >1
40π
s in the blue regions. As the radius of ore grains relates to the relaxation time
τpeak
(e.g., [
5
,
12
,
51
,
52
]), the spectral characteristics captured by our conversion approach can
be interpreted in terms of the expected ore grain size. Furthermore, the spectral behavior
displayed in Figure 14 is consistent with our interpretation of the
σ00
image (see Figure
13). Note that parts of the blue region in the difference plot coincide with areas of lower
coverage (see Figure 12).
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
(a)
| |1[
Hz
]
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
(b)
1[
Hz
]
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
(c)
5.0
2.5
0.0
log
10(Coverage)
1.6
2.4
3.2
log
10([
m
])
60
40
20
0
[mrad]
Figure 12.
Tomographic inversion results for spectral data extracted at a frequency of 1 Hz. (
a
)
Estimated magnitude of the complex resistivity
|ρ|
. (
b
) Phase of the complex resistivity
φ
. (
c
)
Coverage achieved by the measurement.
Minerals 2023,13, 955 15 of 20
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
Sulfidized rocks
Quartzites Quartzites
00
1[
Hz
]
4.8
4.0
log10 [S/m]
Figure 13. Image of σ00 obtained from the tomographic inversion at 1 Hz.
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
(a)
20[
Hz
]
1000 1500 2000 2500 3000
x [m]
0
200
400
z [m]
(b)
|20[
Hz
]| | 1[
Hz
]|
10
0
10
[mrad]
60
40
20
0
[mrad]
Figure 14.
Comparison of the tomographic inversion results at 20 Hz and 1 Hz. Due to the negligible
frequency dependence of the magnitude, only the phase is compared here. (
a
) Phase distribution
obtained from the tomographic inversion of spectral data extracted at a frequency of 20 Hz. (
b
)
Differences between the phase estimates at 1 Hz and 20 Hz.
6. Discussion
The presented work shows that a quantitative TD to FD conversion of IP transients
using Debye decomposition is possible and can be applied to tomographic TD IP data sets.
The major precondition for the application of the described approach is the suitability of
the DD to describe the polarization process at hand. Other limitations of the approach can
be traced back to the limited information content in the TD IP measurements themselves.
A symptom of this is the high phase error, caused primarily by the relatively small number
of time gates used to discretize the transients, which is often
N=
20 or less for TD IP field
instruments. The phase error can be drastically reduced by a more accurate sampling of the
transients, with the noise level on the TD transient remaining unchanged. Furthermore, the
presented approach is unable to yield accurate FD estimates for relaxation processes that are
not resolved by the measurement, as shown in Figure 1. For relaxation processes far beyond
the measured time interval, or in the case of highly erratic behavior, our approach can fail
to invert a given transient into an appropriate RTD. A prior assessment of a given data
set is therefore necessary, forcing the user to make a judgment on whether a quantitative
analysis and interpretation of the data set is appropriate or not.
During the error propagation from the TD to the FD, we exclusively focus on statis-
tical errors. As the TD to FD conversion is non-linear, the analytical error propagation
is not mathematically exact, and the estimate of the FD uncertainty will always be an
approximation of the real error. Numerical approaches may be more suited for exact error
propagation, but may potentially increase the computational effort to a level exceeding
what is reasonable when applying the conversion to larger-scale tomographic TD IP data
sets. For the inversion of a TD transient into an RTD, we set up our algorithm to estimate
Minerals 2023,13, 955 16 of 20
the natural logarithms of
γk
. This worked well for all the cases we encountered during
our work since we avoided negative estimates for
γk
and achieved a higher consistency in
the estimated RTDs. However, the following inconsistencies were introduced with regard
to the assumptions underlying the error propagation: We inverted the discrete and linear
transients of
di
by minimizing a cost function that assumed they were subject to normal
noise. In the model space, we used
CE
as a covariance matrix to describe the uncertainty of
the estimated model parameters
ln(γk)
, with
γk
and
di
both having the unit of resistance.
This is mathematically inexact since
ln(γk)
and
di
cannot both be normally distributed.
The same inconsistency arose in the conversion from the model domain to the FD. Still,
the presented synthetic validation study suggests that the propagated FD uncertainty is a
reasonable estimate that can be used during further processing and tomographic inversions.
To account for the uncertainty of the FD error estimate during a tomographic inversion,
it is advised to run multiple tomographic inversions with slightly changed error settings
to ensure the stability of the features that are being interpreted. Correlations between
the errors of
ln |Z|
and
φ
can appear as off-diagonal elements in Equation (20). These
are typically several orders of magnitude smaller than the variances of the parameters.
Since the tomographic inversion algorithm we use is unable to account for correlations
between
ln |Z|
and
φ
, we neglect them during the tomographic inversion of the field data
set. However, including these correlations in the tomographic inversion improves the
consistency of the overall analysis and therefore should be done if possible.
The data set used for the demonstration of the conversion’s applicability to tomo-
graphic field measurements was obtained using a pole-dipole electrode configuration.
For practical reasons, it was unfeasible to perform reciprocal measurements due to the
way the measurements were realized in the field. Although electrode configurations like
the dipole-dipole make it easier to collect reciprocal measurements in the field, using the
pole-dipole configuration provides us with superior IP data quality. Since the conversion
approach relies on high-quality transients, we value IP data quality more than the ability to
perform reciprocal measurements in this study. The TD to FD conversion presented in this
work is generally independent of the approach that is used for the estimation of the TD
error. One can use any method to estimate the TD error that is suitable for the application
at hand, e.g., normal-reciprocal measurements or standard deviations provided by the
measurement instrument. We adapted the method described by Flores Orozco et al. [
46
],
which came with some benefits that fit well into an adequate pre-processing of the data set
prior to the TD to FD conversion. It does not require a reciprocal data set, which is favorable
since a reciprocal data set might not always be available as in our case for the reasons
explained above. Furthermore, the fitted power-law models provided a valuable initial
characterization of the data set, on the basis of which a detailed assessment of the data
and filtering was possible. Using power-law models as a reference potentially introduces
a bias against more complex relaxation behaviors. The first critical point is filtering on
the basis of the Pearson correlation coefficient between the measured TD transient and
the fitted power-law response. Choosing a threshold value that is too high can result in
the exclusion of transients that represent a more complex relaxation behavior. Therefore,
a visual inspection of the excluded transients is advisable. Regarding the estimation of
the TD error, transient readings with larger deviations from the power-law model result
in larger residuals and contribute to larger standard deviation estimates during the bin
analysis. This introduces a potential bias since data sets with many decays that are more
complex than what can be described by the power-law model will be assigned larger TD
error estimates. The bias is reduced with the use of an error model since the standard
deviation of a specific transient reading
di
is not directly tied to the corresponding residual.
We adjusted the fitted error model slightly in order to correct for an overestimation of the
error due to the simplicity of the reference transients.
For specific measurement geometries and subsurface scenarios, a negative IP effect
can be measured (e.g., [
4
,
53
]). The transients associated with this phenomenon take the
form of negative decays. Since we used a logarithmic parameterization during the DD, a
Minerals 2023,13, 955 17 of 20
direct application to negative transients is not possible, as they have to be fitted with values
γk<
0. To solve this problem, negative transients must be identified prior to the TD to FD
conversion, multiplied by
−
1, and then inverted into an RTD. The FD response can now
be calculated according to Equation (2) using the estimated values of
γk
with a changed
sign:
γk← −γk
. This results in values
Z0>R0
,
Z00 >
0 and
φ>
0. The identification
of negative transients can be achieved through the information provided by the fit of the
reference transients.
We classify our approach as a general-purpose compromise between neglecting the
frequency dependence of the phase and using a strong FD parameterization of the subsur-
face. Both of these extremes can lead to a misrepresentation of the spectral behavior of an
unknown target, causing inaccurate tomographic inversion results.
7. Conclusions
We have introduced a general-purpose approach that can quantitatively convert
TD IP data to the FD using Debye decomposition. The conversion approach provides a
basis for the analysis of TD IP measurements in the FD. Quantitative relations between
geoelectric and petrophysical parameters established for the FD can be used to deduce
quantitative information on properties of interest that shape the IP characteristics of the
subsurface. Gaussian error propagation is used to propagate a TD error estimate to the FD,
thereby providing the uncertainty quantification needed for the inversion of tomographic
TD IP measurements to the FD. The implemented algorithm automatically chooses the
regularization strength to achieve the appropriate data fit, assuming the existence of a
reasonable TD error estimate. We demonstrated the conversion of transients to accurate FD
estimates in a synthetic validation study, during which we transformed the input signals of
varying relaxation times. To validate the assumptions made during the error propagation,
we inverted a set of noise realizations and investigated the resulting distributions of all
parameters involved during the TD to FD conversion. Based on the scatter of the FD
estimates, we calculated reference standard deviations, which we used to investigate the
accuracy of the propagated FD error estimates. To demonstrate the practical application of
the conversion to real-world data, we applied it to a tomographic TD IP data set measured
in Kamchatka (Russia). The propagated standard deviations of the field data showed
systematic behavior expected from previous studies. Inverting the FD data into subsurface
models of complex resistivity at frequencies of 1 Hz and 20 Hz showed the ability of the
conversion scheme to recover spectral information from tomographic TD IP data sets.
Author Contributions:
Conceptualization, J.H. and A.K.; methodology, J.H.; software, J.H. and A.K.;
validation, J.H.; formal analysis, J.H.; investigation, J.H., G.G., and A.K.; resources, G.G., K.T., and
A.K.; data curation, J.H. and G.G.; writing—original draft preparation, J.H.; writing—review and
editing, J.H. and A.K.; visualization, J.H.; supervision, A.K. and K.T; funding acquisition, A.K. All
authors have read and agreed to the published version of the manuscript.
Funding:
This research was partially funded by Interreg Euregio Meuse-Rhine within the framework
of the E-Test (Einstein Telescope EMR Site and Technology) project.
Data Availability Statement:
The code used during the synthetic validation study is available at
https://doi.org/10.5281/zenodo.7970987. For further information, please contact the first author.
Acknowledgments:
The first author thankfully acknowledges the discussions with Maximilian
Weigand, which were very helpful during the early stages of the presented work.
Conflicts of Interest: The authors declare no conflicts of interest.
Minerals 2023,13, 955 18 of 20
Abbreviations
The following abbreviations are used in this manuscript:
CPA constant phase angle
DC direct current
DD Debye decomposition
FD frequency domain
HS high sulfidation
IP induced polarization
MAP maximum a posteriori
MOF Maletoyvaemskoie ore field
RMSE root-mean-square error
RTD relaxation-time distribution
TD time domain
Appendix A
In this work, we follow the underlying idea of an Occam-type choice of the regulariza-
tion strength, meaning that we aim to find the simplest model that can fit the measurements
within the context of their uncertainties. This choice of regularization strength has been
employed for the solution of different types of geophysical inverse problems in the past
(e.g., [
20
,
38
,
39
]). Our algorithm for finding a suitable regularization strength is initialized
with a high
λ0
. Performing a pseudo-Newton inversion yields the MAP solution and the
optimized step length for
λ0
. Given that
λ0
is large, the MAP solution should yield
ε>
1,
meaning that the measurements should be underfitted. If this is not the case, we abort the
algorithm and start with a larger initial regularization strength. In the pseudo-Newton
inversions that follow, we reduce the value of
λ
step by step, which leads to a better data fit
and smaller
ε
calculated from the responses of the estimated MAP solutions. We update
the regularization strength according to
λ←λ
ε+ξ, (A1)
where
ξ
is some non-negative constant. A main exit criterion is implemented, causing a ter-
mination of the optimization if the desired data fit is achieved. The closer the optimization
gets to the target
ε=
1, the smaller the updates of
λ
become. To increase the stability of
the algorithm, we implement a secondary exit criterion that terminates the optimization
if an update of
λ
does not lead to a significant update of
ε
. Implementing the secondary
exit criterion has been shown to be essential in practice for increasing the stability of the
algorithm. Setting
ξ>
0 avoids interference between excessively small updates of
λ
and
the secondary exit criterion. If an update of
λ
causes the inversion to significantly overfit
the measurements, we increase
λ
by 10% during the next pseudo-Newton inversion. To
ensure consistency in the obtained inversion results, we perform a final improvement of the
regularization optimization, during which
λ
is increased by 50% until
ε
starts to increase
significantly. In practice, it has been shown that from a found MAP solution with
ε≈
1,
the regularization strength can sometimes be drastically increased while still fitting the
measurements well in the context of their respective errors.
References
1. Bleil, D.F. Induced polarization: A method of geophysical prospecting. Geophysics 1953,18, 636–661.
2. Seigel, H.O. Mathematical formulation and type curves for induced polarization. Geophysics 1959,24, 547–565.
3.
Van Voorhis, G.; Nelson, P.; Drake, T. Complex resistivity spectra of porphyry copper mineralization. Geophysics
1973
,38, 49–60.
4. Nabighian, M.N.; Elliot, C.L. Negative induced-polarization effects from layered media. Geophysics 1976,41, 1236–1255.
5.
Pelton, W.H.; Ward, S.; Hallof, P.; Sill, W.; Nelson, P.H. Mineral discrimination and removal of inductive coupling with
multifrequency IP. Geophysics 1978,43, 588–609.
6.
Yuval, D.; Oldenburg, W. DC resistivity and IP methods in acid mine drainage problems: Results from the Copper Cliff mine
tailings impoundments. J. Appl. Geophys. 1996,34, 187–198.
Minerals 2023,13, 955 19 of 20
7.
Seigel, H.; Nabighian, M.; Parasnis, D.S.; Vozoff, K. The early history of the induced polarization method. Lead. Edge
2007
,
26, 312–321.
8.
Vanhala, H. Mapping oil-contaminated sand and till with the spectral induced polarization (SIP) method. Geophys. Prospect.
1997
,
45, 303–326
9.
Binley, A.; Slater, L.D.; Fukes, M.; Cassiani, G. Relationship between spectral induced polarization and hydraulic properties of
saturated and unsaturated sandstone. Water Resour. Res. 2005,41, 1–13
10.
Slater, L.; Ntarlagiannis, D.; Wishart, D. On the relationship between induced polarization and surface area in metal-sand and
clay-sand mixtures. Geophysics 2006,71, A1–A5.
11.
Placencia-Gómez, E.; Slater, L.; Ntarlagiannis, D.; Binley, A. Laboratory SIP signatures associated with oxidation of disseminated
metal sulfides. J. Contam. Hydrol. 2013,148, 25–38.
12.
Gurin, G.; Tarasov, A.; Ilyin, Y.; Titov, K. Time domain spectral induced polarization of disseminated electronic conductors:
Laboratory data analysis through the Debye decomposition approach. J. Appl. Geophys. 2013,98, 44–53.
13.
Revil, A.; Florsch, N.; Mao, D. Induced polarization response of porous media with metallic particles—Part 1: A theory for
disseminated semiconductors. Geophysics 2015,80, D525–D538.
14.
Martin, T.; Weller, A.; Behling, L. Desaturation effects of pyrite–sand mixtures on induced polarization signals. Geophys. J. Int.
2022,228, 275–290.
15. Vinegar, H.; Waxman, M. Induced polarization of shaly sands. Geophysics 1984,49, 1267–1287.
16.
Kemna, A.; Binley, A.; Slater, L. Crosshole IP imaging for engineering and environmental applications. Geophysics
2004
,69, 97–107.
17.
Weller, A.; Slater, L.; Nordsiek, S.; Ntarlagiannis, D. On the estimation of specific surface per unit pore volume from induced
polarization: A robust empirical relation fits multiple data sets. Geophysics 2010,75, WA105–WA112.
18.
Weller, A.; Slater, L.; Nordsiek, S. On the relationship between induced polarization and surface conductivity: Implications for
petrophysical interpretation of electrical measurements. Geophysics 2013,78, D315. https://doi.org/10.1190/GEO2013-0076.1.
19.
Weller, A.; Slater, L.; Huisman, J.A.; Esser, O.; Haegel, F.H. On the specific polarizability of sands and sand-clay mixtures.
Geophysics 2015,80, A57–A61.
20.
LaBrecque, D.J.; Miletto, M.; Daily, W.; Ramirez, A.; Owen, E. The effects of noise on Occam’s inversion of resistivity tomography
data. Geophysics 1996,61, 538–548.
21.
Kemna, A.; Binley, A.; Ramirez, A.; Daily, W. Complex resistivity tomography for environmental applications. Chem. Eng. J.
2000
,
77, 11–18.
22.
Kemna, A.; Räkers, E.; Binley, A. Application of complex resistivity tomography to field data from a kerosene-contaminated site.
In Proceedings of the 3rd EEGS Meeting, Aarhus, Denmark, 9–11 August 1997; EAGE Publications BV: Houten, The Netherlands,
1997; pp. 151–154.
23.
Binley, A.; Keery, J.; Slater, L.; Barrash, W.; Cardiff, M. The hydrogeologic information in cross-borehole complex conductivity
data from an unconsolidated conglomeratic sedimentary aquifer. Geophysics 2016,81, E409–E421.
24.
Hönig, M.; Tezkan, B. 1D and 2D Cole-Cole-inversion of time-domain induced-polarization data. Geophys. Prospect.
2007
,
55, 117–133.
25.
Fiandaca, G.; Auken, E.; Christiansen, A.V.; Gazoty, A. Time-domain-induced polarization: Full-decay forward modeling and 1D
laterally constrained inversion of Cole-Cole parameters. Geophysics 2012,77, E213–E225.
26.
Fiandaca, G.; Ramm, J.; Binley, A.; Gazoty, A.; Christiansen, A.V.; Auken, E. Resolving spectral information from time domain
induced polarization data through 2-D inversion. Geophys. J. Int. 2013,192, 631–646.
27.
Tezkan, B. 2D Cole Cole Inversion of Time Domain Induced Polarization Data-Model Studies and Field Measurements. In
Proceedings of the Near Surface Geoscience 2014-20th European Meeting of Environmental and Engineering Geophysics, Athens,
Greece, 14–18 September 2014; EAGE Publications BV: Houten, The Netherlands, 2014; Volume 2014, pp. 1–5.
28.
Revil, A.; Koch, K.; Holliger, K. Is it the grain size or the characteristic pore size that controls the induced polarization relaxation
time of clean sands and sandstones? Water Resour. Res. 2012,48, 1–7
29. Morgan, F.D.; Lesmes, D.P. Inversion for dielectric relaxation spectra. J. Chem. Phys. 1994,100, 671–681.
30.
Tong, M.; Li, L.; Wang, W.; Jiang, Y. A time-domain induced-polarization method for estimating permeability in a shaly sand
reservoir. Geophys. Prospect. 2006,54, 623–631.
31. Nordsiek, S.; Weller, A. A new approach to fitting induced-polarization spectra. Geophysics 2008,73, F235–F245.
32.
Martin, T.; Titov, K.; Tarasov, A.; Weller, A. Spectral induced polarization: Frequency domain versus time domain laboratory data.
Geophys. J. Int. 2021,225, 1982–2000.
33.
Tong, M.; Li, L.; Wang, W.; Jiang, Y. Determining capillary-pressure curve, pore-size distribution, and permeability from induced
polarization of shaley sand. Geophysics 2006,71, N33–N40.
34.
Tarasov, A.; Titov, K. Relaxation time distribution from time domain induced polarization measurements. Geophys. J. Int.
2007
,
170, 31–43.
35.
Weigand, M.; Kemna, A. Debye decomposition of time-lapse spectral induced polarisation data. Comput. Geosci.
2016
,86, 34–45.
36.
Kumar, I.; Kumar, B.V.; Babu, R.V.; Dash, J.K.; Chaturvedi, A.K. Relaxation time distribution approach of mineral discrimination
from time domain-induced polarisation data. Explor. Geophys. 2019,50, 337–350.
37. Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation; SIAM Philadelphia, USA: 2005; Volume 89.
Minerals 2023,13, 955 20 of 20
38.
Kemna, A. Tomographic Inversion of Complex Resistivity: Theory and Application. Ph.D. Thesis, Ruhr Universität Bochum, Bochum,
Germany, 2000.
39.
deGroot Hedlin, C.; Constable, S. Occam’s inversion to generate smooth, two-dimensional models from magnetotelluric data.
Geophysics 1990,55, 1613–1624.
40. Gubbins, D. Time Series Analysis and Inverse Theory for Geophysicists; Cambridge University Press: Cambridge, UK, 2004.
41.
Volchkov, A.; Zvezdov, V. The nature of Cu-As mineralization occurrences in the Maletoyvayamsky stratovolcano. RUDY I
METALLY 1997, pp. 44–51.
42.
Tolstykh, N.; Vymazalová, A.; Tuh
`
y, M.; Shapovalova, M. Conditions of formation of Au–Se–Te mineralization in the Gaching ore
occurrence (Maletoyvayam ore field), Kamchatka, Russia. Mineral. Mag. 2018,82, 649–674.
43.
Gurin, G. Geophysical prospecting for epithermal gold deposits (a case study from the Maletoivayam gold ore field, Kamchatka
Peninsula). Lithosphere 2021,21, 116–132.
44.
Gurin, G.; Tarasov, A.; Titov, K. Analysis of Transient Characteristics of Induced Polarization in Innovative Mineral Exploration
Solutions (from Case Studies of Gold Deposits). Russ. Geol. Geophys. 2023,64, 348–356.
45.
OpenStreetMap Contributors. Planet Dump. 2017. Available online: https://www.openstreetmap.org (accessed on 16 July 2023).
46.
Flores Orozco, A.; Gallistl, J.; Bücker, M.; Williams, K.H. Decay curve analysis for data error quantification in time-domain
induced polarization imaging. Geophysics 2018,83, E75–E86.
47.
Koestel, J.; Kemna, A.; Javaux, M.; Binley, A.; Vereecken, H. Quantitative imaging of solute transport in an unsaturated and
undisturbed soil monolith with 3-D ERT and TDR. Water Resour. Res. 2008,44, 1–17
48.
Flores Orozco, A.; Kemna, A.; Zimmermann, E. Data error quantification in spectral induced polarization imaging, Geophysics
2012,77, E227–E237.
49. Marshall, D.J.; Madden, T.R. Induced polarization, a study of its causes. Geophysics 1959,24, 790–816.
50.
Hallof, P.G. A comparison of the various parameters employed in the variable-frequency induced-polarization method. Geophysics
1964,29, 425–433.
51.
Wong, J. An electrochemical model of the induced-polarization phenomenon in disseminated sulfide ores. Geophysics
1979
,
44, 1245–1265.
52.
Revil, A.; Vaudelet, P.; Su, Z.; Chen, R. Induced polarization as a tool to assess mineral deposits: A review. Minerals
2022
,12, 571.
53.
Dahlin, T.; Loke, M.H. Negative apparent chargeability in time-domain induced polarisation data. J. Appl. Geophys.
2015
,
123, 322–332.
Disclaimer/Publisher’s Note:
The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
