Journal ArticleDOI

# 3-D time-domain induced polarization tomography: a new approach based on a source current density formulation

18 Dec 2017-Geophysical Journal International (Oxford Academic)-Vol. 213, Iss: 1, pp 244-260

### Introduction

• Received 2017 December 13; in original form 2017 April 17 S U M M A R Y Induced polarization (IP) of porous rocks can be associated with a secondary source current density, which is proportional to both the intrinsic chargeability and the primary current density.
• IP effects were first discovered during the last century (e.g. Schlumberger 1920; Dakhnov 1962).
• The advantage of formulating the IP problem as an equivalent self-potential problem is that the authors avoid solving a nonlinear inverse problem because retrieving the current density JS(t) from the recorded electrical field is a linear problem.
• To the best of their knowledge, this is the first attempt to recover the Cole–Cole parameters in 3-D using such an approach.

### 2.1 Electrical conductivity and chargeability

• The authors consider below time scale and length scales such as the induction effect can be neglected.
• This potential ψσ∞ is instantaneously recorded when the current injection is turned on.
• (6) Similarly, an apparent chargeability can be computed as Ma = ψσ∞ − ψσ0 ψσ∞ = F (σ∞) − F (σ∞ (1 − M)) F (σ∞) . (7) Therefore, in the classical approach, modelling the time domain IP requires solving the electrical conductivity equation (1) twice with two distinct electrical resistivity distributions (as proposed by Siegel 1959).
• Note that their notations are pretty standard.

### 2.2 Forward modelling in the charging phase

• During the injection of the primary current, each cell of the discretized subsurface will see a secondary current building up.
• If each cell is characterized by four Cole–Cole parameters (σ0, M, τ, c), the secondary source current density is determined by (Seigel 1959), JS(t) = −M(t)Jp. (11) Eq. (11) means that the dipole moment associated with the polarization of a grain (see Fig. 1b) is antiparallel to the applied current density, explaining the sign ‘−’ in this equation.
• In the quasi-static limit of the electromagnetic equations (taking all time derivatives to zero in the continuity equations), for each cell the constitutive equation and conservation equations are simply, J = Jp + JS, (12) ∇ · J = I δ3(r − r0). (13).
• When the primary current has been applied for long time so that the material is entirely polarized, the authors have M(t) = M, and they have.

### 2.3 Forward modelling in the discharging phase

• The authors first assume that the primary current has been applied from −∞ to time 0 so that the material has been entirely polarized.
• Alternatively, other formulations which have a faster convergence rate when t > τ can be used (e.g. Lee 1981; Hilfer 2002).the authors.
• In order to achieve this, the authors discretize the simulation domain into m cells (they use the same cells as for the electrical conductivity problem) and to each cell they assign a source current density.
• For each cell (numbered from 1 to m), the authors assign in Comsol Multiphysics (using the finite elements method) an elementary dipole in the three orthonormal directions (x, y, z) (so three elementary dipoles in total) and they compute the resulting distributions of the potential at each of the nφ voltage electrodes recording the secondary voltages.
• The inverse problem needs to be constrained to reduce the number of solutions and then to pick the optimal solution that reproduces the observed potential data and reflects the main structures of the medium as well.

### 5.1 Experiment setup

• The experiment consists in mixing two materials, a clean sand and some burning coal.
• This water gives the saturated sand a resistivity of around 68 Ohm.m.
• These IP data were acquired like in a self-potential survey, that is, the authors inject current between two electrodes A and B and they measure the potential between all the remaining electrodes with respect to the reference electrode (electrode #1).
• This rapid acquisition protocol is particularly suitable for this kind of coal experiment, because the coal combustion is fast and measurements must be recorded before the extinguishment of the burning coal.
• Then the secondary voltage decay is measured using ten time windows of different lengths (see Table 2).

### 5.2 Inversion

• The authors estimate for each cell a value of the electrical conductivity, the source current density, the time dependent intrinsic chargeability, the relaxation time, and the Cole–Cole frequency exponent.
• As described in Section 2, the authors use this observed resistance data to recover the electrical conductivity spatial heterogeneities.
• Processing and Modeling of Time Domain Induced Polarization Data, 23rd European Meeting of Environmental and Engineering Geophysics, extended abstract, doi: 10.3997/2214-4609.201702085.
• Recent advances and applications in complex resistivity measurements, Geophysics, 40(5), 851–864.

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3D time-domain induced polarization tomography: a
new approach based on a source current density
formulation
A. Soueid Ahmed, A. Revil
To cite this version:
A. Soueid Ahmed, A. Revil. 3D time-domain induced polarization tomography: a new approach based
on a source current density formulation. Geophysical Journal International, Oxford University Press
(OUP), 2017, 213, pp.244-260. �10.1093/gji/ggx547�. �hal-02324216� Geophysical Journal International
Geophys. J. Int. (2018) 213, 244–260 doi: 10.1093/gji/ggx547
Advance Access publication 2017 December 18
GJI Marine geosciences and applied geophysics
3-D time-domain induced polarization tomography: a new approach
based on a source current density formulation
A. Soueid Ahmed
1,2
and A. Revil
1,2
1
Universit
´
e Grenoble Alpes, CNRS, IRD, IFSTTAR, ISTerre, F-38000 Grenoble, France. E-mail: andre.revil@univ-smb.fr
2
Universit
´
e Savoie Mont Blanc, ISTerre, F-73000 Chamb
´
ery, France
Accepted 2017 December 17. Received 2017 December 13; in original form 2017 April 17
SUMMARY
Induced polarization (IP) of porous rocks can be associated with a secondary source current
density, which is proportional to both the intrinsic chargeability and the primary (applied)
current density. This gives the possibility of reformulating the time domain induced polar-
ization (TDIP) problem as a time-dependent self-potential-type problem. This new approach
implies a change of strategy regarding data acquisition and inversion, allowing major time
savings for both. For inverting TDIP data, we ﬁrst retrieve the electrical resistivity distribution.
Then, we use this electrical resistivity distribution to reconstruct the primary current density
during the injection/retrieval of the (primary) current between the current electrodes A and B.
The time-lapse secondary source current density distribution is determined given the primary
source current density and a distribution of chargeability (forward modelling step). The in-
verse problem is linear between the secondary voltages (measured at all the electrodes) and the
computed secondary source current density. A kernel matrix relating the secondary observed
voltages data to the source current density model is computed once (using the electrical con-
ductivity distribution), and then used throughout the inversion process. This recovered source
current density model is in turn used to estimate the time-dependent chargeability (normalized
voltages) in each cell of the domain of interest. Assuming a Cole–Cole model for simplicity,
we can reconstruct the 3-D distributions of the relaxation time τ and the Cole–Cole exponent
c by ﬁtting the intrinsic chargeability decay curve to a Cole–Cole relaxation model for each
cell. Two simple cases are studied in details to explain this new approach. In the ﬁrst case, we
estimate the Cole–Cole parameters as well as the source current density ﬁeld from a synthetic
TDIP data set. Our approach is successfully able to reveal the presence of the anomaly and to
invert its Cole–Cole parameters. In the second case, we perform a laboratory sandbox exper-
iment in which we mix a volume of burning coal and sand. The algorithm is able to localize
the burning coal both in terms of electrical conductivity and chargeability.
Key words: Electrical properties; Hydrogeophysics; Electrical resistivity tomography (ERT);
Inverse theory; Numerical modelling; Tomography.
1 INTRODUCTION
In hydrogeophysics, the characterization of subsurface geological
structures (geometry and petrophysical properties) is nowadays
routinely performed by the means of geoelectrical methods such
as Electrical resistivity tomography (ERT) and induced polariza-
tion (IP) techniques (e.g. Michot et al. 2003; Comte et al. 2010;
Fiandaca et al. 2012, 2013; Kemna et al. 2012; Binley et al. 2015).
ERT is restricted to the mapping of the electrical resistivity ﬁeld
only. Electrical resistivity depends on many factors such as salin-
ity, temperature, water content, and the cation exchange capacity
(CEC) of the material (e.g. Waxman & Smits 1968; Shainberg et al.
1980;Revilet al. 1998). While resistivity monitoring can be applied
to the monitoring of the water content, for instance in agriculture
(e.g. Michot et al. 2003), the interpretation of DC resistivity data
alone is notoriously difﬁcult. IP goes further by being able to map,
in addition to the electrical resistivity, other physical parameters of
interest such as the chargeability and a distribution of relaxation
times (Kemna et al. 2012). In that sense, IP can be considered as a
useful and fruitful extension of the conventional ERT.
IP effects were ﬁrst discovered during the last century (e.g.
Schlumberger 1920; Dakhnov 1962). It was observed that when
injecting a primary current into the ground, and then suddenly shut-
ting it off, porous soils and rocks are able to store reversibly electrical
charges and produce a secondary voltage that is decaying over time.
This secondary voltage can last for few seconds to few minutes
depending on the duration of the impressed primary current and IP
characteristics of the subsurface (Kemna et al. 2012). At the pore
244
The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society Induced polarization tomography 245
scale, this process is diffusive and the charge carriers go back to
their equilibrium state driven by chemical potential gradients once
they have accumulated at some polarization length scales (e.g. at
the edges of grains or pores).
Historically, the IP method was mainly used in mineral explo-
ration for the detection of ore deposits because the chargeability of
these targets is generally very strong (e.g. Bleil 1953; Van Voorhis
et al. 1973; Zonge & Wynn 1975; Pelton et al. 1978; Telford et al.
1990). Later on, thanks to the technological progresses made in
data acquisition, sensitivity of the instruments, and computers per-
formances (e.g. Zimmermann et al. 2008), IP has become a very
important method to investigate a broad spectrum of environmen-
tal applications. One can cite, for instance, the study of contami-
nants plumes (e.g. Olhoeft 1984, 1985, 1986; Vanhala et al. 1992,
Vanhala 1997; Slater & Lesmes 2002; Kemna et al. 2004;Wain-
wright et al. 2014), the characterization of permeability and pore
size distribution (e.g. Sturrock et al. 1999;Revilet al. 2015c;Joseph
et al. 2016;Ostermanet al. 2016), and recently coal seam ﬁres de-
tection and localization (e.g. Shao et al. 2017) just to cite few
examples among a very rich literature.
The conventional time-domain induced polarization (TDIP) is
restricted to the evaluation of the DC electrical conductivity σ
0
and
the chargeability M. However, the IP properties of soils and rocks
can be represented by a distribution of relaxation times as well. This
distribution can be sometimes simpliﬁed by a mean and a standard
deviation. For instance, in a classical representation model known
as the Cole–Cole model (Cole & Cole 1941), the distribution of re-
laxation times is described by two parameters namely the relaxation
time τ , which describes a mean relaxation time, and the Cole–Cole
exponent c, which describes the broadness of the distribution. The
relaxation timeτ refers to the main time taken by a material that has
been submitted to an electrical ﬁeld or an electrical current, to go
back to its equilibrium state. In a Debye model the relaxation time
is the time required to see the secondary voltage falling down by
a factor exp(1) from its nominal value. Thanks to physical mod-
els such as the dynamic Stern layer model (Rosen & Saville 1991;
Rosen et al. 1993;Revil2012, 2013), these Cole–Cole parameters
can be interpreted in terms of textural and electrochemical proper-
ties of the material under consideration. The dynamic Stern layer
model has proven indeed to be an efﬁcient model to explain various
empirical trends observed in the literature for rocks in absence of
metallic particles (see for instance the works by Weller et al. 2011,
2013, 2015a,b, who developed a series of empirical relationships
all explainable by the dynamic Stern layer model). In the absence
of metallic particles, this includes a mean pore (grain) size, a pore
(grain) size distribution and the CEC of the material, which proper-
ties can be independently measured in the laboratory (experimental
checks are for instance provided by Revil et al. 2014; Niu & Revil
2016). Induced polarization can also bring information regarding
the presence of semi-conductors, metals, and semi-metals (Pelton
et al. 1978;Revilet al. 2015a,b;Mao&Revil2016;Maoet al.
2016;Revilet al. 2017a,b). This is due to the very strong IP re-
sponse associated with the presence of metallic particles embedded
in a porous material, which can be also affected by redox processes
(Wong 1979) and the polarization of the pore water around the
metallic particles (Misra et al. 2016a,b).
A number of published works have been conducted to image
the Cole–Cole parameters in the subsurface. For instance, Loke
et al. (2006) used a 2-D least square inversion to recover the Cole–
Cole parameters distributions in a laboratory sandbox experiment.
Ghorbani et al.(2007
) inver ted the Cole–Cole parameters using a 1-
D Bayesian inference approach and they applied their methodology
on synthetic homogenous half spaces. Yuval & Oldenburg (1997)
estimated, in 2-D, the Cole–Cole parameters from TDIP data using
a very fast simulated annealing approach. They successfully recov-
ered these parameters on synthetic and real ﬁeld data sets. Fiandaca
et al.(2012) developed a forward and inverse code, which takes into
account the modelling of the transmitter waveform and the receiver
transfer function. Such methodology allowed for improving the res-
olution of the estimated Cole–Cole parameters. Recently, Nivorlis
et al. (2017) proposed a 3-D computation scheme to retrieve in 3-D
the Cole–Cole parameters. The last step of their work is accom-
plished using a particle swarm optimization algorithm. All these
works follow the same path in describing induced polarization in
terms of a time dependent electrical conductivity problem. In this
approach, conductivity changes from an instantaneous conductivity
(all the charge carriers are mobiles) to a DC conductivity for which
some of the charge carriers are blocked at some polarization lengths
scales and do not participate anymore to the conduction process.
This approach ﬁnds its roots in the seminal work of Siegel (1959),
who proposed to model the IP effects as a perturbation of the electri-
cal conductivity ﬁeld by the chargeability. Following this approach,
an apparent chargeability is obtained by solving the Poisson’s equa-
tion twice: once with the DC electrical conductivity σ
as input
(σ
denotes the instantaneous conductivity, induction effects being
neglected) and the other by taking σ
0
= σ
(1 M) as input (i.e.
using the DC conductivity distribution). This method has the merit
of being straightforward and uses the same forward operator cor-
responding to the Poisson equation to solve the conductivity and
chargeability problems. This formulation has been widely taken up
and used by a majority of geophysicists.
Our approach follows a quite different path, which can also be
traced to the seminal work of Siegel (1959). Indeed, Siegel demon-
strated that the (primary) current injection creates a secondary cur-
rent density J
S
(t) in the conductive ground. This secondary current
is related to the primary current J
p
through the chargeability evo-
lution once the primary current has been shut off. This point, ﬁrst
raised by Siegel (1959) to our knowledge, has not been used by IP
practitioners (despite the advantage that comes with it as discussed
below). Since the secondary source current density is formally simi-
lar to a diffusion source current density in self-potential studies, time
domain induced polarization can be described as a time-dependent
equivalent self-potential-type problem (see Mao & Revil 2016,fora
preliminary s tep in this direction). We can be more explicit here. The
secondary source current density is driven in induced polarization
by chemical potential gradients exactly like diffusion potentials in
self-potential studies (e.g. Ikard et al. 2012). The only formal differ-
ence is that in induced polarization, the chemical potential gradient
of the charge carriers have been ‘actively’ set up by the injection
of the primary current (through cross-coupling effects, see for in-
stance Revil 2017a,b), while in classical self-potential studies, the
chemical potential gradients can come from the injection of a salt
tracer in the environment (e.g. Jardani et al. 2013).
The advantage of formulating the IP problem as an equivalent
self-potential problem is that we avoid solving a nonlinear in-
verse problem because retrieving the current density J
S
(t) from the
recorded electrical ﬁeld is a linear problem. Examples of such linear
problem in self-potential tomography can be found in Mahardika
et al.(2012) and Haas et al.(2013) and in electroencephalography
for instance by Trujillo-Barreto et al.(2004). A linear inverse prob-
lem does not require the use of an iterative process. This means that
notable computational time savings can be made for the tomogra-
phy of the intrinsic chargeability ﬁeld. New instrumentations that
are massively multichannels such as the IRIS Full waver instrument 246 A. Soueid Ahmed and A. Revil
can operate the way we advocate: all the stations measure simultane-
ously the electrical ﬁeld (i.e. the gradient of the electrical potential
distribution along the curvilinear coordinates of the ground surface)
for each injection bipole [A, B].
In this work, the IP data collection is performed in a self-potential
‘fashion’, this means that a limited number of primary current in-
jections is performed and the secondary voltage measurements are
recorded at all remaining electrodes like in a self-potential survey
can save a lot of time with modern multi-channel technologies. Con-
sidering the secondary voltages as pseudo self-potential data is also
correct from a physical point of view since these secondary volt-
ages are driven by chemical potential gradients. They are therefore
identical in nature to diffusion potentials as mentioned above.
The goal of this paper is to develop the novel approach mentioned
above by formulating the TDIP forward and inverse problems as
an equivalent self-potential problem. We present a 3-D framework
for recovering the Cole–Cole parameters spatial distributions from
TDIP data. To the best of our knowledge, this is the ﬁrst attempt to
recover the Cole–Cole parameters in 3-D using such an approach.
The proposed algorithm is validated on two cases. (i) A synthetic
model where the Cole–Cole parameters true distributions are known
and will be compared to the estimated ones. (ii). A laboratory exper-
iment is performed with some coal burning in a sandbox. Our goal
is to develop a proof-of-concept of the method before to explore
complex geometries in future contributions and to be didactic in
describing the step-by-step procedure in getting the end-results.
2 FORWARD MODELLING
2.1 Electrical conductivity and chargeability
We consider below time scale and length scales such as the in-
duction effect can be neglected. The primary electrical potential
ﬁeld generated just after a current injection in a 3-D heterogeneous
isotropic medium can be described by the Poisson equation as
∇·
(
σ
ψ
)
+ I δ
3
(r r
0
) = 0, (1)
where σ
(in S m
1
) denotes the 3-D high frequency electrical con-
ductivity ﬁeld (i.e. the instantaneous conductivity of a medium
submitted to an electrical ﬁeld, electromagnetic induction ef-
fects neglected, see Fig. 1), ψ (in V) is the electrical poten-
tial ﬁeld generated by the injection of the current I (in A),
δ
3
(r r
0
) = δ(x x
0
) δ(y y
0
) δ(z z
0
) is the Dirac distribution,
x, y, z represent the space locations and x
0
, y
0
, z
0
are the spatial co-
ordinates of the current sources locations. Eq. (1) is subject to the
following boundary conditions
ψ = α on
1
, (2)
σ
ψ · n = β on
2
, (3)
with
1
2
= ∂ where ∂ denotes the simulation domain
boundaries, n is the outward unit vector perpendicular to
2
.In
case of α = 0andβ = 0, we refer to these boundary conditions as
homogenous Dirichlet and homogenous Neumann boundary condi-
tions, respectively. If the subsurface is going to inﬁnity, we can take
the potential going to zero at the external boundary of the domain
(this is easily done with a ﬁnite elements solver using coarse mesh-
ing outside the area of interest and performing some benchmark
testing).
Once eq. (1) is solved, one can compute the apparent re-
sistivity associated with the instantaneous conductivity through
Figure 1. Polarization of a porous material. (a) Recorded voltage at the two
voltage electrodes M and N as a function of time. The primary current is
applied for the period T
on
(from T to 0). The secondary voltage is measured
after the primary current is shut down (during T
off
). (b) Polarization of
the grains. The instantaneous conductivity σ
is deﬁned right after the
application of the primary current. All the charge carriers are mobile. After
a long time, the some of the charge carriers are blocked. This deﬁned the
direct current conductivity σ
0
.
ρ
a
= K ψ
MN
/I ,whereK is the geometric factor (depending on
the position of A, B, M, and N and the topography), ψ
MN
is the
potential difference between two measuring electrodes M and N
(Fig. 1a). Furthermore, eq. (1) can be seen as a nonlinear map-
ping operator F(.) associating the electrical potential ψ
σ
(Fig. 1)
to the electrical conductivity σ
,thatis,ψ
σ
= F(σ
). This po-
tential ψ
σ
is instantaneously recorded when the current injection
is turned on. Similarly, a potential ψ
σ
0
= F(σ
0
) can be registered
when the primary current has been applied long enough (Fig. 1).
The concepts of instantaneous and DC conductivities are explained
physically in Fig. 1(b) in the context of the dynamic Stern layer
model.
The chargeability distribution is deﬁned as
M = 1
σ
0
σ
=
σ
σ
0
σ
, (4)
which is equivalent to
σ
0
= σ
(
1 M
)
. (5)
This equation has a clear physical meaning to the light of
Fig. 1(b): in steady-state conditions, some of the charge carriers
are blocked and the DC conductivity is reduced by a factor (1
M) with respect to the instantaneous conductivity for which all
the charge carriers are mobile. This reduction is also thoroughly Induced polarization tomography 247
discussed by Seigel (1949, 1959). The DC conductivity is obtained
with
ψ
σ
0
= F
(
σ
(
1 M
))
. (6)
Similarly, an apparent chargeability can be computed as
M
a
=
ψ
σ
ψ
σ
0
ψ
σ
=
F
(
σ
)
F
(
σ
(
1 M
))
F
(
σ
)
. (7)
Therefore, in the classical approach, modelling the time domain
IP requires solving the electrical conductivity equation (1) twice
with two distinct electrical resistivity distributions (as proposed by
Siegel 1959).
In time domain IP surveys, we generally compute the partial
chargeability (expressed in milliseconds) and deﬁned as
M
t
i
,t
i+1
=
1
ψ
0
t
i+1
t
i
ψ
MN
(
t
)
dt, (8)
where M
t
i
,t
i+1
is the partial chargeability measured during the time
window [t
i
, t
i+1
], ψ
MN
(t) describes the decaying voltage measured
just after the current is shut off (reference time) and ψ
0
denotes the
primary voltage between the potential electrodes M and N at the
end of the current injection (Fig. 1a).
Likewise, partial chargeability (unitless) can be expressed in
mV V
1
as:
M
t
i
,t
i+1
=
1
ψ
0
(
t
i+1
t
i
)
t
i+1
t
i
ψ
MN
(
t
)
dt, (9)
where the primary voltage is given in V and the secondary voltage in
mV. Partial chargeability can be related to the apparent chargeability
M
a
through the approximation:
M
t
i
,t
i+1
M
a
(
t
i+1
t
i
)
. (10)
Note that this approximation is valid only under the condition that
t
i
, t
i+1
τ (see Florsch et al. 2011). This equation can be used
to determine for each quadripole ABMN (A and B being the cur-
rent electrodes and M and N the potential electrodes), an apparent
chargeability M
a
. Note that our notations are pretty standard. Some
authors used sometimes the letters m or η to denote the chargeability.
2.2 Forward modelling in the charging phase
We assume now that the primary current J
p
= σ
0
E (at low frequen-
cies ∇×E = 0 and therefore E =−ψ) has been applied from
T to time 0 (Fig. 1a). During the injection of the primary cur-
rent, each cell of the discretized subsurface will see a secondary
current building up. If each cell is characterized by four Cole–Cole
parameters (σ
0
, M,c), the secondary source current density is
determined by (Seigel 1959),
J
S
(t) =−M(t )J
p
. (11)
Eq. (11) means that the dipole moment associated with the polar-
ization of a grain (see Fig. 1b) is antiparallel to the applied current
eq. (11) can be found in the work of Seigel (1949, 1959) and will
not be repeated here. Note that we are assuming a Cole–Cole com-
plex resistivity model below which, in turn, can be easily related
to a Cole–Cole complex conductivity model using a relationship
between the time constants (see Florsch et al. 2012; Tarasov &
Titov 2013). Eq. (11) is the key equation of this paper. The source
(secondary) current density (index s) is intrinsically associated with
cross-coupling phenomena associated with the existence of ionic
chemical potential gradients at the scale of the grains or the pores
(see Fig. 1b).
In the quasi-static limit of the electromagnetic equations (taking
all time derivatives to zero in the continuity equations), for each cell
the constitutive equation and conservation equations are simply,
J = J
p
+ J
S
, (12)
∇·J = I δ
3
(r r
0
). (13)
The fact that the total current density is solenoidal (i.e. divergence
free) outside the source of primary current sources or sinks (i.e. at
electrodes A and B) was extensively discussed by Seigel (1959).
For each cell, the total current density can be written as,
J =
[
1 M(t)
]
J
p
, (14)
J =−
[
1 M(t)
]
σ
ψ. (15)
When the primary current has been applied for long time so that the
material is entirely polarized, we have M(t) = M, and we have.
J =−σ
0
ψ, (16)
that is, the conductivity has reached its steady sate value σ
0
=
σ
(1 M). In this case, resistivity tomography provides the DC
conductivity distribution of the subsurface. Our goal is to build the
function M(t) during the charging phase obeying to the following
properties,
M(0) = 0, (17)
M(+∞) = M. (18)
For a Debye model, we have
M(t) = M
1 exp
t
τ

, (19)
which is the classical behaviour for an RC circuit. For a Cole–Cole
model, we have
M(t) = MG
t
τ
; c
, (20)
G
t
τ
0
; c
= 1
n=0
(1)
n
t
τ
0
nc
(1 + nc)
, (21)
G
t
τ
0
; c
=
n=1
(1)
n+1
t
τ
0
nc
(1 + nc)
, (22)
where ( . ) denotes the Euler gamma function deﬁned by,
(x) =
0
u
x1
e
u
du, (23)
where x > 0. Note that the series development in eq. (22) converges
very slo wly for t > 10 τ and c < 1. It is easy to show that for c = 1,
we recover the Debye model,
G
t
τ
; c = 1
= 1
n=0
(1)
n
t
τ
n
(1 + n)
, (24)
G
t
τ
; c = 1
= 1
n=0
t
τ
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TL;DR: In this paper, a 3D tomogram of the relative variation in water content (before leakage and during leakage) was estimated and 2.5D time lapse tomography of the electrical conductivity and normalized chargeability was also performed and evidences the position of the preferential flow paths below the profile.
Abstract: During an induced polarization survey, both electrical conductivity and chargeability can be imaged. Recent petrophysical models have been developed to provide a consistent picture of these two parameters in terms of water and clay contents of soils. We test the ability of this method at a test site in which a controlled artificial leakage can be generated in an embankment surrounding an experimental basin. 3D tomography of the conductivity and normalized chargeability are performed during such a controlled leakage. Conductivity and induced polarization measurements were also performed on a core sample from the site. The sample was also characterized in terms of porosity and cation exchange capacity. Combining the 3D survey and these laboratory measurements, a 3D tomogram of the relative variation in water content (before leakage and during leakage) was estimated. It clearly exhibits the ground water flow path through the embankment from the outlet of the tube used to generate the leak to the bottom of the embankment. In addition, a self-potential survey was performed over the zone of leakage. This survey evidences also the projection of the ground water flow path over the ground surface. Both methods are found to provide a consistent picture. A 2.5D time lapse tomography of the electrical conductivity and normalized chargeability was also performed and evidences the position of the preferential flow paths below the profile. These results confirm the ability and efficiency of induced polarization to provide reliable information pertaining to the detection of leakages in dams and embankments.

29 citations

01 Jan 1948
TL;DR: In this paper, the induced polarization susceptibility of a homogeneous, uniformly mineralized earth was defined and a method of analyzing field data was described, and the results of the laboratory experiments provided an explanation of the induction of induced polarization potential.
Abstract: Laboratory experiments have shown certain fundamental relationships concerning the induction of a polarization potential on a metallic body in an electrolyte. The potential induced is a linear function of the potential drop across the body in the energizing field up to a saturation potential of 1.2 volts. Diffusion of ions and chemical action are the predominant factors which determine the rate of growth or decay of the polarization potential. Polarization occurs only at the boundaries of electrically conducting minerals. The results of the laboratory experiments provide an explanation of the induced polarization potential of a homogeneous, uniformly mineralized earth. This potential falls off as 1/r from a point electrode. Induced polarization susceptibility is defined and a method of analyzing field data is described. Field measurements over two mineralized zones (pyrrhotite and magnetite) substantiate the theory as developed.

11 citations

01 Jan 1997
TL;DR: In this article, the effects of waste oil and motor oil on the phase and amplitude spectra of the resistivity were studied using artificially contaminated sand and till samples and mineral soil samples from real waste sites.
Abstract: The laboratory and field results from an environmental application of the spectral induced polarization (SIP) method are presented. The phase spectra of the resistivity of uncontaminated glacial till, silt, sand and gravel were measured in the laboratory. The effects of waste oil and motor oil on the phase and amplitude spectra of the resistivity were studied using artificially contaminated sand and till samples and mineral soil samples from real waste sites. Field IP and SIP measurements were also made at the waste sites. The laboratory phase spectra of sands and tills were straight or slightly concave upwards in a log-log plot. The phase angle varies between 0.1 and 20 mrad at 1 Hz frequency and increases towards higher frequencies with a slope of 0.15-0.25. In laboratory tests, motor oil and waste oil changed the phase and amplitude spectra of sand and till. At first, the amplitude and phase decreased due to oil contamination. Later, during continued maturation, both the amplitude and phase increased. After a few days or weeks of maturation, some of the contaminated samples showed a convex-upwards phase spectrum. The features observed in artificially contaminated samples were also detected in the sample material from real waste sites. Furthermore, the in situ results from the waste sites were in agreement with the laboratory results. In laboratory tests, the phase spectra of clean sand and till remained stable with time, whereas the phase spectra of oil-contaminated samples changed with increasing maturation time. This, together with the field results, suggests that differences between the spectra of clean and polluted soils, and also changes occurring in the phase spectra of contaminated soils with time, can be indicative of contamination.

9 citations

Journal ArticleDOI
TL;DR: In this article, the authors investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation.
Abstract: Physically natural assumption says that any relaxation process taking place in the time interval [ t 0 , t 2 ] , t 2 > t 0 ≥ 0 may be represented as a composition of processes taking place during time intervals [ t 0 , t 1 ] and [ t 1 , t 2 ] where t 1 is an arbitrary instant of time such that t 0 ≤ t 1 ≤ t 2 . For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain.

8 citations

Journal ArticleDOI
30 Apr 2022-Minerals
TL;DR: In this article , a petrophysical model of the induced polarization of metallic ores immersed in a porous conductive and polarizable material is reviewed, and its predictions are compared to a large dataset of experimental data.
Abstract: Disseminated ores in porous or fractured media can be polarized under the application of an external low-frequency electrical field. This polarization is characterized by a dimensionless property that is called the “chargeability”. Induced polarization is a nonintrusive geophysical sensing technique that be used in the field to image both the electrical conductivity and the chargeability of porous rocks together with a characteristic relaxation time. A petrophysical model of the induced polarization of metallic ores immersed in a porous conductive and polarizable material is reviewed, and its predictions are compared to a large dataset of experimental data. The model shows that the chargeability of the material is linearly dependent on the volume fraction of the ore and the chargeability of the background material, which can, in turn, be related to the conductivity of the pore water and the cation exchange capacity of the clay fraction. The relaxation time depends on the grain sizes of the ores and on the conductivity of the background material, which is close to the conductivity of the porous rock itself. Five applications of the induced-polarization method to ore and metallic bodies are discussed in order to show the usefulness of this technique. These applications include: (i) A sandbox experiment, in which cubes of pyrite are located in a specific area of the tank; (ii) The tomography of an iron slag at an archeological site in France; (iii) A study of partially frozen graphitic schists in the French Alps; (iv) The detection of a metallic tank through the tomography of the relaxation times; and (v) The detection and localization of a deep ore body that is associated with a tectonic fault. We also discuss the possibility of combining self-potential and induced-polarization tomography to better characterize ore bodies below the seafloor.

8 citations

##### References
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Journal ArticleDOI
TL;DR: In this paper, the locus of the dielectric constant in the complex plane was defined to be a circular arc with end points on the axis of reals and center below this axis.
Abstract: The dispersion and absorption of a considerable number of liquid and dielectrics are represented by the empirical formula e*−e∞=(e0−e∞)/[1+(iωτ0)1−α]. In this equation, e* is the complex dielectric constant, e0 and e∞ are the static'' and infinite frequency'' dielectric constants, ω=2π times the frequency, and τ0 is a generalized relaxation time. The parameter α can assume values between 0 and 1, the former value giving the result of Debye for polar dielectrics. The expression (1) requires that the locus of the dielectric constant in the complex plane be a circular arc with end points on the axis of reals and center below this axis.If a distribution of relaxation times is assumed to account for Eq. (1), it is possible to calculate the necessary distribution function by the method of Fuoss and Kirkwood. It is, however, difficult to understand the physical significance of this formal result.If a dielectric satisfying Eq. (1) is represented by a three‐element electrical circuit, the mechanism responsible...

8,409 citations

### "3-D time-domain induced polarizatio..." refers methods in this paper

• ...For instance, in a classical representation model known as the Cole–Cole model (Cole & Cole 1941), the distribution of relaxation times is described by two parameters namely the relaxation time τ , which describes a mean relaxation time, and the Cole–Cole exponent c, which describes the broadness…...

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Book
01 Jan 1977

8,009 citations

### "3-D time-domain induced polarizatio..." refers methods in this paper

• ...We formulate the inverse problem as an optimization problem, for which we seek to minimize the following objective function (Tikhonov & Arsenin 1977): L Js = ∥∥Wd (GmJS − dobs)∥∥22 + β∥∥Wm(mJS − mJ 0S )∥∥22, (43) where Wd is the diagonal nϕ × nϕ data covariance matrix, mJS = (JSx, JSy, JSz ) is the…...

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Journal ArticleDOI
TL;DR: This survey intends to relate the model selection performances of cross-validation procedures to the most recent advances of model selection theory, with a particular emphasis on distinguishing empirical statements from rigorous theoretical results.
Abstract: Used to estimate the risk of an estimator or to perform model selection, cross-validation is a widespread strategy because of its simplicity and its apparent universality. Many results exist on the model selection performances of cross-validation procedures. This survey intends to relate these results to the most recent advances of model selection theory, with a particular emphasis on distinguishing empirical statements from rigorous theoretical results. As a conclusion, guidelines are provided for choosing the best cross-validation procedure according to the particular features of the problem in hand.

2,980 citations

### "3-D time-domain induced polarizatio..." refers methods in this paper

• ...…× nσ ) data covariance matrix, s0 is an a priori conductivity model, λ is the regularization parameter and can be chosen using a trial-and-error process, or some approaches such as, the L-curve approach (e.g. Hansen 1998), the Generalized Cross-validation (GCV) approach (e.g. Arlot & Celisse 2010)....

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• ...This is a classical non-linear inverse problem for which we minimize the following objective function: Lσ = ( dobsσ −Fσ (s) ) R−1σ ( dobsσ −Fσ (s) )T +λ (s−s0) C(s−s0)T , (36) with dobsσ denotes the (nσ × 1)observed data vector, where nσ is the number of measurements, in this case it represents the measured resistances or the apparent conductivities, Fσ (.) is the forward problem operator given by the Poisson equation, s denotes the (m × 1) model vector (unknown DC conductivities s = log10(σ0)), and m denotes the number of unknown cells, in our case, Rσ is the (nσ × nσ ) data covariance matrix, s0 is an a priori conductivity model, λ is the regularization parameter and can be chosen using a trial-and-error process, or some approaches such as, the L-curve approach (e.g. Hansen 1998), the Generalized Cross-validation (GCV) approach (e.g. Arlot & Celisse 2010)....

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Journal ArticleDOI
TL;DR: In this paper, a survey on the model selection performances of cross-validation procedures is presented, with a particular emphasis on distinguishing empirical statements from rigorous theoretical results, and guidelines are provided for choosing the best crossvalidation procedure according to the particular features of the problem in hand.
Abstract: Used to estimate the risk of an estimator or to perform model selection, cross-validation is a widespread strategy because of its simplicity and its apparent universality. Many results exist on the model selection performances of cross-validation procedures. This survey intends to relate these results to the most recent advances of model selection theory, with a particular emphasis on distinguishing empirical statements from rigorous theoretical results. As a conclusion, guidelines are provided for choosing the best cross-validation procedure according to the particular features of the problem in hand.

2,720 citations

Book
01 Jan 1987
TL;DR: In this article, the authors present a survey of regularization tools for rank-deficient problems and problems with ill-conditioned and inverse problems, as well as a comparison of the methods in action.
Abstract: Preface Symbols and Acronyms 1. Setting the Stage. Problems With Ill-Conditioned Matrices Ill-Posed and Inverse Problems Prelude to Regularization Four Test Problems 2. Decompositions and Other Tools. The SVD and its Generalizations Rank-Revealing Decompositions Transformation to Standard Form Computation of the SVE 3. Methods for Rank-Deficient Problems. Numerical Rank Truncated SVD and GSVD Truncated Rank-Revealing Decompositions Truncated Decompositions in Action 4. Problems with Ill-Determined Rank. Characteristics of Discrete Ill-Posed Problems Filter Factors Working with Seminorms The Resolution Matrix, Bias, and Variance The Discrete Picard Condition L-Curve Analysis Random Test Matrices for Regularization Methods The Analysis Tools in Action 5. Direct Regularization Methods. Tikhonov Regularization The Regularized General Gauss-Markov Linear Model Truncated SVD and GSVD Again Algorithms Based on Total Least Squares Mollifier Methods Other Direct Methods Characterization of Regularization Methods Direct Regularization Methods in Action 6. Iterative Regularization Methods. Some Practicalities Classical Stationary Iterative Methods Regularizing CG Iterations Convergence Properties of Regularizing CG Iterations The LSQR Algorithm in Finite Precision Hybrid Methods Iterative Regularization Methods in Action 7. Parameter-Choice Methods. Pragmatic Parameter Choice The Discrepancy Principle Methods Based on Error Estimation Generalized Cross-Validation The L-Curve Criterion Parameter-Choice Methods in Action Experimental Comparisons of the Methods 8. Regularization Tools Bibliography Index.

2,634 citations

### "3-D time-domain induced polarizatio..." refers methods in this paper

• ...…× nσ ) data covariance matrix, s0 is an a priori conductivity model, λ is the regularization parameter and can be chosen using a trial-and-error process, or some approaches such as, the L-curve approach (e.g. Hansen 1998), the Generalized Cross-validation (GCV) approach (e.g. Arlot & Celisse 2010)....

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