Geophysical Prospecting, 2007, 55, 589–605
Bayesian inference of the Cole–Cole parameters from time- and
frequency-domain induced polarization
A. Ghorbani,1,∗ C. Camerlynck,1 N. Florsch,1 P. Cosenza1 and A. Revil2
1 UMR
7619 “Sisyphe”, Université Pierre et Marie Curie, Paris, France, and 2 CNRS-CEREGE, Université d’Aix-Marseille III,
Aix-en-Provence, France
Received June 2006, revision accepted November 2006
ABSTRACT
The inversion of induced-polarization parameters is important in the characterization
of the frequency electrical response of porous rocks. A Bayesian approach is developed
to invert these parameters assuming the electrical response is described by a Cole–Cole
model in the time or frequency domain. We show that the Bayesian approach provides
a better analysis of the uncertainty associated with the parameters of the Cole–Cole
model compared with more conventional methods based on the minimization of a cost
function using the least-squares criterion. This is due to the strong non-linearity of the
inverse problem and non-uniqueness of the solution in the time domain. The Bayesian
approach consists of propagating the information provided by the measurements
through the model and combining this information with a priori knowledge of the
data. Our analysis demonstrates that the uncertainty in estimating the Cole–Cole
model parameters from induced-polarization data is much higher for measurements
performed in the time domain than in the frequency domain. Our conclusion is that
it is very difficult, if not impossible, to retrieve the correct value of the Cole–Cole
parameters from time-domain induced-polarization data using standard least-squares
methods. In contrast, the Cole–Cole parameters can be more correctly inverted in the
frequency domain. These results are also valid for other models describing the inducedpolarization spectral response, such as the Cole–Davidson or power law models.
1
INTRODUCTION
When a constant electrical current is injected either through
a water-saturated porous rock or at the ground surface of
a porous soil, part of the electrical power is stored in the
medium. When the current is stopped, this stored energy dissipates. This dissipation can be followed by monitoring the voltage either through the core sample or at the ground surface of
the earth, using non-polarizing electrodes (e.g. Schlumberger
1920; Seigel 1959). This voltage exhibits a typical relaxation
with a mean relaxation time usually ranging from few milliseconds to thousands of seconds (e.g. Seigel, Vanhala and
Sheard 1997). This phenomenon is called ‘induced polarization’ (IP) and can be also observed in the frequency domain.
∗ E-mail:
C
ghorbani@ccr.jussieu.fr
2007 European Association of Geoscientists & Engineers
In the frequency domain, IP is characterized by a phase shift
between the injected current and the measured voltage. In the
frequency domain, the apparent conductivity of the porous
material is therefore written as a complex number σ ∗ (ω), given
by
σ ∗ (ω) =
1
= σ (ω) + iσ (ω),
ρ ∗ (ω)
(1)
where ρ ∗ (ω) is the complex resistivity of the material, ω is
the frequency of the excitation current, i2 = −1, and σ and
σ are the measured real and imaginary components of the
conductivity, respectively.
Historically, induced polarization has been extensively used
to locate ore deposits (e.g. Madden and Cantwell 1967). More
recently, it has been applied in hydrogeophysics (e.g. Kemna,
Räkers and Dresen 1999; Kemna, Binley and Slater 2004).
589
590 A. Ghorbani et al.
For example, in the case of water-saturated sedimentary and
granular materials and soils, the Cole–Cole parameters are
closely related to the grain-size distribution and mean grain
size of the medium (see Chelidze, Derevjanko and Kurilenko
1977; Chelidze and Guéguen 1999; Kemna 2000). Therefore,
a 3D estimation of the grain-size distribution is possible using
inversion of the induced-polarization data obtained with an
array on non-polarizing electrodes. It was also found that the
mean relaxation time of induced polarization is also closely related to the specific surface area of the porous medium. When
combined with the electrical formation factor, the mean relaxation time can be used to determine the hydraulic conductivity
of soils and rocks (Schön 1996; Slater and Lesmes 2002). This
opens exciting perspectives in determining non-invasively the
distribution of the hydraulic conductivity of aquifers, with
numerous potential applications in the domain of water resources. The Cole–Cole function has also been used recently
to model the deformation, in the time domain, of porous sandstones (Revil et al. 2006).
The simplest model to represent IP phenomena is the Debye
model, which is characterized by a single relaxation time.
However, the range of observed relaxation times of watersaturated rocks is relatively wide. Consequently, there have
been many attempts to generalize the Debye model in the literature (Wait 1959; Dias 1972, 2000; Pelton 1977; Pelton et al.
1978; Wong 1979). Among the existing models, the Cole–Cole
model (Cole and Cole 1941) is the most successful (see Taherian, Kenyon and Safinya (1990) for a statistical comparison
of the merits of different models). The Cole–Cole model can
be described by the equivalent model with resistors and capacitances (Fig. 1). According to this model, the resistivity and the
conductivity of a porous rock are given by
ρ ∗ (ω) = ρ0 1 − m 1 −
1
1 + (iωτ )c
respectively. The Cole–Cole model depends on four fundamental parameters. They are the DC-resistivity ρ0 = 1/σ0 (σ0
is the DC-conductivity), the chargeability m, the mean relaxation time τ and the Cole–Cole exponent c. In general in mineralized rocks, m and τ depend on the quantity of polarizable
elements and their size, respectively (Pelton et al. 1978; Luo
and Zhang 1998). The exponent c depends on the size distribution of the polarizable elements (Vanhala 1997; Luo and
Zhang 1998). Klein and Sill (1982) found that the time constants were approximately equal to the grain size of the glass
beads. Binley et al. (2005) showed that there is a strong relationship between Cole–Cole parameters determined in the
frequency domain and the hydraulic conductivity of saturated
and unsaturated sandstone cores.
Several papers have been published in the last decade on the
inversion of the Cole–Cole parameters in both the time and frequency domains. Examples include works by Luo and Zhang
(1998), Routh, Oldenburg and Li (1998), Kemna (2000) and
Xiang et al. (2001) to cite just a few. These classical methods are all based on minimization of a cost function, the determination of a single value for each of the four Cole–Cole
parameters and, in some cases, an estimate of the uncertainty
associated with the values of the inverted parameters. If the
inverse problem is non-linear, the solution is non-unique, and
if the a posteriori probability distributions of the Cole–Cole
parameters do not follow Gaussian distributions, optimization approaches cannot yield the correct result. The goal of
our paper is to use a Bayesian approach to invert the four
(2)
and
σ ∗ (ω) = σ0 1 + m
(iωτ )c
1 + (iωτ )c (1 − m)
,
(3)
Figure 1 An equivalent circuit model corresponding to the Cole–Cole
model.
C
Figure 2 Ill-posed correspondence between the chargeability m of the
Cole–Cole model and its normalized version, m ≡ log(m/(1 − m)),
used for the Bayesian inversion.
2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 591
Cole–Cole parameters and to determine the probability densities in estimating these parameters. The inversion is performed both in the time domain and in the frequency domain
in order to compare the merits of inverting the parameters
in these two domains. Finally, we discuss the limitation of
the least-squares methods by comparing them with our novel
approach.
2
T H E B AY E S I A N I N F E R E N C E A P P R O A C H
Since the formulation of the Bayesian inversion by Tarantola
and Valette (1982a, b) (see also Tarantola 1987; Mosegaard
and Tarantola 1995; Scales and Sneider 1997), the Bayes theorem has been widely used to invert geophysical or hydrogeological data for a variety of applications. The philosophy
of this approach is closely related to the notion of ‘information theory’. It consists of propagating the ‘information’ (or
knowledge) provided by the measurements through the physical law involved (perfectly or probabilistically known), and including the a priori knowledge of the model parameters. Both
the data and the model parameters are described with probability distributions. The Bayesian approach preserves the full
knowledge provided by the data, combined with the physical
law and the a priori information on the data and model parameters. Therefore, it is the most suitable method to perform
the inversion of non-linear problems (Tarantola and Valette
1982a,b).
The validity of some assumptions reduces the Bayesian
method to the use of simple equations and rules (e.g. Florsch
and Hinderer 2000; Robain, Lajarthe and Florsch 2001).
These assumptions are the following:
1 All the measured data and the unknown model parameters
are assumed to be independent parameters. The law connecting the data to the model parameters is written,
d = G(θ),
(4)
where d is the vector of data, θ is the vector of the unknown
model parameters, and G is the function describing the for-
ward problem. In our case, G is based on the Cole–Cole model
used to characterize the induced-polarization problem.
2 The physical law is assumed to be exact and therefore the
probability distribution of the physical law is a Dirac function.
An alternative choice would be to consider that the Cole–
Cole model is not entirely appropriate to describe inducedpolarization data. In such a case, a probability density, taking
for example the form of a Gaussian distribution, could be considered, with a standard deviation reflecting the confidence
given to such a model. This standard deviation could be determined using a large data base of IP measurements performed
over a wide frequency spectrum and ascertaining how well the
Cole–Cole model fits the data (e.g. Taherian et al. 1990). In
the following, we assume, however, that the Cole–Cole model
is correct.
3 The measurements are assumed to follow a Gaussian distribution. This is generally true as long as the noise affecting the
data is random and results from applying the superposition
principle (Tarantola and Valette 1982a,b).
The a posteriori probability density function p(θ) combines
the information related to the Gaussian data, the a priori probability density function of the model parameters, and the forward model d = G(θ). This probability distribution is given
by (Tarantola and Valette 1982a,b)
−1
π(θ) exp −0.5∗ xT Cdd
x
p(θ) =
,
(5)
μ (θ)
where x ≡ d − G (θ), the superscript T denotes the transpose
of the vector, d is a vector of N measurements, θ is the vector
of unknown Cole–Cole model parameters to be inverted, Cdd
is the N × N data covariance matrix, π (θ) is the a priori
probability density of the model parameters, and μ (θ) is the
homogeneous probability measure. As p(θ) is a probability
density, we must have
p(θ)dθ = 1.
(6)
The a priori density π (θ) is used to incorporate a priori knowledge in the model parameters. This probability density could
be a Gaussian distribution, a multimodal distribution, or any
Table 1 Number of chargeability slices and time-interval sampling of decay curve in time-domain method in Cole mode of a Syscal Pro®
instrument. V dly and Mdly are the delay times (in ms), from which the samples (sampling rate of 10 ms) are taken, after injection and after the
current cut-off, respectively. The period of current signal injection is 2 s
V dly
Mdly
1
2-4
5-6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1260
20
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
180
200
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
592 A. Ghorbani et al.
probabilistic description of the parameter. A locally uniform
law (the probability distribution is constant over an interval
[θ 1 , θ 2 ] and vanishes elsewhere) is often used to describe this
a priori density.
The interpretation of μ (θ) is more challenging, especially in
the present case. It has been introduced in the general Bayesian
approach to generalize Shannon’s definition of information entropy (Tarantola and Valette 1982a). It represents the density
probability (or measure) of a given parameter that follows naturally when no a priori knowledge about this model parameter exists. Tarantola (2005) called this term the ‘homogeneous
probability measure’.
Numerous physical parameters that are positive (such as
distances, volumes, densities, electrical conductivities, absolute temperatures, etc.) usually follow non-informative laws of
the form μ (θ) = a/θ, where a is constant. These types of parameters are called ‘Jeffreys parameters’ (Jeffreys 1939). These
laws show scale-invariance properties: any multiplication or
power of a Jeffreys parameter is a Jeffreys parameter, and a
fractal transform also conserves their forms. Considering the
four Cole–Cole parameters (ρ 0 , m, τ , c), only ρ 0 and τ are
Jeffreys parameters.
Since all the four parameters involved in the Cole–Cole
model are independent, the homogeneous probability measure of the Cole–Cole parameters vector, θ = [ρ 0 , m, τ , c]T
is
a
.
(7)
μ(θ) =
m(1 − m)ρ0 τ
Hence, the Bayesian probability distribution is
α (θ) =
−1
aτρ0 m(1−m)π (θ) exp −0.5∗ (d − G (θ))T Cdd
(d − G (θ)) .
(8)
It is not easy to plot and interpret the four parameter
Bayesian probability distributions given by equation (8). Instead, we compute the marginal probability density functions
for one or two parameters. This approach is fundamentally
different and more informative than plotting the objective
function,
T
d − G (ρ0 , m, τ, c) C−1
.
dd d − G (ρ0 , m, τ, c)
Indeed, it is impossible to derive full parameter information
by slicing the objective function. Marginal laws preserve the
Figure 3 Time-domain inversion model with parameters, m = 0.8, c = 0.25, τ = 0.01 s. (a) 3D space of a posteriori pdf. (b) a posteriori pdf
versus interval parameter m with parameters, τ and c, fixed. (c), (d), (e) Marginal a posteriori pdfs of pairs of parameters.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 593
full ‘probability density function’ (pdf) because they involve
the integration of probability over parameters selected as integration variables. For example, the marginal pdf, integrated
with respect to ρ0 and c, is defined by
(9)
p (m, τ ) =
p(ρ0 , m, τ, c)dρ0∗ dc∗ .
However, special care must be taken when performing this integration. In the parameters space, dρ0 dc must be a volume
dV that preserves the information within intervals of ranges
dρ0 and dc. Therefore any change of variable must involve
the corresponding Jacobian determinant encapsulated within
the volume element. It follows that equation (9) is correct
(in this form) only when a constant homogeneous probability
measure is used. For instance, when considering Jeffreys parameters, equation (9) can be used only if suitable variables
are used, such as s’ = log(s) instead of s. Tarantola (2004)
strongly recommended using ‘volumetric probabilities’ instead
of ‘probability density’. It follows that the simplest way to use
‘natural’ variables (those for which the homogeneous probability is constant) is to make change of variable so that all
the homogeneous probabilities become constant. Therefore,
we performed the following change of variable:
⎧
⎪
⎨ m → m ≡ log(m/(1 − m)),
(10)
ρ0 → ρ0 ≡ log(ρ0 ),
⎪
⎩
τ → τ ≡ log(τ ),
and kept c unchanged. The change affecting m is explained
in detail in the Appendix. Another possibility that yields the
same result is to consider that the transform,
∂m (11)
μ (m ) = μ(m) ,
∂m
is constant. μ(m) and μ (m ) are homogeneous probability measures of m and m , respectively and |∂m/∂m| represents the absolute value of the Jacobian of the transformation (Tarantola and Valette 1982b). Equation (11) yields
a differential equation and m = log(m/(1 − m)) is a solution of this equation. Consequently, as in electrical resistance tomography where log(ρ) is a better parametrization than ρ, log(m/(1 − m)) is a better parametrization
than m.
Figure 4 Time-domain inversion model with parameters, m = 0.8; c = 0.75 and τ = 10 s. (a) 3D space of a posteriori pdf. (b), (c), (d) Marginal
a posteriori pdfs of pairs of parameters.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
594 A. Ghorbani et al.
The same type of approach was previously applied to chemical concentrations (Tarantola 2005). Indeed, a chemical concentration (mass fraction) C lies between 0 and 1. Consequently, this is not a Jeffreys parameter. A concentration
parameterCi is the ratio between a given product mass wi
with respect to the total mass M, i.e. Ci ≡ wi /M. Tarantola
(2005) introduced an ‘eigenconcentration’ (in his terminology) defined by the ratio Ci ≡ wi /(M − wi ), so thatCi is now
a Jeffreys parameter andμ log(Ci ) = a. We adopted a similar
approach here. In addition, the chargeability m can be seen as
a ‘concentration of polarizable elements’ (Pelton et al. 1978),
with the case m = 0 corresponding to the absence of polarizable elements and m = 1 corresponding to the saturation
of the medium in polarizable elements. Therefore, the chargeability m shows accumulation effects at m = 0 and m = 1.
Figure 2 shows the correspondence between the parameter m
and the more meaningful parameter m = log(m/(1 − m)) that
illustrates this accumulation effect (ill-posed correspondence
between m and m ).
Finally in our computations, we used the parameter transform described by equation (10) and the integrations required
to estimate the marginal pdf can be simply obtained by summing the sampled probability laws over a regular grid.
3
3.1
INVERSION IN THE TIME DOMAIN
The forward problem
In the time domain, the chargeability m is determined from the
residual voltage measured immediately after the shut-down of
an infinitely long, impressed current, divided by the observed
voltage just before the shut-down of the current (Seigel 1959;
Seigel et al. 1997). For non-metallic media, m ranges between
0 and 0.1. The time constant τ determines the rate of decay of
the residual voltage over time. In practice, this parameter has
a very broad range, from a few milliseconds to thousands of
seconds. The exponent c controls the curvature of the decaying
voltage in a log-log space voltage versus time (Seigel et al.
1997).
Figure 5 Time-domain inversion model with parameters, m = 0.2, c = 0.25, τ = 0.01 s. (a) 3D space of a posteriori pdf. (b), (c), (d) Marginal
a posteriori pdfs of pairs of parameters.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 595
Solving the parametric inversion problem in the time
domain requires a numerical solution for the forward problem. Several methods can be used to compute the residual
voltage assuming a Cole–Cole model. In the time domain, the
following transmitted current cycle (I0 , 0, –I0 , 0), with current amplitude I0 and characteristic duration T, is used. This
transmitted current cycle may be expressed in terms of Fourier
series as (Tombs 1981)
∞
nπ 3nπ
nπt
2
I(t) = I0
cos
− cos
sin
. (12)
nπ
4
4
2T
n=1
Assuming a Cole–Cole impedance model, the voltage drop
measured at a pair of electrodes is (Tombs 1981)
V(t) =
∞
2
nπ
3nπ
cos
− cos
ρ(ωn ) exp (iωn t) ,
I0 Im
nπ
4
4
n=1
(13)
where Im(a) corresponds to the imaginary part of the complex
parameter a and ωn ≡ nπ/2T are characteristic frequencies.
Note that equation (13) converges slowly. An alternative expression of the potential drop V(t) can be derived by applying
the Laplace transform to the Cole–Cole equation. This leads
to a series of positive or negative powers of (iω). Pelton et al.
(1978) determined the voltage response corresponding to a
unit positive step of applied current. This yields
V(t) = ρ0 m
∞
(−1)n (t/τ )nc
n=0
(1 + nc)
,
(14)
where (x) is the gamma function. This expression of V(t) has
an extremely slow convergence for t/τ > 10 and values of c
less than 1. For a better convergence, equation (14) can be
replaced by the following expression (Hilfer 2002):
V(t) = ρ0 m
∞
(−1)n+1 (t/τ c)−nc
n=1
(1 − nc)
.
(15)
Guptasarma (1982) introduced a digital linear filter to transform the frequency-domain response of polarized ground into
the time domain. The computation is easy to perform, very
fast, and the relative error of this approach is below 1%. We
Figure 6 Time-domain inversion model with parameters, m = 0.2, c = 0.75, τ = 0.01 s. (a) 3D space of a posteriori pdf. (b) Marginal a posteriori
pdf law of parameter c alone. (c), (d), (e) Marginal a posteriori pdf of pairs of parameters.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
596 A. Ghorbani et al.
used equation (15) and Guptasarma’s approach, separately, to
perform the forward model. No differences between the two
approaches were observed.
3.2
Classical inversion in the time domain
Yuval and Oldenburg (1997) solved the parametric inverse
problem by using the approach of Guptasarma (1982). As
explained above, we consider a sequence of current flows (I0 ,
0, −I0 , 0). The response of the infinite pulse train S(t) can be
represented as a sequence of step functions, each delayed by
multiples of the switching time T < 4 (T is the period of the
full waveform). This sequence is given by
T
T
+V t+
S(t) = V(t) − V t +
4
2
5T
3T
+ V (t + T) − V t +
+ ···
+ V t+
4
4
(16)
Consequently, using the superposition principle, the forward
modelling of this pulse-train response requires superposition of the series of positive and negative step responses
(Madden and Cantwell 1967). To accelerate the convergence of the infinite pulse-train response, we apply an alternating series using an Euler transformation (Press et al.
1992).
For conventional time-domain IP receivers, it is common to
sample the decay through a sequence of N slices or windows.
The value recorded for each slice is given by (Johnson 1984,
1990)
Mi =
C
ti+1
V(t)dt.
(17)
ti
We determine the chargeability numerically from the forwardmodelling approach described above. The relaxation time
curve is recorded with a minimum delay time of 20 ms; the
width of each partial window lasts at least 20 ms and 20 windows are used (Table 1). This corresponds to classical values used for field investigations. The rate of sampling on the
delay curve is 5 ms. We record V p , the potential at the cutoff of the electrical current, which is normally close to the
initial value of the decay curve V(t). Since the chargeability
is given by the normalized ratio of V(t)dt with respect to
V p (ti+1− ti ), this ratio does not depend on ρ 0 . Therefore, in the
inversion process, we used the normalized transient potential
V(t)/V p .
When using a Fourier series, at least 104 harmonics should
be used to ensure the convergence of the series. This involves
frequencies from 0.125 Hz to 1250 Hz. However, in order
to reproduce the acquisition of data in the field, we consider
a sampling rate of 5 ms, as in most field acquisitions. It is
clear that this choice affects the recorded voltage signal (see
Table 1).
3.3
Figure 7 Partial chargeability curve for the points a, b, c and d represented on Fig. 6.
103
Vp (ti+1 − ti )
Bayesian inversion in the time domain
The computation of the Bayesian solution of the inverse problem is based on equation (8). We record d, the vector of the
N sampled normalized residual voltage data assumed to be
independent of each other, while θ is the vector of the three
remaining unknown parameters (m , c, τ ). All the a priori
distributions of the changed variables were taken as uniform
within given intervals. The parameter m lies in the range –2.3
to +2.3 and consequently m, given by m = 10m /(1 + 10m ),
lies in the range 0.005 to 0.995. The parameter c lies in the
range 0 to 1 while τ = log (τ ) lies in the range –4 to 4 or
alternatively τ lies in the range 10–4 to 104 s.
To represent the 3D a posteriori pdfs, we use 3D plots in
the space of the model parameters (m , c, τ ). We plot the
marginal laws for the pairs of parameters in the spaces (m ,
2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 597
c), (m , τ ) and (c, τ ), respectively. To provide a clear insight
into the results of the inversion, we study a set of eight examples: each of the three Cole–Cole parameters m, c, and
τ can take two distinct values: m = 0.2 and 0.8, c = 0.25
and 0.75 and τ = 0.01 and 10. However, because some of
these eight cases present similar results in terms of marginal
laws in the spaces (m , c), (m , τ ), and (c, τ ), we choose to
present four cases exhibiting very distinct behaviours. They are
shown in Figs. 3–7. The results are discussed in the following
section.
3.4
Presentation of the results
Case 1
This case corresponds to the following set of parameters: m =
0.8, c = 0.25, τ = 0.01 s). In this first case, m is large but both
c and τ are small. Figure 3 shows that the solution crosses the
entire investigated domain. The marginal plots are shown in
Fig. 3(c,d,e). The marginal plots are not simply slices of the full
solution domain. Figure 3(b) shows the marginal pdf in the (τ ,
c) plane (after integration, the diagrams are plotted using the
parameter m instead of m ).
The ‘equivalence domain’ shown in Fig. 3(c) is rather complex and is very different from those that would be given
by Gaussian distributions. Several observations can be made
about Fig. 3. First, it must be remembered that the marginal
laws are constructed to provide probabilistic estimations of
the inverted parameters. Let us consider, for instance, Fig. 3(e),
which shows the pdf of m and τ = log (τ ); we denote this pmτ .
If this probability is properly normalized, we have
4 1
p m, τ dmdτ = 1,
(18)
−4
0
which can be used to determine the following probability:
p m, τ dmdτ .
(19)
p(m, τ ∈ D1 ) =
D1
The maximum of the pdf must be considered carefully. Indeed,
considering the domains D2 and D3 in Fig. 3(e), it is possible
that p(m, τ ∈ D3 ) p(m, τ ∈ D2 ), due to the fact that it is
the pdf integrals that are meaningful, not the pdfs themselves.
Figure 8 The marginal pdf of the Cole–Cole parameters, ρ 0 = 100 m, m = 0.8, c = 0.25, τ = 0.01 s, for synthetic data.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
598 A. Ghorbani et al.
Figure 3(e) shows the marginal law (m , τ ) for which several
maxima exist. This explains the failure of the classical leastsquares methods to determine the Cole–Cole parameters from
time-domain IP data. It is also important to realize that, in
such a case, the a posteriori conditional pdf, when integrated
within a given box, can have a maximum in a region where
the pdf has no maxima. This depends on the sharpness of the
pdf in the box. We will discuss this point below.
Case 3
This case corresponds to the following set of parameters: m =
0.2, c = 0.25, τ = 0.01 s. In this case, the marginal laws give an
equivalence between the parameters that is more pronounced
than in the previous cases (Fig. 5). For instance, there is more
than one value of τ for one value of m that fits the data.
Case 4
Case 2
This case corresponds to the following set of parameters: m =
0.8, c = 0.75, τ = 10 s. In this case, the equivalence domain
is simple (Fig. 4). Consideration of the values of the model
parameters (m, τ , c), according to the relationships between
them that are represented on these diagrams, will all fit the
data equally well. For example, if we take τ = 1000 s (τ =
3), m = 0.97, c = 0.6, the data is fitted equally well as when
taking m = 0.8, c = 0.75, τ = 10 s.
This case corresponds to the following set of parameters: m =
0.2, c = 0.75, τ = 0.01 s. In the marginal law of (m, c) (see
Fig. 6d), two branches appear, corresponding to two possible
solution areas. From this pdf, we can compute the marginal
law of parameter c alone, as shown in Fig. 6(b). It appears that
the left-hand part of the (m, c) pdf has a smaller volume than
the large ‘mount’ on the right-hand side of the plot, although
the left-hand part reach a higher pdf in two or three dimensions. In Fig. 6(a), the pdf has a narrow shape. In such a case,
the least-squares methods or the use of simulated annealing
Figure 9 The marginal pdf of the Cole–Cole parameters, ρ 0 = 100 m, m =0.8, c = 0.25, τ = 10 s, for synthetic data.
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Bayesian inference of IP parameters 599
algorithms reach the absolute maximum of the pdf in the 3D
parameter space domain, but these methods would still fail to
retrieve the correct values of the model parameters. Indeed,
the best fit will be found on the left-hand ridge, even though
the initial synthetic parameter belongs to the right-hand side
of the plot (c = 0.75). This illustrates very well the suitability of the Bayesian approach for strongly non-linear inverse
problems. There are equivalence laws such as the one shown
in the case corresponding to m = 0.8, c = 0.75, τ = 0.01 s.
These diagrams show that the chargeability cannot be correctly recovered. Indeed, any value of m between 0 and 1 will
fit the data equally well. Catalogues of the curves used in some
software to select a solution could lead to serious errors in estimating the model parameters. Note that whatever the real
value of m, inversion of the model parameters in the time domain also yields an acceptable solution with m ∼
= 1. In other
words, for any value of m, good fits of the time relaxation
curve can be obtained for suitable values of c and τ . For example, Fig. 7 shows, for the set of model parameters: m = 0.2,
c = 0.75, τ =0.01 s (curve a in Fig. 7), the curve obtained with
this set of values and an alternative (and very different) solution with model parameters: m = 0.99, c = 0.75, τ = 0.0028 s
(curve c in Fig. 7). Figures 6 and 7 also show that two remote
points in the space parameter domain (see cases a and c in
Fig. 6) yield time responses that cannot be distinguished from
each other. Moreover, when error bars are taken into account
properly, even case b cannot be distinguished from cases a and
c. For comparison, the curve with the fully non-compatible solution (m = 0.6, c = 0.2, τ = 10 s) is also plotted (case d in
both figures).
4 INVERSION IN THE FREQUENCY
DOMAIN
Various inversion algorithms based on the non-linear iterative least-squares method have been proposed to invert spectral induced-polarization measurements using the Cole–Cole
model (see Pelton et al. 1978, 1984; Jaggar and Fell 1988;
Luo and Zhang 1998; Kemna 2000). Xiang et al. (2001) proposed an alternative algorithm, called the ‘direct inversion’
Figure 10 The marginal pdf of the Cole–Cole parameters, ρ 0 = 100 m, m = 0.8, c = 0.75, τ = 0.01 s, for synthetic data.
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600 A. Ghorbani et al.
algorithm, which directly identifies the values of the Cole–
Cole parameters. However, their algorithm does not provide
a full covariance analysis of the inverted values of the model
parameters. In this section, we apply the Bayesian algorithm
to analyse the information that can be retrieved from spectral
IPdata. In addition, we compare the uncertainties of the model
parameters with those obtained in the time domain.
We first analyse the influence of the Cole–Cole parameters
on the spectral IP response in the frequency domain. Increasing
the Cole–Cole exponent c increases both the steepness of the
phase peak and the slope of the amplitude curve response. The
DC-resistivity ρ 0 shifts the amplitude curve vertically, and has
no effect on the phase curve. The DC-resistivity is related to
the formation factor of the sample, the conductivity of the
pore fluid, and the cation exchange capacity of the medium
(e.g. Revil et al. 1998).
Accounting for the additional parameter ρ 0 in the present
case, the results consist of the evaluation of six marginal pdfs,
one for each pair of parameters. We use the same set of syn-
thetic data as proposed in the previous section and we investigate the frequency range, 1.43 mHz to 12 kHz, with
a sampling rate given by 12kHz/2 N where N is number of
frequencies used in the SIP FUCHS-II equipment (Radic Research). The a priori range of the model parameters is the
same as in the time-domain case investigated in Section 3
except for the additional DC-resistivity, ρ 0 . For this parameter, we consider the a priori range, 30–300 m, while the
real value of this parameter is 100 m for all the synthetic
cases analysed below. Since ρ 0 is a Jeffreys parameter, we use
the uniform probability density,
μ(ρ0 ) =
a
⇔ μ log(ρ0 ) = a.
ρ0
(20)
In the time domain, we observed that the application of
the inversion scheme requires the computation of slowly converging series. The inversion of the Cole–Cole parameters in
the frequency domain is hopefully simpler than in the time
domain. Indeed, the Cole–Cole model given by equations (2)
Figure 11 The marginal pdf of the Cole–Cole parameters, ρ 0 = 100 m, m = 0.2, c = 0.25, τ = 0.01 s, for synthetic data.
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2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 601
and (3) corresponds to an analytical complex function transfer,
and is consequently easier to invert than the slowly converging
series arising in the time domain.
The cases are analysed in Figs. 8–11. Figure 8 shows inversion of synthetic data in the spectral domain with ρ 0 =
100 m, m = 0.8, c = 0.25, τ = 0.01 s. The parameters are
well determined, only m has a small uncertainty of ±0.1. Figure 9 shows results obtained with parameters: ρ 0 = 100 m,
m = 0.8, c = 0.25, τ = 10 s. Large equivalence laws appear
between all the parameters: for example, a strong correlation
does exist between the values of τ and ρ 0 , and ρ 0 = 125 m
(log ρ 0 = 2.1) and τ = 100 s are also compatible with the
synthetic data. Figure 10 shows the results for the parameters:
ρ 0 = 100 m, m = 0.8, c = 0.75, τ = 0.01 s. The model
parameters c and m are widely distributed, while τ and ρ 0
can be accurately determined. There is an ill-posed correspondence between m and m = log(m/(1 − m)) that appears in the
m-direction. Results for the model parameters: ρ 0 = 100 m,
m = 0.2, c = 0.25, τ = 0.01 s, are shown in Fig. 11. In this
case, all the model parameters can be determined accurately.
However, note that the uncertainties in c and m are correlated.
Finally in this section, we perform a test using a set of real
data. We use the data published by Xiang et al. (2003). The
results of the Bayesian analysis are shown in Fig. 12. This
figure shows the marginal pdf when data are analysed in the
frequency interval, 10 mHz – 10 kHz. The space of the model
parameters appears much simpler than in the time domain
(there is only one pdf maximum and the pdfs appear nearly
Gaussian). The three model parameters (ρ 0 , m, τ ) are well determined, but the Cole–Cole exponent c is poorly determined.
The maxima of the pdf obtained by the marginal laws for
the pairs of parameters, m, c and τ (i.e. the coordinates of the
contour centres) are very close to the parameters derived using
a classical least-squares method. Thus our approach validates
the use of the classical least-squares method for that problem.
5 USE OF HARMONICS OF SQUARE
CURRENT SIGNALS
An alternative way of performing field or laboratory measurements consists of injecting square signal currents with different
periods and measuring the associated voltage difference using
Figure 12 The marginal pdf of the Cole–Cole parameters, ρ 0 = 22 m, m = 0.13, c = 0.69, τ = 0.001 s, for the data of Xiang et al. (2003).
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602 A. Ghorbani et al.
Table 2 Fundamental frequencies and their harmonics used by the
Zonge instrument in its complex resistivity mode
Harmonics frequencies (in Hz)
3
5
7
9
0.125
1
8
0.375
3
24
0.625
5
40
0.875
7
56
1.125
9
72
a pair of non-polarizing electrodes (e.g. Zonge Engineering
and Research Organization). In this case, the spectrum of the
measured potential difference is the superposition of the signal
corresponding to each square signal. To analyse this case, we
use square current signals with the following periods: 0.125,
1.0 and 8.0 s. We then calculate the spectrum of the frequencies, 0.125, 1.0, and 8.0 Hz, including the 3rd, 5th, 7th and
9th harmonics (Table 2) for each fundamental frequency that
is shown in Fig. 13(a). The transfer function is
ρ(nω0 ) = k
V(ω0 , n)
,
I(n)
(21)
where n is the harmonic number of each signal, ω0 is the fundamental frequency and ρ, V and I are the apparent resistivity,
the voltage drop and the injected current, respectively. This
function has uncertainties that increase with the number n of
harmonics of the fundamental frequency. These errors are incorporated in our Bayesian inversion approach (Fig. 13b). We
assume that there is no error associated with the value of the
injected current and that there is only a constant deviation σx
associated with the measured potential drop. It follows that
the spectral deviation is the deviation σ X (Florsch, Legros and
Hinderer 1995), given by
σx
σX = √ ,
N
(22)
Figure 13 (a) Spectrum with fundamental frequencies, 0.125, 1.0 and
8.0 Hz, including the 3rd, 5th, 7th and 9th harmonics for each fundamental frequency. (b) Spectrum of response divided by reference, and
the error bars for each harmonic.
where N is the number of points in the time sequence. The
standard deviation of data can be written,
σ X(ω) = nσ (ω0 ).
(23)
We compute the a posteriori probability density function given
by equation (5) using the covariance matrix associated with
the data given by equation (23). A synthetic example of the
inversion of the model parameters is shown in Fig. 14. The
values of the model parameters are ρ 0 = 100 m, m = 0.2,
c = 0.75, τ = 0.001 s. From the results plotted in Fig. 14,
we conclude that when the harmonics of a fundamental frequency are used to perform induced polarization, only ρ 0 is
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well-determined, while the uncertainties associated with the
other parameters (m, c, τ ) are still very strong.
6
CONCLUSION
The Bayesian approach provides the information (or probability densities) associated with the Cole–Cole parameters, determined from induced-polarization measurements in the time
or frequency domains. Because the Cole–Cole parameters can
be used to determine the hydraulic conductivity of aquifers,
2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
Bayesian inference of IP parameters 603
Figure 14 The marginal pdf of the Cole–Cole parameters, ρ 0 = 100 m, m = 0.2, c = 0.75, τ = 0.001 s, for synthetic data. The spectrum is
created using the fundamental frequencies 0.125, 1.0, and 8.0 Hz, including the 3rd, 5th, 7th and 9th harmonics for each fundamental frequency.
this approach has strong implications in hydrogeophysics for
determining the permeability using a Bayesian analysis. In this
paper, the Bayesian approach has been applied to the determination of the Cole–Cole parameters using measurements simulated both in the time domain and in the frequency domain, for
which a wider range of frequencies were investigated. While
the use of the Cole–Cole model could reveal limitations of our
approach, we showed that the Bayesian approach developed
here could be used to investigate the uncertainty associated
with the inversion of alternative models such as the Cole–
Davidson model.
The Bayesian procedure results in an explanation of why the
classical time-domain approach cannot lead to a proper estimate of the Cole–Cole parameters. For completeness, we investigated the frequency-domain induced polarization, using
the harmonics of square current signals in a low-frequency
range, and theoretically reaching a broader spectrum than
that obtained using the classical time-domain relaxation associated with a single current injection (corresponding to a
single frequency). However, we showed that error bars in-
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crease for the harmonics of each fundamental frequency. The
Bayesian approach shows that, even in this case, the correct
estimation of the Cole–Cole parameters using classical leastsquares methods is difficult. Finally, the Bayesian approach
allows the delineation of ‘equivalent domains’ in the space of
the model parameters, thus accounting for the uncertainty in
the measurements.
ACKNOWLEDGEMENT
We are indebted to the ANR-CNRS-INSU-ECCO program
(project Polaris II, 2005-2007) for supporting this work. We
thank the referees and the editor for their useful comments.
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propcr.htm
APPENDIX
Determination of m
To derive the non-informative structure of m, we consider an
electrical equivalent circuit where two resistors are involved
(RA and RB ). In this case, the chargeability m can be expressed
as m = (1 + RB /RA )−1 (Fig. 1). Since RA and RB are both
Jeffreys parameters, their ratio S = RB /RA is also a Jeffreys
parameter. The homogeneous probability distribution (HPD)
of m is given by
a
μ(m) =
,
(A1)
m(1 − m)
where a is a constant and m ∈ (0, 1). Alternatively, the chargeability can be written as
C
⎧
⎨ limω→0 ρ ≡ ρ0
⎩ limω→+∞ ρ ≡ ρ∞
m=1−
ρ∞
.
ρ0
(A2)
(A3)
However, ρ∞ is not a Jeffreys parameter since, necessarily, we
have ρ∞ ≤ ρ0 . However, the HPD of the normalized chargeability,
m∗ =
ρ∞
1−m
,
=
ρ0 − ρ∞
m
(A4)
a
,
m∗ (1 − m∗ )
(A5)
is
μ(m∗ ) =
where a is a constant. The HPD of m∗ has the same form as
the HPD of m.
We look for an expression for the chargeability m satisfying
the condition that μ(m ) is constant. If we write m = 1/(1 + S)
where S is a Jeffreys parameter, this yields
1−m
,
(A6)
μ S (S) = μ S
m
and
1−m
,
μ S (log S) = μlog S log
m
(A7)
is a constant. If X is a Jeffreys parameter, 1/X is also a Jeffreys
parameter. It follows that
m
μm log
= a,
(A8)
1−m
where a is a constant. Therefore the transform,
m
,
m = log
1−m
is a suitable solution.
2007 European Association of Geoscientists & Engineers, Geophysical Prospecting, 55, 589–605
(A9)