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

## Summary (2 min read)

### 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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### "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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### "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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### "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)....

[...]

...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)....

[...]

2,720 citations

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)....

[...]