Re-parameterization of the Cole-Cole Model for Improved Spectral Inversion of Induced Polarization Data

Abstract

The induced polarization phenomenon, both in time domain and frequency domain, is often parameterised using the empirical Cole–Cole model. To improve the resolution of model parameters and to decrease the parameter correlations in the inversion process of induced polarization data, we suggest here three re‐parameterisations of the Cole–Cole model, namely the maximum phase angle Cole–Cole model, the maximum imaginary conductivity Cole–Cole model, and the minimum imaginary resistivity Cole–Cole model. The maximum phase angle Cole–Cole model uses the maximum phase φmax and the inverse of the phase peak frequency, τφ, instead of the intrinsic charge‐ability m0 and the time constant adopted in the classic Cole–Cole model. The maximum imaginary conductivity Cole–Cole model uses the maximum imaginary conductivity σmax′′ instead of m0 and the time constant τσ of the Cole–Cole model in its conductivity form. The minimum imaginary resistivity Cole–Cole model uses the minimum imaginary resistivity ρmin′′ instead of m0 and the time constant τρ of the Cole–Cole model in its resistivity form. The effects of the three re‐parameterisations have been tested on synthetic time‐domain and frequency‐domain data using a Markov chain Monte Carlo inversion method, which allows for easy quantification of parameter uncertainty, and on field data using 2D gradient‐based inversion. In comparison with the classic Cole–Cole model, it was found that for all the three re‐parameterisations, the model parameters are less correlated with each other and, consequently, better resolved for both time‐domain and frequency‐domain data. The increase in model resolution is particularly significant for models that are poorly resolved using the classic Cole–Cole parameterisation, for instance, for low values of the frequency exponent or with low signal‐to‐noise ratio. In general, this leads to a significantly deeper depth of investigation for the ϕmax, σmax′′, and ρmin′′ parameters, when compared with the classic m0 parameter, which is shown with a field example. We believe that the use of re‐parameterisations for inverting field data will contribute to narrow the gap between induced polarization theory, laboratory findings, and field applications.

Permeability Estimation Directly From Logging‐While‐Drilling Induced Polarization Data

“…The Cole‐Cole model in its conductivity form is expressed as (e.g., Tarasov & Titov, ) where is the complex conductivity, is the DC conductivity, is the intrinsic chargeability, is the relaxation time, is the frequency exponent, is the frequency, and is the imaginary unit. In equation , the and parameters are strongly correlated, and thus poorly resolved in the inversion (Fiandaca et al, ). In order to reduce the parameter correlation, and hence improve the inversion results, Fiandaca et al () introduced the maximum imaginary conductivity (MIC) model, in which is replaced by maximum imaginary conductivity of the Cole‐Cole spectrum (Figure ) and the model space becomes …”

“…In equation , the and parameters are strongly correlated, and thus poorly resolved in the inversion (Fiandaca et al, ). In order to reduce the parameter correlation, and hence improve the inversion results, Fiandaca et al () introduced the maximum imaginary conductivity (MIC) model, in which is replaced by maximum imaginary conductivity of the Cole‐Cole spectrum (Figure ) and the model space becomes …”

Subsurface imaging of water electrical conductivity, hydraulic permeability and lithology at contaminated sites by induced polarization

“…63 ′ determined from multisalinity resistivity measurements and Furthermore, Weller et al (2013) found that the salinity dependency of the real surface 146 conductivity parallels the salinity dependency of the imaginary surface conductivity, which can be 147 expressed as (Weller et al, 2011;Weller et al, 2015): 148 where * is the complex conductivity, 0 is the DC conductivity, 0 is the intrinsic chargeability, 182 is the relaxation time, is the frequency exponent, is the frequency and is the imaginary 183 unit. Using MCMC (Monte Carlo Markov Chain) analysis, Fiandaca et al (2017a) (Fig. 1b).…”

Tracking Magmatic Hydrogen Sulfur Circulations Using Electrical Impedance: Complex Electrical Properties of Core Samples at the Krafla Volcano, Iceland

“…We consider the MPA to be the simplest parameter, which can describe the amplitude of polarization in a comparable manner for all samples. It can be obtained directly from any phase spectrum and can also be inverted from field time domain IP measurements (Fiandaca et al, ; Madsen et al, ). We calculate the MPA, based on the phase angle value at the summit of spectral peaks (Figure ), if a clear peak is observed in the phase spectrum, or the average phase angle of a linear/flat section when that is observed (e.g., L75).…”

“…The peak frequency at which the MPA is reached, is, however, missing in this description. Since it is a more complicated and ambiguous parameter to calculate, both in field and in time domain IP studies, we do not analyze it quantitatively here (Fiandaca et al, ; Maurya et al, ).…”