Cole-Cole Model Parameters Research Articles

Relationship between Cole–Cole model parameters in permittivity and conductivity formulation

SUMMARY For the analysis of spectral induced polarization (SIP) measurements and for the description of frequency-dependent electrical relaxation responses, so-called Cole–Cole models (CCMs) are widely used. Typically, CCM formulations in terms of complex electrical conductivity or complex electrical resistivity are used in geophysical applications. The differences between these model descriptions, in particular between the respective time constants, and their conversion have been studied. A third variant of the model is formulated in terms of complex permittivity, commonly used in materials science. In general, all these model formulations can be used equivalently for fitting SIP data, which, however, results in differing values for some of the model parameters. For a meaningful comparison of CCM parameters of different samples or measurements, it is necessary that they are based on the same model formulation. In this work, the relationships between the Debye model (DM) and CCM parameters in the formulation for complex permittivity and complex conductivity are studied. A direct analytical conversion is possible for generalized DM formulations, both in single- and multi-term model formulations, resulting in relationships between the respective relaxation time distributions (RTDs). Such a direct conversion for CCM formulations is not possible. We however derived an approximate relationship between $\log$-normal RTD and CCM formulations and respective permittivity and conductivity parameter values. Our study also highlights the significance of using consistent model formulations when experimental data are compared in terms of DM or CCM parameters, as parameters used to predict ice temperature are incorrect if the conductivity time constant is used to predict the temperature from interpolation of a permittivity time constant-temperature relationship.

Spectral induced polarization tomography inversion: Hybridizing homotopic continuation with Bayesian inversion

Induced polarization tomography offers the potential to better characterize the subsurface structures by considering spectral content from data acquisition over a broad frequency range. Spectral induced polarization (SIP) tomography is generally defined as a nonlinear inverse problem commonly solved through deterministic gradient-based methods. To this end, the spectral parameters, i.e., direct current resistivity, chargeability, relaxation time, and frequency exponent, are resolved by individually or simultaneously inverting all frequency data, followed by fitting a generalized Cole-Cole model to the inverted complex resistivities. Due to the high correlation between the Cole-Cole model parameters and a lack of knowledge about the initial approximation of the spectral parameters, using the classical least-squares methods may lead to inaccurate solutions and impede reliable uncertainty analysis. To cope with these limitations, we introduce a new approach based on a hybrid application of a globally convergent homotopic continuation method and a Bayesian inference to reconstruct the distribution of the subsurface spectral parameters. The homotopic optimization, owing to its fast and global convergence, is first implemented to invert multifrequency SIP data sets aimed at retrieving the complex-valued resistivity. Then, Bayesian inversion based on a Markov chain Monte Carlo (MCMC) sampling method, along with a priori information including the lower and upper bounds of the prior distributions, is used to invert the complex resistivity for the Cole-Cole model parameters. By applying the MCMC inversion algorithm, a full nonlinear uncertainty appraisal can be provided. We numerically evaluate the performance of our method using synthetic and real data examples in the presence of topographical effects. Numerical results prove that the homotopic continuation method outperforms the classic, smooth inversion algorithm in the sense of approximation accuracy and computational efficiency. In addition, we determine that our hybrid inversion strategy provides reliable representations of the main features and structure of the earth’s subsurface in terms of the spectral parameters.

Cole-Cole Model Parameter Estimation from Multi-frequency Complex Resistivity Spectrum Based on the Artificial Neural Network

In near surface electrical exploration, it is often necessary to estimate the Cole-Cole model parameters according to the measured multi-frequency complex resistivity spectrum of ore and rock samples in advance. Parameter estimation is a nonlinear optimization problem, and the common method is least square fitting. The disadvantage of this method is that it relies on initial value and the result is unstable when data is confronted with noise interference. To further improve the accuracy of parameter estimation, this paper applied artificial neural network (ANN) method to the Cole-Cole model estimation. Firstly, a large number of forward models are generated as samples to train the neural network and when the data fitting error is lower than the error threshold, the training ends. The trained neural network is directly used to efficiently estimate the parameters of vast amounts of new data. The efficiency of the artificial neural network is analyzed by using simulated and measured spectral induced polarization data. The results show that artificial neural network method has a faster computing speed and higher accuracy in Cole-Cole model parameter estimation.

Inductively Induced Polarization and Its Overall Assessment Using a Normalized Transient Response

Abstract —The paper considers the basic regularities of inductively induced polarization (IIP) manifestation in the transient electromagnetic method. When calculating inductive transient responses of a polarizable ground, we assumed that the electrical conductivity of geologic materials is described by the Cole-Cole model. To present and analyze the results of computer-based simulation of the IIP phenomenon, we used a normalized transient response. It is defined as the ratio of the transient electromotive force (EMF) response of a polarizable ground to the EMF response of a ground that differs from the polarizable one only in that it has zero chargeability. The coordinates of the minimum of the normalized transient response are useful for an overall assessment of IIP manifestation. We show, by the example of a homogeneous polarizable half-space, how the inductive transient response is affected by the Cole-Cole model parameters and the size of transmitter and receiver loops. For a two-layer ground, IIP manifestation depends also on whether the base or upper layer is polarizable, as well as on the layer thickness. Inductively induced polarization is most pronounced when a conductive polarizable layer overlies a resistive nonpolarizable base. In this case, at a certain thickness of the layer, the IIP effect far exceeds that observed in the presence of a thick polarizable layer and even of a homogeneous polarizable half-space.

Development of a novel approach for recovering SIP effects from 1-D inversion of HEM data: Case study from the Alut area, northwest of Iran

We have developed a novel approach for recovering the induced polarization (IP) effects from one-dimensional (1-D) inversion of helicopter-borne electromagnetic data (HEM). To achieve this aim, we incorporate both electromagnetic and spectral induced polarization effects within the HEM data by taking the frequency dependence behaviour of resistivity/conductivity of polarizable formation into account. Furthermore, we have included the EM sensor altitude, magnetic permeability and dielectric permittivity of subsurface layers in order to conduct the full solution of forward operator. In this regard, we have calculated the magnetic permeability from the inversion of aeromagnetic data and incorporated the sensor altitude and dielectric permittivity as additional model parameters to be recovered using the inversion of the HEM data. We have carried out both forward and inverse modelling of the HEM data by implementing fast Hankel transform and hybrid algorithm including very fast simulated annealing (VFSA) and Marquardt-Levenberg (ML) strategies, respectively. This approach enables us to recover model parameters comprising of relaxation parameters of Cole-Cole model, dielectric permittivity and thickness of layered half space model and sensor altitude above the earth surface using 1-D inverse modelling of HEM data.We have tested our approach on both synthetic and field DIGHEM data. Implementing the forward solution indicates that incorporation of the IP effects may shift the frequency domain electromagnetic (FDEM) response so that it yields the sign reversal signature in the in-phase component at low frequencies. Moreover, the results of the inverse modelling of the HEM data show that the IP effect could be recovered using our approach effectively. It provides fast and cost-effective quantitative tool for recovering the IP effect from HEM data for the sake of more precise characterization of buried polarizable formations in the wide area. In addition, incorporation the IP effect along with magnetic permeability and sensor altitude provides a significant degree of improvement in the inversion and interpretation of negative responses occurred in the in-phase component at low frequencies.

A Weighting Inversion Method of Spectrum Induced Polarization

The inversion results of complex resistivity method are four Cole-Cole model parameters. Among the four parameters, the frequency dependence and the time constant are more difficult to invert. It is necessary to study an algorithm that can invert the four Cole-Cole model parameters at the same time. In this paper, the least squares and OCCAM inversion algorithms are used to invert four Cole-Cole model parameters. In other words, two model constraints are added to the objective function. When inverting different Cole-Cole model parameters, the real and imaginary parts of the data are weighted to adjust the proportion of real part and imaginary part of data in inversion. Firstly, the formula of the algorithm is deduced. Then the theoretical models are designed for inversion trial calculation. In the inversion process, the inversion converges steadily by adjusting the Lagrange factor, and the inversion effect is improved by adjusting the weighting coefficients of real part and imaginary part. This method can get better inversion results by adjusting the proportion of the real and imaginary parts of the data in the inversion. The model trial results show that the weighting inversion algorithm significantly improves the results of the inversion of the four Cole-Cole parameters.

A thorough synthetic study on IP effects in AEM data from different systems

IP effects in AEM data are subject of current research around the world, due to the recent recognition of their significance for exploration and general (hydro)geological mapping. There have been success stories and it is now practical to model AIP on thousands of line kms of AEM data. However there still is a need to study more accurately the boundaries of the effect and of its relevance, beyond common past acceptance. In this paper we present a systematic, extended analysis of AIP effect in different AEM (TEM) systems’ data, , based on synthetic modelling of different pseudo geologies. Its goal is to provide a clear overview of possible AIP effects in the data space, without imposing simplistic assumptions (e.g., fixing some parameters to arbitrary values or limited boundaries). We produce 1D forward responses with dispersive resistivity for hundreds of thousands of combinations of Cole-Cole model parameters and AEM system transfer functions. The results are analyzed using various metrics (e.g., sum of negative voltages, exponential fitting) that capture different AIP signatures in the transients. Experiments include half spaces, 2 and 3 layer models, combined with different waveforms, Rx types (dB/dt and B), Tx-Rx geometries, flying heights, transients’ binning, base frequencies. The results, presented as 4D hyperspaces, each with 104transients obtained from the combinations of 4 x 10 different Cole-Cole parameters, allow a clear assessment of the AIP effects over a wide range of geophysical situations. Some of the main observations are: AIP effects are increased most often by the presence of a resistive bedrock, often using slow turn-off of the waveform, are generally better observed recording the B field instead of its derivative, and in any most cases lowering base frequencies to 12.5 Hz. In general, they are more pervasive than previously thought and should be carefully considered in virtually any AEM survey. If present, they can often be sensitive to model parameters down to depth.

Cell concentrations and metabolites enhance the SIP response to biofilm matrix components

We investigated the relative effect of different bacterial biofilm components to influence spectral induced polarization (SIP) signals and Cole-Cole model parameters (i.e. normalized chargeability (S m−1) and relaxation time (τ)) in order to reduce the ambiguity associated with interpretation of SIP signatures in biostimulated environments. SIP measurements were collected over a frequency range, (i.e. 0.1 Hz–1 kHz) from experimental columns containing Pseudomonas aeruginosa PAO1 (non-mucoid bacterial strain) cell suspensions alone and cell suspensions mixed with different molecules associated with P. aeruginosa biofilm matrix (alginate, DNA and phenazine). Results from SIP measurements of cell density investigations suggest an increase in phase (~ 200%), imaginary conductivity (σ″ S m−1), and normalized chargeability (S m−1) values as cell densities increase (1.50–5.55 × 109 CFUml−1), yet no significant change in the real conductivity (σ′ S m−1) or relaxation time was observed. The addition of alginate resulted in a suppression of phase (φ mrads), imaginary conductivity (σ″ S m−1), and normalized chargeability (S m−1) response. However, a mixture of cells and alginate increased all parameters but with no significant changes in real conductivity (σ′ S m−1) or relaxation time. Similar trend was observed in the DNA treatments; increasing concentrations of DNA reduced the imaginary conductivity (σ″ S m−1), phase (φ mrads), and normalized chargeability (S m−1) response. In contrast, the addition of phenazine, a secondary metabolite of P. aeruginosa, resulted in a significant increase in both imaginary conductivity (σ″ S m−1), phase (φ mrads), and normalized chargeability (S m−1) values especially at larger concentrations (>50 mM). Our work suggests that cell density and metabolites are significant factors increasing the polarization response in biofilms and advances the interpretation of electrical signals associated with biofilms in the subsurface.

Extraction of Cole-Cole model parameters through low-frequency measurements

A novel in-direct method for extracting the four parameters of the single dispersion Cole-Cole model is introduced in this Letter. Compared with the corresponding already introduced methods, the main offered advantage is the relaxed unity-gain bandwidth requirement of the employed active element. This is originated from the fact that only measurements from dc to the cutoff frequency of the realized filter function are required. Experimental results using apples are provided and their comparison with those obtained using a LCR tester confirm the accuracy of the proposed method.

Dielectric variations of potato induced by irreversible electroporation under different pulses based on the cole-cole model

To investigate dielectric property variations induced by electric pulses, equivalent doses of conventional irreversible electroporation (IRE) and high-frequency IRE were used to treat potato slices. The dielectric property of the potatoes were measured before and after treatment in the frequency band of 0.1 Hz-10 MHz, and then the Cole-Cole model was used to fit the experimental data in the frequency band of 1 kHz-10 MHz to explain the β dispersion mechanism. The Cole-Cole model parameters showed that under an equivalent dose, the conventional IRE pulses were more efficient than the high-frequency IRE pulses for electroporation. After treatment with conventional IRE pulses, the DC conductivity greatly increased at time 0 after the pulses and showed little recovery over time. The electroporation effect of the high-frequency IRE pulses improved with increasing pulse duration. This study presents a potential new method for evaluating the effects of IRE during and immediately after the treatment process.

Electromagnetic dispersion and sensitivity characteristics of carbonate reservoirs

The need to develop a new measurement method for the identification of favorable hydrocarbon-producing zones in carbonate reservoirs is of great importance due to their complex pore structure and lithology. We have evaluated fundamental experimental data acquired from carbonate formation samples in a buried-hill reservoir to demonstrate the magnitude of the complex resistivity (CR) dispersion effect in oil-saturated carbonate rocks and the possibility of identifying hydrocarbon-bearing carbonate sections. Our experimental data set included CR measurements on four carbonate core samples with various degrees of oil saturation, and at each saturation level, CR measurements were acquired in a frequency range between 10 Hz and 100 kHz. The experimental data indicated that the intensity of the CR dispersion effect increased with oil saturation above a specific critical excitation frequency. The experimental spectra of the carbonate samples were fitted with the Debye model and the Cole-Cole model (CCM), which can be used to interpret the dispersion effect over a wide frequency range. Based on the experimental data and the inverted CCM parameters, a sensitivity study was carried out with the crosswell frequency-domain electromagnetic (EM) modeling. Multifrequency EM data were acquired directly by calculating the Maxwell equation with the CR. Our experimental data and forward modeling results indicated that measurement of induced polarization and EM coupling effects can be an effective means for identifying carbonate hydrocarbon reservoirs and for quantitatively evaluating the effect of formation saturation on favorable conditions.

Observation Frequency Selection Based on Dynamic and Electric Field Excitation Method for Advanced Detection in Coal Mine Roadway

The dynamic and electric field excitation method is a new advanced detection method. In this method, the observation frequency and of emission electrode must be reasonably selected according to the actual geological conditions of coal mine roadway to obtain the apparent frequency with adequate Induced Polarization (IP) information of geological anomalies and improve the measurement accuracy of the instrument. This paper completed the simulation of apparent frequency varying with the observation frequency and Cole-Cole model parameters with MATLAB software. The results show that the key to selecting the observation frequency mainly depends on the charge and discharge time constant .Then the paper analyzed the effect of the measured apparent frequency on the measurement accuracy. According to time constant values​​ of geological conditions in coal mine roadway, considering the efficiency and electromagnetic coupling effects on the measurement, it is best to select the observation frequency within a frequency range of 0.05~0.1Hz and the frequency ratio coefficient of 13 or 15. Now apparent IP effect is obtained.

Characterization of bimodal facies distributions using effective anisotropic complex resistivity: A 2D numerical study based on Cole-Cole models

Subsurface heterogeneity characteristics are of major importance in hydrologic modeling, and likely result in anisotropic electrical properties. We computed the anisotropic effective complex resistivity of 2D bimodal facies distributions numerically. Complex resistivities of individual facies are described in terms of the Cole-Cole relaxation model. First, we determined that effective DC resistivities of the distributions can be reasonably well described by power averaging the properties of individual facies. We found a clear relationship between the mixing parameter and correlation lengths of the facies distributions with respect to horizontal and vertical directions. Then, we used the power-law mixing model to invert for individual Cole-Cole model parameters by fitting predicted electrical responses to simulated spectral effective complex-resistivity data for the two perpendicular directions. Thus, it is possible to derive the electrical properties of individual facies as well as structural parameters describing bimodal facies distribution by means of a noninvasive measurement approach. In particular, anisotropy of the spectral complex-resistivity response provides information on correlation lengths of the distribution. This finding is relevant for all applications of electrical-impedance spectroscopy where anisotropy might be encountered.

Rheological and molecular characterization of linear backbone flexible polymers with the Cole-Cole model relaxation spectrum

The empirical Cole-Cole distribution is an analytical three parameter model of the relaxation spectra that provides accurate fits to experimental dynamic viscosity data for many systems of commercial linear backbone flexible polymers. We demonstrate that for disparate systems of polyethylenes, the three Cole-Cole model parameters have simple power law relationships to moments of the molecular weight distribution enabling direct molecular interpretation of the mechanical relaxation spectrum. A simple relationship between the Cole-Cole distribution and the Cross model for the non-linear flow curve can be deduced utilizing the empirical Cox-Merz rule. Accurately representing the linear viscoelastic material functions with empirical analytical relaxation spectra containing relatively few fitting parameters that can be readily interpreted is a major advance in polymer characterization. The three Cole-Cole parameters effectively replace single point material characterizations such as melt flow index. Development of higher resolution polymer characterization methods is imperative with the advent of metallocene catalyst technology, which enables the molecular weight and backbone architecture to be carefully controlled.