The Cole‐Cole model in time domain induced polarization

Geophysics ◽  
1981 ◽  
Vol 46 (6) ◽  
pp. 932-933 ◽  
Author(s):  
T. Lee
Keyword(s):  
Equivalent Circuit ◽  
Time Domain ◽  
Induced Polarization ◽  
Relaxation Model ◽  
Trivial Case ◽  
Alternative Expression ◽  
Transient Voltage ◽  
Transient Voltages ◽  
Cole Model ◽  
The Given

Recently Pelton et al. (1978) used a Cole‐Cole relaxation model to simulate the transient voltages that are observed during an induced‐polarization survey. These authors took the impedance of the equivalent circuit Z(ω) to be [Formula: see text]They then gave the expression for the transient voltage [Formula: see text] as [Formula: see text]In equation (2), [Formula: see text] was misprinted as [Formula: see text]. In these equations, [Formula: see text] and [Formula: see text], [Formula: see text] and τ are constants to be determined for the given model. [Formula: see text] is the height of the step current that will flow in the transmitter. A disadvantage of equation (2) is that it is only slowly convergent for large t/τ. Pelton et al. (1978) used a τ which ranged from [Formula: see text] to [Formula: see text]. The purpose of this note is to provide an alternative expression for [Formula: see text] that is valid only at the later stages but which does not have this disadvantage. The trivial case of c = 1.0 is ignored.

Time-domain induced polarization forward modeling and an apparent spectral parameter solution based on the Weibull growth model

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. D145-D155
Author(s):  
Qingxin Meng ◽  
Xiangyun Hu ◽  
Heping Pan ◽  
Huolin Ma ◽  
Miao Luo
Keyword(s):  
Growth Model ◽  
Time Domain ◽  
Forward Modeling ◽  
Induced Polarization ◽  
Forward Model ◽  
Model Parameters ◽  
Time Varying ◽  
Basic Algorithm ◽  
Cole Model ◽  
Physical Principles

The application of the Cole-Cole model within time-domain induced polarization (TDIP) forward field modeling shows that the model parameters can characterize time-varying states of the TDIP field and support observed data analysis. The Cole-Cole model contains real and imaginary parts, and it requires a frequency-to-time conversion for TDIP forward modeling. However, the TDIP field is usually expressed by a real number, and its intuitive time-varying states field intensity increases with charging time. Therefore, the forward model should be constructed in a simpler form. We have aimed to develop a forward model using mathematical functions not based on physical principles. The Weibull (WB) growth model, which is primarily used to describe the time-varying curve features in regression analysis, is introduced into the basic algorithm of the TDIP forward model. Subsequently, a forward expression of the TDIP effect is established. Based on the time-varying shape and scale parameters, this expression describes the time-varying rate and relaxation states of the TDIP fields. Furthermore, based on the extensively used conjugate gradient optimization, an apparent WB parameter scheme is initiated to calculate the spectral parameters that represent the relaxation and time-varying rate obtained from the multi-time-channel TDIP data. Finally, this scheme is applied to interpret the different simulated and actual TDIP data. The results demonstrate that the WB growth model can be used for the TDIP forward model without involving physical principles, the model parameters without specific physical significance can be used to represent the time-varying states of TDIP fields, and apparent WB parameters can be used to discern different TDIP observed data. The setting of the TDIP forward model and model parameters can actually be more flexible and diverse, so as to obtain simpler forward expressions and ensure a highly efficient inverse solution.

3D numerical modeling of induced-polarization electromagnetic response based on the finite-difference time-domain method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. E385-E398 ◽  
Author(s):  
Yanju Ji ◽  
Yanqi Wu ◽  
Shanshan Guan ◽  
Xuejiao Zhao
Keyword(s):  
Frequency Dependence ◽  
Impulse Response ◽  
Time Constant ◽  
Time Domain ◽  
Characteristic Time ◽  
Induced Polarization ◽  
Order System ◽  
Ohm’S Law ◽  
Ohm's Law ◽  
Cole Model

Induced-polarization (IP) effects have a significant influence on transient electromagnetic (TEM) data, which commonly manifest a reversed sign. Polarization media usually have a very high economic value. To study the IP effects, a new method for modeling the time-domain electromagnetic signals of 3D dispersive materials is developed. Due to the fractional time derivatives, two main difficulties are needed to be conquered: the derivation of Cole-Cole model impulse response function and the discrete recursion of convolution in Ohm’s law. We use a frequency-domain rational approximation method and the linear programming technique to transfer the fractional order system into an integer order system. This method enables us to achieve a relatively simple and high-precision solution of the Cole-Cole model impulse response. A discrete recursion method for Ohm’s law convolution is proposed to realize an efficient numerical simulation of 3D polarization media by eliminating the prohibitive computing demands. Comparisons with published methods demonstrate the accuracy and efficiency of our algorithm. The characteristic time constant and chargeability have monotonic influences on the IP effects, whereas the frequency dependence indicates a nonmonotonic influence on the IP effects. The negative response is more significant when the frequency dependence is in the midrange. For a 3D low-resistivity chargeable body, a larger size reduces the decay rate of the induced field, which contributes to the obscuration of the polarization field. The middle-sized chargeable body can be detected under certain conditions: high chargeability, millisecond characteristic time constant, and middle frequency dependence. Small-sized chargeable bodies cannot be recognized at all by using the current forward-modeling method and instrument, which highlights the significance of precision improvement.

Modeling induced polarization effects in helicopter time-domain electromagnetic data: Field case studies

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. B49-B61 ◽  
Author(s):  
Vladislav Kaminski ◽  
Andrea Viezzoli
Keyword(s):  
Time Domain ◽  
Signal To Noise Ratio ◽  
Mineral Exploration ◽  
Induced Polarization ◽  
Electromagnetic Transients ◽  
Additional Information ◽  
Time Domain Electromagnetic ◽  
Airborne Electromagnetic ◽  
Cole Model ◽  
Inversion Techniques

Induced polarization (IP) effects are becoming more evident in time-domain helicopter airborne electromagnetic (AEM) data thanks to advances in instrumentation, mainly due to improvements in the signal-to-noise ratio and hence better data quality. Although the IP effects are often manifested as negative receiver voltage values, which are easy to detect, in some cases, IP effects can distort recovered transients in other ways so they may be less obvious and require careful data analysis and processing. These effects represent a challenge for modeling and inversion of the AEM data. For proper modeling of electromagnetic transients, the chargeability of the subsurface and other parameters describing the dispersion also need to be taken into consideration. We use the Cole-Cole model to characterize the dispersion and for modeling of the IP effects in field AEM data, collected by different airborne systems over different geologies and exploration targets, including examples from diamond, gold, and base metal exploration. We determined how multiparametric inversion techniques can simultaneously recover all four Cole-Cole parameters, including resistivity [Formula: see text], chargeability [Formula: see text], relaxation time [Formula: see text], and frequency parameter [Formula: see text]. The results obtained are in good agreement with the ancillary information available. Interpretation of the IP effects in AEM data is therefore seen by the authors as providing corrected electrical resistivity distributions, as well as additional information that could assist in mineral exploration.

Time Domain and Frequency Domain Induced Polarization Modeling for Three-dimensional Anisotropic Medium

2017 ◽  
Vol 22 (4) ◽  
pp. 435-439
Author(s):  
Weiqiang Liu ◽  
Pinrong Lin ◽  
Qingtian Lü ◽  
Rujun Chen ◽  
Hongzhu Cai ◽  
...  
Keyword(s):  
Frequency Domain ◽  
Half Space ◽  
Anisotropic Medium ◽  
Time Domain ◽  
Three Dimensional ◽  
Induced Polarization ◽  
Model Parameters ◽  
Complex Resistivity ◽  
Cole Model ◽  
Synthetic Models

Time domain induced polarization (TDIP) and frequency domain induced polarization (FDIP) synthetic models, incorporating three-dimensional (3D) anisotropic medium, were tested. In TDIP modeling, both resistivity and chargeability of the medium were anisotropic, and the apparent chargeability values were calculated by carrying out two resistivity forward calculations using resistivity with and without an IP effect. We analyzed the TDIP response of a 3D isotropic cube model embedded in the anisotropic subsurface half-space. In FDIP modeling, the complex resistivity of the medium at various frequencies was anisotropic. The complex resistivity was determined by a Cole-Cole model with anisotropic model parameters. We then analyzed the FDIP response of a 3D anisotropic cube model embedded in an isotropic subsurface half-space. Both of the TDIP and FDIP simulation results suggest that IP responses acquired in two orthogonal directions on the surface are different when the same arrays are used and acquisition in orthogonal directions helps resolve the presence of anisotropy. The anisotropy should be taken into account in practice for TDIP and FDIP exploration.

Some case histories of source discrimination using time‐domain spectral IP

Geophysics ◽  
1997 ◽  
Vol 62 (5) ◽  
pp. 1394-1408 ◽  
Author(s):  
Harold O. Seigel ◽  
Heikki Vanhala ◽  
S. Nicholas Sheard
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Cost Effective ◽  
Case Histories ◽  
Source Discrimination ◽  
Mean Grain Size ◽  
On Line ◽  
The Time Domain ◽  
Transient Voltages ◽  
Cole Model

It has long been known that induced polarization (IP) transient voltages decay with time at a rate that is dependent strongly on the mean grain size of the metallic conducting particles in the IP source. The Cole‐Cole model provides a three‐parameter representation (M, τ and C) for IP responses. The time‐constant (τ), in particular, has been found to be very useful in resolving IP sources with differing mean particle size. The characterization of IP responses in terms of these parameters has been termed “spectral IP.” These parameters may be determined either through the analysis of the response of the earth to sequential transmission of ac currents of different frequencies (i.e., frequency‐domain IP), or through the analysis of the transient decays resulting from the transmission of interrupted square‐wave currents (i.e., time‐domain IP). The latter approach offers the major convenience of being applicable to data obtained in the course of routine production surveys, with no increase in survey time. In practice, spectral IP parameters are determined most readily from time‐domain transients through the computer matching of the observed data to the best fit in a family of precalculated Cole‐Cole curves. This may be done, off‐line, using a PC, or, in a recent receiver, essentially on‐line, using software imbedded in the receiver. Field case histories from Canada, Finland, and Australia are given to illustrate the application of spectral IP to the resolution of IP sources in the time domain. It is recommended that this approach to the processing and presentation of time‐domain IP data should be applied routinely, as a very cost‐effective enhancement to the exploration value of such data. An additional benefit from this presentation is that it will facilitate a sharing of experimental results with workers in the frequency domain who may also use the spectral IP approach.

Reliability of Time Domain Induced Polarization Data

2011 ◽  
Author(s):  
Aurélie Gazoty ◽  
Esben Auken ◽  
Jesper Pedersen ◽  
Gianluca Fiandaca ◽  
Anders Vest Christiansen
Keyword(s):  
Time Domain ◽  
Induced Polarization ◽  
Polarization Data
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