Analyzing Dependency and Correlation of Cole-Cole Model Parameters in Spectral Induced Polarization Using Bayesian Inference

Document Type : Research Article

Authors

Department of Earth Physics, Institute of Geophysics, University of Tehran, Tehran, Iran.

Abstract

Spectral induced polarization (SIP) is a useful tool in geophysical exploration for understanding the capacitive properties of materials beneath the surface. Unlike conventional methods, SIP analysis can be done in both the time domain and frequency domain. In the time domain, it measures the decay of electrical potential after transmitting a direct current pulse, while in the frequency domain, it measures the phase shift of an alternating current. The Cole-Cole model (CCM) is widely used to analyze SIP data, aiding in the comprehension of subsurface properties in different geological settings. Initially introduced by Cole and Cole (1941) and subsequently expanded upon by Pelton et al (1978), this model provides a description of the complex resistivity of materials. While initially developed for mineralized rock, CCM has been successfully adapted to characterize sedimentary formations lacking electronically conducting components. In such cases, the polarization arises from interactions between pore fluids and electrically charged mineral surfaces, forming an electric double layer. It is well established that frequency-dependent induced polarization measurements offer additional spectral information beyond a single measure of induced polarization amplitude, even though the universal mechanism is not fully understood. This spectral information, derived from the shape of the frequency response, can be linked to petrophysical and geochemical properties of the Earth’s subsurface, such as soil texture, water saturation, hydraulic conductivity, pH, and the dissemination of metallic minerals, through empirical relationships. In addition to advances in the fundamental understanding of induced polarization phenomena, the SIP method has seen significant progress and development across various research areas in recent years, including forward modeling, inversion, and equipment. However, the success of the SIP method is strongly dependent on providing a reliable and precise inversion algorithm aimed at retrieving the CCM parameters. Inverse problem theory refers to a mathematical framework that addresses the extraction of information about a parameterized physical system using observational data, theoretical relationships between model parameters and data (i.e., forward problem), and prior knowledge. To ensure accurate interpretation of the estimated models, it is crucial to understand the correlation between the parameters in the subsurface models. This research is significant because it explores the dependency and correlation of the CCM parameters using a Bayesian approach in a 2.5D inversion framework specifically designed for SIP data. The motivation for studying correlation analysis between model parameters arises from the challenges that high parameter correlation can pose to Markov chain Monte Carlo (McMC) sampling algorithms in probabilistic models. In other words, the objective is to enhance the understanding of subsurface properties and provide a more reliable interpretation of the estimated models by thoroughly analyzing parameter interdependencies. A novel 2.5D inversion code specifically developed for SIP data is introduced, leveraging Python-based libraries and advanced statistical methods. Through synthetic modeling and McMC sampling, the robustness of this approach across various subsurface scenarios is evaluated, including a homogeneous earth model, a two-layer medium, and a model featuring two anomalies within a homogeneous background. Our method enables the extraction of CCM parameters that reflect electrical properties, offering deeper insights into complex geological formations. Visualizations of McMC chains and corner plots effectively reveal the interdependencies among CCM parameters, illustrating the convergence and reliability of the parameter estimates. Validation against synthetic models highlights the precision and effectiveness of the proposed methodology. Overall, this study demonstrates the potential of Bayesian inversion to improve the interpretation of geophysical data and offers valuable insights into the correlation between CCM parameters across different geological environments.

Keywords

Main Subjects

References

Bérubé, C. L., Chouteau, M., Shamsipou, M., Enkin, P., & Olivo, G. R. (2017). Bayesian inference of spectral induced polarization parameters for laboratory complex resistivity measurements of rocks and soils. Computers and Geosciences, 105, 51-64.
Binley, A., & Slater, L. (2020). Resistivity and induced polarization: Theory and applications to the near-surface earth. Cambridge University Press.
Binley, A., Slater, L. D., Fukes, M., & Cassiani, G. (2005). Relationship between spectral induced polarization and hydraulic properties of saturated and unsaturated sandstone. Water Resources Research, 41(12), W12417.
Blaschek, R., Hördt, A., & Kemna, A. (2008). A new sensitivity-controlled focusing regularization scheme for the inversion of induced polarization data based on the minimum gradient support. Geophysics, 73(2), F45–F54.
Boadu, F. K., & Seabrook, B. (2000). Estimating hydraulic conductivity and porosity of soils from spectral electrical response measurements. Journal of Environmental and Engineering Geophysics, 5(1), 1–9.
Chen, J., Kemna, A., & Hubbard, S. (2008). A comparison between Gauss-Newton and Markov-chain Monte Carlo-based methods for inverting spectral induced-polarization data for Cole-Cole parameters. Geophysics, 73, F247-F259.
Chib, S., & Greenberg, E. (1995). Understanding the Metropolis-Hastings Algorithm. The American Statistician, 49 (4), 327–335.
Cole, Kenneth S., Cole, & Robert H. (1941). Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics. The Journal of Chemical Physics, 9(4), 341-351.
De Pasquale, G., & Linde, N., (2017). On Structure-Based Priors in Bayesian Geophysical Inversion. Geophysical Journal International, 208, 1342–1358.
Dey, A., & Morrison, M. F. (1979). Resistivity modeling for arbitrarily shaped two-dimensional structures. Geophysical Prospecting, 27, 106-136.
Fiandaca, G., Auken, E., Gazoty, A., & Christiansen, A. V. (2012). Time-domain induced polarization: Full-decay forward modeling and 1D laterally constrained inversion of Cole-Cole parameters. Geophysics, 77, E213-E225.
Fiandaca, G., Madsen, L., & Maurya, P. (2017). Re-parameterization of the Cole-Cole Model for Improved Spectral Inversion of Induced Polarization Data. Near Surface Geophysics. 2018, 16, 385–399.
Fiandaca, G., Ramm, J., Binley, A., Gazoty, A., Christiansen, A. V., & Auken, E. (2013). Resolving spectral information from time domain induced polarization data through 2-D inversion. Geophysical Journal International, 192, 631-646.
Foreman-Mackey, D., Farr, W. M., Sinha, M., Archibald, A. M., Hogg, D. W., Sanders, J. S., Zuntz, J., Williams, P. K. G., Nelson, A. R. J., de Val-Borro, M., Erhardt, T., Pashchenko, I., & Abril Pla, O. A. (2019). emcee v3: A Python ensemble sampling toolkit for affine-invariant MCMC. Journal of Open Source Software, 4(43), 1864.
Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. (2013). emcee: The MCMC Hammer. Publications of the Astronomical Society of the Pacific, 125(925), 306.
Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2014). Bayesian data analysis. 3rd ed. Chapman and Hall.
Gelman, A., Roberts, G. O., & Gilks, W. R. (1994). Bayesian Statistics 5, Oxford University Press, 599–607.
Ghanati, R., & Müller-Petke, M. (2021). A homotopy continuation inversion of geoelectrical sounding data. Journal of Applied Geophysics, 191, p.104356.
Ghorbani, A., Camerlynck, C., Florsch, N., Cosenza, P., & Revil, A. (2007). Bayesian inference of the Cole-Cole parameters from time-and frequency-domain induced polarization. Geophysical Prospecting, 55, 589–605.
Goodman, J., & Weare, J. (2010). Ensemble samplers with affine invariance. Communications in Applied Mathematics and Computational Science, 5, 65–80.
Günther, T., & Martin, T. (2016). Spectral two-dimensional inversion of frequency-domain induced polarisation data from a mining slag heap. Journal of Applied Geophysics, 135, 436-448.
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.
Hoffman, M. D., & Gelman, A. (2014). The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), 1593–1623.
Jaggar, S. R., & Fell, P. A. (1988). Forward and inverse Cole–Cole modelling in the analysis of frequency domain electrical impedance data. Exploration Geophysics, 19, 463–470.
Jardani, A., Revil, A., & Dupont, J. P. (2013). Stochastic joint inversion of hydrogeophysical data for salt tracer test monitoring and hydraulic conductivity imaging. Advances in Water Resources, 52, 62-77.
Johnson, T. C., & Thomle, J. (2018). 3-D decoupled inversion of complex conductivity data in the real number domain. Geophysical Journal International, 212, 284–296.
Keery, J., Binley, A., Elshenawy, A., & Clifford, J. (2012). Markov-chain Monte Carlo estimation of distributed Debye relaxations in spectral induced polarization. Geophysics, 77(2), E159-E170.
Kemna, A., Binley, A., & Slater, L. (2004). Crosshole IP imaging for engineering and environmental applications. Geophysics, 69, 97–107.
Luo, Y., & Zhang, G., (1998). Theory and application of spectral induced polarization. SEG, Geophysical Monograph Series, no. 8.
MacKay, D. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press, Cambridge.
Madsen, L. M., Fiandaca, G., Auken, E., & Christiansen, A. V. (2017). Time-domain induced polarization – an analysis of Cole–Cole parameter resolution and correlation using Markov Chain Monte Carlo inversion. Geophysical Journal International, 211, 1341-1353.
Metropolis, N., Rosenbluth, M. N., Rosenbluth, A. W., & Teller, A. H. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1987–1092.
Mosegaard, K., & Tarantola, A. (1995). Monte Carlo sampling of solutions to inverse problems. Journal of Geophysical Research, Atmospheres, 100, 12431–12448.
Pelton, W. H., Ward, S. H., Hallof, P. G., Sill, W. R., & Nelson, P. H. (1978). Mineral discrimination and removal of inductive coupling with multifrequency IP. Geophysics, 43(3), 588–609.
Roudsari, M. S., Ghanati, R., & Bérubé, C. L. (2024). Spectral induced polarization tomography inversion: Hybridizing homotopic continuation with Bayesian inversion. Geophysics, 89(5), 1-63.
Rücker, C., Günther, T., & Spitzer, K. (2006). Three-dimensional modeling and inversion of DC resistivity data incorporating topography—part I: modeling. Geophysical Journal International, 166, 495–505.
Rücker, C., Günther, T., & Wagner, F. M. (2017). pyGIMLi: An open-source library for modelling and inversion in geophysics. Computers & Geosciences, 109, 106–123.
Sambridge, M., & Mosegaard, K. (2002) Monte Carlo methods in geophysical inverse problems. Reviews of Geophysics, 40(3), 3.13.29
Tarantola, A. (2004). Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics, USA.
Vanhala, H. (1997). Mapping oil-contaminated sand and till with the spectral induced polarization (SIP) method. Geophysical Prospecting, 45(2), 303326.
Vrügt, J. A. (2016). Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation. Environmental Modelling and Software, 75, 273316.
Weller, A., & Slater, L. (2023). Ambiguity in induced polarization time constants and the advantage of the Pelton model, Geophysics, 87(6), E393E399.
Weller, A., M. Seichter, & Kampke, A. (1996). Induced-polarization modelling using complex electrical conductivities, Geophysical Journal International, 127, 387398,
Wohling, T., & Vrügt, J. A. (2011). Multiresponse multilayer vadose zone model calibration using Markov chain Monte Carlo simulation and field water retention data. Water Resources Research, 47, 19.
Journal of the Earth and Space Physics
Volume 51, Issue 1
online publication date: 10 June 2025
June 2025
Pages 45-64

Files

Share

How to cite

Statistics