معکوس‌سازی پارامترهای مدل کول-کول با استفاده از الگوریتم شبیه‌سازی بازپخت برای داده‌های قطبش القایی طیفی (SIP)

نوع مقاله: مقاله تحقیقی‌ (پژوهشی‌)

نویسندگان

دانشکده مهندسی معدن و متالورژی، دانشگاه یزد، یزد، ایران

چکیده

قطبش القایی طیفی (SIP) شاخه‌ای از روش‌‌های ژئوفیزیکی است که به‌طور گسترده در پی‌جویی‌های معدنی و زیست‌محیطی مورد استفاده قرار می‌گیرد. برای تفسیر و وارون‌سازی داده‌های قطبش القایی طیفی با مدل کول-کول، چهار پارامتر r0، m، t و c بازیابی می‌شود. تحقیقات زیادی در زمینه بازیابی پارامترهای کول-کول از داده‌های قطبش القایی طیفی صورت گرفته است که اکثراً براساس روش‌های کمترین مربعات خطا استوار است. در پژوهش حاضر از توسعه استنباط بیزی (Bayesian) برای برآورد پارامترهای دو کول-کول استفاده شده است. در این استنباط دو روشِ مرسوم برای نمونه‌برداری از تابع توزیع پسین وجود دارد: 1.روش شبیه‌سازی بازپخت‌ (Simulated Annealing (SA)) 2. روش نمونه‌برداری گیبس(Gibss Sampling (GS)) . در مقاله حاضر، ابتدا نمونه‌برداری با استفاده از روش شبیه‌سازی بازپخت صورت گرفته است‌. این الگوریتم در محیط نرم‌افزار مَت‌لَب برنامه‌نویسی شده و با رایانه‌ای به بسامد 2.53GHz و حافظه‌ زنده 4Gb اجرا شده است. سپس الگوریتم نمونه‌برداری گیبس، که چن و همکاران آن را گسترش داده‌اند، بازسازی شده است. همچنین، نتایج وارون‌سازی حاصل از دو الگوریتم برای داده‌های قطبش القایی طیفی مصنوعی و واقعی به‌دست آمده در آزمایشگاه، با هم مقایسه شده‌ است.
    نتایج نشان می‌دهد که هر دو روش یک رهیافت کلی برای وارون‌سازی پارامترهای مدل کول-کول از داده‌های قطبش القایی طیفی تامین می‌کنند، در یافتن کمینه‌‌ واقعی موفق بوده‌اند، درگیر کمینه‌‌‌های محلی نمی‌شوند و برآورد‌های به‌دست آمده از آنها مستقل از مقادیر اولیه‌ پارامترها است. برای داده‌های مصنوعی با نوفه‌‌ تصادفی 10% و زمان 50 ثانیه، نتایج شبیه‌سازی بازپخت نسبت به نمونه‌برداری گیبس به داده‌های واقعی نزدیک‌تر است، درحالی‌که روش نمونه‌برداری گیبس برای رسیدن به چنین تقریبی به 6 دقیقه زمان نیاز دارد. برای داده‌های آزمایشگاهی نیز روش شبیه‌سازی بازپخت نسبت به روش نمونه‌برداری گیبس در زمان کمتر، برازش مناسب‌تری بر داده‌ها به‌دست می‌دهد. در نتیجه برای مسئله‌ای با پارامترهای بیشتر، زمان صرف شده در روش نمونه‌برداری گیبس به مراتب افزایش می‌یابد، درصورتی‌که روش شبیه‌سازی بازپخت، این زمان را به حداقل می‌رساند.
 
 

کلیدواژه‌ها


عنوان مقاله [English]

Inversion of spectral induced polarization data for Cole–Cole parameters using simulated annealing algorithm

نویسندگان [English]

  • Hosseinali Ghari
  • Ahmad Ghorbani
  • Abdol hamid Ansari
چکیده [English]

Spectral induced polarization (SIP) method is an exploration geophysical method. The induced polarization (IP) method has been used extensively in mine prospecting and increasingly in environmental investigations because IP measurements are very sensitive to the low frequency capacitive properties of rocks and soils.
Many different models have been proposed for the description of the dispersive behaviour of the IP. However, the most widely used model is the Cole-Cole model. This model describes the resistivity dispersion observed in field data from areas with metallic mineral content. It is also used to estimate various subsurface properties of nonmetallic soil and rocks in IP frequency domain investigations (SIP). A multiple Cole-Cole model is typically a more general and proper model than a single Cole-Cole model for describing IP data with various dispersion ranges caused either by multiple-length scales in sediments or by coupling effects in the IP measurements.
The Cole-Cole model parameters are widely used to interpret both time- and frequency-domain induced polarization data. Among many studies in which the Cole-Cole parameters are estimated from SIP measurements on soils and rocks, a majority of them use least squares (deterministic) methods. In this work, we have developed a Bayesian method with simulated annealing sampling algorithm to invert single and double Cole-Cole parameters from SIP data. We have also reproduced the Gibbs sampling algorithm developed by Chen et al in 2008.
The Bayesian approach consists of propagating the information provided by measurements throughout the model and combining this information with a priori knowledge of the data. A major limitation towards a more widespread implementation of Bayesian approaches is that obtaining the posterior distribution often requires the integration of high-dimensional functions. This can be computationally very difficult, but several approaches have been proposed which fall short of direct integration functions. Here, we focused on Markov Chain Monte Carlo (MCMC) methods, which attempt to simulate direct draws from some complex distribution of interest. The simulated annealing and Gibbs sampling are particular MCMC methods widely applicable to a broad class of Bayesian problems and have sparked a major increase in the application of Bayesian analysis. This interest is likely to continue in the future. In the Gibbs sampling method, the acceptance probability is 1; that is, Gibbs sampler candidates are always accepted. On the other hand, the acceptance probability is between 0 and 1 for the simulated annealing method.The MCMC-based inversion method provides extensive global information on unknown parameters such as the marginal probability distribution functions from which we can obtain better estimates and tighter uncertainty bounds of the parameters compared to deterministic methods.
We compared performance of the the simulated annealing method with the Gibbs sampling algorithm method through an inversion of synthetic and real data. Both methods provide a global approach to invert SIP data for the Cole-Cole parameters in which the obtained estimates are independent of the initial values. In addition, this approach has the ability to avoid getting trapped in a local optimum.
For synthetic data with 10% random noise and 50 second run-time, the results of the simulated annealing are more similar to the results of the Gibbs sampling method. The latter needs 6 minutes of time. Also, in the case of the real data obtained in laboratory, the simulated annealing method has provided a more suitable fit in comparison with the Gibbs sampling method. As a result, for a problem with more parameters, the required time is increased in the Gibbs sampling method, while the simulated annealing method reduces the required time.
 

کلیدواژه‌ها [English]

  • Spectral induced polarization
  • Cole-Cole model
  • Inversion
  • Bayesian
  • Simulated Annealing
  • Sampling
Binley, A., Slater, L. D., Fukes, M., and Cassiani, G., 2005, Relationship between spectral induced polarization and hydraulic properties of saturated and unsaturated sandstone: Water Resources Research, 41, W12417.

Cao, Z., Chang, Y., and Luo, Y., 2005, Inversion study of spectral induced polarization based on improved genetic algorithm: Progress in Electromagnetics: Research Symposium (PIERS) Online, 1, 266-270, http://piers.mit.edu/piersonline/piers.php, doi: 10.2529/piers04120094920.

Chen, J., Kemna, A., and 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(6), 247-259.

Cole, K. S., and Cole, R. H., 1941, Dispersion and adsorption in dielectrics I: Alternating current characteristics: Journal of Chemical Physics, 1, 341–351.

Duflo, 1996, Applications of Mathematics I: Springer-Verlag, Berlin, New York, 34.

Gelman, A., and Rubin, D., 1992, Inference from iterative simulation using multiple sequences: Statistical Science, 7, 457–472.

Geman, S., and Geman, D., 1984, Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images: IEEE Trans. Pattern Anal. Mach. Intell, 6, 721-741.

Geweke, J., 1992, Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, J. M. Bernardo, J. O. Berger, A. P. David, and A. F. M. Smith, eds., Bayesian statistics 4: Oxford University Press, 169–193.

Ghorbani, A., Camerlynck, C., Florsch, N., Cosenza, P., and Revil, A., 2007, Bayesian inference of the Cole-Cole parameters from time and frequency domain induced polarization: Geophysical Prospecting, 55, 589-605.

Ghorbani, A., Cosenza, Ph., Revil, A., Zamora, M., Schmutz, M., Florsch, N., Jougnot, D., 2009, Non-invasive monitoring of water content and textural changes in clay-rocks using spectral induced polarization: A laboratory investigation: Appl. Clay Sci, 43, 493–502.

Jaggar, S. R., and Fell., P. A., 1988, Forward and inverse Cole-Cole modeling in the analysis of frequency domain electrical impedance data: Exploration Geophysics, 19, 463–470.

Pelton, W. H., Ward, S. H., Hallof, P. G., Sill, W. R., and Nelson, P. H., 1978, Mineral discrimination and removal of inductive coupling with multifrequency IP: Geophysics, 43, 588-609.

Pelton, W. H., Smith, B. D., and Sill, W. R., 1984. Interpretation of complex resistivity and dielectric data, Part II: Geophysical Transactions, 29(4), 11–45.

Ryan, A. J., Glaser, D. R., Werkema J, D. D., and Atekwana, E. A., 2012, Spectral induced polarization response to nanoparticles in a saturated sand matrix: J. Appl. Geophys, 77, 63-71.

Scales, J. A., and Sneider, R., 1997, To Bayes or not to Bayes: Geophysics, 62, 1045-1046.

Schlumberger, C., 1920, Etude sur la Prospection Electrique du Soussol. Gauthier–Villars, Paris.

Seigel, H. O., Vanhala, H., and Sheard, S. N., 1997, Some case histories of source discrimination using time-domain spectral IP: Geophysics, 62, 1394–1408.

Slater, L. D., and Lesmes., D., 2002, IP interpretation in environmental investigations: Geophysics, 67, 77–88.

Slater, L., 2007, Near surface electrical characterization of hydraulic conductivity: From petrophysical properties to aquifer geometries —A review: Surveys in Geophysics, 28, 169–197.

Tarantola, A., 1987, Inverse Problem Theory, Methods for Data Fitting and Model Parameter Estimation: Elsevier Science Publishing Co.

Tarantola, A., and Valette, B., 1982a, Generalized nonlinear inverse problems solved using the least square criterion: Reviews of Geophysics, 20(2), 219-232.

Tarantola, A., and Valette, B., 1982b, Inverse problems - Quest for information: Journal of Geophysics, 50, 159-170.

Vanhala, H., 1997, Laboratory and Field Studies of Environmental and Exploration Applications of the Spectral Induced Polarization (SIP) Method: PhD thesis, Geological survey of Finland.

Xiang J., Jones N. B., Cheng D., and Schlindwein, F. S., 2001, Direct inversion of the apparent complex-resistivity spectrum: Geophysics, 66, 1399–1404.

Kemna, A., 2000, Tomographic Inversion of Complex Resistivity: Theory and Application: Ph.D. thesis, Ruhr-University Bochum.

Kemna, A., Binley, A., Ramirez, and Daily., W., 2000, Complex resistivity tomography for environmental applications: Chemical Engineering Journal, 77, 11 18.

Kemna, A. H. M., Münch, K., Titov, Zimmermann, E., and Vereecken., H., 2005, Relation of SIP relaxation time of sands to salinity, grain size and hydraulic conductivity: 11th European Meeting of Environmental and Engineering Geophysics, EAGE, Extended Abstracts, P05.

Kirkpatrick, S., Gelatt, C. D., and Vecchi, Jr., M. P., 1983, Optimization by simulated annealing: Science, 220, 671-680.

Klein, J. D., and Sill, W. R., 1982, Electrical properties of artificial clay-bearing sandstone: Geophysics, 47(11), 1593-1605.

Luo, Y., and Zhang, G., 1998, Theory and Application of Spectral Induced polarization: SEG, Geophysical Monograph Series, no. 8.

Madden, T. R., and Cantwell, T., 1967, Induced polarization, a review. In: Mining Geophysics, Society of Exploration Geophysicists, 2, 916–931.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., andTeller, E., 1953, Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21(6), 1087–1092.

Mosegaard, K., and Tarantola, A., 1995, Monte Carlo sampling of solutions to inverse problems: Journal of Geophysical Research, 100(B7), 12431-12447.

Neal, R. M., 1993, An improved acceptance procedure for the hybrid Monte Carlo algorithm: Journal of Computational Physics. 111, 194-203

Orozco, F. A., Williams, K., Long, P., Hubbard, S., and Kemna, A., 2011, Using spectral induced polarization (SIP) to infer biogeochemical processes associated with bioremediation of a uranium-contaminated aquifer: Journal of Geophysical Research, 116, PP 17.

Pelton, W. H., 1977, Interpretation of Complex Resistivity and Dielectric Data: PhD thesis, University of Utah