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

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.