A comparison between Gauss-Newton and Markov-chain Monte Carlo-based methods for inverting spectral induced-polarization data for Cole-Cole parameters

We have developed a Bayesian model to invert spectral induced-polarization (SIP) data for Cole-Cole parameters using Markov-chain Monte Carlo (MCMC) sampling methods. We compared the performance of the MCMC-based stochastic method with an iterative Gauss-Newton-based deterministic method for Cole-Cole parameter estimation through inversion of synthetic and laboratory SIP data. The Gauss-Newton-based method can provide an optimal solution for given objective functions under constraints, but the obtained optimal solution generally depends on the choice of initial values and the estimated uncertainty information often is inaccurate or insufficient. In contrast, 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 than with the deterministic method. In addition, the results obtained with the MCMC method are independent of the choice of initial values. Because the MCMC-based method does not explicitly offer a single optimal solution for given objective functions, the deterministic and stochastic methods can complement each other. For example, the stochastic method can be used first to obtain the medians of unknown parameters by starting from an arbitrary set of initial values. The deterministic method then can be initiated using the medians as starting values to obtain the optimal estimates of the Cole-Cole parameters.