Differential evolution adaptive metropolis sampling method to provide model uncertainty and model selection criteria to determine optimal model for Rayleigh wave dispersion

Abstract

The near-surface S-wave velocity is important tool for environmental studies. This parameter can be derived by inverting of Rayleigh wave dispersion. Inversion of Rayleigh wave dispersion has a nonunique solution. Thus, solving inverse problems is not only done to find the fittest model but also to characterize the uncertainty of the model result. In this paper, we applied and tested a Bayesian inversion method using a developed differential evolution adaptive metropolis (DREAM(ZS)) approach to provide posterior distribution of model parameters (PDMPs). This method consists of Markov chain Monte Carlo (MCMC) simulation method which rapidly estimates the PDMP. After obtaining the resulted posterior, we could estimate representative model (such as mean, mode, median, covariance, and percentile model, the maximum posterior model, and uncertainty model), the probability distributions for individual parameters, and the dispersion curve uncertainty of these models. For inversion of real data, the number of model parameters or the layer number (degrees of freedom (DoF)) which can be propagated by Rayleigh wave is unknown. Therefore, this layer number is needed to accurately estimate subsurface model parameters. For this problem, membership function of fuzzy (MFF) criteria is proposed and applied to the model selection criteria and the various model selection criteria such as the Bayesian information criteria (BIC), the Akaike’s information criterion (AIC), the generalized cross-validation (GCV), the Kullback information criterion (KIC), the finite prediction error (FPE), and information complexity (ICOMP) are compared to the proposed method for selection of the optimal model. The DREAM(ZS) method as well as the seven model selection criteria methods to select the optimal model are used to investigate the influence of noise on Rayleigh wave dispersion on the uncertainty of model parameter values for three synthetics, such as model with a linear velocity increase, model with a low-velocity layer (LVL), and model with a high-velocity layer (HVL). Our results demonstrated that the DREAM(ZS) method is effective to estimate the S-wave velocity and thickness of each layer and to quantify the uncertainty on the estimates, while all the model selection criteria approaches are able to determine the optimal model from different number of layers for noise-free and slightly noisy data, and only the MFF criteria is able to obtain the optimal model for noise-free and noisy data. The optimal model is typically close to the true model. Therefore, inverting Rayleigh wave dispersion using both approaches can produce the best model. We also applied these methods to Rayleigh wave dispersion data collected from the Ljubljana site in Slovenia. We compare our results with those estimated from the number of blow count in the standard penetrating test (N-SPT) data; it shows a good correlation toward each other.