Bayesian Approaches for Probabilistic Prediction of Debris-Flow Runout Using Limited Site-Specific Data Sets

Accurate and reliable predictions of debris-flow runout are essential for debris-flow hazard assessment. Debris-flow runout distances are commonly estimated by empirical relationships for their simplicity. However, existing empirical models have been developed by traditional regression analysis, which generally require a large amount of data to guarantee their accuracy and predictability. Owing to the unpredictability of debris flows and lack of data referencing historic events, the amount of debris-flow data is usually very limited and associated with measurement errors. How to develop a reliable runout prediction model and simultaneously quantify the uncertainties still remains a difficult task. This paper proposed Bayesian approaches to develop the runout model of site-specific debris flow in basin-based limited investigation data and prior information. The proposed approaches can select the most suitable model of debris-flow runout among various alternatives based on field data and prior information, and simultaneously characterize the predictive uncertainty of runout distance. To overcome the limitation of the Metropolis-Hastings algorithm in inefficient sampling, a multichain method, specifically the DREAM((ZS)) algorithm, is used to obtain the posterior distribution to solve the sampling problem on complex and high-dimensional target distributions. Copula theory is applied to calculate the model evidence based on the random samples of model parameters generated from the DREAM((ZS)) algorithm, providing a promising tool for calculating the model evidence in Bayesian inference. The proposed approaches are illustrated using the debris-flow data in the Wenchuan area. Results show that the proposed approaches accurately estimate the runout distances in the study area, and reasonably consider the measurement noise and modeling errors in the empirical relationship. Based on the limited site-specific data sets, the Bayesian approaches perform better than the preexisting empirical relationship and favor a simple model in terms of balance between data fitting and model complexity.