Bayesian modeling and characterization of underwater explosion shock wave loads with parameter uncertainty

The shock wave load generated by underwater explosions exhibits significant variability and uncertainty. To address the prediction bias caused by classical deterministic empirical models that ignore this uncertainty, an uncertainty analysis of both model parameters and model errors was conducted for key load model parameters including peak pressure p_m, time constant θ, impulse I, and shock wave specific energy density e_s, based on 682 sets of underwater explosion test data. Within the framework of the empirical model Cole, a Bayesian probabilistic model for underwater explosion shock wave loads was developed. Bayesian inference methods were employed to update and calibrate the model parameters, enabling a probabilistic characterization of the explosion shock wave load. The results show that the coefficient of variation for the calculated parameters of the model Cole ranges from 0.03 to 0.48, while the coefficient of variation for model errors lies between 0.19 and 0.38. Among these, only the modelling error for peak pressure approximately follows a normal distribution. In contrast, the modelling errors for the time constant, impulse, and shock wave specific energy density exhibit distinctly skewed distributions. Moreover, the model errors gradually stabilize as the scaled distance increases. Under the condition of limited experimental samples, the Bayesian probabilistic model significantly improves parameter estimation accuracy, effectively reduces model uncertainty, and achieves a reasonable balance between model precision and experimental cost. The analysis demonstrates that the developed Bayesian probabilistic model for underwater explosion shock wave loads can reasonably characterize the uncertainty of the loads. It provides stochastic inputs that explicitly account for load variability for the reliability-based blast-resistant design of underwater structures, and offers a more comprehensive basis for engineering risk assessment and probabilistic analysis.