Uncertainty Quantification and Bayesian Probabilistic Modeling of Underwater Explosion Shock Wave Loads

Underwater explosion shock wave loads exhibit significant variability and uncertainty in their physical characteristics. Classical deterministic empirical models such as the Cole formula neglect these attributes, leading to marked discrepancies between computed outcomes and experimental measurements. This investigation analyzes 682 sets of underwater explosion test data to quantify uncertainties in key blast load parameters: peak pressure(pm), time constant(θ), impulse(I), and specific shock wave energy density(es). A Bayesian probabilistic framework integrating Cole’s empirical model was developed, with parameters calibrated through Bayesian inference to enable probabilistic characterization of shock wave loads. The results demonstrate that Model parameters exhibit variation coefficients of 0.03–0.48, while modeling errors demonstrate coefficients spanning 0.19–0.38. Only pm modeling errors follow approximately normal distribution, while θ, I, and es errors manifest skewed distributions. Remarkably, all modeling errors stabilize significantly with increasing scaled distance. The Bayesian probabilistic approach demonstrates superior sample efficiency in engineering applications. As sample sizes increase, posterior variances for θ, pm, I, and es parameters exhibit systematic contraction. Sampling optimization thresholds were identified: θ, I, es models achieve optimal accuracy at 20% data usage, while pm models require 30–60% sampling. Despite minimal Cov fluctuation in most models under varying samples, es Bayesian models displayed significant Cov reduction compared to Cole's baseline. The developed Bayesian model comprehensively characterizes blast load uncertainty through joint point estimates and uncertainty quantification, surpassing Cole’s prior model. This approach generates stochastic input fields accommodating load variability for blast-resistant structural reliability designs. The extensible modeling framework facilitates probabilistic risk assessment across diverse explosion scenarios, providing enhanced information completeness for engineering decision-making. Methodologically, Bayesian inference achieves improved parameter estimation accuracy under limited experimental data conditions while effectively controlling model uncertainty. The probabilistic characterization demonstrates practical utility by balancing computational precision with experimental resource optimization requirements.