Improvement of empirical formulas to estimate the reduction effects by berms on irregular wave runup over a dune-berm coast

Abstract

This study formulates the reduction effects of a sandy berm on irregular wave runup over a dune-berm coast. The numerical experiments by Park and Cox (2016) are closely re-examined to develop an empirical formula describing the variability of reduction effects of a sandy berm over a broad range of conditions. Based on a sequence of regression analyses, the reduction effects are expressed as a reduction factor in terms of normalized berm width, normalized surge level, and wave steepness in deep water. The comparison with the numerical experiments demonstrates that the regression formula can satisfactorily reproduce the variability of the reduction effects over the range of numerical experiments. The analysis of prediction uncertainty demonstrated that the derived formula reproduced the reduction effects observed in the numerical experiments with negligible bias and a 90% confidence interval of approximately ±20% relative error. In addition, conversion formulas between representative runup values based on different statistical definitions are derived to enable consistent comparisons between them. The proposed reduction formula is implemented into three empirical runup models that are applicable to the quick estimations of irregular wave runup on a dune-berm coast: the models by Park and Cox (2016), Stockdon et al. (2006), and Mase et al. (2013). Consistent comparisons were conducted among the empirical predictions and numerical experiments based on the statistical conversion formulas. Combined with the proposed reduction formula, all three models well reproduced the normalized 2% runup in the numerical experiments over a wide range of conditions. On the other hand, the uncertainty in the runup prediction appeared in different forms depending on the selected model. When the proposed reduction formula was implemented in the modified Park and Cox (2016) and modified Stockdon et al. (2006) models, the uncertainty was described by a log-normal distribution of the error ratio between the empirical predictions and numerical experiments. Quantitatively, these two models predicted 90% of the normalized runup on a dune within a range of relative error of less than approximately 20–30%. When the proposed reduction formula was combined with the model by Mase et al. (2013), the uncertainty followed a normal distribution of the residual error between the empirical predictions and numerical experiments. On the normalized runup, the model prediction indicated a small conservative bias (+0.05) and a root-mean-square error of 0.13.