Overtopping risk of high-hazard embankment dam under climate change condition

frequently used in the calculation of the Probable Maximum Flood (PMF) [8]. It is found that the NAHRIM Km envelope curve is more reliable compared to Hershfied method because it provides a more precise PMP estimate [39]. Thus, considering the worst-case scenario, the NAHRIM Km Envelope Curve approach was used to develop a frequency equation for PMP precipitation. It is often used for rainfall stations that have extensive records [40]. The PMP is calculated for different time periods, including hourly (1, 3, 6, and 12 hours) and daily (1, 2, 3, and 5 days). Historical rainfall intensity duration frequency (IDF) curve. The IDF relationship for a given return period is a special case of the generalized formula in Eq (3). In this equation, ω, v, θ, and η are positive coefficients, with the condition that vη � 1. However, if we assume that v = 1, the resulting error is significantly lower than the regular estimation error de-scribed in Eq (4). Intensity ¼ o ðdv þ yÞZ ð3Þ Intensity ¼ o ðd þ yÞZ ð4Þ For any two return periods T1 and T2, where T1 < T2 yield a set of limitation in Eq 4 in which θ1 = θ2 = θ � 0, 0 < η1 = η2 = η < 1, and ω1 > ω2> 0. ω is considered to be an increas- ing function of the return period T. This leads to a generalised IDF relationship, as shown in Eq 5, which captures the dependence of I on T and d. In this function, b(d) = (d+θ) η where θ and η are the parameters to be estimated (θ>0, 0<η<1). i ¼ aðTÞ bðdÞ ð5Þ The function a(T) is determined by the probability distribution function of the maximum

rainfall in-tensity, I(d). In this equation, the intensity I(d) represents a specific time with a dis- tribution FI(d)(i;d), resulting in a variable distribution X�I(d)b(d) that closely approximates the intensity re-scaled by b(d). Mathematically, this can be expressed as Eq 6. XT ¼ a T ð Þ ¼ F 1 Y 1 1 T � � ð6Þ To simplify the parameter of the GEV function, a(T), can be represented using: Eq 7 aðTÞ ¼ lTk ð7Þ and finally, Eq 8 was transformed from Eq 5 and Eq 8 in general term as: iHistorical ¼ lTk ðd þ yÞZ ð8Þ Development of future IDF curve. The ongoing trend of global warming makes it imper- ative to recognize that the current IDF curves, which are exclusively based on historical data, may not be sufficient for precisely projecting future patterns of rainfall [41]. The future IDF curves were developed using observed data and bias-corrected simulated datasets obtained from the climate change model used to simulate future climate conditions. A climate change factor (CCF) was calculated and applied to the IDF curves to account for the impact of climate change. The CCF allows for estimating new rainfall intensities that PLOS ONE Climate change PLOS ONE | https://doi.org/10.1371/journal.pone.0311181 February 27, 2025 8 / 32