Mathematical Formulation of Intensity–Duration–Frequency Curves and Their Hydrological Risk Implications in Civil Engineering Design

Intensity–duration–frequency (IDF) curves, which relate rainfall intensity (i), storm duration (d), and return period (T), are cornerstone tools for planning, designing, and operating hydraulic works. Since Sherman’s pioneering formulation in 1931, many modern implementations have systematically omitted the duration-shifting parameter C, causing predicted intensities to diverge to infinity as d→0. This mathematical paradox becomes especially problematic under extreme hydrological regimes and convective storms exceeding 300 mm/h, where an accurate curve fit is critical. Here, we first review conventional IDF curve fitting techniques and their limitations. We then introduce IDF-GtzLo, a novel, intuitive formulation that reinstates and calibrates C directly from observed storm statistics, ensuring finite intensities for all durations. Applied to 36 automatic weather stations across Mexico, our method reduces the root mean square error by 23 % compared to the classical model. By eliminating the infinite intensity paradox and improving statistical performance, IDF-GtzLo offers a more reliable foundation for hydrological risk assessment and the design of infrastructure resilient to climate-driven extremes.