Intra-annual Structure of the Seasonal Cycle of Extreme Precipitation in Mashhad, Iran (1951–2025)



Abstract:

Accurate modeling of extreme precipitation is critical for urban flood management in Mashhad. This study analyzes monthly maximum daily precipitation (1951–2025) using a non-stationary GEV framework with Fourier harmonic expansions. Results reveal distinct seasonal behaviors for location (μ) and scale (σ) parameters, which remained statistically stable over the study period. The location parameter peaked at 11.29 mm (day 68), while the scale peaked at 7.30 mm (day 45), revealing a 23-day phase lag. Seasonal return levels for the peak period (late winter–early spring) were estimated at 35, 68, and 88 mm for 10-, 50-, and 100-year return periods. BIC-based model selection confirmed no significant inter-annual trend or interaction, suggesting stability in the seasonal pattern of extremes. These findings necessitate revising urban hydraulic design criteria toward seasonal return levels, prioritizing the critical late-winter-to-early-spring window.

Keywords: Extreme Precipitation, Non-stationary GEV, Seasonal Cycle, Mashhad, Climate Change.

Introduction

Extreme precipitation causes significant urban flooding. While existing literature often focuses on long-term magnitude trends, the intra-annual dynamics—specifically the seasonal variation of location and scale parameters—are frequently overlooked. In arid regions like Mashhad, understanding this seasonal cycle is essential for flood risk management. This study provides a robust characterization of the seasonal cycle of monthly extreme rainfall using a non-stationary GEV framework, evaluating whether this structure exhibits stability or long-term trends over the 75-year record.



Data and Methodology

The dataset consists of daily precipitation records from the Mashhad synoptic station over a 75-year period (1951–2025). The Block Maxima method was applied to extract monthly maximum daily precipitation, yielding 900 data points; 143 zero-valued maxima (mostly from the dry summer season) were excluded due to the estimation instability they introduced, leaving 757 positive monthly maxima. The GEV model is accordingly interpreted as conditional on a positive monthly maximum.

The location (µ) and scale (σ) parameters were each modeled as g(θ)=θ_0+f_1 (season)+f_2 (year)+f_3 (season,year), where f_1 is a Fourier harmonic expansion of the within-year seasonal cycle (as a function of the calendar-day midpoint of each monthly block), f_2 is an inter-annual trend modeled with orthogonal Legendre polynomials of the rescaled year, and f_3 is their interaction, capturing potential changes in seasonal amplitude and phase over time. The shape parameter (ξ) was held time-invariant. Model selection followed a forward stepwise approach based on BIC (up to order 5). Parameters were jointly re-estimated at each step, and harmonic sine–cosine pairs were added in tandem to preserve phase consistency. For each parameter, f_1, f_2, and f_3 were tested sequentially, first for location and then for scale, conditional on the final location structure. The sensitivity of the inter-annual and interaction tests on location to the stationary-scale assumption was verified by repeating them under the final seasonal scale structure, with consistent conclusions.

Seasonal T-year seasonal return level were computed as

x_T (t)=μ(t)+σ(t)/ξ[(-〖ln⁡(1-1/T))〗^(-ξ)-1],

representing the magnitude expected to be exceeded on average once every T years under the recurring climatological conditions of day t.

Results and Conclusion

Model selection confirmed a second-order harmonic expansion (H=2) for location and a third-order expansion (H=3) for scale as the optimal seasonal structure, significantly improving the fit over the stationary GEV model (lower BIC; ξ>0, Fréchet-type heavy tail. Contrary to expectations of long-term climatic shifts, no significant inter-annual trend or seasonal–annual interaction was detected in either parameter—a null result that proved robust across alternative model specifications, indicating a high degree of stability in the seasonal pattern of extremes over the 75-year record.

A critical finding is the internal phase structure: the scale parameter peaked 23 days earlier than the location parameter (day 45 vs. day 68), suggesting that the climatic drivers of extreme variance differ in timing from those driving mean intensity—an effect obscured by simple annual aggregates.



Conclusion

The extreme precipitation regime in Mashhad is characterized by robust seasonal cyclicity rather than long-term inter-annual change. The phase lag between location and scale underscores the complex internal structure of seasonal extremes in late winter and early spring. The estimated seasonal return levels offer a more precise planning tool than traditional stationary models. Hydraulic infrastructure design should therefore transition from annual stationarity assumptions to a seasonal design approach, focused on the critical window from mid-February to early March, providing a scientific basis for enhancing the resilience of Mashhad's urban drainage systems against seasonal flood risk.