Probabilistic subseasonal rainfall emulation with diffusion models: evaluating tail-risk calibration for flood early-warning triggers across data-sparse river basins

Subseasonal rainfall forecasts underpin flood early-warning triggers in river basins where rain-gauge networks are sparse, yet the probabilistic calibration of such forecasts in the upper tail — the region that determines whether a trigger fires — remains poorly characterised. The emulator is formalised as a score-based generative model whose reverse-time dynamics are conditioned on large-scale predictors, and tail calibration is defined through a threshold-exceedance reliability functional that separates sharpness from exceedance bias. A theoretical analysis establishes conditions under which diffusion emulators inherit the tail behaviour of the training distribution and identifies the sample-size regime in which tail calibration degrades. The framework is illustrated with a reanalysis-derived temperature series for Harare, Zimbabwe (2000–2023, 312 monthly observations, mean 18.914 °C, warming trend 0.2001 °C per decade), used here as a calibration benchmark for the emulator's marginal-distribution fidelity rather than as a rainfall dataset. The contribution is a calibration-first evaluation protocol that treats tail reliability, not ensemble spread, as the primary criterion for trigger-grade emulation, together with explicit boundary conditions under which diffusion emulators are and are not adequate for flood early-warning use.